# Correctness report — tbs_neural_causal_pi **Verdict: ERRORS FOUND — REPAIRED BY MINIMAL PATCHES** **Paper: MINOR ISSUES — no main result is shown false or weakened** Minor: - Proposition C.1: weakened: The requirement that each entire non-Gumbel latent law satisfy Assumption~\ref{assumption:lower_bound}; only its normalized connected-component laws are required to satisfy it.; error repaired; condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned - Proposition B.1: weakened: Preservation of the original global Lipschitz regularity for unnormalized categorical functions on unrestricted domains, including cases where their total mass approaches zero.; error repaired; condition added (bookkeeping or a pinned reading) - Theorem 3.3: weakened: Uniformity of the comparison constant over unbounded continuous output dimensions and categorical cardinalities at fixed $L,K$ and abstract causal graph. (the referee found the key results and the contribution kept); error repaired; condition added (bookkeeping or a pinned reading) - Proposition 4.1: weakened: The requirement that each entire non-Gumbel latent law satisfy Assumption~\ref{assumption:lower_bound}; only its normalized connected-component laws are required to satisfy it. (the referee found the key results and the contribution kept); corrected; condition added (bookkeeping or a pinned reading); ambiguous; the owner settled a reading - Lemma B.3: weakened: The assertion of compatible face-adjacent orderings for arbitrary non-dyadic coordinate-box refinements satisfying property (*).; error repaired; condition added (bookkeeping or a pinned reading) - Corollary 3.8: weakened: A single neural model with the displayed error bound uniformly over unbounded continuous treatment values, and the displayed-rate statement when $d_{\max}^{\text{in}}=0$ or $d_{\max}^U=0$. (the referee found the key results and the contribution kept); error repaired; condition added (bookkeeping or a pinned reading) - Theorem 4.2: weakened: The guarantee for the printed optimizer is replaced by a guarantee for candidates whose continuous structural functions have uniform output $1$-norm at most $K$ and whose normalized categorical propensities have Lipschitz constant at most $\hat L_f$. True models whose continuous structural functions violate that output $1$-norm bound on the input domain are excluded, even when every output coordinate is bounded by $K$. The sampling orders based only on $d_{\max}^U$ and the rate for the original $\alpha_n$ are given up. The displayed-order version excludes $d_{\max}^{\text{in}}=0$ and $d_{\max}^U=0$. (the referee found the key results and the contribution kept); error repaired; condition added (bookkeeping or a pinned reading); ambiguous; the owner settled a reading - Corollary 4.4: weakened: The corrected statement excludes uniform intervention laws that are not absolutely continuous with respect to $P$ or whose density is not square-integrable, true models or continuous candidates whose structural functions violate $\sup_x\|f_i(x)\|_1\leqslant K$ on their full input domains even when every coordinate is bounded by $K$, categorical candidates whose normalized propensity functions exceed $\hat L_f$, and candidate-specific independently resampled simulation experiments; it replaces the printed sampling orders determined only by $d_{\max}^U$ with orders accounting for the joint latent dimension and does not retain the sample-size guarantee for the original estimator and radius. The displayed-order version also excludes $d_{\max}^{\text{in}}=0$ and $d_{\max}^U=0$, and the uniform intervention law requires $t_10\}}\left(W_1^{-1/D_C}+L_2^{-2/D_C}\right)+(\tau-\tau\log\tau)\right). \end{equation*} \end{enumerate} - **Proof.** Put $p_i=\mathbb P(C_i)$ and $\mu_i=\mathbb P|_{C_i}/p_i$. Each component is compact because it is homeomorphic to a compact cube. Distinct components have positive distance because they are disjoint compact sets and there are finitely many of them. A sufficiently small neighborhood of a support point in $C_i$ meets no other component and has positive measure; hence $p_i>0$. If $d_i^C=0$, write $C_i=\{c_i\}$, use the unique map $\hat g_1^i:[0,1]\rightarrow[0,1]^0$, and set $\hat g_2^i=c_i$. This produces $\mu_i$ exactly. For $q=d_i^C\geqslant1$, choose $g_2^i\in\mathcal H([0,1]^q,C_i)\cap\mathcal F_L([0,1]^q,C_i)$ from Assumption~\ref{assump:support}, and put … (the corrected paper gives it in full) - **Split.** The correction states this result as the statement above and 1 corollary, each a result of its own. Nothing moved to one of them counts as given up. - **Corollary 3.6.1.** Given any probability measure $\mathbb P$ on $\mathbb R^d$ that satisfies \cref{assump:support}, let $\lambda$ be the Lebesgue measure on $[0,1]^{N_C}$, where $N_C$ is defined in \cref{assump:support}, and write $D_C=\max\{1,\max_i d_i^C\}$. Suppose in addition that $\mathbb P$ constrained to each component satisfies \cref{assumption:lower_bound}. There exists a neural network $\hat g\in\mathcal{NN}(\Theta(D_C),L_1,\Theta(d\cdot D_C),L_2,\tau)$ with the architecture of \cref{fig:wide_nn} such that \[ W(\hat g_{\#}\lambda,\mathbb P)\leqslant O\left(\mathbb I_{\{\max_i d_i^C>0\}}\left(L_1^{-2/D_C}+L_2^{-2/D_C}\right)+(\tau-\tau\log\tau)\right). \] Formal check: proved in Lean, given 2 cited result(s), under 26 condition(s) made explicit — `Papers.TbsNeuralCausalPi.deep_generator_approximation` (`TCorFix3c402872.lean:21`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). - **Conditions.** The corrected statement states the conditions it uses itself; it no longer refers to 3 condition(s) of the paper it used as printed. - **Formal check.** proved in Lean (the corrected statement), given 2 cited result(s), under 17 condition(s) made explicit — `Papers.TbsNeuralCausalPi.wide_generator_approximation` (`TWideNn.lean:112`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Corollary 3.8 — corrected; the corrected statement is weaker - **As printed.** \label{corollary:deep_nn} Given a causal model $ \mathcal{M}^* \in \mathcal{M}(\mathcal{G} ,\mathcal{F}_L ,\boldsymbol{U})$, suppose that Assumptions \ref{assump:independence_u}-\ref{assump:Lip_and_bounded} hold and the distributions of $U_C$ for all $C^2$-component satisfy the assumptions of the second conclusion of \cref{thm:wide_nn}. Let $ d_{\max}^{\text{in}} $ and $d_{\max}^{\text{out}}$ be the largest input and output dimension of $f_i$ in (\ref{eq:structure_equation}) and $d_{\max}^U$ be the largest dimension of all latent variables. There exists a neural causal model $$\hat{\mathcal{M}} \in \text{NCM}_{\mathcal{G}} ( \mathcal{NN}(\Theta( d^{\text{in}}_{\max}d^{\text{out}}_{\max}) ,L_{0}) , \mathcal{NN}(\Theta(d_{\max}^U),L_1,\Theta((d_{\max}^U)^2),L_2))$$ with structure equations (\ref{eq:structure_equation_hat}). For any intervention $\boldsymbol{T} = \boldsymbol{t}$, $\hat{\mathcal{M}}$ satisfies \begin{equation*} W\left( P^{\mathcal{M}^*}(\boldsymbol{V}(\boldsymbol{t})),P^{\hat{\mathcal{M}}}(\boldsymbol{V}(\boldsymbol{t}))\right) \leqslant O(L_{0}^{-2/d^\text{in}_{\max}} +L_{1}^{-2/d^U_{\max}} +L_{2}^{-2/d^U_{\max}} +(\tau - \tau\log\tau) ). \end{equation*} - **What is wrong.** ``\dots \[ \begin{array}{@{}c@{}}\displaystyle \hat{f}_{i} = \begin{cases} \arg\min_{\hat{f}_{i} \in \mathcal{NN}_{d_{i}^{\text{in}} ,d_{i}^{\text{out}}} (d_{i}^{\text{out}} (2d_{i}^{\text{in}} +10),L_{0} )} \| \hat{f}_{i} -f_{i} \| _{\infty }, V_i \ \text{continuous}, \\ \displaystyle \max\{0,\arg\min_{\hat{f}_{i} \in \mathcal{NN}_{d_{i}^{\text{in}} ,d_{i}^{\text{out}}} (d_{i}^{\text{out}} (2d_{i}^{\text{in}} +10),L_{0} )} \| \hat{f}_{i} -f_{i} \| _{\infty }\}, V_i \ \text{categorical}. \end{cases}\end{array}\mbox{''} \] This step uses uniform approximation of the structural functions by the selected neural networks, which the paper assumes only for the bounded cubes on which the approximation errors are minimized, but applies it to structural inputs containing generated latent values, generated parent values, and arbitrary real intervention values. ``By \cite[Theorem 2]{Yarotsky2018OptimalAO}, we have, if $V_i$ is continuous, $ \| \hat{f}_{i} -f_{i} \| _{\infty } \leqslant O(L_0^{-1/d_{i}^{\text{in}}}) \leqslant O(L_0^{-1/d_{\max}^{\text{in}}}) $.'' The stated exponent gives only $O(L_0^{-1/d_i^{\text{in}}})$, whereas the final estimate requires $O(L_0^{-2/d_i^{\text{in}}})$. ``\dots $\boldsymbol{p} = {f}\left(\text{pa} (v_{k} ),\boldsymbol{u}_{V_{k}}\right) ,\hat{\boldsymbol{p}} = \hat{f}\left(\text{pa} (v_{k} ),\boldsymbol{u}_{V_{k}}\right)/\|\hat{f}\|_1$.'' Truncation at zero permits the entire approximating propensity vector to vanish outside its approximation cube, making division by its $1$-norm undefined. ``\dots \[ \begin{array}{@{}c@{}}\displaystyle \|\boldsymbol{p} -\hat{\boldsymbol{p}} \|_\infty \leqslant \frac{1}{\| \hat{f} \| _{1}} \| \hat{f}\left(\text{pa} (v_{k} ),\boldsymbol{u}_{V_{k}}\right) - f\left(\text{pa} (v_{k} ),\boldsymbol{u}_{V_{k}}\right) \| _{\infty } +\frac{1}{\| \hat{f} \| _{1}} |\| \hat{f} \| _{1} -1|\|f \|_\infty \\ \displaystyle \leqslant \frac{( n_{i} +1)}{\| \hat{f} \| _{1}} S\leqslant \frac{( n_{i} +1)}{1-n_{i} S} S \leqslant 2(K + 1) S\end{array} \] where we use \cref{assump:Lip_and_bounded} that $ n_{i} \leqslant K$ and $ S \leqslant 1/(2K)$.'' A bound on realized categorical values does not imply $n_i\leqslant K$ when some categories have zero probability. In the step quoted in (b): The displayed power divides by zero for $d_i^{\text{in}}=0$, and the final powers are undefined when a maximum input or latent dimension is zero. ``\dots $ d_j^U $ is the dimension of latent variables $ U_j $ and $L_1,L_2,\tau$ are hyperparameters of the neural network defined in \cref{thm:wide_nn}.'' The architectural width orders admit either absolute multiplicative factors or factors depending on the fixed causal model and $R$. - **Counterexample.** Let $\mathcal G$ consist of $T\to Y$, with independent latent variables $U_T,U_Y\sim\operatorname{Unif}([0,1])$. Set $f_T(u)=u$ and $f_Y(t,u)=\sin t$, so that $T=U_T$ and $Y=\sin T$, and take $L=K=1$. Use the identity support chart for each latent law, with domination constant $1$ and zero boundary mass. For $t\in\mathbb R$, intervene at $T=t$, giving $P^{\mathcal M^*}(Y(t))=\delta_{\sin t}$. For any fixed finite neural causal model of the displayed architecture, write $a(t)=\mathbb E[\hat Y(t)]$. For this same model, consider the family of structural width allowances $W_0(R)=\lceil R\rceil d_{\max}^{\text{in}}d_{\max}^{\text{out}}$ for $R\geqslant1$. The generated latent support is compact, so the finitely many affine regions of the structural ReLU network make $a(t)$ affine for all sufficiently large $t$, implying $\sup_{t\in\mathbb R}W(P^{\mathcal M^*}(Y(t)),P^{\hat{\mathcal M}}(\hat Y(t)))\geqslant1$, whereas the claimed uniform bound tends to zero. The allowances $W_0(R)$ have the stated order for each fixed $R$ under the model-dependent reading, but their multiplicative factors are unbounded as $R\to\infty$, distinguishing that reading from an absolute width factor. - **Rests on.** Lemma B.3, Lemma E.1, Proposition B.2, Theorem 3.3, Theorem 3.6, also found wrong as printed. - **Correction.** Given a causal model $\mathcal M^*\in\mathcal M(\mathcal G,\mathcal F_L,\boldsymbol U)$, suppose that Assumptions \ref{assump:independence_u}-\ref{assump:Lip_and_bounded} hold and the distributions of $U_C$ for all $C^2$-component satisfy the assumptions of the proposition entitled ``Deep distribution approximation''. Let $d_{\max}^{\text{in}}$ and $d_{\max}^{\text{out}}$ be the largest input and output dimension of $f_i$ in (\ref{eq:structure_equation}) and $d_{\max}^U$ be the largest dimension of all latent variables. Suppose in addition that $d_{\max}^{\text{in}}>0$ and $d_{\max}^U>0$. Fix any $R>0$. There exists a neural causal model $$\hat{\mathcal{M}} \in \text{NCM}_{\mathcal{G}} ( \mathcal{NN}(\Theta( d^{\text{in}}_{\max}d^{\text{out}}_{\max}) ,L_{0}) , \mathcal{NN}(\Theta(d_{\max}^U),L_1,\Theta((d_{\max}^U)^2),L_2))$$ with structure equations (\ref{eq:structure_equation_hat}). For any intervention $\boldsymbol{T}=\boldsymbol{t}$ such that $\|t_i\|_\infty\leqslant R$ for each continuous treatment variable $T_i$, $\hat{\mathcal M}$ satisfies \begin{equation*} W\left(P^{\mathcal M^*}(\boldsymbol V(\boldsymbol t)),P^{\hat{\mathcal M}}(\boldsymbol V(\boldsymbol t))\right)\leqslant O\left(L_0^{-2/d_{\max}^{\text{in}}}+L_1^{-2/d_{\max}^U}+L_2^{-2/d_{\max}^U}+(\tau-\tau\log\tau)\right). \end{equation*} The implicit constant is independent of the intervention and the network budgets, but may depend on $R$ and the fixed causal model. - **Proof.** Suppose that structure equations of $ \mathcal{M}$ are \begin{equation*} V_{i} =\begin{cases} f_{i}\left(\text{Pa} (V_{i} ),\boldsymbol{U}_{V_{i}}\right) , & V_{i} \ \text{is continuous} ,\\ \arg\max_{k\in [ n_{i}]}\left\{g_{k}^{V_{i}} +\log\left( f_{i}\left(\text{Pa}( V_{i}) ,\boldsymbol{U}_{V_{i}}\right)\right)_{k}\right\} , & V_{i} \ \text{is categorical} , \|f_i\|_1= 1 \end{cases}, \quad f_{i} \in \mathcal{F}_L. \end{equation*} Put $m_i=d_i^{\text{in}}$, let $d_i^{\text{out}}$ be the output dimension of $f_i$, and put $m=d_{\max}^{\text{in}}>0$. In each structural input, distinguish the real coordinates, consisting of continuous parents and non-Gumbel latent coordinates, from the … (the corrected paper gives it in full) - **What the correction gives up.** A single neural model with the displayed error bound uniformly over unbounded continuous treatment values, and the displayed-rate statement when $d_{\max}^{\text{in}}=0$ or $d_{\max}^U=0$. (The paper's owner accepted the weaker statement.) - **Conditions.** The corrected statement states the conditions it uses itself; it no longer refers to 2 condition(s) of the paper it used as printed. - **Formal check.** proved in Lean (the corrected statement), given 3 cited result(s), under 45 condition(s) made explicit — `Papers.TbsNeuralCausalPi.deep_ncm_approximation` (`TDeepNn.lean:303`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Theorem 4.2 — corrected; the corrected statement is weaker - **As printed.** \label{thm:consistency} Let $ \mathcal{M^*}$ be any SCM satisfying the assumptions of Corollary \ref{corollary:deep_nn}. Suppose that the Lipschitz constant of functions in $\mathcal{M}$ is $ L_f $, $ F:\mathbb{R}^{n_{V}}\rightarrow \mathbb{R}$ in (\ref{eq:opt_empiri}) is Lipschitz continuous and $\tau_n > 0 $, let $K>0$ be the constant in \cref{assump:Lip_and_bounded}, $\hat{L}_f = \sqrt{d_{\max}^{\text{in}}d_{\max}^{\text{out}}}L_f$, \begin{align*} \mathcal{F}_{0,n} &= \mathcal{NN}^{\hat{L}_{f} ,K}\left( W_{0,n} , \Theta\left(\log d^{\text{in}}_{\max}\right)\right) , \mathcal{F}_{1,n} =\mathcal{NN}^{\infty ,K}( \Theta( d^{U}_{\max}) ,L_{1,n}, \Theta( (d^{U}_{\max})^2) ,L_{2,n},\tau_n), \end{align*} where $ d_{\max}^U, d_{\max}^{\text{in}} $ and $d_{\max}^{\text{out}}$ are defined in Corollary \ref{corollary:deep_nn}, take the radius to be $\alpha_n = \epsilon_n+s_n, \epsilon_n = O(W_{0,n}^{-1/d^{\text{in}}_{\max}}+ \sum_{i=1}^2 L_{i,n}^{-2/d^U_{\max}}+ \tau _{n} \log \tau _{n}),$ $$ s_n = O(m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n} +n^{-1/{\max\{2,d_{\max}^U\}}}\log^2(n) + \log(nm_n)\lambda_n).$$ If $ m_n = \Omega(n), L_{i,n} = \Theta(m_{n}^{d_{\max}^{U} /\left( 2d_{\max}^{U} +4\right)}), i=1,2, \lim _{n\rightarrow \infty }\min\{\tau_n^{-1}, W_{0,n},(\log(nm_n)\lambda_n)^{-1}\}= \infty$ , then with probability 1, $ [\liminf _{n\rightarrow \infty } F_{n},\limsup _{n\rightarrow \infty } F_{n}] \subset [\underline{F}^{\hat{L}_f} ,F_{*}]$. - **What is wrong.** ``\dots Thus, $ \mathcal{P}\left([ -K,K]^{d_{j}^{U}}\right)$ are compact and the space $ ( M,d)$ is compact space. Closed set $ \Theta _{\infty } \subset M$ is also compact.'' The raw-network Lipschitz restriction does not bound the Lipschitz constant of $q/\|q\|_1$, so the categorical parameter class is not equicontinuous and its limits need not satisfy the population bound $\hat L_f$. ``\dots \[ \sup_{\theta \in \Theta_\infty}|g_{n}( \theta ) -g( \theta ) |\leqslant O\left(\log( nm_{n})\lambda _{n} +\delta _{n} +m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n}\right) = s_n, \] where $s_n$ is defined in \cref{thm:consistency}.'' This step uses Lemma~\ref{lemma:product_err}, which the paper assumes only for comparison measures that are products of their coordinate measures, but applies it to the paired empirical law $P_{m_n}^{\theta}(\boldsymbol U)$, which is not the product of its empirical marginals. In the step quoted in (b): The empirical observation error includes categorical Gumbel fluctuations that are absent from the retained non-Gumbel latent marginal errors. In the step quoted in (b): A positive-temperature Gumbel-softmax generator has a soft mixture output, rather than the displayed hard selected population mixture. In the step quoted in (b): Even a hard selector has random empirical component counts, so its empirical law is not a mixture of component empirical laws with the unchanged population weights. ``Let \[ \Theta _{\infty } =\overline{\cup _{n=1}^{\infty } \Theta _{n}} , g_{n}( \theta ) =S_{\lambda _{n}}\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) , f( \theta ) =\mathbb{E}_{t\sim \mu _{T}}\mathbb{E}_{\mathcal{M}^{\theta }} [F(V_{1} (t),\cdots ,V_{n_{V}} (t))] \] and $\tilde{\Theta}_\infty = \{\theta^*\} $.'' Common-draw empirical laws depend on the generator representation, which is not determined by the latent population laws in $\theta$, so the displayed definition does not determine $g_n(\theta)$. In the step quoted in (b): The population comparison bound does not control the discrepancies between representation-dependent empirical laws in the population metric $d$. The step quoted in (b) uses the current-depth pseudo-dimension bound and Lemma~\ref{lemma:sup_wasserstein}, which the paper assumes only for the current fixed generator architecture, but applies it to $\Theta_\infty$, which contains the closures of arbitrarily deep future generator classes. ``\dots \[ \mathcal{F}_L^K = \{f: \|f\|_\infty \leqslant K, f\ \text{Lipschitz continuous}\} \] and $ \mathcal{P}( K)$ is the probability space on $ K$.'' The displayed definition of $\mathcal F_L^K$ requires each function to be Lipschitz but supplies no common Lipschitz bound, so functions $x\mapsto\sin(rx)$ with arbitrarily large $r$ belong to it when $K=1$. ``Take the Wasserstein radius to be $\alpha_n = O(s_n + \epsilon_n)$, \cref{prop:convg_opt} implies the conclusion.'' The coordinatewise output bound used for compactness permits $f(x)=(K,K)$, which violates the uniform output $1$-norm bound $\sup_x\|f(x)\|_1\leqslant K$ in (\ref{eq:consistent_limiting}), so that compactness argument does not place every limit in its feasible class. ``\dots take the radius to be $\alpha_n = \epsilon_n+s_n, \epsilon_n = O(W_{0,n}^{-1/d^{\text{in}}_{\max}}+ \sum_{i=1}^2 L_{i,n}^{-2/d^U_{\max}}+ \tau _{n} \log \tau _{n}),$ \[ s_n = O(m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n} +n^{-1/{\max\{2,d_{\max}^U\}}}\log^2(n) + \log(nm_n)\lambda_n).\mbox{''} \] The reciprocal-dimensional exponents are undefined when $d_{\max}^{\text{in}}=0$ or $d_{\max}^U=0$. ``\dots we have verified the third assumption in Proposition \ref{prop:convg_opt}.'' The coordinatewise bound in \cref{assump:Lip_and_bounded} permits a continuous structural output $(K,K)$, whereas a uniform approximating sequence whose output $1$-norm is at most $K$ cannot converge to that output. In the step quoted in (b): The uniform sampling argument does not verify the class measurability condition required by the concentration theorem for its neural and categorical selector families; individual measurability alone does not justify this invocation. - **Counterexample.** Take the graph $T\rightarrow Y$ without bidirected edges, independent $U_T,U_Y\sim\operatorname{Unif}([-1,1])$, and independent standard categorical Gumbel noises. Set $T=U_T$ and $f_Y(t,u)=(1-r(t),r(t))$, where $r(t)=1/2+|t|/8$, with $Y\in\{1,2\}$, $K=2$, $L_f=1$, $F(T,Y)=Y$, and $\mu_T=\delta_0$. For $0<\delta\leqslant1$, let $q_\delta(t,u)=(\delta(1/2-|t|/8),\delta|t|/8+\min\{|t|,\delta\}/2)$ be the raw categorical network. Use exact affine generators for the latent laws, independent observational and common simulation samples, $m_n=n$, $W_{0,n}=n$, $L_{i,n}=\lceil n^{1/6}\rceil$, and $\tau_n=\lambda_n=n^{-2}$, with positive tolerances and sufficiently large sampling-radius constants of the printed orders. For each finite common simulation sample, choose $\delta_n>0$ smaller than every simulated $|T|$, which is possible almost surely. The true network is eventually feasible almost surely, and $q_{\delta_n}/\|q_{\delta_n}\|_1$ produces the identical simulated observations but objective value $1$ under treatment zero, so $F_n=1$ eventually, whereas $\underline F^2=F_*=3/2$. - **Rests on.** Lemma B.3, Lemma E.1, Lemma E.5, Lemma E.6, Proposition B.2, Theorem 3.3, Corollary 3.8, Theorem 3.6, also found wrong as printed. - **Correction.** Let $ \mathcal{M^*}$ be any SCM satisfying the assumptions of Corollary \ref{corollary:deep_nn}. Suppose that the Lipschitz constant of functions in $\mathcal{M}$ is $L_f$, $F:\mathbb{R}^{n_V}\rightarrow\mathbb{R}$ in (\ref{eq:opt_empiri}) is Lipschitz continuous and $\tau_n>0$, let $K>0$ be the constant in \cref{assump:Lip_and_bounded}, $\hat L_f=\sqrt{d_{\max}^{\text{in}}d_{\max}^{\text{out}}}L_f$, \begin{align*} \mathcal F_{0,n}&=\mathcal{NN}^{\hat L_f,K}\left(W_{0,n},\Theta\left(\log d_{\max}^{\text{in}}\right)\right),\qquad \mathcal F_{1,n}=\mathcal{NN}^{\infty,K}\left(\Theta(d_{\max}^U),L_{1,n},\Theta((d_{\max}^U)^2),L_{2,n},\tau_n\right), \end{align*} where $d_{\max}^U$, $d_{\max}^{\text{in}}$ and $d_{\max}^{\text{out}}$ are defined in Corollary \ref{corollary:deep_nn}. Suppose in addition that $d_{\max}^{\text{in}}\geqslant1$ and $d_{\max}^U\geqslant1$. In (\ref{eq:opt_empiri}), restrict each categorical raw structural network $q\in\mathcal F_{0,n}$ to satisfy $q(x)\in[0,\infty)^{n_i}$, $\|q(x)\|_1>0$ for every structural input $x$, and \[ \operatorname{Lip}\left(x\mapsto\frac{q(x)}{\|q(x)\|_1}\right)\leqslant\hat L_f. \] Put \[ D_U=\sum_{j=1}^{n_U}d_j^U,\qquad D=\max\{1,D_U\}, \] where the sum defining $D_U$ includes the non-Gumbel latent variables only. Take the radius to be $\alpha_n=\epsilon_n+s_n$, with positive tolerances chosen as sufficiently large constant multiples of the respective quantities in \[ \epsilon_n=O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+\sum_{i=1}^2L_{i,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right), \] \[ s_n=O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2 n+\log(nm_n)\lambda_n\right). \] For every continuous structural function $f_i$ of $\mathcal M^*$, suppose in addition that $\sup_x\|f_i(x)\|_1\leqslant K$ on its full structural input domain. Restrict every continuous structural network $\hat f_i\in\mathcal F_{0,n}$ in (\ref{eq:opt_empiri}) to satisfy this same uniform output $1$-norm bound. Retain the output bound in (\ref{eq:consistent_limiting}), namely \[ \mathcal F_L^K=\{f:\sup_x\|f(x)\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}. \] Let $F_n$ be the optimal value of the empirical optimization with the objective and Sinkhorn constraint of (\ref{eq:opt_empiri}) and both the continuous and categorical restrictions just specified, and let $\underline F^{\hat L_f}$ be the optimal value of (\ref{eq:consistent_limiting}) with this bounded Lipschitz class. If $ m_n = \Omega(n), L_{i,n} = \Theta(m_{n}^{d_{\max}^{U} /\left( 2d_{\max}^{U} +4\right)}), i=1,2, \lim _{n\rightarrow \infty }\min\{\tau_n^{-1}, W_{0,n},(\log(nm_n)\lambda_n)^{-1}\}= \infty$ , then with probability 1, $ [\liminf _{n\rightarrow \infty } F_{n},\limsup _{n\rightarrow \infty } F_{n}] \subset [\underline{F}^{\hat{L}_f} ,F_{*}]$. - **Proof.** \label{proof:consistency} We begin by defining a proper metric space. By \cref{assump:Lip_and_bounded}, all random variables are bounded. Suppose that $ \max_{i,j}\{\| V_{i} \| _{\infty} ,\| U_{j} \| _{\infty}\} \leqslant K$. A canonical causal model $ \mathcal{M} \in \mathcal{M}(\mathcal{G} ,\mathcal{F} ,\boldsymbol{U})$ is decided by \begin{equation*} \theta _{\mathcal{M}} =( f_{1} ,\cdots ,f_{n_{V}} ,{P}({U_{1}}) ,\cdots , {P}({U_{n_{U}}})) , \end{equation*} where $ f_{i}$ are functions in the structure equations (\ref{eq:structure_equation}) and $ U_{j}$ are uniform parts of the latent variables. We denote $ \mathcal{M}^{\theta }$ to be the SCM represented by $ \theta $, the underlying … (the corrected paper gives it in full) - **What the correction gives up.** The guarantee for the printed optimizer is replaced by a guarantee for candidates whose continuous structural functions have uniform output $1$-norm at most $K$ and whose normalized categorical propensities have Lipschitz constant at most $\hat L_f$. True models whose continuous structural functions violate that output $1$-norm bound on the input domain are excluded, even when every output coordinate is bounded by $K$. The sampling orders based only on $d_{\max}^U$ and the rate for the original $\alpha_n$ are given up. The displayed-order version excludes $d_{\max}^{\text{in}}=0$ and $d_{\max}^U=0$. (The paper's owner accepted the weaker statement.) - **Conditions.** The corrected statement states the conditions it uses itself; it no longer refers to 10 condition(s) of the paper it used as printed. - **Formal check.** proved in Lean (the corrected statement), given 6 cited result(s), under 66 condition(s) made explicit — `Papers.TbsNeuralCausalPi.consistency` (`TConsistency.lean:189`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Proposition 4.3 — corrected - **As printed.** \label{prop:Lip_ATE} Given two causal models $ \mathcal{M} ,\hat{\mathcal{M}} \in\mathcal{M}(\mathcal{G},\mathcal{F}_L,\boldsymbol{U})$ satisfying \cref{assump:independence_u} and \cref{assump:Lip_and_bounded}, let their observational distributions be $ \nu ,\mu $. Suppose the norms of all variables are bounded by $K>0$. If (1) (Overlap) $ \nu $ is absolutely continuous with respect to one probability measure $P$ and the density $ p_{\nu }\left( t|\text{Pa}( T) =x\right) \geqslant \delta >0 $ for x almost surely and $ t\in [ t_{1} ,t_{2}]$ and (2) (No Confounding) there is no hidden confounder in the causal graph $\mathcal{G}$, we have \begin{equation*} \begin{aligned} \int _{t_{1}}^{t_{2}}(\mathbb{E}_{\mathcal{M}}[ Y( t)] -\mathbb{E}_{\hat{\mathcal{M}} }[\hat{Y}( t)])^{2} P(dt) & \leqslant \frac{2C_{W}}{\delta } W( \mu ,\nu ) +2(L+1)^{n_{V}} W^{2}( \mu ,\nu )( t_{2} -t_{1}) , \end{aligned} \end{equation*} where $ C_{W} =4(L+1)^{n_{V}} K+2K\max\left\{(L+1)^{n_{V}} ,1\right\}$. - **What is wrong.** ``Note that $ f_{y}( x,t) =\mathbb{E}_{{\mathcal{M}} }[ Y|X=x,T=t]$ is $ L_{y}$-Lipschitz continuous with Lipschitz constant $ L_{y} \leqslant ( L+1)^{n_{V}}$.'' Lipschitz categorical propensity vectors can produce an outcome regression whose slope exceeds $(L+1)^{n_V}$ because distances between categorical labels enter the outcome mean. ``This is because $Y$ can be written as $ Y=F_{0}( X,T,\boldsymbol{U}_{y})$ where $ \boldsymbol{U}_{y}$ are latent variables independent of $ T,X$ and $ F_{0}$ is composition of $ f_{i}$ in structure equations.'' An observed ancestor can affect both $X=\text{Pa}(T)$ and $Y$, so its latent input to $Y$ need not be independent of $X$ even without hidden confounding. ``\dots \[ \begin{array}{@{}c@{}}\displaystyle \left(\int f_{y}( x,t) \nu ( dx) -\int \hat{f}_{y}( x,t) \nu ( dx)\right)^{2} \geqslant \frac{1}{2}\left(\int f_{y}( x,t) \nu ( dx) -\int \hat{f}_{y}( x,t) \mu ( dx)\right)^{2} \\ \displaystyle \ \ \ \ \ \ \ \ \ \ \ \ -\left(\int \hat{f}_{y}( x,t) \mu ( dx) -\int \hat{f}_{y}( x,t) \nu ( dx)\right)^{2} \\ \displaystyle \geqslant \frac{1}{2}\left(\int f_{y}( x,t) \nu ( dx) -\int \hat{f}_{y}( x,t) \mu ( dx)\right)^{2} -L_{y}^{2} W^{2}( \mu ,\nu ) \\ \displaystyle =\frac{1}{2}(\mathbb{E}_{\mathcal{M} }[ Y( t)] -\mathbb{E}_{\hat{\mathcal{M}} }[\hat{Y}( t)])^{2} -L_{y}^{2} W^{2}( \mu ,\nu ) .\end{array}\mbox{''} \] The observational bound $|\hat Y|\leqslant K$ bounds its regression only almost surely under its observational input law and need not bound a structural regression version that represents intervention means at inputs outside that law's support. ``\dots we get \[ \int _{t_{1}}^{t_{2}}(\mathbb{E}_{\mathcal{M} }[ Y( t)] -\mathbb{E}_{\hat{\mathcal{M}} }[\hat{Y}( t)])^{2} P(dt) \leqslant \frac{2C_W}{\delta } W( \mu ,\nu ) +2L_{y}^{2} W^{2}( \mu ,\nu )( t_{2} -t_{1}) .\mbox{''} \] The quadratic coefficient obtained here is $2L_y^2$, and the bound $L_y\leqslant(L+1)^{n_V}$ does not justify the statement's coefficient $2(L+1)^{n_V}$. In the step quoted in (d): Integrating a constant against $P(dt)$ on $[t_1,t_2]$ gives its multiple by $P([t_1,t_2])$, whereas the displayed remainder uses $t_2-t_1$ and the nondescendant calculation switches to Lebesgue measure. - **Counterexample.** Take the unconfounded continuous graph $T\rightarrow V_2\rightarrow V_3\rightarrow Y$, with $n_V=4$, $L=100$, $K=1$, $p=10^{-8}$, and $\eta=10^{-9}$. Let $U_1$ have law $P=(1-p)\operatorname{Unif}([0,\eta])+p\operatorname{Unif}([1-\eta,1])$, and let the other three singleton canonical noises be independent uniform variables on $[0,1]$, independent of $U_1$ and unused in the structural equations. Define $\phi(x)=\min\{\max\{100x,0\},1\}$. In $\mathcal{M}$, set $T=U_1$, $V_2=\phi(T)$, $V_3=\phi(V_2)$, and $Y=\phi(V_3)$. In $\hat{\mathcal{M}}$, use the same latent law and set $\hat T=\min\{U_1,\eta\}$, $\hat V_2=100\hat T$, $\hat V_3=100\hat V_2$, and $\hat Y=100\hat V_3$. Take $t_1=0$, $t_2=1$, and $\delta=1$, so that $T$ has law $P$ and the conditional density relative to $P$ is $p_\nu(t)=1$. Couple the models using their common latent variables, so their observed vectors agree on the first mixture component and each coordinate difference is at most $1$ on the second component, giving $W(\mu,\nu)\leqslant4p$. All observational variables and non-Gumbel latents are bounded by $1$ and all structural functions are at most $100$-Lipschitz, but on $[1-\eta,1]$ the intervention means are $1$ and $10^6t$, so the left-hand side is at least $p(10^6(1-\eta)-1)^2>9999$, whereas the printed right-hand side is at most $8C_Wp+32(101^4)p^2<50$ with $C_W=6(101^4)=624362406$. - **Correction.** Given two causal models $ \mathcal{M} ,\hat{\mathcal{M}} \in\mathcal{M}(\mathcal{G},\mathcal{F}_L,\boldsymbol{U})$ satisfying \cref{assump:independence_u} and \cref{assump:Lip_and_bounded}, let their observational distributions be $ \nu ,\mu $. Suppose the norms of all variables are bounded by $K>0$. If (1) (Overlap) $\nu$ is absolutely continuous with respect to one probability measure $P$ and the density $p_{\nu}\left(t|\text{Pa}(T)=x\right)\geqslant\delta>0$ for $x$ almost surely and $t\in[t_1,t_2]$ and (2) (No Confounding) there is no hidden confounder in the causal graph $\mathcal{G}$, we have \begin{equation*} \begin{aligned} \int_{t_1}^{t_2}(\mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}(t)])^2 P(dt) &\leqslant\frac{2C_W}{\delta}W(\mu,\nu)+2L_y^2W^2(\mu,\nu)P([t_1,t_2]), \end{aligned} \end{equation*} where \[ D=\max\left(\{1\}\cup\left\{\frac{n_i-1}{2}:V_i\text{ is categorical}\right\}\right),\qquad L_y=(LD+1)^{n_V}, \] \[ q=\sum_{i:V_i\text{ is continuous}}d_i+\left|\{i:V_i\text{ is categorical}\}\right|,\qquad B=K(1+2qL_y), \] and $C_W=4L_yB+2B\max\{L_y,1\}$. - **Proof.** If $Y$ is not a descendant of $T$, its structural equation and all the equations of its ancestors are unaffected by the intervention, so $\mathbb{E}_{\mathcal{M}}[Y(t)]=\mathbb{E}_{\mathcal{M}}[Y]$ and $\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}(t)]=\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}]$. For any coupling of the observational laws, the absolute difference of the outcome means is at most the expected absolute difference of the outcome coordinates, which is at most the expected $1$-norm distance of the observed vectors. Taking the infimum over couplings gives \begin{align*} \int_{t_1}^{t_2}(\mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}(t)])^2P(dt) … (the corrected paper gives it in full) - **Conditions.** The corrected statement states the conditions it uses itself; it no longer refers to 1 condition(s) of the paper it used as printed. - **Formal check.** proved in Lean (the corrected statement), under 8 condition(s) made explicit — `Papers.TbsNeuralCausalPi.lipschitz_dose_response_gap` (`TLipAte.lean:48`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Corollary 4.4 — corrected; the corrected statement is weaker - **As printed.** \label{cor:rate} Let $F, \mu_T$ in \cref{thm:consistency} to be $ F(\boldsymbol{V}) = Y, \mu_T \sim \text{Unif}([t_1, t_2]), \epsilon >0 $. Suppose that the assumptions in Proposition \ref{prop:Lip_ATE} and \cref{thm:consistency} hold, with probability at least $1-O(n^{-2})$, we have $ | F_n - F_*| \leqslant O(\sqrt{\alpha_n})$, where $F_n,F_*,\alpha_n$ are defined in \cref{thm:consistency}. - **What is wrong.** ``We get that \[ W(P^{\mathcal{M^*}}(\boldsymbol{V}),P^{\theta_n}(\boldsymbol{V}))\leqslant W(P_n^{\mathcal{M^*}}(\boldsymbol{V}),P^{\mathcal{M^*}}(\boldsymbol{V})) + W(P^{\mathcal{M^*}}(\boldsymbol{V}),P^{\theta_n}(\boldsymbol{V})) + W(P_{m_n}^{\theta_n}(\boldsymbol{V}),P^{\theta_n}(\boldsymbol{V})) \leqslant O(\alpha_n).\mbox{''} \] The middle term repeats the population Wasserstein distance instead of the distance between the two empirical laws, so the displayed inequality does not establish the asserted upper bound. ``By \cref{prop:Lip_ATE}, we get $ | F_n - F_*| \leqslant O(\sqrt{\alpha_n})$.'' Proposition~\ref{prop:Lip_ATE} bounds the squared response difference integrated against $P$, whereas the objective integrates against $\mu_T=\operatorname{Unif}([t_1,t_2])$, and the assumptions impose no relation between these measures. ``\dots (\ref{eq:opt_empiri}) has feasible solutions and \[ W(P_n^{\mathcal{M^*}}(\boldsymbol{V}),P^{\mathcal{M^*}}(\boldsymbol{V})) \leqslant O(\alpha_n),\quad W(P_{m_n}^{\theta_n}(\boldsymbol{V}),P^{\theta_n}(\boldsymbol{V})) \leqslant O(\alpha_n), \] where notations $\theta$ and $ P^\theta$ are defined in the proof of \cref{thm:consistency} and $\theta_n$ is one of the minimizers of (\ref{eq:opt_empiri}).'' The sampling assertion invokes uniform concentration without verifying its standing class measurability condition for the neural, normalized square-root and categorical indicator families used to establish feasibility and the uniform simulation estimate. - **Counterexample.** Take the unconfounded graph $T\rightarrow Y$ and independent $U_T,U_Y\sim\operatorname{Unif}([0,1])$. Let the true structural equations be $T=U_T$ and $Y=U_Y$, and take $K=2$, $L_f=1$, $t_1=0$, and $t_2=2$. Use the identity homeomorphism for each latent support, with $C_f=1$, so both latent laws satisfy the support and lower-bound assumptions. Take $P=\operatorname{Unif}([0,1])$ and the conditional density version $p_\nu(t)=1$ on $[0,2]$, which represents the treatment law relative to $P$ and satisfies overlap with $\delta=1$. Take the candidate equations $\hat T=\hat U_T$ and $\hat Y=\max\{-2,\min\{\hat U_Y-\max\{\hat T-1,0\},2\}\}$, with independent uniform latent laws generated by identity networks. Choose $m_n=n$, $W_{0,n}=n$, $L_{1,n},L_{2,n}=\Theta(n^{1/6})$, $\tau_n=n^{-1}$, and $\lambda_n=n^{-2}$, and take $\epsilon_n=c_1n^{-1/3}$ and $s_n=c_2n^{-1/3}\log n$ with sufficiently large fixed positive $c_1,c_2$, so that $d_{\max}^U=1$, $d_{\max}^{\text{in}}=2$, and the printed radius prescription holds with $\alpha_n=\epsilon_n+s_n=\Theta(n^{-1/3}\log n)$. Let $\rho=\operatorname{Unif}([0,1]^2)$ be the common observational law of the true and candidate models. Form the true and candidate empirical observational laws from independent samples of $n$ independent complete observations with law $\rho$. Let $x_n=Cn^{-1/2}\log^2 n$ for a fixed sufficiently large $C>0$. Let $E_n$ be the event that both $W(P_n^{\mathcal M^*}(\boldsymbol V),\rho)\leqslant x_n$ and $W(P_n^{\hat{\mathcal M}}(\boldsymbol V),\rho)\leqslant x_n$. The bounded two-dimensional concentration inequality in Theorem 2 of \cite{fournier2015rate} gives $\mathbb P(E_n^c)\leqslant C_0\exp(-c_0n(x_n/\log(2+1/x_n))^2)=O(n^{-3})$, and on $E_n$ the triangle inequality and Lemma~\ref{lemma:sinkhorn_approx} give $S_{\lambda_n}(P_n^{\mathcal M^*}(\boldsymbol V),P_n^{\hat{\mathcal M}}(\boldsymbol V))\leqslant2x_n+2(1+2\log n)n^{-2}=o(n^{-1/3}\log n)\leqslant\alpha_n$ eventually, so the candidate is feasible with probability at least $1-O(n^{-2})$ while its uniform intervention objective is $1/2-\frac12\int_1^2(t-1)\,dt=1/4$ and $F_*=1/2$, whence $F_n\leqslant1/4$ and $|F_n-F_*|\geqslant1/4$ contradict the asserted rate as $\alpha_n\rightarrow0$; the response difference vanishes $P$-almost surely but is nonzero on $(1,2]$. - **Rests on.** Lemma B.3, Lemma E.1, Lemma E.5, Lemma E.6, Proposition B.2, Proposition 4.3, Theorem 3.3, Theorem 4.2, Corollary 3.8, Theorem 3.6, also found wrong as printed. - **Correction.** Let $F,\mu_T$ in \cref{thm:consistency} be $F(\boldsymbol V)=Y$, $\mu_T\sim\operatorname{Unif}([t_1,t_2])$, and $\epsilon>0$. Suppose that $t_10$ for every structural input $x$, and \[ \operatorname{Lip}\left(x\mapsto\frac{q(x)}{\|q(x)\|_1}\right)\leqslant\hat L_f. \] For each sample size, obtain every candidate empirical law by evaluating the candidate on the same $m_n$ independent draws of its uniform and Gumbel exogenous inputs, independent of the observational sample. Let $F_n$ be the optimal value of this restricted optimization, retaining the objective and Sinkhorn constraint of (\ref{eq:opt_empiri}). Put \[ D_U=\sum_{j=1}^{n_U}d_j^U,\qquad D=\max\{1,D_U\}, \] where the sum includes only the non-Gumbel latent variables. Take $\alpha_n=\epsilon_n+s_n$, with positive tolerances chosen as sufficiently large constant multiples of the respective quantities in \[ \epsilon_n=O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+\sum_{i=1}^2L_{i,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right), \] \[ s_n=O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2n+\log(nm_n)\lambda_n\right). \] Then, with probability at least $1-O(n^{-2})$, we have $|F_n-F_*|\leqslant O(\sqrt{\alpha_n})$, where $F_*$ is defined in \cref{thm:consistency} and $F_n,\alpha_n$ are specified here. - **Proof.** We establish feasibility and the complete-observation sampling bounds for the specified restricted optimization. Write $\eta$ for a network representation, $\theta_\eta$ for its population SCM parameter, and $\theta^*$ for the true parameter. For two population parameters define \[ d(\theta,\theta')=\sum_{i=1}^{n_V}\|f_i-f'_i\|_\infty+\sum_{j=1}^{n_U}W(P(U_j),P(U'_j)), \] using normalized categorical propensity functions in the first sum. Let $\mathcal A_n$ be the set of admissible representations. With common independent exogenous draws $\xi_1,\ldots,\xi_{m_n}$, let $H_\eta(\xi_l)$ be the complete observed vector and put \[ … (the corrected paper gives it in full) - **What the correction gives up.** The corrected statement excludes uniform intervention laws that are not absolutely continuous with respect to $P$ or whose density is not square-integrable, true models or continuous candidates whose structural functions violate $\sup_x\|f_i(x)\|_1\leqslant K$ on their full input domains even when every coordinate is bounded by $K$, categorical candidates whose normalized propensity functions exceed $\hat L_f$, and candidate-specific independently resampled simulation experiments; it replaces the printed sampling orders determined only by $d_{\max}^U$ with orders accounting for the joint latent dimension and does not retain the sample-size guarantee for the original estimator and radius. The displayed-order version also excludes $d_{\max}^{\text{in}}=0$ and $d_{\max}^U=0$, and the uniform intervention law requires $t_10$, and coordinate regrouping preserves any global Lipschitz regularity these normalized functions possess. - **Proof.** Let $\boldsymbol G=\{G_{V_i}:V_i\text{ is categorical}\}$. By \cref{assump:independence_u} and \cref{assump:categorical_rv}, the remaining latent variables $U_1,\ldots,U_{n_U}$ and the Gumbel coordinates are independent. For a categorical variable, put $s_i(x)=\|f_i(x)\|_1$ and $\pi_i(x)=f_i(x)/s_i(x)$. The categorical generation assumption requires nonnegative coordinates and $s_i(x)>0$ on its structural input domain. For positive coordinates, \[ \log\pi_{i,j}(x)=\log f_{i,j}(x)-\log s_i(x). \] The subtracted term is common to every coordinate, so replacing $f_i$ by $\pi_i$ leaves its Gumbel argmax unchanged. Zero coordinates remain zero and have extended logarithm $-\infty$. Thus this … (the corrected paper gives it in full) - **What the correction gives up.** Preservation of the original global Lipschitz regularity for unnormalized categorical functions on unrestricted domains, including cases where their total mass approaches zero. (An intermediate result: its strongest true version was adopted.) - **Formal check.** proved in Lean (the corrected statement), under 7 condition(s) made explicit — `Papers.TbsNeuralCausalPi.exists_canonical_representation` (`TEquivalentGConstrianed.lean:39`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Proposition B.2 — statement unchanged; proof corrected - **As printed.** \label{prop:holder_curve} Let $ \lambda $ be the Lebesgue measure on $ [ 0,1]$. Given any probability measure $ \mathbb{P}$ that satisfies \cref{assumption:lower_bound}, there exists a continuous curve $ \gamma : [0,1]\rightarrow \text{supp}(\mathbb{P})$ such that $\gamma _{\#} \lambda =\mathbb{P}$. Furthermore, if $d \geqslant 1$, $\gamma$ is $ 1/d$-Hölder continuous. - **What is wrong.** ``$ C_{i_{1} ,i_{2} \cdots ,i_{n-1}} =\bigcup _{i_{n}} C_{i_{1} ,i_{2} \cdots ,i_{n}}$ and $ \text{diam}( C_{i_{1} ,i_{2} \cdots ,i_{n-1}}) \leqslant 2/3\cdot \text{diam}( C_{i_{1} ,i_{2} \cdots ,i_{n}})$.'' The contraction condition admits a literal parent-smaller-than-child reading and an intended child-contraction reading, while the later child-contraction display and uniform-convergence argument determine the latter. ``$ \mathbb{P}( \partial C_{i_{1} ,i_{2} \cdots ,i_{n}}) =0$ and $ R(\mathcal{C}_{n+1}) \leqslant \left( 1+2^{-n}\right) R(\mathcal{C}_{n})$.'' The condition $\mathbb P(\partial C)=0$ need not hold for the standard dyadic cells, because the hypotheses allow atoms on their internal faces. ``\par \textbf{Step 2: Construct Hilbert Curve.} By \cref{lemma:hilbert}, there exist an sequence of ordering $ \{O_{n}\}$ of $ \{\mathcal{C}_{n}\}$ that satisfies'' This step uses the compatible face-adjacent orderings of Lemma~\ref{lemma:hilbert}, which the paper assumes only for standard dyadic grids of closed cubes, but applies it to the cells obtained by perturbing cutting planes to make their boundaries have zero $\mathbb P$-measure. ``\dots where $p_i = \mathbb{P}(C_{O_{n+1}(i)})$.'' Assigning interval lengths $\mathbb P(C)$ generally gives unequal lengths, whereas the uniform Hilbert-curve integral and the same-or-successive-interval bound require lengths $2^{-nd}$. ``$ \mathcal{I}_{n}$ and $ \mathcal{C}_{n}$ in the above construction correspond to developments $ \alpha _{k} ,\beta _{k}$ in \cite[Theorem 1]{Shchepin2010OnHM}.'' This step uses the development hypothesis of Theorem 1 in \cite{Shchepin2010OnHM}, which the paper assumes only for sequences of covers whose maximum diameter tends to zero, but applies it to the mass-assigned interval covers $\mathcal I_n$, which may contain intervals of uniformly positive length when $\mathbb P$ has an interior atom. ``Since $ \mathcal{I}_{k}$ consists of one-dimensional intervals, we have $ \text{diam}(\mathcal{I}_{k}) =\text{gap}(\mathcal{I}_{k})$.'' The maximum interval diameter is not the gap between disjoint intervals, and lower domination alone does not bound their ratio. - **Counterexample.** Take $d=D=1$, $K=[0,1]$, $f(x)=x$, $L=1$, and $C_f=1/2$. Let $\mathbb P_0=(\lambda+\delta_{1/2})/2$. Let $\mathbb P_1=(\lambda+\delta_{1/3})/2$. Let $\mathbb P_2$ have density $1/2+1/(4\sqrt{x})$ on $00$ and all measurable sets $B\in[0,1]^d$. Suppose we have proven the conclusion for this case. By \cref{assumption:lower_bound}, there exists $ f\in \mathcal{H}([0,1]^{d},K) \cap \mathcal{F}_L([0,1]^{d},K)$ such that $f^{-1}_{\#} \mathbb{P}$ satisfies these conditions. There exists continuous function $\gamma$ such that $f^{-1}_{\#}\mathbb{P}=\gamma_{\#}\lambda$, which implies $\mathbb{P} = (f\circ\gamma)_{\#}\lambda$. In particular, if $\gamma$ is $1/d$-Holder continuous, $f\circ\gamma$ … (the corrected paper gives it in full) - **Conditions.** The corrected statement states the conditions it uses itself; it no longer refers to 1 condition(s) of the paper it used as printed. - **Formal check.** proved in Lean, under 10 condition(s) made explicit — `Papers.TbsNeuralCausalPi.exists_measure_representing_holder_curve` (`Helpers/HilbertCurve.lean:386`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Lemma B.3 — corrected; the corrected statement is weaker - **As printed.** \label{lemma:hilbert} Given a sequence of partitions of the unit cube $ \{\mathcal{C}_{n}\}$ that has property (*), there exist a ordering for each partition $ \mathcal{C}_{n}$, i.e., $$ O_{n} :\left\{1,\cdots ,2^{nd}\right\}\rightarrow \left\{\overline{i_{1} i_{2} \cdots i_{n}} :i_{j} =0,\cdots ,2^{d} -1\right\},$$ such that the following two conclusions hold. \begin{enumerate} \item If $ ( j-1) \cdotp 2^{d} < i\leqslant j\cdotp 2^{d}$, the first $ n$ digits of $ O_{n+1}( i)$ are the same as $ O_{n}( j)$. \item Adjacent cubes in the ordering intersect, i.e., $ C_{O_{n}( i)} \cap C_{O_{n}( i+1)} \neq \emptyset $. Furthermore, $ C_{O_{n}( i)} \cap C_{O_{n}( i+1)}$ is a $d-1$ dimensional cube. \end{enumerate} - **What is wrong.** ``\dots i.e., \[ O_{n} :\left\{1,\cdots ,2^{nd}\right\}\rightarrow \left\{\overline{i_{1} i_{2} \cdots i_{n}} :i_{j} =0,\cdots ,2^{d} -1\right\}, \] such that the following two conclusions hold.'' Property (*) allows refinement in only one coordinate at a time, whereas the cited Hilbert construction supplies the asserted compatible orderings on dyadic lattices. - **Counterexample.** Take $d=2$. Let $\mathcal C_n$ be the closed rectangular grid with $4^{\lceil n/2\rceil}$ columns and $4^{\lfloor n/2\rfloor}$ rows. These grids satisfy property (*), but at levels $1$, $2$, and $3$ the required horizontal and vertical traversals force an interior first-level column to enter and exit on the same horizontal extreme, contradicting the required passage between its two neighboring columns. - **Correction.** Given a sequence of partitions of the unit cube $\{\mathcal C_n\}$ that has property (*), suppose that, when $d\geqslant1$, $\mathcal C_n$ is the standard dyadic grid of closed cubes \[ \mathcal C_n=\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}. \] There exists an ordering for each partition $\mathcal C_n$, i.e., \[ O_n:\{1,\ldots,2^{nd}\}\longrightarrow\{\overline{i_1\cdots i_n}:i_j=0,\ldots,2^d-1\}, \] such that the following two conclusions hold. \begin{enumerate} \item If $ ( j-1) \cdotp 2^{d} < i\leqslant j\cdotp 2^{d}$, the first $ n$ digits of $ O_{n+1}( i)$ are the same as $ O_{n}( j)$. \item Adjacent cubes in the ordering intersect, i.e., $ C_{O_{n}( i)} \cap C_{O_{n}( i+1)} \neq \emptyset $. Furthermore, $ C_{O_{n}( i)} \cap C_{O_{n}( i+1)}$ is a $d-1$ dimensional cube. \end{enumerate} - **Proof.** For $d=0$, every partition has its single point as its only cell. Choose the unique ordering; the first conclusion holds and there are no adjacent pairs. For $d\geqslant1$, use the recursive Hilbert ordering on the dyadic lattice described in Section 2 of \cite{hamiltonCompactHilbertIndices2007}. That construction visits every cell once, visits all the children of a cell consecutively before moving to another cell, and joins successive cells across a coordinate face. Label each child by its position among the $2^d$ children of its parent, and define $O_n(i)$ to be the resulting address of the $i$th cell at level $n$. Visiting every cell once makes $O_n$ a bijection. Completing the children … (the corrected paper gives it in full) - **What the correction gives up.** The assertion of compatible face-adjacent orderings for arbitrary non-dyadic coordinate-box refinements satisfying property (*). (An intermediate result: its strongest true version was adopted.) - **Formal check.** proved in Lean (the corrected statement), under 5 condition(s) made explicit — `Papers.TbsNeuralCausalPi.hilbert_ordering` (`Helpers/HilbertCurve.lean:121`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Proposition C.1 — corrected; the corrected statement is weaker - **As printed.** \label{prop:counterexample_foral} There exists a constant $c>0$ and an SCM $\mathcal{M}^*$ satisfying Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound} with a backdoor causal graph such that there is no unobserved confounding in $\mathcal{M}^*$. For any $\epsilon >0$, there exists an SCM $\mathcal{M}_\epsilon$ with the same causal graph such that $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V})) \leq \epsilon $ and $|\text{ATE}_{\mathcal{M}^*} - \text{ATE}_{\mathcal{M}_\epsilon}| > c $. - **What is wrong.** ``It is easy to see that $\mathcal{M}^* $ satisfies Assumption \ref{assump:independence_u}-\ref{assumption:lower_bound}.'' The assertion does not specify whether Assumption~\ref{assumption:lower_bound} concerns each whole latent law or its normalized restrictions to connected support components, and these readings differ for the discrete latent laws in the construction. - **Counterexample.** Take the latent variable $U_1$ in the displayed construction, with $P(U_1=1)=3/5$ and $P(U_1=0)=2/5$. Its support components are $\{0\}$ and $\{1\}$, and its normalized restrictions to them are the corresponding point masses. Each normalized component law satisfies Assumption~\ref{assumption:lower_bound} through a chart from $[0,1]^0$, whereas the whole law cannot satisfy it because its disconnected support cannot be homeomorphic to any cube $[0,1]^d$. - **Correction.** \label{prop:counterexample_foral} There exists a constant $c>0$ and an SCM $\mathcal{M}^*$ satisfying Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound} with a backdoor causal graph such that there is no unobserved confounding in $\mathcal{M}^*$. For any $\epsilon >0$, there exists an SCM $\mathcal{M}_\epsilon$ with the same causal graph such that $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V})) \leq \epsilon $ and $|\text{ATE}_{\mathcal{M}^*} - \text{ATE}_{\mathcal{M}_\epsilon}| > c $. Here, for each non-Gumbel latent law $\mathbb P$ with connected support components $C_i$, Assumption~\ref{assumption:lower_bound} is required of each normalized component law $\mathbb P|_{C_i}/\mathbb P(C_i)$, with $K=C_i$. - **Proof.** Let $ \boldsymbol{V} = \{ X,T,Y\} $ be the covariate, treatment and outcome respectively. $ T$ is a binary variable. Let structure equations of causal model $ \mathcal{M}_{\delta }$ be \begin{align*} X & =\begin{cases} -1 & ,\text{w.p.} \ 1/2\\ 0 & ,\text{w.p.} \ 1/4\\ \delta & ,\text{w.p.} \ 1/4, \end{cases}\\ P( T=1|X=x) & =\begin{cases} 1/2 & ,x=-1\ \text{or} \ 0,\\ 3/4 & ,x=\delta , \end{cases}\\ Y & =\begin{cases} T+U_{y} & ,X< 0\\ X/\delta +U_{y} & ,X\geqslant 0 \end{cases} \end{align*}where all latent variables are independent and $ U_{y}$ is mean-zero noise. The distribution of $ \mathcal{M}_{\delta }$ is \begin{align*} P_{X,T,Y}( 0,1,y) =p_{U_{y}}( y) /8, & \ P_{X,T,Y}( 0,0,y) … (the corrected paper gives it in full) - **What the correction gives up.** The requirement that each entire non-Gumbel latent law satisfy Assumption~\ref{assumption:lower_bound}; only its normalized connected-component laws are required to satisfy it. (The paper's owner accepted the weaker statement.) - **Formal check.** proved in Lean (the corrected statement), under 7 condition(s) made explicit — `Papers.TbsNeuralCausalPi.counterexample_backdoor` (`TCounterexampleForal.lean:60`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Lemma E.1 — corrected - **As printed.** \label{lemma:push-forward_err} Let $ \mu ,\hat{\mu }$ be two measures on compact set $ K\subset \mathbb{R}^{d_{1}}$, for any measurable functions $ F,\hat{F} :K\rightarrow \mathbb{R}^{d_{2}}$, if $ F$ is $ L$-Lipschitz continuous, then \begin{equation*} W( F_{\#} \mu ,\hat{F}_{\#}\hat{\mu }) \leqslant LW( \mu ,\hat{\mu }) +\| F_{1} -F_{2} \| _{\infty } . \end{equation*} - **What is wrong.** ``\dots \[ \begin{array}{@{}c@{}}\displaystyle \mathbb{E}_{X\sim \hat{\mu }}[ g\circ F( X)] -\mathbb{E}_{X\sim \hat{\mu }}[ g\circ \hat{F}( X)] \leqslant \mathbb{E}_{X\sim \hat{\mu }}[ |g\circ F( X) -g\circ \hat{F}( X) |] \\ \displaystyle \leqslant \mathbb{E}_{X\sim \hat{\mu }}[ \| F( X) -\hat{F}( X) \| ] \\ \displaystyle \leqslant \| F( X) -\hat{F}( X) \| _{\infty }.\end{array}\mbox{''} \] The scalar Lipschitz test function bounds the output $1$-norm, whose comparison with the coordinatewise infinity norm requires the factor $d_2$. ``Combine (\ref{eq:first_term}) and (\ref{eq:second_term}), we have the conclusion.'' Combining the two calculated terms compares $F$ with $\hat F$, and cannot establish a bound involving the undefined maps $F_1,F_2$ in the statement. - **Counterexample.** Let $K=[0,1]$ and let $\mu=\hat\mu$ be the uniform probability measure on $K$. Set $F(x)=(x,x)$ and $\hat F(x)=(x+1,x+1)$. Then $F$ is $2$-Lipschitz, $W(F_{\#}\mu,\hat F_{\#}\hat\mu)=2$, and $\|F-\hat F\|_\infty=1$, so even identifying the intended maps leaves the printed coefficient false. - **Correction.** Let $\mu,\hat\mu$ be two measures on compact set $K\subset\mathbb R^{d_1}$, for any measurable functions $F,\hat F:K\rightarrow\mathbb R^{d_2}$, if $F$ is $L$-Lipschitz continuous, then \begin{equation*} W(F_{\#}\mu,\hat F_{\#}\hat\mu)\leqslant LW(\mu,\hat\mu)+d_2\|F-\hat F\|_\infty. \end{equation*} - **Proof.** For any $ 1$-Lipschitz function $ g:\mathbb{R}^{d_{2}}\rightarrow \mathbb{R}$, \begin{align*} \mathbb{E}_{X\sim F_{\#} \mu }[ g( X)] -\mathbb{E}_{X\sim \hat{F}_{\#}\hat{\mu }}[ g( X)] & =\mathbb{E}_{X\sim \mu }[ g\circ F( X)] -\mathbb{E}_{X\sim \hat{\mu }}[ g\circ \hat{F}( X)]\\ & =(\mathbb{E}_{X\sim \mu }[ g\circ F( X)] -\mathbb{E}_{X\sim \hat{\mu }}[ g\circ F( X)]) \\ &\quad +(\mathbb{E}_{X\sim \hat{\mu }}[ g\circ F( X)] -\mathbb{E}_{X\sim \hat{\mu }}[ g\circ \hat{F}( X)]) . \end{align*} Note that for any $x,y\in K$, \begin{align*} |g\circ F(x)-g\circ F(y)|\leqslant\|F(x)-F(y)\|\leqslant L\|x-y\|. \end{align*} Thus $g\circ F$ is $L$-Lipschitz continuous and … (the corrected paper gives it in full) - **Conditions.** The corrected statement states the conditions it uses itself; it no longer refers to 1 condition(s) of the paper it used as printed. - **Formal check.** proved in Lean (the corrected statement), under 2 condition(s) made explicit — `Papers.TbsNeuralCausalPi.wasserstein_map_le` (`Helpers/WassersteinBounds.lean:154`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Lemma E.5 — corrected - **As printed.** \label{lemma:sinkhorn_approx} For any $ \mu \in \Delta _{n} ,\nu \in \Delta _{m}$ and $ \lambda >0$, we have \begin{align*} 0\leqslant W_{1,\lambda}( \mu ,\nu ) -W( \mu ,\nu ) & \leqslant (\log( mn) +1) \lambda, \end{align*} where $ W_{1,\lambda}(\cdot,\cdot) $ is the entropy regularized Wasserstein-1 distance. Moreover, we have the following estimation of approximation error. \begin{align*} |S_{\lambda }( \mu ,\nu ) -W( \mu ,\nu ) | & \leqslant 2(\log( mn) +1) \lambda . \end{align*} - **What is wrong.** ``\dots we have \[ 0\leqslant W_{1,\lambda }( \mu ,\nu ) -W( \mu ,\nu ) \leqslant c\lambda , \] where $ c=\max\left\{h( T) :\text{transportation plan} \ T\ \text{is achieve optimal loss} \ W( \mu ,\nu )\right\}$.'' The cited work distinguishes the entropy-penalized optimal value from the transport cost at its optimizer, and its full regularized value has the opposite sign of error from the asserted lower bound. - **Counterexample.** Take support points $0,1$ with ground distance $|x-y|$. Let $\mu=(3/4,1/4)$ and $\nu=(1/4,3/4)$. Let $T^*=\left(\begin{smallmatrix}1/4&1/2\\0&1/4\end{smallmatrix}\right)$. Then $W(\mu,\nu)=1/2$ and $h(T^*)=1+(3/2)\log2$, so the source's full regularized value satisfies $W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)\leqslant-\lambda(1+(3/2)\log2)<0$, contrary to the printed lower bound. - **Correction.** For any $\mu\in\Delta_n$, $\nu\in\Delta_m$, and $\lambda>0$, let $\mu$ and $\nu$ be the weights of finite probability measures on points $x_1,\ldots,x_n$ and $y_1,\ldots,y_m$ in a common Euclidean space with the paper's $1$-norm. Put $c_{ij}=\|x_i-y_j\|$ and \[ \Pi(\mu,\nu)=\left\{T\in[0,\infty)^{n\times m}:\sum_jT_{ij}=\mu_i,\ \sum_iT_{ij}=\nu_j\right\},\qquad h(T)=1-\sum_{i,j}T_{ij}\log T_{ij}, \] with $0\log0=0$. Here the entropy regularized Wasserstein-1 distance is \[ W_{1,\lambda}(\mu,\nu)=\min_{T\in\Pi(\mu,\nu)}\left\{\sum_{i,j}c_{ij}T_{ij}-\lambda h(T)\right\}, \] and $W(\mu,\nu)=\min_{T\in\Pi(\mu,\nu)}\sum_{i,j}c_{ij}T_{ij}$. Then \begin{align*} -(\log(mn)+1)\lambda\leqslant W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)\leqslant0. \end{align*} Moreover, we have the following estimation of approximation error. \begin{align*} |S_{\lambda }( \mu ,\nu ) -W( \mu ,\nu ) | & \leqslant 2(\log( mn) +1) \lambda . \end{align*} - **Proof.** The transportation polytope is nonempty and compact, and both objectives are continuous under $0\log0=0$, so their minima exist. Let $T^*$ minimize the unregularized transport cost and let $T_\lambda$ minimize the regularized objective. Since $h(T)\geqslant0$ for every probability plan, \[ W_{1,\lambda}(\mu,\nu)\leqslant\sum_{i,j}c_{ij}T^*_{ij}-\lambda h(T^*)\leqslant W(\mu,\nu). \] Also $\sum_{i,j}c_{ij}(T_\lambda)_{ij}\geqslant W(\mu,\nu)$. It follows that \begin{align}\label{eq:err_reg_wass} -\lambda\max_{T\in\Pi(\mu,\nu)}h(T)\leqslant W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)\leqslant0. \end{align} Since $ T_{ij} \in [ 0,1]$ and $ f( x) =-x\log x$ is concave, we have \begin{align*} h( T) = & … (the corrected paper gives it in full) - **Conditions.** The corrected statement states the conditions it uses itself; it no longer refers to 2 condition(s) of the paper it used as printed. - **Formal check.** proved in Lean (the corrected statement), under 1 condition(s) made explicit — `Papers.TbsNeuralCausalPi.sinkhorn_approx` (`Helpers/Sinkhorn.lean:164`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Lemma E.6 — corrected - **As printed.** \label{lemma:sup_wasserstein} Given a measure $ \mu $ on $ \mathbb{R}^{d}$ and a real function class $ \mathcal{F}$ with output dimension $ d$, suppose that the pseudo-dimension of $ \mathcal{F}$ is less than $ d_{\mathcal{F}} < \infty $ and there exists $K>0$ such that $ |f( x) |\leqslant K,\forall f\in \mathcal{F} ,x\in \mathbb{R}^{d'}$, then with probability at least $ 1-\exp\left( O\left( \delta ^{-d}\log\left( \delta ^{-1}\right) +d_{\mathcal{F}}\log \epsilon ^{-1} -n\epsilon ^{2}\right)\right)$, \begin{align*} \sup _{f\in \mathcal{F}} W( f_{\#} \mu ,f_{\#} \mu _{n}) & \leqslant \delta +\epsilon \end{align*} for all $ \delta ,\epsilon >0$, where $ \mu _{n}$ is the empirical distribution of $ \mu $. - **What is wrong.** ``\dots \[ P\left(\sup _{f\in \mathcal{F}}\mathbb{E}_{X\sim \mu _{n}}[ h\circ f( X)] -\mathbb{E}_{X\sim \mu }[ h\circ f( X)] >\epsilon \right) \leqslant \exp\left( O\left( d_{\mathcal{F}}\log \epsilon ^{-1} -n\epsilon ^{2}\right)\right) .\mbox{''} \] The scalar source theorem does not identify whether the vector pseudo-dimension means the maximum coordinate pseudo-dimension or the pseudo-dimension on the augmented domain of coordinate indices. In the step quoted in (a): The displayed covering bound omits the factor of sample size required by the explicitly unnormalized empirical $1$-norm. The step quoted in (a) uses Theorem 18.4 of \cite{anthony1999neural}, which the paper assumes only for scalar functions with range $[0,1]$ in the averaged empirical metric, but applies it to the unnormalized evaluation vectors of functions bounded by an arbitrary $K$. In the step quoted in (a): The invocation of Theorem 17.1 of \cite{anthony1999neural} omits its standing measurability condition on the function class, which does not follow from measurability of each function and finite pseudo-dimension. The step quoted in (a) uses Theorem 17.1 of \cite{anthony1999neural}, which the paper assumes only for squared losses of $[0,1]$-valued functions with responses in $[0,1]$, but applies it to the unnormalized scalar expectations of $h\circ f$. - **Counterexample.** Let $d=d'=1$, $\mu=\operatorname{Unif}([-1,1])$, $K=1$, and $h(x)=x$. Let $\mathcal F=\{f_c:c\in[0,1/2]\}$, where $f_c(x)=c+\tfrac12\mathbb I_{\{x\geqslant0\}}$, and take $d_{\mathcal F}=2$. Take $n=10000$ and $\epsilon=1/4$. Evaluation vectors have distance $n|c-c'|$, so $\mathcal N_1(1/4,\mathcal F,10000)\geqslant10000$, whereas the attributed bound is $192e^3<4000$; increasing the range also violates that bound even in the averaged metric. - **Correction.** Given a probability measure $\mu$ on $\mathbb R^{d'}$ and a class $\mathcal F$ of measurable functions $f:\mathbb R^{d'}\rightarrow\mathbb R^d$, suppose that each scalar coordinate class $\mathcal F_j=\{x\mapsto f_j(x):f\in\mathcal F\}$, $j\in[d]$, has pseudo-dimension less than $d_{\mathcal F}<\infty$ and there exists $K>0$ such that $\|f(x)\|_\infty\leqslant K$ for all $f\in\mathcal F$ and $x\in\mathbb R^{d'}$. Suppose in addition that there are a standard Borel space $T$ and a jointly Borel map $G:T\times\mathbb R^{d'}\rightarrow[-K,K]^d$ such that $\mathcal F=\{G(t,\cdot):t\in T\}$. There are positive constants $C,c$ depending only on $d,K$ such that, with probability at least \[ 1-\exp\left(C\left((1+\delta^{-d})\log(2+\delta^{-1})+d_{\mathcal F}\log(2+\epsilon^{-1})\right)-cn\epsilon^2\right), \] \begin{align*} \sup_{f\in\mathcal F}W(f_{\#}\mu,f_{\#}\mu_n)\leqslant\delta+\epsilon \end{align*} for all $\delta,\epsilon>0$, where $\mu_n$ is the empirical distribution of $\mu$. - **Proof.** The empty class satisfies the uniform assertion vacuously, so assume $T\ne\varnothing$. Let $X_1,\ldots,X_n$ be independent observations with law $\mu$. By the dual formulation of Wasserstein distance, \begin{align*} \sup_{f\in\mathcal F}W(f_{\#}\mu,f_{\#}\mu_n) =\sup_{h\in\mathcal H}\sup_{f\in\mathcal F}\left(\mathbb E_{X\sim\mu_n}h(f(X))-\mathbb E_{X\sim\mu}h(f(X))\right), \end{align*} where \[ \mathcal H=\{h:[-K,K]^d\rightarrow\mathbb R:h(0)=0,\ |h(x)-h(y)|\leqslant\|x-y\|\}. \] Every such $h$ takes values in $[-dK,dK]$. Retain the unnormalized empirical covering number \begin{align*} \mathcal N_1(r,\mathcal F,N)=\sup_{x_1,\ldots,x_N}\mathcal … (the corrected paper gives it in full) - **Conditions.** The corrected statement states the conditions it uses itself; it no longer refers to 3 condition(s) of the paper it used as printed. - **Formal check.** proved in Lean (the corrected statement), given 2 cited result(s), under 3 condition(s) made explicit — `Papers.TbsNeuralCausalPi.uniform_empirical_wasserstein` (`Helpers/EmpiricalWasserstein.lean:190`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Conditions and definitions restated Conditions or definitions the paper states too loosely for the results that use them; the corrected results now state what they use. - The cited form of Theorem 17.1 of \cite{anthony1999neural} asserts a uniform squared-loss concentration bound for every class of measurable $[0,1]$-valued functions on any measurable input space. ``\dots \[ P\left(\sup _{f\in \mathcal{F}}\mathbb{E}_{X\sim \mu _{n}}[ h\circ f( X)] -\mathbb{E}_{X\sim \mu }[ h\circ f( X)] >\epsilon \right) \leqslant \exp\left( O\left( d_{\mathcal{F}}\log \epsilon ^{-1} -n\epsilon ^{2}\right)\right) .\mbox{''} \] This step uses Theorem 17.1 of \cite{anthony1999neural}, which the paper assumes only for bounded scalar squared loss on a joint input-response law satisfying the standing measurability conditions, but applies it to the unnormalized expectation class in the printed proof. In the step quoted in (a): Individual measurability of the functions does not supply the additional class measurability conditions required by the standing caveat on printed page 15 of \cite{anthony1999neural}. Corrected text: The cited assertion used in Lemma~\ref{lemma:sup_wasserstein} is the following restricted form of Theorem 17.1 in \cite{anthony1999neural}, with its standing measurability condition. Let $X$ and $T$ be standard Borel spaces, let $g:T\times X\rightarrow[0,1]$ be jointly Borel, and put $\mathcal G=\{g(t,\cdot):t\in T\}\ne\varnothing$. For every Borel probability distribution $P$ on $X\times[0,1]$, every $0<\epsilon<1$ and every positive integer $m$, \[ P^m\left\{\text{some }t\in T\text{ has }\left|\mathbb E_{(x,y)\sim P}(g(t,x)-y)^2-\frac1m\sum_{i=1}^m(g(t,x_i)-y_i)^2\right|\geqslant\epsilon\right\} \leqslant4\mathcal N_1^{\mathrm{avg}}(\epsilon/16,\mathcal G,2m)\exp(-\epsilon^2m/32). \] Here the sample consists of independent draws from $P$, the probability is taken in its completion, and $\mathcal N_1^{\mathrm{avg}}(r,\mathcal G,N)$ is the supremum over $x_1,\ldots,x_N\in X$ of the covering number of the evaluation vectors in the metric $N^{-1}\sum_i|u_i-v_i|$. - A Neural Causal Model has i.i.d. multivariate uniform latent inputs and neural structural functions. ``\dots all $ U_i $ are i.i.d. multivariate uniform variables, $G_{V_i}$ are i.i.d Gumblel variables and functions in (\ref{eq:structure_eq}) are Neural Networks (NNs).'' The phrase ``i.i.d. multivariate uniform variables'' admits either identical block laws of one common dimension or independent unit-cube blocks of their indicated dimensions. Corrected text: (Neural Causal Model) A Neural Causal Model (NCM) is a special kind of SCM where $\boldsymbol U=\{U_1,\ldots,U_{n_U}\}\cup\{G_{V_i}:V_i\ \text{categorical}\}$, the scalar coordinates of all the $U_i$ are mutually independent with common $\operatorname{Unif}([0,1])$ law, and $U_i$ groups the indicated finite number $d_i^U$ of these coordinates, so that $U_i\sim\operatorname{Unif}([0,1]^{d_i^U})$. All coordinates of the $G_{V_i}$ are independent standard Gumbel variables, independent of all uniform coordinates. The continuous structural maps and the categorical propensity-weight maps $f_i$ in (\ref{eq:structure_eq}) are Neural Networks (NNs); categorical values are generated from these weights by the Gumbel argmax rule. The weights are nonnegative and have strictly positive sum, with $\log0=-\infty$. - The radius specification in Theorem~\ref{thm:consistency} defines the approximation and sampling tolerances used by Theorem~\ref{thm:consistency} and Corollary~\ref{cor:rate}. ``\dots take the radius to be $\alpha_n = \epsilon_n+s_n, \epsilon_n = O(W_{0,n}^{-1/d^{\text{in}}_{\max}}+ \sum_{i=1}^2 L_{i,n}^{-2/d^U_{\max}}+ \tau _{n} \log \tau _{n}),$ \[ s_n = O(m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n} +n^{-1/{\max\{2,d_{\max}^U\}}}\log^2(n) + \log(nm_n)\lambda_n).\mbox{''} \] The sampling expression uses the largest individual latent dimension instead of the total non-Gumbel latent dimension and omits the network-depth sampling term, while the approximation expression uses $\tau_n\log\tau_n$ instead of the positive small-temperature expression $\tau_n(1-\log\tau_n)$. Identify the complete restatements in Theorem~\ref{thm:consistency} and Corollary~\ref{cor:rate} as superseding this radius specification, retaining exactly their total non-Gumbel dimension, network-depth sampling term, and positive temperature expression; this protects both results with their accepted hypotheses, quantifiers, and rates. ### Restated to follow a correction above No error of their own: these results use a corrected result and are restated accordingly. - **Proposition 4.1.** \label{prop:counterexample} There exists a constant $c>0$ and an identifiable SCM $\mathcal{M}^*$ satisfying Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound} such that for any $\epsilon >0$, there exists an SCM $\mathcal{M}_\epsilon$ satisfying $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V})) \leq \epsilon $ and $|\text{ATE}_{\mathcal{M}^*} - \text{ATE}_{\mathcal{M}_\epsilon}| > c $. Here, for each non-Gumbel latent law $\mathbb P$ with connected support components $C_i$, Assumption~\ref{assumption:lower_bound} is required of each normalized component law $\mathbb P|_{C_i}/\mathbb P(C_i)$, with $K=C_i$. - **Lemma E.3.** Given two measures $\mu=\mu_1\otimes\cdots\otimes\mu_n$, $\hat\mu=\hat\mu_1\otimes\cdots\otimes\hat\mu_n$, where every factor $\mu_k$ and $\hat\mu_k$ is a probability measure, we have \begin{equation*} W(\mu,\hat\mu)\leqslant\sum_{k=1}^n W(\mu_k,\hat\mu_k). \end{equation*} ## Statements made precise (not errors) Nothing in this section was shown false. As printed, each statement did not determine what is claimed; it now states the reading adopted, which an independent referee pass confirmed. ### Conditions and definitions made precise - The local $n$ can be fixed or can increase with $W_1$, giving different quantization requirements; neither the passage nor the host's amended proof determines its dependence on width. Independently, the stated quantization substitution does not instantiate the supplied source. The source separates its depth parameter from a quantization mesh $\delta/n$ and requires $\delta<1/(n(n-1))$. Interpreting the printed fraction as either $\delta$ or the mesh violates that requirement. The host's amended proof repeats the paper's quantization choice without establishing admissible source parameters. This finding concerns the citation application; the asserted rate can be obtained with appropriate source parameters. - The full entropy-penalized optimal value and the transport cost evaluated at an entropy-regularized optimizer are different readings of $W_{1,\lambda}$; their differences from $W$ can have different signs. The host does not identify either functional, its entropy convention, or the ground-cost data. The cited proposition gives absolute approximation bounds for two separately defined functionals, with an inverse regularization parameter. It does not justify the host's signed assertion for its unspecified functional; in particular, the source's full regularized value violates the asserted lower bound. - (a) For vector-valued $\mathcal F$, its pseudo-dimension could mean $\max_{j\in[d]}\operatorname{Pdim}(\{f_j:f\in\mathcal F\})$ or $\operatorname{Pdim}(\{(x,j)\mapsto f_j(x):f\in\mathcal F\})$. These impose different bounds, and neither the paper nor the scalar cited theorem specifies the choice. The dual formulation does determine that $h$ is scalar-valued. (b) The attributed covering inequality is false for the paper's explicitly unnormalized empirical $\ell_1$ metric: its bound omits sample-size scaling. (c) It also omits range scaling: Theorem 18.4 requires scalar functions valued in $[0,1]$, whereas the host only bounds their magnitude by arbitrary $K$. The fixed-$h$ concentration order can be recovered from Theorem 17.1 by range normalization and a square-root transformation with zero responses, but that does not validate the printed covering inequality. Now: By Theorem 18.4 in \cite{anthony1999neural}, if $\mathcal G$ is a nonempty class of functions from a set $X$ into $[0,1]$ with finite pseudo-dimension $v$, then for every $0<\epsilon\leqslant1$ and every positive integer $m$, \begin{align*} \mathcal{N}_{1}^{\mathrm{avg}}(\epsilon,\mathcal G,m)\leqslant\mathcal{M}_{1}^{\mathrm{avg}}(\epsilon,\mathcal G,m)0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\|V_i\|_\infty,\|U_j\|_\infty\}\leqslant K$ almost surely under the SCM law. - **Theorem 3.3.** For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. - **Theorem 3.3.** In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. - **Theorem 3.3.** Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). - **Theorem 3.3.** The structural uniform infinity norm takes the supremum of the coordinatewise infinity norm over the common structural input domain. - **Theorem 3.3.** For a categorical propensity coordinate equal to zero, $\log0=-\infty$ in the canonical argmax construction. - **Theorem 3.6.** A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. - **Theorem 3.6.** The gap event is $E_\delta=\{\exists\,1\leqslant i0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. - **Theorem 3.6.** In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. - **Theorem 3.6.** $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. - **Theorem 3.6.** For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. - **Theorem 3.6.** The law $\hat g_{\#}\lambda$ includes the independent standard Gumbel variables used in the mixing layer. - **Theorem 3.6.** For each component of positive dimension $q=d_i^C$, the application of Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and quantization mesh $\delta/n$. - **Theorem 3.6.** The error estimates are asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of these budgets and of $\tau$. - **Theorem 3.6.** The measures $\mu$ and $\hat\mu$ are probability measures. - **Theorem 3.6.** A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. - **Theorem 3.6.** $W$ is Wasserstein-1 distance for the ground distance $\|x-y\|$. - **Theorem 3.6.** The refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. - **Theorem 3.6.** $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. - **Theorem 3.6.** In the proposition entitled ``Deep distribution approximation'', the multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. - **Theorem 3.6.** The uniform infinity norm is $\|F-\hat F\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}|F_j(x)-\hat F_j(x)|$. - **Corollary 3.6.1.** The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. - **Corollary 3.6.1.** The mass assigned to $I_{O_n(i)}$ is $\lambda_d(C_{O_n(i)})=2^{-nd}$. - **Corollary 3.6.1.** In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal C_n$ denotes the standard dyadic collection $\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}$. - **Corollary 3.6.1.** In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal I_n$ denotes $\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$ with $I_{O_n(i)}=[(i-1)2^{-nd},i2^{-nd}]$. - **Corollary 3.6.1.** For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. - **Corollary 3.6.1.** For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. - **Corollary 3.6.1.** The reference measure $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$. - **Corollary 3.6.1.** Each $O_n$ is a bijection. - **Corollary 3.6.1.** The cells of $\mathcal C_n$ are closed and may share boundaries. - **Corollary 3.8.** The phrase ``uniform cubes'' denotes the unit cube domains $[0,1]^m$ used for deterministic uniform function approximation; it imposes no uniform probability law on structural inputs. - **Corollary 3.8.** Membership in $\text{NCM}_{\mathcal G}(\mathcal F_0,\mathcal F_1)$ requires $\hat f_i\in\mathcal F_0$ for every observed node and $\hat g_j\in\mathcal F_1$ for every latent component, with the compatible input and output dimensions understood separately for each network. - **Corollary 3.8.** $d_{\max}^U=\max_{j\in[n_U]}d_j^U$ ranges over non-Gumbel latent variables, while $d_{\max}^{\mathrm{in}}=\max_i d_i^{\mathrm{in}}$ and $d_{\max}^{\mathrm{out}}=\max_i d_i^{\mathrm{out}}$, with $d_i^{\mathrm{out}}=n_i$ for categorical nodes. - **Corollary 3.8.** For each categorical $V_i$, the noise in equation (\ref{eq:structure_equation_hat}) satisfies $g_k=g_k^{V_i}$ for $k\in[n_i]$. - **Corollary 3.8.** The multiplicative width factors in $\Theta(d_{\max}^{\text{in}}d_{\max}^{\text{out}})$, $\Theta(d_{\max}^U)$, and $\Theta((d_{\max}^U)^2)$ can be chosen as absolute numerical constants, independent of the causal model, $R$, all dimensions, and all category and component counts. - **Corollary 3.8.** For each latent law $P(U_C)$, the assumptions of the proposition entitled ``Deep distribution approximation'' require the support charts of \cref{assump:support} and \cref{assumption:lower_bound} on each normalized component law $\mu_i=P(U_C)|_{C_i}/P(U_C\in C_i)$, with $(f^{-1})_{\#}\mu_i(\partial[0,1]^{d_i^C})=0$ when $d_i^C>0$. - **Corollary 3.8.** In Corollary~\ref{corollary:deep_nn}, for $m_i=d_i^{\text{in}}>0$, the symbol $\hat f_i$ denotes $q_i\circ\Pi_i$ for a continuous variable and the vector $\bigl(\max\{\epsilon_i,(q_i\circ\Pi_i)_k\}\bigr)_{k=1}^{n_i}$ for a categorical variable, where $q_i$, $\Pi_i$, and $\epsilon_i=A_iL_0^{-2/m_i}$ are chosen in its proof. - **Corollary 3.8.** The latent generator class in the statement uses the same temperature $\tau$ as the displayed error estimate. - **Corollary 3.8.** The error estimate is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, for fixed $R$ and fixed causal model. - **Corollary 3.8.** In Corollary~\ref{corollary:deep_nn}, for $m_i=0$, the symbol $\hat f_i$ denotes the exact constant function $f_i$. - **Proposition 4.1.** The second Bernoulli law in the construction has $P(U_2=0)=2/3$. - **Theorem 4.2.** The bound $K$ applies throughout the optimization class to the observed variables $V_i$ and indexed non-Gumbel latent variables $U_j$, with $n_i\leqslant K$ for categorical nodes and admissible intervention coordinates in $[-K,K]$; it does not bound the separately supplied standard Gumbel noises. - **Theorem 4.2.** Truncation means coordinatewise clipping of the final output to $[-K,K]$; in $\mathcal{NN}^{\infty,K}$ no finite Lipschitz bound is imposed. - **Theorem 4.2.** $\underline{F}^{L}$ denotes the constrained infimum of the displayed scalar objective over the stated canonical SCM class and its admissible latent laws, rather than a minimizing SCM. - **Theorem 4.2.** The coordinates $P(U_j)$ in $\theta_{\mathcal M}$ are the population laws of the non-Gumbel latent variables, not necessarily uniform laws; the independent standard Gumbel laws are fixed and omitted from this population tuple. - **Theorem 4.2.** The boundedness condition means $\|f(x)\|_{\infty}\leqslant K$ for every $f\in\mathcal{F}$ and $x\in\mathbb{R}^{d'}$. - **Theorem 4.2.** $F_n=\inf\left\{\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\hat{\mathcal{M}}}[F(V_1(t),\ldots,V_{n_V}(t))]:\hat{\mathcal{M}}\in\text{NCM}_{\mathcal{G}}(\mathcal{F}_{0,n},\mathcal{F}_{1,n}),\ S_{\lambda_n}(P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V}),P_n^{\mathcal{M}^*}(\boldsymbol{V}))\leqslant\alpha_n\right\}$. - **Theorem 4.2.** The observational empirical law $P_n^{\mathcal M^*}(\boldsymbol V)$ is formed from $n$ independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. - **Theorem 4.2.** The empirical optimization ranges over network representations, including distinct representations with the same latent population laws. - **Theorem 4.2.** The common simulation draws are independent of the observational sample. - **Theorem 4.2.** The common exogenous draws are independent and identically distributed according to the specified product law of the exogenous inputs. - **Theorem 4.2.** Probabilities of uniform deviation events are evaluated in the completion of the sample law. - **Theorem 4.2.** For each $n$, every candidate is evaluated on the same $m_n$ draws of all its exogenous inputs, including its uniform inputs, generator Gumbel inputs, and categorical Gumbel inputs. - **Theorem 4.2.** The finite component counts and component cube dimensions of the latent generator architectures in $\mathcal F_{1,n}$ are fixed as $n$ varies. - **Theorem 4.2.** The self terms in $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$ use the same $1$-norm ground distance on the corresponding support points. - **Theorem 4.2.** $F_n$ denotes the infimum of objective values, with an attained minimum having the same value. - **Theorem 4.2.** For a zero categorical propensity, $\log0=-\infty$. - **Theorem 4.2.** Ties in a categorical argmax are resolved by the smallest label. - **Theorem 4.2.** The orders for $\epsilon_n$ and $s_n$ are asymptotic as $n\rightarrow\infty$. - **Theorem 4.2.** The architectural $\Theta$ orders use sufficiently large fixed constant factors supplied by the network constructions in the proof. - **Theorem 4.2.** The probability bound is for each fixed pair $\delta,\epsilon>0$. - **Theorem 4.2.** The depth expression $\Theta(\log d_{\max}^{\text{in}})$ means $\Theta(1+\log d_{\max}^{\text{in}})$ at $d_{\max}^{\text{in}}=1$. - **Proposition 4.3.** For each of $\mathcal{M}$ and $\hat{\mathcal{M}}$, $\max_{i\in[n_V],j\in[n_U]}\{\|V_i\|_\infty,\|U_j\|_\infty\}\leqslant K$ almost surely, where the indexed $U_j$ are the non-Gumbel latent variables. - **Proposition 4.3.** The functions $f,\hat{f}:[-K,K]\rightarrow\mathbb{R}$ are $L$-Lipschitz regression versions satisfying $f(T)=\mathbb{E}_{\nu}[Y\mid T]$ almost surely under $\nu$ and $\hat{f}(\hat{T})=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}]$ almost surely under $\mu$. - **Proposition 4.3.** The measure $P$ is the reference probability measure on the treatment space: for $X=\text{Pa}(T)$, the conditional density notation means $\nu(dt|X=x)=p_\nu(t|X=x)P(dt)$. - **Corollary 4.4.** The continuous structural functions and categorical propensity functions of $\mathcal M^*$ have Lipschitz constants at most $L_f$ in the paper's $1$-norm. - **Corollary 4.4.** The variable bound inherited from Assumption~\ref{assump:Lip_and_bounded} uses $\|\cdot\|_\infty$. - **Corollary 4.4.** $\mu_T=\text{Unif}([t_1,t_2])$ is the deterministic intervention probability measure. - **Corollary 4.4.** The measure $P$ is a probability measure on the treatment space, and for almost every parent value $x$, $\nu(dt\mid\text{Pa}(T)=x)=p_\nu(t\mid\text{Pa}(T)=x)P(dt)$, with the stated density version satisfying $p_\nu(t\mid\text{Pa}(T)=x)\geqslant\delta$ for every $t\in[t_1,t_2]$. - **Corollary 4.4.** The observational empirical law $P_n^{\mathcal M^*}(\boldsymbol V)$ is formed from $n$ independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. - **Proposition B.1.** The categorical generation formula requires $f_i(x)\in[0,\infty)^{n_i}$ and $\|f_i(x)\|_1>0$ on its structural input domain. - **Proposition B.1.** A zero categorical coordinate has extended logarithm $\log0=-\infty$. - **Proposition C.2.** Equation~\ref{eq:aprox_lip} holds for every $n\in\mathbb N$ and every $\theta_1,\theta_2\in\Theta_\infty$, with the same finite constant $L_g$. - **Proposition C.2.** There exists a compact $\tilde\Theta_\infty\subseteq\Theta_\infty$ such that $\forall n\in\mathbb N\;\forall\theta_0\in\tilde\Theta_\infty\;\exists\theta\in\Theta_n$, $d(\theta,\theta_0)\leqslant\epsilon_n$. - **Proposition C.2.** In (\ref{eq:aprox_lip}), $\Theta=\Theta_\infty$. **Side conditions the proofs use without stating them.** - **Theorem 3.6.** For every $k\in[n]$, $\int_{\mathbb R^d}\|x\|\,\mu_k(dx)+\int_{\mathbb R^d}\|x\|\,\hat\mu_k(dx)<\infty$. - **Proposition C.2.** For every $n\in\mathbb N$, the feasible set of (\ref{eq:opt_n}) contains a minimizer of $f$. **Notation.** - $V_i$: $V_i$, for $i\in[n_V]$, denotes the observed variable at node $i$, and $V_i(\boldsymbol{t})$ denotes its value under the intervention $\boldsymbol{T}=\boldsymbol{t}$, obtained by replacing treatment equations by their assigned values and recursively evaluating the remaining structural equations. In Proposition~\ref{prop:counterexample_foral}: In the proof, the observed variables are $X,T,Y$ and the binary treatment $T$ uses the labels $0,1$. - $n_i$: For categorical $V_i$, $n_i$ is its positive category count, with category labels $[n_i]=\{1,\ldots,n_i\}$; it is also the length of its propensity vector and $G_{V_i}$. - $\boldsymbol{V}$: $\boldsymbol{V}=\{V_i\}_{i=1}^{n_V}$ is the collection of observed variables; joint-law and function-evaluation notation uses their indexed joint values, and $\boldsymbol{V}(\boldsymbol{t})$ is the corresponding collection under intervention $\boldsymbol{T}=\boldsymbol{t}$. In Proposition~\ref{prop:counterexample_foral}: In the proof, $\boldsymbol{V}=\{X,T,Y\}$ consists of the covariate, treatment, and outcome. - $U_j$: $U_j$, $j\in[n_U]$, denotes a member of the indexed latent family; under the categorical convention this family is separate from the categorical noise vectors $G_{V_i}$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}, \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: $U_j$, $j\in[n_U]$, indexes the non-Gumbel latent coordinates, with $U_j\in[-K,K]^{d_j^U}$ and $D_U=\sum_{j=1}^{n_U}d_j^U$. Their population laws are not required to be uniform; the uniform variables supplied to the latent generators are separate inputs. - $\boldsymbol{U}$: $\boldsymbol{U}$ denotes the full latent collection of the SCM, including the separately introduced categorical generation noises when the categorical convention is used; $U_1,\cdots,U_{n_U}$ then index the non-Gumbel part. In Proposition~\ref{prop:equivalent_g_constrianed}: The original model uses $\boldsymbol{U}=\{U_1,\cdots,U_{n_U}\}\cup\boldsymbol{G}$, with $\boldsymbol{G}=\{G_{V_i}:V_i\text{ is categorical}\}$; canonicalization merges the non-Gumbel variables into component blocks and retains the categorical noises. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: For canonical SCMs, $\boldsymbol{U}=\{U_C:C\text{ is a }C^2\text{-component of }\mathcal{G}\}\cup\{G_{V_i}:V_i\text{ is categorical}\}$; the bounded latent coordinates subject to Assumption~\ref{assump:Lip_and_bounded} are the non-Gumbel $U_j$. - $n_U$: $n_U$ counts the indexed family $\{U_j\}_{j=1}^{n_U}$, with separately adjoined categorical noises $G_{V_i}$ excluded from that count when the categorical convention is used. In Proposition~\ref{prop:equivalent_g_constrianed}: $n_U$ counts the original indexed latent variables before merging; the merged latent family is instead indexed by the $n_C$ components. - $\boldsymbol{U}_{V_i}$: $\boldsymbol{U}_{V_i}$ is the designated collection of latent inputs to $f_i$; under the categorical propensity convention it is a subset of $\{U_1,\cdots,U_{n_U}\}$ and excludes the separately displayed categorical Gumbel noise. In Proposition~\ref{prop:equivalent_g_constrianed}: For the original SCM, $\boldsymbol{U}_{V_i}$ is its original latent input subset; the constructed canonical SCM uses merged latent blocks, and the reused $f_i$ ignores coordinates outside that original subset. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: For the canonical SCMs in $\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})$, $\boldsymbol{U}_{V_i}=\{U_C:V_i\in C,\ C\text{ is a }C^2\text{-component of }\mathcal{G}\}$. - $\text{Pa}$: $\text{Pa}(V_i)$ is the set of directed parents of $V_i$ in $\mathcal{G}_0$, excludes $V_i$, and in structural-function arguments or conditioning expressions denotes the corresponding collection of parent values. - $f_i$: $f_i$, for $i\in[n_V]$, is the function associated with node $V_i$, evaluated on the values of its observed parents and designated latent inputs; under the categorical convention it denotes the nonnegative propensity vector rather than the categorical label. In Corollary~\ref{corollary:deep_nn}, \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Corollary~\ref{corollary:deep_nn}: The neural approximation argument uses the stated cube-domain convention for the inputs of $f_i$, with $d_i^{\text{in}}$ and $d_i^{\text{out}}$ its input and output dimensions. In Proposition~\ref{prop:equivalent_g_constrianed}: When the proof reuses $f_i$ after merging latent variables into blocks, it denotes the original function applied to the original coordinates and ignores added block coordinates that were not in $\boldsymbol{U}_{V_i}$. For categorical nodes the original weights satisfy $f_i(x)\in[0,\infty)^{n_i}$ and $\|f_i(x)\|_1>0$ on their structural input domain, with $\log0=-\infty$; the constructed canonical categorical function is the normalized vector $f_i/\|f_i\|_1$ with coordinates regrouped. Its smoothness is that of this normalized vector: normalization preserves differentiability order and local Lipschitz regularity wherever the total weight is positive, and regrouping preserves any global Lipschitz regularity the normalized vector possesses. In Theorem~\ref{thm:approximation}: In $\|f_i-\hat f_i\|_\infty$, the structural uniform infinity norm is the supremum of the coordinatewise infinity norm over the common structural input domain; zero categorical coordinates use $\log0=-\infty$ in the canonical argmax construction. In Theorem~\ref{thm:consistency}: The proof treats $f_i$ as an element of $\mathcal{F}_{V_i}$ on $[-K,K]^{d_{i,\text{in}}^V}$ with bounded output cube $[-K,K]^{d_{i,\text{out}}^V}$, imposing $\|f_i(x)\|_1=1$ pointwise for categorical nodes; the population bounded Lipschitz class is read as $\mathcal F_L^K=\{f:\sup_x\|f(x)\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}, \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Corollary~\ref{corollary:deep_nn}: For the canonical SCM functions in (\ref{eq:structure_equation}), categorical $f_i$ are normalized: $(f_i)_k\geqslant0$ and $\|f_i\|_1=1$, so their values lie in $\Delta_{n_i}$ and their coordinates are the conditional category probabilities given $\text{Pa}(V_i),\boldsymbol{U}_{V_i}$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: For every continuous structural function $f_i$ of $\mathcal M^*$, $\sup_x\|f_i(x)\|_1\leqslant K$ on its full structural input domain, and every continuous candidate structural network $\hat f_i\in\mathcal F_{0,n}$ satisfies the same uniform output $1$-norm bound. - $F$: $F$ is functional notation whose tuple, scalar-objective, and push-forward-map meanings are distinguished by context. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:Lip_ATE}: In the SCM notation, $F=(f_1,\cdots,f_{n_V})$ is the tuple of functions in the structural equations. In Theorem~\ref{thm:consistency}: In (\ref{eq:opt_empiri}) and (\ref{eq:consistent_limiting}), $F:\mathbb{R}^{n_V}\to\mathbb{R}$ is the Lipschitz continuous causal objective evaluated on the intervened observed variables, rather than the tuple of structural functions. In Corollary~\ref{cor:rate}: The causal objective is the outcome coordinate projection $F(\boldsymbol{V})=Y$. In Lemma~\ref{lemma:push-forward_err}: $F:K\to\mathbb{R}^{d_2}$ is a measurable, $L$-Lipschitz continuous map, where $K\subset\mathbb{R}^{d_1}$ is compact. In Proposition~\ref{prop:equivalent_g_constrianed}: The SCM tuple uses $F=(f_1,\cdots,f_{n_V})$; in the proof's expression $\boldsymbol{V}=F(\boldsymbol{U})$, $F$ instead denotes the map obtained by recursively evaluating the structural equations, with its intervention version denoted $F_t$. - $P$: $P(X)$ denotes the probability law of $X$, and $P(A)$ denotes the probability of an event $A$ under the law in context. In Proposition~\ref{prop:Lip_ATE}: Bare $P$ is the reference probability measure on the treatment space used in $P(dt)$; $p_\nu(t|\text{Pa}(T)=x)$ is the density of the conditional treatment law $\nu(dt|x)$ with respect to this same measure. In Corollary~\ref{cor:rate}: Bare $P$ is the reference probability measure on the treatment space with respect to which $\nu(dt|\text{Pa}(T)=x)=p_\nu(t|\text{Pa}(T)=x)P(dt)$. The intervention measure satisfies $\mu_T\ll P$ and $\int_{t_1}^{t_2}(d\mu_T/dP(t))^2P(dt)<\infty$. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: Bare $P$ is an arbitrary Borel probability distribution on $X\times[0,1]$, where $X$ is standard Borel. The population squared-loss expectation is taken under this $P$, and the deviation probability is evaluated in the completion of $P^m$. - $\mathcal{M}$: Bare $\mathcal{M}$ denotes an individual SCM tuple, whereas $\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})$ denotes the class of canonical SCMs with causal graph $\mathcal{G}$, the indicated latent-variable structure, and structural functions $f_i\in\mathcal{F}$ for $i\in[n_V]$. In Theorem~\ref{thm:consistency}: In the population optimization defining $\underline F^{\hat L_f}$, $\mathcal M(\mathcal G,\mathcal F_{\hat L_f}^K,\boldsymbol U)$ uses $\mathcal F_L^K=\{f:\sup_x\|f(x)\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$, with nonnegative normalized categorical propensity vectors; the latent-law range consists of independent non-Gumbel laws supported on $[-K,K]^{d_j^U}$ and fixed independent standard Gumbel laws for categorical variables. - $\mathcal{G}$: $\mathcal G$ denotes the graph or function class specified by its local occurrence. In Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: $\mathcal{G}$ has no hidden confounder. In Proposition~\ref{prop:counterexample_foral}: $\mathcal{G}$ is the common backdoor causal graph of the constructed models, with no unobserved confounding. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}, Proposition~\ref{prop:equivalent_g_constrianed}, \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Corollary~\ref{corollary:deep_nn}: $\mathcal G$ is the causal ADMG on $\boldsymbol V$, retaining the structural parent arrows and adding a bidirected arrow between $V_i,V_j$ when correlated latent inputs affect the two nodes. In \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $\mathcal G$ is a nonempty class of functions $g:X\rightarrow[0,1]$ with finite pseudo-dimension $v$. The covering and packing numbers use evaluations at arbitrary $m$ points of $X$ in the averaged metric $d_1(u,w)=m^{-1}\sum_{i=1}^m|u_i-w_i|$. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $\mathcal G=\{g(t,\cdot):t\in T\}\ne\varnothing$, where $X,T$ are standard Borel spaces and $g:T\times X\rightarrow[0,1]$ is jointly Borel; no pseudo-dimension assumption is imposed. The covering number $\mathcal N_1^{\mathrm{avg}}(r,\mathcal G,2m)$ is the supremum over $2m$ points of $X$ of the covering number of the evaluation vectors in the averaged metric $(2m)^{-1}\sum_i|u_i-v_i|$. - $\hat{\mathcal{F}}$: $\hat{\mathcal{F}}$ is the specified class containing the comparison model's structural functions on their respective node domains; categorical members are normalized propensity functions, and no neural architecture or Lipschitz bound is imposed merely by the hat. - $\hat{\mathcal{M}}$: $\hat{\mathcal{M}}$ is a model on the same causal graph $\mathcal{G}$ as the model with which it is compared; the hat alone does not denote a fitted estimator or an optimization minimizer. In Theorem~\ref{thm:approximation}: $\hat{\mathcal{M}}$ is any comparison model in $\mathcal{M}(\mathcal{G},\hat{\mathcal{F}},\hat{\boldsymbol{U}})$; it is not required by this notation to be a neural model. In Corollary~\ref{corollary:deep_nn}: $\hat{\mathcal{M}}$ is the constructed neural causal model with the displayed structural and latent-generator architectures and structure equations~\ref{eq:structure_equation_hat}, chosen after fixing $R>0$; it may depend on $R$ and the network budgets, but the same model applies to every intervention with $\|t_i\|_\infty\leqslant R$ for continuous treatment variables and $t_i\in[n_i]$ for categorical treatment variables. In Proposition~\ref{prop:Lip_ATE}: $\hat{\mathcal{M}}\in\mathcal{M}(\mathcal{G},\mathcal{F}_L,\boldsymbol{U})$ is the second comparison SCM, with observational law $\mu$. In Proposition~\ref{prop:equivalent_g_constrianed}: $\hat{\mathcal{M}}$ is the canonical representation obtained by merging the original non-Gumbel latent variables and retaining the categorical Gumbel noises; its continuous structural functions are the original $f_i$ under coordinate regrouping, and its categorical propensity functions are $f_i/\|f_i\|_1$ under the same regrouping, ignoring additional input coordinates, so its intervention laws equal those of $\mathcal{M}$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In equation~\ref{eq:opt_empiri}, $\hat{\mathcal M}$ ranges over network representations in $\text{NCM}_{\mathcal G}(\mathcal F_{0,n},\mathcal F_{1,n})$, with each categorical raw structural network $q$ satisfying $q(x)\in[0,\infty)^{n_i}$, $\|q(x)\|_1>0$, and $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$ on its full structural input domain. Different representations with the same latent population laws may have different common-draw empirical distributions. In Theorem~\ref{thm:consistency}: Every continuous structural network $\hat f_i$ additionally satisfies $\sup_x\|\hat f_i(x)\|_1\leqslant K$ on its full structural input domain. In equation~\ref{eq:consistent_limiting}, $\hat{\mathcal M}$ instead ranges over canonical SCMs in $\mathcal M(\mathcal G,\mathcal F_L^K,\boldsymbol U)$ and all latent distributions $P(\boldsymbol U)$, with $\mathcal F_L^K=\{f:\sup_x\|f(x)\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$ and normalized categorical propensity functions. In Corollary~\ref{cor:rate}: Every continuous structural network $\hat f_i$ additionally satisfies $\sup_x\|\hat f_i(x)\|_1\leqslant K$ on its full structural input domain. The population comparison models use the bounded Lipschitz structural class $\mathcal F_L^K=\{f:\sup_x\|f(x)\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$ and normalized categorical propensity functions. - $\hat{\boldsymbol{U}}$: $\hat{\boldsymbol{U}}$ denotes the full canonical latent collection of $\hat{\mathcal{M}}$, including categorical Gumbel noises, rather than only the non-Gumbel inputs to its structural functions. - $\epsilon$: $\epsilon(\mathcal{F}_1,\mathcal{F}_2)$ denotes the directed approximation error $\sup_{f_2\in\mathcal{F}_2}\inf_{f_1\in\mathcal{F}_1}\|f_2-f_1\|$, whereas a bare $\epsilon>0$ denotes a local positive real tolerance. In Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:counterexample_foral}: $\epsilon$ is arbitrary and positive, and specifies the requested upper bound on $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V}))$. In Lemma~\ref{lemma:product_err}: In the proof, $\epsilon$ is the positive slack in each chosen coupling's transportation cost and is subsequently taken to zero. In Proposition~\ref{prop:convg_opt}: In the proof, $\epsilon$ is a positive gap witnessing a hypothetical subsequence with $f_{n_k}^*0$ without assigning it any further role. In the empirical concentration argument of the proof, a local deviation tolerance is chosen as $\epsilon=Cr_n$, together with $\delta=r_n$, for sufficiently large $n$. In Lemma~\ref{lemma:sup_wasserstein}: $\epsilon>0$ is the deviation tolerance and also the unnormalized covering radius when it appears in $\mathcal{N}_1(\epsilon,\mathcal{F},n)$; the probability bound is for each fixed pair $\delta,\epsilon>0$. In the proof, for $0<\epsilon<2dK$, the normalized squared-loss tolerance is $\eta=\epsilon/(2dK)$. In \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $0<\epsilon\leqslant1$ is the covering and packing radius in the averaged metric $d_1(u,w)=m^{-1}\sum_{i=1}^m|u_i-w_i|$. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $0<\epsilon<1$ is the deviation tolerance for the absolute difference between population and empirical squared loss; the covering radius is $\epsilon/16$ on $2m$ evaluation points. In Corollary~\ref{corollary:deep_nn}: In the scalar inequality $|\max\{\epsilon,z\}-p|\leqslant\max\{\epsilon,|z-p|\}$, $\epsilon\geqslant0$ is an arbitrary floor parameter; the inequality is then applied coordinatewise with $\epsilon=\epsilon_i$. In Theorem~\ref{thm:consistency}: In the empirical concentration argument, the local deviation tolerance is chosen as $\epsilon=A_0r_n$, together with $\delta=r_n$, for sufficiently large $n$. - $\mathcal{M}_\epsilon$: $\mathcal{M}_\epsilon$ denotes an SCM chosen for each tolerance $\epsilon>0$ in the counterexample family, rather than a single model independent of $\epsilon$. In Proposition~\ref{prop:counterexample_foral}: $\mathcal{M}_\epsilon$ has the same causal graph as $\mathcal{M}^*$; in the proof it is supplied by $\mathcal{M}_\delta$ with $\delta>0$ small enough that $\delta/2\leqslant\epsilon$. - $\mathcal{C}$: $\mathcal{C}$ denotes the locally specified collection, consisting of causal models in partial identification and cubes in the geometric construction. In Proposition~\ref{prop:holder_curve}: $\mathcal{C}=\{C_1,\ldots,C_m\}$ is a finite collection of axis-aligned cubes; $\mathcal{C}_n$ is the collection of closed cells in the level-$n$ partition of $[0,1]^d$, and $\mathcal{C}^{*}=\bigcup_{n=1}^{\infty}\mathcal{C}_n$. - $\theta$: $\theta$ denotes the causal functional when applied to an SCM in (\ref{prob:pid}), and a parameter point in the later optimization and parameterization notation. In Proposition~\ref{prop:convg_opt}: $\theta$ is a point of the metric space $(M,d)$, ranged over $\Theta_n$, $\Theta_\infty$, or $\tilde{\Theta}_\infty$ in the respective optimization problems. In Theorem~\ref{thm:consistency}: In the population causal-model parameterization, $\theta=(f_1,\ldots,f_{n_V},P(U_1),\ldots,P(U_{n_U}))\in M$, $\mathcal{M}^{\theta}$ denotes the represented SCM, and $P^{\theta}$ its population distribution; $\theta_{i,j}$ denotes the parameters of the corresponding latent-generator neural network. In the empirical optimization, a candidate index additionally specifies its network representation: different representations with the same latent population laws can give different $P_{m_n}^{\theta}$ on the common simulation inputs. In Corollary~\ref{cor:rate}: In the population causal-model parameterization, $\theta=(f_1,\ldots,f_{n_V},P(U_1),\ldots,P(U_{n_U}))\in M$, $\mathcal{M}^{\theta}$ denotes the represented SCM, and $P^{\theta}$ its population distribution; $\theta_{i,j}$ denotes the parameters of the corresponding latent-generator neural network. In the empirical optimization, a candidate index additionally specifies its network representation: different representations with the same latent population laws can give different $P_{m_n}^{\theta}$ on the common simulation inputs. - $\hat{\boldsymbol{V}}$: $\hat{\boldsymbol{V}}=\{\hat{V}_i\}_{i=1}^{n_V}$ is the collection of observed variables of $\hat{\mathcal{M}}$ corresponding to $\boldsymbol{V}$, and $\hat{\boldsymbol{V}}(\boldsymbol{t})$ denotes this collection under intervention $\boldsymbol{T}=\boldsymbol{t}$. - $\hat{V}_i$: $\hat{V}_i$ is the observed variable corresponding to $V_i$ in $\hat{\mathcal{M}}$, with $i\in[n_V]$ and the same continuous or categorical state space; $\hat{V}_i(\boldsymbol{t})$ denotes its value after replacing treatment equations by $\boldsymbol{T}=\boldsymbol{t}$. In Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}: The structural equations are (\ref{eq:structure_equation_hat}), with latent inputs $\hat{U}_{C_j}=\hat{g}_j(Z_{C_j})$ and structural functions $\hat{f}_i$. In Theorem~\ref{thm:approximation}, Theorem~\ref{thm:consistency}: For a zero categorical propensity coordinate, the categorical argmax uses the extended logarithm $\log0=-\infty$. In Theorem~\ref{thm:consistency}: For a categorical node, the raw structural network $q$ has nonnegative coordinates and $\|q(x)\|_1>0$ at every structural input, and its normalized propensity vector is $q(x)/\|q(x)\|_1$; ties in the categorical argmax are resolved by the smallest label. - $C$: The meaning of $C$ is determined by its local definition: a graph component, a geometric set, a numerical constant, or the complete categorical label vector. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}, Proposition~\ref{prop:equivalent_g_constrianed}: In canonical latent indexing, $C\subset\boldsymbol{V}$ is maximal subject to every pair of distinct vertices in $C$ being joined by a bi-directed arrow in $\mathcal{G}$. In Theorem~\ref{thm:wide_nn}, Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the distribution-generator construction, $C_i$ denotes a connected component of $\operatorname{supp}(\mathbb{P})$, with $\operatorname{supp}(\mathbb{P})=\bigcup_{i=1}^{N_C}C_i$ and $\mathcal{H}([0,1]^{d_i^C},C_i)\cap\mathcal{F}_L([0,1]^{d_i^C},C_i)\neq\emptyset$. In Proposition~\ref{prop:holder_curve}: In the partition construction, $C=\prod_{i=1}^{n}[a_i,b_i]$ is an axis-aligned cube; indexed sets $C_{i_1,\ldots,i_n}$ are the closed cells of the constructed partitions of $[0,1]^d$. In Lemma~\ref{lemma:sup_wasserstein}: $C>0$ is a deterministic numerical constant depending only on $d,K$ in the probability bound. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In numerical probability and tolerance bounds, $C>0$ denotes a deterministic constant rather than a set of observed vertices. In Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:counterexample_foral}: In the lower-bound premise for each non-Gumbel latent law $\mathbb P$, $C_i$ denotes a connected component of its support, and the assumption is applied with $K=C_i$ to the normalized component law $\mathbb P|_{C_i}/\mathbb P(C_i)$ wherever this normalization is defined. In Theorem~\ref{thm:consistency}, \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: In the latent empirical concentration argument, $C$ denotes the complete categorical label vector, with values $c\in\prod_{i:V_i\text{ is categorical}}[n_i]$; $p_c=P(C=c)$, $N_c$ is its observational sample count, and $\mu_c=P((U_1,\ldots,U_{n_U})\mid C=c)$ for $p_c>0$. - $U_C$: $U_C$ denotes the canonical latent vector indexed by a $C^2$-component $C$ of $\mathcal{G}$, and $\boldsymbol{U}_{V_i}=\{U_C:V_i\in C,\ C\text{ is a }C^2\text{-component of }\mathcal{G}\}$. In Corollary~\ref{corollary:deep_nn}: For every $C^2$-component $C$, the law $P(U_C)$ satisfies Assumption~\ref{assump:support}; for each connected support component $C_i$ of positive mass, the normalized law $\mu_i=P(U_C)|_{C_i}/P(U_C\in C_i)$ satisfies Assumption~\ref{assumption:lower_bound} with $K=C_i$, and its support chart $f$ satisfies $(f^{-1})_{\#}\mu_i(\partial[0,1]^{d_i^C})=0$ when $d_i^C>0$. - $G_{V_i}$: $G_{V_i}=(g_1^{V_i},\ldots,g_{n_i}^{V_i})$ is present for categorical $V_i$ and consists of standard Gumbel coordinates furnishing that node's categorical-generation noise, separately from the latent inputs $\boldsymbol{U}_{V_i}$ of its propensity function. - $g_k^{V_i}$: For categorical $V_i$ and $k\in[n_i]$, $g_k^{V_i}$ is the $k$th standard Gumbel coordinate of $G_{V_i}$, used in $\arg\max_{k\in[n_i]}\{g_k^{V_i}+\log(f_i(\text{Pa}(V_i),\boldsymbol{U}_{V_i}))_k\}$. - $G_k$: $G_k$ is the $k$th standard Gumbel coordinate, with density $x\mapsto\exp(-x-\exp(-x))$ on $\mathbb{R}$, in the relevant Gumbel-softmax vector. In Lemma~\ref{lemma:gumbel_err}: The coordinate index is $k\in[n]$. In Theorem~\ref{thm:wide_nn}: $G_k$, $k\in[N_C]$, are the independent standard Gumbel coordinates of the mixing layer, one per connected support component, independent of the uniform inputs used by the deterministic blocks; their randomness is included in $\hat g_{\#}\lambda$. In the result split from Theorem~\ref{thm:wide_nn}: $G_k$, $k\in[N_C]$, are the independent standard Gumbel coordinates of the mixing layer, one per connected support component, independent of the uniform inputs used by the deterministic blocks; their randomness is included in $\hat g_{\#}\lambda$. - $\hat{f}_i$: For $i\in[n_V]$, $\hat{f}_i$ is the comparison model's structural map for a continuous node and its categorical propensity or weight map for a categorical node, evaluated on parent values and the corresponding latent coordinates. In Theorem~\ref{thm:approximation}: The functions are those of $\hat{\mathcal{M}}\in\mathcal{M}(\mathcal{G},\hat{\mathcal{F}},\hat{\boldsymbol{U}})$; categorical structural propensities are read under the canonical representation, including its normalization convention and $\log0=-\infty$ for zero coordinates. The quantity $\|f_i-\hat f_i\|_\infty$ takes the supremum of the coordinatewise infinity norm over the common structural input domain. In Corollary~\ref{corollary:deep_nn}, \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Corollary~\ref{corollary:deep_nn}: In the construction, let $q_i$ be a minimizer of $\|q_i-f_i\|_\infty$ over $\mathcal{NN}_{d_i^{\text{in}},d_i^{\text{out}}}(d_i^{\text{out}}(2d_i^{\text{in}}+10),L_0)$. The chosen network is $\hat{f}_i=q_i$ for a continuous node and $\hat{f}_i=\max\{0,q_i\}$ coordinatewise for a categorical node; the latter's normalized propensity is $\hat{f}_i/\|\hat{f}_i\|_1$ wherever the denominator is nonzero. In Theorem~\ref{thm:consistency}: Continuous structural networks range over $\mathcal{F}_{0,n}$ subject additionally to $\sup_x\|\hat f_i(x)\|_1\leqslant K$ on the full structural input domain. For a categorical node, the raw network $q\in\mathcal{F}_{0,n}$ satisfies $q(x)\in[0,\infty)^{n_i}$ and $\|q(x)\|_1>0$ for every structural input $x$, and its normalized propensity satisfies $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$; the proof's canonical structural-function coordinate is $q/\|q\|_1$. In Corollary~\ref{cor:rate}: Continuous structural networks range over $\mathcal F_{0,n}$ subject additionally to $\sup_x\|\hat f_i(x)\|_1\leqslant K$ on the full structural input domain. For a categorical node, the raw network $q\in\mathcal F_{0,n}$ satisfies $q(x)\in[0,\infty)^{n_i}$ and $\|q(x)\|_1>0$ for every structural input $x$, and its normalized propensity satisfies $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$. - $f$: $f$ denotes a locally specified function, with domain, codomain and regularity supplied by its occurrence, rather than one common map shared across the paper. In Proposition~\ref{prop:convg_opt}: $f:\Theta_\infty\rightarrow\mathbb{R}$ is the continuous objective minimized in all three optimization problems. In Lemma~\ref{lemma:Lip_condition}: $f(t)=\mathbb{E}_{\nu}[Y\mid T=t]$ is the population regression function, taken in the $L$-Lipschitz version specified by the hypothesis, with $|f(t)|\leqslant K$. In Lemma~\ref{lemma:combin_err}: In the proof, $f:\mathbb{R}^d\rightarrow\mathbb{R}$ ranges over $1$-Lipschitz test functions in the Wasserstein dual formulation. In Lemma~\ref{lemma:sinkhorn_approx}: In the proof, $f(x)=-x\log x$ is the concave scalar function used on transportation-plan entries in $[0,1]$, with its continuous value $f(0)=0$. In Theorem~\ref{thm:approximation}: In the categorical step of the proof, the unindexed $f$ denotes the structural propensity function at the node currently being compared, and $p=f(\text{pa}(v_k),\boldsymbol{u}_{V_k})$. In Lemma~\ref{lemma:sup_wasserstein}: $f$ ranges over the fixed class $\mathcal F$ of measurable maps $\mathbb R^{d'}\rightarrow\mathbb R^d$, with $\|f(x)\|_\infty\leqslant K$ for every $x$; each scalar coordinate class $\mathcal F_j=\{x\mapsto f_j(x):f\in\mathcal F\}$, $j\in[d]$, has pseudo-dimension less than $d_{\mathcal F}<\infty$. There are a standard Borel space $T$ and a jointly Borel map $G:T\times\mathbb R^{d'}\rightarrow[-K,K]^d$ such that $\mathcal F=\{G(t,\cdot):t\in T\}$. Probabilities of uniform deviation events are evaluated in the completion of the sample law. In the result split from Theorem~\ref{thm:wide_nn}: In the lower-bound assumption, $f\in\mathcal{H}([0,1]^d,K)\cap\mathcal{F}_L([0,1]^d,K)$ is a Lipschitz homeomorphism satisfying $\mathbb{P}(f(B))\geqslant C_f\lambda(B)$ for measurable $B\subset[0,1]^d$; when applied componentwise, the support and measure are those of the component. In Proposition~\ref{prop:holder_curve}, Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the lower-bound assumption, $f\in\mathcal{H}([0,1]^d,K)\cap\mathcal{F}_L([0,1]^d,K)$ is a Lipschitz homeomorphism satisfying $\mathbb{P}(f(B))\geqslant C_f\lambda(B)$ for measurable $B\subset[0,1]^d$; when applied componentwise, the support and measure are those of the component. In Theorem~\ref{thm:consistency}: In the proof's optimization parameterization, $f(\theta)=\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\mathcal{M}^{\theta}}[F(V_1(t),\ldots,V_{n_V}(t))]$ is the population intervention objective on the model parameter space, distinct from the structural functions $f_i$ and the homeomorphism in the lower-bound assumption. In Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:counterexample_foral}: For each non-Gumbel latent law $\mathbb P$ and each connected support component $C_i$ of positive mass, the lower-bound assumption uses a Lipschitz homeomorphism $f\in\mathcal H([0,1]^d,C_i)\cap\mathcal F_L([0,1]^d,C_i)$ satisfying $\mathbb P(f(B))/\mathbb P(C_i)\geqslant C_f\lambda(B)$ for every measurable $B\subset[0,1]^d$. - $\hat{f}$: $\hat{f}$ is a function on the domain of its paired function $f$, with the particular function and codomain specified by the local comparison. In Lemma~\ref{lemma:Lip_condition}: $\hat{f}(t)=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}=t]$ is a population regression version, $L$-Lipschitz continuous and bounded by $K$, defined on the common treatment domain on which it is compared with $f$. - $\boldsymbol{t}$: $\boldsymbol{t}=(t_i)_{T_i\in\boldsymbol{T}}$ assigns treatment values to the chosen set $\boldsymbol{T}\subset\boldsymbol{V}$ by replacing each treatment structural equation with $T_i=t_i$. In Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}: For the displayed ATE contrast, $\boldsymbol{T}=\{T\}$ and the assigned treatment value is 1. In Proposition~\ref{prop:Lip_ATE}: The intervention is on the single treatment $T$, and its value $t$ ranges over $[t_1,t_2]$ in the displayed integral. In Corollary~\ref{corollary:deep_nn}: For the fixed $R>0$ chosen before $\hat{\mathcal M}$, each continuous treatment coordinate satisfies $\|t_i\|_\infty\leqslant R$, and each categorical treatment coordinate belongs to its declared set $[n_i]$. - $\boldsymbol{t}_0$: $\boldsymbol{t}_0$ assigns the reference values to the same treatment set $\boldsymbol{T}$ as $\boldsymbol{t}$ in $\mathbb{E}_{\boldsymbol{u}\sim P(\boldsymbol{U})}[Y(\boldsymbol{t})-Y(\boldsymbol{t}_0)]$. In Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}: The reference intervention is $T=0$. - $T$: $T$ is locally specified as a treatment variable, a transportation map, a transportation plan, or a standard Borel parameter space. In Lemma~\ref{lemma:Lip_condition}: $T$ is the treatment coordinate of $(T,Y)\sim\nu$; conditioning in $f(t)$ is on $T=t$. In Proposition~\ref{prop:counterexample_foral}: $T$ is the binary treatment in the proof, taking values $0$ and $1$, with the conditional probabilities specified by the displayed structural equations. In Lemma~\ref{lemma:gumbel_err}: $T:\Delta_n\rightarrow\Delta_n$ sends $x$ to the one-hot vector at the first coordinate attaining $\max_j x_j$; it is a transportation map, rather than a treatment variable. In Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: $T$ denotes the treatment node, and $Y(t)$ denotes the outcome under the fixed intervention $T=t$. In Lemma~\ref{lemma:sinkhorn_approx}: $T=(T_{ij})\in\Pi(\mu,\nu)$ is a nonnegative $n\times m$ transportation plan satisfying $\sum_jT_{ij}=\mu_i$ and $\sum_iT_{ij}=\nu_j$. The minimizations defining the transport costs range over $\Pi(\mu,\nu)$; the entropy maximum displayed in the proof ranges over plans attaining the unregularized optimal cost $W(\mu,\nu)$. In Lemma~\ref{lemma:sup_wasserstein}: $T$ is a standard Borel parameter space, and $G:T\times\mathbb R^{d'}\rightarrow[-K,K]^d$ is jointly Borel with $\mathcal F=\{G(t,\cdot):t\in T\}$. It is not a treatment variable. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: In the cited squared-loss formulation, $T$ is a standard Borel parameter space for a jointly Borel map $g:T\times X\rightarrow[0,1]$, with $\mathcal G=\{g(t,\cdot):t\in T\}\ne\varnothing$. - $Y$: $Y$ denotes the random quantity specified by the local model or joint law, with the outcome and coupling uses distinguished below. In Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: $Y$ is the scalar outcome node of the SCM, and $Y(t)$ is its value after replacing the treatment equation by $T=t$. In Lemma~\ref{lemma:Lip_condition}: $Y$ is the second coordinate of $(T,Y)\sim\nu$, and $f(t)$ is a version of $\mathbb{E}_{\nu}[Y\mid T=t]$. In Corollary~\ref{cor:rate}: The objective selects the outcome coordinate, $F(\boldsymbol{V})=Y$. In Lemma~\ref{lemma:product_err}: In the proof, $Y=(Y_1,\ldots,Y_n)$ is the second product-coupled vector; $(X_k,Y_k)\sim\pi_k$, where $\pi_k$ couples $\mu_k$ and $\hat{\mu}_k$, so $Y$ has law $\hat{\mu}$. - $\mu$: $\mu$ denotes a measure on the domain specified by its local use, rather than a single source measure on a globally fixed Euclidean space. In Lemma~\ref{lemma:push-forward_err}: $\mu$ is a probability measure supported on the compact set $K\subset\mathbb{R}^{d_1}$. In Lemma~\ref{lemma:product_err}: $\mu=\mu_1\otimes\cdots\otimes\mu_n$ is the product of the stated component measures. In Lemma~\ref{lemma:Lip_condition}: $\mu$ is the joint law of $(\hat{T},\hat{Y})$, and $\hat{f}(t)$ is the conditional expectation under this law. In Proposition~\ref{prop:Lip_ATE}: $\mu$ is the observational distribution of $\hat{\mathcal{M}}$. In Lemma~\ref{lemma:sinkhorn_approx}: $\mu\in\Delta_n$ is the weight vector of a finite probability measure on the supplied points $x_1,\ldots,x_n$ in the common Euclidean ground space. In Lemma~\ref{lemma:sup_wasserstein}: $\mu$ is the population probability measure on $\mathbb{R}^{d'}$ whose empirical distribution is $\mu_n$. - $\hat{T}$: $\hat{T}$ is the real-valued treatment coordinate of $(\hat{T},\hat{Y})\sim\mu$, and $\hat{f}(t)=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}=t]$. In Lemma~\ref{lemma:Lip_condition}: The proof treats $(\hat{T},\hat{Y})$ as supported in $\{(t,y):\|(t,y)\|_\infty\leqslant K\}$. - $\hat{Y}$: $\hat{Y}$ denotes the real-valued outcome in the comparison distribution or comparison causal model. In Lemma~\ref{lemma:Lip_condition}: $(\hat{T},\hat{Y})\sim\mu$, with $\hat{f}(t)=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}=t]$; the proof treats this pair as supported in $\{(t,y):\|(t,y)\|_\infty\leqslant K\}$. In Proposition~\ref{prop:Lip_ATE}: $\hat{Y}$ is the outcome coordinate generated by $\hat{\mathcal{M}}$, whose observational law is $\mu$; $\hat{Y}(t)$ denotes its outcome under intervention $T=t$, and $|\hat{Y}|\leqslant K$. - $\text{ATE}_{\mathcal{M}}$: $\text{ATE}_{\mathcal{M}}(\boldsymbol{t})=\mathbb{E}_{\boldsymbol{u}\sim P(\boldsymbol{U})}[Y(\boldsymbol{t})-Y(\boldsymbol{t}_0)]$ is a population contrast between the two intervened outcomes in the same SCM. In Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}: The argument suppressed in $\text{ATE}_{\mathcal{M}}$ is $T=1$, and the reference intervention is $T=0$. - $P^{\mathcal{M}}$: $P^{\mathcal{M}}$ denotes the population-law notation associated with $\mathcal{M}$, evaluated at the variables written in its argument. In Theorem~\ref{thm:approximation}: $P^{\mathcal{M}}(\boldsymbol{V}(\boldsymbol{t}))$ is the joint interventional law under $\boldsymbol{T}=\boldsymbol{t}$, whereas $P^{\mathcal{M}}(\boldsymbol{U})$ is the joint latent law. - $\mathbb{P}$: $\mathbb{P}$ denotes the probability measure appropriate to the local distribution or events. In Proposition~\ref{prop:holder_curve}: The statement uses $\mathbb{P}$ satisfying Assumption~\ref{assumption:lower_bound} on $K\subset\mathbb{R}^D$. The proof reduces to a measure on $[0,1]^d$ that vanishes on the boundary and satisfies $\mathbb{P}(B)\geqslant C_1\lambda(B)$ for all measurable $B\subset[0,1]^d$; the reduced measure is $f^{-1}_{\#}\mathbb{P}$ for the homeomorphism in that assumption. Absolute continuity with respect to Lebesgue measure is assumed only in the illustrative proof sketch. In Theorem~\ref{thm:wide_nn}, \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: In the distribution-generator construction, $\mathbb P$ is the target probability measure on the ambient Euclidean space; expressions such as $\mathbb P(X_i=1)=p_i$ in the selector construction denote probabilities under the selector's law. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: In the categorical propensity definition, $\mathbb{P}(V_i=j\mid\text{Pa}(V_i),\boldsymbol{U}_{V_i})$ denotes conditional probability under the relevant SCM law, rather than evaluation of the distribution-generator target measure. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: $\mathbb P$ is the target probability measure on $\mathbb R^d$ satisfying Assumption~\ref{assump:support}; its normalized restrictions to the connected support components are the laws $\mu_i$. In Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:counterexample_foral}: In the componentwise lower-bound requirement, $\mathbb P$ denotes each non-Gumbel latent law; Assumption~\ref{assumption:lower_bound} is imposed on $\mathbb P|_{C_i}/\mathbb P(C_i)$ for every connected support component $C_i$, with $K=C_i$. - $\nu$: $\nu$ denotes a probability distribution, represented by its probability vector when its domain is finite, with the ambient domain specified locally. In Proposition~\ref{prop:Lip_ATE}: $\nu=P^{\mathcal M}(\boldsymbol V)$ is the observational law of $\mathcal M$. In the proof, $\nu_A$ is its marginal on the observed nondescendants $A$ of $T$, excluding $T$, and $X=\text{Pa}(T)$ is a subvector of $A$. The conditional treatment laws satisfy $\nu(dt\mid A=a)=\nu(dt\mid X=x)=p_\nu(t\mid X=x)P(dt)$, where $x$ is the $X$ subvector of $a$. In Lemma~\ref{lemma:Lip_condition}: $\nu=P(T,Y)$ is the joint law of $(T,Y)$, while $\nu(dt)$ in the integral denotes its treatment marginal. In Lemma~\ref{lemma:sinkhorn_approx}: $\nu=(\nu_1,\ldots,\nu_m)\in\Delta_m$ is the weight vector representing the finite probability measure $\sum_{j=1}^{m}\nu_j\delta_{y_j}$; the points $y_j$ and the points $x_i$ representing $\mu$ lie in a common Euclidean space, with ground distance $\|x_i-y_j\|$ given by the $1$-norm. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: $\nu^\tau$ is the distribution of the Gumbel-softmax weight vector $w^\tau$, and $\nu^0$ is the distribution of its limiting one-hot vector. - $\hat{\mu}$: $\hat{\mu}$ denotes the comparison probability measure on the same space as $\mu$. In Lemma~\ref{lemma:push-forward_err}: $\hat{\mu}$ is supported on the compact set $K\subset\mathbb{R}^{d_1}$. In Lemma~\ref{lemma:product_err}: $\hat{\mu}=\hat{\mu}_1\otimes\cdots\otimes\hat{\mu}_n$ is the product of the comparison factor measures. - $C_i$: $C_i$ denotes a subset of the ambient Euclidean space in the local support or partition construction. In Theorem~\ref{thm:wide_nn}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}, \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Theorem~\ref{thm:wide_nn}: $C_i$, $i\in[N_C]$, are the finitely many connected components of $\operatorname{supp}(\mathbb{P})$, with $\operatorname{supp}(\mathbb{P})=\bigcup_{i=1}^{N_C}C_i$ and $\mathcal{H}([0,1]^{d_i^C},C_i)\cap\mathcal{F}_L([0,1]^{d_i^C},C_i)\neq\emptyset$ for some $d_i^C\geqslant0$. In the result split from Theorem~\ref{thm:wide_nn}: $C_i$, $i\in[N_C]$, are the finitely many connected components of $\operatorname{supp}(\mathbb P)$, with $\operatorname{supp}(\mathbb P)=\bigcup_{i=1}^{N_C}C_i$ and $\mathcal H([0,1]^{d_i^C},C_i)\cap\mathcal F_L([0,1]^{d_i^C},C_i)\neq\emptyset$ for some $d_i^C\geqslant0$. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: If $d_i^C=0$, then $C_i$ is a singleton, since it is homeomorphic to $[0,1]^0$. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: $C_i$, $i\in[N_C]$, are the finitely many connected components of $\operatorname{supp}(\mathbb{P})$, with $\operatorname{supp}(\mathbb{P})=\bigcup_{i=1}^{N_C}C_i$ and $\mathcal{H}([0,1]^{d_i^C},C_i)\cap\mathcal{F}_L([0,1]^{d_i^C},C_i)\neq\emptyset$ for some $d_i^C\geqslant0$; the quantized construction applies to components with $d_i^C>0$. In Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:counterexample_foral}: For each non-Gumbel latent law $\mathbb P$, every normalized component law $\mathbb P|_{C_i}/\mathbb P(C_i)$ satisfies Assumption~\ref{assumption:lower_bound} with compact support set $K=C_i$. - $\mu_i$: $\mu_i$ denotes the component measure or probability-vector coordinate specified by its local decomposition. In Theorem~\ref{thm:wide_nn}: $i\in[N_C]$ and $\mu_i=\mathbb{P}_{C_i}/\mathbb{P}(C_i)$, where $\mathbb{P}_{C_i}$ is the restriction of $\mathbb{P}$ to $C_i$, whenever $\mathbb{P}(C_i)>0$. In Lemma~\ref{lemma:combin_err}: $i\in[n]$ and $\mu_i$ is an arbitrary measure on $\mathbb{R}^d$ entering the mixture $\sum_{i=1}^n p_i\mu_i$. In Lemma~\ref{lemma:product_err}: $i\in[n]$ and $\mu_i$ is the $i$th factor of $\mu=\mu_1\otimes\cdots\otimes\mu_n$, paired with $\hat{\mu}_i$ on its factor space. In the result split from Theorem~\ref{thm:wide_nn}: $i\in[N_C]$ and $\mu_i=\mathbb{P}_{C_i}/\mathbb{P}(C_i)$, where $\mathbb{P}_{C_i}$ is the restriction of $\mathbb{P}$ to $C_i$, whenever $\mathbb{P}(C_i)>0$. In Lemma~\ref{lemma:sinkhorn_approx}: $i\in[n]$ and $\mu_i\in[0,1]$ is the mass at $x_i$ in the finite probability measure with weight vector $\mu\in\Delta_n$; a transportation plan satisfies $\sum_jT_{ij}=\mu_i$. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: $i\in[N_C]$ and $\mu_i=\mathbb{P}_{C_i}/\mathbb{P}(C_i)$, where $\mathbb{P}_{C_i}$ is the restriction of $\mathbb{P}$ to $C_i$, whenever $\mathbb{P}(C_i)>0$. - $\hat{\mu}_i$: $\hat{\mu}_i$ denotes a component probability measure paired with $\mu_i$, without an intrinsic neural-network construction. In Lemma~\ref{lemma:combin_err}: $i\in[n]$ and $\hat{\mu}_i$ is a measure on $\mathbb{R}^d$ entering the comparison mixture $\sum_{i=1}^n p_i\hat{\mu}_i$. In Lemma~\ref{lemma:product_err}: $i\in[n]$ and $\hat{\mu}_i$ is the $i$th factor of $\hat{\mu}=\hat{\mu}_1\otimes\cdots\otimes\hat{\mu}_n$, on the same factor space as $\mu_i$. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: $i\in[N_C]$ and $\hat{\mu}_i=P(\hat{g}_2^i\circ\hat{g}_1^i(U_i))$, where $U_i$ is uniform on $[0,1]$; this is the law of the two deterministic component blocks before the Gumbel mixing layer. For $d_i^C=0$, it is the point mass at the singleton $C_i$. - $X^\tau$: $X^\tau$ uses positive weights $p_1,\ldots,p_n$, which need not sum to one, and $0<\tau<1$, with coordinates indexed by $k\in[n]$; all coordinates are positive and sum to one. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: In the mixing construction, $n=N_C$ and $p_i=\mathbb{P}(C_i)>0$ for $i\in[N_C]$; $X^\tau$ is the vector of Gumbel-softmax weights assigned to these support components. - $\tau$: $\tau>0$ is the Gumbel-softmax temperature hyperparameter; a superscript $0$ on a softmax output or layer denotes the limit as $\tau\rightarrow0$, rather than evaluation of the positive-temperature formula at zero. In Lemma~\ref{lemma:gumbel_err}: The temperature satisfies $0<\tau<1$. In Theorem~\ref{thm:wide_nn}: The error estimate is asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of the budgets and $\tau$; the proof uses the Gumbel-softmax estimate for $0<\tau<1$. In Theorem~\ref{thm:consistency}: The temperature is the positive sequence $\tau_n$ appearing in $\mathcal{F}_{1,n}$, with $\tau_n\rightarrow0$. In the result split from Theorem~\ref{thm:wide_nn}: The error estimate is asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of the budgets and $\tau$; the proof uses the Gumbel-softmax estimate for $0<\tau<1$. In Corollary~\ref{cor:rate}: The generator temperature is the positive sequence $\tau_n$ in $\mathcal F_{1,n}$ and in the approximation term $\tau_n(1-\log\tau_n)$. In Corollary~\ref{corollary:deep_nn}: The displayed error bound is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, with $R>0$ and the causal model fixed. The implicit constant is independent of the intervention and the network budgets, but may depend on $R$ and the fixed causal model. - $X^0$: $X^0=\lim_{\tau\downarrow0}X^\tau=e_I$ almost surely, where $I=\arg\max_{k\in[n]}(\log p_k+G_k)$, using the same Gumbel draws throughout the limit; ties may be assigned to the first maximizing coordinate. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: In the mixing construction, $n=N_C$ and $p_i=\mathbb{P}(C_i)>0$ for $i\in[N_C]$; $X^0$ is the one-hot limit of these component weights, with $P(X^0=e_i)=\mathbb{P}(C_i)$. - $\mu^0$: $\mu^0$ is the law of the one-hot vector $X^0$, supported on the vertices of $\Delta_n$, with $\mu^0(\{e_k\})=p_k/\sum_{j=1}^n p_j$ for $k\in[n]$. - $n$: $n$ is a local integer parameter whose role is determined by the indexed objects in the surrounding passage. In Lemma~\ref{lemma:hilbert}, Proposition~\ref{prop:holder_curve}: In the partition notation $\mathcal{C}_n$ and the curve and refinement sequences, $n$ is a level index beginning at $0$, with $\mathcal{C}_0=\{C_{-1}\}$ and $C_{-1}=[0,1]^d$. In Lemma~\ref{lemma:combin_err}: $n\geqslant1$ is the number of mixture components, with component indices $k=1,\cdots,n$. In Lemma~\ref{lemma:product_err}: $n\geqslant1$ is the number of factors in the product measures. In Lemma~\ref{lemma:gumbel_err}: $n\geqslant1$ is the dimension of $X^\tau$ and the number of positive weights $p_k$ and Gumbel coordinates $G_k$. In Proposition~\ref{prop:convg_opt}: $n\in\mathbb{N}$ indexes the nested sets, constraint functions, tolerances, and optimization values, and varies in the limit $n\rightarrow\infty$. In Proposition~\ref{prop:holder_curve}: In the local cube-metric definition $C=\prod_{i=1}^{n}[a_i,b_i]$ and $C_k=\prod_{i=1}^{n}[a_i^k,b_i^k]$, $n$ is the ambient dimension and $i$ ranges from $1$ to $n$. In Lemma~\ref{lemma:sinkhorn_approx}: $n\geqslant1$ is the length of the probability vector $\mu\in\Delta_n$, and a transportation plan has $n$ rows and $m$ columns. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: $n$ is the number of observations from the true distribution and also indexes the approximation and tolerance sequences; the asymptotic limit is $n\rightarrow\infty$. In Lemma~\ref{lemma:sup_wasserstein}: $n\geqslant1$ is the sample size defining $\mu_n$ and the number of evaluation points in the empirical covering-number notation $\mathcal{N}_1(\epsilon,\mathcal{F},n)$. In Theorem~\ref{thm:wide_nn}: For a component of positive dimension $q=d_i^C$, the refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor\geqslant2$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$; the quantization parameter is $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$ and the mesh is $\delta/n$. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: For a component of positive dimension $q=d_i^C$, the refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor\geqslant2$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$; the quantization parameter is $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$ and the mesh is $\delta/n$. - $\mu_n$: $\mu_n$ denotes the locally specified measure, with $n$ serving either as a component index or as the empirical sample size. In Lemma~\ref{lemma:combin_err}: $\mu_n$ is the component measure with index $n$ in the mixture, among measures $\mu_k$ on $\mathbb{R}^d$ for $k\in[n]$. In Lemma~\ref{lemma:product_err}: $\mu_n$ is the final component measure in the product $\mu=\mu_1\otimes\cdots\otimes\mu_n$. In Lemma~\ref{lemma:sup_wasserstein}: $\mu_n=n^{-1}\sum_{i=1}^n\delta_{X_i}$ is the empirical distribution of the sample with law $\mu$, on $\mathbb{R}^{d'}$. - $W$: The notation $W(\mu,\nu)$ denotes Wasserstein-1 distance between measures on the same underlying space, whereas $W$ in a network architecture argument denotes an integer width. In \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Corollary~\ref{corollary:deep_nn}: In $a\,\omega(cW^{-2/d_i^{\text{in}}})$, $W$ is the number of weights of the scalar approximating network, rather than its width; at fixed width $2d_i^{\text{in}}+10$, this weight count grows linearly in the depth $L_0$. - $\lambda$: $\lambda$ is a positive scalar when it indexes a regularized transport functional and a reference measure when it occurs in a push-forward or measure-domination expression, with its reference domain specified locally. In Theorem~\ref{thm:wide_nn}: In the push-forward expression, $\lambda$ is Lebesgue measure on $[0,1]^{N_C}$; in the componentwise construction, it is the uniform measure on $[0,1]$. In Proposition~\ref{prop:holder_curve}: For the curve's push-forward, $\lambda$ is Lebesgue measure on $[0,1]$; in expressions $\lambda(B)$ with $B\subset[0,1]^d$, it is $d$-dimensional Lebesgue measure. In Lemma~\ref{lemma:sinkhorn_approx}: $\lambda>0$ is the scalar regularization parameter. In the result split from Theorem~\ref{thm:wide_nn}: In the push-forward expression, $\lambda$ is Lebesgue measure on $[0,1]^{N_C}$; in the componentwise construction, it is the uniform measure on $[0,1]$, and in the lower-bound premise it is Lebesgue measure on the corresponding cube. - $S_\lambda$: $S_\lambda$ is the debiased regularized transport functional obtained by subtracting half of each self-comparison from $W_{1,\lambda}(\mu,\nu)$. In Lemma~\ref{lemma:sinkhorn_approx}: The arguments are finite probability measures represented by their weights and support points, with $c_{ij}=\|x_i-y_j\|$ for the $1$-norm. Here $W_{1,\lambda}(\mu,\nu)=\min_{T\in\Pi(\mu,\nu)}\{\sum_{i,j}c_{ij}T_{ij}-\lambda h(T)\}$, where $\Pi(\mu,\nu)$ is the set of nonnegative transportation plans with marginals $\mu,\nu$, and $h(T)=1-\sum_{i,j}T_{ij}\log T_{ij}$ with $0\log0=0$. Both self terms use this same convention and the same $1$-norm ground distance on their corresponding support points. - $W_{1,\lambda}$: $W_{1,\lambda}$ uses the supplied finite support points, the paper's $1$-norm ground cost, and the entropy subtraction $-\lambda h(T)$ with $h(T)=1-\sum_{i,j}T_{ij}\log T_{ij}$ and $0\log0=0$. - $L$: $L$ denotes a nonnegative Lipschitz bound with respect to the paper's $1$-norm when used in a Lipschitz condition or $\mathcal{F}_L$, and an integer depth when used as the depth argument of a neural network class. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:Lip_ATE}: In the SCM class $\mathcal{M}(\mathcal{G},\mathcal{F}_L,\boldsymbol{U})$, $L$ bounds the continuous structural functions and the categorical propensity functions, rather than the categorical $\arg\max$ maps. In Lemma~\ref{lemma:push-forward_err}: $L$ is a Lipschitz bound for $F$; the hypothesis does not impose that bound on $\hat{F}$. In Lemma~\ref{lemma:Lip_condition}: $L$ is a common Lipschitz bound for the regression functions $f$ and $\hat{f}$. In Proposition~\ref{prop:holder_curve}, Theorem~\ref{thm:wide_nn}: In the support assumptions, $L$ bounds the homeomorphisms from the unit cubes to the support sets; it is not a Lipschitz bound for the resulting space-filling curves. In Theorem~\ref{thm:consistency}: In $\mathcal F_L^K$ and $\underline F^L$, $L$ bounds the Lipschitz constants of the continuous structural functions and normalized categorical propensity vectors of population candidate SCMs, with respect to the paper's $1$-norm; the class also requires $\sup_x\|f(x)\|_1\leqslant K$ on the full structural input domain. The conclusion evaluates this parameter at $\hat L_f$. In the result split from Theorem~\ref{thm:wide_nn}: In the support assumptions, $L$ bounds the homeomorphisms from the unit cubes to the support sets; it is not a Lipschitz bound for the resulting space-filling curves. - $\mathcal{F}_L$: $\mathcal{F}_L(K_1,K_2)$ denotes the class of maps $f:K_1\rightarrow K_2$ with Lipschitz constant at most $L$, using the paper's $1$-norm convention, with the arguments supplied by context when omitted. - $\mathcal{H}$: $\mathcal{H}(K_1,K_2)$ denotes homeomorphisms from $K_1$ onto $K_2$, while an explicit local definition of unparameterized $\mathcal{H}$ supplies its function class. In Lemma~\ref{lemma:sup_wasserstein}: In the proof, $\mathcal{H}=\left\{h:h\in\mathcal{F}_1([-K,K]^d,[-K,K]^d),\ h(0)=0\right\}$, and $\mathcal{H}_\delta$ is a $\delta$-net of this class in the $\ell_\infty$ norm. - $\mathcal{F}$: $\mathcal{F}$ denotes a function class whose domains and codomains are supplied by its local use. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Proposition~\ref{prop:equivalent_g_constrianed}: In $\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})$, $\mathcal{F}$ specifies the allowed structural functions, with $f_i\in\mathcal{F}$ for $i\in[n_V]$; each function has the input and output dimensions of its structural equation. In Lemma~\ref{lemma:sup_wasserstein}: $\mathcal F$ is a class of measurable maps $f:\mathbb R^{d'}\rightarrow\mathbb R^d$. Each scalar coordinate class $\mathcal F_j=\{x\mapsto f_j(x):f\in\mathcal F\}$, $j\in[d]$, has pseudo-dimension less than $d_{\mathcal F}<\infty$, and $\|f(x)\|_\infty\leqslant K$ for every $f\in\mathcal F$ and $x\in\mathbb R^{d'}$. There are a standard Borel space $T$ and a jointly Borel map $G:T\times\mathbb R^{d'}\rightarrow[-K,K]^d$ such that $\mathcal F=\{G(t,\cdot):t\in T\}$. - $K$: $K$ is a context-dependent bound or compact set: numerical occurrences are positive bounds, whereas set-valued occurrences specify the compact domain or support of a measure. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: In the boundedness assumption, $K>0$ satisfies $\max_{i\in[n_V],j\in[n_U]}\{\|V_i\|_\infty,\|U_j\|_\infty\}\leqslant K$; the indexed $U_j$ are the latent variables separated from the categorical Gumbel noises. In Lemma~\ref{lemma:push-forward_err}: $K\subset\mathbb{R}^{d_1}$ is the compact set carrying $\mu$ and $\hat{\mu}$ and the common domain of $F$ and $\hat{F}$. In Lemma~\ref{lemma:Lip_condition}: $K$ bounds the regression functions: $|f(t)|\leqslant K$ and $|\hat{f}(t)|\leqslant K$. In Proposition~\ref{prop:holder_curve}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}, the result split from Theorem~\ref{thm:wide_nn}: In the lower-bound support assumption, $K\subset\mathbb{R}^{D}$ is compact and supports the measure under consideration; $f\in\mathcal{H}([0,1]^d,K)\cap\mathcal{F}_L([0,1]^d,K)$. This set-valued occurrence is distinct from a numerical boundedness constant. In Theorem~\ref{thm:consistency}: In the explanatory notation $\mathcal{P}(K)$, $K$ is the set on which probability measures are defined; in the displayed factors $\mathcal{P}([-K,K]^{d_j^U})$, the $K$ inside the cube endpoints is the numerical boundedness constant. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the truncated neural classes, numerical $K$ is the clipping level in $\max\{-K,\min\{f,K\}\}$ and is the boundedness constant chosen for the underlying SCM. Every true continuous structural function and every candidate continuous structural network additionally satisfies $\sup_x\|f_i(x)\|_1\leqslant K$ and $\sup_x\|\hat f_i(x)\|_1\leqslant K$, respectively, on its full structural input domain. The limiting structural class retains $\mathcal F_L^K=\{f:\sup_x\|f(x)\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$. In Lemma~\ref{lemma:sup_wasserstein}: $K>0$ is the uniform output bound $\|f(x)\|_\infty\leqslant K$ for every $f\in\mathcal F$ and every $x\in\mathbb R^{d'}$; consequently the outputs lie in $[-K,K]^d$. In Proposition~\ref{prop:counterexample_foral}: For each non-Gumbel latent law $\mathbb P$ with support components $C_i$, Assumption~\ref{assumption:lower_bound} applies to every normalized component law $\mu_i=\mathbb P|_{C_i}/\mathbb P(C_i)$ with the compact support set $K=C_i$. In Proposition~\ref{prop:counterexample}: For each non-Gumbel latent law $\mathbb P$ with support components $C_i$, Assumption~\ref{assumption:lower_bound} applies to every normalized component law $\mu_i=\mathbb P|_{C_i}/\mathbb P(C_i)$ with the compact support set $K=C_i$. In Corollary~\ref{corollary:deep_nn}: For each non-Gumbel latent law $\mathbb P=P(U_C)$ with support components $C_i$, the lower-bound support assumption applies to each normalized component law $\mu_i=\mathbb P|_{C_i}/\mathbb P(C_i)$ with compact support set $K=C_i$. Its chart $f:[0,1]^{d_i^C}\rightarrow C_i$ satisfies $(f^{-1})_{\#}\mu_i(\partial[0,1]^{d_i^C})=0$ when $d_i^C>0$. - $\mathcal{F}_L^K$: $\mathcal{F}_L^K(K_1,K_2)$ denotes a class of bounded Lipschitz continuous maps $f:K_1\rightarrow K_2$, with the norm bound and Lipschitz restriction given by its local definition. In Theorem~\ref{thm:consistency}: In the statement and the limiting optimization, $\mathcal F_L^K=\{f:\sup_x\|f(x)\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$, with the supremum over the full structural input domain; in the proof's displayed population-function definition, $\mathcal F_L^K=\{f:\|f\|_\infty\leqslant K,\ f\ \text{Lipschitz continuous}\}$, where the function infinity norm is the uniform coordinatewise bound and that display supplies no numerical Lipschitz bound. In Corollary~\ref{cor:rate}: The population bounded Lipschitz class is $\mathcal F_L^K=\{f:\sup_x\|f(x)\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$, with the supremum over the full structural input domain. - $\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})$: $\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})$ consists of canonical SCMs on $\mathcal{G}$ with $f_i\in\mathcal{F}$ for $i\in[n_V]$ and the indicated canonical latent coordinates; the argument $\boldsymbol{U}$ does not fix a common latent probability law across the class. - $\mathcal{NN}$: The two-argument notation $\mathcal{NN}_{k_1,k_2}(W,L)$ specifies a ReLU network class, with omitted dimensions supplied by context; the five-argument notation specifies the separable three-part generator architecture including its Gumbel-Softmax layer. In Corollary~\ref{corollary:deep_nn}: The four-argument generator class $\mathcal{NN}(\Theta(d_{\max}^U),L_1,\Theta((d_{\max}^U)^2),L_2)$ suppresses the temperature $\tau$ of the same third Gumbel-Softmax layer, which remains present in the approximation bound. The multiplicative width factors in $\Theta(d_{\max}^{\text{in}}d_{\max}^{\text{out}})$, $\Theta(d_{\max}^U)$, and $\Theta((d_{\max}^U)^2)$ can be chosen as absolute numerical constants, independent of the causal model, $R$, all dimensions, and all category and component counts. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: In the generator architecture, a component with $d_i^C=0$ uses the unique map to $[0,1]^0$ as its first deterministic block and a constant affine map to its singleton support component as its second deterministic block. In the result split from Theorem~\ref{thm:wide_nn}: The multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. - $\mathcal{NN}(W_1,L_1,W_2,L_2,\tau)$: This is the three-part generator class with component count $N_C$, component cube dimensions $d_j^C$, and output dimension $d$ suppressed from its arguments; $W_1,L_1$ and $W_2,L_2$ specify the respective component-network widths and depths, and $\tau>0$ specifies the mixing temperature. In Corollary~\ref{corollary:deep_nn}: The four-argument generator-class occurrence suppresses the temperature argument; the latent generators retain the Gumbel-softmax temperature $\tau$ appearing in the bound. The multiplicative width factors in $\Theta(d_{\max}^U)$ and $\Theta((d_{\max}^U)^2)$ can be chosen as absolute numerical constants, independent of the causal model, $R$, all dimensions, and all category and component counts. In Theorem~\ref{thm:consistency}: The class used is its truncated version $\mathcal{NN}^{\infty,K}(\Theta(d_{\max}^U),L_{1,n},\Theta((d_{\max}^U)^2),L_{2,n},\tau_n)$; the superscript $\infty$ imposes no finite Lipschitz bound, while $K$ specifies coordinatewise output truncation, and the finite component counts and component cube dimensions of the latent generator architectures are fixed as $n$ varies. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: The architecture includes components with $d_i^C=0$: their first deterministic block is the unique map to $[0,1]^0$, and their second deterministic block is a constant affine map to the corresponding singleton. In Corollary~\ref{cor:rate}: The latent generator class is the truncated class $\mathcal F_{1,n}=\mathcal{NN}^{\infty,K}(\Theta(d_{\max}^U),L_{1,n},\Theta((d_{\max}^U)^2),L_{2,n},\tau_n)$; the superscript $\infty$ imposes no finite Lipschitz bound, $K$ specifies coordinatewise output truncation, and the finite component counts and component cube dimensions are fixed as $n$ varies. In the result split from Theorem~\ref{thm:wide_nn}: The multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. - $\mathcal{NN}^{L_f,K}$: $\mathcal{NN}^{L_f,K}(W,L)$ abbreviates $\mathcal{NN}_{d_1,d_2}^{L_f,K}(W,L)=\{\max\{-K,\min\{f,K\}\}:f\in\mathcal{NN}_{d_1,d_2}(W,L),\ \text{Lip}(f)\leqslant L_f\}$, with coordinatewise clipping and dimensions supplied by the function being represented; here $L$ is network depth. In Theorem~\ref{thm:consistency}: The ambient structural class is $\mathcal F_{0,n}=\mathcal{NN}^{\hat L_f,K}(W_{0,n},\Theta(\log d_{\max}^{\text{in}}))$, where the depth expression uses $\Theta(1+\log d_{\max}^{\text{in}})$ when $d_{\max}^{\text{in}}=1$; categorical raw structural members $q$ admitted to the empirical optimization additionally satisfy $q(x)\in[0,\infty)^{n_i}$, $\|q(x)\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$, while each admitted continuous structural network $\hat f_i$ additionally satisfies $\sup_x\|\hat f_i(x)\|_1\leqslant K$ on its full structural input domain. In Corollary~\ref{cor:rate}: The ambient structural class is $\mathcal F_{0,n}=\mathcal{NN}^{\hat L_f,K}(W_{0,n},\Theta(\log d_{\max}^{\text{in}}))$, with the depth convention inherited from its specification; each categorical raw structural network $q$ admitted to the empirical optimization additionally satisfies $q(x)\in[0,\infty)^{n_i}$, $\|q(x)\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$, while each admitted continuous structural network $\hat f_i$ additionally satisfies $\sup_x\|\hat f_i(x)\|_1\leqslant K$ on its full structural input domain. - $W_1$: $W_1$ is the integer width of each component network $\hat{g}_1^j\in\mathcal{NN}_{1,d_j^C}(W_1,L_1)$, $j\in[N_C]$, rather than the combined width of all parallel component networks. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: For each component of positive dimension $q=d_i^C$, the construction takes $W_1\geqslant A_q2^q$ and $n=\lfloor(W_1/A_q)^{1/q}\rfloor$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. - $W_2$: $W_2$ is the integer width of each component network $\hat{g}_2^j\in\mathcal{NN}_{d_j^C,d}(W_2,L_2)$, $j\in[N_C]$. - $L_1$: $L_1$ is the integer depth of each component network $\hat{g}_1^j\in\mathcal{NN}_{1,d_j^C}(W_1,L_1)$, $j\in[N_C]$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: At sample size $n$, the first-block depth is $L_{1,n}$, with $L_{1,n}=\Theta(m_n^{d_{\max}^U/(2d_{\max}^U+4)})$; the architectural order uses sufficiently large fixed constant factors supplied by the network constructions. In Corollary~\ref{corollary:deep_nn}: The displayed error bound is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, with $R$ and the causal model fixed; $R>0$ is fixed before choosing $\hat{\mathcal M}$. The implicit constant is independent of the intervention and the network budgets, but may depend on $R$ and the fixed causal model. - $L_2$: $L_2$ is the integer depth of each component network $\hat{g}_2^j\in\mathcal{NN}_{d_j^C,d}(W_2,L_2)$, $j\in[N_C]$. In \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Theorem~\ref{thm:wide_nn}: $L_2$ is also the depth of each scalar coordinate approximant $\hat{g}_2^{i,j}$ used to form $\hat{g}_2^i=(\hat{g}_2^{i,1},\ldots,\hat{g}_2^{i,d})$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: At sample size $n$, the second-block depth is $L_{2,n}$, with $L_{2,n}=\Theta(m_n^{d_{\max}^U/(2d_{\max}^U+4)})$; the architectural order uses sufficiently large fixed constant factors supplied by the network constructions. In Corollary~\ref{corollary:deep_nn}: The displayed error bound is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, with $R$ and the causal model fixed; $R>0$ is fixed before choosing $\hat{\mathcal M}$. The implicit constant is independent of the intervention and the network budgets, but may depend on $R$ and the fixed causal model. - $\mathcal{NN}^{\infty,K}$: $\mathcal{NN}^{\infty,K}$ clips the final generator output coordinatewise to $[-K,K]$, with no imposed Lipschitz bound; its suppressed dimensions and component structure are those of the corresponding generator class. - $\hat{g}$: The notation $\hat{g}_{\#}\lambda$ denotes the output law of the generator after integrating both its uniform input and its Gumbel-softmax randomness, rather than a push-forward conditional on a realized mixing vector. In Theorem~\ref{thm:wide_nn}: The generator has uniform-input dimension $N_C$, output dimension $d$, and target law $\mathbb{P}$; its mixing weights are the target component masses. A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. In Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}: In the NCM class notation, $\hat{g}\in\mathcal{F}_1$ is shorthand for the separate latent generators $\hat{g}_j$, whose compatible dimensions depend on the corresponding latent variable. In Theorem~\ref{thm:consistency}: At sample size $n$, generators range over $\mathcal{F}_{1,n}$ and have coordinatewise output truncation to $[-K,K]$, with finite component counts and component cube dimensions fixed as $n$ varies. The optimization ranges over network representations, including distinct representations inducing the same latent population laws; each representation is evaluated on the same $m_n$ exogenous input draws. In the result split from Theorem~\ref{thm:wide_nn}: The generator has uniform-input dimension $N_C$, output dimension $d$, and target law $\mathbb{P}$; its mixing weights are the target component masses. A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. - $\text{NCM}_{\mathcal{G}}$: $\text{NCM}_{\mathcal{G}}(\mathcal{F}_0,\mathcal{F}_1)$ is the collection of models on $\mathcal{G}$ with structural equations (\ref{eq:structure_equation_hat}), in which every structural network belongs to $\mathcal{F}_0$ and every component-specific latent generator belongs to $\mathcal{F}_1$, with compatible dimensions suppressed. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: At sample size $n$, the candidate class is $\text{NCM}_{\mathcal G}(\mathcal F_{0,n},\mathcal F_{1,n})$, with the architecture and truncation specified in Theorem~\ref{thm:consistency}, restricted so that every categorical raw structural network $q$ satisfies $q(x)\in[0,\infty)^{n_i}$, $\|q(x)\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$. In Theorem~\ref{thm:consistency}: The empirical optimization ranges over network representations in the restricted candidate class; representations with the same latent population laws remain distinct candidates and may yield different common-draw empirical distributions. Every admitted continuous structural network additionally satisfies $\sup_x\|\hat f_i(x)\|_1\leqslant K$ on its full structural input domain. In Corollary~\ref{cor:rate}: Candidates are evaluated as network representations on the same $m_n$ exogenous simulation draws; representations with identical latent population laws can yield different empirical distributions on these common draws. Every admitted continuous structural network additionally satisfies $\sup_x\|\hat f_i(x)\|_1\leqslant K$ on its full structural input domain. - $\mathcal{F}_0$: $\mathcal{F}_0$ is the structural-network class supplied to $\text{NCM}_{\mathcal{G}}(\mathcal{F}_0,\mathcal{F}_1)$, with the input and output dimensions appropriate to each node omitted from the notation. In Theorem~\ref{thm:consistency}: The class is $\mathcal{F}_{0,n}=\mathcal{NN}^{\hat L_f,K}(W_{0,n},\Theta(\log d_{\max}^{\text{in}}))$, where $\hat L_f=\sqrt{d_{\max}^{\text{in}}d_{\max}^{\text{out}}}L_f$; it consists of coordinatewise truncations to $[-K,K]$ of networks whose Lipschitz constant is at most $\hat L_f$, and varies with $n$. When $d_{\max}^{\text{in}}=1$, the depth expression uses $\Theta(1+\log d_{\max}^{\text{in}})$. Continuous structural networks admitted to the optimization additionally satisfy $\sup_x\|\hat f_i(x)\|_1\leqslant K$ on their full structural input domains. Categorical raw networks admitted to the optimization additionally satisfy $q(x)\in[0,\infty)^{n_i}$, $\|q(x)\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$. - $\mathcal{F}_1$: $\mathcal{F}_1$ is function-class notation whose subscript can be a class identifier or the Lipschitz constant in the previously defined notation $\mathcal{F}_L(K_1,K_2)$. In Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}: As the second argument of $\text{NCM}_{\mathcal{G}}(\mathcal{F}_0,\mathcal{F}_1)$, the class restricts the latent generators, with input and output dimensions omitted from the notation. In Theorem~\ref{thm:consistency}: The latent-generator class is $\mathcal{F}_{1,n}=\mathcal{NN}^{\infty,K}(\Theta(d_{\max}^U),L_{1,n},\Theta((d_{\max}^U)^2),L_{2,n},\tau_n)$, a class varying with $n$. - $\hat{g}_1$: Each coordinate block $\hat{g}_1^j$ uses one scalar input to generate coordinates in the cube $[0,1]^{d_j^C}$ associated with component $j$, without mixing inputs from different components. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: Writing $\mu_j=\mathbb{P}_{C_j}/\mathbb{P}(C_j)$, for $d_j^C\geqslant1$ the first block approximates the cube law $((g_2^j)^{-1})_{\#}\mu_j$ by the law of $\hat{g}_1^j(U_j)$; for $d_j^C=0$, it is the unique map to $[0,1]^0$. - $\hat{g}_2$: Each coordinate block $\hat{g}_2^j$ maps the $d_j^C$-dimensional cube coordinates for component $j$ into $\mathbb{R}^d$, without mixing inputs from different components. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: For $d_j^C\geqslant1$, the block $\hat{g}_2^j$ approximates a map $g_2^j\in\mathcal{H}([0,1]^{d_j^C},C_j)\cap\mathcal{F}_L([0,1]^{d_j^C},C_j)$; for $d_j^C=0$, it is a constant affine map to the singleton $C_j$. - $\hat{g}_3^\tau$: $\hat{g}_3^\tau$ mixes the $N_C$ component outputs using the Gumbel-softmax coefficient vector, with $\tau>0$; the notation $\hat{g}_3^0$ denotes its almost-sure hard-selection limit as $\tau\rightarrow0$. In Theorem~\ref{thm:wide_nn}: The mixing weights are $p_k=\mathbb{P}(C_k)$ for the connected components $C_k$ of the target support. In Corollary~\ref{corollary:deep_nn}: The layer is used separately for each latent generator, with mixing weights given by the masses of the connected components of that latent variable's target law. In Theorem~\ref{thm:consistency}: At sample size $n$, the generator class uses temperature $\tau_n>0$ and truncates the resulting generator output coordinatewise to $[-K,K]$. In the result split from Theorem~\ref{thm:wide_nn}: The mixing weights are $p_k=\mathbb{P}(C_k)$ for the connected components $C_k$ of the target support. - $g_2^i$: For $i\in[N_C]$, $g_2^i$ is a chosen map in $\mathcal{H}([0,1]^{d_i^C},C_i)\cap\mathcal{F}_L([0,1]^{d_i^C},C_i)$, whose inverse pulls the component law back to the unit cube. In the result split from Theorem~\ref{thm:wide_nn}: $g_2^i$ is chosen so that $(g_2^i)^{-1}_{\#}\mu_i(B)\geqslant C_g\lambda(B)/\mathbb{P}(C_i)$ for every measurable $B\subset[0,1]^{d_i^C}$, where $\mu_i=\mathbb{P}_{C_i}/\mathbb{P}(C_i)$ and $\lambda$ is cube Lebesgue measure. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: For $d_i^C=0$, $g_2^i$ is the unique map from $[0,1]^0$ to the singleton $C_i$. - $\hat{g}_2^{i,j}$: For $i\in[N_C]$ and $j\in[d]$, $\hat{g}_2^{i,j}:[0,1]^{d_i^C}\rightarrow\mathbb{R}$ denotes a coordinate network chosen to approximate $(g_2^i)_j$ uniformly on $[0,1]^{d_i^C}$. In Theorem~\ref{thm:wide_nn}: When $d_i^C=0$, $\hat{g}_2^{i,j}$ is the $j$th coordinate of the constant affine map to the singleton $C_i$, so its approximation error is zero. - $\gamma_i$: For $i\in[N_C]$, $\gamma_i$ represents the pullback of the normalized component law $\mu_i$ through $g_2^i$, wherever $\mathbb{P}(C_i)>0$. - $\hat{g}_1^{i,j}$: For $i\in[N_C]$ and $j\in[d_i^C]$, $\hat{g}_1^{i,j}:[0,1]\rightarrow\mathbb{R}$ denotes a coordinate network chosen to approximate $(\gamma_i)_j$ uniformly on $[0,1]$. - $Z_{C_j}$: $Z_{C_j}$ denotes the uniform and Gumbel input vector for the generator of the latent variable associated with graph $C^2$-component $C_j$, with $N_{C,j}$ uniform coordinates and $N_{C,j}$ Gumbel coordinates, where $N_{C,j}$ counts the connected components of $\operatorname{supp}(P(U_{C_j}))$. In Theorem~\ref{thm:consistency}: At each sample size $n$, the simulation inputs $Z_{C_j}^{(r)}$, $r\in[m_n]$, are shared across all candidate network representations: each candidate latent generator is evaluated as $\hat g_j(Z_{C_j}^{(r)})$. These inputs are drawn as part of the common independent and identically distributed exogenous product-law draws, independently of the observational sample. - $\hat{U}_{C_j}$: For a $C^2$-component $C_j$ of $\mathcal{G}$, $\hat{U}_{C_j}=\hat{g}_j(Z_{C_j})$ denotes the generated latent vector corresponding to $U_{C_j}$, with law $(\hat{g}_j)_{\#}P(Z_{C_j})$. - $N_C$: $N_C$ denotes the finite number of connected components of $\operatorname{supp}(\mathbb P)$, indexed by $i\in[N_C]$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: The finite component count of each latent generator architecture in $\mathcal F_{1,n}$ is fixed as $n$ varies. - $d_i^C$: For $i\in[N_C]$, $d_i^C$ is a nonnegative integer such that $\mathcal{H}([0,1]^{d_i^C},C_i)\cap\mathcal{F}_L([0,1]^{d_i^C},C_i)\neq\emptyset$; $d_i^C=0$ corresponds to a singleton component. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: The component cube dimensions of each latent generator architecture in $\mathcal F_{1,n}$ are fixed as $n$ varies. - $C_f$: $C_f$ is a positive domination constant associated with the particular measure $\mathbb{P}$ and chosen homeomorphism $f$ in the lower-bound assumption. - $\gamma$: $\gamma$ denotes a continuous measure-representing curve parametrized by $[0,1]$, with parameter measure $\lambda$ equal to Lebesgue measure on $[0,1]$. In Proposition~\ref{prop:holder_curve}: In the statement, $\gamma:[0,1]\rightarrow\operatorname{supp}(\mathbb{P})$ satisfies $\gamma_{\#}\lambda=\mathbb{P}$ and is $1/d$-H\"older continuous when $d\geqslant1$. In the proof's change of coordinates, the locally denoted $\gamma$ instead satisfies $\gamma_{\#}\lambda=f^{-1}_{\#}\mathbb{P}$; the curve representing the original measure is $f\circ\gamma$. For $d=0$, the representing curve is constant. - $\mathcal{C}_n$: For $n\geqslant0$, $\mathcal{C}_0=\{[0,1]^d\}$ and $\mathcal{C}_n=\{C_{i_1,\ldots,i_n}:i_j\in\{0,\ldots,2^d-1\}\}$ is a coordinate-plane partition satisfying property (*), with each cell the union of its $2^d$ children. In Proposition~\ref{prop:holder_curve}: For $d\geqslant1$, the full cube-coordinate construction recursively refines closed coordinate cells by cuts near their coordinate midpoints, perturbing the cuts to obtain $\mathbb{P}(\partial C)=0$ for each cell. It requires $R(\mathcal{C}_{n+1})\leqslant(1+2^{-n})R(\mathcal{C}_n)$; its cells need not be the standard dyadic cubes. Here $\mathbb{P}$ is the measure after the proof's change to cube coordinates. In Lemma~\ref{lemma:hilbert}: When $d\geqslant1$, $\mathcal{C}_n=\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\}$; these cells are closed and may share boundaries. - $O_n$: For each positive refinement level $n$, $O_n$ is an ordering, hence a bijection, from $\{1,\ldots,2^{nd}\}$ onto $\{\overline{i_1\cdots i_n}:i_j\in\{0,\ldots,2^d-1\}\}$. In Lemma~\ref{lemma:hilbert}: The chosen sequence preserves parent prefixes: the first $n$ digits of $O_{n+1}(i)$ equal $O_n(j)$ when $(j-1)2^d0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$. For each sample size, every candidate empirical law is obtained by evaluating the candidate on the same $m_n$ independent draws of its uniform and Gumbel exogenous inputs, independent of the observational sample. In Theorem~\ref{thm:consistency}: The optimization ranges over network representations using $\mathcal F_{0,n}$ and $\mathcal F_{1,n}$. Each continuous structural network $\hat f_i$ satisfies $\sup_x\|\hat f_i(x)\|_1\leqslant K$ on its full structural input domain. Each categorical raw structural network $q\in\mathcal F_{0,n}$ satisfies $q(x)\in[0,\infty)^{n_i}$, $\|q(x)\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$. For each $n$, all candidates are evaluated on the same $m_n$ independent and identically distributed draws of all exogenous inputs according to their specified product law, including uniform inputs, generator Gumbel inputs, and categorical Gumbel inputs, independent of the observational sample; representations with the same latent population laws may yield different empirical constraints. - $\mu_T$: $\mu_T$ is the given measure on treatment intervention values used in the outer expectation $\mathbb{E}_{t\sim\mu_T}$. In Corollary~\ref{cor:rate}: $\mu_T=\operatorname{Unif}([t_1,t_2])$, $\mu_T\ll P$, and $A_T^2=\int_{t_1}^{t_2}(d\mu_T/dP(t))^2P(dt)<\infty$, where $P$ is the reference probability measure on the treatment space. - $F_*$: $F_*=\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\mathcal{M}^*}[F(V_1(t),\ldots,V_{n_V}(t))]$ denotes the population objective evaluated at the true SCM. In Corollary~\ref{cor:rate}: $F_*=\frac{1}{t_2-t_1}\int_{t_1}^{t_2}\mathbb{E}_{\mathcal{M}^*}[Y(t)]\,dt$, since $F(\boldsymbol{V})=Y$ and $\mu_T=\operatorname{Unif}([t_1,t_2])$. - $\underline{F}^L$: $\underline{F}^L$ is the population optimal objective value in equation~\ref{eq:consistent_limiting}, minimizing $\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\hat{\mathcal{M}}}[F(V_1(t),\ldots,V_{n_V}(t))]$ over canonical SCMs on $\mathcal{G}$ with structural functions in $\mathcal{F}_L^K$ and all latent distributions $P(\boldsymbol{U})$, subject to $W(P^{\hat{\mathcal{M}}}(\boldsymbol{V}),P^{\mathcal{M}^*}(\boldsymbol{V}))=0$. In Theorem~\ref{thm:consistency}: $\underline F^L$ uses the class $\mathcal F_L^K=\{f:\sup_x\|f(x)\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$ in equation~\ref{eq:consistent_limiting}, with the supremum taken over the full structural input domain; categorical structural functions are normalized propensity vectors. The conclusion evaluates this population optimal value at $L=\hat L_f$. - $P_n^{\mathcal{M}^*}$: $P_n^{\mathcal{M}^*}(\boldsymbol V)=n^{-1}\sum_{a=1}^n\delta_{\boldsymbol V^{(a)}}$ is the empirical observational law of the $n$ observed joint values from $P^{\mathcal{M}^*}(\boldsymbol V)$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: The $n$ observations defining $P_n^{\mathcal M^*}(\boldsymbol V)$ are independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. - $m_n$: $m_n$ is a prescribed positive integer sequence, indexed by the observational sample size $n$, giving the simulation sample size for each candidate model. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: The simulation sizes satisfy $m_n=\Omega(n)$; equality $m_n=n$ is not required. - $P_{m_n}^{\hat{\mathcal{M}}}$: $P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V})$ denotes the empirical observational law for the candidate $\hat{\mathcal{M}}$, namely $m_n^{-1}\sum_{r=1}^{m_n}\delta_{\boldsymbol{V}^{(r)}}$, and is indexed by the candidate rather than denoting only the law at an optimizer. In Theorem~\ref{thm:consistency}: For each $n$, $P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V})$ is obtained by evaluating every candidate network representation on the same $m_n$ independent draws of all exogenous inputs, including uniform inputs, generator Gumbel inputs, and categorical Gumbel inputs; these draws are independent of the observational sample. Different network representations with the same latent population laws may give different common-draw empirical distributions. In Corollary~\ref{cor:rate}: For each $n$, $P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V})$ is obtained by evaluating every candidate network representation on the same $m_n$ independent draws of all exogenous inputs, including uniform inputs, generator Gumbel inputs, and categorical Gumbel inputs; these draws are independent of the observational sample. Different network representations with the same latent population laws may give different common-draw empirical distributions. - $\mathcal{F}_{0,n}$: $\mathcal{F}_{0,n}$ is the prescribed structural network class at sample size $n$, with the input and output dimensions appropriate to the node and with truncation applied coordinatewise. In Theorem~\ref{thm:consistency}: In the parameter-space construction, a continuous-node candidate is a function $\hat f_i\in\mathcal{F}_{0,n}$ satisfying $\sup_x\|\hat f_i(x)\|_1\leqslant K$ on its full structural input domain. A categorical-node propensity is represented by $x\mapsto q(x)/\|q(x)\|_1$ with $q\in\mathcal{F}_{0,n}$ restricted to satisfy $q(x)\in[0,\infty)^{n_i}$ and $\|q(x)\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$. The depth expression $\Theta(\log d_{\max}^{\text{in}})$ uses $\Theta(1+\log d_{\max}^{\text{in}})$ when $d_{\max}^{\text{in}}=1$. In Corollary~\ref{cor:rate}: A continuous-node candidate is a function $\hat f_i\in\mathcal{F}_{0,n}$ satisfying $\sup_x\|\hat f_i(x)\|_1\leqslant K$ on its full structural input domain. A categorical-node propensity is represented by $x\mapsto q(x)/\|q(x)\|_1$ with $q\in\mathcal{F}_{0,n}$ restricted to satisfy $q(x)\in[0,\infty)^{n_i}$ and $\|q(x)\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$. The depth expression $\Theta(\log d_{\max}^{\text{in}})$ uses $\Theta(1+\log d_{\max}^{\text{in}})$ when $d_{\max}^{\text{in}}=1$. - $\mathcal{F}_{1,n}$: $\mathcal{F}_{1,n}$ is the latent-generator network class indexed by $n$, with architecture $\mathcal{NN}^{\infty,K}(\Theta(d_{\max}^{U}),L_{1,n},\Theta((d_{\max}^{U})^2),L_{2,n},\tau_n)$ and coordinatewise output truncation to $[-K,K]$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: The finite component counts and component cube dimensions of the latent generator architectures are fixed as $n$ varies. - $\epsilon_n$: $\epsilon_n$ denotes a positive deterministic approximation-tolerance sequence indexed by $n\in\mathbb N$, with its approximation role specified locally. In Proposition~\ref{prop:convg_opt}: $\epsilon_n>0$ for $n\in\mathbb N$, $\epsilon_n\to0$, and it bounds the distance from each point of $\tilde\Theta_\infty$ to an available point of $\Theta_n$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the radius definition, $\alpha_n=\epsilon_n+s_n$, where $\epsilon_n>0$ is chosen with sufficiently large constants in $\epsilon_n=O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+\sum_{i=1}^2L_{i,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right)$. In \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: In the empirical generator concentration argument, $\epsilon_n$ is a local deterministic deviation tolerance in the bound $\sup_{\theta_{i,j}}W(P(f_{\theta_{i,j}}(Z_{i,j})),P_{m_n}(f_{\theta_{i,j}}(Z_{i,j})))\leqslant\epsilon_n+\epsilon'_n$; it is chosen as $\epsilon_n=Cm_n^{-1/(d_{\max}^U+2)}\log m_n$ with fixed $C>0$ sufficiently large. This local choice is distinct from the architecture approximation tolerance in the constraint radius. - $s_n$: $s_n$ is a deterministic tolerance sequence indexed by $n$, specified by its displayed asymptotic order rather than by a unique numerical formula. In Theorem~\ref{thm:consistency}: $s_n>0$ is chosen as a sufficiently large constant multiple of the quantity in the displayed order $O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2 n+\log(nm_n)\lambda_n\right)$, where $D_U=\sum_{j=1}^{n_U}d_j^U$ includes the non-Gumbel latent variables only and $D=\max\{1,D_U\}$. In Corollary~\ref{cor:rate}: $s_n>0$ is chosen as a sufficiently large constant multiple of the quantity in the displayed order $O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2 n+\log(nm_n)\lambda_n\right)$, where $D_U=\sum_{j=1}^{n_U}d_j^U$ includes the non-Gumbel latent variables only and $D=\max\{1,D_U\}$. - $\alpha_n$: $\alpha_n=\epsilon_n+s_n$ is a deterministic constraint tolerance sequence indexed by the observational sample size $n$. In Theorem~\ref{thm:consistency}: $\alpha_n=\epsilon_n+s_n>0$, where the positive tolerances $\epsilon_n$ and $s_n$ are chosen with sufficiently large constants in their displayed asymptotic orders. In Corollary~\ref{cor:rate}: $\alpha_n=\epsilon_n+s_n>0$, where the positive tolerances $\epsilon_n$ and $s_n$ are chosen as sufficiently large constant multiples of the respective quantities in their displayed asymptotic orders. - $\tau_n$: $\tau_n$ is a sequence indexed by $n$, used locally either as a generator temperature or as an additive tolerance in an approximate Lipschitz bound. In Proposition~\ref{prop:convg_opt}: $\tau_n$ is the additive tolerance in equation~\ref{eq:aprox_lip}, and $\tau_n\rightarrow0$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the network architecture and growth conditions, $\tau_n>0$ is the Gumbel-softmax generator temperature and $\tau_n\rightarrow0$; in the proof's application of the optimization proposition, its additive tolerance is assigned the separate value $4(1+\log(nm_n))\lambda_n$. In \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: When verifying equation~\ref{eq:aprox_lip}, $\tau_n$ denotes the additive tolerance $4(1+\log(nm_n))\lambda_n$, which tends to zero under the stated regularization condition; this assignment does not identify it with the generator temperature. - $\lambda_n$: $\lambda_n>0$, indexed by the observational sample size $n$, is the regularization parameter in $S_{\lambda_n}(P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V}),P_n^{\mathcal{M}^*}(\boldsymbol{V}))$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: $\log(nm_n)\lambda_n\rightarrow0$ as $n\rightarrow\infty$. - $d$: $d$ is local notation for a fixed dimension, a fixed metric, or a regression-difference function, with its role determined by its defining expression. In Proposition~\ref{prop:holder_curve}: $d$ is the cube dimension in the homeomorphism $f:[0,1]^d\rightarrow K$ from Assumption~\ref{assumption:lower_bound}, rather than the ambient dimension $D$; the proof treats $d=0$ separately and uses the exponent $1/d$ only for $d\geqslant1$. In Lemma~\ref{lemma:hilbert}: $d$ is the dimension of the cube $[0,1]^d$ being partitioned. In Theorem~\ref{thm:wide_nn}: $d$ is the ambient dimension of the target law on $\mathbb{R}^d$, distinct from the component cube dimensions $d_i^C$. In Proposition~\ref{prop:convg_opt}: $d:M\times M\rightarrow[0,\infty)$ is the metric of the given space $(M,d)$. In Theorem~\ref{thm:consistency}: In the proof's parameter space, $d(\theta,\theta')=\sum_{k=1}^{n_V}\|f_k-f_k'\|_\infty+\sum_{k=1}^{n_U}W(P(U_k),P(U_k'))$; occurrences of $d$ in imported distribution bounds retain their local dimension meaning. In Lemma~\ref{lemma:sup_wasserstein}: $d$ is the common output dimension of the functions in $\mathcal F$ and the dimension used in the output test-function covering bound; the input law $\mu$ is on $\mathbb R^{d'}$. In the result split from Theorem~\ref{thm:wide_nn}: $d$ is the ambient dimension of the target law on $\mathbb R^d$, distinct from the component cube dimensions $d_i^C$. In Proposition~\ref{prop:Lip_ATE}: In the proof's test-function calculation, $s=(a,t)$ and $d(s)=\hat f_y(s)-f_y(s)$ on the allowed covariate domain $Q$; $f_y$ and $\hat f_y$ are the structural regression versions for the respective models. - $d_1$: $d_1$ denotes a locally specified input or source dimension, including the input dimension in $\mathcal{NN}_{d_1,d_2}$, or an explicitly specified averaged metric on evaluation vectors. In Lemma~\ref{lemma:push-forward_err}: $K\subset\mathbb{R}^{d_1}$ is the common domain supporting $\mu$ and $\hat{\mu}$. In Theorem~\ref{thm:wide_nn}: In the proof's notation for the combined maps $\hat{g}_i$, $d_1=1$. In \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $d_1$ is the averaged metric on length-$m$ evaluation vectors, $d_1(u,w)=\frac1m\sum_{i=1}^m|u_i-w_i|$, with $m$ a positive integer. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $d_1$ is the averaged metric on length-$N$ evaluation vectors, $d_1(u,w)=\frac1N\sum_{i=1}^N|u_i-w_i|$; in the covering number appearing in the probability bound, $N=2m$. - $d_2$: $d_2$ denotes the locally specified output or target dimension of a map, including the output dimension in $\mathcal{NN}_{d_1,d_2}$. In Lemma~\ref{lemma:push-forward_err}: $F,\hat{F}:K\rightarrow\mathbb{R}^{d_2}$, so $d_2$ is their common target dimension. In Theorem~\ref{thm:wide_nn}: In the proof's notation for the combined maps $\hat{g}_i$, $d_2=d$. - $D$: $D$ is a deterministic geometric parameter specified by its local definition. In Theorem~\ref{thm:approximation}, Proposition~\ref{prop:holder_curve}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}, the result split from Theorem~\ref{thm:wide_nn}: In the compact-support assumption, $D$ is the ambient Euclidean dimension in $K\subset\mathbb{R}^{D}$, rather than the cube dimension $d$. In Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: In the constants controlling the treatment-effect comparison, $D=\max\left(\{1\}\cup\left\{\frac{n_i-1}{2}:V_i\text{ is categorical}\right\}\right)$ and $L_y=(LD+1)^{n_V}$; this $D$ can be nonintegral and equals $1$ when there are no categorical nodes. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the sampling tolerance $s_n$, $D=\max\{1,D_U\}$, where $D_U=\sum_{j=1}^{n_U}d_j^U$ includes the non-Gumbel latent variables only; this local $D$ is distinct from the ambient dimension in the compact-support assumption and from the categorical factor in the treatment-effect comparison. In Theorem~\ref{thm:approximation}: In the proof's error recurrence, $D=\max\left(\{0\}\cup\{n_i-1:V_i\text{ is categorical}\}\right)$ and $A=\max\{1+L,1+LD/2\}$; this local $D$ is zero when there are no categorical nodes. - $d_{\max}^{\text{in}}$: $d_{\max}^{\text{in}}=\max_{i\in[n_V]}d_i^{\text{in}}$, where $d_i^{\text{in}}$ is the input dimension of $f_i(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$ in the canonical structural equations. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: $d_{\max}^{\text{in}}\geqslant1$; when $d_{\max}^{\text{in}}=1$, the structural-network depth expression $\Theta(\log d_{\max}^{\text{in}})$ means $\Theta(1+\log d_{\max}^{\text{in}})$. In Corollary~\ref{corollary:deep_nn}: $d_{\max}^{\text{in}}>0$. - $d_{\max}^{\text{out}}$: $d_{\max}^{\text{out}}=\max_{i\in[n_V]}d_i^{\text{out}}$, where $d_i^{\text{out}}=d_i$ for a continuous structural function and $d_i^{\text{out}}=n_i$ for a categorical propensity vector. - $d_{\max}^U$: $d_{\max}^U=\max_{j\in[n_U]}d_j^U$, where $d_j^U$ is the dimension of a canonical latent variable $U_j$ whose distribution is approximated by a generator, distinct from the separately supplied categorical Gumbel noises. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: $d_{\max}^U\geqslant1$; it remains the maximum individual non-Gumbel latent dimension, distinct from the total dimension $D_U=\sum_{j=1}^{n_U}d_j^U$ and $D=\max\{1,D_U\}$. In Corollary~\ref{corollary:deep_nn}: $d_{\max}^U>0$. - $d_{\mathcal{F}}$: $d_{\mathcal F}<\infty$ is a strict upper bound on the pseudo-dimensions of the scalar function classes specified in its local use. In Lemma~\ref{lemma:sup_wasserstein}: For every $j\in[d]$, $\mathcal F_j=\{x\mapsto f_j(x):f\in\mathcal F\}$ satisfies $\operatorname{Pdim}(\mathcal F_j)0$ are arbitrary positive softmax weights; normalization is not required in the statement. The proof normalizes them without loss of generality, and the limiting categorical probabilities are $p_i/\sum_{k=1}^n p_k$. In Lemma~\ref{lemma:sinkhorn_approx}: In the probability-simplex notation, $p_i$ is a coordinate satisfying $p_i\geqslant0$ and $\sum_{i=1}^n p_i=1$. In Theorem~\ref{thm:approximation}: In the categorical case of the proof, $(p_1,\ldots,p_{n_i})=f(\text{pa}(v_k),\boldsymbol{u}_{V_k})$ is the conditional categorical probability vector evaluated at the displayed parent and latent values; its coordinates are nonnegative and sum to one. In Proposition~\ref{prop:equivalent_g_constrianed}: In the probability-simplex notation, $p_i$ is a coordinate satisfying $p_i\geqslant0$ and $\sum_{i=1}^n p_i=1$. In the result split from Theorem~\ref{thm:wide_nn}: $p_i=\mathbb P(C_i)$ for $i\in[N_C]$, and $\sum_{i=1}^{N_C}p_i=1$; these are the probabilities used to combine the support components. In Proposition~\ref{prop:holder_curve}: In the printed partition construction at level $n+1$, $p_i=\mathbb P(C_{O_{n+1}(i)})$ for $1\leqslant i\leqslant2^{(n+1)d}$. Here $\mathbb P$ is the reduced cube measure and $C_{O_{n+1}(i)}$ is an ordered partition cube. These masses give the lengths of the corresponding parameter intervals and the denominators in the polygonal curve; domination by Lebesgue measure makes them positive. In Theorem~\ref{thm:consistency}: In the latent approximation construction, $p_i$ denotes the probability mass of component $C_i$ under the particular latent target law being approximated. These component probabilities sum to one and weight the component approximation errors and the hard selected mixture. In the categorical measurability argument, $p_i$ instead denotes coordinate $i$ of the categorical propensity vector evaluated at the structural input; these coordinates are nonnegative and sum to one, and zero coordinates are allowed. - $p_\nu$: Writing $X=\text{Pa}(T)$, $p_\nu(t\mid X=x)$ is the conditional treatment density of $\nu(dt\mid x)$ with respect to the comparison measure $P(dt)$, so that $\nu(dt\mid x)=p_\nu(t\mid x)P(dt)$; its overlap lower bound holds for $\nu_X$-almost every $x$ and $t\in[t_1,t_2]$. - $\delta$: $\delta>0$ denotes a locally specified positive real parameter. In Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: $\delta$ in the overlap premise is the lower bound satisfying $p_\nu(t\mid\text{Pa}(T)=x)\geqslant\delta$ for $x$ almost surely and $t\in[t_1,t_2]$. In Lemma~\ref{lemma:gumbel_err}: $\delta$ is the auxiliary threshold for the gaps between $G_i+\log p_i$; at the end of the proof it is chosen as $\delta=-\tau\log\tau$. In Proposition~\ref{prop:counterexample_foral}: In the proof, $\delta$ parameterizes $\mathcal{M}_\delta$: the covariate has an atom at $\delta$, and the outcome equation contains $X/\delta$ for $X\geqslant0$; the construction takes $\delta\rightarrow0$. In Lemma~\ref{lemma:sup_wasserstein}: $\delta$ is an arbitrary positive tolerance; the probability bound is for each fixed pair $\delta,\epsilon>0$. In the proof, $\mathcal H_\delta$ has uniform-norm covering radius $\delta/2$, so replacing a test function by its representative contributes at most $\delta$ to the difference of its empirical and population integrals. In Theorem~\ref{thm:consistency}: In the simulation-concentration argument, the empirical Wasserstein bound is applied with $\delta=r_n$, where $H_n=1+L_{1,n}+L_{2,n}$ and $r_n=m_n^{-1/(D+2)}\log m_n+H_nm_n^{-1/2}\log m_n$; its other tolerance is $\epsilon=A_0r_n$. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: For a component of positive dimension $q=d_i^C$, $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$ is the source quantization parameter, and the quantization mesh is $\delta/n$; here $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, with $A_q\geqslant1$ depending only on $q$. In Corollary~\ref{cor:rate}: Separately from the overlap parameter, the proof's simulation-concentration argument chooses the local tolerance $\delta=r_n$, where $H_n=1+L_{1,n}+L_{2,n}$ and $r_n=m_n^{-1/(D+2)}\log m_n+H_nm_n^{-1/2}\log m_n$, for sufficiently large $n$; the other tolerance is $\epsilon=Cr_n$ for a sufficiently large fixed positive constant $C$. - $c$: $c$ denotes a locally specified scalar, categorical configuration, or real-valued function, according to its defining expression. In Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:counterexample_foral}: $c>0$ is chosen together with $\mathcal{M}^*$ before quantifying over $\epsilon>0$, and is independent of $\epsilon$; it is a strict lower bound for $|\text{ATE}_{\mathcal{M}^*}-\text{ATE}_{\mathcal{M}_\epsilon}|$. In Lemma~\ref{lemma:sinkhorn_approx}: In the proof's final self-term calculation, $c=W_{1,\lambda}(\nu,\nu)$ is the entropy-regularized self-transport value, with $|c|\leqslant(1+2\log m)\lambda$; it is nonpositive under the specified entropy convention. In Lemma~\ref{lemma:sup_wasserstein}: $c>0$ is the concentration constant multiplying $n\epsilon^2$ in the negative term of the probability exponent; it depends only on $d,K$ and is independent of $n,\delta,\epsilon$. In Proposition~\ref{prop:Lip_ATE}: In the proof, $c(t)=\int\hat f_y(a,t)\nu_A(da)-\int\hat f_y(a,t)\mu_A(da)$ is the contribution from changing the observational marginal of $A$, with $\nu_A$ and $\mu_A$ the respective observational marginals. In Theorem~\ref{thm:consistency}: In the observational concentration argument, $c$ ranges over the fixed finite set of possible complete categorical label vectors: $p_c=P(C=c)$ and $\mu_c=P((U_1,\ldots,U_{n_U})\mid C=c)$, the latter being used for $p_c>0$. In Corollary~\ref{cor:rate}: In categorical types $(u,c)$ and conditioning expressions, $c$ is a complete categorical label vector, with $p_c=P(C=c)$ and $\mu_c=P((U_1,\ldots,U_{n_U})\mid C=c)$ for $p_c>0$. Separately, in the concentration exponent $-cC^2m_nr_n^2$, $c>0$ is a fixed concentration constant. - $t_1$: $t_1$ is a real scalar whose role is local to the intervention interval or the Lipschitz comparison. In Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: $t_1$ is the fixed lower endpoint of the intervention interval $[t_1,t_2]$. In Corollary~\ref{cor:rate}: $t_10$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$. Here $\mathcal P_n=\{P(\boldsymbol U):\boldsymbol U\text{ is the latent distribution of }\hat{\mathcal M}\in\text{NCM}_{\mathcal G}(\mathcal F_{0,n},\mathcal F_{1,n})\}$, identified with its tuple of independent latent marginal laws. This product records population model parameters; the empirical optimization ranges over their network representations, including distinct representations inducing identical latent population laws. - $\Theta_\infty$: $\Theta_\infty=\overline{\bigcup_{n=1}^{\infty}\Theta_n}$, with closure taken in the ambient metric space $(M,d)$. In Proposition~\ref{prop:convg_opt}: $\Theta_\infty$ is compact and contains the nested sets $\Theta_n$; $\{\theta\in\Theta_\infty:g(\theta)=0\}$ is nonempty. In Theorem~\ref{thm:consistency}: The closure is that of the neural-model parameter sets in the metric $d(\theta,\theta')=\sum_{k=1}^{n_V}\|f_k-f_k'\|_\infty+\sum_{k=1}^{n_U}W(P(U_k),P(U_k'))$ on the product space of structural functions and latent probability measures. - $\tilde{\Theta}_\infty$: $\tilde\Theta_\infty$ is a compact subset of $\Theta_\infty$ approximated by $\Theta_n$: for every $\theta_0\in\tilde\Theta_\infty$ and every $n$, some $\theta\in\Theta_n$ satisfies $d(\theta,\theta_0)\leqslant\epsilon_n$. In Proposition~\ref{prop:convg_opt}: $\{\theta\in\tilde\Theta_\infty:g(\theta)=0\}$ is nonempty. - $\delta_n$: $\delta_n$ denotes a deterministic error tolerance indexed by $n$ and tending to zero. In Proposition~\ref{prop:convg_opt}: $\delta_n>0$ for $n\in\mathbb N$, and $\sup_{\theta\in\Theta_\infty}|g_n(\theta)-g(\theta)|\leqslant\delta_n$. In \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: For sufficiently large $n$, the local observational latent deviation tolerance is $\delta_n=C_1n^{-1/\max\{d_{\max}^U,2\}}\log^2 n$, with fixed $C_1>0$ chosen sufficiently large; this is distinct from the abstract uniform constraint-error tolerance. - $g_n$: $\{g_n\}_{n\in\mathbb{N}}$ is a sequence of real-valued constraint functions on $\Theta_\infty$. In Proposition~\ref{prop:convg_opt}: Each $g_n$ is continuous and nonnegative, with $\sup_{\theta\in\Theta_\infty}|g_n(\theta)-g(\theta)|\leqslant\delta_n$, and the approximate Lipschitz bound has common coefficient $L_g$ and additive tolerance $\tau_n$. In \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: The notation is the empirical constraint $g_n(\theta)=S_{\lambda_n}(P_n^{\theta^*}(\boldsymbol V),P_{m_n}^{\theta}(\boldsymbol V))$, compared with the population constraint $g(\theta)=W(P^{\theta^*}(\boldsymbol V),P^{\theta}(\boldsymbol V))$. - $g$: $g$ denotes a real-valued function on the domain specified by its local use. In Theorem~\ref{thm:approximation}: In the proof, $g$ ranges over real-valued $1$-Lipschitz Wasserstein dual test functions of $(\boldsymbol{u},\boldsymbol{t},v_1,\ldots,v_k)$. In Lemma~\ref{lemma:push-forward_err}: In the proof, $g:\mathbb{R}^{d_2}\rightarrow\mathbb{R}$ is an arbitrary $1$-Lipschitz test function. In Lemma~\ref{lemma:Lip_condition}: In the proof, $g$ ranges over real-valued $1$-Lipschitz test functions of $(t,y)$ in the Wasserstein dual formula. In Proposition~\ref{prop:convg_opt}: $g:\Theta_\infty\rightarrow\mathbb{R}$ is the limiting constraint, with $\sup_{\theta\in\Theta_\infty}|g_n(\theta)-g(\theta)|\leqslant\delta_n$; its zero set defines the limiting feasible regions. In Theorem~\ref{thm:consistency}: $g(\theta)=W(P^{\theta^*}(\boldsymbol{V}),P^\theta(\boldsymbol{V}))$ is the population observational discrepancy, rather than an empirical discrepancy. In \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $g$ ranges over functions in a nonempty class $\mathcal G\subset\{g:X\rightarrow[0,1]\}$ with finite pseudo-dimension $v$. No Lipschitz condition on $g$ is specified. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $g:T\times X\rightarrow[0,1]$ is jointly Borel, where $T$ and $X$ are standard Borel spaces, and $\mathcal G=\{g(t,\cdot):t\in T\}\ne\varnothing$. For each $t\in T$, the squared loss is $(g(t,x)-y)^2$. No pseudo-dimension bound or Lipschitz condition on $g$ is imposed. - $L_g$: $L_g$ is a finite nonnegative Lipschitz bound. In Proposition~\ref{prop:convg_opt}: $L_g$ is independent of $n$ and is the coefficient in $|g_n(\theta_1)-g_n(\theta_2)|\leqslant L_g d(\theta_1,\theta_2)+\tau_n$; the limiting function $g$ is $L_g$-Lipschitz. In Lemma~\ref{lemma:Lip_condition}: In the proof, $L_g=2K\max\{L,1\}$ bounds the Lipschitz constant of $g_0(t,y)=(\hat f(t)-f(t))y$ on $\{(t,y):\|(t,y)\|_\infty\leqslant K\}$. - $B$: $B$ is a locally specified measurable test set or positive scalar bound. In the result split from Theorem~\ref{thm:wide_nn}: In the componentwise lower-bound premise, $B$ ranges over measurable subsets of $[0,1]^{d_i^C}$; the component measure and its parameterizing map satisfy the domination inequality of Assumption~\ref{assumption:lower_bound} on these sets. Separately, in the proof's clipping and mixing estimates, $B>0$ is a fixed scalar chosen so that every support component lies in $[-B,B]^d$, and the second-block outputs are clipped into this box. In Proposition~\ref{prop:holder_curve}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the lower-bound premise, $B$ ranges over measurable subsets of $[0,1]^d$ and $\mathbb{P}(f(B))\geqslant C_f\lambda(B)$; for a component with dimension $d_i^C$, the corresponding test sets lie in $[0,1]^{d_i^C}$. In Proposition~\ref{prop:Lip_ATE}: $B$ is the scalar bound $B=K(1+2qL_y)$, where $q=\sum_{i:V_i\text{ is continuous}}d_i+|\{i:V_i\text{ is categorical}\}|$ and $L_y=(LD+1)^{n_V}$; it enters $C_W=4L_yB+2B\max\{L_y,1\}$. In Theorem~\ref{thm:approximation}, Lemma~\ref{lemma:push-forward_err}, Theorem~\ref{thm:wide_nn}, Lemma~\ref{lemma:sup_wasserstein}, \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: In a push-forward expression $f_\#\mu(B)$, $B$ is an arbitrary measurable subset of the codomain of $f$. In Theorem~\ref{thm:approximation}: In the proof's error recurrence, $B=\max\left(\{1\}\cup\{d_i:V_i\text{ is continuous}\}\cup\{n_i(n_i-1)/2:V_i\text{ is categorical}\}\right)$. In Theorem~\ref{thm:consistency}: In the proof's model-comparison recurrence, $B$ is a fixed positive bound on the coefficients of the structural approximation errors, chosen together with $A\geqslant1$ so that $e_{k+1}\leqslant Ae_k+B d(\theta,\theta')$. Separately, in the derivation of the pseudo-dimension bound, $B=CH_n^2$, where $H_n=1+L_{1,n}+L_{2,n}$ and $C$ is the constant in that derivation; this latter scalar may vary with $n$. In Corollary~\ref{corollary:deep_nn}: In the structural extension and approximation construction, $B=B_0H^{n_V}$, where $m=d_{\max}^{\text{in}}$, $H=1+2Lm$, and $B_0=\max\left(\{1,R,K\}\cup\{n_i:V_i\text{ is categorical}\}\cup\{\|f_i(a_i)\|_\infty:V_i\text{ is continuous}\}\right)$; $a_i$ has zero in each real input coordinate and one in each categorical input coordinate. This fixed positive scalar is independent of the intervention and network budgets and defines the structural input cubes $Q_i=[-B,B]^{m_i}$. In Corollary~\ref{cor:rate}: In the proof's model-comparison recurrence, $B$ is a fixed positive bound on the coefficients of the structural approximation errors, chosen together with $A\geqslant1$ so that $e_{k+1}\leqslant Ae_k+B d(\theta,\theta')$; it is independent of the sample size and the compared model parameters. - $\mathcal{N}_1$: $\mathcal{N}_1(\epsilon,\mathcal{F},n)=\sup_{x_1,\ldots,x_n}\mathcal{N}(\epsilon,\{(f(x_1),\ldots,f(x_n)):f\in\mathcal{F}\},\|\cdot\|_1)$, where the points range over the domain of $\mathcal{F}$ and the norm is the printed $\ell_1$ norm without a factor $1/n$. In \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $\mathcal N_1^{\mathrm{avg}}(\epsilon,\mathcal G,m)$ is the supremum over $x_1,\ldots,x_m\in X$ of the covering numbers of $\{(g(x_1),\ldots,g(x_m)):g\in\mathcal G\}$ in $d_1(u,w)=m^{-1}\sum_{i=1}^m|u_i-w_i|$, for $\epsilon>0$ and positive integers $m$. This averaged covering number is distinct from the covering number defined with the unnormalized $\ell_1$ norm. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $\mathcal N_1^{\mathrm{avg}}(r,\mathcal G,N)$ is the supremum over $x_1,\ldots,x_N\in X$ of the covering number of $\{(g(x_1),\ldots,g(x_N)):g\in\mathcal G\}$ in $d_1(u,w)=N^{-1}\sum_{i=1}^N|u_i-w_i|$. The probability bound uses $r=\epsilon/16$ and $N=2m$. This averaged covering number is distinct from the covering number defined with the unnormalized $\ell_1$ norm. - $\mathcal{H}_\delta$: $\mathcal H_\delta\subseteq\mathcal H$ is a finite $\delta/2$-net in the uniform norm of $\mathcal H=\{h:[-K,K]^d\rightarrow\mathbb R:h(0)=0,\ |h(x)-h(y)|\leqslant\|x-y\|\}$, for $\delta>0$. - $h$: $h$ denotes the auxiliary function or positive grid spacing defined in its surrounding passage. In Theorem~\ref{thm:wide_nn}: $h(w,X)=w^{\mathsf{T}}X\in\mathbb{R}^d$, where $w$ has $N_C$ coordinates, $X$ collects the $N_C$ component outputs, and the stated region is $\|w\|_1=1$, $\|X\|_\infty\leqslant K$. In Lemma~\ref{lemma:Lip_condition}: $h(t)=(\hat{f}(t)-f(t))\hat{f}(t)$ is the real-valued auxiliary function in the proof, with the stated Lipschitz constant $L_h=4LK$. In Lemma~\ref{lemma:sinkhorn_approx}: $h(T)=1-\sum_{i=1}^n\sum_{j=1}^m T_{ij}\log T_{ij}$ is the entropy expression for $T\in\Pi(\mu,\nu)$, with $0\log0=0$. In Lemma~\ref{lemma:sup_wasserstein}: $h$ is a scalar test function in $\mathcal H=\{h:[-K,K]^d\rightarrow\mathbb R:h(0)=0,\ |h(x)-h(y)|\leqslant\|x-y\|\}$, so its values lie in $[-dK,dK]$; the concentration argument first fixes such an $h$ and uses $z_f=\sqrt{(h\circ f+dK)/(2dK)}$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the structural interpolation construction, $h=2K/R>0$ is the side length of a grid cell obtained by dividing each coordinate of $[-K,K]^m$ into $R$ equal intervals; $I_hf$ is the resulting piecewise affine interpolant. In Proposition~\ref{prop:Lip_ATE}: In the proof, $s=(a,t)$ and $h(s)=(\hat f_y(s)-f_y(s))\hat f_y(s)$ on the bounded covariate domain $Q$. The structural regression versions satisfy $|f_y|,|\hat f_y|\leqslant B$ there, and $|h(s)-h(s')|\leqslant4BL_y\|s-s'\|$. - $m$: $m$ is a local integer parameter whose role is determined by the objects in the surrounding passage. In Proposition~\ref{prop:holder_curve}: In $\gamma_m(t)\in C_{k,n-1}$ for all $m\geqslant n$, $m$ is a later curve level. In Theorem~\ref{thm:consistency}: In the Sinkhorn comparison, the second probability-vector length is the simulation count $m_n$, giving the factor $\log(nm_n)$. In the structural interpolation argument, $m$ is the input dimension of $f:[-K,K]^m\rightarrow\mathbb R^{m'}$; the construction treats $m\geqslant1$ by triangulation and $m=0$ by exact constant representation. In Lemma~\ref{lemma:sinkhorn_approx}: $m\geqslant1$ is the length of $\nu\in\Delta_m$, indexing the points $y_1,\ldots,y_m$, and the number of columns of its transportation plan with $\mu\in\Delta_n$; it need not count only positive coordinates or distinct points. In \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $m\geqslant1$ is an arbitrary integer counting evaluation points in the averaged empirical covering and packing numbers. In Corollary~\ref{cor:rate}: In the structural interpolation argument, $m$ is the input dimension of $f:[-K,K]^m\rightarrow\mathbb R^{m'}$; the construction treats $m\geqslant1$ by triangulation and $m=0$ by exact constant representation. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $m\geqslant1$ is the number of independent observations $(x_i,y_i)$ with law $P$ in the empirical squared loss; $\mathcal N_1^{\mathrm{avg}}(\epsilon/16,\mathcal G,2m)$ uses $2m$ evaluation points. In Theorem~\ref{thm:approximation}: In the recurrence argument, $m=n_V-n_T$ is the number of nonintervened observed nodes. The argument includes $m=0$, with an empty geometric sum. In Corollary~\ref{corollary:deep_nn}: In the structural extension and approximation construction, $m=d_{\max}^{\text{in}}>0$ is the largest structural input dimension, while $m_i=d_i^{\text{in}}$ is the input dimension at node $i$. - $\|\cdot\|_\infty$: $\|\cdot\|_\infty$ denotes the infinity norm on vectors and its uniform extension to functions: $\|f\|_\infty=\sup_{x\in\operatorname{dom}(f)}\|f(x)\|_\infty$. - $R$: $R$ is a locally specified shape-ratio functional or fixed positive intervention bound. In Proposition~\ref{prop:holder_curve}: $R(\mathcal C)=\max_{i,k}|a_i^k-b_i^k|/\min_{i,k}|a_i^k-b_i^k|$ for the finite collection $\mathcal C=\{C_1,\ldots,C_m\}$ with $C_k=\prod_{i=1}^n[a_i^k,b_i^k]$, wherever the denominator is positive. In Corollary~\ref{corollary:deep_nn}: $R>0$ is fixed before choosing $\hat{\mathcal M}$ and bounds each continuous intervention value by $\|t_i\|_\infty\leqslant R$; the implicit approximation constant may depend on $R$, which remains fixed as the network budgets tend to infinity and $\tau\downarrow0$. - $S$: $S$ denotes the auxiliary object defined locally in its surrounding argument. In Theorem~\ref{thm:approximation}: In the proof, $S=\sum_{i=1}^{n_V}\|f_i-\hat{f}_i\|_\infty$ is the summed structural-function approximation error. In Corollary~\ref{corollary:deep_nn}: In the categorical normalization argument, at the structural input $x=(\text{pa}(v_k),\boldsymbol{u}_{V_k})$, $S=\|\hat f_i(x)-f_i(x)\|_\infty$, where $f_i(x)$ is the normalized true propensity vector and $\hat f_i(x)$ is the unnormalized nonnegative approximant. The comparison is with $\hat f_i(x)/\|\hat f_i(x)\|_1$ wherever its denominator is nonzero. In Lemma~\ref{lemma:sup_wasserstein}, \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $S$ in $\mathcal{N}(\epsilon,S,\|\cdotp\|_1)$ is the set to be covered; the unnormalized empirical construction uses $\{(f(x_1),\ldots,f(x_N)):f\in\mathcal F\}$ for specified input points. For $\mathcal{N}_1^{\mathrm{avg}}(\epsilon,\mathcal G,m)$, the evaluation set is instead $\{(g(x_1),\ldots,g(x_m)):g\in\mathcal G\}$, equipped with $d_1(u,w)=m^{-1}\sum_{i=1}^m|u_i-w_i|$. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: The set covered in $\mathcal{N}_1^{\mathrm{avg}}(r,\mathcal G,N)$ is $S=\{(g(x_1),\ldots,g(x_N)):g\in\mathcal G\}$ for specified evaluation points, equipped with $d_1(u,w)=N^{-1}\sum_{i=1}^N|u_i-w_i|$; the bound uses $N=2m$. - $\mathcal{N}$: $\mathcal{N}$ denotes either the covering-number functional when supplied with covering arguments, or a locally quantified causal model. In Lemma~\ref{lemma:sup_wasserstein}, \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $\mathcal{N}(\epsilon,S,\|\cdotp\|_1)$ denotes the covering number of the evaluation set $S$ in the summed $1$-norm; the separately marked averaged covering numbers use the averaged empirical metric. In Proposition~\ref{prop:Lip_ATE}: In the proof, $\mathcal{N}\in\{\mathcal{M},\hat{\mathcal{M}}\}$ ranges over the two causal models when bounding their structural regression functions. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $\mathcal{N}_1^{\mathrm{avg}}(r,\mathcal G,N)$ is the supremum over $x_1,\ldots,x_N\in X$ of the covering number of $\{(g(x_1),\ldots,g(x_N)):g\in\mathcal G\}$ in $d_1(u,w)=N^{-1}\sum_{i=1}^N|u_i-w_i|$; the concentration bound uses $r=\epsilon/16$ and $N=2m$. - $\theta_{\mathcal{M}}$: $\theta_{\mathcal{M}}=(f_1,\ldots,f_{n_V},P(U_1),\ldots,P(U_{n_U}))$ represents a canonical SCM through the functions in (\ref{eq:structure_equation}) and the population marginal laws of its non-Gumbel latent coordinates, called the uniform parts in the parameterization; the categorical standard Gumbel laws are not additional tuple coordinates. - $\mathcal{M}^{\theta}$: $\mathcal{M}^\theta$ is the canonical SCM with structural functions and non-Gumbel latent marginal laws specified by $\theta$, using the given causal graph and canonical latent incidence structure; the standard Gumbel laws in categorical equations are supplied by the canonical construction. - $\theta^*$: $\theta^*$ denotes a distinguished parameter point whose role is specified by the local optimization or causal-model parameterization. In Proposition~\ref{prop:convg_opt}: $\theta^*\in\Theta_\infty$ is a minimizer of (\ref{eq:opt*}), satisfying $g(\theta^*)=0$ and $f(\theta^*)=f^*$; it is distinct in notation and role from $\tilde\theta^*$, the minimizer over $\tilde\Theta_\infty$. In Theorem~\ref{thm:consistency}: $\theta^*$ is the canonical parameter tuple of the underlying SCM, containing its structural functions and non-Gumbel latent marginal laws; $\tilde\Theta_\infty=\{\theta^*\}$. In Corollary~\ref{cor:rate}: $\theta^*$ is the canonical parameter tuple representing the underlying SCM $\mathcal M^*$, containing its structural functions and non-Gumbel latent population marginal laws. - $\theta'$: $\theta'=(f_1',\ldots,f_{n_V}',P(U_1'),\ldots,P(U_{n_U}'))\in M$ is a second canonical model parameter tuple, with $f_i'\in\mathcal{F}_{V_i}$ and non-Gumbel latent marginal laws $P(U_j')\in\mathcal{P}([-K,K]^{d_j^U})$, compared with $\theta$ through $d(\theta,\theta')$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: When $\theta'$ indexes an empirical candidate law, its candidate index additionally specifies a network representation; the population tuple alone need not determine $P_{m_n}^{\theta'}$ on the common simulation inputs. - $P^\theta$: $P^\theta$ denotes the population law induced by the canonical SCM $\mathcal{M}^\theta$ represented by the structural functions and latent marginal laws in $\theta$; its argument specifies the variables or intervention whose law is taken, and it is distinct from the empirical laws $P_n^\theta$ and $P_{m_n}^\theta$. - $\mathcal{F}_{V_i}$: $\mathcal{F}_{V_i}$ is the population structural-function class for $i\in[n_V]$ on the indicated input cube, with the continuous or categorical branch selected by the type of $V_i$; in the categorical branch, $\|f(x)\|_1=1$ at each structural input $x$. In Theorem~\ref{thm:consistency}: The bounded Lipschitz class used in the population optimization has $\sup_x\|f(x)\|_1\leqslant K$ and $\operatorname{Lip}(f)\leqslant L$ on the full structural input domain; categorical structural functions are nonnegative normalized propensity vectors. - $d_{i,\text{in}}^{V}$: $d_{i,\text{in}}^{V}$, for $i\in[n_V]$, is the input coordinate dimension of the structural function $f_i$, whose arguments are $\text{Pa}(V_i)$ and $\boldsymbol{U}_{V_i}$; the population function-class construction uses the domain $[-K,K]^{d_{i,\text{in}}^{V}}$. - $d_{i,\text{out}}^{V}$: $d_{i,\text{out}}^{V}$ is the output dimension of the canonical structural function $f_i$, for $i\in[n_V]$, equal to $d_i$ when $V_i$ is continuous and to $n_i$ when $V_i$ is categorical. - $d_j^U$: $d_j^U$, for $j\in[n_U]$, is the Euclidean coordinate dimension of the indexed latent variable $U_j$, excluding the separately introduced categorical Gumbel noises. - $\mathcal{P}$: $\mathcal{P}(K)$ is the space of Borel probability measures on the set $K$, equipped with Wasserstein-1 distance, which metrizes weak convergence on the compact domains used here. - $\mathcal{P}_n$: $\mathcal{P}_n=\{P(\boldsymbol{U}):\boldsymbol{U}\text{ is the latent collection of }\hat{\mathcal{M}}\in\text{NCM}_{\mathcal{G}}(\mathcal{F}_{0,n},\mathcal{F}_{1,n})\}$ is the population latent-law class indexed by $n$, rather than a class of empirical latent measures. In Theorem~\ref{thm:consistency}: In the proof's population parameterization $\Theta_n$, $\mathcal{P}_n$ is identified with $\mathcal{P}_{n,1}\times\cdots\times\mathcal{P}_{n,n_U}$, the tuple of generated non-Gumbel marginal laws on $[-K,K]^{d_j^U}$, $j\in[n_U]$; the separately prescribed categorical Gumbel noises are not additional coordinates of this tuple. Each generated marginal law integrates the generator's uniform input and Gumbel-softmax randomness. This population-law tuple does not determine a candidate's common-draw empirical distribution: different network representations of the same latent laws can have different empirical distributions. - $\hat{\mathcal{F}}_{V_i,n}$: $\hat{\mathcal{F}}_{V_i,n}$ is the node-specific structural function class at index $n$, for $i\in[n_V]$, with categorical normalization interpreted pointwise as $x\mapsto f(x)/\|f(x)\|_1$ wherever $\|f(x)\|_1>0$. In Theorem~\ref{thm:consistency}: For categorical $V_i$, $\hat{\mathcal{F}}_{V_i,n}$ consists of the maps $x\mapsto q(x)/\|q(x)\|_1$ with $q\in\mathcal{F}_{0,n}$ satisfying $q(x)\in[0,\infty)^{n_i}$ and $\|q(x)\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\|q(x)\|_1)\leqslant\hat L_f$. For continuous $V_i$, the candidate structural class is $\{\hat f_i\in\mathcal F_{0,n}:\sup_x\|\hat f_i(x)\|_1\leqslant K\}$, where the supremum is over the full structural input domain. - $\hat{g}_1^i$: For $i\in[N_C]$, $\hat{g}_1^i$ denotes the chosen first generator block for component $C_i$, with scalar input and $d_i^C$ output coordinates. In the result split from Theorem~\ref{thm:wide_nn}: For $d_i^C>0$, $\hat{g}_1^i$ is the coordinatewise deep ReLU approximation of the curve $\gamma_i:[0,1]\rightarrow[0,1]^{d_i^C}$ satisfying $(\gamma_i)_{\#}\lambda=(g_2^i)^{-1}_{\#}\mu_i$; its coordinate networks have width $12$ and depth $L_1$, its combined width is $\Theta(d_i^C)$, and $\|\hat{g}_1^i-\gamma_i\|_\infty\leqslant O(L_1^{-2/d_i^C})$. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: For $d_i^C=0$, $\hat{g}_1^i$ is the unique map to $[0,1]^0$. In Theorem~\ref{thm:wide_nn}: For $d_i^C>0$, $\hat{g}_1^i$ is a quantized ReLU network of width $W_1$ and depth $\Theta(d_i^C)$, chosen so that $W((g_2^i)^{-1}_{\#}\mu_i,P(\hat{g}_1^i(U_i)))\leqslant O(W_1^{-1/d_i^C})$. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: For a component of positive dimension $q=d_i^C$, $\hat{g}_1^i$ is the quantized ReLU component generator of width $W_1$ and depth $\Theta(q)$ satisfying $W((g_2^i)^{-1}_{\#}\mu_i,P(\hat{g}_1^i(U_i)))\leqslant O(W_1^{-1/q})$. The source construction uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and mesh $\delta/n$, where $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$ and $A_q\geqslant1$ depends only on $q$. - $U_i$: $U_i$ denotes the index-$i$ random variable in the locally specified latent or generator-input family. In Corollary~\ref{corollary:deep_nn}: In the SCM notation, $U_i$ is a non-Gumbel latent variable of the model being approximated, with its own latent dimension and population law; scalar uniform variables appearing in the component-generator construction are inputs used to approximate that law. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: In the component-generator construction, $U_i\sim\operatorname{Unif}([0,1])$ for $i\in[N_C]$; $i$ indexes connected support components, and $U_i$ is the scalar input to $\hat g_1^i$. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: In the component-generator construction, $U_i\sim\operatorname{Unif}([0,1])$ for $i\in[N_C]$; $i$ indexes connected support components, and $U_i$ is the scalar input to $\hat g_1^i$. In \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: $U_i$, $i\in[n_U]$, is a non-Gumbel latent variable with values in $[-K,K]^{d_i^U}$ under the locally specified model law $P^\theta$ or $P^{\theta^*}$; the scalar uniform generator inputs are $Z_{i,j}$. The vector $(U_1,\ldots,U_{n_U})$ has total dimension $D_U=\sum_{i=1}^{n_U}d_i^U$. - $s$: $s$ denotes the fixed integer depth parameter $s=1$, distinct from the quantization parameter $\delta$ and the quantization mesh $\delta/n$. **Objections raised during the review and withdrawn.** The paper's owner showed that each misreads the paper; the printed statement stands under the reading given. - **a condition or definition of the paper.** A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden layers, each of width at most $W$, followed by the affine output layer (width at most $W$ and exactly $W$ give the same class of functions: a narrower layer is padded with zero units). The covering number $\mathcal{N}(\epsilon,S,\|\cdot\|_1)$ is that of the cited source (Anthony and Bartlett): the least size of a subset $C\subseteq S$ such that every point of $S$ lies at $\ell_1$ distance strictly less than $\epsilon$ from some point of $C$. - **a condition or definition of the paper.** A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden layers, each of width at most $W$, followed by the affine output layer (width at most $W$ and exactly $W$ give the same class of functions: a narrower layer is padded with zero units). The covering number $\mathcal{N}(\epsilon,S,\|\cdot\|_1)$ is that of the cited source (Anthony and Bartlett): the least size of a subset $C\subseteq S$ such that every point of $S$ lies at $\ell_1$ distance strictly less than $\epsilon$ from some point of $C$. ## What was proved in Lean, and what was not A result is proved in Lean when a Lean declaration reviewed as stating it has a complete proof using no axiom beyond the standard ones. For a corrected result, the corrected statement is the one formalized. | Result | Formal status | Lean declaration | Axioms | | --- | --- | --- | --- | | Theorem 3.3 | proved in Lean (the corrected statement), under 9 condition(s) made explicit | `Papers.TbsNeuralCausalPi.approximation_wasserstein_bound` (`TApproximation.lean:30`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Theorem 3.6 | proved in Lean (the corrected statement), given 2 cited result(s), under 17 condition(s) made explicit | `Papers.TbsNeuralCausalPi.wide_generator_approximation` (`TWideNn.lean:112`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Corollary 3.6.1 | proved in Lean, given 2 cited result(s), under 26 condition(s) made explicit | `Papers.TbsNeuralCausalPi.deep_generator_approximation` (`TCorFix3c402872.lean:21`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Corollary 3.8 | proved in Lean (the corrected statement), given 3 cited result(s), under 45 condition(s) made explicit | `Papers.TbsNeuralCausalPi.deep_ncm_approximation` (`TDeepNn.lean:303`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Proposition 4.1 | proved in Lean (the corrected statement), under 7 condition(s) made explicit | `Papers.TbsNeuralCausalPi.counterexample_identifiable` (`TCounterexample.lean:53`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Theorem 4.2 | proved in Lean (the corrected statement), given 6 cited result(s), under 66 condition(s) made explicit | `Papers.TbsNeuralCausalPi.consistency` (`TConsistency.lean:189`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Proposition 4.3 | proved in Lean (the corrected statement), under 8 condition(s) made explicit | `Papers.TbsNeuralCausalPi.lipschitz_dose_response_gap` (`TLipAte.lean:48`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Corollary 4.4 | proved in Lean (the corrected statement), given 6 cited result(s), under 74 condition(s) made explicit | `Papers.TbsNeuralCausalPi.rate_of_convergence` (`TRate.lean:87`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Proposition B.1 | proved in Lean (the corrected statement), under 7 condition(s) made explicit | `Papers.TbsNeuralCausalPi.exists_canonical_representation` (`TEquivalentGConstrianed.lean:39`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Proposition B.2 | proved in Lean, under 10 condition(s) made explicit | `Papers.TbsNeuralCausalPi.exists_measure_representing_holder_curve` (`Helpers/HilbertCurve.lean:386`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Lemma B.3 | proved in Lean (the corrected statement), under 5 condition(s) made explicit | `Papers.TbsNeuralCausalPi.hilbert_ordering` (`Helpers/HilbertCurve.lean:121`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Proposition C.1 | proved in Lean (the corrected statement), under 7 condition(s) made explicit | `Papers.TbsNeuralCausalPi.counterexample_backdoor` (`TCounterexampleForal.lean:60`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Proposition C.2 | proved in Lean, under 4 condition(s) made explicit | `Papers.TbsNeuralCausalPi.convgOpt_limits_mem` (`Helpers/ConvgOpt.lean:193`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Lemma C.3 | proved in Lean, under 1 condition(s) made explicit | `Papers.TbsNeuralCausalPi.regression_gap_le_wasserstein` (`Helpers/RegressionComparison.lean:335`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Lemma E.1 | proved in Lean (the corrected statement), under 2 condition(s) made explicit | `Papers.TbsNeuralCausalPi.wasserstein_map_le` (`Helpers/WassersteinBounds.lean:154`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Lemma E.2 | proved in Lean, under 2 condition(s) made explicit | `Papers.TbsNeuralCausalPi.wasserstein_mixture_le` (`Helpers/WassersteinBounds.lean:303`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Lemma E.3 | proved in Lean (the corrected statement), under 2 condition(s) made explicit | `Papers.TbsNeuralCausalPi.wasserstein_pi_le` (`Helpers/WassersteinBounds.lean:467`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Lemma E.4 | proved in Lean, under 2 condition(s) made explicit | `Papers.TbsNeuralCausalPi.wasserstein_gumbelSoftmax_le` (`Helpers/GumbelSoftmax.lean:515`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Lemma E.5 | proved in Lean (the corrected statement), under 1 condition(s) made explicit | `Papers.TbsNeuralCausalPi.sinkhorn_approx` (`Helpers/Sinkhorn.lean:164`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Lemma E.6 | proved in Lean (the corrected statement), given 2 cited result(s), under 3 condition(s) made explicit | `Papers.TbsNeuralCausalPi.uniform_empirical_wasserstein` (`Helpers/EmpiricalWasserstein.lean:190`) | the standard ones only (propext, Classical.choice, Quot.sound) | **Taken from the literature, not proved here.** - perekrestenkoHighDimensionalDistributionGeneration2022a, Theorem VIII.1 — checked against source supplied by the user at discovery/cited_sources/perekrestenkoHighDimensionalDistributionGeneration2022a.txt What Lean assumes for it: `Papers.TbsNeuralCausalPi.PerekrestenkoQuantizedGeneration` (`Setup/CitedResults.lean:73`) ```lean def PerekrestenkoQuantizedGeneration : Sort 0 := -- @realizes s(natural-number source depth fixed by let s := 1 below; distinct from delta and delta / n) ∃ c : ℕ, 0 < c ∧ ∀ q : ℕ, 0 < q → ∃ A : ℝ, 1 ≤ A ∧ -- @realizes \Theta(d_i^C)(depth c * q with one multiplier c for every dimension q; the source's depth (s+3)q-s-1 = 4q-2 at s = 1) ∀ (d N : ℕ) (P : ProbabilityMeasure (L1Vec d)) (S : Fin N → Set (L1Vec d)) (dC : Fin N → ℕ) (L : ℝ≥0), SupportFiniteComponents (P : Measure _) S → ComponentsLipschitzHomeomorphic S dC L → ∀ (i : Fin N), dC i = q → ∀ f : homeoClass (unitCube (dC i)) (S i), LipschitzWith L f → ∃ B : ℝ, 0 < B ∧ ∀ W₁ : ℕ, A * (2 : ℝ)^q ≤ W₁ → -- @realizes W_1(component width budget above the dimension-dependent threshold) let n : ℕ := ⌊((W₁ : ℝ) / A) ^ (1 / (q : ℝ))⌋₊ -- @realizes n(refinement floor chosen from W_1 and the dimension-only width factor A) let s : ℕ := 1 -- @realizes s(carrier ℕ and fixed value s = 1; source depth, distinct from quantization delta and mesh delta / n) let delta : ℝ := (⌈Real.sqrt q * (q + 1) * n * ((n : ℝ) - 1)⌉₊ : ℝ)⁻¹ -- @realizes \delta(prescribed quantization parameter at the pinned refinement) 2 ≤ n ∧ A * (n : ℝ)^q ≤ W₁ ∧ ∃ Phi : ReLUNet 1 (dC i) W₁ (c * q), -- @realizes \hat{g}_1^i(quantized component network of width W_1 and depth proportional to its positive dimension) (∀ l ∈ Phi.hidden, (l.output : ℝ) ≤ A * (n : ℝ)^q) ∧ Phi.paramsQuantized (delta / n) ∧ wasserstein1 ((ProbabilityTheory.cond (P : Measure _) (S i)).map (chartInverse f)) ((uniformCubeLaw 1).map Phi.eval) ≤ ENNReal.ofReal (B * (W₁ : ℝ) ^ (-(1 : ℝ) / q)) ``` The passage of the proof: > By \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, there exists quantized ReLu network $ \hat{g}_{1}^{i}$ of width $ W_{1}$ and depth $ \Theta( d_i^C)$ (let $ s=\frac{1}{\lceil n( n-1)\rceil }$ in \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}), such that \begin{equation*} > W\left(\left( (g_{2}^{i})^{-1}\right)_{\#} \mu _{i} ,P(\hat{g}_{1}^{i}(U_{i}))\right) \leqslant O\left( W_{1}^{-1/d_i^C}\right) , > \end{equation*} > where $ U_{i}$ are i.i.d. uniform random variables on $ [ 0,1]$. - Yarotsky2018OptimalAO, Theorem 2 — checked against source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt What Lean assumes for it: `Papers.TbsNeuralCausalPi.YarotskyCoordinateApprox` (`Setup/CitedResults.lean:115`) ```lean def YarotskyCoordinateApprox : Sort 0 := ∀ d : ℕ, 0 < d → ∃ a c : ℝ, 0 < a ∧ 0 < c ∧ ∀ (D : ℕ) (S : Set (L1Vec D)), IsCompact S → ∀ (f : homeoClass (unitCube d) S) (L : ℝ≥0), LipschitzWith L f → ∀ j : Fin D, ∀ depth : ℕ, 1 ≤ depth → ∃ q : ReLUNet d 1 (12*d) depth, ∀ x ∈ unitCube d, |(if hx : x ∈ unitCube d then (f ⟨x,hx⟩).val.ofLp j else 0) - (q.eval x).ofLp 0| ≤ a * euclidModulus (fun y => if hy : y ∈ unitCube d then (f ⟨y,hy⟩).val.ofLp j else 0) (c * (depth : ℝ) ^ (-(2 : ℝ) / d)) ``` The passage of the proof: > By \cite[Theorem 2]{Yarotsky2018OptimalAO}, there exist a deep ReLu network $ \hat{g}_{2}^{i,j}$ of width $ \Theta\left( d_{i}^{C}\right)$ and depth $ L_{2}$ such that > \begin{equation*} > \| \left(g_{2}^{i}\right)_{j} -\hat{g}_{2}^{i,j} \| _{\infty } \leqslant O\left( L_{2}^{-2/d_{i}^{C}}\right) , > \end{equation*} > Let $ \hat{g}_{2}^{i}( x) =\left(\hat{g}_{2}^{i,1}( x) ,\cdots ,\hat{g}_{2}^{i,d}( x)\right)$, we get > \begin{equation*} > \| g_{2}^{i} -\hat{g}_{2}^{i} \|_\infty \leqslant O\left( L_{2}^{-2/d_{i}^{C}}\right) . > \end{equation*} - Yarotsky2018OptimalAO, Theorem 2 — checked against source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt What Lean assumes for it: `Papers.TbsNeuralCausalPi.YarotskyStructuralApprox` (`Setup/CitedResults.lean:140`) ```lean def YarotskyStructuralApprox : Sort 0 := ∀ d : ℕ, 0 < d → ∃ a c : ℝ, 0 < a ∧ 0 < c ∧ ∀ (n b : ℕ) (kind : Fin n → NodeKind) (dims : Fin b → ℕ) (dag : Causalean.Graph.DAG (Fin n)) (G : SimpleGraph (Fin n)) (C : Fin b → Set (Fin n)) (M : CanonicalSCM kind dims dag G C) (i : Fin n) (hd : structuralInputSize M.toEuclideanSCM i = d), let f := fun x : L1Vec d => M.f i (structuralFiniteCoordinates M.toEuclideanSCM i (WithLp.toLp 1 (fun k => x.ofLp (Fin.cast hd k)))) ∀ L : ℝ≥0, LipschitzOnWith L f (unitCube d) → ∀ j : Fin (outputDim (kind i)), ∀ depth : ℕ, 1 ≤ depth → ∃ q : ReLUNet d 1 (2*d+10) depth, ∀ x ∈ unitCube d, |(f x).ofLp j - (q.eval x).ofLp 0| ≤ a * euclidModulus (fun y => (f y).ofLp j) (c * (depth : ℝ) ^ (-(2 : ℝ) / d)) ``` The passage of the proof: > By \cite[Theorem 2]{Yarotsky2018OptimalAO}, part b), applied with $p=2/d_{i}^{\text{in}}$ to each output coordinate of $f_i$, which is Lipschitz continuous on the input cube by \cref{assump:Lip_and_bounded}: there are narrow fully-connected ReLU networks of constant width $2d_{i}^{\text{in}}+10$ and depth $L_0$ whose sup-norm error is at most $a\,\omega(cW^{-2/d_{i}^{\text{in}}})$, where $W$, the number of weights, grows linearly in $L_0$ at fixed width; stacking the $d_{i}^{\text{out}}$ coordinates gives, if $V_i$ is continuous, $ \| \hat{f}_{i} -f_{i} \| _{\infty } \leqslant O(L_0^{-2/d_{i}^{\text{in}}}) \leqslant O(L_0^{-2/d_{\max}^{\text{in}}}) $. - fournier2015rate, Theorem 2 — checked against source supplied by the user at discovery/cited_sources/fournier2015rate.txt What Lean assumes for it: `Papers.TbsNeuralCausalPi.FournierGuillinSmallDeviation` (`Setup/CitedResults.lean:213`) ```lean def FournierGuillinSmallDeviation : Sort 0 := ∀ d : ℕ, 1 ≤ d → -- @realizes d(positive source Euclidean dimension) ∀ alpha gamma e : ℝ, 1 < alpha → 0 < gamma → ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ -- @realizes C(positive source prefactor fixed before the law and sample size) -- @realizes c(positive source exponent constant fixed before the law and sample size) ∀ mu : ProbabilityMeasure (EuclideanSpace ℝ (Fin d)), -- @realizes \mu(probability law on Euclidean space with the source two-norm) Integrable (fun z => Real.exp (gamma * ‖z‖ ^ alpha)) (mu : Measure (EuclideanSpace ℝ (Fin d))) → (∫ z, Real.exp (gamma * ‖z‖ ^ alpha) ∂(mu : Measure (EuclideanSpace ℝ (Fin d)))) = e → ∀ N : ℕ, 1 ≤ N → -- @realizes n(positive IID source sample size N) ∀ x : ℝ, 0 < x → x ≤ 1 → (Measure.pi (fun _ : Fin N => (mu : Measure (EuclideanSpace ℝ (Fin d))))) {z | ENNReal.ofReal x ≤ wasserstein1 ((N : ℝ≥0∞)⁻¹ • ∑ i : Fin N, Measure.dirac (z i)) (mu : Measure (EuclideanSpace ℝ (Fin d)))} ≤ ENNReal.ofReal (C * Real.exp (-c * N * (if d = 1 then x^2 else if d = 2 then (x / Real.log (2 + x⁻¹))^2 else x^d))) ``` The passage of the proof: > \item By definition of $ g_{n}$, > \begin{align*} > |g_{n}( \theta ) -g_{n}( \theta') | & =|S_{\lambda _{n}}\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -S_{\lambda _{n}}\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta'}(\boldsymbol{V})\right) |\\ > & \leqslant |W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta'}(\boldsymbol{V})\right) |+ 4(\log( mn) +1) \lambda _{n}\\ > & \leqslant W\left( P_{m_{n}}^{\theta }(\boldsymbol{V}) ,P_{m_{n}}^{\theta'}(\boldsymbol{V})\right) + 4(\log( mn) +1) \lambda _{n}\\ > & \leqslant O( d( \theta ,\theta')) +4(1+\log( nm_{n})) \lambda _{n} , > \end{align*} > where we use Lemma \ref{lemma:sinkhorn_approx} in the second inequality and triangle inequality and \cref{thm:approximation} in the third inequality. By the condition in \cref{thm:consistency}, the second term $ (1+\log( nm_{n})) \lambda _{n} \rightarrow 0 $ > as $n \rightarrow \infty$. Therefore, (\ref{eq:aprox_lip}) is verified with $\tau_n = 4(1+\log( nm_{n})) \lambda _{n} $. > > We then verify the uniform convergence of $g_n$. By triangle inequality, > \begin{align*} > |g_{n}( \theta ) -g( \theta ) | & =|S_{\lambda _{n}}\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}(\boldsymbol{V}) ,P^{\theta }(\boldsymbol{V})\right) |\\ > & \leqslant |S_{\lambda _{n}}\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) |\\ > & \ \ \ \ \ \ \ \ +|W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}(\boldsymbol{V}) ,P^{\theta }(\boldsymbol{V})\right) |. > \end{align*} > By Lemma \ref{lemma:sinkhorn_approx}, $ |S_{\lambda _{n}}\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) |\leqslant 2(\log( mn) +1) \lambda _{n}$ and we get > \begin{align} > |g_{n}( \theta ) -g( \theta ) | & \leqslant 2(\log( mn) +1) \lambda _{n} +|W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P^{\theta }(\boldsymbol{V})\right) |\notag\\ > & \quad \ \ \ \ +|W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}(\boldsymbol{V}) ,P^{\theta }(\boldsymbol{V})\right) |\notag\\ > & \leqslant 2(\log( mn) +1) \lambda _{n} +W\left( P_{m_{n}}^{\theta }(\boldsymbol{V}) ,P^{\theta }(\boldsymbol{V})\right) +W( P^{\theta^*}_{n}(\boldsymbol{V}) ,P^{\theta^*}(\boldsymbol{V}))\notag\\ > & \leqslant 2(\log( mn) +1) \lambda _{n} +O\left(W( P^{\theta^*}_{n}(\boldsymbol{U}) ,P^{\theta^*}(\boldsymbol{U})) + W\left( P^{\theta }(\boldsymbol{U}) ,P^{\theta}_{m_n}(\boldsymbol{U})\right)\right) ,\label{eq:uniform_app_err} > \end{align} > where we use \cref{thm:approximation} in the last inequality. > > We first bound $ \sup _{\theta } W\left( P^{\theta }(\boldsymbol{U}) ,P_{m_{n}}^{\theta }(\boldsymbol{U})\right)$ using standard VC dimension argument. Note that $ \{U_{1} ,\cdots ,U_{n_{U}}\}$ are independent. By \cref{lemma:product_err}, > \begin{equation}\label{eq:decomp_u} > W\left( P^{\theta }(\boldsymbol{U}) ,P_{m_{n}}^{\theta }(\boldsymbol{U})\right) \leqslant \sum _{i=1}^{n_{U}} W\left( P^{\theta }( U_{i}) ,P_{m_{n}}^{\theta }( U_{i})\right). > \end{equation} > By the construction in \cref{thm:wide_nn}, $ P^{\theta }( U_{i}) \sim \sum _{j\in[N_{C,i}]} p_{j}P( f_{\theta _{i,j}}( Z_{i,j})) ,Z_{i,j} \sim \text{Unif}( 0,1)$, where $ f_{\theta _{i,j}}$ are neural networks with constant width and depth $ L_{1,n} + L_{2,n}$, $N_{C,i}$ is the number of connected components of $\text{supp}(P(U_i))$ and $\theta_{i,j} $ are the parameters of the neural networks. By \cref{lemma:combin_err}, > \begin{align*} > \sup _{\theta } W\left( P^{\theta }( U_{i}) ,P_{m_{n}}^{\theta }( U_{i})\right) & \leqslant O(\max_{j\in[N_{C,j}]}\sup _{\theta _{i,j}} W(P( f_{\theta _{i,j}}( Z_{i,j})) ,P_{m_{n}}( f_{\theta _{i,j}}( Z_{i,j}))) . > \end{align*} > By \cite{bartlett2019nearly}, the pseudo-dimension of $ \mathcal{NN}(\Theta(1) ,L)$ is $ O\left( L^{2}\log L\right)$. By boundness of all the neural networks and \cref{lemma:sup_wasserstein} \footnote{Note that we truncate the neural network to ensure boundness in \cref{thm:consistency}. This extra truncate operation $\tilde{f} = \max\{-K,\min\{K,f\}\} $ can be viewed as two extra ReLu layers, so \cref{lemma:sup_wasserstein} is still applicable. }, with probability at least $$ 1-\exp\left( O\left( (\epsilon^{'}_{n})^{-d_{\max}^{U}}\log (\epsilon^{'}_{n})^{-1} + (L_{n,1}+L_{n,2})^{2}\log(L_{n,1}+L_{n,2})\log \epsilon _{n}^{-1} -m_{n} \epsilon _{n}^{2}\right)\right) ,$$ > the following event happens. > \begin{align*} > \sup _{\theta _{i,j}} W(P( f_{\theta _{i,j}}( Z_{i,j})) ,P_{m_{n}}( f_{\theta _{i,j}}( Z_{i,j})) & \leqslant \epsilon _{n} +\epsilon^{'}_{n} . > \end{align*} > Let $$ \epsilon^{'}_{n} =m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n} ,L_{n,i} =m_{n}^{d_{\max}^{U} /\left( 2d_{\max}^{U} +4\right)}\log m_{n} ,\epsilon _{n} =Cm_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n}$$ > with constant $ C >0$ sufficiently large, we get > \begin{align*} > P\left(\sup _{\theta _{i,j}} W(P( f_{\theta _{i,j}}( Z_{i,j})) ,P_{m_{n}}( f_{\theta _{i,j}}( Z_{i,j}))) \leqslant O\left( m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n}\right)\right) & \geqslant 1-O\left( m_{n}^{-2}\right) . > \end{align*} > As long as $ m_{n} = \Omega(n)$, the Borel-Cantelli lemma implies that almost surely, there exists $ N > 0 $ such that when $ n > N$, $ \sup _{\theta _{i,j}} W(P( f_{\theta _{i,j}}( Z_{i,j})) ,P_{m_{n}}( f_{\theta _{i,j}}( Z_{i,j}))) \leqslant O\left( m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n}\right)$ for all $ i,j$. Therefore, almost surely, for sufficiently large $ n$, > \begin{align} > \sup _{\theta } W\left( P^{\theta }( U_{i}) ,P_{m_{n}}^{\theta }( U_{i})\right) & \leqslant O\left( m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n}\right),\label{eq:uniform_app_err2} > \end{align} > > \cite[Theorem 2]{fournier2015rate} shows that for sufficiently large $ n $, if $\delta_n = C_1 n^{-1/\max\{ d^U_{\max},2\}}\log^2(n) $, where $ C_1 >0$ is a constant, > \begin{align*} > P( W( P^{\theta^*}({U}_i) ,P^{\theta^*}_{n}({U}_i)) >\delta _{n}) & \leqslant \begin{cases} > \exp( -CC_{1}\log^2( n)) & d^U_{\max} =1,\\ > \exp\left(\frac{-CC_{1}\log^{4} (n)}{\log^2\left( 2+C_{1}^{-1} n^{1/2}\log^{-2} (n)\right)}\right) & d^U_{\max} = 2,\\ > \exp( -CC_{1}\log^{2/d^U_{\max}} (n)) & d^U_{\max} >2, > \end{cases} > \end{align*} > where $C > 0$ is a constant. Let $ C_{1}$ sufficiently large such that > \begin{equation*} > P( W( P^{\theta^*}({U}_{i}) ,P^{\theta^*}_{n}({U}_{i})) >\delta _{n}) \leqslant O\left( n^{-2}\right) . > \end{equation*} > Therefore, $ \sum _{n=1}^{\infty }P( W( P^{\theta^*}({U}_{i}) ,P^{\theta^*}_{n}({U}_{i})) >\delta _{n}) < \infty .$ By Borel-Cantelli lemma, with probability 1, there exist $ N >0$ such that $ W( P^{\theta^*}({U}_{i}) ,P^{\theta^*}_{n}({U}_{i})) \leqslant \delta _{n} ,\forall n >N$. By \cref{eq:decomp_u}, with probability 1, there exist $ N >0$ such that > \begin{equation} > W( P^{\theta^*}(\boldsymbol{U}) ,P^{\theta^*}_{n}(\boldsymbol{U})) \leqslant O(\delta _{n}) ,\forall n >N. \label{eq:uniform_app_err3} > \end{equation} > > Therefore, by \cref{eq:uniform_app_err,eq:uniform_app_err2,eq:uniform_app_err3}, almost surely, the following inequality holds eventually, > \begin{equation*} > \sup_{\theta \in \Theta_\infty}|g_{n}( \theta ) -g( \theta ) |\leqslant O\left(\log( nm_{n})\lambda _{n} +\delta _{n} +m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n}\right) = s_n, > \end{equation*} > where $s_n$ is defined in \cref{thm:consistency}. - anthony1999neural, Theorem 17.1 — checked against source supplied by the user at discovery/cited_sources/anthony1999neural.txt What Lean assumes for it: `Papers.TbsNeuralCausalPi.AnthonyBartlettSquaredLoss` (`Setup/CitedResults.lean:183`) ```lean def AnthonyBartlettSquaredLoss : Sort 0 := ∀ (X : Type) (_mX : MeasurableSpace X) [StandardBorelSpace X] (G : Set (X → ℝ)), G.Nonempty → BorelParametrizedClass G → ∀ hG : (∀ g ∈ G, Measurable g ∧ ∀ x, g x ∈ Set.Icc (0:ℝ) 1), ∀ P : ProbabilityMeasure (X × Set.Icc (0 : ℝ) 1), ∀ (m : ℕ), 1 ≤ m → ∀ eps : ℝ, 0 < eps → eps < 1 → (Measure.pi (fun _ : Fin m => (P : Measure (X × Set.Icc (0 : ℝ) 1)))).completion {z : Fin m → X × Set.Icc (0 : ℝ) 1 | ∃ g ∈ G, |(∫ xy : X × Set.Icc (0 : ℝ) 1, (g xy.1 - xy.2.val)^2 ∂(P : Measure (X × Set.Icc (0 : ℝ) 1))) - (m : ℝ)⁻¹ * ∑ i, (g (z i).1 - (z i).2.val)^2| ≥ eps} ≤ 4 * (N1avg (eps/16) G (2*m) (fun g hg => (hG g hg).2)).toENNReal * ENNReal.ofReal (Real.exp (-(eps^2 * m)/32)) ``` The passage of the proof: > The cited assertion used in Lemma~\ref{lemma:sup_wasserstein} is the following restricted form of Theorem 17.1 in \cite{anthony1999neural}, with its standing measurability condition. Let $X$ and $T$ be standard Borel spaces, let $g:T\times X\rightarrow[0,1]$ be jointly Borel, and put $\mathcal G=\{g(t,\cdot):t\in T\}\ne\varnothing$. For every Borel probability distribution $P$ on $X\times[0,1]$, every $0<\epsilon<1$ and every positive integer $m$, > \[ > P^m\left\{\text{some }t\in T\text{ has }\left|\mathbb E_{(x,y)\sim P}(g(t,x)-y)^2-\frac1m\sum_{i=1}^m(g(t,x_i)-y_i)^2\right|\geqslant\epsilon\right\} > \leqslant4\mathcal N_1^{\mathrm{avg}}(\epsilon/16,\mathcal G,2m)\exp(-\epsilon^2m/32). > \] > Here the sample consists of independent draws from $P$, the probability is taken in its completion, and $\mathcal N_1^{\mathrm{avg}}(r,\mathcal G,N)$ is the supremum over $x_1,\ldots,x_N\in X$ of the covering number of the evaluation vectors in the metric $N^{-1}\sum_i|u_i-v_i|$. - anthony1999neural, Theorem 18.4 — checked against source supplied by the user at discovery/cited_sources/anthony1999neural.txt What Lean assumes for it: `Papers.TbsNeuralCausalPi.AnthonyBartlettPacking` (`Setup/CitedResults.lean:164`) ```lean def AnthonyBartlettPacking : Sort 0 := ∀ (X : Type) (G : Set (X → ℝ)), G.Nonempty → ∀ hG : (∀ g ∈ G, ∀ x, g x ∈ Set.Icc (0:ℝ) 1), ∀ v : ℕ, Causalean.Stat.Concentration.HasPseudoDimAtMost (fun g : G => g.val) v → ∀ (m : ℕ), 1 ≤ m → ∀ eps : ℝ, 0 < eps → eps ≤ 1 → (M1avg eps G m hG).toENNReal < ENNReal.ofReal (Real.exp 1 * (v+1) * (2 * Real.exp 1 / eps) ^ v) ``` The passage of the proof: > By Theorem 18.4 in \cite{anthony1999neural}, if $\mathcal G$ is a nonempty class of functions from a set $X$ into $[0,1]$ with finite pseudo-dimension $v$, then for every $0<\epsilon\leqslant1$ and every positive integer $m$, > \begin{align*} > \mathcal{N}_{1}^{\mathrm{avg}}(\epsilon,\mathcal G,m)\leqslant\mathcal{M}_{1}^{\mathrm{avg}}(\epsilon,\mathcal G,m) \end{align*} > where $\mathcal{N}_{1}^{\mathrm{avg}}(\epsilon,\mathcal G,m)$ and $\mathcal{M}_{1}^{\mathrm{avg}}(\epsilon,\mathcal G,m)$ are the largest, over $x_1,\ldots,x_m\in X$, covering and packing numbers of $\{(g(x_1),\ldots,g(x_m)):g\in\mathcal G\}$ in the averaged metric $d_1(u,w)=\frac1m\sum_{i=1}^m|u_i-w_i|$. The Lean development builds; it contains no `sorry`, `admit` or added axiom. ## The corrected paper `bundle/corrected_paper/` holds the paper as it reads after the corrections, as plain source in the author's own class and layout, compiled to `bundle/corrected_paper/Main.pdf` (129 pages). `bundle/MATCH.md` sets each result beside its Lean declaration; `bundle/lean/` holds the Lean code. **18 item(s) remain before the paper is ready to submit** (0 blocking), each with its place in the source: `bundle/corrected_paper/TODO.md`. What is checked is the paper's results; the prose around them is the authors', as printed. # Technical appendix The records behind the report above, by the run's own identifiers. - paper: `Main.tex` (source sha256 `f7e36bd33e0269af426cb564f74032199c6274c291caf5bc32937b1195e895c3`) - stage completed: `V6` - results in scope: 20 | bucket | count | | --- | --- | | verified | 0 | | verified_with_bookkeeping | 6 | | conditional | 1 | | repaired | 13 | | unresolved | 0 | | not_checked | 0 | | not_determined | 0 | | refuted | 0 | > This is a conditional report. A `verified` result is verified only up to the cited results it assumes, the substrate gates left assumed, and the conditions added during formalization — all listed in full below. `unresolved` means the formalization did not close; it is NOT a claim that the result is wrong. A `refuted` row is a result shown false by a positive witness with no confirmed repair; a `repaired` row was false as printed, and an independent referee confirmed its minimal repair. A `not_determined` row could not be checked as printed: the paper does not say enough to determine the claim, and no reading or repair was adopted; it is NOT a claim that the result is wrong. ## Results ### 1. `thm:approximation` | field | value | | --- | --- | | bucket | `repaired` | | source | `Main.tex:240-246` (`theorem`, label `thm:approximation`) | | statement | \label{thm:approximation} Given any SCM model $ \mathcal{M} \in \mathcal{M}(\mathcal{G} ,\mathcal{F}_L ,\boldsymbol{U})$, let the treatment variable set be $ \boldsymbol{T} =\{T_k\}_{k=1}^{n_t}$ and suppose that \cref{assump:independence_u}, \ref{assump:categorical_rv} and \ref{assump:Lip_and_bounded} hold for $\mathcal{M}$ with Lipschitz constant $L$, constant $K$. For any intervention $ \boldsymbol{T}= \boldsymbol{t}$ and $ \hat{\mathcal{M}} \in \mathcal{M}(\mathcal{G} ,\hat{\mathcal{F}} ,\hat{\boldsymbol{U}})$ , we have \begin{equation} W\left( P^{\mathcal{M}}(\boldsymbol{V}(\boldsymbol{t})),P^{\hat{\mathcal{M}}}(\hat{\boldsymbol{V}}(\boldsymbol{t}))\right) \leqslant C_{\mathcal{G}}(L,K)\left( \sum_{i=1}^{n_V} \\| f_i - \hat{f}_i \\|_\infty +W\left(P^{\mathcal{M}}(\boldsymbol{U}),P^{\hat{\mathcal{M}}}(\hat{\boldsymbol{U}})\right)\right) , \end{equation} where $ C_{\mathcal{G}}(L,K)$ is a constant that only depends on $ L,K $ and the causal graph $ \mathcal{G} $ and $f_i , \hat{f}_i$ are structural functions of $\mathcal{M}$ and $\hat{\mathcal{M}}$ respectively. | | appended hypothesis list | ass:assump-independence-u, ass:assump-categorical-rv-latents, ass:assump-categorical-rv-generation, ass:assump-lip-and-bounded-continuous, ass:assump-lip-and-bounded-categorical, ass:assump-lip-and-bounded-bound | | Lean | `TApproximation.lean:30` — `Papers.TbsNeuralCausalPi.approximation_wasserstein_bound` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-177`, `ass:pin-181`, `ass:pin-182`, `ass:pin-183`, `ass:pin-189`, `ass:pin-190`, `ass:pin-191`, `ass:pin-fix-4a86c2eb`, `ass:pin-fix-ab17be7a` | | cited results | — | | assumed substrate gates | — | | findings | [refuted, V1.5] (a) In (\ref{eq:approximation_1}), the Lipschitz property gives the output $1$-norm, not its coordinatewise infinity norm. The missing dimension factor invalidates the coefficient supplied by the proof; the hypotheses do not bound continuous output dimensions through $L$, $K$, or the graph. (b) In (\ref{eq:cate_2}), the probabilities $p_j$ vary with the variables being integrated. The omitted term is $\sum_j(p_j(x)-p_j(y))g(y,j)$, so the asserted kernel Lipschitz bound and (\ref{eq:approximation_middle_step}) are false. A bound using $1+KL$ can close this categorical step, but does not justify the printed coefficient. Zero propensities could alternatively be excluded or interpreted through an extended logarithm; the closed-simplex canonical convention determines the latter. witness: (a) Take the graph with two isolated roots $T$ and $V_1\in\mathbb{R}^{100}$, independent $U_T,U_1\sim\mathrm{Unif}([-1,1])$, $T=U_T$, and $f_1(u)=(u/100,\ldots,u/100)$. Let $\hat f_1(u)=f_1(u)+(1/100,\ldots,1/100)$ and keep all latent laws and the treatment function unchanged. Both true structural functions are $1$-Lipschitz in the paper's $1$-norm, and the true variables satisfy the bound $K=1$. Under $T=0$, $S=1/100$ and the latent Wasserstein distance is zero, whereas the output Wasserstein distance is $1$. The proof chooses $C_{\mathcal G}(1,1)=4$, giving the false inequality $1\leqslant4/100$. More generally, translating an unrestricted $D$-dimensional output by $(a,\ldots,a)$ gives Wasserstein error $Da$ but structural infinity error $a$, precluding a dimension-independent coefficient for the fixed abstract graph. (b) Take a four-category root with $p(u)=(u,1/10,1/10,4/5-u)$, true $U\sim\mathrm{Unif}([1/4,1/2])$, and comparison $\hat U\sim\mathrm{Unif}([1/4+h,1/2+h])$, where $00\}}\left(W_1^{-1/D_C}+L_2^{-2/D_C}\right)+(\tau-\tau\log\tau)\right). \end{equation*} \end{enumerate} · proof change: Put $p_i=\mathbb P(C_i)$ and $\mu_i=\mathbb P\|_{C_i}/p_i$. Each component is compact because it is homeomorphic to a compact cube. Distinct components have positive distance because they are disjoint compact sets and there are finitely many of them. A sufficiently small neighborhood of a support point in $C_i$ meets no other component and has positive measure; hence $p_i>0$. If $d_i^C=0$, write $C_i=\{c_i\}$, use the unique map $\hat g_1^i:[0,1]\rightarrow[0,1]^0$, and set $\hat g_2^i=c_i$. This produces $\mu_i$ exactly. For $q=d_i^C\geqslant1$, choose $g_2^i\in\mathcal H([0,1]^q,C_i)\cap\mathcal F_L([0,1]^q,C_i)$ from Assumption~\ref{assump:support}, and put $\nu_i=((g_2^i)^{-1})_{\#}\mu_i$. We instantiate Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} with its positive integer depth parameter $s=1$. For a refinement integer $n\geqslant2$, set \[ \delta=\frac{1}{\lceil\sqrt q(q+1)n(n-1)\rceil}. \] Since $\sqrt q(q+1)\geqslant2$, this choice satisfies \[ 0<\delta\leqslant\frac{1}{\sqrt q(q+1)n(n-1)}<\frac{1}{n(n-1)}. \] The quantization mesh in the cited theorem is $\delta/n$. Its conclusion supplies a one-input, $q$-output ReLU network $\Phi$ with at most $C_q(n^q+n^{q-1})$ nonzero weights and biases, depth $4q-2$ in the source's convention, and Euclidean Wasserstein error at most \[ \frac{\sqrt q}{2n}+\frac{2\sqrt q}{n}+\frac{q(q+1)}{2}(n-1)\delta \leqslant\frac{\sqrt q}{2n}+\frac{2\sqrt q}{n}+\frac{\sqrt q}{2n} =\frac{3\sqrt q}{n}. \] Here $C_q$ depends only on $q$. The pointwise inequality between the $1$-norm and the Euclidean norm multiplies this Wasserstein bound by at most $\sqrt q$, giving error at most $3q/n$ in the paper's norm. A hidden unit with zero incoming weights and zero bias has identically zero output and can be deleted. After deleting these units, every remaining hidden unit accounts for a nonzero weight or bias, so every hidden layer has width at most $C_q(n^q+n^{q-1})\leqslant2C_qn^q$. Clip the output coordinates into $[0,1]$ using $t\mapsto\max\{t,0\}-\max\{t-1,0\}$. This requires width $2q$ and a fixed number of additional layers. Clipping fixes the target cube and is $1$-Lipschitz in the $1$-norm, so its application to the first marginal of any coupling cannot increase the error. There is consequently a constant $A_q\geqslant1$, depending only on $q$, such that the clipped network has width at most $A_qn^q$ and hidden depth $\Theta(q)$. For every sufficiently large width budget, specifically $W_1\geqslant A_q2^q$, choose \[ n=\left\lfloor(W_1/A_q)^{1/q}\right\rfloor. \] Then $n\geqslant2$, $A_qn^q\leqslant W_1$, and $n\geqslant\tfrac12(W_1/A_q)^{1/q}$. With this choice the clipped network $\hat g_1^i$ satisfies \[ W(P(\hat g_1^i(U_i)),\nu_i)\leqslant\frac{3q}{n}\leqslant6qA_q^{1/q}W_1^{-1/q},\qquad U_i\sim\operatorname{Unif}([0,1]). \] Thus the refinement integer increases with width, while the depth parameter stays equal to $1$; the mesh $\delta/n$ is $\Theta_q(W_1^{-3/q})$. Padding with zero units realizes width $W_1$ without altering the function. For each scalar coordinate of $g_2^i$, apply Theorem 2(b) of \cite{Yarotsky2018OptimalAO} with positive input dimension $q$. It supplies width $2q+10$ and error bounded by a constant times the coordinate modulus of continuity at a constant times the number of weights raised to $-2/q$. At this fixed width the number of weights is proportional to depth. The Lipschitz modulus therefore gives coordinate error $O(L_2^{-2/q})$. Parallelizing the $d$ coordinate networks gives width $\Theta(dq)$ and a map $\hat g_2^i$ with $\\|g_2^i-\hat g_2^i\\|_\infty=O(L_2^{-2/q})$. Reserve a fixed number of layers for clipping into a box $[-B,B]^d$ containing all components; the remaining depth is proportional to $L_2$ for sufficiently large $L_2$, and clipping cannot increase any coordinate error. Lemma~\ref{lemma:push-forward_err} gives \[ W(\mu_i,P(\hat g_2^i\circ\hat g_1^i(U_i)))\leqslant L_iO(W_1^{-1/q})+d\,O(L_2^{-2/q})=O(W_1^{-1/q}+L_2^{-2/q}), \] where $L_i$ is the fixed chart Lipschitz constant. Take independent uniform inputs $U_i$. Pad the first-block hidden depths to a common value $\Theta(D_C)$ and the second-block widths to $\Theta(dD_C)$. Identity layers on nonnegative hidden outputs preserve the maps. Let $\hat\mu_i=P(\hat g_2^i\circ\hat g_1^i(U_i))$. For budgets at least $1$, each positive $q\leqslant D_C$ satisfies $W_1^{-1/q}\leqslant W_1^{-1/D_C}$ and $L_2^{-2/q}\leqslant L_2^{-2/D_C}$. Singleton errors are zero. Lemma~\ref{lemma:combin_err} hence yields \[ W\left(\mathbb P,\sum_i p_i\hat\mu_i\right)\leqslant\sum_i p_iW(\mu_i,\hat\mu_i)\leqslant O\left(\mathbb I_{\{\max_i d_i^C>0\}}(W_1^{-1/D_C}+L_2^{-2/D_C})\right). \] Let $X^\tau$ be the Gumbel-softmax weights with probabilities $p_i$, independent of the block outputs, and let $X^0$ be their one-hot limit. The output with weights $X^0$ has law $\sum_i p_i\hat\mu_i$. Lemma~\ref{lemma:gumbel_err} gives $W(P(X^\tau),P(X^0))=O(\tau-\tau\log\tau)$ for $0<\tau<1$. Couple the weights and use identical independent block outputs $v_i$ in the two constructions. Since $\\|v_i\\|_\infty\leqslant B$, \[ \left\\|\sum_i(X_i^\tau-X_i^0)v_i\right\\|\leqslant\sum_i\|X_i^\tau-X_i^0\|\\|v_i\\|\leqslant dB\\|X^\tau-X^0\\|. \] Taking expectations and the infimum over weight couplings bounds the additional output Wasserstein error by $O(\tau-\tau\log\tau)$. The triangle inequality proves the stated estimate. If every component is a singleton, both deterministic blocks are exact and only the mixing error remains. · split off as results of their own: thm:cor-fix-3c402872 | | inherits refutation from | `lem:push-forward-err` | ### 3. Corollary of thm:wide_nn (split off by its repair) `thm:cor-fix-3c402872` | field | value | | --- | --- | | bucket | `conditional` | | source | `Approximation.tex:462-474` (`theorem`, label —) | | statement | Given any probability measure $\mathbb P$ on $\mathbb R^d$ that satisfies \cref{assump:support}, let $\lambda$ be the Lebesgue measure on $[0,1]^{N_C}$, where $N_C$ is defined in \cref{assump:support}, and write $D_C=\max\{1,\max_i d_i^C\}$. Suppose in addition that $\mathbb P$ constrained to each component satisfies \cref{assumption:lower_bound}. There exists a neural network $\hat g\in\mathcal{NN}(\Theta(D_C),L_1,\Theta(d\cdot D_C),L_2,\tau)$ with the architecture of \cref{fig:wide_nn} such that \[ W(\hat g_{\#}\lambda,\mathbb P)\leqslant O\left(\mathbb I_{\{\max_i d_i^C>0\}}\left(L_1^{-2/D_C}+L_2^{-2/D_C}\right)+(\tau-\tau\log\tau)\right). \] | | Lean | `TCorFix3c402872.lean:21` — `Papers.TbsNeuralCausalPi.deep_generator_approximation` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:bk-1`, `ass:pin-192`, `ass:pin-203`, `ass:pin-209`, `ass:pin-211`, `ass:pin-33`, `ass:pin-75`, `ass:pin-fix-00130002`, `ass:pin-fix-0bcb2cef`, `ass:pin-fix-2a1e7b01`, `ass:pin-fix-2fe04b0e`, `ass:pin-fix-507a5848`, `ass:pin-fix-5f02b8a6`, `ass:pin-fix-72c9e86d`, `ass:pin-fix-7a79e01d`, `ass:pin-fix-8cc0dba5`, `ass:pin-fix-99ac9bb3`, `ass:pin-fix-9bbf998f`, `ass:pin-fix-b83fa4d1`, `ass:pin-fix-c514425b`, `ass:pin-fix-c778b424`, `ass:pin-fix-c85b9186`, `ass:pin-fix-ca667e13`, `ass:pin-fix-db6ed772`, `ass:pin-fix-e896f793`, `ass:pin-fix-f6f485b1` | | cited results | `lem:wide-nn-perekrestenko-theorem-viii-1` (perekrestenkoHighDimensionalDistributionGeneration2022a, Theorem VIII.1); `lem:wide-nn-yarotsky-theorem-2` (Yarotsky2018OptimalAO, Theorem 2) | | assumed substrate gates | — | | findings | — | ### 4. corollary:deep_nn `thm:deep-nn` | field | value | | --- | --- | | bucket | `repaired` | | source | `Approximation.tex:484-490` (`corollary`, label `corollary:deep_nn`) | | statement | \label{corollary:deep_nn} Given a causal model $ \mathcal{M}^* \in \mathcal{M}(\mathcal{G} ,\mathcal{F}_L ,\boldsymbol{U})$, suppose that Assumptions \ref{assump:independence_u}-\ref{assump:Lip_and_bounded} hold and the distributions of $U_C$ for all $C^2$-component satisfy the assumptions of the second conclusion of \cref{thm:wide_nn}. Let $ d_{\max}^{\text{in}} $ and $d_{\max}^{\text{out}}$ be the largest input and output dimension of $f_i$ in (\ref{eq:structure_equation}) and $d_{\max}^U$ be the largest dimension of all latent variables. There exists a neural causal model $$\hat{\mathcal{M}} \in \text{NCM}_{\mathcal{G}} ( \mathcal{NN}(\Theta( d^{\text{in}}_{\max}d^{\text{out}}_{\max}) ,L_{0}) , \mathcal{NN}(\Theta(d_{\max}^U),L_1,\Theta((d_{\max}^U)^2),L_2))$$ with structure equations (\ref{eq:structure_equation_hat}). For any intervention $\boldsymbol{T} = \boldsymbol{t}$, $\hat{\mathcal{M}}$ satisfies \begin{equation*} W\left( P^{\mathcal{M}^*}(\boldsymbol{V}(\boldsymbol{t})),P^{\hat{\mathcal{M}}}(\boldsymbol{V}(\boldsymbol{t}))\right) \leqslant O(L_{0}^{-2/d^\text{in}_{\max}} +L_{1}^{-2/d^U_{\max}} +L_{2}^{-2/d^U_{\max}} +(\tau - \tau\log\tau) ). \end{equation*} | | appended hypothesis list | ass:assump-independence-u, ass:assump-categorical-rv-latents, ass:assump-categorical-rv-generation, ass:assump-lip-and-bounded-continuous, ass:assump-lip-and-bounded-categorical, ass:assump-lip-and-bounded-bound, ass:assump-support-components, ass:assump-support-homeomorphisms, ass:assumption-lower-bound-domination, ass:assumption-lower-bound-boundary, ass:corollary-deep-nn-latent-laws | | Lean | `TDeepNn.lean:303` — `Papers.TbsNeuralCausalPi.deep_ncm_approximation` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:bk-1`, `ass:pin-177`, `ass:pin-181`, `ass:pin-182`, `ass:pin-183`, `ass:pin-186`, `ass:pin-189`, `ass:pin-190`, `ass:pin-191`, `ass:pin-192`, `ass:pin-194`, `ass:pin-199`, `ass:pin-203`, `ass:pin-208`, `ass:pin-209`, `ass:pin-211`, `ass:pin-33`, `ass:pin-75`, `ass:pin-fix-00130002`, `ass:pin-fix-0bcb2cef`, `ass:pin-fix-2a1e7b01`, `ass:pin-fix-2fe04b0e`, `ass:pin-fix-419eb8e3`, `ass:pin-fix-43f1bb71`, `ass:pin-fix-4a86c2eb`, `ass:pin-fix-507a5848`, `ass:pin-fix-5f02b8a6`, `ass:pin-fix-72c9e86d`, `ass:pin-fix-73abef8c`, `ass:pin-fix-7a79e01d`, `ass:pin-fix-8429b12c`, `ass:pin-fix-8cc0dba5`, `ass:pin-fix-945d85ed`, `ass:pin-fix-99ac9bb3`, `ass:pin-fix-9bbf998f`, `ass:pin-fix-ab17be7a`, `ass:pin-fix-b83fa4d1`, `ass:pin-fix-bbf1c78a`, `ass:pin-fix-c514425b`, `ass:pin-fix-c778b424`, `ass:pin-fix-c85b9186`, `ass:pin-fix-ca667e13`, `ass:pin-fix-db6ed772`, `ass:pin-fix-e896f793`, `ass:pin-fix-f6f485b1` | | cited results | `lem:deep-nn-yarotsky-theorem-2` (Yarotsky2018OptimalAO, Theorem 2); `lem:wide-nn-perekrestenko-theorem-viii-1` (perekrestenkoHighDimensionalDistributionGeneration2022a, Theorem VIII.1); `lem:wide-nn-yarotsky-theorem-2` (Yarotsky2018OptimalAO, Theorem 2) | | assumed substrate gates | — | | findings | [refuted, V1.5] [F2.5 reviewer] [claim-false] Cube-only minimization does not imply the claimed causal-law approximation for arbitrary selected minimizers when latent inputs lie outside the cube. witness: [claim-false] Cube-only minimization does not imply the claimed causal-law approximation for arbitrary selected minimizers when latent inputs lie outside the cube. minimal patch (confirmed): corrected statement: Given a causal model $\mathcal M^*\in\mathcal M(\mathcal G,\mathcal F_L,\boldsymbol U)$, suppose that Assumptions \ref{assump:independence_u}-\ref{assump:Lip_and_bounded} hold and the distributions of $U_C$ for all $C^2$-component satisfy the assumptions of the proposition entitled ``Deep distribution approximation''. Let $d_{\max}^{\text{in}}$ and $d_{\max}^{\text{out}}$ be the largest input and output dimension of $f_i$ in (\ref{eq:structure_equation}) and $d_{\max}^U$ be the largest dimension of all latent variables. Suppose in addition that $d_{\max}^{\text{in}}>0$ and $d_{\max}^U>0$. Fix any $R>0$. There exists a neural causal model $$\hat{\mathcal{M}} \in \text{NCM}_{\mathcal{G}} ( \mathcal{NN}(\Theta( d^{\text{in}}_{\max}d^{\text{out}}_{\max}) ,L_{0}) , \mathcal{NN}(\Theta(d_{\max}^U),L_1,\Theta((d_{\max}^U)^2),L_2))$$ with structure equations (\ref{eq:structure_equation_hat}). For any intervention $\boldsymbol{T}=\boldsymbol{t}$ such that $\\|t_i\\|_\infty\leqslant R$ for each continuous treatment variable $T_i$, $\hat{\mathcal M}$ satisfies \begin{equation*} W\left(P^{\mathcal M^*}(\boldsymbol V(\boldsymbol t)),P^{\hat{\mathcal M}}(\boldsymbol V(\boldsymbol t))\right)\leqslant O\left(L_0^{-2/d_{\max}^{\text{in}}}+L_1^{-2/d_{\max}^U}+L_2^{-2/d_{\max}^U}+(\tau-\tau\log\tau)\right). \end{equation*} The implicit constant is independent of the intervention and the network budgets, but may depend on $R$ and the fixed causal model. · proof change: Suppose that structure equations of $ \mathcal{M}$ are \begin{equation*} V_{i} =\begin{cases} f_{i}\left(\text{Pa} (V_{i} ),\boldsymbol{U}_{V_{i}}\right) , & V_{i} \ \text{is continuous} ,\\ \arg\max_{k\in [ n_{i}]}\left\{g_{k}^{V_{i}} +\log\left( f_{i}\left(\text{Pa}( V_{i}) ,\boldsymbol{U}_{V_{i}}\right)\right)_{k}\right\} , & V_{i} \ \text{is categorical} , \\|f_i\\|_1= 1 \end{cases}, \quad f_{i} \in \mathcal{F}_L. \end{equation*} Put $m_i=d_i^{\text{in}}$, let $d_i^{\text{out}}$ be the output dimension of $f_i$, and put $m=d_{\max}^{\text{in}}>0$. In each structural input, distinguish the real coordinates, consisting of continuous parents and non-Gumbel latent coordinates, from the categorical parent coordinates. Let $a_i$ have value zero in every real coordinate and value one in every categorical coordinate. Define the fixed constants \[ B_0=\max\left(\{1,R,K\}\cup\{n_i:V_i\text{ is categorical}\}\cup\{\\|f_i(a_i)\\|_\infty:V_i\text{ is continuous}\}\right),\qquad H=1+2Lm,\qquad B=B_0H^{n_V}. \] These constants do not depend on the intervention or on the network budgets. We first bound all true structural inputs for the interventions in the statement. Work on the probability-one event where every non-Gumbel latent coordinate has absolute value at most $K$. Order the observed nodes topologically. We claim that the node in position $\ell$ has infinity norm at most $B_0H^\ell$. An intervened continuous node has norm at most $R\leqslant B_0$, and a categorical node has value at most its cardinality, which is at most $B_0$. For a nonintervened continuous node in position $\ell$, the induction hypothesis bounds all its parents and latent coordinates by $b=B_0H^{\ell-1}$. Every coordinate of $a_i$ has absolute value at most one, so its actual input $x_i$ satisfies $\\|x_i-a_i\\|\leqslant2m_i b\leqslant2mb$. Consequently, \[ \\|f_i(x_i)\\|_\infty\leqslant\\|f_i(a_i)\\|_\infty+L\\|x_i-a_i\\|\leqslant B_0+2Lmb\leqslant Hb=B_0H^\ell. \] This proves the induction, including its first position. In particular, every true structural input belongs to $Q_i=[-B,B]^{m_i}$, simultaneously for all permitted interventions. For $m_i=0$, interpret $Q_i$ as the unique point of $\mathbb R^0$. We next define a Lipschitz extension to real values of the categorical input coordinates. A categorical input coordinate with only one category is always replaced by one. For each remaining categorical input coordinate $r$, with category count $c_r\geqslant2$, clip its value $x_r$ to $t_r\in[1,c_r]$, and set \[ k_r=\min\{c_r-1,\lfloor t_r\rfloor\},\qquad \theta_r=t_r-k_r\in[0,1]. \] Let $s_i$ be the number of these remaining categorical coordinates. For $\beta\in\{0,1\}^{s_i}$, let $x^\beta$ retain the real coordinates of $x$, replace coordinate $r$ by $k_r+\beta_r$, and replace every one-category coordinate by one. With $w_{r,0}=1-\theta_r$ and $w_{r,1}=\theta_r$, define \[ f_i^\dagger(x)=\sum_{\beta\in\{0,1\}^{s_i}}\left(\prod_{r=1}^{s_i}w_{r,\beta_r}\right)f_i(x^\beta). \] When $s_i=0$, the sum has one term with weight one. Thus this definition includes functions with no categorical parents. At valid categorical labels it agrees with $f_i$. This interpolation preserves the $L$-Lipschitz bound in the paper's $1$-norm. Changing one real coordinate leaves the interpolation weights fixed, so the difference is a convex combination of differences bounded in $1$-norm by $L$ times the coordinate change. With all other coordinates fixed, on each interval between consecutive labels a categorical coordinate gives a linear interpolation between two vectors whose $1$-norm difference is at most $L$. Its slope therefore has $1$-norm at most $L$; outside the label interval the function is constant in that coordinate. Splitting any coordinate change at its finitely many intervening labels proves the same bound for an arbitrary change of that coordinate. Changing all coordinates one at a time and adding the bounds gives \[ \\|f_i^\dagger(x)-f_i^\dagger(y)\\|\leqslant L\\|x-y\\|. \] For a categorical output, the interpolation weights are nonnegative and sum to one, so $f_i^\dagger(x)\in\Delta_{n_i}$ for every real input $x$. Let $\Pi_i:\mathbb R^{m_i}\to Q_i$ be coordinatewise projection, explicitly \[ (\Pi_i x)_r=-B+\max\{x_r+B,0\}-\max\{x_r-B,0\}, \] and define $\bar f_i=f_i^\dagger\circ\Pi_i$. Clipping each coordinate cannot increase its distance from another clipped coordinate; summing proves $\\|\Pi_i x-\Pi_i y\\|\leqslant\\|x-y\\|$. Hence $\bar f_i$ is globally $L$-Lipschitz. Since $a_i\in Q_i$, for every $z\in Q_i$ and every continuous output, \[ \\|f_i^\dagger(z)\\|_\infty\leqslant\\|f_i(a_i)\\|_\infty+L\\|z-a_i\\|\leqslant B_0+2LmB\leqslant HB. \] Thus $\bar f_i$ is globally bounded by $HB$ for continuous outputs; categorical outputs remain probability vectors. Use the same latent laws and Gumbel noises to form the model with structural functions $\bar f_i$. Couple it with the original model using identical draws. Intervened nodes agree by their assigned values. If the parents of a nonintervened node agree, their common original input belongs to $Q_i$ and has valid categorical coordinates. Projection fixes this input and interpolation agrees there with $f_i$, so the structural outputs agree; for a categorical node use the identical Gumbel draws and the convention $\log0=-\infty$. Induction in topological order proves equality of the two interventional laws for every intervention in the statement. The extended true model satisfies the boundedness assumption with the fixed enlarged constant $HB$, because its nonintervened continuous outputs are globally bounded by $HB$, its categorical labels are at most $B_0$, its assigned treatment coordinates are at most $B_0$, and its non-Gumbel latent coordinates are at most $B_0$. We now choose the neural approximants in place of the exact-minimizer construction. Reserve $c=4$ hidden layers of the structural budget for projection, categorical floors, and padding. For each $m_i>0$, apply \cite[Theorem 2]{Yarotsky2018OptimalAO}, part b), to every coordinate of $f_i^\dagger$ on $Q_i$, after the affine rescaling $z\mapsto2Bz-B\boldsymbol 1$ from $[0,1]^{m_i}$. The rescaled coordinate functions have Lipschitz moduli bounded by a fixed constant times their argument. The source gives scalar width $2m_i+10$ and error at most a fixed constant times the modulus at a constant times the number of weights raised to $-2/m_i$. At fixed width that number is proportional to hidden depth. Parallelizing the $d_i^{\text{out}}$ coordinates therefore gives, for every sufficiently large $L_0$, a network $q_i$ of width at most $d_i^{\text{out}}(2m_i+10)$ and hidden depth at most $L_0-c$ such that \[ \sup_{z\in Q_i}\\|q_i(z)-f_i^\dagger(z)\\|_\infty\leqslant C_i(L_0-c)^{-2/m_i}. \] The constant $C_i$ is fixed independently of $L_0$. For $L_0\geqslant2c$, choose a fixed $A_i\geqslant\max\{1,2^{2/m_i}C_i\}$ and set \[ \epsilon_i=A_iL_0^{-2/m_i}. \] Then choose one of the networks just supplied, so that its uniform error on $Q_i$ is at most $\epsilon_i$. Define the structural networks by \[ \hat f_i(x)= \begin{cases} q_i(\Pi_i x),&V_i\text{ is continuous},\\ \bigl(\max\{\epsilon_i,(q_i(\Pi_i x))_k\}\bigr)_{k=1}^{n_i},&V_i\text{ is categorical}. \end{cases} \] For $m_i=0$, instead set $\epsilon_i=0$ and let $\hat f_i=f_i$ be the exact constant affine network. These definitions specify $\hat f_i$ without any choice of a minimizer. Projection uses one hidden layer with at most $2m_i$ units, and each categorical floor is $z\mapsto\epsilon_i+\max\{z-\epsilon_i,0\}$, using one further hidden layer. Affine maps between these layers and the network $q_i$ are absorbed into the adjacent affine transformations. If signed coordinates must be carried through a hidden layer, use $z=\max\{z,0\}-\max\{-z,0\}$ with two units per coordinate. Thus the reserved four layers suffice, with width at most $4m_i+2d_i^{\text{out}}(2m_i+10)+2d_i^{\text{out}}$. For $m_i\geqslant1$ and $d_i^{\text{out}}\geqslant1$, this is at most $30m_i d_i^{\text{out}}\leqslant64d_{\max}^{\text{in}}d_{\max}^{\text{out}}$, because $2m_i+10\leqslant12m_i$. A constant map with $m_i=0$ uses at most $2d_i^{\text{out}}$ units per hidden layer by carrying its positive and negative parts, which also fits this universal width since $d_{\max}^{\text{in}}\geqslant1$. A zero-dimensional output needs no units. Zero padding of units and insertion of identity layers on nonnegative hidden outputs give the structural architecture and hidden depth $L_0$ in the statement. The continuous global approximation error is at most $\epsilon_i$, because $\Pi_i x\in Q_i$ for every input $x$. For categorical outputs, if $p\geqslant0$ and $\epsilon\geqslant0$, then \[ \|\max\{\epsilon,z\}-p\|\leqslant\max\{\epsilon,\|z-p\|\}. \] Indeed, for $z\geqslant\epsilon$ the left side is $\|z-p\|$; for $z<\epsilon$ and $p\leqslant\epsilon$ it is at most $\epsilon$; and for $z<\epsilon0$, every categorical coordinate is at least $\epsilon_i>0$. For $m_i=0$, the exact constant vector is a probability vector. In both cases its $1$-norm is strictly positive. The global error bound holds also when generated latent or parent inputs lie outside $Q_i$. For a categorical variable, at an arbitrary input $x$, put $p=\bar f_i(x)$, $q=\hat f_i(x)$, $a=\\|q\\|_1$, and $S=\\|q-p\\|_\infty$. The construction gives $a>0$ and $S\leqslant\epsilon_i$. Since $p$ is a probability vector and $q$ is nonnegative, \[ \|a-1\|\leqslant\sum_{j=1}^{n_i}\|q_j-p_j\|\leqslant n_iS. \] For sufficiently large $L_0$, simultaneously for all nodes with $m_i>0$, we have $n_i\epsilon_i\leqslant1/2$, and hence $a\geqslant1/2$. Since $\\|p\\|_\infty\leqslant1$, \begin{align*} \left\\|\frac q a-p\right\\|_\infty &\leqslant\frac{\\|q-p\\|_\infty+\|1-a\|\\|p\\|_\infty}{a}\\ &\leqslant2(n_i+1)S\leqslant2(n_i+1)\epsilon_i. \end{align*} For $m_i=0$, normalization is exact because $q=p$ and $a=1$. For $m_i>0$, the inequalities $m_i\leqslant d_{\max}^{\text{in}}$ and $L_0\geqslant1$ imply $L_0^{-2/m_i}\leqslant L_0^{-2/d_{\max}^{\text{in}}}$. Taking the supremum over every real structural input therefore gives \[ \left\\|\bar f_i-\frac{\hat f_i}{\\|\hat f_i\\|_1}\right\\|_\infty\leqslant O(L_0^{-2/d_{\max}^{\text{in}}}). \] The continuous structural error has the same order without normalization. These bounds use the fixed cardinalities $n_i$ and do not require $n_i\leqslant K$. For each non-Gumbel latent variable $U_j$ with positive dimension $d_j^U$, apply the proposition entitled ``Deep distribution approximation'' to its law. Choose the resulting generators with independent uniform and Gumbel inputs $Z_j$ and put $\hat U_j=\hat g_j(Z_j)$. Each component cube dimension is at most the ambient dimension $d_j^U$. For budgets at least one, the component bounds therefore give \[ W(P(U_j),P(\hat U_j))\leqslant O\left(L_1^{-2/d_j^U}+L_2^{-2/d_j^U}+(\tau-\tau\log\tau)\right). \] For a component of positive cube dimension $q\leqslant d_j^U$, the scalar first-block networks have width $12$, and the scalar second-block networks have width $2q+10\leqslant12q$. Parallelization gives widths $12q$ and $12d_j^Uq$. Coordinate clipping needs at most $2q$ or $2d_j^U$ units, and carrying signed values requires at most twice the corresponding width; reserving four layers therefore gives widths at most $64q$ and $64d_j^Uq$. A singleton component uses its exact constant blocks; carrying a signed constant second-block output needs at most $2d_j^U$ units. Since $d_{\max}^U\geqslant1$, every component fits widths $64d_{\max}^U$ and $64(d_{\max}^U)^2$. These are the per-component widths in the separable architecture, so the number of components does not multiply either width budget. Zero padding and identity layers on nonnegative hidden outputs give the architecture in the statement. The numerical factor $64$ is independent of the model, $R$, dimensions, and cardinalities; only the error constants and minimum depth budgets may depend on the fixed model. A latent vector of dimension zero is the unique point of $\mathbb R^0$ and is represented exactly, with error zero. Independence of the original latent variables and independence of the chosen generator inputs allow application of \cref{lemma:product_err}, yielding \begin{align*} W(P(U_1,\ldots,U_{n_U}),P(\hat U_1,\ldots,\hat U_{n_U})) &\leqslant\sum_{j=1}^{n_U}W(P(U_j),P(\hat U_j))\\ &\leqslant O\left(L_1^{-2/d_{\max}^U}+L_2^{-2/d_{\max}^U}+(\tau-\tau\log\tau)\right). \end{align*} Indeed, for $d_j^U>0$, the inequalities $d_j^U\leqslant d_{\max}^U$ and $L_r\geqslant1$ give $L_r^{-2/d_j^U}\leqslant L_r^{-2/d_{\max}^U}$ for $r=1,2$, and there are only finitely many latent variables. Identical categorical Gumbel noises may be adjoined to both latent vectors without changing this bound: couple these noises identically for one inequality and project onto the non-Gumbel coordinates for the reverse inequality. Let $ \hat{\mathcal{M}}$ be the casual model with the following structure equations \begin{equation*} \hat{V}_{i} =\begin{cases} \hat{f}_{i}\left(\text{Pa} (\hat{V}_{i} ),(\hat{g}_{j} (Z_{C_{j}} ))_{U_{C_{j} }\in \boldsymbol{U}_{V_{i}}}\right), & V_{i} \text{ is continuous} ,\\ \arg\max_{k\in [ n_{i}]}\left\{g_{k} +\log\left(\hat{f}_{i}\left(\text{Pa} (\hat{V}_{i} ),(\hat{g}_{j} (Z_{C_{j}} ))_{U_{C_{j} } \in \boldsymbol{U}_{V_{i}}}\right) \right)_{k}\right\} , & V_{i} \text{ is categorical} . \end{cases} \end{equation*} Replacing a categorical propensity vector $\hat f_i(x)$ by $\hat f_i(x)/\\|\hat f_i(x)\\|_1$ subtracts the same finite constant $\log\\|\hat f_i(x)\\|_1$ from every finite log propensity. It therefore preserves the categorical structural equation, with $\log0=-\infty$ for zero coordinates. Apply \cref{thm:approximation} to the globally $L$-Lipschitz true functions $\bar f_i$, the continuous approximate functions $\hat f_i$, and the normalized approximate categorical functions. The extended true model has boundedness constant $HB$ and, by the coupling induction in this proof, the same interventional laws as $\mathcal M^*$ for every intervention in the statement. Consequently, \begin{align*} &W\left(P^{\mathcal M^*}(\boldsymbol V(\boldsymbol t)),P^{\hat{\mathcal M}}(\boldsymbol V(\boldsymbol t))\right)\\ &\quad\leqslant C\left(\sum_{V_i\ \text{continuous}}\\|\bar f_i-\hat f_i\\|_\infty +\sum_{V_i\ \text{categorical}}\left\\|\bar f_i-\frac{\hat f_i}{\\|\hat f_i\\|_1}\right\\|_\infty +W(P(U_1,\ldots,U_{n_U}),P(\hat U_1,\ldots,\hat U_{n_U}))\right)\\ &\quad\leqslant O\left(L_0^{-2/d_{\max}^{\text{in}}}+L_1^{-2/d_{\max}^U}+L_2^{-2/d_{\max}^U}+(\tau-\tau\log\tau)\right). \end{align*} For the second inequality, every term of the first sum has the continuous structural error bound proved here, every term of the second sum has the normalized categorical error bound proved here, and the latent term has the bound obtained using \cref{lemma:product_err}; the two sums contain only finitely many nodes. The comparison constant $C$ depends only on the fixed model, its dimensions and cardinalities, and $R$. The cubes, extensions, and networks were chosen independently of the intervention within the prescribed region. This proves the estimate uniformly throughout that region. · contribution narrowed (accepted by the owner): A single neural model with the displayed error bound uniformly over unbounded continuous treatment values, and the displayed-rate statement when $d_{\max}^{\text{in}}=0$ or $d_{\max}^U=0$. | | inherits refutation from | `lem:hilbert`, `lem:push-forward-err`, `prop:holder-curve`, `thm:approximation`, `thm:wide-nn` | ### 5. Informal, see Proposition~\ref{prop:counterexample_foral} for a formal version `prop:counterexample` | field | value | | --- | --- | | bucket | `verified_with_bookkeeping` | | source | `Main.tex:282-284` (`proposition`, label `prop:counterexample`) | | statement | \label{prop:counterexample} There exists a constant $c>0$ and an identifiable SCM $\mathcal{M}^*$ satisfying Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound} such that for any $\epsilon >0$, there exists an SCM $\mathcal{M}_\epsilon$ satisfying $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V})) \leq \epsilon $ and $\|\text{ATE}_{\mathcal{M}^*} - \text{ATE}_{\mathcal{M}_\epsilon}\| > c $. | | Lean | `TCounterexample.lean:53` — `Papers.TbsNeuralCausalPi.counterexample_identifiable` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-177`, `ass:pin-181`, `ass:pin-182`, `ass:pin-183`, `ass:pin-189`, `ass:pin-190`, `ass:pin-204` | | cited results | — | | assumed substrate gates | — | | findings | — | ### 6. `thm:consistency` | field | value | | --- | --- | | bucket | `repaired` | | source | `Main.tex:295-302` (`theorem`, label `thm:consistency`) | | statement | \label{thm:consistency} Let $ \mathcal{M^*}$ be any SCM satisfying the assumptions of Corollary \ref{corollary:deep_nn}. Suppose that the Lipschitz constant of functions in $\mathcal{M}$ is $ L_f $, $ F:\mathbb{R}^{n_{V}}\rightarrow \mathbb{R}$ in (\ref{eq:opt_empiri}) is Lipschitz continuous and $\tau_n > 0 $, let $K>0$ be the constant in \cref{assump:Lip_and_bounded}, $\hat{L}_f = \sqrt{d_{\max}^{\text{in}}d_{\max}^{\text{out}}}L_f$, \begin{align*} \mathcal{F}_{0,n} &= \mathcal{NN}^{\hat{L}_{f} ,K}\left( W_{0,n} , \Theta\left(\log d^{\text{in}}_{\max}\right)\right) , \mathcal{F}_{1,n} =\mathcal{NN}^{\infty ,K}( \Theta( d^{U}_{\max}) ,L_{1,n}, \Theta( (d^{U}_{\max})^2) ,L_{2,n},\tau_n), \end{align*} where $ d_{\max}^U, d_{\max}^{\text{in}} $ and $d_{\max}^{\text{out}}$ are defined in Corollary \ref{corollary:deep_nn}, take the radius to be $\alpha_n = \epsilon_n+s_n, \epsilon_n = O(W_{0,n}^{-1/d^{\text{in}}_{\max}}+ \sum_{i=1}^2 L_{i,n}^{-2/d^U_{\max}}+ \tau _{n} \log \tau _{n}),$ $$ s_n = O(m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n} +n^{-1/{\max\{2,d_{\max}^U\}}}\log^2(n) + \log(nm_n)\lambda_n).$$ If $ m_n = \Omega(n), L_{i,n} = \Theta(m_{n}^{d_{\max}^{U} /\left( 2d_{\max}^{U} +4\right)}), i=1,2, \lim _{n\rightarrow \infty }\min\{\tau_n^{-1}, W_{0,n},(\log(nm_n)\lambda_n)^{-1}\}= \infty$ , then with probability 1, $ [\liminf _{n\rightarrow \infty } F_{n},\limsup _{n\rightarrow \infty } F_{n}] \subset [\underline{F}^{\hat{L}_f} ,F_{*}]$. | | appended hypothesis list | ass:assump-independence-u, ass:assump-categorical-rv-latents, ass:assump-categorical-rv-generation, ass:assump-lip-and-bounded-continuous, ass:assump-lip-and-bounded-categorical, ass:assump-lip-and-bounded-bound, ass:assump-support-components, ass:assump-support-homeomorphisms, ass:assumption-lower-bound-domination, ass:assumption-lower-bound-boundary, ass:corollary-deep-nn-latent-laws, ass:thm-consistency-inherited, ass:thm-consistency-structural-lipschitz, ass:thm-consistency-objective-lipschitz, ass:thm-consistency-positive-temperature, ass:thm-consistency-growth | | Lean | `TConsistency.lean:189` — `Papers.TbsNeuralCausalPi.consistency` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:bk-1`, `ass:pin-177`, `ass:pin-181`, `ass:pin-182`, `ass:pin-183`, `ass:pin-186`, `ass:pin-187`, `ass:pin-189`, `ass:pin-190`, `ass:pin-191`, `ass:pin-192`, `ass:pin-194`, `ass:pin-196`, `ass:pin-197`, `ass:pin-199`, `ass:pin-201`, `ass:pin-203`, `ass:pin-207`, `ass:pin-208`, `ass:pin-209`, `ass:pin-210`, `ass:pin-211`, `ass:pin-33`, `ass:pin-75`, `ass:pin-fix-00130002`, `ass:pin-fix-09991de8`, `ass:pin-fix-0a7b21b0`, `ass:pin-fix-0bcb2cef`, `ass:pin-fix-2a1e7b01`, `ass:pin-fix-2fe04b0e`, `ass:pin-fix-30413b86`, `ass:pin-fix-3b195d8d`, `ass:pin-fix-419eb8e3`, `ass:pin-fix-43f1bb71`, `ass:pin-fix-448b19a0`, `ass:pin-fix-4a86c2eb`, `ass:pin-fix-507a5848`, `ass:pin-fix-59a01041`, `ass:pin-fix-5f02b8a6`, `ass:pin-fix-676b96c3`, `ass:pin-fix-72c9e86d`, `ass:pin-fix-73abef8c`, `ass:pin-fix-761562d5`, `ass:pin-fix-7a79e01d`, `ass:pin-fix-8429b12c`, `ass:pin-fix-84ccd033`, `ass:pin-fix-8cc0dba5`, `ass:pin-fix-945d85ed`, `ass:pin-fix-9463b114`, `ass:pin-fix-9675bba5`, `ass:pin-fix-99ac9bb3`, `ass:pin-fix-9bbf998f`, `ass:pin-fix-9d74053b`, `ass:pin-fix-ab17be7a`, `ass:pin-fix-b83fa4d1`, `ass:pin-fix-bbf1c78a`, `ass:pin-fix-bd620122`, `ass:pin-fix-c514425b`, `ass:pin-fix-c778b424`, `ass:pin-fix-c85b9186`, `ass:pin-fix-ca667e13`, `ass:pin-fix-d19c63b5`, `ass:pin-fix-db6ed772`, `ass:pin-fix-e896f793`, `ass:pin-fix-f66ad243`, `ass:pin-fix-f6f485b1` | | cited results | `lem:consistency-fournier-theorem-2` (fournier2015rate, Theorem 2); `lem:deep-nn-yarotsky-theorem-2` (Yarotsky2018OptimalAO, Theorem 2); `lem:sup-wasserstein-anthony-theorem-17-1` (anthony1999neural, Theorem 17.1); `lem:sup-wasserstein-anthony-theorems` (anthony1999neural, Theorem 18.4); `lem:wide-nn-perekrestenko-theorem-viii-1` (perekrestenkoHighDimensionalDistributionGeneration2022a, Theorem VIII.1); `lem:wide-nn-yarotsky-theorem-2` (Yarotsky2018OptimalAO, Theorem 2) | | assumed substrate gates | — | | findings | [refuted, V1.5] (a) The statement and its compactness argument fail for categorical nodes. The restriction $\operatorname{Lip}(f)\leqslant\hat L_f$ on a raw network does not imply that $f/\\|f\\|_1$ has that restriction. The printed categorical parameter class therefore contains arbitrarily steep propensity functions, is not equicontinuous, and need not have compact closure in the printed metric. Its feasible limits cannot be compared with $\underline F^{\hat L_f}$ as claimed; the witness gives an empirical minimum strictly below that bound. (b) Independent resampling for each candidate and evaluation on common exogenous draws are different readings of the empirical optimization. The uniform push-forward argument determines the common-draw reading. All inherited independence, categorical-generation, support and lower-bound assumptions flagged by lint are supplied through the corollary's assumptions. The corollary reference supplies assumptions and dimensions, rather than requiring its conclusion as an additional dependency. The optimization proposition's premises are conditions the proof attempts to verify, rather than additional standing hypotheses. The finite-measure notation is explicit in the Sinkhorn lemma used here; mixture weights are component probabilities; and the empirical lemma's boundedness and pseudo-dimension conditions are verified in the cited argument. Those remaining lint observations do not establish transcription drift. witness: (a) Take the graph $T\rightarrow Y$ with no bidirected edge, independent $U_T,U_Y\sim\operatorname{Unif}([-1,1])$, and independent categorical Gumbel noises. Set $T=U_T$ and give $Y\in\{1,2\}$ the normalized propensity $f_Y(t,u)=(1-r(t),r(t))$, where $r(t)=1/2+\|t\|/8$ on the input cube. Choose $K=2$ and $L_f=1$. Then $d_{\max}^{\text{in}}=d_{\max}^{\text{out}}=2$, $d_{\max}^U=1$, and $\hat L_f=2$. Both non-Gumbel latent laws satisfy the required support, domination and boundary assumptions. Let $F(T,Y)=Y$ and $\mu_T=\delta_0$. Every observationally equivalent canonical SCM on this graph with propensity Lipschitz bound $2$ has a continuous conditional category probability agreeing with $r(t)$ almost everywhere on $[-1,1]$. Consequently it agrees at zero, so $\underline F^2=F_*=3/2$. For $0<\delta\leqslant1$, consider the raw categorical network \[ q_\delta(t,u)=\left(\delta\left(\frac12-\frac{\|t\|}{8}\right),\frac{\delta\|t\|}{8}+\frac12\min\{\|t\|,\delta\}\right). \] This is a fixed-width, fixed-depth ReLU function. It is nonnegative on $[-2,2]^2$, bounded by $K$, and has Lipschitz constant at most $1/2+\delta/4<2$. Its coordinate sum is $\delta/2+\min\{\|t\|,\delta\}/2>0$. Thus it belongs to the printed structural class for sufficiently large network widths. Its normalized propensity equals $f_Y$ whenever $\|t\|\geqslant\delta$, but at zero it assigns category $1$ probability one. Moreover, \[ \left\\|\frac{q_\delta(\delta,u)}{\\|q_\delta(\delta,u)\\|_1}-\frac{q_\delta(0,u)}{\\|q_\delta(0,u)\\|_1}\right\\|_1=1+\frac\delta4, \] so the normalized Lipschitz constants diverge. Their pointwise limit is discontinuous at zero, demonstrating the failure of the asserted compactness. Use exact affine generators for the two uniform latent laws. Take $m_n=n$, $W_{0,n}=n$, $L_{i,n}=\lceil n^{1/6}\rceil$, and $\tau_n=\lambda_n=n^{-2}$, with positive approximation tolerances and sufficiently large sampling-radius constants of the printed orders. These sequences satisfy the theorem's growth conditions. The exact true network is eventually feasible almost surely by the supplied empirical error bound. For each finite common simulation sample, choose $\delta_n>0$ smaller than every simulated $\|T\|$. Almost surely this is possible. The model using $q_{\delta_n}$ then produces exactly the true network's simulated observations, including categorical outputs under the same Gumbel draws, and is equally feasible. Its objective under treatment zero is $1$. Since every admissible categorical outcome is at least $1$, the empirical minimum is attained at $F_n=1$ eventually. Hence $\liminf F_n=\limsup F_n=1<\underline F^2=3/2$, contradicting the printed conclusion. minimal patch (confirmed): corrected statement: Let $ \mathcal{M^*}$ be any SCM satisfying the assumptions of Corollary \ref{corollary:deep_nn}. Suppose that the Lipschitz constant of functions in $\mathcal{M}$ is $L_f$, $F:\mathbb{R}^{n_V}\rightarrow\mathbb{R}$ in (\ref{eq:opt_empiri}) is Lipschitz continuous and $\tau_n>0$, let $K>0$ be the constant in \cref{assump:Lip_and_bounded}, $\hat L_f=\sqrt{d_{\max}^{\text{in}}d_{\max}^{\text{out}}}L_f$, \begin{align*} \mathcal F_{0,n}&=\mathcal{NN}^{\hat L_f,K}\left(W_{0,n},\Theta\left(\log d_{\max}^{\text{in}}\right)\right),\qquad \mathcal F_{1,n}=\mathcal{NN}^{\infty,K}\left(\Theta(d_{\max}^U),L_{1,n},\Theta((d_{\max}^U)^2),L_{2,n},\tau_n\right), \end{align*} where $d_{\max}^U$, $d_{\max}^{\text{in}}$ and $d_{\max}^{\text{out}}$ are defined in Corollary \ref{corollary:deep_nn}. Suppose in addition that $d_{\max}^{\text{in}}\geqslant1$ and $d_{\max}^U\geqslant1$. In (\ref{eq:opt_empiri}), restrict each categorical raw structural network $q\in\mathcal F_{0,n}$ to satisfy $q(x)\in[0,\infty)^{n_i}$, $\\|q(x)\\|_1>0$ for every structural input $x$, and \[ \operatorname{Lip}\left(x\mapsto\frac{q(x)}{\\|q(x)\\|_1}\right)\leqslant\hat L_f. \] Put \[ D_U=\sum_{j=1}^{n_U}d_j^U,\qquad D=\max\{1,D_U\}, \] where the sum defining $D_U$ includes the non-Gumbel latent variables only. Take the radius to be $\alpha_n=\epsilon_n+s_n$, with positive tolerances chosen as sufficiently large constant multiples of the respective quantities in \[ \epsilon_n=O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+\sum_{i=1}^2L_{i,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right), \] \[ s_n=O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2 n+\log(nm_n)\lambda_n\right). \] For every continuous structural function $f_i$ of $\mathcal M^*$, suppose in addition that $\sup_x\\|f_i(x)\\|_1\leqslant K$ on its full structural input domain. Restrict every continuous structural network $\hat f_i\in\mathcal F_{0,n}$ in (\ref{eq:opt_empiri}) to satisfy this same uniform output $1$-norm bound. Retain the output bound in (\ref{eq:consistent_limiting}), namely \[ \mathcal F_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}. \] Let $F_n$ be the optimal value of the empirical optimization with the objective and Sinkhorn constraint of (\ref{eq:opt_empiri}) and both the continuous and categorical restrictions just specified, and let $\underline F^{\hat L_f}$ be the optimal value of (\ref{eq:consistent_limiting}) with this bounded Lipschitz class. If $ m_n = \Omega(n), L_{i,n} = \Theta(m_{n}^{d_{\max}^{U} /\left( 2d_{\max}^{U} +4\right)}), i=1,2, \lim _{n\rightarrow \infty }\min\{\tau_n^{-1}, W_{0,n},(\log(nm_n)\lambda_n)^{-1}\}= \infty$ , then with probability 1, $ [\liminf _{n\rightarrow \infty } F_{n},\limsup _{n\rightarrow \infty } F_{n}] \subset [\underline{F}^{\hat{L}_f} ,F_{*}]$. · proof change: \label{proof:consistency} We begin by defining a proper metric space. By \cref{assump:Lip_and_bounded}, all random variables are bounded. Suppose that $ \max_{i,j}\{\\| V_{i} \\| _{\infty} ,\\| U_{j} \\| _{\infty}\} \leqslant K$. A canonical causal model $ \mathcal{M} \in \mathcal{M}(\mathcal{G} ,\mathcal{F} ,\boldsymbol{U})$ is decided by \begin{equation*} \theta _{\mathcal{M}} =( f_{1} ,\cdots ,f_{n_{V}} ,{P}({U_{1}}) ,\cdots , {P}({U_{n_{U}}})) , \end{equation*} where $ f_{i}$ are functions in the structure equations (\ref{eq:structure_equation}) and $ U_{j}$ are uniform parts of the latent variables. We denote $ \mathcal{M}^{\theta }$ to be the SCM represented by $ \theta $, the underlying SCM be $ \mathcal{M}^{\theta^*}$ and $P^\theta$ to be the distribution of $ \mathcal{M}^\theta$. We consider the space \[ M=\mathcal F_{V_1}\times\cdots\times\mathcal F_{V_{n_V}}\times\prod_{j=1}^{n_U}\mathcal P([-K,K]^{d_j^U}), \] where \[ \mathcal F_{V_i}=\begin{cases} \mathcal F_{\hat L_f}^K([-K,K]^{d_{i,\text{in}}^V},[-K,K]^{d_{i,\text{out}}^V}),&V_i\text{ continuous},\\ \{p:[-K,K]^{d_{i,\text{in}}^V}\rightarrow\Delta_{n_i}:\operatorname{Lip}(p)\leqslant\hat L_f\},&V_i\text{ categorical}. \end{cases} \] Here $d_{i,\text{in}}^V,d_{i,\text{out}}^V$ are the input and output dimensions of $f_i$, $\mathcal F_{\hat L_f}^K$ has uniform output $1$-norm bound $K$ and common Lipschitz bound $\hat L_f$, and $\mathcal P(A)$ denotes the probability measures on $A$. For $ \theta =\left( f_{1} ,\cdots ,f_{n_{V}} ,{P}({U_{1}}) ,\cdots , {P}({U_{n_{U}}})\right) ,\theta' =\left( f_{1}' ,\cdots ,f_{n_{V}}' ,{P}({U_{1}'}) ,\cdots ,{P}({U_{n_{U}}'})\right)$, we define a metric on $ M$ \begin{equation*} d( \theta ,\theta') =\sum _{k=1}^{n_{V}} \\| f_{k} -f_{k}' \\| _{\infty } +\sum _{k=1}^{n_{U}} W(P(U_{k}),{P}({U_{k}'})) . \end{equation*} We derive the population continuity bound in the fixed dimensions of this theorem. Couple corresponding non-Gumbel latent laws optimally and take the product of these couplings, using their independence. The expected joint latent $1$-norm discrepancy is the sum of their marginal Wasserstein distances. Generate the observed nodes in a topological order. At a continuous node the expected output discrepancy is at most the structural output $1$-norm approximation error plus $\hat L_f$ times the expected input discrepancy. At a categorical node with conditional propensity vectors $p,p'$, a maximal coupling has disagreement probability $\\|p-p'\\|_1/2$; its expected label distance is at most $(n_i-1)\\|p-p'\\|_1/2$. Such a coupling assigns common mass $\min\{p_k,p'_k\}$ to label $k$ and distributes the remaining masses on the disjoint residual supports. It preserves each model's conditional categorical law. If $e$ is the sum of the expected latent and previously generated observed discrepancies, adjoining a continuous node increases it to at most $(1+\hat L_f)e+d_i\\|f_i-f'_i\\|_\infty$; adjoining a categorical node increases it to at most $(1+(n_i-1)\hat L_f/2)e+n_i(n_i-1)\\|f_i-f'_i\\|_\infty/2$. Choose fixed constants $A\geqslant1$ and $B$ bounding these finitely many coefficients. Iterating $e_{k+1}\leqslant Ae_k+Bd(\theta,\theta')$ gives \[ e_k\leqslant A^ke_0+Bd(\theta,\theta')\sum_{r=0}^{k-1}A^r. \] Indeed, the formula holds for $k=0$, and multiplying its right side by $A$ and adding $Bd(\theta,\theta')$ gives the formula for $k+1$. Identical intervention values contribute zero discrepancy and remove their generation steps. Projection to the observed coordinates therefore proves \[ W(P^\theta(\boldsymbol V(\boldsymbol t)),P^{\theta'}(\boldsymbol V(\boldsymbol t)))\leqslant C d(\theta,\theta') \] uniformly over the intervention values, with $C$ depending on the fixed graph, dimensions, cardinalities and regularity bounds. The triangle inequality consequently makes $g(\theta)=W(P^{\theta^*}(\boldsymbol V),P^\theta(\boldsymbol V))$ Lipschitz in $d$; denote a Lipschitz constant by $C_g$. Now, we define $ \Theta _{n}$. Let \begin{equation*} \mathcal{P}_{n} =\left\{{P}({\boldsymbol{U}}):\boldsymbol{U} \ \text{is the latent distribution of} \ \hat{\mathcal{M}} \in \text{NCM}_{\mathcal{G}} (\mathcal{F}_{0,n} ,\mathcal{F}_{1,n})\right\}. \end{equation*} Thus $\mathcal P_n$ consists of the latent population laws obtained by pushing forward the independent uniform and generator Gumbel inputs through the current positive-temperature generator networks. We denote $\Theta_n=\hat{\mathcal F}_{V_1,n}\times\cdots\times\hat{\mathcal F}_{V_{n_V},n}\times\mathcal P_n$, where \[ \hat{\mathcal F}_{V_i,n}=\begin{cases} \{f\in\mathcal F_{0,n}:\sup_x\\|f(x)\\|_1\leqslant K\},&V_i\text{ continuous},\\ \{q/\\|q\\|_1:q\in\mathcal F_{0,n},\ q(x)\in[0,\infty)^{n_i},\ \\|q(x)\\|_1>0\ \forall x,\ \operatorname{Lip}(q/\\|q\\|_1)\leqslant\hat L_f\},&V_i\text{ categorical}. \end{cases} \] The norms and Lipschitz bounds are evaluated on the structural input domains, and $\mathcal F_{0,n}$ is defined in \cref{thm:consistency}. Thus $\Theta_n\subset M$. Note that by construction, latent variables are independent and $ \mathcal{P}_{n}$ can be decomposed into direct produce $ \mathcal{P}_{n,1} \times \cdots \times \mathcal{P}_{n,n_{U}}$. Let $\mathcal A_n$ be the set of restricted structural and generator network representations at sample size $n$, and let $\theta_\eta\in\Theta_n$ be the population parameter of a representation $\eta\in\mathcal A_n$. Let $\xi_1,\ldots,\xi_{m_n}$ be the common independent exogenous draws, including the categorical Gumbel coordinates, and write $H_\eta(\xi_l)$ for the resulting complete observed vector. Define \[ \Theta_\infty=\overline{\bigcup_n\Theta_n},\qquad P_{m_n}^\eta=\frac1{m_n}\sum_{l=1}^{m_n}\delta_{H_\eta(\xi_l)},\qquad h_n(\eta)=S_{\lambda_n}(P_n^{\theta^*},P_{m_n}^\eta), \] \[ f(\theta)=\mathbb E_{t\sim\mu_T}\mathbb E_{\mathcal M^\theta}[F(V_1(t),\ldots,V_{n_V}(t))]. \] The empirical constraint is a function of the representation $\eta$, whereas the population objective and $g(\theta)$ are functions of $\theta$. The optimization minimizes $f(\theta_\eta)$ over representations satisfying $h_n(\eta)\leqslant\alpha_n$. For fixed $n$, pad the finitely many network architectures to their stated widths and encode each representation by its finitely many real weights and mixture probabilities. Finite products of Euclidean spaces and simplices are standard Borel spaces. Affine maps, ReLU, clipping and finite composition have jointly continuous evaluation in their weights and inputs. At the fixed temperature $\tau_n>0$, mixture weights can be written \[ \frac{p_j^{1/\tau_n}\exp(G_j/\tau_n)}{\sum_k p_k^{1/\tau_n}\exp(G_k/\tau_n)},\qquad p\in\Delta, \] whose denominator is positive even when some $p_j=0$; hence generator evaluation is jointly Borel. The restrictions defining $\mathcal A_n$ are Borel restrictions on these parameters. To check this, choose a countable dense set in each continuous structural input domain. Continuity makes the output bound, coordinatewise nonnegativity and every Lipschitz inequality equivalent to their inequalities on that dense set or its Cartesian square. Each such inequality is Borel in the weights. For a nonnegative categorical raw network $q$, positivity on a compact input box $D_0$ is equivalent to membership in \[ \bigcup_{l=1}^\infty\bigcap_{x\in Q_0}\{\eta:\\|q_\eta(x)\\|_1\geqslant1/l\}, \] where $Q_0$ is countable and dense in $D_0$; continuity and attainment of a minimum prove both implications. Use the intersection of these conditions over a countable compact exhaustion when the full input domain is unbounded. On this Borel parameter set $q/\\|q\\|_1$ is jointly Borel and continuous in the input, so its Lipschitz restriction is again a countable family of Borel inequalities. The same dense-input tests apply to the raw Lipschitz restrictions before clipping. Thus $\mathcal A_n$ is standard Borel. These tests use dense input sets only. The exogenous input space is a finite product of uniform cubes and real Gumbel coordinates, hence standard Borel. For a categorical propensity vector $p$, put $r_i=\exp(-g_i)/p_i$ when $p_i>0$ and $r_i=\infty$ when $p_i=0$. These extended-real functions are Borel. The smallest minimizing label is $i$ exactly when $r_ii$; this is a finite intersection of Borel comparisons. Induction through the finite topological order therefore proves joint Borel evaluation of $H_\eta(\xi)$ and of its interventional version $H_\eta(t,\xi)$. The function $F$ is bounded on the bounded observed range, so integration over the fixed exogenous law and $\mu_T$ makes $\eta\mapsto f(\theta_\eta)$ Borel. Finite empirical optimal transport and entropic transport values are continuous in their cost matrices: changing every cost by at most $a$ changes every transport objective, and hence its infimum, by at most $a$. Therefore the empirical Sinkhorn constraint $h_n(\eta)$ is jointly Borel in the representation and data. For every real $a$, the event $\{F_na$ and \[ \frac{E_i}{q_i-a}<\min_{j\ne i}\frac{E_j}{q_j+a}. \] A robust label wins for every propensity vector whose coordinates differ from $q$ by at most $a$. The robust-label events are disjoint. For $q_i>a$, integrating the density of $E_i$ and the survival probabilities of the other exponentials gives \[ \mathbb P(i\text{ robust})=\int_0^\infty\exp\left(-s\left(1+\frac{\sum_{j\ne i}(q_j+a)}{q_i-a}\right)\right)\,ds =\frac{q_i-a}{q_i-a+\sum_{j\ne i}(q_j+a)}. \] For $q_i\leqslant a$ this probability is zero. Consequently, \[ \mathbb P(i\text{ robust})\geqslant\frac{(q_i-a)_+}{1+Na}. \] Summing and using $\sum_i(q_i-a)_+\geqslant1-Na$ bounds the probability of no robust label by $2Na/(1+Na)$ when $Na<1$, and the trivial bound $1\leqslant2Na$ covers $Na\geqslant1$. For each table and generator representation let $b$ be the indicator that a nonrobust choice occurs at any table prefix. Independence of the categorical noise from the generated latent vector and a union bound over the finitely many prefixes give $\mathbb E b\leqslant C\rho$. If $b=0$, induction over the categorical nodes forces the original and table prefixes to agree. Consequently, writing $\widetilde H$ for the table observed map and $B_V$ for a common bound on the observed $1$-norm diameter, \[ \\|H_\eta(\xi)-\widetilde H(\xi)\\|_1\leqslant C\rho+B_V b(\xi). \] This is a pointwise bound, including zero propensities. We bound the complexity of finite-type indicators and ambiguity indicators. A table simulation has type $(u,c)$, where $u$ is its latent grid index and $c$ its complete categorical label vector; its observed output is a fixed vector for each type. There are at most $J=C\rho^{-D_U}$ types. Include the indicators of membership in every subset of these types, and the ambiguity indicators $b$, in a scalar class $\mathcal B_{n,\rho}$. Here is an explicit standard Borel parameterization of this entire indicator class. Let $\Gamma_n$ be the finite-dimensional generator weight space with its mixture simplex factors, and let generator parameters range over all of $\Gamma_n$. This supplies an envelope for every restricted generator used in the optimization. A parameter consists of $\gamma\in\Gamma_n$, a table in the finite set $\mathcal T_\rho$, and a finite tag specifying either a subset of the at most $J$ types or the ambiguity indicator. This is a finite union of products of standard Borel spaces. Generator evaluation is jointly Borel by the positive-denominator mixture formula. Round its clipped latent output using fixed half-open grid cells, with the last cell closed at the endpoint; the grid index is a finite-valued Borel function. For each fixed table and grid index, the table simulation uses finitely many categorical race comparisons. In the notation $r_i=\exp(-g_i)/q_i$, with $r_i=\infty$ at $q_i=0$, the smallest-label rule is the finite system $r_ii$, so its selected label is Borel. The robust-label events are Borel too: when $q_i>a$ all their denominators are positive, and when $q_i\leqslant a$ the event is empty. Their finite unions and complements make $b$ jointly Borel. Type-subset membership is a finite union of these Borel events. Thus $\mathcal B_{n,\rho}$ has jointly Borel evaluation on a standard Borel parameter space, including zero propensities, grid boundaries and all ties. When using the ambient Euclidean input space in Lemma~\ref{lemma:sup_wasserstein}, clip the uniform input coordinates to their cubes; this changes no sample value and preserves joint Borel evaluation. The parameter envelope has precisely the same architecture and finite tags as those counted in the growth bound. For each continuous scalar dual test $h$, its normalized square-root transform is jointly Borel by composition. Consequently the structural condition of Lemma~\ref{lemma:sup_wasserstein} holds for these families, without approximation of selector parameters. On $M_0$ fixed exogenous inputs, the latent grid indices are determined by at most $C M_0/\rho$ scalar generator comparisons. The ReLU blocks have $P\leqslant C H_n$ parameters and at most $C H_n$ layers. Within a fixed activation pattern, their outputs are polynomials in the parameters of degree at most $C H_n$. For a positive-temperature mixture write its weights as $a_j R_j/\sum_k a_kR_k$, where $a_j=p_j^{1/\tau_n}\geqslant0$, $\sum_j a_j>0$, and $R_j=\exp(G_j/\tau_n)$ is fixed on each input. Treat the $a_j$ as additional free parameters. Clearing the positive denominator in a grid comparison leaves a polynomial of degree at most $C H_n+1$. Truncation adds only fixed-degree comparisons. The polynomial sign bound in Lemma 17 of \cite{bartlett2019nearly}, applied successively to the activation comparisons in each layer and then to these grid comparisons, gives at most \[ (C(M_0+H_n)H_n/\rho)^{C H_n^2} \] latent grid patterns: at each layer the sign bound contributes at most $(C(M_0+H_n)H_n/\rho)^{C P}$, and there are at most $C H_n$ layers. For a fixed table and grid pattern, the categorical race comparisons and every table output on these inputs are determined. The subgraph growth of $\mathcal B_{n,\rho}$ is therefore at most this number times $\|\mathcal T_\rho\|2^J$: a subset of the types has $2^J$ choices, and an ambiguity indicator needs no additional choices. Solving the shattering inequality \[ 2^{M_0}\leqslant\|\mathcal T_\rho\|2^J(C(M_0+H_n)H_n/\rho)^{C H_n^2} \] bounds their pseudo-dimensions by \[ v_n(\rho)\leqslant C\left(\rho^{-D_U}\log(2+\rho^{-1})+H_n^2\log(2+H_n/\rho)\right). \] To derive this bound, put $A=\log\|\mathcal T_\rho\|+J\log2$ and $B=C H_n^2$. The shattering inequality gives $M_0\log2\leqslant A+B\log(C(M_0+H_n)H_n/\rho)$. If $M_0\leqslant4A/\log2$, the first term suffices. Otherwise, absorbing $A$ and solving the remaining logarithmic inequality gives $M_0\leqslant C B\log(2+BH_n/\rho)+C B\log(2+A/B)$; this follows from $\log x\leqslant x/2$ after rescaling $M_0/B$. The bound on $A$ makes this at most the displayed expression. Put \[ r_n=m_n^{-1/(D+2)}\log m_n+H_n m_n^{-1/2}\log m_n, \] and take $\rho=r_n$ for sufficiently large $n$. The depth schedule implies $H_n=O(m_n^{d_{\max}^U/(2d_{\max}^U+4)})$, so $r_n\rightarrow0$. Apply Lemma~\ref{lemma:sup_wasserstein} to $\mathcal B_{n,r_n}$ with scalar output. Its functions take values in $\{0,1\}$, its pseudo-dimension is at most $v_n(r_n)$, and Wasserstein distance between its Bernoulli output laws equals the absolute difference of their means. Choose the lemma's tolerances $\delta=r_n$ and $\epsilon=A_0r_n$. Its exponent is bounded above by \[ C\left((1+r_n^{-1})\log(2+r_n^{-1})+r_n^{-D_U}\log^2(2+r_n^{-1})+H_n^2\log(2+H_n/r_n)\log(2+r_n^{-1})\right)-cA_0^2m_nr_n^2. \] Here $\log(2+H_n/r_n)=O(\log m_n)$ and \[ m_nr_n^2\geqslant m_n^{D/(D+2)}\log^2m_n+H_n^2\log^2m_n, \] whereas $r_n^{-D_U}\leqslant m_n^{D_U/(D+2)}(\log m_n)^{-D_U}$ and $r_n^{-1}\leqslant m_n^{1/(D+2)}/\log m_n$. Since $D\geqslant D_U,1$, a sufficiently large fixed $A_0$ makes the failure probability at most $O(m_n^{-3})$. On the complementary event every type-subset probability and every ambiguity-indicator mean differs from its empirical counterpart by at most $Cr_n$. The supremum over type subsets is the total variation distance between the population and empirical type laws, so that distance is at most $Cr_n$. Couple their common type masses identically and their residual masses arbitrarily. Because table outputs have diameter at most $B_V$, their Wasserstein distance is at most $B_VCr_n$. For each original representation, coupling original and table outputs on the same draw bounds their population error by $C\rho+B_V\mathbb E b$ and their empirical error by $C\rho+B_Vm_n^{-1}\sum_l b(\xi_l)$. The ambiguity estimate bounds both by $Cr_n$. The triangle inequality therefore gives \[ \sup_{\eta\in\mathcal A_n}W(P^{\theta_\eta}(\boldsymbol V),P_{m_n}^\eta(\boldsymbol V))\leqslant Cr_n \] with probability at least $1-O(m_n^{-3})$. This retains the positive-temperature generator, paired simulation observations and every categorical noise coordinate. For the observational sample let $C$ denote the complete categorical label vector, which has a fixed finite set of possible values. Conditional on $C=c$, all continuous observations equal a deterministic map $\Phi_c(U_1,\ldots,U_{n_U})$: fix the categorical values to $c$ and evaluate the continuous structural equations in topological order. Induction through their Lipschitz bounds makes $\Phi_c$ Lipschitz with a constant uniform over these finitely many $c$. The conditional joint latent laws $\mu_c=P((U_1,\ldots,U_{n_U})\mid C=c)$ are supported on the bounded $D_U$-dimensional latent cube; they need not be products. Write $p_c=P(C=c)$ and $N_c$ for the observational sample counts. For $p_c=0$, $N_c=0$ almost surely. For every $p_c>0$, a binomial concentration bound gives $N_c\geqslant np_c/2$ except on an exponentially small event. Conditional on the label vector of the sample, the latents of the observations carrying label $c$ are independent with law $\mu_c$. When $D_U\geqslant1$, Theorem 2 of \cite{fournier2015rate}, applied to each conditional joint latent empirical measure of size $N_c$, bounds its Wasserstein error by $C_c n^{-1/\max\{2,D_U\}}\log^2n$ with failure probability $O(n^{-4})$ whenever $N_c\geqslant np_c/2$. Its small-tolerance exponents are constant multiples of $N_c x^2$ for $D_U=1$, $N_c(x/\log(2+1/x))^2$ for $D_U=2$, and $N_c x^{D_U}$ for $D_U>2$. At the indicated $x$ these grow at least as $\log^4n$, $\log^2n$ and $\log^{2D_U}n$, respectively. Bounded support supplies the exponential-moment condition; norm equivalence changes only constants. Conditioning on the counts preserves the bound, since it holds uniformly for $np_c/2\leqslant N_c\leqslant n$. Lipschitz push-forward by $\Phi_c$ bounds each conditional observed Wasserstein error by the same order. If $D_U=0$, each conditional observed law is a point mass and its conditional sampling error is zero. Binomial concentration also bounds $\sum_c\|N_c/n-p_c\|$ by $C\sqrt{\log n/n}$ with probability at least $1-O(n^{-4})$. Couple common categorical-component masses using the conditional couplings and unmatched masses arbitrarily within the observed support. This gives \[ W(P_n^{\theta^*}(\boldsymbol V),P^{\theta^*}(\boldsymbol V))\leqslant\sum_c\min\{p_c,N_c/n\}W((\Phi_c)_\#\mu_c,(\Phi_c)_\#(\mu_c)_{N_c})+\frac{B_V}{2}\sum_c\|N_c/n-p_c\|\leqslant t_n, \] \[ t_n=C_1n^{-1/\max\{2,D_U\}}\log^2n, \] with probability at least $1-O(n^{-3})$, after a union bound over the finitely many categories. Terms with $N_c=0$ have zero common mass and are omitted. Finally Lemma~\ref{lemma:sinkhorn_approx} and the Wasserstein triangle inequality give, on the combined event, \[ \sup_{\eta\in\mathcal A_n}\|h_n(\eta)-g(\theta_\eta)\|\leqslant2(1+\log(nm_n))\lambda_n+Cr_n+t_n\leqslant s_n. \] This event has probability at least $1-O(n^{-3}+m_n^{-3})$. Since $m_n=\Omega(n)$, the failure probabilities are summable; Borel--Cantelli gives the estimate eventually almost surely. It applies to the current representations. \item We construct structural approximants satisfying both output and Lipschitz restrictions. Let $f:[-K,K]^m\rightarrow\mathbb R^{m'}$ have Lipschitz constant $L_f$, with $m\geqslant1$. Divide each coordinate into $R$ equal intervals of length $h=2K/R$, and triangulate each cell into simplices with successive vertices $v_0,v_1=v_0+he_{\pi(1)},\ldots,v_m=v_{m-1}+he_{\pi(m)}$, where $\pi$ runs over the coordinate permutations. Let $I_hf$ be affine on each simplex and equal $f$ at its vertices. Its derivative column in direction $e_{\pi(j)}$ is $(f(v_j)-f(v_{j-1}))/h$, of $1$-norm at most $L_f$. Splitting a line segment at simplex boundaries and summing the derivative bound proves $\operatorname{Lip}(I_hf)\leqslant L_f$; segments in boundaries follow by continuity. If $a_j$ are the barycentric coordinates of $x$, then \[ \\|I_hf(x)-f(x)\\|_1\leqslant\sum_{j=0}^ma_j\\|f(v_j)-f(x)\\|_1\leqslant L_fmh. \] For categorical $f$, the interpolated values are convex combinations in $\Delta_{n_i}$, so they are nonnegative and sum exactly to one. For continuous $f$, the additional uniform output bound on the true structural function gives \[ \\|I_hf(x)\\|_1\leqslant\sum_j a_j\\|f(v_j)\\|_1\leqslant K. \] Thus each continuous approximant satisfies the continuous network restriction. The nodal basis functions are translates and rescalings of \[ \phi(z)=\max\left\{0,\min\left(\{1+z_k,1-z_k:k\in[m]\}\cup\{1+z_k-z_l:k,l\in[m]\}\right)\right\}. \] On each simplex in the star of the central vertex, the active affine expression is its barycentric coordinate: the inequalities defining the simplex order its coordinates, and the minimum selects the difference between the two coordinates bordering that vertex. It equals one at the central vertex and zero at the other vertices; outside the star the minimum is nonpositive. Therefore \[ I_hf(x)=\sum_v f(v)\phi((x-v)/h) \] on the input cube. Each minimum is computed by a binary tree using $\min(a,b)=a-\max\{a-b,0\}$. There are $O(m^2)$ affine inputs, so each term has width $O(m^2)$ and depth $O(1+\log m)$. Parallelizing the terms and summing their outputs gives width $O(R^m)$ in fixed dimensions. Coordinatewise input clipping to the cube is a contraction and makes the network globally defined with the same Lipschitz and output bounds. Choosing $R$ proportional to $W_{0,n}^{1/m}$ gives error $O(W_{0,n}^{-1/m})$. When $m=0$, represent the constant function exactly. For a categorical function take $q=I_hf$; its normalization equals itself, so both Lipschitz restrictions hold. The output clipping in $\mathcal F_{0,n}$ acts identically on these bounded approximants. We obtain the latent approximants from the inherited component assumptions. For each normalized support component measure $\mu_i$, let $\psi_i:[0,1]^{q_i}\rightarrow C_i$ be its Lipschitz chart. For $q_i>0$, Proposition~\ref{prop:holder_curve} applied to the pullback measure supplies a curve $\gamma_i:[0,1]\rightarrow[0,1]^{q_i}$ of Holder exponent $1/q_i$ with that pullback law. Theorem 2(b) of \cite{Yarotsky2018OptimalAO} applied to its coordinates gives first-block uniform error $O(L_{1,n}^{-2/q_i})$; applied to the Lipschitz coordinates of $\psi_i$ it gives second-block error $O(L_{2,n}^{-2/q_i})$. Parallelize the coordinate networks and clip the first outputs to their input cubes. Their widths fit the stated fixed-width budgets. Lemma~\ref{lemma:push-forward_err}, including its fixed output-dimension factor, bounds the component Wasserstein error by $O(L_{1,n}^{-2/q_i}+L_{2,n}^{-2/q_i})$. For $q_i=0$, represent the singleton component by its constant value with zero error. With component probabilities $p_i$, Lemma~\ref{lemma:combin_err} bounds the hard selected population mixture error by the weighted sum of these component errors. The positive-temperature construction uses independent Gumbel-softmax weights; the hard mixture is its zero-temperature comparison law. Clip component outputs to a fixed box containing their supports. For outputs $v_i$ in that box, \[ \left\\|\sum_i(X_i^{\tau_n}-X_i^0)v_i\right\\|_1\leqslant\left(\max_i\\|v_i\\|_1\right)\sum_i\|X_i^{\tau_n}-X_i^0\|\leqslant C\\|X^{\tau_n}-X^0\\|_1. \] Couple these weights using identical Gumbel draws and retain identical independent component outputs. The coupling estimate in Lemma~\ref{lemma:gumbel_err} bounds the additional population error by $O(\tau_n(1-\log\tau_n))$ for sufficiently large $n$. The triangle inequality proves, for each non-Gumbel latent $U_j$, a generator $g_{j,n}$ with \[ W(P(U_j),P(g_{j,n}(Z_j)))\leqslant O\left(L_{1,n}^{-2/d_{\max}^U}+L_{2,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right). \] Coordinatewise output truncation $T_K$ cannot increase this error: $T_KU_j=U_j$ and $\\|U_j-T_Kg_{j,n}(Z_j)\\|_1\leqslant\\|U_j-g_{j,n}(Z_j)\\|_1$ in every coupling. Choose the generators independently. Summing their marginal errors and the structural errors in $d$ gives a representation $\eta_n\in\mathcal A_n$ with \[ a_n=d(\theta^*,\theta_{\eta_n})\leqslant O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+L_{1,n}^{-2/d_{\max}^U}+L_{2,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right)\rightarrow0. \] Choose the constant in $\epsilon_n$ large enough that $C_ga_n\leqslant\epsilon_n$. \end{enumerate} Work on the probability-one event where the current-class sampling estimate holds eventually. Since $g(\theta^*)=0$, the constructed representations satisfy \[ h_n(\eta_n)\leqslant g(\theta_{\eta_n})+s_n\leqslant C_ga_n+s_n\leqslant\epsilon_n+s_n=\alpha_n. \] Thus the restricted optimization is eventually feasible. Its objective satisfies $f(\theta_{\eta_n})\rightarrow f(\theta^*)=F_*$, whence $\limsup_nF_n\leqslant F_*$. Choose a feasible representation $\bar\eta_n$ with $F_n\leqslant f(\theta_{\bar\eta_n})\leqslant F_n+1/n$. The bounded objective on compact $M$ makes $F_n$ eventually bounded. Along a subsequence attaining its lower limit, compactness gives a further subsequence on which $\theta_{\bar\eta_n}\rightarrow\bar\theta$. Feasibility and the current-class bound imply \[ 0\leqslant g(\theta_{\bar\eta_n})\leqslant h_n(\bar\eta_n)+s_n\leqslant\alpha_n+s_n\rightarrow0. \] Continuity yields $g(\bar\theta)=0$. Its continuous structural functions satisfy $\sup_x\\|f_i(x)\\|_1\leqslant K$, all structural functions have Lipschitz bound $\hat L_f$, its categorical functions are simplex-valued, and its latent marginal laws define an independent bounded canonical latent distribution. Since categorical variables occur only when $K\geqslant1$, their normalized propensity vectors also satisfy the output $1$-norm bound. Hence the limit is an admissible observationally equivalent model in (\ref{eq:consistent_limiting}), so $f(\bar\theta)\geqslant\underline F^{\hat L_f}$. Continuity and $\|f(\theta_{\bar\eta_n})-F_n\|\leqslant1/n$ give $\liminf_nF_n=f(\bar\theta)$. This proves \[ [\liminf_nF_n,\limsup_nF_n]\subset[\underline F^{\hat L_f},F_*] \] almost surely. On the finite-sample event used in the sampling proof, every feasible representation also satisfies \[ W(P^{\theta^*}(\boldsymbol V),P^{\theta_\eta}(\boldsymbol V))\leqslant h_n(\eta)+s_n\leqslant\alpha_n+s_n\leqslant2\alpha_n. \] This event has probability at least $1-O(n^{-3}+m_n^{-3})$ and supplies the complete-observation error and feasibility estimates used in Corollary~\ref{cor:rate} for the same restricted optimizer and revised radius. · contribution narrowed (accepted by the owner): The guarantee for the printed optimizer is replaced by a guarantee for candidates whose continuous structural functions have uniform output $1$-norm at most $K$ and whose normalized categorical propensities have Lipschitz constant at most $\hat L_f$. True models whose continuous structural functions violate that output $1$-norm bound on the input domain are excluded, even when every output coordinate is bounded by $K$. The sampling orders based only on $d_{\max}^U$ and the rate for the original $\alpha_n$ are given up. The displayed-order version excludes $d_{\max}^{\text{in}}=0$ and $d_{\max}^U=0$. | | inherits refutation from | `lem:hilbert`, `lem:push-forward-err`, `lem:sinkhorn-approx`, `lem:sup-wasserstein`, `prop:holder-curve`, `thm:approximation`, `thm:deep-nn`, `thm:wide-nn` | ### 7. Hölder continuity of ATE `prop:lip-ate` | field | value | | --- | --- | | bucket | `repaired` | | source | `Main.tex:309-317` (`proposition`, label `prop:Lip_ATE`) | | statement | \label{prop:Lip_ATE} Given two causal models $ \mathcal{M} ,\hat{\mathcal{M}} \in\mathcal{M}(\mathcal{G},\mathcal{F}_L,\boldsymbol{U})$ satisfying \cref{assump:independence_u} and \cref{assump:Lip_and_bounded}, let their observational distributions be $ \nu ,\mu $. Suppose the norms of all variables are bounded by $K>0$. If (1) (Overlap) $ \nu $ is absolutely continuous with respect to one probability measure $P$ and the density $ p_{\nu }\left( t\|\text{Pa}( T) =x\right) \geqslant \delta >0 $ for x almost surely and $ t\in [ t_{1} ,t_{2}]$ and (2) (No Confounding) there is no hidden confounder in the causal graph $\mathcal{G}$, we have \begin{equation*} \begin{aligned} \int _{t_{1}}^{t_{2}}(\mathbb{E}_{\mathcal{M}}[ Y( t)] -\mathbb{E}_{\hat{\mathcal{M}} }[\hat{Y}( t)])^{2} P(dt) & \leqslant \frac{2C_{W}}{\delta } W( \mu ,\nu ) +2(L+1)^{n_{V}} W^{2}( \mu ,\nu )( t_{2} -t_{1}) , \end{aligned} \end{equation*} where $ C_{W} =4(L+1)^{n_{V}} K+2K\max\left\{(L+1)^{n_{V}} ,1\right\}$. | | appended hypothesis list | ass:assump-independence-u, ass:assump-lip-and-bounded-continuous, ass:assump-lip-and-bounded-categorical, ass:assump-lip-and-bounded-bound, ass:prop-lip-ate-bounded, ass:prop-lip-ate-overlap, ass:prop-lip-ate-no-confounding | | Lean | `TLipAte.lean:48` — `Papers.TbsNeuralCausalPi.lipschitz_dose_response_gap` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-154`, `ass:pin-182`, `ass:pin-183`, `ass:pin-189`, `ass:pin-190`, `ass:pin-191`, `ass:pin-206`, `ass:pin-fix-19bccf89` | | cited results | — | | assumed substrate gates | — | | findings | [refuted, V1.5] (a) The statement is false for mixed variables: Lipschitz categorical propensities do not make the categorical generation map Lipschitz, and category labels can amplify the outcome regression beyond $(L+1)^{n_V}$. (b) Independently, even with continuous variables, conditioning on $X=\text{Pa}(T)$ need not leave the latent inputs needed for $Y$ independent of $X$; an observed ancestor can enter both $X$ and $Y$, invalidating the asserted composition argument and regression bound. (c) The proof derives a quadratic coefficient $2L_y^2$, but the statement replaces it by $2(L+1)^{n_V}$; the stated bound on $L_y$ does not justify this substitution. (d) Integrating the constant remainder against $P(dt)$ produces $P([t_1,t_2])$, not $t_2-t_1$; the nondescendant branch likewise switches to $dt$. Dismiss the wiring lint: the regression and boundedness conditions of Lemma~\ref{lemma:Lip_condition} are conditions this proof attempts to derive, rather than additional standing hypotheses. The failed regression derivation is a correctness defect. witness: (a) Set $N=10000$, $K=N+1$, $L=1$, and use the unconfounded graph $T\rightarrow Z\rightarrow Y$, with $Z\in[N]$ categorical and $Y$ continuous. Take independent non-Gumbel latent variables uniform on $[0,1]$, set $T=U_T$ and $Y=Z+U_Y/2$, and use independent standard Gumbel noises for $Z$. Put $a=1/(4\pi)$ and $b=1/4+1/(2N)$. In $\mathcal M$, the propensity coordinates are $(f_Z(t))_1=b-a\sin(2\pi t)$, $(f_Z(t))_N=b+a\sin(2\pi t)$, and $(f_Z(t))_j=1/(2N)$ for $2\leqslant j\leqslant N-1$. In $\hat{\mathcal M}$ reverse the two sine signs. These are strictly positive normalized vectors, with $1$-norm Lipschitz constant at most $4\pi a=1$. The remaining structural functions are also $1$-Lipschitz, and all bounded variables satisfy the bound $K$. Choose $P$ uniform on $[0,1]$, $\delta=1$, $t_1=0$, and $t_2=1$; overlap holds because $T$ has law $P$. Couple the models by $\hat T=T+1/2$ modulo $1$, the same categorical draw, and the same $U_Y$. The propensity vectors agree at these coupled inputs, so $Z$ and $Y$ agree and $W(\mu,\nu)\leqslant1/2$. Nevertheless, the intervention-mean difference is $2a(N-1)\sin(2\pi t)$, giving an integral of $(N-1)^2/(8\pi^2)>1200000$. Here $n_V=3$ and $C_W=48K$, so the asserted right-hand side is at most $48K+4=480052$. The false load-bearing step is the claimed regression Lipschitz bound: the actual slope bound is $(N-1)/2$, exceeding $8$. (b) For a separate all-continuous example, take independent $U_Z\sim\operatorname{Unif}([-1,1])$ and $U_T\sim\operatorname{Unif}([0,1])$, with $Z=U_Z$, $X=Z/100$, $T=U_T+X/4$, and $Y=(Z+T)/2$. The graph has arrows $Z\rightarrow X$, $X\rightarrow T$, $Z\rightarrow Y$, and $T\rightarrow Y$, without hidden confounding. All structural functions have Lipschitz constant at most $1$, and $K=2$ bounds the variables. With $P$ uniform on $[-1,2]$, overlap holds on $[1/10,9/10]$ with $\delta=1$. But $f_y(x,t)=50x+t/2$ has Lipschitz constant at least $50$, whereas $(L+1)^{n_V}=16$. The relevant ancestor noise $U_Z=100X$ is not independent of $X$. (c) In the unconfounded continuous chain $T=U_T$, $Z=4T$, $Y=4Z$, with $U_T$ uniform on $[0,1]$, take $L=4$, $K=17$, $P$ uniform on $[0,1]$, and $\delta=1$. Then $n_V=3$, $L_y=16$, and $L_y^2=256>125=(L+1)^{n_V}$. A comparison model with $\hat T=U_T/2$ and the same downstream functions has positive Wasserstein distance, so replacing the quadratic coefficient by the printed one is not a valid bound. (d) Take $P$ uniform on $[0,1/2]$, $t_1=0$, and $t_2=1/2$. Then $\int_{t_1}^{t_2}1\,P(dt)=1$, while $t_2-t_1=1/2$. This comparison measure is compatible with every premise, for example with independent $T\sim P$, $U_Y\sim\operatorname{Unif}([0,1])$, and the two unconfounded models $Y=T+U_Y/10$ and $\hat Y=T+U_Y/10+1/10$, using $L=K=\delta=1$. Thus the constant remainder cannot be integrated as the printed proof does. minimal patch (confirmed): corrected statement: Given two causal models $ \mathcal{M} ,\hat{\mathcal{M}} \in\mathcal{M}(\mathcal{G},\mathcal{F}_L,\boldsymbol{U})$ satisfying \cref{assump:independence_u} and \cref{assump:Lip_and_bounded}, let their observational distributions be $ \nu ,\mu $. Suppose the norms of all variables are bounded by $K>0$. If (1) (Overlap) $\nu$ is absolutely continuous with respect to one probability measure $P$ and the density $p_{\nu}\left(t\|\text{Pa}(T)=x\right)\geqslant\delta>0$ for $x$ almost surely and $t\in[t_1,t_2]$ and (2) (No Confounding) there is no hidden confounder in the causal graph $\mathcal{G}$, we have \begin{equation*} \begin{aligned} \int_{t_1}^{t_2}(\mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}(t)])^2 P(dt) &\leqslant\frac{2C_W}{\delta}W(\mu,\nu)+2L_y^2W^2(\mu,\nu)P([t_1,t_2]), \end{aligned} \end{equation*} where \[ D=\max\left(\{1\}\cup\left\{\frac{n_i-1}{2}:V_i\text{ is categorical}\right\}\right),\qquad L_y=(LD+1)^{n_V}, \] \[ q=\sum_{i:V_i\text{ is continuous}}d_i+\left\|\{i:V_i\text{ is categorical}\}\right\|,\qquad B=K(1+2qL_y), \] and $C_W=4L_yB+2B\max\{L_y,1\}$. · proof change: If $Y$ is not a descendant of $T$, its structural equation and all the equations of its ancestors are unaffected by the intervention, so $\mathbb{E}_{\mathcal{M}}[Y(t)]=\mathbb{E}_{\mathcal{M}}[Y]$ and $\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}(t)]=\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}]$. For any coupling of the observational laws, the absolute difference of the outcome means is at most the expected absolute difference of the outcome coordinates, which is at most the expected $1$-norm distance of the observed vectors. Taking the infimum over couplings gives \begin{align*} \int_{t_1}^{t_2}(\mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}(t)])^2P(dt) &=P([t_1,t_2])(\mathbb{E}_{\mathcal{M}}[Y]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}])^2\\ &\leqslant P([t_1,t_2])W^2(\mu,\nu)\\ &\leqslant\frac{2C_W}{\delta}W(\mu,\nu)+2L_y^2P([t_1,t_2])W^2(\mu,\nu), \end{align*} since $L_y\geqslant1$. Now, suppose that $Y$ is a descendant of $T$. Let $X=\text{Pa}(T)$, and let $A$ be the vector of all observed nodes other than $T$ that are not descendants of $T$. In particular, $X$ is a subvector of $A$. Every observed parent of a node in $A$ also belongs to $A$: a parent equal to $T$, or descended from $T$, would make that node a descendant of $T$. By the canonical representation and the absence of hidden confounding, the noises belonging to distinct observed nodes are independent. Thus $A$ is generated without the noise of $T$ or any descendant of $T$, and the descendant noises are independent of $(A,T)$. Conditional on $A=a$, the equation of $T$ uses only $X=x$ and its own independent noise. Consequently, \[ \nu(dt\|A=a)=\nu(dt\|X=x). \] For each $\mathcal{N}\in\{\mathcal{M},\hat{\mathcal{M}}\}$, fix $A=a$ and $T=t$ and generate the descendants of $T$ in topological order using their independent noises with their laws in $\mathcal{N}$. Denote the resulting mean of the outcome by $g_{\mathcal{N}}(a,t)$. These means are finite: categorical outputs have finite ranges, and the Lipschitz bound for a continuous structural function bounds its output by its value at a fixed input plus $L$ times the input distance; induction through the finite graph, using bounded non-Gumbel noises, gives finite first moments at every fixed $(a,t)$. Independence of the descendant noises from $(A,T)$ shows that $g_{\mathcal{N}}$ is a version of the observational outcome regression given $(A,T)$. The intervention leaves the equations generating $A$ unchanged, so its distribution is its observational marginal. Therefore, \[ \mathbb{E}_{\mathcal{M}}[Y(t)]=\int g_{\mathcal{M}}(a,t)\nu_A(da),\qquad \mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}(t)]=\int g_{\hat{\mathcal{M}}}(a,t)\mu_A(da), \] where $\nu_A$ and $\mu_A$ are the observational marginals of $A$. To bound the Lipschitz constants, take an arbitrary $\mathcal{N}\in\{\mathcal{M},\hat{\mathcal{M}}\}$. For a continuous descendant $V_i$, couple its outputs at two parent histories $h,h'$ using the same private noise. The structural Lipschitz bound gives \[ \mathbb{E}\\|V_i(h)-V_i(h')\\|\leqslant L\\|h-h'\\|. \] For a categorical descendant, let $\pi_i(h)$ be its propensity vector after integrating its private non-Gumbel noise. The canonical Gumbel rule has categorical probabilities equal to this vector. Integration of the propensity Lipschitz bound gives \[ \\|\pi_i(h)-\pi_i(h')\\|\leqslant L\\|h-h'\\|. \] Place mass $\min\{\pi_{i,j}(h),\pi_{i,j}(h')\}$ on each equal pair $(j,j)$. The remaining masses in the two marginals have the same total mass and disjoint supports; coupling these remaining masses gives disagreement probability \[ 1-\sum_{j=1}^{n_i}\min\{\pi_{i,j}(h),\pi_{i,j}(h')\} =\frac12\sum_{j=1}^{n_i}\|\pi_{i,j}(h)-\pi_{i,j}(h')\|. \] Two labels in $[n_i]$ differ by at most $n_i-1$, so this coupling satisfies \[ \mathbb{E}\|V_i(h)-V_i(h')\|\leqslant\frac{n_i-1}{2}\\|\pi_i(h)-\pi_i(h')\\|\leqslant LD\\|h-h'\\|. \] The continuous bound is also at most $LD\\|h-h'\\|$, since $D\geqslant1$. Starting at fixed inputs $(a,t)$ and $(a',t')$, use these couplings successively for the descendants, conditional on the accumulated histories. Let $E_r$ be the expected $1$-norm distance between these histories after $r$ descendants have been generated. Projection onto a parent history cannot increase this distance. Hence \[ E_0=\\|a-a'\\|+\|t-t'\|,\qquad E_{r+1}\leqslant E_r+LD E_r=(1+LD)E_r. \] Induction gives $E_r\leqslant(1+LD)^rE_0$: the assertion holds at $r=0$, and the recurrence multiplies its bound by $1+LD$ at the next step. There are at most $n_V$ descendants. Projection onto the outcome coordinate, followed by $\|\mathbb{E}Y-\mathbb{E}Y'\|\leqslant\mathbb{E}\|Y-Y'\|$, therefore gives \[ \|g_{\mathcal{N}}(a,t)-g_{\mathcal{N}}(a',t')\|\leqslant L_y(\\|a-a'\\|+\|t-t'\|). \] This argument applies to each of the two models without conditioning on latent inputs shared with observed ancestors. Let $Q$ be the set of allowed inputs $s=(a,t)$ whose numerical coordinates all lie in $[-K,K]$. Categorical coordinates retain their categorical ranges. The numerical dimension of $(A,T)$ is at most $q$, so any $s,s'\in Q$ satisfy $\\|s-s'\\|\leqslant2Kq$. Both observational laws of $(A,T)$ give $Q$ probability one. For each $\mathcal{N}\in\{\mathcal{M},\hat{\mathcal{M}}\}$, the observational outcome bound and the conditional expectation identity give $\|g_{\mathcal{N}}(A,T)\|\leqslant K$ almost surely under its own observational law. Choose an input $s_{\mathcal{N}}\in Q$ where this inequality holds. For every $s\in Q$, the Lipschitz inequality then yields \[ \|g_{\mathcal{N}}(s)\|\leqslant\|g_{\mathcal{N}}(s_{\mathcal{N}})\|+L_y\\|s-s_{\mathcal{N}}\\|\leqslant K+2KqL_y=B. \] Thus the structural regression versions are bounded by $B$ on $Q$, including inputs outside their observational supports. Put $f_y(a,t)=g_{\mathcal{M}}(a,t)$. Recall that $ \nu,\mu $ are the observation distributions of $\mathcal{M}$ and $\hat{\mathcal{M}}$ respectively in \cref{prop:Lip_ATE}. Put $\hat f_y(a,t)=g_{\hat{\mathcal{M}}}(a,t)$. Its construction gives a version of $\mathbb{E}_{\hat{\mathcal{M}}}[\hat Y\|A=a,T=t]$, the $L_y$-Lipschitz inequality, the interventional identity, and the bound $\|\hat f_y(a,t)\|\leqslant B$ for $(a,t)\in Q$. We use the test-function argument of Lemma~\ref{lemma:Lip_condition} on the vector covariate $s=(a,t)$, with regression bound $B$, and give the calculation explicitly. Write \[ d(s)=\hat f_y(s)-f_y(s),\qquad G(s,y)=d(s)y,\qquad h(s)=d(s)\hat f_y(s). \] On $Q$, the established bounds imply $\|d(s)\|\leqslant2B$ and $\|d(s)-d(s')\|\leqslant2L_y\\|s-s'\\|$. For $s,s'\in Q$ and $\|y\|,\|y'\|\leqslant K\leqslant B$, \begin{align*} \|G(s,y)-G(s',y')\| &\leqslant2B\|y-y'\|+2KL_y\\|s-s'\\|\\ &\leqslant2B\max\{L_y,1\}(\\|s-s'\\|+\|y-y'\|),\\ \|h(s)-h(s')\| &\leqslant \|d(s)\|\,\|\hat f_y(s)-\hat f_y(s')\|+\|\hat f_y(s')\|\,\|d(s)-d(s')\|\\ &\leqslant4BL_y\\|s-s'\\|. \end{align*} Conditioning on the covariate gives $\mathbb{E}_{\mu}[G]=\mathbb{E}_{\mu}[h]$ and $\mathbb{E}_{\nu}[G]=\mathbb{E}_{\nu}[df_y]$. Since $h-df_y=d^2$, \begin{align*} \mathbb{E}_{\nu}[d(A,T)^2] &=(\mathbb{E}_{\mu}[G]-\mathbb{E}_{\nu}[G])+(\mathbb{E}_{\nu}[h]-\mathbb{E}_{\mu}[h])\\ &\leqslant\bigl(2B\max\{L_y,1\}+4BL_y\bigr)W(\mu,\nu) =C_WW(\mu,\nu). \end{align*} For each Wasserstein bound, integrate the corresponding displayed Lipschitz inequality against any coupling of the observational laws and take the infimum of its expected $1$-norm cost. Both endpoints belong to the domains on which these inequalities were proved. Projection onto $(A,T,Y)$ or $(A,T)$ cannot increase that cost. Set $I=[t_1,t_2]$. Observational boundedness gives $\nu_T(\mathbb{R}\setminus[-K,K])=0$. The conditional treatment law given $A$ equals the law given $X=\text{Pa}(T)$. Integrating the overlap bound therefore gives \[ 0=\nu_T(I\setminus[-K,K])\geqslant\delta P(I\setminus[-K,K]), \] so $P(I\setminus[-K,K])=0$. Thus, for $P$-almost every $t\in I$ and for either observational marginal of $A$, the input $(A,t)$ belongs to $Q$. The overlap assumption now yields \begin{align*} C_WW(\mu,\nu) &\geqslant\int (f_y(a,t)-\hat f_y(a,t))^2\nu(dt\|A=a)\nu_A(da)\\ &\geqslant\delta\int_I\int(f_y(a,t)-\hat f_y(a,t))^2\nu_A(da)P(dt)\\ &\geqslant\delta\int_I b(t)^2P(dt), \end{align*} where $b(t)=\int(f_y(a,t)-\hat f_y(a,t))\nu_A(da)$. The last inequality is Jensen's inequality for the probability measure $\nu_A$. Put \[ c(t)=\int\hat f_y(a,t)\nu_A(da)-\int\hat f_y(a,t)\mu_A(da). \] For $P$-almost every $t\in I$, integrate the Lipschitz bound of $\hat f_y$ in $a$ against a coupling of the two marginals and take the infimum to obtain $\|c(t)\|\leqslant L_yW(\nu_A,\mu_A)\leqslant L_yW(\mu,\nu)$. The interventional identities give \[ \mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat Y(t)]=b(t)+c(t). \] Since $2b(t)c(t)\leqslant b(t)^2+c(t)^2$, it follows, for $P$-almost every $t\in I$, that \[ (\mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat Y(t)])^2 \leqslant2b(t)^2+2L_y^2W^2(\mu,\nu). \] Integrating this inequality against $P(dt)$ on $[t_1,t_2]$ and using the bound on the integral of $b(t)^2$ gives \begin{align*} \int_{t_1}^{t_2}(\mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat Y(t)])^2P(dt) &\leqslant2\int_{t_1}^{t_2}b(t)^2P(dt)+2L_y^2W^2(\mu,\nu)\int_{t_1}^{t_2}1\,P(dt)\\ &\leqslant\frac{2C_W}{\delta}W(\mu,\nu)+2L_y^2W^2(\mu,\nu)P([t_1,t_2]). \end{align*} | ### 8. cor:rate `thm:rate` | field | value | | --- | --- | | bucket | `repaired` | | source | `Main.tex:319-321` (`corollary`, label `cor:rate`) | | statement | \label{cor:rate} Let $F, \mu_T$ in \cref{thm:consistency} to be $ F(\boldsymbol{V}) = Y, \mu_T \sim \text{Unif}([t_1, t_2]), \epsilon >0 $. Suppose that the assumptions in Proposition \ref{prop:Lip_ATE} and \cref{thm:consistency} hold, with probability at least $1-O(n^{-2})$, we have $ \| F_n - F_*\| \leqslant O(\sqrt{\alpha_n})$, where $F_n,F_*,\alpha_n$ are defined in \cref{thm:consistency}. | | appended hypothesis list | ass:assump-independence-u, ass:assump-categorical-rv-latents, ass:assump-categorical-rv-generation, ass:assump-lip-and-bounded-continuous, ass:assump-lip-and-bounded-categorical, ass:assump-lip-and-bounded-bound, ass:assump-support-components, ass:assump-support-homeomorphisms, ass:assumption-lower-bound-domination, ass:assumption-lower-bound-boundary, ass:corollary-deep-nn-latent-laws, ass:thm-consistency-inherited, ass:thm-consistency-structural-lipschitz, ass:thm-consistency-objective-lipschitz, ass:thm-consistency-positive-temperature, ass:thm-consistency-growth, ass:prop-lip-ate-bounded, ass:prop-lip-ate-overlap, ass:prop-lip-ate-no-confounding, ass:cor-rate-inherited | | Lean | `TRate.lean:87` — `Papers.TbsNeuralCausalPi.rate_of_convergence` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:bk-1`, `ass:pin-154`, `ass:pin-177`, `ass:pin-181`, `ass:pin-182`, `ass:pin-183`, `ass:pin-185`, `ass:pin-186`, `ass:pin-187`, `ass:pin-188`, `ass:pin-189`, `ass:pin-190`, `ass:pin-191`, `ass:pin-192`, `ass:pin-194`, `ass:pin-196`, `ass:pin-197`, `ass:pin-198`, `ass:pin-199`, `ass:pin-201`, `ass:pin-203`, `ass:pin-205`, `ass:pin-206`, `ass:pin-207`, `ass:pin-208`, `ass:pin-209`, `ass:pin-210`, `ass:pin-211`, `ass:pin-33`, `ass:pin-75`, `ass:pin-fix-00130002`, `ass:pin-fix-09991de8`, `ass:pin-fix-0a7b21b0`, `ass:pin-fix-0bcb2cef`, `ass:pin-fix-19bccf89`, `ass:pin-fix-2a1e7b01`, `ass:pin-fix-2fe04b0e`, `ass:pin-fix-30413b86`, `ass:pin-fix-3b195d8d`, `ass:pin-fix-419eb8e3`, `ass:pin-fix-43f1bb71`, `ass:pin-fix-448b19a0`, `ass:pin-fix-4a86c2eb`, `ass:pin-fix-507a5848`, `ass:pin-fix-59a01041`, `ass:pin-fix-5f02b8a6`, `ass:pin-fix-676b96c3`, `ass:pin-fix-72c9e86d`, `ass:pin-fix-73abef8c`, `ass:pin-fix-761562d5`, `ass:pin-fix-7a79e01d`, `ass:pin-fix-8429b12c`, `ass:pin-fix-84ccd033`, `ass:pin-fix-8cc0dba5`, `ass:pin-fix-91573abf`, `ass:pin-fix-945d85ed`, `ass:pin-fix-9463b114`, `ass:pin-fix-9675bba5`, `ass:pin-fix-99ac9bb3`, `ass:pin-fix-9bbf998f`, `ass:pin-fix-9d74053b`, `ass:pin-fix-ab17be7a`, `ass:pin-fix-b83fa4d1`, `ass:pin-fix-bbf1c78a`, `ass:pin-fix-bd620122`, `ass:pin-fix-c514425b`, `ass:pin-fix-c778b424`, `ass:pin-fix-c85b9186`, `ass:pin-fix-ca667e13`, `ass:pin-fix-d19c63b5`, `ass:pin-fix-db6ed772`, `ass:pin-fix-e896f793`, `ass:pin-fix-f66ad243`, `ass:pin-fix-f6f485b1` | | cited results | `lem:consistency-fournier-theorem-2` (fournier2015rate, Theorem 2); `lem:deep-nn-yarotsky-theorem-2` (Yarotsky2018OptimalAO, Theorem 2); `lem:sup-wasserstein-anthony-theorem-17-1` (anthony1999neural, Theorem 17.1); `lem:sup-wasserstein-anthony-theorems` (anthony1999neural, Theorem 18.4); `lem:wide-nn-perekrestenko-theorem-viii-1` (perekrestenkoHighDimensionalDistributionGeneration2022a, Theorem VIII.1); `lem:wide-nn-yarotsky-theorem-2` (Yarotsky2018OptimalAO, Theorem 2) | | assumed substrate gates | — | | findings | [refuted, V1.5] The inherited-assumption lint warnings are dismissed: independence, both categorical clauses, all three Lipschitz/boundedness clauses, both support clauses, both lower-bound clauses, and the latent-law premise are inherited through \cref{thm:consistency}. The regularized-notation warning is also dismissed: the corollary uses the stated Sinkhorn comparison without independently invoking $W_{1,\lambda}$. The triangle inequality requires three distances: population to observational empirical law, observational empirical to candidate empirical law, and candidate empirical to population law. The printed middle term repeats the population distance, but substituting the empirical distance bounded immediately above closes this proof slip. The substantive defect is the final invocation of Proposition \ref{prop:Lip_ATE}: it controls the squared response difference integrated against $P$, whereas $F_n-F_*$ averages against $\mu_T=\text{Unif}([t_1,t_2])$. No stated hypothesis relates these measures. The counterexample satisfies the referenced assumptions but makes the displayed rate false. witness: Take the unconfounded graph $T\rightarrow Y$, independent $U_T,U_Y\sim\text{Unif}([0,1])$, and the true equations $T=U_T$, $Y=U_Y$. Set $K=2$, $L_f=1$, $t_1=0$, and $t_2=2$. Both latent laws satisfy the support and lower-bound assumptions using the identity homeomorphism and $C_f=1$. Choose $P=\text{Unif}([0,1])$. The conditional treatment law equals $P$; the density version $p_\nu(t)=1$ on $[0,2]$ satisfies overlap with $\delta=1$, since its values outside the support of $P$ do not affect the density identity. Consider the candidate equations $\hat T=\hat U_T$ and $\hat Y=\max\{-2,\min\{\hat U_Y-\max\{\hat T-1,0\},2\}\}$, with independent uniform latent laws generated exactly by identity networks. Its raw structural functions are $1$-Lipschitz in the paper's norm and belong to the prescribed classes eventually. Its observational law is exactly the true uniform law on $[0,1]^2$. Nevertheless, $F_*=1/2$ and its uniform intervention objective is $1/2-\frac12\int_1^2(t-1)\,dt=1/4$. Choose $m_n=n$, $W_{0,n}=n$, $L_{1,n},L_{2,n}=\Theta(n^{1/6})$, $\tau_n=n^{-1}$, and $\lambda_n=n^{-2}$. The stipulated radius has $\alpha_n=\Theta(n^{-1/4}\log n)\rightarrow0$. Partitioning the observational square into $\lceil n^{1/4}\rceil^2$ cells and concentrating the empirical cell masses bounds both empirical-to-population Wasserstein distances by $O(n^{-1/4})$ with probability $1-O(n^{-2})$. The Sinkhorn comparison therefore makes this candidate feasible on that event. Hence $F_n\leqslant1/4$ and $\|F_n-F_*\|\geqslant1/4$, contradicting $O(\sqrt{\alpha_n})$. The false printed step is the final transfer from the $P$-integrated bound to the uniform intervention average; the response difference vanishes on the support of $P$ but not on $(1,2]$. minimal patch (confirmed): corrected statement: Let $F,\mu_T$ in \cref{thm:consistency} be $F(\boldsymbol V)=Y$, $\mu_T\sim\operatorname{Unif}([t_1,t_2])$, and $\epsilon>0$. Suppose that $t_10$ for every structural input $x$, and \[ \operatorname{Lip}\left(x\mapsto\frac{q(x)}{\\|q(x)\\|_1}\right)\leqslant\hat L_f. \] For each sample size, obtain every candidate empirical law by evaluating the candidate on the same $m_n$ independent draws of its uniform and Gumbel exogenous inputs, independent of the observational sample. Let $F_n$ be the optimal value of this restricted optimization, retaining the objective and Sinkhorn constraint of (\ref{eq:opt_empiri}). Put \[ D_U=\sum_{j=1}^{n_U}d_j^U,\qquad D=\max\{1,D_U\}, \] where the sum includes only the non-Gumbel latent variables. Take $\alpha_n=\epsilon_n+s_n$, with positive tolerances chosen as sufficiently large constant multiples of the respective quantities in \[ \epsilon_n=O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+\sum_{i=1}^2L_{i,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right), \] \[ s_n=O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2n+\log(nm_n)\lambda_n\right). \] Then, with probability at least $1-O(n^{-2})$, we have $\|F_n-F_*\|\leqslant O(\sqrt{\alpha_n})$, where $F_*$ is defined in \cref{thm:consistency} and $F_n,\alpha_n$ are specified here. · proof change: We establish feasibility and the complete-observation sampling bounds for the specified restricted optimization. Write $\eta$ for a network representation, $\theta_\eta$ for its population SCM parameter, and $\theta^*$ for the true parameter. For two population parameters define \[ d(\theta,\theta')=\sum_{i=1}^{n_V}\\|f_i-f'_i\\|_\infty+\sum_{j=1}^{n_U}W(P(U_j),P(U'_j)), \] using normalized categorical propensity functions in the first sum. Let $\mathcal A_n$ be the set of admissible representations. With common independent exogenous draws $\xi_1,\ldots,\xi_{m_n}$, let $H_\eta(\xi_l)$ be the complete observed vector and put \[ P_{m_n}^\eta=\frac1{m_n}\sum_{l=1}^{m_n}\delta_{H_\eta(\xi_l)},\qquad h_n(\eta)=S_{\lambda_n}(P_n^{\theta^*},P_{m_n}^\eta),\qquad g(\theta)=W(P^{\theta^*},P^\theta). \] All distributions in these expressions are distributions of the complete observed vector. Write $f(\theta)=\mathbb E_{t\sim\mu_T}\mathbb E_{\mathcal M^\theta}[F(\boldsymbol V(t))]$ for the objective. For fixed $n$, pad the finitely many network architectures to their stated widths and encode each representation by its finitely many real weights and mixture probabilities. Finite products of Euclidean spaces and simplices are standard Borel spaces. Affine maps, ReLU, clipping and finite composition have jointly continuous evaluation in their weights and inputs. At the fixed temperature $\tau_n>0$, mixture weights can be written \[ \frac{p_j^{1/\tau_n}\exp(G_j/\tau_n)}{\sum_k p_k^{1/\tau_n}\exp(G_k/\tau_n)},\qquad p\in\Delta, \] whose denominator is positive even when some $p_j=0$; hence generator evaluation is jointly Borel. The restrictions defining $\mathcal A_n$ are Borel restrictions on these parameters. Choose a countable dense set in each continuous structural input domain. Continuity makes the output bound, coordinatewise nonnegativity and every Lipschitz inequality equivalent to their inequalities on that dense set or its Cartesian square, and each such inequality is Borel in the weights. For a nonnegative categorical raw network $q$, positivity on a compact input box $D_0$ is equivalent to membership in \[ \bigcup_{l=1}^\infty\bigcap_{x\in Q_0}\{\eta:\\|q_\eta(x)\\|_1\geqslant1/l\}, \] where $Q_0$ is countable and dense in $D_0$. Positivity implies a positive minimum on the compact box; conversely the dense-set lower bound extends by continuity. Intersect these conditions over a countable compact exhaustion if the full domain is unbounded. On the resulting Borel parameter set $q/\\|q\\|_1$ is jointly Borel and continuous in the input, so its Lipschitz restriction is another countable family of Borel inequalities. The raw Lipschitz restrictions before clipping have the same dense-input characterization. Thus $\mathcal A_n$ is standard Borel. The exogenous space is a finite product of uniform cubes and real Gumbel coordinates. For categorical propensities $p$, define $r_i=\exp(-g_i)/p_i$ for $p_i>0$ and $r_i=\infty$ for $p_i=0$. The smallest minimizing label is $i$ precisely when $r_ii$. These are Borel comparisons of extended-real Borel functions. Induction in topological order proves that both $H_\eta(\xi)$ and $H_\eta(t,\xi)$ are jointly Borel. Integration of the bounded objective on the observed range makes $f(\theta_\eta)$ Borel in $\eta$: parameter integrals of bounded Borel functions are Borel by the monotone class theorem. Empirical transport values are continuous in the cost matrix, since a uniform change of at most $a$ in costs changes every coupling objective and its infimum by at most $a$. Hence $h_n$ is jointly Borel in parameters and data. The event $\{F_na$ and \[ \frac{E_i}{q_i-a}<\min_{j\ne i}\frac{E_j}{q_j+a}. \] A robust label wins for every propensity vector whose coordinates differ from $q$ by at most $a$. The robust-label events are disjoint. When $q_i>a$, integration of the independent exponential densities gives \[ \mathbb P(i\text{ robust})=\frac{q_i-a}{q_i-a+\sum_{j\ne i}(q_j+a)}. \] When $q_i\leqslant a$, the probability is zero. In either case it is at least $(q_i-a)_+/(1+Na)$. Summing and using $\sum_i(q_i-a)_+\geqslant1-Na$ bounds the probability of no robust label by $2Na/(1+Na)$ when $Na<1$; when $Na\geqslant1$, the bound $1\leqslant2Na$ suffices. For each table and generator representation let $b$ indicate that a nonrobust choice occurs at any table prefix. Independence of categorical noise from the generated latent vector and a union bound over the finitely many prefixes give $\mathbb E b\leqslant C\rho$. If $b=0$, induction over the categorical nodes forces the original and table prefixes to agree. Thus, with $\widetilde H$ the table observed map and $B_V$ a common observed $1$-norm diameter bound, \[ \\|H_\eta(\xi)-\widetilde H(\xi)\\|\leqslant C\rho+B_V b(\xi). \] This bound includes zero propensities. A table simulation has type $(u,c)$, where $u$ is its latent grid index and $c$ its complete categorical label vector; its observed output is fixed for each type. There are at most $J=C\rho^{-D_U}$ types. Include the indicators of membership in every subset of these types, and the ambiguity indicators $b$, in a scalar class $\mathcal B_{n,\rho}$. To verify the structural condition of Lemma~\ref{lemma:sup_wasserstein} for this class, let $\Gamma_n$ be the entire finite-dimensional generator weight space with its mixture simplex factors. This gives a measurable envelope for the restricted generator representations. Index indicators by a parameter in $\Gamma_n$, a table in the finite set $\mathcal T_\rho$, and a finite tag specifying either a subset of the at most $J$ types or the ambiguity indicator. This is a standard Borel space. Generator evaluation is jointly Borel by the positive-denominator mixture formula. Its clipped output has a Borel grid index when cells are half-open and the final cell includes its endpoint. Given the table and grid index, each categorical choice is a finite Borel comparison using $r_ii$, with $r_i=\infty$ for zero table propensity. Each robust event is a Borel comparison with positive denominators on $q_i>a$, and is empty on $q_i\leqslant a$. Finite unions and complements therefore give jointly Borel ambiguity and type-subset indicators. This includes all ties, grid boundaries and zero propensities. Clip uniform exogenous coordinates outside their cubes to view these functions on the ambient Euclidean input space; the values on the sampled inputs do not change. Composing a jointly Borel indicator with a continuous scalar dual function and its normalized square-root transform preserves joint Borel evaluation. Thus all the classes to which the lemma's concentration argument is applied have the required standard Borel parameterization. The envelope uses the same architectural parameters and finite tags as the growth calculation; no density assertion for discontinuous selectors is needed. On $M_0$ fixed exogenous inputs, the latent grid indices are determined by at most $CM_0/\rho$ scalar generator comparisons. The ReLU blocks have at most $CH_n$ parameters and $CH_n$ layers. Within a fixed activation pattern their outputs are polynomials in the parameters of degree at most $CH_n$. For a positive-temperature mixture write its weights as $a_jR_j/\sum_k a_kR_k$, where $a_j=p_j^{1/\tau_n}\geqslant0$, $\sum_j a_j>0$, and $R_j=\exp(G_j/\tau_n)$ is fixed on each input. Treat the $a_j$ as additional free parameters. Clearing the positive denominator in a grid comparison leaves a polynomial of degree at most $CH_n+1$. Truncation adds fixed-degree comparisons. The polynomial sign bound in Lemma 17 of \cite{bartlett2019nearly}, applied successively to activation comparisons in each layer and then to grid comparisons, gives at most \[ (C(M_0+H_n)H_n/\rho)^{CH_n^2} \] latent grid patterns: each layer contributes at most $(C(M_0+H_n)H_n/\rho)^{CH_n}$, and there are at most $CH_n$ layers. For a fixed table and grid pattern all categorical race comparisons and table outputs on these inputs are determined. The growth of $\mathcal B_{n,\rho}$ is therefore at most this number times $\|\mathcal T_\rho\|2^J$. Hence its pseudo-dimension satisfies \[ v_n(\rho)\leqslant C\left(\rho^{-D_U}\log(2+\rho^{-1})+H_n^2\log(2+H_n/\rho)\right). \] To obtain this bound explicitly, put $A_0=\log\|\mathcal T_\rho\|+J\log2$ and $B_0=CH_n^2$ in the shattering inequality \[ M_0\log2\leqslant A_0+B_0\log(C(M_0+H_n)H_n/\rho). \] If $M_0\leqslant4A_0/\log2$, the first term suffices. Otherwise the inequality bounds $M_0$ by $CB_0\log(2+B_0H_n/\rho)+CB_0\log(2+A_0/B_0)$; the established bound on $A_0$ gives the displayed estimate. Put \[ r_n=m_n^{-1/(D+2)}\log m_n+H_nm_n^{-1/2}\log m_n. \] The depth schedule in \cref{thm:consistency} gives $H_n=O(m_n^{d_{\max}^U/(2d_{\max}^U+4)})$, so $r_n\rightarrow0$. For sufficiently large $n$ take $\rho=r_n$ and apply Lemma~\ref{lemma:sup_wasserstein} to $\mathcal B_{n,r_n}$ with scalar output. These functions take values in $\{0,1\}$, and the Wasserstein distance of their Bernoulli output laws equals the absolute difference of their means. With tolerances $\delta=r_n$ and $\epsilon=Cr_n$, its failure exponent is bounded above by \[ C_1\left((1+r_n^{-1})\log(2+r_n^{-1})+r_n^{-D_U}\log^2(2+r_n^{-1})+H_n^2\log(2+H_n/r_n)\log(2+r_n^{-1})\right)-cC^2m_nr_n^2. \] Here $\log(2+H_n/r_n)=O(\log m_n)$ and \[ m_nr_n^2\geqslant m_n^{D/(D+2)}\log^2m_n+H_n^2\log^2m_n, \] whereas $r_n^{-D_U}\leqslant m_n^{D_U/(D+2)}(\log m_n)^{-D_U}$ and $r_n^{-1}\leqslant m_n^{1/(D+2)}/\log m_n$. Since $D\geqslant\max\{D_U,1\}$, a sufficiently large fixed $C$ makes the failure probability $O(m_n^{-3})$. On the complementary event every type-subset probability and ambiguity-indicator mean differs from its empirical counterpart by at most $Cr_n$. The type-subset bound controls total variation between population and empirical type laws. Couple common type masses identically and residual masses arbitrarily; their table-output Wasserstein distance is at most $B_VCr_n$. Coupling original and table outputs on the same draw bounds their population discrepancy by $C\rho+B_V\mathbb E b$ and their empirical discrepancy by $C\rho+B_Vm_n^{-1}\sum_l b(\xi_l)$. Both are at most $Cr_n$ by the ambiguity estimate. The triangle inequality therefore gives \[ \sup_{\eta\in\mathcal A_n}W(P^{\theta_\eta},P_{m_n}^\eta)\leqslant Cr_n \] with probability at least $1-O(m_n^{-3})$. For the observational sample let $C$ be its complete categorical label vector, taking values in a fixed finite set. Conditional on $C=c$, all continuous observations equal a deterministic map $\Phi_c(U_1,\ldots,U_{n_U})$, obtained by fixing the categorical labels and evaluating the continuous equations in topological order. Their Lipschitz bounds make $\Phi_c$ Lipschitz with a constant uniform over these finitely many labels. The conditional joint latent laws $\mu_c=P((U_1,\ldots,U_{n_U})\mid C=c)$ are supported on the bounded $D_U$-dimensional cube and need not be products. Write $p_c=P(C=c)$ and $N_c$ for sample counts. If $p_c=0$, then $N_c=0$ almost surely. For $p_c>0$, binomial concentration gives $N_c\geqslant np_c/2$ except on an exponentially small event. Conditional on the sample labels, latents in category $c$ are independent with law $\mu_c$. For $D_U\geqslant1$, Theorem 2 of \cite{fournier2015rate} applied to these conditional joint latent laws bounds their empirical Wasserstein errors by $C_cn^{-1/\max\{2,D_U\}}\log^2n$ with failure probability $O(n^{-4})$, uniformly for $np_c/2\leqslant N_c\leqslant n$. Indeed, its small-tolerance exponents are constant multiples of $N_cx^2$ for $D_U=1$, $N_c(x/\log(2+1/x))^2$ for $D_U=2$, and $N_cx^{D_U}$ for $D_U>2$. At the specified tolerance these grow at least as $\log^4n$, $\log^2n$, and $\log^{2D_U}n$, respectively. Bounded support supplies the exponential-moment assumption. A coupling pushed through $\Phi_c$ multiplies its cost by at most the Lipschitz constant of $\Phi_c$. For $D_U=0$, each conditional observed law is a point mass and its conditional empirical error is zero. Binomial concentration also gives $\sum_c\|N_c/n-p_c\|\leqslant C\sqrt{\log n/n}$ with probability at least $1-O(n^{-4})$. Couple common categorical-component masses by their conditional couplings and unmatched masses arbitrarily within the observed support. After a union bound over the finitely many categories, \[ W(P_n^{\theta^*},P^{\theta^*})\leqslant\sum_c\min\{p_c,N_c/n\}W((\Phi_c)_\#\mu_c,(\Phi_c)_\#(\mu_c)_{N_c})+\frac{B_V}{2}\sum_c\|N_c/n-p_c\|\leqslant t_n, \] \[ t_n=C_2n^{-1/\max\{2,D_U\}}\log^2n, \] with probability at least $1-O(n^{-3})$. Terms with zero common mass are omitted. Lemma~\ref{lemma:sinkhorn_approx} and the triangle inequality give, on the combined event, \[ \sup_{\eta\in\mathcal A_n}\|h_n(\eta)-g(\theta_\eta)\|\leqslant2(1+\log(nm_n))\lambda_n+Cr_n+t_n\leqslant s_n. \] The event has probability at least $1-O(n^{-3}+m_n^{-3})$. We construct a feasible approximation to the true model. For a structural function $f:[-K,K]^m\rightarrow\mathbb R^{m'}$ with $m\geqslant1$, divide every coordinate into $R$ intervals of length $h=2K/R$ and triangulate each cell into simplices with successive vertices $v_0,v_1=v_0+he_{\pi(1)},\ldots,v_m=v_{m-1}+he_{\pi(m)}$. Let $I_hf$ interpolate the vertex values affinely. Its derivative column in direction $e_{\pi(j)}$ is $(f(v_j)-f(v_{j-1}))/h$, of $1$-norm at most $L_f$. Subdividing a segment at simplex boundaries and summing these derivative bounds proves $\operatorname{Lip}(I_hf)\leqslant L_f$. With barycentric coordinates $a_j$, \[ \\|I_hf(x)-f(x)\\|\leqslant\sum_{j=0}^ma_j\\|f(v_j)-f(x)\\|\leqslant L_fmh. \] Convex interpolation preserves the bound $K$ for continuous outputs and preserves the simplex for categorical propensities. Its nodal basis functions are translates and rescalings of \[ \phi(z)=\max\left\{0,\min\left(\{1+z_k,1-z_k:k\in[m]\}\cup\{1+z_k-z_l:k,l\in[m]\}\right)\right\}. \] On each simplex of a nodal star this function is affine, equals one at the central vertex, and equals zero at its other vertices. Hence $I_hf(x)=\sum_v f(v)\phi((x-v)/h)$. Compute each minimum by a binary tree using $\min(a,b)=a-\max\{a-b,0\}$. Each term has width $O(m^2)$ and depth $O(1+\log m)$. Parallelizing $O(R^m)$ terms and summing their outputs gives the required fixed-dimensional depth and width $O(R^m)$. Coordinatewise input clipping is a contraction and preserves the Lipschitz bound. Taking $R$ proportional to $W_{0,n}^{1/m}$ gives error $O(W_{0,n}^{-1/m})$. A function with $m=0$ is constant and represented exactly. For categorical functions take $q=I_hf$; it is nonnegative and normalized, and its normalized Lipschitz constant equals its original one. Thus these approximants satisfy the categorical restrictions. Continuous output clipping preserves the error bound and cannot increase the Lipschitz constant. For each normalized latent support component measure $\mu_i$, let $\psi_i:[0,1]^{q_i}\rightarrow C_i$ be its Lipschitz chart. For $q_i>0$, Proposition~\ref{prop:holder_curve} applied to its pullback measure gives a curve $\gamma_i:[0,1]\rightarrow[0,1]^{q_i}$ of Holder exponent $1/q_i$ with that pullback law. Theorem 2(b) of \cite{Yarotsky2018OptimalAO} applied to its coordinates gives first-block uniform error $O(L_{1,n}^{-2/q_i})$ and, applied to the Lipschitz coordinates of $\psi_i$, gives second-block error $O(L_{2,n}^{-2/q_i})$. Parallelize coordinate networks and clip first-block outputs to their input cubes. Their widths fit the specified budgets. Lemma~\ref{lemma:push-forward_err} bounds the resulting component Wasserstein error by $O(L_{1,n}^{-2/q_i}+L_{2,n}^{-2/q_i})$. For $q_i=0$, use the exact constant component. Lemma~\ref{lemma:combin_err} bounds the hard-selected mixture error by the weighted component errors. Couple positive-temperature and hard Gumbel weights, retaining identical independent bounded component outputs $v_i$. Then \[ \left\\|\sum_i(X_i^{\tau_n}-X_i^0)v_i\right\\|\leqslant C\\|X^{\tau_n}-X^0\\|_1. \] Lemma~\ref{lemma:gumbel_err} bounds the additional error by $O(\tau_n(1-\log\tau_n))$ for sufficiently large $n$. Output truncation cannot increase the latent error because the true latents lie in $[-K,K]^{d_j^U}$ and truncation contracts distance to every point of this cube. Choose the latent generators independently. Summing their marginal errors and the structural errors gives a representation $\eta_n\in\mathcal A_n$ satisfying \[ a_n=d(\theta^*,\theta_{\eta_n})\leqslant O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+\sum_{i=1}^2L_{i,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right). \] Choose the constant in $\epsilon_n$ so that $C_ga_n\leqslant\epsilon_n$. On the combined sampling event, \[ h_n(\eta_n)\leqslant g(\theta_{\eta_n})+s_n\leqslant C_ga_n+s_n\leqslant\epsilon_n+s_n=\alpha_n. \] Thus the optimization is feasible. Since $m_n=\Omega(n)$, the same event has probability at least $1-O(n^{-2})$ and gives \[ W(P_n^{\mathcal M^*},P^{\mathcal M^*})\leqslant O(\alpha_n),\qquad\sup_{\eta\in\mathcal A_n}W(P_{m_n}^\eta,P^{\theta_\eta})\leqslant O(\alpha_n). \] For every feasible representation $\eta$, Lemma~\ref{lemma:sinkhorn_approx} and the empirical constraint give \[ W(P_n^{\mathcal M^*}(\boldsymbol V),P_{m_n}^\eta(\boldsymbol V))\leqslant S_{\lambda_n}(P_n^{\mathcal M^*}(\boldsymbol V),P_{m_n}^\eta(\boldsymbol V))+2(1+\log(nm_n))\lambda_n\leqslant O(\alpha_n), \] since the regularization term is included in $s_n\leqslant\alpha_n$. On the same sampling event, the triangle inequality gives, for every feasible representation $\eta$, \begin{align*} w_\eta&=W(P^{\mathcal M^*}(\boldsymbol V),P^{\theta_\eta}(\boldsymbol V))\\ &\leqslant W(P^{\mathcal M^*}(\boldsymbol V),P_n^{\mathcal M^*}(\boldsymbol V))+W(P_n^{\mathcal M^*}(\boldsymbol V),P_{m_n}^\eta(\boldsymbol V))+W(P_{m_n}^\eta(\boldsymbol V),P^{\theta_\eta}(\boldsymbol V))\\ &\leqslant O(\alpha_n). \end{align*} All feasible models have bounded continuous structural functions and normalized categorical propensity functions with Lipschitz constant at most $\hat L_f$. The true model also belongs to the corresponding bounded Lipschitz class. Apply Proposition~\ref{prop:Lip_ATE} to the true model and each feasible model, using their common unconfounded graph and the overlap assumption of the true model. Write \[ b_\eta(t)=\mathbb E_{\mathcal M^*}[Y(t)]-\mathbb E_{\mathcal M^{\theta_\eta}}[Y(t)],\qquad r(t)=\frac{d\mu_T}{dP}(t). \] With the fixed constants $C_W,L_y$ supplied by Proposition~\ref{prop:Lip_ATE} at Lipschitz bound $\hat L_f$, it gives \[ \int_{t_1}^{t_2}b_\eta(t)^2P(dt)\leqslant\frac{2C_W}{\delta}w_\eta+2L_y^2w_\eta^2P([t_1,t_2]). \] Absolute continuity and Cauchy--Schwarz therefore give \begin{align*} \left\|\mathbb E_{t\sim\mu_T}\mathbb E_{\mathcal M^{\theta_\eta}}[Y(t)]-F_*\right\| &=\left\|\int_{t_1}^{t_2}b_\eta(t)r(t)P(dt)\right\|\\ &\leqslant\left(\int_{t_1}^{t_2}b_\eta(t)^2P(dt)\right)^{1/2}\left(\int_{t_1}^{t_2}r(t)^2P(dt)\right)^{1/2}\\ &\leqslant A_T\left(\frac{2C_W}{\delta}w_\eta+2L_y^2w_\eta^2P([t_1,t_2])\right)^{1/2}\\ &\leqslant C\sqrt{\alpha_n}. \end{align*} For the last inequality, the growth assumptions imply $\alpha_n\rightarrow0$, so $\alpha_n\leqslant1$ eventually and $w_\eta^2=O(\alpha_n^2)=O(\alpha_n)$; all other factors are fixed. The bound is uniform over feasible representations. Feasibility was established on the same event, so every feasible objective lies in $[F_*-C\sqrt{\alpha_n},F_*+C\sqrt{\alpha_n}]$. Taking its infimum proves $\|F_n-F_*\|\leqslant C\sqrt{\alpha_n}$, whether or not the infimum is attained. The event has probability at least $1-O(n^{-2})$. · contribution narrowed (accepted by the owner): The corrected statement excludes uniform intervention laws that are not absolutely continuous with respect to $P$ or whose density is not square-integrable, true models or continuous candidates whose structural functions violate $\sup_x\\|f_i(x)\\|_1\leqslant K$ on their full input domains even when every coordinate is bounded by $K$, categorical candidates whose normalized propensity functions exceed $\hat L_f$, and candidate-specific independently resampled simulation experiments; it replaces the printed sampling orders determined only by $d_{\max}^U$ with orders accounting for the joint latent dimension and does not retain the sample-size guarantee for the original estimator and radius. The displayed-order version also excludes $d_{\max}^{\text{in}}=0$ and $d_{\max}^U=0$, and the uniform intervention law requires $t_10$, and coordinate regrouping preserves any global Lipschitz regularity these normalized functions possess. · proof change: Let $\boldsymbol G=\{G_{V_i}:V_i\text{ is categorical}\}$. By \cref{assump:independence_u} and \cref{assump:categorical_rv}, the remaining latent variables $U_1,\ldots,U_{n_U}$ and the Gumbel coordinates are independent. For a categorical variable, put $s_i(x)=\\|f_i(x)\\|_1$ and $\pi_i(x)=f_i(x)/s_i(x)$. The categorical generation assumption requires nonnegative coordinates and $s_i(x)>0$ on its structural input domain. For positive coordinates, \[ \log\pi_{i,j}(x)=\log f_{i,j}(x)-\log s_i(x). \] The subtracted term is common to every coordinate, so replacing $f_i$ by $\pi_i$ leaves its Gumbel argmax unchanged. Zero coordinates remain zero and have extended logarithm $-\infty$. Thus this replacement preserves the categorical structural output for every input and the normalized vector belongs to the probability simplex. Let $D_1,\ldots,D_{n_C}$ be the maximal $C^2$-components. For a non-Gumbel latent variable define $I(U_k)=\{V_j:U_k\in\boldsymbol U_{V_j}\}$. Independence implies that every nonempty $I(U_k)$ is a bidirected clique, so it is contained in a maximal $C^2$-component. Assign $U_k$ to one such component, choosing one when there is more than one choice. Unused latent variables may be assigned arbitrarily. Let $\hat U_C$ be the tuple of variables assigned to $C$; for an empty tuple use a constant latent variable. Disjoint tuples are independent. Their coordinate projections recover the original joint law of the non-Gumbel latent variables, and they are independent of $\boldsymbol G$. For each $V_i$, supply all $\hat U_C$ with $V_i\in C$ as canonical inputs. Project these tuples onto the original inputs $\boldsymbol U_{V_i}$ and ignore the other coordinates. Define its continuous function by applying $f_i$ to these projected inputs and its categorical function by applying $\pi_i$. Together with the unchanged parent relations and Gumbel noise, these are exactly the normalized structural equations in (\ref{eq:structure_equation}). Every original bidirected edge is generated by a shared original latent variable and remains present after that variable is assigned to a component containing both endpoints. Every newly declared shared tuple has its affected vertices inside its clique $C$, so it introduces no edge outside the original graph. Hence the canonical causal graph is $\mathcal G$. Couple each tuple with its original coordinates and retain the same Gumbel variables. Under any intervention, the treated variables agree by construction. In a topological order, suppose all parents of the next untreated variable agree. Its projected latent inputs agree; its continuous structural function is unchanged, or its categorical argmax agrees by the normalization identity. Its value therefore agrees as well. Induction over this order proves equality of all observed variables under every intervention, and hence equality of their intervention laws. Coordinate projections preserve differentiability order and are $1$-Lipschitz for the $1$-norm, so they preserve global Lipschitz constants of the functions to which they are applied. Continuous functions therefore retain their original regularity, and categorical functions retain the regularity of $\pi_i$. To verify the normalization claims, if $f_i$ is differentiable to a given order and $s_i>0$, the quotient by the positive differentiable function $s_i$ has that same differentiability order. If $f_i$ is locally Lipschitz, fix an input where $s_i>0$ and a neighborhood on which $s_i\geqslant a>0$. For two points $x,y$ in that neighborhood, nonnegativity gives $\|s_i(x)-s_i(y)\|\leqslant\\|f_i(x)-f_i(y)\\|$, and \[ \left\\|\frac{f_i(x)}{s_i(x)}-\frac{f_i(y)}{s_i(y)}\right\\| \leqslant\frac{\\|f_i(x)-f_i(y)\\|}{s_i(x)}+\frac{\|s_i(x)-s_i(y)\|}{s_i(x)} \leqslant\frac{2}{a}\\|f_i(x)-f_i(y)\\|. \] This proves local Lipschitz regularity of the normalized propensity. If that propensity is globally Lipschitz, composing it with the coordinate projections retains its global bound, completing the stated regularity conclusions. · narrowed: Preservation of the original global Lipschitz regularity for unnormalized categorical functions on unrestricted domains, including cases where their total mass approaches zero. | ### 10. prop:holder_curve `prop:holder-curve` | field | value | | --- | --- | | bucket | `repaired` | | source | `proof.tex:1238-1240` (`proposition`, label `prop:holder_curve`) | | statement | \label{prop:holder_curve} Let $ \lambda $ be the Lebesgue measure on $ [ 0,1]$. Given any probability measure $ \mathbb{P}$ that satisfies \cref{assumption:lower_bound}, there exists a continuous curve $ \gamma : [0,1]\rightarrow \text{supp}(\mathbb{P})$ such that $\gamma _{\#} \lambda =\mathbb{P}$. Furthermore, if $d \geqslant 1$, $\gamma$ is $ 1/d$-Hölder continuous. | | appended hypothesis list | ass:assumption-lower-bound-domination, ass:assumption-lower-bound-boundary | | Lean | `Helpers/HilbertCurve.lean:386` — `Papers.TbsNeuralCausalPi.exists_measure_representing_holder_curve` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-33`, `ass:pin-fix-0bcb2cef`, `ass:pin-fix-2a1e7b01`, `ass:pin-fix-2fe04b0e`, `ass:pin-fix-7a79e01d`, `ass:pin-fix-8cc0dba5`, `ass:pin-fix-99ac9bb3`, `ass:pin-fix-c778b424`, `ass:pin-fix-c85b9186`, `ass:pin-fix-e896f793` | | cited results | — | | assumed substrate gates | — | | findings | [refuted, V1.5] the step its proof takes from [Shchepin2010OnHM, Theorem 1] (lem:holder-curve-shchepin-theorem-1) does not hold as the proof uses it: The partition definition prints a parent-smaller-than-child inequality, whereas the subsequent construction and uniform convergence argument determine child contraction. Under that reading, two citation hypotheses still fail. (a) The interval partitions need not be developments: the host assumptions allow interior atoms, whose corresponding intervals never shrink. (b) Even for atom-free admissible measures, the printed identity $\text{diam}(\mathcal I_k)=\text{gap}(\mathcal I_k)$ is false, and the uniform comparison required by Shchepin need not hold. Maximum interval length is controlled by maximum cell mass, while a gap between disjoint intervals can be the mass of a much smaller intervening cell. The lower domination assumption supplies no uniform upper bound on their ratio. witness: (a) Take $d=1$, the identity chart, and $\mathbb P=(\lambda+\delta_{1/3})/2$ on $[0,1]$. This measure dominates $\lambda/2$ and vanishes on the boundary. Dyadic cuts avoid the atom and satisfy the cube construction. At every level the cell containing $1/3$ has mass at least $1/2$, so its interval has length at least $1/2$. Consequently $\text{diam}(\mathcal I_k)$ does not tend to zero, contrary to the cited definition of a development. (b) Take instead $d\mathbb P(x)=(1/2+1/(4\sqrt{x}))\,dx$ on $00$ and all measurable sets $B\in[0,1]^d$. Suppose we have proven the conclusion for this case. By \cref{assumption:lower_bound}, there exists $ f\in \mathcal{H}([0,1]^{d},K) \cap \mathcal{F}_L([0,1]^{d},K)$ such that $f^{-1}_{\#} \mathbb{P}$ satisfies these conditions. There exists continuous function $\gamma$ such that $f^{-1}_{\#}\mathbb{P}=\gamma_{\#}\lambda$, which implies $\mathbb{P} = (f\circ\gamma)_{\#}\lambda$. In particular, if $\gamma$ is $1/d$-Holder continuous, $f\circ\gamma$ is also $1/d$-Holder continuous because $f$ is Lipschitz continuous. When $d = 0$, the constant map satisfies the requirement. We focus on the case $d\geqslant 1$ in the following. The proof is to modify the construction of the Hilbert space-filling curve. By changing the velocity of the curve, the push-forward measure of the Hilbert space-filling curve can simulate a great number of distributions. We use $ \lambda $ to represent the Lebesgue measure $ [0,1] $. \par We prove the cube case by first constructing a uniform Hilbert curve and then changing its parameter. Write $\lambda_d$ for Lebesgue measure on $[0,1]^d$ and retain $\lambda$ for Lebesgue measure on $[0,1]$. The cube probability measure satisfies $\mathbb P\geqslant C_1\lambda_d$, so $00$, the equivalence $Q(t)\leqslant x$ if and only if $t\leqslant A(x)$. Therefore $\lambda\{t:Q(t)\leqslant x\}=A(x)$, which proves $Q_{\#}\lambda=\rho$. If $u0$ and an SCM $\mathcal{M}^*$ satisfying Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound} with a backdoor causal graph such that there is no unobserved confounding in $\mathcal{M}^*$. For any $\epsilon >0$, there exists an SCM $\mathcal{M}_\epsilon$ with the same causal graph such that $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V})) \leq \epsilon $ and $\|\text{ATE}_{\mathcal{M}^*} - \text{ATE}_{\mathcal{M}_\epsilon}\| > c $. | | Lean | `TCounterexampleForal.lean:60` — `Papers.TbsNeuralCausalPi.counterexample_backdoor` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity pinned · correctness ok | | added conditions | `ass:pin-177`, `ass:pin-181`, `ass:pin-182`, `ass:pin-183`, `ass:pin-189`, `ass:pin-190`, `ass:pin-204` | | cited results | — | | assumed substrate gates | — | | findings | [note, V1.5] pinned reading: The second Bernoulli law in the construction has $P(U_2=0)=2/3$. (The displayed value $P(U_2=1)=1/3$ and the joint masses for $X=0,T=0$ in the proof of Proposition~\ref{prop:counterexample_foral} both determine this entry.) ／ [refuted, V1.5] [F2.5 reviewer] [statement-wrong] The requirement to satisfy Assumption lower_bound does not settle whole-latent-law versus normalized-component-law regularity; Lean selects the latter. witness: [statement-wrong] The requirement to satisfy Assumption lower_bound does not settle whole-latent-law versus normalized-component-law regularity; Lean selects the latter. minimal patch (confirmed): corrected statement: \label{prop:counterexample_foral} There exists a constant $c>0$ and an SCM $\mathcal{M}^*$ satisfying Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound} with a backdoor causal graph such that there is no unobserved confounding in $\mathcal{M}^*$. For any $\epsilon >0$, there exists an SCM $\mathcal{M}_\epsilon$ with the same causal graph such that $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V})) \leq \epsilon $ and $\|\text{ATE}_{\mathcal{M}^*} - \text{ATE}_{\mathcal{M}_\epsilon}\| > c $. Here, for each non-Gumbel latent law $\mathbb P$ with connected support components $C_i$, Assumption~\ref{assumption:lower_bound} is required of each normalized component law $\mathbb P\|_{C_i}/\mathbb P(C_i)$, with $K=C_i$. · proof change: Let $ \boldsymbol{V} = \{ X,T,Y\} $ be the covariate, treatment and outcome respectively. $ T$ is a binary variable. Let structure equations of causal model $ \mathcal{M}_{\delta }$ be \begin{align*} X & =\begin{cases} -1 & ,\text{w.p.} \ 1/2\\ 0 & ,\text{w.p.} \ 1/4\\ \delta & ,\text{w.p.} \ 1/4, \end{cases}\\ P( T=1\|X=x) & =\begin{cases} 1/2 & ,x=-1\ \text{or} \ 0,\\ 3/4 & ,x=\delta , \end{cases}\\ Y & =\begin{cases} T+U_{y} & ,X< 0\\ X/\delta +U_{y} & ,X\geqslant 0 \end{cases} \end{align*}where all latent variables are independent and $ U_{y}$ is mean-zero noise. The distribution of $ \mathcal{M}_{\delta }$ is \begin{align*} P_{X,T,Y}( 0,1,y) =p_{U_{y}}( y) /8, & \ P_{X,T,Y}( 0,0,y) =p_{U_{y}}( y) /8\\ P_{X,T,Y}( \delta ,1,y) =3p_{U_{y}}( y-1) /16, & \ P_{X,T,Y}( \delta ,0,y) =p_{U_{y}}( y-1) /16.\\ P_{X,T,Y}( -1,1,y) =p_{U_{y}}( y-1) /4, & P_{X,T,Y}( -1,0,y) =p_{U_{y}}( y) /4. \end{align*} As $ \delta \rightarrow 0$, distribution of $ \mathcal{M}_{\delta }\xrightarrow{d}\mathcal{M}^*$, where structure equations of $ \mathcal{M}^*$ are \begin{align*} P( U_{1} =1) =3/5,P(U_{1} =0) =2/5 & ,P( U_{2} =1) =1/3,P( U_{1} =0) =2/3,\\ X & =\begin{cases} -1 & ,\text{w.p.} \ 1/2,\\ 0 & ,\text{w.p.} \ 1/2, \end{cases}\\ P( T=1\|X=x) & \ =\begin{cases} 1/2 & ,x=-1,\\ 5/8 & ,x=0, \end{cases}\\ Y & \ =\begin{cases} T+U_{y} & ,X=-1,\\ U_{1} +U_{y} & ,X=0,T=1,\\ U_{2} +U_{y} & ,X=0,T=0, \end{cases} \end{align*} where $ U_1,U_2,U_y $ are independent. The distribution of $ \mathcal{M}^*$ is \begin{align*} P_{X,T,Y}( 0,1,y) & \ =p_{U_{y}}( y) /8+3p_{U_{y}}( y-1) /16,\\ P_{X,T,Y}( 0,0,y) & =p_{U_{y}}( y) /8+p_{U_{y}}( y-1) /16,\\ P_{X,T,Y}( -1,1,y) & =p_{U_{y}}( y-1) /4,\\ P_{X,T,Y}( -1,0,y) & =p_{U_{y}}( y) /4. \end{align*} Choose $U_y=0$ almost surely. Generate $X$ by $X=U_X$, where $P(U_X=-1)=P(U_X=0)=1/2$, independently of $U_1,U_2$ and the categorical Gumbel variables. Every non-Gumbel latent law, including any vector formed by merging latents affecting the same observed variable, now has finite support. For such a law $\mathbb P$, write its support components as $C_i=\{a_i\}$ and put $\mu_i=\mathbb P\|_{C_i}/\mathbb P(C_i)$. The map $f:[0,1]^0\rightarrow C_i$ sending the unique point to $a_i$ is a Lipschitz homeomorphism. If $\lambda_0$ denotes the probability measure on $[0,1]^0$, then for every measurable $B\subset[0,1]^0$, \[ \mu_i(f(B))=\lambda_0(B). \] Thus each normalized component law satisfies Assumption~\ref{assumption:lower_bound} with $d=0$ and $C_f=1$; its boundary requirement applies only when $d>0$. These singleton components also verify Assumption~\ref{assump:support}. All displayed structural inputs range over finite sets. Their structural and normalized propensity maps have Lipschitz extensions: form the finite Cartesian grid of their input coordinate values, retain the prescribed values at the specified inputs, and assign bounded values at the remaining vertices, using probability vectors for propensity maps. Interpolate multilinearly on each grid cell and clamp each input coordinate to the grid's range. This preserves the prescribed values and the probability simplex. Each cell has finite coordinate slopes; taking their maximum and subdividing a line segment at cell boundaries gives a finite global Lipschitz constant. Choose $L_f$ to bound these finitely many constants. The resulting observed variables and non-Gumbel latent variables are bounded, so a finite $K$ exists. Independence and categorical generation hold by construction. Hence $\mathcal M^*$ satisfies Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound}. Some calculation gives $ W( P^{\mathcal{M}^*}(\boldsymbol{V}) ,P^{\mathcal{M}_{\delta }}(\boldsymbol{V})) \leqslant \delta /2$. It is easy to calculate the ATE of the two models. \begin{align*} \text{ATE}_{\mathcal{M}^*} = 19/30, & \ \text{ATE}_{\mathcal{M}_{\delta }} = 1/2. \end{align*} · contribution narrowed (accepted by the owner): The requirement that each entire non-Gumbel latent law satisfy Assumption~\ref{assumption:lower_bound}; only its normalized connected-component laws are required to satisfy it. | ### 13. prop:convg_opt `prop:convg-opt` | field | value | | --- | --- | | bucket | `verified_with_bookkeeping` | | source | `proof.tex:1605-1631` (`proposition`, label `prop:convg_opt`) | | statement | \label{prop:convg_opt} Given a metric space $ (M,d) $ and sets $ \Theta_1\subset\Theta_2\cdots\subset\Theta_n\subset\cdots\subset \Theta_\infty \subset M$, where $\Theta_\infty = \overline{\cup_{n=1}^\infty \Theta_n}$ , positive sequences $ \{\epsilon_{n}\}_{n\in \mathbb{N}},\{\delta_{n}\}_{n\in \mathbb{N}}$ such that $ \lim _{n\rightarrow \infty } \epsilon _{n}=\lim _{n\rightarrow \infty }\tau_n=\lim _{n\rightarrow \infty } \delta_{n} =0$ and continuous functions $ f,g_{n} :\Theta_\infty \rightarrow \mathbb{R}$, suppose that \begin{enumerate} \item $\Theta_\infty$ is compact. \item $g_n$ satisfies \begin{equation} \label{eq:aprox_lip} \\|g_n(\theta_1) - g_n(\theta_2)\\| \leqslant L_g d(\theta_1,\theta_2) + \tau_n, \quad \forall \theta_1,\theta_2\in\Theta \end{equation} and $ \sup_{\theta\in\Theta_\infty}\\|g_{n}(\theta) - g(\theta)\\|\leqslant \delta_n, g_n \geqslant 0$. \item There exists a compact subset $ \tilde{\Theta}_\infty \subset \Theta_\infty $ such that for any $\theta_0 \in \tilde{\Theta}_\infty$, there exists $\theta \in\Theta_n$ such that $ d(\theta,\theta_0 )\leqslant \epsilon_n.$ \end{enumerate} Consider the following optimization problems: \begin{align} \min_{\theta \in \Theta_n } & f( \theta ) , \notag\\ s.t. &\ g_{n}( \theta ) \leqslant L_g\epsilon_{n} + \delta_n, \label{eq:opt_n} \end{align} \begin{align} \min_{\theta \in \Theta_\infty } & f( \theta ) ,\notag\\ s.t. &\ g( \theta ) = 0. \label{eq:opt*} \end{align} and \begin{align} \min_{\theta \in \tilde{\Theta}_\infty } & f( \theta ) ,\notag\\ s.t. &\ g( \theta ) = 0. \label{eq:opt*_sub} \end{align} We assume that the feasible region of (\ref{eq:opt*}) and (\ref{eq:opt*_sub}) are nonempty. Let $ f_{n}^{*}, f^*, \tilde{f}^* $ be the minimal of (\ref{eq:opt_n}), (\ref{eq:opt*}) and (\ref{eq:opt*_sub}) respectively. Then, $ [\liminf _{k\rightarrow \infty } f_{n}^{*}, \limsup_{k\rightarrow \infty } f_{n}^{*}] \subset [f^{*}, \tilde{f}^*]$. | | appended hypothesis list | ass:prop-convg-opt-setting, ass:prop-convg-opt-compact, ass:prop-convg-opt-approximate-lipschitz, ass:prop-convg-opt-uniform-error, ass:prop-convg-opt-approximation, ass:prop-convg-opt-nonempty | | Lean | `Helpers/ConvgOpt.lean:193` — `Papers.TbsNeuralCausalPi.convgOpt_limits_mem` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity pinned · correctness bookkeeping | | added conditions | `ass:bk-6`, `ass:pin-159`, `ass:pin-172`, `ass:pin-52` | | cited results | — | | assumed substrate gates | — | | findings | [note, V1.5] pinned reading: In (\ref{eq:aprox_lip}), $\Theta=\Theta_\infty$. (Proposition~\ref{prop:convg_opt} defines the functions on $\Theta_\infty$ and uses this inequality there to establish the limiting Lipschitz bound.) | ### 14. lemma:Lip_condition `lem:lip-condition` | field | value | | --- | --- | | bucket | `verified_with_bookkeeping` | | source | `proof.tex:1799-1805` (`lemma`, label `lemma:Lip_condition`) | | statement | \label{lemma:Lip_condition} Let $ (\hat{T} ,\hat{Y}) \sim \mu ,( T,Y) \sim \nu $ and suppose that $ f( t) =\mathbb{E}_{\nu }[ Y\|T] ,\hat{f}( t) =\mathbb{E}_{\mu }[\hat{Y} \|\hat{T}]$ are $ L$-Lipschitz continuous and $\|f(t)\|\leqslant K, \|\hat{f}(t)\| \leqslant K$, then we have \begin{equation*} \int ( f( t) -\hat{f}( t))^{2} d\nu ( dt) \leqslant C_{W} W( \mu ,\nu ) , \end{equation*} where $ C_{W} =4LK+2K\max\{L,1\}$. | | appended hypothesis list | ass:lemma-lip-condition-regression, ass:lemma-lip-condition-bounded | | Lean | `Helpers/RegressionComparison.lean:335` — `Papers.TbsNeuralCausalPi.regression_gap_le_wasserstein` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-206` | | cited results | — | | assumed substrate gates | — | | findings | — | ### 15. lemma:push-forward_err `lem:push-forward-err` | field | value | | --- | --- | | bucket | `repaired` | | source | `proof.tex:1918-1922` (`lemma`, label `lemma:push-forward_err`) | | statement | \label{lemma:push-forward_err} Let $ \mu ,\hat{\mu }$ be two measures on compact set $ K\subset \mathbb{R}^{d_{1}}$, for any measurable functions $ F,\hat{F} :K\rightarrow \mathbb{R}^{d_{2}}$, if $ F$ is $ L$-Lipschitz continuous, then \begin{equation*} W( F_{\#} \mu ,\hat{F}_{\#}\hat{\mu }) \leqslant LW( \mu ,\hat{\mu }) +\\| F_{1} -F_{2} \\| _{\infty } . \end{equation*} | | appended hypothesis list | ass:lemma-push-forward-err-lipschitz | | Lean | `Helpers/WassersteinBounds.lean:154` — `Papers.TbsNeuralCausalPi.wasserstein_map_le` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-fix-72c9e86d`, `ass:pin-fix-f6f485b1` | | cited results | — | | assumed substrate gates | — | | findings | [refuted, V1.5] (a) The printed conclusion uses $F_1,F_2$, although only $F,\hat F$ are defined; the proof derives a comparison of the latter maps and supplies no identification with the former. This is a slip in the asserted bound. (b) Independently, the last inequality in the second-term calculation replaces the $1$-norm by the infinity norm without its dimensional factor. With the binding uniform infinity norm, reproduction gives $LW(\mu,\hat\mu)+\sup_{x\in K}\\|F(x)-\hat F(x)\\|$, which is at most $LW(\mu,\hat\mu)+d_2\\|F-\hat F\\|_\infty$, rather than the asserted coefficient. witness: (a) The step combining the two terms yields an expression involving $F-\hat F$; it cannot imply a bound involving unrelated $F_1,F_2$. Under a reading of these as coordinates of $F$, take the instance below: $F_1=F_2$, so the printed right-hand side is zero while the left-hand side is $2$. (b) Let $K=[0,1]$, let $\mu=\hat\mu$ be uniform on $K$, and set $F(x)=(x,x)$ and $\hat F(x)=(x+1,x+1)$. The map $F$ is $2$-Lipschitz in the paper's $1$-norm. Translation gives $W(F_\#\mu,\hat F_\#\hat\mu)=2$, whereas $W(\mu,\hat\mu)=0$ and $\\|F-\hat F\\|_\infty=1$. Thus even identifying the conclusion's two map names with the intended maps leaves the bound false. minimal patch (confirmed): corrected statement: Let $\mu,\hat\mu$ be two measures on compact set $K\subset\mathbb R^{d_1}$, for any measurable functions $F,\hat F:K\rightarrow\mathbb R^{d_2}$, if $F$ is $L$-Lipschitz continuous, then \begin{equation*} W(F_{\#}\mu,\hat F_{\#}\hat\mu)\leqslant LW(\mu,\hat\mu)+d_2\\|F-\hat F\\|_\infty. \end{equation*} · proof change: For any $ 1$-Lipschitz function $ g:\mathbb{R}^{d_{2}}\rightarrow \mathbb{R}$, \begin{align*} \mathbb{E}_{X\sim F_{\#} \mu }[ g( X)] -\mathbb{E}_{X\sim \hat{F}_{\#}\hat{\mu }}[ g( X)] & =\mathbb{E}_{X\sim \mu }[ g\circ F( X)] -\mathbb{E}_{X\sim \hat{\mu }}[ g\circ \hat{F}( X)]\\ & =(\mathbb{E}_{X\sim \mu }[ g\circ F( X)] -\mathbb{E}_{X\sim \hat{\mu }}[ g\circ F( X)]) \\ &\quad +(\mathbb{E}_{X\sim \hat{\mu }}[ g\circ F( X)] -\mathbb{E}_{X\sim \hat{\mu }}[ g\circ \hat{F}( X)]) . \end{align*} Note that for any $x,y\in K$, \begin{align*} \|g\circ F(x)-g\circ F(y)\|\leqslant\\|F(x)-F(y)\\|\leqslant L\\|x-y\\|. \end{align*} Thus $g\circ F$ is $L$-Lipschitz continuous and \begin{align}\label{eq:first_term} \mathbb E_{X\sim\mu}[g\circ F(X)]-\mathbb E_{X\sim\hat\mu}[g\circ F(X)]\leqslant LW(\mu,\hat\mu). \end{align} For the second term, \begin{align}\label{eq:second_term} \mathbb E_{X\sim\hat\mu}[g\circ F(X)]-\mathbb E_{X\sim\hat\mu}[g\circ\hat F(X)] &\leqslant\mathbb E_{X\sim\hat\mu}\|g(F(X))-g(\hat F(X))\|\notag\\ &\leqslant\mathbb E_{X\sim\hat\mu}\\|F(X)-\hat F(X)\\|\notag\\ &\leqslant\sup_{x\in K}\\|F(x)-\hat F(x)\\|\notag\\ &\leqslant d_2\\|F-\hat F\\|_\infty. \end{align} The last inequality follows by summing the $d_2$ coordinatewise bounds. If the uniform error is infinite, the asserted bound holds in the extended nonnegative reals. Combining (\ref{eq:first_term}) and (\ref{eq:second_term}) bounds the difference of expectations by $LW(\mu,\hat\mu)+d_2\\|F-\hat F\\|_\infty$ for every scalar $1$-Lipschitz function $g$. Taking the supremum over these functions gives the asserted bound for $W(F_{\#}\mu,\hat F_{\#}\hat\mu)$. | ### 16. lemma:combin_err `lem:combin-err` | field | value | | --- | --- | | bucket | `verified_with_bookkeeping` | | source | `proof.tex:1945-1949` (`lemma`, label `lemma:combin_err`) | | statement | \label{lemma:combin_err} Let $ \mu _{1} ,\cdots ,\mu _{n} ,\hat{\mu }_{1} ,\cdots ,\hat{\mu }_{n}$ be measure on $ \mathbb{R}^{d}$, for any $ p_{1} ,\cdots ,p_{n} \in [ 0,1]$ such that $ \sum _{k=1}^{n} p_{k} =1$, we have \begin{equation*} W\left(\sum _{k=1}^{n} p_{k} \mu _{k} ,\sum _{k=1}^{n} p_{k}\hat{\mu }_{k}\right) \leqslant \sum _{k=1}^{n} p_{k} W( \mu _{k} ,\hat{\mu }_{k}) . \end{equation*} | | appended hypothesis list | ass:lemma-combin-err-weights | | Lean | `Helpers/WassersteinBounds.lean:303` — `Papers.TbsNeuralCausalPi.wasserstein_mixture_le` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity pinned · correctness bookkeeping | | added conditions | `ass:bk-1`, `ass:pin-75` | | cited results | — | | assumed substrate gates | — | | findings | [note, V1.5] pinned reading: For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. (Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula.) | ### 17. lemma:product_err `lem:product-err` | field | value | | --- | --- | | bucket | `verified_with_bookkeeping` | | source | `proof.tex:1963-1968` (`lemma`, label `lemma:product_err`) | | statement | \label{lemma:product_err} Given two measures $ \mu =\mu _{1} \otimes \cdots \otimes \mu _{n} ,\hat{\mu } =\hat{\mu }_{1} \otimes \cdots \otimes \hat{\mu }_{n}$, we have \begin{equation*} W( \mu ,\hat{\mu }) \leqslant \sum _{k=1}^{n} W( \mu _{k} ,\hat{\mu }_{k}) . \end{equation*} | | Lean | `Helpers/WassersteinBounds.lean:467` — `Papers.TbsNeuralCausalPi.wasserstein_pi_le` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-fix-b83fa4d1`, `ass:pin-fix-ca667e13` | | cited results | — | | assumed substrate gates | — | | findings | [note, V1.5] clarified — the reading the statement means is now stated: [F2.5 reviewer] [statement-wrong] “Measures” does not specify probability laws versus equal-mass finite measures, whose transport costs differ from Lean’s probability-coupling-only extension. Orchestrator: explicitly require probability laws in each factor, as in the proof and every main-result use; correct the final proof line to the sum of factor costs plus n times epsilon. The finite-measure reading is false: factors 2*delta_0 versus 2*delta_1 and 2*delta_0 versus itself have product transport cost 4 and sum 2. No added main hypothesis. Reading(s) the referee pinned for this statement, to be written into it: $W$ is Wasserstein-1 distance for the ground distance $\\|x-y\\|$, and $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. (the reading of def:distribution-distances) minimal patch (confirmed): corrected statement: Given two measures $\mu=\mu_1\otimes\cdots\otimes\mu_n$, $\hat\mu=\hat\mu_1\otimes\cdots\otimes\hat\mu_n$, where every factor $\mu_k$ and $\hat\mu_k$ is a probability measure, we have \begin{equation*} W(\mu,\hat\mu)\leqslant\sum_{k=1}^n W(\mu_k,\hat\mu_k). \end{equation*} · proof change: If $n=0$, both empty products are the probability measure on the one-point space and both sides are zero. Suppose $n\geqslant1$. If any $W(\mu_k,\hat\mu_k)=\infty$, the inequality holds in the extended nonnegative reals. Otherwise, for any $\epsilon>0$ and every $k\in[n]$, choose a probability coupling $\pi_k$ of $\mu_k$ and $\hat\mu_k$ such that \begin{equation*} \mathbb E_{(X_k,Y_k)\sim\pi_k}[\\|X_k-Y_k\\|_1]\leqslant W(\mu_k,\hat\mu_k)+\epsilon. \end{equation*} Such a coupling exists by the definition of the finite infimum defining $W$. Under the product probability law $\pi_1\otimes\cdots\otimes\pi_n$, put $X=(X_1,\ldots,X_n)$ and $Y=(Y_1,\ldots,Y_n)$. The marginal laws of $X$ and $Y$ are respectively $\mu$ and $\hat\mu$, because the coordinate pairs are independent and have the prescribed marginal laws. Thus this construction is a coupling of the product measures. The product $1$-norm is the sum of the coordinate $1$-norms, so \begin{align*} W(\mu,\hat\mu) &\leqslant\mathbb E_{\pi_1\otimes\cdots\otimes\pi_n}[\\|X-Y\\|]\\ &=\sum_{k=1}^n\mathbb E_{(X_k,Y_k)\sim\pi_k}[\\|X_k-Y_k\\|_1]\\ &\leqslant\sum_{k=1}^n W(\mu_k,\hat\mu_k)+n\epsilon. \end{align*} Letting $\epsilon\downarrow0$ proves the result. | ### 18. Approximation error of Gumbel-Softmax layer `lem:gumbel-err` | field | value | | --- | --- | | bucket | `verified_with_bookkeeping` | | source | `proof.tex:1981-1990` (`lemma`, label `lemma:gumbel_err`) | | statement | \label{lemma:gumbel_err} Given $1 >\tau >0$ and $p_{1} ,\cdots ,p_{n} >0$, let $\mu ^{\tau } \sim X^{\tau } =\left( X_{1}^{\tau } ,\cdots ,X_{n}^{\tau }\right)$ be \begin{equation} X_{k} =\frac{\exp((\log p_{k} +G_{k}) /\tau )}{\sum _{k=1}^{n}\exp((\log p_{k} +G_{k}) /\tau )} ,\label{eq:gumbel} \end{equation} where $G_{k} \sim \exp( -x-\exp( -x))$ are i.i.d. standard Gumbel variables. Let $\mu ^{0}$ be the distribution of $X^{0} =\lim _{\tau \rightarrow 0} X^{\tau } =\left( X_{1}^{0} ,\cdots ,X_{n}^{0}\right)$. Then, \begin{equation*} W\left( \mu ^{\tau } ,\mu ^{0}\right) \leqslant 2( n-1) \tau -2n( n-1) e^{-1} \tau \log \tau . \end{equation*} | | appended hypothesis list | ass:lemma-gumbel-err-parameters | | Lean | `Helpers/GumbelSoftmax.lean:515` — `Papers.TbsNeuralCausalPi.wasserstein_gumbelSoftmax_le` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity pinned · correctness ok | | added conditions | `ass:pin-203`, `ass:pin-211` | | cited results | — | | assumed substrate gates | — | | findings | [note, V1.5] pinned reading: The gap event is $E_\delta=\{\exists\,1\leqslant i0$, we have \begin{align*} 0\leqslant W_{1,\lambda}( \mu ,\nu ) -W( \mu ,\nu ) & \leqslant (\log( mn) +1) \lambda, \end{align*} where $ W_{1,\lambda}(\cdot,\cdot) $ is the entropy regularized Wasserstein-1 distance. Moreover, we have the following estimation of approximation error. \begin{align*} \|S_{\lambda }( \mu ,\nu ) -W( \mu ,\nu ) \| & \leqslant 2(\log( mn) +1) \lambda . \end{align*} | | Lean | `Helpers/Sinkhorn.lean:164` — `Papers.TbsNeuralCausalPi.sinkhorn_approx` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-fix-761562d5` | | cited results | — | | assumed substrate gates | — | | findings | [refuted, V1.5] the step its proof takes from [luiseDifferentialPropertiesSinkhorn2018, Proposition 1] (lem:sinkhorn-luise-proposition-1) does not hold as the proof uses it: The full entropy-penalized optimal value and the transport cost evaluated at an entropy-regularized optimizer are different readings of $W_{1,\lambda}$; their differences from $W$ can have different signs. The host does not identify either functional, its entropy convention, or the ground-cost data. The cited proposition gives absolute approximation bounds for two separately defined functionals, with an inverse regularization parameter. It does not justify the host's signed assertion for its unspecified functional; in particular, the source's full regularized value violates the asserted lower bound. witness: The source defines its full regularized value by minimizing transport cost minus entropy divided by its parameter. Even after reciprocal reparameterization to the paper's $\lambda$, this is transport cost minus $\lambda h(T)$. On support $\{0,1\}$ with ground distance $\|x-y\|$, take $\mu=(3/4,1/4)$ and $\nu=(1/4,3/4)$. Then $W(\mu,\nu)=1/2$, and the optimal unregularized plan $T^*=\left(\begin{smallmatrix}1/4&1/2\\0&1/4\end{smallmatrix}\right)$ has $h(T^*)=1+(3/2)\log2$. Evaluating the regularized objective at this feasible plan gives $W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)\leqslant-\lambda(1+(3/2)\log2)<0$. Thus the cited regularized functional cannot supply the printed step $0\leqslant W_{1,\lambda}-W$. Selecting the sharp transport-cost functional would require an identification the host does not provide. minimal patch (confirmed): corrected statement: For any $\mu\in\Delta_n$, $\nu\in\Delta_m$, and $\lambda>0$, let $\mu$ and $\nu$ be the weights of finite probability measures on points $x_1,\ldots,x_n$ and $y_1,\ldots,y_m$ in a common Euclidean space with the paper's $1$-norm. Put $c_{ij}=\\|x_i-y_j\\|$ and \[ \Pi(\mu,\nu)=\left\{T\in[0,\infty)^{n\times m}:\sum_jT_{ij}=\mu_i,\ \sum_iT_{ij}=\nu_j\right\},\qquad h(T)=1-\sum_{i,j}T_{ij}\log T_{ij}, \] with $0\log0=0$. Here the entropy regularized Wasserstein-1 distance is \[ W_{1,\lambda}(\mu,\nu)=\min_{T\in\Pi(\mu,\nu)}\left\{\sum_{i,j}c_{ij}T_{ij}-\lambda h(T)\right\}, \] and $W(\mu,\nu)=\min_{T\in\Pi(\mu,\nu)}\sum_{i,j}c_{ij}T_{ij}$. Then \begin{align*} -(\log(mn)+1)\lambda\leqslant W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)\leqslant0. \end{align*} Moreover, we have the following estimation of approximation error. \begin{align*} \|S_{\lambda }( \mu ,\nu ) -W( \mu ,\nu ) \| & \leqslant 2(\log( mn) +1) \lambda . \end{align*} · proof change: The transportation polytope is nonempty and compact, and both objectives are continuous under $0\log0=0$, so their minima exist. Let $T^*$ minimize the unregularized transport cost and let $T_\lambda$ minimize the regularized objective. Since $h(T)\geqslant0$ for every probability plan, \[ W_{1,\lambda}(\mu,\nu)\leqslant\sum_{i,j}c_{ij}T^*_{ij}-\lambda h(T^*)\leqslant W(\mu,\nu). \] Also $\sum_{i,j}c_{ij}(T_\lambda)_{ij}\geqslant W(\mu,\nu)$. It follows that \begin{align}\label{eq:err_reg_wass} -\lambda\max_{T\in\Pi(\mu,\nu)}h(T)\leqslant W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)\leqslant0. \end{align} Since $ T_{ij} \in [ 0,1]$ and $ f( x) =-x\log x$ is concave, we have \begin{align*} h( T) = & -nm\cdot \frac{1}{nm}\sum _{i,j=1}^{n,m} T_{ij}\log T_{ij} +1.\\ & \leqslant -nm\cdot \left(\frac{1}{nm}\sum _{i,j=1}^{n,m} T_{ij}\right)\log\left(\frac{1}{nm}\sum _{i,j=1}^{n,m} T_{ij}\right) +1\\ & =1+\log( nm) . \end{align*} By definition of Sinkhorn distance, \begin{align*} S_{\lambda }( \mu ,\nu ) & = W_{1,\lambda }( \mu ,\nu ) -W_{1,\lambda }( \mu ,\mu )/2 -W_{1,\lambda }( \nu ,\nu )/2 . \end{align*} Let $a=W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)$, $b=W_{1,\lambda}(\mu,\mu)$, and $c=W_{1,\lambda}(\nu,\nu)$. The self-transport Wasserstein costs are zero. By (\ref{eq:err_reg_wass}) and the entropy calculation, \[ \|a\|\leqslant(1+\log(nm))\lambda,\quad \|b\|\leqslant(1+2\log n)\lambda,\quad \|c\|\leqslant(1+2\log m)\lambda. \] Consequently \begin{align*} \|S_\lambda(\mu,\nu)-W(\mu,\nu)\| &=\|a-b/2-c/2\|\\ &\leqslant\|a\|+\|b\|/2+\|c\|/2\\ &\leqslant2(1+\log(nm))\lambda. \end{align*} | ### 20. lemma:sup_wasserstein `lem:sup-wasserstein` | field | value | | --- | --- | | bucket | `repaired` | | source | `proof.tex:2061-2067` (`lemma`, label `lemma:sup_wasserstein`) | | statement | \label{lemma:sup_wasserstein} Given a measure $ \mu $ on $ \mathbb{R}^{d}$ and a real function class $ \mathcal{F}$ with output dimension $ d$, suppose that the pseudo-dimension of $ \mathcal{F}$ is less than $ d_{\mathcal{F}} < \infty $ and there exists $K>0$ such that $ \|f( x) \|\leqslant K,\forall f\in \mathcal{F} ,x\in \mathbb{R}^{d'}$, then with probability at least $ 1-\exp\left( O\left( \delta ^{-d}\log\left( \delta ^{-1}\right) +d_{\mathcal{F}}\log \epsilon ^{-1} -n\epsilon ^{2}\right)\right)$, \begin{align*} \sup _{f\in \mathcal{F}} W( f_{\#} \mu ,f_{\#} \mu _{n}) & \leqslant \delta +\epsilon \end{align*} for all $ \delta ,\epsilon >0$, where $ \mu _{n}$ is the empirical distribution of $ \mu $. | | appended hypothesis list | ass:lemma-sup-wasserstein-pseudodimension, ass:lemma-sup-wasserstein-bounded | | Lean | `Helpers/EmpiricalWasserstein.lean:190` — `Papers.TbsNeuralCausalPi.uniform_empirical_wasserstein` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-207`, `ass:pin-fix-448b19a0`, `ass:pin-fix-d19c63b5` | | cited results | `lem:sup-wasserstein-anthony-theorem-17-1` (anthony1999neural, Theorem 17.1); `lem:sup-wasserstein-anthony-theorems` (anthony1999neural, Theorem 18.4) | | assumed substrate gates | — | | findings | [refuted, V1.5] the step its proof takes from [anthony1999neural, Theorems 18.4 and 17.1] (lem:sup-wasserstein-anthony-theorems) does not hold as the proof uses it: (a) For vector-valued $\mathcal F$, its pseudo-dimension could mean $\max_{j\in[d]}\operatorname{Pdim}(\{f_j:f\in\mathcal F\})$ or $\operatorname{Pdim}(\{(x,j)\mapsto f_j(x):f\in\mathcal F\})$. These impose different bounds, and neither the paper nor the scalar cited theorem specifies the choice. The dual formulation does determine that $h$ is scalar-valued. (b) The attributed covering inequality is false for the paper's explicitly unnormalized empirical $\ell_1$ metric: its bound omits sample-size scaling. (c) It also omits range scaling: Theorem 18.4 requires scalar functions valued in $[0,1]$, whereas the host only bounds their magnitude by arbitrary $K$. The fixed-$h$ concentration order can be recovered from Theorem 17.1 by range normalization and a square-root transformation with zero responses, but that does not validate the printed covering inequality. witness: (b) Take $d=d'=1$, $\mu=\mathrm{Unif}([-1,1])$, $K=1$, $h(x)=x$, and $\mathcal F=\{f_c:c\in[0,1/2]\}$, where $f_c(x)=c+\tfrac12\mathbb{I}_{x\geqslant0}$. This nonconstant class has pseudo-dimension $1$, so choose $d_{\mathcal F}=2$. Its evaluation vectors have distance $n\|c-c'\|$ in the printed metric. With $n=10000$ and $\epsilon=1/4$, every covering ball covers a parameter interval of length at most $2\epsilon/n$, so $\mathcal N_1(\epsilon,\mathcal F,n)\geqslant10000$. The asserted upper bound is $3e(8e)^2=192e^3<4000$, a contradiction. (c) Independently take $K=10000$, $n=2$, and $f_c(x)=c+(K/2)\mathbb{I}_{x\geqslant0}$ for $c\in[0,K/2]$, keeping the other choices. The host's hypotheses again hold with pseudo-dimension $1$. At $\epsilon=1/4$, its printed covering number is at least $20000$; even the averaged empirical metric gives a lower bound of $10000$. Both exceed the same asserted bound $192e^3$. Rescaling the range to $[0,1]$ changes the covering radius and introduces the omitted dependence on $K$. minimal patch (confirmed): corrected statement: Given a probability measure $\mu$ on $\mathbb R^{d'}$ and a class $\mathcal F$ of measurable functions $f:\mathbb R^{d'}\rightarrow\mathbb R^d$, suppose that each scalar coordinate class $\mathcal F_j=\{x\mapsto f_j(x):f\in\mathcal F\}$, $j\in[d]$, has pseudo-dimension less than $d_{\mathcal F}<\infty$ and there exists $K>0$ such that $\\|f(x)\\|_\infty\leqslant K$ for all $f\in\mathcal F$ and $x\in\mathbb R^{d'}$. Suppose in addition that there are a standard Borel space $T$ and a jointly Borel map $G:T\times\mathbb R^{d'}\rightarrow[-K,K]^d$ such that $\mathcal F=\{G(t,\cdot):t\in T\}$. There are positive constants $C,c$ depending only on $d,K$ such that, with probability at least \[ 1-\exp\left(C\left((1+\delta^{-d})\log(2+\delta^{-1})+d_{\mathcal F}\log(2+\epsilon^{-1})\right)-cn\epsilon^2\right), \] \begin{align*} \sup_{f\in\mathcal F}W(f_{\#}\mu,f_{\#}\mu_n)\leqslant\delta+\epsilon \end{align*} for all $\delta,\epsilon>0$, where $\mu_n$ is the empirical distribution of $\mu$. · proof change: The empty class satisfies the uniform assertion vacuously, so assume $T\ne\varnothing$. Let $X_1,\ldots,X_n$ be independent observations with law $\mu$. By the dual formulation of Wasserstein distance, \begin{align*} \sup_{f\in\mathcal F}W(f_{\#}\mu,f_{\#}\mu_n) =\sup_{h\in\mathcal H}\sup_{f\in\mathcal F}\left(\mathbb E_{X\sim\mu_n}h(f(X))-\mathbb E_{X\sim\mu}h(f(X))\right), \end{align*} where \[ \mathcal H=\{h:[-K,K]^d\rightarrow\mathbb R:h(0)=0,\ \|h(x)-h(y)\|\leqslant\\|x-y\\|\}. \] Every such $h$ takes values in $[-dK,dK]$. Retain the unnormalized empirical covering number \begin{align*} \mathcal N_1(r,\mathcal F,N)=\sup_{x_1,\ldots,x_N}\mathcal N\left(r,\{(f(x_1),\ldots,f(x_N)):f\in\mathcal F\},\\|\cdotp\\|_1\right), \end{align*} where the $1$-norm on the evaluation vector sums over all sample points and output coordinates. With the uniform norm, $\mathcal H$ is closed, uniformly bounded and equicontinuous on the compact cube, hence is a compact metric space by Arzela--Ascoli. Evaluation $(h,y)\mapsto h(y)$ is continuous: its increment is bounded by $\\|h-h'\\|_\infty+\\|y-y'\\|$. Consequently $(t,h,x)\mapsto h(G(t,x))$ is jointly Borel on $T\times\mathcal H\times\mathbb R^{d'}$. Its bounded population integral is Borel in $(t,h)$: this holds for indicators of measurable rectangles, is preserved by bounded monotone limits and linear combinations, and extends to bounded Borel functions by the monotone class theorem. Every strict deviation event is the projection of a Borel set from this standard Borel parameter space and is therefore analytic and universally measurable; weak supremum events are countable intersections of strict ones. The same reasoning applies to two samples and finite coordinate swaps. All probabilities below are thus defined in the completion of the sample law. Obviously, if $ h$ is 1-Lipschitz, \begin{align*} \mathcal{N}_{1}( \epsilon ,h\circ \mathcal{F} ,n) & \leqslant \mathcal{N}_{1}( \epsilon ,\mathcal{F} ,n) . \end{align*} Put $v=\lfloor d_{\mathcal F}\rfloor$. Each coordinate class has pseudo-dimension at most $v$. Normalize its range by $f_j\mapsto(f_j+K)/(2K)$. Applying Theorem 18.4 of \cite{anthony1999neural} in its averaged empirical metric, and then undoing the normalization, bounds a scalar unnormalized cover of radius $r/(2d)$ on $N$ points by \[ e(v+1)\left(2+\frac{8eKdN}{r}\right)^v. \] The additional $2$ allows radii beyond the normalized unit range. Take the product of these $d$ covers. For every product ball meeting the evaluation vectors of $\mathcal F$, choose one such vector as its center. Two vectors in that product ball differ by at most $r/d$ in each scalar unnormalized metric, hence by at most $r$ in the summed vector metric. Thus \[ \mathcal N_1(r,\mathcal F,N)\leqslant\left[e(v+1)\left(2+\frac{8eKdN}{r}\right)^v\right]^d. \] This exhibits both sample-size and range scaling. Fix $h\in\mathcal H$, and define \[ a_f(x)=\frac{h(f(x))+dK}{2dK},\qquad z_f(x)=\sqrt{a_f(x)}. \] Both functions take values in $[0,1]$. On $N$ points, the inequality $\|\sqrt a-\sqrt b\|\leqslant\sqrt{\|a-b\|}$ and the Cauchy--Schwarz inequality give \[ \frac1N\sum_{i=1}^N\|z_f(x_i)-z_g(x_i)\|\leqslant\left(\frac{1}{2dKN}\sum_{i=1}^N\\|f(x_i)-g(x_i)\\|\right)^{1/2}. \] Consequently an unnormalized vector cover of radius $2dKNr^2$ gives an averaged scalar cover of the class $\{z_f:f\in\mathcal F\}$ of radius $r$, whose cardinality is at most \[ \left[e(v+1)(2+4e/r^2)^v\right]^d. \] By Theorem 17.1 in \cite{anthony1999neural}, with its standing measurability condition, let $X$ and $T_0$ be standard Borel spaces, let $g:T_0\times X\rightarrow[0,1]$ be jointly Borel, and put $\mathcal G=\{g(t,\cdot):t\in T_0\}\ne\varnothing$. For every Borel probability distribution $P$ on $X\times[0,1]$, every $0<\zeta<1$ and every positive integer $m$, \[ P^m\left\{\text{some }t\in T_0\text{ has }\left\|\mathbb E_{(x,y)\sim P}(g(t,x)-y)^2-\frac1m\sum_{i=1}^m(g(t,x_i)-y_i)^2\right\|\geqslant\zeta\right\} \leqslant4\mathcal N_1^{\mathrm{avg}}(\zeta/16,\mathcal G,2m)\exp(-\zeta^2m/32). \] Here the draws are independent, the probability is taken in the completed sample law, and $\mathcal N_1^{\mathrm{avg}}$ uses the averaged empirical metric. For the present class, $(t,x)\mapsto\sqrt{(h(G(t,x))+dK)/(2dK)}$ is jointly Borel, since $h$ is continuous and the square root is continuous on $[0,1]$. Its parameter space is the same standard Borel space $T$, and its constant-response input law is $\mu\otimes\delta_0$. Apply this form of Theorem 17.1 to the class with response identically zero. Its squared loss is $z_f^2=a_f$. For $0<\epsilon<2dK$, put $\eta=\epsilon/(2dK)$ and use the averaged cover at radius $\eta/16$ on $2n$ points. The theorem gives \[ \mathbb P\left(\sup_{f\in\mathcal F}\|(\mu_n-\mu)(h\circ f)\|>\epsilon\right) \leqslant4\left[e(v+1)(2+1024e/\eta^2)^v\right]^d\exp(-n\eta^2/32). \] Since $v\leqslant d_{\mathcal F}$ and $\log(v+1)\leqslant v$, the logarithm of the right-hand side is bounded by $C_0+C_1d_{\mathcal F}\log(2+\epsilon^{-1})-c_1n\epsilon^2$, with constants depending only on $d,K$. If $\epsilon\geqslant2dK$, the deviation is deterministically at most $2dK$, and the desired deviation bound is immediate. Let $\mathcal H_\delta$ be a finite $\delta/2$-net of $\mathcal H$ in the uniform norm. Such a net has \[ \log\|\mathcal H_\delta\|\leqslant C_2(1+\delta^{-d})\log(2+\delta^{-1}), \] where $C_2$ depends only on $d,K$. To see this, take a grid in $[-K,K]^d$ with $1$-norm mesh at most $\delta/8$ and at most $(1+C_3/\delta)^d$ points. Quantize the values of $h$ at these points into bins of width at most $\delta/8$ in $[-dK,dK]$. There are at most $2+C_4/\delta$ bins at each point. Choose one member of $\mathcal H$ from each nonempty quantization pattern. Two functions with the same pattern differ by at most $\delta/8$ at grid points and by at most $3\delta/8$ everywhere, using their Lipschitz bounds and a nearest grid point. These representatives form the required net. Counting patterns gives the displayed entropy bound. Apply the fixed-$h$ concentration bound to every $h\in\mathcal H_\delta$ and take a union bound. For $0<\epsilon<2dK$, after enlarging a constant depending only on $d,K$, \[ \mathbb P\left(\sup_{h\in\mathcal H_\delta,f\in\mathcal F}(\mu_n-\mu)(h\circ f)>\epsilon\right) \leqslant\exp\left(C\left((1+\delta^{-d})\log(2+\delta^{-1})+d_{\mathcal F}\log(2+\epsilon^{-1})\right)-cn\epsilon^2\right). \] For every $h\in\mathcal H$, choose $\tilde h\in\mathcal H_\delta$ with $\\|h-\tilde h\\|_\infty\leqslant\delta/2$. Since $\mu_n$ and $\mu$ are probability measures, \[ \|(\mu_n-\mu)((h-\tilde h)\circ f)\|\leqslant\delta. \] Therefore \[ \sup_{h\in\mathcal H,f\in\mathcal F}(\mu_n-\mu)(h\circ f) \leqslant\delta+\sup_{\tilde h\in\mathcal H_\delta,f\in\mathcal F}(\mu_n-\mu)(\tilde h\circ f). \] The dual formula yields the stated Wasserstein bound. If $\epsilon\geqslant2dK$, both push-forward measures are supported in a set of $1$-norm diameter $2dK$, so their Wasserstein distance is at most $2dK\leqslant\delta+\epsilon$ deterministically. This completes the proof for all positive $\delta,\epsilon$. | ## Errors by root cause 8 independent error(s) — 8 repaired by a minimal patch, 0 not repaired, 0 unfixable and left out of Lean; 5 result(s) inherit one. ### Errors found — repaired The result as printed is false (the witness shows why). The smallest change that keeps the paper's key contribution was confirmed by an independent referee pass; the F stages formalize it in place of the printed statement. Key contribution, as judged by the confirming referee: The key-result list is complete. The repairs of thm:wide-nn, including its split deep-approximation result, prop:holder-curve, thm:deep-nn, prop:counterexample, thm:consistency, and thm:rate remain mutually compatible. Their accepted scope restrictions are reflected in key_contribution. The Fournier invocation noted for def:thm-consistency-architecture-radius creates no wiring defect. The second proof block for thm:wide-nn supplies its second-part argument and does not change the statement's meaning. - **`thm:approximation`** (`Main.tex:240-246`): (a) In (\ref{eq:approximation_1}), the Lipschitz property gives the output $1$-norm, not its coordinatewise infinity norm. The missing dimension factor invalidates the coefficient supplied by the proof; the hypotheses do not bound continuous output dimensions through $L$, $K$, or the graph. (b) In (\ref{eq:cate_2}), the probabilities $p_j$ vary with the variables being integrated. The omitted term is $\sum_j(p_j(x)-p_j(y))g(y,j)$, so the asserted kernel Lipschitz bound and (\ref{eq:approximation_middle_step}) are false. A bound using $1+KL$ can close this categorical step, but does not justify the printed coefficient. Zero propensities could alternatively be excluded or interpreted through an extended logarithm; the closed-simplex canonical convention determines the latter. - witness: (a) Take the graph with two isolated roots $T$ and $V_1\in\mathbb{R}^{100}$, independent $U_T,U_1\sim\mathrm{Unif}([-1,1])$, $T=U_T$, and $f_1(u)=(u/100,\ldots,u/100)$. Let $\hat f_1(u)=f_1(u)+(1/100,\ldots,1/100)$ and keep all latent laws and the treatment function unchanged. Both true structural functions are $1$-Lipschitz in the paper's $1$-norm, and the true variables satisfy the bound $K=1$. Under $T=0$, $S=1/100$ and the latent Wasserstein distance is zero, whereas the output Wasserstein distance is $1$. The proof chooses $C_{\mathcal G}(1,1)=4$, giving the false inequality $1\leqslant4/100$. More generally, translating an unrestricted $D$-dimensional output by $(a,\ldots,a)$ gives Wasserstein error $Da$ but structural infinity error $a$, precluding a dimension-independent coefficient for the fixed abstract graph. (b) Take a four-category root with $p(u)=(u,1/10,1/10,4/5-u)$, true $U\sim\mathrm{Unif}([1/4,1/2])$, and comparison $\hat U\sim\mathrm{Unif}([1/4+h,1/2+h])$, where $01200000$. Here $n_V=3$ and $C_W=48K$, so the asserted right-hand side is at most $48K+4=480052$. The false load-bearing step is the claimed regression Lipschitz bound: the actual slope bound is $(N-1)/2$, exceeding $8$. (b) For a separate all-continuous example, take independent $U_Z\sim\operatorname{Unif}([-1,1])$ and $U_T\sim\operatorname{Unif}([0,1])$, with $Z=U_Z$, $X=Z/100$, $T=U_T+X/4$, and $Y=(Z+T)/2$. The graph has arrows $Z\rightarrow X$, $X\rightarrow T$, $Z\rightarrow Y$, and $T\rightarrow Y$, without hidden confounding. All structural functions have Lipschitz constant at most $1$, and $K=2$ bounds the variables. With $P$ uniform on $[-1,2]$, overlap holds on $[1/10,9/10]$ with $\delta=1$. But $f_y(x,t)=50x+t/2$ has Lipschitz constant at least $50$, whereas $(L+1)^{n_V}=16$. The relevant ancestor noise $U_Z=100X$ is not independent of $X$. (c) In the unconfounded continuous chain $T=U_T$, $Z=4T$, $Y=4Z$, with $U_T$ uniform on $[0,1]$, take $L=4$, $K=17$, $P$ uniform on $[0,1]$, and $\delta=1$. Then $n_V=3$, $L_y=16$, and $L_y^2=256>125=(L+1)^{n_V}$. A comparison model with $\hat T=U_T/2$ and the same downstream functions has positive Wasserstein distance, so replacing the quadratic coefficient by the printed one is not a valid bound. (d) Take $P$ uniform on $[0,1/2]$, $t_1=0$, and $t_2=1/2$. Then $\int_{t_1}^{t_2}1\,P(dt)=1$, while $t_2-t_1=1/2$. This comparison measure is compatible with every premise, for example with independent $T\sim P$, $U_Y\sim\operatorname{Unif}([0,1])$, and the two unconfounded models $Y=T+U_Y/10$ and $\hat Y=T+U_Y/10+1/10$, using $L=K=\delta=1$. Thus the constant remainder cannot be integrated as the printed proof does. - Claim: Proposition~\ref{prop:Lip_ATE} bounds the integrated squared difference of interventional outcome means by the displayed linear and quadratic Wasserstein terms under overlap, no confounding, and the structural Lipschitz and boundedness assumptions. - What is wrong: (a) ``Note that $ f_{y}( x,t) =\mathbb{E}_{{\mathcal{M}} }[ Y|X=x,T=t]$ is $ L_{y}$-Lipschitz continuous with Lipschitz constant $ L_{y} \leqslant ( L+1)^{n_{V}}$.'' Lipschitz categorical propensity vectors can produce an outcome regression whose slope exceeds $(L+1)^{n_V}$ because distances between categorical labels enter the outcome mean. (b) ``This is because $Y$ can be written as $ Y=F_{0}( X,T,\boldsymbol{U}_{y})$ where $ \boldsymbol{U}_{y}$ are latent variables independent of $ T,X$ and $ F_{0}$ is composition of $ f_{i}$ in structure equations.'' An observed ancestor can affect both $X=\text{Pa}(T)$ and $Y$, so its latent input to $Y$ need not be independent of $X$ even without hidden confounding. (c) ``\dots \vsDisplay{\begin{array}{@{}c@{}}\displaystyle \left(\int f_{y}( x,t) \nu ( dx) -\int \hat{f}_{y}( x,t) \nu ( dx)\right)^{2} \geqslant \frac{1}{2}\left(\int f_{y}( x,t) \nu ( dx) -\int \hat{f}_{y}( x,t) \mu ( dx)\right)^{2} \\ \displaystyle \ \ \ \ \ \ \ \ \ \ \ \ -\left(\int \hat{f}_{y}( x,t) \mu ( dx) -\int \hat{f}_{y}( x,t) \nu ( dx)\right)^{2} \\ \displaystyle \geqslant \frac{1}{2}\left(\int f_{y}( x,t) \nu ( dx) -\int \hat{f}_{y}( x,t) \mu ( dx)\right)^{2} -L_{y}^{2} W^{2}( \mu ,\nu ) \\ \displaystyle =\frac{1}{2}(\mathbb{E}_{\mathcal{M} }[ Y( t)] -\mathbb{E}_{\hat{\mathcal{M}} }[\hat{Y}( t)])^{2} -L_{y}^{2} W^{2}( \mu ,\nu ) .\end{array}\mbox{''}} The observational bound $|\hat Y|\leqslant K$ bounds its regression only almost surely under its observational input law and need not bound a structural regression version that represents intervention means at inputs outside that law's support. (d) ``\dots we get \vsDisplay{\int _{t_{1}}^{t_{2}}(\mathbb{E}_{\mathcal{M} }[ Y( t)] -\mathbb{E}_{\hat{\mathcal{M}} }[\hat{Y}( t)])^{2} P(dt) \leqslant \frac{2C_W}{\delta } W( \mu ,\nu ) +2L_{y}^{2} W^{2}( \mu ,\nu )( t_{2} -t_{1}) .\mbox{''}} The quadratic coefficient obtained here is $2L_y^2$, and the bound $L_y\leqslant(L+1)^{n_V}$ does not justify the statement's coefficient $2(L+1)^{n_V}$. (e) In the step quoted in (d): Integrating a constant against $P(dt)$ on $[t_1,t_2]$ gives its multiple by $P([t_1,t_2])$, whereas the displayed remainder uses $t_2-t_1$ and the nondescendant calculation switches to Lebesgue measure. Counterexample: Take the unconfounded continuous graph $T\rightarrow V_2\rightarrow V_3\rightarrow Y$, with $n_V=4$, $L=100$, $K=1$, $p=10^{-8}$, and $\eta=10^{-9}$. Let $U_1$ have law $P=(1-p)\operatorname{Unif}([0,\eta])+p\operatorname{Unif}([1-\eta,1])$, and let the other three singleton canonical noises be independent uniform variables on $[0,1]$, independent of $U_1$ and unused in the structural equations. Define $\phi(x)=\min\{\max\{100x,0\},1\}$. In $\mathcal{M}$, set $T=U_1$, $V_2=\phi(T)$, $V_3=\phi(V_2)$, and $Y=\phi(V_3)$. In $\hat{\mathcal{M}}$, use the same latent law and set $\hat T=\min\{U_1,\eta\}$, $\hat V_2=100\hat T$, $\hat V_3=100\hat V_2$, and $\hat Y=100\hat V_3$. Take $t_1=0$, $t_2=1$, and $\delta=1$, so that $T$ has law $P$ and the conditional density relative to $P$ is $p_\nu(t)=1$. Couple the models using their common latent variables, so their observed vectors agree on the first mixture component and each coordinate difference is at most $1$ on the second component, giving $W(\mu,\nu)\leqslant4p$. All observational variables and non-Gumbel latents are bounded by $1$ and all structural functions are at most $100$-Lipschitz, but on $[1-\eta,1]$ the intervention means are $1$ and $10^6t$, so the left-hand side is at least $p(10^6(1-\eta)-1)^2>9999$, whereas the printed right-hand side is at most $8C_Wp+32(101^4)p^2<50$ with $C_W=6(101^4)=624362406$. - Proposed fix: Condition on all observed nondescendants of $T$ and couple categorical transitions to obtain $L_y=(LD+1)^{n_V}$, where $D=\max(\{1\}\cup\{(n_i-1)/2:V_i\text{ is categorical}\})$. Derive the structural regression bound $B=K(1+2qL_y)$ on the bounded input domain from an observational regression value and the Lipschitz inequality, where $q$ is the numerical dimension of $\boldsymbol{V}$, and replace the linear constant by $C_W=4L_yB+2B\max\{L_y,1\}$. Retain the quadratic coefficient $2L_y^2$ and integrate its remainder as $P([t_1,t_2])$. No additional hypothesis is needed; these constants protect the full $O(\sqrt{\alpha_n})$ rate of Corollary~\ref{cor:rate} under its existing assumptions. - Gives up: The linear coefficient based on a bound of $K$ for structural regressions at every intervention input, the regression coefficient independent of categorical cardinalities, the unsquared quadratic coefficient $2(L+1)^{n_V}$, and interval-length scaling of the remainder for arbitrary probability measures $P$. - repair confirmed by an independent referee pass (see Corrected version); the corrected statement is proved in Lean - corrected statement: Given two causal models $ \mathcal{M} ,\hat{\mathcal{M}} \in\mathcal{M}(\mathcal{G},\mathcal{F}_L,\boldsymbol{U})$ satisfying \cref{assump:independence_u} and \cref{assump:Lip_and_bounded}, let their observational distributions be $ \nu ,\mu $. Suppose the norms of all variables are bounded by $K>0$. If (1) (Overlap) $\nu$ is absolutely continuous with respect to one probability measure $P$ and the density $p_{\nu}\left(t|\text{Pa}(T)=x\right)\geqslant\delta>0$ for $x$ almost surely and $t\in[t_1,t_2]$ and (2) (No Confounding) there is no hidden confounder in the causal graph $\mathcal{G}$, we have \begin{equation*} \begin{aligned} \int_{t_1}^{t_2}(\mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}(t)])^2 P(dt) &\leqslant\frac{2C_W}{\delta}W(\mu,\nu)+2L_y^2W^2(\mu,\nu)P([t_1,t_2]), \end{aligned} \end{equation*} where \[ D=\max\left(\{1\}\cup\left\{\frac{n_i-1}{2}:V_i\text{ is categorical}\right\}\right),\qquad L_y=(LD+1)^{n_V}, \] \[ q=\sum_{i:V_i\text{ is continuous}}d_i+\left|\{i:V_i\text{ is categorical}\}\right|,\qquad B=K(1+2qL_y), \] and $C_W=4L_yB+2B\max\{L_y,1\}$. - inherited by: `thm:rate` - **`prop:equivalent-g-constrianed`** (`proof.tex:600-602`): ``Same smoothness'' could mean only the same differentiability order, or preservation of the original regularity through coordinate regrouping, including global Lipschitz regularity. The proof's explicit retention of the original functions determines the latter reading. (a) The original categorical assumption permits unnormalized functions, but the printed construction retains them unchanged while claiming the normalized canonical form. (b) Normalizing those functions does not generally preserve global Lipschitz regularity, so normalization is not a harmless missing line under the proposition's unrestricted domains. - witness: (a) The categorical assumption gives probabilities $f_i/\|f_i\|_1$, whereas the canonical definition requires $\|f_i\|_1=1$. Regrouping coordinates does not change this norm, so retaining an unnormalized $f_i$ does not produce the claimed canonical form. (b) Let independent $U_1\sim\mathcal{N}(0,1)$ and $U_2\sim\operatorname{Unif}(-1,1)$ generate $T=U_1$ and a binary categorical $Y$ with its independent Gumbel noise. Set $a(u)=(3+\sin u)/4$ and $f_Y(t,u)=e^{-t^2}(2+a(u)\sin(t^2),\,2-a(u)\sin(t^2))$. These positive functions are smooth and globally Lipschitz, but $\|f_Y(t,u)\|_1=4e^{-t^2}$. The graph is $T\rightarrow Y$, with no bidirected arrow. Its intervention probability is $\mathbb{P}(Y(t)=1)=1/2+(3/16)\sin(t^2)$, whose derivative $(3/8)t\cos(t^2)$ is unbounded. In a canonical model on this graph, a globally Lipschitz normalized propensity averaged over its fixed independent latent law would give a globally Lipschitz intervention probability. Hence no such canonical model can preserve the original global Lipschitz regularity and all intervention laws. - Claim: Proposition~\ref{prop:equivalent_g_constrianed} asserts a normalized canonical representation with the same intervention laws and the original structural regularity, including global Lipschitz regularity. - What is wrong: (a) ``Let $ \hat{\mathcal{M}} \in \mathcal{M}(\mathcal{G} ,{\mathcal{F}} ,\hat{\boldsymbol{U}})$ the SCM with structure equations \vsDisplay{\begin{array}{@{}c@{}}\displaystyle \hat{V}_{i} =\begin{cases} f_{i}\left(\text{Pa} (\hat{V}_{i} ),\hat{U}_{k_{1}} ,\cdots ,\hat{U}_{k_{n_{i}}}\right) , V_{i} \ \text{is continuous} , \\ \displaystyle \arg\max_{k\in [ n_{i}]}\left\{g_{k}^{V_{i}} +\log\left( f_{i}\left(\text{Pa}(\hat{V}_{i}) ,\hat{U}_{k_{1}} ,\cdots ,\hat{U}_{k_{n_{i}}}\right)\right)_{k}\right\} , V_{i} \ \text{is categorical} , \end{cases} ,\hat{\boldsymbol{U}}_{k_{i}} \cap \boldsymbol{U}_{V_i} \neq \emptyset ,\ i\in [n_V].\end{array}\mbox{''}} Regrouping latent coordinates retains the original categorical functions, whose coordinate sums need not equal one, so the displayed construction need not have the claimed normalized canonical form. (b) ``In addition, the functions in the new model $\hat{\mathcal{M}}$ has the same smoothness as the original model.\par'' Global Lipschitz regularity of an unnormalized categorical function does not imply global Lipschitz regularity of its normalized propensity on an unrestricted domain. Counterexample: Let independent $U_1\sim\mathcal N(0,1)$ and $U_2\sim\operatorname{Unif}(-1,1)$ generate $T=U_1$ and a binary categorical $Y$ with independent standard Gumbel noise. Take the graph $T\rightarrow Y$ with no bidirected arrow. Put $a(u)=(3+\sin u)/4$ and $f_Y(t,u)=e^{-t^2}(2+a(u)\sin(t^2),2-a(u)\sin(t^2))$. These positive functions are smooth and globally Lipschitz, but their intervention probability is $\mathbb P(Y(t)=1)=1/2+(3/16)\sin(t^2)$ with unbounded derivative $(3/8)t\cos(t^2)$, whereas averaging a globally Lipschitz normalized propensity over the fixed independent latent law would give a globally Lipschitz intervention probability. - Proposed fix: The suggested version normalizes categorical functions before regrouping coordinates, retains every intervention law and the continuous functions' regularity, and states the categorical regularity in terms of the normalized propensity functions. - Gives up: Preservation of the original global Lipschitz regularity for unnormalized categorical functions on unrestricted domains, including cases where their total mass approaches zero. - repair confirmed by an independent referee pass (see Corrected version); the corrected statement is proved in Lean - corrected statement: Suppose that \cref{assump:independence_u} and \cref{assump:categorical_rv} hold, given any SCM $\mathcal{M} $ with causal graph $\mathcal{G}$ and latent variables $\boldsymbol{U}$, we can construct a canonical SCM $\hat{\mathcal{M}}$ of the form (\ref{eq:structure_equation}) by merging the latent variables in $\boldsymbol{U}$ such that $\mathcal{M}$ and $\hat{\mathcal{M}}$ produce the same intervention results. Besides, the continuous structural functions in $\hat{\mathcal M}$ have the same smoothness as those in $\mathcal M$. The categorical functions in $\hat{\mathcal M}$ have the smoothness of the normalized propensity functions $f_i/\|f_i\|_1$: normalization preserves differentiability order and local Lipschitz regularity wherever $\|f_i\|_1>0$, and coordinate regrouping preserves any global Lipschitz regularity these normalized functions possess. - narrowed: Preservation of the original global Lipschitz regularity for unnormalized categorical functions on unrestricted domains, including cases where their total mass approaches zero. (an intermediate result: its strongest true version was applied automatically) - **`lem:hilbert`** (`proof.tex:1249-1255`): The terms ``partition'' and ``cube'' could mean disjoint cells and equal-sided cubes, or closed coordinate boxes with shared boundaries. The preparation and construction in the proof of Proposition \ref{prop:holder_curve} determine the latter reading. Under it, property (*) permits refinement patterns for which the claimed nested face-adjacent orderings do not exist. The cited source constructs Hilbert orderings on dyadic lattices and does not establish this generalization. - witness: Take $d=2$ and let $\mathcal{C}_n$ be the rectangular grid with $4^{\lceil n/2\rceil}$ columns and $4^{\lfloor n/2\rfloor}$ rows, including cell boundaries. Every cell has four children, so property (*) holds. The first three grids have sizes $4\times1$, $4\times4$, and $16\times4$. At the first level, adjacency forces the four columns to be visited consecutively. At the second level, the four horizontal cells within each column must be visited vertically, in one of the two directions. At the third level, each such cell has four vertical-strip children, whose ordering must run horizontally from one extreme strip to the other. Across successive vertically adjacent second-level cells, these horizontal directions must alternate. Consequently, after four such cells, the first and last third-level cells within a first-level column lie on the same horizontal extreme. An interior first-level column must instead enter from one neighboring column and exit toward the other, requiring opposite extremes. This contradiction rules out the asserted orderings. - Claim: Lemma~\ref{lemma:hilbert} asserts nested face-adjacent orderings for every sequence satisfying property (*). - What is wrong: ``\dots i.e., \vsDisplay{O_{n} :\left\{1,\cdots ,2^{nd}\right\}\rightarrow \left\{\overline{i_{1} i_{2} \cdots i_{n}} :i_{j} =0,\cdots ,2^{d} -1\right\},} such that the following two conclusions hold.'' Property (*) allows refinement in only one coordinate at a time, whereas the cited Hilbert construction supplies the asserted compatible orderings on dyadic lattices. Counterexample: Take $d=2$. Let $\mathcal C_n$ be the closed rectangular grid with $4^{\lceil n/2\rceil}$ columns and $4^{\lfloor n/2\rfloor}$ rows. These grids satisfy property (*), but at levels $1$, $2$, and $3$ the required horizontal and vertical traversals force an interior first-level column to enter and exit on the same horizontal extreme, contradicting the required passage between its two neighboring columns. - Proposed fix: State the dyadic-grid setting supplied by the cited construction. The proof of Proposition~\ref{prop:holder_curve} now uses this setting only for a uniform Hilbert curve, protecting the full distributional scope and rates of Theorem~\ref{thm:wide_nn}, Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}, and Corollary~\ref{cor:rate}. - Gives up: The assertion of compatible face-adjacent orderings for arbitrary non-dyadic coordinate-box refinements satisfying property (*). - repair confirmed by an independent referee pass (see Corrected version); the corrected statement is proved in Lean - corrected statement: Given a sequence of partitions of the unit cube $\{\mathcal C_n\}$ that has property (*), suppose that, when $d\geqslant1$, $\mathcal C_n$ is the standard dyadic grid of closed cubes \[ \mathcal C_n=\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}. \] There exists an ordering for each partition $\mathcal C_n$, i.e., \[ O_n:\{1,\ldots,2^{nd}\}\longrightarrow\{\overline{i_1\cdots i_n}:i_j=0,\ldots,2^d-1\}, \] such that the following two conclusions hold. \begin{enumerate} \item If $ ( j-1) \cdotp 2^{d} < i\leqslant j\cdotp 2^{d}$, the first $ n$ digits of $ O_{n+1}( i)$ are the same as $ O_{n}( j)$. \item Adjacent cubes in the ordering intersect, i.e., $ C_{O_{n}( i)} \cap C_{O_{n}( i+1)} \neq \emptyset $. Furthermore, $ C_{O_{n}( i)} \cap C_{O_{n}( i+1)}$ is a $d-1$ dimensional cube. \end{enumerate} - narrowed: The assertion of compatible face-adjacent orderings for arbitrary non-dyadic coordinate-box refinements satisfying property (*). (an intermediate result: its strongest true version was applied automatically) - inherited by: `thm:deep-nn`, `thm:consistency`, `thm:rate`, `prop:holder-curve` - **`prop:counterexample-foral`** (`proof.tex:1542-1544`): [F2.5 reviewer] [statement-wrong] The requirement to satisfy Assumption lower_bound does not settle whole-latent-law versus normalized-component-law regularity; Lean selects the latter. - witness: [statement-wrong] The requirement to satisfy Assumption lower_bound does not settle whole-latent-law versus normalized-component-law regularity; Lean selects the latter. - Claim: Proposition~\ref{prop:counterexample_foral} asserts that an unconfounded SCM satisfying Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound} admits models on the same graph with arbitrarily close observational laws and an ATE gap bounded away from zero. - What is wrong: ``It is easy to see that $\mathcal{M}^* $ satisfies Assumption \ref{assump:independence_u}-\ref{assumption:lower_bound}.'' The assertion does not specify whether Assumption~\ref{assumption:lower_bound} concerns each whole latent law or its normalized restrictions to connected support components, and these readings differ for the discrete latent laws in the construction. Counterexample: Take the latent variable $U_1$ in the displayed construction, with $P(U_1=1)=3/5$ and $P(U_1=0)=2/5$. Its support components are $\{0\}$ and $\{1\}$, and its normalized restrictions to them are the corresponding point masses. Each normalized component law satisfies Assumption~\ref{assumption:lower_bound} through a chart from $[0,1]^0$, whereas the whole law cannot satisfy it because its disconnected support cannot be homeomorphic to any cube $[0,1]^d$. - Proposed fix: State the owner-accepted normalized-component interpretation of Assumption~\ref{assumption:lower_bound} explicitly in both counterexample results, and retain the singleton-chart verification in the proof. - Gives up: The requirement that each entire non-Gumbel latent law satisfy Assumption~\ref{assumption:lower_bound}; only its normalized connected-component laws are required to satisfy it. - repair confirmed by an independent referee pass (see Corrected version); the corrected statement is proved in Lean - corrected statement: \label{prop:counterexample_foral} There exists a constant $c>0$ and an SCM $\mathcal{M}^*$ satisfying Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound} with a backdoor causal graph such that there is no unobserved confounding in $\mathcal{M}^*$. For any $\epsilon >0$, there exists an SCM $\mathcal{M}_\epsilon$ with the same causal graph such that $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V})) \leq \epsilon $ and $|\text{ATE}_{\mathcal{M}^*} - \text{ATE}_{\mathcal{M}_\epsilon}| > c $. Here, for each non-Gumbel latent law $\mathbb P$ with connected support components $C_i$, Assumption~\ref{assumption:lower_bound} is required of each normalized component law $\mathbb P|_{C_i}/\mathbb P(C_i)$, with $K=C_i$. - contribution narrowed: The requirement that each entire non-Gumbel latent law satisfy Assumption~\ref{assumption:lower_bound}; only its normalized connected-component laws are required to satisfy it. (the owner accepted the weaker version) - **`lem:push-forward-err`** (`proof.tex:1918-1922`): (a) The printed conclusion uses $F_1,F_2$, although only $F,\hat F$ are defined; the proof derives a comparison of the latter maps and supplies no identification with the former. This is a slip in the asserted bound. (b) Independently, the last inequality in the second-term calculation replaces the $1$-norm by the infinity norm without its dimensional factor. With the binding uniform infinity norm, reproduction gives $LW(\mu,\hat\mu)+\sup_{x\in K}\|F(x)-\hat F(x)\|$, which is at most $LW(\mu,\hat\mu)+d_2\|F-\hat F\|_\infty$, rather than the asserted coefficient. - witness: (a) The step combining the two terms yields an expression involving $F-\hat F$; it cannot imply a bound involving unrelated $F_1,F_2$. Under a reading of these as coordinates of $F$, take the instance below: $F_1=F_2$, so the printed right-hand side is zero while the left-hand side is $2$. (b) Let $K=[0,1]$, let $\mu=\hat\mu$ be uniform on $K$, and set $F(x)=(x,x)$ and $\hat F(x)=(x+1,x+1)$. The map $F$ is $2$-Lipschitz in the paper's $1$-norm. Translation gives $W(F_\#\mu,\hat F_\#\hat\mu)=2$, whereas $W(\mu,\hat\mu)=0$ and $\|F-\hat F\|_\infty=1$. Thus even identifying the conclusion's two map names with the intended maps leaves the bound false. - Claim: Lemma~\ref{lemma:push-forward_err} asserts a push-forward comparison with coefficient $1$ on the uniform infinity error. - What is wrong: (a) ``\dots \vsDisplay{\begin{array}{@{}c@{}}\displaystyle \mathbb{E}_{X\sim \hat{\mu }}[ g\circ F( X)] -\mathbb{E}_{X\sim \hat{\mu }}[ g\circ \hat{F}( X)] \leqslant \mathbb{E}_{X\sim \hat{\mu }}[ |g\circ F( X) -g\circ \hat{F}( X) |] \\ \displaystyle \leqslant \mathbb{E}_{X\sim \hat{\mu }}[ \| F( X) -\hat{F}( X) \| ] \\ \displaystyle \leqslant \| F( X) -\hat{F}( X) \| _{\infty }.\end{array}\mbox{''}} The scalar Lipschitz test function bounds the output $1$-norm, whose comparison with the coordinatewise infinity norm requires the factor $d_2$. (b) ``Combine (\ref{eq:first_term}) and (\ref{eq:second_term}), we have the conclusion.'' Combining the two calculated terms compares $F$ with $\hat F$, and cannot establish a bound involving the undefined maps $F_1,F_2$ in the statement. Counterexample: Let $K=[0,1]$ and let $\mu=\hat\mu$ be the uniform probability measure on $K$. Set $F(x)=(x,x)$ and $\hat F(x)=(x+1,x+1)$. Then $F$ is $2$-Lipschitz, $W(F_{\#}\mu,\hat F_{\#}\hat\mu)=2$, and $\|F-\hat F\|_\infty=1$, so even identifying the intended maps leaves the printed coefficient false. - Proposed fix: Replace $F_1-F_2$ by $F-\hat F$ and multiply its uniform infinity norm by $d_2$. The dimension factor is absorbed into the fixed constants in Theorem~\ref{thm:wide_nn}, Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}, and Corollary~\ref{cor:rate}, protecting their approximation and convergence exponents. - Gives up: The asserted coefficient $1$ on the coordinatewise uniform infinity error for output dimension $d_2>1$. - repair confirmed by an independent referee pass (see Corrected version); the corrected statement is proved in Lean - corrected statement: Let $\mu,\hat\mu$ be two measures on compact set $K\subset\mathbb R^{d_1}$, for any measurable functions $F,\hat F:K\rightarrow\mathbb R^{d_2}$, if $F$ is $L$-Lipschitz continuous, then \begin{equation*} W(F_{\#}\mu,\hat F_{\#}\hat\mu)\leqslant LW(\mu,\hat\mu)+d_2\|F-\hat F\|_\infty. \end{equation*} - inherited by: `thm:wide-nn`, `thm:deep-nn`, `thm:consistency`, `thm:rate` - **`lem:sinkhorn-approx`** (`proof.tex:2030-2039`): the step its proof takes from [luiseDifferentialPropertiesSinkhorn2018, Proposition 1] (lem:sinkhorn-luise-proposition-1) does not hold as the proof uses it: The full entropy-penalized optimal value and the transport cost evaluated at an entropy-regularized optimizer are different readings of $W_{1,\lambda}$; their differences from $W$ can have different signs. The host does not identify either functional, its entropy convention, or the ground-cost data. The cited proposition gives absolute approximation bounds for two separately defined functionals, with an inverse regularization parameter. It does not justify the host's signed assertion for its unspecified functional; in particular, the source's full regularized value violates the asserted lower bound. - witness: The source defines its full regularized value by minimizing transport cost minus entropy divided by its parameter. Even after reciprocal reparameterization to the paper's $\lambda$, this is transport cost minus $\lambda h(T)$. On support $\{0,1\}$ with ground distance $|x-y|$, take $\mu=(3/4,1/4)$ and $\nu=(1/4,3/4)$. Then $W(\mu,\nu)=1/2$, and the optimal unregularized plan $T^*=\left(\begin{smallmatrix}1/4&1/2\\0&1/4\end{smallmatrix}\right)$ has $h(T^*)=1+(3/2)\log2$. Evaluating the regularized objective at this feasible plan gives $W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)\leqslant-\lambda(1+(3/2)\log2)<0$. Thus the cited regularized functional cannot supply the printed step $0\leqslant W_{1,\lambda}-W$. Selecting the sharp transport-cost functional would require an identification the host does not provide. - Claim: Lemma~\ref{lemma:sinkhorn_approx} asserts a nonnegative regularization error and the stated absolute approximation bound for $S_\lambda$. - What is wrong: ``\dots we have \vsDisplay{0\leqslant W_{1,\lambda }( \mu ,\nu ) -W( \mu ,\nu ) \leqslant c\lambda ,} where $ c=\max\left\{h( T) :\text{transportation plan} \ T\ \text{is achieve optimal loss} \ W( \mu ,\nu )\right\}$.'' The cited work distinguishes the entropy-penalized optimal value from the transport cost at its optimizer, and its full regularized value has the opposite sign of error from the asserted lower bound. Counterexample: Take support points $0,1$ with ground distance $|x-y|$. Let $\mu=(3/4,1/4)$ and $\nu=(1/4,3/4)$. Let $T^*=\left(\begin{smallmatrix}1/4&1/2\\0&1/4\end{smallmatrix}\right)$. Then $W(\mu,\nu)=1/2$ and $h(T^*)=1+(3/2)\log2$, so the source's full regularized value satisfies $W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)\leqslant-\lambda(1+(3/2)\log2)<0$, contrary to the printed lower bound. - Proposed fix: Specify the finite supports, $1$-norm cost, entropy convention, and full entropy-penalized optimal value, and reverse the signed interval for its error. Derive the bound directly from the two minimization problems. The unchanged absolute bound for $S_\lambda$ protects Theorem~\ref{thm:consistency} and Corollary~\ref{cor:rate}. - Gives up: The assertion that the full entropy-penalized optimal value is at least $W(\mu,\nu)$. - repair confirmed by an independent referee pass (see Corrected version); the corrected statement is proved in Lean - corrected statement: For any $\mu\in\Delta_n$, $\nu\in\Delta_m$, and $\lambda>0$, let $\mu$ and $\nu$ be the weights of finite probability measures on points $x_1,\ldots,x_n$ and $y_1,\ldots,y_m$ in a common Euclidean space with the paper's $1$-norm. Put $c_{ij}=\|x_i-y_j\|$ and \[ \Pi(\mu,\nu)=\left\{T\in[0,\infty)^{n\times m}:\sum_jT_{ij}=\mu_i,\ \sum_iT_{ij}=\nu_j\right\},\qquad h(T)=1-\sum_{i,j}T_{ij}\log T_{ij}, \] with $0\log0=0$. Here the entropy regularized Wasserstein-1 distance is \[ W_{1,\lambda}(\mu,\nu)=\min_{T\in\Pi(\mu,\nu)}\left\{\sum_{i,j}c_{ij}T_{ij}-\lambda h(T)\right\}, \] and $W(\mu,\nu)=\min_{T\in\Pi(\mu,\nu)}\sum_{i,j}c_{ij}T_{ij}$. Then \begin{align*} -(\log(mn)+1)\lambda\leqslant W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)\leqslant0. \end{align*} Moreover, we have the following estimation of approximation error. \begin{align*} |S_{\lambda }( \mu ,\nu ) -W( \mu ,\nu ) | & \leqslant 2(\log( mn) +1) \lambda . \end{align*} - inherited by: `thm:consistency`, `thm:rate` - **`lem:sup-wasserstein`** (`proof.tex:2061-2067`): the step its proof takes from [anthony1999neural, Theorems 18.4 and 17.1] (lem:sup-wasserstein-anthony-theorems) does not hold as the proof uses it: (a) For vector-valued $\mathcal F$, its pseudo-dimension could mean $\max_{j\in[d]}\operatorname{Pdim}(\{f_j:f\in\mathcal F\})$ or $\operatorname{Pdim}(\{(x,j)\mapsto f_j(x):f\in\mathcal F\})$. These impose different bounds, and neither the paper nor the scalar cited theorem specifies the choice. The dual formulation does determine that $h$ is scalar-valued. (b) The attributed covering inequality is false for the paper's explicitly unnormalized empirical $\ell_1$ metric: its bound omits sample-size scaling. (c) It also omits range scaling: Theorem 18.4 requires scalar functions valued in $[0,1]$, whereas the host only bounds their magnitude by arbitrary $K$. The fixed-$h$ concentration order can be recovered from Theorem 17.1 by range normalization and a square-root transformation with zero responses, but that does not validate the printed covering inequality. - witness: (b) Take $d=d'=1$, $\mu=\mathrm{Unif}([-1,1])$, $K=1$, $h(x)=x$, and $\mathcal F=\{f_c:c\in[0,1/2]\}$, where $f_c(x)=c+\tfrac12\mathbb{I}_{x\geqslant0}$. This nonconstant class has pseudo-dimension $1$, so choose $d_{\mathcal F}=2$. Its evaluation vectors have distance $n|c-c'|$ in the printed metric. With $n=10000$ and $\epsilon=1/4$, every covering ball covers a parameter interval of length at most $2\epsilon/n$, so $\mathcal N_1(\epsilon,\mathcal F,n)\geqslant10000$. The asserted upper bound is $3e(8e)^2=192e^3<4000$, a contradiction. (c) Independently take $K=10000$, $n=2$, and $f_c(x)=c+(K/2)\mathbb{I}_{x\geqslant0}$ for $c\in[0,K/2]$, keeping the other choices. The host's hypotheses again hold with pseudo-dimension $1$. At $\epsilon=1/4$, its printed covering number is at least $20000$; even the averaged empirical metric gives a lower bound of $10000$. Both exceed the same asserted bound $192e^3$. Rescaling the range to $[0,1]$ changes the covering radius and introduces the omitted dependence on $K$. - Claim: Lemma~\ref{lemma:sup_wasserstein} asserts a uniform empirical Wasserstein bound controlled by $d_{\mathcal F}$ and a bounded vector output range. - What is wrong: (a) ``\dots \vsDisplay{P\left(\sup _{f\in \mathcal{F}}\mathbb{E}_{X\sim \mu _{n}}[ h\circ f( X)] -\mathbb{E}_{X\sim \mu }[ h\circ f( X)] >\epsilon \right) \leqslant \exp\left( O\left( d_{\mathcal{F}}\log \epsilon ^{-1} -n\epsilon ^{2}\right)\right) .\mbox{''}} The scalar source theorem does not identify whether the vector pseudo-dimension means the maximum coordinate pseudo-dimension or the pseudo-dimension on the augmented domain of coordinate indices. (b) In the step quoted in (a): The displayed covering bound omits the factor of sample size required by the explicitly unnormalized empirical $1$-norm. (c) The step quoted in (a) uses Theorem 18.4 of \cite{anthony1999neural}, which the paper assumes only for scalar functions with range $[0,1]$ in the averaged empirical metric, but applies it to the unnormalized evaluation vectors of functions bounded by an arbitrary $K$. (d) In the step quoted in (a): The invocation of Theorem 17.1 of \cite{anthony1999neural} omits its standing measurability condition on the function class, which does not follow from measurability of each function and finite pseudo-dimension. (e) The step quoted in (a) uses Theorem 17.1 of \cite{anthony1999neural}, which the paper assumes only for squared losses of $[0,1]$-valued functions with responses in $[0,1]$, but applies it to the unnormalized scalar expectations of $h\circ f$. Counterexample: Let $d=d'=1$, $\mu=\operatorname{Unif}([-1,1])$, $K=1$, and $h(x)=x$. Let $\mathcal F=\{f_c:c\in[0,1/2]\}$, where $f_c(x)=c+\tfrac12\mathbb I_{\{x\geqslant0\}}$, and take $d_{\mathcal F}=2$. Take $n=10000$ and $\epsilon=1/4$. Evaluation vectors have distance $n|c-c'|$, so $\mathcal N_1(1/4,\mathcal F,10000)\geqslant10000$, whereas the attributed bound is $192e^3<4000$; increasing the range also violates that bound even in the averaged metric. - Proposed fix: State the coordinatewise pseudo-dimension condition explicitly, retain the unnormalized covering definition with its sample-size and range scaling, and apply the squared-loss concentration theorem to the square root of a normalized scalar test function. Use scalar dual functions and a $\delta/2$-net. State that $\mathcal F$ has a standard Borel parameter space with jointly Borel evaluation, and verify this property for the scalar dual and normalized square-root families. Take uniform-event probabilities in the completed sample law. This supplies the standing measurability condition without assuming a concentration inequality and preserves the sampling order required by Theorem~\ref{thm:consistency} and Corollary~\ref{cor:rate}. - repair confirmed by an independent referee pass (see Corrected version); the corrected statement is proved in Lean - corrected statement: Given a probability measure $\mu$ on $\mathbb R^{d'}$ and a class $\mathcal F$ of measurable functions $f:\mathbb R^{d'}\rightarrow\mathbb R^d$, suppose that each scalar coordinate class $\mathcal F_j=\{x\mapsto f_j(x):f\in\mathcal F\}$, $j\in[d]$, has pseudo-dimension less than $d_{\mathcal F}<\infty$ and there exists $K>0$ such that $\|f(x)\|_\infty\leqslant K$ for all $f\in\mathcal F$ and $x\in\mathbb R^{d'}$. Suppose in addition that there are a standard Borel space $T$ and a jointly Borel map $G:T\times\mathbb R^{d'}\rightarrow[-K,K]^d$ such that $\mathcal F=\{G(t,\cdot):t\in T\}$. There are positive constants $C,c$ depending only on $d,K$ such that, with probability at least \[ 1-\exp\left(C\left((1+\delta^{-d})\log(2+\delta^{-1})+d_{\mathcal F}\log(2+\epsilon^{-1})\right)-cn\epsilon^2\right), \] \begin{align*} \sup_{f\in\mathcal F}W(f_{\#}\mu,f_{\#}\mu_n)\leqslant\delta+\epsilon \end{align*} for all $\delta,\epsilon>0$, where $\mu_n$ is the empirical distribution of $\mu$. - inherited by: `thm:consistency`, `thm:rate` ## Corrected version 20 of 20 result(s) differ from the version you provided — 15 repaired, 0 with the proof completed (no error), 5 only with a side condition or reading made explicit; the others are as printed. The Lean formalization is of the corrected version. ### `thm:approximation` (`Main.tex:240-246`) - as printed: \label{thm:approximation} Given any SCM model $ \mathcal{M} \in \mathcal{M}(\mathcal{G} ,\mathcal{F}_L ,\boldsymbol{U})$, let the treatment variable set be $ \boldsymbol{T} =\{T_k\}_{k=1}^{n_t}$ and suppose that \cref{assump:independence_u}, \ref{assump:categorical_rv} and \ref{assump:Lip_and_bounded} hold for $\mathcal{M}$ with Lipschitz constant $L$, constant $K$. For any intervention $ \boldsymbol{T}= \boldsymbol{t}$ and $ \hat{\mathcal{M}} \in \mathcal{M}(\mathcal{G} ,\hat{\mathcal{F}} ,\hat{\boldsymbol{U}})$ , we have \begin{equation} W\left( P^{\mathcal{M}}(\boldsymbol{V}(\boldsymbol{t})),P^{\hat{\mathcal{M}}}(\hat{\boldsymbol{V}}(\boldsymbol{t}))\right) \leqslant C_{\mathcal{G}}(L,K)\left( \sum_{i=1}^{n_V} \| f_i - \hat{f}_i \|_\infty +W\left(P^{\mathcal{M}}(\boldsymbol{U}),P^{\hat{\mathcal{M}}}(\hat{\boldsymbol{U}})\right)\right) , \end{equation} where $ C_{\mathcal{G}}(L,K)$ is a constant that only depends on $ L,K $ and the causal graph $ \mathcal{G} $ and $f_i , \hat{f}_i$ are structural functions of $\mathcal{M}$ and $\hat{\mathcal{M}}$ respectively. - corrected: Given any SCM model $ \mathcal{M} \in \mathcal{M}(\mathcal{G} ,\mathcal{F}_L ,\boldsymbol{U})$, let the treatment variable set be $ \boldsymbol{T} =\{T_k\}_{k=1}^{n_t}$ and suppose that \cref{assump:independence_u}, \ref{assump:categorical_rv} and \ref{assump:Lip_and_bounded} hold for $\mathcal{M}$ with Lipschitz constant $L$, constant $K$. For any intervention $\boldsymbol T=\boldsymbol t$ and $\hat{\mathcal M}\in\mathcal M(\mathcal G,\hat{\mathcal F},\hat{\boldsymbol U})$, we have \begin{equation*} W\left(P^{\mathcal M}(\boldsymbol V(\boldsymbol t)),P^{\hat{\mathcal M}}(\hat{\boldsymbol V}(\boldsymbol t))\right) \leqslant C_{\mathcal G}(L,K;\{d_i,n_i\})\left(\sum_{i=1}^{n_V}\|f_i-\hat f_i\|_\infty+W\left(P^{\mathcal M}(\boldsymbol U),P^{\hat{\mathcal M}}(\hat{\boldsymbol U})\right)\right), \end{equation*} where $C_{\mathcal G}(L,K;\{d_i,n_i\})$ is a constant that only depends on $L,K$, the causal graph $\mathcal G$, the continuous output dimensions $d_i$, and the categorical cardinalities $n_i$, and $f_i,\hat f_i$ are structural functions of $\mathcal M$ and $\hat{\mathcal M}$ respectively. - side condition added by the confirming referee `ass:pin-177`: $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. — pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality. - side condition added by the confirming referee `ass:pin-181`: Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\|f_i(x)\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. — pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities. - side condition added by the confirming referee `ass:pin-182`: For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\|f_i(x)-f_i(x')\|\leqslant L_f\|x-x'\|$ on its structural input domain, using the paper's $1$-norm. — pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$. - side condition added by the confirming referee `ass:pin-183`: There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\|V_i\|_\infty,\|U_j\|_\infty\}\leqslant K$ almost surely under the SCM law. — pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation. - side condition added by the confirming referee `ass:pin-189`: For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. — pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition. - side condition added by the confirming referee `ass:pin-190`: In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. — pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction. - side condition added by the confirming referee `ass:pin-191`: Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). — pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations. - side condition added by the confirming referee `ass:pin-fix-4a86c2eb`: The structural uniform infinity norm takes the supremum of the coordinatewise infinity norm over the common structural input domain. — pins reading of thm:approximation: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ab17be7a`: For a categorical propensity coordinate equal to zero, $\log0=-\infty$ in the canonical argmax construction. — pins reading of thm:approximation: stated by the repair - proof change: the printed proof, revised step by step] \label{proof:approximation} The structure equations of $ \mathcal{M}$ have the form \begin{equation*} V_{i} =\begin{cases} f_{i}\left(\text{Pa} (V_{i} ),\boldsymbol{U}_{V_{i}}\right) , & V_{i} \ \text{is continuous} ,\\ \arg\max_{k\in [ n_{i}]}\left\{g_{k}^{V_{i}} +\log\left( f_{i}\left(\text{Pa}( V_{i}) ,\boldsymbol{U}_{V_{i}}\right)\right)_{k}\right\} , & V_{i} \ \text{is categorical} , \|f_i\|_1 = 1, \end{cases} \end{equation*}where $ g_{k}^{V_{i}}$ are i.i.d. standard Gumbel variables and $ \hat{\mathcal{M}} \in \mathcal{M}(\mathcal{G} ,\hat{\mathcal{F}} ,\hat{\boldsymbol{U}})$ has structure equations \begin{equation*} \hat{V}_{i} =\begin{cases} \hat{f}_{i}\left(\text{Pa} (\hat{V}_{i} ),\hat{\boldsymbol{U}}_{V_{i}}\right) , & V_{i} \ \text{is continuous} ,\\ \arg\max_{k\in [ n_{i}]}\left\{\hat{g}_{k}^{V_{i}} +\log\left(\hat{f}_{i}\left(\text{Pa}(\hat{V}_{i}) ,\hat{\boldsymbol{U}}_{V_{i}}\right)\right)_{k}\right\} , & V_{i} \ \text{is categorical} , \|\hat{f}_1\|_1 = 1, \end{cases} \end{equation*} where $ \hat{g}_{k}^{V_{i}}$ are i.i.d. standard Gumbel variables. Let the treatment variables set be $ \boldsymbol{T} =\{T_{1} ,\cdots ,T_{n_{T}}\} $. We may give all vertices and $ \boldsymbol{U}_0 = \{ U_1,\cdots,U_{n_U}\}$, an topology ordering (rearrange the subscript if necessary) \begin{equation*} U_{1} ,\cdots ,U_{n_{U}} ,T_{1} ,\cdots ,T_{n_{T}} ,V_{1}(\boldsymbol{t}) ,\cdots ,V_{n_{V} - n_{T}}(\boldsymbol{t}) \end{equation*} such that for each directed edge $(V_{i}(\boldsymbol{t}) ,V_{j}(\boldsymbol{t}) ) $, vertex $V_{i}(\boldsymbol{t})$ lies before vertex $V_{j}(\boldsymbol{t})$ in the ordering, ensuring that all edges start from vertices that appear early in the order and end at vertices that appear later in the order. We put $ \boldsymbol{U}_0 $ and $\boldsymbol{T}$ at the beginning because they are the root nodes of the intervened model. We denote $ \mu _{\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k}^{\boldsymbol{T}}$ (resp. $ \hat{\mu }_{\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k}^{\boldsymbol{T}}$) to be the distribution of $ \boldsymbol{U}_0 ,\boldsymbol{T} ,V_{1}(\boldsymbol{t}) ,\cdots ,V_{k}(\boldsymbol{t})$ (resp. $ \hat{\boldsymbol{U}}_0 ,\boldsymbol{T} ,\hat{V}_{1}(\boldsymbol{t}) ,\cdots ,\hat{V}_{k}(\boldsymbol{t})$), $ \mu _{V_{k} |\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}}$ the distribution of $ V_{k}$ given $ \boldsymbol{U}_0 ,\boldsymbol{T} ,V_{1}(\boldsymbol{t}) ,\cdots ,V_{k-1}(\boldsymbol{t})$. Let $S=\sum_{i=1}^{n_V}\|f_i-\hat f_i\|_\infty$. Put \[ D=\max\left(\{0\}\cup\{n_i-1:V_i\text{ is categorical}\}\right), \] \[ A=\max\{1+L,1+LD/2\},\qquad B=\max\left(\{1\}\cup\{d_i:V_i\text{ is continuous}\}\cup\{n_i(n_i-1)/2:V_i\text{ is categorical}\}\right). \] Write $x=(\boldsymbol u,\boldsymbol t,v_1,\ldots,v_{k-1})$ for a history, and let $E_k$ denote the Wasserstein distance between the two history laws through $V_k$. Next, we prove that \begin{equation}\label{eq:approximation_middle_step} W\left(\mu_{\boldsymbol U_0,\boldsymbol T,1:k}^{\boldsymbol T},\hat\mu_{\hat{\boldsymbol U}_0,\boldsymbol T,1:k}^{\boldsymbol T}\right) \leqslant A W\left(\mu_{\boldsymbol U_0,\boldsymbol T,1:k-1}^{\boldsymbol T},\hat\mu_{\hat{\boldsymbol U}_0,\boldsymbol T,1:k-1}^{\boldsymbol T}\right)+BS. \end{equation} By definition of Wasserstein-1 distance, \begin{align*} W\left( \mu _{\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k}^{\boldsymbol{T}} ,\hat{\mu }_{\hat{\boldsymbol{U}}_0 ,\boldsymbol{T} ,1:k}^{\boldsymbol{T}}\right) & =\sup _{g\in Lip( 1)}\int g(\boldsymbol{u} ,\boldsymbol{t} ,v_{1} ,\cdots ,v_{k}) d\left( \mu _{\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k}^{\boldsymbol{T}} -\hat{\mu }_{\hat{\boldsymbol{U}}_0 ,\boldsymbol{T} ,1:k}^{\boldsymbol{T}}\right)\\ & =\sup _{g\in Lip( 1)}\int g(\boldsymbol{u} ,\boldsymbol{t} ,v_{1} ,\cdots ,v_{k}) \ d\mu _{V_{k} |\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}} \ d\mu _{{\boldsymbol{U}}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}}\\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\int g(\boldsymbol{u} ,\boldsymbol{t} ,v_{1} ,\cdots ,v_{k}) \ d\hat{\mu }_{\hat{V}_{k} |\hat{\boldsymbol{U}}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}} \ d\hat{\mu }_{\hat{\boldsymbol{U}}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}}\\ & = \sup _{g\in Lip( 1)}\underbrace{\int g(\boldsymbol{u} ,\boldsymbol{t} ,v_{1} ,\cdots ,v_{k}) \ d\left( \mu _{V_{k} |\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}} -\hat{\mu }_{\hat{V}_{k} |\hat{\boldsymbol{U}}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}}\right) \ d\hat{\mu }_{\hat{\boldsymbol{U}}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}}}_{(1)}\\ & \ \ \ +\ \underbrace{\int g(\boldsymbol{u} ,\boldsymbol{t} ,v_{1} ,\cdots ,v_{k}) \ d\mu _{V_{k} |\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}} \ d\left( \mu _{\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}} -\hat{\mu }_{\hat{\boldsymbol{U}}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}}\right)}_{( 2)} . \end{align*} For $(1)$, if $V_k$ is continuous, the two conditional laws at history $x$ are point masses at $f_k(x)$ and $\hat f_k(x)$. Since $g$ is $1$-Lipschitz in the paper's $1$-norm, \begin{align}\label{eq:approximation_1} \left|\int g(x,v_k)\,d(\mu_{V_k|x}^{\boldsymbol T}-\hat\mu_{\hat V_k|x}^{\boldsymbol T})\right| &=|g(x,f_k(x))-g(x,\hat f_k(x))|\notag\\ &\leqslant\|f_k(x)-\hat f_k(x)\|\notag\\ &\leqslant d_k\|f_k-\hat f_k\|_\infty\leqslant BS. \end{align} If $V_k$ is categorical, write $p(x)=f_k(x)$ and $\hat p(x)=\hat f_k(x)$, with $n_k$ coordinates. These vectors belong to $\Delta_{n_k}$. Put $c(x)=(\max_jg(x,j)+\min_jg(x,j))/2$. The range of $j\mapsto g(x,j)$ is at most $n_k-1$, and $\sum_j(p_j(x)-\hat p_j(x))=0$. Therefore \begin{align}\label{eq:cate_1} \left|\sum_{j=1}^{n_k}(p_j(x)-\hat p_j(x))g(x,j)\right| &=\left|\sum_{j=1}^{n_k}(p_j(x)-\hat p_j(x))(g(x,j)-c(x))\right|\notag\\ &\leqslant\frac{n_k-1}{2}\sum_{j=1}^{n_k}|p_j(x)-\hat p_j(x)|\notag\\ &\leqslant\frac{n_k(n_k-1)}2\|f_k-\hat f_k\|_\infty\leqslant BS. \end{align} For (2), if $V_i$ is continuous, \begin{align} & \int g(\boldsymbol{u} ,\boldsymbol{t} ,v_{1} ,\cdots ,v_{k}) \ d\mu _{V_{k} |\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}} \ d\left( \mu _{\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}} -\hat{\mu }_{\hat{\boldsymbol{U}}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}}\right)\notag\\ & =\int g\left(\boldsymbol{u} ,\boldsymbol{t} ,v_{1} ,\cdots ,v_{k-1} ,{f}_{k}\left(\text{pa}( v_{k}) ,\boldsymbol{u}_{V_{k}}\right)\right) \ d\left( \mu _{\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}} -\hat{\mu }_{\hat{\boldsymbol{U}}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}}\right) .\label{eq:approximation_2} \end{align} Since $ g,{f}_{k}$ are Lipschitz continuous functions, $ g\left(\boldsymbol{u} ,\boldsymbol{t} ,v_{1} ,\cdots ,v_{k-1} ,{f}_{k}\left(\text{pa}( v_{k}) ,\boldsymbol{u}_{V_{k}}\right)\right)$ is $(L+1)$-Lipschitz continuous with respect to $ (\boldsymbol{u} ,\boldsymbol{t} ,v_{1} ,\cdots ,v_{k-1}) $. We have \begin{align} \int g\left(\boldsymbol{u} ,\boldsymbol{t} ,v_{1} ,\cdots ,v_{k-1} ,{f}_{k}\left(\text{pa}( v_{k}) ,\boldsymbol{u}_{V_{k}}\right)\right) & d\left( \mu _{\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}} -\hat{\mu }_{\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}}\right) \notag\\ &\leqslant ( L+1) W\left( \mu _{\boldsymbol{U}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}} ,\hat{\mu }_{\hat{\boldsymbol{U}}_0 ,\boldsymbol{T} ,1:k-1}^{\boldsymbol{T}}\right). \label{eq:approximation_3} \end{align} If $V_k$ is categorical, the true conditional expectation of the test function is $H(x)=\sum_{j=1}^{n_k}p_j(x)g(x,j)$. For two histories $x,y$, subtracting and adding $\sum_jp_j(x)g(y,j)$ gives \[ H(x)-H(y)=\sum_jp_j(x)(g(x,j)-g(y,j))+\sum_j(p_j(x)-p_j(y))g(y,j). \] The first sum has absolute value at most $\|x-y\|$. In the second sum subtract the midpoint of the range of $j\mapsto g(y,j)$; the probabilities sum to one and that range has length at most $n_k-1$. Because the propensity vector is $L$-Lipschitz in the $1$-norm, \[ |H(x)-H(y)|\leqslant\|x-y\|+\frac{n_k-1}{2}\|p(x)-p(y)\|\leqslant\left(1+\frac{(n_k-1)L}{2}\right)\|x-y\|\leqslant A\|x-y\|. \] It follows from the Wasserstein dual bound that \begin{align}\label{eq:cate_2} \left|\int H\,d\left(\mu_{\boldsymbol U_0,\boldsymbol T,1:k-1}^{\boldsymbol T}-\hat\mu_{\hat{\boldsymbol U}_0,\boldsymbol T,1:k-1}^{\boldsymbol T}\right)\right|\leqslant A E_{k-1}. \end{align} Combine (\ref{eq:approximation_1})-(\ref{eq:cate_2}), we prove \cref{eq:approximation_middle_step}. The recurrence gives, for $m=n_V-n_T$, \[ E_m\leqslant A^mE_0+BS\sum_{r=0}^{m-1}A^r. \] Indeed, at $m=0$ the sum is empty and this is equality. If it holds at $m$, then \[ E_{m+1}\leqslant AE_m+BS\leqslant A^{m+1}E_0+BS\left(1+\sum_{r=1}^{m}A^r\right)=A^{m+1}E_0+BS\sum_{r=0}^{m}A^r. \] This induction does not divide by $L$ and includes $L=0$. Define \[ C_{\mathcal G}(L,K;\{d_i,n_i\})=\max\left\{A^{n_V},B\sum_{r=0}^{n_V-1}A^r\right\}. \] The identical intervention values contribute zero distance. The independent Gumbel coordinates have the same law in the two models: coupling them identically gives an upper bound by the Wasserstein distance of the non-Gumbel latent laws, while projection onto those latents gives the reverse bound. Thus \[ E_0=W\left(P^{\mathcal M}(\boldsymbol U),P^{\hat{\mathcal M}}(\hat{\boldsymbol U})\right). \] Projection from the joint history to the observed variables is $1$-Lipschitz, and $m\leqslant n_V$. Consequently \begin{align}\label{eq:conclu} W\left(P^{\mathcal M}(\boldsymbol V(\boldsymbol t)),P^{\hat{\mathcal M}}(\hat{\boldsymbol V}(\boldsymbol t))\right) &\leqslant E_m\notag\\ &\leqslant C_{\mathcal G}(L,K;\{d_i,n_i\})\left(S+W\left(P^{\mathcal M}(\boldsymbol U),P^{\hat{\mathcal M}}(\hat{\boldsymbol U})\right)\right). \end{align} - why: (a) In (\ref{eq:approximation_1}), the Lipschitz property gives the output $1$-norm, not its coordinatewise infinity norm. The missing dimension factor invalidates the coefficient supplied by the proof; the hypotheses do not bound continuous output dimensions through $L$, $K$, or the graph. (b) In (\ref{eq:cate_2}), the probabilities $p_j$ vary with the variables being integrated. The omitted term is $\sum_j(p_j(x)-p_j(y))g(y,j)$, so the asserted kernel Lipschitz bound and (\ref{eq:approximation_middle_step}) are false. A bound using $1+KL$ can close this categorical step, but does not justify the printed coefficient. Zero propensities could alternatively be excluded or interpreted through an extended logarithm; the closed-simplex canonical convention determines the latter. ### `thm:wide-nn` (`Approximation.tex:462-474`) - as printed: \label{thm:wide_nn} Given any probability measure $ \mathbb{P}$ on $ \mathbb{R}^{d}$ that satisfies \cref{assump:support}, let $ \lambda $ be the Lebesgue measure on $ [ 0,1]^{N_{C}}$, where $N_C$ is defined in \cref{assump:support}. \begin{enumerate} \item There exists a neural network $ \hat{g} \in \mathcal{NN}(W_1,\Theta(\max_j d_j^C),\Theta(d\cdot\max_j d_j^C),L_2,\tau)$ with the above architecture (\cref{fig:wide_nn}) such that \begin{equation*} W(\hat{g}_{\#} \lambda ,\mathbb{P}) \leqslant O\left( W_{1}^{-1/\max_i\{d_i^C\}} +L_{2}^{-2/\max_i\{d_i^C\}} +(\tau - \tau\log\tau) \right) . \end{equation*} \item Furthermore, if in addition, $ \mathbb{P}$ constrained to each component satisfies \cref{assumption:lower_bound}, there exists a neural network $ \hat{g} \in \mathcal{NN}(\Theta(\max_j d_j^C),L_1,\Theta(d\cdot\max_j d_j^C),L_2,\tau)$ with the above architecture such that \begin{equation*} W(\hat{g}_{\#} \lambda ,\mathbb{P}) \leqslant O\left( L_{1}^{-2/\max_i\{d_i^C\}} +L_{2}^{-2/\max_i\{d_i^C\}}+(\tau - \tau\log\tau)\right). \end{equation*} \end{enumerate} - corrected: Given any probability measure $\mathbb P$ on $\mathbb R^d$ that satisfies \cref{assump:support}, let $\lambda$ be the Lebesgue measure on $[0,1]^{N_C}$, where $N_C$ is defined in \cref{assump:support}. Write $D_C=\max\{1,\max_i d_i^C\}$. \begin{enumerate} \item There exists a neural network $\hat g\in\mathcal{NN}(W_1,\Theta(D_C),\Theta(d\cdot D_C),L_2,\tau)$ with the above architecture (\cref{fig:wide_nn}) such that \begin{equation*} W(\hat g_{\#}\lambda,\mathbb P)\leqslant O\left(\mathbb I_{\{\max_i d_i^C>0\}}\left(W_1^{-1/D_C}+L_2^{-2/D_C}\right)+(\tau-\tau\log\tau)\right). \end{equation*} \end{enumerate} - no longer uses `prop:holder-curve` - no longer uses `ass:assumption-lower-bound-domination` - no longer uses `ass:assumption-lower-bound-boundary` - side condition added by the confirming referee `ass:bk-1`: For every $k\in[n]$, $\int_{\mathbb R^d}\|x\|\,\mu_k(dx)+\int_{\mathbb R^d}\|x\|\,\hat\mu_k(dx)<\infty$. — The dual proof separately evaluates expectations of unbounded Lipschitz functions. - side condition added by the confirming referee `ass:pin-192`: A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. — pins reading of def:relu-networks: The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths. - side condition added by the confirming referee `ass:pin-203`: The gap event is $E_\delta=\{\exists\,1\leqslant i0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. — pins reading of def:gumbel-softmax-layer: The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit. - side condition added by the confirming referee `ass:pin-211`: In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. — pins reading of def:lemma-gumbel-err-softmax: Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores. - side condition added by the confirming referee `ass:pin-33`: $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. — pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes. - side condition added by the confirming referee `ass:pin-75`: For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. — pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula. - side condition added by the confirming referee `ass:pin-fix-00130002`: The law $\hat g_{\#}\lambda$ includes the independent standard Gumbel variables used in the mixing layer. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-507a5848`: For each component of positive dimension $q=d_i^C$, the application of Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and quantization mesh $\delta/n$. — pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair - side condition added by the confirming referee `ass:pin-fix-5f02b8a6`: The error estimates are asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of these budgets and of $\tau$. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-72c9e86d`: The measures $\mu$ and $\hat\mu$ are probability measures. — pins reading of lem:push-forward-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9bbf998f`: A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-b83fa4d1`: $W$ is Wasserstein-1 distance for the ground distance $\|x-y\|$. — pins reading of lem:product-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c514425b`: The refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. — pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ca667e13`: $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. — pins reading of lem:product-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-db6ed772`: In the proposition entitled ``Deep distribution approximation'', the multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f6f485b1`: The uniform infinity norm is $\|F-\hat F\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}|F_j(x)-\hat F_j(x)|$. — pins reading of lem:push-forward-err: stated by the repair - proof change: Put $p_i=\mathbb P(C_i)$ and $\mu_i=\mathbb P|_{C_i}/p_i$. Each component is compact because it is homeomorphic to a compact cube. Distinct components have positive distance because they are disjoint compact sets and there are finitely many of them. A sufficiently small neighborhood of a support point in $C_i$ meets no other component and has positive measure; hence $p_i>0$. If $d_i^C=0$, write $C_i=\{c_i\}$, use the unique map $\hat g_1^i:[0,1]\rightarrow[0,1]^0$, and set $\hat g_2^i=c_i$. This produces $\mu_i$ exactly. For $q=d_i^C\geqslant1$, choose $g_2^i\in\mathcal H([0,1]^q,C_i)\cap\mathcal F_L([0,1]^q,C_i)$ from Assumption~\ref{assump:support}, and put $\nu_i=((g_2^i)^{-1})_{\#}\mu_i$. We instantiate Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} with its positive integer depth parameter $s=1$. For a refinement integer $n\geqslant2$, set \[ \delta=\frac{1}{\lceil\sqrt q(q+1)n(n-1)\rceil}. \] Since $\sqrt q(q+1)\geqslant2$, this choice satisfies \[ 0<\delta\leqslant\frac{1}{\sqrt q(q+1)n(n-1)}<\frac{1}{n(n-1)}. \] The quantization mesh in the cited theorem is $\delta/n$. Its conclusion supplies a one-input, $q$-output ReLU network $\Phi$ with at most $C_q(n^q+n^{q-1})$ nonzero weights and biases, depth $4q-2$ in the source's convention, and Euclidean Wasserstein error at most \[ \frac{\sqrt q}{2n}+\frac{2\sqrt q}{n}+\frac{q(q+1)}{2}(n-1)\delta \leqslant\frac{\sqrt q}{2n}+\frac{2\sqrt q}{n}+\frac{\sqrt q}{2n} =\frac{3\sqrt q}{n}. \] Here $C_q$ depends only on $q$. The pointwise inequality between the $1$-norm and the Euclidean norm multiplies this Wasserstein bound by at most $\sqrt q$, giving error at most $3q/n$ in the paper's norm. A hidden unit with zero incoming weights and zero bias has identically zero output and can be deleted. After deleting these units, every remaining hidden unit accounts for a nonzero weight or bias, so every hidden layer has width at most $C_q(n^q+n^{q-1})\leqslant2C_qn^q$. Clip the output coordinates into $[0,1]$ using $t\mapsto\max\{t,0\}-\max\{t-1,0\}$. This requires width $2q$ and a fixed number of additional layers. Clipping fixes the target cube and is $1$-Lipschitz in the $1$-norm, so its application to the first marginal of any coupling cannot increase the error. There is consequently a constant $A_q\geqslant1$, depending only on $q$, such that the clipped network has width at most $A_qn^q$ and hidden depth $\Theta(q)$. For every sufficiently large width budget, specifically $W_1\geqslant A_q2^q$, choose \[ n=\left\lfloor(W_1/A_q)^{1/q}\right\rfloor. \] Then $n\geqslant2$, $A_qn^q\leqslant W_1$, and $n\geqslant\tfrac12(W_1/A_q)^{1/q}$. With this choice the clipped network $\hat g_1^i$ satisfies \[ W(P(\hat g_1^i(U_i)),\nu_i)\leqslant\frac{3q}{n}\leqslant6qA_q^{1/q}W_1^{-1/q},\qquad U_i\sim\operatorname{Unif}([0,1]). \] Thus the refinement integer increases with width, while the depth parameter stays equal to $1$; the mesh $\delta/n$ is $\Theta_q(W_1^{-3/q})$. Padding with zero units realizes width $W_1$ without altering the function. For each scalar coordinate of $g_2^i$, apply Theorem 2(b) of \cite{Yarotsky2018OptimalAO} with positive input dimension $q$. It supplies width $2q+10$ and error bounded by a constant times the coordinate modulus of continuity at a constant times the number of weights raised to $-2/q$. At this fixed width the number of weights is proportional to depth. The Lipschitz modulus therefore gives coordinate error $O(L_2^{-2/q})$. Parallelizing the $d$ coordinate networks gives width $\Theta(dq)$ and a map $\hat g_2^i$ with $\|g_2^i-\hat g_2^i\|_\infty=O(L_2^{-2/q})$. Reserve a fixed number of layers for clipping into a box $[-B,B]^d$ containing all components; the remaining depth is proportional to $L_2$ for sufficiently large $L_2$, and clipping cannot increase any coordinate error. Lemma~\ref{lemma:push-forward_err} gives \[ W(\mu_i,P(\hat g_2^i\circ\hat g_1^i(U_i)))\leqslant L_iO(W_1^{-1/q})+d\,O(L_2^{-2/q})=O(W_1^{-1/q}+L_2^{-2/q}), \] where $L_i$ is the fixed chart Lipschitz constant. Take independent uniform inputs $U_i$. Pad the first-block hidden depths to a common value $\Theta(D_C)$ and the second-block widths to $\Theta(dD_C)$. Identity layers on nonnegative hidden outputs preserve the maps. Let $\hat\mu_i=P(\hat g_2^i\circ\hat g_1^i(U_i))$. For budgets at least $1$, each positive $q\leqslant D_C$ satisfies $W_1^{-1/q}\leqslant W_1^{-1/D_C}$ and $L_2^{-2/q}\leqslant L_2^{-2/D_C}$. Singleton errors are zero. Lemma~\ref{lemma:combin_err} hence yields \[ W\left(\mathbb P,\sum_i p_i\hat\mu_i\right)\leqslant\sum_i p_iW(\mu_i,\hat\mu_i)\leqslant O\left(\mathbb I_{\{\max_i d_i^C>0\}}(W_1^{-1/D_C}+L_2^{-2/D_C})\right). \] Let $X^\tau$ be the Gumbel-softmax weights with probabilities $p_i$, independent of the block outputs, and let $X^0$ be their one-hot limit. The output with weights $X^0$ has law $\sum_i p_i\hat\mu_i$. Lemma~\ref{lemma:gumbel_err} gives $W(P(X^\tau),P(X^0))=O(\tau-\tau\log\tau)$ for $0<\tau<1$. Couple the weights and use identical independent block outputs $v_i$ in the two constructions. Since $\|v_i\|_\infty\leqslant B$, \[ \left\|\sum_i(X_i^\tau-X_i^0)v_i\right\|\leqslant\sum_i|X_i^\tau-X_i^0|\|v_i\|\leqslant dB\|X^\tau-X^0\|. \] Taking expectations and the infimum over weight couplings bounds the additional output Wasserstein error by $O(\tau-\tau\log\tau)$. The triangle inequality proves the stated estimate. If every component is a singleton, both deterministic blocks are exact and only the mixing error remains. - why: the step its proof takes from [Yarotsky2018OptimalAO, Theorem 2] (lem:wide-nn-yarotsky-theorem-2) does not hold as the proof uses it: For $d_i^C\geqslant1$, the citation is valid: Theorem 2(b) supplies width $2d_i^C+10$, its weight count is proportional to $L_2$ for fixed dimension, and the Lipschitz modulus gives the asserted rate. However, the host permits $d_i^C=0$ and applies the citation to every component without establishing the source's positive input dimension. The source gives neither the displayed exponent nor the claimed width in that case. ### `thm:cor-fix-3c402872` (`Approximation.tex:462-474`) - as printed: Given any probability measure $\mathbb P$ on $\mathbb R^d$ that satisfies \cref{assump:support}, let $\lambda$ be the Lebesgue measure on $[0,1]^{N_C}$, where $N_C$ is defined in \cref{assump:support}, and write $D_C=\max\{1,\max_i d_i^C\}$. Suppose in addition that $\mathbb P$ constrained to each component satisfies \cref{assumption:lower_bound}. There exists a neural network $\hat g\in\mathcal{NN}(\Theta(D_C),L_1,\Theta(d\cdot D_C),L_2,\tau)$ with the architecture of \cref{fig:wide_nn} such that \[ W(\hat g_{\#}\lambda,\mathbb P)\leqslant O\left(\mathbb I_{\{\max_i d_i^C>0\}}\left(L_1^{-2/D_C}+L_2^{-2/D_C}\right)+(\tau-\tau\log\tau)\right). \] - corrected: statement unchanged - no error: only a side condition or reading the referee made explicit (see below) - side condition added by the confirming referee `ass:bk-1`: For every $k\in[n]$, $\int_{\mathbb R^d}\|x\|\,\mu_k(dx)+\int_{\mathbb R^d}\|x\|\,\hat\mu_k(dx)<\infty$. — The dual proof separately evaluates expectations of unbounded Lipschitz functions. - side condition added by the confirming referee `ass:pin-192`: A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. — pins reading of def:relu-networks: The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths. - side condition added by the confirming referee `ass:pin-203`: The gap event is $E_\delta=\{\exists\,1\leqslant i0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. — pins reading of def:gumbel-softmax-layer: The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit. - side condition added by the confirming referee `ass:pin-211`: In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. — pins reading of def:lemma-gumbel-err-softmax: Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores. - side condition added by the confirming referee `ass:pin-33`: $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. — pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes. - side condition added by the confirming referee `ass:pin-75`: For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. — pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula. - side condition added by the confirming referee `ass:pin-fix-00130002`: The law $\hat g_{\#}\lambda$ includes the independent standard Gumbel variables used in the mixing layer. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-0bcb2cef`: The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-2a1e7b01`: The mass assigned to $I_{O_n(i)}$ is $\lambda_d(C_{O_n(i)})=2^{-nd}$. — pins reading of def:prop-holder-curve-interval-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-2fe04b0e`: In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal C_n$ denotes the standard dyadic collection $\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}$. — pins reading of def:prop-holder-curve-cube-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-507a5848`: For each component of positive dimension $q=d_i^C$, the application of Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and quantization mesh $\delta/n$. — pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair - side condition added by the confirming referee `ass:pin-fix-5f02b8a6`: The error estimates are asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of these budgets and of $\tau$. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-72c9e86d`: The measures $\mu$ and $\hat\mu$ are probability measures. — pins reading of lem:push-forward-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-7a79e01d`: In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal I_n$ denotes $\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$ with $I_{O_n(i)}=[(i-1)2^{-nd},i2^{-nd}]$. — pins reading of def:prop-holder-curve-interval-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8cc0dba5`: For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. — pins reading of def:hilbert-partitions: stated by the repair - side condition added by the confirming referee `ass:pin-fix-99ac9bb3`: For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. — pins reading of def:hilbert-partitions: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9bbf998f`: A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-b83fa4d1`: $W$ is Wasserstein-1 distance for the ground distance $\|x-y\|$. — pins reading of lem:product-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c514425b`: The refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. — pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c778b424`: The reference measure $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$. — pins reading of def:prop-holder-curve-cube-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c85b9186`: Each $O_n$ is a bijection. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ca667e13`: $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. — pins reading of lem:product-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-db6ed772`: In the proposition entitled ``Deep distribution approximation'', the multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-e896f793`: The cells of $\mathcal C_n$ are closed and may share boundaries. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f6f485b1`: The uniform infinity norm is $\|F-\hat F\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}|F_j(x)-\hat F_j(x)|$. — pins reading of lem:push-forward-err: stated by the repair ### `thm:deep-nn` (`Approximation.tex:484-490`) - as printed: \label{corollary:deep_nn} Given a causal model $ \mathcal{M}^* \in \mathcal{M}(\mathcal{G} ,\mathcal{F}_L ,\boldsymbol{U})$, suppose that Assumptions \ref{assump:independence_u}-\ref{assump:Lip_and_bounded} hold and the distributions of $U_C$ for all $C^2$-component satisfy the assumptions of the second conclusion of \cref{thm:wide_nn}. Let $ d_{\max}^{\text{in}} $ and $d_{\max}^{\text{out}}$ be the largest input and output dimension of $f_i$ in (\ref{eq:structure_equation}) and $d_{\max}^U$ be the largest dimension of all latent variables. There exists a neural causal model $$\hat{\mathcal{M}} \in \text{NCM}_{\mathcal{G}} ( \mathcal{NN}(\Theta( d^{\text{in}}_{\max}d^{\text{out}}_{\max}) ,L_{0}) , \mathcal{NN}(\Theta(d_{\max}^U),L_1,\Theta((d_{\max}^U)^2),L_2))$$ with structure equations (\ref{eq:structure_equation_hat}). For any intervention $\boldsymbol{T} = \boldsymbol{t}$, $\hat{\mathcal{M}}$ satisfies \begin{equation*} W\left( P^{\mathcal{M}^*}(\boldsymbol{V}(\boldsymbol{t})),P^{\hat{\mathcal{M}}}(\boldsymbol{V}(\boldsymbol{t}))\right) \leqslant O(L_{0}^{-2/d^\text{in}_{\max}} +L_{1}^{-2/d^U_{\max}} +L_{2}^{-2/d^U_{\max}} +(\tau - \tau\log\tau) ). \end{equation*} - corrected: Given a causal model $\mathcal M^*\in\mathcal M(\mathcal G,\mathcal F_L,\boldsymbol U)$, suppose that Assumptions \ref{assump:independence_u}-\ref{assump:Lip_and_bounded} hold and the distributions of $U_C$ for all $C^2$-component satisfy the assumptions of the proposition entitled ``Deep distribution approximation''. Let $d_{\max}^{\text{in}}$ and $d_{\max}^{\text{out}}$ be the largest input and output dimension of $f_i$ in (\ref{eq:structure_equation}) and $d_{\max}^U$ be the largest dimension of all latent variables. Suppose in addition that $d_{\max}^{\text{in}}>0$ and $d_{\max}^U>0$. Fix any $R>0$. There exists a neural causal model $$\hat{\mathcal{M}} \in \text{NCM}_{\mathcal{G}} ( \mathcal{NN}(\Theta( d^{\text{in}}_{\max}d^{\text{out}}_{\max}) ,L_{0}) , \mathcal{NN}(\Theta(d_{\max}^U),L_1,\Theta((d_{\max}^U)^2),L_2))$$ with structure equations (\ref{eq:structure_equation_hat}). For any intervention $\boldsymbol{T}=\boldsymbol{t}$ such that $\|t_i\|_\infty\leqslant R$ for each continuous treatment variable $T_i$, $\hat{\mathcal M}$ satisfies \begin{equation*} W\left(P^{\mathcal M^*}(\boldsymbol V(\boldsymbol t)),P^{\hat{\mathcal M}}(\boldsymbol V(\boldsymbol t))\right)\leqslant O\left(L_0^{-2/d_{\max}^{\text{in}}}+L_1^{-2/d_{\max}^U}+L_2^{-2/d_{\max}^U}+(\tau-\tau\log\tau)\right). \end{equation*} The implicit constant is independent of the intervention and the network budgets, but may depend on $R$ and the fixed causal model. - no longer uses `ass:corollary-deep-nn-latent-laws` - no longer uses `def:corollary-deep-nn-structural-approximants` - side condition added by the confirming referee `ass:bk-1`: For every $k\in[n]$, $\int_{\mathbb R^d}\|x\|\,\mu_k(dx)+\int_{\mathbb R^d}\|x\|\,\hat\mu_k(dx)<\infty$. — The dual proof separately evaluates expectations of unbounded Lipschitz functions. - side condition added by the confirming referee `ass:pin-177`: $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. — pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality. - side condition added by the confirming referee `ass:pin-181`: Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\|f_i(x)\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. — pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities. - side condition added by the confirming referee `ass:pin-182`: For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\|f_i(x)-f_i(x')\|\leqslant L_f\|x-x'\|$ on its structural input domain, using the paper's $1$-norm. — pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$. - side condition added by the confirming referee `ass:pin-183`: There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\|V_i\|_\infty,\|U_j\|_\infty\}\leqslant K$ almost surely under the SCM law. — pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation. - side condition added by the confirming referee `ass:pin-186`: The phrase ``uniform cubes'' denotes the unit cube domains $[0,1]^m$ used for deterministic uniform function approximation; it imposes no uniform probability law on structural inputs. — pins reading of ass:relu-approximation-domain: The same paragraph displays approximation of functions on $[0,1]^{d_1}$ and separately says that similar arguments apply to any bounded cubes. - side condition added by the confirming referee `ass:pin-189`: For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. — pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition. - side condition added by the confirming referee `ass:pin-190`: In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. — pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction. - side condition added by the confirming referee `ass:pin-191`: Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). — pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations. - side condition added by the confirming referee `ass:pin-192`: A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. — pins reading of def:relu-networks: The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths. - side condition added by the confirming referee `ass:pin-194`: Membership in $\text{NCM}_{\mathcal G}(\mathcal F_0,\mathcal F_1)$ requires $\hat f_i\in\mathcal F_0$ for every observed node and $\hat g_j\in\mathcal F_1$ for every latent component, with the compatible input and output dimensions understood separately for each network. — pins reading of def:neural-causal-class: The source paragraph explicitly omits input and output dimensions and describes separate structural approximants and latent generators in equation~\ref{eq:structure_equation_hat}. - side condition added by the confirming referee `ass:pin-199`: $d_{\max}^U=\max_{j\in[n_U]}d_j^U$ ranges over non-Gumbel latent variables, while $d_{\max}^{\mathrm{in}}=\max_i d_i^{\mathrm{in}}$ and $d_{\max}^{\mathrm{out}}=\max_i d_i^{\mathrm{out}}$, with $d_i^{\mathrm{out}}=n_i$ for categorical nodes. — pins reading of def:corollary-deep-nn-dimensions: The NCM construction separately approximates non-Gumbel latent distributions and the functions in the canonical structural equations; those functions output categorical propensity vectors. The binding dimension symbols specify these same meanings. - side condition added by the confirming referee `ass:pin-203`: The gap event is $E_\delta=\{\exists\,1\leqslant i0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. — pins reading of def:gumbel-softmax-layer: The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit. - side condition added by the confirming referee `ass:pin-211`: In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. — pins reading of def:lemma-gumbel-err-softmax: Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores. - side condition added by the confirming referee `ass:pin-33`: $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. — pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes. - side condition added by the confirming referee `ass:pin-75`: For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. — pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula. - side condition added by the confirming referee `ass:pin-fix-00130002`: The law $\hat g_{\#}\lambda$ includes the independent standard Gumbel variables used in the mixing layer. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-0bcb2cef`: The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-2a1e7b01`: The mass assigned to $I_{O_n(i)}$ is $\lambda_d(C_{O_n(i)})=2^{-nd}$. — pins reading of def:prop-holder-curve-interval-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-2fe04b0e`: In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal C_n$ denotes the standard dyadic collection $\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}$. — pins reading of def:prop-holder-curve-cube-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-419eb8e3`: The multiplicative width factors in $\Theta(d_{\max}^{\text{in}}d_{\max}^{\text{out}})$, $\Theta(d_{\max}^U)$, and $\Theta((d_{\max}^U)^2)$ can be chosen as absolute numerical constants, independent of the causal model, $R$, all dimensions, and all category and component counts. — pins reading of thm:deep-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-43f1bb71`: For each latent law $P(U_C)$, the assumptions of the proposition entitled ``Deep distribution approximation'' require the support charts of \cref{assump:support} and \cref{assumption:lower_bound} on each normalized component law $\mu_i=P(U_C)|_{C_i}/P(U_C\in C_i)$, with $(f^{-1})_{\#}\mu_i(\partial[0,1]^{d_i^C})=0$ when $d_i^C>0$. — pins reading of thm:deep-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-4a86c2eb`: The structural uniform infinity norm takes the supremum of the coordinatewise infinity norm over the common structural input domain. — pins reading of thm:approximation: stated by the repair - side condition added by the confirming referee `ass:pin-fix-507a5848`: For each component of positive dimension $q=d_i^C$, the application of Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and quantization mesh $\delta/n$. — pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair - side condition added by the confirming referee `ass:pin-fix-5f02b8a6`: The error estimates are asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of these budgets and of $\tau$. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-72c9e86d`: The measures $\mu$ and $\hat\mu$ are probability measures. — pins reading of lem:push-forward-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-73abef8c`: In Corollary~\ref{corollary:deep_nn}, for $m_i=d_i^{\text{in}}>0$, the symbol $\hat f_i$ denotes $q_i\circ\Pi_i$ for a continuous variable and the vector $\bigl(\max\{\epsilon_i,(q_i\circ\Pi_i)_k\}\bigr)_{k=1}^{n_i}$ for a categorical variable, where $q_i$, $\Pi_i$, and $\epsilon_i=A_iL_0^{-2/m_i}$ are chosen in its proof. — pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair - side condition added by the confirming referee `ass:pin-fix-7a79e01d`: In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal I_n$ denotes $\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$ with $I_{O_n(i)}=[(i-1)2^{-nd},i2^{-nd}]$. — pins reading of def:prop-holder-curve-interval-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8429b12c`: The latent generator class in the statement uses the same temperature $\tau$ as the displayed error estimate. — pins reading of thm:deep-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8cc0dba5`: For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. — pins reading of def:hilbert-partitions: stated by the repair - side condition added by the confirming referee `ass:pin-fix-945d85ed`: The error estimate is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, for fixed $R$ and fixed causal model. — pins reading of thm:deep-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-99ac9bb3`: For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. — pins reading of def:hilbert-partitions: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9bbf998f`: A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ab17be7a`: For a categorical propensity coordinate equal to zero, $\log0=-\infty$ in the canonical argmax construction. — pins reading of thm:approximation: stated by the repair - side condition added by the confirming referee `ass:pin-fix-b83fa4d1`: $W$ is Wasserstein-1 distance for the ground distance $\|x-y\|$. — pins reading of lem:product-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-bbf1c78a`: In Corollary~\ref{corollary:deep_nn}, for $m_i=0$, the symbol $\hat f_i$ denotes the exact constant function $f_i$. — pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c514425b`: The refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. — pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c778b424`: The reference measure $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$. — pins reading of def:prop-holder-curve-cube-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c85b9186`: Each $O_n$ is a bijection. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ca667e13`: $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. — pins reading of lem:product-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-db6ed772`: In the proposition entitled ``Deep distribution approximation'', the multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-e896f793`: The cells of $\mathcal C_n$ are closed and may share boundaries. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f6f485b1`: The uniform infinity norm is $\|F-\hat F\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}|F_j(x)-\hat F_j(x)|$. — pins reading of lem:push-forward-err: stated by the repair - proof change: the printed proof, revised step by step] Suppose that structure equations of $ \mathcal{M}$ are \begin{equation*} V_{i} =\begin{cases} f_{i}\left(\text{Pa} (V_{i} ),\boldsymbol{U}_{V_{i}}\right) , & V_{i} \ \text{is continuous} ,\\ \arg\max_{k\in [ n_{i}]}\left\{g_{k}^{V_{i}} +\log\left( f_{i}\left(\text{Pa}( V_{i}) ,\boldsymbol{U}_{V_{i}}\right)\right)_{k}\right\} , & V_{i} \ \text{is categorical} , \|f_i\|_1= 1 \end{cases}, \quad f_{i} \in \mathcal{F}_L. \end{equation*} Put $m_i=d_i^{\text{in}}$, let $d_i^{\text{out}}$ be the output dimension of $f_i$, and put $m=d_{\max}^{\text{in}}>0$. In each structural input, distinguish the real coordinates, consisting of continuous parents and non-Gumbel latent coordinates, from the categorical parent coordinates. Let $a_i$ have value zero in every real coordinate and value one in every categorical coordinate. Define the fixed constants \[ B_0=\max\left(\{1,R,K\}\cup\{n_i:V_i\text{ is categorical}\}\cup\{\|f_i(a_i)\|_\infty:V_i\text{ is continuous}\}\right),\qquad H=1+2Lm,\qquad B=B_0H^{n_V}. \] These constants do not depend on the intervention or on the network budgets. We first bound all true structural inputs for the interventions in the statement. Work on the probability-one event where every non-Gumbel latent coordinate has absolute value at most $K$. Order the observed nodes topologically. We claim that the node in position $\ell$ has infinity norm at most $B_0H^\ell$. An intervened continuous node has norm at most $R\leqslant B_0$, and a categorical node has value at most its cardinality, which is at most $B_0$. For a nonintervened continuous node in position $\ell$, the induction hypothesis bounds all its parents and latent coordinates by $b=B_0H^{\ell-1}$. Every coordinate of $a_i$ has absolute value at most one, so its actual input $x_i$ satisfies $\|x_i-a_i\|\leqslant2m_i b\leqslant2mb$. Consequently, \[ \|f_i(x_i)\|_\infty\leqslant\|f_i(a_i)\|_\infty+L\|x_i-a_i\|\leqslant B_0+2Lmb\leqslant Hb=B_0H^\ell. \] This proves the induction, including its first position. In particular, every true structural input belongs to $Q_i=[-B,B]^{m_i}$, simultaneously for all permitted interventions. For $m_i=0$, interpret $Q_i$ as the unique point of $\mathbb R^0$. We next define a Lipschitz extension to real values of the categorical input coordinates. A categorical input coordinate with only one category is always replaced by one. For each remaining categorical input coordinate $r$, with category count $c_r\geqslant2$, clip its value $x_r$ to $t_r\in[1,c_r]$, and set \[ k_r=\min\{c_r-1,\lfloor t_r\rfloor\},\qquad \theta_r=t_r-k_r\in[0,1]. \] Let $s_i$ be the number of these remaining categorical coordinates. For $\beta\in\{0,1\}^{s_i}$, let $x^\beta$ retain the real coordinates of $x$, replace coordinate $r$ by $k_r+\beta_r$, and replace every one-category coordinate by one. With $w_{r,0}=1-\theta_r$ and $w_{r,1}=\theta_r$, define \[ f_i^\dagger(x)=\sum_{\beta\in\{0,1\}^{s_i}}\left(\prod_{r=1}^{s_i}w_{r,\beta_r}\right)f_i(x^\beta). \] When $s_i=0$, the sum has one term with weight one. Thus this definition includes functions with no categorical parents. At valid categorical labels it agrees with $f_i$. This interpolation preserves the $L$-Lipschitz bound in the paper's $1$-norm. Changing one real coordinate leaves the interpolation weights fixed, so the difference is a convex combination of differences bounded in $1$-norm by $L$ times the coordinate change. With all other coordinates fixed, on each interval between consecutive labels a categorical coordinate gives a linear interpolation between two vectors whose $1$-norm difference is at most $L$. Its slope therefore has $1$-norm at most $L$; outside the label interval the function is constant in that coordinate. Splitting any coordinate change at its finitely many intervening labels proves the same bound for an arbitrary change of that coordinate. Changing all coordinates one at a time and adding the bounds gives \[ \|f_i^\dagger(x)-f_i^\dagger(y)\|\leqslant L\|x-y\|. \] For a categorical output, the interpolation weights are nonnegative and sum to one, so $f_i^\dagger(x)\in\Delta_{n_i}$ for every real input $x$. Let $\Pi_i:\mathbb R^{m_i}\to Q_i$ be coordinatewise projection, explicitly \[ (\Pi_i x)_r=-B+\max\{x_r+B,0\}-\max\{x_r-B,0\}, \] and define $\bar f_i=f_i^\dagger\circ\Pi_i$. Clipping each coordinate cannot increase its distance from another clipped coordinate; summing proves $\|\Pi_i x-\Pi_i y\|\leqslant\|x-y\|$. Hence $\bar f_i$ is globally $L$-Lipschitz. Since $a_i\in Q_i$, for every $z\in Q_i$ and every continuous output, \[ \|f_i^\dagger(z)\|_\infty\leqslant\|f_i(a_i)\|_\infty+L\|z-a_i\|\leqslant B_0+2LmB\leqslant HB. \] Thus $\bar f_i$ is globally bounded by $HB$ for continuous outputs; categorical outputs remain probability vectors. Use the same latent laws and Gumbel noises to form the model with structural functions $\bar f_i$. Couple it with the original model using identical draws. Intervened nodes agree by their assigned values. If the parents of a nonintervened node agree, their common original input belongs to $Q_i$ and has valid categorical coordinates. Projection fixes this input and interpolation agrees there with $f_i$, so the structural outputs agree; for a categorical node use the identical Gumbel draws and the convention $\log0=-\infty$. Induction in topological order proves equality of the two interventional laws for every intervention in the statement. The extended true model satisfies the boundedness assumption with the fixed enlarged constant $HB$, because its nonintervened continuous outputs are globally bounded by $HB$, its categorical labels are at most $B_0$, its assigned treatment coordinates are at most $B_0$, and its non-Gumbel latent coordinates are at most $B_0$. We now choose the neural approximants in place of the exact-minimizer construction. Reserve $c=4$ hidden layers of the structural budget for projection, categorical floors, and padding. For each $m_i>0$, apply \cite[Theorem 2]{Yarotsky2018OptimalAO}, part b), to every coordinate of $f_i^\dagger$ on $Q_i$, after the affine rescaling $z\mapsto2Bz-B\boldsymbol 1$ from $[0,1]^{m_i}$. The rescaled coordinate functions have Lipschitz moduli bounded by a fixed constant times their argument. The source gives scalar width $2m_i+10$ and error at most a fixed constant times the modulus at a constant times the number of weights raised to $-2/m_i$. At fixed width that number is proportional to hidden depth. Parallelizing the $d_i^{\text{out}}$ coordinates therefore gives, for every sufficiently large $L_0$, a network $q_i$ of width at most $d_i^{\text{out}}(2m_i+10)$ and hidden depth at most $L_0-c$ such that \[ \sup_{z\in Q_i}\|q_i(z)-f_i^\dagger(z)\|_\infty\leqslant C_i(L_0-c)^{-2/m_i}. \] The constant $C_i$ is fixed independently of $L_0$. For $L_0\geqslant2c$, choose a fixed $A_i\geqslant\max\{1,2^{2/m_i}C_i\}$ and set \[ \epsilon_i=A_iL_0^{-2/m_i}. \] Then choose one of the networks just supplied, so that its uniform error on $Q_i$ is at most $\epsilon_i$. Define the structural networks by \[ \hat f_i(x)= \begin{cases} q_i(\Pi_i x),&V_i\text{ is continuous},\\ \bigl(\max\{\epsilon_i,(q_i(\Pi_i x))_k\}\bigr)_{k=1}^{n_i},&V_i\text{ is categorical}. \end{cases} \] For $m_i=0$, instead set $\epsilon_i=0$ and let $\hat f_i=f_i$ be the exact constant affine network. These definitions specify $\hat f_i$ without any choice of a minimizer. Projection uses one hidden layer with at most $2m_i$ units, and each categorical floor is $z\mapsto\epsilon_i+\max\{z-\epsilon_i,0\}$, using one further hidden layer. Affine maps between these layers and the network $q_i$ are absorbed into the adjacent affine transformations. If signed coordinates must be carried through a hidden layer, use $z=\max\{z,0\}-\max\{-z,0\}$ with two units per coordinate. Thus the reserved four layers suffice, with width at most $4m_i+2d_i^{\text{out}}(2m_i+10)+2d_i^{\text{out}}$. For $m_i\geqslant1$ and $d_i^{\text{out}}\geqslant1$, this is at most $30m_i d_i^{\text{out}}\leqslant64d_{\max}^{\text{in}}d_{\max}^{\text{out}}$, because $2m_i+10\leqslant12m_i$. A constant map with $m_i=0$ uses at most $2d_i^{\text{out}}$ units per hidden layer by carrying its positive and negative parts, which also fits this universal width since $d_{\max}^{\text{in}}\geqslant1$. A zero-dimensional output needs no units. Zero padding of units and insertion of identity layers on nonnegative hidden outputs give the structural architecture and hidden depth $L_0$ in the statement. The continuous global approximation error is at most $\epsilon_i$, because $\Pi_i x\in Q_i$ for every input $x$. For categorical outputs, if $p\geqslant0$ and $\epsilon\geqslant0$, then \[ |\max\{\epsilon,z\}-p|\leqslant\max\{\epsilon,|z-p|\}. \] Indeed, for $z\geqslant\epsilon$ the left side is $|z-p|$; for $z<\epsilon$ and $p\leqslant\epsilon$ it is at most $\epsilon$; and for $z<\epsilon0$, every categorical coordinate is at least $\epsilon_i>0$. For $m_i=0$, the exact constant vector is a probability vector. In both cases its $1$-norm is strictly positive. The global error bound holds also when generated latent or parent inputs lie outside $Q_i$. For a categorical variable, at an arbitrary input $x$, put $p=\bar f_i(x)$, $q=\hat f_i(x)$, $a=\|q\|_1$, and $S=\|q-p\|_\infty$. The construction gives $a>0$ and $S\leqslant\epsilon_i$. Since $p$ is a probability vector and $q$ is nonnegative, \[ |a-1|\leqslant\sum_{j=1}^{n_i}|q_j-p_j|\leqslant n_iS. \] For sufficiently large $L_0$, simultaneously for all nodes with $m_i>0$, we have $n_i\epsilon_i\leqslant1/2$, and hence $a\geqslant1/2$. Since $\|p\|_\infty\leqslant1$, \begin{align*} \left\|\frac q a-p\right\|_\infty &\leqslant\frac{\|q-p\|_\infty+|1-a|\|p\|_\infty}{a}\\ &\leqslant2(n_i+1)S\leqslant2(n_i+1)\epsilon_i. \end{align*} For $m_i=0$, normalization is exact because $q=p$ and $a=1$. For $m_i>0$, the inequalities $m_i\leqslant d_{\max}^{\text{in}}$ and $L_0\geqslant1$ imply $L_0^{-2/m_i}\leqslant L_0^{-2/d_{\max}^{\text{in}}}$. Taking the supremum over every real structural input therefore gives \[ \left\|\bar f_i-\frac{\hat f_i}{\|\hat f_i\|_1}\right\|_\infty\leqslant O(L_0^{-2/d_{\max}^{\text{in}}}). \] The continuous structural error has the same order without normalization. These bounds use the fixed cardinalities $n_i$ and do not require $n_i\leqslant K$. For each non-Gumbel latent variable $U_j$ with positive dimension $d_j^U$, apply the proposition entitled ``Deep distribution approximation'' to its law. Choose the resulting generators with independent uniform and Gumbel inputs $Z_j$ and put $\hat U_j=\hat g_j(Z_j)$. Each component cube dimension is at most the ambient dimension $d_j^U$. For budgets at least one, the component bounds therefore give \[ W(P(U_j),P(\hat U_j))\leqslant O\left(L_1^{-2/d_j^U}+L_2^{-2/d_j^U}+(\tau-\tau\log\tau)\right). \] For a component of positive cube dimension $q\leqslant d_j^U$, the scalar first-block networks have width $12$, and the scalar second-block networks have width $2q+10\leqslant12q$. Parallelization gives widths $12q$ and $12d_j^Uq$. Coordinate clipping needs at most $2q$ or $2d_j^U$ units, and carrying signed values requires at most twice the corresponding width; reserving four layers therefore gives widths at most $64q$ and $64d_j^Uq$. A singleton component uses its exact constant blocks; carrying a signed constant second-block output needs at most $2d_j^U$ units. Since $d_{\max}^U\geqslant1$, every component fits widths $64d_{\max}^U$ and $64(d_{\max}^U)^2$. These are the per-component widths in the separable architecture, so the number of components does not multiply either width budget. Zero padding and identity layers on nonnegative hidden outputs give the architecture in the statement. The numerical factor $64$ is independent of the model, $R$, dimensions, and cardinalities; only the error constants and minimum depth budgets may depend on the fixed model. A latent vector of dimension zero is the unique point of $\mathbb R^0$ and is represented exactly, with error zero. Independence of the original latent variables and independence of the chosen generator inputs allow application of \cref{lemma:product_err}, yielding \begin{align*} W(P(U_1,\ldots,U_{n_U}),P(\hat U_1,\ldots,\hat U_{n_U})) &\leqslant\sum_{j=1}^{n_U}W(P(U_j),P(\hat U_j))\\ &\leqslant O\left(L_1^{-2/d_{\max}^U}+L_2^{-2/d_{\max}^U}+(\tau-\tau\log\tau)\right). \end{align*} Indeed, for $d_j^U>0$, the inequalities $d_j^U\leqslant d_{\max}^U$ and $L_r\geqslant1$ give $L_r^{-2/d_j^U}\leqslant L_r^{-2/d_{\max}^U}$ for $r=1,2$, and there are only finitely many latent variables. Identical categorical Gumbel noises may be adjoined to both latent vectors without changing this bound: couple these noises identically for one inequality and project onto the non-Gumbel coordinates for the reverse inequality. Let $ \hat{\mathcal{M}}$ be the casual model with the following structure equations \begin{equation*} \hat{V}_{i} =\begin{cases} \hat{f}_{i}\left(\text{Pa} (\hat{V}_{i} ),(\hat{g}_{j} (Z_{C_{j}} ))_{U_{C_{j} }\in \boldsymbol{U}_{V_{i}}}\right), & V_{i} \text{ is continuous} ,\\ \arg\max_{k\in [ n_{i}]}\left\{g_{k} +\log\left(\hat{f}_{i}\left(\text{Pa} (\hat{V}_{i} ),(\hat{g}_{j} (Z_{C_{j}} ))_{U_{C_{j} } \in \boldsymbol{U}_{V_{i}}}\right) \right)_{k}\right\} , & V_{i} \text{ is categorical} . \end{cases} \end{equation*} Replacing a categorical propensity vector $\hat f_i(x)$ by $\hat f_i(x)/\|\hat f_i(x)\|_1$ subtracts the same finite constant $\log\|\hat f_i(x)\|_1$ from every finite log propensity. It therefore preserves the categorical structural equation, with $\log0=-\infty$ for zero coordinates. Apply \cref{thm:approximation} to the globally $L$-Lipschitz true functions $\bar f_i$, the continuous approximate functions $\hat f_i$, and the normalized approximate categorical functions. The extended true model has boundedness constant $HB$ and, by the coupling induction in this proof, the same interventional laws as $\mathcal M^*$ for every intervention in the statement. Consequently, \begin{align*} &W\left(P^{\mathcal M^*}(\boldsymbol V(\boldsymbol t)),P^{\hat{\mathcal M}}(\boldsymbol V(\boldsymbol t))\right)\\ &\quad\leqslant C\left(\sum_{V_i\ \text{continuous}}\|\bar f_i-\hat f_i\|_\infty +\sum_{V_i\ \text{categorical}}\left\|\bar f_i-\frac{\hat f_i}{\|\hat f_i\|_1}\right\|_\infty +W(P(U_1,\ldots,U_{n_U}),P(\hat U_1,\ldots,\hat U_{n_U}))\right)\\ &\quad\leqslant O\left(L_0^{-2/d_{\max}^{\text{in}}}+L_1^{-2/d_{\max}^U}+L_2^{-2/d_{\max}^U}+(\tau-\tau\log\tau)\right). \end{align*} For the second inequality, every term of the first sum has the continuous structural error bound proved here, every term of the second sum has the normalized categorical error bound proved here, and the latent term has the bound obtained using \cref{lemma:product_err}; the two sums contain only finitely many nodes. The comparison constant $C$ depends only on the fixed model, its dimensions and cardinalities, and $R$. The cubes, extensions, and networks were chosen independently of the intervention within the prescribed region. This proves the estimate uniformly throughout that region. - why: [F2.5 reviewer] [claim-false] Cube-only minimization does not imply the claimed causal-law approximation for arbitrary selected minimizers when latent inputs lie outside the cube. ### `prop:counterexample` (`Main.tex:282-284`) - as printed: \label{prop:counterexample} There exists a constant $c>0$ and an identifiable SCM $\mathcal{M}^*$ satisfying Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound} such that for any $\epsilon >0$, there exists an SCM $\mathcal{M}_\epsilon$ satisfying $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V})) \leq \epsilon $ and $|\text{ATE}_{\mathcal{M}^*} - \text{ATE}_{\mathcal{M}_\epsilon}| > c $. - corrected: \label{prop:counterexample} There exists a constant $c>0$ and an identifiable SCM $\mathcal{M}^*$ satisfying Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound} such that for any $\epsilon >0$, there exists an SCM $\mathcal{M}_\epsilon$ satisfying $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V})) \leq \epsilon $ and $|\text{ATE}_{\mathcal{M}^*} - \text{ATE}_{\mathcal{M}_\epsilon}| > c $. Here, for each non-Gumbel latent law $\mathbb P$ with connected support components $C_i$, Assumption~\ref{assumption:lower_bound} is required of each normalized component law $\mathbb P|_{C_i}/\mathbb P(C_i)$, with $K=C_i$. - side condition added by the confirming referee `ass:pin-177`: $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. — pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality. - side condition added by the confirming referee `ass:pin-181`: Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\|f_i(x)\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. — pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities. - side condition added by the confirming referee `ass:pin-182`: For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\|f_i(x)-f_i(x')\|\leqslant L_f\|x-x'\|$ on its structural input domain, using the paper's $1$-norm. — pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$. - side condition added by the confirming referee `ass:pin-183`: There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\|V_i\|_\infty,\|U_j\|_\infty\}\leqslant K$ almost surely under the SCM law. — pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation. - side condition added by the confirming referee `ass:pin-189`: For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. — pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition. - side condition added by the confirming referee `ass:pin-190`: In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. — pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction. - side condition added by the confirming referee `ass:pin-204`: The second Bernoulli law in the construction has $P(U_2=0)=2/3$. — pins reading of prop:counterexample-foral: The displayed value $P(U_2=1)=1/3$ and the joint masses for $X=0,T=0$ in the proof of Proposition~\ref{prop:counterexample_foral} both determine this entry. - proof change: The proof as printed proves the repaired statement; no step changes. ### `thm:consistency` (`Main.tex:295-302`) - as printed: \label{thm:consistency} Let $ \mathcal{M^*}$ be any SCM satisfying the assumptions of Corollary \ref{corollary:deep_nn}. Suppose that the Lipschitz constant of functions in $\mathcal{M}$ is $ L_f $, $ F:\mathbb{R}^{n_{V}}\rightarrow \mathbb{R}$ in (\ref{eq:opt_empiri}) is Lipschitz continuous and $\tau_n > 0 $, let $K>0$ be the constant in \cref{assump:Lip_and_bounded}, $\hat{L}_f = \sqrt{d_{\max}^{\text{in}}d_{\max}^{\text{out}}}L_f$, \begin{align*} \mathcal{F}_{0,n} &= \mathcal{NN}^{\hat{L}_{f} ,K}\left( W_{0,n} , \Theta\left(\log d^{\text{in}}_{\max}\right)\right) , \mathcal{F}_{1,n} =\mathcal{NN}^{\infty ,K}( \Theta( d^{U}_{\max}) ,L_{1,n}, \Theta( (d^{U}_{\max})^2) ,L_{2,n},\tau_n), \end{align*} where $ d_{\max}^U, d_{\max}^{\text{in}} $ and $d_{\max}^{\text{out}}$ are defined in Corollary \ref{corollary:deep_nn}, take the radius to be $\alpha_n = \epsilon_n+s_n, \epsilon_n = O(W_{0,n}^{-1/d^{\text{in}}_{\max}}+ \sum_{i=1}^2 L_{i,n}^{-2/d^U_{\max}}+ \tau _{n} \log \tau _{n}),$ $$ s_n = O(m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n} +n^{-1/{\max\{2,d_{\max}^U\}}}\log^2(n) + \log(nm_n)\lambda_n).$$ If $ m_n = \Omega(n), L_{i,n} = \Theta(m_{n}^{d_{\max}^{U} /\left( 2d_{\max}^{U} +4\right)}), i=1,2, \lim _{n\rightarrow \infty }\min\{\tau_n^{-1}, W_{0,n},(\log(nm_n)\lambda_n)^{-1}\}= \infty$ , then with probability 1, $ [\liminf _{n\rightarrow \infty } F_{n},\limsup _{n\rightarrow \infty } F_{n}] \subset [\underline{F}^{\hat{L}_f} ,F_{*}]$. - corrected: Let $ \mathcal{M^*}$ be any SCM satisfying the assumptions of Corollary \ref{corollary:deep_nn}. Suppose that the Lipschitz constant of functions in $\mathcal{M}$ is $L_f$, $F:\mathbb{R}^{n_V}\rightarrow\mathbb{R}$ in (\ref{eq:opt_empiri}) is Lipschitz continuous and $\tau_n>0$, let $K>0$ be the constant in \cref{assump:Lip_and_bounded}, $\hat L_f=\sqrt{d_{\max}^{\text{in}}d_{\max}^{\text{out}}}L_f$, \begin{align*} \mathcal F_{0,n}&=\mathcal{NN}^{\hat L_f,K}\left(W_{0,n},\Theta\left(\log d_{\max}^{\text{in}}\right)\right),\qquad \mathcal F_{1,n}=\mathcal{NN}^{\infty,K}\left(\Theta(d_{\max}^U),L_{1,n},\Theta((d_{\max}^U)^2),L_{2,n},\tau_n\right), \end{align*} where $d_{\max}^U$, $d_{\max}^{\text{in}}$ and $d_{\max}^{\text{out}}$ are defined in Corollary \ref{corollary:deep_nn}. Suppose in addition that $d_{\max}^{\text{in}}\geqslant1$ and $d_{\max}^U\geqslant1$. In (\ref{eq:opt_empiri}), restrict each categorical raw structural network $q\in\mathcal F_{0,n}$ to satisfy $q(x)\in[0,\infty)^{n_i}$, $\|q(x)\|_1>0$ for every structural input $x$, and \[ \operatorname{Lip}\left(x\mapsto\frac{q(x)}{\|q(x)\|_1}\right)\leqslant\hat L_f. \] Put \[ D_U=\sum_{j=1}^{n_U}d_j^U,\qquad D=\max\{1,D_U\}, \] where the sum defining $D_U$ includes the non-Gumbel latent variables only. Take the radius to be $\alpha_n=\epsilon_n+s_n$, with positive tolerances chosen as sufficiently large constant multiples of the respective quantities in \[ \epsilon_n=O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+\sum_{i=1}^2L_{i,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right), \] \[ s_n=O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2 n+\log(nm_n)\lambda_n\right). \] For every continuous structural function $f_i$ of $\mathcal M^*$, suppose in addition that $\sup_x\|f_i(x)\|_1\leqslant K$ on its full structural input domain. Restrict every continuous structural network $\hat f_i\in\mathcal F_{0,n}$ in (\ref{eq:opt_empiri}) to satisfy this same uniform output $1$-norm bound. Retain the output bound in (\ref{eq:consistent_limiting}), namely \[ \mathcal F_L^K=\{f:\sup_x\|f(x)\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}. \] Let $F_n$ be the optimal value of the empirical optimization with the objective and Sinkhorn constraint of (\ref{eq:opt_empiri}) and both the continuous and categorical restrictions just specified, and let $\underline F^{\hat L_f}$ be the optimal value of (\ref{eq:consistent_limiting}) with this bounded Lipschitz class. If $ m_n = \Omega(n), L_{i,n} = \Theta(m_{n}^{d_{\max}^{U} /\left( 2d_{\max}^{U} +4\right)}), i=1,2, \lim _{n\rightarrow \infty }\min\{\tau_n^{-1}, W_{0,n},(\log(nm_n)\lambda_n)^{-1}\}= \infty$ , then with probability 1, $ [\liminf _{n\rightarrow \infty } F_{n},\limsup _{n\rightarrow \infty } F_{n}] \subset [\underline{F}^{\hat{L}_f} ,F_{*}]$. - no longer uses `def:thm-consistency-architecture-radius` - no longer uses `thm:approximation` - no longer uses `prop:convg-opt` - no longer uses `ass:corollary-deep-nn-latent-laws` - no longer uses `ass:thm-consistency-structural-lipschitz` - no longer uses `ass:thm-consistency-objective-lipschitz` - no longer uses `ass:thm-consistency-positive-temperature` - no longer uses `def:thm-consistency-population-parameter-space` - no longer uses `def:thm-consistency-population-function-classes` - no longer uses `def:thm-consistency-neural-parameter-space` - side condition added by the confirming referee `ass:bk-1`: For every $k\in[n]$, $\int_{\mathbb R^d}\|x\|\,\mu_k(dx)+\int_{\mathbb R^d}\|x\|\,\hat\mu_k(dx)<\infty$. — The dual proof separately evaluates expectations of unbounded Lipschitz functions. - side condition added by the confirming referee `ass:pin-177`: $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. — pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality. - side condition added by the confirming referee `ass:pin-181`: Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\|f_i(x)\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. — pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities. - side condition added by the confirming referee `ass:pin-182`: For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\|f_i(x)-f_i(x')\|\leqslant L_f\|x-x'\|$ on its structural input domain, using the paper's $1$-norm. — pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$. - side condition added by the confirming referee `ass:pin-183`: There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\|V_i\|_\infty,\|U_j\|_\infty\}\leqslant K$ almost surely under the SCM law. — pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation. - side condition added by the confirming referee `ass:pin-186`: The phrase ``uniform cubes'' denotes the unit cube domains $[0,1]^m$ used for deterministic uniform function approximation; it imposes no uniform probability law on structural inputs. — pins reading of ass:relu-approximation-domain: The same paragraph displays approximation of functions on $[0,1]^{d_1}$ and separately says that similar arguments apply to any bounded cubes. - side condition added by the confirming referee `ass:pin-187`: The bound $K$ applies throughout the optimization class to the observed variables $V_i$ and indexed non-Gumbel latent variables $U_j$, with $n_i\leqslant K$ for categorical nodes and admissible intervention coordinates in $[-K,K]$; it does not bound the separately supplied standard Gumbel noises. — pins reading of ass:bk-owner-1: The boundedness assumption explicitly indexes $V_i$ and $U_j$, whereas the categorical convention separately adjoins $G_{V_i}$. The binding boundedness symbols preserve this distinction. - side condition added by the confirming referee `ass:pin-189`: For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. — pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition. - side condition added by the confirming referee `ass:pin-190`: In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. — pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction. - side condition added by the confirming referee `ass:pin-191`: Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). — pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations. - side condition added by the confirming referee `ass:pin-192`: A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. — pins reading of def:relu-networks: The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths. - side condition added by the confirming referee `ass:pin-194`: Membership in $\text{NCM}_{\mathcal G}(\mathcal F_0,\mathcal F_1)$ requires $\hat f_i\in\mathcal F_0$ for every observed node and $\hat g_j\in\mathcal F_1$ for every latent component, with the compatible input and output dimensions understood separately for each network. — pins reading of def:neural-causal-class: The source paragraph explicitly omits input and output dimensions and describes separate structural approximants and latent generators in equation~\ref{eq:structure_equation_hat}. - side condition added by the confirming referee `ass:pin-196`: Truncation means coordinatewise clipping of the final output to $[-K,K]$; in $\mathcal{NN}^{\infty,K}$ no finite Lipschitz bound is imposed. — pins reading of def:truncated-neural-class: The defining minimum and maximum and the word ``Similarly'' specify the operation; the binding neural-class symbol conventions fix its application to the generator output. - side condition added by the confirming referee `ass:pin-197`: $\underline{F}^{L}$ denotes the constrained infimum of the displayed scalar objective over the stated canonical SCM class and its admissible latent laws, rather than a minimizing SCM. — pins reading of def:eq-consistent-limiting: The sentence introducing the display explicitly says ``optimal value'', and the following sentence calls it the sharp lower bound when the Lipschitz bounds agree. - side condition added by the confirming referee `ass:pin-199`: $d_{\max}^U=\max_{j\in[n_U]}d_j^U$ ranges over non-Gumbel latent variables, while $d_{\max}^{\mathrm{in}}=\max_i d_i^{\mathrm{in}}$ and $d_{\max}^{\mathrm{out}}=\max_i d_i^{\mathrm{out}}$, with $d_i^{\mathrm{out}}=n_i$ for categorical nodes. — pins reading of def:corollary-deep-nn-dimensions: The NCM construction separately approximates non-Gumbel latent distributions and the functions in the canonical structural equations; those functions output categorical propensity vectors. The binding dimension symbols specify these same meanings. - side condition added by the confirming referee `ass:pin-201`: The coordinates $P(U_j)$ in $\theta_{\mathcal M}$ are the population laws of the non-Gumbel latent variables, not necessarily uniform laws; the independent standard Gumbel laws are fixed and omitted from this population tuple. — pins reading of def:thm-consistency-model-parameterization: Immediately after the tuple, the population parameter space contains factors $\mathcal P([-K,K]^{d_j^U})$. The canonical representation supplies the categorical Gumbel laws separately. - side condition added by the confirming referee `ass:pin-203`: The gap event is $E_\delta=\{\exists\,1\leqslant i0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. — pins reading of def:gumbel-softmax-layer: The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit. - side condition added by the confirming referee `ass:pin-210`: $F_n=\inf\left\{\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\hat{\mathcal{M}}}[F(V_1(t),\ldots,V_{n_V}(t))]:\hat{\mathcal{M}}\in\text{NCM}_{\mathcal{G}}(\mathcal{F}_{0,n},\mathcal{F}_{1,n}),\ S_{\lambda_n}(P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V}),P_n^{\mathcal{M}^*}(\boldsymbol{V}))\leqslant\alpha_n\right\}$. — pins reading of def:eq-opt-empiri: The prose introducing equation (\ref{eq:opt_empiri}) calls $F_n$ an estimator of optimal values, and the following paragraph asks whether it is a good lower bound. The supplied symbol convention likewise specifies a scalar infimum. - side condition added by the confirming referee `ass:pin-211`: In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. — pins reading of def:lemma-gumbel-err-softmax: Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores. - side condition added by the confirming referee `ass:pin-33`: $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. — pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes. - side condition added by the confirming referee `ass:pin-75`: For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. — pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula. - side condition added by the confirming referee `ass:pin-fix-00130002`: The law $\hat g_{\#}\lambda$ includes the independent standard Gumbel variables used in the mixing layer. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-09991de8`: The observational empirical law $P_n^{\mathcal M^*}(\boldsymbol V)$ is formed from $n$ independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-0a7b21b0`: The empirical optimization ranges over network representations, including distinct representations with the same latent population laws. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-0bcb2cef`: The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-2a1e7b01`: The mass assigned to $I_{O_n(i)}$ is $\lambda_d(C_{O_n(i)})=2^{-nd}$. — pins reading of def:prop-holder-curve-interval-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-2fe04b0e`: In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal C_n$ denotes the standard dyadic collection $\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}$. — pins reading of def:prop-holder-curve-cube-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-30413b86`: The common simulation draws are independent of the observational sample. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-3b195d8d`: The common exogenous draws are independent and identically distributed according to the specified product law of the exogenous inputs. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-419eb8e3`: The multiplicative width factors in $\Theta(d_{\max}^{\text{in}}d_{\max}^{\text{out}})$, $\Theta(d_{\max}^U)$, and $\Theta((d_{\max}^U)^2)$ can be chosen as absolute numerical constants, independent of the causal model, $R$, all dimensions, and all category and component counts. — pins reading of thm:deep-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-43f1bb71`: For each latent law $P(U_C)$, the assumptions of the proposition entitled ``Deep distribution approximation'' require the support charts of \cref{assump:support} and \cref{assumption:lower_bound} on each normalized component law $\mu_i=P(U_C)|_{C_i}/P(U_C\in C_i)$, with $(f^{-1})_{\#}\mu_i(\partial[0,1]^{d_i^C})=0$ when $d_i^C>0$. — pins reading of thm:deep-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-448b19a0`: Probabilities of uniform deviation events are evaluated in the completion of the sample law. — pins reading of lem:sup-wasserstein: stated by the repair - side condition added by the confirming referee `ass:pin-fix-4a86c2eb`: The structural uniform infinity norm takes the supremum of the coordinatewise infinity norm over the common structural input domain. — pins reading of thm:approximation: stated by the repair - side condition added by the confirming referee `ass:pin-fix-507a5848`: For each component of positive dimension $q=d_i^C$, the application of Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and quantization mesh $\delta/n$. — pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair - side condition added by the confirming referee `ass:pin-fix-59a01041`: For each $n$, every candidate is evaluated on the same $m_n$ draws of all its exogenous inputs, including its uniform inputs, generator Gumbel inputs, and categorical Gumbel inputs. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-5f02b8a6`: The error estimates are asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of these budgets and of $\tau$. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-676b96c3`: The finite component counts and component cube dimensions of the latent generator architectures in $\mathcal F_{1,n}$ are fixed as $n$ varies. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-72c9e86d`: The measures $\mu$ and $\hat\mu$ are probability measures. — pins reading of lem:push-forward-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-73abef8c`: In Corollary~\ref{corollary:deep_nn}, for $m_i=d_i^{\text{in}}>0$, the symbol $\hat f_i$ denotes $q_i\circ\Pi_i$ for a continuous variable and the vector $\bigl(\max\{\epsilon_i,(q_i\circ\Pi_i)_k\}\bigr)_{k=1}^{n_i}$ for a categorical variable, where $q_i$, $\Pi_i$, and $\epsilon_i=A_iL_0^{-2/m_i}$ are chosen in its proof. — pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair - side condition added by the confirming referee `ass:pin-fix-761562d5`: The self terms in $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$ use the same $1$-norm ground distance on the corresponding support points. — pins reading of lem:sinkhorn-approx: stated by the repair - side condition added by the confirming referee `ass:pin-fix-7a79e01d`: In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal I_n$ denotes $\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$ with $I_{O_n(i)}=[(i-1)2^{-nd},i2^{-nd}]$. — pins reading of def:prop-holder-curve-interval-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8429b12c`: The latent generator class in the statement uses the same temperature $\tau$ as the displayed error estimate. — pins reading of thm:deep-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-84ccd033`: $F_n$ denotes the infimum of objective values, with an attained minimum having the same value. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8cc0dba5`: For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. — pins reading of def:hilbert-partitions: stated by the repair - side condition added by the confirming referee `ass:pin-fix-945d85ed`: The error estimate is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, for fixed $R$ and fixed causal model. — pins reading of thm:deep-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9463b114`: For a zero categorical propensity, $\log0=-\infty$. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9675bba5`: Ties in a categorical argmax are resolved by the smallest label. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-99ac9bb3`: For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. — pins reading of def:hilbert-partitions: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9bbf998f`: A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9d74053b`: The orders for $\epsilon_n$ and $s_n$ are asymptotic as $n\rightarrow\infty$. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ab17be7a`: For a categorical propensity coordinate equal to zero, $\log0=-\infty$ in the canonical argmax construction. — pins reading of thm:approximation: stated by the repair - side condition added by the confirming referee `ass:pin-fix-b83fa4d1`: $W$ is Wasserstein-1 distance for the ground distance $\|x-y\|$. — pins reading of lem:product-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-bbf1c78a`: In Corollary~\ref{corollary:deep_nn}, for $m_i=0$, the symbol $\hat f_i$ denotes the exact constant function $f_i$. — pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair - side condition added by the confirming referee `ass:pin-fix-bd620122`: The architectural $\Theta$ orders use sufficiently large fixed constant factors supplied by the network constructions in the proof. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c514425b`: The refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. — pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c778b424`: The reference measure $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$. — pins reading of def:prop-holder-curve-cube-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c85b9186`: Each $O_n$ is a bijection. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ca667e13`: $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. — pins reading of lem:product-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-d19c63b5`: The probability bound is for each fixed pair $\delta,\epsilon>0$. — pins reading of lem:sup-wasserstein: stated by the repair - side condition added by the confirming referee `ass:pin-fix-db6ed772`: In the proposition entitled ``Deep distribution approximation'', the multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-e896f793`: The cells of $\mathcal C_n$ are closed and may share boundaries. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f66ad243`: The depth expression $\Theta(\log d_{\max}^{\text{in}})$ means $\Theta(1+\log d_{\max}^{\text{in}})$ at $d_{\max}^{\text{in}}=1$. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f6f485b1`: The uniform infinity norm is $\|F-\hat F\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}|F_j(x)-\hat F_j(x)|$. — pins reading of lem:push-forward-err: stated by the repair - proof change: the printed proof, revised step by step] \label{proof:consistency} We begin by defining a proper metric space. By \cref{assump:Lip_and_bounded}, all random variables are bounded. Suppose that $ \max_{i,j}\{\| V_{i} \| _{\infty} ,\| U_{j} \| _{\infty}\} \leqslant K$. A canonical causal model $ \mathcal{M} \in \mathcal{M}(\mathcal{G} ,\mathcal{F} ,\boldsymbol{U})$ is decided by \begin{equation*} \theta _{\mathcal{M}} =( f_{1} ,\cdots ,f_{n_{V}} ,{P}({U_{1}}) ,\cdots , {P}({U_{n_{U}}})) , \end{equation*} where $ f_{i}$ are functions in the structure equations (\ref{eq:structure_equation}) and $ U_{j}$ are uniform parts of the latent variables. We denote $ \mathcal{M}^{\theta }$ to be the SCM represented by $ \theta $, the underlying SCM be $ \mathcal{M}^{\theta^*}$ and $P^\theta$ to be the distribution of $ \mathcal{M}^\theta$. We consider the space \[ M=\mathcal F_{V_1}\times\cdots\times\mathcal F_{V_{n_V}}\times\prod_{j=1}^{n_U}\mathcal P([-K,K]^{d_j^U}), \] where \[ \mathcal F_{V_i}=\begin{cases} \mathcal F_{\hat L_f}^K([-K,K]^{d_{i,\text{in}}^V},[-K,K]^{d_{i,\text{out}}^V}),&V_i\text{ continuous},\\ \{p:[-K,K]^{d_{i,\text{in}}^V}\rightarrow\Delta_{n_i}:\operatorname{Lip}(p)\leqslant\hat L_f\},&V_i\text{ categorical}. \end{cases} \] Here $d_{i,\text{in}}^V,d_{i,\text{out}}^V$ are the input and output dimensions of $f_i$, $\mathcal F_{\hat L_f}^K$ has uniform output $1$-norm bound $K$ and common Lipschitz bound $\hat L_f$, and $\mathcal P(A)$ denotes the probability measures on $A$. For $ \theta =\left( f_{1} ,\cdots ,f_{n_{V}} ,{P}({U_{1}}) ,\cdots , {P}({U_{n_{U}}})\right) ,\theta' =\left( f_{1}' ,\cdots ,f_{n_{V}}' ,{P}({U_{1}'}) ,\cdots ,{P}({U_{n_{U}}'})\right)$, we define a metric on $ M$ \begin{equation*} d( \theta ,\theta') =\sum _{k=1}^{n_{V}} \| f_{k} -f_{k}' \| _{\infty } +\sum _{k=1}^{n_{U}} W(P(U_{k}),{P}({U_{k}'})) . \end{equation*} We derive the population continuity bound in the fixed dimensions of this theorem. Couple corresponding non-Gumbel latent laws optimally and take the product of these couplings, using their independence. The expected joint latent $1$-norm discrepancy is the sum of their marginal Wasserstein distances. Generate the observed nodes in a topological order. At a continuous node the expected output discrepancy is at most the structural output $1$-norm approximation error plus $\hat L_f$ times the expected input discrepancy. At a categorical node with conditional propensity vectors $p,p'$, a maximal coupling has disagreement probability $\|p-p'\|_1/2$; its expected label distance is at most $(n_i-1)\|p-p'\|_1/2$. Such a coupling assigns common mass $\min\{p_k,p'_k\}$ to label $k$ and distributes the remaining masses on the disjoint residual supports. It preserves each model's conditional categorical law. If $e$ is the sum of the expected latent and previously generated observed discrepancies, adjoining a continuous node increases it to at most $(1+\hat L_f)e+d_i\|f_i-f'_i\|_\infty$; adjoining a categorical node increases it to at most $(1+(n_i-1)\hat L_f/2)e+n_i(n_i-1)\|f_i-f'_i\|_\infty/2$. Choose fixed constants $A\geqslant1$ and $B$ bounding these finitely many coefficients. Iterating $e_{k+1}\leqslant Ae_k+Bd(\theta,\theta')$ gives \[ e_k\leqslant A^ke_0+Bd(\theta,\theta')\sum_{r=0}^{k-1}A^r. \] Indeed, the formula holds for $k=0$, and multiplying its right side by $A$ and adding $Bd(\theta,\theta')$ gives the formula for $k+1$. Identical intervention values contribute zero discrepancy and remove their generation steps. Projection to the observed coordinates therefore proves \[ W(P^\theta(\boldsymbol V(\boldsymbol t)),P^{\theta'}(\boldsymbol V(\boldsymbol t)))\leqslant C d(\theta,\theta') \] uniformly over the intervention values, with $C$ depending on the fixed graph, dimensions, cardinalities and regularity bounds. The triangle inequality consequently makes $g(\theta)=W(P^{\theta^*}(\boldsymbol V),P^\theta(\boldsymbol V))$ Lipschitz in $d$; denote a Lipschitz constant by $C_g$. Now, we define $ \Theta _{n}$. Let \begin{equation*} \mathcal{P}_{n} =\left\{{P}({\boldsymbol{U}}):\boldsymbol{U} \ \text{is the latent distribution of} \ \hat{\mathcal{M}} \in \text{NCM}_{\mathcal{G}} (\mathcal{F}_{0,n} ,\mathcal{F}_{1,n})\right\}. \end{equation*} Thus $\mathcal P_n$ consists of the latent population laws obtained by pushing forward the independent uniform and generator Gumbel inputs through the current positive-temperature generator networks. We denote $\Theta_n=\hat{\mathcal F}_{V_1,n}\times\cdots\times\hat{\mathcal F}_{V_{n_V},n}\times\mathcal P_n$, where \[ \hat{\mathcal F}_{V_i,n}=\begin{cases} \{f\in\mathcal F_{0,n}:\sup_x\|f(x)\|_1\leqslant K\},&V_i\text{ continuous},\\ \{q/\|q\|_1:q\in\mathcal F_{0,n},\ q(x)\in[0,\infty)^{n_i},\ \|q(x)\|_1>0\ \forall x,\ \operatorname{Lip}(q/\|q\|_1)\leqslant\hat L_f\},&V_i\text{ categorical}. \end{cases} \] The norms and Lipschitz bounds are evaluated on the structural input domains, and $\mathcal F_{0,n}$ is defined in \cref{thm:consistency}. Thus $\Theta_n\subset M$. Note that by construction, latent variables are independent and $ \mathcal{P}_{n}$ can be decomposed into direct produce $ \mathcal{P}_{n,1} \times \cdots \times \mathcal{P}_{n,n_{U}}$. Let $\mathcal A_n$ be the set of restricted structural and generator network representations at sample size $n$, and let $\theta_\eta\in\Theta_n$ be the population parameter of a representation $\eta\in\mathcal A_n$. Let $\xi_1,\ldots,\xi_{m_n}$ be the common independent exogenous draws, including the categorical Gumbel coordinates, and write $H_\eta(\xi_l)$ for the resulting complete observed vector. Define \[ \Theta_\infty=\overline{\bigcup_n\Theta_n},\qquad P_{m_n}^\eta=\frac1{m_n}\sum_{l=1}^{m_n}\delta_{H_\eta(\xi_l)},\qquad h_n(\eta)=S_{\lambda_n}(P_n^{\theta^*},P_{m_n}^\eta), \] \[ f(\theta)=\mathbb E_{t\sim\mu_T}\mathbb E_{\mathcal M^\theta}[F(V_1(t),\ldots,V_{n_V}(t))]. \] The empirical constraint is a function of the representation $\eta$, whereas the population objective and $g(\theta)$ are functions of $\theta$. The optimization minimizes $f(\theta_\eta)$ over representations satisfying $h_n(\eta)\leqslant\alpha_n$. For fixed $n$, pad the finitely many network architectures to their stated widths and encode each representation by its finitely many real weights and mixture probabilities. Finite products of Euclidean spaces and simplices are standard Borel spaces. Affine maps, ReLU, clipping and finite composition have jointly continuous evaluation in their weights and inputs. At the fixed temperature $\tau_n>0$, mixture weights can be written \[ \frac{p_j^{1/\tau_n}\exp(G_j/\tau_n)}{\sum_k p_k^{1/\tau_n}\exp(G_k/\tau_n)},\qquad p\in\Delta, \] whose denominator is positive even when some $p_j=0$; hence generator evaluation is jointly Borel. The restrictions defining $\mathcal A_n$ are Borel restrictions on these parameters. To check this, choose a countable dense set in each continuous structural input domain. Continuity makes the output bound, coordinatewise nonnegativity and every Lipschitz inequality equivalent to their inequalities on that dense set or its Cartesian square. Each such inequality is Borel in the weights. For a nonnegative categorical raw network $q$, positivity on a compact input box $D_0$ is equivalent to membership in \[ \bigcup_{l=1}^\infty\bigcap_{x\in Q_0}\{\eta:\|q_\eta(x)\|_1\geqslant1/l\}, \] where $Q_0$ is countable and dense in $D_0$; continuity and attainment of a minimum prove both implications. Use the intersection of these conditions over a countable compact exhaustion when the full input domain is unbounded. On this Borel parameter set $q/\|q\|_1$ is jointly Borel and continuous in the input, so its Lipschitz restriction is again a countable family of Borel inequalities. The same dense-input tests apply to the raw Lipschitz restrictions before clipping. Thus $\mathcal A_n$ is standard Borel. These tests use dense input sets only. The exogenous input space is a finite product of uniform cubes and real Gumbel coordinates, hence standard Borel. For a categorical propensity vector $p$, put $r_i=\exp(-g_i)/p_i$ when $p_i>0$ and $r_i=\infty$ when $p_i=0$. These extended-real functions are Borel. The smallest minimizing label is $i$ exactly when $r_ii$; this is a finite intersection of Borel comparisons. Induction through the finite topological order therefore proves joint Borel evaluation of $H_\eta(\xi)$ and of its interventional version $H_\eta(t,\xi)$. The function $F$ is bounded on the bounded observed range, so integration over the fixed exogenous law and $\mu_T$ makes $\eta\mapsto f(\theta_\eta)$ Borel. Finite empirical optimal transport and entropic transport values are continuous in their cost matrices: changing every cost by at most $a$ changes every transport objective, and hence its infimum, by at most $a$. Therefore the empirical Sinkhorn constraint $h_n(\eta)$ is jointly Borel in the representation and data. For every real $a$, the event $\{F_na$ and \[ \frac{E_i}{q_i-a}<\min_{j\ne i}\frac{E_j}{q_j+a}. \] A robust label wins for every propensity vector whose coordinates differ from $q$ by at most $a$. The robust-label events are disjoint. For $q_i>a$, integrating the density of $E_i$ and the survival probabilities of the other exponentials gives \[ \mathbb P(i\text{ robust})=\int_0^\infty\exp\left(-s\left(1+\frac{\sum_{j\ne i}(q_j+a)}{q_i-a}\right)\right)\,ds =\frac{q_i-a}{q_i-a+\sum_{j\ne i}(q_j+a)}. \] For $q_i\leqslant a$ this probability is zero. Consequently, \[ \mathbb P(i\text{ robust})\geqslant\frac{(q_i-a)_+}{1+Na}. \] Summing and using $\sum_i(q_i-a)_+\geqslant1-Na$ bounds the probability of no robust label by $2Na/(1+Na)$ when $Na<1$, and the trivial bound $1\leqslant2Na$ covers $Na\geqslant1$. For each table and generator representation let $b$ be the indicator that a nonrobust choice occurs at any table prefix. Independence of the categorical noise from the generated latent vector and a union bound over the finitely many prefixes give $\mathbb E b\leqslant C\rho$. If $b=0$, induction over the categorical nodes forces the original and table prefixes to agree. Consequently, writing $\widetilde H$ for the table observed map and $B_V$ for a common bound on the observed $1$-norm diameter, \[ \|H_\eta(\xi)-\widetilde H(\xi)\|_1\leqslant C\rho+B_V b(\xi). \] This is a pointwise bound, including zero propensities. We bound the complexity of finite-type indicators and ambiguity indicators. A table simulation has type $(u,c)$, where $u$ is its latent grid index and $c$ its complete categorical label vector; its observed output is a fixed vector for each type. There are at most $J=C\rho^{-D_U}$ types. Include the indicators of membership in every subset of these types, and the ambiguity indicators $b$, in a scalar class $\mathcal B_{n,\rho}$. Here is an explicit standard Borel parameterization of this entire indicator class. Let $\Gamma_n$ be the finite-dimensional generator weight space with its mixture simplex factors, and let generator parameters range over all of $\Gamma_n$. This supplies an envelope for every restricted generator used in the optimization. A parameter consists of $\gamma\in\Gamma_n$, a table in the finite set $\mathcal T_\rho$, and a finite tag specifying either a subset of the at most $J$ types or the ambiguity indicator. This is a finite union of products of standard Borel spaces. Generator evaluation is jointly Borel by the positive-denominator mixture formula. Round its clipped latent output using fixed half-open grid cells, with the last cell closed at the endpoint; the grid index is a finite-valued Borel function. For each fixed table and grid index, the table simulation uses finitely many categorical race comparisons. In the notation $r_i=\exp(-g_i)/q_i$, with $r_i=\infty$ at $q_i=0$, the smallest-label rule is the finite system $r_ii$, so its selected label is Borel. The robust-label events are Borel too: when $q_i>a$ all their denominators are positive, and when $q_i\leqslant a$ the event is empty. Their finite unions and complements make $b$ jointly Borel. Type-subset membership is a finite union of these Borel events. Thus $\mathcal B_{n,\rho}$ has jointly Borel evaluation on a standard Borel parameter space, including zero propensities, grid boundaries and all ties. When using the ambient Euclidean input space in Lemma~\ref{lemma:sup_wasserstein}, clip the uniform input coordinates to their cubes; this changes no sample value and preserves joint Borel evaluation. The parameter envelope has precisely the same architecture and finite tags as those counted in the growth bound. For each continuous scalar dual test $h$, its normalized square-root transform is jointly Borel by composition. Consequently the structural condition of Lemma~\ref{lemma:sup_wasserstein} holds for these families, without approximation of selector parameters. On $M_0$ fixed exogenous inputs, the latent grid indices are determined by at most $C M_0/\rho$ scalar generator comparisons. The ReLU blocks have $P\leqslant C H_n$ parameters and at most $C H_n$ layers. Within a fixed activation pattern, their outputs are polynomials in the parameters of degree at most $C H_n$. For a positive-temperature mixture write its weights as $a_j R_j/\sum_k a_kR_k$, where $a_j=p_j^{1/\tau_n}\geqslant0$, $\sum_j a_j>0$, and $R_j=\exp(G_j/\tau_n)$ is fixed on each input. Treat the $a_j$ as additional free parameters. Clearing the positive denominator in a grid comparison leaves a polynomial of degree at most $C H_n+1$. Truncation adds only fixed-degree comparisons. The polynomial sign bound in Lemma 17 of \cite{bartlett2019nearly}, applied successively to the activation comparisons in each layer and then to these grid comparisons, gives at most \[ (C(M_0+H_n)H_n/\rho)^{C H_n^2} \] latent grid patterns: at each layer the sign bound contributes at most $(C(M_0+H_n)H_n/\rho)^{C P}$, and there are at most $C H_n$ layers. For a fixed table and grid pattern, the categorical race comparisons and every table output on these inputs are determined. The subgraph growth of $\mathcal B_{n,\rho}$ is therefore at most this number times $|\mathcal T_\rho|2^J$: a subset of the types has $2^J$ choices, and an ambiguity indicator needs no additional choices. Solving the shattering inequality \[ 2^{M_0}\leqslant|\mathcal T_\rho|2^J(C(M_0+H_n)H_n/\rho)^{C H_n^2} \] bounds their pseudo-dimensions by \[ v_n(\rho)\leqslant C\left(\rho^{-D_U}\log(2+\rho^{-1})+H_n^2\log(2+H_n/\rho)\right). \] To derive this bound, put $A=\log|\mathcal T_\rho|+J\log2$ and $B=C H_n^2$. The shattering inequality gives $M_0\log2\leqslant A+B\log(C(M_0+H_n)H_n/\rho)$. If $M_0\leqslant4A/\log2$, the first term suffices. Otherwise, absorbing $A$ and solving the remaining logarithmic inequality gives $M_0\leqslant C B\log(2+BH_n/\rho)+C B\log(2+A/B)$; this follows from $\log x\leqslant x/2$ after rescaling $M_0/B$. The bound on $A$ makes this at most the displayed expression. Put \[ r_n=m_n^{-1/(D+2)}\log m_n+H_n m_n^{-1/2}\log m_n, \] and take $\rho=r_n$ for sufficiently large $n$. The depth schedule implies $H_n=O(m_n^{d_{\max}^U/(2d_{\max}^U+4)})$, so $r_n\rightarrow0$. Apply Lemma~\ref{lemma:sup_wasserstein} to $\mathcal B_{n,r_n}$ with scalar output. Its functions take values in $\{0,1\}$, its pseudo-dimension is at most $v_n(r_n)$, and Wasserstein distance between its Bernoulli output laws equals the absolute difference of their means. Choose the lemma's tolerances $\delta=r_n$ and $\epsilon=A_0r_n$. Its exponent is bounded above by \[ C\left((1+r_n^{-1})\log(2+r_n^{-1})+r_n^{-D_U}\log^2(2+r_n^{-1})+H_n^2\log(2+H_n/r_n)\log(2+r_n^{-1})\right)-cA_0^2m_nr_n^2. \] Here $\log(2+H_n/r_n)=O(\log m_n)$ and \[ m_nr_n^2\geqslant m_n^{D/(D+2)}\log^2m_n+H_n^2\log^2m_n, \] whereas $r_n^{-D_U}\leqslant m_n^{D_U/(D+2)}(\log m_n)^{-D_U}$ and $r_n^{-1}\leqslant m_n^{1/(D+2)}/\log m_n$. Since $D\geqslant D_U,1$, a sufficiently large fixed $A_0$ makes the failure probability at most $O(m_n^{-3})$. On the complementary event every type-subset probability and every ambiguity-indicator mean differs from its empirical counterpart by at most $Cr_n$. The supremum over type subsets is the total variation distance between the population and empirical type laws, so that distance is at most $Cr_n$. Couple their common type masses identically and their residual masses arbitrarily. Because table outputs have diameter at most $B_V$, their Wasserstein distance is at most $B_VCr_n$. For each original representation, coupling original and table outputs on the same draw bounds their population error by $C\rho+B_V\mathbb E b$ and their empirical error by $C\rho+B_Vm_n^{-1}\sum_l b(\xi_l)$. The ambiguity estimate bounds both by $Cr_n$. The triangle inequality therefore gives \[ \sup_{\eta\in\mathcal A_n}W(P^{\theta_\eta}(\boldsymbol V),P_{m_n}^\eta(\boldsymbol V))\leqslant Cr_n \] with probability at least $1-O(m_n^{-3})$. This retains the positive-temperature generator, paired simulation observations and every categorical noise coordinate. For the observational sample let $C$ denote the complete categorical label vector, which has a fixed finite set of possible values. Conditional on $C=c$, all continuous observations equal a deterministic map $\Phi_c(U_1,\ldots,U_{n_U})$: fix the categorical values to $c$ and evaluate the continuous structural equations in topological order. Induction through their Lipschitz bounds makes $\Phi_c$ Lipschitz with a constant uniform over these finitely many $c$. The conditional joint latent laws $\mu_c=P((U_1,\ldots,U_{n_U})\mid C=c)$ are supported on the bounded $D_U$-dimensional latent cube; they need not be products. Write $p_c=P(C=c)$ and $N_c$ for the observational sample counts. For $p_c=0$, $N_c=0$ almost surely. For every $p_c>0$, a binomial concentration bound gives $N_c\geqslant np_c/2$ except on an exponentially small event. Conditional on the label vector of the sample, the latents of the observations carrying label $c$ are independent with law $\mu_c$. When $D_U\geqslant1$, Theorem 2 of \cite{fournier2015rate}, applied to each conditional joint latent empirical measure of size $N_c$, bounds its Wasserstein error by $C_c n^{-1/\max\{2,D_U\}}\log^2n$ with failure probability $O(n^{-4})$ whenever $N_c\geqslant np_c/2$. Its small-tolerance exponents are constant multiples of $N_c x^2$ for $D_U=1$, $N_c(x/\log(2+1/x))^2$ for $D_U=2$, and $N_c x^{D_U}$ for $D_U>2$. At the indicated $x$ these grow at least as $\log^4n$, $\log^2n$ and $\log^{2D_U}n$, respectively. Bounded support supplies the exponential-moment condition; norm equivalence changes only constants. Conditioning on the counts preserves the bound, since it holds uniformly for $np_c/2\leqslant N_c\leqslant n$. Lipschitz push-forward by $\Phi_c$ bounds each conditional observed Wasserstein error by the same order. If $D_U=0$, each conditional observed law is a point mass and its conditional sampling error is zero. Binomial concentration also bounds $\sum_c|N_c/n-p_c|$ by $C\sqrt{\log n/n}$ with probability at least $1-O(n^{-4})$. Couple common categorical-component masses using the conditional couplings and unmatched masses arbitrarily within the observed support. This gives \[ W(P_n^{\theta^*}(\boldsymbol V),P^{\theta^*}(\boldsymbol V))\leqslant\sum_c\min\{p_c,N_c/n\}W((\Phi_c)_\#\mu_c,(\Phi_c)_\#(\mu_c)_{N_c})+\frac{B_V}{2}\sum_c|N_c/n-p_c|\leqslant t_n, \] \[ t_n=C_1n^{-1/\max\{2,D_U\}}\log^2n, \] with probability at least $1-O(n^{-3})$, after a union bound over the finitely many categories. Terms with $N_c=0$ have zero common mass and are omitted. Finally Lemma~\ref{lemma:sinkhorn_approx} and the Wasserstein triangle inequality give, on the combined event, \[ \sup_{\eta\in\mathcal A_n}|h_n(\eta)-g(\theta_\eta)|\leqslant2(1+\log(nm_n))\lambda_n+Cr_n+t_n\leqslant s_n. \] This event has probability at least $1-O(n^{-3}+m_n^{-3})$. Since $m_n=\Omega(n)$, the failure probabilities are summable; Borel--Cantelli gives the estimate eventually almost surely. It applies to the current representations. \item We construct structural approximants satisfying both output and Lipschitz restrictions. Let $f:[-K,K]^m\rightarrow\mathbb R^{m'}$ have Lipschitz constant $L_f$, with $m\geqslant1$. Divide each coordinate into $R$ equal intervals of length $h=2K/R$, and triangulate each cell into simplices with successive vertices $v_0,v_1=v_0+he_{\pi(1)},\ldots,v_m=v_{m-1}+he_{\pi(m)}$, where $\pi$ runs over the coordinate permutations. Let $I_hf$ be affine on each simplex and equal $f$ at its vertices. Its derivative column in direction $e_{\pi(j)}$ is $(f(v_j)-f(v_{j-1}))/h$, of $1$-norm at most $L_f$. Splitting a line segment at simplex boundaries and summing the derivative bound proves $\operatorname{Lip}(I_hf)\leqslant L_f$; segments in boundaries follow by continuity. If $a_j$ are the barycentric coordinates of $x$, then \[ \|I_hf(x)-f(x)\|_1\leqslant\sum_{j=0}^ma_j\|f(v_j)-f(x)\|_1\leqslant L_fmh. \] For categorical $f$, the interpolated values are convex combinations in $\Delta_{n_i}$, so they are nonnegative and sum exactly to one. For continuous $f$, the additional uniform output bound on the true structural function gives \[ \|I_hf(x)\|_1\leqslant\sum_j a_j\|f(v_j)\|_1\leqslant K. \] Thus each continuous approximant satisfies the continuous network restriction. The nodal basis functions are translates and rescalings of \[ \phi(z)=\max\left\{0,\min\left(\{1+z_k,1-z_k:k\in[m]\}\cup\{1+z_k-z_l:k,l\in[m]\}\right)\right\}. \] On each simplex in the star of the central vertex, the active affine expression is its barycentric coordinate: the inequalities defining the simplex order its coordinates, and the minimum selects the difference between the two coordinates bordering that vertex. It equals one at the central vertex and zero at the other vertices; outside the star the minimum is nonpositive. Therefore \[ I_hf(x)=\sum_v f(v)\phi((x-v)/h) \] on the input cube. Each minimum is computed by a binary tree using $\min(a,b)=a-\max\{a-b,0\}$. There are $O(m^2)$ affine inputs, so each term has width $O(m^2)$ and depth $O(1+\log m)$. Parallelizing the terms and summing their outputs gives width $O(R^m)$ in fixed dimensions. Coordinatewise input clipping to the cube is a contraction and makes the network globally defined with the same Lipschitz and output bounds. Choosing $R$ proportional to $W_{0,n}^{1/m}$ gives error $O(W_{0,n}^{-1/m})$. When $m=0$, represent the constant function exactly. For a categorical function take $q=I_hf$; its normalization equals itself, so both Lipschitz restrictions hold. The output clipping in $\mathcal F_{0,n}$ acts identically on these bounded approximants. We obtain the latent approximants from the inherited component assumptions. For each normalized support component measure $\mu_i$, let $\psi_i:[0,1]^{q_i}\rightarrow C_i$ be its Lipschitz chart. For $q_i>0$, Proposition~\ref{prop:holder_curve} applied to the pullback measure supplies a curve $\gamma_i:[0,1]\rightarrow[0,1]^{q_i}$ of Holder exponent $1/q_i$ with that pullback law. Theorem 2(b) of \cite{Yarotsky2018OptimalAO} applied to its coordinates gives first-block uniform error $O(L_{1,n}^{-2/q_i})$; applied to the Lipschitz coordinates of $\psi_i$ it gives second-block error $O(L_{2,n}^{-2/q_i})$. Parallelize the coordinate networks and clip the first outputs to their input cubes. Their widths fit the stated fixed-width budgets. Lemma~\ref{lemma:push-forward_err}, including its fixed output-dimension factor, bounds the component Wasserstein error by $O(L_{1,n}^{-2/q_i}+L_{2,n}^{-2/q_i})$. For $q_i=0$, represent the singleton component by its constant value with zero error. With component probabilities $p_i$, Lemma~\ref{lemma:combin_err} bounds the hard selected population mixture error by the weighted sum of these component errors. The positive-temperature construction uses independent Gumbel-softmax weights; the hard mixture is its zero-temperature comparison law. Clip component outputs to a fixed box containing their supports. For outputs $v_i$ in that box, \[ \left\|\sum_i(X_i^{\tau_n}-X_i^0)v_i\right\|_1\leqslant\left(\max_i\|v_i\|_1\right)\sum_i|X_i^{\tau_n}-X_i^0|\leqslant C\|X^{\tau_n}-X^0\|_1. \] Couple these weights using identical Gumbel draws and retain identical independent component outputs. The coupling estimate in Lemma~\ref{lemma:gumbel_err} bounds the additional population error by $O(\tau_n(1-\log\tau_n))$ for sufficiently large $n$. The triangle inequality proves, for each non-Gumbel latent $U_j$, a generator $g_{j,n}$ with \[ W(P(U_j),P(g_{j,n}(Z_j)))\leqslant O\left(L_{1,n}^{-2/d_{\max}^U}+L_{2,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right). \] Coordinatewise output truncation $T_K$ cannot increase this error: $T_KU_j=U_j$ and $\|U_j-T_Kg_{j,n}(Z_j)\|_1\leqslant\|U_j-g_{j,n}(Z_j)\|_1$ in every coupling. Choose the generators independently. Summing their marginal errors and the structural errors in $d$ gives a representation $\eta_n\in\mathcal A_n$ with \[ a_n=d(\theta^*,\theta_{\eta_n})\leqslant O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+L_{1,n}^{-2/d_{\max}^U}+L_{2,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right)\rightarrow0. \] Choose the constant in $\epsilon_n$ large enough that $C_ga_n\leqslant\epsilon_n$. \end{enumerate} Work on the probability-one event where the current-class sampling estimate holds eventually. Since $g(\theta^*)=0$, the constructed representations satisfy \[ h_n(\eta_n)\leqslant g(\theta_{\eta_n})+s_n\leqslant C_ga_n+s_n\leqslant\epsilon_n+s_n=\alpha_n. \] Thus the restricted optimization is eventually feasible. Its objective satisfies $f(\theta_{\eta_n})\rightarrow f(\theta^*)=F_*$, whence $\limsup_nF_n\leqslant F_*$. Choose a feasible representation $\bar\eta_n$ with $F_n\leqslant f(\theta_{\bar\eta_n})\leqslant F_n+1/n$. The bounded objective on compact $M$ makes $F_n$ eventually bounded. Along a subsequence attaining its lower limit, compactness gives a further subsequence on which $\theta_{\bar\eta_n}\rightarrow\bar\theta$. Feasibility and the current-class bound imply \[ 0\leqslant g(\theta_{\bar\eta_n})\leqslant h_n(\bar\eta_n)+s_n\leqslant\alpha_n+s_n\rightarrow0. \] Continuity yields $g(\bar\theta)=0$. Its continuous structural functions satisfy $\sup_x\|f_i(x)\|_1\leqslant K$, all structural functions have Lipschitz bound $\hat L_f$, its categorical functions are simplex-valued, and its latent marginal laws define an independent bounded canonical latent distribution. Since categorical variables occur only when $K\geqslant1$, their normalized propensity vectors also satisfy the output $1$-norm bound. Hence the limit is an admissible observationally equivalent model in (\ref{eq:consistent_limiting}), so $f(\bar\theta)\geqslant\underline F^{\hat L_f}$. Continuity and $|f(\theta_{\bar\eta_n})-F_n|\leqslant1/n$ give $\liminf_nF_n=f(\bar\theta)$. This proves \[ [\liminf_nF_n,\limsup_nF_n]\subset[\underline F^{\hat L_f},F_*] \] almost surely. On the finite-sample event used in the sampling proof, every feasible representation also satisfies \[ W(P^{\theta^*}(\boldsymbol V),P^{\theta_\eta}(\boldsymbol V))\leqslant h_n(\eta)+s_n\leqslant\alpha_n+s_n\leqslant2\alpha_n. \] This event has probability at least $1-O(n^{-3}+m_n^{-3})$ and supplies the complete-observation error and feasibility estimates used in Corollary~\ref{cor:rate} for the same restricted optimizer and revised radius. - why: (a) The statement and its compactness argument fail for categorical nodes. The restriction $\operatorname{Lip}(f)\leqslant\hat L_f$ on a raw network does not imply that $f/\|f\|_1$ has that restriction. The printed categorical parameter class therefore contains arbitrarily steep propensity functions, is not equicontinuous, and need not have compact closure in the printed metric. Its feasible limits cannot be compared with $\underline F^{\hat L_f}$ as claimed; the witness gives an empirical minimum strictly below that bound. (b) Independent resampling for each candidate and evaluation on common exogenous draws are different readings of the empirical optimization. The uniform push-forward argument determines the common-draw reading. All inherited independence, categorical-generation, support and lower-bound assumptions flagged by lint are supplied through the corollary's assumptions. The corollary reference supplies assumptions and dimensions, rather than requiring its conclusion as an additional dependency. The optimization proposition's premises are conditions the proof attempts to verify, rather than additional standing hypotheses. The finite-measure notation is explicit in the Sinkhorn lemma used here; mixture weights are component probabilities; and the empirical lemma's boundedness and pseudo-dimension conditions are verified in the cited argument. Those remaining lint observations do not establish transcription drift. ### `prop:lip-ate` (`Main.tex:309-317`) - as printed: \label{prop:Lip_ATE} Given two causal models $ \mathcal{M} ,\hat{\mathcal{M}} \in\mathcal{M}(\mathcal{G},\mathcal{F}_L,\boldsymbol{U})$ satisfying \cref{assump:independence_u} and \cref{assump:Lip_and_bounded}, let their observational distributions be $ \nu ,\mu $. Suppose the norms of all variables are bounded by $K>0$. If (1) (Overlap) $ \nu $ is absolutely continuous with respect to one probability measure $P$ and the density $ p_{\nu }\left( t|\text{Pa}( T) =x\right) \geqslant \delta >0 $ for x almost surely and $ t\in [ t_{1} ,t_{2}]$ and (2) (No Confounding) there is no hidden confounder in the causal graph $\mathcal{G}$, we have \begin{equation*} \begin{aligned} \int _{t_{1}}^{t_{2}}(\mathbb{E}_{\mathcal{M}}[ Y( t)] -\mathbb{E}_{\hat{\mathcal{M}} }[\hat{Y}( t)])^{2} P(dt) & \leqslant \frac{2C_{W}}{\delta } W( \mu ,\nu ) +2(L+1)^{n_{V}} W^{2}( \mu ,\nu )( t_{2} -t_{1}) , \end{aligned} \end{equation*} where $ C_{W} =4(L+1)^{n_{V}} K+2K\max\left\{(L+1)^{n_{V}} ,1\right\}$. - corrected: Given two causal models $ \mathcal{M} ,\hat{\mathcal{M}} \in\mathcal{M}(\mathcal{G},\mathcal{F}_L,\boldsymbol{U})$ satisfying \cref{assump:independence_u} and \cref{assump:Lip_and_bounded}, let their observational distributions be $ \nu ,\mu $. Suppose the norms of all variables are bounded by $K>0$. If (1) (Overlap) $\nu$ is absolutely continuous with respect to one probability measure $P$ and the density $p_{\nu}\left(t|\text{Pa}(T)=x\right)\geqslant\delta>0$ for $x$ almost surely and $t\in[t_1,t_2]$ and (2) (No Confounding) there is no hidden confounder in the causal graph $\mathcal{G}$, we have \begin{equation*} \begin{aligned} \int_{t_1}^{t_2}(\mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}(t)])^2 P(dt) &\leqslant\frac{2C_W}{\delta}W(\mu,\nu)+2L_y^2W^2(\mu,\nu)P([t_1,t_2]), \end{aligned} \end{equation*} where \[ D=\max\left(\{1\}\cup\left\{\frac{n_i-1}{2}:V_i\text{ is categorical}\right\}\right),\qquad L_y=(LD+1)^{n_V}, \] \[ q=\sum_{i:V_i\text{ is continuous}}d_i+\left|\{i:V_i\text{ is categorical}\}\right|,\qquad B=K(1+2qL_y), \] and $C_W=4L_yB+2B\max\{L_y,1\}$. - no longer uses `ass:prop-lip-ate-overlap` - side condition added by the confirming referee `ass:pin-154`: For each of $\mathcal{M}$ and $\hat{\mathcal{M}}$, $\max_{i\in[n_V],j\in[n_U]}\{\|V_i\|_\infty,\|U_j\|_\infty\}\leqslant K$ almost surely, where the indexed $U_j$ are the non-Gumbel latent variables. — pins reading of ass:prop-lip-ate-bounded: Proposition \ref{prop:Lip_ATE} explicitly inherits Assumption \ref{assump:Lip_and_bounded}, which prints this maximum; Assumption \ref{assump:categorical_rv} distinguishes the indexed $U_j$ from the categorical noises $G_{V_i}$. - side condition added by the confirming referee `ass:pin-182`: For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\|f_i(x)-f_i(x')\|\leqslant L_f\|x-x'\|$ on its structural input domain, using the paper's $1$-norm. — pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$. - side condition added by the confirming referee `ass:pin-183`: There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\|V_i\|_\infty,\|U_j\|_\infty\}\leqslant K$ almost surely under the SCM law. — pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation. - side condition added by the confirming referee `ass:pin-189`: For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. — pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition. - side condition added by the confirming referee `ass:pin-190`: In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. — pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction. - side condition added by the confirming referee `ass:pin-191`: Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). — pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations. - side condition added by the confirming referee `ass:pin-206`: The functions $f,\hat{f}:[-K,K]\rightarrow\mathbb{R}$ are $L$-Lipschitz regression versions satisfying $f(T)=\mathbb{E}_{\nu}[Y\mid T]$ almost surely under $\nu$ and $\hat{f}(\hat{T})=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}]$ almost surely under $\mu$. — pins reading of ass:lemma-lip-condition-regression: The proof of Lemma~\ref{lemma:Lip_condition} uses both cross-evaluations and bounds its test function on the common box; the owner-supplied support condition specifies that box. - side condition added by the confirming referee `ass:pin-fix-19bccf89`: The measure $P$ is the reference probability measure on the treatment space: for $X=\text{Pa}(T)$, the conditional density notation means $\nu(dt|X=x)=p_\nu(t|X=x)P(dt)$. — pins reading of prop:lip-ate: stated by the repair - proof change: the printed proof, revised step by step] If $Y$ is not a descendant of $T$, its structural equation and all the equations of its ancestors are unaffected by the intervention, so $\mathbb{E}_{\mathcal{M}}[Y(t)]=\mathbb{E}_{\mathcal{M}}[Y]$ and $\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}(t)]=\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}]$. For any coupling of the observational laws, the absolute difference of the outcome means is at most the expected absolute difference of the outcome coordinates, which is at most the expected $1$-norm distance of the observed vectors. Taking the infimum over couplings gives \begin{align*} \int_{t_1}^{t_2}(\mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}(t)])^2P(dt) &=P([t_1,t_2])(\mathbb{E}_{\mathcal{M}}[Y]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}])^2\\ &\leqslant P([t_1,t_2])W^2(\mu,\nu)\\ &\leqslant\frac{2C_W}{\delta}W(\mu,\nu)+2L_y^2P([t_1,t_2])W^2(\mu,\nu), \end{align*} since $L_y\geqslant1$. Now, suppose that $Y$ is a descendant of $T$. Let $X=\text{Pa}(T)$, and let $A$ be the vector of all observed nodes other than $T$ that are not descendants of $T$. In particular, $X$ is a subvector of $A$. Every observed parent of a node in $A$ also belongs to $A$: a parent equal to $T$, or descended from $T$, would make that node a descendant of $T$. By the canonical representation and the absence of hidden confounding, the noises belonging to distinct observed nodes are independent. Thus $A$ is generated without the noise of $T$ or any descendant of $T$, and the descendant noises are independent of $(A,T)$. Conditional on $A=a$, the equation of $T$ uses only $X=x$ and its own independent noise. Consequently, \[ \nu(dt|A=a)=\nu(dt|X=x). \] For each $\mathcal{N}\in\{\mathcal{M},\hat{\mathcal{M}}\}$, fix $A=a$ and $T=t$ and generate the descendants of $T$ in topological order using their independent noises with their laws in $\mathcal{N}$. Denote the resulting mean of the outcome by $g_{\mathcal{N}}(a,t)$. These means are finite: categorical outputs have finite ranges, and the Lipschitz bound for a continuous structural function bounds its output by its value at a fixed input plus $L$ times the input distance; induction through the finite graph, using bounded non-Gumbel noises, gives finite first moments at every fixed $(a,t)$. Independence of the descendant noises from $(A,T)$ shows that $g_{\mathcal{N}}$ is a version of the observational outcome regression given $(A,T)$. The intervention leaves the equations generating $A$ unchanged, so its distribution is its observational marginal. Therefore, \[ \mathbb{E}_{\mathcal{M}}[Y(t)]=\int g_{\mathcal{M}}(a,t)\nu_A(da),\qquad \mathbb{E}_{\hat{\mathcal{M}}}[\hat{Y}(t)]=\int g_{\hat{\mathcal{M}}}(a,t)\mu_A(da), \] where $\nu_A$ and $\mu_A$ are the observational marginals of $A$. To bound the Lipschitz constants, take an arbitrary $\mathcal{N}\in\{\mathcal{M},\hat{\mathcal{M}}\}$. For a continuous descendant $V_i$, couple its outputs at two parent histories $h,h'$ using the same private noise. The structural Lipschitz bound gives \[ \mathbb{E}\|V_i(h)-V_i(h')\|\leqslant L\|h-h'\|. \] For a categorical descendant, let $\pi_i(h)$ be its propensity vector after integrating its private non-Gumbel noise. The canonical Gumbel rule has categorical probabilities equal to this vector. Integration of the propensity Lipschitz bound gives \[ \|\pi_i(h)-\pi_i(h')\|\leqslant L\|h-h'\|. \] Place mass $\min\{\pi_{i,j}(h),\pi_{i,j}(h')\}$ on each equal pair $(j,j)$. The remaining masses in the two marginals have the same total mass and disjoint supports; coupling these remaining masses gives disagreement probability \[ 1-\sum_{j=1}^{n_i}\min\{\pi_{i,j}(h),\pi_{i,j}(h')\} =\frac12\sum_{j=1}^{n_i}|\pi_{i,j}(h)-\pi_{i,j}(h')|. \] Two labels in $[n_i]$ differ by at most $n_i-1$, so this coupling satisfies \[ \mathbb{E}|V_i(h)-V_i(h')|\leqslant\frac{n_i-1}{2}\|\pi_i(h)-\pi_i(h')\|\leqslant LD\|h-h'\|. \] The continuous bound is also at most $LD\|h-h'\|$, since $D\geqslant1$. Starting at fixed inputs $(a,t)$ and $(a',t')$, use these couplings successively for the descendants, conditional on the accumulated histories. Let $E_r$ be the expected $1$-norm distance between these histories after $r$ descendants have been generated. Projection onto a parent history cannot increase this distance. Hence \[ E_0=\|a-a'\|+|t-t'|,\qquad E_{r+1}\leqslant E_r+LD E_r=(1+LD)E_r. \] Induction gives $E_r\leqslant(1+LD)^rE_0$: the assertion holds at $r=0$, and the recurrence multiplies its bound by $1+LD$ at the next step. There are at most $n_V$ descendants. Projection onto the outcome coordinate, followed by $|\mathbb{E}Y-\mathbb{E}Y'|\leqslant\mathbb{E}|Y-Y'|$, therefore gives \[ |g_{\mathcal{N}}(a,t)-g_{\mathcal{N}}(a',t')|\leqslant L_y(\|a-a'\|+|t-t'|). \] This argument applies to each of the two models without conditioning on latent inputs shared with observed ancestors. Let $Q$ be the set of allowed inputs $s=(a,t)$ whose numerical coordinates all lie in $[-K,K]$. Categorical coordinates retain their categorical ranges. The numerical dimension of $(A,T)$ is at most $q$, so any $s,s'\in Q$ satisfy $\|s-s'\|\leqslant2Kq$. Both observational laws of $(A,T)$ give $Q$ probability one. For each $\mathcal{N}\in\{\mathcal{M},\hat{\mathcal{M}}\}$, the observational outcome bound and the conditional expectation identity give $|g_{\mathcal{N}}(A,T)|\leqslant K$ almost surely under its own observational law. Choose an input $s_{\mathcal{N}}\in Q$ where this inequality holds. For every $s\in Q$, the Lipschitz inequality then yields \[ |g_{\mathcal{N}}(s)|\leqslant|g_{\mathcal{N}}(s_{\mathcal{N}})|+L_y\|s-s_{\mathcal{N}}\|\leqslant K+2KqL_y=B. \] Thus the structural regression versions are bounded by $B$ on $Q$, including inputs outside their observational supports. Put $f_y(a,t)=g_{\mathcal{M}}(a,t)$. Recall that $ \nu,\mu $ are the observation distributions of $\mathcal{M}$ and $\hat{\mathcal{M}}$ respectively in \cref{prop:Lip_ATE}. Put $\hat f_y(a,t)=g_{\hat{\mathcal{M}}}(a,t)$. Its construction gives a version of $\mathbb{E}_{\hat{\mathcal{M}}}[\hat Y|A=a,T=t]$, the $L_y$-Lipschitz inequality, the interventional identity, and the bound $|\hat f_y(a,t)|\leqslant B$ for $(a,t)\in Q$. We use the test-function argument of Lemma~\ref{lemma:Lip_condition} on the vector covariate $s=(a,t)$, with regression bound $B$, and give the calculation explicitly. Write \[ d(s)=\hat f_y(s)-f_y(s),\qquad G(s,y)=d(s)y,\qquad h(s)=d(s)\hat f_y(s). \] On $Q$, the established bounds imply $|d(s)|\leqslant2B$ and $|d(s)-d(s')|\leqslant2L_y\|s-s'\|$. For $s,s'\in Q$ and $|y|,|y'|\leqslant K\leqslant B$, \begin{align*} |G(s,y)-G(s',y')| &\leqslant2B|y-y'|+2KL_y\|s-s'\|\\ &\leqslant2B\max\{L_y,1\}(\|s-s'\|+|y-y'|),\\ |h(s)-h(s')| &\leqslant |d(s)|\,|\hat f_y(s)-\hat f_y(s')|+|\hat f_y(s')|\,|d(s)-d(s')|\\ &\leqslant4BL_y\|s-s'\|. \end{align*} Conditioning on the covariate gives $\mathbb{E}_{\mu}[G]=\mathbb{E}_{\mu}[h]$ and $\mathbb{E}_{\nu}[G]=\mathbb{E}_{\nu}[df_y]$. Since $h-df_y=d^2$, \begin{align*} \mathbb{E}_{\nu}[d(A,T)^2] &=(\mathbb{E}_{\mu}[G]-\mathbb{E}_{\nu}[G])+(\mathbb{E}_{\nu}[h]-\mathbb{E}_{\mu}[h])\\ &\leqslant\bigl(2B\max\{L_y,1\}+4BL_y\bigr)W(\mu,\nu) =C_WW(\mu,\nu). \end{align*} For each Wasserstein bound, integrate the corresponding displayed Lipschitz inequality against any coupling of the observational laws and take the infimum of its expected $1$-norm cost. Both endpoints belong to the domains on which these inequalities were proved. Projection onto $(A,T,Y)$ or $(A,T)$ cannot increase that cost. Set $I=[t_1,t_2]$. Observational boundedness gives $\nu_T(\mathbb{R}\setminus[-K,K])=0$. The conditional treatment law given $A$ equals the law given $X=\text{Pa}(T)$. Integrating the overlap bound therefore gives \[ 0=\nu_T(I\setminus[-K,K])\geqslant\delta P(I\setminus[-K,K]), \] so $P(I\setminus[-K,K])=0$. Thus, for $P$-almost every $t\in I$ and for either observational marginal of $A$, the input $(A,t)$ belongs to $Q$. The overlap assumption now yields \begin{align*} C_WW(\mu,\nu) &\geqslant\int (f_y(a,t)-\hat f_y(a,t))^2\nu(dt|A=a)\nu_A(da)\\ &\geqslant\delta\int_I\int(f_y(a,t)-\hat f_y(a,t))^2\nu_A(da)P(dt)\\ &\geqslant\delta\int_I b(t)^2P(dt), \end{align*} where $b(t)=\int(f_y(a,t)-\hat f_y(a,t))\nu_A(da)$. The last inequality is Jensen's inequality for the probability measure $\nu_A$. Put \[ c(t)=\int\hat f_y(a,t)\nu_A(da)-\int\hat f_y(a,t)\mu_A(da). \] For $P$-almost every $t\in I$, integrate the Lipschitz bound of $\hat f_y$ in $a$ against a coupling of the two marginals and take the infimum to obtain $|c(t)|\leqslant L_yW(\nu_A,\mu_A)\leqslant L_yW(\mu,\nu)$. The interventional identities give \[ \mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat Y(t)]=b(t)+c(t). \] Since $2b(t)c(t)\leqslant b(t)^2+c(t)^2$, it follows, for $P$-almost every $t\in I$, that \[ (\mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat Y(t)])^2 \leqslant2b(t)^2+2L_y^2W^2(\mu,\nu). \] Integrating this inequality against $P(dt)$ on $[t_1,t_2]$ and using the bound on the integral of $b(t)^2$ gives \begin{align*} \int_{t_1}^{t_2}(\mathbb{E}_{\mathcal{M}}[Y(t)]-\mathbb{E}_{\hat{\mathcal{M}}}[\hat Y(t)])^2P(dt) &\leqslant2\int_{t_1}^{t_2}b(t)^2P(dt)+2L_y^2W^2(\mu,\nu)\int_{t_1}^{t_2}1\,P(dt)\\ &\leqslant\frac{2C_W}{\delta}W(\mu,\nu)+2L_y^2W^2(\mu,\nu)P([t_1,t_2]). \end{align*} - why: (a) The statement is false for mixed variables: Lipschitz categorical propensities do not make the categorical generation map Lipschitz, and category labels can amplify the outcome regression beyond $(L+1)^{n_V}$. (b) Independently, even with continuous variables, conditioning on $X=\text{Pa}(T)$ need not leave the latent inputs needed for $Y$ independent of $X$; an observed ancestor can enter both $X$ and $Y$, invalidating the asserted composition argument and regression bound. (c) The proof derives a quadratic coefficient $2L_y^2$, but the statement replaces it by $2(L+1)^{n_V}$; the stated bound on $L_y$ does not justify this substitution. (d) Integrating the constant remainder against $P(dt)$ produces $P([t_1,t_2])$, not $t_2-t_1$; the nondescendant branch likewise switches to $dt$. Dismiss the wiring lint: the regression and boundedness conditions of Lemma~\ref{lemma:Lip_condition} are conditions this proof attempts to derive, rather than additional standing hypotheses. The failed regression derivation is a correctness defect. ### `thm:rate` (`Main.tex:319-321`) - as printed: \label{cor:rate} Let $F, \mu_T$ in \cref{thm:consistency} to be $ F(\boldsymbol{V}) = Y, \mu_T \sim \text{Unif}([t_1, t_2]), \epsilon >0 $. Suppose that the assumptions in Proposition \ref{prop:Lip_ATE} and \cref{thm:consistency} hold, with probability at least $1-O(n^{-2})$, we have $ | F_n - F_*| \leqslant O(\sqrt{\alpha_n})$, where $F_n,F_*,\alpha_n$ are defined in \cref{thm:consistency}. - corrected: Let $F,\mu_T$ in \cref{thm:consistency} be $F(\boldsymbol V)=Y$, $\mu_T\sim\operatorname{Unif}([t_1,t_2])$, and $\epsilon>0$. Suppose that $t_10$ for every structural input $x$, and \[ \operatorname{Lip}\left(x\mapsto\frac{q(x)}{\|q(x)\|_1}\right)\leqslant\hat L_f. \] For each sample size, obtain every candidate empirical law by evaluating the candidate on the same $m_n$ independent draws of its uniform and Gumbel exogenous inputs, independent of the observational sample. Let $F_n$ be the optimal value of this restricted optimization, retaining the objective and Sinkhorn constraint of (\ref{eq:opt_empiri}). Put \[ D_U=\sum_{j=1}^{n_U}d_j^U,\qquad D=\max\{1,D_U\}, \] where the sum includes only the non-Gumbel latent variables. Take $\alpha_n=\epsilon_n+s_n$, with positive tolerances chosen as sufficiently large constant multiples of the respective quantities in \[ \epsilon_n=O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+\sum_{i=1}^2L_{i,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right), \] \[ s_n=O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2n+\log(nm_n)\lambda_n\right). \] Then, with probability at least $1-O(n^{-2})$, we have $|F_n-F_*|\leqslant O(\sqrt{\alpha_n})$, where $F_*$ is defined in \cref{thm:consistency} and $F_n,\alpha_n$ are specified here. - no longer uses `def:thm-consistency-architecture-radius` - no longer uses `ass:corollary-deep-nn-latent-laws` - no longer uses `def:thm-consistency-population-function-classes` - no longer uses `def:thm-consistency-neural-parameter-space` - side condition added by the confirming referee `ass:bk-1`: For every $k\in[n]$, $\int_{\mathbb R^d}\|x\|\,\mu_k(dx)+\int_{\mathbb R^d}\|x\|\,\hat\mu_k(dx)<\infty$. — The dual proof separately evaluates expectations of unbounded Lipschitz functions. - side condition added by the confirming referee `ass:pin-154`: For each of $\mathcal{M}$ and $\hat{\mathcal{M}}$, $\max_{i\in[n_V],j\in[n_U]}\{\|V_i\|_\infty,\|U_j\|_\infty\}\leqslant K$ almost surely, where the indexed $U_j$ are the non-Gumbel latent variables. — pins reading of ass:prop-lip-ate-bounded: Proposition \ref{prop:Lip_ATE} explicitly inherits Assumption \ref{assump:Lip_and_bounded}, which prints this maximum; Assumption \ref{assump:categorical_rv} distinguishes the indexed $U_j$ from the categorical noises $G_{V_i}$. - side condition added by the confirming referee `ass:pin-177`: $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. — pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality. - side condition added by the confirming referee `ass:pin-181`: Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\|f_i(x)\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. — pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities. - side condition added by the confirming referee `ass:pin-182`: For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\|f_i(x)-f_i(x')\|\leqslant L_f\|x-x'\|$ on its structural input domain, using the paper's $1$-norm. — pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$. - side condition added by the confirming referee `ass:pin-183`: There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\|V_i\|_\infty,\|U_j\|_\infty\}\leqslant K$ almost surely under the SCM law. — pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation. - side condition added by the confirming referee `ass:pin-185`: The continuous structural functions and categorical propensity functions of $\mathcal M^*$ have Lipschitz constants at most $L_f$ in the paper's $1$-norm. — pins reading of ass:thm-consistency-structural-lipschitz: Theorem~\ref{thm:consistency} begins with $\mathcal M^*$ and permits candidate networks with bound $\hat L_f$; Assumption~\ref{assump:Lip_and_bounded} identifies the functions bounded by $L_f$. - side condition added by the confirming referee `ass:pin-186`: The phrase ``uniform cubes'' denotes the unit cube domains $[0,1]^m$ used for deterministic uniform function approximation; it imposes no uniform probability law on structural inputs. — pins reading of ass:relu-approximation-domain: The same paragraph displays approximation of functions on $[0,1]^{d_1}$ and separately says that similar arguments apply to any bounded cubes. - side condition added by the confirming referee `ass:pin-187`: The bound $K$ applies throughout the optimization class to the observed variables $V_i$ and indexed non-Gumbel latent variables $U_j$, with $n_i\leqslant K$ for categorical nodes and admissible intervention coordinates in $[-K,K]$; it does not bound the separately supplied standard Gumbel noises. — pins reading of ass:bk-owner-1: The boundedness assumption explicitly indexes $V_i$ and $U_j$, whereas the categorical convention separately adjoins $G_{V_i}$. The binding boundedness symbols preserve this distinction. - side condition added by the confirming referee `ass:pin-188`: The variable bound inherited from Assumption~\ref{assump:Lip_and_bounded} uses $\|\cdot\|_\infty$. — pins reading of ass:bk-owner-2: The owner condition extends the existing bound $K$, which Assumption~\ref{assump:Lip_and_bounded} explicitly states using infinity norms of observed and non-Gumbel latent variables. - side condition added by the confirming referee `ass:pin-189`: For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. — pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition. - side condition added by the confirming referee `ass:pin-190`: In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. — pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction. - side condition added by the confirming referee `ass:pin-191`: Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). — pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations. - side condition added by the confirming referee `ass:pin-192`: A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. — pins reading of def:relu-networks: The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths. - side condition added by the confirming referee `ass:pin-194`: Membership in $\text{NCM}_{\mathcal G}(\mathcal F_0,\mathcal F_1)$ requires $\hat f_i\in\mathcal F_0$ for every observed node and $\hat g_j\in\mathcal F_1$ for every latent component, with the compatible input and output dimensions understood separately for each network. — pins reading of def:neural-causal-class: The source paragraph explicitly omits input and output dimensions and describes separate structural approximants and latent generators in equation~\ref{eq:structure_equation_hat}. - side condition added by the confirming referee `ass:pin-196`: Truncation means coordinatewise clipping of the final output to $[-K,K]$; in $\mathcal{NN}^{\infty,K}$ no finite Lipschitz bound is imposed. — pins reading of def:truncated-neural-class: The defining minimum and maximum and the word ``Similarly'' specify the operation; the binding neural-class symbol conventions fix its application to the generator output. - side condition added by the confirming referee `ass:pin-197`: $\underline{F}^{L}$ denotes the constrained infimum of the displayed scalar objective over the stated canonical SCM class and its admissible latent laws, rather than a minimizing SCM. — pins reading of def:eq-consistent-limiting: The sentence introducing the display explicitly says ``optimal value'', and the following sentence calls it the sharp lower bound when the Lipschitz bounds agree. - side condition added by the confirming referee `ass:pin-198`: $\mu_T=\text{Unif}([t_1,t_2])$ is the deterministic intervention probability measure. — pins reading of def:cor-rate-objective: Corollary~\ref{cor:rate} specializes the intervention measure used in the outer expectation $\mathbb E_{t\sim\mu_T}$ of Theorem~\ref{thm:consistency}. - side condition added by the confirming referee `ass:pin-199`: $d_{\max}^U=\max_{j\in[n_U]}d_j^U$ ranges over non-Gumbel latent variables, while $d_{\max}^{\mathrm{in}}=\max_i d_i^{\mathrm{in}}$ and $d_{\max}^{\mathrm{out}}=\max_i d_i^{\mathrm{out}}$, with $d_i^{\mathrm{out}}=n_i$ for categorical nodes. — pins reading of def:corollary-deep-nn-dimensions: The NCM construction separately approximates non-Gumbel latent distributions and the functions in the canonical structural equations; those functions output categorical propensity vectors. The binding dimension symbols specify these same meanings. - side condition added by the confirming referee `ass:pin-201`: The coordinates $P(U_j)$ in $\theta_{\mathcal M}$ are the population laws of the non-Gumbel latent variables, not necessarily uniform laws; the independent standard Gumbel laws are fixed and omitted from this population tuple. — pins reading of def:thm-consistency-model-parameterization: Immediately after the tuple, the population parameter space contains factors $\mathcal P([-K,K]^{d_j^U})$. The canonical representation supplies the categorical Gumbel laws separately. - side condition added by the confirming referee `ass:pin-203`: The gap event is $E_\delta=\{\exists\,1\leqslant i0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. — pins reading of def:gumbel-softmax-layer: The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit. - side condition added by the confirming referee `ass:pin-210`: $F_n=\inf\left\{\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\hat{\mathcal{M}}}[F(V_1(t),\ldots,V_{n_V}(t))]:\hat{\mathcal{M}}\in\text{NCM}_{\mathcal{G}}(\mathcal{F}_{0,n},\mathcal{F}_{1,n}),\ S_{\lambda_n}(P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V}),P_n^{\mathcal{M}^*}(\boldsymbol{V}))\leqslant\alpha_n\right\}$. — pins reading of def:eq-opt-empiri: The prose introducing equation (\ref{eq:opt_empiri}) calls $F_n$ an estimator of optimal values, and the following paragraph asks whether it is a good lower bound. The supplied symbol convention likewise specifies a scalar infimum. - side condition added by the confirming referee `ass:pin-211`: In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. — pins reading of def:lemma-gumbel-err-softmax: Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores. - side condition added by the confirming referee `ass:pin-33`: $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. — pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes. - side condition added by the confirming referee `ass:pin-75`: For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. — pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula. - side condition added by the confirming referee `ass:pin-fix-00130002`: The law $\hat g_{\#}\lambda$ includes the independent standard Gumbel variables used in the mixing layer. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-09991de8`: The observational empirical law $P_n^{\mathcal M^*}(\boldsymbol V)$ is formed from $n$ independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-0a7b21b0`: The empirical optimization ranges over network representations, including distinct representations with the same latent population laws. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-0bcb2cef`: The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-19bccf89`: The measure $P$ is the reference probability measure on the treatment space: for $X=\text{Pa}(T)$, the conditional density notation means $\nu(dt|X=x)=p_\nu(t|X=x)P(dt)$. — pins reading of prop:lip-ate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-2a1e7b01`: The mass assigned to $I_{O_n(i)}$ is $\lambda_d(C_{O_n(i)})=2^{-nd}$. — pins reading of def:prop-holder-curve-interval-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-2fe04b0e`: In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal C_n$ denotes the standard dyadic collection $\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}$. — pins reading of def:prop-holder-curve-cube-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-30413b86`: The common simulation draws are independent of the observational sample. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-3b195d8d`: The common exogenous draws are independent and identically distributed according to the specified product law of the exogenous inputs. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-419eb8e3`: The multiplicative width factors in $\Theta(d_{\max}^{\text{in}}d_{\max}^{\text{out}})$, $\Theta(d_{\max}^U)$, and $\Theta((d_{\max}^U)^2)$ can be chosen as absolute numerical constants, independent of the causal model, $R$, all dimensions, and all category and component counts. — pins reading of thm:deep-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-43f1bb71`: For each latent law $P(U_C)$, the assumptions of the proposition entitled ``Deep distribution approximation'' require the support charts of \cref{assump:support} and \cref{assumption:lower_bound} on each normalized component law $\mu_i=P(U_C)|_{C_i}/P(U_C\in C_i)$, with $(f^{-1})_{\#}\mu_i(\partial[0,1]^{d_i^C})=0$ when $d_i^C>0$. — pins reading of thm:deep-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-448b19a0`: Probabilities of uniform deviation events are evaluated in the completion of the sample law. — pins reading of lem:sup-wasserstein: stated by the repair - side condition added by the confirming referee `ass:pin-fix-4a86c2eb`: The structural uniform infinity norm takes the supremum of the coordinatewise infinity norm over the common structural input domain. — pins reading of thm:approximation: stated by the repair - side condition added by the confirming referee `ass:pin-fix-507a5848`: For each component of positive dimension $q=d_i^C$, the application of Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and quantization mesh $\delta/n$. — pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair - side condition added by the confirming referee `ass:pin-fix-59a01041`: For each $n$, every candidate is evaluated on the same $m_n$ draws of all its exogenous inputs, including its uniform inputs, generator Gumbel inputs, and categorical Gumbel inputs. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-5f02b8a6`: The error estimates are asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of these budgets and of $\tau$. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-676b96c3`: The finite component counts and component cube dimensions of the latent generator architectures in $\mathcal F_{1,n}$ are fixed as $n$ varies. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-72c9e86d`: The measures $\mu$ and $\hat\mu$ are probability measures. — pins reading of lem:push-forward-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-73abef8c`: In Corollary~\ref{corollary:deep_nn}, for $m_i=d_i^{\text{in}}>0$, the symbol $\hat f_i$ denotes $q_i\circ\Pi_i$ for a continuous variable and the vector $\bigl(\max\{\epsilon_i,(q_i\circ\Pi_i)_k\}\bigr)_{k=1}^{n_i}$ for a categorical variable, where $q_i$, $\Pi_i$, and $\epsilon_i=A_iL_0^{-2/m_i}$ are chosen in its proof. — pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair - side condition added by the confirming referee `ass:pin-fix-761562d5`: The self terms in $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$ use the same $1$-norm ground distance on the corresponding support points. — pins reading of lem:sinkhorn-approx: stated by the repair - side condition added by the confirming referee `ass:pin-fix-7a79e01d`: In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal I_n$ denotes $\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$ with $I_{O_n(i)}=[(i-1)2^{-nd},i2^{-nd}]$. — pins reading of def:prop-holder-curve-interval-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8429b12c`: The latent generator class in the statement uses the same temperature $\tau$ as the displayed error estimate. — pins reading of thm:deep-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-84ccd033`: $F_n$ denotes the infimum of objective values, with an attained minimum having the same value. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8cc0dba5`: For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. — pins reading of def:hilbert-partitions: stated by the repair - side condition added by the confirming referee `ass:pin-fix-91573abf`: The observational empirical law $P_n^{\mathcal M^*}(\boldsymbol V)$ is formed from $n$ independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. — pins reading of thm:rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-945d85ed`: The error estimate is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, for fixed $R$ and fixed causal model. — pins reading of thm:deep-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9463b114`: For a zero categorical propensity, $\log0=-\infty$. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9675bba5`: Ties in a categorical argmax are resolved by the smallest label. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-99ac9bb3`: For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. — pins reading of def:hilbert-partitions: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9bbf998f`: A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9d74053b`: The orders for $\epsilon_n$ and $s_n$ are asymptotic as $n\rightarrow\infty$. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ab17be7a`: For a categorical propensity coordinate equal to zero, $\log0=-\infty$ in the canonical argmax construction. — pins reading of thm:approximation: stated by the repair - side condition added by the confirming referee `ass:pin-fix-b83fa4d1`: $W$ is Wasserstein-1 distance for the ground distance $\|x-y\|$. — pins reading of lem:product-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-bbf1c78a`: In Corollary~\ref{corollary:deep_nn}, for $m_i=0$, the symbol $\hat f_i$ denotes the exact constant function $f_i$. — pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair - side condition added by the confirming referee `ass:pin-fix-bd620122`: The architectural $\Theta$ orders use sufficiently large fixed constant factors supplied by the network constructions in the proof. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c514425b`: The refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. — pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c778b424`: The reference measure $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$. — pins reading of def:prop-holder-curve-cube-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c85b9186`: Each $O_n$ is a bijection. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ca667e13`: $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. — pins reading of lem:product-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-d19c63b5`: The probability bound is for each fixed pair $\delta,\epsilon>0$. — pins reading of lem:sup-wasserstein: stated by the repair - side condition added by the confirming referee `ass:pin-fix-db6ed772`: In the proposition entitled ``Deep distribution approximation'', the multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. — pins reading of thm:wide-nn: stated by the repair - side condition added by the confirming referee `ass:pin-fix-e896f793`: The cells of $\mathcal C_n$ are closed and may share boundaries. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f66ad243`: The depth expression $\Theta(\log d_{\max}^{\text{in}})$ means $\Theta(1+\log d_{\max}^{\text{in}})$ at $d_{\max}^{\text{in}}=1$. — pins reading of thm:consistency: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f6f485b1`: The uniform infinity norm is $\|F-\hat F\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}|F_j(x)-\hat F_j(x)|$. — pins reading of lem:push-forward-err: stated by the repair - proof change: the printed proof, revised step by step] We establish feasibility and the complete-observation sampling bounds for the specified restricted optimization. Write $\eta$ for a network representation, $\theta_\eta$ for its population SCM parameter, and $\theta^*$ for the true parameter. For two population parameters define \[ d(\theta,\theta')=\sum_{i=1}^{n_V}\|f_i-f'_i\|_\infty+\sum_{j=1}^{n_U}W(P(U_j),P(U'_j)), \] using normalized categorical propensity functions in the first sum. Let $\mathcal A_n$ be the set of admissible representations. With common independent exogenous draws $\xi_1,\ldots,\xi_{m_n}$, let $H_\eta(\xi_l)$ be the complete observed vector and put \[ P_{m_n}^\eta=\frac1{m_n}\sum_{l=1}^{m_n}\delta_{H_\eta(\xi_l)},\qquad h_n(\eta)=S_{\lambda_n}(P_n^{\theta^*},P_{m_n}^\eta),\qquad g(\theta)=W(P^{\theta^*},P^\theta). \] All distributions in these expressions are distributions of the complete observed vector. Write $f(\theta)=\mathbb E_{t\sim\mu_T}\mathbb E_{\mathcal M^\theta}[F(\boldsymbol V(t))]$ for the objective. For fixed $n$, pad the finitely many network architectures to their stated widths and encode each representation by its finitely many real weights and mixture probabilities. Finite products of Euclidean spaces and simplices are standard Borel spaces. Affine maps, ReLU, clipping and finite composition have jointly continuous evaluation in their weights and inputs. At the fixed temperature $\tau_n>0$, mixture weights can be written \[ \frac{p_j^{1/\tau_n}\exp(G_j/\tau_n)}{\sum_k p_k^{1/\tau_n}\exp(G_k/\tau_n)},\qquad p\in\Delta, \] whose denominator is positive even when some $p_j=0$; hence generator evaluation is jointly Borel. The restrictions defining $\mathcal A_n$ are Borel restrictions on these parameters. Choose a countable dense set in each continuous structural input domain. Continuity makes the output bound, coordinatewise nonnegativity and every Lipschitz inequality equivalent to their inequalities on that dense set or its Cartesian square, and each such inequality is Borel in the weights. For a nonnegative categorical raw network $q$, positivity on a compact input box $D_0$ is equivalent to membership in \[ \bigcup_{l=1}^\infty\bigcap_{x\in Q_0}\{\eta:\|q_\eta(x)\|_1\geqslant1/l\}, \] where $Q_0$ is countable and dense in $D_0$. Positivity implies a positive minimum on the compact box; conversely the dense-set lower bound extends by continuity. Intersect these conditions over a countable compact exhaustion if the full domain is unbounded. On the resulting Borel parameter set $q/\|q\|_1$ is jointly Borel and continuous in the input, so its Lipschitz restriction is another countable family of Borel inequalities. The raw Lipschitz restrictions before clipping have the same dense-input characterization. Thus $\mathcal A_n$ is standard Borel. The exogenous space is a finite product of uniform cubes and real Gumbel coordinates. For categorical propensities $p$, define $r_i=\exp(-g_i)/p_i$ for $p_i>0$ and $r_i=\infty$ for $p_i=0$. The smallest minimizing label is $i$ precisely when $r_ii$. These are Borel comparisons of extended-real Borel functions. Induction in topological order proves that both $H_\eta(\xi)$ and $H_\eta(t,\xi)$ are jointly Borel. Integration of the bounded objective on the observed range makes $f(\theta_\eta)$ Borel in $\eta$: parameter integrals of bounded Borel functions are Borel by the monotone class theorem. Empirical transport values are continuous in the cost matrix, since a uniform change of at most $a$ in costs changes every coupling objective and its infimum by at most $a$. Hence $h_n$ is jointly Borel in parameters and data. The event $\{F_na$ and \[ \frac{E_i}{q_i-a}<\min_{j\ne i}\frac{E_j}{q_j+a}. \] A robust label wins for every propensity vector whose coordinates differ from $q$ by at most $a$. The robust-label events are disjoint. When $q_i>a$, integration of the independent exponential densities gives \[ \mathbb P(i\text{ robust})=\frac{q_i-a}{q_i-a+\sum_{j\ne i}(q_j+a)}. \] When $q_i\leqslant a$, the probability is zero. In either case it is at least $(q_i-a)_+/(1+Na)$. Summing and using $\sum_i(q_i-a)_+\geqslant1-Na$ bounds the probability of no robust label by $2Na/(1+Na)$ when $Na<1$; when $Na\geqslant1$, the bound $1\leqslant2Na$ suffices. For each table and generator representation let $b$ indicate that a nonrobust choice occurs at any table prefix. Independence of categorical noise from the generated latent vector and a union bound over the finitely many prefixes give $\mathbb E b\leqslant C\rho$. If $b=0$, induction over the categorical nodes forces the original and table prefixes to agree. Thus, with $\widetilde H$ the table observed map and $B_V$ a common observed $1$-norm diameter bound, \[ \|H_\eta(\xi)-\widetilde H(\xi)\|\leqslant C\rho+B_V b(\xi). \] This bound includes zero propensities. A table simulation has type $(u,c)$, where $u$ is its latent grid index and $c$ its complete categorical label vector; its observed output is fixed for each type. There are at most $J=C\rho^{-D_U}$ types. Include the indicators of membership in every subset of these types, and the ambiguity indicators $b$, in a scalar class $\mathcal B_{n,\rho}$. To verify the structural condition of Lemma~\ref{lemma:sup_wasserstein} for this class, let $\Gamma_n$ be the entire finite-dimensional generator weight space with its mixture simplex factors. This gives a measurable envelope for the restricted generator representations. Index indicators by a parameter in $\Gamma_n$, a table in the finite set $\mathcal T_\rho$, and a finite tag specifying either a subset of the at most $J$ types or the ambiguity indicator. This is a standard Borel space. Generator evaluation is jointly Borel by the positive-denominator mixture formula. Its clipped output has a Borel grid index when cells are half-open and the final cell includes its endpoint. Given the table and grid index, each categorical choice is a finite Borel comparison using $r_ii$, with $r_i=\infty$ for zero table propensity. Each robust event is a Borel comparison with positive denominators on $q_i>a$, and is empty on $q_i\leqslant a$. Finite unions and complements therefore give jointly Borel ambiguity and type-subset indicators. This includes all ties, grid boundaries and zero propensities. Clip uniform exogenous coordinates outside their cubes to view these functions on the ambient Euclidean input space; the values on the sampled inputs do not change. Composing a jointly Borel indicator with a continuous scalar dual function and its normalized square-root transform preserves joint Borel evaluation. Thus all the classes to which the lemma's concentration argument is applied have the required standard Borel parameterization. The envelope uses the same architectural parameters and finite tags as the growth calculation; no density assertion for discontinuous selectors is needed. On $M_0$ fixed exogenous inputs, the latent grid indices are determined by at most $CM_0/\rho$ scalar generator comparisons. The ReLU blocks have at most $CH_n$ parameters and $CH_n$ layers. Within a fixed activation pattern their outputs are polynomials in the parameters of degree at most $CH_n$. For a positive-temperature mixture write its weights as $a_jR_j/\sum_k a_kR_k$, where $a_j=p_j^{1/\tau_n}\geqslant0$, $\sum_j a_j>0$, and $R_j=\exp(G_j/\tau_n)$ is fixed on each input. Treat the $a_j$ as additional free parameters. Clearing the positive denominator in a grid comparison leaves a polynomial of degree at most $CH_n+1$. Truncation adds fixed-degree comparisons. The polynomial sign bound in Lemma 17 of \cite{bartlett2019nearly}, applied successively to activation comparisons in each layer and then to grid comparisons, gives at most \[ (C(M_0+H_n)H_n/\rho)^{CH_n^2} \] latent grid patterns: each layer contributes at most $(C(M_0+H_n)H_n/\rho)^{CH_n}$, and there are at most $CH_n$ layers. For a fixed table and grid pattern all categorical race comparisons and table outputs on these inputs are determined. The growth of $\mathcal B_{n,\rho}$ is therefore at most this number times $|\mathcal T_\rho|2^J$. Hence its pseudo-dimension satisfies \[ v_n(\rho)\leqslant C\left(\rho^{-D_U}\log(2+\rho^{-1})+H_n^2\log(2+H_n/\rho)\right). \] To obtain this bound explicitly, put $A_0=\log|\mathcal T_\rho|+J\log2$ and $B_0=CH_n^2$ in the shattering inequality \[ M_0\log2\leqslant A_0+B_0\log(C(M_0+H_n)H_n/\rho). \] If $M_0\leqslant4A_0/\log2$, the first term suffices. Otherwise the inequality bounds $M_0$ by $CB_0\log(2+B_0H_n/\rho)+CB_0\log(2+A_0/B_0)$; the established bound on $A_0$ gives the displayed estimate. Put \[ r_n=m_n^{-1/(D+2)}\log m_n+H_nm_n^{-1/2}\log m_n. \] The depth schedule in \cref{thm:consistency} gives $H_n=O(m_n^{d_{\max}^U/(2d_{\max}^U+4)})$, so $r_n\rightarrow0$. For sufficiently large $n$ take $\rho=r_n$ and apply Lemma~\ref{lemma:sup_wasserstein} to $\mathcal B_{n,r_n}$ with scalar output. These functions take values in $\{0,1\}$, and the Wasserstein distance of their Bernoulli output laws equals the absolute difference of their means. With tolerances $\delta=r_n$ and $\epsilon=Cr_n$, its failure exponent is bounded above by \[ C_1\left((1+r_n^{-1})\log(2+r_n^{-1})+r_n^{-D_U}\log^2(2+r_n^{-1})+H_n^2\log(2+H_n/r_n)\log(2+r_n^{-1})\right)-cC^2m_nr_n^2. \] Here $\log(2+H_n/r_n)=O(\log m_n)$ and \[ m_nr_n^2\geqslant m_n^{D/(D+2)}\log^2m_n+H_n^2\log^2m_n, \] whereas $r_n^{-D_U}\leqslant m_n^{D_U/(D+2)}(\log m_n)^{-D_U}$ and $r_n^{-1}\leqslant m_n^{1/(D+2)}/\log m_n$. Since $D\geqslant\max\{D_U,1\}$, a sufficiently large fixed $C$ makes the failure probability $O(m_n^{-3})$. On the complementary event every type-subset probability and ambiguity-indicator mean differs from its empirical counterpart by at most $Cr_n$. The type-subset bound controls total variation between population and empirical type laws. Couple common type masses identically and residual masses arbitrarily; their table-output Wasserstein distance is at most $B_VCr_n$. Coupling original and table outputs on the same draw bounds their population discrepancy by $C\rho+B_V\mathbb E b$ and their empirical discrepancy by $C\rho+B_Vm_n^{-1}\sum_l b(\xi_l)$. Both are at most $Cr_n$ by the ambiguity estimate. The triangle inequality therefore gives \[ \sup_{\eta\in\mathcal A_n}W(P^{\theta_\eta},P_{m_n}^\eta)\leqslant Cr_n \] with probability at least $1-O(m_n^{-3})$. For the observational sample let $C$ be its complete categorical label vector, taking values in a fixed finite set. Conditional on $C=c$, all continuous observations equal a deterministic map $\Phi_c(U_1,\ldots,U_{n_U})$, obtained by fixing the categorical labels and evaluating the continuous equations in topological order. Their Lipschitz bounds make $\Phi_c$ Lipschitz with a constant uniform over these finitely many labels. The conditional joint latent laws $\mu_c=P((U_1,\ldots,U_{n_U})\mid C=c)$ are supported on the bounded $D_U$-dimensional cube and need not be products. Write $p_c=P(C=c)$ and $N_c$ for sample counts. If $p_c=0$, then $N_c=0$ almost surely. For $p_c>0$, binomial concentration gives $N_c\geqslant np_c/2$ except on an exponentially small event. Conditional on the sample labels, latents in category $c$ are independent with law $\mu_c$. For $D_U\geqslant1$, Theorem 2 of \cite{fournier2015rate} applied to these conditional joint latent laws bounds their empirical Wasserstein errors by $C_cn^{-1/\max\{2,D_U\}}\log^2n$ with failure probability $O(n^{-4})$, uniformly for $np_c/2\leqslant N_c\leqslant n$. Indeed, its small-tolerance exponents are constant multiples of $N_cx^2$ for $D_U=1$, $N_c(x/\log(2+1/x))^2$ for $D_U=2$, and $N_cx^{D_U}$ for $D_U>2$. At the specified tolerance these grow at least as $\log^4n$, $\log^2n$, and $\log^{2D_U}n$, respectively. Bounded support supplies the exponential-moment assumption. A coupling pushed through $\Phi_c$ multiplies its cost by at most the Lipschitz constant of $\Phi_c$. For $D_U=0$, each conditional observed law is a point mass and its conditional empirical error is zero. Binomial concentration also gives $\sum_c|N_c/n-p_c|\leqslant C\sqrt{\log n/n}$ with probability at least $1-O(n^{-4})$. Couple common categorical-component masses by their conditional couplings and unmatched masses arbitrarily within the observed support. After a union bound over the finitely many categories, \[ W(P_n^{\theta^*},P^{\theta^*})\leqslant\sum_c\min\{p_c,N_c/n\}W((\Phi_c)_\#\mu_c,(\Phi_c)_\#(\mu_c)_{N_c})+\frac{B_V}{2}\sum_c|N_c/n-p_c|\leqslant t_n, \] \[ t_n=C_2n^{-1/\max\{2,D_U\}}\log^2n, \] with probability at least $1-O(n^{-3})$. Terms with zero common mass are omitted. Lemma~\ref{lemma:sinkhorn_approx} and the triangle inequality give, on the combined event, \[ \sup_{\eta\in\mathcal A_n}|h_n(\eta)-g(\theta_\eta)|\leqslant2(1+\log(nm_n))\lambda_n+Cr_n+t_n\leqslant s_n. \] The event has probability at least $1-O(n^{-3}+m_n^{-3})$. We construct a feasible approximation to the true model. For a structural function $f:[-K,K]^m\rightarrow\mathbb R^{m'}$ with $m\geqslant1$, divide every coordinate into $R$ intervals of length $h=2K/R$ and triangulate each cell into simplices with successive vertices $v_0,v_1=v_0+he_{\pi(1)},\ldots,v_m=v_{m-1}+he_{\pi(m)}$. Let $I_hf$ interpolate the vertex values affinely. Its derivative column in direction $e_{\pi(j)}$ is $(f(v_j)-f(v_{j-1}))/h$, of $1$-norm at most $L_f$. Subdividing a segment at simplex boundaries and summing these derivative bounds proves $\operatorname{Lip}(I_hf)\leqslant L_f$. With barycentric coordinates $a_j$, \[ \|I_hf(x)-f(x)\|\leqslant\sum_{j=0}^ma_j\|f(v_j)-f(x)\|\leqslant L_fmh. \] Convex interpolation preserves the bound $K$ for continuous outputs and preserves the simplex for categorical propensities. Its nodal basis functions are translates and rescalings of \[ \phi(z)=\max\left\{0,\min\left(\{1+z_k,1-z_k:k\in[m]\}\cup\{1+z_k-z_l:k,l\in[m]\}\right)\right\}. \] On each simplex of a nodal star this function is affine, equals one at the central vertex, and equals zero at its other vertices. Hence $I_hf(x)=\sum_v f(v)\phi((x-v)/h)$. Compute each minimum by a binary tree using $\min(a,b)=a-\max\{a-b,0\}$. Each term has width $O(m^2)$ and depth $O(1+\log m)$. Parallelizing $O(R^m)$ terms and summing their outputs gives the required fixed-dimensional depth and width $O(R^m)$. Coordinatewise input clipping is a contraction and preserves the Lipschitz bound. Taking $R$ proportional to $W_{0,n}^{1/m}$ gives error $O(W_{0,n}^{-1/m})$. A function with $m=0$ is constant and represented exactly. For categorical functions take $q=I_hf$; it is nonnegative and normalized, and its normalized Lipschitz constant equals its original one. Thus these approximants satisfy the categorical restrictions. Continuous output clipping preserves the error bound and cannot increase the Lipschitz constant. For each normalized latent support component measure $\mu_i$, let $\psi_i:[0,1]^{q_i}\rightarrow C_i$ be its Lipschitz chart. For $q_i>0$, Proposition~\ref{prop:holder_curve} applied to its pullback measure gives a curve $\gamma_i:[0,1]\rightarrow[0,1]^{q_i}$ of Holder exponent $1/q_i$ with that pullback law. Theorem 2(b) of \cite{Yarotsky2018OptimalAO} applied to its coordinates gives first-block uniform error $O(L_{1,n}^{-2/q_i})$ and, applied to the Lipschitz coordinates of $\psi_i$, gives second-block error $O(L_{2,n}^{-2/q_i})$. Parallelize coordinate networks and clip first-block outputs to their input cubes. Their widths fit the specified budgets. Lemma~\ref{lemma:push-forward_err} bounds the resulting component Wasserstein error by $O(L_{1,n}^{-2/q_i}+L_{2,n}^{-2/q_i})$. For $q_i=0$, use the exact constant component. Lemma~\ref{lemma:combin_err} bounds the hard-selected mixture error by the weighted component errors. Couple positive-temperature and hard Gumbel weights, retaining identical independent bounded component outputs $v_i$. Then \[ \left\|\sum_i(X_i^{\tau_n}-X_i^0)v_i\right\|\leqslant C\|X^{\tau_n}-X^0\|_1. \] Lemma~\ref{lemma:gumbel_err} bounds the additional error by $O(\tau_n(1-\log\tau_n))$ for sufficiently large $n$. Output truncation cannot increase the latent error because the true latents lie in $[-K,K]^{d_j^U}$ and truncation contracts distance to every point of this cube. Choose the latent generators independently. Summing their marginal errors and the structural errors gives a representation $\eta_n\in\mathcal A_n$ satisfying \[ a_n=d(\theta^*,\theta_{\eta_n})\leqslant O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+\sum_{i=1}^2L_{i,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right). \] Choose the constant in $\epsilon_n$ so that $C_ga_n\leqslant\epsilon_n$. On the combined sampling event, \[ h_n(\eta_n)\leqslant g(\theta_{\eta_n})+s_n\leqslant C_ga_n+s_n\leqslant\epsilon_n+s_n=\alpha_n. \] Thus the optimization is feasible. Since $m_n=\Omega(n)$, the same event has probability at least $1-O(n^{-2})$ and gives \[ W(P_n^{\mathcal M^*},P^{\mathcal M^*})\leqslant O(\alpha_n),\qquad\sup_{\eta\in\mathcal A_n}W(P_{m_n}^\eta,P^{\theta_\eta})\leqslant O(\alpha_n). \] For every feasible representation $\eta$, Lemma~\ref{lemma:sinkhorn_approx} and the empirical constraint give \[ W(P_n^{\mathcal M^*}(\boldsymbol V),P_{m_n}^\eta(\boldsymbol V))\leqslant S_{\lambda_n}(P_n^{\mathcal M^*}(\boldsymbol V),P_{m_n}^\eta(\boldsymbol V))+2(1+\log(nm_n))\lambda_n\leqslant O(\alpha_n), \] since the regularization term is included in $s_n\leqslant\alpha_n$. On the same sampling event, the triangle inequality gives, for every feasible representation $\eta$, \begin{align*} w_\eta&=W(P^{\mathcal M^*}(\boldsymbol V),P^{\theta_\eta}(\boldsymbol V))\\ &\leqslant W(P^{\mathcal M^*}(\boldsymbol V),P_n^{\mathcal M^*}(\boldsymbol V))+W(P_n^{\mathcal M^*}(\boldsymbol V),P_{m_n}^\eta(\boldsymbol V))+W(P_{m_n}^\eta(\boldsymbol V),P^{\theta_\eta}(\boldsymbol V))\\ &\leqslant O(\alpha_n). \end{align*} All feasible models have bounded continuous structural functions and normalized categorical propensity functions with Lipschitz constant at most $\hat L_f$. The true model also belongs to the corresponding bounded Lipschitz class. Apply Proposition~\ref{prop:Lip_ATE} to the true model and each feasible model, using their common unconfounded graph and the overlap assumption of the true model. Write \[ b_\eta(t)=\mathbb E_{\mathcal M^*}[Y(t)]-\mathbb E_{\mathcal M^{\theta_\eta}}[Y(t)],\qquad r(t)=\frac{d\mu_T}{dP}(t). \] With the fixed constants $C_W,L_y$ supplied by Proposition~\ref{prop:Lip_ATE} at Lipschitz bound $\hat L_f$, it gives \[ \int_{t_1}^{t_2}b_\eta(t)^2P(dt)\leqslant\frac{2C_W}{\delta}w_\eta+2L_y^2w_\eta^2P([t_1,t_2]). \] Absolute continuity and Cauchy--Schwarz therefore give \begin{align*} \left|\mathbb E_{t\sim\mu_T}\mathbb E_{\mathcal M^{\theta_\eta}}[Y(t)]-F_*\right| &=\left|\int_{t_1}^{t_2}b_\eta(t)r(t)P(dt)\right|\\ &\leqslant\left(\int_{t_1}^{t_2}b_\eta(t)^2P(dt)\right)^{1/2}\left(\int_{t_1}^{t_2}r(t)^2P(dt)\right)^{1/2}\\ &\leqslant A_T\left(\frac{2C_W}{\delta}w_\eta+2L_y^2w_\eta^2P([t_1,t_2])\right)^{1/2}\\ &\leqslant C\sqrt{\alpha_n}. \end{align*} For the last inequality, the growth assumptions imply $\alpha_n\rightarrow0$, so $\alpha_n\leqslant1$ eventually and $w_\eta^2=O(\alpha_n^2)=O(\alpha_n)$; all other factors are fixed. The bound is uniform over feasible representations. Feasibility was established on the same event, so every feasible objective lies in $[F_*-C\sqrt{\alpha_n},F_*+C\sqrt{\alpha_n}]$. Taking its infimum proves $|F_n-F_*|\leqslant C\sqrt{\alpha_n}$, whether or not the infimum is attained. The event has probability at least $1-O(n^{-2})$. - why: The inherited-assumption lint warnings are dismissed: independence, both categorical clauses, all three Lipschitz/boundedness clauses, both support clauses, both lower-bound clauses, and the latent-law premise are inherited through \cref{thm:consistency}. The regularized-notation warning is also dismissed: the corollary uses the stated Sinkhorn comparison without independently invoking $W_{1,\lambda}$. The triangle inequality requires three distances: population to observational empirical law, observational empirical to candidate empirical law, and candidate empirical to population law. The printed middle term repeats the population distance, but substituting the empirical distance bounded immediately above closes this proof slip. The substantive defect is the final invocation of Proposition \ref{prop:Lip_ATE}: it controls the squared response difference integrated against $P$, whereas $F_n-F_*$ averages against $\mu_T=\text{Unif}([t_1,t_2])$. No stated hypothesis relates these measures. The counterexample satisfies the referenced assumptions but makes the displayed rate false. ### `prop:equivalent-g-constrianed` (`proof.tex:600-602`) - as printed: \label{prop:equivalent_g_constrianed} Suppose that \cref{assump:independence_u} and \cref{assump:categorical_rv} hold, given any SCM $\mathcal{M} $ with causal graph $\mathcal{G}$ and latent variables $\boldsymbol{U}$, we can construct a canonical SCM $\hat{\mathcal{M}}$ of the form (\ref{eq:structure_equation}) by merging the latent variables in $\boldsymbol{U}$ such that $\mathcal{M}$ and $\hat{\mathcal{M}}$ produce the same intervention results. Besides, functions in $\hat{\mathcal{M}}$ have the same smoothness as $\mathcal{M}$. - corrected: Suppose that \cref{assump:independence_u} and \cref{assump:categorical_rv} hold, given any SCM $\mathcal{M} $ with causal graph $\mathcal{G}$ and latent variables $\boldsymbol{U}$, we can construct a canonical SCM $\hat{\mathcal{M}}$ of the form (\ref{eq:structure_equation}) by merging the latent variables in $\boldsymbol{U}$ such that $\mathcal{M}$ and $\hat{\mathcal{M}}$ produce the same intervention results. Besides, the continuous structural functions in $\hat{\mathcal M}$ have the same smoothness as those in $\mathcal M$. The categorical functions in $\hat{\mathcal M}$ have the smoothness of the normalized propensity functions $f_i/\|f_i\|_1$: normalization preserves differentiability order and local Lipschitz regularity wherever $\|f_i\|_1>0$, and coordinate regrouping preserves any global Lipschitz regularity these normalized functions possess. - side condition added by the confirming referee `ass:pin-177`: $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. — pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality. - side condition added by the confirming referee `ass:pin-181`: Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\|f_i(x)\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. — pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities. - side condition added by the confirming referee `ass:pin-189`: For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. — pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition. - side condition added by the confirming referee `ass:pin-190`: In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. — pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction. - side condition added by the confirming referee `ass:pin-191`: Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). — pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations. - side condition added by the confirming referee `ass:pin-fix-c0174608`: The categorical generation formula requires $f_i(x)\in[0,\infty)^{n_i}$ and $\|f_i(x)\|_1>0$ on its structural input domain. — pins reading of prop:equivalent-g-constrianed: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c356d336`: A zero categorical coordinate has extended logarithm $\log0=-\infty$. — pins reading of prop:equivalent-g-constrianed: stated by the repair - proof change: the printed proof, revised step by step] Let $\boldsymbol G=\{G_{V_i}:V_i\text{ is categorical}\}$. By \cref{assump:independence_u} and \cref{assump:categorical_rv}, the remaining latent variables $U_1,\ldots,U_{n_U}$ and the Gumbel coordinates are independent. For a categorical variable, put $s_i(x)=\|f_i(x)\|_1$ and $\pi_i(x)=f_i(x)/s_i(x)$. The categorical generation assumption requires nonnegative coordinates and $s_i(x)>0$ on its structural input domain. For positive coordinates, \[ \log\pi_{i,j}(x)=\log f_{i,j}(x)-\log s_i(x). \] The subtracted term is common to every coordinate, so replacing $f_i$ by $\pi_i$ leaves its Gumbel argmax unchanged. Zero coordinates remain zero and have extended logarithm $-\infty$. Thus this replacement preserves the categorical structural output for every input and the normalized vector belongs to the probability simplex. Let $D_1,\ldots,D_{n_C}$ be the maximal $C^2$-components. For a non-Gumbel latent variable define $I(U_k)=\{V_j:U_k\in\boldsymbol U_{V_j}\}$. Independence implies that every nonempty $I(U_k)$ is a bidirected clique, so it is contained in a maximal $C^2$-component. Assign $U_k$ to one such component, choosing one when there is more than one choice. Unused latent variables may be assigned arbitrarily. Let $\hat U_C$ be the tuple of variables assigned to $C$; for an empty tuple use a constant latent variable. Disjoint tuples are independent. Their coordinate projections recover the original joint law of the non-Gumbel latent variables, and they are independent of $\boldsymbol G$. For each $V_i$, supply all $\hat U_C$ with $V_i\in C$ as canonical inputs. Project these tuples onto the original inputs $\boldsymbol U_{V_i}$ and ignore the other coordinates. Define its continuous function by applying $f_i$ to these projected inputs and its categorical function by applying $\pi_i$. Together with the unchanged parent relations and Gumbel noise, these are exactly the normalized structural equations in (\ref{eq:structure_equation}). Every original bidirected edge is generated by a shared original latent variable and remains present after that variable is assigned to a component containing both endpoints. Every newly declared shared tuple has its affected vertices inside its clique $C$, so it introduces no edge outside the original graph. Hence the canonical causal graph is $\mathcal G$. Couple each tuple with its original coordinates and retain the same Gumbel variables. Under any intervention, the treated variables agree by construction. In a topological order, suppose all parents of the next untreated variable agree. Its projected latent inputs agree; its continuous structural function is unchanged, or its categorical argmax agrees by the normalization identity. Its value therefore agrees as well. Induction over this order proves equality of all observed variables under every intervention, and hence equality of their intervention laws. Coordinate projections preserve differentiability order and are $1$-Lipschitz for the $1$-norm, so they preserve global Lipschitz constants of the functions to which they are applied. Continuous functions therefore retain their original regularity, and categorical functions retain the regularity of $\pi_i$. To verify the normalization claims, if $f_i$ is differentiable to a given order and $s_i>0$, the quotient by the positive differentiable function $s_i$ has that same differentiability order. If $f_i$ is locally Lipschitz, fix an input where $s_i>0$ and a neighborhood on which $s_i\geqslant a>0$. For two points $x,y$ in that neighborhood, nonnegativity gives $|s_i(x)-s_i(y)|\leqslant\|f_i(x)-f_i(y)\|$, and \[ \left\|\frac{f_i(x)}{s_i(x)}-\frac{f_i(y)}{s_i(y)}\right\| \leqslant\frac{\|f_i(x)-f_i(y)\|}{s_i(x)}+\frac{|s_i(x)-s_i(y)|}{s_i(x)} \leqslant\frac{2}{a}\|f_i(x)-f_i(y)\|. \] This proves local Lipschitz regularity of the normalized propensity. If that propensity is globally Lipschitz, composing it with the coordinate projections retains its global bound, completing the stated regularity conclusions. - why: ``Same smoothness'' could mean only the same differentiability order, or preservation of the original regularity through coordinate regrouping, including global Lipschitz regularity. The proof's explicit retention of the original functions determines the latter reading. (a) The original categorical assumption permits unnormalized functions, but the printed construction retains them unchanged while claiming the normalized canonical form. (b) Normalizing those functions does not generally preserve global Lipschitz regularity, so normalization is not a harmless missing line under the proposition's unrestricted domains. ### `prop:holder-curve` (`proof.tex:1238-1240`) - as printed: \label{prop:holder_curve} Let $ \lambda $ be the Lebesgue measure on $ [ 0,1]$. Given any probability measure $ \mathbb{P}$ that satisfies \cref{assumption:lower_bound}, there exists a continuous curve $ \gamma : [0,1]\rightarrow \text{supp}(\mathbb{P})$ such that $\gamma _{\#} \lambda =\mathbb{P}$. Furthermore, if $d \geqslant 1$, $\gamma$ is $ 1/d$-Hölder continuous. - corrected: statement unchanged - no longer uses `lem:holder-curve-shchepin-theorem-1` - side condition added by the confirming referee `ass:pin-33`: $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. — pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes. - side condition added by the confirming referee `ass:pin-fix-0bcb2cef`: The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-2a1e7b01`: The mass assigned to $I_{O_n(i)}$ is $\lambda_d(C_{O_n(i)})=2^{-nd}$. — pins reading of def:prop-holder-curve-interval-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-2fe04b0e`: In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal C_n$ denotes the standard dyadic collection $\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}$. — pins reading of def:prop-holder-curve-cube-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-7a79e01d`: In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal I_n$ denotes $\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$ with $I_{O_n(i)}=[(i-1)2^{-nd},i2^{-nd}]$. — pins reading of def:prop-holder-curve-interval-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8cc0dba5`: For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. — pins reading of def:hilbert-partitions: stated by the repair - side condition added by the confirming referee `ass:pin-fix-99ac9bb3`: For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. — pins reading of def:hilbert-partitions: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c778b424`: The reference measure $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$. — pins reading of def:prop-holder-curve-cube-construction: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c85b9186`: Each $O_n$ is a bijection. — pins reading of lem:hilbert: stated by the repair - side condition added by the confirming referee `ass:pin-fix-e896f793`: The cells of $\mathcal C_n$ are closed and may share boundaries. — pins reading of lem:hilbert: stated by the repair - proof change: the printed proof, revised step by step] \label{proof:holder_curve} Without loss of generality, we can assume that $ \mathbb{P}$ is supported on $[0,1]^d$, vanishes at the boundary and $\mathbb{P}(B) \geqslant C_1\lambda(B)$ for some constant $C_1>0$ and all measurable sets $B\in[0,1]^d$. Suppose we have proven the conclusion for this case. By \cref{assumption:lower_bound}, there exists $ f\in \mathcal{H}([0,1]^{d},K) \cap \mathcal{F}_L([0,1]^{d},K)$ such that $f^{-1}_{\#} \mathbb{P}$ satisfies these conditions. There exists continuous function $\gamma$ such that $f^{-1}_{\#}\mathbb{P}=\gamma_{\#}\lambda$, which implies $\mathbb{P} = (f\circ\gamma)_{\#}\lambda$. In particular, if $\gamma$ is $1/d$-Holder continuous, $f\circ\gamma$ is also $1/d$-Holder continuous because $f$ is Lipschitz continuous. When $d = 0$, the constant map satisfies the requirement. We focus on the case $d\geqslant 1$ in the following. The proof is to modify the construction of the Hilbert space-filling curve. By changing the velocity of the curve, the push-forward measure of the Hilbert space-filling curve can simulate a great number of distributions. We use $ \lambda $ to represent the Lebesgue measure $ [0,1] $. \par We prove the cube case by first constructing a uniform Hilbert curve and then changing its parameter. Write $\lambda_d$ for Lebesgue measure on $[0,1]^d$ and retain $\lambda$ for Lebesgue measure on $[0,1]$. The cube probability measure satisfies $\mathbb P\geqslant C_1\lambda_d$, so $00$, the equivalence $Q(t)\leqslant x$ if and only if $t\leqslant A(x)$. Therefore $\lambda\{t:Q(t)\leqslant x\}=A(x)$, which proves $Q_{\#}\lambda=\rho$. If $u0$ and an SCM $\mathcal{M}^*$ satisfying Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound} with a backdoor causal graph such that there is no unobserved confounding in $\mathcal{M}^*$. For any $\epsilon >0$, there exists an SCM $\mathcal{M}_\epsilon$ with the same causal graph such that $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V})) \leq \epsilon $ and $|\text{ATE}_{\mathcal{M}^*} - \text{ATE}_{\mathcal{M}_\epsilon}| > c $. - corrected: \label{prop:counterexample_foral} There exists a constant $c>0$ and an SCM $\mathcal{M}^*$ satisfying Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound} with a backdoor causal graph such that there is no unobserved confounding in $\mathcal{M}^*$. For any $\epsilon >0$, there exists an SCM $\mathcal{M}_\epsilon$ with the same causal graph such that $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V})) \leq \epsilon $ and $|\text{ATE}_{\mathcal{M}^*} - \text{ATE}_{\mathcal{M}_\epsilon}| > c $. Here, for each non-Gumbel latent law $\mathbb P$ with connected support components $C_i$, Assumption~\ref{assumption:lower_bound} is required of each normalized component law $\mathbb P|_{C_i}/\mathbb P(C_i)$, with $K=C_i$. - side condition added by the confirming referee `ass:pin-177`: $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. — pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality. - side condition added by the confirming referee `ass:pin-181`: Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\|f_i(x)\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. — pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities. - side condition added by the confirming referee `ass:pin-182`: For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\|f_i(x)-f_i(x')\|\leqslant L_f\|x-x'\|$ on its structural input domain, using the paper's $1$-norm. — pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$. - side condition added by the confirming referee `ass:pin-183`: There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\|V_i\|_\infty,\|U_j\|_\infty\}\leqslant K$ almost surely under the SCM law. — pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation. - side condition added by the confirming referee `ass:pin-189`: For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. — pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition. - side condition added by the confirming referee `ass:pin-190`: In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. — pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction. - side condition added by the confirming referee `ass:pin-204`: The second Bernoulli law in the construction has $P(U_2=0)=2/3$. — pins reading of prop:counterexample-foral: The displayed value $P(U_2=1)=1/3$ and the joint masses for $X=0,T=0$ in the proof of Proposition~\ref{prop:counterexample_foral} both determine this entry. - proof change: the printed proof, revised step by step] Let $ \boldsymbol{V} = \{ X,T,Y\} $ be the covariate, treatment and outcome respectively. $ T$ is a binary variable. Let structure equations of causal model $ \mathcal{M}_{\delta }$ be \begin{align*} X & =\begin{cases} -1 & ,\text{w.p.} \ 1/2\\ 0 & ,\text{w.p.} \ 1/4\\ \delta & ,\text{w.p.} \ 1/4, \end{cases}\\ P( T=1|X=x) & =\begin{cases} 1/2 & ,x=-1\ \text{or} \ 0,\\ 3/4 & ,x=\delta , \end{cases}\\ Y & =\begin{cases} T+U_{y} & ,X< 0\\ X/\delta +U_{y} & ,X\geqslant 0 \end{cases} \end{align*}where all latent variables are independent and $ U_{y}$ is mean-zero noise. The distribution of $ \mathcal{M}_{\delta }$ is \begin{align*} P_{X,T,Y}( 0,1,y) =p_{U_{y}}( y) /8, & \ P_{X,T,Y}( 0,0,y) =p_{U_{y}}( y) /8\\ P_{X,T,Y}( \delta ,1,y) =3p_{U_{y}}( y-1) /16, & \ P_{X,T,Y}( \delta ,0,y) =p_{U_{y}}( y-1) /16.\\ P_{X,T,Y}( -1,1,y) =p_{U_{y}}( y-1) /4, & P_{X,T,Y}( -1,0,y) =p_{U_{y}}( y) /4. \end{align*} As $ \delta \rightarrow 0$, distribution of $ \mathcal{M}_{\delta }\xrightarrow{d}\mathcal{M}^*$, where structure equations of $ \mathcal{M}^*$ are \begin{align*} P( U_{1} =1) =3/5,P(U_{1} =0) =2/5 & ,P( U_{2} =1) =1/3,P( U_{1} =0) =2/3,\\ X & =\begin{cases} -1 & ,\text{w.p.} \ 1/2,\\ 0 & ,\text{w.p.} \ 1/2, \end{cases}\\ P( T=1|X=x) & \ =\begin{cases} 1/2 & ,x=-1,\\ 5/8 & ,x=0, \end{cases}\\ Y & \ =\begin{cases} T+U_{y} & ,X=-1,\\ U_{1} +U_{y} & ,X=0,T=1,\\ U_{2} +U_{y} & ,X=0,T=0, \end{cases} \end{align*} where $ U_1,U_2,U_y $ are independent. The distribution of $ \mathcal{M}^*$ is \begin{align*} P_{X,T,Y}( 0,1,y) & \ =p_{U_{y}}( y) /8+3p_{U_{y}}( y-1) /16,\\ P_{X,T,Y}( 0,0,y) & =p_{U_{y}}( y) /8+p_{U_{y}}( y-1) /16,\\ P_{X,T,Y}( -1,1,y) & =p_{U_{y}}( y-1) /4,\\ P_{X,T,Y}( -1,0,y) & =p_{U_{y}}( y) /4. \end{align*} Choose $U_y=0$ almost surely. Generate $X$ by $X=U_X$, where $P(U_X=-1)=P(U_X=0)=1/2$, independently of $U_1,U_2$ and the categorical Gumbel variables. Every non-Gumbel latent law, including any vector formed by merging latents affecting the same observed variable, now has finite support. For such a law $\mathbb P$, write its support components as $C_i=\{a_i\}$ and put $\mu_i=\mathbb P|_{C_i}/\mathbb P(C_i)$. The map $f:[0,1]^0\rightarrow C_i$ sending the unique point to $a_i$ is a Lipschitz homeomorphism. If $\lambda_0$ denotes the probability measure on $[0,1]^0$, then for every measurable $B\subset[0,1]^0$, \[ \mu_i(f(B))=\lambda_0(B). \] Thus each normalized component law satisfies Assumption~\ref{assumption:lower_bound} with $d=0$ and $C_f=1$; its boundary requirement applies only when $d>0$. These singleton components also verify Assumption~\ref{assump:support}. All displayed structural inputs range over finite sets. Their structural and normalized propensity maps have Lipschitz extensions: form the finite Cartesian grid of their input coordinate values, retain the prescribed values at the specified inputs, and assign bounded values at the remaining vertices, using probability vectors for propensity maps. Interpolate multilinearly on each grid cell and clamp each input coordinate to the grid's range. This preserves the prescribed values and the probability simplex. Each cell has finite coordinate slopes; taking their maximum and subdividing a line segment at cell boundaries gives a finite global Lipschitz constant. Choose $L_f$ to bound these finitely many constants. The resulting observed variables and non-Gumbel latent variables are bounded, so a finite $K$ exists. Independence and categorical generation hold by construction. Hence $\mathcal M^*$ satisfies Assumptions \ref{assump:independence_u}-\ref{assumption:lower_bound}. Some calculation gives $ W( P^{\mathcal{M}^*}(\boldsymbol{V}) ,P^{\mathcal{M}_{\delta }}(\boldsymbol{V})) \leqslant \delta /2$. It is easy to calculate the ATE of the two models. \begin{align*} \text{ATE}_{\mathcal{M}^*} = 19/30, & \ \text{ATE}_{\mathcal{M}_{\delta }} = 1/2. \end{align*} - why: [F2.5 reviewer] [statement-wrong] The requirement to satisfy Assumption lower_bound does not settle whole-latent-law versus normalized-component-law regularity; Lean selects the latter. ### `prop:convg-opt` (`proof.tex:1605-1631`) - as printed: \label{prop:convg_opt} Given a metric space $ (M,d) $ and sets $ \Theta_1\subset\Theta_2\cdots\subset\Theta_n\subset\cdots\subset \Theta_\infty \subset M$, where $\Theta_\infty = \overline{\cup_{n=1}^\infty \Theta_n}$ , positive sequences $ \{\epsilon_{n}\}_{n\in \mathbb{N}},\{\delta_{n}\}_{n\in \mathbb{N}}$ such that $ \lim _{n\rightarrow \infty } \epsilon _{n}=\lim _{n\rightarrow \infty }\tau_n=\lim _{n\rightarrow \infty } \delta_{n} =0$ and continuous functions $ f,g_{n} :\Theta_\infty \rightarrow \mathbb{R}$, suppose that \begin{enumerate} \item $\Theta_\infty$ is compact. \item $g_n$ satisfies \begin{equation} \label{eq:aprox_lip} \|g_n(\theta_1) - g_n(\theta_2)\| \leqslant L_g d(\theta_1,\theta_2) + \tau_n, \quad \forall \theta_1,\theta_2\in\Theta \end{equation} and $ \sup_{\theta\in\Theta_\infty}\|g_{n}(\theta) - g(\theta)\|\leqslant \delta_n, g_n \geqslant 0$. \item There exists a compact subset $ \tilde{\Theta}_\infty \subset \Theta_\infty $ such that for any $\theta_0 \in \tilde{\Theta}_\infty$, there exists $\theta \in\Theta_n$ such that $ d(\theta,\theta_0 )\leqslant \epsilon_n.$ \end{enumerate} Consider the following optimization problems: \begin{align} \min_{\theta \in \Theta_n } & f( \theta ) , \notag\\ s.t. &\ g_{n}( \theta ) \leqslant L_g\epsilon_{n} + \delta_n, \label{eq:opt_n} \end{align} \begin{align} \min_{\theta \in \Theta_\infty } & f( \theta ) ,\notag\\ s.t. &\ g( \theta ) = 0. \label{eq:opt*} \end{align} and \begin{align} \min_{\theta \in \tilde{\Theta}_\infty } & f( \theta ) ,\notag\\ s.t. &\ g( \theta ) = 0. \label{eq:opt*_sub} \end{align} We assume that the feasible region of (\ref{eq:opt*}) and (\ref{eq:opt*_sub}) are nonempty. Let $ f_{n}^{*}, f^*, \tilde{f}^* $ be the minimal of (\ref{eq:opt_n}), (\ref{eq:opt*}) and (\ref{eq:opt*_sub}) respectively. Then, $ [\liminf _{k\rightarrow \infty } f_{n}^{*}, \limsup_{k\rightarrow \infty } f_{n}^{*}] \subset [f^{*}, \tilde{f}^*]$. - corrected: statement unchanged - no error: only a side condition or reading the referee made explicit (see below) - side condition added by the confirming referee `ass:bk-6`: For every $n\in\mathbb N$, the feasible set of (\ref{eq:opt_n}) contains a minimizer of $f$. — The finite values are defined as minima, and the lower-limit argument selects attaining minimizers. - side condition added by the confirming referee `ass:pin-159`: Equation~\ref{eq:aprox_lip} holds for every $n\in\mathbb N$ and every $\theta_1,\theta_2\in\Theta_\infty$, with the same finite constant $L_g$. — pins reading of ass:prop-convg-opt-approximate-lipschitz: Proposition~\ref{prop:convg_opt} defines the functions on $\Theta_\infty$ and uses this inequality to conclude that the limiting function is $L_g$-Lipschitz. - side condition added by the confirming referee `ass:pin-172`: There exists a compact $\tilde\Theta_\infty\subseteq\Theta_\infty$ such that $\forall n\in\mathbb N\;\forall\theta_0\in\tilde\Theta_\infty\;\exists\theta\in\Theta_n$, $d(\theta,\theta_0)\leqslant\epsilon_n$. — pins reading of ass:prop-convg-opt-approximation: The proof of Proposition \ref{prop:convg_opt} applies condition 3 to construct a sequence $\theta_n$ approximating $\tilde\theta^*$. - side condition added by the confirming referee `ass:pin-52`: In (\ref{eq:aprox_lip}), $\Theta=\Theta_\infty$. — pins reading of ass:prop-convg-opt-approximate-lipschitz: Proposition~\ref{prop:convg_opt} defines the functions on $\Theta_\infty$ and uses this inequality there to establish the limiting Lipschitz bound. ### `lem:lip-condition` (`proof.tex:1799-1805`) - as printed: \label{lemma:Lip_condition} Let $ (\hat{T} ,\hat{Y}) \sim \mu ,( T,Y) \sim \nu $ and suppose that $ f( t) =\mathbb{E}_{\nu }[ Y|T] ,\hat{f}( t) =\mathbb{E}_{\mu }[\hat{Y} |\hat{T}]$ are $ L$-Lipschitz continuous and $|f(t)|\leqslant K, |\hat{f}(t)| \leqslant K$, then we have \begin{equation*} \int ( f( t) -\hat{f}( t))^{2} d\nu ( dt) \leqslant C_{W} W( \mu ,\nu ) , \end{equation*} where $ C_{W} =4LK+2K\max\{L,1\}$. - corrected: statement unchanged - no error: only a side condition or reading the referee made explicit (see below) - side condition added by the confirming referee `ass:pin-206`: The functions $f,\hat{f}:[-K,K]\rightarrow\mathbb{R}$ are $L$-Lipschitz regression versions satisfying $f(T)=\mathbb{E}_{\nu}[Y\mid T]$ almost surely under $\nu$ and $\hat{f}(\hat{T})=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}]$ almost surely under $\mu$. — pins reading of ass:lemma-lip-condition-regression: The proof of Lemma~\ref{lemma:Lip_condition} uses both cross-evaluations and bounds its test function on the common box; the owner-supplied support condition specifies that box. ### `lem:push-forward-err` (`proof.tex:1918-1922`) - as printed: \label{lemma:push-forward_err} Let $ \mu ,\hat{\mu }$ be two measures on compact set $ K\subset \mathbb{R}^{d_{1}}$, for any measurable functions $ F,\hat{F} :K\rightarrow \mathbb{R}^{d_{2}}$, if $ F$ is $ L$-Lipschitz continuous, then \begin{equation*} W( F_{\#} \mu ,\hat{F}_{\#}\hat{\mu }) \leqslant LW( \mu ,\hat{\mu }) +\| F_{1} -F_{2} \| _{\infty } . \end{equation*} - corrected: Let $\mu,\hat\mu$ be two measures on compact set $K\subset\mathbb R^{d_1}$, for any measurable functions $F,\hat F:K\rightarrow\mathbb R^{d_2}$, if $F$ is $L$-Lipschitz continuous, then \begin{equation*} W(F_{\#}\mu,\hat F_{\#}\hat\mu)\leqslant LW(\mu,\hat\mu)+d_2\|F-\hat F\|_\infty. \end{equation*} - no longer uses `ass:lemma-push-forward-err-lipschitz` - side condition added by the confirming referee `ass:pin-fix-72c9e86d`: The measures $\mu$ and $\hat\mu$ are probability measures. — pins reading of lem:push-forward-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f6f485b1`: The uniform infinity norm is $\|F-\hat F\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}|F_j(x)-\hat F_j(x)|$. — pins reading of lem:push-forward-err: stated by the repair - proof change: the printed proof, revised step by step] For any $ 1$-Lipschitz function $ g:\mathbb{R}^{d_{2}}\rightarrow \mathbb{R}$, \begin{align*} \mathbb{E}_{X\sim F_{\#} \mu }[ g( X)] -\mathbb{E}_{X\sim \hat{F}_{\#}\hat{\mu }}[ g( X)] & =\mathbb{E}_{X\sim \mu }[ g\circ F( X)] -\mathbb{E}_{X\sim \hat{\mu }}[ g\circ \hat{F}( X)]\\ & =(\mathbb{E}_{X\sim \mu }[ g\circ F( X)] -\mathbb{E}_{X\sim \hat{\mu }}[ g\circ F( X)]) \\ &\quad +(\mathbb{E}_{X\sim \hat{\mu }}[ g\circ F( X)] -\mathbb{E}_{X\sim \hat{\mu }}[ g\circ \hat{F}( X)]) . \end{align*} Note that for any $x,y\in K$, \begin{align*} |g\circ F(x)-g\circ F(y)|\leqslant\|F(x)-F(y)\|\leqslant L\|x-y\|. \end{align*} Thus $g\circ F$ is $L$-Lipschitz continuous and \begin{align}\label{eq:first_term} \mathbb E_{X\sim\mu}[g\circ F(X)]-\mathbb E_{X\sim\hat\mu}[g\circ F(X)]\leqslant LW(\mu,\hat\mu). \end{align} For the second term, \begin{align}\label{eq:second_term} \mathbb E_{X\sim\hat\mu}[g\circ F(X)]-\mathbb E_{X\sim\hat\mu}[g\circ\hat F(X)] &\leqslant\mathbb E_{X\sim\hat\mu}|g(F(X))-g(\hat F(X))|\notag\\ &\leqslant\mathbb E_{X\sim\hat\mu}\|F(X)-\hat F(X)\|\notag\\ &\leqslant\sup_{x\in K}\|F(x)-\hat F(x)\|\notag\\ &\leqslant d_2\|F-\hat F\|_\infty. \end{align} The last inequality follows by summing the $d_2$ coordinatewise bounds. If the uniform error is infinite, the asserted bound holds in the extended nonnegative reals. Combining (\ref{eq:first_term}) and (\ref{eq:second_term}) bounds the difference of expectations by $LW(\mu,\hat\mu)+d_2\|F-\hat F\|_\infty$ for every scalar $1$-Lipschitz function $g$. Taking the supremum over these functions gives the asserted bound for $W(F_{\#}\mu,\hat F_{\#}\hat\mu)$. - why: (a) The printed conclusion uses $F_1,F_2$, although only $F,\hat F$ are defined; the proof derives a comparison of the latter maps and supplies no identification with the former. This is a slip in the asserted bound. (b) Independently, the last inequality in the second-term calculation replaces the $1$-norm by the infinity norm without its dimensional factor. With the binding uniform infinity norm, reproduction gives $LW(\mu,\hat\mu)+\sup_{x\in K}\|F(x)-\hat F(x)\|$, which is at most $LW(\mu,\hat\mu)+d_2\|F-\hat F\|_\infty$, rather than the asserted coefficient. ### `lem:combin-err` (`proof.tex:1945-1949`) - as printed: \label{lemma:combin_err} Let $ \mu _{1} ,\cdots ,\mu _{n} ,\hat{\mu }_{1} ,\cdots ,\hat{\mu }_{n}$ be measure on $ \mathbb{R}^{d}$, for any $ p_{1} ,\cdots ,p_{n} \in [ 0,1]$ such that $ \sum _{k=1}^{n} p_{k} =1$, we have \begin{equation*} W\left(\sum _{k=1}^{n} p_{k} \mu _{k} ,\sum _{k=1}^{n} p_{k}\hat{\mu }_{k}\right) \leqslant \sum _{k=1}^{n} p_{k} W( \mu _{k} ,\hat{\mu }_{k}) . \end{equation*} - corrected: statement unchanged - no error: only a side condition or reading the referee made explicit (see below) - side condition added by the confirming referee `ass:bk-1`: For every $k\in[n]$, $\int_{\mathbb R^d}\|x\|\,\mu_k(dx)+\int_{\mathbb R^d}\|x\|\,\hat\mu_k(dx)<\infty$. — The dual proof separately evaluates expectations of unbounded Lipschitz functions. - side condition added by the confirming referee `ass:pin-75`: For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. — pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula. ### `lem:product-err` (`proof.tex:1963-1968`) - as printed: \label{lemma:product_err} Given two measures $ \mu =\mu _{1} \otimes \cdots \otimes \mu _{n} ,\hat{\mu } =\hat{\mu }_{1} \otimes \cdots \otimes \hat{\mu }_{n}$, we have \begin{equation*} W( \mu ,\hat{\mu }) \leqslant \sum _{k=1}^{n} W( \mu _{k} ,\hat{\mu }_{k}) . \end{equation*} - corrected: Given two measures $\mu=\mu_1\otimes\cdots\otimes\mu_n$, $\hat\mu=\hat\mu_1\otimes\cdots\otimes\hat\mu_n$, where every factor $\mu_k$ and $\hat\mu_k$ is a probability measure, we have \begin{equation*} W(\mu,\hat\mu)\leqslant\sum_{k=1}^n W(\mu_k,\hat\mu_k). \end{equation*} - side condition added by the confirming referee `ass:pin-fix-b83fa4d1`: $W$ is Wasserstein-1 distance for the ground distance $\|x-y\|$. — pins reading of lem:product-err: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ca667e13`: $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. — pins reading of lem:product-err: stated by the repair - proof change: the printed proof, revised step by step] If $n=0$, both empty products are the probability measure on the one-point space and both sides are zero. Suppose $n\geqslant1$. If any $W(\mu_k,\hat\mu_k)=\infty$, the inequality holds in the extended nonnegative reals. Otherwise, for any $\epsilon>0$ and every $k\in[n]$, choose a probability coupling $\pi_k$ of $\mu_k$ and $\hat\mu_k$ such that \begin{equation*} \mathbb E_{(X_k,Y_k)\sim\pi_k}[\|X_k-Y_k\|_1]\leqslant W(\mu_k,\hat\mu_k)+\epsilon. \end{equation*} Such a coupling exists by the definition of the finite infimum defining $W$. Under the product probability law $\pi_1\otimes\cdots\otimes\pi_n$, put $X=(X_1,\ldots,X_n)$ and $Y=(Y_1,\ldots,Y_n)$. The marginal laws of $X$ and $Y$ are respectively $\mu$ and $\hat\mu$, because the coordinate pairs are independent and have the prescribed marginal laws. Thus this construction is a coupling of the product measures. The product $1$-norm is the sum of the coordinate $1$-norms, so \begin{align*} W(\mu,\hat\mu) &\leqslant\mathbb E_{\pi_1\otimes\cdots\otimes\pi_n}[\|X-Y\|]\\ &=\sum_{k=1}^n\mathbb E_{(X_k,Y_k)\sim\pi_k}[\|X_k-Y_k\|_1]\\ &\leqslant\sum_{k=1}^n W(\mu_k,\hat\mu_k)+n\epsilon. \end{align*} Letting $\epsilon\downarrow0$ proves the result. ### `lem:gumbel-err` (`proof.tex:1981-1990`) - as printed: \label{lemma:gumbel_err} Given $1 >\tau >0$ and $p_{1} ,\cdots ,p_{n} >0$, let $\mu ^{\tau } \sim X^{\tau } =\left( X_{1}^{\tau } ,\cdots ,X_{n}^{\tau }\right)$ be \begin{equation} X_{k} =\frac{\exp((\log p_{k} +G_{k}) /\tau )}{\sum _{k=1}^{n}\exp((\log p_{k} +G_{k}) /\tau )} ,\label{eq:gumbel} \end{equation} where $G_{k} \sim \exp( -x-\exp( -x))$ are i.i.d. standard Gumbel variables. Let $\mu ^{0}$ be the distribution of $X^{0} =\lim _{\tau \rightarrow 0} X^{\tau } =\left( X_{1}^{0} ,\cdots ,X_{n}^{0}\right)$. Then, \begin{equation*} W\left( \mu ^{\tau } ,\mu ^{0}\right) \leqslant 2( n-1) \tau -2n( n-1) e^{-1} \tau \log \tau . \end{equation*} - corrected: statement unchanged - no error: only a side condition or reading the referee made explicit (see below) - side condition added by the confirming referee `ass:pin-203`: The gap event is $E_\delta=\{\exists\,1\leqslant i0$, we have \begin{align*} 0\leqslant W_{1,\lambda}( \mu ,\nu ) -W( \mu ,\nu ) & \leqslant (\log( mn) +1) \lambda, \end{align*} where $ W_{1,\lambda}(\cdot,\cdot) $ is the entropy regularized Wasserstein-1 distance. Moreover, we have the following estimation of approximation error. \begin{align*} |S_{\lambda }( \mu ,\nu ) -W( \mu ,\nu ) | & \leqslant 2(\log( mn) +1) \lambda . \end{align*} - corrected: For any $\mu\in\Delta_n$, $\nu\in\Delta_m$, and $\lambda>0$, let $\mu$ and $\nu$ be the weights of finite probability measures on points $x_1,\ldots,x_n$ and $y_1,\ldots,y_m$ in a common Euclidean space with the paper's $1$-norm. Put $c_{ij}=\|x_i-y_j\|$ and \[ \Pi(\mu,\nu)=\left\{T\in[0,\infty)^{n\times m}:\sum_jT_{ij}=\mu_i,\ \sum_iT_{ij}=\nu_j\right\},\qquad h(T)=1-\sum_{i,j}T_{ij}\log T_{ij}, \] with $0\log0=0$. Here the entropy regularized Wasserstein-1 distance is \[ W_{1,\lambda}(\mu,\nu)=\min_{T\in\Pi(\mu,\nu)}\left\{\sum_{i,j}c_{ij}T_{ij}-\lambda h(T)\right\}, \] and $W(\mu,\nu)=\min_{T\in\Pi(\mu,\nu)}\sum_{i,j}c_{ij}T_{ij}$. Then \begin{align*} -(\log(mn)+1)\lambda\leqslant W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)\leqslant0. \end{align*} Moreover, we have the following estimation of approximation error. \begin{align*} |S_{\lambda }( \mu ,\nu ) -W( \mu ,\nu ) | & \leqslant 2(\log( mn) +1) \lambda . \end{align*} - no longer uses `def:lemma-sinkhorn-approx-regularized-notation` - no longer uses `lem:sinkhorn-luise-proposition-1` - side condition added by the confirming referee `ass:pin-fix-761562d5`: The self terms in $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$ use the same $1$-norm ground distance on the corresponding support points. — pins reading of lem:sinkhorn-approx: stated by the repair - proof change: the printed proof, revised step by step] The transportation polytope is nonempty and compact, and both objectives are continuous under $0\log0=0$, so their minima exist. Let $T^*$ minimize the unregularized transport cost and let $T_\lambda$ minimize the regularized objective. Since $h(T)\geqslant0$ for every probability plan, \[ W_{1,\lambda}(\mu,\nu)\leqslant\sum_{i,j}c_{ij}T^*_{ij}-\lambda h(T^*)\leqslant W(\mu,\nu). \] Also $\sum_{i,j}c_{ij}(T_\lambda)_{ij}\geqslant W(\mu,\nu)$. It follows that \begin{align}\label{eq:err_reg_wass} -\lambda\max_{T\in\Pi(\mu,\nu)}h(T)\leqslant W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)\leqslant0. \end{align} Since $ T_{ij} \in [ 0,1]$ and $ f( x) =-x\log x$ is concave, we have \begin{align*} h( T) = & -nm\cdot \frac{1}{nm}\sum _{i,j=1}^{n,m} T_{ij}\log T_{ij} +1.\\ & \leqslant -nm\cdot \left(\frac{1}{nm}\sum _{i,j=1}^{n,m} T_{ij}\right)\log\left(\frac{1}{nm}\sum _{i,j=1}^{n,m} T_{ij}\right) +1\\ & =1+\log( nm) . \end{align*} By definition of Sinkhorn distance, \begin{align*} S_{\lambda }( \mu ,\nu ) & = W_{1,\lambda }( \mu ,\nu ) -W_{1,\lambda }( \mu ,\mu )/2 -W_{1,\lambda }( \nu ,\nu )/2 . \end{align*} Let $a=W_{1,\lambda}(\mu,\nu)-W(\mu,\nu)$, $b=W_{1,\lambda}(\mu,\mu)$, and $c=W_{1,\lambda}(\nu,\nu)$. The self-transport Wasserstein costs are zero. By (\ref{eq:err_reg_wass}) and the entropy calculation, \[ |a|\leqslant(1+\log(nm))\lambda,\quad |b|\leqslant(1+2\log n)\lambda,\quad |c|\leqslant(1+2\log m)\lambda. \] Consequently \begin{align*} |S_\lambda(\mu,\nu)-W(\mu,\nu)| &=|a-b/2-c/2|\\ &\leqslant|a|+|b|/2+|c|/2\\ &\leqslant2(1+\log(nm))\lambda. \end{align*} - why: the step its proof takes from [luiseDifferentialPropertiesSinkhorn2018, Proposition 1] (lem:sinkhorn-luise-proposition-1) does not hold as the proof uses it: The full entropy-penalized optimal value and the transport cost evaluated at an entropy-regularized optimizer are different readings of $W_{1,\lambda}$; their differences from $W$ can have different signs. The host does not identify either functional, its entropy convention, or the ground-cost data. The cited proposition gives absolute approximation bounds for two separately defined functionals, with an inverse regularization parameter. It does not justify the host's signed assertion for its unspecified functional; in particular, the source's full regularized value violates the asserted lower bound. ### `lem:sup-wasserstein` (`proof.tex:2061-2067`) - as printed: \label{lemma:sup_wasserstein} Given a measure $ \mu $ on $ \mathbb{R}^{d}$ and a real function class $ \mathcal{F}$ with output dimension $ d$, suppose that the pseudo-dimension of $ \mathcal{F}$ is less than $ d_{\mathcal{F}} < \infty $ and there exists $K>0$ such that $ |f( x) |\leqslant K,\forall f\in \mathcal{F} ,x\in \mathbb{R}^{d'}$, then with probability at least $ 1-\exp\left( O\left( \delta ^{-d}\log\left( \delta ^{-1}\right) +d_{\mathcal{F}}\log \epsilon ^{-1} -n\epsilon ^{2}\right)\right)$, \begin{align*} \sup _{f\in \mathcal{F}} W( f_{\#} \mu ,f_{\#} \mu _{n}) & \leqslant \delta +\epsilon \end{align*} for all $ \delta ,\epsilon >0$, where $ \mu _{n}$ is the empirical distribution of $ \mu $. - corrected: Given a probability measure $\mu$ on $\mathbb R^{d'}$ and a class $\mathcal F$ of measurable functions $f:\mathbb R^{d'}\rightarrow\mathbb R^d$, suppose that each scalar coordinate class $\mathcal F_j=\{x\mapsto f_j(x):f\in\mathcal F\}$, $j\in[d]$, has pseudo-dimension less than $d_{\mathcal F}<\infty$ and there exists $K>0$ such that $\|f(x)\|_\infty\leqslant K$ for all $f\in\mathcal F$ and $x\in\mathbb R^{d'}$. Suppose in addition that there are a standard Borel space $T$ and a jointly Borel map $G:T\times\mathbb R^{d'}\rightarrow[-K,K]^d$ such that $\mathcal F=\{G(t,\cdot):t\in T\}$. There are positive constants $C,c$ depending only on $d,K$ such that, with probability at least \[ 1-\exp\left(C\left((1+\delta^{-d})\log(2+\delta^{-1})+d_{\mathcal F}\log(2+\epsilon^{-1})\right)-cn\epsilon^2\right), \] \begin{align*} \sup_{f\in\mathcal F}W(f_{\#}\mu,f_{\#}\mu_n)\leqslant\delta+\epsilon \end{align*} for all $\delta,\epsilon>0$, where $\mu_n$ is the empirical distribution of $\mu$. - no longer uses `ass:lemma-sup-wasserstein-pseudodimension` - no longer uses `ass:lemma-sup-wasserstein-bounded` - no longer uses `lem:sup-wasserstein-gottlieb-lemma-6` - side condition added by the confirming referee `ass:pin-207`: The boundedness condition means $\|f(x)\|_{\infty}\leqslant K$ for every $f\in\mathcal{F}$ and $x\in\mathbb{R}^{d'}$. — pins reading of ass:lemma-sup-wasserstein-bounded: The proof of Lemma~\ref{lemma:sup_wasserstein} places the outputs in $[-K,K]^d$, consistently with the supplied function-class interpretation. - side condition added by the confirming referee `ass:pin-fix-448b19a0`: Probabilities of uniform deviation events are evaluated in the completion of the sample law. — pins reading of lem:sup-wasserstein: stated by the repair - side condition added by the confirming referee `ass:pin-fix-d19c63b5`: The probability bound is for each fixed pair $\delta,\epsilon>0$. — pins reading of lem:sup-wasserstein: stated by the repair - proof change: the printed proof, revised step by step] The empty class satisfies the uniform assertion vacuously, so assume $T\ne\varnothing$. Let $X_1,\ldots,X_n$ be independent observations with law $\mu$. By the dual formulation of Wasserstein distance, \begin{align*} \sup_{f\in\mathcal F}W(f_{\#}\mu,f_{\#}\mu_n) =\sup_{h\in\mathcal H}\sup_{f\in\mathcal F}\left(\mathbb E_{X\sim\mu_n}h(f(X))-\mathbb E_{X\sim\mu}h(f(X))\right), \end{align*} where \[ \mathcal H=\{h:[-K,K]^d\rightarrow\mathbb R:h(0)=0,\ |h(x)-h(y)|\leqslant\|x-y\|\}. \] Every such $h$ takes values in $[-dK,dK]$. Retain the unnormalized empirical covering number \begin{align*} \mathcal N_1(r,\mathcal F,N)=\sup_{x_1,\ldots,x_N}\mathcal N\left(r,\{(f(x_1),\ldots,f(x_N)):f\in\mathcal F\},\|\cdotp\|_1\right), \end{align*} where the $1$-norm on the evaluation vector sums over all sample points and output coordinates. With the uniform norm, $\mathcal H$ is closed, uniformly bounded and equicontinuous on the compact cube, hence is a compact metric space by Arzela--Ascoli. Evaluation $(h,y)\mapsto h(y)$ is continuous: its increment is bounded by $\|h-h'\|_\infty+\|y-y'\|$. Consequently $(t,h,x)\mapsto h(G(t,x))$ is jointly Borel on $T\times\mathcal H\times\mathbb R^{d'}$. Its bounded population integral is Borel in $(t,h)$: this holds for indicators of measurable rectangles, is preserved by bounded monotone limits and linear combinations, and extends to bounded Borel functions by the monotone class theorem. Every strict deviation event is the projection of a Borel set from this standard Borel parameter space and is therefore analytic and universally measurable; weak supremum events are countable intersections of strict ones. The same reasoning applies to two samples and finite coordinate swaps. All probabilities below are thus defined in the completion of the sample law. Obviously, if $ h$ is 1-Lipschitz, \begin{align*} \mathcal{N}_{1}( \epsilon ,h\circ \mathcal{F} ,n) & \leqslant \mathcal{N}_{1}( \epsilon ,\mathcal{F} ,n) . \end{align*} Put $v=\lfloor d_{\mathcal F}\rfloor$. Each coordinate class has pseudo-dimension at most $v$. Normalize its range by $f_j\mapsto(f_j+K)/(2K)$. Applying Theorem 18.4 of \cite{anthony1999neural} in its averaged empirical metric, and then undoing the normalization, bounds a scalar unnormalized cover of radius $r/(2d)$ on $N$ points by \[ e(v+1)\left(2+\frac{8eKdN}{r}\right)^v. \] The additional $2$ allows radii beyond the normalized unit range. Take the product of these $d$ covers. For every product ball meeting the evaluation vectors of $\mathcal F$, choose one such vector as its center. Two vectors in that product ball differ by at most $r/d$ in each scalar unnormalized metric, hence by at most $r$ in the summed vector metric. Thus \[ \mathcal N_1(r,\mathcal F,N)\leqslant\left[e(v+1)\left(2+\frac{8eKdN}{r}\right)^v\right]^d. \] This exhibits both sample-size and range scaling. Fix $h\in\mathcal H$, and define \[ a_f(x)=\frac{h(f(x))+dK}{2dK},\qquad z_f(x)=\sqrt{a_f(x)}. \] Both functions take values in $[0,1]$. On $N$ points, the inequality $|\sqrt a-\sqrt b|\leqslant\sqrt{|a-b|}$ and the Cauchy--Schwarz inequality give \[ \frac1N\sum_{i=1}^N|z_f(x_i)-z_g(x_i)|\leqslant\left(\frac{1}{2dKN}\sum_{i=1}^N\|f(x_i)-g(x_i)\|\right)^{1/2}. \] Consequently an unnormalized vector cover of radius $2dKNr^2$ gives an averaged scalar cover of the class $\{z_f:f\in\mathcal F\}$ of radius $r$, whose cardinality is at most \[ \left[e(v+1)(2+4e/r^2)^v\right]^d. \] By Theorem 17.1 in \cite{anthony1999neural}, with its standing measurability condition, let $X$ and $T_0$ be standard Borel spaces, let $g:T_0\times X\rightarrow[0,1]$ be jointly Borel, and put $\mathcal G=\{g(t,\cdot):t\in T_0\}\ne\varnothing$. For every Borel probability distribution $P$ on $X\times[0,1]$, every $0<\zeta<1$ and every positive integer $m$, \[ P^m\left\{\text{some }t\in T_0\text{ has }\left|\mathbb E_{(x,y)\sim P}(g(t,x)-y)^2-\frac1m\sum_{i=1}^m(g(t,x_i)-y_i)^2\right|\geqslant\zeta\right\} \leqslant4\mathcal N_1^{\mathrm{avg}}(\zeta/16,\mathcal G,2m)\exp(-\zeta^2m/32). \] Here the draws are independent, the probability is taken in the completed sample law, and $\mathcal N_1^{\mathrm{avg}}$ uses the averaged empirical metric. For the present class, $(t,x)\mapsto\sqrt{(h(G(t,x))+dK)/(2dK)}$ is jointly Borel, since $h$ is continuous and the square root is continuous on $[0,1]$. Its parameter space is the same standard Borel space $T$, and its constant-response input law is $\mu\otimes\delta_0$. Apply this form of Theorem 17.1 to the class with response identically zero. Its squared loss is $z_f^2=a_f$. For $0<\epsilon<2dK$, put $\eta=\epsilon/(2dK)$ and use the averaged cover at radius $\eta/16$ on $2n$ points. The theorem gives \[ \mathbb P\left(\sup_{f\in\mathcal F}|(\mu_n-\mu)(h\circ f)|>\epsilon\right) \leqslant4\left[e(v+1)(2+1024e/\eta^2)^v\right]^d\exp(-n\eta^2/32). \] Since $v\leqslant d_{\mathcal F}$ and $\log(v+1)\leqslant v$, the logarithm of the right-hand side is bounded by $C_0+C_1d_{\mathcal F}\log(2+\epsilon^{-1})-c_1n\epsilon^2$, with constants depending only on $d,K$. If $\epsilon\geqslant2dK$, the deviation is deterministically at most $2dK$, and the desired deviation bound is immediate. Let $\mathcal H_\delta$ be a finite $\delta/2$-net of $\mathcal H$ in the uniform norm. Such a net has \[ \log|\mathcal H_\delta|\leqslant C_2(1+\delta^{-d})\log(2+\delta^{-1}), \] where $C_2$ depends only on $d,K$. To see this, take a grid in $[-K,K]^d$ with $1$-norm mesh at most $\delta/8$ and at most $(1+C_3/\delta)^d$ points. Quantize the values of $h$ at these points into bins of width at most $\delta/8$ in $[-dK,dK]$. There are at most $2+C_4/\delta$ bins at each point. Choose one member of $\mathcal H$ from each nonempty quantization pattern. Two functions with the same pattern differ by at most $\delta/8$ at grid points and by at most $3\delta/8$ everywhere, using their Lipschitz bounds and a nearest grid point. These representatives form the required net. Counting patterns gives the displayed entropy bound. Apply the fixed-$h$ concentration bound to every $h\in\mathcal H_\delta$ and take a union bound. For $0<\epsilon<2dK$, after enlarging a constant depending only on $d,K$, \[ \mathbb P\left(\sup_{h\in\mathcal H_\delta,f\in\mathcal F}(\mu_n-\mu)(h\circ f)>\epsilon\right) \leqslant\exp\left(C\left((1+\delta^{-d})\log(2+\delta^{-1})+d_{\mathcal F}\log(2+\epsilon^{-1})\right)-cn\epsilon^2\right). \] For every $h\in\mathcal H$, choose $\tilde h\in\mathcal H_\delta$ with $\|h-\tilde h\|_\infty\leqslant\delta/2$. Since $\mu_n$ and $\mu$ are probability measures, \[ |(\mu_n-\mu)((h-\tilde h)\circ f)|\leqslant\delta. \] Therefore \[ \sup_{h\in\mathcal H,f\in\mathcal F}(\mu_n-\mu)(h\circ f) \leqslant\delta+\sup_{\tilde h\in\mathcal H_\delta,f\in\mathcal F}(\mu_n-\mu)(\tilde h\circ f). \] The dual formula yields the stated Wasserstein bound. If $\epsilon\geqslant2dK$, both push-forward measures are supported in a set of $1$-norm diameter $2dK$, so their Wasserstein distance is at most $2dK\leqslant\delta+\epsilon$ deterministically. This completes the proof for all positive $\delta,\epsilon$. - why: the step its proof takes from [anthony1999neural, Theorems 18.4 and 17.1] (lem:sup-wasserstein-anthony-theorems) does not hold as the proof uses it: (a) For vector-valued $\mathcal F$, its pseudo-dimension could mean $\max_{j\in[d]}\operatorname{Pdim}(\{f_j:f\in\mathcal F\})$ or $\operatorname{Pdim}(\{(x,j)\mapsto f_j(x):f\in\mathcal F\})$. These impose different bounds, and neither the paper nor the scalar cited theorem specifies the choice. The dual formulation does determine that $h$ is scalar-valued. (b) The attributed covering inequality is false for the paper's explicitly unnormalized empirical $\ell_1$ metric: its bound omits sample-size scaling. (c) It also omits range scaling: Theorem 18.4 requires scalar functions valued in $[0,1]$, whereas the host only bounds their magnitude by arbitrary $K$. The fixed-$h$ concentration order can be recovered from Theorem 17.1 by range normalization and a square-root transformation with zero responses, but that does not validate the printed covering inequality. ## Findings withdrawn by the owner Not errors: the paper's owner showed each finding misreads the paper. The printed statement stands, under the owner's reading (or side condition), which the formalization adopts. - `def:relu-networks` (raised at F2.5): the owner's reading (`ass:pin-owner-3`): A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden layers, each of width at most $W$, followed by the affine output layer (width at most $W$ and exactly $W$ give the same class of functions: a narrower layer is padded with zero units). The covering number $\mathcal{N}(\epsilon,S,\\|\cdot\\|_1)$ is that of the cited source (Anthony and Bartlett): the least size of a subset $C\subseteq S$ such that every point of $S$ lies at $\ell_1$ distance strictly less than $\epsilon$ from some point of $C$. — the withdrawn finding: [statement-wrong] The frozen phrase 'width W and depth L' does not distinguish exact sizes from budgets or specify whether depth counts hidden layers or includes the output layer. - `def:lemma-sup-wasserstein-empirical-covering` (raised at F2.5): the owner's reading (`ass:pin-owner-4`): A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden layers, each of width at most $W$, followed by the affine output layer (width at most $W$ and exactly $W$ give the same class of functions: a narrower layer is padded with zero units). The covering number $\mathcal{N}(\epsilon,S,\\|\cdot\\|_1)$ is that of the cited source (Anthony and Bartlett): the least size of a subset $C\subseteq S$ such that every point of $S$ lies at $\ell_1$ distance strictly less than $\epsilon$ from some point of $C$. — the withdrawn finding: [statement-wrong] The frozen definition does not specify strict versus closed covering balls or internal versus ambient centers; Lean selects strict internal covers. ## Suggestions to the authors Not errors: readings the paper determines but does not state, and side conditions the proofs use silently. - clarification (`ass:pin-177`, used by `thm:approximation`): $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. — pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality. - clarification (`ass:pin-181`, used by `thm:approximation`): Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\\|f_i(x)\\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. — pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities. - clarification (`ass:pin-182`, used by `thm:approximation`): For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\\|f_i(x)-f_i(x')\\|\leqslant L_f\\|x-x'\\|$ on its structural input domain, using the paper's $1$-norm. — pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$. - clarification (`ass:pin-183`, used by `thm:approximation`): There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely under the SCM law. — pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation. - clarification (`ass:pin-189`, used by `thm:approximation`): For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. — pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition. - clarification (`ass:pin-190`, used by `thm:approximation`): In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. — pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction. - clarification (`ass:pin-191`, used by `thm:approximation`): Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). — pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations. - clarification (`ass:pin-fix-4a86c2eb`, used by `thm:approximation`): The structural uniform infinity norm takes the supremum of the coordinatewise infinity norm over the common structural input domain. — pins reading of thm:approximation: stated by the repair - clarification (`ass:pin-fix-ab17be7a`, used by `thm:approximation`): For a categorical propensity coordinate equal to zero, $\log0=-\infty$ in the canonical argmax construction. — pins reading of thm:approximation: stated by the repair - clarification (`ass:pin-192`, used by `thm:wide-nn`): A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. — pins reading of def:relu-networks: The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths. - clarification (`ass:pin-203`, used by `thm:wide-nn`): The gap event is $E_\delta=\{\exists\,1\leqslant i0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. — pins reading of def:gumbel-softmax-layer: The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit. - clarification (`ass:pin-211`, used by `thm:wide-nn`): In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. — pins reading of def:lemma-gumbel-err-softmax: Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores. - clarification (`ass:pin-33`, used by `thm:wide-nn`): $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. — pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes. - clarification (`ass:pin-75`, used by `thm:wide-nn`): For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. — pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula. - clarification (`ass:pin-fix-00130002`, used by `thm:wide-nn`): The law $\hat g_{\#}\lambda$ includes the independent standard Gumbel variables used in the mixing layer. — pins reading of thm:wide-nn: stated by the repair - clarification (`ass:pin-fix-507a5848`, used by `thm:wide-nn`): For each component of positive dimension $q=d_i^C$, the application of Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and quantization mesh $\delta/n$. — pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair - clarification (`ass:pin-fix-5f02b8a6`, used by `thm:wide-nn`): The error estimates are asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of these budgets and of $\tau$. — pins reading of thm:wide-nn: stated by the repair - clarification (`ass:pin-fix-72c9e86d`, used by `thm:wide-nn`): The measures $\mu$ and $\hat\mu$ are probability measures. — pins reading of lem:push-forward-err: stated by the repair - clarification (`ass:pin-fix-9bbf998f`, used by `thm:wide-nn`): A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. — pins reading of thm:wide-nn: stated by the repair - clarification (`ass:pin-fix-b83fa4d1`, used by `thm:wide-nn`): $W$ is Wasserstein-1 distance for the ground distance $\\|x-y\\|$. — pins reading of lem:product-err: stated by the repair - clarification (`ass:pin-fix-c514425b`, used by `thm:wide-nn`): The refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. — pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair - clarification (`ass:pin-fix-ca667e13`, used by `thm:wide-nn`): $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. — pins reading of lem:product-err: stated by the repair - clarification (`ass:pin-fix-db6ed772`, used by `thm:wide-nn`): In the proposition entitled ``Deep distribution approximation'', the multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. — pins reading of thm:wide-nn: stated by the repair - clarification (`ass:pin-fix-f6f485b1`, used by `thm:wide-nn`): The uniform infinity norm is $\\|F-\hat F\\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}\|F_j(x)-\hat F_j(x)\|$. — pins reading of lem:push-forward-err: stated by the repair - clarification (`ass:pin-fix-0bcb2cef`, used by `thm:cor-fix-3c402872`): The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. — pins reading of lem:hilbert: stated by the repair - clarification (`ass:pin-fix-2a1e7b01`, used by `thm:cor-fix-3c402872`): The mass assigned to $I_{O_n(i)}$ is $\lambda_d(C_{O_n(i)})=2^{-nd}$. — pins reading of def:prop-holder-curve-interval-construction: stated by the repair - clarification (`ass:pin-fix-2fe04b0e`, used by `thm:cor-fix-3c402872`): In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal C_n$ denotes the standard dyadic collection $\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}$. — pins reading of def:prop-holder-curve-cube-construction: stated by the repair - clarification (`ass:pin-fix-7a79e01d`, used by `thm:cor-fix-3c402872`): In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal I_n$ denotes $\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$ with $I_{O_n(i)}=[(i-1)2^{-nd},i2^{-nd}]$. — pins reading of def:prop-holder-curve-interval-construction: stated by the repair - clarification (`ass:pin-fix-8cc0dba5`, used by `thm:cor-fix-3c402872`): For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. — pins reading of def:hilbert-partitions: stated by the repair - clarification (`ass:pin-fix-99ac9bb3`, used by `thm:cor-fix-3c402872`): For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. — pins reading of def:hilbert-partitions: stated by the repair - clarification (`ass:pin-fix-c778b424`, used by `thm:cor-fix-3c402872`): The reference measure $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$. — pins reading of def:prop-holder-curve-cube-construction: stated by the repair - clarification (`ass:pin-fix-c85b9186`, used by `thm:cor-fix-3c402872`): Each $O_n$ is a bijection. — pins reading of lem:hilbert: stated by the repair - clarification (`ass:pin-fix-e896f793`, used by `thm:cor-fix-3c402872`): The cells of $\mathcal C_n$ are closed and may share boundaries. — pins reading of lem:hilbert: stated by the repair - clarification (`ass:pin-186`, used by `thm:deep-nn`): The phrase ``uniform cubes'' denotes the unit cube domains $[0,1]^m$ used for deterministic uniform function approximation; it imposes no uniform probability law on structural inputs. — pins reading of ass:relu-approximation-domain: The same paragraph displays approximation of functions on $[0,1]^{d_1}$ and separately says that similar arguments apply to any bounded cubes. - clarification (`ass:pin-194`, used by `thm:deep-nn`): Membership in $\text{NCM}_{\mathcal G}(\mathcal F_0,\mathcal F_1)$ requires $\hat f_i\in\mathcal F_0$ for every observed node and $\hat g_j\in\mathcal F_1$ for every latent component, with the compatible input and output dimensions understood separately for each network. — pins reading of def:neural-causal-class: The source paragraph explicitly omits input and output dimensions and describes separate structural approximants and latent generators in equation~\ref{eq:structure_equation_hat}. - clarification (`ass:pin-199`, used by `thm:deep-nn`): $d_{\max}^U=\max_{j\in[n_U]}d_j^U$ ranges over non-Gumbel latent variables, while $d_{\max}^{\mathrm{in}}=\max_i d_i^{\mathrm{in}}$ and $d_{\max}^{\mathrm{out}}=\max_i d_i^{\mathrm{out}}$, with $d_i^{\mathrm{out}}=n_i$ for categorical nodes. — pins reading of def:corollary-deep-nn-dimensions: The NCM construction separately approximates non-Gumbel latent distributions and the functions in the canonical structural equations; those functions output categorical propensity vectors. The binding dimension symbols specify these same meanings. - clarification (`ass:pin-208`, used by `thm:deep-nn`): For each categorical $V_i$, the noise in equation (\ref{eq:structure_equation_hat}) satisfies $g_k=g_k^{V_i}$ for $k\in[n_i]$. — pins reading of def:eq-structure-equation-hat: The canonical representation in equation (\ref{eq:structure_equation}) assigns separate independent standard Gumbel coordinates to each categorical node; the approximated equation retains that categorical-generation rule. - clarification (`ass:pin-fix-419eb8e3`, used by `thm:deep-nn`): The multiplicative width factors in $\Theta(d_{\max}^{\text{in}}d_{\max}^{\text{out}})$, $\Theta(d_{\max}^U)$, and $\Theta((d_{\max}^U)^2)$ can be chosen as absolute numerical constants, independent of the causal model, $R$, all dimensions, and all category and component counts. — pins reading of thm:deep-nn: stated by the repair - clarification (`ass:pin-fix-43f1bb71`, used by `thm:deep-nn`): For each latent law $P(U_C)$, the assumptions of the proposition entitled ``Deep distribution approximation'' require the support charts of \cref{assump:support} and \cref{assumption:lower_bound} on each normalized component law $\mu_i=P(U_C)\|_{C_i}/P(U_C\in C_i)$, with $(f^{-1})_{\#}\mu_i(\partial[0,1]^{d_i^C})=0$ when $d_i^C>0$. — pins reading of thm:deep-nn: stated by the repair - clarification (`ass:pin-fix-73abef8c`, used by `thm:deep-nn`): In Corollary~\ref{corollary:deep_nn}, for $m_i=d_i^{\text{in}}>0$, the symbol $\hat f_i$ denotes $q_i\circ\Pi_i$ for a continuous variable and the vector $\bigl(\max\{\epsilon_i,(q_i\circ\Pi_i)_k\}\bigr)_{k=1}^{n_i}$ for a categorical variable, where $q_i$, $\Pi_i$, and $\epsilon_i=A_iL_0^{-2/m_i}$ are chosen in its proof. — pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair - clarification (`ass:pin-fix-8429b12c`, used by `thm:deep-nn`): The latent generator class in the statement uses the same temperature $\tau$ as the displayed error estimate. — pins reading of thm:deep-nn: stated by the repair - clarification (`ass:pin-fix-945d85ed`, used by `thm:deep-nn`): The error estimate is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, for fixed $R$ and fixed causal model. — pins reading of thm:deep-nn: stated by the repair - clarification (`ass:pin-fix-bbf1c78a`, used by `thm:deep-nn`): In Corollary~\ref{corollary:deep_nn}, for $m_i=0$, the symbol $\hat f_i$ denotes the exact constant function $f_i$. — pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair - clarification (`ass:pin-204`, used by `prop:counterexample`): The second Bernoulli law in the construction has $P(U_2=0)=2/3$. — pins reading of prop:counterexample-foral: The displayed value $P(U_2=1)=1/3$ and the joint masses for $X=0,T=0$ in the proof of Proposition~\ref{prop:counterexample_foral} both determine this entry. - clarification (`ass:pin-187`, used by `thm:consistency`): The bound $K$ applies throughout the optimization class to the observed variables $V_i$ and indexed non-Gumbel latent variables $U_j$, with $n_i\leqslant K$ for categorical nodes and admissible intervention coordinates in $[-K,K]$; it does not bound the separately supplied standard Gumbel noises. — pins reading of ass:bk-owner-1: The boundedness assumption explicitly indexes $V_i$ and $U_j$, whereas the categorical convention separately adjoins $G_{V_i}$. The binding boundedness symbols preserve this distinction. - clarification (`ass:pin-196`, used by `thm:consistency`): Truncation means coordinatewise clipping of the final output to $[-K,K]$; in $\mathcal{NN}^{\infty,K}$ no finite Lipschitz bound is imposed. — pins reading of def:truncated-neural-class: The defining minimum and maximum and the word ``Similarly'' specify the operation; the binding neural-class symbol conventions fix its application to the generator output. - clarification (`ass:pin-197`, used by `thm:consistency`): $\underline{F}^{L}$ denotes the constrained infimum of the displayed scalar objective over the stated canonical SCM class and its admissible latent laws, rather than a minimizing SCM. — pins reading of def:eq-consistent-limiting: The sentence introducing the display explicitly says ``optimal value'', and the following sentence calls it the sharp lower bound when the Lipschitz bounds agree. - clarification (`ass:pin-201`, used by `thm:consistency`): The coordinates $P(U_j)$ in $\theta_{\mathcal M}$ are the population laws of the non-Gumbel latent variables, not necessarily uniform laws; the independent standard Gumbel laws are fixed and omitted from this population tuple. — pins reading of def:thm-consistency-model-parameterization: Immediately after the tuple, the population parameter space contains factors $\mathcal P([-K,K]^{d_j^U})$. The canonical representation supplies the categorical Gumbel laws separately. - clarification (`ass:pin-207`, used by `thm:consistency`): The boundedness condition means $\\|f(x)\\|_{\infty}\leqslant K$ for every $f\in\mathcal{F}$ and $x\in\mathbb{R}^{d'}$. — pins reading of ass:lemma-sup-wasserstein-bounded: The proof of Lemma~\ref{lemma:sup_wasserstein} places the outputs in $[-K,K]^d$, consistently with the supplied function-class interpretation. - clarification (`ass:pin-210`, used by `thm:consistency`): $F_n=\inf\left\{\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\hat{\mathcal{M}}}[F(V_1(t),\ldots,V_{n_V}(t))]:\hat{\mathcal{M}}\in\text{NCM}_{\mathcal{G}}(\mathcal{F}_{0,n},\mathcal{F}_{1,n}),\ S_{\lambda_n}(P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V}),P_n^{\mathcal{M}^*}(\boldsymbol{V}))\leqslant\alpha_n\right\}$. — pins reading of def:eq-opt-empiri: The prose introducing equation (\ref{eq:opt_empiri}) calls $F_n$ an estimator of optimal values, and the following paragraph asks whether it is a good lower bound. The supplied symbol convention likewise specifies a scalar infimum. - clarification (`ass:pin-fix-09991de8`, used by `thm:consistency`): The observational empirical law $P_n^{\mathcal M^*}(\boldsymbol V)$ is formed from $n$ independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. — pins reading of thm:consistency: stated by the repair - clarification (`ass:pin-fix-0a7b21b0`, used by `thm:consistency`): The empirical optimization ranges over network representations, including distinct representations with the same latent population laws. — pins reading of thm:consistency: stated by the repair - clarification (`ass:pin-fix-30413b86`, used by `thm:consistency`): The common simulation draws are independent of the observational sample. — pins reading of thm:consistency: stated by the repair - clarification (`ass:pin-fix-3b195d8d`, used by `thm:consistency`): The common exogenous draws are independent and identically distributed according to the specified product law of the exogenous inputs. — pins reading of thm:consistency: stated by the repair - clarification (`ass:pin-fix-448b19a0`, used by `thm:consistency`): Probabilities of uniform deviation events are evaluated in the completion of the sample law. — pins reading of lem:sup-wasserstein: stated by the repair - clarification (`ass:pin-fix-59a01041`, used by `thm:consistency`): For each $n$, every candidate is evaluated on the same $m_n$ draws of all its exogenous inputs, including its uniform inputs, generator Gumbel inputs, and categorical Gumbel inputs. — pins reading of thm:consistency: stated by the repair - clarification (`ass:pin-fix-676b96c3`, used by `thm:consistency`): The finite component counts and component cube dimensions of the latent generator architectures in $\mathcal F_{1,n}$ are fixed as $n$ varies. — pins reading of thm:consistency: stated by the repair - clarification (`ass:pin-fix-761562d5`, used by `thm:consistency`): The self terms in $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$ use the same $1$-norm ground distance on the corresponding support points. — pins reading of lem:sinkhorn-approx: stated by the repair - clarification (`ass:pin-fix-84ccd033`, used by `thm:consistency`): $F_n$ denotes the infimum of objective values, with an attained minimum having the same value. — pins reading of thm:consistency: stated by the repair - clarification (`ass:pin-fix-9463b114`, used by `thm:consistency`): For a zero categorical propensity, $\log0=-\infty$. — pins reading of thm:consistency: stated by the repair - clarification (`ass:pin-fix-9675bba5`, used by `thm:consistency`): Ties in a categorical argmax are resolved by the smallest label. — pins reading of thm:consistency: stated by the repair - clarification (`ass:pin-fix-9d74053b`, used by `thm:consistency`): The orders for $\epsilon_n$ and $s_n$ are asymptotic as $n\rightarrow\infty$. — pins reading of thm:consistency: stated by the repair - clarification (`ass:pin-fix-bd620122`, used by `thm:consistency`): The architectural $\Theta$ orders use sufficiently large fixed constant factors supplied by the network constructions in the proof. — pins reading of thm:consistency: stated by the repair - clarification (`ass:pin-fix-d19c63b5`, used by `thm:consistency`): The probability bound is for each fixed pair $\delta,\epsilon>0$. — pins reading of lem:sup-wasserstein: stated by the repair - clarification (`ass:pin-fix-f66ad243`, used by `thm:consistency`): The depth expression $\Theta(\log d_{\max}^{\text{in}})$ means $\Theta(1+\log d_{\max}^{\text{in}})$ at $d_{\max}^{\text{in}}=1$. — pins reading of thm:consistency: stated by the repair - clarification (`ass:pin-154`, used by `prop:lip-ate`): For each of $\mathcal{M}$ and $\hat{\mathcal{M}}$, $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely, where the indexed $U_j$ are the non-Gumbel latent variables. — pins reading of ass:prop-lip-ate-bounded: Proposition \ref{prop:Lip_ATE} explicitly inherits Assumption \ref{assump:Lip_and_bounded}, which prints this maximum; Assumption \ref{assump:categorical_rv} distinguishes the indexed $U_j$ from the categorical noises $G_{V_i}$. - clarification (`ass:pin-206`, used by `prop:lip-ate`): The functions $f,\hat{f}:[-K,K]\rightarrow\mathbb{R}$ are $L$-Lipschitz regression versions satisfying $f(T)=\mathbb{E}_{\nu}[Y\mid T]$ almost surely under $\nu$ and $\hat{f}(\hat{T})=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}]$ almost surely under $\mu$. — pins reading of ass:lemma-lip-condition-regression: The proof of Lemma~\ref{lemma:Lip_condition} uses both cross-evaluations and bounds its test function on the common box; the owner-supplied support condition specifies that box. - clarification (`ass:pin-fix-19bccf89`, used by `prop:lip-ate`): The measure $P$ is the reference probability measure on the treatment space: for $X=\text{Pa}(T)$, the conditional density notation means $\nu(dt\|X=x)=p_\nu(t\|X=x)P(dt)$. — pins reading of prop:lip-ate: stated by the repair - clarification (`ass:pin-185`, used by `thm:rate`): The continuous structural functions and categorical propensity functions of $\mathcal M^*$ have Lipschitz constants at most $L_f$ in the paper's $1$-norm. — pins reading of ass:thm-consistency-structural-lipschitz: Theorem~\ref{thm:consistency} begins with $\mathcal M^*$ and permits candidate networks with bound $\hat L_f$; Assumption~\ref{assump:Lip_and_bounded} identifies the functions bounded by $L_f$. - clarification (`ass:pin-188`, used by `thm:rate`): The variable bound inherited from Assumption~\ref{assump:Lip_and_bounded} uses $\\|\cdot\\|_\infty$. — pins reading of ass:bk-owner-2: The owner condition extends the existing bound $K$, which Assumption~\ref{assump:Lip_and_bounded} explicitly states using infinity norms of observed and non-Gumbel latent variables. - clarification (`ass:pin-198`, used by `thm:rate`): $\mu_T=\text{Unif}([t_1,t_2])$ is the deterministic intervention probability measure. — pins reading of def:cor-rate-objective: Corollary~\ref{cor:rate} specializes the intervention measure used in the outer expectation $\mathbb E_{t\sim\mu_T}$ of Theorem~\ref{thm:consistency}. - clarification (`ass:pin-205`, used by `thm:rate`): The measure $P$ is a probability measure on the treatment space, and for almost every parent value $x$, $\nu(dt\mid\text{Pa}(T)=x)=p_\nu(t\mid\text{Pa}(T)=x)P(dt)$, with the stated density version satisfying $p_\nu(t\mid\text{Pa}(T)=x)\geqslant\delta$ for every $t\in[t_1,t_2]$. — pins reading of ass:prop-lip-ate-overlap: The overlap clause of Proposition~\ref{prop:Lip_ATE} names a conditional treatment density, and its displayed conclusion integrates over treatment values against $P(dt)$; the supplied meanings of $P$ and $p_\nu$ agree with this interpretation. - clarification (`ass:pin-fix-91573abf`, used by `thm:rate`): The observational empirical law $P_n^{\mathcal M^*}(\boldsymbol V)$ is formed from $n$ independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. — pins reading of thm:rate: stated by the repair - clarification (`ass:pin-fix-c0174608`, used by `prop:equivalent-g-constrianed`): The categorical generation formula requires $f_i(x)\in[0,\infty)^{n_i}$ and $\\|f_i(x)\\|_1>0$ on its structural input domain. — pins reading of prop:equivalent-g-constrianed: stated by the repair - clarification (`ass:pin-fix-c356d336`, used by `prop:equivalent-g-constrianed`): A zero categorical coordinate has extended logarithm $\log0=-\infty$. — pins reading of prop:equivalent-g-constrianed: stated by the repair - clarification (`ass:pin-159`, used by `prop:convg-opt`): Equation~\ref{eq:aprox_lip} holds for every $n\in\mathbb N$ and every $\theta_1,\theta_2\in\Theta_\infty$, with the same finite constant $L_g$. — pins reading of ass:prop-convg-opt-approximate-lipschitz: Proposition~\ref{prop:convg_opt} defines the functions on $\Theta_\infty$ and uses this inequality to conclude that the limiting function is $L_g$-Lipschitz. - clarification (`ass:pin-172`, used by `prop:convg-opt`): There exists a compact $\tilde\Theta_\infty\subseteq\Theta_\infty$ such that $\forall n\in\mathbb N\;\forall\theta_0\in\tilde\Theta_\infty\;\exists\theta\in\Theta_n$, $d(\theta,\theta_0)\leqslant\epsilon_n$. — pins reading of ass:prop-convg-opt-approximation: The proof of Proposition \ref{prop:convg_opt} applies condition 3 to construct a sequence $\theta_n$ approximating $\tilde\theta^*$. - clarification (`ass:pin-52`, used by `prop:convg-opt`): In (\ref{eq:aprox_lip}), $\Theta=\Theta_\infty$. — pins reading of ass:prop-convg-opt-approximate-lipschitz: Proposition~\ref{prop:convg_opt} defines the functions on $\Theta_\infty$ and uses this inequality there to establish the limiting Lipschitz bound. - `ass:assump-categorical-rv-generation`: pinned reading: Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\\|f_i(x)\\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. (The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities.) - `ass:assump-lip-and-bounded-bound`: pinned reading: There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely under the SCM law. (Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation.) - `ass:assump-lip-and-bounded-categorical`: pinned reading: For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\\|f_i(x)-f_i(x')\\|\leqslant L_f\\|x-x'\\|$ on its structural input domain, using the paper's $1$-norm. (Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$.) - `ass:bk-owner-1`: pinned reading: The bound $K$ applies throughout the optimization class to the observed variables $V_i$ and indexed non-Gumbel latent variables $U_j$, with $n_i\leqslant K$ for categorical nodes and admissible intervention coordinates in $[-K,K]$; it does not bound the separately supplied standard Gumbel noises. (The boundedness assumption explicitly indexes $V_i$ and $U_j$, whereas the categorical convention separately adjoins $G_{V_i}$. The binding boundedness symbols preserve this distinction.) - `ass:bk-owner-2`: pinned reading: The variable bound inherited from Assumption~\ref{assump:Lip_and_bounded} uses $\\|\cdot\\|_\infty$. (The owner condition extends the existing bound $K$, which Assumption~\ref{assump:Lip_and_bounded} explicitly states using infinity norms of observed and non-Gumbel latent variables.) - `ass:lemma-lip-condition-regression`: pinned reading: The functions $f,\hat{f}:[-K,K]\rightarrow\mathbb{R}$ are $L$-Lipschitz regression versions satisfying $f(T)=\mathbb{E}_{\nu}[Y\mid T]$ almost surely under $\nu$ and $\hat{f}(\hat{T})=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}]$ almost surely under $\mu$. (The proof of Lemma~\ref{lemma:Lip_condition} uses both cross-evaluations and bounds its test function on the common box; the owner-supplied support condition specifies that box.) - `ass:lemma-sup-wasserstein-bounded`: pinned reading: The boundedness condition means $\\|f(x)\\|_{\infty}\leqslant K$ for every $f\in\mathcal{F}$ and $x\in\mathbb{R}^{d'}$. (The proof of Lemma~\ref{lemma:sup_wasserstein} places the outputs in $[-K,K]^d$, consistently with the supplied function-class interpretation.) - `ass:prop-convg-opt-approximate-lipschitz`: pinned reading: Equation~\ref{eq:aprox_lip} holds for every $n\in\mathbb N$ and every $\theta_1,\theta_2\in\Theta_\infty$, with the same finite constant $L_g$. (Proposition~\ref{prop:convg_opt} defines the functions on $\Theta_\infty$ and uses this inequality to conclude that the limiting function is $L_g$-Lipschitz.) - `ass:prop-convg-opt-approximation`: pinned reading: There exists a compact $\tilde\Theta_\infty\subseteq\Theta_\infty$ such that $\forall n\in\mathbb N\;\forall\theta_0\in\tilde\Theta_\infty\;\exists\theta\in\Theta_n$, $d(\theta,\theta_0)\leqslant\epsilon_n$. (The proof of Proposition \ref{prop:convg_opt} applies condition 3 to construct a sequence $\theta_n$ approximating $\tilde\theta^*$.) - `ass:prop-lip-ate-bounded`: pinned reading: For each of $\mathcal{M}$ and $\hat{\mathcal{M}}$, $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely, where the indexed $U_j$ are the non-Gumbel latent variables. (Proposition \ref{prop:Lip_ATE} explicitly inherits Assumption \ref{assump:Lip_and_bounded}, which prints this maximum; Assumption \ref{assump:categorical_rv} distinguishes the indexed $U_j$ from the categorical noises $G_{V_i}$.) - `ass:prop-lip-ate-overlap`: pinned reading: The measure $P$ is a probability measure on the treatment space, and for almost every parent value $x$, $\nu(dt\mid\text{Pa}(T)=x)=p_\nu(t\mid\text{Pa}(T)=x)P(dt)$, with the stated density version satisfying $p_\nu(t\mid\text{Pa}(T)=x)\geqslant\delta$ for every $t\in[t_1,t_2]$. (The overlap clause of Proposition~\ref{prop:Lip_ATE} names a conditional treatment density, and its displayed conclusion integrates over treatment values against $P(dt)$; the supplied meanings of $P$ and $p_\nu$ agree with this interpretation.) - `ass:relu-approximation-domain`: pinned reading: The phrase ``uniform cubes'' denotes the unit cube domains $[0,1]^m$ used for deterministic uniform function approximation; it imposes no uniform probability law on structural inputs. (The same paragraph displays approximation of functions on $[0,1]^{d_1}$ and separately says that similar arguments apply to any bounded cubes.) - `ass:thm-consistency-structural-lipschitz`: pinned reading: The continuous structural functions and categorical propensity functions of $\mathcal M^*$ have Lipschitz constants at most $L_f$ in the paper's $1$-norm. (Theorem~\ref{thm:consistency} begins with $\mathcal M^*$ and permits candidate networks with bound $\hat L_f$; Assumption~\ref{assump:Lip_and_bounded} identifies the functions bounded by $L_f$.) - `def:cor-rate-objective`: pinned reading: $\mu_T=\text{Unif}([t_1,t_2])$ is the deterministic intervention probability measure. (Corollary~\ref{cor:rate} specializes the intervention measure used in the outer expectation $\mathbb E_{t\sim\mu_T}$ of Theorem~\ref{thm:consistency}.) - `def:corollary-deep-nn-dimensions`: pinned reading: $d_{\max}^U=\max_{j\in[n_U]}d_j^U$ ranges over non-Gumbel latent variables, while $d_{\max}^{\mathrm{in}}=\max_i d_i^{\mathrm{in}}$ and $d_{\max}^{\mathrm{out}}=\max_i d_i^{\mathrm{out}}$, with $d_i^{\mathrm{out}}=n_i$ for categorical nodes. (The NCM construction separately approximates non-Gumbel latent distributions and the functions in the canonical structural equations; those functions output categorical propensity vectors. The binding dimension symbols specify these same meanings.) - `def:definition-2`: pinned reading: $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. (The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality.) - `def:eq-consistent-limiting`: pinned reading: $\underline{F}^{L}$ denotes the constrained infimum of the displayed scalar objective over the stated canonical SCM class and its admissible latent laws, rather than a minimizing SCM. (The sentence introducing the display explicitly says ``optimal value'', and the following sentence calls it the sharp lower bound when the Lipschitz bounds agree.) - `def:eq-opt-empiri`: pinned reading: $F_n=\inf\left\{\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\hat{\mathcal{M}}}[F(V_1(t),\ldots,V_{n_V}(t))]:\hat{\mathcal{M}}\in\text{NCM}_{\mathcal{G}}(\mathcal{F}_{0,n},\mathcal{F}_{1,n}),\ S_{\lambda_n}(P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V}),P_n^{\mathcal{M}^*}(\boldsymbol{V}))\leqslant\alpha_n\right\}$. (The prose introducing equation (\ref{eq:opt_empiri}) calls $F_n$ an estimator of optimal values, and the following paragraph asks whether it is a good lower bound. The supplied symbol convention likewise specifies a scalar infimum.) - `def:eq-structure-eq`: pinned reading: For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. (The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition.) - `def:eq-structure-equation`: pinned reading: In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. (The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction.) - `def:eq-structure-equation-hat`: pinned reading: For each categorical $V_i$, the noise in equation (\ref{eq:structure_equation_hat}) satisfies $g_k=g_k^{V_i}$ for $k\in[n_i]$. (The canonical representation in equation (\ref{eq:structure_equation}) assigns separate independent standard Gumbel coordinates to each categorical node; the approximated equation retains that categorical-generation rule.) - `def:gumbel-softmax-layer`: pinned reading: The layer uses $V=[\Vec{v}_1,\ldots,\Vec{v}_{N_C}]$ and the same Gumbel draws for all $\tau>0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. (The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit.) - `def:homeomorphisms`: pinned reading: $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. (The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes.) - `def:intervention`: pinned reading: Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). (The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations.) - `def:lemma-gumbel-err-softmax`: pinned reading: In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. (Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores.) - `def:neural-causal-class`: pinned reading: Membership in $\text{NCM}_{\mathcal G}(\mathcal F_0,\mathcal F_1)$ requires $\hat f_i\in\mathcal F_0$ for every observed node and $\hat g_j\in\mathcal F_1$ for every latent component, with the compatible input and output dimensions understood separately for each network. (The source paragraph explicitly omits input and output dimensions and describes separate structural approximants and latent generators in equation~\ref{eq:structure_equation_hat}.) - `def:relu-networks`: pinned reading: A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. (The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths.) - `def:thm-consistency-model-parameterization`: pinned reading: The coordinates $P(U_j)$ in $\theta_{\mathcal M}$ are the population laws of the non-Gumbel latent variables, not necessarily uniform laws; the independent standard Gumbel laws are fixed and omitted from this population tuple. (Immediately after the tuple, the population parameter space contains factors $\mathcal P([-K,K]^{d_j^U})$. The canonical representation supplies the categorical Gumbel laws separately.) - `def:truncated-neural-class`: pinned reading: Truncation means coordinatewise clipping of the final output to $[-K,K]$; in $\mathcal{NN}^{\infty,K}$ no finite Lipschitz bound is imposed. (The defining minimum and maximum and the word ``Similarly'' specify the operation; the binding neural-class symbol conventions fix its application to the generator output.) - `lem:combin-err`: pinned reading: For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. (Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula.) - `lem:gumbel-err`: pinned reading: The gap event is $E_\delta=\{\exists\,1\leqslant i0$ on their structural input domain, with $\log0=-\infty$; the constructed canonical categorical function is the normalized vector $f_i/\\|f_i\\|_1$ with coordinates regrouped. Its smoothness is that of this normalized vector: normalization preserves differentiability order and local Lipschitz regularity wherever the total weight is positive, and regrouping preserves any global Lipschitz regularity the normalized vector possesses. In Theorem~\ref{thm:approximation}: In $\\|f_i-\hat f_i\\|_\infty$, the structural uniform infinity norm is the supremum of the coordinatewise infinity norm over the common structural input domain; zero categorical coordinates use $\log0=-\infty$ in the canonical argmax construction. In Theorem~\ref{thm:consistency}: The proof treats $f_i$ as an element of $\mathcal{F}_{V_i}$ on $[-K,K]^{d_{i,\text{in}}^V}$ with bounded output cube $[-K,K]^{d_{i,\text{out}}^V}$, imposing $\\|f_i(x)\\|_1=1$ pointwise for categorical nodes; the population bounded Lipschitz class is read as $\mathcal F_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}, \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Corollary~\ref{corollary:deep_nn}: For the canonical SCM functions in (\ref{eq:structure_equation}), categorical $f_i$ are normalized: $(f_i)_k\geqslant0$ and $\\|f_i\\|_1=1$, so their values lie in $\Delta_{n_i}$ and their coordinates are the conditional category probabilities given $\text{Pa}(V_i),\boldsymbol{U}_{V_i}$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: For every continuous structural function $f_i$ of $\mathcal M^*$, $\sup_x\\|f_i(x)\\|_1\leqslant K$ on its full structural input domain, and every continuous candidate structural network $\hat f_i\in\mathcal F_{0,n}$ satisfies the same uniform output $1$-norm bound. — Theorem~\ref{thm:consistency} specifies $\sup_x\\|f_i(x)\\|_1\leqslant K$ on the full structural input domain for continuous functions of $\mathcal M^*$, imposes the same bound on continuous candidate networks, and retains this bound in $\mathcal F_L^K$; Corollary~\ref{cor:rate} assumes its hypotheses and uses its network classes. - reading of the symbol `F`: $F$ is functional notation whose tuple, scalar-objective, and push-forward-map meanings are distinguished by context. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:Lip_ATE}: In the SCM notation, $F=(f_1,\cdots,f_{n_V})$ is the tuple of functions in the structural equations. In Theorem~\ref{thm:consistency}: In (\ref{eq:opt_empiri}) and (\ref{eq:consistent_limiting}), $F:\mathbb{R}^{n_V}\to\mathbb{R}$ is the Lipschitz continuous causal objective evaluated on the intervened observed variables, rather than the tuple of structural functions. In Corollary~\ref{cor:rate}: The causal objective is the outcome coordinate projection $F(\boldsymbol{V})=Y$. In Lemma~\ref{lemma:push-forward_err}: $F:K\to\mathbb{R}^{d_2}$ is a measurable, $L$-Lipschitz continuous map, where $K\subset\mathbb{R}^{d_1}$ is compact. In Proposition~\ref{prop:equivalent_g_constrianed}: The SCM tuple uses $F=(f_1,\cdots,f_{n_V})$; in the proof's expression $\boldsymbol{V}=F(\boldsymbol{U})$, $F$ instead denotes the map obtained by recursively evaluating the structural equations, with its intervention version denoted $F_t$. — The SCM definition following (\ref{eq:structure_eq}) defines the tuple; (\ref{eq:opt_empiri}) and Theorem~\ref{thm:consistency} introduce the scalar objective; Corollary~\ref{cor:rate} specifies its coordinate projection; the proof of Proposition~\ref{prop:equivalent_g_constrianed} writes $\boldsymbol{V}=F(\boldsymbol{U})$; Lemma~\ref{lemma:push-forward_err} explicitly specifies the measurable-map domain and codomain. - reading of the symbol `P`: $P(X)$ denotes the probability law of $X$, and $P(A)$ denotes the probability of an event $A$ under the law in context. In Proposition~\ref{prop:Lip_ATE}: Bare $P$ is the reference probability measure on the treatment space used in $P(dt)$; $p_\nu(t\|\text{Pa}(T)=x)$ is the density of the conditional treatment law $\nu(dt\|x)$ with respect to this same measure. In Corollary~\ref{cor:rate}: Bare $P$ is the reference probability measure on the treatment space with respect to which $\nu(dt\|\text{Pa}(T)=x)=p_\nu(t\|\text{Pa}(T)=x)P(dt)$. The intervention measure satisfies $\mu_T\ll P$ and $\int_{t_1}^{t_2}(d\mu_T/dP(t))^2P(dt)<\infty$. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: Bare $P$ is an arbitrary Borel probability distribution on $X\times[0,1]$, where $X$ is standard Borel. The population squared-loss expectation is taken under this $P$, and the deviation probability is evaluated in the completion of $P^m$. — Lemma lem:sup-wasserstein-anthony-theorem-17-1 quantifies over Borel probability distributions on $X\times[0,1]$ and explicitly evaluates its deviation probability in the completion of the product law. - reading of the symbol `\mathcal{M}`: Bare $\mathcal{M}$ denotes an individual SCM tuple, whereas $\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})$ denotes the class of canonical SCMs with causal graph $\mathcal{G}$, the indicated latent-variable structure, and structural functions $f_i\in\mathcal{F}$ for $i\in[n_V]$. In Theorem~\ref{thm:consistency}: In the population optimization defining $\underline F^{\hat L_f}$, $\mathcal M(\mathcal G,\mathcal F_{\hat L_f}^K,\boldsymbol U)$ uses $\mathcal F_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$, with nonnegative normalized categorical propensity vectors; the latent-law range consists of independent non-Gumbel laws supported on $[-K,K]^{d_j^U}$ and fixed independent standard Gumbel laws for categorical variables. — Theorem~\ref{thm:consistency} specifies $\underline F^{\hat L_f}$ through (\ref{eq:consistent_limiting}) and explicitly retains the uniform output 1-norm bounded Lipschitz class. The canonical parameterization in its proof specifies the non-Gumbel marginal laws on their cubes. - reading of the symbol `\mathcal{G}`: $\mathcal G$ denotes the graph or function class specified by its local occurrence. In Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: $\mathcal{G}$ has no hidden confounder. In Proposition~\ref{prop:counterexample_foral}: $\mathcal{G}$ is the common backdoor causal graph of the constructed models, with no unobserved confounding. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}, Proposition~\ref{prop:equivalent_g_constrianed}, \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Corollary~\ref{corollary:deep_nn}: $\mathcal G$ is the causal ADMG on $\boldsymbol V$, retaining the structural parent arrows and adding a bidirected arrow between $V_i,V_j$ when correlated latent inputs affect the two nodes. In \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $\mathcal G$ is a nonempty class of functions $g:X\rightarrow[0,1]$ with finite pseudo-dimension $v$. The covering and packing numbers use evaluations at arbitrary $m$ points of $X$ in the averaged metric $d_1(u,w)=m^{-1}\sum_{i=1}^m\|u_i-w_i\|$. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $\mathcal G=\{g(t,\cdot):t\in T\}\ne\varnothing$, where $X,T$ are standard Borel spaces and $g:T\times X\rightarrow[0,1]$ is jointly Borel; no pseudo-dimension assumption is imposed. The covering number $\mathcal N_1^{\mathrm{avg}}(r,\mathcal G,2m)$ is the supremum over $2m$ points of $X$ of the covering number of the evaluation vectors in the averaged metric $(2m)^{-1}\sum_i\|u_i-v_i\|$. — Lemma lem:sup-wasserstein-anthony-theorem-17-1 specifies standard Borel spaces $X,T$ and a jointly Borel map $g:T\times X\rightarrow[0,1]$ defining the nonempty class. - reading of the symbol `\hat{\mathcal{F}}`: $\hat{\mathcal{F}}$ is the specified class containing the comparison model's structural functions on their respective node domains; categorical members are normalized propensity functions, and no neural architecture or Lipschitz bound is imposed merely by the hat. — The canonical-representation definition specifies membership of every structural function in its class and categorical normalization in equation~\ref{eq:structure_equation}. Theorem~\ref{thm:approximation} places the comparison model in $\mathcal{M}(\mathcal{G},\hat{\mathcal{F}},\hat{\boldsymbol{U}})$ without specifying a neural architecture or a Lipschitz bound for that class. - reading of the symbol `\hat{\mathcal{M}}`: $\hat{\mathcal{M}}$ is a model on the same causal graph $\mathcal{G}$ as the model with which it is compared; the hat alone does not denote a fitted estimator or an optimization minimizer. In Theorem~\ref{thm:approximation}: $\hat{\mathcal{M}}$ is any comparison model in $\mathcal{M}(\mathcal{G},\hat{\mathcal{F}},\hat{\boldsymbol{U}})$; it is not required by this notation to be a neural model. In Corollary~\ref{corollary:deep_nn}: $\hat{\mathcal{M}}$ is the constructed neural causal model with the displayed structural and latent-generator architectures and structure equations~\ref{eq:structure_equation_hat}, chosen after fixing $R>0$; it may depend on $R$ and the network budgets, but the same model applies to every intervention with $\\|t_i\\|_\infty\leqslant R$ for continuous treatment variables and $t_i\in[n_i]$ for categorical treatment variables. In Proposition~\ref{prop:Lip_ATE}: $\hat{\mathcal{M}}\in\mathcal{M}(\mathcal{G},\mathcal{F}_L,\boldsymbol{U})$ is the second comparison SCM, with observational law $\mu$. In Proposition~\ref{prop:equivalent_g_constrianed}: $\hat{\mathcal{M}}$ is the canonical representation obtained by merging the original non-Gumbel latent variables and retaining the categorical Gumbel noises; its continuous structural functions are the original $f_i$ under coordinate regrouping, and its categorical propensity functions are $f_i/\\|f_i\\|_1$ under the same regrouping, ignoring additional input coordinates, so its intervention laws equal those of $\mathcal{M}$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In equation~\ref{eq:opt_empiri}, $\hat{\mathcal M}$ ranges over network representations in $\text{NCM}_{\mathcal G}(\mathcal F_{0,n},\mathcal F_{1,n})$, with each categorical raw structural network $q$ satisfying $q(x)\in[0,\infty)^{n_i}$, $\\|q(x)\\|_1>0$, and $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$ on its full structural input domain. Different representations with the same latent population laws may have different common-draw empirical distributions. In Theorem~\ref{thm:consistency}: Every continuous structural network $\hat f_i$ additionally satisfies $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$ on its full structural input domain. In equation~\ref{eq:consistent_limiting}, $\hat{\mathcal M}$ instead ranges over canonical SCMs in $\mathcal M(\mathcal G,\mathcal F_L^K,\boldsymbol U)$ and all latent distributions $P(\boldsymbol U)$, with $\mathcal F_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$ and normalized categorical propensity functions. In Corollary~\ref{cor:rate}: Every continuous structural network $\hat f_i$ additionally satisfies $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$ on its full structural input domain. The population comparison models use the bounded Lipschitz structural class $\mathcal F_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$ and normalized categorical propensity functions. — Corollary~\ref{cor:rate} explicitly restricts every continuous structural network to the uniform output $1$-norm bound and uses the population structural classes specified in Theorem~\ref{thm:consistency}. - reading of the symbol `\hat{\boldsymbol{U}}`: $\hat{\boldsymbol{U}}$ denotes the full canonical latent collection of $\hat{\mathcal{M}}$, including categorical Gumbel noises, rather than only the non-Gumbel inputs to its structural functions. — The canonical-representation definition includes both component latents and categorical Gumbel vectors, and Theorem~\ref{thm:approximation} retains the full latent-law comparison. Proposition~\ref{prop:equivalent_g_constrianed} is no longer a listed result using this symbol. - reading of the symbol `\epsilon`: $\epsilon(\mathcal{F}_1,\mathcal{F}_2)$ denotes the directed approximation error $\sup_{f_2\in\mathcal{F}_2}\inf_{f_1\in\mathcal{F}_1}\\|f_2-f_1\\|$, whereas a bare $\epsilon>0$ denotes a local positive real tolerance. In Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:counterexample_foral}: $\epsilon$ is arbitrary and positive, and specifies the requested upper bound on $W(P^{\mathcal{M}^*}(\boldsymbol{V}),P^{\mathcal{M}_\epsilon}(\boldsymbol{V}))$. In Lemma~\ref{lemma:product_err}: In the proof, $\epsilon$ is the positive slack in each chosen coupling's transportation cost and is subsequently taken to zero. In Proposition~\ref{prop:convg_opt}: In the proof, $\epsilon$ is a positive gap witnessing a hypothetical subsequence with $f_{n_k}^*0$ without assigning it any further role. In the empirical concentration argument of the proof, a local deviation tolerance is chosen as $\epsilon=Cr_n$, together with $\delta=r_n$, for sufficiently large $n$. In Lemma~\ref{lemma:sup_wasserstein}: $\epsilon>0$ is the deviation tolerance and also the unnormalized covering radius when it appears in $\mathcal{N}_1(\epsilon,\mathcal{F},n)$; the probability bound is for each fixed pair $\delta,\epsilon>0$. In the proof, for $0<\epsilon<2dK$, the normalized squared-loss tolerance is $\eta=\epsilon/(2dK)$. In \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $0<\epsilon\leqslant1$ is the covering and packing radius in the averaged metric $d_1(u,w)=m^{-1}\sum_{i=1}^m\|u_i-w_i\|$. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $0<\epsilon<1$ is the deviation tolerance for the absolute difference between population and empirical squared loss; the covering radius is $\epsilon/16$ on $2m$ evaluation points. In Corollary~\ref{corollary:deep_nn}: In the scalar inequality $\|\max\{\epsilon,z\}-p\|\leqslant\max\{\epsilon,\|z-p\|\}$, $\epsilon\geqslant0$ is an arbitrary floor parameter; the inequality is then applied coordinatewise with $\epsilon=\epsilon_i$. In Theorem~\ref{thm:consistency}: In the empirical concentration argument, the local deviation tolerance is chosen as $\epsilon=A_0r_n$, together with $\delta=r_n$, for sufficiently large $n$. — The proof of Corollary~\ref{corollary:deep_nn} states its scalar floor inequality for $\epsilon\geqslant0$. The proofs of Theorem~\ref{thm:consistency} and Corollary~\ref{cor:rate} choose local deviation tolerances when applying Lemma~\ref{lemma:sup_wasserstein}. - reading of the symbol `\mathcal{M}_\epsilon`: $\mathcal{M}_\epsilon$ denotes an SCM chosen for each tolerance $\epsilon>0$ in the counterexample family, rather than a single model independent of $\epsilon$. In Proposition~\ref{prop:counterexample_foral}: $\mathcal{M}_\epsilon$ has the same causal graph as $\mathcal{M}^*$; in the proof it is supplied by $\mathcal{M}_\delta$ with $\delta>0$ small enough that $\delta/2\leqslant\epsilon$. — Flat.tex lines 775 and 2710 quantify the model after the tolerance; the construction at lines 2715--2770 uses $\mathcal{M}_\delta$ and establishes $W(P^{\mathcal{M}^*},P^{\mathcal{M}_\delta})\leqslant\delta/2$. - reading of the symbol `\mathcal{C}`: $\mathcal{C}$ denotes the locally specified collection, consisting of causal models in partial identification and cubes in the geometric construction. In Proposition~\ref{prop:holder_curve}: $\mathcal{C}=\{C_1,\ldots,C_m\}$ is a finite collection of axis-aligned cubes; $\mathcal{C}_n$ is the collection of closed cells in the level-$n$ partition of $[0,1]^d$, and $\mathcal{C}^{*}=\bigcup_{n=1}^{\infty}\mathcal{C}_n$. — The cube-metric definition at flat.tex lines 2536--2539 defines the geometric collection used in Proposition~\ref{prop:holder_curve}. - reading of the symbol `\theta`: $\theta$ denotes the causal functional when applied to an SCM in (\ref{prob:pid}), and a parameter point in the later optimization and parameterization notation. In Proposition~\ref{prop:convg_opt}: $\theta$ is a point of the metric space $(M,d)$, ranged over $\Theta_n$, $\Theta_\infty$, or $\tilde{\Theta}_\infty$ in the respective optimization problems. In Theorem~\ref{thm:consistency}: In the population causal-model parameterization, $\theta=(f_1,\ldots,f_{n_V},P(U_1),\ldots,P(U_{n_U}))\in M$, $\mathcal{M}^{\theta}$ denotes the represented SCM, and $P^{\theta}$ its population distribution; $\theta_{i,j}$ denotes the parameters of the corresponding latent-generator neural network. In the empirical optimization, a candidate index additionally specifies its network representation: different representations with the same latent population laws can give different $P_{m_n}^{\theta}$ on the common simulation inputs. In Corollary~\ref{cor:rate}: In the population causal-model parameterization, $\theta=(f_1,\ldots,f_{n_V},P(U_1),\ldots,P(U_{n_U}))\in M$, $\mathcal{M}^{\theta}$ denotes the represented SCM, and $P^{\theta}$ its population distribution; $\theta_{i,j}$ denotes the parameters of the corresponding latent-generator neural network. In the empirical optimization, a candidate index additionally specifies its network representation: different representations with the same latent population laws can give different $P_{m_n}^{\theta}$ on the common simulation inputs. — The proof of Corollary~\ref{cor:rate}, flat.tex lines 3028--3036, imports the parameter notation from the proof of Theorem~\ref{thm:consistency}. Its empirical optimization evaluates network representations on common simulation inputs, so the population tuple alone does not determine a candidate empirical law. - reading of the symbol `\hat{\boldsymbol{V}}`: $\hat{\boldsymbol{V}}=\{\hat{V}_i\}_{i=1}^{n_V}$ is the collection of observed variables of $\hat{\mathcal{M}}$ corresponding to $\boldsymbol{V}$, and $\hat{\boldsymbol{V}}(\boldsymbol{t})$ denotes this collection under intervention $\boldsymbol{T}=\boldsymbol{t}$. — Theorem~\ref{thm:approximation} at flat.tex lines 240--246 uses the intervened collection; the proof of Proposition~\ref{prop:equivalent_g_constrianed} at line 1796 explicitly writes $\hat{\boldsymbol{V}}(t)=F_t(\hat{\boldsymbol{U}})$. - reading of the symbol `\hat{V}_i`: $\hat{V}_i$ is the observed variable corresponding to $V_i$ in $\hat{\mathcal{M}}$, with $i\in[n_V]$ and the same continuous or categorical state space; $\hat{V}_i(\boldsymbol{t})$ denotes its value after replacing treatment equations by $\boldsymbol{T}=\boldsymbol{t}$. In Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}: The structural equations are (\ref{eq:structure_equation_hat}), with latent inputs $\hat{U}_{C_j}=\hat{g}_j(Z_{C_j})$ and structural functions $\hat{f}_i$. In Theorem~\ref{thm:approximation}, Theorem~\ref{thm:consistency}: For a zero categorical propensity coordinate, the categorical argmax uses the extended logarithm $\log0=-\infty$. In Theorem~\ref{thm:consistency}: For a categorical node, the raw structural network $q$ has nonnegative coordinates and $\\|q(x)\\|_1>0$ at every structural input, and its normalized propensity vector is $q(x)/\\|q(x)\\|_1$; ties in the categorical argmax are resolved by the smallest label. — Equation~\ref{eq:structure_equation_hat} supplies the generated latent inputs. Theorem~\ref{thm:approximation} specifies the extended logarithm for zero categorical propensities; Theorem~\ref{thm:consistency} additionally specifies normalized raw structural networks and the smallest-label tie convention. - reading of the symbol `C`: The meaning of $C$ is determined by its local definition: a graph component, a geometric set, a numerical constant, or the complete categorical label vector. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}, Proposition~\ref{prop:equivalent_g_constrianed}: In canonical latent indexing, $C\subset\boldsymbol{V}$ is maximal subject to every pair of distinct vertices in $C$ being joined by a bi-directed arrow in $\mathcal{G}$. In Theorem~\ref{thm:wide_nn}, Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the distribution-generator construction, $C_i$ denotes a connected component of $\operatorname{supp}(\mathbb{P})$, with $\operatorname{supp}(\mathbb{P})=\bigcup_{i=1}^{N_C}C_i$ and $\mathcal{H}([0,1]^{d_i^C},C_i)\cap\mathcal{F}_L([0,1]^{d_i^C},C_i)\neq\emptyset$. In Proposition~\ref{prop:holder_curve}: In the partition construction, $C=\prod_{i=1}^{n}[a_i,b_i]$ is an axis-aligned cube; indexed sets $C_{i_1,\ldots,i_n}$ are the closed cells of the constructed partitions of $[0,1]^d$. In Lemma~\ref{lemma:sup_wasserstein}: $C>0$ is a deterministic numerical constant depending only on $d,K$ in the probability bound. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In numerical probability and tolerance bounds, $C>0$ denotes a deterministic constant rather than a set of observed vertices. In Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:counterexample_foral}: In the lower-bound premise for each non-Gumbel latent law $\mathbb P$, $C_i$ denotes a connected component of its support, and the assumption is applied with $K=C_i$ to the normalized component law $\mathbb P\|_{C_i}/\mathbb P(C_i)$ wherever this normalization is defined. In Theorem~\ref{thm:consistency}, \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: In the latent empirical concentration argument, $C$ denotes the complete categorical label vector, with values $c\in\prod_{i:V_i\text{ is categorical}}[n_i]$; $p_c=P(C=c)$, $N_c$ is its observational sample count, and $\mu_c=P((U_1,\ldots,U_{n_U})\mid C=c)$ for $p_c>0$. — The canonical representation defines graph-component indexing, and the partition preparation defines geometric cubes. In the Fournier--Guillin invocation in the proof of Theorem~\ref{thm:consistency} and in lem:consistency-fournier-theorem-2, $C$ is the complete categorical label vector, with $p_c=P(C=c)$ and $\mu_c=P((U_1,\ldots,U_{n_U})\mid C=c)$ for $p_c>0$. - reading of the symbol `U_C`: $U_C$ denotes the canonical latent vector indexed by a $C^2$-component $C$ of $\mathcal{G}$, and $\boldsymbol{U}_{V_i}=\{U_C:V_i\in C,\ C\text{ is a }C^2\text{-component of }\mathcal{G}\}$. In Corollary~\ref{corollary:deep_nn}: For every $C^2$-component $C$, the law $P(U_C)$ satisfies Assumption~\ref{assump:support}; for each connected support component $C_i$ of positive mass, the normalized law $\mu_i=P(U_C)\|_{C_i}/P(U_C\in C_i)$ satisfies Assumption~\ref{assumption:lower_bound} with $K=C_i$, and its support chart $f$ satisfies $(f^{-1})_{\#}\mu_i(\partial[0,1]^{d_i^C})=0$ when $d_i^C>0$. — Corollary~\ref{corollary:deep_nn} explicitly requires support charts, the lower-bound assumption on normalized component laws, and zero mass on positive-dimensional chart boundaries. - reading of the symbol `G_{V_i}`: $G_{V_i}=(g_1^{V_i},\ldots,g_{n_i}^{V_i})$ is present for categorical $V_i$ and consists of standard Gumbel coordinates furnishing that node's categorical-generation noise, separately from the latent inputs $\boldsymbol{U}_{V_i}$ of its propensity function. — The paragraph preceding Assumption~\ref{assump:categorical_rv} distinguishes shared propensity inputs from categorical-generation noise; the assumption defines $G_{V_i}$ and restricts $\boldsymbol{U}_{V_i}$ to the other latent variables. The NCM and canonical definitions retain the same vector. - reading of the symbol `g_k^{V_i}`: For categorical $V_i$ and $k\in[n_i]$, $g_k^{V_i}$ is the $k$th standard Gumbel coordinate of $G_{V_i}$, used in $\arg\max_{k\in[n_i]}\{g_k^{V_i}+\log(f_i(\text{Pa}(V_i),\boldsymbol{U}_{V_i}))_k\}$. — Assumption~\ref{assump:categorical_rv} defines the coordinate vector and its categorical-generation role; the canonical representation repeats it in equation~\ref{eq:structure_equation}. - reading of the symbol `G_k`: $G_k$ is the $k$th standard Gumbel coordinate, with density $x\mapsto\exp(-x-\exp(-x))$ on $\mathbb{R}$, in the relevant Gumbel-softmax vector. In Lemma~\ref{lemma:gumbel_err}: The coordinate index is $k\in[n]$. In Theorem~\ref{thm:wide_nn}: $G_k$, $k\in[N_C]$, are the independent standard Gumbel coordinates of the mixing layer, one per connected support component, independent of the uniform inputs used by the deterministic blocks; their randomness is included in $\hat g_{\#}\lambda$. In the result split from Theorem~\ref{thm:wide_nn}: $G_k$, $k\in[N_C]$, are the independent standard Gumbel coordinates of the mixing layer, one per connected support component, independent of the uniform inputs used by the deterministic blocks; their randomness is included in $\hat g_{\#}\lambda$. — The Gumbel-softmax definition at flat.tex lines 722--724 assigns one standard Gumbel coordinate to each support component. The binding law convention for Theorem~\ref{thm:wide_nn} and the proofs in discovery/core.json for Theorem~\ref{thm:wide_nn} and thm:cor-fix-3c402872 include the mixing randomness and make the mixing weights independent of all block outputs. - reading of the symbol `\hat{f}_i`: For $i\in[n_V]$, $\hat{f}_i$ is the comparison model's structural map for a continuous node and its categorical propensity or weight map for a categorical node, evaluated on parent values and the corresponding latent coordinates. In Theorem~\ref{thm:approximation}: The functions are those of $\hat{\mathcal{M}}\in\mathcal{M}(\mathcal{G},\hat{\mathcal{F}},\hat{\boldsymbol{U}})$; categorical structural propensities are read under the canonical representation, including its normalization convention and $\log0=-\infty$ for zero coordinates. The quantity $\\|f_i-\hat f_i\\|_\infty$ takes the supremum of the coordinatewise infinity norm over the common structural input domain. In Corollary~\ref{corollary:deep_nn}, \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Corollary~\ref{corollary:deep_nn}: In the construction, let $q_i$ be a minimizer of $\\|q_i-f_i\\|_\infty$ over $\mathcal{NN}_{d_i^{\text{in}},d_i^{\text{out}}}(d_i^{\text{out}}(2d_i^{\text{in}}+10),L_0)$. The chosen network is $\hat{f}_i=q_i$ for a continuous node and $\hat{f}_i=\max\{0,q_i\}$ coordinatewise for a categorical node; the latter's normalized propensity is $\hat{f}_i/\\|\hat{f}_i\\|_1$ wherever the denominator is nonzero. In Theorem~\ref{thm:consistency}: Continuous structural networks range over $\mathcal{F}_{0,n}$ subject additionally to $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$ on the full structural input domain. For a categorical node, the raw network $q\in\mathcal{F}_{0,n}$ satisfies $q(x)\in[0,\infty)^{n_i}$ and $\\|q(x)\\|_1>0$ for every structural input $x$, and its normalized propensity satisfies $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$; the proof's canonical structural-function coordinate is $q/\\|q\\|_1$. In Corollary~\ref{cor:rate}: Continuous structural networks range over $\mathcal F_{0,n}$ subject additionally to $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$ on the full structural input domain. For a categorical node, the raw network $q\in\mathcal F_{0,n}$ satisfies $q(x)\in[0,\infty)^{n_i}$ and $\\|q(x)\\|_1>0$ for every structural input $x$, and its normalized propensity satisfies $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$. — Corollary~\ref{cor:rate} uses the network and population structural classes specified in Theorem~\ref{thm:consistency} and explicitly imposes the continuous output bound and categorical normalized-propensity restrictions. - reading of the symbol `f`: $f$ denotes a locally specified function, with domain, codomain and regularity supplied by its occurrence, rather than one common map shared across the paper. In Proposition~\ref{prop:convg_opt}: $f:\Theta_\infty\rightarrow\mathbb{R}$ is the continuous objective minimized in all three optimization problems. In Lemma~\ref{lemma:Lip_condition}: $f(t)=\mathbb{E}_{\nu}[Y\mid T=t]$ is the population regression function, taken in the $L$-Lipschitz version specified by the hypothesis, with $\|f(t)\|\leqslant K$. In Lemma~\ref{lemma:combin_err}: In the proof, $f:\mathbb{R}^d\rightarrow\mathbb{R}$ ranges over $1$-Lipschitz test functions in the Wasserstein dual formulation. In Lemma~\ref{lemma:sinkhorn_approx}: In the proof, $f(x)=-x\log x$ is the concave scalar function used on transportation-plan entries in $[0,1]$, with its continuous value $f(0)=0$. In Theorem~\ref{thm:approximation}: In the categorical step of the proof, the unindexed $f$ denotes the structural propensity function at the node currently being compared, and $p=f(\text{pa}(v_k),\boldsymbol{u}_{V_k})$. In Lemma~\ref{lemma:sup_wasserstein}: $f$ ranges over the fixed class $\mathcal F$ of measurable maps $\mathbb R^{d'}\rightarrow\mathbb R^d$, with $\\|f(x)\\|_\infty\leqslant K$ for every $x$; each scalar coordinate class $\mathcal F_j=\{x\mapsto f_j(x):f\in\mathcal F\}$, $j\in[d]$, has pseudo-dimension less than $d_{\mathcal F}<\infty$. There are a standard Borel space $T$ and a jointly Borel map $G:T\times\mathbb R^{d'}\rightarrow[-K,K]^d$ such that $\mathcal F=\{G(t,\cdot):t\in T\}$. Probabilities of uniform deviation events are evaluated in the completion of the sample law. In the result split from Theorem~\ref{thm:wide_nn}: In the lower-bound assumption, $f\in\mathcal{H}([0,1]^d,K)\cap\mathcal{F}_L([0,1]^d,K)$ is a Lipschitz homeomorphism satisfying $\mathbb{P}(f(B))\geqslant C_f\lambda(B)$ for measurable $B\subset[0,1]^d$; when applied componentwise, the support and measure are those of the component. In Proposition~\ref{prop:holder_curve}, Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the lower-bound assumption, $f\in\mathcal{H}([0,1]^d,K)\cap\mathcal{F}_L([0,1]^d,K)$ is a Lipschitz homeomorphism satisfying $\mathbb{P}(f(B))\geqslant C_f\lambda(B)$ for measurable $B\subset[0,1]^d$; when applied componentwise, the support and measure are those of the component. In Theorem~\ref{thm:consistency}: In the proof's optimization parameterization, $f(\theta)=\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\mathcal{M}^{\theta}}[F(V_1(t),\ldots,V_{n_V}(t))]$ is the population intervention objective on the model parameter space, distinct from the structural functions $f_i$ and the homeomorphism in the lower-bound assumption. In Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:counterexample_foral}: For each non-Gumbel latent law $\mathbb P$ and each connected support component $C_i$ of positive mass, the lower-bound assumption uses a Lipschitz homeomorphism $f\in\mathcal H([0,1]^d,C_i)\cap\mathcal F_L([0,1]^d,C_i)$ satisfying $\mathbb P(f(B))/\mathbb P(C_i)\geqslant C_f\lambda(B)$ for every measurable $B\subset[0,1]^d$. — Lemma~\ref{lemma:sup_wasserstein} specifies a standard Borel parameter space, a jointly Borel parameterization of $\mathcal F$, and evaluation of uniform deviation probabilities in the completion of the sample law. - reading of the symbol `\hat{f}`: $\hat{f}$ is a function on the domain of its paired function $f$, with the particular function and codomain specified by the local comparison. In Lemma~\ref{lemma:Lip_condition}: $\hat{f}(t)=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}=t]$ is a population regression version, $L$-Lipschitz continuous and bounded by $K$, defined on the common treatment domain on which it is compared with $f$. — Lemma~\ref{lemma:Lip_condition} still defines $\hat{f}$ as the population regression. Corollary~\ref{corollary:deep_nn} is no longer among the results using this symbol. - reading of the symbol `\boldsymbol{t}`: $\boldsymbol{t}=(t_i)_{T_i\in\boldsymbol{T}}$ assigns treatment values to the chosen set $\boldsymbol{T}\subset\boldsymbol{V}$ by replacing each treatment structural equation with $T_i=t_i$. In Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}: For the displayed ATE contrast, $\boldsymbol{T}=\{T\}$ and the assigned treatment value is 1. In Proposition~\ref{prop:Lip_ATE}: The intervention is on the single treatment $T$, and its value $t$ ranges over $[t_1,t_2]$ in the displayed integral. In Corollary~\ref{corollary:deep_nn}: For the fixed $R>0$ chosen before $\hat{\mathcal M}$, each continuous treatment coordinate satisfies $\\|t_i\\|_\infty\leqslant R$, and each categorical treatment coordinate belongs to its declared set $[n_i]$. — Corollary~\ref{corollary:deep_nn} fixes $R>0$ before choosing $\hat{\mathcal M}$ and restricts continuous intervention coordinates by $\\|t_i\\|_\infty\leqslant R$; its binding reading specifies categorical values in $[n_i]$. - reading of the symbol `\boldsymbol{t}_0`: $\boldsymbol{t}_0$ assigns the reference values to the same treatment set $\boldsymbol{T}$ as $\boldsymbol{t}$ in $\mathbb{E}_{\boldsymbol{u}\sim P(\boldsymbol{U})}[Y(\boldsymbol{t})-Y(\boldsymbol{t}_0)]$. In Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}: The reference intervention is $T=0$. — The Average Treatment Effect definition uses the same $\boldsymbol{T}$ for both interventions. The proof of Proposition~\ref{prop:counterexample_foral} calculates the binary-treatment contrast used also in Proposition~\ref{prop:counterexample}. - reading of the symbol `T`: $T$ is locally specified as a treatment variable, a transportation map, a transportation plan, or a standard Borel parameter space. In Lemma~\ref{lemma:Lip_condition}: $T$ is the treatment coordinate of $(T,Y)\sim\nu$; conditioning in $f(t)$ is on $T=t$. In Proposition~\ref{prop:counterexample_foral}: $T$ is the binary treatment in the proof, taking values $0$ and $1$, with the conditional probabilities specified by the displayed structural equations. In Lemma~\ref{lemma:gumbel_err}: $T:\Delta_n\rightarrow\Delta_n$ sends $x$ to the one-hot vector at the first coordinate attaining $\max_j x_j$; it is a transportation map, rather than a treatment variable. In Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: $T$ denotes the treatment node, and $Y(t)$ denotes the outcome under the fixed intervention $T=t$. In Lemma~\ref{lemma:sinkhorn_approx}: $T=(T_{ij})\in\Pi(\mu,\nu)$ is a nonnegative $n\times m$ transportation plan satisfying $\sum_jT_{ij}=\mu_i$ and $\sum_iT_{ij}=\nu_j$. The minimizations defining the transport costs range over $\Pi(\mu,\nu)$; the entropy maximum displayed in the proof ranges over plans attaining the unregularized optimal cost $W(\mu,\nu)$. In Lemma~\ref{lemma:sup_wasserstein}: $T$ is a standard Borel parameter space, and $G:T\times\mathbb R^{d'}\rightarrow[-K,K]^d$ is jointly Borel with $\mathcal F=\{G(t,\cdot):t\in T\}$. It is not a treatment variable. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: In the cited squared-loss formulation, $T$ is a standard Borel parameter space for a jointly Borel map $g:T\times X\rightarrow[0,1]$, with $\mathcal G=\{g(t,\cdot):t\in T\}\ne\varnothing$. — Lemma~\ref{lemma:sinkhorn_approx} defines $\Pi(\mu,\nu)$ explicitly; the entropy maximum displayed in its proof is restricted to transportation plans attaining $W(\mu,\nu)$. - reading of the symbol `Y`: $Y$ denotes the random quantity specified by the local model or joint law, with the outcome and coupling uses distinguished below. In Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: $Y$ is the scalar outcome node of the SCM, and $Y(t)$ is its value after replacing the treatment equation by $T=t$. In Lemma~\ref{lemma:Lip_condition}: $Y$ is the second coordinate of $(T,Y)\sim\nu$, and $f(t)$ is a version of $\mathbb{E}_{\nu}[Y\mid T=t]$. In Corollary~\ref{cor:rate}: The objective selects the outcome coordinate, $F(\boldsymbol{V})=Y$. In Lemma~\ref{lemma:product_err}: In the proof, $Y=(Y_1,\ldots,Y_n)$ is the second product-coupled vector; $(X_k,Y_k)\sim\pi_k$, where $\pi_k$ couples $\mu_k$ and $\hat{\mu}_k$, so $Y$ has law $\hat{\mu}$. — The ATE definition uses the intervened outcome $Y(\boldsymbol{t})$; Lemma~\ref{lemma:Lip_condition} specifies $(T,Y)\sim\nu$; Corollary~\ref{cor:rate} sets $F(\boldsymbol{V})=Y$; the proof of Lemma~\ref{lemma:product_err} uses $(X_k,Y_k)\sim\pi_k$ and the product-coupled vectors $X,Y$. - reading of the symbol `\mu`: $\mu$ denotes a measure on the domain specified by its local use, rather than a single source measure on a globally fixed Euclidean space. In Lemma~\ref{lemma:push-forward_err}: $\mu$ is a probability measure supported on the compact set $K\subset\mathbb{R}^{d_1}$. In Lemma~\ref{lemma:product_err}: $\mu=\mu_1\otimes\cdots\otimes\mu_n$ is the product of the stated component measures. In Lemma~\ref{lemma:Lip_condition}: $\mu$ is the joint law of $(\hat{T},\hat{Y})$, and $\hat{f}(t)$ is the conditional expectation under this law. In Proposition~\ref{prop:Lip_ATE}: $\mu$ is the observational distribution of $\hat{\mathcal{M}}$. In Lemma~\ref{lemma:sinkhorn_approx}: $\mu\in\Delta_n$ is the weight vector of a finite probability measure on the supplied points $x_1,\ldots,x_n$ in the common Euclidean ground space. In Lemma~\ref{lemma:sup_wasserstein}: $\mu$ is the population probability measure on $\mathbb{R}^{d'}$ whose empirical distribution is $\mu_n$. — Lemma~\ref{lemma:sup_wasserstein} specifies the population probability measure $\mu$ on $\mathbb R^{d'}$. Lemma lem:sup-wasserstein-anthony-theorems now states a covering and packing bound on function evaluations and specifies neither $\mu$ nor a sampling distribution $P$. - reading of the symbol `\hat{T}`: $\hat{T}$ is the real-valued treatment coordinate of $(\hat{T},\hat{Y})\sim\mu$, and $\hat{f}(t)=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}=t]$. In Lemma~\ref{lemma:Lip_condition}: The proof treats $(\hat{T},\hat{Y})$ as supported in $\{(t,y):\\|(t,y)\\|_\infty\leqslant K\}$. — Lemma~\ref{lemma:Lip_condition} specifies the joint law and conditional regression; its proof evaluates a Lipschitz test function on a bounded joint support. - reading of the symbol `\hat{Y}`: $\hat{Y}$ denotes the real-valued outcome in the comparison distribution or comparison causal model. In Lemma~\ref{lemma:Lip_condition}: $(\hat{T},\hat{Y})\sim\mu$, with $\hat{f}(t)=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}=t]$; the proof treats this pair as supported in $\{(t,y):\\|(t,y)\\|_\infty\leqslant K\}$. In Proposition~\ref{prop:Lip_ATE}: $\hat{Y}$ is the outcome coordinate generated by $\hat{\mathcal{M}}$, whose observational law is $\mu$; $\hat{Y}(t)$ denotes its outcome under intervention $T=t$, and $\|\hat{Y}\|\leqslant K$. — Lemma~\ref{lemma:Lip_condition} specifies the joint comparison law; Proposition~\ref{prop:Lip_ATE} and its proof identify the comparison model and its intervened outcome. - reading of the symbol `\text{ATE}_{\mathcal{M}}`: $\text{ATE}_{\mathcal{M}}(\boldsymbol{t})=\mathbb{E}_{\boldsymbol{u}\sim P(\boldsymbol{U})}[Y(\boldsymbol{t})-Y(\boldsymbol{t}_0)]$ is a population contrast between the two intervened outcomes in the same SCM. In Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}: The argument suppressed in $\text{ATE}_{\mathcal{M}}$ is $T=1$, and the reference intervention is $T=0$. — The Average Treatment Effect definition specifies the latent-law expectation. The construction in the proof of Proposition~\ref{prop:counterexample_foral}, also supporting Proposition~\ref{prop:counterexample}, uses binary treatment and calculates the contrast between treatment 1 and treatment 0. - reading of the symbol `P^{\mathcal{M}}`: $P^{\mathcal{M}}$ denotes the population-law notation associated with $\mathcal{M}$, evaluated at the variables written in its argument. In Theorem~\ref{thm:approximation}: $P^{\mathcal{M}}(\boldsymbol{V}(\boldsymbol{t}))$ is the joint interventional law under $\boldsymbol{T}=\boldsymbol{t}$, whereas $P^{\mathcal{M}}(\boldsymbol{U})$ is the joint latent law. — The introduction defines the observational law, and Theorem~\ref{thm:approximation} uses the notation for both interventional and latent laws. - reading of the symbol `\mathbb{P}`: $\mathbb{P}$ denotes the probability measure appropriate to the local distribution or events. In Proposition~\ref{prop:holder_curve}: The statement uses $\mathbb{P}$ satisfying Assumption~\ref{assumption:lower_bound} on $K\subset\mathbb{R}^D$. The proof reduces to a measure on $[0,1]^d$ that vanishes on the boundary and satisfies $\mathbb{P}(B)\geqslant C_1\lambda(B)$ for all measurable $B\subset[0,1]^d$; the reduced measure is $f^{-1}_{\#}\mathbb{P}$ for the homeomorphism in that assumption. Absolute continuity with respect to Lebesgue measure is assumed only in the illustrative proof sketch. In Theorem~\ref{thm:wide_nn}, \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: In the distribution-generator construction, $\mathbb P$ is the target probability measure on the ambient Euclidean space; expressions such as $\mathbb P(X_i=1)=p_i$ in the selector construction denote probabilities under the selector's law. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: In the categorical propensity definition, $\mathbb{P}(V_i=j\mid\text{Pa}(V_i),\boldsymbol{U}_{V_i})$ denotes conditional probability under the relevant SCM law, rather than evaluation of the distribution-generator target measure. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: $\mathbb P$ is the target probability measure on $\mathbb R^d$ satisfying Assumption~\ref{assump:support}; its normalized restrictions to the connected support components are the laws $\mu_i$. In Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:counterexample_foral}: In the componentwise lower-bound requirement, $\mathbb P$ denotes each non-Gumbel latent law; Assumption~\ref{assumption:lower_bound} is imposed on $\mathbb P\|_{C_i}/\mathbb P(C_i)$ for every connected support component $C_i$, with $K=C_i$. — Propositions~\ref{prop:counterexample} and~\ref{prop:counterexample_foral} explicitly impose Assumption~\ref{assumption:lower_bound} on normalized restrictions of each non-Gumbel latent law. - reading of the symbol `\nu`: $\nu$ denotes a probability distribution, represented by its probability vector when its domain is finite, with the ambient domain specified locally. In Proposition~\ref{prop:Lip_ATE}: $\nu=P^{\mathcal M}(\boldsymbol V)$ is the observational law of $\mathcal M$. In the proof, $\nu_A$ is its marginal on the observed nondescendants $A$ of $T$, excluding $T$, and $X=\text{Pa}(T)$ is a subvector of $A$. The conditional treatment laws satisfy $\nu(dt\mid A=a)=\nu(dt\mid X=x)=p_\nu(t\mid X=x)P(dt)$, where $x$ is the $X$ subvector of $a$. In Lemma~\ref{lemma:Lip_condition}: $\nu=P(T,Y)$ is the joint law of $(T,Y)$, while $\nu(dt)$ in the integral denotes its treatment marginal. In Lemma~\ref{lemma:sinkhorn_approx}: $\nu=(\nu_1,\ldots,\nu_m)\in\Delta_m$ is the weight vector representing the finite probability measure $\sum_{j=1}^{m}\nu_j\delta_{y_j}$; the points $y_j$ and the points $x_i$ representing $\mu$ lie in a common Euclidean space, with ground distance $\\|x_i-y_j\\|$ given by the $1$-norm. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: $\nu^\tau$ is the distribution of the Gumbel-softmax weight vector $w^\tau$, and $\nu^0$ is the distribution of its limiting one-hot vector. — The proof of Proposition~\ref{prop:Lip_ATE} defines $A$ as all observed nondescendants of $T$ other than $T$, sets $X=\text{Pa}(T)$, and writes $\nu(dt\mid A=a)=\nu(dt\mid X=x)$ and the marginal $\nu_A$. - reading of the symbol `\hat{\mu}`: $\hat{\mu}$ denotes the comparison probability measure on the same space as $\mu$. In Lemma~\ref{lemma:push-forward_err}: $\hat{\mu}$ is supported on the compact set $K\subset\mathbb{R}^{d_1}$. In Lemma~\ref{lemma:product_err}: $\hat{\mu}=\hat{\mu}_1\otimes\cdots\otimes\hat{\mu}_n$ is the product of the comparison factor measures. — Lemma~\ref{lemma:push-forward_err} specifies a compact common support; Lemma~\ref{lemma:product_err} separately defines the comparison product measure. - reading of the symbol `C_i`: $C_i$ denotes a subset of the ambient Euclidean space in the local support or partition construction. In Theorem~\ref{thm:wide_nn}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}, \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Theorem~\ref{thm:wide_nn}: $C_i$, $i\in[N_C]$, are the finitely many connected components of $\operatorname{supp}(\mathbb{P})$, with $\operatorname{supp}(\mathbb{P})=\bigcup_{i=1}^{N_C}C_i$ and $\mathcal{H}([0,1]^{d_i^C},C_i)\cap\mathcal{F}_L([0,1]^{d_i^C},C_i)\neq\emptyset$ for some $d_i^C\geqslant0$. In the result split from Theorem~\ref{thm:wide_nn}: $C_i$, $i\in[N_C]$, are the finitely many connected components of $\operatorname{supp}(\mathbb P)$, with $\operatorname{supp}(\mathbb P)=\bigcup_{i=1}^{N_C}C_i$ and $\mathcal H([0,1]^{d_i^C},C_i)\cap\mathcal F_L([0,1]^{d_i^C},C_i)\neq\emptyset$ for some $d_i^C\geqslant0$. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: If $d_i^C=0$, then $C_i$ is a singleton, since it is homeomorphic to $[0,1]^0$. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: $C_i$, $i\in[N_C]$, are the finitely many connected components of $\operatorname{supp}(\mathbb{P})$, with $\operatorname{supp}(\mathbb{P})=\bigcup_{i=1}^{N_C}C_i$ and $\mathcal{H}([0,1]^{d_i^C},C_i)\cap\mathcal{F}_L([0,1]^{d_i^C},C_i)\neq\emptyset$ for some $d_i^C\geqslant0$; the quantized construction applies to components with $d_i^C>0$. In Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:counterexample_foral}: For each non-Gumbel latent law $\mathbb P$, every normalized component law $\mathbb P\|_{C_i}/\mathbb P(C_i)$ satisfies Assumption~\ref{assumption:lower_bound} with compact support set $K=C_i$. — Propositions~\ref{prop:counterexample} and~\ref{prop:counterexample_foral} require Assumption~\ref{assumption:lower_bound} for each normalized component law, with $K=C_i$. - reading of the symbol `\mu_i`: $\mu_i$ denotes the component measure or probability-vector coordinate specified by its local decomposition. In Theorem~\ref{thm:wide_nn}: $i\in[N_C]$ and $\mu_i=\mathbb{P}_{C_i}/\mathbb{P}(C_i)$, where $\mathbb{P}_{C_i}$ is the restriction of $\mathbb{P}$ to $C_i$, whenever $\mathbb{P}(C_i)>0$. In Lemma~\ref{lemma:combin_err}: $i\in[n]$ and $\mu_i$ is an arbitrary measure on $\mathbb{R}^d$ entering the mixture $\sum_{i=1}^n p_i\mu_i$. In Lemma~\ref{lemma:product_err}: $i\in[n]$ and $\mu_i$ is the $i$th factor of $\mu=\mu_1\otimes\cdots\otimes\mu_n$, paired with $\hat{\mu}_i$ on its factor space. In the result split from Theorem~\ref{thm:wide_nn}: $i\in[N_C]$ and $\mu_i=\mathbb{P}_{C_i}/\mathbb{P}(C_i)$, where $\mathbb{P}_{C_i}$ is the restriction of $\mathbb{P}$ to $C_i$, whenever $\mathbb{P}(C_i)>0$. In Lemma~\ref{lemma:sinkhorn_approx}: $i\in[n]$ and $\mu_i\in[0,1]$ is the mass at $x_i$ in the finite probability measure with weight vector $\mu\in\Delta_n$; a transportation plan satisfies $\sum_jT_{ij}=\mu_i$. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: $i\in[N_C]$ and $\mu_i=\mathbb{P}_{C_i}/\mathbb{P}(C_i)$, where $\mathbb{P}_{C_i}$ is the restriction of $\mathbb{P}$ to $C_i$, whenever $\mathbb{P}(C_i)>0$. — The component construction at flat.tex line 1883 defines the normalized restriction used by lem:wide-nn-perekrestenko-theorem-viii-1. - reading of the symbol `\hat{\mu}_i`: $\hat{\mu}_i$ denotes a component probability measure paired with $\mu_i$, without an intrinsic neural-network construction. In Lemma~\ref{lemma:combin_err}: $i\in[n]$ and $\hat{\mu}_i$ is a measure on $\mathbb{R}^d$ entering the comparison mixture $\sum_{i=1}^n p_i\hat{\mu}_i$. In Lemma~\ref{lemma:product_err}: $i\in[n]$ and $\hat{\mu}_i$ is the $i$th factor of $\hat{\mu}=\hat{\mu}_1\otimes\cdots\otimes\hat{\mu}_n$, on the same factor space as $\mu_i$. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: $i\in[N_C]$ and $\hat{\mu}_i=P(\hat{g}_2^i\circ\hat{g}_1^i(U_i))$, where $U_i$ is uniform on $[0,1]$; this is the law of the two deterministic component blocks before the Gumbel mixing layer. For $d_i^C=0$, it is the point mass at the singleton $C_i$. — The proof of Theorem~\ref{thm:wide_nn} at flat.tex line 1904 explicitly defines the component comparison law; the proof of thm:cor-fix-3c402872 uses the same remaining construction. - reading of the symbol `X^\tau`: $X^\tau$ uses positive weights $p_1,\ldots,p_n$, which need not sum to one, and $0<\tau<1$, with coordinates indexed by $k\in[n]$; all coordinates are positive and sum to one. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: In the mixing construction, $n=N_C$ and $p_i=\mathbb{P}(C_i)>0$ for $i\in[N_C]$; $X^\tau$ is the vector of Gumbel-softmax weights assigned to these support components. — The proof of Theorem~\ref{thm:wide_nn} and the proof of thm:cor-fix-3c402872 set $p_i=\mathbb{P}(C_i)$ and use $X^\tau$ as the Gumbel-softmax mixing weights. - reading of the symbol `\tau`: $\tau>0$ is the Gumbel-softmax temperature hyperparameter; a superscript $0$ on a softmax output or layer denotes the limit as $\tau\rightarrow0$, rather than evaluation of the positive-temperature formula at zero. In Lemma~\ref{lemma:gumbel_err}: The temperature satisfies $0<\tau<1$. In Theorem~\ref{thm:wide_nn}: The error estimate is asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of the budgets and $\tau$; the proof uses the Gumbel-softmax estimate for $0<\tau<1$. In Theorem~\ref{thm:consistency}: The temperature is the positive sequence $\tau_n$ appearing in $\mathcal{F}_{1,n}$, with $\tau_n\rightarrow0$. In the result split from Theorem~\ref{thm:wide_nn}: The error estimate is asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of the budgets and $\tau$; the proof uses the Gumbel-softmax estimate for $0<\tau<1$. In Corollary~\ref{cor:rate}: The generator temperature is the positive sequence $\tau_n$ in $\mathcal F_{1,n}$ and in the approximation term $\tau_n(1-\log\tau_n)$. In Corollary~\ref{corollary:deep_nn}: The displayed error bound is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, with $R>0$ and the causal model fixed. The implicit constant is independent of the intervention and the network budgets, but may depend on $R$ and the fixed causal model. — Corollary~\ref{corollary:deep_nn} fixes $R>0$ before choosing the approximating model, and its binding reading specifies the asymptotic regime with $R$ and the causal model fixed. - reading of the symbol `X^0`: $X^0=\lim_{\tau\downarrow0}X^\tau=e_I$ almost surely, where $I=\arg\max_{k\in[n]}(\log p_k+G_k)$, using the same Gumbel draws throughout the limit; ties may be assigned to the first maximizing coordinate. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: In the mixing construction, $n=N_C$ and $p_i=\mathbb{P}(C_i)>0$ for $i\in[N_C]$; $X^0$ is the one-hot limit of these component weights, with $P(X^0=e_i)=\mathbb{P}(C_i)$. — The proof of Theorem~\ref{thm:wide_nn} and the proof of thm:cor-fix-3c402872 set $p_i=\mathbb{P}(C_i)$ and use $X^0$ as the zero-temperature component selector. - reading of the symbol `\mu^0`: $\mu^0$ is the law of the one-hot vector $X^0$, supported on the vertices of $\Delta_n$, with $\mu^0(\{e_k\})=p_k/\sum_{j=1}^n p_j$ for $k\in[n]$. — Lemma~\ref{lemma:gumbel_err} defines the limiting law; its proof normalizes the positive weights without changing the softmax and identifies this law with the argmax push-forward. - reading of the symbol `n`: $n$ is a local integer parameter whose role is determined by the indexed objects in the surrounding passage. In Lemma~\ref{lemma:hilbert}, Proposition~\ref{prop:holder_curve}: In the partition notation $\mathcal{C}_n$ and the curve and refinement sequences, $n$ is a level index beginning at $0$, with $\mathcal{C}_0=\{C_{-1}\}$ and $C_{-1}=[0,1]^d$. In Lemma~\ref{lemma:combin_err}: $n\geqslant1$ is the number of mixture components, with component indices $k=1,\cdots,n$. In Lemma~\ref{lemma:product_err}: $n\geqslant1$ is the number of factors in the product measures. In Lemma~\ref{lemma:gumbel_err}: $n\geqslant1$ is the dimension of $X^\tau$ and the number of positive weights $p_k$ and Gumbel coordinates $G_k$. In Proposition~\ref{prop:convg_opt}: $n\in\mathbb{N}$ indexes the nested sets, constraint functions, tolerances, and optimization values, and varies in the limit $n\rightarrow\infty$. In Proposition~\ref{prop:holder_curve}: In the local cube-metric definition $C=\prod_{i=1}^{n}[a_i,b_i]$ and $C_k=\prod_{i=1}^{n}[a_i^k,b_i^k]$, $n$ is the ambient dimension and $i$ ranges from $1$ to $n$. In Lemma~\ref{lemma:sinkhorn_approx}: $n\geqslant1$ is the length of the probability vector $\mu\in\Delta_n$, and a transportation plan has $n$ rows and $m$ columns. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: $n$ is the number of observations from the true distribution and also indexes the approximation and tolerance sequences; the asymptotic limit is $n\rightarrow\infty$. In Lemma~\ref{lemma:sup_wasserstein}: $n\geqslant1$ is the sample size defining $\mu_n$ and the number of evaluation points in the empirical covering-number notation $\mathcal{N}_1(\epsilon,\mathcal{F},n)$. In Theorem~\ref{thm:wide_nn}: For a component of positive dimension $q=d_i^C$, the refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor\geqslant2$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$; the quantization parameter is $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$ and the mesh is $\delta/n$. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: For a component of positive dimension $q=d_i^C$, the refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor\geqslant2$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$; the quantization parameter is $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$ and the mesh is $\delta/n$. — The binding reading of lem:wide-nn-perekrestenko-theorem-viii-1 specifies the refinement integer, its width threshold, and the dimension-dependent quantization parameter in the component construction used in Theorem~\ref{thm:wide_nn}. - reading of the symbol `\mu_n`: $\mu_n$ denotes the locally specified measure, with $n$ serving either as a component index or as the empirical sample size. In Lemma~\ref{lemma:combin_err}: $\mu_n$ is the component measure with index $n$ in the mixture, among measures $\mu_k$ on $\mathbb{R}^d$ for $k\in[n]$. In Lemma~\ref{lemma:product_err}: $\mu_n$ is the final component measure in the product $\mu=\mu_1\otimes\cdots\otimes\mu_n$. In Lemma~\ref{lemma:sup_wasserstein}: $\mu_n=n^{-1}\sum_{i=1}^n\delta_{X_i}$ is the empirical distribution of the sample with law $\mu$, on $\mathbb{R}^{d'}$. — Lemma~\ref{lemma:sup_wasserstein} specifies $\mu_n$ as the empirical distribution of $\mu$ on $\mathbb R^{d'}$. Lemmas~\ref{lemma:combin_err} and~\ref{lemma:product_err} use the same notation for an indexed component measure. Lemma lem:sup-wasserstein-anthony-theorems now states only a covering and packing bound and does not define $\mu_n$. - reading of the symbol `W`: The notation $W(\mu,\nu)$ denotes Wasserstein-1 distance between measures on the same underlying space, whereas $W$ in a network architecture argument denotes an integer width. In \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Corollary~\ref{corollary:deep_nn}: In $a\,\omega(cW^{-2/d_i^{\text{in}}})$, $W$ is the number of weights of the scalar approximating network, rather than its width; at fixed width $2d_i^{\text{in}}+10$, this weight count grows linearly in the depth $L_0$. — The approximation expression in lem:deep-nn-yarotsky-theorem-2 explicitly identifies $W$ as the number of weights and states that it grows linearly in $L_0$ at fixed width. The architecture definition continues to use $W$ for width, and the notation paragraph continues to define $W(\mu,\nu)$ as Wasserstein-1 distance. - reading of the symbol `\lambda`: $\lambda$ is a positive scalar when it indexes a regularized transport functional and a reference measure when it occurs in a push-forward or measure-domination expression, with its reference domain specified locally. In Theorem~\ref{thm:wide_nn}: In the push-forward expression, $\lambda$ is Lebesgue measure on $[0,1]^{N_C}$; in the componentwise construction, it is the uniform measure on $[0,1]$. In Proposition~\ref{prop:holder_curve}: For the curve's push-forward, $\lambda$ is Lebesgue measure on $[0,1]$; in expressions $\lambda(B)$ with $B\subset[0,1]^d$, it is $d$-dimensional Lebesgue measure. In Lemma~\ref{lemma:sinkhorn_approx}: $\lambda>0$ is the scalar regularization parameter. In the result split from Theorem~\ref{thm:wide_nn}: In the push-forward expression, $\lambda$ is Lebesgue measure on $[0,1]^{N_C}$; in the componentwise construction, it is the uniform measure on $[0,1]$, and in the lower-bound premise it is Lebesgue measure on the corresponding cube. — Theorem~\ref{thm:wide_nn} specifies Lebesgue measure on $[0,1]^{N_C}$ and assumes only Assumption~\ref{assump:support}; the componentwise construction at flat.tex line 720 uses uniform measure on $[0,1]$. - reading of the symbol `S_\lambda`: $S_\lambda$ is the debiased regularized transport functional obtained by subtracting half of each self-comparison from $W_{1,\lambda}(\mu,\nu)$. In Lemma~\ref{lemma:sinkhorn_approx}: The arguments are finite probability measures represented by their weights and support points, with $c_{ij}=\\|x_i-y_j\\|$ for the $1$-norm. Here $W_{1,\lambda}(\mu,\nu)=\min_{T\in\Pi(\mu,\nu)}\{\sum_{i,j}c_{ij}T_{ij}-\lambda h(T)\}$, where $\Pi(\mu,\nu)$ is the set of nonnegative transportation plans with marginals $\mu,\nu$, and $h(T)=1-\sum_{i,j}T_{ij}\log T_{ij}$ with $0\log0=0$. Both self terms use this same convention and the same $1$-norm ground distance on their corresponding support points. — Lemma~\ref{lemma:sinkhorn_approx} specifies $W_{1,\lambda}$ as the minimum of transport cost minus $\lambda h(T)$ and specifies the ground distance and entropy convention. - reading of the symbol `W_{1,\lambda}`: $W_{1,\lambda}$ uses the supplied finite support points, the paper's $1$-norm ground cost, and the entropy subtraction $-\lambda h(T)$ with $h(T)=1-\sum_{i,j}T_{ij}\log T_{ij}$ and $0\log0=0$. — Lemma~\ref{lemma:sinkhorn_approx} explicitly supplies the support points, transportation constraints, ground cost, and entropy convention. - reading of the symbol `L`: $L$ denotes a nonnegative Lipschitz bound with respect to the paper's $1$-norm when used in a Lipschitz condition or $\mathcal{F}_L$, and an integer depth when used as the depth argument of a neural network class. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:Lip_ATE}: In the SCM class $\mathcal{M}(\mathcal{G},\mathcal{F}_L,\boldsymbol{U})$, $L$ bounds the continuous structural functions and the categorical propensity functions, rather than the categorical $\arg\max$ maps. In Lemma~\ref{lemma:push-forward_err}: $L$ is a Lipschitz bound for $F$; the hypothesis does not impose that bound on $\hat{F}$. In Lemma~\ref{lemma:Lip_condition}: $L$ is a common Lipschitz bound for the regression functions $f$ and $\hat{f}$. In Proposition~\ref{prop:holder_curve}, Theorem~\ref{thm:wide_nn}: In the support assumptions, $L$ bounds the homeomorphisms from the unit cubes to the support sets; it is not a Lipschitz bound for the resulting space-filling curves. In Theorem~\ref{thm:consistency}: In $\mathcal F_L^K$ and $\underline F^L$, $L$ bounds the Lipschitz constants of the continuous structural functions and normalized categorical propensity vectors of population candidate SCMs, with respect to the paper's $1$-norm; the class also requires $\sup_x\\|f(x)\\|_1\leqslant K$ on the full structural input domain. The conclusion evaluates this parameter at $\hat L_f$. In the result split from Theorem~\ref{thm:wide_nn}: In the support assumptions, $L$ bounds the homeomorphisms from the unit cubes to the support sets; it is not a Lipschitz bound for the resulting space-filling curves. — Theorem~\ref{thm:consistency} retains the population class $\mathcal F_L^K$ in equation~\ref{eq:consistent_limiting}, explicitly defines its full-domain output $1$-norm and Lipschitz bounds, and uses $\underline F^{\hat L_f}$ in its conclusion. - reading of the symbol `\mathcal{F}_L`: $\mathcal{F}_L(K_1,K_2)$ denotes the class of maps $f:K_1\rightarrow K_2$ with Lipschitz constant at most $L$, using the paper's $1$-norm convention, with the arguments supplied by context when omitted. — The notation paragraph at flat.tex line 194 defines the class, identifies $\\|\cdot\\|$ as the $1$-norm, and explicitly permits omission of the domain arguments. - reading of the symbol `\mathcal{H}`: $\mathcal{H}(K_1,K_2)$ denotes homeomorphisms from $K_1$ onto $K_2$, while an explicit local definition of unparameterized $\mathcal{H}$ supplies its function class. In Lemma~\ref{lemma:sup_wasserstein}: In the proof, $\mathcal{H}=\left\{h:h\in\mathcal{F}_1([-K,K]^d,[-K,K]^d),\ h(0)=0\right\}$, and $\mathcal{H}_\delta$ is a $\delta$-net of this class in the $\ell_\infty$ norm. — The notation paragraph defines homeomorphisms; the proof of Lemma~\ref{lemma:sup_wasserstein} explicitly defines the unparameterized Lipschitz class at flat.tex line 3253. - reading of the symbol `\mathcal{F}`: $\mathcal{F}$ denotes a function class whose domains and codomains are supplied by its local use. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Proposition~\ref{prop:equivalent_g_constrianed}: In $\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})$, $\mathcal{F}$ specifies the allowed structural functions, with $f_i\in\mathcal{F}$ for $i\in[n_V]$; each function has the input and output dimensions of its structural equation. In Lemma~\ref{lemma:sup_wasserstein}: $\mathcal F$ is a class of measurable maps $f:\mathbb R^{d'}\rightarrow\mathbb R^d$. Each scalar coordinate class $\mathcal F_j=\{x\mapsto f_j(x):f\in\mathcal F\}$, $j\in[d]$, has pseudo-dimension less than $d_{\mathcal F}<\infty$, and $\\|f(x)\\|_\infty\leqslant K$ for every $f\in\mathcal F$ and $x\in\mathbb R^{d'}$. There are a standard Borel space $T$ and a jointly Borel map $G:T\times\mathbb R^{d'}\rightarrow[-K,K]^d$ such that $\mathcal F=\{G(t,\cdot):t\in T\}$. — Lemma~\ref{lemma:sup_wasserstein} additionally requires a representation by a jointly Borel map on a standard Borel parameter space. - reading of the symbol `K`: $K$ is a context-dependent bound or compact set: numerical occurrences are positive bounds, whereas set-valued occurrences specify the compact domain or support of a measure. In Theorem~\ref{thm:approximation}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: In the boundedness assumption, $K>0$ satisfies $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$; the indexed $U_j$ are the latent variables separated from the categorical Gumbel noises. In Lemma~\ref{lemma:push-forward_err}: $K\subset\mathbb{R}^{d_1}$ is the compact set carrying $\mu$ and $\hat{\mu}$ and the common domain of $F$ and $\hat{F}$. In Lemma~\ref{lemma:Lip_condition}: $K$ bounds the regression functions: $\|f(t)\|\leqslant K$ and $\|\hat{f}(t)\|\leqslant K$. In Proposition~\ref{prop:holder_curve}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}, the result split from Theorem~\ref{thm:wide_nn}: In the lower-bound support assumption, $K\subset\mathbb{R}^{D}$ is compact and supports the measure under consideration; $f\in\mathcal{H}([0,1]^d,K)\cap\mathcal{F}_L([0,1]^d,K)$. This set-valued occurrence is distinct from a numerical boundedness constant. In Theorem~\ref{thm:consistency}: In the explanatory notation $\mathcal{P}(K)$, $K$ is the set on which probability measures are defined; in the displayed factors $\mathcal{P}([-K,K]^{d_j^U})$, the $K$ inside the cube endpoints is the numerical boundedness constant. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the truncated neural classes, numerical $K$ is the clipping level in $\max\{-K,\min\{f,K\}\}$ and is the boundedness constant chosen for the underlying SCM. Every true continuous structural function and every candidate continuous structural network additionally satisfies $\sup_x\\|f_i(x)\\|_1\leqslant K$ and $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$, respectively, on its full structural input domain. The limiting structural class retains $\mathcal F_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$. In Lemma~\ref{lemma:sup_wasserstein}: $K>0$ is the uniform output bound $\\|f(x)\\|_\infty\leqslant K$ for every $f\in\mathcal F$ and every $x\in\mathbb R^{d'}$; consequently the outputs lie in $[-K,K]^d$. In Proposition~\ref{prop:counterexample_foral}: For each non-Gumbel latent law $\mathbb P$ with support components $C_i$, Assumption~\ref{assumption:lower_bound} applies to every normalized component law $\mu_i=\mathbb P\|_{C_i}/\mathbb P(C_i)$ with the compact support set $K=C_i$. In Proposition~\ref{prop:counterexample}: For each non-Gumbel latent law $\mathbb P$ with support components $C_i$, Assumption~\ref{assumption:lower_bound} applies to every normalized component law $\mu_i=\mathbb P\|_{C_i}/\mathbb P(C_i)$ with the compact support set $K=C_i$. In Corollary~\ref{corollary:deep_nn}: For each non-Gumbel latent law $\mathbb P=P(U_C)$ with support components $C_i$, the lower-bound support assumption applies to each normalized component law $\mu_i=\mathbb P\|_{C_i}/\mathbb P(C_i)$ with compact support set $K=C_i$. Its chart $f:[0,1]^{d_i^C}\rightarrow C_i$ satisfies $(f^{-1})_{\#}\mu_i(\partial[0,1]^{d_i^C})=0$ when $d_i^C>0$. — The binding support convention for Corollary~\ref{corollary:deep_nn} applies Assumption~\ref{assumption:lower_bound} to each normalized latent component law and specifies the boundary condition for its pullback. - reading of the symbol `\mathcal{F}_L^K`: $\mathcal{F}_L^K(K_1,K_2)$ denotes a class of bounded Lipschitz continuous maps $f:K_1\rightarrow K_2$, with the norm bound and Lipschitz restriction given by its local definition. In Theorem~\ref{thm:consistency}: In the statement and the limiting optimization, $\mathcal F_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$, with the supremum over the full structural input domain; in the proof's displayed population-function definition, $\mathcal F_L^K=\{f:\\|f\\|_\infty\leqslant K,\ f\ \text{Lipschitz continuous}\}$, where the function infinity norm is the uniform coordinatewise bound and that display supplies no numerical Lipschitz bound. In Corollary~\ref{cor:rate}: The population bounded Lipschitz class is $\mathcal F_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$, with the supremum over the full structural input domain. — Corollary~\ref{cor:rate} explicitly uses the population structural classes specified in Theorem~\ref{thm:consistency}, whose statement defines the uniform output $1$-norm bound and numerical Lipschitz bound. - reading of the symbol `\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})`: $\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})$ consists of canonical SCMs on $\mathcal{G}$ with $f_i\in\mathcal{F}$ for $i\in[n_V]$ and the indicated canonical latent coordinates; the argument $\boldsymbol{U}$ does not fix a common latent probability law across the class. — The canonical-representation definition at flat.tex lines 215--233 defines the class; the paragraph following equation~\ref{eq:consistent_limiting} explicitly optimizes over all latent distributions, and the proof of Theorem~\ref{thm:consistency} includes $P(U_j)$ among model parameters. - reading of the symbol `\mathcal{NN}`: The two-argument notation $\mathcal{NN}_{k_1,k_2}(W,L)$ specifies a ReLU network class, with omitted dimensions supplied by context; the five-argument notation specifies the separable three-part generator architecture including its Gumbel-Softmax layer. In Corollary~\ref{corollary:deep_nn}: The four-argument generator class $\mathcal{NN}(\Theta(d_{\max}^U),L_1,\Theta((d_{\max}^U)^2),L_2)$ suppresses the temperature $\tau$ of the same third Gumbel-Softmax layer, which remains present in the approximation bound. The multiplicative width factors in $\Theta(d_{\max}^{\text{in}}d_{\max}^{\text{out}})$, $\Theta(d_{\max}^U)$, and $\Theta((d_{\max}^U)^2)$ can be chosen as absolute numerical constants, independent of the causal model, $R$, all dimensions, and all category and component counts. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: In the generator architecture, a component with $d_i^C=0$ uses the unique map to $[0,1]^0$ as its first deterministic block and a constant affine map to its singleton support component as its second deterministic block. In the result split from Theorem~\ref{thm:wide_nn}: The multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. — Corollary~\ref{corollary:deep_nn} specifies absolute numerical factors for all three displayed widths, and thm:cor-fix-3c402872 specifies absolute factors for its two generator widths. The existing zero-dimensional convention follows the architecture reading accompanying Theorem~\ref{thm:wide_nn}. - reading of the symbol `\mathcal{NN}(W_1,L_1,W_2,L_2,\tau)`: This is the three-part generator class with component count $N_C$, component cube dimensions $d_j^C$, and output dimension $d$ suppressed from its arguments; $W_1,L_1$ and $W_2,L_2$ specify the respective component-network widths and depths, and $\tau>0$ specifies the mixing temperature. In Corollary~\ref{corollary:deep_nn}: The four-argument generator-class occurrence suppresses the temperature argument; the latent generators retain the Gumbel-softmax temperature $\tau$ appearing in the bound. The multiplicative width factors in $\Theta(d_{\max}^U)$ and $\Theta((d_{\max}^U)^2)$ can be chosen as absolute numerical constants, independent of the causal model, $R$, all dimensions, and all category and component counts. In Theorem~\ref{thm:consistency}: The class used is its truncated version $\mathcal{NN}^{\infty,K}(\Theta(d_{\max}^U),L_{1,n},\Theta((d_{\max}^U)^2),L_{2,n},\tau_n)$; the superscript $\infty$ imposes no finite Lipschitz bound, while $K$ specifies coordinatewise output truncation, and the finite component counts and component cube dimensions of the latent generator architectures are fixed as $n$ varies. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: The architecture includes components with $d_i^C=0$: their first deterministic block is the unique map to $[0,1]^0$, and their second deterministic block is a constant affine map to the corresponding singleton. In Corollary~\ref{cor:rate}: The latent generator class is the truncated class $\mathcal F_{1,n}=\mathcal{NN}^{\infty,K}(\Theta(d_{\max}^U),L_{1,n},\Theta((d_{\max}^U)^2),L_{2,n},\tau_n)$; the superscript $\infty$ imposes no finite Lipschitz bound, $K$ specifies coordinatewise output truncation, and the finite component counts and component cube dimensions are fixed as $n$ varies. In the result split from Theorem~\ref{thm:wide_nn}: The multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. — The binding architecture conventions for Corollary~\ref{corollary:deep_nn} and thm:cor-fix-3c402872 specify absolute numerical width factors. The convention accompanying Theorem~\ref{thm:wide_nn} specifies the zero-dimensional component blocks. - reading of the symbol `\mathcal{NN}^{L_f,K}`: $\mathcal{NN}^{L_f,K}(W,L)$ abbreviates $\mathcal{NN}_{d_1,d_2}^{L_f,K}(W,L)=\{\max\{-K,\min\{f,K\}\}:f\in\mathcal{NN}_{d_1,d_2}(W,L),\ \text{Lip}(f)\leqslant L_f\}$, with coordinatewise clipping and dimensions supplied by the function being represented; here $L$ is network depth. In Theorem~\ref{thm:consistency}: The ambient structural class is $\mathcal F_{0,n}=\mathcal{NN}^{\hat L_f,K}(W_{0,n},\Theta(\log d_{\max}^{\text{in}}))$, where the depth expression uses $\Theta(1+\log d_{\max}^{\text{in}})$ when $d_{\max}^{\text{in}}=1$; categorical raw structural members $q$ admitted to the empirical optimization additionally satisfy $q(x)\in[0,\infty)^{n_i}$, $\\|q(x)\\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$, while each admitted continuous structural network $\hat f_i$ additionally satisfies $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$ on its full structural input domain. In Corollary~\ref{cor:rate}: The ambient structural class is $\mathcal F_{0,n}=\mathcal{NN}^{\hat L_f,K}(W_{0,n},\Theta(\log d_{\max}^{\text{in}}))$, with the depth convention inherited from its specification; each categorical raw structural network $q$ admitted to the empirical optimization additionally satisfies $q(x)\in[0,\infty)^{n_i}$, $\\|q(x)\\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$, while each admitted continuous structural network $\hat f_i$ additionally satisfies $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$ on its full structural input domain. — Corollary~\ref{cor:rate} uses the structural network class specified in Theorem~\ref{thm:consistency} and explicitly imposes both continuous output $1$-norm and categorical normalized-propensity restrictions on members admitted to the empirical optimization. - reading of the symbol `W_1`: $W_1$ is the integer width of each component network $\hat{g}_1^j\in\mathcal{NN}_{1,d_j^C}(W_1,L_1)$, $j\in[N_C]$, rather than the combined width of all parallel component networks. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: For each component of positive dimension $q=d_i^C$, the construction takes $W_1\geqslant A_q2^q$ and $n=\lfloor(W_1/A_q)^{1/q}\rfloor$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. — The binding reading of lem:wide-nn-perekrestenko-theorem-viii-1 specifies the refinement integer and its width threshold. - reading of the symbol `W_2`: $W_2$ is the integer width of each component network $\hat{g}_2^j\in\mathcal{NN}_{d_j^C,d}(W_2,L_2)$, $j\in[N_C]$. — The caption of Figure~\ref{fig:wide_nn} and flat.tex line 726 specify the width of each second-part block. - reading of the symbol `L_1`: $L_1$ is the integer depth of each component network $\hat{g}_1^j\in\mathcal{NN}_{1,d_j^C}(W_1,L_1)$, $j\in[N_C]$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: At sample size $n$, the first-block depth is $L_{1,n}$, with $L_{1,n}=\Theta(m_n^{d_{\max}^U/(2d_{\max}^U+4)})$; the architectural order uses sufficiently large fixed constant factors supplied by the network constructions. In Corollary~\ref{corollary:deep_nn}: The displayed error bound is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, with $R$ and the causal model fixed; $R>0$ is fixed before choosing $\hat{\mathcal M}$. The implicit constant is independent of the intervention and the network budgets, but may depend on $R$ and the fixed causal model. — Corollary~\ref{corollary:deep_nn} fixes $R>0$ before choosing the approximating model and specifies the asymptotic regime and dependence of the implicit constant. - reading of the symbol `L_2`: $L_2$ is the integer depth of each component network $\hat{g}_2^j\in\mathcal{NN}_{d_j^C,d}(W_2,L_2)$, $j\in[N_C]$. In \cite[Theorem 2]{Yarotsky2018OptimalAO}, used in the proof of Theorem~\ref{thm:wide_nn}: $L_2$ is also the depth of each scalar coordinate approximant $\hat{g}_2^{i,j}$ used to form $\hat{g}_2^i=(\hat{g}_2^{i,1},\ldots,\hat{g}_2^{i,d})$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: At sample size $n$, the second-block depth is $L_{2,n}$, with $L_{2,n}=\Theta(m_n^{d_{\max}^U/(2d_{\max}^U+4)})$; the architectural order uses sufficiently large fixed constant factors supplied by the network constructions. In Corollary~\ref{corollary:deep_nn}: The displayed error bound is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, with $R$ and the causal model fixed; $R>0$ is fixed before choosing $\hat{\mathcal M}$. The implicit constant is independent of the intervention and the network budgets, but may depend on $R$ and the fixed causal model. — Corollary~\ref{corollary:deep_nn} fixes $R>0$ before choosing the approximating model and specifies the asymptotic regime and dependence of the implicit constant. - reading of the symbol `\mathcal{NN}^{\infty,K}`: $\mathcal{NN}^{\infty,K}$ clips the final generator output coordinatewise to $[-K,K]$, with no imposed Lipschitz bound; its suppressed dimensions and component structure are those of the corresponding generator class. — flat.tex lines 726 and 778--781 define the generator and its truncated version; the proof at lines 2940--2945 defines $T_K$ explicitly. - reading of the symbol `\hat{g}`: The notation $\hat{g}_{\#}\lambda$ denotes the output law of the generator after integrating both its uniform input and its Gumbel-softmax randomness, rather than a push-forward conditional on a realized mixing vector. In Theorem~\ref{thm:wide_nn}: The generator has uniform-input dimension $N_C$, output dimension $d$, and target law $\mathbb{P}$; its mixing weights are the target component masses. A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. In Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}: In the NCM class notation, $\hat{g}\in\mathcal{F}_1$ is shorthand for the separate latent generators $\hat{g}_j$, whose compatible dimensions depend on the corresponding latent variable. In Theorem~\ref{thm:consistency}: At sample size $n$, generators range over $\mathcal{F}_{1,n}$ and have coordinatewise output truncation to $[-K,K]$, with finite component counts and component cube dimensions fixed as $n$ varies. The optimization ranges over network representations, including distinct representations inducing the same latent population laws; each representation is evaluated on the same $m_n$ exogenous input draws. In the result split from Theorem~\ref{thm:wide_nn}: The generator has uniform-input dimension $N_C$, output dimension $d$, and target law $\mathbb{P}$; its mixing weights are the target component masses. A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. — Theorem~\ref{thm:consistency} specifies the truncated generator architecture; its binding readings specify optimization over network representations, common exogenous evaluation draws, and fixed finite component counts and cube dimensions. - reading of the symbol `\text{NCM}_{\mathcal{G}}`: $\text{NCM}_{\mathcal{G}}(\mathcal{F}_0,\mathcal{F}_1)$ is the collection of models on $\mathcal{G}$ with structural equations (\ref{eq:structure_equation_hat}), in which every structural network belongs to $\mathcal{F}_0$ and every component-specific latent generator belongs to $\mathcal{F}_1$, with compatible dimensions suppressed. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: At sample size $n$, the candidate class is $\text{NCM}_{\mathcal G}(\mathcal F_{0,n},\mathcal F_{1,n})$, with the architecture and truncation specified in Theorem~\ref{thm:consistency}, restricted so that every categorical raw structural network $q$ satisfies $q(x)\in[0,\infty)^{n_i}$, $\\|q(x)\\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$. In Theorem~\ref{thm:consistency}: The empirical optimization ranges over network representations in the restricted candidate class; representations with the same latent population laws remain distinct candidates and may yield different common-draw empirical distributions. Every admitted continuous structural network additionally satisfies $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$ on its full structural input domain. In Corollary~\ref{cor:rate}: Candidates are evaluated as network representations on the same $m_n$ exogenous simulation draws; representations with identical latent population laws can yield different empirical distributions on these common draws. Every admitted continuous structural network additionally satisfies $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$ on its full structural input domain. — Corollary~\ref{cor:rate} explicitly restricts every continuous structural network to satisfy the uniform output $1$-norm bound on its full structural input domain, while retaining the common-draw evaluation of candidate network representations. - reading of the symbol `\mathcal{F}_0`: $\mathcal{F}_0$ is the structural-network class supplied to $\text{NCM}_{\mathcal{G}}(\mathcal{F}_0,\mathcal{F}_1)$, with the input and output dimensions appropriate to each node omitted from the notation. In Theorem~\ref{thm:consistency}: The class is $\mathcal{F}_{0,n}=\mathcal{NN}^{\hat L_f,K}(W_{0,n},\Theta(\log d_{\max}^{\text{in}}))$, where $\hat L_f=\sqrt{d_{\max}^{\text{in}}d_{\max}^{\text{out}}}L_f$; it consists of coordinatewise truncations to $[-K,K]$ of networks whose Lipschitz constant is at most $\hat L_f$, and varies with $n$. When $d_{\max}^{\text{in}}=1$, the depth expression uses $\Theta(1+\log d_{\max}^{\text{in}})$. Continuous structural networks admitted to the optimization additionally satisfy $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$ on their full structural input domains. Categorical raw networks admitted to the optimization additionally satisfy $q(x)\in[0,\infty)^{n_i}$, $\\|q(x)\\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$. — Theorem~\ref{thm:consistency} retains its architecture, depth convention and categorical restrictions, and additionally restricts continuous networks admitted to the empirical optimization by their uniform output $1$-norm. - reading of the symbol `\mathcal{F}_1`: $\mathcal{F}_1$ is function-class notation whose subscript can be a class identifier or the Lipschitz constant in the previously defined notation $\mathcal{F}_L(K_1,K_2)$. In Corollary~\ref{corollary:deep_nn}, Theorem~\ref{thm:consistency}: As the second argument of $\text{NCM}_{\mathcal{G}}(\mathcal{F}_0,\mathcal{F}_1)$, the class restricts the latent generators, with input and output dimensions omitted from the notation. In Theorem~\ref{thm:consistency}: The latent-generator class is $\mathcal{F}_{1,n}=\mathcal{NN}^{\infty,K}(\Theta(d_{\max}^U),L_{1,n},\Theta((d_{\max}^U)^2),L_{2,n},\tau_n)$, a class varying with $n$. — The notations paragraph and flat.tex line 746 retain the general function-class and latent-generator meanings. Lemma~\ref{lemma:sup_wasserstein} is no longer listed as a result using this symbol. - reading of the symbol `\hat{g}_1`: Each coordinate block $\hat{g}_1^j$ uses one scalar input to generate coordinates in the cube $[0,1]^{d_j^C}$ associated with component $j$, without mixing inputs from different components. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: Writing $\mu_j=\mathbb{P}_{C_j}/\mathbb{P}(C_j)$, for $d_j^C\geqslant1$ the first block approximates the cube law $((g_2^j)^{-1})_{\#}\mu_j$ by the law of $\hat{g}_1^j(U_j)$; for $d_j^C=0$, it is the unique map to $[0,1]^0$. — Flat.tex line 726 specifies the separable architecture; lines 1883--1886 and 2643--2650 give the two constructions. The binding singleton convention specifies the unique first block into $[0,1]^0$. - reading of the symbol `\hat{g}_2`: Each coordinate block $\hat{g}_2^j$ maps the $d_j^C$-dimensional cube coordinates for component $j$ into $\mathbb{R}^d$, without mixing inputs from different components. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: For $d_j^C\geqslant1$, the block $\hat{g}_2^j$ approximates a map $g_2^j\in\mathcal{H}([0,1]^{d_j^C},C_j)\cap\mathcal{F}_L([0,1]^{d_j^C},C_j)$; for $d_j^C=0$, it is a constant affine map to the singleton $C_j$. — Flat.tex line 726 gives the separable architecture; lines 1883--1890 construct component-map approximants. The binding singleton convention specifies constant affine second blocks. - reading of the symbol `\hat{g}_3^\tau`: $\hat{g}_3^\tau$ mixes the $N_C$ component outputs using the Gumbel-softmax coefficient vector, with $\tau>0$; the notation $\hat{g}_3^0$ denotes its almost-sure hard-selection limit as $\tau\rightarrow0$. In Theorem~\ref{thm:wide_nn}: The mixing weights are $p_k=\mathbb{P}(C_k)$ for the connected components $C_k$ of the target support. In Corollary~\ref{corollary:deep_nn}: The layer is used separately for each latent generator, with mixing weights given by the masses of the connected components of that latent variable's target law. In Theorem~\ref{thm:consistency}: At sample size $n$, the generator class uses temperature $\tau_n>0$ and truncates the resulting generator output coordinatewise to $[-K,K]$. In the result split from Theorem~\ref{thm:wide_nn}: The mixing weights are $p_k=\mathbb{P}(C_k)$ for the connected components $C_k$ of the target support. — Flat.tex lines 722--726 and 1921--1924 define the mixing layer; thm:cor-fix-3c402872 retains the same component-mixture architecture. - reading of the symbol `g_2^i`: For $i\in[N_C]$, $g_2^i$ is a chosen map in $\mathcal{H}([0,1]^{d_i^C},C_i)\cap\mathcal{F}_L([0,1]^{d_i^C},C_i)$, whose inverse pulls the component law back to the unit cube. In the result split from Theorem~\ref{thm:wide_nn}: $g_2^i$ is chosen so that $(g_2^i)^{-1}_{\#}\mu_i(B)\geqslant C_g\lambda(B)/\mathbb{P}(C_i)$ for every measurable $B\subset[0,1]^{d_i^C}$, where $\mu_i=\mathbb{P}_{C_i}/\mathbb{P}(C_i)$ and $\lambda$ is cube Lebesgue measure. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: For $d_i^C=0$, $g_2^i$ is the unique map from $[0,1]^0$ to the singleton $C_i$. — Flat.tex lines 1883 and 2639--2642 distinguish the homeomorphism construction from its additional pullback domination property; the latter construction now belongs to thm:cor-fix-3c402872. - reading of the symbol `\hat{g}_2^{i,j}`: For $i\in[N_C]$ and $j\in[d]$, $\hat{g}_2^{i,j}:[0,1]^{d_i^C}\rightarrow\mathbb{R}$ denotes a coordinate network chosen to approximate $(g_2^i)_j$ uniformly on $[0,1]^{d_i^C}$. In Theorem~\ref{thm:wide_nn}: When $d_i^C=0$, $\hat{g}_2^{i,j}$ is the $j$th coordinate of the constant affine map to the singleton $C_i$, so its approximation error is zero. — Flat.tex lines 1886--1890 give the positive-dimensional coordinate approximants; the binding reading of Theorem~\ref{thm:wide_nn} specifies constant affine second blocks for singleton components. - reading of the symbol `\gamma_i`: For $i\in[N_C]$, $\gamma_i$ represents the pullback of the normalized component law $\mu_i$ through $g_2^i$, wherever $\mathbb{P}(C_i)>0$. — The second part of the proof of Theorem~\ref{thm:wide_nn}, flat.tex lines 2639-2646, defines $\mu_i$ and the exact pushforward identity. - reading of the symbol `\hat{g}_1^{i,j}`: For $i\in[N_C]$ and $j\in[d_i^C]$, $\hat{g}_1^{i,j}:[0,1]\rightarrow\mathbb{R}$ denotes a coordinate network chosen to approximate $(\gamma_i)_j$ uniformly on $[0,1]$. — The second part of the proof of Theorem~\ref{thm:wide_nn}, flat.tex lines 2643-2650, specifies the coordinate target, width, depth, and uniform approximation bound. - reading of the symbol `Z_{C_j}`: $Z_{C_j}$ denotes the uniform and Gumbel input vector for the generator of the latent variable associated with graph $C^2$-component $C_j$, with $N_{C,j}$ uniform coordinates and $N_{C,j}$ Gumbel coordinates, where $N_{C,j}$ counts the connected components of $\operatorname{supp}(P(U_{C_j}))$. In Theorem~\ref{thm:consistency}: At each sample size $n$, the simulation inputs $Z_{C_j}^{(r)}$, $r\in[m_n]$, are shared across all candidate network representations: each candidate latent generator is evaluated as $\hat g_j(Z_{C_j}^{(r)})$. These inputs are drawn as part of the common independent and identically distributed exogenous product-law draws, independently of the observational sample. — The binding input-evaluation convention accompanying Theorem~\ref{thm:consistency} specifies common exogenous draws across candidate representations, independent of the observational sample. - reading of the symbol `\hat{U}_{C_j}`: For a $C^2$-component $C_j$ of $\mathcal{G}$, $\hat{U}_{C_j}=\hat{g}_j(Z_{C_j})$ denotes the generated latent vector corresponding to $U_{C_j}$, with law $(\hat{g}_j)_{\#}P(Z_{C_j})$. — The canonical representation preceding equation~\ref{eq:structure_equation} associates latent variables with graph $C^2$-components; flat.tex lines 248 and 258 define the generated vectors; Corollary~\ref{corollary:deep_nn}'s proof compares their laws at lines 2684-2689. - reading of the symbol `N_C`: $N_C$ denotes the finite number of connected components of $\operatorname{supp}(\mathbb P)$, indexed by $i\in[N_C]$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: The finite component count of each latent generator architecture in $\mathcal F_{1,n}$ is fixed as $n$ varies. — Assumption~\ref{assump:support} defines the finite component count. The binding reading of Theorem~\ref{thm:consistency} fixes the latent generator architecture component counts as $n$ varies; Corollary~\ref{cor:rate} uses those same classes. - reading of the symbol `d_i^C`: For $i\in[N_C]$, $d_i^C$ is a nonnegative integer such that $\mathcal{H}([0,1]^{d_i^C},C_i)\cap\mathcal{F}_L([0,1]^{d_i^C},C_i)\neq\emptyset$; $d_i^C=0$ corresponds to a singleton component. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: The component cube dimensions of each latent generator architecture in $\mathcal F_{1,n}$ are fixed as $n$ varies. — The binding reading of Theorem~\ref{thm:consistency} fixes the component cube dimensions of the latent generator architectures as $n$ varies; Corollary~\ref{cor:rate} uses those same network classes. - reading of the symbol `C_f`: $C_f$ is a positive domination constant associated with the particular measure $\mathbb{P}$ and chosen homeomorphism $f$ in the lower-bound assumption. — Assumption~\ref{assumption:lower_bound}, flat.tex line 716, existentially chooses $C_f$ together with $f$ and specifies the inequality for every measurable cube subset. - reading of the symbol `\gamma`: $\gamma$ denotes a continuous measure-representing curve parametrized by $[0,1]$, with parameter measure $\lambda$ equal to Lebesgue measure on $[0,1]$. In Proposition~\ref{prop:holder_curve}: In the statement, $\gamma:[0,1]\rightarrow\operatorname{supp}(\mathbb{P})$ satisfies $\gamma_{\#}\lambda=\mathbb{P}$ and is $1/d$-H\"older continuous when $d\geqslant1$. In the proof's change of coordinates, the locally denoted $\gamma$ instead satisfies $\gamma_{\#}\lambda=f^{-1}_{\#}\mathbb{P}$; the curve representing the original measure is $f\circ\gamma$. For $d=0$, the representing curve is constant. — Proposition~\ref{prop:holder_curve} and its change-of-coordinates proof retain the existing curve meanings. Theorem~\ref{thm:wide_nn} now states only the first approximation conclusion; the componentwise curve construction belongs to the separately stated lower-bound conclusion. - reading of the symbol `\mathcal{C}_n`: For $n\geqslant0$, $\mathcal{C}_0=\{[0,1]^d\}$ and $\mathcal{C}_n=\{C_{i_1,\ldots,i_n}:i_j\in\{0,\ldots,2^d-1\}\}$ is a coordinate-plane partition satisfying property (*), with each cell the union of its $2^d$ children. In Proposition~\ref{prop:holder_curve}: For $d\geqslant1$, the full cube-coordinate construction recursively refines closed coordinate cells by cuts near their coordinate midpoints, perturbing the cuts to obtain $\mathbb{P}(\partial C)=0$ for each cell. It requires $R(\mathcal{C}_{n+1})\leqslant(1+2^{-n})R(\mathcal{C}_n)$; its cells need not be the standard dyadic cubes. Here $\mathbb{P}$ is the measure after the proof's change to cube coordinates. In Lemma~\ref{lemma:hilbert}: When $d\geqslant1$, $\mathcal{C}_n=\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\}$; these cells are closed and may share boundaries. — The proof of Proposition~\ref{prop:holder_curve}, Step 1, constructs closed coordinate cells recursively and perturbs midpoint cutting planes to ensure $\mathbb{P}(\partial C)=0$ at flat.tex lines 2542--2564. Lemma~\ref{lemma:hilbert} separately imposes the standard dyadic grid. - reading of the symbol `O_n`: For each positive refinement level $n$, $O_n$ is an ordering, hence a bijection, from $\{1,\ldots,2^{nd}\}$ onto $\{\overline{i_1\cdots i_n}:i_j\in\{0,\ldots,2^d-1\}\}$. In Lemma~\ref{lemma:hilbert}: The chosen sequence preserves parent prefixes: the first $n$ digits of $O_{n+1}(i)$ equal $O_n(j)$ when $(j-1)2^d0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$. For each sample size, every candidate empirical law is obtained by evaluating the candidate on the same $m_n$ independent draws of its uniform and Gumbel exogenous inputs, independent of the observational sample. In Theorem~\ref{thm:consistency}: The optimization ranges over network representations using $\mathcal F_{0,n}$ and $\mathcal F_{1,n}$. Each continuous structural network $\hat f_i$ satisfies $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$ on its full structural input domain. Each categorical raw structural network $q\in\mathcal F_{0,n}$ satisfies $q(x)\in[0,\infty)^{n_i}$, $\\|q(x)\\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$. For each $n$, all candidates are evaluated on the same $m_n$ independent and identically distributed draws of all exogenous inputs according to their specified product law, including uniform inputs, generator Gumbel inputs, and categorical Gumbel inputs, independent of the observational sample; representations with the same latent population laws may yield different empirical constraints. — Corollary~\ref{cor:rate} explicitly restricts each continuous structural network in the empirical optimization to satisfy the uniform output $1$-norm bound on its full structural input domain, in addition to its categorical restrictions. - reading of the symbol `\mu_T`: $\mu_T$ is the given measure on treatment intervention values used in the outer expectation $\mathbb{E}_{t\sim\mu_T}$. In Corollary~\ref{cor:rate}: $\mu_T=\operatorname{Unif}([t_1,t_2])$, $\mu_T\ll P$, and $A_T^2=\int_{t_1}^{t_2}(d\mu_T/dP(t))^2P(dt)<\infty$, where $P$ is the reference probability measure on the treatment space. — Corollary~\ref{cor:rate} specifies the uniform intervention law and additionally requires absolute continuity with respect to the reference treatment probability measure $P$ of Proposition~\ref{prop:Lip_ATE}, with square-integrable density. - reading of the symbol `F_*`: $F_*=\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\mathcal{M}^*}[F(V_1(t),\ldots,V_{n_V}(t))]$ denotes the population objective evaluated at the true SCM. In Corollary~\ref{cor:rate}: $F_*=\frac{1}{t_2-t_1}\int_{t_1}^{t_2}\mathbb{E}_{\mathcal{M}^*}[Y(t)]\,dt$, since $F(\boldsymbol{V})=Y$ and $\mu_T=\operatorname{Unif}([t_1,t_2])$. — The definition at flat.tex line 781 gives the population objective. Corollary~\ref{cor:rate} at line 812 specializes $F$ and $\mu_T$. - reading of the symbol `\underline{F}^L`: $\underline{F}^L$ is the population optimal objective value in equation~\ref{eq:consistent_limiting}, minimizing $\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\hat{\mathcal{M}}}[F(V_1(t),\ldots,V_{n_V}(t))]$ over canonical SCMs on $\mathcal{G}$ with structural functions in $\mathcal{F}_L^K$ and all latent distributions $P(\boldsymbol{U})$, subject to $W(P^{\hat{\mathcal{M}}}(\boldsymbol{V}),P^{\mathcal{M}^*}(\boldsymbol{V}))=0$. In Theorem~\ref{thm:consistency}: $\underline F^L$ uses the class $\mathcal F_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$ in equation~\ref{eq:consistent_limiting}, with the supremum taken over the full structural input domain; categorical structural functions are normalized propensity vectors. The conclusion evaluates this population optimal value at $L=\hat L_f$. — The prose introducing equation~\ref{eq:consistent_limiting} defines the population optimal value. Theorem~\ref{thm:consistency} explicitly retains $\mathcal F_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$ and uses $\underline F^{\hat L_f}$ in its conclusion. - reading of the symbol `P_n^{\mathcal{M}^*}`: $P_n^{\mathcal{M}^*}(\boldsymbol V)=n^{-1}\sum_{a=1}^n\delta_{\boldsymbol V^{(a)}}$ is the empirical observational law of the $n$ observed joint values from $P^{\mathcal{M}^*}(\boldsymbol V)$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: The $n$ observations defining $P_n^{\mathcal M^*}(\boldsymbol V)$ are independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. — The binding readings of Theorem~\ref{thm:consistency} and Corollary~\ref{cor:rate} both specify that the observational empirical law is formed from $n$ independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. - reading of the symbol `m_n`: $m_n$ is a prescribed positive integer sequence, indexed by the observational sample size $n$, giving the simulation sample size for each candidate model. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: The simulation sizes satisfy $m_n=\Omega(n)$; equality $m_n=n$ is not required. — Theorem~\ref{thm:consistency} assumes $m_n=\Omega(n)$, and Corollary~\ref{cor:rate} inherits this assumption. - reading of the symbol `P_{m_n}^{\hat{\mathcal{M}}}`: $P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V})$ denotes the empirical observational law for the candidate $\hat{\mathcal{M}}$, namely $m_n^{-1}\sum_{r=1}^{m_n}\delta_{\boldsymbol{V}^{(r)}}$, and is indexed by the candidate rather than denoting only the law at an optimizer. In Theorem~\ref{thm:consistency}: For each $n$, $P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V})$ is obtained by evaluating every candidate network representation on the same $m_n$ independent draws of all exogenous inputs, including uniform inputs, generator Gumbel inputs, and categorical Gumbel inputs; these draws are independent of the observational sample. Different network representations with the same latent population laws may give different common-draw empirical distributions. In Corollary~\ref{cor:rate}: For each $n$, $P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V})$ is obtained by evaluating every candidate network representation on the same $m_n$ independent draws of all exogenous inputs, including uniform inputs, generator Gumbel inputs, and categorical Gumbel inputs; these draws are independent of the observational sample. Different network representations with the same latent population laws may give different common-draw empirical distributions. — Corollary~\ref{cor:rate} explicitly evaluates every candidate on common independent uniform and Gumbel exogenous draws, independent of the observational sample, and inherits the assumptions of Theorem~\ref{thm:consistency}. - reading of the symbol `\mathcal{F}_{0,n}`: $\mathcal{F}_{0,n}$ is the prescribed structural network class at sample size $n$, with the input and output dimensions appropriate to the node and with truncation applied coordinatewise. In Theorem~\ref{thm:consistency}: In the parameter-space construction, a continuous-node candidate is a function $\hat f_i\in\mathcal{F}_{0,n}$ satisfying $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$ on its full structural input domain. A categorical-node propensity is represented by $x\mapsto q(x)/\\|q(x)\\|_1$ with $q\in\mathcal{F}_{0,n}$ restricted to satisfy $q(x)\in[0,\infty)^{n_i}$ and $\\|q(x)\\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$. The depth expression $\Theta(\log d_{\max}^{\text{in}})$ uses $\Theta(1+\log d_{\max}^{\text{in}})$ when $d_{\max}^{\text{in}}=1$. In Corollary~\ref{cor:rate}: A continuous-node candidate is a function $\hat f_i\in\mathcal{F}_{0,n}$ satisfying $\sup_x\\|\hat f_i(x)\\|_1\leqslant K$ on its full structural input domain. A categorical-node propensity is represented by $x\mapsto q(x)/\\|q(x)\\|_1$ with $q\in\mathcal{F}_{0,n}$ restricted to satisfy $q(x)\in[0,\infty)^{n_i}$ and $\\|q(x)\\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$. The depth expression $\Theta(\log d_{\max}^{\text{in}})$ uses $\Theta(1+\log d_{\max}^{\text{in}})$ when $d_{\max}^{\text{in}}=1$. — Theorem~\ref{thm:consistency} defines the architecture, depth convention, categorical restrictions, and continuous output restriction. Corollary~\ref{cor:rate} uses these network classes and assumes the hypotheses of Theorem~\ref{thm:consistency}. - reading of the symbol `\mathcal{F}_{1,n}`: $\mathcal{F}_{1,n}$ is the latent-generator network class indexed by $n$, with architecture $\mathcal{NN}^{\infty,K}(\Theta(d_{\max}^{U}),L_{1,n},\Theta((d_{\max}^{U})^2),L_{2,n},\tau_n)$ and coordinatewise output truncation to $[-K,K]$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: The finite component counts and component cube dimensions of the latent generator architectures are fixed as $n$ varies. — The architecture display in Theorem~\ref{thm:consistency} defines the class, and its binding reading fixes the finite component counts and component cube dimensions as $n$ varies. Corollary~\ref{cor:rate} uses these same classes and assumes Theorem~\ref{thm:consistency}. - reading of the symbol `\epsilon_n`: $\epsilon_n$ denotes a positive deterministic approximation-tolerance sequence indexed by $n\in\mathbb N$, with its approximation role specified locally. In Proposition~\ref{prop:convg_opt}: $\epsilon_n>0$ for $n\in\mathbb N$, $\epsilon_n\to0$, and it bounds the distance from each point of $\tilde\Theta_\infty$ to an available point of $\Theta_n$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the radius definition, $\alpha_n=\epsilon_n+s_n$, where $\epsilon_n>0$ is chosen with sufficiently large constants in $\epsilon_n=O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+\sum_{i=1}^2L_{i,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right)$. In \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: In the empirical generator concentration argument, $\epsilon_n$ is a local deterministic deviation tolerance in the bound $\sup_{\theta_{i,j}}W(P(f_{\theta_{i,j}}(Z_{i,j})),P_{m_n}(f_{\theta_{i,j}}(Z_{i,j})))\leqslant\epsilon_n+\epsilon'_n$; it is chosen as $\epsilon_n=Cm_n^{-1/(d_{\max}^U+2)}\log m_n$ with fixed $C>0$ sufficiently large. This local choice is distinct from the architecture approximation tolerance in the constraint radius. — The displayed radius definitions in Theorem~\ref{thm:consistency} and Corollary~\ref{cor:rate} specify the architecture approximation tolerance. The printed concentration argument at lines 2897--2902 separately chooses a local deviation tolerance. The current texts do not specify the representation $\eta_n\in\mathcal A_n$, the error $a_n$, or the inequality $C_ga_n\leqslant\epsilon_n$. - reading of the symbol `s_n`: $s_n$ is a deterministic tolerance sequence indexed by $n$, specified by its displayed asymptotic order rather than by a unique numerical formula. In Theorem~\ref{thm:consistency}: $s_n>0$ is chosen as a sufficiently large constant multiple of the quantity in the displayed order $O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2 n+\log(nm_n)\lambda_n\right)$, where $D_U=\sum_{j=1}^{n_U}d_j^U$ includes the non-Gumbel latent variables only and $D=\max\{1,D_U\}$. In Corollary~\ref{cor:rate}: $s_n>0$ is chosen as a sufficiently large constant multiple of the quantity in the displayed order $O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2 n+\log(nm_n)\lambda_n\right)$, where $D_U=\sum_{j=1}^{n_U}d_j^U$ includes the non-Gumbel latent variables only and $D=\max\{1,D_U\}$. — Corollary~\ref{cor:rate} explicitly specifies the order involving $D$, $D_U$, and the network-depth term, with positive tolerances chosen as sufficiently large constant multiples. - reading of the symbol `\alpha_n`: $\alpha_n=\epsilon_n+s_n$ is a deterministic constraint tolerance sequence indexed by the observational sample size $n$. In Theorem~\ref{thm:consistency}: $\alpha_n=\epsilon_n+s_n>0$, where the positive tolerances $\epsilon_n$ and $s_n$ are chosen with sufficiently large constants in their displayed asymptotic orders. In Corollary~\ref{cor:rate}: $\alpha_n=\epsilon_n+s_n>0$, where the positive tolerances $\epsilon_n$ and $s_n$ are chosen as sufficiently large constant multiples of the respective quantities in their displayed asymptotic orders. — Corollary~\ref{cor:rate} explicitly specifies $\alpha_n=\epsilon_n+s_n$, with positive tolerances chosen as sufficiently large constant multiples of the respective displayed quantities. - reading of the symbol `\tau_n`: $\tau_n$ is a sequence indexed by $n$, used locally either as a generator temperature or as an additive tolerance in an approximate Lipschitz bound. In Proposition~\ref{prop:convg_opt}: $\tau_n$ is the additive tolerance in equation~\ref{eq:aprox_lip}, and $\tau_n\rightarrow0$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the network architecture and growth conditions, $\tau_n>0$ is the Gumbel-softmax generator temperature and $\tau_n\rightarrow0$; in the proof's application of the optimization proposition, its additive tolerance is assigned the separate value $4(1+\log(nm_n))\lambda_n$. In \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: When verifying equation~\ref{eq:aprox_lip}, $\tau_n$ denotes the additive tolerance $4(1+\log(nm_n))\lambda_n$, which tends to zero under the stated regularization condition; this assignment does not identify it with the generator temperature. — The printed argument in lem:consistency-fournier-theorem-2 assigns $\tau_n=4(1+\log(nm_n))\lambda_n$ when verifying equation~\ref{eq:aprox_lip}. This is the additive tolerance of Proposition~\ref{prop:convg_opt}, distinct from the generator temperature in Theorem~\ref{thm:consistency}. - reading of the symbol `\lambda_n`: $\lambda_n>0$, indexed by the observational sample size $n$, is the regularization parameter in $S_{\lambda_n}(P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V}),P_n^{\mathcal{M}^*}(\boldsymbol{V}))$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: $\log(nm_n)\lambda_n\rightarrow0$ as $n\rightarrow\infty$. — Line 194 defines Sinkhorn regularization with $\lambda>0$; equation~\ref{eq:opt_empiri} uses $\lambda_n$ for the two empirical laws. The growth condition of Theorem~\ref{thm:consistency} requires $(\log(nm_n)\lambda_n)^{-1}\rightarrow\infty$, and Corollary~\ref{cor:rate} assumes that theorem's conditions. - reading of the symbol `d`: $d$ is local notation for a fixed dimension, a fixed metric, or a regression-difference function, with its role determined by its defining expression. In Proposition~\ref{prop:holder_curve}: $d$ is the cube dimension in the homeomorphism $f:[0,1]^d\rightarrow K$ from Assumption~\ref{assumption:lower_bound}, rather than the ambient dimension $D$; the proof treats $d=0$ separately and uses the exponent $1/d$ only for $d\geqslant1$. In Lemma~\ref{lemma:hilbert}: $d$ is the dimension of the cube $[0,1]^d$ being partitioned. In Theorem~\ref{thm:wide_nn}: $d$ is the ambient dimension of the target law on $\mathbb{R}^d$, distinct from the component cube dimensions $d_i^C$. In Proposition~\ref{prop:convg_opt}: $d:M\times M\rightarrow[0,\infty)$ is the metric of the given space $(M,d)$. In Theorem~\ref{thm:consistency}: In the proof's parameter space, $d(\theta,\theta')=\sum_{k=1}^{n_V}\\|f_k-f_k'\\|_\infty+\sum_{k=1}^{n_U}W(P(U_k),P(U_k'))$; occurrences of $d$ in imported distribution bounds retain their local dimension meaning. In Lemma~\ref{lemma:sup_wasserstein}: $d$ is the common output dimension of the functions in $\mathcal F$ and the dimension used in the output test-function covering bound; the input law $\mu$ is on $\mathbb R^{d'}$. In the result split from Theorem~\ref{thm:wide_nn}: $d$ is the ambient dimension of the target law on $\mathbb R^d$, distinct from the component cube dimensions $d_i^C$. In Proposition~\ref{prop:Lip_ATE}: In the proof's test-function calculation, $s=(a,t)$ and $d(s)=\hat f_y(s)-f_y(s)$ on the allowed covariate domain $Q$; $f_y$ and $\hat f_y$ are the structural regression versions for the respective models. — The proof of Proposition~\ref{prop:Lip_ATE} in discovery/core.json explicitly defines $d(s)=\hat f_y(s)-f_y(s)$. - reading of the symbol `d_1`: $d_1$ denotes a locally specified input or source dimension, including the input dimension in $\mathcal{NN}_{d_1,d_2}$, or an explicitly specified averaged metric on evaluation vectors. In Lemma~\ref{lemma:push-forward_err}: $K\subset\mathbb{R}^{d_1}$ is the common domain supporting $\mu$ and $\hat{\mu}$. In Theorem~\ref{thm:wide_nn}: In the proof's notation for the combined maps $\hat{g}_i$, $d_1=1$. In \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $d_1$ is the averaged metric on length-$m$ evaluation vectors, $d_1(u,w)=\frac1m\sum_{i=1}^m\|u_i-w_i\|$, with $m$ a positive integer. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $d_1$ is the averaged metric on length-$N$ evaluation vectors, $d_1(u,w)=\frac1N\sum_{i=1}^N\|u_i-w_i\|$; in the covering number appearing in the probability bound, $N=2m$. — lem:sup-wasserstein-anthony-theorem-17-1 defines $d_1$ on length-$N$ evaluation vectors and uses the covering number with $N=2m$. - reading of the symbol `d_2`: $d_2$ denotes the locally specified output or target dimension of a map, including the output dimension in $\mathcal{NN}_{d_1,d_2}$. In Lemma~\ref{lemma:push-forward_err}: $F,\hat{F}:K\rightarrow\mathbb{R}^{d_2}$, so $d_2$ is their common target dimension. In Theorem~\ref{thm:wide_nn}: In the proof's notation for the combined maps $\hat{g}_i$, $d_2=d$. — The push-forward definition at line 194 specifies $f:\mathbb{R}^{d_1}\rightarrow\mathbb{R}^{d_2}$; the network notation at lines 255 and 779 uses input and output dimensions. The construction at line 1896 assigns $d_2=d$ locally. - reading of the symbol `D`: $D$ is a deterministic geometric parameter specified by its local definition. In Theorem~\ref{thm:approximation}, Proposition~\ref{prop:holder_curve}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}, the result split from Theorem~\ref{thm:wide_nn}: In the compact-support assumption, $D$ is the ambient Euclidean dimension in $K\subset\mathbb{R}^{D}$, rather than the cube dimension $d$. In Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: In the constants controlling the treatment-effect comparison, $D=\max\left(\{1\}\cup\left\{\frac{n_i-1}{2}:V_i\text{ is categorical}\right\}\right)$ and $L_y=(LD+1)^{n_V}$; this $D$ can be nonintegral and equals $1$ when there are no categorical nodes. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the sampling tolerance $s_n$, $D=\max\{1,D_U\}$, where $D_U=\sum_{j=1}^{n_U}d_j^U$ includes the non-Gumbel latent variables only; this local $D$ is distinct from the ambient dimension in the compact-support assumption and from the categorical factor in the treatment-effect comparison. In Theorem~\ref{thm:approximation}: In the proof's error recurrence, $D=\max\left(\{0\}\cup\{n_i-1:V_i\text{ is categorical}\}\right)$ and $A=\max\{1+L,1+LD/2\}$; this local $D$ is zero when there are no categorical nodes. — The proof of Theorem~\ref{thm:approximation} in discovery/core.json defines an additional local categorical-cardinality factor $D$. - reading of the symbol `d_{\max}^{\text{in}}`: $d_{\max}^{\text{in}}=\max_{i\in[n_V]}d_i^{\text{in}}$, where $d_i^{\text{in}}$ is the input dimension of $f_i(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$ in the canonical structural equations. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: $d_{\max}^{\text{in}}\geqslant1$; when $d_{\max}^{\text{in}}=1$, the structural-network depth expression $\Theta(\log d_{\max}^{\text{in}})$ means $\Theta(1+\log d_{\max}^{\text{in}})$. In Corollary~\ref{corollary:deep_nn}: $d_{\max}^{\text{in}}>0$. — Corollary~\ref{corollary:deep_nn} explicitly assumes $d_{\max}^{\text{in}}>0$. - reading of the symbol `d_{\max}^{\text{out}}`: $d_{\max}^{\text{out}}=\max_{i\in[n_V]}d_i^{\text{out}}$, where $d_i^{\text{out}}=d_i$ for a continuous structural function and $d_i^{\text{out}}=n_i$ for a categorical propensity vector. — Corollary~\ref{corollary:deep_nn} defines the maximum over the functions in equation~\ref{eq:structure_equation}, whose categorical outputs are propensity vectors indexed by $[n_i]$; the proof defines their individual output dimensions. - reading of the symbol `d_{\max}^U`: $d_{\max}^U=\max_{j\in[n_U]}d_j^U$, where $d_j^U$ is the dimension of a canonical latent variable $U_j$ whose distribution is approximated by a generator, distinct from the separately supplied categorical Gumbel noises. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: $d_{\max}^U\geqslant1$; it remains the maximum individual non-Gumbel latent dimension, distinct from the total dimension $D_U=\sum_{j=1}^{n_U}d_j^U$ and $D=\max\{1,D_U\}$. In Corollary~\ref{corollary:deep_nn}: $d_{\max}^U>0$. — Corollary~\ref{corollary:deep_nn} explicitly assumes $d_{\max}^U>0$. - reading of the symbol `d_{\mathcal{F}}`: $d_{\mathcal F}<\infty$ is a strict upper bound on the pseudo-dimensions of the scalar function classes specified in its local use. In Lemma~\ref{lemma:sup_wasserstein}: For every $j\in[d]$, $\mathcal F_j=\{x\mapsto f_j(x):f\in\mathcal F\}$ satisfies $\operatorname{Pdim}(\mathcal F_j)0$ are arbitrary positive softmax weights; normalization is not required in the statement. The proof normalizes them without loss of generality, and the limiting categorical probabilities are $p_i/\sum_{k=1}^n p_k$. In Lemma~\ref{lemma:sinkhorn_approx}: In the probability-simplex notation, $p_i$ is a coordinate satisfying $p_i\geqslant0$ and $\sum_{i=1}^n p_i=1$. In Theorem~\ref{thm:approximation}: In the categorical case of the proof, $(p_1,\ldots,p_{n_i})=f(\text{pa}(v_k),\boldsymbol{u}_{V_k})$ is the conditional categorical probability vector evaluated at the displayed parent and latent values; its coordinates are nonnegative and sum to one. In Proposition~\ref{prop:equivalent_g_constrianed}: In the probability-simplex notation, $p_i$ is a coordinate satisfying $p_i\geqslant0$ and $\sum_{i=1}^n p_i=1$. In the result split from Theorem~\ref{thm:wide_nn}: $p_i=\mathbb P(C_i)$ for $i\in[N_C]$, and $\sum_{i=1}^{N_C}p_i=1$; these are the probabilities used to combine the support components. In Proposition~\ref{prop:holder_curve}: In the printed partition construction at level $n+1$, $p_i=\mathbb P(C_{O_{n+1}(i)})$ for $1\leqslant i\leqslant2^{(n+1)d}$. Here $\mathbb P$ is the reduced cube measure and $C_{O_{n+1}(i)}$ is an ordered partition cube. These masses give the lengths of the corresponding parameter intervals and the denominators in the polygonal curve; domination by Lebesgue measure makes them positive. In Theorem~\ref{thm:consistency}: In the latent approximation construction, $p_i$ denotes the probability mass of component $C_i$ under the particular latent target law being approximated. These component probabilities sum to one and weight the component approximation errors and the hard selected mixture. In the categorical measurability argument, $p_i$ instead denotes coordinate $i$ of the categorical propensity vector evaluated at the structural input; these coordinates are nonnegative and sum to one, and zero coordinates are allowed. — The proof of Theorem~\ref{thm:consistency} uses component probabilities in the latent approximation construction and separately introduces a categorical propensity vector $p$ when defining $r_i=\exp(-g_i)/p_i$ for $p_i>0$ and $r_i=\infty$ for $p_i=0$. - reading of the symbol `p_\nu`: Writing $X=\text{Pa}(T)$, $p_\nu(t\mid X=x)$ is the conditional treatment density of $\nu(dt\mid x)$ with respect to the comparison measure $P(dt)$, so that $\nu(dt\mid x)=p_\nu(t\mid x)P(dt)$; its overlap lower bound holds for $\nu_X$-almost every $x$ and $t\in[t_1,t_2]$. — Proposition~\ref{prop:Lip_ATE} states the conditional overlap bound. Its proof uses $\int a(x,t)\nu(dt\mid x)\nu(dx)\geqslant\delta\int_{t_1}^{t_2}\int a(x,t)\nu(dx)P(dt)$, identifying the reference measure and the covariate marginal. - reading of the symbol `\delta`: $\delta>0$ denotes a locally specified positive real parameter. In Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: $\delta$ in the overlap premise is the lower bound satisfying $p_\nu(t\mid\text{Pa}(T)=x)\geqslant\delta$ for $x$ almost surely and $t\in[t_1,t_2]$. In Lemma~\ref{lemma:gumbel_err}: $\delta$ is the auxiliary threshold for the gaps between $G_i+\log p_i$; at the end of the proof it is chosen as $\delta=-\tau\log\tau$. In Proposition~\ref{prop:counterexample_foral}: In the proof, $\delta$ parameterizes $\mathcal{M}_\delta$: the covariate has an atom at $\delta$, and the outcome equation contains $X/\delta$ for $X\geqslant0$; the construction takes $\delta\rightarrow0$. In Lemma~\ref{lemma:sup_wasserstein}: $\delta$ is an arbitrary positive tolerance; the probability bound is for each fixed pair $\delta,\epsilon>0$. In the proof, $\mathcal H_\delta$ has uniform-norm covering radius $\delta/2$, so replacing a test function by its representative contributes at most $\delta$ to the difference of its empirical and population integrals. In Theorem~\ref{thm:consistency}: In the simulation-concentration argument, the empirical Wasserstein bound is applied with $\delta=r_n$, where $H_n=1+L_{1,n}+L_{2,n}$ and $r_n=m_n^{-1/(D+2)}\log m_n+H_nm_n^{-1/2}\log m_n$; its other tolerance is $\epsilon=A_0r_n$. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: For a component of positive dimension $q=d_i^C$, $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$ is the source quantization parameter, and the quantization mesh is $\delta/n$; here $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, with $A_q\geqslant1$ depending only on $q$. In Corollary~\ref{cor:rate}: Separately from the overlap parameter, the proof's simulation-concentration argument chooses the local tolerance $\delta=r_n$, where $H_n=1+L_{1,n}+L_{2,n}$ and $r_n=m_n^{-1/(D+2)}\log m_n+H_nm_n^{-1/2}\log m_n$, for sufficiently large $n$; the other tolerance is $\epsilon=Cr_n$ for a sufficiently large fixed positive constant $C$. — Lemma~\ref{lemma:sup_wasserstein} retains the tolerance $\delta$ in its conclusion, while its proof explicitly chooses $\mathcal H_\delta$ as a $\delta/2$-net. - reading of the symbol `c`: $c$ denotes a locally specified scalar, categorical configuration, or real-valued function, according to its defining expression. In Proposition~\ref{prop:counterexample}, Proposition~\ref{prop:counterexample_foral}: $c>0$ is chosen together with $\mathcal{M}^*$ before quantifying over $\epsilon>0$, and is independent of $\epsilon$; it is a strict lower bound for $\|\text{ATE}_{\mathcal{M}^*}-\text{ATE}_{\mathcal{M}_\epsilon}\|$. In Lemma~\ref{lemma:sinkhorn_approx}: In the proof's final self-term calculation, $c=W_{1,\lambda}(\nu,\nu)$ is the entropy-regularized self-transport value, with $\|c\|\leqslant(1+2\log m)\lambda$; it is nonpositive under the specified entropy convention. In Lemma~\ref{lemma:sup_wasserstein}: $c>0$ is the concentration constant multiplying $n\epsilon^2$ in the negative term of the probability exponent; it depends only on $d,K$ and is independent of $n,\delta,\epsilon$. In Proposition~\ref{prop:Lip_ATE}: In the proof, $c(t)=\int\hat f_y(a,t)\nu_A(da)-\int\hat f_y(a,t)\mu_A(da)$ is the contribution from changing the observational marginal of $A$, with $\nu_A$ and $\mu_A$ the respective observational marginals. In Theorem~\ref{thm:consistency}: In the observational concentration argument, $c$ ranges over the fixed finite set of possible complete categorical label vectors: $p_c=P(C=c)$ and $\mu_c=P((U_1,\ldots,U_{n_U})\mid C=c)$, the latter being used for $p_c>0$. In Corollary~\ref{cor:rate}: In categorical types $(u,c)$ and conditioning expressions, $c$ is a complete categorical label vector, with $p_c=P(C=c)$ and $\mu_c=P((U_1,\ldots,U_{n_U})\mid C=c)$ for $p_c>0$. Separately, in the concentration exponent $-cC^2m_nr_n^2$, $c>0$ is a fixed concentration constant. — The proof of Corollary~\ref{cor:rate} uses categorical types $(u,c)$, conditions on $C=c$, and uses a positive constant $c$ in its concentration exponent. - reading of the symbol `t_1`: $t_1$ is a real scalar whose role is local to the intervention interval or the Lipschitz comparison. In Proposition~\ref{prop:Lip_ATE}, Corollary~\ref{cor:rate}: $t_1$ is the fixed lower endpoint of the intervention interval $[t_1,t_2]$. In Corollary~\ref{cor:rate}: $t_10$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$. Here $\mathcal P_n=\{P(\boldsymbol U):\boldsymbol U\text{ is the latent distribution of }\hat{\mathcal M}\in\text{NCM}_{\mathcal G}(\mathcal F_{0,n},\mathcal F_{1,n})\}$, identified with its tuple of independent latent marginal laws. This product records population model parameters; the empirical optimization ranges over their network representations, including distinct representations inducing identical latent population laws. — The proof of Theorem~\ref{thm:consistency} defines the product parameter set. Its statement restricts continuous structural networks by the uniform output $1$-norm bound and categorical networks by positivity and the Lipschitz bound on normalized propensities; its binding reading distinguishes network representations with identical latent population laws. - reading of the symbol `\Theta_\infty`: $\Theta_\infty=\overline{\bigcup_{n=1}^{\infty}\Theta_n}$, with closure taken in the ambient metric space $(M,d)$. In Proposition~\ref{prop:convg_opt}: $\Theta_\infty$ is compact and contains the nested sets $\Theta_n$; $\{\theta\in\Theta_\infty:g(\theta)=0\}$ is nonempty. In Theorem~\ref{thm:consistency}: The closure is that of the neural-model parameter sets in the metric $d(\theta,\theta')=\sum_{k=1}^{n_V}\\|f_k-f_k'\\|_\infty+\sum_{k=1}^{n_U}W(P(U_k),P(U_k'))$ on the product space of structural functions and latent probability measures. — The proof of Theorem~\ref{thm:consistency} defines the closure after specifying the product parameter space and its metric. - reading of the symbol `\tilde{\Theta}_\infty`: $\tilde\Theta_\infty$ is a compact subset of $\Theta_\infty$ approximated by $\Theta_n$: for every $\theta_0\in\tilde\Theta_\infty$ and every $n$, some $\theta\in\Theta_n$ satisfies $d(\theta,\theta_0)\leqslant\epsilon_n$. In Proposition~\ref{prop:convg_opt}: $\{\theta\in\tilde\Theta_\infty:g(\theta)=0\}$ is nonempty. — Proposition~\ref{prop:convg_opt} retains its compact approximated subset. The proof of Theorem~\ref{thm:consistency} in discovery/core.json instead constructs representations $\eta_n$ approximating $\theta^*$ and does not define $\tilde\Theta_\infty$. - reading of the symbol `\delta_n`: $\delta_n$ denotes a deterministic error tolerance indexed by $n$ and tending to zero. In Proposition~\ref{prop:convg_opt}: $\delta_n>0$ for $n\in\mathbb N$, and $\sup_{\theta\in\Theta_\infty}\|g_n(\theta)-g(\theta)\|\leqslant\delta_n$. In \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: For sufficiently large $n$, the local observational latent deviation tolerance is $\delta_n=C_1n^{-1/\max\{d_{\max}^U,2\}}\log^2 n$, with fixed $C_1>0$ chosen sufficiently large; this is distinct from the abstract uniform constraint-error tolerance. — The explicit choice beginning the Fournier--Guillin invocation in lem:consistency-fournier-theorem-2 fixes the local sequence. - reading of the symbol `g_n`: $\{g_n\}_{n\in\mathbb{N}}$ is a sequence of real-valued constraint functions on $\Theta_\infty$. In Proposition~\ref{prop:convg_opt}: Each $g_n$ is continuous and nonnegative, with $\sup_{\theta\in\Theta_\infty}\|g_n(\theta)-g(\theta)\|\leqslant\delta_n$, and the approximate Lipschitz bound has common coefficient $L_g$ and additive tolerance $\tau_n$. In \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: The notation is the empirical constraint $g_n(\theta)=S_{\lambda_n}(P_n^{\theta^*}(\boldsymbol V),P_{m_n}^{\theta}(\boldsymbol V))$, compared with the population constraint $g(\theta)=W(P^{\theta^*}(\boldsymbol V),P^{\theta}(\boldsymbol V))$. — The displayed empirical-constraint expressions in lem:consistency-fournier-theorem-2 specify this local use. The proof of Theorem~\ref{thm:consistency} instead defines $h_n(\eta)$ on network representations. - reading of the symbol `g`: $g$ denotes a real-valued function on the domain specified by its local use. In Theorem~\ref{thm:approximation}: In the proof, $g$ ranges over real-valued $1$-Lipschitz Wasserstein dual test functions of $(\boldsymbol{u},\boldsymbol{t},v_1,\ldots,v_k)$. In Lemma~\ref{lemma:push-forward_err}: In the proof, $g:\mathbb{R}^{d_2}\rightarrow\mathbb{R}$ is an arbitrary $1$-Lipschitz test function. In Lemma~\ref{lemma:Lip_condition}: In the proof, $g$ ranges over real-valued $1$-Lipschitz test functions of $(t,y)$ in the Wasserstein dual formula. In Proposition~\ref{prop:convg_opt}: $g:\Theta_\infty\rightarrow\mathbb{R}$ is the limiting constraint, with $\sup_{\theta\in\Theta_\infty}\|g_n(\theta)-g(\theta)\|\leqslant\delta_n$; its zero set defines the limiting feasible regions. In Theorem~\ref{thm:consistency}: $g(\theta)=W(P^{\theta^*}(\boldsymbol{V}),P^\theta(\boldsymbol{V}))$ is the population observational discrepancy, rather than an empirical discrepancy. In \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $g$ ranges over functions in a nonempty class $\mathcal G\subset\{g:X\rightarrow[0,1]\}$ with finite pseudo-dimension $v$. No Lipschitz condition on $g$ is specified. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $g:T\times X\rightarrow[0,1]$ is jointly Borel, where $T$ and $X$ are standard Borel spaces, and $\mathcal G=\{g(t,\cdot):t\in T\}\ne\varnothing$. For each $t\in T$, the squared loss is $(g(t,x)-y)^2$. No pseudo-dimension bound or Lipschitz condition on $g$ is imposed. — lem:sup-wasserstein-anthony-theorem-17-1 specifies standard Borel spaces $T,X$, a jointly Borel map $g:T\times X\rightarrow[0,1]$, and the section class $\mathcal G=\{g(t,\cdot):t\in T\}\ne\varnothing$. - reading of the symbol `L_g`: $L_g$ is a finite nonnegative Lipschitz bound. In Proposition~\ref{prop:convg_opt}: $L_g$ is independent of $n$ and is the coefficient in $\|g_n(\theta_1)-g_n(\theta_2)\|\leqslant L_g d(\theta_1,\theta_2)+\tau_n$; the limiting function $g$ is $L_g$-Lipschitz. In Lemma~\ref{lemma:Lip_condition}: In the proof, $L_g=2K\max\{L,1\}$ bounds the Lipschitz constant of $g_0(t,y)=(\hat f(t)-f(t))y$ on $\{(t,y):\\|(t,y)\\|_\infty\leqslant K\}$. — Equation~\ref{eq:aprox_lip} uses a common coefficient for the sequence $g_n$; the proof of Lemma~\ref{lemma:Lip_condition} explicitly sets $L_g=2K\max\{L,1\}$. - reading of the symbol `B`: $B$ is a locally specified measurable test set or positive scalar bound. In the result split from Theorem~\ref{thm:wide_nn}: In the componentwise lower-bound premise, $B$ ranges over measurable subsets of $[0,1]^{d_i^C}$; the component measure and its parameterizing map satisfy the domination inequality of Assumption~\ref{assumption:lower_bound} on these sets. Separately, in the proof's clipping and mixing estimates, $B>0$ is a fixed scalar chosen so that every support component lies in $[-B,B]^d$, and the second-block outputs are clipped into this box. In Proposition~\ref{prop:holder_curve}, Corollary~\ref{corollary:deep_nn}, Proposition~\ref{prop:counterexample_foral}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the lower-bound premise, $B$ ranges over measurable subsets of $[0,1]^d$ and $\mathbb{P}(f(B))\geqslant C_f\lambda(B)$; for a component with dimension $d_i^C$, the corresponding test sets lie in $[0,1]^{d_i^C}$. In Proposition~\ref{prop:Lip_ATE}: $B$ is the scalar bound $B=K(1+2qL_y)$, where $q=\sum_{i:V_i\text{ is continuous}}d_i+\|\{i:V_i\text{ is categorical}\}\|$ and $L_y=(LD+1)^{n_V}$; it enters $C_W=4L_yB+2B\max\{L_y,1\}$. In Theorem~\ref{thm:approximation}, Lemma~\ref{lemma:push-forward_err}, Theorem~\ref{thm:wide_nn}, Lemma~\ref{lemma:sup_wasserstein}, \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: In a push-forward expression $f_\#\mu(B)$, $B$ is an arbitrary measurable subset of the codomain of $f$. In Theorem~\ref{thm:approximation}: In the proof's error recurrence, $B=\max\left(\{1\}\cup\{d_i:V_i\text{ is continuous}\}\cup\{n_i(n_i-1)/2:V_i\text{ is categorical}\}\right)$. In Theorem~\ref{thm:consistency}: In the proof's model-comparison recurrence, $B$ is a fixed positive bound on the coefficients of the structural approximation errors, chosen together with $A\geqslant1$ so that $e_{k+1}\leqslant Ae_k+B d(\theta,\theta')$. Separately, in the derivation of the pseudo-dimension bound, $B=CH_n^2$, where $H_n=1+L_{1,n}+L_{2,n}$ and $C$ is the constant in that derivation; this latter scalar may vary with $n$. In Corollary~\ref{corollary:deep_nn}: In the structural extension and approximation construction, $B=B_0H^{n_V}$, where $m=d_{\max}^{\text{in}}$, $H=1+2Lm$, and $B_0=\max\left(\{1,R,K\}\cup\{n_i:V_i\text{ is categorical}\}\cup\{\\|f_i(a_i)\\|_\infty:V_i\text{ is continuous}\}\right)$; $a_i$ has zero in each real input coordinate and one in each categorical input coordinate. This fixed positive scalar is independent of the intervention and network budgets and defines the structural input cubes $Q_i=[-B,B]^{m_i}$. In Corollary~\ref{cor:rate}: In the proof's model-comparison recurrence, $B$ is a fixed positive bound on the coefficients of the structural approximation errors, chosen together with $A\geqslant1$ so that $e_{k+1}\leqslant Ae_k+B d(\theta,\theta')$; it is independent of the sample size and the compared model parameters. — The proof of thm:cor-fix-3c402872 clips second-block outputs into a fixed box $[-B,B]^d$ containing all components and subsequently uses this bound in the mixing estimate. - reading of the symbol `\mathcal{N}_1`: $\mathcal{N}_1(\epsilon,\mathcal{F},n)=\sup_{x_1,\ldots,x_n}\mathcal{N}(\epsilon,\{(f(x_1),\ldots,f(x_n)):f\in\mathcal{F}\},\\|\cdot\\|_1)$, where the points range over the domain of $\mathcal{F}$ and the norm is the printed $\ell_1$ norm without a factor $1/n$. In \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $\mathcal N_1^{\mathrm{avg}}(\epsilon,\mathcal G,m)$ is the supremum over $x_1,\ldots,x_m\in X$ of the covering numbers of $\{(g(x_1),\ldots,g(x_m)):g\in\mathcal G\}$ in $d_1(u,w)=m^{-1}\sum_{i=1}^m\|u_i-w_i\|$, for $\epsilon>0$ and positive integers $m$. This averaged covering number is distinct from the covering number defined with the unnormalized $\ell_1$ norm. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $\mathcal N_1^{\mathrm{avg}}(r,\mathcal G,N)$ is the supremum over $x_1,\ldots,x_N\in X$ of the covering number of $\{(g(x_1),\ldots,g(x_N)):g\in\mathcal G\}$ in $d_1(u,w)=N^{-1}\sum_{i=1}^N\|u_i-w_i\|$. The probability bound uses $r=\epsilon/16$ and $N=2m$. This averaged covering number is distinct from the covering number defined with the unnormalized $\ell_1$ norm. — The definition in lem:sup-wasserstein-anthony-theorem-17-1 explicitly specifies the averaged metric on length-$N$ evaluation vectors. - reading of the symbol `\mathcal{H}_\delta`: $\mathcal H_\delta\subseteq\mathcal H$ is a finite $\delta/2$-net in the uniform norm of $\mathcal H=\{h:[-K,K]^d\rightarrow\mathbb R:h(0)=0,\ \|h(x)-h(y)\|\leqslant\\|x-y\\|\}$, for $\delta>0$. — The proof of Lemma~\ref{lemma:sup_wasserstein} defines the scalar class $\mathcal H$ and explicitly chooses a finite $\delta/2$-net, with representatives belonging to $\mathcal H$. - reading of the symbol `h`: $h$ denotes the auxiliary function or positive grid spacing defined in its surrounding passage. In Theorem~\ref{thm:wide_nn}: $h(w,X)=w^{\mathsf{T}}X\in\mathbb{R}^d$, where $w$ has $N_C$ coordinates, $X$ collects the $N_C$ component outputs, and the stated region is $\\|w\\|_1=1$, $\\|X\\|_\infty\leqslant K$. In Lemma~\ref{lemma:Lip_condition}: $h(t)=(\hat{f}(t)-f(t))\hat{f}(t)$ is the real-valued auxiliary function in the proof, with the stated Lipschitz constant $L_h=4LK$. In Lemma~\ref{lemma:sinkhorn_approx}: $h(T)=1-\sum_{i=1}^n\sum_{j=1}^m T_{ij}\log T_{ij}$ is the entropy expression for $T\in\Pi(\mu,\nu)$, with $0\log0=0$. In Lemma~\ref{lemma:sup_wasserstein}: $h$ is a scalar test function in $\mathcal H=\{h:[-K,K]^d\rightarrow\mathbb R:h(0)=0,\ \|h(x)-h(y)\|\leqslant\\|x-y\\|\}$, so its values lie in $[-dK,dK]$; the concentration argument first fixes such an $h$ and uses $z_f=\sqrt{(h\circ f+dK)/(2dK)}$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: In the structural interpolation construction, $h=2K/R>0$ is the side length of a grid cell obtained by dividing each coordinate of $[-K,K]^m$ into $R$ equal intervals; $I_hf$ is the resulting piecewise affine interpolant. In Proposition~\ref{prop:Lip_ATE}: In the proof, $s=(a,t)$ and $h(s)=(\hat f_y(s)-f_y(s))\hat f_y(s)$ on the bounded covariate domain $Q$. The structural regression versions satisfy $\|f_y\|,\|\hat f_y\|\leqslant B$ there, and $\|h(s)-h(s')\|\leqslant4BL_y\\|s-s'\\|$. — The proof of Lemma~\ref{lemma:sup_wasserstein} uses the scalar Lipschitz test function and its squared-loss transformation. The separate citation statements quantify over $\mathcal G$ and do not define or use $h$. - reading of the symbol `m`: $m$ is a local integer parameter whose role is determined by the objects in the surrounding passage. In Proposition~\ref{prop:holder_curve}: In $\gamma_m(t)\in C_{k,n-1}$ for all $m\geqslant n$, $m$ is a later curve level. In Theorem~\ref{thm:consistency}: In the Sinkhorn comparison, the second probability-vector length is the simulation count $m_n$, giving the factor $\log(nm_n)$. In the structural interpolation argument, $m$ is the input dimension of $f:[-K,K]^m\rightarrow\mathbb R^{m'}$; the construction treats $m\geqslant1$ by triangulation and $m=0$ by exact constant representation. In Lemma~\ref{lemma:sinkhorn_approx}: $m\geqslant1$ is the length of $\nu\in\Delta_m$, indexing the points $y_1,\ldots,y_m$, and the number of columns of its transportation plan with $\mu\in\Delta_n$; it need not count only positive coordinates or distinct points. In \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $m\geqslant1$ is an arbitrary integer counting evaluation points in the averaged empirical covering and packing numbers. In Corollary~\ref{cor:rate}: In the structural interpolation argument, $m$ is the input dimension of $f:[-K,K]^m\rightarrow\mathbb R^{m'}$; the construction treats $m\geqslant1$ by triangulation and $m=0$ by exact constant representation. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $m\geqslant1$ is the number of independent observations $(x_i,y_i)$ with law $P$ in the empirical squared loss; $\mathcal N_1^{\mathrm{avg}}(\epsilon/16,\mathcal G,2m)$ uses $2m$ evaluation points. In Theorem~\ref{thm:approximation}: In the recurrence argument, $m=n_V-n_T$ is the number of nonintervened observed nodes. The argument includes $m=0$, with an empty geometric sum. In Corollary~\ref{corollary:deep_nn}: In the structural extension and approximation construction, $m=d_{\max}^{\text{in}}>0$ is the largest structural input dimension, while $m_i=d_i^{\text{in}}$ is the input dimension at node $i$. — The proof of Theorem~\ref{thm:approximation} sets $m=n_V-n_T$ in its recurrence argument. The proof of Corollary~\ref{corollary:deep_nn} explicitly sets $m=d_{\max}^{\text{in}}>0$. - reading of the symbol `\|\cdot\|_\infty`: $\\|\cdot\\|_\infty$ denotes the infinity norm on vectors and its uniform extension to functions: $\\|f\\|_\infty=\sup_{x\in\operatorname{dom}(f)}\\|f(x)\\|_\infty$. — Line 194 defines the infinity norm; lines 1886-1902 use it for scalar and vector function approximation; lines 2835-2838 and 2861 use it as the norm on bounded continuous functions. - reading of the symbol `R`: $R$ is a locally specified shape-ratio functional or fixed positive intervention bound. In Proposition~\ref{prop:holder_curve}: $R(\mathcal C)=\max_{i,k}\|a_i^k-b_i^k\|/\min_{i,k}\|a_i^k-b_i^k\|$ for the finite collection $\mathcal C=\{C_1,\ldots,C_m\}$ with $C_k=\prod_{i=1}^n[a_i^k,b_i^k]$, wherever the denominator is positive. In Corollary~\ref{corollary:deep_nn}: $R>0$ is fixed before choosing $\hat{\mathcal M}$ and bounds each continuous intervention value by $\\|t_i\\|_\infty\leqslant R$; the implicit approximation constant may depend on $R$, which remains fixed as the network budgets tend to infinity and $\tau\downarrow0$. — The preparation in the proof of Proposition~\ref{prop:holder_curve} defines the edge-length ratio explicitly. Corollary~\ref{corollary:deep_nn} specifies "Fix any $R>0$", bounds continuous treatment values by $R$, and permits the approximation constant to depend on $R$. - reading of the symbol `S`: $S$ denotes the auxiliary object defined locally in its surrounding argument. In Theorem~\ref{thm:approximation}: In the proof, $S=\sum_{i=1}^{n_V}\\|f_i-\hat{f}_i\\|_\infty$ is the summed structural-function approximation error. In Corollary~\ref{corollary:deep_nn}: In the categorical normalization argument, at the structural input $x=(\text{pa}(v_k),\boldsymbol{u}_{V_k})$, $S=\\|\hat f_i(x)-f_i(x)\\|_\infty$, where $f_i(x)$ is the normalized true propensity vector and $\hat f_i(x)$ is the unnormalized nonnegative approximant. The comparison is with $\hat f_i(x)/\\|\hat f_i(x)\\|_1$ wherever its denominator is nonzero. In Lemma~\ref{lemma:sup_wasserstein}, \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $S$ in $\mathcal{N}(\epsilon,S,\\|\cdotp\\|_1)$ is the set to be covered; the unnormalized empirical construction uses $\{(f(x_1),\ldots,f(x_N)):f\in\mathcal F\}$ for specified input points. For $\mathcal{N}_1^{\mathrm{avg}}(\epsilon,\mathcal G,m)$, the evaluation set is instead $\{(g(x_1),\ldots,g(x_m)):g\in\mathcal G\}$, equipped with $d_1(u,w)=m^{-1}\sum_{i=1}^m\|u_i-w_i\|$. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: The set covered in $\mathcal{N}_1^{\mathrm{avg}}(r,\mathcal G,N)$ is $S=\{(g(x_1),\ldots,g(x_N)):g\in\mathcal G\}$ for specified evaluation points, equipped with $d_1(u,w)=N^{-1}\sum_{i=1}^N\|u_i-w_i\|$; the bound uses $N=2m$. — The proof of Corollary~\ref{corollary:deep_nn} defines $S$ directly as the coordinatewise infinity norm of the difference between the unnormalized approximant and the true propensity vector at the displayed structural input. - reading of the symbol `\mathcal{N}`: $\mathcal{N}$ denotes either the covering-number functional when supplied with covering arguments, or a locally quantified causal model. In Lemma~\ref{lemma:sup_wasserstein}, \cite[Theorem 18.4]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $\mathcal{N}(\epsilon,S,\\|\cdotp\\|_1)$ denotes the covering number of the evaluation set $S$ in the summed $1$-norm; the separately marked averaged covering numbers use the averaged empirical metric. In Proposition~\ref{prop:Lip_ATE}: In the proof, $\mathcal{N}\in\{\mathcal{M},\hat{\mathcal{M}}\}$ ranges over the two causal models when bounding their structural regression functions. In \cite[Theorem 17.1]{anthony1999neural}, used in the proof of Lemma~\ref{lemma:sup_wasserstein}: $\mathcal{N}_1^{\mathrm{avg}}(r,\mathcal G,N)$ is the supremum over $x_1,\ldots,x_N\in X$ of the covering number of $\{(g(x_1),\ldots,g(x_N)):g\in\mathcal G\}$ in $d_1(u,w)=N^{-1}\sum_{i=1}^N\|u_i-w_i\|$; the concentration bound uses $r=\epsilon/16$ and $N=2m$. — The statement of lem:sup-wasserstein-anthony-theorem-17-1 defines the averaged empirical covering number and uses it on $2m$ evaluation points. - reading of the symbol `\theta_{\mathcal{M}}`: $\theta_{\mathcal{M}}=(f_1,\ldots,f_{n_V},P(U_1),\ldots,P(U_{n_U}))$ represents a canonical SCM through the functions in (\ref{eq:structure_equation}) and the population marginal laws of its non-Gumbel latent coordinates, called the uniform parts in the parameterization; the categorical standard Gumbel laws are not additional tuple coordinates. — Flat.tex lines 2820--2823 display the tuple and say that $f_i$ are the functions in (\ref{eq:structure_equation}) and $U_j$ are the uniform parts of the latent variables; the canonical representation separately specifies the standard Gumbel variables. - reading of the symbol `\mathcal{M}^{\theta}`: $\mathcal{M}^\theta$ is the canonical SCM with structural functions and non-Gumbel latent marginal laws specified by $\theta$, using the given causal graph and canonical latent incidence structure; the standard Gumbel laws in categorical equations are supplied by the canonical construction. — The canonical representation preceding (\ref{eq:structure_equation}) specifies the latent structure and Gumbel laws; flat.tex lines 2820--2824 specify the model parameter tuple and the SCM it represents. - reading of the symbol `\theta^*`: $\theta^*$ denotes a distinguished parameter point whose role is specified by the local optimization or causal-model parameterization. In Proposition~\ref{prop:convg_opt}: $\theta^*\in\Theta_\infty$ is a minimizer of (\ref{eq:opt*}), satisfying $g(\theta^*)=0$ and $f(\theta^*)=f^*$; it is distinct in notation and role from $\tilde\theta^*$, the minimizer over $\tilde\Theta_\infty$. In Theorem~\ref{thm:consistency}: $\theta^*$ is the canonical parameter tuple of the underlying SCM, containing its structural functions and non-Gumbel latent marginal laws; $\tilde\Theta_\infty=\{\theta^*\}$. In Corollary~\ref{cor:rate}: $\theta^*$ is the canonical parameter tuple representing the underlying SCM $\mathcal M^*$, containing its structural functions and non-Gumbel latent population marginal laws. — The proof of Corollary~\ref{cor:rate} explicitly imports the parameter notation from the proof of Theorem~\ref{thm:consistency}; there $\mathcal M^{\theta^*}$ is the underlying SCM. - reading of the symbol `\theta'`: $\theta'=(f_1',\ldots,f_{n_V}',P(U_1'),\ldots,P(U_{n_U}'))\in M$ is a second canonical model parameter tuple, with $f_i'\in\mathcal{F}_{V_i}$ and non-Gumbel latent marginal laws $P(U_j')\in\mathcal{P}([-K,K]^{d_j^U})$, compared with $\theta$ through $d(\theta,\theta')$. In Theorem~\ref{thm:consistency}, Corollary~\ref{cor:rate}: When $\theta'$ indexes an empirical candidate law, its candidate index additionally specifies a network representation; the population tuple alone need not determine $P_{m_n}^{\theta'}$ on the common simulation inputs. — The proof of Theorem~\ref{thm:consistency} defines the primed population tuple and uses it in empirical-law comparisons. The empirical optimization in Theorem~\ref{thm:consistency} and Corollary~\ref{cor:rate} ranges over network representations evaluated on common exogenous inputs. - reading of the symbol `P^\theta`: $P^\theta$ denotes the population law induced by the canonical SCM $\mathcal{M}^\theta$ represented by the structural functions and latent marginal laws in $\theta$; its argument specifies the variables or intervention whose law is taken, and it is distinct from the empirical laws $P_n^\theta$ and $P_{m_n}^\theta$. — Flat.tex line 2824 defines $P^\theta$ through $\mathcal{M}^\theta$; lines 2852--2854 and 2873--2885 distinguish population and empirical laws, and lines 2890--2915 use the non-Gumbel latent marginals. The proof of Corollary~\ref{cor:rate} explicitly imports the same notation. - reading of the symbol `\mathcal{F}_{V_i}`: $\mathcal{F}_{V_i}$ is the population structural-function class for $i\in[n_V]$ on the indicated input cube, with the continuous or categorical branch selected by the type of $V_i$; in the categorical branch, $\\|f(x)\\|_1=1$ at each structural input $x$. In Theorem~\ref{thm:consistency}: The bounded Lipschitz class used in the population optimization has $\sup_x\\|f(x)\\|_1\leqslant K$ and $\operatorname{Lip}(f)\leqslant L$ on the full structural input domain; categorical structural functions are nonnegative normalized propensity vectors. — The proof of Theorem~\ref{thm:consistency} displays the continuous and categorical branches; its statement explicitly retains $\mathcal F_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$ in (\ref{eq:consistent_limiting}) and restricts categorical networks to nonnegative normalized propensities. - reading of the symbol `d_{i,\text{in}}^{V}`: $d_{i,\text{in}}^{V}$, for $i\in[n_V]$, is the input coordinate dimension of the structural function $f_i$, whose arguments are $\text{Pa}(V_i)$ and $\boldsymbol{U}_{V_i}$; the population function-class construction uses the domain $[-K,K]^{d_{i,\text{in}}^{V}}$. — The proof of Theorem~\ref{thm:consistency} explicitly identifies the input and output dimensions of $f_i$ immediately after displaying $\mathcal{F}_{V_i}$; equation~\ref{eq:structure_equation} identifies its structural arguments. - reading of the symbol `d_{i,\text{out}}^{V}`: $d_{i,\text{out}}^{V}$ is the output dimension of the canonical structural function $f_i$, for $i\in[n_V]$, equal to $d_i$ when $V_i$ is continuous and to $n_i$ when $V_i$ is categorical. — The proof of Theorem~\ref{thm:consistency} explicitly defines the input and output dimensions of $f_i$ in its population function classes; equation (\ref{eq:structure_equation}) makes the continuous output a value of $V_i$ and the categorical output its normalized propensity vector. - reading of the symbol `d_j^U`: $d_j^U$, for $j\in[n_U]$, is the Euclidean coordinate dimension of the indexed latent variable $U_j$, excluding the separately introduced categorical Gumbel noises. — The proof of Corollary~\ref{corollary:deep_nn} explicitly calls $d_j^U$ the dimension of $U_j$; the proof of Theorem~\ref{thm:consistency} uses it in $\mathcal{P}([-K,K]^{d_j^U})$. Assumption~\ref{assump:categorical_rv} separates the indexed latent variables from $G_{V_i}$. - reading of the symbol `\mathcal{P}`: $\mathcal{P}(K)$ is the space of Borel probability measures on the set $K$, equipped with Wasserstein-1 distance, which metrizes weak convergence on the compact domains used here. — The proof of Theorem~\ref{thm:consistency} uses $\mathcal{P}([-K,K]^{d_j^U})$ as a parameter-space factor and explicitly discusses its measures, weak topology, and Wasserstein metric. - reading of the symbol `\mathcal{P}_n`: $\mathcal{P}_n=\{P(\boldsymbol{U}):\boldsymbol{U}\text{ is the latent collection of }\hat{\mathcal{M}}\in\text{NCM}_{\mathcal{G}}(\mathcal{F}_{0,n},\mathcal{F}_{1,n})\}$ is the population latent-law class indexed by $n$, rather than a class of empirical latent measures. In Theorem~\ref{thm:consistency}: In the proof's population parameterization $\Theta_n$, $\mathcal{P}_n$ is identified with $\mathcal{P}_{n,1}\times\cdots\times\mathcal{P}_{n,n_U}$, the tuple of generated non-Gumbel marginal laws on $[-K,K]^{d_j^U}$, $j\in[n_U]$; the separately prescribed categorical Gumbel noises are not additional coordinates of this tuple. Each generated marginal law integrates the generator's uniform input and Gumbel-softmax randomness. This population-law tuple does not determine a candidate's common-draw empirical distribution: different network representations of the same latent laws can have different empirical distributions. — The displayed definition and product decomposition at flat.tex lines 2840--2850 describe population latent laws; Theorem~\ref{thm:consistency} specifies optimization over network representations, which need not be determined by those laws. - reading of the symbol `\hat{\mathcal{F}}_{V_i,n}`: $\hat{\mathcal{F}}_{V_i,n}$ is the node-specific structural function class at index $n$, for $i\in[n_V]$, with categorical normalization interpreted pointwise as $x\mapsto f(x)/\\|f(x)\\|_1$ wherever $\\|f(x)\\|_1>0$. In Theorem~\ref{thm:consistency}: For categorical $V_i$, $\hat{\mathcal{F}}_{V_i,n}$ consists of the maps $x\mapsto q(x)/\\|q(x)\\|_1$ with $q\in\mathcal{F}_{0,n}$ satisfying $q(x)\in[0,\infty)^{n_i}$ and $\\|q(x)\\|_1>0$ for every structural input $x$, and $\operatorname{Lip}(x\mapsto q(x)/\\|q(x)\\|_1)\leqslant\hat L_f$. For continuous $V_i$, the candidate structural class is $\{\hat f_i\in\mathcal F_{0,n}:\sup_x\\|\hat f_i(x)\\|_1\leqslant K\}$, where the supremum is over the full structural input domain. — Theorem~\ref{thm:consistency} specifies both the categorical normalization restrictions and the uniform output $1$-norm restriction for continuous structural networks. - reading of the symbol `\hat{g}_1^i`: For $i\in[N_C]$, $\hat{g}_1^i$ denotes the chosen first generator block for component $C_i$, with scalar input and $d_i^C$ output coordinates. In the result split from Theorem~\ref{thm:wide_nn}: For $d_i^C>0$, $\hat{g}_1^i$ is the coordinatewise deep ReLU approximation of the curve $\gamma_i:[0,1]\rightarrow[0,1]^{d_i^C}$ satisfying $(\gamma_i)_{\#}\lambda=(g_2^i)^{-1}_{\#}\mu_i$; its coordinate networks have width $12$ and depth $L_1$, its combined width is $\Theta(d_i^C)$, and $\\|\hat{g}_1^i-\gamma_i\\|_\infty\leqslant O(L_1^{-2/d_i^C})$. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: For $d_i^C=0$, $\hat{g}_1^i$ is the unique map to $[0,1]^0$. In Theorem~\ref{thm:wide_nn}: For $d_i^C>0$, $\hat{g}_1^i$ is a quantized ReLU network of width $W_1$ and depth $\Theta(d_i^C)$, chosen so that $W((g_2^i)^{-1}_{\#}\mu_i,P(\hat{g}_1^i(U_i)))\leqslant O(W_1^{-1/d_i^C})$. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: For a component of positive dimension $q=d_i^C$, $\hat{g}_1^i$ is the quantized ReLU component generator of width $W_1$ and depth $\Theta(q)$ satisfying $W((g_2^i)^{-1}_{\#}\mu_i,P(\hat{g}_1^i(U_i)))\leqslant O(W_1^{-1/q})$. The source construction uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and mesh $\delta/n$, where $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$ and $A_q\geqslant1$ depends only on $q$. — The defining passage at flat.tex lines 1883--1886 and the binding quantization conventions of lem:wide-nn-perekrestenko-theorem-viii-1 specify this construction. - reading of the symbol `U_i`: $U_i$ denotes the index-$i$ random variable in the locally specified latent or generator-input family. In Corollary~\ref{corollary:deep_nn}: In the SCM notation, $U_i$ is a non-Gumbel latent variable of the model being approximated, with its own latent dimension and population law; scalar uniform variables appearing in the component-generator construction are inputs used to approximate that law. In Theorem~\ref{thm:wide_nn}, the result split from Theorem~\ref{thm:wide_nn}: In the component-generator construction, $U_i\sim\operatorname{Unif}([0,1])$ for $i\in[N_C]$; $i$ indexes connected support components, and $U_i$ is the scalar input to $\hat g_1^i$. In \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, used in the proof of Theorem~\ref{thm:wide_nn}: In the component-generator construction, $U_i\sim\operatorname{Unif}([0,1])$ for $i\in[N_C]$; $i$ indexes connected support components, and $U_i$ is the scalar input to $\hat g_1^i$. In \cite[Theorem 2]{fournier2015rate}, used in the proof of Theorem~\ref{thm:consistency}: $U_i$, $i\in[n_U]$, is a non-Gumbel latent variable with values in $[-K,K]^{d_i^U}$ under the locally specified model law $P^\theta$ or $P^{\theta^*}$; the scalar uniform generator inputs are $Z_{i,j}$. The vector $(U_1,\ldots,U_{n_U})$ has total dimension $D_U=\sum_{i=1}^{n_U}d_i^U$. — The consistency proof parameterizes latent laws by $P(U_i)$ on $[-K,K]^{d_i^U}$ and denotes scalar uniform generator inputs separately by $Z_{i,j}$. The definition of $D_U$ in Theorem~\ref{thm:consistency} includes only non-Gumbel latent variables. - reading of the symbol `s`: $s$ denotes the fixed integer depth parameter $s=1$, distinct from the quantization parameter $\delta$ and the quantization mesh $\delta/n$. — The binding reading of lem:wide-nn-perekrestenko-theorem-viii-1 specifies integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$ for $q=d_i^C>0$, and mesh $\delta/n$. - added condition (`ass:bk-1`, `thm:wide-nn`): For every $k\in[n]$, $\int_{\mathbb R^d}\\|x\\|\,\mu_k(dx)+\int_{\mathbb R^d}\\|x\\|\,\hat\mu_k(dx)<\infty$. — The dual proof separately evaluates expectations of unbounded Lipschitz functions. - added condition (`ass:bk-1`, `thm:cor-fix-3c402872`): For every $k\in[n]$, $\int_{\mathbb R^d}\\|x\\|\,\mu_k(dx)+\int_{\mathbb R^d}\\|x\\|\,\hat\mu_k(dx)<\infty$. — The dual proof separately evaluates expectations of unbounded Lipschitz functions. - added condition (`ass:bk-1`, `thm:deep-nn`): For every $k\in[n]$, $\int_{\mathbb R^d}\\|x\\|\,\mu_k(dx)+\int_{\mathbb R^d}\\|x\\|\,\hat\mu_k(dx)<\infty$. — The dual proof separately evaluates expectations of unbounded Lipschitz functions. - added condition (`ass:bk-1`, `thm:consistency`): For every $k\in[n]$, $\int_{\mathbb R^d}\\|x\\|\,\mu_k(dx)+\int_{\mathbb R^d}\\|x\\|\,\hat\mu_k(dx)<\infty$. — The dual proof separately evaluates expectations of unbounded Lipschitz functions. - added condition (`ass:bk-1`, `thm:rate`): For every $k\in[n]$, $\int_{\mathbb R^d}\\|x\\|\,\mu_k(dx)+\int_{\mathbb R^d}\\|x\\|\,\hat\mu_k(dx)<\infty$. — The dual proof separately evaluates expectations of unbounded Lipschitz functions. - added condition (`ass:bk-6`, `prop:convg-opt`): For every $n\in\mathbb N$, the feasible set of (\ref{eq:opt_n}) contains a minimizer of $f$. — The finite values are defined as minima, and the lower-limit argument selects attaining minimizers. - added condition (`ass:bk-1`, `lem:combin-err`): For every $k\in[n]$, $\int_{\mathbb R^d}\\|x\\|\,\mu_k(dx)+\int_{\mathbb R^d}\\|x\\|\,\hat\mu_k(dx)<\infty$. — The dual proof separately evaluates expectations of unbounded Lipschitz functions. ## Added conditions | id | where | class | condition | | --- | --- | --- | --- | | `ass:pin-177` | `thm:approximation` | V1.5 bookkeeping | $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. (pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality.) | | `ass:pin-181` | `thm:approximation` | V1.5 bookkeeping | Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\\|f_i(x)\\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. (pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities.) | | `ass:pin-182` | `thm:approximation` | V1.5 bookkeeping | For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\\|f_i(x)-f_i(x')\\|\leqslant L_f\\|x-x'\\|$ on its structural input domain, using the paper's $1$-norm. (pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$.) | | `ass:pin-183` | `thm:approximation` | V1.5 bookkeeping | There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely under the SCM law. (pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation.) | | `ass:pin-189` | `thm:approximation` | V1.5 bookkeeping | For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. (pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition.) | | `ass:pin-190` | `thm:approximation` | V1.5 bookkeeping | In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. (pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction.) | | `ass:pin-191` | `thm:approximation` | V1.5 bookkeeping | Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). (pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations.) | | `ass:pin-fix-4a86c2eb` | `thm:approximation` | V1.5 bookkeeping | The structural uniform infinity norm takes the supremum of the coordinatewise infinity norm over the common structural input domain. (pins reading of thm:approximation: stated by the repair) | | `ass:pin-fix-ab17be7a` | `thm:approximation` | V1.5 bookkeeping | For a categorical propensity coordinate equal to zero, $\log0=-\infty$ in the canonical argmax construction. (pins reading of thm:approximation: stated by the repair) | | `ass:bk-1` | `thm:wide-nn` | V1.5 bookkeeping | For every $k\in[n]$, $\int_{\mathbb R^d}\\|x\\|\,\mu_k(dx)+\int_{\mathbb R^d}\\|x\\|\,\hat\mu_k(dx)<\infty$. (The dual proof separately evaluates expectations of unbounded Lipschitz functions.) | | `ass:pin-192` | `thm:wide-nn` | V1.5 bookkeeping | A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. (pins reading of def:relu-networks: The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths.) | | `ass:pin-203` | `thm:wide-nn` | V1.5 bookkeeping | The gap event is $E_\delta=\{\exists\,1\leqslant i0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. (pins reading of def:gumbel-softmax-layer: The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit.) | | `ass:pin-211` | `thm:wide-nn` | V1.5 bookkeeping | In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. (pins reading of def:lemma-gumbel-err-softmax: Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores.) | | `ass:pin-33` | `thm:wide-nn` | V1.5 bookkeeping | $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. (pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes.) | | `ass:pin-75` | `thm:wide-nn` | V1.5 bookkeeping | For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. (pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula.) | | `ass:pin-fix-00130002` | `thm:wide-nn` | V1.5 bookkeeping | The law $\hat g_{\#}\lambda$ includes the independent standard Gumbel variables used in the mixing layer. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-507a5848` | `thm:wide-nn` | V1.5 bookkeeping | For each component of positive dimension $q=d_i^C$, the application of Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and quantization mesh $\delta/n$. (pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair) | | `ass:pin-fix-5f02b8a6` | `thm:wide-nn` | V1.5 bookkeeping | The error estimates are asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of these budgets and of $\tau$. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-72c9e86d` | `thm:wide-nn` | V1.5 bookkeeping | The measures $\mu$ and $\hat\mu$ are probability measures. (pins reading of lem:push-forward-err: stated by the repair) | | `ass:pin-fix-9bbf998f` | `thm:wide-nn` | V1.5 bookkeeping | A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-b83fa4d1` | `thm:wide-nn` | V1.5 bookkeeping | $W$ is Wasserstein-1 distance for the ground distance $\\|x-y\\|$. (pins reading of lem:product-err: stated by the repair) | | `ass:pin-fix-c514425b` | `thm:wide-nn` | V1.5 bookkeeping | The refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. (pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair) | | `ass:pin-fix-ca667e13` | `thm:wide-nn` | V1.5 bookkeeping | $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. (pins reading of lem:product-err: stated by the repair) | | `ass:pin-fix-db6ed772` | `thm:wide-nn` | V1.5 bookkeeping | In the proposition entitled ``Deep distribution approximation'', the multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-f6f485b1` | `thm:wide-nn` | V1.5 bookkeeping | The uniform infinity norm is $\\|F-\hat F\\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}\|F_j(x)-\hat F_j(x)\|$. (pins reading of lem:push-forward-err: stated by the repair) | | `ass:bk-1` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | For every $k\in[n]$, $\int_{\mathbb R^d}\\|x\\|\,\mu_k(dx)+\int_{\mathbb R^d}\\|x\\|\,\hat\mu_k(dx)<\infty$. (The dual proof separately evaluates expectations of unbounded Lipschitz functions.) | | `ass:pin-192` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. (pins reading of def:relu-networks: The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths.) | | `ass:pin-203` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | The gap event is $E_\delta=\{\exists\,1\leqslant i0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. (pins reading of def:gumbel-softmax-layer: The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit.) | | `ass:pin-211` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. (pins reading of def:lemma-gumbel-err-softmax: Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores.) | | `ass:pin-33` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. (pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes.) | | `ass:pin-75` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. (pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula.) | | `ass:pin-fix-00130002` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | The law $\hat g_{\#}\lambda$ includes the independent standard Gumbel variables used in the mixing layer. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-0bcb2cef` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-2a1e7b01` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | The mass assigned to $I_{O_n(i)}$ is $\lambda_d(C_{O_n(i)})=2^{-nd}$. (pins reading of def:prop-holder-curve-interval-construction: stated by the repair) | | `ass:pin-fix-2fe04b0e` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal C_n$ denotes the standard dyadic collection $\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}$. (pins reading of def:prop-holder-curve-cube-construction: stated by the repair) | | `ass:pin-fix-507a5848` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | For each component of positive dimension $q=d_i^C$, the application of Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and quantization mesh $\delta/n$. (pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair) | | `ass:pin-fix-5f02b8a6` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | The error estimates are asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of these budgets and of $\tau$. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-72c9e86d` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | The measures $\mu$ and $\hat\mu$ are probability measures. (pins reading of lem:push-forward-err: stated by the repair) | | `ass:pin-fix-7a79e01d` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal I_n$ denotes $\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$ with $I_{O_n(i)}=[(i-1)2^{-nd},i2^{-nd}]$. (pins reading of def:prop-holder-curve-interval-construction: stated by the repair) | | `ass:pin-fix-8cc0dba5` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. (pins reading of def:hilbert-partitions: stated by the repair) | | `ass:pin-fix-99ac9bb3` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. (pins reading of def:hilbert-partitions: stated by the repair) | | `ass:pin-fix-9bbf998f` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-b83fa4d1` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | $W$ is Wasserstein-1 distance for the ground distance $\\|x-y\\|$. (pins reading of lem:product-err: stated by the repair) | | `ass:pin-fix-c514425b` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | The refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. (pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair) | | `ass:pin-fix-c778b424` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | The reference measure $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$. (pins reading of def:prop-holder-curve-cube-construction: stated by the repair) | | `ass:pin-fix-c85b9186` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | Each $O_n$ is a bijection. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-ca667e13` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. (pins reading of lem:product-err: stated by the repair) | | `ass:pin-fix-db6ed772` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | In the proposition entitled ``Deep distribution approximation'', the multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-e896f793` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | The cells of $\mathcal C_n$ are closed and may share boundaries. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-f6f485b1` | `thm:cor-fix-3c402872` | V1.5 bookkeeping | The uniform infinity norm is $\\|F-\hat F\\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}\|F_j(x)-\hat F_j(x)\|$. (pins reading of lem:push-forward-err: stated by the repair) | | `ass:bk-1` | `thm:deep-nn` | V1.5 bookkeeping | For every $k\in[n]$, $\int_{\mathbb R^d}\\|x\\|\,\mu_k(dx)+\int_{\mathbb R^d}\\|x\\|\,\hat\mu_k(dx)<\infty$. (The dual proof separately evaluates expectations of unbounded Lipschitz functions.) | | `ass:pin-177` | `thm:deep-nn` | V1.5 bookkeeping | $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. (pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality.) | | `ass:pin-181` | `thm:deep-nn` | V1.5 bookkeeping | Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\\|f_i(x)\\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. (pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities.) | | `ass:pin-182` | `thm:deep-nn` | V1.5 bookkeeping | For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\\|f_i(x)-f_i(x')\\|\leqslant L_f\\|x-x'\\|$ on its structural input domain, using the paper's $1$-norm. (pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$.) | | `ass:pin-183` | `thm:deep-nn` | V1.5 bookkeeping | There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely under the SCM law. (pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation.) | | `ass:pin-186` | `thm:deep-nn` | V1.5 bookkeeping | The phrase ``uniform cubes'' denotes the unit cube domains $[0,1]^m$ used for deterministic uniform function approximation; it imposes no uniform probability law on structural inputs. (pins reading of ass:relu-approximation-domain: The same paragraph displays approximation of functions on $[0,1]^{d_1}$ and separately says that similar arguments apply to any bounded cubes.) | | `ass:pin-189` | `thm:deep-nn` | V1.5 bookkeeping | For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. (pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition.) | | `ass:pin-190` | `thm:deep-nn` | V1.5 bookkeeping | In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. (pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction.) | | `ass:pin-191` | `thm:deep-nn` | V1.5 bookkeeping | Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). (pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations.) | | `ass:pin-192` | `thm:deep-nn` | V1.5 bookkeeping | A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. (pins reading of def:relu-networks: The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths.) | | `ass:pin-194` | `thm:deep-nn` | V1.5 bookkeeping | Membership in $\text{NCM}_{\mathcal G}(\mathcal F_0,\mathcal F_1)$ requires $\hat f_i\in\mathcal F_0$ for every observed node and $\hat g_j\in\mathcal F_1$ for every latent component, with the compatible input and output dimensions understood separately for each network. (pins reading of def:neural-causal-class: The source paragraph explicitly omits input and output dimensions and describes separate structural approximants and latent generators in equation~\ref{eq:structure_equation_hat}.) | | `ass:pin-199` | `thm:deep-nn` | V1.5 bookkeeping | $d_{\max}^U=\max_{j\in[n_U]}d_j^U$ ranges over non-Gumbel latent variables, while $d_{\max}^{\mathrm{in}}=\max_i d_i^{\mathrm{in}}$ and $d_{\max}^{\mathrm{out}}=\max_i d_i^{\mathrm{out}}$, with $d_i^{\mathrm{out}}=n_i$ for categorical nodes. (pins reading of def:corollary-deep-nn-dimensions: The NCM construction separately approximates non-Gumbel latent distributions and the functions in the canonical structural equations; those functions output categorical propensity vectors. The binding dimension symbols specify these same meanings.) | | `ass:pin-203` | `thm:deep-nn` | V1.5 bookkeeping | The gap event is $E_\delta=\{\exists\,1\leqslant i0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. (pins reading of def:gumbel-softmax-layer: The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit.) | | `ass:pin-211` | `thm:deep-nn` | V1.5 bookkeeping | In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. (pins reading of def:lemma-gumbel-err-softmax: Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores.) | | `ass:pin-33` | `thm:deep-nn` | V1.5 bookkeeping | $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. (pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes.) | | `ass:pin-75` | `thm:deep-nn` | V1.5 bookkeeping | For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. (pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula.) | | `ass:pin-fix-00130002` | `thm:deep-nn` | V1.5 bookkeeping | The law $\hat g_{\#}\lambda$ includes the independent standard Gumbel variables used in the mixing layer. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-0bcb2cef` | `thm:deep-nn` | V1.5 bookkeeping | The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-2a1e7b01` | `thm:deep-nn` | V1.5 bookkeeping | The mass assigned to $I_{O_n(i)}$ is $\lambda_d(C_{O_n(i)})=2^{-nd}$. (pins reading of def:prop-holder-curve-interval-construction: stated by the repair) | | `ass:pin-fix-2fe04b0e` | `thm:deep-nn` | V1.5 bookkeeping | In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal C_n$ denotes the standard dyadic collection $\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}$. (pins reading of def:prop-holder-curve-cube-construction: stated by the repair) | | `ass:pin-fix-419eb8e3` | `thm:deep-nn` | V1.5 bookkeeping | The multiplicative width factors in $\Theta(d_{\max}^{\text{in}}d_{\max}^{\text{out}})$, $\Theta(d_{\max}^U)$, and $\Theta((d_{\max}^U)^2)$ can be chosen as absolute numerical constants, independent of the causal model, $R$, all dimensions, and all category and component counts. (pins reading of thm:deep-nn: stated by the repair) | | `ass:pin-fix-43f1bb71` | `thm:deep-nn` | V1.5 bookkeeping | For each latent law $P(U_C)$, the assumptions of the proposition entitled ``Deep distribution approximation'' require the support charts of \cref{assump:support} and \cref{assumption:lower_bound} on each normalized component law $\mu_i=P(U_C)\|_{C_i}/P(U_C\in C_i)$, with $(f^{-1})_{\#}\mu_i(\partial[0,1]^{d_i^C})=0$ when $d_i^C>0$. (pins reading of thm:deep-nn: stated by the repair) | | `ass:pin-fix-4a86c2eb` | `thm:deep-nn` | V1.5 bookkeeping | The structural uniform infinity norm takes the supremum of the coordinatewise infinity norm over the common structural input domain. (pins reading of thm:approximation: stated by the repair) | | `ass:pin-fix-507a5848` | `thm:deep-nn` | V1.5 bookkeeping | For each component of positive dimension $q=d_i^C$, the application of Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and quantization mesh $\delta/n$. (pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair) | | `ass:pin-fix-5f02b8a6` | `thm:deep-nn` | V1.5 bookkeeping | The error estimates are asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of these budgets and of $\tau$. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-72c9e86d` | `thm:deep-nn` | V1.5 bookkeeping | The measures $\mu$ and $\hat\mu$ are probability measures. (pins reading of lem:push-forward-err: stated by the repair) | | `ass:pin-fix-73abef8c` | `thm:deep-nn` | V1.5 bookkeeping | In Corollary~\ref{corollary:deep_nn}, for $m_i=d_i^{\text{in}}>0$, the symbol $\hat f_i$ denotes $q_i\circ\Pi_i$ for a continuous variable and the vector $\bigl(\max\{\epsilon_i,(q_i\circ\Pi_i)_k\}\bigr)_{k=1}^{n_i}$ for a categorical variable, where $q_i$, $\Pi_i$, and $\epsilon_i=A_iL_0^{-2/m_i}$ are chosen in its proof. (pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair) | | `ass:pin-fix-7a79e01d` | `thm:deep-nn` | V1.5 bookkeeping | In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal I_n$ denotes $\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$ with $I_{O_n(i)}=[(i-1)2^{-nd},i2^{-nd}]$. (pins reading of def:prop-holder-curve-interval-construction: stated by the repair) | | `ass:pin-fix-8429b12c` | `thm:deep-nn` | V1.5 bookkeeping | The latent generator class in the statement uses the same temperature $\tau$ as the displayed error estimate. (pins reading of thm:deep-nn: stated by the repair) | | `ass:pin-fix-8cc0dba5` | `thm:deep-nn` | V1.5 bookkeeping | For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. (pins reading of def:hilbert-partitions: stated by the repair) | | `ass:pin-fix-945d85ed` | `thm:deep-nn` | V1.5 bookkeeping | The error estimate is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, for fixed $R$ and fixed causal model. (pins reading of thm:deep-nn: stated by the repair) | | `ass:pin-fix-99ac9bb3` | `thm:deep-nn` | V1.5 bookkeeping | For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. (pins reading of def:hilbert-partitions: stated by the repair) | | `ass:pin-fix-9bbf998f` | `thm:deep-nn` | V1.5 bookkeeping | A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-ab17be7a` | `thm:deep-nn` | V1.5 bookkeeping | For a categorical propensity coordinate equal to zero, $\log0=-\infty$ in the canonical argmax construction. (pins reading of thm:approximation: stated by the repair) | | `ass:pin-fix-b83fa4d1` | `thm:deep-nn` | V1.5 bookkeeping | $W$ is Wasserstein-1 distance for the ground distance $\\|x-y\\|$. (pins reading of lem:product-err: stated by the repair) | | `ass:pin-fix-bbf1c78a` | `thm:deep-nn` | V1.5 bookkeeping | In Corollary~\ref{corollary:deep_nn}, for $m_i=0$, the symbol $\hat f_i$ denotes the exact constant function $f_i$. (pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair) | | `ass:pin-fix-c514425b` | `thm:deep-nn` | V1.5 bookkeeping | The refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. (pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair) | | `ass:pin-fix-c778b424` | `thm:deep-nn` | V1.5 bookkeeping | The reference measure $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$. (pins reading of def:prop-holder-curve-cube-construction: stated by the repair) | | `ass:pin-fix-c85b9186` | `thm:deep-nn` | V1.5 bookkeeping | Each $O_n$ is a bijection. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-ca667e13` | `thm:deep-nn` | V1.5 bookkeeping | $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. (pins reading of lem:product-err: stated by the repair) | | `ass:pin-fix-db6ed772` | `thm:deep-nn` | V1.5 bookkeeping | In the proposition entitled ``Deep distribution approximation'', the multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-e896f793` | `thm:deep-nn` | V1.5 bookkeeping | The cells of $\mathcal C_n$ are closed and may share boundaries. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-f6f485b1` | `thm:deep-nn` | V1.5 bookkeeping | The uniform infinity norm is $\\|F-\hat F\\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}\|F_j(x)-\hat F_j(x)\|$. (pins reading of lem:push-forward-err: stated by the repair) | | `ass:pin-177` | `prop:counterexample` | V1.5 bookkeeping | $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. (pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality.) | | `ass:pin-181` | `prop:counterexample` | V1.5 bookkeeping | Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\\|f_i(x)\\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. (pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities.) | | `ass:pin-182` | `prop:counterexample` | V1.5 bookkeeping | For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\\|f_i(x)-f_i(x')\\|\leqslant L_f\\|x-x'\\|$ on its structural input domain, using the paper's $1$-norm. (pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$.) | | `ass:pin-183` | `prop:counterexample` | V1.5 bookkeeping | There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely under the SCM law. (pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation.) | | `ass:pin-189` | `prop:counterexample` | V1.5 bookkeeping | For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. (pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition.) | | `ass:pin-190` | `prop:counterexample` | V1.5 bookkeeping | In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. (pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction.) | | `ass:pin-204` | `prop:counterexample` | V1.5 bookkeeping | The second Bernoulli law in the construction has $P(U_2=0)=2/3$. (pins reading of prop:counterexample-foral: The displayed value $P(U_2=1)=1/3$ and the joint masses for $X=0,T=0$ in the proof of Proposition~\ref{prop:counterexample_foral} both determine this entry.) | | `ass:bk-1` | `thm:consistency` | V1.5 bookkeeping | For every $k\in[n]$, $\int_{\mathbb R^d}\\|x\\|\,\mu_k(dx)+\int_{\mathbb R^d}\\|x\\|\,\hat\mu_k(dx)<\infty$. (The dual proof separately evaluates expectations of unbounded Lipschitz functions.) | | `ass:pin-177` | `thm:consistency` | V1.5 bookkeeping | $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. (pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality.) | | `ass:pin-181` | `thm:consistency` | V1.5 bookkeeping | Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\\|f_i(x)\\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. (pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities.) | | `ass:pin-182` | `thm:consistency` | V1.5 bookkeeping | For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\\|f_i(x)-f_i(x')\\|\leqslant L_f\\|x-x'\\|$ on its structural input domain, using the paper's $1$-norm. (pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$.) | | `ass:pin-183` | `thm:consistency` | V1.5 bookkeeping | There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely under the SCM law. (pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation.) | | `ass:pin-186` | `thm:consistency` | V1.5 bookkeeping | The phrase ``uniform cubes'' denotes the unit cube domains $[0,1]^m$ used for deterministic uniform function approximation; it imposes no uniform probability law on structural inputs. (pins reading of ass:relu-approximation-domain: The same paragraph displays approximation of functions on $[0,1]^{d_1}$ and separately says that similar arguments apply to any bounded cubes.) | | `ass:pin-187` | `thm:consistency` | V1.5 bookkeeping | The bound $K$ applies throughout the optimization class to the observed variables $V_i$ and indexed non-Gumbel latent variables $U_j$, with $n_i\leqslant K$ for categorical nodes and admissible intervention coordinates in $[-K,K]$; it does not bound the separately supplied standard Gumbel noises. (pins reading of ass:bk-owner-1: The boundedness assumption explicitly indexes $V_i$ and $U_j$, whereas the categorical convention separately adjoins $G_{V_i}$. The binding boundedness symbols preserve this distinction.) | | `ass:pin-189` | `thm:consistency` | V1.5 bookkeeping | For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. (pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition.) | | `ass:pin-190` | `thm:consistency` | V1.5 bookkeeping | In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. (pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction.) | | `ass:pin-191` | `thm:consistency` | V1.5 bookkeeping | Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). (pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations.) | | `ass:pin-192` | `thm:consistency` | V1.5 bookkeeping | A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. (pins reading of def:relu-networks: The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths.) | | `ass:pin-194` | `thm:consistency` | V1.5 bookkeeping | Membership in $\text{NCM}_{\mathcal G}(\mathcal F_0,\mathcal F_1)$ requires $\hat f_i\in\mathcal F_0$ for every observed node and $\hat g_j\in\mathcal F_1$ for every latent component, with the compatible input and output dimensions understood separately for each network. (pins reading of def:neural-causal-class: The source paragraph explicitly omits input and output dimensions and describes separate structural approximants and latent generators in equation~\ref{eq:structure_equation_hat}.) | | `ass:pin-196` | `thm:consistency` | V1.5 bookkeeping | Truncation means coordinatewise clipping of the final output to $[-K,K]$; in $\mathcal{NN}^{\infty,K}$ no finite Lipschitz bound is imposed. (pins reading of def:truncated-neural-class: The defining minimum and maximum and the word ``Similarly'' specify the operation; the binding neural-class symbol conventions fix its application to the generator output.) | | `ass:pin-197` | `thm:consistency` | V1.5 bookkeeping | $\underline{F}^{L}$ denotes the constrained infimum of the displayed scalar objective over the stated canonical SCM class and its admissible latent laws, rather than a minimizing SCM. (pins reading of def:eq-consistent-limiting: The sentence introducing the display explicitly says ``optimal value'', and the following sentence calls it the sharp lower bound when the Lipschitz bounds agree.) | | `ass:pin-199` | `thm:consistency` | V1.5 bookkeeping | $d_{\max}^U=\max_{j\in[n_U]}d_j^U$ ranges over non-Gumbel latent variables, while $d_{\max}^{\mathrm{in}}=\max_i d_i^{\mathrm{in}}$ and $d_{\max}^{\mathrm{out}}=\max_i d_i^{\mathrm{out}}$, with $d_i^{\mathrm{out}}=n_i$ for categorical nodes. (pins reading of def:corollary-deep-nn-dimensions: The NCM construction separately approximates non-Gumbel latent distributions and the functions in the canonical structural equations; those functions output categorical propensity vectors. The binding dimension symbols specify these same meanings.) | | `ass:pin-201` | `thm:consistency` | V1.5 bookkeeping | The coordinates $P(U_j)$ in $\theta_{\mathcal M}$ are the population laws of the non-Gumbel latent variables, not necessarily uniform laws; the independent standard Gumbel laws are fixed and omitted from this population tuple. (pins reading of def:thm-consistency-model-parameterization: Immediately after the tuple, the population parameter space contains factors $\mathcal P([-K,K]^{d_j^U})$. The canonical representation supplies the categorical Gumbel laws separately.) | | `ass:pin-203` | `thm:consistency` | V1.5 bookkeeping | The gap event is $E_\delta=\{\exists\,1\leqslant i0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. (pins reading of def:gumbel-softmax-layer: The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit.) | | `ass:pin-210` | `thm:consistency` | V1.5 bookkeeping | $F_n=\inf\left\{\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\hat{\mathcal{M}}}[F(V_1(t),\ldots,V_{n_V}(t))]:\hat{\mathcal{M}}\in\text{NCM}_{\mathcal{G}}(\mathcal{F}_{0,n},\mathcal{F}_{1,n}),\ S_{\lambda_n}(P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V}),P_n^{\mathcal{M}^*}(\boldsymbol{V}))\leqslant\alpha_n\right\}$. (pins reading of def:eq-opt-empiri: The prose introducing equation (\ref{eq:opt_empiri}) calls $F_n$ an estimator of optimal values, and the following paragraph asks whether it is a good lower bound. The supplied symbol convention likewise specifies a scalar infimum.) | | `ass:pin-211` | `thm:consistency` | V1.5 bookkeeping | In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. (pins reading of def:lemma-gumbel-err-softmax: Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores.) | | `ass:pin-33` | `thm:consistency` | V1.5 bookkeeping | $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. (pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes.) | | `ass:pin-75` | `thm:consistency` | V1.5 bookkeeping | For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. (pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula.) | | `ass:pin-fix-00130002` | `thm:consistency` | V1.5 bookkeeping | The law $\hat g_{\#}\lambda$ includes the independent standard Gumbel variables used in the mixing layer. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-09991de8` | `thm:consistency` | V1.5 bookkeeping | The observational empirical law $P_n^{\mathcal M^*}(\boldsymbol V)$ is formed from $n$ independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-0a7b21b0` | `thm:consistency` | V1.5 bookkeeping | The empirical optimization ranges over network representations, including distinct representations with the same latent population laws. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-0bcb2cef` | `thm:consistency` | V1.5 bookkeeping | The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-2a1e7b01` | `thm:consistency` | V1.5 bookkeeping | The mass assigned to $I_{O_n(i)}$ is $\lambda_d(C_{O_n(i)})=2^{-nd}$. (pins reading of def:prop-holder-curve-interval-construction: stated by the repair) | | `ass:pin-fix-2fe04b0e` | `thm:consistency` | V1.5 bookkeeping | In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal C_n$ denotes the standard dyadic collection $\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}$. (pins reading of def:prop-holder-curve-cube-construction: stated by the repair) | | `ass:pin-fix-30413b86` | `thm:consistency` | V1.5 bookkeeping | The common simulation draws are independent of the observational sample. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-3b195d8d` | `thm:consistency` | V1.5 bookkeeping | The common exogenous draws are independent and identically distributed according to the specified product law of the exogenous inputs. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-419eb8e3` | `thm:consistency` | V1.5 bookkeeping | The multiplicative width factors in $\Theta(d_{\max}^{\text{in}}d_{\max}^{\text{out}})$, $\Theta(d_{\max}^U)$, and $\Theta((d_{\max}^U)^2)$ can be chosen as absolute numerical constants, independent of the causal model, $R$, all dimensions, and all category and component counts. (pins reading of thm:deep-nn: stated by the repair) | | `ass:pin-fix-43f1bb71` | `thm:consistency` | V1.5 bookkeeping | For each latent law $P(U_C)$, the assumptions of the proposition entitled ``Deep distribution approximation'' require the support charts of \cref{assump:support} and \cref{assumption:lower_bound} on each normalized component law $\mu_i=P(U_C)\|_{C_i}/P(U_C\in C_i)$, with $(f^{-1})_{\#}\mu_i(\partial[0,1]^{d_i^C})=0$ when $d_i^C>0$. (pins reading of thm:deep-nn: stated by the repair) | | `ass:pin-fix-448b19a0` | `thm:consistency` | V1.5 bookkeeping | Probabilities of uniform deviation events are evaluated in the completion of the sample law. (pins reading of lem:sup-wasserstein: stated by the repair) | | `ass:pin-fix-4a86c2eb` | `thm:consistency` | V1.5 bookkeeping | The structural uniform infinity norm takes the supremum of the coordinatewise infinity norm over the common structural input domain. (pins reading of thm:approximation: stated by the repair) | | `ass:pin-fix-507a5848` | `thm:consistency` | V1.5 bookkeeping | For each component of positive dimension $q=d_i^C$, the application of Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and quantization mesh $\delta/n$. (pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair) | | `ass:pin-fix-59a01041` | `thm:consistency` | V1.5 bookkeeping | For each $n$, every candidate is evaluated on the same $m_n$ draws of all its exogenous inputs, including its uniform inputs, generator Gumbel inputs, and categorical Gumbel inputs. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-5f02b8a6` | `thm:consistency` | V1.5 bookkeeping | The error estimates are asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of these budgets and of $\tau$. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-676b96c3` | `thm:consistency` | V1.5 bookkeeping | The finite component counts and component cube dimensions of the latent generator architectures in $\mathcal F_{1,n}$ are fixed as $n$ varies. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-72c9e86d` | `thm:consistency` | V1.5 bookkeeping | The measures $\mu$ and $\hat\mu$ are probability measures. (pins reading of lem:push-forward-err: stated by the repair) | | `ass:pin-fix-73abef8c` | `thm:consistency` | V1.5 bookkeeping | In Corollary~\ref{corollary:deep_nn}, for $m_i=d_i^{\text{in}}>0$, the symbol $\hat f_i$ denotes $q_i\circ\Pi_i$ for a continuous variable and the vector $\bigl(\max\{\epsilon_i,(q_i\circ\Pi_i)_k\}\bigr)_{k=1}^{n_i}$ for a categorical variable, where $q_i$, $\Pi_i$, and $\epsilon_i=A_iL_0^{-2/m_i}$ are chosen in its proof. (pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair) | | `ass:pin-fix-761562d5` | `thm:consistency` | V1.5 bookkeeping | The self terms in $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$ use the same $1$-norm ground distance on the corresponding support points. (pins reading of lem:sinkhorn-approx: stated by the repair) | | `ass:pin-fix-7a79e01d` | `thm:consistency` | V1.5 bookkeeping | In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal I_n$ denotes $\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$ with $I_{O_n(i)}=[(i-1)2^{-nd},i2^{-nd}]$. (pins reading of def:prop-holder-curve-interval-construction: stated by the repair) | | `ass:pin-fix-8429b12c` | `thm:consistency` | V1.5 bookkeeping | The latent generator class in the statement uses the same temperature $\tau$ as the displayed error estimate. (pins reading of thm:deep-nn: stated by the repair) | | `ass:pin-fix-84ccd033` | `thm:consistency` | V1.5 bookkeeping | $F_n$ denotes the infimum of objective values, with an attained minimum having the same value. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-8cc0dba5` | `thm:consistency` | V1.5 bookkeeping | For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. (pins reading of def:hilbert-partitions: stated by the repair) | | `ass:pin-fix-945d85ed` | `thm:consistency` | V1.5 bookkeeping | The error estimate is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, for fixed $R$ and fixed causal model. (pins reading of thm:deep-nn: stated by the repair) | | `ass:pin-fix-9463b114` | `thm:consistency` | V1.5 bookkeeping | For a zero categorical propensity, $\log0=-\infty$. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-9675bba5` | `thm:consistency` | V1.5 bookkeeping | Ties in a categorical argmax are resolved by the smallest label. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-99ac9bb3` | `thm:consistency` | V1.5 bookkeeping | For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. (pins reading of def:hilbert-partitions: stated by the repair) | | `ass:pin-fix-9bbf998f` | `thm:consistency` | V1.5 bookkeeping | A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-9d74053b` | `thm:consistency` | V1.5 bookkeeping | The orders for $\epsilon_n$ and $s_n$ are asymptotic as $n\rightarrow\infty$. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-ab17be7a` | `thm:consistency` | V1.5 bookkeeping | For a categorical propensity coordinate equal to zero, $\log0=-\infty$ in the canonical argmax construction. (pins reading of thm:approximation: stated by the repair) | | `ass:pin-fix-b83fa4d1` | `thm:consistency` | V1.5 bookkeeping | $W$ is Wasserstein-1 distance for the ground distance $\\|x-y\\|$. (pins reading of lem:product-err: stated by the repair) | | `ass:pin-fix-bbf1c78a` | `thm:consistency` | V1.5 bookkeeping | In Corollary~\ref{corollary:deep_nn}, for $m_i=0$, the symbol $\hat f_i$ denotes the exact constant function $f_i$. (pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair) | | `ass:pin-fix-bd620122` | `thm:consistency` | V1.5 bookkeeping | The architectural $\Theta$ orders use sufficiently large fixed constant factors supplied by the network constructions in the proof. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-c514425b` | `thm:consistency` | V1.5 bookkeeping | The refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. (pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair) | | `ass:pin-fix-c778b424` | `thm:consistency` | V1.5 bookkeeping | The reference measure $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$. (pins reading of def:prop-holder-curve-cube-construction: stated by the repair) | | `ass:pin-fix-c85b9186` | `thm:consistency` | V1.5 bookkeeping | Each $O_n$ is a bijection. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-ca667e13` | `thm:consistency` | V1.5 bookkeeping | $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. (pins reading of lem:product-err: stated by the repair) | | `ass:pin-fix-d19c63b5` | `thm:consistency` | V1.5 bookkeeping | The probability bound is for each fixed pair $\delta,\epsilon>0$. (pins reading of lem:sup-wasserstein: stated by the repair) | | `ass:pin-fix-db6ed772` | `thm:consistency` | V1.5 bookkeeping | In the proposition entitled ``Deep distribution approximation'', the multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-e896f793` | `thm:consistency` | V1.5 bookkeeping | The cells of $\mathcal C_n$ are closed and may share boundaries. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-f66ad243` | `thm:consistency` | V1.5 bookkeeping | The depth expression $\Theta(\log d_{\max}^{\text{in}})$ means $\Theta(1+\log d_{\max}^{\text{in}})$ at $d_{\max}^{\text{in}}=1$. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-f6f485b1` | `thm:consistency` | V1.5 bookkeeping | The uniform infinity norm is $\\|F-\hat F\\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}\|F_j(x)-\hat F_j(x)\|$. (pins reading of lem:push-forward-err: stated by the repair) | | `ass:pin-154` | `prop:lip-ate` | V1.5 bookkeeping | For each of $\mathcal{M}$ and $\hat{\mathcal{M}}$, $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely, where the indexed $U_j$ are the non-Gumbel latent variables. (pins reading of ass:prop-lip-ate-bounded: Proposition \ref{prop:Lip_ATE} explicitly inherits Assumption \ref{assump:Lip_and_bounded}, which prints this maximum; Assumption \ref{assump:categorical_rv} distinguishes the indexed $U_j$ from the categorical noises $G_{V_i}$.) | | `ass:pin-182` | `prop:lip-ate` | V1.5 bookkeeping | For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\\|f_i(x)-f_i(x')\\|\leqslant L_f\\|x-x'\\|$ on its structural input domain, using the paper's $1$-norm. (pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$.) | | `ass:pin-183` | `prop:lip-ate` | V1.5 bookkeeping | There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely under the SCM law. (pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation.) | | `ass:pin-189` | `prop:lip-ate` | V1.5 bookkeeping | For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. (pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition.) | | `ass:pin-190` | `prop:lip-ate` | V1.5 bookkeeping | In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. (pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction.) | | `ass:pin-191` | `prop:lip-ate` | V1.5 bookkeeping | Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). (pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations.) | | `ass:pin-206` | `prop:lip-ate` | V1.5 bookkeeping | The functions $f,\hat{f}:[-K,K]\rightarrow\mathbb{R}$ are $L$-Lipschitz regression versions satisfying $f(T)=\mathbb{E}_{\nu}[Y\mid T]$ almost surely under $\nu$ and $\hat{f}(\hat{T})=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}]$ almost surely under $\mu$. (pins reading of ass:lemma-lip-condition-regression: The proof of Lemma~\ref{lemma:Lip_condition} uses both cross-evaluations and bounds its test function on the common box; the owner-supplied support condition specifies that box.) | | `ass:pin-fix-19bccf89` | `prop:lip-ate` | V1.5 bookkeeping | The measure $P$ is the reference probability measure on the treatment space: for $X=\text{Pa}(T)$, the conditional density notation means $\nu(dt\|X=x)=p_\nu(t\|X=x)P(dt)$. (pins reading of prop:lip-ate: stated by the repair) | | `ass:bk-1` | `thm:rate` | V1.5 bookkeeping | For every $k\in[n]$, $\int_{\mathbb R^d}\\|x\\|\,\mu_k(dx)+\int_{\mathbb R^d}\\|x\\|\,\hat\mu_k(dx)<\infty$. (The dual proof separately evaluates expectations of unbounded Lipschitz functions.) | | `ass:pin-154` | `thm:rate` | V1.5 bookkeeping | For each of $\mathcal{M}$ and $\hat{\mathcal{M}}$, $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely, where the indexed $U_j$ are the non-Gumbel latent variables. (pins reading of ass:prop-lip-ate-bounded: Proposition \ref{prop:Lip_ATE} explicitly inherits Assumption \ref{assump:Lip_and_bounded}, which prints this maximum; Assumption \ref{assump:categorical_rv} distinguishes the indexed $U_j$ from the categorical noises $G_{V_i}$.) | | `ass:pin-177` | `thm:rate` | V1.5 bookkeeping | $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. (pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality.) | | `ass:pin-181` | `thm:rate` | V1.5 bookkeeping | Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\\|f_i(x)\\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. (pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities.) | | `ass:pin-182` | `thm:rate` | V1.5 bookkeeping | For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\\|f_i(x)-f_i(x')\\|\leqslant L_f\\|x-x'\\|$ on its structural input domain, using the paper's $1$-norm. (pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$.) | | `ass:pin-183` | `thm:rate` | V1.5 bookkeeping | There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely under the SCM law. (pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation.) | | `ass:pin-185` | `thm:rate` | V1.5 bookkeeping | The continuous structural functions and categorical propensity functions of $\mathcal M^*$ have Lipschitz constants at most $L_f$ in the paper's $1$-norm. (pins reading of ass:thm-consistency-structural-lipschitz: Theorem~\ref{thm:consistency} begins with $\mathcal M^*$ and permits candidate networks with bound $\hat L_f$; Assumption~\ref{assump:Lip_and_bounded} identifies the functions bounded by $L_f$.) | | `ass:pin-186` | `thm:rate` | V1.5 bookkeeping | The phrase ``uniform cubes'' denotes the unit cube domains $[0,1]^m$ used for deterministic uniform function approximation; it imposes no uniform probability law on structural inputs. (pins reading of ass:relu-approximation-domain: The same paragraph displays approximation of functions on $[0,1]^{d_1}$ and separately says that similar arguments apply to any bounded cubes.) | | `ass:pin-187` | `thm:rate` | V1.5 bookkeeping | The bound $K$ applies throughout the optimization class to the observed variables $V_i$ and indexed non-Gumbel latent variables $U_j$, with $n_i\leqslant K$ for categorical nodes and admissible intervention coordinates in $[-K,K]$; it does not bound the separately supplied standard Gumbel noises. (pins reading of ass:bk-owner-1: The boundedness assumption explicitly indexes $V_i$ and $U_j$, whereas the categorical convention separately adjoins $G_{V_i}$. The binding boundedness symbols preserve this distinction.) | | `ass:pin-188` | `thm:rate` | V1.5 bookkeeping | The variable bound inherited from Assumption~\ref{assump:Lip_and_bounded} uses $\\|\cdot\\|_\infty$. (pins reading of ass:bk-owner-2: The owner condition extends the existing bound $K$, which Assumption~\ref{assump:Lip_and_bounded} explicitly states using infinity norms of observed and non-Gumbel latent variables.) | | `ass:pin-189` | `thm:rate` | V1.5 bookkeeping | For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. (pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition.) | | `ass:pin-190` | `thm:rate` | V1.5 bookkeeping | In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. (pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction.) | | `ass:pin-191` | `thm:rate` | V1.5 bookkeeping | Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). (pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations.) | | `ass:pin-192` | `thm:rate` | V1.5 bookkeeping | A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. (pins reading of def:relu-networks: The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths.) | | `ass:pin-194` | `thm:rate` | V1.5 bookkeeping | Membership in $\text{NCM}_{\mathcal G}(\mathcal F_0,\mathcal F_1)$ requires $\hat f_i\in\mathcal F_0$ for every observed node and $\hat g_j\in\mathcal F_1$ for every latent component, with the compatible input and output dimensions understood separately for each network. (pins reading of def:neural-causal-class: The source paragraph explicitly omits input and output dimensions and describes separate structural approximants and latent generators in equation~\ref{eq:structure_equation_hat}.) | | `ass:pin-196` | `thm:rate` | V1.5 bookkeeping | Truncation means coordinatewise clipping of the final output to $[-K,K]$; in $\mathcal{NN}^{\infty,K}$ no finite Lipschitz bound is imposed. (pins reading of def:truncated-neural-class: The defining minimum and maximum and the word ``Similarly'' specify the operation; the binding neural-class symbol conventions fix its application to the generator output.) | | `ass:pin-197` | `thm:rate` | V1.5 bookkeeping | $\underline{F}^{L}$ denotes the constrained infimum of the displayed scalar objective over the stated canonical SCM class and its admissible latent laws, rather than a minimizing SCM. (pins reading of def:eq-consistent-limiting: The sentence introducing the display explicitly says ``optimal value'', and the following sentence calls it the sharp lower bound when the Lipschitz bounds agree.) | | `ass:pin-198` | `thm:rate` | V1.5 bookkeeping | $\mu_T=\text{Unif}([t_1,t_2])$ is the deterministic intervention probability measure. (pins reading of def:cor-rate-objective: Corollary~\ref{cor:rate} specializes the intervention measure used in the outer expectation $\mathbb E_{t\sim\mu_T}$ of Theorem~\ref{thm:consistency}.) | | `ass:pin-199` | `thm:rate` | V1.5 bookkeeping | $d_{\max}^U=\max_{j\in[n_U]}d_j^U$ ranges over non-Gumbel latent variables, while $d_{\max}^{\mathrm{in}}=\max_i d_i^{\mathrm{in}}$ and $d_{\max}^{\mathrm{out}}=\max_i d_i^{\mathrm{out}}$, with $d_i^{\mathrm{out}}=n_i$ for categorical nodes. (pins reading of def:corollary-deep-nn-dimensions: The NCM construction separately approximates non-Gumbel latent distributions and the functions in the canonical structural equations; those functions output categorical propensity vectors. The binding dimension symbols specify these same meanings.) | | `ass:pin-201` | `thm:rate` | V1.5 bookkeeping | The coordinates $P(U_j)$ in $\theta_{\mathcal M}$ are the population laws of the non-Gumbel latent variables, not necessarily uniform laws; the independent standard Gumbel laws are fixed and omitted from this population tuple. (pins reading of def:thm-consistency-model-parameterization: Immediately after the tuple, the population parameter space contains factors $\mathcal P([-K,K]^{d_j^U})$. The canonical representation supplies the categorical Gumbel laws separately.) | | `ass:pin-203` | `thm:rate` | V1.5 bookkeeping | The gap event is $E_\delta=\{\exists\,1\leqslant i0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. (pins reading of def:gumbel-softmax-layer: The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit.) | | `ass:pin-210` | `thm:rate` | V1.5 bookkeeping | $F_n=\inf\left\{\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\hat{\mathcal{M}}}[F(V_1(t),\ldots,V_{n_V}(t))]:\hat{\mathcal{M}}\in\text{NCM}_{\mathcal{G}}(\mathcal{F}_{0,n},\mathcal{F}_{1,n}),\ S_{\lambda_n}(P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V}),P_n^{\mathcal{M}^*}(\boldsymbol{V}))\leqslant\alpha_n\right\}$. (pins reading of def:eq-opt-empiri: The prose introducing equation (\ref{eq:opt_empiri}) calls $F_n$ an estimator of optimal values, and the following paragraph asks whether it is a good lower bound. The supplied symbol convention likewise specifies a scalar infimum.) | | `ass:pin-211` | `thm:rate` | V1.5 bookkeeping | In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. (pins reading of def:lemma-gumbel-err-softmax: Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores.) | | `ass:pin-33` | `thm:rate` | V1.5 bookkeeping | $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. (pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes.) | | `ass:pin-75` | `thm:rate` | V1.5 bookkeeping | For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. (pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula.) | | `ass:pin-fix-00130002` | `thm:rate` | V1.5 bookkeeping | The law $\hat g_{\#}\lambda$ includes the independent standard Gumbel variables used in the mixing layer. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-09991de8` | `thm:rate` | V1.5 bookkeeping | The observational empirical law $P_n^{\mathcal M^*}(\boldsymbol V)$ is formed from $n$ independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-0a7b21b0` | `thm:rate` | V1.5 bookkeeping | The empirical optimization ranges over network representations, including distinct representations with the same latent population laws. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-0bcb2cef` | `thm:rate` | V1.5 bookkeeping | The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-19bccf89` | `thm:rate` | V1.5 bookkeeping | The measure $P$ is the reference probability measure on the treatment space: for $X=\text{Pa}(T)$, the conditional density notation means $\nu(dt\|X=x)=p_\nu(t\|X=x)P(dt)$. (pins reading of prop:lip-ate: stated by the repair) | | `ass:pin-fix-2a1e7b01` | `thm:rate` | V1.5 bookkeeping | The mass assigned to $I_{O_n(i)}$ is $\lambda_d(C_{O_n(i)})=2^{-nd}$. (pins reading of def:prop-holder-curve-interval-construction: stated by the repair) | | `ass:pin-fix-2fe04b0e` | `thm:rate` | V1.5 bookkeeping | In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal C_n$ denotes the standard dyadic collection $\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}$. (pins reading of def:prop-holder-curve-cube-construction: stated by the repair) | | `ass:pin-fix-30413b86` | `thm:rate` | V1.5 bookkeeping | The common simulation draws are independent of the observational sample. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-3b195d8d` | `thm:rate` | V1.5 bookkeeping | The common exogenous draws are independent and identically distributed according to the specified product law of the exogenous inputs. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-419eb8e3` | `thm:rate` | V1.5 bookkeeping | The multiplicative width factors in $\Theta(d_{\max}^{\text{in}}d_{\max}^{\text{out}})$, $\Theta(d_{\max}^U)$, and $\Theta((d_{\max}^U)^2)$ can be chosen as absolute numerical constants, independent of the causal model, $R$, all dimensions, and all category and component counts. (pins reading of thm:deep-nn: stated by the repair) | | `ass:pin-fix-43f1bb71` | `thm:rate` | V1.5 bookkeeping | For each latent law $P(U_C)$, the assumptions of the proposition entitled ``Deep distribution approximation'' require the support charts of \cref{assump:support} and \cref{assumption:lower_bound} on each normalized component law $\mu_i=P(U_C)\|_{C_i}/P(U_C\in C_i)$, with $(f^{-1})_{\#}\mu_i(\partial[0,1]^{d_i^C})=0$ when $d_i^C>0$. (pins reading of thm:deep-nn: stated by the repair) | | `ass:pin-fix-448b19a0` | `thm:rate` | V1.5 bookkeeping | Probabilities of uniform deviation events are evaluated in the completion of the sample law. (pins reading of lem:sup-wasserstein: stated by the repair) | | `ass:pin-fix-4a86c2eb` | `thm:rate` | V1.5 bookkeeping | The structural uniform infinity norm takes the supremum of the coordinatewise infinity norm over the common structural input domain. (pins reading of thm:approximation: stated by the repair) | | `ass:pin-fix-507a5848` | `thm:rate` | V1.5 bookkeeping | For each component of positive dimension $q=d_i^C$, the application of Theorem VIII.1 of \cite{perekrestenkoHighDimensionalDistributionGeneration2022a} uses integer depth parameter $s=1$, quantization parameter $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, and quantization mesh $\delta/n$. (pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair) | | `ass:pin-fix-59a01041` | `thm:rate` | V1.5 bookkeeping | For each $n$, every candidate is evaluated on the same $m_n$ draws of all its exogenous inputs, including its uniform inputs, generator Gumbel inputs, and categorical Gumbel inputs. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-5f02b8a6` | `thm:rate` | V1.5 bookkeeping | The error estimates are asymptotic as the network budgets tend to infinity and $\tau\downarrow0$, with constants independent of these budgets and of $\tau$. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-676b96c3` | `thm:rate` | V1.5 bookkeeping | The finite component counts and component cube dimensions of the latent generator architectures in $\mathcal F_{1,n}$ are fixed as $n$ varies. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-72c9e86d` | `thm:rate` | V1.5 bookkeeping | The measures $\mu$ and $\hat\mu$ are probability measures. (pins reading of lem:push-forward-err: stated by the repair) | | `ass:pin-fix-73abef8c` | `thm:rate` | V1.5 bookkeeping | In Corollary~\ref{corollary:deep_nn}, for $m_i=d_i^{\text{in}}>0$, the symbol $\hat f_i$ denotes $q_i\circ\Pi_i$ for a continuous variable and the vector $\bigl(\max\{\epsilon_i,(q_i\circ\Pi_i)_k\}\bigr)_{k=1}^{n_i}$ for a categorical variable, where $q_i$, $\Pi_i$, and $\epsilon_i=A_iL_0^{-2/m_i}$ are chosen in its proof. (pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair) | | `ass:pin-fix-761562d5` | `thm:rate` | V1.5 bookkeeping | The self terms in $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$ use the same $1$-norm ground distance on the corresponding support points. (pins reading of lem:sinkhorn-approx: stated by the repair) | | `ass:pin-fix-7a79e01d` | `thm:rate` | V1.5 bookkeeping | In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal I_n$ denotes $\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$ with $I_{O_n(i)}=[(i-1)2^{-nd},i2^{-nd}]$. (pins reading of def:prop-holder-curve-interval-construction: stated by the repair) | | `ass:pin-fix-8429b12c` | `thm:rate` | V1.5 bookkeeping | The latent generator class in the statement uses the same temperature $\tau$ as the displayed error estimate. (pins reading of thm:deep-nn: stated by the repair) | | `ass:pin-fix-84ccd033` | `thm:rate` | V1.5 bookkeeping | $F_n$ denotes the infimum of objective values, with an attained minimum having the same value. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-8cc0dba5` | `thm:rate` | V1.5 bookkeeping | For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. (pins reading of def:hilbert-partitions: stated by the repair) | | `ass:pin-fix-91573abf` | `thm:rate` | V1.5 bookkeeping | The observational empirical law $P_n^{\mathcal M^*}(\boldsymbol V)$ is formed from $n$ independent complete observations with common law $P^{\mathcal M^*}(\boldsymbol V)$. (pins reading of thm:rate: stated by the repair) | | `ass:pin-fix-945d85ed` | `thm:rate` | V1.5 bookkeeping | The error estimate is asymptotic as $L_0,L_1,L_2\to\infty$ and $\tau\downarrow0$, for fixed $R$ and fixed causal model. (pins reading of thm:deep-nn: stated by the repair) | | `ass:pin-fix-9463b114` | `thm:rate` | V1.5 bookkeeping | For a zero categorical propensity, $\log0=-\infty$. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-9675bba5` | `thm:rate` | V1.5 bookkeeping | Ties in a categorical argmax are resolved by the smallest label. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-99ac9bb3` | `thm:rate` | V1.5 bookkeeping | For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. (pins reading of def:hilbert-partitions: stated by the repair) | | `ass:pin-fix-9bbf998f` | `thm:rate` | V1.5 bookkeeping | A component with $d_i^C=0$ is a singleton, and its two deterministic blocks are the unique map to $[0,1]^0$ and a constant affine map to that singleton. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-9d74053b` | `thm:rate` | V1.5 bookkeeping | The orders for $\epsilon_n$ and $s_n$ are asymptotic as $n\rightarrow\infty$. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-ab17be7a` | `thm:rate` | V1.5 bookkeeping | For a categorical propensity coordinate equal to zero, $\log0=-\infty$ in the canonical argmax construction. (pins reading of thm:approximation: stated by the repair) | | `ass:pin-fix-b83fa4d1` | `thm:rate` | V1.5 bookkeeping | $W$ is Wasserstein-1 distance for the ground distance $\\|x-y\\|$. (pins reading of lem:product-err: stated by the repair) | | `ass:pin-fix-bbf1c78a` | `thm:rate` | V1.5 bookkeeping | In Corollary~\ref{corollary:deep_nn}, for $m_i=0$, the symbol $\hat f_i$ denotes the exact constant function $f_i$. (pins reading of def:corollary-deep-nn-structural-approximants: stated by the repair) | | `ass:pin-fix-bd620122` | `thm:rate` | V1.5 bookkeeping | The architectural $\Theta$ orders use sufficiently large fixed constant factors supplied by the network constructions in the proof. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-c514425b` | `thm:rate` | V1.5 bookkeeping | The refinement integer is $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for $W_1\geqslant A_q2^q$, where $A_q\geqslant1$ depends only on $q$ and bounds the clipped network width by $A_qn^q$. (pins reading of lem:wide-nn-perekrestenko-theorem-viii-1: stated by the repair) | | `ass:pin-fix-c778b424` | `thm:rate` | V1.5 bookkeeping | The reference measure $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$. (pins reading of def:prop-holder-curve-cube-construction: stated by the repair) | | `ass:pin-fix-c85b9186` | `thm:rate` | V1.5 bookkeeping | Each $O_n$ is a bijection. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-ca667e13` | `thm:rate` | V1.5 bookkeeping | $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. (pins reading of lem:product-err: stated by the repair) | | `ass:pin-fix-d19c63b5` | `thm:rate` | V1.5 bookkeeping | The probability bound is for each fixed pair $\delta,\epsilon>0$. (pins reading of lem:sup-wasserstein: stated by the repair) | | `ass:pin-fix-db6ed772` | `thm:rate` | V1.5 bookkeeping | In the proposition entitled ``Deep distribution approximation'', the multiplicative width factors in $\Theta(D_C)$ and $\Theta(dD_C)$ are absolute numerical constants, independent of the measure, chart constants, dimensions, and component count. (pins reading of thm:wide-nn: stated by the repair) | | `ass:pin-fix-e896f793` | `thm:rate` | V1.5 bookkeeping | The cells of $\mathcal C_n$ are closed and may share boundaries. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-f66ad243` | `thm:rate` | V1.5 bookkeeping | The depth expression $\Theta(\log d_{\max}^{\text{in}})$ means $\Theta(1+\log d_{\max}^{\text{in}})$ at $d_{\max}^{\text{in}}=1$. (pins reading of thm:consistency: stated by the repair) | | `ass:pin-fix-f6f485b1` | `thm:rate` | V1.5 bookkeeping | The uniform infinity norm is $\\|F-\hat F\\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}\|F_j(x)-\hat F_j(x)\|$. (pins reading of lem:push-forward-err: stated by the repair) | | `ass:pin-177` | `prop:equivalent-g-constrianed` | V1.5 bookkeeping | $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. (pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality.) | | `ass:pin-181` | `prop:equivalent-g-constrianed` | V1.5 bookkeeping | Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\\|f_i(x)\\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. (pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities.) | | `ass:pin-189` | `prop:equivalent-g-constrianed` | V1.5 bookkeeping | For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. (pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition.) | | `ass:pin-190` | `prop:equivalent-g-constrianed` | V1.5 bookkeeping | In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. (pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction.) | | `ass:pin-191` | `prop:equivalent-g-constrianed` | V1.5 bookkeeping | Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). (pins reading of def:intervention: The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations.) | | `ass:pin-fix-c0174608` | `prop:equivalent-g-constrianed` | V1.5 bookkeeping | The categorical generation formula requires $f_i(x)\in[0,\infty)^{n_i}$ and $\\|f_i(x)\\|_1>0$ on its structural input domain. (pins reading of prop:equivalent-g-constrianed: stated by the repair) | | `ass:pin-fix-c356d336` | `prop:equivalent-g-constrianed` | V1.5 bookkeeping | A zero categorical coordinate has extended logarithm $\log0=-\infty$. (pins reading of prop:equivalent-g-constrianed: stated by the repair) | | `ass:pin-33` | `prop:holder-curve` | V1.5 bookkeeping | $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. (pins reading of def:homeomorphisms: The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes.) | | `ass:pin-fix-0bcb2cef` | `prop:holder-curve` | V1.5 bookkeeping | The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-2a1e7b01` | `prop:holder-curve` | V1.5 bookkeeping | The mass assigned to $I_{O_n(i)}$ is $\lambda_d(C_{O_n(i)})=2^{-nd}$. (pins reading of def:prop-holder-curve-interval-construction: stated by the repair) | | `ass:pin-fix-2fe04b0e` | `prop:holder-curve` | V1.5 bookkeeping | In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal C_n$ denotes the standard dyadic collection $\left\{\prod_{r=1}^d[a_r2^{-n},(a_r+1)2^{-n}]:a_r\in\{0,\ldots,2^n-1\}\right\}$. (pins reading of def:prop-holder-curve-cube-construction: stated by the repair) | | `ass:pin-fix-7a79e01d` | `prop:holder-curve` | V1.5 bookkeeping | In the proof of Proposition~\ref{prop:holder_curve}, $\mathcal I_n$ denotes $\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$ with $I_{O_n(i)}=[(i-1)2^{-nd},i2^{-nd}]$. (pins reading of def:prop-holder-curve-interval-construction: stated by the repair) | | `ass:pin-fix-8cc0dba5` | `prop:holder-curve` | V1.5 bookkeeping | For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. (pins reading of def:hilbert-partitions: stated by the repair) | | `ass:pin-fix-99ac9bb3` | `prop:holder-curve` | V1.5 bookkeeping | For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. (pins reading of def:hilbert-partitions: stated by the repair) | | `ass:pin-fix-c778b424` | `prop:holder-curve` | V1.5 bookkeeping | The reference measure $\lambda_d$ denotes Lebesgue measure on $[0,1]^d$. (pins reading of def:prop-holder-curve-cube-construction: stated by the repair) | | `ass:pin-fix-c85b9186` | `prop:holder-curve` | V1.5 bookkeeping | Each $O_n$ is a bijection. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-e896f793` | `prop:holder-curve` | V1.5 bookkeeping | The cells of $\mathcal C_n$ are closed and may share boundaries. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-0bcb2cef` | `lem:hilbert` | V1.5 bookkeeping | The assertion about adjacent cubes is quantified over $i$ satisfying $1\leqslant i<2^{nd}$. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-8cc0dba5` | `lem:hilbert` | V1.5 bookkeeping | For every $n$, the cells of $\mathcal C_n$ are closed coordinate boxes covering $[0,1]^d$; a cell may have zero side lengths. (pins reading of def:hilbert-partitions: stated by the repair) | | `ass:pin-fix-99ac9bb3` | `lem:hilbert` | V1.5 bookkeeping | For distinct cells $C,C'\in\mathcal C_n$, the partition condition is $\operatorname{int}(C)\cap\operatorname{int}(C')=\emptyset$, with interiors taken in $\mathbb R^d$. (pins reading of def:hilbert-partitions: stated by the repair) | | `ass:pin-fix-c85b9186` | `lem:hilbert` | V1.5 bookkeeping | Each $O_n$ is a bijection. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-fix-e896f793` | `lem:hilbert` | V1.5 bookkeeping | The cells of $\mathcal C_n$ are closed and may share boundaries. (pins reading of lem:hilbert: stated by the repair) | | `ass:pin-177` | `prop:counterexample-foral` | V1.5 bookkeeping | $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. (pins reading of def:definition-2: The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality.) | | `ass:pin-181` | `prop:counterexample-foral` | V1.5 bookkeeping | Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\\|f_i(x)\\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. (pins reading of ass:assump-categorical-rv-generation: The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities.) | | `ass:pin-182` | `prop:counterexample-foral` | V1.5 bookkeeping | For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\\|f_i(x)-f_i(x')\\|\leqslant L_f\\|x-x'\\|$ on its structural input domain, using the paper's $1$-norm. (pins reading of ass:assump-lip-and-bounded-categorical: Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$.) | | `ass:pin-183` | `prop:counterexample-foral` | V1.5 bookkeeping | There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely under the SCM law. (pins reading of ass:assump-lip-and-bounded-bound: Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation.) | | `ass:pin-189` | `prop:counterexample-foral` | V1.5 bookkeeping | For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. (pins reading of def:eq-structure-eq: The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition.) | | `ass:pin-190` | `prop:counterexample-foral` | V1.5 bookkeeping | In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. (pins reading of def:eq-structure-equation: The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction.) | | `ass:pin-204` | `prop:counterexample-foral` | V1.5 bookkeeping | The second Bernoulli law in the construction has $P(U_2=0)=2/3$. (pins reading of prop:counterexample-foral: The displayed value $P(U_2=1)=1/3$ and the joint masses for $X=0,T=0$ in the proof of Proposition~\ref{prop:counterexample_foral} both determine this entry.) | | `ass:bk-6` | `prop:convg-opt` | V1.5 bookkeeping | For every $n\in\mathbb N$, the feasible set of (\ref{eq:opt_n}) contains a minimizer of $f$. (The finite values are defined as minima, and the lower-limit argument selects attaining minimizers.) | | `ass:pin-159` | `prop:convg-opt` | V1.5 bookkeeping | Equation~\ref{eq:aprox_lip} holds for every $n\in\mathbb N$ and every $\theta_1,\theta_2\in\Theta_\infty$, with the same finite constant $L_g$. (pins reading of ass:prop-convg-opt-approximate-lipschitz: Proposition~\ref{prop:convg_opt} defines the functions on $\Theta_\infty$ and uses this inequality to conclude that the limiting function is $L_g$-Lipschitz.) | | `ass:pin-172` | `prop:convg-opt` | V1.5 bookkeeping | There exists a compact $\tilde\Theta_\infty\subseteq\Theta_\infty$ such that $\forall n\in\mathbb N\;\forall\theta_0\in\tilde\Theta_\infty\;\exists\theta\in\Theta_n$, $d(\theta,\theta_0)\leqslant\epsilon_n$. (pins reading of ass:prop-convg-opt-approximation: The proof of Proposition \ref{prop:convg_opt} applies condition 3 to construct a sequence $\theta_n$ approximating $\tilde\theta^*$.) | | `ass:pin-52` | `prop:convg-opt` | V1.5 bookkeeping | In (\ref{eq:aprox_lip}), $\Theta=\Theta_\infty$. (pins reading of ass:prop-convg-opt-approximate-lipschitz: Proposition~\ref{prop:convg_opt} defines the functions on $\Theta_\infty$ and uses this inequality there to establish the limiting Lipschitz bound.) | | `ass:pin-206` | `lem:lip-condition` | V1.5 bookkeeping | The functions $f,\hat{f}:[-K,K]\rightarrow\mathbb{R}$ are $L$-Lipschitz regression versions satisfying $f(T)=\mathbb{E}_{\nu}[Y\mid T]$ almost surely under $\nu$ and $\hat{f}(\hat{T})=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}]$ almost surely under $\mu$. (pins reading of ass:lemma-lip-condition-regression: The proof of Lemma~\ref{lemma:Lip_condition} uses both cross-evaluations and bounds its test function on the common box; the owner-supplied support condition specifies that box.) | | `ass:pin-fix-72c9e86d` | `lem:push-forward-err` | V1.5 bookkeeping | The measures $\mu$ and $\hat\mu$ are probability measures. (pins reading of lem:push-forward-err: stated by the repair) | | `ass:pin-fix-f6f485b1` | `lem:push-forward-err` | V1.5 bookkeeping | The uniform infinity norm is $\\|F-\hat F\\|_\infty=\sup_{x\in K}\max_{j\in[d_2]}\|F_j(x)-\hat F_j(x)\|$. (pins reading of lem:push-forward-err: stated by the repair) | | `ass:bk-1` | `lem:combin-err` | V1.5 bookkeeping | For every $k\in[n]$, $\int_{\mathbb R^d}\\|x\\|\,\mu_k(dx)+\int_{\mathbb R^d}\\|x\\|\,\hat\mu_k(dx)<\infty$. (The dual proof separately evaluates expectations of unbounded Lipschitz functions.) | | `ass:pin-75` | `lem:combin-err` | V1.5 bookkeeping | For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. (pins reading of lem:combin-err: Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula.) | | `ass:pin-fix-b83fa4d1` | `lem:product-err` | V1.5 bookkeeping | $W$ is Wasserstein-1 distance for the ground distance $\\|x-y\\|$. (pins reading of lem:product-err: stated by the repair) | | `ass:pin-fix-ca667e13` | `lem:product-err` | V1.5 bookkeeping | $S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$. (pins reading of lem:product-err: stated by the repair) | | `ass:pin-203` | `lem:gumbel-err` | V1.5 bookkeeping | The gap event is $E_\delta=\{\exists\,1\leqslant i0$. (pins reading of lem:sup-wasserstein: stated by the repair) | ## Conditions assumed ### Cited results | id | used by | citation | locator | checked against | | --- | --- | --- | --- | --- | | `lem:wide-nn-perekrestenko-theorem-viii-1` | `thm:wide-nn` | perekrestenkoHighDimensionalDistributionGeneration2022a | Theorem VIII.1 | source supplied by the user at discovery/cited_sources/perekrestenkoHighDimensionalDistributionGeneration2022a.txt | | `lem:wide-nn-yarotsky-theorem-2` | `thm:wide-nn` | Yarotsky2018OptimalAO | Theorem 2 | source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt | | `lem:wide-nn-perekrestenko-theorem-viii-1` | `thm:cor-fix-3c402872` | perekrestenkoHighDimensionalDistributionGeneration2022a | Theorem VIII.1 | source supplied by the user at discovery/cited_sources/perekrestenkoHighDimensionalDistributionGeneration2022a.txt | | `lem:wide-nn-yarotsky-theorem-2` | `thm:cor-fix-3c402872` | Yarotsky2018OptimalAO | Theorem 2 | source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt | | `lem:deep-nn-yarotsky-theorem-2` | `thm:deep-nn` | Yarotsky2018OptimalAO | Theorem 2 | source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt | | `lem:wide-nn-perekrestenko-theorem-viii-1` | `thm:deep-nn` | perekrestenkoHighDimensionalDistributionGeneration2022a | Theorem VIII.1 | source supplied by the user at discovery/cited_sources/perekrestenkoHighDimensionalDistributionGeneration2022a.txt | | `lem:wide-nn-yarotsky-theorem-2` | `thm:deep-nn` | Yarotsky2018OptimalAO | Theorem 2 | source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt | | `lem:consistency-fournier-theorem-2` | `thm:consistency` | fournier2015rate | Theorem 2 | source supplied by the user at discovery/cited_sources/fournier2015rate.txt | | `lem:deep-nn-yarotsky-theorem-2` | `thm:consistency` | Yarotsky2018OptimalAO | Theorem 2 | source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt | | `lem:sup-wasserstein-anthony-theorem-17-1` | `thm:consistency` | anthony1999neural | Theorem 17.1 | source supplied by the user at discovery/cited_sources/anthony1999neural.txt | | `lem:sup-wasserstein-anthony-theorems` | `thm:consistency` | anthony1999neural | Theorem 18.4 | source supplied by the user at discovery/cited_sources/anthony1999neural.txt | | `lem:wide-nn-perekrestenko-theorem-viii-1` | `thm:consistency` | perekrestenkoHighDimensionalDistributionGeneration2022a | Theorem VIII.1 | source supplied by the user at discovery/cited_sources/perekrestenkoHighDimensionalDistributionGeneration2022a.txt | | `lem:wide-nn-yarotsky-theorem-2` | `thm:consistency` | Yarotsky2018OptimalAO | Theorem 2 | source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt | | `lem:consistency-fournier-theorem-2` | `thm:rate` | fournier2015rate | Theorem 2 | source supplied by the user at discovery/cited_sources/fournier2015rate.txt | | `lem:deep-nn-yarotsky-theorem-2` | `thm:rate` | Yarotsky2018OptimalAO | Theorem 2 | source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt | | `lem:sup-wasserstein-anthony-theorem-17-1` | `thm:rate` | anthony1999neural | Theorem 17.1 | source supplied by the user at discovery/cited_sources/anthony1999neural.txt | | `lem:sup-wasserstein-anthony-theorems` | `thm:rate` | anthony1999neural | Theorem 18.4 | source supplied by the user at discovery/cited_sources/anthony1999neural.txt | | `lem:wide-nn-perekrestenko-theorem-viii-1` | `thm:rate` | perekrestenkoHighDimensionalDistributionGeneration2022a | Theorem VIII.1 | source supplied by the user at discovery/cited_sources/perekrestenkoHighDimensionalDistributionGeneration2022a.txt | | `lem:wide-nn-yarotsky-theorem-2` | `thm:rate` | Yarotsky2018OptimalAO | Theorem 2 | source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt | | `lem:sup-wasserstein-anthony-theorem-17-1` | `lem:sup-wasserstein` | anthony1999neural | Theorem 17.1 | source supplied by the user at discovery/cited_sources/anthony1999neural.txt | | `lem:sup-wasserstein-anthony-theorems` | `lem:sup-wasserstein` | anthony1999neural | Theorem 18.4 | source supplied by the user at discovery/cited_sources/anthony1999neural.txt | `lem:wide-nn-perekrestenko-theorem-viii-1` (perekrestenkoHighDimensionalDistributionGeneration2022a, Theorem VIII.1), invoked in the proof of `thm:wide-nn`: What Lean assumes for it: `Papers.TbsNeuralCausalPi.PerekrestenkoQuantizedGeneration` (`Setup/CitedResults.lean:73`) ```lean def PerekrestenkoQuantizedGeneration : Sort 0 := -- @realizes s(natural-number source depth fixed by let s := 1 below; distinct from delta and delta / n) ∃ c : ℕ, 0 < c ∧ ∀ q : ℕ, 0 < q → ∃ A : ℝ, 1 ≤ A ∧ -- @realizes \Theta(d_i^C)(depth c * q with one multiplier c for every dimension q; the source's depth (s+3)q-s-1 = 4q-2 at s = 1) ∀ (d N : ℕ) (P : ProbabilityMeasure (L1Vec d)) (S : Fin N → Set (L1Vec d)) (dC : Fin N → ℕ) (L : ℝ≥0), SupportFiniteComponents (P : Measure _) S → ComponentsLipschitzHomeomorphic S dC L → ∀ (i : Fin N), dC i = q → ∀ f : homeoClass (unitCube (dC i)) (S i), LipschitzWith L f → ∃ B : ℝ, 0 < B ∧ ∀ W₁ : ℕ, A * (2 : ℝ)^q ≤ W₁ → -- @realizes W_1(component width budget above the dimension-dependent threshold) let n : ℕ := ⌊((W₁ : ℝ) / A) ^ (1 / (q : ℝ))⌋₊ -- @realizes n(refinement floor chosen from W_1 and the dimension-only width factor A) let s : ℕ := 1 -- @realizes s(carrier ℕ and fixed value s = 1; source depth, distinct from quantization delta and mesh delta / n) let delta : ℝ := (⌈Real.sqrt q * (q + 1) * n * ((n : ℝ) - 1)⌉₊ : ℝ)⁻¹ -- @realizes \delta(prescribed quantization parameter at the pinned refinement) 2 ≤ n ∧ A * (n : ℝ)^q ≤ W₁ ∧ ∃ Phi : ReLUNet 1 (dC i) W₁ (c * q), -- @realizes \hat{g}_1^i(quantized component network of width W_1 and depth proportional to its positive dimension) (∀ l ∈ Phi.hidden, (l.output : ℝ) ≤ A * (n : ℝ)^q) ∧ Phi.paramsQuantized (delta / n) ∧ wasserstein1 ((ProbabilityTheory.cond (P : Measure _) (S i)).map (chartInverse f)) ((uniformCubeLaw 1).map Phi.eval) ≤ ENNReal.ofReal (B * (W₁ : ℝ) ^ (-(1 : ℝ) / q)) ``` The passage of the proof: > By \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, there exists quantized ReLu network $ \hat{g}_{1}^{i}$ of width $ W_{1}$ and depth $ \Theta( d_i^C)$ (let $ s=\frac{1}{\lceil n( n-1)\rceil }$ in \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}), such that \begin{equation*} > W\left(\left( (g_{2}^{i})^{-1}\right)_{\#} \mu _{i} ,P(\hat{g}_{1}^{i}(U_{i}))\right) \leqslant O\left( W_{1}^{-1/d_i^C}\right) , > \end{equation*} > where $ U_{i}$ are i.i.d. uniform random variables on $ [ 0,1]$. `lem:wide-nn-yarotsky-theorem-2` (Yarotsky2018OptimalAO, Theorem 2), invoked in the proof of `thm:wide-nn`: What Lean assumes for it: `Papers.TbsNeuralCausalPi.YarotskyCoordinateApprox` (`Setup/CitedResults.lean:115`) ```lean def YarotskyCoordinateApprox : Sort 0 := ∀ d : ℕ, 0 < d → ∃ a c : ℝ, 0 < a ∧ 0 < c ∧ ∀ (D : ℕ) (S : Set (L1Vec D)), IsCompact S → ∀ (f : homeoClass (unitCube d) S) (L : ℝ≥0), LipschitzWith L f → ∀ j : Fin D, ∀ depth : ℕ, 1 ≤ depth → ∃ q : ReLUNet d 1 (12*d) depth, ∀ x ∈ unitCube d, |(if hx : x ∈ unitCube d then (f ⟨x,hx⟩).val.ofLp j else 0) - (q.eval x).ofLp 0| ≤ a * euclidModulus (fun y => if hy : y ∈ unitCube d then (f ⟨y,hy⟩).val.ofLp j else 0) (c * (depth : ℝ) ^ (-(2 : ℝ) / d)) ``` The passage of the proof: > By \cite[Theorem 2]{Yarotsky2018OptimalAO}, there exist a deep ReLu network $ \hat{g}_{2}^{i,j}$ of width $ \Theta\left( d_{i}^{C}\right)$ and depth $ L_{2}$ such that > \begin{equation*} > \| \left(g_{2}^{i}\right)_{j} -\hat{g}_{2}^{i,j} \| _{\infty } \leqslant O\left( L_{2}^{-2/d_{i}^{C}}\right) , > \end{equation*} > Let $ \hat{g}_{2}^{i}( x) =\left(\hat{g}_{2}^{i,1}( x) ,\cdots ,\hat{g}_{2}^{i,d}( x)\right)$, we get > \begin{equation*} > \| g_{2}^{i} -\hat{g}_{2}^{i} \|_\infty \leqslant O\left( L_{2}^{-2/d_{i}^{C}}\right) . > \end{equation*} `lem:deep-nn-yarotsky-theorem-2` (Yarotsky2018OptimalAO, Theorem 2), invoked in the proof of `thm:deep-nn`: What Lean assumes for it: `Papers.TbsNeuralCausalPi.YarotskyStructuralApprox` (`Setup/CitedResults.lean:140`) ```lean def YarotskyStructuralApprox : Sort 0 := ∀ d : ℕ, 0 < d → ∃ a c : ℝ, 0 < a ∧ 0 < c ∧ ∀ (n b : ℕ) (kind : Fin n → NodeKind) (dims : Fin b → ℕ) (dag : Causalean.Graph.DAG (Fin n)) (G : SimpleGraph (Fin n)) (C : Fin b → Set (Fin n)) (M : CanonicalSCM kind dims dag G C) (i : Fin n) (hd : structuralInputSize M.toEuclideanSCM i = d), let f := fun x : L1Vec d => M.f i (structuralFiniteCoordinates M.toEuclideanSCM i (WithLp.toLp 1 (fun k => x.ofLp (Fin.cast hd k)))) ∀ L : ℝ≥0, LipschitzOnWith L f (unitCube d) → ∀ j : Fin (outputDim (kind i)), ∀ depth : ℕ, 1 ≤ depth → ∃ q : ReLUNet d 1 (2*d+10) depth, ∀ x ∈ unitCube d, |(f x).ofLp j - (q.eval x).ofLp 0| ≤ a * euclidModulus (fun y => (f y).ofLp j) (c * (depth : ℝ) ^ (-(2 : ℝ) / d)) ``` The passage of the proof: > By \cite[Theorem 2]{Yarotsky2018OptimalAO}, part b), applied with $p=2/d_{i}^{\text{in}}$ to each output coordinate of $f_i$, which is Lipschitz continuous on the input cube by \cref{assump:Lip_and_bounded}: there are narrow fully-connected ReLU networks of constant width $2d_{i}^{\text{in}}+10$ and depth $L_0$ whose sup-norm error is at most $a\,\omega(cW^{-2/d_{i}^{\text{in}}})$, where $W$, the number of weights, grows linearly in $L_0$ at fixed width; stacking the $d_{i}^{\text{out}}$ coordinates gives, if $V_i$ is continuous, $ \| \hat{f}_{i} -f_{i} \| _{\infty } \leqslant O(L_0^{-2/d_{i}^{\text{in}}}) \leqslant O(L_0^{-2/d_{\max}^{\text{in}}}) $. `lem:consistency-fournier-theorem-2` (fournier2015rate, Theorem 2), invoked in the proof of `thm:consistency`: What Lean assumes for it: `Papers.TbsNeuralCausalPi.FournierGuillinSmallDeviation` (`Setup/CitedResults.lean:213`) ```lean def FournierGuillinSmallDeviation : Sort 0 := ∀ d : ℕ, 1 ≤ d → -- @realizes d(positive source Euclidean dimension) ∀ alpha gamma e : ℝ, 1 < alpha → 0 < gamma → ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ -- @realizes C(positive source prefactor fixed before the law and sample size) -- @realizes c(positive source exponent constant fixed before the law and sample size) ∀ mu : ProbabilityMeasure (EuclideanSpace ℝ (Fin d)), -- @realizes \mu(probability law on Euclidean space with the source two-norm) Integrable (fun z => Real.exp (gamma * ‖z‖ ^ alpha)) (mu : Measure (EuclideanSpace ℝ (Fin d))) → (∫ z, Real.exp (gamma * ‖z‖ ^ alpha) ∂(mu : Measure (EuclideanSpace ℝ (Fin d)))) = e → ∀ N : ℕ, 1 ≤ N → -- @realizes n(positive IID source sample size N) ∀ x : ℝ, 0 < x → x ≤ 1 → (Measure.pi (fun _ : Fin N => (mu : Measure (EuclideanSpace ℝ (Fin d))))) {z | ENNReal.ofReal x ≤ wasserstein1 ((N : ℝ≥0∞)⁻¹ • ∑ i : Fin N, Measure.dirac (z i)) (mu : Measure (EuclideanSpace ℝ (Fin d)))} ≤ ENNReal.ofReal (C * Real.exp (-c * N * (if d = 1 then x^2 else if d = 2 then (x / Real.log (2 + x⁻¹))^2 else x^d))) ``` The passage of the proof: > \item By definition of $ g_{n}$, > \begin{align*} > |g_{n}( \theta ) -g_{n}( \theta') | & =|S_{\lambda _{n}}\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -S_{\lambda _{n}}\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta'}(\boldsymbol{V})\right) |\\ > & \leqslant |W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta'}(\boldsymbol{V})\right) |+ 4(\log( mn) +1) \lambda _{n}\\ > & \leqslant W\left( P_{m_{n}}^{\theta }(\boldsymbol{V}) ,P_{m_{n}}^{\theta'}(\boldsymbol{V})\right) + 4(\log( mn) +1) \lambda _{n}\\ > & \leqslant O( d( \theta ,\theta')) +4(1+\log( nm_{n})) \lambda _{n} , > \end{align*} > where we use Lemma \ref{lemma:sinkhorn_approx} in the second inequality and triangle inequality and \cref{thm:approximation} in the third inequality. By the condition in \cref{thm:consistency}, the second term $ (1+\log( nm_{n})) \lambda _{n} \rightarrow 0 $ > as $n \rightarrow \infty$. Therefore, (\ref{eq:aprox_lip}) is verified with $\tau_n = 4(1+\log( nm_{n})) \lambda _{n} $. > > We then verify the uniform convergence of $g_n$. By triangle inequality, > \begin{align*} > |g_{n}( \theta ) -g( \theta ) | & =|S_{\lambda _{n}}\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}(\boldsymbol{V}) ,P^{\theta }(\boldsymbol{V})\right) |\\ > & \leqslant |S_{\lambda _{n}}\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) |\\ > & \ \ \ \ \ \ \ \ +|W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}(\boldsymbol{V}) ,P^{\theta }(\boldsymbol{V})\right) |. > \end{align*} > By Lemma \ref{lemma:sinkhorn_approx}, $ |S_{\lambda _{n}}\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) |\leqslant 2(\log( mn) +1) \lambda _{n}$ and we get > \begin{align} > |g_{n}( \theta ) -g( \theta ) | & \leqslant 2(\log( mn) +1) \lambda _{n} +|W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P_{m_{n}}^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P^{\theta }(\boldsymbol{V})\right) |\notag\\ > & \quad \ \ \ \ +|W\left( P^{\theta^*}_{n}(\boldsymbol{V}) ,P^{\theta }(\boldsymbol{V})\right) -W\left( P^{\theta^*}(\boldsymbol{V}) ,P^{\theta }(\boldsymbol{V})\right) |\notag\\ > & \leqslant 2(\log( mn) +1) \lambda _{n} +W\left( P_{m_{n}}^{\theta }(\boldsymbol{V}) ,P^{\theta }(\boldsymbol{V})\right) +W( P^{\theta^*}_{n}(\boldsymbol{V}) ,P^{\theta^*}(\boldsymbol{V}))\notag\\ > & \leqslant 2(\log( mn) +1) \lambda _{n} +O\left(W( P^{\theta^*}_{n}(\boldsymbol{U}) ,P^{\theta^*}(\boldsymbol{U})) + W\left( P^{\theta }(\boldsymbol{U}) ,P^{\theta}_{m_n}(\boldsymbol{U})\right)\right) ,\label{eq:uniform_app_err} > \end{align} > where we use \cref{thm:approximation} in the last inequality. > > We first bound $ \sup _{\theta } W\left( P^{\theta }(\boldsymbol{U}) ,P_{m_{n}}^{\theta }(\boldsymbol{U})\right)$ using standard VC dimension argument. Note that $ \{U_{1} ,\cdots ,U_{n_{U}}\}$ are independent. By \cref{lemma:product_err}, > \begin{equation}\label{eq:decomp_u} > W\left( P^{\theta }(\boldsymbol{U}) ,P_{m_{n}}^{\theta }(\boldsymbol{U})\right) \leqslant \sum _{i=1}^{n_{U}} W\left( P^{\theta }( U_{i}) ,P_{m_{n}}^{\theta }( U_{i})\right). > \end{equation} > By the construction in \cref{thm:wide_nn}, $ P^{\theta }( U_{i}) \sim \sum _{j\in[N_{C,i}]} p_{j}P( f_{\theta _{i,j}}( Z_{i,j})) ,Z_{i,j} \sim \text{Unif}( 0,1)$, where $ f_{\theta _{i,j}}$ are neural networks with constant width and depth $ L_{1,n} + L_{2,n}$, $N_{C,i}$ is the number of connected components of $\text{supp}(P(U_i))$ and $\theta_{i,j} $ are the parameters of the neural networks. By \cref{lemma:combin_err}, > \begin{align*} > \sup _{\theta } W\left( P^{\theta }( U_{i}) ,P_{m_{n}}^{\theta }( U_{i})\right) & \leqslant O(\max_{j\in[N_{C,j}]}\sup _{\theta _{i,j}} W(P( f_{\theta _{i,j}}( Z_{i,j})) ,P_{m_{n}}( f_{\theta _{i,j}}( Z_{i,j}))) . > \end{align*} > By \cite{bartlett2019nearly}, the pseudo-dimension of $ \mathcal{NN}(\Theta(1) ,L)$ is $ O\left( L^{2}\log L\right)$. By boundness of all the neural networks and \cref{lemma:sup_wasserstein} \footnote{Note that we truncate the neural network to ensure boundness in \cref{thm:consistency}. This extra truncate operation $\tilde{f} = \max\{-K,\min\{K,f\}\} $ can be viewed as two extra ReLu layers, so \cref{lemma:sup_wasserstein} is still applicable. }, with probability at least $$ 1-\exp\left( O\left( (\epsilon^{'}_{n})^{-d_{\max}^{U}}\log (\epsilon^{'}_{n})^{-1} + (L_{n,1}+L_{n,2})^{2}\log(L_{n,1}+L_{n,2})\log \epsilon _{n}^{-1} -m_{n} \epsilon _{n}^{2}\right)\right) ,$$ > the following event happens. > \begin{align*} > \sup _{\theta _{i,j}} W(P( f_{\theta _{i,j}}( Z_{i,j})) ,P_{m_{n}}( f_{\theta _{i,j}}( Z_{i,j})) & \leqslant \epsilon _{n} +\epsilon^{'}_{n} . > \end{align*} > Let $$ \epsilon^{'}_{n} =m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n} ,L_{n,i} =m_{n}^{d_{\max}^{U} /\left( 2d_{\max}^{U} +4\right)}\log m_{n} ,\epsilon _{n} =Cm_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n}$$ > with constant $ C >0$ sufficiently large, we get > \begin{align*} > P\left(\sup _{\theta _{i,j}} W(P( f_{\theta _{i,j}}( Z_{i,j})) ,P_{m_{n}}( f_{\theta _{i,j}}( Z_{i,j}))) \leqslant O\left( m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n}\right)\right) & \geqslant 1-O\left( m_{n}^{-2}\right) . > \end{align*} > As long as $ m_{n} = \Omega(n)$, the Borel-Cantelli lemma implies that almost surely, there exists $ N > 0 $ such that when $ n > N$, $ \sup _{\theta _{i,j}} W(P( f_{\theta _{i,j}}( Z_{i,j})) ,P_{m_{n}}( f_{\theta _{i,j}}( Z_{i,j}))) \leqslant O\left( m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n}\right)$ for all $ i,j$. Therefore, almost surely, for sufficiently large $ n$, > \begin{align} > \sup _{\theta } W\left( P^{\theta }( U_{i}) ,P_{m_{n}}^{\theta }( U_{i})\right) & \leqslant O\left( m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n}\right),\label{eq:uniform_app_err2} > \end{align} > > \cite[Theorem 2]{fournier2015rate} shows that for sufficiently large $ n $, if $\delta_n = C_1 n^{-1/\max\{ d^U_{\max},2\}}\log^2(n) $, where $ C_1 >0$ is a constant, > \begin{align*} > P( W( P^{\theta^*}({U}_i) ,P^{\theta^*}_{n}({U}_i)) >\delta _{n}) & \leqslant \begin{cases} > \exp( -CC_{1}\log^2( n)) & d^U_{\max} =1,\\ > \exp\left(\frac{-CC_{1}\log^{4} (n)}{\log^2\left( 2+C_{1}^{-1} n^{1/2}\log^{-2} (n)\right)}\right) & d^U_{\max} = 2,\\ > \exp( -CC_{1}\log^{2/d^U_{\max}} (n)) & d^U_{\max} >2, > \end{cases} > \end{align*} > where $C > 0$ is a constant. Let $ C_{1}$ sufficiently large such that > \begin{equation*} > P( W( P^{\theta^*}({U}_{i}) ,P^{\theta^*}_{n}({U}_{i})) >\delta _{n}) \leqslant O\left( n^{-2}\right) . > \end{equation*} > Therefore, $ \sum _{n=1}^{\infty }P( W( P^{\theta^*}({U}_{i}) ,P^{\theta^*}_{n}({U}_{i})) >\delta _{n}) < \infty .$ By Borel-Cantelli lemma, with probability 1, there exist $ N >0$ such that $ W( P^{\theta^*}({U}_{i}) ,P^{\theta^*}_{n}({U}_{i})) \leqslant \delta _{n} ,\forall n >N$. By \cref{eq:decomp_u}, with probability 1, there exist $ N >0$ such that > \begin{equation} > W( P^{\theta^*}(\boldsymbol{U}) ,P^{\theta^*}_{n}(\boldsymbol{U})) \leqslant O(\delta _{n}) ,\forall n >N. \label{eq:uniform_app_err3} > \end{equation} > > Therefore, by \cref{eq:uniform_app_err,eq:uniform_app_err2,eq:uniform_app_err3}, almost surely, the following inequality holds eventually, > \begin{equation*} > \sup_{\theta \in \Theta_\infty}|g_{n}( \theta ) -g( \theta ) |\leqslant O\left(\log( nm_{n})\lambda _{n} +\delta _{n} +m_{n}^{-1/\left( d_{\max}^{U} +2\right)}\log m_{n}\right) = s_n, > \end{equation*} > where $s_n$ is defined in \cref{thm:consistency}. `lem:sup-wasserstein-anthony-theorem-17-1` (anthony1999neural, Theorem 17.1), invoked in the proof of `lem:sup-wasserstein`: What Lean assumes for it: `Papers.TbsNeuralCausalPi.AnthonyBartlettSquaredLoss` (`Setup/CitedResults.lean:183`) ```lean def AnthonyBartlettSquaredLoss : Sort 0 := ∀ (X : Type) (_mX : MeasurableSpace X) [StandardBorelSpace X] (G : Set (X → ℝ)), G.Nonempty → BorelParametrizedClass G → ∀ hG : (∀ g ∈ G, Measurable g ∧ ∀ x, g x ∈ Set.Icc (0:ℝ) 1), ∀ P : ProbabilityMeasure (X × Set.Icc (0 : ℝ) 1), ∀ (m : ℕ), 1 ≤ m → ∀ eps : ℝ, 0 < eps → eps < 1 → (Measure.pi (fun _ : Fin m => (P : Measure (X × Set.Icc (0 : ℝ) 1)))).completion {z : Fin m → X × Set.Icc (0 : ℝ) 1 | ∃ g ∈ G, |(∫ xy : X × Set.Icc (0 : ℝ) 1, (g xy.1 - xy.2.val)^2 ∂(P : Measure (X × Set.Icc (0 : ℝ) 1))) - (m : ℝ)⁻¹ * ∑ i, (g (z i).1 - (z i).2.val)^2| ≥ eps} ≤ 4 * (N1avg (eps/16) G (2*m) (fun g hg => (hG g hg).2)).toENNReal * ENNReal.ofReal (Real.exp (-(eps^2 * m)/32)) ``` The passage of the proof: > The cited assertion used in Lemma~\ref{lemma:sup_wasserstein} is the following restricted form of Theorem 17.1 in \cite{anthony1999neural}, with its standing measurability condition. Let $X$ and $T$ be standard Borel spaces, let $g:T\times X\rightarrow[0,1]$ be jointly Borel, and put $\mathcal G=\{g(t,\cdot):t\in T\}\ne\varnothing$. For every Borel probability distribution $P$ on $X\times[0,1]$, every $0<\epsilon<1$ and every positive integer $m$, > \[ > P^m\left\{\text{some }t\in T\text{ has }\left|\mathbb E_{(x,y)\sim P}(g(t,x)-y)^2-\frac1m\sum_{i=1}^m(g(t,x_i)-y_i)^2\right|\geqslant\epsilon\right\} > \leqslant4\mathcal N_1^{\mathrm{avg}}(\epsilon/16,\mathcal G,2m)\exp(-\epsilon^2m/32). > \] > Here the sample consists of independent draws from $P$, the probability is taken in its completion, and $\mathcal N_1^{\mathrm{avg}}(r,\mathcal G,N)$ is the supremum over $x_1,\ldots,x_N\in X$ of the covering number of the evaluation vectors in the metric $N^{-1}\sum_i|u_i-v_i|$. `lem:sup-wasserstein-anthony-theorems` (anthony1999neural, Theorem 18.4), invoked in the proof of `lem:sup-wasserstein`: What Lean assumes for it: `Papers.TbsNeuralCausalPi.AnthonyBartlettPacking` (`Setup/CitedResults.lean:164`) ```lean def AnthonyBartlettPacking : Sort 0 := ∀ (X : Type) (G : Set (X → ℝ)), G.Nonempty → ∀ hG : (∀ g ∈ G, ∀ x, g x ∈ Set.Icc (0:ℝ) 1), ∀ v : ℕ, Causalean.Stat.Concentration.HasPseudoDimAtMost (fun g : G => g.val) v → ∀ (m : ℕ), 1 ≤ m → ∀ eps : ℝ, 0 < eps → eps ≤ 1 → (M1avg eps G m hG).toENNReal < ENNReal.ofReal (Real.exp 1 * (v+1) * (2 * Real.exp 1 / eps) ^ v) ``` The passage of the proof: > By Theorem 18.4 in \cite{anthony1999neural}, if $\mathcal G$ is a nonempty class of functions from a set $X$ into $[0,1]$ with finite pseudo-dimension $v$, then for every $0<\epsilon\leqslant1$ and every positive integer $m$, > \begin{align*} > \mathcal{N}_{1}^{\mathrm{avg}}(\epsilon,\mathcal G,m)\leqslant\mathcal{M}_{1}^{\mathrm{avg}}(\epsilon,\mathcal G,m) \end{align*} > where $\mathcal{N}_{1}^{\mathrm{avg}}(\epsilon,\mathcal G,m)$ and $\mathcal{M}_{1}^{\mathrm{avg}}(\epsilon,\mathcal G,m)$ are the largest, over $x_1,\ldots,x_m\in X$, covering and packing numbers of $\{(g(x_1),\ldots,g(x_m)):g\in\mathcal G\}$ in the averaged metric $d_1(u,w)=\frac1m\sum_{i=1}^m|u_i-w_i|$. ### Deferred substrate gates None: every substrate this run needed was built and proved. ## Unresolved / notes - run-level `ass:assump-categorical-rv-generation`: [note, V1.5] pinned reading: Writing $x=(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$, the displayed law means $P(V_i=k\mid x)=(f_i(x))_k/\\|f_i(x)\\|_1$ for the nonnegative weight vector defining the categorical distribution, with zero coordinates interpreted using $\log 0=-\infty$. (The paragraph introducing Assumption \ref{assump:categorical_rv} separates shared inputs influencing propensity functions from independent categorical-generation noise. Assumption \ref{assump:Lip_and_bounded} explicitly identifies normalized propensity coordinates with conditional category probabilities.) - run-level `ass:assump-lip-and-bounded-categorical`: [note, V1.5] pinned reading: For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\\|f_i(x)-f_i(x')\\|\leqslant L_f\\|x-x'\\|$ on its structural input domain, using the paper's $1$-norm. (Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$.) - run-level `ass:assump-lip-and-bounded-bound`: [note, V1.5] pinned reading: There is a deterministic $K>0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely under the SCM law. (Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation.) - run-level `ass:thm-consistency-structural-lipschitz`: [note, V1.5] pinned reading: The continuous structural functions and categorical propensity functions of $\mathcal M^*$ have Lipschitz constants at most $L_f$ in the paper's $1$-norm. (Theorem~\ref{thm:consistency} begins with $\mathcal M^*$ and permits candidate networks with bound $\hat L_f$; Assumption~\ref{assump:Lip_and_bounded} identifies the functions bounded by $L_f$.) - run-level `ass:prop-lip-ate-bounded`: [note, V1.5] pinned reading: For each of $\mathcal{M}$ and $\hat{\mathcal{M}}$, $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely, where the indexed $U_j$ are the non-Gumbel latent variables. (Proposition \ref{prop:Lip_ATE} explicitly inherits Assumption \ref{assump:Lip_and_bounded}, which prints this maximum; Assumption \ref{assump:categorical_rv} distinguishes the indexed $U_j$ from the categorical noises $G_{V_i}$.) - run-level `ass:prop-lip-ate-overlap`: [note, V1.5] pinned reading: The measure $P$ is a probability measure on the treatment space, and for almost every parent value $x$, $\nu(dt\mid\text{Pa}(T)=x)=p_\nu(t\mid\text{Pa}(T)=x)P(dt)$, with the stated density version satisfying $p_\nu(t\mid\text{Pa}(T)=x)\geqslant\delta$ for every $t\in[t_1,t_2]$. (The overlap clause of Proposition~\ref{prop:Lip_ATE} names a conditional treatment density, and its displayed conclusion integrates over treatment values against $P(dt)$; the supplied meanings of $P$ and $p_\nu$ agree with this interpretation.) - run-level `ass:prop-convg-opt-approximate-lipschitz`: [note, V1.5] pinned reading: Equation~\ref{eq:aprox_lip} holds for every $n\in\mathbb N$ and every $\theta_1,\theta_2\in\Theta_\infty$, with the same finite constant $L_g$. (Proposition~\ref{prop:convg_opt} defines the functions on $\Theta_\infty$ and uses this inequality to conclude that the limiting function is $L_g$-Lipschitz.) - run-level `ass:prop-convg-opt-approximation`: [note, V1.5] pinned reading: There exists a compact $\tilde\Theta_\infty\subseteq\Theta_\infty$ such that $\forall n\in\mathbb N\;\forall\theta_0\in\tilde\Theta_\infty\;\exists\theta\in\Theta_n$, $d(\theta,\theta_0)\leqslant\epsilon_n$. (The proof of Proposition \ref{prop:convg_opt} applies condition 3 to construct a sequence $\theta_n$ approximating $\tilde\theta^*$.) - run-level `ass:lemma-lip-condition-regression`: [note, V1.5] pinned reading: The functions $f,\hat{f}:[-K,K]\rightarrow\mathbb{R}$ are $L$-Lipschitz regression versions satisfying $f(T)=\mathbb{E}_{\nu}[Y\mid T]$ almost surely under $\nu$ and $\hat{f}(\hat{T})=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}]$ almost surely under $\mu$. (The proof of Lemma~\ref{lemma:Lip_condition} uses both cross-evaluations and bounds its test function on the common box; the owner-supplied support condition specifies that box.) - run-level `ass:lemma-sup-wasserstein-bounded`: [note, V1.5] pinned reading: The boundedness condition means $\\|f(x)\\|_{\infty}\leqslant K$ for every $f\in\mathcal{F}$ and $x\in\mathbb{R}^{d'}$. (The proof of Lemma~\ref{lemma:sup_wasserstein} places the outputs in $[-K,K]^d$, consistently with the supplied function-class interpretation.) - run-level `ass:relu-approximation-domain`: [note, V1.5] pinned reading: The phrase ``uniform cubes'' denotes the unit cube domains $[0,1]^m$ used for deterministic uniform function approximation; it imposes no uniform probability law on structural inputs. (The same paragraph displays approximation of functions on $[0,1]^{d_1}$ and separately says that similar arguments apply to any bounded cubes.) - run-level `ass:bk-owner-1`: [note, V1.5] pinned reading: The bound $K$ applies throughout the optimization class to the observed variables $V_i$ and indexed non-Gumbel latent variables $U_j$, with $n_i\leqslant K$ for categorical nodes and admissible intervention coordinates in $[-K,K]$; it does not bound the separately supplied standard Gumbel noises. (The boundedness assumption explicitly indexes $V_i$ and $U_j$, whereas the categorical convention separately adjoins $G_{V_i}$. The binding boundedness symbols preserve this distinction.) - run-level `ass:bk-owner-2`: [note, V1.5] pinned reading: The variable bound inherited from Assumption~\ref{assump:Lip_and_bounded} uses $\\|\cdot\\|_\infty$. (The owner condition extends the existing bound $K$, which Assumption~\ref{assump:Lip_and_bounded} explicitly states using infinity norms of observed and non-Gumbel latent variables.) - run-level `def:eq-structure-eq`: [note, V1.5] pinned reading: For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. (The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition.) - run-level `def:definition-2`: [note, V1.5] pinned reading: $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. (The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality.) - run-level `def:eq-structure-equation`: [note, V1.5] pinned reading: In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. (The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction.) - run-level `def:intervention`: [note, V1.5] pinned reading: Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). (The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations.) - run-level `def:homeomorphisms`: [note, V1.5] pinned reading: $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. (The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes.) - run-level `def:relu-networks`: [note, V1.5] pinned reading: A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. (The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths.) - run-level `def:eq-structure-equation-hat`: [note, V1.5] pinned reading: For each categorical $V_i$, the noise in equation (\ref{eq:structure_equation_hat}) satisfies $g_k=g_k^{V_i}$ for $k\in[n_i]$. (The canonical representation in equation (\ref{eq:structure_equation}) assigns separate independent standard Gumbel coordinates to each categorical node; the approximated equation retains that categorical-generation rule.) - run-level `def:gumbel-softmax-layer`: [note, V1.5] pinned reading: The layer uses $V=[\Vec{v}_1,\ldots,\Vec{v}_{N_C}]$ and the same Gumbel draws for all $\tau>0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. (The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit.) - run-level `def:neural-causal-class`: [note, V1.5] pinned reading: Membership in $\text{NCM}_{\mathcal G}(\mathcal F_0,\mathcal F_1)$ requires $\hat f_i\in\mathcal F_0$ for every observed node and $\hat g_j\in\mathcal F_1$ for every latent component, with the compatible input and output dimensions understood separately for each network. (The source paragraph explicitly omits input and output dimensions and describes separate structural approximants and latent generators in equation~\ref{eq:structure_equation_hat}.) - run-level `def:eq-opt-empiri`: [note, V1.5] pinned reading: $F_n=\inf\left\{\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\hat{\mathcal{M}}}[F(V_1(t),\ldots,V_{n_V}(t))]:\hat{\mathcal{M}}\in\text{NCM}_{\mathcal{G}}(\mathcal{F}_{0,n},\mathcal{F}_{1,n}),\ S_{\lambda_n}(P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V}),P_n^{\mathcal{M}^*}(\boldsymbol{V}))\leqslant\alpha_n\right\}$. (The prose introducing equation (\ref{eq:opt_empiri}) calls $F_n$ an estimator of optimal values, and the following paragraph asks whether it is a good lower bound. The supplied symbol convention likewise specifies a scalar infimum.) - run-level `def:truncated-neural-class`: [note, V1.5] pinned reading: Truncation means coordinatewise clipping of the final output to $[-K,K]$; in $\mathcal{NN}^{\infty,K}$ no finite Lipschitz bound is imposed. (The defining minimum and maximum and the word ``Similarly'' specify the operation; the binding neural-class symbol conventions fix its application to the generator output.) - run-level `def:eq-consistent-limiting`: [note, V1.5] pinned reading: $\underline{F}^{L}$ denotes the constrained infimum of the displayed scalar objective over the stated canonical SCM class and its admissible latent laws, rather than a minimizing SCM. (The sentence introducing the display explicitly says ``optimal value'', and the following sentence calls it the sharp lower bound when the Lipschitz bounds agree.) - run-level `def:cor-rate-objective`: [note, V1.5] pinned reading: $\mu_T=\text{Unif}([t_1,t_2])$ is the deterministic intervention probability measure. (Corollary~\ref{cor:rate} specializes the intervention measure used in the outer expectation $\mathbb E_{t\sim\mu_T}$ of Theorem~\ref{thm:consistency}.) - run-level `def:lemma-gumbel-err-softmax`: [note, V1.5] pinned reading: In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. (Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores.) - run-level `def:corollary-deep-nn-dimensions`: [note, V1.5] pinned reading: $d_{\max}^U=\max_{j\in[n_U]}d_j^U$ ranges over non-Gumbel latent variables, while $d_{\max}^{\mathrm{in}}=\max_i d_i^{\mathrm{in}}$ and $d_{\max}^{\mathrm{out}}=\max_i d_i^{\mathrm{out}}$, with $d_i^{\mathrm{out}}=n_i$ for categorical nodes. (The NCM construction separately approximates non-Gumbel latent distributions and the functions in the canonical structural equations; those functions output categorical propensity vectors. The binding dimension symbols specify these same meanings.) - run-level `def:thm-consistency-model-parameterization`: [note, V1.5] pinned reading: The coordinates $P(U_j)$ in $\theta_{\mathcal M}$ are the population laws of the non-Gumbel latent variables, not necessarily uniform laws; the independent standard Gumbel laws are fixed and omitted from this population tuple. (Immediately after the tuple, the population parameter space contains factors $\mathcal P([-K,K]^{d_j^U})$. The canonical representation supplies the categorical Gumbel laws separately.) - run-level `symbol \boldsymbol{U}`: [note, V1.5] symbol table corrected: space "The joint value space of the latent collection: $\prod_{j=1}^{n_U}\mathbb{R}^{d_j^U}$ in the initial SCM notation, with the additional factors $\prod_{V_i\text{ categorical}}\mathbb{R}^{n_i}$ under the categorical-noise convention.", def "Initially $\boldsymbol{U}=\{U_j\}_{j=1}^{n_U}$; under the categorical convention, $\boldsymbol{U}=\{U_1,\cdots,U_{n_U}\}\cup\{G_{V_i}:V_i\text{ is categorical}\}$. In canonical form its non-Gumbel members are indexed by the $C^2$-components." (The recorded collection and space omit categorical Gumbel vectors. The existing interpretation remains valid, with its canonical-model clause retagged to the results still using the symbol.) - run-level `symbol n_U`: [note, V1.5] symbol table corrected: def "$n_U$ is the number of indexed latent variables $U_1,\ldots,U_{n_U}$; initially $\boldsymbol{U}=\{U_j\}_{j=1}^{n_U}$, while under the categorical convention $\boldsymbol{U}=\{U_1,\ldots,U_{n_U}\}\cup\{G_{V_i}:V_i\text{ is categorical}\}$." (Counting every member of the full latent collection would include the separately adjoined Gumbel vectors. The paper's later indexed sums, bounds, and parameter tuples count the non-Gumbel family.) - run-level `symbol f_i`: [note, V1.5] symbol table corrected: space "Functions on the joint value space of $\text{Pa}(V_i)$ and $\boldsymbol{U}_{V_i}$, with codomain $\mathbb{R}^{d_i}$ for continuous variables, $[n_i]$ in (\ref{eq:structure_eq}) for categorical variables, and $\mathbb{R}^{n_i}$ under the categorical propensity convention.", def "In (\ref{eq:structure_eq}), $V_i=f_i(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$. Under the categorical convention, $V_i=\arg\max_{k\in[n_i]}\{g_k^{V_i}+\log(f_i(\text{Pa}(V_i),\boldsymbol{U}_{V_i}))_k\}\sim\text{Categorical}(f_i/\\|f_i\\|_1)$." (The recorded space confuses parent and latent collections with their value spaces and omits the categorical-label codomain of the preliminary structural equation; the categorical convention instead uses propensity vectors. The existing scoped readings remain applicable, with the consistency population class and the continuous true and candidate functions now subject to the specified uniform output $1$-norm bound.) - run-level `symbol F`: [note, V1.5] symbol table corrected: type "tuple of structural functions, causal objective function, or measurable map, according to context", space "The structural-function tuple space, $\mathbb{R}^{n_V}\to\mathbb{R}$ for the causal objective, or the map space specified by the local push-forward definition.", def "In the SCM definition, $F=(f_1,\cdots,f_{n_V})$. In the optimization objectives, $F$ is evaluated as $F(V_1(t),\cdots,V_{n_V}(t))$. In push-forward uses, $F$ is the map specified locally." (The recorded tuple-only type and singleton-product space cannot describe the scalar objective or either push-forward-map use. These are explicit local overloads, not an unresolved choice of meaning; the printed scalar objective domain is retained without silently replacing it by a higher-dimensional product.) - run-level `symbol P`: [note, V1.5] symbol table corrected: type "probability-law notation, also used for a locally specified probability measure", def "We use $P(X)$ to represent the distribution of random variable $X$; bare $P$ also denotes a locally specified probability measure." (The law-notation entry misses bare reference measures. The existing deviation-bound reading must additionally specify that $P$ is Borel and that its product-law probability is completed.) - run-level `symbol \mathcal{M}`: [note, V1.5] symbol table corrected: type "structural causal model notation, also used for a canonical SCM class", space "$\mathcal{C}\ \text{or}\ \mathcal{P}(\mathcal{C})$", def "An individual SCM is $\mathcal{M}=(\boldsymbol{V},\boldsymbol{U},F,P(\boldsymbol{U}),\mathcal{G}_0)$. Given a function class $\mathcal{F}$, the SCM class $\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})$ consists of all canonical SCM models with causal graph $\mathcal{G}$ such that $f_i\in\mathcal{F}$, $i\in[n_V]$." (The individual-SCM and SCM-class distinction still requires the existing fix. The previous scope names a different population value and class and uses an infinity-norm bound; the current statement instead uses $\underline F^{\hat L_f}$ and the output 1-norm bounded class $\mathcal F_L^K$.) - run-level `symbol \mathcal{G}`: [note, V1.5] symbol table corrected: type "causal graph or function class, according to context", space "Causal ADMGs on $\boldsymbol V$, or classes of functions $X\rightarrow[0,1]$, according to context.", def "$\mathcal G$ denotes the locally specified causal graph; in the empirical covering and deviation citation it denotes a nonempty class of functions from a set $X$ into $[0,1]$. The covering and packing bound assumes finite pseudo-dimension $v$; the deviation bound imposes the stated measurable parameterization instead." (The graph-only entry misses the function-class use. The existing deviation-bound clause is now incomplete: the class must have the specified jointly Borel parameterization on standard Borel spaces.) - run-level `symbol \hat{\mathcal{F}}`: [note, V1.5] symbol table corrected: space "A class of structural functions on the node-specific parent and latent input domains, with continuous-node or categorical-propensity output spaces.", def "$\hat{\mathcal{F}}$ is the function class in $\mathcal{M}(\mathcal{G},\hat{\mathcal{F}},\hat{\boldsymbol{U}})$: the structural functions satisfy $\hat{f}_i\in\hat{\mathcal{F}}$ for $i\in[n_V]$." (The recorded space is a set of individual maps with a single unspecified domain and codomain, rather than a function class accommodating the node-specific structural maps. The comparison class is not inherently a neural or Lipschitz class.) - run-level `symbol \hat{\mathcal{M}}`: [note, V1.5] symbol table corrected: space "The canonical SCM class on $\mathcal{G}$ specified locally.", def "$\hat{\mathcal{M}}$ denotes a second canonical SCM with causal graph $\mathcal{G}$, with its structural functions and latent variables specified in the surrounding construction or optimization." (The recorded class is too narrow for the symbol's several local uses. The existing readings remain applicable; the additional scope records the continuous output bound and population structural class explicitly specified in Corollary~\ref{cor:rate}, alongside the retained categorical restrictions.) - run-level `symbol \hat{\boldsymbol{U}}`: [note, V1.5] symbol table corrected: space "The product of the component latent spaces together with $\prod_{V_i\ \text{categorical}}\mathbb{R}^{n_i}$ for the categorical Gumbel vectors.", def "$\hat{\boldsymbol{U}}$ is the full latent collection of the comparison canonical SCM, comprising one latent variable for each $C^2$-component of $\mathcal{G}$ and a Gumbel vector for each categorical observed node." (The full canonical latent collection, including categorical Gumbel vectors, remains the correct definition. The obsolete grouping scope is removed without imposing that construction on the arbitrary comparison model.) - run-level `symbol \epsilon`: [note, V1.5] symbol table corrected: type "approximation-error functional or local tolerance or floor parameter", space "$[0,\infty]$ for the approximation-error functional; $[0,\infty)$ for local scalar parameters, with strict positivity where specified.", def "$\epsilon(\mathcal{F}_1,\mathcal{F}_2)=\sup_{f_2\in\mathcal{F}_2}\inf_{f_1\in\mathcal{F}_1}\\|f_2-f_1\\|$ for function classes $\mathcal{F}_1,\mathcal{F}_2$; a bare $\epsilon$ is a locally quantified scalar tolerance or floor parameter." (The recorded positive tolerance omits the approximation-error functional and the nonnegative floor parameter now used in the structural approximation proof. Existing tolerance scopes remain valid, with the additional local uses specified here.) - run-level `symbol \mathcal{M}_\epsilon`: [note, V1.5] symbol table corrected: space "$\{\text{SCMs}\}$" (The paper does not assert membership in the unspecified introductory search class $\mathcal{C}$. The model varies with the tolerance; the regularity assumptions stated for $\mathcal{M}^*$ are not thereby imposed on this family.) - run-level `symbol \mathcal{C}`: [note, V1.5] symbol table corrected: type "context-dependent collection of causal models or cubes", space "Collections of SCMs or finite collections of axis-aligned cubes, as locally specified.", def "$\mathcal{C}$ is a collection of causal models in (\ref{prob:pid}); in the geometric construction, $\mathcal{C}=\{C_1,\ldots,C_m\}$ with $C_k=\prod_{i=1}^{n}[a_i^k,b_i^k]$ is a collection of cubes." (The causal-model-only entry misses the geometric reuse. The existing general reading and the clause for prop:holder-curve remain valid; the obsolete scope is removed.) - run-level `symbol \theta`: [note, V1.5] symbol table corrected: type "context-dependent causal functional or parameter", space "$\mathcal{C}\to\mathbb{R}$ in (\ref{prob:pid}); $M$ in the population optimization and model-parameter notation; network representations in the empirical optimization", def "$\theta(\mathcal{M})$ is the causal quantity of interest in (\ref{prob:pid}); elsewhere $\theta$ denotes a parameter point, including $\theta_{\mathcal{M}}=(f_1,\ldots,f_{n_V},P(U_1),\ldots,P(U_{n_U}))$ in the population causal-model parameterization. Empirical candidates are indexed by network representations, which need not be determined by their population parameter tuples." (The recorded causal-functional entry misses the later parameter meanings. The existing general reading and other scoped clauses remain valid; the clause for Corollary~\ref{cor:rate} must also distinguish empirical network representations from their population parameter tuples.) - run-level `symbol \hat{V}_i`: [note, V1.5] symbol table corrected: def "For $i\in[n_V]$, $\hat{V}_i$ is generated by the structural functions and latent inputs of $\hat{\mathcal{M}}$: $\hat{V}_i=\hat{f}_i(\text{Pa}(\hat{V}_i),\hat{\boldsymbol{U}}_{V_i})$ for continuous variables, and $\hat{V}_i=\arg\max_{k\in[n_i]}\{\hat{g}_k^{V_i}+\log(\hat{f}_i(\text{Pa}(\hat{V}_i),\hat{\boldsymbol{U}}_{V_i}))_k\}$ for categorical variables." (The recorded label omits the structural generation formula, so the existing definition fix remains necessary. The general reading remains unchanged, and the existing joint scoped clause is explicitly retained alongside the additional categorical-generation conventions.) - run-level `symbol C`: [note, V1.5] symbol table corrected: type "context-dependent node set, geometric set, numerical constant, or categorical label vector", space "$2^{\boldsymbol{V}}$ in causal-graph notation; subsets of the ambient Euclidean space in geometric and support-component notation; $(0,\infty)$ for numerical constants $C>0$; $\prod_{i:V_i\text{ is categorical}}[n_i]$ for the complete categorical label vector", def "$C$ denotes a $C^2$-component in causal-graph notation; in geometric notation it denotes a cube, and $C_i$ can denote a connected component of a measure's support; in numerical bounds $C>0$ denotes a constant; in the latent empirical concentration argument, $C$ denotes the complete categorical label vector." (The entry on file covers graph components, geometric sets and constants, but omits the complete categorical label vector used in latent empirical concentration. This additional local meaning coexists with the existing meanings in the consistency proof; the three overlapping clauses are explicitly retained.) - run-level `symbol \hat{f}`: [note, V1.5] symbol table corrected: def "$\hat{f}$ denotes the function paired with $f$ in the local comparison; its structural-function, neural-approximation, or population-regression meaning is specified by the local definition." (The comparison-or-neural-approximation entry omits the population-regression meaning. The existing general fix and regression clause remain applicable; only the obsolete approximation clause is removed.) - run-level `symbol \mathcal{G}_{\mathcal{M}}`: [note, V1.5] symbol table corrected: def "$\mathcal{G}_{\mathcal{M}}$ extends $\mathcal{G}_0$ by adding bi-directed arrows between $V_i,V_j$ if there exists a correlated latent variable pair $(U_k,U_l)$ with $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$." (The recorded definition refers vaguely to specified arrows without recording their defining criterion. The SCM definition explicitly gives the correlated-latent-pair rule, including the shared-latent interpretation explained immediately afterward.) - run-level `symbol T`: [note, V1.5] symbol table corrected: type "local treatment variable, transportation map, transportation plan, or parameter space", space "Treatment value spaces, maps $\Delta_n\rightarrow\Delta_n$, nonnegative transportation matrices, or standard Borel spaces, as specified locally.", def "$T$ denotes a treatment variable in the causal passages, the transportation map in the Gumbel-softmax proof, a transportation plan in the Sinkhorn proof, or a standard Borel parameter space for a jointly Borel function family." (The local roles of $T$ remain settled. The previous Sinkhorn clause incorrectly extended the proof's entropy maximum to all feasible plans; its displayed domain consists of unregularized optimal plans.) - run-level `symbol Y`: [note, V1.5] symbol table corrected: type "random variable or random vector", space "The outcome space is $\mathbb{R}$; in the product-coupling notation, the space is the product of the coordinate spaces.", def "$Y$ denotes the outcome in causal uses and the second coordinate of $(T,Y)$ in the regression comparison; in product-coupling notation it denotes the vector with coordinates $Y_k$." (The scalar-outcome entry does not cover the auxiliary vector in the product-measure proof. The different structural equations for outcomes in different SCMs are model specifications, rather than changes to the meaning of an outcome node; the product-coupling use is a separate local use of the notation.) - run-level `symbol \mu`: [note, V1.5] symbol table corrected: space "The space of measures on the locally stated domain, including finite probability vectors $\Delta_n$", def "$\mu$ denotes the measure specified locally; in the preliminary push-forward notation it is a measure on $\mathbb{R}^{d_1}$." (The local measure meanings and existing scoped clauses remain valid. Only the evidence concerning the separated covering citation changes.) - run-level `symbol P^{\mathcal{M}}`: [note, V1.5] symbol table corrected: space "Probability laws of the indicated variables in $\mathcal{M}$.", def "$P^{\mathcal{M}}(X)$ denotes the distribution of the indicated random variable or collection $X$ in SCM $\mathcal{M}$." (The notation applies to the variables in its argument, including latent and intervened variables, rather than exclusively to the observational distribution. The general reading and the clause for Theorem~\ref{thm:approximation} remain unchanged.) - run-level `symbol \mathbb{P}`: [note, V1.5] symbol table corrected: space "Probability measures on the measurable space of the local distribution or events.", def "$\mathbb P$ denotes the probability measure in the local context; in the distribution-generator construction it is the target distribution." (The local probability-measure meanings remain valid. The counterexamples now explicitly distinguish a non-Gumbel latent law from its normalized component laws, to which the lower-bound assumption applies.) - run-level `symbol \hat{\mu}`: [note, V1.5] symbol table corrected: space "Probability measures on the stated Euclidean space." (The comparison-measure definition is correct, but the recorded space imposes a compact set $K$ even on the product lemma. Compact support and the product definition belong to separate scopes.) - run-level `symbol C_i`: [note, V1.5] symbol table corrected: type "support component or partition cell", def "$C_i$ denotes a connected support component in the distribution-generator construction, or a cube in the partition construction." (The existing support-component and partition-cell meanings remain valid. Both counterexamples additionally use each component as the support set for the lower-bound assumption on its normalized component law.) - run-level `symbol \mu_i`: [note, V1.5] symbol table corrected: type "component probability measure or probability-vector coordinate", space "Probability measures on the locally specified component space, or $[0,1]$ for a probability-vector coordinate.", def "$\mu_i$ denotes the locally specified component probability measure, or the $i$th coordinate of $\mu\in\Delta_n$." (The existing local measure and coordinate meanings remain valid. The extracted generator lemma uses the normalized component law.) - run-level `symbol \hat{\mu}_i`: [note, V1.5] symbol table corrected: space "Probability measures on the respective Euclidean component spaces.", def "$\hat{\mu}_i$ is the comparison component measure paired with $\mu_i$." (The general comparison-component meaning remains valid, but the generator proofs additionally define an explicit push-forward law. That construction belongs in those scopes rather than in the general definition.) - run-level `symbol X^\tau`: [note, V1.5] symbol table corrected: def "$X^\tau=(X_1^\tau,\ldots,X_n^\tau)$, where $X_k^\tau=\frac{\exp((\log p_k+G_k)/\tau)}{\sum_{j=1}^n\exp((\log p_j+G_j)/\tau)}$ and $G_k$ are i.i.d. standard Gumbel variables." (The recorded label still omits the defining softmax formula. The general reading remains valid, and the additional uses require identifying the dimension and weights with the target measure's support components.) - run-level `symbol \mu_n`: [note, V1.5] symbol table corrected: type "context-dependent measure", space "The space of measures on the locally stated domain", def "$\mu_n$ denotes either the component measure with index $n$ in a list $\mu_1,\ldots,\mu_n$, or the empirical distribution of $\mu$, according to the local definition." (The component and empirical meanings remain unchanged. The evidence is updated because the covering citation no longer specifies observations from a probability distribution.) - run-level `symbol W`: [note, V1.5] symbol table corrected: type "overloaded distance functional, network width and network weight count", space "$\bigl(\mathcal{P}\times\mathcal{P}\rightarrow[0,\infty]\bigr)\ \cup\ \mathbb{N}$", def "$W(\mu,\nu)$ denotes the Wasserstein-1 distance for the paper's $1$-norm; in $\mathcal{NN}_{k_1,k_2}(W,L)$ and its truncated variants, $W$ denotes network width; in the approximation expression $a\,\omega(cW^{-2/d_i^{\text{in}}})$, $W$ denotes the number of network weights." (The recorded distance-only entry omits both architectural width and the weight count explicitly used in the scalar approximation expression. The existing distinction between distance and width remains valid; the additional scoped clause distinguishes the weight count from width.) - run-level `symbol \lambda`: [note, V1.5] symbol table corrected: type "context-dependent regularization parameter or reference measure", space "$(0,\infty)$ for regularization; the space of measures on the stated reference domain for push-forwards", def "$\lambda$ denotes a positive regularization parameter in $S_\lambda$ and $W_{1,\lambda}$, and a reference measure in push-forward and measure-domination expressions." (The scalar and reference-measure meanings remain settled. The previous scoped clause attributed a lower-bound premise to Theorem~\ref{thm:wide_nn}; that qualification now belongs only to thm:cor-fix-3c402872, whose existing clause remains applicable.) - run-level `symbol S_\lambda`: [note, V1.5] symbol table corrected: def "$S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$." (The recorded label omits the defining self-comparison subtraction. The existing subtraction formula remains valid; the scoped reading now also records the explicitly specified minus-entropy convention.) - run-level `symbol W_{1,\lambda}`: [note, V1.5] symbol table corrected: space "$\Delta_n\times\Delta_m\to\mathbb{R}$, with the support points supplied", def "For finite probability measures with weights $\mu\in\Delta_n$, $\nu\in\Delta_m$ on points $x_i,y_j$ in a common Euclidean space, put $c_{ij}=\\|x_i-y_j\\|$, $\Pi(\mu,\nu)=\left\{T\in[0,\infty)^{n\times m}:\sum_jT_{ij}=\mu_i,\ \sum_iT_{ij}=\nu_j\right\}$ and $h(T)=1-\sum_{i,j}T_{ij}\log T_{ij}$, with $0\log0=0$. Then $W_{1,\lambda}(\mu,\nu)=\min_{T\in\Pi(\mu,\nu)}\left\{\sum_{i,j}c_{ij}T_{ij}-\lambda h(T)\right\}$." (The recorded nonnegative range is wrong for the specified entropy-subtracted objective, and the label alone omits its defining formula. Probability weights alone do not determine the functional without their support points.) - run-level `symbol L`: [note, V1.5] symbol table corrected: type "Lipschitz bound or neural network depth", def "$L$ is the Lipschitz bound in $\mathcal{F}_L(K_1,K_2)$; in $\mathcal{NN}_{k_1,k_2}(W,L)$ it is the neural network depth." (The existing distinction between Lipschitz bound and network depth still applies. The population-class clause must replace $\mathcal B_L^K$ and $\underline F_\infty^L$ by the retained $\mathcal F_L^K$ and $\underline F^L$, including the full-domain output $1$-norm bound.) - run-level `symbol \mathcal{H}`: [note, V1.5] symbol table corrected: space "$\{f:K_1\to K_2\}$, with domain and codomain supplied by the local definition", def "$\mathcal{H}$ denotes the locally specified function class; $\mathcal{H}(K_1,K_2)$ is the set of homeomorphisms from $K_1$ to $K_2$." (The homeomorphism-only entry misses the explicitly defined local Lipschitz class. Its printed vector-valued definition is retained as written; the changed statement does not replace that proof definition.) - run-level `symbol \mathcal{F}`: [note, V1.5] symbol table corrected: space "A set of functions with the domains and codomains specified locally.", def "$\mathcal{F}$ denotes a function class; in $\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})$ it is the class to which the structural functions $f_i$, $i\in[n_V]$, belong." (The local function-class meaning remains valid. Its empirical Wasserstein scope additionally requires the stated jointly Borel parameterization, which the reading on file omits.) - run-level `symbol K`: [note, V1.5] symbol table corrected: type "numerical bound or compact support set", space "$(0,\infty)$ for a numerical bound, or a compact subset of the indicated Euclidean space for a support set", def "$K$ denotes the numerical bound specified by a boundedness hypothesis or truncation, or the compact support set specified by a measure hypothesis." (The numerical-bound entry does not cover the compact-set occurrences. The additional scope identifies the compact set with each latent support component and specifies the normalized law and pullback boundary condition; both overlapping clauses on file remain unchanged.) - run-level `symbol \mathcal{NN}`: [note, V1.5] symbol table corrected: type "overloaded ReLU network and three-part generator class", space "$\{\text{ReLU networks or three-part neural generators}\}$", def "$\mathcal{NN}_{k_1,k_2}(W,L)$ is the set of ReLU networks with input dimension $k_1$, output dimension $k_2$, width $W$ and depth $L$; $\mathcal{NN}(W_1,L_1,W_2,L_2,\tau)$ denotes generators $\hat{g}=\hat{g}_3^\tau\circ\hat{g}_2\circ\hat{g}_1$, where $\hat{g}_i=(\hat{g}_i^1(x_1),\ldots,\hat{g}_i^{N_C}(x_{N_C}))$, $\hat{g}_1^j\in\mathcal{NN}_{1,d_j^C}(W_1,L_1)$, $\hat{g}_2^j\in\mathcal{NN}_{d_j^C,d}(W_2,L_2)$ for $j\in[N_C]$, and $\hat{g}_3^\tau$ is the Gumbel-Softmax layer with temperature $\tau>0$." (The notation denotes both ordinary ReLU networks and the composite generator class. The additional scoped conventions specify absolute width factors while retaining the existing temperature and zero-dimensional readings.) - run-level `symbol \mathcal{NN}(W_1,L_1,W_2,L_2,\tau)`: [note, V1.5] symbol table corrected: def "The class consists of generators $\hat{g}=\hat{g}_3^\tau\circ\hat{g}_2\circ\hat{g}_1$ with separable blocks $\hat{g}_i(x_1,\ldots,x_{N_C})=(\hat{g}_i^1(x_1),\ldots,\hat{g}_i^{N_C}(x_{N_C}))$, $i=1,2$, where $\hat{g}_1^j\in\mathcal{NN}_{1,d_j^C}(W_1,L_1)$ and $\hat{g}_2^j\in\mathcal{NN}_{d_j^C,d}(W_2,L_2)$ for $j\in[N_C]$, and $\hat{g}_3^\tau$ is the Gumbel-softmax mixing layer." (The recorded architecture omits its defining component networks. The existing zero-dimensional convention remains in force; the additional scoped readings specify the absolute width factors and the suppressed temperature.) - run-level `symbol \mathcal{NN}^{\infty,K}`: [note, V1.5] symbol table corrected: def "$\mathcal{NN}^{\infty,K}(W_1,L_1,W_2,L_2,\tau)=\{T_K\hat{g}:\hat{g}\in\mathcal{NN}(W_1,L_1,W_2,L_2,\tau)\}$, where $T_K\hat{g}=\min\{K,\max\{-K,\hat{g}\}\}$ coordinatewise." (The recorded definition leaves the truncation operation unspecified. The paper defines output clipping, rather than clipping each intermediate block, and the superscript $\infty$ imposes no finite Lipschitz restriction.) - run-level `symbol \hat{g}`: [note, V1.5] symbol table corrected: type "random neural generator", def "The displayed generator is $\hat{g}=\hat{g}_3^\tau\circ\hat{g}_2\circ\hat{g}_1$, with separable ReLU blocks $\hat{g}_1,\hat{g}_2$ followed by the Gumbel-softmax mixing layer." (The recorded entry still omits the three-block construction and its Gumbel randomness, so its existing fix remains necessary. The empirical reading additionally distinguishes network representations having equal population laws, since their evaluations use common exogenous inputs.) - run-level `symbol \mathcal{F}_1`: [note, V1.5] symbol table corrected: space "$\{\text{function classes}\}$", def "The notation $\mathcal{F}_1$ denotes a function class, with its role and domain determined by context." (The general function-class correction and latent-generator scopes remain applicable. The obsolete Wasserstein-proof scope is explicitly dropped.) - run-level `symbol \hat{g}_1`: [note, V1.5] symbol table corrected: def "The first block has the separable form $\hat{g}_1(x_1,\ldots,x_{N_C})=(\hat{g}_1^1(x_1),\ldots,\hat{g}_1^{N_C}(x_{N_C}))$, where $\hat{g}_1^j\in\mathcal{NN}_{1,d_j^C}(W_1,L_1)$ for $j\in[N_C]$." (The separable-block definition remains necessary. Its pullback-law scope now covers both distribution-generator results and specifies the zero-dimensional output.) - run-level `symbol \hat{g}_2`: [note, V1.5] symbol table corrected: def "The second block has the separable form $\hat{g}_2(x_1,\ldots,x_{N_C})=(\hat{g}_2^1(x_1),\ldots,\hat{g}_2^{N_C}(x_{N_C}))$, where $\hat{g}_2^j\in\mathcal{NN}_{d_j^C,d}(W_2,L_2)$ for $j\in[N_C]$." (The separable-block definition remains necessary. Its component approximation scope now covers both distribution-generator results and explicitly specifies singleton outputs.) - run-level `symbol \hat{g}_3^\tau`: [note, V1.5] symbol table corrected: def "The layer is $\hat{g}_3^\tau(x_1,\ldots,x_{N_C})=\sum_{k=1}^{N_C}w_k^\tau x_k$, where $x_k\in\mathbb{R}^d$ and $w_k^\tau=\exp((\log p_k+G_k)/\tau)/\sum_{j=1}^{N_C}\exp((\log p_j+G_j)/\tau)$." (The explicit weighted-sum definition remains necessary. The component-mass reading also applies to the separately stated distribution-generator result.) - run-level `symbol \hat{g}_2^{i,j}`: [note, V1.5] symbol table corrected: type "deep ReLU network", def "For $d_i^C\geqslant1$, $\hat{g}_2^{i,j}$ has width $\Theta(d_i^C)$ and depth $L_2$ and satisfies $\\|(g_2^i)_j-\hat{g}_2^{i,j}\\|_{\infty}\leqslant O(L_2^{-2/d_i^C})$; for $d_i^C=0$, it is the constant affine coordinate map to the singleton $C_i$." (The positive-dimensional approximation definition remains applicable, but its width and inverse-dimension exponent must not be applied to singleton components.) - run-level `symbol \gamma_i`: [note, V1.5] symbol table corrected: def "$\gamma_i:[0,1]\rightarrow[0,1]^{d_i^C}$ satisfies $(\gamma_i)_{\#}\lambda=(g_2^i)^{-1}_{\#}\mu_i$, where $\mu_i=\mathbb{P}_{C_i}/\mathbb{P}(C_i)$ and $\lambda$ is Lebesgue measure on $[0,1]$; for $d_i^C\geqslant1$, it is $1/d_i^C$-H\"older continuous." (The vague transformed-distribution description omits normalization, the chosen pullback map, the exact pushforward identity, and the stated regularity. The component measure represented is a probability law obtained by dividing by $\mathbb{P}(C_i)$.) - run-level `symbol \hat{g}_1^{i,j}`: [note, V1.5] symbol table corrected: type "deep ReLU network", def "$\hat{g}_1^{i,j}$ has width $12$ and depth $L_1$ and satisfies $\\|\hat{g}_1^{i,j}-(\gamma_i)_j\\|_{\infty}\leqslant O(L_1^{-2/d_i^C})$." (The entry omits the specified ReLU architecture and approximation bound. The coordinate target determines $j\in[d_i^C]$; the subsequent displayed assembly ending at $d$ is inconsistent when $d\neq d_i^C$, but does not give the individual coordinate network another meaning.) - run-level `symbol C_f`: [note, V1.5] symbol table corrected: def "$C_f>0$ satisfies $\mathbb{P}(f(B))\geqslant C_f\lambda(B)$ for every measurable $B\subset[0,1]^d$, where $f\in\mathcal{H}([0,1]^d,K)\cap\mathcal{F}_L([0,1]^d,K)$ and $\lambda$ is Lebesgue measure on $[0,1]^d$." (The recorded label omits the defining domination inequality. The assumption chooses the constant for the measure and homeomorphism under consideration; it does not specify a uniform constant across different measures or choices of homeomorphism.) - run-level `symbol \mathcal{I}_n`: [note, V1.5] symbol table corrected: def "$\mathcal{I}_0=\{[0,1]\}$ and $\mathcal{I}_n=\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$, where $I_{O_n(i)}=[\sum_{j=1}^{i-1}p_{n,j},\sum_{j=1}^{i}p_{n,j}]$ and $p_{n,j}=\mathbb{P}(C_{O_n(j)})$; each parent interval is the union of its $2^d$ ordered child intervals." (The recorded entry omits the defining interval lengths. The grade on file incorrectly imposes equal lengths and a quantile change of parameter; the printed construction uses cumulative target masses directly.) - run-level `symbol f_k`: [note, V1.5] symbol table corrected: type "locally indexed map", space "A node-specific structural-function space or the partition-map space $\{f:\mathcal{I}_k\to\mathcal{C}_k\}$, according to scope.", def "$f_k$ denotes a structural or categorical-propensity function at node $V_k$ in SCM arguments, and the map $f_k(I_{O_k(i)})=C_{O_k(i)}$, $i=1,\ldots,2^{kd}$, in the partition construction." (The paper still overloads $f_k$ between structural functions and a partition correspondence, so the general fix remains applicable. The obsolete partition-result clause is dropped; the structural clause on file remains applicable to both currently listed results.) - run-level `symbol \mathcal{F}_{0,n}`: [note, V1.5] symbol table corrected: def "$\mathcal{F}_{0,n}=\mathcal{NN}^{\hat{L}_f,K}\left(W_{0,n},\Theta(\log d_{\max}^{\text{in}})\right)$, where $\hat{L}_f=\sqrt{d_{\max}^{\text{in}}d_{\max}^{\text{out}}}L_f$ and $\mathcal{NN}_{d_1,d_2}^{L_f,K}(W,L)=\left\{\max\{-K,\min\{f,K\}\}:f\in\mathcal{NN}_{d_1,d_2}(W,L),\text{Lip}(f)\leqslant L_f\right\}$." (The vague recorded definition still requires the explicit architecture formula. The class itself retains coordinatewise clipping; continuous candidates in the optimization additionally satisfy the uniform output $1$-norm bound.) - run-level `symbol \mathcal{F}_{1,n}`: [note, V1.5] symbol table corrected: def "$\mathcal{F}_{1,n}=\mathcal{NN}^{\infty,K}(\Theta(d_{\max}^{U}),L_{1,n},\Theta((d_{\max}^{U})^2),L_{2,n},\tau_n)$, where $\mathcal{NN}^{\infty,K}$ is the truncated version of the specified generator architecture." (The explicit architecture fix still applies. The former ambiguity grade omits the now binding convention that component counts and cube dimensions remain fixed along the sequence.) - run-level `symbol \epsilon_n`: [note, V1.5] symbol table corrected: def "$\epsilon_n$ denotes a positive approximation tolerance indexed by $n$, with its role specified locally." (The local approximation and deviation meanings remain as previously pinned. Remove only the two clauses asserting the representation construction and its $C_ga_n$ bound, which the current texts do not establish as definitions of this symbol.) - run-level `symbol s_n`: [note, V1.5] symbol table corrected: def "$s_n=O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2 n+\log(nm_n)\lambda_n\right)$ is the sampling and Sinkhorn approximation tolerance, where $D_U=\sum_{j=1}^{n_U}d_j^U$ includes the non-Gumbel latent variables only and $D=\max\{1,D_U\}$." (The vague recorded label omits the specified tolerance order; the existing fix remains applicable. Corollary~\ref{cor:rate} now explicitly makes the same positive, sufficiently large choice, which requires an additional scoped clause.) - run-level `symbol \tau_n`: [note, V1.5] symbol table corrected: type "local temperature or additive tolerance sequence", space "$\mathbb{R}$, with $\tau_n>0$ in temperature uses", def "$\tau_n$ denotes the generator temperature or the additive tolerance in $\\|g_n(\theta_1)-g_n(\theta_2)\\|\leqslant L_gd(\theta_1,\theta_2)+\tau_n$, according to its local use." (The temperature-only entry misses the separate additive-tolerance use. The existing scopes remain valid, but the printed auxiliary argument also requires its own scope.) - run-level `symbol d`: [note, V1.5] symbol table corrected: type "local dimension, metric, or regression-difference function", space "$\mathbb Z_{\geqslant0}$ for dimension uses; metrics $M\times M\rightarrow[0,\infty)$ for metric uses; real-valued functions on the local covariate domain for regression-difference uses.", def "$d$ denotes the locally specified Euclidean or cube dimension, the metric of a locally specified metric space $(M,d)$, or the proof function $d(s)=\hat f_y(s)-f_y(s)$." (All existing dimension and metric clauses remain. The treatment-comparison proof additionally uses d for a real-valued regression difference, which the previous general reading omitted.) - run-level `symbol d_1`: [note, V1.5] symbol table corrected: type "local input dimension or averaged metric", space "$\mathbb N$ for dimension uses; a metric on $\mathbb R^m$ for the averaged covering-number use.", def "$d_1$ denotes a locally specified input or source dimension, or the averaged metric $d_1(u,w)=\frac1m\sum_{i=1}^m\|u_i-w_i\|$." (The dimension and metric overload remains settled. The separate squared-loss citation additionally uses the averaged metric with denominator $N=2m$, rather than the number $m$ of sampled pairs.) - run-level `symbol D`: [note, V1.5] symbol table corrected: type "context-dependent geometric parameter", space "$[0,\infty)$", def "$D$ denotes the ambient dimension in $K\subset\mathbb R^D$, a locally defined categorical-cardinality factor, or the latent-dimension parameter $\max\{1,D_U\}$ with $D_U=\sum_{j=1}^{n_U}d_j^U$ over non-Gumbel latent variables." (The existing ambient, treatment-comparison, and latent-dimension meanings remain. The approximation proof additionally defines D as the maximum categorical label diameter, which differs from the treatment-comparison factor.) - run-level `symbol d_{\mathcal{F}}`: [note, V1.5] symbol table corrected: def "$d_{\mathcal F}<\infty$ is a strict pseudo-dimension bound for the scalar function classes specified locally." (The strict coordinatewise interpretation remains necessary. The old scalar-citation clause no longer applies because that citation now uses a separately named pseudo-dimension.) - run-level `symbol C_{\mathcal{G}}(L,K)`: [note, V1.5] symbol table corrected: def "$C_{\mathcal G}(L,K;\{d_i,n_i\})$ is the intervention comparison constant that only depends on $L,K$, the causal graph $\mathcal G$, the continuous output dimensions $d_i$, and the categorical cardinalities $n_i$." (The recorded dependence and the general reading on file omit the output dimensions and categorical cardinalities now explicitly permitted. The printed proof's old coefficient cannot be adopted as the definition of the newly specified coefficient.) - run-level `symbol C_W`: [note, V1.5] symbol table corrected: def "$C_W$ denotes the comparison coefficient specified in the applicable bound." (The recorded formula and the scoped formula on file omit the new bound B. The proposition uses B in both terms of C_W; the lemma retains its separate formula using K.) - run-level `symbol L_{i,n}`: [note, V1.5] symbol table corrected: def "$L_{1,n}$ and $L_{2,n}$ are the two depth parameters in $\mathcal{F}_{1,n}=\mathcal{NN}^{\infty,K}(\Theta(d^U_{\max}),L_{1,n},\Theta((d^U_{\max})^2),L_{2,n},\tau_n)$." (The recorded generator-block index can be read as indexing latent variables or support components. Here $i=1,2$ distinguishes the two successive generator parts.) - run-level `symbol p_i`: [note, V1.5] symbol table corrected: type "nonnegative weight", space "$[0,\infty)$", def "$p_i$ denotes a nonnegative scalar weight, with its normalization and interpretation specified locally." (The general weight definition remains appropriate. The existing consistency clause covers component masses but omits the categorical propensity coordinates now used in that proof.) - run-level `symbol \delta`: [note, V1.5] symbol table corrected: def "$\delta>0$ is a locally specified positive parameter, used as an overlap lower bound, an approximation tolerance, an auxiliary separation parameter, or a quantization parameter." (The general local-parameter meaning remains valid. The net used in the empirical Wasserstein proof now has radius $\delta/2$, rather than the radius $\delta$ recorded on file.) - run-level `symbol c`: [note, V1.5] symbol table corrected: type "locally specified scalar, categorical configuration, or scalar-valued function", space "The locally specified scalar space, finite categorical configuration space, or real-valued function space.", def "$c$ is local notation for a scalar constant, a categorical configuration in conditioning expressions, or the function $c(t)=\int\hat f_y(a,t)\nu_A(da)-\int\hat f_y(a,t)\mu_A(da)$." (The counterexample gap is only one local meaning. The rate proof additionally uses categorical configurations and a concentration constant; the existing other scopes remain applicable.) - run-level `symbol g`: [note, V1.5] symbol table corrected: type "real-valued function with locally specified role", space "Real-valued functions on the locally specified domain.", def "$g$ denotes a locally specified real-valued function, used as a Wasserstein dual test function, a limiting constraint function, a member of the scalar class $\mathcal G$, or a jointly Borel map parametrizing that class." (The limiting-constraint entry omits the other local roles. In the cited probability bound, $g$ now denotes the jointly Borel parameterizing map on $T\times X$, while its sections belong to $\mathcal G$.) - run-level `symbol L_g`: [note, V1.5] symbol table corrected: def "$L_g$ denotes a Lipschitz bound, with its function and value specified locally." (The recorded definition restricts $L_g$ to the approximate constraint sequence. The paper also defines it explicitly for a different function in the Lipschitz proof; these are separate local constants.) - run-level `symbol B`: [note, V1.5] symbol table corrected: type "locally specified measurable test set or positive scalar bound", space "Measurable subsets of the locally specified measure space, or $(0,\infty)$.", def "$B$ denotes a locally specified measurable test set or scalar bound: in $f_\#\mu(B)=\mu(f^{-1}(B))$ it is a measurable subset of the codomain of $f$, in a lower-bound premise it is a measurable subset of the parameter cube, and in a proof estimate it is the locally defined positive scalar bound." (The recorded cube-set meaning omits the other local meanings already identified on file. The separately stated distribution approximation proof additionally uses $B$ as a fixed scalar clipping bound.) - run-level `symbol \mathcal{H}_\delta`: [note, V1.5] symbol table corrected: space "$2^{\mathcal H}$", def "$\mathcal H_\delta$ is a finite $\delta/2$-net in the uniform norm of $\mathcal H=\{h:[-K,K]^d\rightarrow\mathbb R:h(0)=0,\ \|h(x)-h(y)\|\leqslant\\|x-y\\|\}$." (The proof now specifies scalar-valued test functions and a net of radius $\delta/2$. The vector-valued class and radius recorded on file no longer describe its definition.) - run-level `symbol h`: [note, V1.5] symbol table corrected: type "local auxiliary function or grid spacing", space "Locally specified function spaces, or $(0,\infty)$ for grid spacing.", def "$h$ denotes a locally defined auxiliary function, or the grid spacing $h=2K/R$ in the structural interpolation construction." (The mixing function, entropy, regression auxiliary functions, and grid spacing remain local meanings. The scalar test-function reading belongs to the Wasserstein proof, rather than the independent covering citation.) - run-level `symbol m`: [note, V1.5] symbol table corrected: type "local integer index, count, vector length, input dimension, or sample size", space "Nonnegative integers for input dimensions and counts of nonintervened nodes; positive integers for vector lengths and sample sizes.", def "$m$ is a local integer parameter, used as a curve index, the number $n_V-n_T$ of nonintervened nodes, the length of $\nu\in\Delta_m$, a structural input dimension, or the sample size and evaluation-point count in empirical covering and concentration bounds." (The entry restricted to a probability-vector length misses the other local roles. The current proofs additionally use $m$ as a possibly zero count of nonintervened nodes and as the positive maximum structural input dimension; the existing scopes remain valid.) - run-level `symbol \|\cdot\|_\infty`: [note, V1.5] symbol table corrected: space "Finite-dimensional vector spaces and bounded function spaces, with values in $[0,\infty)$." (The vector norm is correctly identified, but the recorded space excludes the function arguments repeatedly used by the paper. The domain of the supremum is the domain of the function in the surrounding passage.) - run-level `symbol R`: [note, V1.5] symbol table corrected: type "locally specified shape-ratio functional or intervention bound", space "Shape-ratio functionals on finite collections with positive edge lengths, or $(0,\infty)$.", def "$R$ denotes either the shape-ratio functional $R(\mathcal C)=\max_{i,k}\|a_i^k-b_i^k\|/\min_{i,k}\|a_i^k-b_i^k\|$ for $C_k=\prod_i[a_i^k,b_i^k]$, or a fixed positive bound on continuous intervention values, according to the surrounding passage." (The recorded functional remains the meaning in the cube construction, but does not describe the positive scalar intervention bound introduced in Corollary~\ref{corollary:deep_nn}. These are distinct local meanings.) - run-level `symbol S`: [note, V1.5] symbol table corrected: type "locally defined approximation error or set to be covered", space "$[0,\infty]$ for approximation errors; subsets of the ambient normed space for covering numbers.", def "$S$ is local notation: $S=\sum_{i=1}^{n_V}\\|f_i-\hat f_i\\|_\infty$ in the structural comparison argument; $S=\\|\hat f_i(x)-f_i(x)\\|_\infty$ in the categorical normalization argument; and $S$ is the set covered in $\mathcal{N}(\epsilon,S,\\|\cdotp\\|_1)$." (The table records only the summed approximation error, although the paper also uses $S$ for a pointwise categorical approximation error and a set to be covered. The categorical passage supplies neither the projected extension nor the additional construction guarantees recorded in the previous scoped reading.) - run-level `symbol \mathcal{N}`: [note, V1.5] symbol table corrected: type "covering number or local causal-model variable", space "Covering numbers in $\mathbb{N}\cup\{\infty\}$, or causal models in the locally specified class.", def "$\mathcal{N}(\epsilon,S,\\|\cdotp\\|_1)$ is the covering number of $S$ in the $\ell_1$ norm; a standalone $\mathcal{N}\in\{\mathcal{M},\hat{\mathcal{M}}\}$ denotes either of the two causal models." (The existing distinction between covering numbers and local causal models remains valid. The separate concentration bound additionally requires its averaged metric and doubled evaluation count.) - run-level `symbol \theta^*`: [note, V1.5] symbol table corrected: type "context-dependent parameter point", def "$\theta^*$ denotes a minimizer of (\ref{eq:opt*}) in the abstract optimization proof, and the parameter representing the underlying SCM $\mathcal{M}^{\theta^*}$ in the causal-model parameterization." (The existing distinction between an abstract optimization minimizer and the true-model parameter remains necessary. The rate result imports the latter meaning, which needs its own scope.) - run-level `symbol \mathcal{F}_{V_i}`: [note, V1.5] symbol table corrected: def "$\mathcal{F}_{V_i}=\mathcal{F}_L^K([-K,K]^{d_{i,\text{in}}^V},[-K,K]^{d_{i,\text{out}}^V})$ for continuous $V_i$, and $\mathcal{F}_{V_i}=\{f:\\|f\\|_1=1,\ f\in\mathcal{F}_L^K([-K,K]^{d_{i,\text{in}}^V},[-K,K]^{d_{i,\text{out}}^V})\}$ for categorical $V_i$, with $\mathcal{F}_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$." (The vague recorded class omits its defining branches and bounds. The earlier fix used the proof's infinity-norm display without a specified Lipschitz bound; the current statement specifies a uniform output 1-norm bound and an explicit Lipschitz bound.) - run-level `symbol \mathcal{P}`: [note, V1.5] symbol table corrected: type "space of probability measures", def "$\mathcal{P}(K)$ denotes the set of probability measures on $K$." (This is a space whose elements are probability measures, rather than a probability space carrying random outcomes. Its arguments in the parameter product are the compact latent-coordinate cubes.) - run-level `symbol \hat{g}_1^i`: [note, V1.5] symbol table corrected: type "ReLU neural network", def "$\hat{g}_1^i:[0,1]\rightarrow\mathbb{R}^{d_i^C}$ is the first component generator block, chosen to approximate the pullback component law $(g_2^i)^{-1}_{\#}\mu_i$ by the law of $\hat{g}_1^i(U_i)$." (The general network definition and existing wide, deep, and singleton scopes remain valid. The extracted lemma additionally specifies the quantized construction and its source parameters.) - run-level `symbol U_i`: [note, V1.5] symbol table corrected: space "The locally specified latent space $\mathbb{R}^{d_i^U}$, or $[0,1]$ for a scalar generator input.", def "$U_i$ denotes a locally indexed latent variable or uniform generator input." (The recorded latent space uses the wrong index and does not cover scalar generator inputs. The additional scope identifies the bounded latent coordinates used in the consistency argument without imposing uniform laws on them or restating the sampling model.) - run-level `symbol s`: [note, V1.5] symbol table corrected: type "integer depth parameter", space "$\mathbb{N}$", def "$s$ is the integer depth parameter of \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, fixed at $s=1$." (The entry calls $s$ a quantization parameter. The binding reading makes it the integer depth parameter fixed at one; the previous grade also incorrectly identified the quantization parameter with $1/\lceil n(n-1)\rceil$.) - run-level `lem:wide-nn-perekrestenko-theorem-viii-1`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/perekrestenkoHighDimensionalDistributionGeneration2022a.txt - run-level `lem:wide-nn-yarotsky-theorem-2`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt - run-level `lem:deep-nn-yarotsky-theorem-2`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt - run-level `lem:consistency-fournier-theorem-2`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/fournier2015rate.txt - run-level `lem:sup-wasserstein-anthony-theorems`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/anthony1999neural.txt - run-level `lem:sup-wasserstein-anthony-theorem-17-1`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/anthony1999neural.txt - run-level `lem:sup-wasserstein-anthony-theorem-17-1`: [refuted, V1.5] [F4 reviewer] [claim-false] Both the frozen node and the Lean carrier state Theorem 17.1 for every class of [0,1]-valued (measurable) functions on any measurable space X. They drop the measurability/permissibility standing condition the uniform-convergence bound needs, so the assumed carrier is refutable in ZFC and makes everything downstream vacuous. Repair: restrict to permissible classes, e.g. a class parametrized by a standard Borel space with jointly measurable evaluation on a standard Borel X. Orchestrator approved repair scope: the book standing measurability footnote on printed p.15 has been copied verbatim into discovery/cited_sources/anthony1999neural.txt; the rendered source page is retained beside it. The ordinal countable/co-countable counterexample checks, including RHS < 0.003 at eps=1/2 and m=2000. Outer probability alone does not fix it. Repair the cited input and the general empirical-Wasserstein lemma with an explicit structural permissibility domain, for example standard Borel inputs and standard Borel parameter families with jointly measurable evaluation. Never define permissibility by assuming the desired concentration bound. Prove the necessary property locally for all concrete neural, normalized square-root test and categorical-indicator families used in the consistency and rate proofs (measurable envelopes are allowed); do not assume pointwise rational-parameter density for discontinuous selectors. Preserve all accepted main assumptions, estimator definitions and rates. No unproved class-admissibility premise may be added to a main result. Carry each condition into its corrected statement or proof and preserve completed work. witness: [claim-false] Both the frozen node and the Lean carrier state Theorem 17.1 for every class of [0,1]-valued (measurable) functions on any measurable space X. They drop the measurability/permissibility standing condition the uniform-convergence bound needs, so the assumed carrier is refutable in ZFC and makes everything downstream vacuous. Repair: restrict to permissible classes, e.g. a class parametrized by a standard Borel space with jointly measurable evaluation on a standard Borel X. Orchestrator approved repair scope: the book standing measurability footnote on printed p.15 has been copied verbatim into discovery/cited_sources/anthony1999neural.txt; the rendered source page is retained beside it. The ordinal countable/co-countable counterexample checks, including RHS < 0.003 at eps=1/2 and m=2000. Outer probability alone does not fix it. Repair the cited input and the general empirical-Wasserstein lemma with an explicit structural permissibility domain, for example standard Borel inputs and standard Borel parameter families with jointly measurable evaluation. Never define permissibility by assuming the desired concentration bound. Prove the necessary property locally for all concrete neural, normalized square-root test and categorical-indicator families used in the consistency and rate proofs (measurable envelopes are allowed); do not assume pointwise rational-parameter density for discontinuous selectors. Preserve all accepted main assumptions, estimator definitions and rates. No unproved class-admissibility premise may be added to a main result. Carry each condition into its corrected statement or proof and preserve completed work. minimal patch (confirmed): corrected statement: The cited assertion used in Lemma~\ref{lemma:sup_wasserstein} is the following restricted form of Theorem 17.1 in \cite{anthony1999neural}, with its standing measurability condition. Let $X$ and $T$ be standard Borel spaces, let $g:T\times X\rightarrow[0,1]$ be jointly Borel, and put $\mathcal G=\{g(t,\cdot):t\in T\}\ne\varnothing$. For every Borel probability distribution $P$ on $X\times[0,1]$, every $0<\epsilon<1$ and every positive integer $m$, \[ P^m\left\{\text{some }t\in T\text{ has }\left\|\mathbb E_{(x,y)\sim P}(g(t,x)-y)^2-\frac1m\sum_{i=1}^m(g(t,x_i)-y_i)^2\right\|\geqslant\epsilon\right\} \leqslant4\mathcal N_1^{\mathrm{avg}}(\epsilon/16,\mathcal G,2m)\exp(-\epsilon^2m/32). \] Here the sample consists of independent draws from $P$, the probability is taken in its completion, and $\mathcal N_1^{\mathrm{avg}}(r,\mathcal G,N)$ is the supremum over $x_1,\ldots,x_N\in X$ of the covering number of the evaluation vectors in the metric $N^{-1}\sum_i\|u_i-v_i\|$. · proof change: The cited assertion used in Lemma~\ref{lemma:sup_wasserstein} is the following restricted form of Theorem 17.1 in \cite{anthony1999neural}, with its standing measurability condition. Let $X$ and $T$ be standard Borel spaces, let $g:T\times X\rightarrow[0,1]$ be jointly Borel, and put $\mathcal G=\{g(t,\cdot):t\in T\}\ne\varnothing$. For every Borel probability distribution $P$ on $X\times[0,1]$, every $0<\epsilon<1$ and every positive integer $m$, \[ P^m\left\{\text{some }t\in T\text{ has }\left\|\mathbb E_{(x,y)\sim P}(g(t,x)-y)^2-\frac1m\sum_{i=1}^m(g(t,x_i)-y_i)^2\right\|\geqslant\epsilon\right\} \leqslant4\mathcal N_1^{\mathrm{avg}}(\epsilon/16,\mathcal G,2m)\exp(-\epsilon^2m/32). \] Here the sample consists of independent draws from $P$, the probability is taken in its completion, and $\mathcal N_1^{\mathrm{avg}}(r,\mathcal G,N)$ is the supremum over $x_1,\ldots,x_N\in X$ of the covering number of the evaluation vectors in the metric $N^{-1}\sum_i\|u_i-v_i\|$. This structural domain supplies the standing measurability condition: the loss is bounded and jointly Borel, and its integral against $P$ is Borel in $t$. The latter assertion holds for indicators of measurable rectangles, is preserved by bounded monotone limits and linear combinations, and hence holds for bounded Borel functions by the monotone class theorem. Each bad event is a projection of a Borel subset of $T\times(X\times[0,1])^m$, so it is analytic and universally measurable. Replacing the sample by two independent samples or applying any of the finitely many coordinate swaps gives Borel functions on finite products and therefore the same measurability conclusion. Thus the sample and ghost-sample events used by Theorem 17.1 are measurable in the completed sample laws, and that theorem gives the displayed inequality. - run-level `ass:corollary-deep-nn-latent-laws`: [note, V1.5] clarified — the reading the statement means is now stated: [F4 reviewer] [statement-wrong] The frozen assumption references a second conclusion deleted from thm:wide_nn; explicitly specify whether latent laws require only support charts or also componentwise domination and boundary vanishing. Orchestrator ruling and repair scope: the original second conclusion was SPLIT into thm:cor-fix-3c402872, not replaced by the support-only first conclusion. Its assumptions were and remain support charts plus componentwise lower domination and the boundary-null clause of assumption:lower_bound on normalized component laws. The accepted corrected thm:deep-nn already directly depends on both lower-bound assumptions and its corrected proof invokes the componentwise deep construction. Correct the dangling reference or spell out those SAME inherited conditions; never add a new latent regularity assumption or fall back to the support-only first conclusion. Preserve all other accepted main clauses and all proofs. This is a consequential cross-reference clarification after a split. This hypothesis of the paper is used by thm:deep-nn, thm:consistency, thm:rate. Repair it by replacing its own printed text; when the paper states it in running prose or inside a result's statement (no units of its own), restate each result using it instead (in that result's own repair when it has a finding, else a patch with `consequence_of`) and give this node a patch with no ops (`gives_up` null: what a restatement gives up is stated on the result). minimal patch (confirmed): proof change: The latent-law condition is restated in Corollary~\ref{corollary:deep_nn} by citing the proposition entitled ``Deep distribution approximation''. Its hypotheses require Assumption~\ref{assump:support} for each latent law and Assumption~\ref{assumption:lower_bound} for each normalized component law, including the boundary condition on its pullback to the component cube. Theorem~\ref{thm:consistency} assumes the conditions of Corollary~\ref{corollary:deep_nn}, and Corollary~\ref{cor:rate} assumes the conditions of Theorem~\ref{thm:consistency}; those citations therefore retain precisely these latent-law conditions. - run-level `def:definition-3`: [refuted, V1.5] [F4 reviewer] [statement-wrong] The phrase 'i.i.d. multivariate uniform variables' admits common-law blocks versus dimension-dependent independent uniform blocks; Lean implements the latter. Orchestrator ruling: the underlying scalar uniform coordinates are independent with common Unif([0,1]) law; the multivariate inputs group finitely many such coordinates into unit-cube blocks of their indicated dimensions. State that coordinate-level convention explicitly. Blocks of different dimensions need not be equated as probability measures on different types. If a common block dimension is desired, independent unused uniform coordinates pad the blocks and the structural networks ignore them, leaving the represented SCM law unchanged. Preserve the current product law, Gumbel independence, structural-network/propensity requirements, and every main architecture and approximation conclusion. This is a clarification of dimension grouping, not permission to enlarge a main conclusion by changing its generated laws. This definition of the paper is used by thm:deep-nn. Repair it by replacing its own printed text; when the paper states it in running prose or inside a result's statement (no units of its own), restate each result using it instead (in that result's own repair when it has a finding, else a patch with `consequence_of`) and give this node a patch with no ops (`gives_up` null: what a restatement gives up is stated on the result). witness: [statement-wrong] The phrase 'i.i.d. multivariate uniform variables' admits common-law blocks versus dimension-dependent independent uniform blocks; Lean implements the latter. Orchestrator ruling: the underlying scalar uniform coordinates are independent with common Unif([0,1]) law; the multivariate inputs group finitely many such coordinates into unit-cube blocks of their indicated dimensions. State that coordinate-level convention explicitly. Blocks of different dimensions need not be equated as probability measures on different types. If a common block dimension is desired, independent unused uniform coordinates pad the blocks and the structural networks ignore them, leaving the represented SCM law unchanged. Preserve the current product law, Gumbel independence, structural-network/propensity requirements, and every main architecture and approximation conclusion. This is a clarification of dimension grouping, not permission to enlarge a main conclusion by changing its generated laws. minimal patch (confirmed): corrected statement: (Neural Causal Model) A Neural Causal Model (NCM) is a special kind of SCM where $\boldsymbol U=\{U_1,\ldots,U_{n_U}\}\cup\{G_{V_i}:V_i\ \text{categorical}\}$, the scalar coordinates of all the $U_i$ are mutually independent with common $\operatorname{Unif}([0,1])$ law, and $U_i$ groups the indicated finite number $d_i^U$ of these coordinates, so that $U_i\sim\operatorname{Unif}([0,1]^{d_i^U})$. All coordinates of the $G_{V_i}$ are independent standard Gumbel variables, independent of all uniform coordinates. The continuous structural maps and the categorical propensity-weight maps $f_i$ in (\ref{eq:structure_eq}) are Neural Networks (NNs); categorical values are generated from these weights by the Gumbel argmax rule. The weights are nonnegative and have strictly positive sum, with $\log0=-\infty$. · proof change: For Borel sets $A_{ir}\subseteq[0,1]$, mutual independence of the scalar coordinates gives \[ \mathbb P\left(\bigcap_i\bigcap_{r=1}^{d_i^U}\{(U_i)_r\in A_{ir}\}\right)=\prod_i\prod_{r=1}^{d_i^U}\lambda_1(A_{ir}), \] where $\lambda_1$ is Lebesgue probability measure on $[0,1]$. Thus the blocks have their unit-cube laws and are independent. This is the product law used in the neural construction, also when block dimensions differ; a zero-dimensional block is the unique point. The independent Gumbel coordinates have the stipulated product law jointly with these blocks. If a common block dimension is used, adjoin independent uniform coordinates to each smaller block and set all weights from those new input coordinates to zero. The structural functions then have exactly the same values for every original input and every value of the adjoined coordinates, so their generated SCM law is unchanged. - run-level `def:hilbert-partitions`: [note, V1.5] clarified — the reading the statement means is now stated: [F4 reviewer] [statement-wrong] Specify whether coordinate cells must be full-dimensional or may be degenerate, and whether disjoint interiors alone or intersection containment in both boundaries is intended; Lean permits degeneracy while imposing the latter condition. Orchestrator ruling: follow the exact binding pin-164: closed coordinate boxes cover the cube and distinct cells have disjoint interiors; shared boundaries are allowed. No strict-positive side-length condition is stated by that definition, so allow degenerate boxes there. For degenerate boxes, intersection containment in BOTH frontiers is stronger than disjoint interiors, so remove that extra requirement from the Lean general property and encode disjoint interiors directly. The repaired Hilbert lemma and Holder proof use the standard DYADIC collection, where side lengths are positive and both properties hold; preserve that existing specialization and its proved construction. Do not add a new assumption to the measure-representation result. This definition of the paper is used by lem:hilbert, prop:holder-curve. Repair it by replacing its own printed text; when the paper states it in running prose or inside a result's statement (no units of its own), restate each result using it instead (in that result's own repair when it has a finding, else a patch with `consequence_of`) and give this node a patch with no ops (`gives_up` null: what a restatement gives up is stated on the result). minimal patch (confirmed): proof change: These conventions specify property (*). Lemma~\ref{lemma:hilbert} uses its stated standard dyadic specialization, and Proposition~\ref{prop:holder_curve} constructs that same specialization. At level $n$, each coordinate interval in that specialization has length $2^{-n}>0$ and the intervals cover $[0,1]$. Taking products covers $[0,1]^d$. If two cells differ, one coordinate has distinct dyadic intervals, whose open interiors are disjoint, so the product interiors are disjoint. Their intersection, when nonempty, lies on an endpoint in such a coordinate and therefore in both cell boundaries. These facts verify the conventions for every cell used by the two results without requiring an additional condition on $\mathbb P$. - run-level `def:thm-consistency-architecture-radius`: [refuted, V1.5] [F2.5 reviewer] [statement-wrong] The frozen radius definition retains the maximum-dimension sampling rate, while the supplied repaired thm:consistency uses total latent dimension and an additional network-depth term; these frozen specifications require reconciliation. Orchestrator: this definition was revived as stale context of the restored Fournier citation. Synchronize it with the already accepted repaired thm:consistency and thm:rate, preserving their exact rates and all main hypotheses and quantifiers: total non-Gumbel dimension DU, D=max(1,DU), added network-depth sampling term, positive temperature expression. No new main weakening is authorized. Keep the published Fournier concentration input separate from the paper-owned conditional-sampling and final-radius deductions, which stay local. This definition of the paper is used by lem:consistency-fournier-theorem-2. Repair it by replacing its own printed text; when the paper states it in running prose or inside a result's statement (no units of its own), restate each result using it instead (in that result's own repair when it has a finding, else a patch with `consequence_of`) and give this node a patch with no ops (`gives_up` null: what a restatement gives up is stated on the result). witness: [statement-wrong] The frozen radius definition retains the maximum-dimension sampling rate, while the supplied repaired thm:consistency uses total latent dimension and an additional network-depth term; these frozen specifications require reconciliation. Orchestrator: this definition was revived as stale context of the restored Fournier citation. Synchronize it with the already accepted repaired thm:consistency and thm:rate, preserving their exact rates and all main hypotheses and quantifiers: total non-Gumbel dimension DU, D=max(1,DU), added network-depth sampling term, positive temperature expression. No new main weakening is authorized. Keep the published Fournier concentration input separate from the paper-owned conditional-sampling and final-radius deductions, which stay local. minimal patch (confirmed): proof change: The network-class and radius construction occurs inside the statement of Theorem~\ref{thm:consistency}, rather than as a separately printed definition. Theorem~\ref{thm:consistency} and Corollary~\ref{cor:rate} already restate this construction in full and no longer use the original radius specification. Their restatements retain their network classes, restrictions, hypotheses, and quantifiers and put \[ D_U=\sum_{j=1}^{n_U}d_j^U,\qquad D=\max\{1,D_U\}, \] where the sum includes the non-Gumbel latent variables only. They take $\alpha_n=\epsilon_n+s_n$, with positive tolerances chosen as sufficiently large constant multiples of the respective quantities in \[ \epsilon_n=O\left(W_{0,n}^{-1/d_{\max}^{\text{in}}}+\sum_{i=1}^2L_{i,n}^{-2/d_{\max}^U}+\tau_n(1-\log\tau_n)\right), \] \[ s_n=O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2n+\log(nm_n)\lambda_n\right). \] These are exactly the radius specifications in those restatements. The concentration input from Theorem 2 of \cite{fournier2015rate} remains separate: the conditional-sampling argument, uniform simulation estimate, and assembly of the final radius are deductions in the proofs of Theorem~\ref{thm:consistency} and Corollary~\ref{cor:rate}. - run-level `lem:wide-nn-perekrestenko-theorem-viii-1`: [note, V1.5] made precise — the printed statement did not determine the claim; the reading its repair states was confirmed: The local $n$ can be fixed or can increase with $W_1$, giving different quantization requirements; neither the passage nor the host's amended proof determines its dependence on width. Independently, the stated quantization substitution does not instantiate the supplied source. The source separates its depth parameter from a quantization mesh $\delta/n$ and requires $\delta<1/(n(n-1))$. Interpreting the printed fraction as either $\delta$ or the mesh violates that requirement. The host's amended proof repeats the paper's quantization choice without establishing admissible source parameters. This finding concerns the citation application; the asserted rate can be obtained with appropriate source parameters. minimal patch (confirmed): proof change: Theorem~\ref{thm:wide_nn} restates this citation application in full in its proof, using $s=1$, $\delta=1/\lceil\sqrt q(q+1)n(n-1)\rceil$, quantization mesh $\delta/n$, and $n=\lfloor(W_1/A_q)^{1/q}\rfloor$ for positive component dimension $q=d_i^C$; that proof derives the admissibility, width bound, norm conversion, and approximation rate. - run-level `lem:sinkhorn-luise-proposition-1`: [note, V1.5] made precise — the printed statement did not determine the claim; the reading its repair states was confirmed: The full entropy-penalized optimal value and the transport cost evaluated at an entropy-regularized optimizer are different readings of $W_{1,\lambda}$; their differences from $W$ can have different signs. The host does not identify either functional, its entropy convention, or the ground-cost data. The cited proposition gives absolute approximation bounds for two separately defined functionals, with an inverse regularization parameter. It does not justify the host's signed assertion for its unspecified functional; in particular, the source's full regularized value violates the asserted lower bound. minimal patch (confirmed): proof change: The attributed passage is replaced in the proof of Lemma~\ref{lemma:sinkhorn_approx}; its signed assertion is no longer used. Indeed, with the full regularized value defined there, evaluating at an unregularized optimizer gives $W_{1,\lambda}\leqslant W$, while the transport cost of a regularized optimizer is at least $W$. Since every plan has mass one, concavity of $-x\log x$ gives $h(T)\leqslant1+\log(nm)$. Therefore $-\lambda(1+\log(nm))\leqslant W_{1,\lambda}-W\leqslant0$. Applying these inequalities also to the two self terms and using the defining subtraction for $S_\lambda$ yields $\|S_\lambda-W\|\leqslant2\lambda(1+\log(nm))$. - run-level `lem:sup-wasserstein-anthony-theorems`: [note, V1.5] made precise — the printed statement did not determine the claim; the reading its repair states was confirmed: (a) For vector-valued $\mathcal F$, its pseudo-dimension could mean $\max_{j\in[d]}\operatorname{Pdim}(\{f_j:f\in\mathcal F\})$ or $\operatorname{Pdim}(\{(x,j)\mapsto f_j(x):f\in\mathcal F\})$. These impose different bounds, and neither the paper nor the scalar cited theorem specifies the choice. The dual formulation does determine that $h$ is scalar-valued. (b) The attributed covering inequality is false for the paper's explicitly unnormalized empirical $\ell_1$ metric: its bound omits sample-size scaling. (c) It also omits range scaling: Theorem 18.4 requires scalar functions valued in $[0,1]$, whereas the host only bounds their magnitude by arbitrary $K$. The fixed-$h$ concentration order can be recovered from Theorem 17.1 by range normalization and a square-root transformation with zero responses, but that does not validate the printed covering inequality. minimal patch (confirmed): corrected statement: By Theorem 18.4 in \cite{anthony1999neural}, if $\mathcal G$ is a nonempty class of functions from a set $X$ into $[0,1]$ with finite pseudo-dimension $v$, then for every $0<\epsilon\leqslant1$ and every positive integer $m$, \begin{align*} \mathcal{N}_{1}^{\mathrm{avg}}(\epsilon,\mathcal G,m)\leqslant\mathcal{M}_{1}^{\mathrm{avg}}(\epsilon,\mathcal G,m)0$ such that $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely under the SCM law. (Assumption \ref{assump:Lip_and_bounded} bounds the model-generated random variables themselves, and the surrounding setup treats their probability laws as the objects of comparison. It does not quantify over exceptional realizations of a probability-space representation.) - note on `ass:assump-lip-and-bounded-categorical`: pinned reading: For categorical $V_i$, $f_i(x)=(\mathbb{P}(V_i=j\mid x))_{j\in[n_i]}$ is the structural propensity vector, and $\\|f_i(x)-f_i(x')\\|\leqslant L_f\\|x-x'\\|$ on its structural input domain, using the paper's $1$-norm. (Equation (\ref{eq:structure_equation}) uses $f_i$ as a normalized vector with indexed coordinates, while the notation paragraph defines the unadorned norm as the $1$-norm. The assumption applies Lipschitz continuity to that same $f_i$.) - note on `ass:bk-owner-1`: pinned reading: The bound $K$ applies throughout the optimization class to the observed variables $V_i$ and indexed non-Gumbel latent variables $U_j$, with $n_i\leqslant K$ for categorical nodes and admissible intervention coordinates in $[-K,K]$; it does not bound the separately supplied standard Gumbel noises. (The boundedness assumption explicitly indexes $V_i$ and $U_j$, whereas the categorical convention separately adjoins $G_{V_i}$. The binding boundedness symbols preserve this distinction.) - note on `ass:bk-owner-2`: pinned reading: The variable bound inherited from Assumption~\ref{assump:Lip_and_bounded} uses $\\|\cdot\\|_\infty$. (The owner condition extends the existing bound $K$, which Assumption~\ref{assump:Lip_and_bounded} explicitly states using infinity norms of observed and non-Gumbel latent variables.) - note on `ass:corollary-deep-nn-latent-laws`: clarified — the reading the statement means is now stated: [F4 reviewer] [statement-wrong] The frozen assumption references a second conclusion deleted from thm:wide_nn; explicitly specify whether latent laws require only support charts or also componentwise domination and boundary vanishing. Orchestrator ruling and repair scope: the original second conclusion was SPLIT into thm:cor-fix-3c402872, not replaced by the support-only first conclusion. Its assumptions were and remain support charts plus componentwise lower domination and the boundary-null clause of assumption:lower_bound on normalized component laws. The accepted corrected thm:deep-nn already directly depends on both lower-bound assumptions and its corrected proof invokes the componentwise deep construction. Correct the dangling reference or spell out those SAME inherited conditions; never add a new latent regularity assumption or fall back to the support-only first conclusion. Preserve all other accepted main clauses and all proofs. This is a consequential cross-reference clarification after a split. This hypothesis of the paper is used by thm:deep-nn, thm:consistency, thm:rate. Repair it by replacing its own printed text; when the paper states it in running prose or inside a result's statement (no units of its own), restate each result using it instead (in that result's own repair when it has a finding, else a patch with `consequence_of`) and give this node a patch with no ops (`gives_up` null: what a restatement gives up is stated on the result). - note on `ass:lemma-lip-condition-regression`: pinned reading: The functions $f,\hat{f}:[-K,K]\rightarrow\mathbb{R}$ are $L$-Lipschitz regression versions satisfying $f(T)=\mathbb{E}_{\nu}[Y\mid T]$ almost surely under $\nu$ and $\hat{f}(\hat{T})=\mathbb{E}_{\mu}[\hat{Y}\mid\hat{T}]$ almost surely under $\mu$. (The proof of Lemma~\ref{lemma:Lip_condition} uses both cross-evaluations and bounds its test function on the common box; the owner-supplied support condition specifies that box.) - note on `ass:lemma-sup-wasserstein-bounded`: pinned reading: The boundedness condition means $\\|f(x)\\|_{\infty}\leqslant K$ for every $f\in\mathcal{F}$ and $x\in\mathbb{R}^{d'}$. (The proof of Lemma~\ref{lemma:sup_wasserstein} places the outputs in $[-K,K]^d$, consistently with the supplied function-class interpretation.) - note on `ass:prop-convg-opt-approximate-lipschitz`: pinned reading: Equation~\ref{eq:aprox_lip} holds for every $n\in\mathbb N$ and every $\theta_1,\theta_2\in\Theta_\infty$, with the same finite constant $L_g$. (Proposition~\ref{prop:convg_opt} defines the functions on $\Theta_\infty$ and uses this inequality to conclude that the limiting function is $L_g$-Lipschitz.) - note on `ass:prop-convg-opt-approximation`: pinned reading: There exists a compact $\tilde\Theta_\infty\subseteq\Theta_\infty$ such that $\forall n\in\mathbb N\;\forall\theta_0\in\tilde\Theta_\infty\;\exists\theta\in\Theta_n$, $d(\theta,\theta_0)\leqslant\epsilon_n$. (The proof of Proposition \ref{prop:convg_opt} applies condition 3 to construct a sequence $\theta_n$ approximating $\tilde\theta^*$.) - note on `ass:prop-lip-ate-bounded`: pinned reading: For each of $\mathcal{M}$ and $\hat{\mathcal{M}}$, $\max_{i\in[n_V],j\in[n_U]}\{\\|V_i\\|_\infty,\\|U_j\\|_\infty\}\leqslant K$ almost surely, where the indexed $U_j$ are the non-Gumbel latent variables. (Proposition \ref{prop:Lip_ATE} explicitly inherits Assumption \ref{assump:Lip_and_bounded}, which prints this maximum; Assumption \ref{assump:categorical_rv} distinguishes the indexed $U_j$ from the categorical noises $G_{V_i}$.) - note on `ass:prop-lip-ate-overlap`: pinned reading: The measure $P$ is a probability measure on the treatment space, and for almost every parent value $x$, $\nu(dt\mid\text{Pa}(T)=x)=p_\nu(t\mid\text{Pa}(T)=x)P(dt)$, with the stated density version satisfying $p_\nu(t\mid\text{Pa}(T)=x)\geqslant\delta$ for every $t\in[t_1,t_2]$. (The overlap clause of Proposition~\ref{prop:Lip_ATE} names a conditional treatment density, and its displayed conclusion integrates over treatment values against $P(dt)$; the supplied meanings of $P$ and $p_\nu$ agree with this interpretation.) - note on `ass:relu-approximation-domain`: pinned reading: The phrase ``uniform cubes'' denotes the unit cube domains $[0,1]^m$ used for deterministic uniform function approximation; it imposes no uniform probability law on structural inputs. (The same paragraph displays approximation of functions on $[0,1]^{d_1}$ and separately says that similar arguments apply to any bounded cubes.) - note on `ass:thm-consistency-structural-lipschitz`: pinned reading: The continuous structural functions and categorical propensity functions of $\mathcal M^*$ have Lipschitz constants at most $L_f$ in the paper's $1$-norm. (Theorem~\ref{thm:consistency} begins with $\mathcal M^*$ and permits candidate networks with bound $\hat L_f$; Assumption~\ref{assump:Lip_and_bounded} identifies the functions bounded by $L_f$.) - note on `def:cor-rate-objective`: pinned reading: $\mu_T=\text{Unif}([t_1,t_2])$ is the deterministic intervention probability measure. (Corollary~\ref{cor:rate} specializes the intervention measure used in the outer expectation $\mathbb E_{t\sim\mu_T}$ of Theorem~\ref{thm:consistency}.) - note on `def:corollary-deep-nn-dimensions`: pinned reading: $d_{\max}^U=\max_{j\in[n_U]}d_j^U$ ranges over non-Gumbel latent variables, while $d_{\max}^{\mathrm{in}}=\max_i d_i^{\mathrm{in}}$ and $d_{\max}^{\mathrm{out}}=\max_i d_i^{\mathrm{out}}$, with $d_i^{\mathrm{out}}=n_i$ for categorical nodes. (The NCM construction separately approximates non-Gumbel latent distributions and the functions in the canonical structural equations; those functions output categorical propensity vectors. The binding dimension symbols specify these same meanings.) - note on `def:definition-2`: pinned reading: $C$ is a $C^2$-component precisely when it is inclusion maximal among subsets of $\boldsymbol V$ whose distinct vertices are pairwise joined by a bidirected arrow. (The definition requires an arrow for each pair and maximality of the subset. Standard graph terminology and the settled meaning of the component symbol fix distinct pairs and inclusion maximality.) - note on `def:eq-consistent-limiting`: pinned reading: $\underline{F}^{L}$ denotes the constrained infimum of the displayed scalar objective over the stated canonical SCM class and its admissible latent laws, rather than a minimizing SCM. (The sentence introducing the display explicitly says ``optimal value'', and the following sentence calls it the sharp lower bound when the Lipschitz bounds agree.) - note on `def:eq-opt-empiri`: pinned reading: $F_n=\inf\left\{\mathbb{E}_{t\sim\mu_T}\mathbb{E}_{\hat{\mathcal{M}}}[F(V_1(t),\ldots,V_{n_V}(t))]:\hat{\mathcal{M}}\in\text{NCM}_{\mathcal{G}}(\mathcal{F}_{0,n},\mathcal{F}_{1,n}),\ S_{\lambda_n}(P_{m_n}^{\hat{\mathcal{M}}}(\boldsymbol{V}),P_n^{\mathcal{M}^*}(\boldsymbol{V}))\leqslant\alpha_n\right\}$. (The prose introducing equation (\ref{eq:opt_empiri}) calls $F_n$ an estimator of optimal values, and the following paragraph asks whether it is a good lower bound. The supplied symbol convention likewise specifies a scalar infimum.) - note on `def:eq-structure-eq`: pinned reading: For distinct observed vertices $V_i,V_j$, the ``correlated latent variable pair'' criterion means that some $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$ are not independent, with $k=l$ permitted. (The paragraph immediately preceding Assumption \ref{assump:independence_u} explains that latent variables may enter several equations and that, under independence of the indexed latent family, a bidirected arrow represents a common latent parent. No covariance or moment convention is imposed in the SCM definition.) - note on `def:eq-structure-equation`: pinned reading: In the categorical branch of (\ref{eq:structure_equation}), zero coordinates of the normalized nonnegative vector $f_i(x)$ are allowed and are evaluated using $\log 0=-\infty$. (The canonical definition normalizes categorical propensity vectors, and Assumption \ref{assump:Lip_and_bounded} identifies their coordinates with conditional probabilities. The probability-simplex convention permits zero probabilities, so strict positivity would impose an additional restriction.) - note on `def:eq-structure-equation-hat`: pinned reading: For each categorical $V_i$, the noise in equation (\ref{eq:structure_equation_hat}) satisfies $g_k=g_k^{V_i}$ for $k\in[n_i]$. (The canonical representation in equation (\ref{eq:structure_equation}) assigns separate independent standard Gumbel coordinates to each categorical node; the approximated equation retains that categorical-generation rule.) - note on `def:gumbel-softmax-layer`: pinned reading: The layer uses $V=[\Vec{v}_1,\ldots,\Vec{v}_{N_C}]$ and the same Gumbel draws for all $\tau>0$, with $\lim_{\tau\downarrow0}\hat{X}^{\tau}_i=\hat{X}^{0}_i=\mathbb{I}_{\{i=\arg\max_{j\in[N_C]}(\log p_j+G_j)\}}$ almost surely for every $i\in[N_C]$. (The displayed softmax formula, the one-hot selector description, and the product with its explicitly specified component count determine these indices and the common-noise limit.) - note on `def:hilbert-partitions`: clarified — the reading the statement means is now stated: [F4 reviewer] [statement-wrong] Specify whether coordinate cells must be full-dimensional or may be degenerate, and whether disjoint interiors alone or intersection containment in both boundaries is intended; Lean permits degeneracy while imposing the latter condition. Orchestrator ruling: follow the exact binding pin-164: closed coordinate boxes cover the cube and distinct cells have disjoint interiors; shared boundaries are allowed. No strict-positive side-length condition is stated by that definition, so allow degenerate boxes there. For degenerate boxes, intersection containment in BOTH frontiers is stronger than disjoint interiors, so remove that extra requirement from the Lean general property and encode disjoint interiors directly. The repaired Hilbert lemma and Holder proof use the standard DYADIC collection, where side lengths are positive and both properties hold; preserve that existing specialization and its proved construction. Do not add a new assumption to the measure-representation result. This definition of the paper is used by lem:hilbert, prop:holder-curve. Repair it by replacing its own printed text; when the paper states it in running prose or inside a result's statement (no units of its own), restate each result using it instead (in that result's own repair when it has a finding, else a patch with `consequence_of`) and give this node a patch with no ops (`gives_up` null: what a restatement gives up is stated on the result). - note on `def:homeomorphisms`: pinned reading: $f\in\mathcal{H}(K_1,K_2)$ means that $f:K_1\rightarrow K_2$ is bijective and both $f$ and $f^{-1}:K_2\rightarrow K_1$ are continuous. (The preliminary notation calls these maps homeomorphisms; the discussion following Assumption Mixed Distribution uses their inverses to transform the entire support components into unit cubes.) - note on `def:intervention`: pinned reading: Under $\boldsymbol{T}=\boldsymbol{t}$, every unreplaced equation is evaluated in topological order at the intervened values of its parents, with the original latent law unchanged; for canonical categorical nodes the unreplaced equation is the categorical branch of (\ref{eq:structure_equation}). (The intervention passage explicitly calls these the structural equations of the intervened model and defines the resulting variables as their values under intervention. The later canonical definition specifies the categorical form of those structural equations.) - note on `def:lemma-gumbel-err-softmax`: pinned reading: In the displayed softmax formula, $X_k$ means $X_k^\tau$, $\mu^\tau=P(X^\tau)$, and $X^0=\lim_{\tau\downarrow0}X^\tau$ almost surely with the same i.i.d. standard Gumbel variables $G_1,\ldots,G_n$ used for every $\tau$. (Lemma~\ref{lemma:gumbel_err} calls these objects distributions and its proof transports each softmax vector to its maximizing coordinate, determined by the same Gumbel scores.) - note on `def:neural-causal-class`: pinned reading: Membership in $\text{NCM}_{\mathcal G}(\mathcal F_0,\mathcal F_1)$ requires $\hat f_i\in\mathcal F_0$ for every observed node and $\hat g_j\in\mathcal F_1$ for every latent component, with the compatible input and output dimensions understood separately for each network. (The source paragraph explicitly omits input and output dimensions and describes separate structural approximants and latent generators in equation~\ref{eq:structure_equation_hat}.) - note on `def:relu-networks`: pinned reading: A network in $\mathcal{NN}_{k_1,k_2}(W,L)$ has $L$ hidden ReLU layers of width at most $W$, followed by an affine output layer of dimension $k_2$. (The owner's binding convention in ass:pin-owner-3 explicitly fixes the layer count and hidden-layer widths.) - note on `def:thm-consistency-model-parameterization`: pinned reading: The coordinates $P(U_j)$ in $\theta_{\mathcal M}$ are the population laws of the non-Gumbel latent variables, not necessarily uniform laws; the independent standard Gumbel laws are fixed and omitted from this population tuple. (Immediately after the tuple, the population parameter space contains factors $\mathcal P([-K,K]^{d_j^U})$. The canonical representation supplies the categorical Gumbel laws separately.) - note on `def:truncated-neural-class`: pinned reading: Truncation means coordinatewise clipping of the final output to $[-K,K]$; in $\mathcal{NN}^{\infty,K}$ no finite Lipschitz bound is imposed. (The defining minimum and maximum and the word ``Similarly'' specify the operation; the binding neural-class symbol conventions fix its application to the generator output.) - note on `lem:combin-err`: pinned reading: For every $k\in[n]$, $\mu_k$ and $\hat\mu_k$ are probability measures on $\mathbb R^d$. (Lemma~\ref{lemma:combin_err} treats each component as a sampling law in its expectation calculation and applies the probability-law Wasserstein dual formula.) - note on `lem:consistency-fournier-theorem-2`: cited source checked: source supplied by the user at discovery/cited_sources/fournier2015rate.txt - note on `lem:deep-nn-yarotsky-theorem-2`: cited source checked: source supplied by the user at discovery/cited_sources/Yarotsky2018OptimalAO.txt - note on `lem:gumbel-err`: pinned reading: The gap event is $E_\delta=\{\exists\,1\leqslant i0$ is a locally specified positive parameter, used as an overlap lower bound, an approximation tolerance, an auxiliary separation parameter, or a quantization parameter." (The general local-parameter meaning remains valid. The net used in the empirical Wasserstein proof now has radius $\delta/2$, rather than the radius $\delta$ recorded on file.) - note on `symbol \epsilon`: symbol table corrected: type "approximation-error functional or local tolerance or floor parameter", space "$[0,\infty]$ for the approximation-error functional; $[0,\infty)$ for local scalar parameters, with strict positivity where specified.", def "$\epsilon(\mathcal{F}_1,\mathcal{F}_2)=\sup_{f_2\in\mathcal{F}_2}\inf_{f_1\in\mathcal{F}_1}\\|f_2-f_1\\|$ for function classes $\mathcal{F}_1,\mathcal{F}_2$; a bare $\epsilon$ is a locally quantified scalar tolerance or floor parameter." (The recorded positive tolerance omits the approximation-error functional and the nonnegative floor parameter now used in the structural approximation proof. Existing tolerance scopes remain valid, with the additional local uses specified here.) - note on `symbol \epsilon_n`: symbol table corrected: def "$\epsilon_n$ denotes a positive approximation tolerance indexed by $n$, with its role specified locally." (The local approximation and deviation meanings remain as previously pinned. Remove only the two clauses asserting the representation construction and its $C_ga_n$ bound, which the current texts do not establish as definitions of this symbol.) - note on `symbol \gamma_i`: symbol table corrected: def "$\gamma_i:[0,1]\rightarrow[0,1]^{d_i^C}$ satisfies $(\gamma_i)_{\#}\lambda=(g_2^i)^{-1}_{\#}\mu_i$, where $\mu_i=\mathbb{P}_{C_i}/\mathbb{P}(C_i)$ and $\lambda$ is Lebesgue measure on $[0,1]$; for $d_i^C\geqslant1$, it is $1/d_i^C$-H\"older continuous." (The vague transformed-distribution description omits normalization, the chosen pullback map, the exact pushforward identity, and the stated regularity. The component measure represented is a probability law obtained by dividing by $\mathbb{P}(C_i)$.) - note on `symbol \hat{\boldsymbol{U}}`: symbol table corrected: space "The product of the component latent spaces together with $\prod_{V_i\ \text{categorical}}\mathbb{R}^{n_i}$ for the categorical Gumbel vectors.", def "$\hat{\boldsymbol{U}}$ is the full latent collection of the comparison canonical SCM, comprising one latent variable for each $C^2$-component of $\mathcal{G}$ and a Gumbel vector for each categorical observed node." (The full canonical latent collection, including categorical Gumbel vectors, remains the correct definition. The obsolete grouping scope is removed without imposing that construction on the arbitrary comparison model.) - note on `symbol \hat{\mathcal{F}}`: symbol table corrected: space "A class of structural functions on the node-specific parent and latent input domains, with continuous-node or categorical-propensity output spaces.", def "$\hat{\mathcal{F}}$ is the function class in $\mathcal{M}(\mathcal{G},\hat{\mathcal{F}},\hat{\boldsymbol{U}})$: the structural functions satisfy $\hat{f}_i\in\hat{\mathcal{F}}$ for $i\in[n_V]$." (The recorded space is a set of individual maps with a single unspecified domain and codomain, rather than a function class accommodating the node-specific structural maps. The comparison class is not inherently a neural or Lipschitz class.) - note on `symbol \hat{\mathcal{M}}`: symbol table corrected: space "The canonical SCM class on $\mathcal{G}$ specified locally.", def "$\hat{\mathcal{M}}$ denotes a second canonical SCM with causal graph $\mathcal{G}$, with its structural functions and latent variables specified in the surrounding construction or optimization." (The recorded class is too narrow for the symbol's several local uses. The existing readings remain applicable; the additional scope records the continuous output bound and population structural class explicitly specified in Corollary~\ref{cor:rate}, alongside the retained categorical restrictions.) - note on `symbol \hat{\mu}`: symbol table corrected: space "Probability measures on the stated Euclidean space." (The comparison-measure definition is correct, but the recorded space imposes a compact set $K$ even on the product lemma. Compact support and the product definition belong to separate scopes.) - note on `symbol \hat{\mu}_i`: symbol table corrected: space "Probability measures on the respective Euclidean component spaces.", def "$\hat{\mu}_i$ is the comparison component measure paired with $\mu_i$." (The general comparison-component meaning remains valid, but the generator proofs additionally define an explicit push-forward law. That construction belongs in those scopes rather than in the general definition.) - note on `symbol \hat{f}`: symbol table corrected: def "$\hat{f}$ denotes the function paired with $f$ in the local comparison; its structural-function, neural-approximation, or population-regression meaning is specified by the local definition." (The comparison-or-neural-approximation entry omits the population-regression meaning. The existing general fix and regression clause remain applicable; only the obsolete approximation clause is removed.) - note on `symbol \hat{g}`: symbol table corrected: type "random neural generator", def "The displayed generator is $\hat{g}=\hat{g}_3^\tau\circ\hat{g}_2\circ\hat{g}_1$, with separable ReLU blocks $\hat{g}_1,\hat{g}_2$ followed by the Gumbel-softmax mixing layer." (The recorded entry still omits the three-block construction and its Gumbel randomness, so its existing fix remains necessary. The empirical reading additionally distinguishes network representations having equal population laws, since their evaluations use common exogenous inputs.) - note on `symbol \hat{g}_1`: symbol table corrected: def "The first block has the separable form $\hat{g}_1(x_1,\ldots,x_{N_C})=(\hat{g}_1^1(x_1),\ldots,\hat{g}_1^{N_C}(x_{N_C}))$, where $\hat{g}_1^j\in\mathcal{NN}_{1,d_j^C}(W_1,L_1)$ for $j\in[N_C]$." (The separable-block definition remains necessary. Its pullback-law scope now covers both distribution-generator results and specifies the zero-dimensional output.) - note on `symbol \hat{g}_1^{i,j}`: symbol table corrected: type "deep ReLU network", def "$\hat{g}_1^{i,j}$ has width $12$ and depth $L_1$ and satisfies $\\|\hat{g}_1^{i,j}-(\gamma_i)_j\\|_{\infty}\leqslant O(L_1^{-2/d_i^C})$." (The entry omits the specified ReLU architecture and approximation bound. The coordinate target determines $j\in[d_i^C]$; the subsequent displayed assembly ending at $d$ is inconsistent when $d\neq d_i^C$, but does not give the individual coordinate network another meaning.) - note on `symbol \hat{g}_1^i`: symbol table corrected: type "ReLU neural network", def "$\hat{g}_1^i:[0,1]\rightarrow\mathbb{R}^{d_i^C}$ is the first component generator block, chosen to approximate the pullback component law $(g_2^i)^{-1}_{\#}\mu_i$ by the law of $\hat{g}_1^i(U_i)$." (The general network definition and existing wide, deep, and singleton scopes remain valid. The extracted lemma additionally specifies the quantized construction and its source parameters.) - note on `symbol \hat{g}_2`: symbol table corrected: def "The second block has the separable form $\hat{g}_2(x_1,\ldots,x_{N_C})=(\hat{g}_2^1(x_1),\ldots,\hat{g}_2^{N_C}(x_{N_C}))$, where $\hat{g}_2^j\in\mathcal{NN}_{d_j^C,d}(W_2,L_2)$ for $j\in[N_C]$." (The separable-block definition remains necessary. Its component approximation scope now covers both distribution-generator results and explicitly specifies singleton outputs.) - note on `symbol \hat{g}_2^{i,j}`: symbol table corrected: type "deep ReLU network", def "For $d_i^C\geqslant1$, $\hat{g}_2^{i,j}$ has width $\Theta(d_i^C)$ and depth $L_2$ and satisfies $\\|(g_2^i)_j-\hat{g}_2^{i,j}\\|_{\infty}\leqslant O(L_2^{-2/d_i^C})$; for $d_i^C=0$, it is the constant affine coordinate map to the singleton $C_i$." (The positive-dimensional approximation definition remains applicable, but its width and inverse-dimension exponent must not be applied to singleton components.) - note on `symbol \hat{g}_3^\tau`: symbol table corrected: def "The layer is $\hat{g}_3^\tau(x_1,\ldots,x_{N_C})=\sum_{k=1}^{N_C}w_k^\tau x_k$, where $x_k\in\mathbb{R}^d$ and $w_k^\tau=\exp((\log p_k+G_k)/\tau)/\sum_{j=1}^{N_C}\exp((\log p_j+G_j)/\tau)$." (The explicit weighted-sum definition remains necessary. The component-mass reading also applies to the separately stated distribution-generator result.) - note on `symbol \hat{V}_i`: symbol table corrected: def "For $i\in[n_V]$, $\hat{V}_i$ is generated by the structural functions and latent inputs of $\hat{\mathcal{M}}$: $\hat{V}_i=\hat{f}_i(\text{Pa}(\hat{V}_i),\hat{\boldsymbol{U}}_{V_i})$ for continuous variables, and $\hat{V}_i=\arg\max_{k\in[n_i]}\{\hat{g}_k^{V_i}+\log(\hat{f}_i(\text{Pa}(\hat{V}_i),\hat{\boldsymbol{U}}_{V_i}))_k\}$ for categorical variables." (The recorded label omits the structural generation formula, so the existing definition fix remains necessary. The general reading remains unchanged, and the existing joint scoped clause is explicitly retained alongside the additional categorical-generation conventions.) - note on `symbol \lambda`: symbol table corrected: type "context-dependent regularization parameter or reference measure", space "$(0,\infty)$ for regularization; the space of measures on the stated reference domain for push-forwards", def "$\lambda$ denotes a positive regularization parameter in $S_\lambda$ and $W_{1,\lambda}$, and a reference measure in push-forward and measure-domination expressions." (The scalar and reference-measure meanings remain settled. The previous scoped clause attributed a lower-bound premise to Theorem~\ref{thm:wide_nn}; that qualification now belongs only to thm:cor-fix-3c402872, whose existing clause remains applicable.) - note on `symbol \mathbb{P}`: symbol table corrected: space "Probability measures on the measurable space of the local distribution or events.", def "$\mathbb P$ denotes the probability measure in the local context; in the distribution-generator construction it is the target distribution." (The local probability-measure meanings remain valid. The counterexamples now explicitly distinguish a non-Gumbel latent law from its normalized component laws, to which the lower-bound assumption applies.) - note on `symbol \mathcal{C}`: symbol table corrected: type "context-dependent collection of causal models or cubes", space "Collections of SCMs or finite collections of axis-aligned cubes, as locally specified.", def "$\mathcal{C}$ is a collection of causal models in (\ref{prob:pid}); in the geometric construction, $\mathcal{C}=\{C_1,\ldots,C_m\}$ with $C_k=\prod_{i=1}^{n}[a_i^k,b_i^k]$ is a collection of cubes." (The causal-model-only entry misses the geometric reuse. The existing general reading and the clause for prop:holder-curve remain valid; the obsolete scope is removed.) - note on `symbol \mathcal{F}`: symbol table corrected: space "A set of functions with the domains and codomains specified locally.", def "$\mathcal{F}$ denotes a function class; in $\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})$ it is the class to which the structural functions $f_i$, $i\in[n_V]$, belong." (The local function-class meaning remains valid. Its empirical Wasserstein scope additionally requires the stated jointly Borel parameterization, which the reading on file omits.) - note on `symbol \mathcal{F}_{0,n}`: symbol table corrected: def "$\mathcal{F}_{0,n}=\mathcal{NN}^{\hat{L}_f,K}\left(W_{0,n},\Theta(\log d_{\max}^{\text{in}})\right)$, where $\hat{L}_f=\sqrt{d_{\max}^{\text{in}}d_{\max}^{\text{out}}}L_f$ and $\mathcal{NN}_{d_1,d_2}^{L_f,K}(W,L)=\left\{\max\{-K,\min\{f,K\}\}:f\in\mathcal{NN}_{d_1,d_2}(W,L),\text{Lip}(f)\leqslant L_f\right\}$." (The vague recorded definition still requires the explicit architecture formula. The class itself retains coordinatewise clipping; continuous candidates in the optimization additionally satisfy the uniform output $1$-norm bound.) - note on `symbol \mathcal{F}_{1,n}`: symbol table corrected: def "$\mathcal{F}_{1,n}=\mathcal{NN}^{\infty,K}(\Theta(d_{\max}^{U}),L_{1,n},\Theta((d_{\max}^{U})^2),L_{2,n},\tau_n)$, where $\mathcal{NN}^{\infty,K}$ is the truncated version of the specified generator architecture." (The explicit architecture fix still applies. The former ambiguity grade omits the now binding convention that component counts and cube dimensions remain fixed along the sequence.) - note on `symbol \mathcal{F}_{V_i}`: symbol table corrected: def "$\mathcal{F}_{V_i}=\mathcal{F}_L^K([-K,K]^{d_{i,\text{in}}^V},[-K,K]^{d_{i,\text{out}}^V})$ for continuous $V_i$, and $\mathcal{F}_{V_i}=\{f:\\|f\\|_1=1,\ f\in\mathcal{F}_L^K([-K,K]^{d_{i,\text{in}}^V},[-K,K]^{d_{i,\text{out}}^V})\}$ for categorical $V_i$, with $\mathcal{F}_L^K=\{f:\sup_x\\|f(x)\\|_1\leqslant K,\ \operatorname{Lip}(f)\leqslant L\}$." (The vague recorded class omits its defining branches and bounds. The earlier fix used the proof's infinity-norm display without a specified Lipschitz bound; the current statement specifies a uniform output 1-norm bound and an explicit Lipschitz bound.) - note on `symbol \mathcal{F}_1`: symbol table corrected: space "$\{\text{function classes}\}$", def "The notation $\mathcal{F}_1$ denotes a function class, with its role and domain determined by context." (The general function-class correction and latent-generator scopes remain applicable. The obsolete Wasserstein-proof scope is explicitly dropped.) - note on `symbol \mathcal{G}`: symbol table corrected: type "causal graph or function class, according to context", space "Causal ADMGs on $\boldsymbol V$, or classes of functions $X\rightarrow[0,1]$, according to context.", def "$\mathcal G$ denotes the locally specified causal graph; in the empirical covering and deviation citation it denotes a nonempty class of functions from a set $X$ into $[0,1]$. The covering and packing bound assumes finite pseudo-dimension $v$; the deviation bound imposes the stated measurable parameterization instead." (The graph-only entry misses the function-class use. The existing deviation-bound clause is now incomplete: the class must have the specified jointly Borel parameterization on standard Borel spaces.) - note on `symbol \mathcal{G}_{\mathcal{M}}`: symbol table corrected: def "$\mathcal{G}_{\mathcal{M}}$ extends $\mathcal{G}_0$ by adding bi-directed arrows between $V_i,V_j$ if there exists a correlated latent variable pair $(U_k,U_l)$ with $U_k\in\boldsymbol{U}_{V_i}$ and $U_l\in\boldsymbol{U}_{V_j}$." (The recorded definition refers vaguely to specified arrows without recording their defining criterion. The SCM definition explicitly gives the correlated-latent-pair rule, including the shared-latent interpretation explained immediately afterward.) - note on `symbol \mathcal{H}`: symbol table corrected: space "$\{f:K_1\to K_2\}$, with domain and codomain supplied by the local definition", def "$\mathcal{H}$ denotes the locally specified function class; $\mathcal{H}(K_1,K_2)$ is the set of homeomorphisms from $K_1$ to $K_2$." (The homeomorphism-only entry misses the explicitly defined local Lipschitz class. Its printed vector-valued definition is retained as written; the changed statement does not replace that proof definition.) - note on `symbol \mathcal{H}_\delta`: symbol table corrected: space "$2^{\mathcal H}$", def "$\mathcal H_\delta$ is a finite $\delta/2$-net in the uniform norm of $\mathcal H=\{h:[-K,K]^d\rightarrow\mathbb R:h(0)=0,\ \|h(x)-h(y)\|\leqslant\\|x-y\\|\}$." (The proof now specifies scalar-valued test functions and a net of radius $\delta/2$. The vector-valued class and radius recorded on file no longer describe its definition.) - note on `symbol \mathcal{I}_n`: symbol table corrected: def "$\mathcal{I}_0=\{[0,1]\}$ and $\mathcal{I}_n=\{I_{O_n(i)}:1\leqslant i\leqslant2^{nd}\}$, where $I_{O_n(i)}=[\sum_{j=1}^{i-1}p_{n,j},\sum_{j=1}^{i}p_{n,j}]$ and $p_{n,j}=\mathbb{P}(C_{O_n(j)})$; each parent interval is the union of its $2^d$ ordered child intervals." (The recorded entry omits the defining interval lengths. The grade on file incorrectly imposes equal lengths and a quantile change of parameter; the printed construction uses cumulative target masses directly.) - note on `symbol \mathcal{M}`: symbol table corrected: type "structural causal model notation, also used for a canonical SCM class", space "$\mathcal{C}\ \text{or}\ \mathcal{P}(\mathcal{C})$", def "An individual SCM is $\mathcal{M}=(\boldsymbol{V},\boldsymbol{U},F,P(\boldsymbol{U}),\mathcal{G}_0)$. Given a function class $\mathcal{F}$, the SCM class $\mathcal{M}(\mathcal{G},\mathcal{F},\boldsymbol{U})$ consists of all canonical SCM models with causal graph $\mathcal{G}$ such that $f_i\in\mathcal{F}$, $i\in[n_V]$." (The individual-SCM and SCM-class distinction still requires the existing fix. The previous scope names a different population value and class and uses an infinity-norm bound; the current statement instead uses $\underline F^{\hat L_f}$ and the output 1-norm bounded class $\mathcal F_L^K$.) - note on `symbol \mathcal{M}_\epsilon`: symbol table corrected: space "$\{\text{SCMs}\}$" (The paper does not assert membership in the unspecified introductory search class $\mathcal{C}$. The model varies with the tolerance; the regularity assumptions stated for $\mathcal{M}^*$ are not thereby imposed on this family.) - note on `symbol \mathcal{N}`: symbol table corrected: type "covering number or local causal-model variable", space "Covering numbers in $\mathbb{N}\cup\{\infty\}$, or causal models in the locally specified class.", def "$\mathcal{N}(\epsilon,S,\\|\cdotp\\|_1)$ is the covering number of $S$ in the $\ell_1$ norm; a standalone $\mathcal{N}\in\{\mathcal{M},\hat{\mathcal{M}}\}$ denotes either of the two causal models." (The existing distinction between covering numbers and local causal models remains valid. The separate concentration bound additionally requires its averaged metric and doubled evaluation count.) - note on `symbol \mathcal{NN}`: symbol table corrected: type "overloaded ReLU network and three-part generator class", space "$\{\text{ReLU networks or three-part neural generators}\}$", def "$\mathcal{NN}_{k_1,k_2}(W,L)$ is the set of ReLU networks with input dimension $k_1$, output dimension $k_2$, width $W$ and depth $L$; $\mathcal{NN}(W_1,L_1,W_2,L_2,\tau)$ denotes generators $\hat{g}=\hat{g}_3^\tau\circ\hat{g}_2\circ\hat{g}_1$, where $\hat{g}_i=(\hat{g}_i^1(x_1),\ldots,\hat{g}_i^{N_C}(x_{N_C}))$, $\hat{g}_1^j\in\mathcal{NN}_{1,d_j^C}(W_1,L_1)$, $\hat{g}_2^j\in\mathcal{NN}_{d_j^C,d}(W_2,L_2)$ for $j\in[N_C]$, and $\hat{g}_3^\tau$ is the Gumbel-Softmax layer with temperature $\tau>0$." (The notation denotes both ordinary ReLU networks and the composite generator class. The additional scoped conventions specify absolute width factors while retaining the existing temperature and zero-dimensional readings.) - note on `symbol \mathcal{NN}(W_1,L_1,W_2,L_2,\tau)`: symbol table corrected: def "The class consists of generators $\hat{g}=\hat{g}_3^\tau\circ\hat{g}_2\circ\hat{g}_1$ with separable blocks $\hat{g}_i(x_1,\ldots,x_{N_C})=(\hat{g}_i^1(x_1),\ldots,\hat{g}_i^{N_C}(x_{N_C}))$, $i=1,2$, where $\hat{g}_1^j\in\mathcal{NN}_{1,d_j^C}(W_1,L_1)$ and $\hat{g}_2^j\in\mathcal{NN}_{d_j^C,d}(W_2,L_2)$ for $j\in[N_C]$, and $\hat{g}_3^\tau$ is the Gumbel-softmax mixing layer." (The recorded architecture omits its defining component networks. The existing zero-dimensional convention remains in force; the additional scoped readings specify the absolute width factors and the suppressed temperature.) - note on `symbol \mathcal{NN}^{\infty,K}`: symbol table corrected: def "$\mathcal{NN}^{\infty,K}(W_1,L_1,W_2,L_2,\tau)=\{T_K\hat{g}:\hat{g}\in\mathcal{NN}(W_1,L_1,W_2,L_2,\tau)\}$, where $T_K\hat{g}=\min\{K,\max\{-K,\hat{g}\}\}$ coordinatewise." (The recorded definition leaves the truncation operation unspecified. The paper defines output clipping, rather than clipping each intermediate block, and the superscript $\infty$ imposes no finite Lipschitz restriction.) - note on `symbol \mathcal{P}`: symbol table corrected: type "space of probability measures", def "$\mathcal{P}(K)$ denotes the set of probability measures on $K$." (This is a space whose elements are probability measures, rather than a probability space carrying random outcomes. Its arguments in the parameter product are the compact latent-coordinate cubes.) - note on `symbol \mu`: symbol table corrected: space "The space of measures on the locally stated domain, including finite probability vectors $\Delta_n$", def "$\mu$ denotes the measure specified locally; in the preliminary push-forward notation it is a measure on $\mathbb{R}^{d_1}$." (The local measure meanings and existing scoped clauses remain valid. Only the evidence concerning the separated covering citation changes.) - note on `symbol \mu_i`: symbol table corrected: type "component probability measure or probability-vector coordinate", space "Probability measures on the locally specified component space, or $[0,1]$ for a probability-vector coordinate.", def "$\mu_i$ denotes the locally specified component probability measure, or the $i$th coordinate of $\mu\in\Delta_n$." (The existing local measure and coordinate meanings remain valid. The extracted generator lemma uses the normalized component law.) - note on `symbol \mu_n`: symbol table corrected: type "context-dependent measure", space "The space of measures on the locally stated domain", def "$\mu_n$ denotes either the component measure with index $n$ in a list $\mu_1,\ldots,\mu_n$, or the empirical distribution of $\mu$, according to the local definition." (The component and empirical meanings remain unchanged. The evidence is updated because the covering citation no longer specifies observations from a probability distribution.) - note on `symbol \tau_n`: symbol table corrected: type "local temperature or additive tolerance sequence", space "$\mathbb{R}$, with $\tau_n>0$ in temperature uses", def "$\tau_n$ denotes the generator temperature or the additive tolerance in $\\|g_n(\theta_1)-g_n(\theta_2)\\|\leqslant L_gd(\theta_1,\theta_2)+\tau_n$, according to its local use." (The temperature-only entry misses the separate additive-tolerance use. The existing scopes remain valid, but the printed auxiliary argument also requires its own scope.) - note on `symbol \theta`: symbol table corrected: type "context-dependent causal functional or parameter", space "$\mathcal{C}\to\mathbb{R}$ in (\ref{prob:pid}); $M$ in the population optimization and model-parameter notation; network representations in the empirical optimization", def "$\theta(\mathcal{M})$ is the causal quantity of interest in (\ref{prob:pid}); elsewhere $\theta$ denotes a parameter point, including $\theta_{\mathcal{M}}=(f_1,\ldots,f_{n_V},P(U_1),\ldots,P(U_{n_U}))$ in the population causal-model parameterization. Empirical candidates are indexed by network representations, which need not be determined by their population parameter tuples." (The recorded causal-functional entry misses the later parameter meanings. The existing general reading and other scoped clauses remain valid; the clause for Corollary~\ref{cor:rate} must also distinguish empirical network representations from their population parameter tuples.) - note on `symbol \theta^*`: symbol table corrected: type "context-dependent parameter point", def "$\theta^*$ denotes a minimizer of (\ref{eq:opt*}) in the abstract optimization proof, and the parameter representing the underlying SCM $\mathcal{M}^{\theta^*}$ in the causal-model parameterization." (The existing distinction between an abstract optimization minimizer and the true-model parameter remains necessary. The rate result imports the latter meaning, which needs its own scope.) - note on `symbol B`: symbol table corrected: type "locally specified measurable test set or positive scalar bound", space "Measurable subsets of the locally specified measure space, or $(0,\infty)$.", def "$B$ denotes a locally specified measurable test set or scalar bound: in $f_\#\mu(B)=\mu(f^{-1}(B))$ it is a measurable subset of the codomain of $f$, in a lower-bound premise it is a measurable subset of the parameter cube, and in a proof estimate it is the locally defined positive scalar bound." (The recorded cube-set meaning omits the other local meanings already identified on file. The separately stated distribution approximation proof additionally uses $B$ as a fixed scalar clipping bound.) - note on `symbol c`: symbol table corrected: type "locally specified scalar, categorical configuration, or scalar-valued function", space "The locally specified scalar space, finite categorical configuration space, or real-valued function space.", def "$c$ is local notation for a scalar constant, a categorical configuration in conditioning expressions, or the function $c(t)=\int\hat f_y(a,t)\nu_A(da)-\int\hat f_y(a,t)\mu_A(da)$." (The counterexample gap is only one local meaning. The rate proof additionally uses categorical configurations and a concentration constant; the existing other scopes remain applicable.) - note on `symbol C`: symbol table corrected: type "context-dependent node set, geometric set, numerical constant, or categorical label vector", space "$2^{\boldsymbol{V}}$ in causal-graph notation; subsets of the ambient Euclidean space in geometric and support-component notation; $(0,\infty)$ for numerical constants $C>0$; $\prod_{i:V_i\text{ is categorical}}[n_i]$ for the complete categorical label vector", def "$C$ denotes a $C^2$-component in causal-graph notation; in geometric notation it denotes a cube, and $C_i$ can denote a connected component of a measure's support; in numerical bounds $C>0$ denotes a constant; in the latent empirical concentration argument, $C$ denotes the complete categorical label vector." (The entry on file covers graph components, geometric sets and constants, but omits the complete categorical label vector used in latent empirical concentration. This additional local meaning coexists with the existing meanings in the consistency proof; the three overlapping clauses are explicitly retained.) - note on `symbol C_{\mathcal{G}}(L,K)`: symbol table corrected: def "$C_{\mathcal G}(L,K;\{d_i,n_i\})$ is the intervention comparison constant that only depends on $L,K$, the causal graph $\mathcal G$, the continuous output dimensions $d_i$, and the categorical cardinalities $n_i$." (The recorded dependence and the general reading on file omit the output dimensions and categorical cardinalities now explicitly permitted. The printed proof's old coefficient cannot be adopted as the definition of the newly specified coefficient.) - note on `symbol C_f`: symbol table corrected: def "$C_f>0$ satisfies $\mathbb{P}(f(B))\geqslant C_f\lambda(B)$ for every measurable $B\subset[0,1]^d$, where $f\in\mathcal{H}([0,1]^d,K)\cap\mathcal{F}_L([0,1]^d,K)$ and $\lambda$ is Lebesgue measure on $[0,1]^d$." (The recorded label omits the defining domination inequality. The assumption chooses the constant for the measure and homeomorphism under consideration; it does not specify a uniform constant across different measures or choices of homeomorphism.) - note on `symbol C_i`: symbol table corrected: type "support component or partition cell", def "$C_i$ denotes a connected support component in the distribution-generator construction, or a cube in the partition construction." (The existing support-component and partition-cell meanings remain valid. Both counterexamples additionally use each component as the support set for the lower-bound assumption on its normalized component law.) - note on `symbol C_W`: symbol table corrected: def "$C_W$ denotes the comparison coefficient specified in the applicable bound." (The recorded formula and the scoped formula on file omit the new bound B. The proposition uses B in both terms of C_W; the lemma retains its separate formula using K.) - note on `symbol d`: symbol table corrected: type "local dimension, metric, or regression-difference function", space "$\mathbb Z_{\geqslant0}$ for dimension uses; metrics $M\times M\rightarrow[0,\infty)$ for metric uses; real-valued functions on the local covariate domain for regression-difference uses.", def "$d$ denotes the locally specified Euclidean or cube dimension, the metric of a locally specified metric space $(M,d)$, or the proof function $d(s)=\hat f_y(s)-f_y(s)$." (All existing dimension and metric clauses remain. The treatment-comparison proof additionally uses d for a real-valued regression difference, which the previous general reading omitted.) - note on `symbol D`: symbol table corrected: type "context-dependent geometric parameter", space "$[0,\infty)$", def "$D$ denotes the ambient dimension in $K\subset\mathbb R^D$, a locally defined categorical-cardinality factor, or the latent-dimension parameter $\max\{1,D_U\}$ with $D_U=\sum_{j=1}^{n_U}d_j^U$ over non-Gumbel latent variables." (The existing ambient, treatment-comparison, and latent-dimension meanings remain. The approximation proof additionally defines D as the maximum categorical label diameter, which differs from the treatment-comparison factor.) - note on `symbol d_{\mathcal{F}}`: symbol table corrected: def "$d_{\mathcal F}<\infty$ is a strict pseudo-dimension bound for the scalar function classes specified locally." (The strict coordinatewise interpretation remains necessary. The old scalar-citation clause no longer applies because that citation now uses a separately named pseudo-dimension.) - note on `symbol d_1`: symbol table corrected: type "local input dimension or averaged metric", space "$\mathbb N$ for dimension uses; a metric on $\mathbb R^m$ for the averaged covering-number use.", def "$d_1$ denotes a locally specified input or source dimension, or the averaged metric $d_1(u,w)=\frac1m\sum_{i=1}^m\|u_i-w_i\|$." (The dimension and metric overload remains settled. The separate squared-loss citation additionally uses the averaged metric with denominator $N=2m$, rather than the number $m$ of sampled pairs.) - note on `symbol F`: symbol table corrected: type "tuple of structural functions, causal objective function, or measurable map, according to context", space "The structural-function tuple space, $\mathbb{R}^{n_V}\to\mathbb{R}$ for the causal objective, or the map space specified by the local push-forward definition.", def "In the SCM definition, $F=(f_1,\cdots,f_{n_V})$. In the optimization objectives, $F$ is evaluated as $F(V_1(t),\cdots,V_{n_V}(t))$. In push-forward uses, $F$ is the map specified locally." (The recorded tuple-only type and singleton-product space cannot describe the scalar objective or either push-forward-map use. These are explicit local overloads, not an unresolved choice of meaning; the printed scalar objective domain is retained without silently replacing it by a higher-dimensional product.) - note on `symbol f_i`: symbol table corrected: space "Functions on the joint value space of $\text{Pa}(V_i)$ and $\boldsymbol{U}_{V_i}$, with codomain $\mathbb{R}^{d_i}$ for continuous variables, $[n_i]$ in (\ref{eq:structure_eq}) for categorical variables, and $\mathbb{R}^{n_i}$ under the categorical propensity convention.", def "In (\ref{eq:structure_eq}), $V_i=f_i(\text{Pa}(V_i),\boldsymbol{U}_{V_i})$. Under the categorical convention, $V_i=\arg\max_{k\in[n_i]}\{g_k^{V_i}+\log(f_i(\text{Pa}(V_i),\boldsymbol{U}_{V_i}))_k\}\sim\text{Categorical}(f_i/\\|f_i\\|_1)$." (The recorded space confuses parent and latent collections with their value spaces and omits the categorical-label codomain of the preliminary structural equation; the categorical convention instead uses propensity vectors. The existing scoped readings remain applicable, with the consistency population class and the continuous true and candidate functions now subject to the specified uniform output $1$-norm bound.) - note on `symbol f_k`: symbol table corrected: type "locally indexed map", space "A node-specific structural-function space or the partition-map space $\{f:\mathcal{I}_k\to\mathcal{C}_k\}$, according to scope.", def "$f_k$ denotes a structural or categorical-propensity function at node $V_k$ in SCM arguments, and the map $f_k(I_{O_k(i)})=C_{O_k(i)}$, $i=1,\ldots,2^{kd}$, in the partition construction." (The paper still overloads $f_k$ between structural functions and a partition correspondence, so the general fix remains applicable. The obsolete partition-result clause is dropped; the structural clause on file remains applicable to both currently listed results.) - note on `symbol g`: symbol table corrected: type "real-valued function with locally specified role", space "Real-valued functions on the locally specified domain.", def "$g$ denotes a locally specified real-valued function, used as a Wasserstein dual test function, a limiting constraint function, a member of the scalar class $\mathcal G$, or a jointly Borel map parametrizing that class." (The limiting-constraint entry omits the other local roles. In the cited probability bound, $g$ now denotes the jointly Borel parameterizing map on $T\times X$, while its sections belong to $\mathcal G$.) - note on `symbol h`: symbol table corrected: type "local auxiliary function or grid spacing", space "Locally specified function spaces, or $(0,\infty)$ for grid spacing.", def "$h$ denotes a locally defined auxiliary function, or the grid spacing $h=2K/R$ in the structural interpolation construction." (The mixing function, entropy, regression auxiliary functions, and grid spacing remain local meanings. The scalar test-function reading belongs to the Wasserstein proof, rather than the independent covering citation.) - note on `symbol K`: symbol table corrected: type "numerical bound or compact support set", space "$(0,\infty)$ for a numerical bound, or a compact subset of the indicated Euclidean space for a support set", def "$K$ denotes the numerical bound specified by a boundedness hypothesis or truncation, or the compact support set specified by a measure hypothesis." (The numerical-bound entry does not cover the compact-set occurrences. The additional scope identifies the compact set with each latent support component and specifies the normalized law and pullback boundary condition; both overlapping clauses on file remain unchanged.) - note on `symbol L`: symbol table corrected: type "Lipschitz bound or neural network depth", def "$L$ is the Lipschitz bound in $\mathcal{F}_L(K_1,K_2)$; in $\mathcal{NN}_{k_1,k_2}(W,L)$ it is the neural network depth." (The existing distinction between Lipschitz bound and network depth still applies. The population-class clause must replace $\mathcal B_L^K$ and $\underline F_\infty^L$ by the retained $\mathcal F_L^K$ and $\underline F^L$, including the full-domain output $1$-norm bound.) - note on `symbol L_{i,n}`: symbol table corrected: def "$L_{1,n}$ and $L_{2,n}$ are the two depth parameters in $\mathcal{F}_{1,n}=\mathcal{NN}^{\infty,K}(\Theta(d^U_{\max}),L_{1,n},\Theta((d^U_{\max})^2),L_{2,n},\tau_n)$." (The recorded generator-block index can be read as indexing latent variables or support components. Here $i=1,2$ distinguishes the two successive generator parts.) - note on `symbol L_g`: symbol table corrected: def "$L_g$ denotes a Lipschitz bound, with its function and value specified locally." (The recorded definition restricts $L_g$ to the approximate constraint sequence. The paper also defines it explicitly for a different function in the Lipschitz proof; these are separate local constants.) - note on `symbol m`: symbol table corrected: type "local integer index, count, vector length, input dimension, or sample size", space "Nonnegative integers for input dimensions and counts of nonintervened nodes; positive integers for vector lengths and sample sizes.", def "$m$ is a local integer parameter, used as a curve index, the number $n_V-n_T$ of nonintervened nodes, the length of $\nu\in\Delta_m$, a structural input dimension, or the sample size and evaluation-point count in empirical covering and concentration bounds." (The entry restricted to a probability-vector length misses the other local roles. The current proofs additionally use $m$ as a possibly zero count of nonintervened nodes and as the positive maximum structural input dimension; the existing scopes remain valid.) - note on `symbol n_U`: symbol table corrected: def "$n_U$ is the number of indexed latent variables $U_1,\ldots,U_{n_U}$; initially $\boldsymbol{U}=\{U_j\}_{j=1}^{n_U}$, while under the categorical convention $\boldsymbol{U}=\{U_1,\ldots,U_{n_U}\}\cup\{G_{V_i}:V_i\text{ is categorical}\}$." (Counting every member of the full latent collection would include the separately adjoined Gumbel vectors. The paper's later indexed sums, bounds, and parameter tuples count the non-Gumbel family.) - note on `symbol P`: symbol table corrected: type "probability-law notation, also used for a locally specified probability measure", def "We use $P(X)$ to represent the distribution of random variable $X$; bare $P$ also denotes a locally specified probability measure." (The law-notation entry misses bare reference measures. The existing deviation-bound reading must additionally specify that $P$ is Borel and that its product-law probability is completed.) - note on `symbol p_i`: symbol table corrected: type "nonnegative weight", space "$[0,\infty)$", def "$p_i$ denotes a nonnegative scalar weight, with its normalization and interpretation specified locally." (The general weight definition remains appropriate. The existing consistency clause covers component masses but omits the categorical propensity coordinates now used in that proof.) - note on `symbol P^{\mathcal{M}}`: symbol table corrected: space "Probability laws of the indicated variables in $\mathcal{M}$.", def "$P^{\mathcal{M}}(X)$ denotes the distribution of the indicated random variable or collection $X$ in SCM $\mathcal{M}$." (The notation applies to the variables in its argument, including latent and intervened variables, rather than exclusively to the observational distribution. The general reading and the clause for Theorem~\ref{thm:approximation} remain unchanged.) - note on `symbol R`: symbol table corrected: type "locally specified shape-ratio functional or intervention bound", space "Shape-ratio functionals on finite collections with positive edge lengths, or $(0,\infty)$.", def "$R$ denotes either the shape-ratio functional $R(\mathcal C)=\max_{i,k}\|a_i^k-b_i^k\|/\min_{i,k}\|a_i^k-b_i^k\|$ for $C_k=\prod_i[a_i^k,b_i^k]$, or a fixed positive bound on continuous intervention values, according to the surrounding passage." (The recorded functional remains the meaning in the cube construction, but does not describe the positive scalar intervention bound introduced in Corollary~\ref{corollary:deep_nn}. These are distinct local meanings.) - note on `symbol s`: symbol table corrected: type "integer depth parameter", space "$\mathbb{N}$", def "$s$ is the integer depth parameter of \cite[Theorem VIII.1]{perekrestenkoHighDimensionalDistributionGeneration2022a}, fixed at $s=1$." (The entry calls $s$ a quantization parameter. The binding reading makes it the integer depth parameter fixed at one; the previous grade also incorrectly identified the quantization parameter with $1/\lceil n(n-1)\rceil$.) - note on `symbol S`: symbol table corrected: type "locally defined approximation error or set to be covered", space "$[0,\infty]$ for approximation errors; subsets of the ambient normed space for covering numbers.", def "$S$ is local notation: $S=\sum_{i=1}^{n_V}\\|f_i-\hat f_i\\|_\infty$ in the structural comparison argument; $S=\\|\hat f_i(x)-f_i(x)\\|_\infty$ in the categorical normalization argument; and $S$ is the set covered in $\mathcal{N}(\epsilon,S,\\|\cdotp\\|_1)$." (The table records only the summed approximation error, although the paper also uses $S$ for a pointwise categorical approximation error and a set to be covered. The categorical passage supplies neither the projected extension nor the additional construction guarantees recorded in the previous scoped reading.) - note on `symbol S_\lambda`: symbol table corrected: def "$S_\lambda(\mu,\nu)=W_{1,\lambda}(\mu,\nu)-\tfrac12W_{1,\lambda}(\mu,\mu)-\tfrac12W_{1,\lambda}(\nu,\nu)$." (The recorded label omits the defining self-comparison subtraction. The existing subtraction formula remains valid; the scoped reading now also records the explicitly specified minus-entropy convention.) - note on `symbol s_n`: symbol table corrected: def "$s_n=O\left(m_n^{-1/(D+2)}\log m_n+\frac{(L_{1,n}+L_{2,n})\log m_n}{\sqrt{m_n}}+n^{-1/\max\{2,D_U\}}\log^2 n+\log(nm_n)\lambda_n\right)$ is the sampling and Sinkhorn approximation tolerance, where $D_U=\sum_{j=1}^{n_U}d_j^U$ includes the non-Gumbel latent variables only and $D=\max\{1,D_U\}$." (The vague recorded label omits the specified tolerance order; the existing fix remains applicable. Corollary~\ref{cor:rate} now explicitly makes the same positive, sufficiently large choice, which requires an additional scoped clause.) - note on `symbol T`: symbol table corrected: type "local treatment variable, transportation map, transportation plan, or parameter space", space "Treatment value spaces, maps $\Delta_n\rightarrow\Delta_n$, nonnegative transportation matrices, or standard Borel spaces, as specified locally.", def "$T$ denotes a treatment variable in the causal passages, the transportation map in the Gumbel-softmax proof, a transportation plan in the Sinkhorn proof, or a standard Borel parameter space for a jointly Borel function family." (The local roles of $T$ remain settled. The previous Sinkhorn clause incorrectly extended the proof's entropy maximum to all feasible plans; its displayed domain consists of unregularized optimal plans.) - note on `symbol U_i`: symbol table corrected: space "The locally specified latent space $\mathbb{R}^{d_i^U}$, or $[0,1]$ for a scalar generator input.", def "$U_i$ denotes a locally indexed latent variable or uniform generator input." (The recorded latent space uses the wrong index and does not cover scalar generator inputs. The additional scope identifies the bounded latent coordinates used in the consistency argument without imposing uniform laws on them or restating the sampling model.) - note on `symbol W`: symbol table corrected: type "overloaded distance functional, network width and network weight count", space "$\bigl(\mathcal{P}\times\mathcal{P}\rightarrow[0,\infty]\bigr)\ \cup\ \mathbb{N}$", def "$W(\mu,\nu)$ denotes the Wasserstein-1 distance for the paper's $1$-norm; in $\mathcal{NN}_{k_1,k_2}(W,L)$ and its truncated variants, $W$ denotes network width; in the approximation expression $a\,\omega(cW^{-2/d_i^{\text{in}}})$, $W$ denotes the number of network weights." (The recorded distance-only entry omits both architectural width and the weight count explicitly used in the scalar approximation expression. The existing distinction between distance and width remains valid; the additional scoped clause distinguishes the weight count from width.) - note on `symbol W_{1,\lambda}`: symbol table corrected: space "$\Delta_n\times\Delta_m\to\mathbb{R}$, with the support points supplied", def "For finite probability measures with weights $\mu\in\Delta_n$, $\nu\in\Delta_m$ on points $x_i,y_j$ in a common Euclidean space, put $c_{ij}=\\|x_i-y_j\\|$, $\Pi(\mu,\nu)=\left\{T\in[0,\infty)^{n\times m}:\sum_jT_{ij}=\mu_i,\ \sum_iT_{ij}=\nu_j\right\}$ and $h(T)=1-\sum_{i,j}T_{ij}\log T_{ij}$, with $0\log0=0$. Then $W_{1,\lambda}(\mu,\nu)=\min_{T\in\Pi(\mu,\nu)}\left\{\sum_{i,j}c_{ij}T_{ij}-\lambda h(T)\right\}$." (The recorded nonnegative range is wrong for the specified entropy-subtracted objective, and the label alone omits its defining formula. Probability weights alone do not determine the functional without their support points.) - note on `symbol X^\tau`: symbol table corrected: def "$X^\tau=(X_1^\tau,\ldots,X_n^\tau)$, where $X_k^\tau=\frac{\exp((\log p_k+G_k)/\tau)}{\sum_{j=1}^n\exp((\log p_j+G_j)/\tau)}$ and $G_k$ are i.i.d. standard Gumbel variables." (The recorded label still omits the defining softmax formula. The general reading remains valid, and the additional uses require identifying the dimension and weights with the target measure's support components.) - note on `symbol Y`: symbol table corrected: type "random variable or random vector", space "The outcome space is $\mathbb{R}$; in the product-coupling notation, the space is the product of the coordinate spaces.", def "$Y$ denotes the outcome in causal uses and the second coordinate of $(T,Y)$ in the regression comparison; in product-coupling notation it denotes the vector with coordinates $Y_k$." (The scalar-outcome entry does not cover the auxiliary vector in the product-measure proof. The different structural equations for outcomes in different SCMs are model specifications, rather than changes to the meaning of an outcome node; the product-coupling use is a separate local use of the notation.) ### V1 lint warnings - thm:wide-nn: thm:wide_nn references \ref{lemma:push-forward_err} (lemma:push-forward_err) but `depends_on` lacks ass:lemma-push-forward-err-lipschitz; V1.5 must confirm whether the paper assumes it or merely mentions it - thm:wide-nn: thm:wide_nn references \ref{lemma:combin_err} (lemma:combin_err) but `depends_on` lacks ass:lemma-combin-err-weights; V1.5 must confirm whether the paper assumes it or merely mentions it - thm:wide-nn: thm:wide_nn references \ref{lemma:gumbel_err} (lemma:gumbel_err) but `depends_on` lacks ass:lemma-gumbel-err-parameters, def:lemma-gumbel-err-softmax; V1.5 must confirm whether the paper assumes it or merely mentions it - thm:wide-nn: thm:wide_nn references \ref{prop:holder_curve} (prop:holder_curve) but `depends_on` lacks def:prop-holder-curve-lebesgue-measure; V1.5 must confirm whether the paper assumes it or merely mentions it - thm:deep-nn: depends on ass:assump-support-components which the environment never references; V1.5 must confirm the paper actually assumes it - thm:deep-nn: depends on ass:assump-support-homeomorphisms which the environment never references; V1.5 must confirm the paper actually assumes it - thm:deep-nn: depends on ass:assumption-lower-bound-domination which the environment never references; V1.5 must confirm the paper actually assumes it - thm:deep-nn: depends on ass:assumption-lower-bound-boundary which the environment never references; V1.5 must confirm the paper actually assumes it - prop:counterexample-foral: prop:counterexample_foral references \ref{thm:approximation} (thm:approximation) but `depends_on` lacks thm:approximation; V1.5 must confirm whether the paper assumes it or merely mentions it - prop:counterexample: prop:counterexample references \ref{thm:approximation} (thm:approximation) but `depends_on` lacks thm:approximation; V1.5 must confirm whether the paper assumes it or merely mentions it - thm:consistency: thm:consistency references \ref{corollary:deep_nn} (corollary:deep_nn) but `depends_on` lacks thm:deep-nn; V1.5 must confirm whether the paper assumes it or merely mentions it - thm:consistency: thm:consistency references \ref{prop:convg_opt} (prop:convg_opt) but `depends_on` lacks ass:prop-convg-opt-setting, ass:prop-convg-opt-compact, ass:prop-convg-opt-approximate-lipschitz, ass:prop-convg-opt-uniform-error, ass:prop-convg-opt-approximation, ass:prop-convg-opt-nonempty, def:prop-convg-opt-problems, def:prop-convg-opt-values; V1.5 must confirm whether the paper assumes it or merely mentions it - thm:consistency: thm:consistency references \ref{lemma:sinkhorn_approx} (lemma:sinkhorn_approx) but `depends_on` lacks def:lemma-sinkhorn-approx-regularized-notation; V1.5 must confirm whether the paper assumes it or merely mentions it - thm:consistency: thm:consistency references \ref{lemma:combin_err} (lemma:combin_err) but `depends_on` lacks ass:lemma-combin-err-weights; V1.5 must confirm whether the paper assumes it or merely mentions it - thm:consistency: thm:consistency references \ref{lemma:sup_wasserstein} (lemma:sup_wasserstein) but `depends_on` lacks ass:lemma-sup-wasserstein-pseudodimension, ass:lemma-sup-wasserstein-bounded; V1.5 must confirm whether the paper assumes it or merely mentions it - thm:consistency: depends on ass:assump-independence-u which the environment never references; V1.5 must confirm the paper actually assumes it - thm:consistency: depends on ass:assump-categorical-rv-latents which the environment never references; V1.5 must confirm the paper actually assumes it - thm:consistency: depends on ass:assump-categorical-rv-generation which the environment never references; V1.5 must confirm the paper actually assumes it - thm:consistency: depends on ass:assump-support-components which the environment never references; V1.5 must confirm the paper actually assumes it - thm:consistency: depends on ass:assump-support-homeomorphisms which the environment never references; V1.5 must confirm the paper actually assumes it - thm:consistency: depends on ass:assumption-lower-bound-domination which the environment never references; V1.5 must confirm the paper actually assumes it - thm:consistency: depends on ass:assumption-lower-bound-boundary which the environment never references; V1.5 must confirm the paper actually assumes it - prop:lip-ate: prop:Lip_ATE references \ref{lemma:Lip_condition} (lemma:Lip_condition) but `depends_on` lacks ass:lemma-lip-condition-regression, ass:lemma-lip-condition-bounded; V1.5 must confirm whether the paper assumes it or merely mentions it - thm:rate: cor:rate references \ref{lemma:sinkhorn_approx} (lemma:sinkhorn_approx) but `depends_on` lacks def:lemma-sinkhorn-approx-regularized-notation; V1.5 must confirm whether the paper assumes it or merely mentions it - thm:rate: depends on ass:assump-independence-u which the environment never references; V1.5 must confirm the paper actually assumes it - thm:rate: depends on ass:assump-categorical-rv-latents which the environment never references; V1.5 must confirm the paper actually assumes it - thm:rate: depends on ass:assump-categorical-rv-generation which the environment never references; V1.5 must confirm the paper actually assumes it - thm:rate: depends on ass:assump-lip-and-bounded-continuous which the environment never references; V1.5 must confirm the paper actually assumes it - thm:rate: depends on ass:assump-lip-and-bounded-categorical which the environment never references; V1.5 must confirm the paper actually assumes it - thm:rate: depends on ass:assump-lip-and-bounded-bound which the environment never references; V1.5 must confirm the paper actually assumes it - thm:rate: depends on ass:assump-support-components which the environment never references; V1.5 must confirm the paper actually assumes it - thm:rate: depends on ass:assump-support-homeomorphisms which the environment never references; V1.5 must confirm the paper actually assumes it - thm:rate: depends on ass:assumption-lower-bound-domination which the environment never references; V1.5 must confirm the paper actually assumes it - thm:rate: depends on ass:assumption-lower-bound-boundary which the environment never references; V1.5 must confirm the paper actually assumes it - thm:rate: depends on ass:corollary-deep-nn-latent-laws which the environment never references; V1.5 must confirm the paper actually assumes it - thm:wide-nn: the proof of thm:wide_nn cites [Yarotsky2018OptimalAO, Theorem 2] also at thm:wide_nn/proof/4/4, outside the passage lem:wide-nn-yarotsky-theorem-2 registers (thm:wide_nn/proof/2/5 … thm:wide_nn/proof/2/6): a published result used there with no cited node covering that use. Not a defect of the node or of its wiring: grade the step as any use of a published result. - thm:consistency: the proof of thm:consistency cites [Yarotsky2018OptimalAO, Proposition 1] (thm:consistency/proof/1/19); the transcription reads the citation as a method or argument borrowed from the cited work, not its result and made no cited node for it (The citation attributes the simplex interpolation construction; the host proof supplies the structural approximation, Lipschitz, categorical-preservation and neural-realization bounds itself.). paper: "\item \cite[Proposition 1]{Yarotsky2018OptimalAO} shows that given a $ L$-Lipschitz…" ### Parser warnings - proof.tex:1470: a second \begin{proof} for thm:wide_nn; appended to that result's proof (the locator names the first proof only) ## Lean evidence - build of the run barrel: **ok** - source scan (sorry / admit / axiom): clean | declaration | axioms | | --- | --- | | `Papers.TbsNeuralCausalPi.approximation_wasserstein_bound` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.wide_generator_approximation` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.deep_generator_approximation` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.deep_ncm_approximation` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.counterexample_identifiable` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.consistency` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.lipschitz_dose_response_gap` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.rate_of_convergence` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.exists_canonical_representation` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.exists_measure_representing_holder_curve` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.hilbert_ordering` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.counterexample_backdoor` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.convgOpt_limits_mem` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.regression_gap_le_wasserstein` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.wasserstein_map_le` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.wasserstein_mixture_le` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.wasserstein_pi_le` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.wasserstein_gumbelSoftmax_le` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.sinkhorn_approx` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsNeuralCausalPi.uniform_empirical_wasserstein` | clean — `propext`, `Classical.choice`, `Quot.sound` | ## Inputs | artifact | status | | --- | --- | | `discovery/v1_scope.json` | present (29 environment(s)) | | `discovery/v1_5_review.json` | present (verdict PASS) | | `formalization/crosswalk_full.json` | present (2941 object(s)) | | `graph.json` | present (2952 node(s)) | | Lean evidence | collected (build ok, 0 scan issue(s), 20 declaration(s)) | ## Annotated paper Annotated paper: `annotated/Main.tex` (314 annotation(s), 2 unlocated; index `annotated/ANNOTATIONS.md`); PDF: `annotated/Main.pdf`