# Correctness report — ts_adaptive_estimation_inference

**Verdict: ERRORS FOUND — REPAIRED BY MINIMAL PATCHES**

**Paper: MINOR ISSUES — no main result is shown false or weakened**

Minor:

- Theorem 3.1: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned
- Proposition 3.5: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned
- Theorem 3.6: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned
- Proposition 3.8: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned
- Theorem 3.9: condition added (bookkeeping or a pinned reading)
- Proposition 3.13: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned
- Theorem 3.14: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned
- Proposition 3.17: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned
- Theorem 3.18: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned
- Corollary 3.19: error repaired; condition added (bookkeeping or a pinned reading)
- Lemma B.1: error repaired; condition added (bookkeeping or a pinned reading)
- Lemma B.2: condition added (bookkeeping or a pinned reading)
- Lemma B.3: condition added (bookkeeping or a pinned reading)
- Lemma B.4: condition added (bookkeeping or a pinned reading)
- Corollary B.5: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned
- Lemma B.6: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned
- Lemma B.8: condition added (bookkeeping or a pinned reading)
- Corollary B.9: condition added (bookkeeping or a pinned reading)
- Lemma B.10: condition added (bookkeeping or a pinned reading)
- Lemma C.1: condition added (bookkeeping or a pinned reading)
- Lemma C.2: corrected; condition added (bookkeeping or a pinned reading)
- Corollary C.3: condition added (bookkeeping or a pinned reading)
- Lemma C.4: condition added (bookkeeping or a pinned reading)
- Lemma C.2: error repaired; condition added (bookkeeping or a pinned reading)
- Corollary C.6: condition added (bookkeeping or a pinned reading)
- Lemma C.7: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned
- `ass:cr-deepiv-measurability`: ambiguous; a reading was pinned; ambiguous; the owner settled a reading
- `ass:mean-sq-continuity`: ambiguous; a reading was pinned
- `ass:rdiv-minima`: ambiguous; a reading was pinned
- `ass:trae-minima`: ambiguous; a reading was pinned
- `def:iter-rdiv-discrepancy-bound`: ambiguous; a reading was pinned
- `def:trae-discrepancy-bound`: ambiguous; a reading was pinned
- `def:iter-trae-discrepancy-bound`: ambiguous; a reading was pinned
- `r_0`: notation ambiguous; a reading was pinned
- `q`: notation ambiguous; a reading was pinned
- `Y`: notation ambiguous; a reading was pinned
- `\mathcal H`: notation ambiguous; a reading was pinned
- `\mathcal F`: notation ambiguous; a reading was pinned
- `\mathcal S`: notation ambiguous; a reading was pinned
- `a_0`: notation ambiguous; a reading was pinned
- `\hat\theta`: notation ambiguous; a reading was pinned
- `\hat h`: notation ambiguous; a reading was pinned
- `\hat q`: notation ambiguous; a reading was pinned
- `\beta`: notation ambiguous; a reading was pinned
- `w_0`: notation ambiguous; a reading was pinned
- `\lambda`: notation ambiguous; a reading was pinned
- `r_\delta`: notation ambiguous; a reading was pinned
- `\delta`: notation ambiguous; a reading was pinned
- `h_\lambda^\delta`: notation ambiguous; a reading was pinned
- `\ell`: notation ambiguous; a reading was pinned
- `\xi`: notation ambiguous; a reading was pinned
- `\rho`: notation ambiguous; a reading was pinned
- `\hat h_\lambda`: notation ambiguous; a reading was pinned
- `D_n`: notation ambiguous; a reading was pinned
- `A_n`: notation ambiguous; a reading was pinned
- `\mathcal S_n`: notation ambiguous; a reading was pinned
- `\mathcal E_n`: notation ambiguous; a reading was pinned
- `\gamma`: notation ambiguous; a reading was pinned
- `\lambda^\circ`: notation ambiguous; a reading was pinned
- `\lambda_{\mathrm{dp}}`: notation ambiguous; a reading was pinned
- `g_0`: notation ambiguous; a reading was pinned
- `\hat r`: notation ambiguous; a reading was pinned
- `I_{\mathrm{pre}}`: notation ambiguous; a reading was pinned
- `I_{\mathrm{fit}}`: notation ambiguous; a reading was pinned
- `I_{\mathrm{cal}}`: notation ambiguous; a reading was pinned
- `L_n`: notation ambiguous; a reading was pinned
- `L`: notation ambiguous; a reading was pinned
- `B`: notation ambiguous; a reading was pinned
- `c_0`: notation ambiguous; a reading was pinned
- `C_G`: notation ambiguous; a reading was pinned
- `\mathcal T_g`: notation ambiguous; a reading was pinned
- `\mathcal G_0`: notation ambiguous; a reading was pinned
- `\eta_n`: notation ambiguous; a reading was pinned
- `\delta_{n,r}`: notation ambiguous; a reading was pinned
- `\delta_{n,\mathcal G}`: notation ambiguous; a reading was pinned
- `\delta_{n,\mathcal H}`: notation ambiguous; a reading was pinned
- `\delta_n`: notation ambiguous; a reading was pinned
- `\nu_n`: notation ambiguous; a reading was pinned
- `\tilde\delta_n`: notation ambiguous; a reading was pinned
- `\alpha`: notation ambiguous; a reading was pinned
- `\lambda_{\mathrm R}^\circ`: notation ambiguous; a reading was pinned
- `\lambda_{\mathrm T}^\circ`: notation ambiguous; a reading was pinned
- `t`: notation ambiguous; a reading was pinned
- `a_t`: notation ambiguous; a reading was pinned
- `\lambda_{\mathrm{IR}}^\circ`: notation ambiguous; a reading was pinned
- `b_t`: notation ambiguous; a reading was pinned
- `\lambda_{\mathrm{IT}}^\circ`: notation ambiguous; a reading was pinned
- `h_{j,\lambda}^*`: notation ambiguous; a reading was pinned
- `\hat h_{j,\lambda}`: notation ambiguous; a reading was pinned
- `a_j`: notation ambiguous; a reading was pinned
- `b_j`: notation ambiguous; a reading was pinned
- `\mathcal E_n^{\mathrm R}`: notation ambiguous; a reading was pinned
- `\mathcal E_{n,t}^{\mathrm{IR}}`: notation ambiguous; a reading was pinned
- `\mathcal E_{n,t}^{\mathrm{IT}}`: notation ambiguous; a reading was pinned
- `b`: notation ambiguous; a reading was pinned
- `r_\lambda`: notation ambiguous; a reading was pinned
- `H^2`: notation ambiguous; a reading was pinned
- `\beta_h`: notation ambiguous; a reading was pinned
- `\beta_q`: notation ambiguous; a reading was pinned
- `\boldsymbol y`: notation ambiguous; a reading was pinned
- `def:rdiv-preliminary`: ambiguous; the owner settled a reading
- `lem:iter-rdiv-stability`: ambiguous; the owner settled a reading
- `ass:critical-radius-boundedness`: ambiguous; the owner settled a reading; error repaired
- `lem:trae-squared-norm-lemma-14`: error repaired

29 result(s) of the paper were examined. 3 were found wrong as printed: 3 corrected, 0 not repaired. In Lean: 28 proved, 1 taken from the literature as cited, 0 not certified, 0 not formalized.

## What was wrong and what was done

### Corollary 3.19 — corrected

- **As printed.** \label{cor:normality} Suppose that the assumptions of \cref{thm:tikhonv_rate} hold for both $h_0$ and $q_0$ with (potentially different) source conditions $\beta_h,\beta_q>0$. Choose a deterministic sequence $\xi_n\downarrow0$, and run the primal and dual tuning rules at confidence levels $\xi_n/2$. Let a common deterministic sequence $\delta_n$ satisfy \eqref{eq:delta_trae} for both problems at these confidence levels. Assume that, for some $\gamma_h,\gamma_q>0$, the score maps satisfy \begin{align*} \mathbb{E}\left[\left(\tilde{m}( W;h_{1}) -\tilde{m}( W;h_{2})\right)^{2}\right] & \leqslant C\| h_{1} -h_{2} \| ^{\gamma_h },\\ \mathbb{E}\left[\left(m( W;q_{1}) -m( W;q_{2})\right)^{2}\right] & \leqslant C\| q_{1} -q_{2} \| ^{\gamma_q } \end{align*} for all $h_1,h_2\in\mathcal H$ and $q_1,q_2\in\mathcal Q$. Let $\beta_m=\max\{\beta_h,\beta_q\}$ and assume that $\delta_n$ satisfies \[ \delta _{n} =o\left( n^{-\alpha _{\mathrm{dr}}}\right) ,\qquad \alpha _{\mathrm{dr}} :=\frac{1+\min\{\beta _{m} ,1\}}{2+4\min\{\beta _{m} ,1\}} . \] Let $\hat h,\hat q$ be the adaptive adversarial Tikhonov estimators, trained and tuned independently of the evaluation sample used in $\hat\theta(\hat h,\hat q)$. Finally, assume that $0<\sigma_0^2:=\mathbb E[\psi_0(W)^2]<\infty$. Then, \[ \sqrt{n}(\hat{\theta }(\hat{h},\hat{q}) -\theta _{0}) =\frac{1}{\sqrt{n}}\sum _{i=1}^{n} \psi _{0}( W_i) +o_{p}( 1) , \] where $\psi _{0}( W) =\tilde{m}( W;h_{0}) +m( W;q_{0}) -q_{0}( Z) h_{0}( X) -\theta _{0}$. Consequently, $\sqrt n(\hat\theta(\hat h,\hat q)-\theta_0)\rightsquigarrow N(0,\sigma_0^2)$.
- **What is wrong.** ``For completeness, define the uncentered score \[ S(W;h,q)=\tilde m(W;h)+m(W;q)-h(X)q(Z), \qquad d_n(W)=S(W;\hat h,\hat q)-S(W;h_0,q_0).\mbox{''} \] Defining the fitted score difference sectionwise does not ensure a jointly measurable version coherent with both its conditional population integrals and its evaluations on the finite evaluation sample.
- **Counterexample.** Let $U$ and $V$ be independent uniform random variables on $[0,1]$, realized on the uncompleted Borel product probability space, with $U$ representing training information and $V$ a fresh observation. Let $A\subset[0,1]$ be a non-Borel set. Consider the sectionwise representatives $g(u,v)=\mathbf 1_A(u)\mathbf 1_{\{0\}}(v)$ and $\bar g(u,v)=0$, which represent the same zero population-$L_2$ function for every fixed $u$. Every fixed-training section of $g$ is measurable and has zero population-$L_2$ norm, but $g(U,V)$ is not measurable on this product space because its nonzero set is $A\times\{0\}$; choosing the coherent measurable version $\bar g$ resolves this version ambiguity.
- **Correction.** Suppose that the assumptions of \cref{thm:tikhonv_rate} hold for both $h_0$ and $q_0$ with (potentially different) source conditions $\beta_h,\beta_q>0$. Choose a deterministic sequence $\xi_n\downarrow0$, and run the primal and dual tuning rules at confidence levels $\xi_n/2$. Let a common deterministic sequence $\delta_n$ satisfy \eqref{eq:delta_trae} for both problems at these confidence levels. Assume that, for some $\gamma_h,\gamma_q>0$, the score maps satisfy \begin{align*} \mathbb{E}\left[\left(\tilde{m}( W;h_{1}) -\tilde{m}( W;h_{2})\right)^{2}\right] & \leqslant C\| h_{1} -h_{2} \| ^{\gamma_h },\\ \mathbb{E}\left[\left(m( W;q_{1}) -m( W;q_{2})\right)^{2}\right] & \leqslant C\| q_{1} -q_{2} \| ^{\gamma_q } \end{align*} for all $h_1,h_2\in\mathcal H$ and $q_1,q_2\in\mathcal Q$. Let $\beta_m=\max\{\beta_h,\beta_q\}$ and assume that $\delta_n$ satisfies \[ \delta _{n} =o\left( n^{-\alpha _{\mathrm{dr}}}\right) ,\qquad \alpha _{\mathrm{dr}} :=\frac{1+\min\{\beta _{m} ,1\}}{2+4\min\{\beta _{m} ,1\}} . \] Let $\hat h,\hat q$ be the adaptive adversarial Tikhonov estimators, trained and tuned independently of the evaluation sample used in $\hat\theta(\hat h,\hat q)$. For each $n$, write $\mathcal G_n$ for the full nuisance-training and tuning sigma-field. Take jointly measurable versions of the fitted nuisance evaluations, representing the fitted functions for almost every training realization and agreeing almost surely with their evaluations at every observation in the finite evaluation sample. Write \[ S(w;h,q)=\tilde m(w;h)+m(w;q)-h(X(w))q(Z(w)),\qquad d_n(w)=S(w;\hat h,\hat q)-S(w;h_0,q_0). \] Suppose that, for each $n$, $d_n$ admits a version jointly measurable in $\mathcal G_n$ and a fresh observation, agreeing with $d_n$ for almost every training realization and almost every fresh observation, and agreeing almost surely with $d_n(W_i)$ simultaneously for $i=1,\ldots,n$ in the evaluation sample. Finally, assume that $0<\sigma_0^2:=\mathbb E[\psi_0(W)^2]<\infty$. Then, \[ \sqrt{n}(\hat{\theta }(\hat{h},\hat{q}) -\theta _{0}) =\frac{1}{\sqrt{n}}\sum _{i=1}^{n} \psi _{0}( W_i) +o_{p}( 1) , \] where $\psi _{0}( W) =\tilde{m}( W;h_{0}) +m( W;q_{0}) -q_{0}( Z) h_{0}( X) -\theta _{0}$. Consequently, $\sqrt n(\hat\theta(\hat h,\hat q)-\theta_0)\rightsquigarrow N(0,\sigma_0^2)$.
- **Proof.** Apply \cref{thm:tikhonv_rate} to both nuisance estimators at the prescribed confidence levels $\xi_n/2$. Their rate bounds then hold on a common event with probability at least $1-\xi_n\to1$. Because $\beta_h,\beta_q>0$ and $\delta_n\to0$, these bounds give \begin{equation}\label{eq:conv_hq} \| \hat{h} -h_{0} \| =o_{p}( 1) ,\qquad \| \hat{q} -q_{0}\| =o_{p}( 1) . \end{equation} The remaining bias condition is \begin{equation}\label{eq:square_root} \sqrt{n}\mathbb{E}[(\hat{q}( Z) -q_{0}( Z))(\hat{h}( X) -h_{0}( X))] =o_{p}( 1) . \end{equation} Suppose first that $\beta_m=\beta_h\geqslant\beta_q$. By the adjoint identity and the Cauchy--Schwarz inequality, \begin{align*} … (the corrected paper gives it in full)
- **Formal check.** proved in Lean (the corrected statement), given 2 cited result(s), under 7 condition(s) made explicit — `Papers.TsAdaptiveEstimationInference.dr_asymptotic_normality` (`TNormality.lean:41`); axioms: the standard ones only (propext, Classical.choice, Quot.sound).

### Lemma B.1 — statement unchanged; proof corrected

- **As printed.** \label{lemma:t_err_dp} Under \cref{asp:re_cd,asp:cr_deepiv}, with probability at least $1-\xi$, \begin{align} \mathbb E_Z\!\left[H^2\{\hat g(\cdot\mid Z),g_0(\cdot\mid Z)\}\right] &\leqslant C\delta_{n,\mathcal G}^2, \label{eq:rdiv_mle_hellinger}\\ \|\hat r-r_0\|_{L_2(Z)}&\leqslant \delta_{n,r},\nonumber\\ \sup_{h\in\mathcal H} \|(\hat{\mathcal T}-\mathcal T)h\|_{L_2(Z)} &\leqslant C\delta_{n,\mathcal G},\label{eq:rdiv_op_global}\\ \|(\hat{\mathcal T}-\mathcal T)(h-h')\|_{L_1(Z)} &\leqslant C\delta_{n,\mathcal G}\|h-h'\|, \qquad h,h'\in\mathcal H.\label{eq:rdiv_op_relative} \end{align} Moreover, if \[ \widehat L(h):=\mathbb E[(Y-\hat{\mathcal T}h(Z))^2\mid I_{\mathrm{pre}}], \] then, uniformly over $h,h'\in\mathcal H$, \begin{equation}\label{eq:rdiv_first_stage_loss} |\{\widehat L(h)-\widehat L(h')\}-\{L(h)-L(h')\}| \leqslant C\delta_{n,\mathcal G}\|h-h'\|. \end{equation}
- **What is wrong.** ``\dots applied with $g_0\in\mathcal G$ and the critical radius of $\mathcal G_0$, gives \eqref{eq:rdiv_mle_hellinger} with probability at least $1-\xi/2$.'' This step uses the conditional-MLE bound of \citet[Theorem~24]{li2024regularized}, which the paper assumes only for the density lower bound and the conditions of the source's referenced RDIV theorem, including its source condition, but applies it to a model satisfying only Assumptions~\ref{asp:re_cd} and~\ref{asp:cr_deepiv}. In the step quoted in (a): The cited theorem states an exponential confidence formula without a negative exponent, and its proof gives $1-\exp(n\delta_n^2)$, which is nonpositive and does not imply the asserted probability $1-\xi/2$.
- **Counterexample.** Let $\mu$ be Lebesgue measure on $[0,1]$, let $Z$ be uniform on $[0,1]$, and let $\phi_j$ be the mean-zero orthonormal Rademacher functions. Set $\sigma_j=2^{-j^2-2}$ and $g_0(x\mid z)=1+\sum_{j\geqslant1}\sigma_j\phi_j(x)\phi_j(z)$; both marginals are uniform, and $1/2\leqslant g_0\leqslant2$. Take $\tilde{\mathcal H}=\overline{\operatorname{span}}\{1,\phi_j:j\geqslant1\}\subseteq L_2(X)$ and $\tilde{\mathcal Q}=L_2(Z)$, so that $\mathcal T\phi_j=\sigma_j\phi_j$ and $\mathcal T^*\mathcal T\phi_j=\sigma_j^2\phi_j$. Set $h_0=\sum_{j\geqslant1}j^{-2}\phi_j$, $r_0=\mathcal T h_0$, $Y=r_0(Z)$, and $\hat r=r_0$. Take $\mathcal H=\{c h_0:1/2\leqslant c\leqslant3/2\}$, $\mathcal G=\{g_0,1\}$, $B=3$, $c_0=1/2$, and $C_G=2$. Choose $\delta_{n,r}=0$ and deterministic, $\beta$-independent critical-radius upper bounds satisfying the logarithmic floors in Assumption~\ref{asp:cr_deepiv}; these exist because the density class is finite and the primal and density-induced classes are bounded scalar intervals. Fix $n\geqslant1$ and $\xi\in(0,1)$. These objects satisfy Assumptions~\ref{asp:re_cd} and~\ref{asp:cr_deepiv}, but $\sum_{j\geqslant1}j^{-4}\sigma_j^{-2\beta}=\infty$ for every $\beta>0$, so the cited source condition fails; moreover, $1-\exp(n\delta_{n,\mathcal G}^2)\leqslant0<1-\xi/2$, so the reported confidence formula supplies no asserted positive probability guarantee even in this model.
- **Correction.** The statement stands as printed; its proof is corrected.
- **Proof.** Allocate failure probability $\xi/2$ to the conditional-density estimator and $\xi/2$ to the pilot regression. As elsewhere, evaluate the corresponding deterministic accuracy bounds at these confidence levels and relabel them $\delta_{n,\mathcal G}$ and $\delta_{n,r}$. We establish the density bound directly using the Localized Concentration lemma of \citet{li2024regularized}. Write $P=\mathbb E_{X,Z}$ and $P_n=\mathbb E_{n,\mathrm{pre}}$, and define \[ f_g(x,z)=\sqrt{\frac{g(x\mid z)+g_0(x\mid z)}{2g_0(x\mid z)}},\qquad \mathcal U=\{f_g-1:g\in\mathcal G\}. \] Realizability gives $f_{g_0}=1$, so $0\in\mathcal U$. The density bounds give \[ 2^{-1/2}\leqslant f_g\leqslant M,\qquad … (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, given 1 cited result(s), under 2 condition(s) made explicit — `Papers.TsAdaptiveEstimationInference.t_err_dp` (`Helpers/RDIV.lean:85`); axioms: the standard ones only (propext, Classical.choice, Quot.sound).

### Lemma C.2 — statement unchanged; proof corrected

- **As printed.** Under the assumptions of \cref{thm:iter_trae_rate}, there is a nondecreasing deterministic sequence with $b_0=0$ and $b_j\geqslant2$, $j=1,\ldots,t$, where each $b_j$ depends only on $j$ and the prescribed boundedness and mean-square-continuity constants, independently of $\beta$ and $\|w_0\|$. This sequence may be calibrated so that, on $\mathcal E_{n,t}^{\mathrm{IT}}$, simultaneously for $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$, \begin{align} \|\hat h_{j,\lambda}-h_{j,\lambda}^*\|^2 &\leqslant C\frac{b_j\delta_n^2}{\lambda^2}, \qquad j=1,\ldots,t, \label{eq:iter_trae_strong_propagation}\\ L(\hat h_{t,\lambda}) &\leqslant C\left[ \delta_n^2+\min\left\{\lambda, \frac{b_t\delta_n^2}{\lambda}\right\} +\|w_0\|^2\lambda^{q_t} \right]. \label{eq:iter_trae_weak_propagation} \end{align}
- **What is wrong.** ``\dots applied at this fixed iteration--grid pair, gives \[ L(\hat h_{j,\lambda})-L(h_{j,\lambda}^*) \leqslant \lambda I_{n,j} +C\left\{ \delta_ne_j\|f_j^*\|_\infty +\delta_nu_j+\delta_n^2 \right\}.\mbox{''} \] This step uses the excess-loss comparison of \citet[Lemma~11]{bennett2023source}, which the paper assumes only for the source's iterative-theorem conditions, including $\mathbb E[h_0(X)-h(X)\mid Z=\cdot]\in\mathcal F$ for every $h\in\mathcal H$, but applies it to a projected operator for which only $\mathcal T(h_0-h)\in\mathcal F$ is assumed. The step quoted in (a) uses the excess-loss comparison of \citet[Lemma~11]{bennett2023source}, which the paper assumes only for the source's standing Realizability and Closedness assumption, which requires convexity of $\mathcal H$, but applies it to the general hypothesis class $\mathcal H$, which need not be convex. ``For the weak bound, the ordinary excess-loss comparison \citep[Lemma~10]{bennett2023source} retains the population bias.'' This step uses the excess-loss comparison of \citet[Lemma~10]{bennett2023source}, which the paper assumes only for the source's one-step-theorem conditions, including $\mathbb E[h_0(X)-h(X)\mid Z=\cdot]\in\mathcal F$ for every $h\in\mathcal H$, but applies it to a projected operator for which only $\mathcal T(h_0-h)\in\mathcal F$ is assumed. The step quoted in (c) uses the excess-loss comparison of \citet[Lemma~10]{bennett2023source}, which the paper assumes only for the source's standing Realizability and Closedness assumption, which requires convexity of $\mathcal H$, but applies it to the general hypothesis class $\mathcal H$, which need not be convex. ``The localized squared-norm comparison for $\operatorname{star}(\mathcal H-\mathcal H)$ then yields \[ I_{n,j}-I_j \leqslant C\delta_n(e_j+e_{j-1})+C\delta_n^2+2Ce_je_{j-1}.\mbox{''} \] The localized concentration application for squares centered at $h_{j-1,\lambda}^*$ or $h_{k,\lambda}^*$ requires measurable localized loss suprema, which do not follow from measurability for unshifted squares centered at $h_0$ or for the specified zero-comparator adversarial process.
- **Counterexample.** For the citation-hypothesis obstruction, let $V$ and $\varepsilon$ be independent Rademacher variables, and set $X=Z=V$, $Y=1/2+\varepsilon/4$, and $W=(Y,X,Z)$. Take $\widetilde{\mathcal H}=\operatorname{span}\{1,x\}$, $\widetilde{\mathcal Q}=\operatorname{span}\{1\}$, and $m(W;q)=Yq(Z)$. Let $\mathcal H=\{x\mapsto a+bx:0\leqslant a\leqslant1/2,\ b\in\{-1/4,0,1/4\}\}$ and $\mathcal F=\{z\mapsto d:|d|\leqslant1\}$. Take $\beta=1$ and $w_0=h_0=1/2$; then $\mathcal T(a+bx)=a$ and $h_{j,\lambda}^*=\{1-(\lambda/(1+\lambda))^j\}/2\in\mathcal H$ for every $j\geqslant1$ and $\lambda>0$. Use independent copies of $W$, fix $t\geqslant1$ and $\xi\in(0,1)$, choose sufficiently large deterministic critical-radius bounds and calibrated scales of order $\sqrt{\{\log\log(en)+\log(t(K_n+1)/\xi)+1\}/n}$, and select measurable empirical minimizers over the compact class $\mathcal H$. For a separate instance of the squared-loss measurability obstruction, let $A\subset(0,1/2)$ be non-Lebesgue-measurable and let $W=X=Z$ have the completed uniform law on $[0,1]$. Take $\widetilde{\mathcal H}=L_2(X)$, $\widetilde{\mathcal Q}=\operatorname{span}\{1\}$, $\mathcal T h=\mathbb E[h(X)]$, $r_0=h_0=w_0=1$, $\beta=1$, $m(W;f)=f$, and $\mathcal F$ equal to the constant functions with values in $[0,1]$. Using the supplied representatives, take $\mathcal H=\{x\mapsto c:0\leqslant c\leqslant1\}\cup\{1-\mathbf1_{\{a\}}:a\in[0,1]\}\cup\{1-\tfrac12\mathbf1_{\{a\}}:a\in A\}$, set $t=2$ and $\rho=1/2$, and choose the constant empirical minimizers $\hat h_{j,\lambda}=h_{j,\lambda}^*=1-\{\lambda/(1+\lambda)\}^j$. Use $B=1$ and deterministic critical-radius bounds of order $\sqrt{\{\log\log(en)+\log(t(K_n+1)/\xi)+1\}/n}$; the localized linear Rademacher suprema are measurable with bounds $O(r/\sqrt n+1/n)$, and the optional singleton increments are dominated by the singleton increments indexed by $[0,1]$ for both the stipulated unshifted squared-loss and zero-comparator adversarial suprema. In the first instance, for $h(x)=x/4$ the unprojected residual $1/2-z/4$ is outside $\mathcal F$, and $0,x/4\in\mathcal H$ while their midpoint $x/8\notin\mathcal H$, so both excess-loss citations lack their required hypotheses; in the separate uniform-law instance, for $\lambda=1$ and sufficiently large $n$, the localized centered squared-loss supremum at $h_{1,1}^*=1/2$ with radius $1/2$ equals $(4n)^{-1}\mathbf1_{\{\exists i:X_i\in A\}}$ off the measurable null set of sample ties and is not measurable under the fitting-sample law.
- **Correction.** The statement stands as printed; its proof is corrected.
- **Proof.** We adapt the iterative argument of \citet[Theorem~6]{bennett2023source} to the finite grid and give the recursive comparison explicitly. Fix $j$ and $\lambda$, and set \[ e_j=\|\hat h_{j,\lambda}-h_{j,\lambda}^*\|, \qquad u_j=\|\mathcal T(\hat h_{j,\lambda}-h_{j,\lambda}^*)\|, \qquad f_j^*=\mathcal T(h_0-h_{j,\lambda}^*). \] Also define \[ M_{\leqslant j} =\max_{1\leqslant\ell\leqslant j} \left\{\lambda,\|f_\ell^*\|_\infty^2\right\}. \] By \cref{asp:iter_realizability_trae,asp:closeness}, $f_\ell^*\in\mathcal F$. The boundedness part of \cref{asp:critical_radius} therefore gives $M_{\leqslant j}\leqslant\max\{2,B^2\}$ for every grid point under consideration. To track the random penalty … (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, given 3 cited result(s), under 4 condition(s) made explicit — `Papers.TsAdaptiveEstimationInference.iter_trae_strong_propagation` (`Helpers/IterTRAEPropagation.lean:38`); 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.

- Assumption~\ref{asp:critical_radius} supplies bounded classes for the localized concentration arguments used in one-step and iterative TRAE. ``For a fixed $B<\infty$, suppose that $|m(W;f)|$, $|h(X)|$, and $|f(Z)|$ are a.s. bounded by $B$, uniformly over $ h\in \mathcal{H} ,f\in \mathcal{F}$.'' Uniform boundedness does not ensure measurability of the localized empirical and Rademacher loss suprema, and measurability for the $h_0$-centered squares and zero-comparator adversarial loss does not ensure it for squares centered at $h_{j-1,\lambda}^*$ or $h_{j,\lambda}^*$. Corrected text: For a fixed $B<\infty$, suppose that $|m(W;f)|$, $|h(X)|$, and $|f(Z)|$ are a.s. bounded by $B$, uniformly over $ h\in \mathcal{H} ,f\in \mathcal{F}$. Suppose in addition that, for every integer $n\geqslant1$ and every $r>0$, the localized squared-loss suprema \[ \sup_{\substack{h\in\mathcal H\\ \|h-h_0\|\leqslant r}} \left|(\mathbb E_n-\mathbb E)[h(X)^2-h_0(X)^2]\right| \quad\text{and}\quad \sup_{\substack{h\in\mathcal H\\ \|h-h_0\|\leqslant r}} \left|\frac1n\sum_{i=1}^n\epsilon_i\{h(X_i)^2-h_0(X_i)^2\}\right| \] are measurable random variables under the fitting-sample law and its product with the independent Rademacher law, respectively. Regard the functions in $\mathcal H\cdotp\mathcal F$ and $\mathcal F$ as functions of $W$ through $(X,Z)$ and $Z$, respectively, and put \[ \mathcal U=\operatorname{star}(m\circ\mathcal F) \times\operatorname{star}(\mathcal H\cdotp\mathcal F) \times\operatorname{star}(\mathcal F), \qquad \|u\|_{L_2(W;\mathbb R^3)}^2 =\mathbb E\!\left[\sum_{j=1}^3u_j(W)^2\right]. \] Define the loss with zero comparator by \[ \operatorname{clip}_B(a)=\max\{-B,\min\{a,B\}\}, \qquad \ell_B(u)=2u_1-2u_2-\operatorname{clip}_B(u_3)^2, \qquad \ell_B(0)=0. \] Suppose also that, for every integer $n\geqslant1$ and every $r>0$, the localized loss suprema \[ \sup_{\substack{u\in\mathcal U\\ \|u\|_{L_2(W;\mathbb R^3)}\leqslant r}} \left|(\mathbb E_n-\mathbb E)[\ell_B(u(W))-\ell_B(0)]\right| \quad\text{and}\quad \sup_{\substack{u\in\mathcal U\\ \|u\|_{L_2(W;\mathbb R^3)}\leqslant r}} \left|\frac1n\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(0)\}\right| \] are measurable random variables under the fitting-sample law and its product with the independent Rademacher law, respectively. Whenever the primal and dual roles are exchanged, suppose that these same two localized loss suprema for the exchanged problem are measurable under its fitting-sample law and its product with the independent Rademacher law, respectively. For iterative TRAE, suppose in addition that the following fixed-population-center squared-loss processes are measurable. For every integer $n\geqslant1$, prescribed integer $t\geqslant1$, grid point $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ for which $\lambda_{\mathrm{IT}}^\circ$ is defined, radius $r>0$, and $j=1,\ldots,t$, put \[ a_{j,\lambda}(h;X) =\{h(X)-h_{j-1,\lambda}^*(X)\}^2 -\{h_{j,\lambda}^*(X)-h_{j-1,\lambda}^*(X)\}^2, \qquad d_{j,\lambda}(h;X)=\{h(X)-h_{j,\lambda}^*(X)\}^2, \] where the population iterates are defined by \eqref{eq:iter_population}. For each $v\in\{a_{j,\lambda},d_{j,\lambda}\}$, suppose that both \[ \sup_{\substack{h\in\mathcal H\\\|h-h_{j,\lambda}^*\|\leqslant r}} \left|(\mathbb E_n-\mathbb E)v(h;X)\right| \quad\text{and}\quad \sup_{\substack{h\in\mathcal H\\\|h-h_{j,\lambda}^*\|\leqslant r}} \left|\frac1n\sum_{i=1}^n\epsilon_i v(h;X_i)\right| \] are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, using the same function representatives as the empirical penalties. The zero-iterate comparison uses $\hat h_{0,\lambda}=h_{0,\lambda}^*=0$.
- The localized squared-norm inequality holds uniformly over $h\in\mathcal H$ under the stated TRAE assumptions. ``\dots uniformly over $h\in\mathcal H$, \[ \left|\{\|h\|_{2,n}^2-\|h_0\|_{2,n}^2\} -\{\|h\|^2-\|h_0\|^2\}\right| \leqslant C\{\delta_n\|h-h_0\|+\delta_n^2\}.\mbox{''} \] The concentration argument requires measurable localized squared-loss suprema, which do not follow from measurability of the empirical loss and its selected minimizer. Add measurability of the localized squared-loss empirical and Rademacher suprema to Assumption~\ref{asp:critical_radius}, and explicitly use this condition in the cited step of Lemma~\ref{lemma:strong_convexity}; this supplies the missing regularity while protecting Theorem~\ref{thm:tikhonv_rate}, Theorem~\ref{thm:iter_trae_rate}, Corollary~\ref{cor:normality}, and Corollary~\ref{cor:trae_discrepancy_bound} at their stated rates.

## Clarifications (not errors)

Nothing in this section is an error of the paper. Each item is something the paper determines but does not say, made explicit so that the statements can be read, and formalized, in one way.

**How a statement is read.**

- **Theorem 3.1.** On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event.
- **Proposition 3.5.** For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions.
- **Proposition 3.5.** For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$.
- **Proposition 3.5.** For each fixed $n\geqslant N_{\mathrm R}(B)$, all probabilistic conclusions, including initial acceptance when $h_0=0$, hold on the same event $\mathcal E_n^{\mathrm R}$ of probability at least $1-\xi$.
- **Proposition 3.5.** The assumptions imported from Theorem~\ref{thm:deepiv_rate} are the general hypotheses of Proposition~\ref{prop:exist_lambda_deepiv}, without imposing $h_0=0$.
- **Proposition 3.5.** For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$.
- **Theorem 3.6.** For each fixed $n\geqslant N_{\mathrm R}(B)$, the displayed error bounds and, when $h_0=0$, both sharper $O(\delta_n^2)$ bounds hold jointly with probability at least $1-\xi$ over the preliminary, fitting, and calibration samples.
- **Theorem 3.6.** The assumptions of \cref{thm:deepiv_rate} mean its standing hypotheses inherited from \cref{prop:exist_lambda_deepiv}; the bounds here hold for arbitrary $h_0$, without imposing $h_0=0$.
- **Proposition 3.8.** The displayed discrepancy inequality holds on $\mathcal E_{n,t}^{\mathrm{IR}}$, with $\mathbb P(\mathcal E_{n,t}^{\mathrm{IR}})\geqslant1-\xi$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$.
- **Proposition 3.8.** For each fixed $n\geqslant N_{\mathrm R}(B)$ and prescribed $t$, there is an event of probability at least $1-\xi$ on which the rule accepts a candidate and satisfies the stated output alternatives, including acceptance of $\lambda_0=2$ when $h_0=0$.
- **Proposition 3.13.** The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$.
- **Proposition 3.13.** For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention.
- **Proposition 3.13.** For each fixed $n\geqslant N_{\mathrm T}(B)$, acceptance within the stated number of optimization problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_n$, with probability at least $1-\xi$ over the fitting sample.
- **Proposition 3.13.** There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$.
- **Proposition 3.13.** The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$.
- **Theorem 3.14.** With $\beta$ and $\xi$ fixed as $n\to\infty$, for each $n\geqslant N_{\mathrm T}(B)$ the displayed strong- and weak-error bounds hold jointly on a fitting-sample event of probability at least $1-\xi$.
- **Proposition 3.17.** For each fixed $n\geqslant N_{\mathrm{IT}}(B)$ and prescribed $t$, acceptance within the stated number of minimax problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_{n,t}^{\mathrm{IT}}$, with probability at least $1-\xi$ over the fitting sample.
- **Theorem 3.18.** Under the assumptions of Theorem~\ref{thm:iter_trae_rate}, the displayed inequality holds simultaneously for all $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ on $\mathcal E_{n,t}^{\mathrm{IT}}$, where $\mathbb P(\mathcal E_{n,t}^{\mathrm{IT}})\geqslant1-\xi$.
- **Theorem 3.18.** The constant $C_t$ may depend on the prescribed $t$ and fixed population quantities, including $\beta$ and $\|w_0\|$, but is independent of $n$, $\lambda$, and the realized fitting observations.
- **Theorem 3.18.** The constants hidden by $O_t(\cdot)$ may depend on the prescribed $t$, fixed population quantities including $\beta$ and $\|w_0\|$, and the fixed grid ratio $\rho$, but are independent of $n$ and the candidate $\lambda$.

**Side conditions the proofs use without stating them.**

- **Theorem 3.18.** With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\|u-u_{j,\lambda}^*\|_{L_2(W;\mathbb R^3)}\leqslant r}|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}|$ and $\sup_{u\in\mathcal U,\,\|u-u_{j,\lambda}^*\|_{L_2(W;\mathbb R^3)}\leqslant r}|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$.

**Notation.**

- $r_0$: $r_0\in\tilde{\mathcal Q}$ is the Riesz representer of the continuous linear functional $q\mapsto\mathbb E[m(W;q)]$, satisfying $\mathbb E[m(W;q)]=\mathbb E[r_0(Z)q(Z)]$ for every $q\in\tilde{\mathcal Q}$. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:t_err_dp}, Lemma~\ref{lemma:err_l_dp}, Lemma~\ref{lemma:err_r_dp}, Lemma~\ref{lemma:str_dp}, Corollary~\ref{cor:rdiv_discrepancy_bound}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $\tilde{\mathcal Q}=L_2(Z)$ and $r_0(z)=\mathbb E[Y\mid Z=z]$.
- $q$: The letter $q$ is locally reused for function-valued arguments in $\tilde{\mathcal Q}$ and for a deterministic scalar bias exponent in $(1,\infty)$; these uses denote distinct objects. In Theorem~\ref{thm:common_dp}: In $B_n(\lambda)=C_b\lambda^q$, $q>1$ is a fixed scalar exponent. In Theorem~\ref{thm:deepiv_rate}, Theorem~\ref{thm:tikhonv_rate}: In the application of the generic discrepancy guarantee, the scalar exponent is $q=p=\min\{\beta+1,2\}$. In Theorem~\ref{thm:iter_rdiv_rate}, Theorem~\ref{thm:iter_trae_rate}: In the proof's scalar abbreviation and application of the generic discrepancy guarantee, $q=q_t=\min\{\beta+1,2t\}$. In Corollary~\ref{cor:normality}: In the dual minimization, $q\in\mathcal Q$ is the candidate function of $Z$; in $\hat\theta(h,q)$ it is the dual nuisance argument.
- $Y$: $Y$ is the scalar outcome, with $r_0(z)=\mathbb E[Y\mid Z=z]$ in the IV setting. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:t_err_dp}, Lemma~\ref{lemma:err_l_dp}, Lemma~\ref{lemma:err_r_dp}, Lemma~\ref{lemma:str_dp}, Corollary~\ref{cor:rdiv_discrepancy_bound}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $|Y|\leqslant B$ almost surely for a fixed $B<\infty$.
- $\mathcal H$: $\mathcal H$ denotes the primal hypothesis class of functions of $X$. In Proposition~\ref{prop:rkhs_qp}: $\mathcal H$ is the full RKHS generated by $k_{\mathcal H}$, before restriction to the retained numerical spectral span; its RKHS norm specifies the minimum-norm representative among functions with the same fitting-sample evaluations.
- $\mathcal F$: $\mathcal F$ is a function class on the input domain used in the surrounding complexity or loss definition. In Corollary~\ref{cor:normality}: For the primal application, $\mathcal F\subset\tilde{\mathcal Q}$ is the adversarial test class; for the dual application the corresponding test class is $\mathcal S\subseteq\tilde{\mathcal H}$, with the primal and dual spaces exchanged. In Proposition~\ref{prop:rkhs_qp}: $\mathcal F$ is the full RKHS generated by $k_{\mathcal F}$, and its empirical supremum includes the penalty $-\eta\|f\|_{\mathcal F}^2$ with $\eta>0$. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:strong_convexity}: $\mathcal F\subset\tilde{\mathcal Q}$ is the test class in $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$ and its population counterpart; it contains $\mathcal T(h_0-h)$ for every $h\in\mathcal H$ under \cref{asp:closeness}. In \cite[Lemma 9 (lem:concentration)]{bennett2023source}, used in the proof of Lemma~\ref{eq:iter_trae_strong_propagation}: $\mathcal F\subset\tilde{\mathcal Q}$ is the test class in $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$ and its population counterpart; it contains $\mathcal T(h_0-h)$ for every $h\in\mathcal H$ under \cref{asp:closeness}.
- $\mathcal S$: $\mathcal S$ is a fixed population function class whose local definition specifies its role and whether its functions have argument $X$ or $(X,Z)$. In Corollary~\ref{cor:normality}: $\mathcal S$ is the dual test class appearing in $L_n^{\mathrm{dual}}(q)=\sup_{s\in\mathcal S}\mathbb E_n[2\tilde m(W;s)-2q(Z)s(X)-s(X)^2]$. In Lemma~\ref{lemma:strong_convexity}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:strong_convexity}: $\mathcal S=\operatorname{star}(\mathcal H-\mathcal H)$ is the star-shaped difference class used to verify the anchored critical-radius condition. In Lemma~\ref{lemma:t_err_dp}: In the likelihood concentration argument, $\mathcal S=\operatorname{star}(\mathcal G_0-\mathcal G_0)$ is the fixed star-shaped difference class of likelihood-transform functions on $(X,Z)$, with norm $\|\cdot\|_{L_2(X,Z)}$. In \cite[Localized Concentration (source label lem:local-concen)]{li2024regularized}, used in the proof of Lemma~\ref{lemma:t_err_dp}: In the likelihood concentration argument, $\mathcal S=\operatorname{star}(\mathcal G_0-\mathcal G_0)$ is the fixed star-shaped difference class of likelihood-transform functions on $(X,Z)$, with norm $\|\cdot\|_{L_2(X,Z)}$.
- $a_0$: The notation $a_0$ is local to either the dual inverse problem or the scalar recursion for $(a_j)_{j\geqslant0}$. In Corollary~\ref{cor:normality}: $a_0\in\tilde{\mathcal H}$ is the Riesz representer satisfying $\mathbb E[\tilde m(W;h)]=\mathbb E[a_0(X)h(X)]$ for every $h\in\tilde{\mathcal H}$, and is the right-hand side of $\mathcal T^*q_0=a_0$. In Lemma~\ref{lemma:iter_rdiv_stability}: In the proof's recursion $a_j=C(1+a_{j-1})$, $a_0=0$ is a scalar initial value, distinct from the Riesz representer in the dual inverse problem.
- $\hat\theta$: $\hat\theta$ denotes an empirical estimate of $\theta_0$, with its nuisance arguments displayed when the doubly robust score functional is intended. In Corollary~\ref{cor:normality}: $\hat\theta(h,q)=\mathbb E_{n,\mathrm{eval}}[\tilde m(W;h)+m(W;q)-h(X)q(Z)]$, and the estimator in the conclusion is its evaluation at $(h,q)=(\hat h,\hat q)$; here $q\in\mathcal Q$ is a nuisance-function argument.
- $\hat h$: $\hat h$ denotes an estimated primal solution, with $\hat h_\lambda$ denoting a candidate at penalty $\lambda$, $\hat h_{j,\lambda}$ denoting an empirical iterate, and $\hat h_{\lambda_{\mathrm{dp}}}$ or $\hat h_{t,\lambda_{\mathrm{dp}}}$ denoting the corresponding discrepancy-selected estimate. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: $\hat h_{0,\lambda}=0$ and $\hat h_{j,\lambda}=\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\|h-\hat h_{j-1,\lambda}\|_{2,n}^2\}$ for $j=1,\ldots,t$, with the RDIV squared loss; the candidate used for discrepancy selection is $\hat h_{t,\lambda}$. In Corollary~\ref{cor:normality}: $\hat h=\hat h_{\lambda_{\mathrm{dp}}}$ is the one-step adaptive adversarial Tikhonov estimate, with the primal tuning rule run at confidence level $\xi_n/2$ and with the common deterministic sequence $\delta_n$ specified for both nuisance problems. Its evaluations use jointly measurable versions relative to the full nuisance-training and tuning sigma-field $\mathcal G_n$ and the observation, representing the fitted function for almost every training realization and agreeing almost surely with its evaluations at every observation in the finite evaluation sample. In Proposition~\ref{prop:exist_lambda_deepiv}: $\hat h_\lambda=\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\mathbb E_n[h(X)^2]\}$ with $L_n(h)=\mathbb E_{n,\mathrm{fit}}[(Y-(\hat{\mathcal T}h)(Z))^2]$. In Proposition~\ref{prop:exist_lambda}: $\hat h_\lambda=\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\mathbb E_n[h(X)^2]\}$, where $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$. In Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}: $\hat h_{0,\lambda}=0$ and $\hat h_{j,\lambda}=\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\|h-\hat h_{j-1,\lambda}\|_{2,n}^2\}$ for $j=1,\ldots,t$, with the adversarial loss; the candidate used for discrepancy selection is $\hat h_{t,\lambda}$.
- $\hat q$: $\hat q$ denotes the dual empirical adversarial Tikhonov fit at the discrepancy-selected dual penalty. In Corollary~\ref{cor:normality}: The dual discrepancy rule is run at confidence level $\xi_n/2$, using the common deterministic sequence $\delta_n$ that satisfies \eqref{eq:delta_trae} for both nuisance problems. Take jointly measurable fitted evaluations of $\hat q$ representing the fitted function for almost every nuisance-training and tuning realization and agreeing almost surely with $\hat q(Z(W_i))$ simultaneously for $i=1,\ldots,n$ in the evaluation sample.
- $\beta$: $\beta>0$ is a source exponent satisfying $h_0=(\mathcal T^*\mathcal T)^{\beta/2}w_0$ for some $w_0\in\widetilde{\mathcal H}$. In Corollary~\ref{cor:normality}: The two applications use potentially different source exponents: $h_0=(\mathcal T^*\mathcal T)^{\beta_h/2}w_h$ and $q_0=(\mathcal T\mathcal T^*)^{\beta_q/2}w_q$, with source elements in their respective Hilbert spaces; $\beta_m=\max\{\beta_h,\beta_q\}$.
- $w_0$: $w_0\in\tilde{\mathcal H}$ is a chosen source witness satisfying $h_0=(\mathcal T^*\mathcal T)^{\beta/2}w_0$, with no uniqueness or unit-norm requirement. In Proposition~\ref{prop:exist_lambda_deepiv}, Proposition~\ref{prop:exist_lambda_iter_rdiv}: When $h_0=0$, take $w_0=0$ in the initial-acceptance argument.
- $\lambda$: $\lambda$ is the scalar regularization parameter. In the classical Hilbert-space construction \eqref{eq:tikh_sol} and Definition~\ref{def:dp}, finite penalties satisfy $0<\lambda<\infty$, with $\lambda=\infty$ denoting the zero-estimator convention. The value $\lambda=0$ is admitted only for the separate finite-dimensional computational convention of Section~\ref{sec:rkhs_computation}, which selects a minimum-Euclidean-norm solution on the retained spectral subspace; it does not define an unregularized solution of \eqref{eq:tikh_sol} for arbitrary $r_\delta$. In Theorem~\ref{thm:common_dp}, Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Corollary~\ref{cor:normality}: The tuning rule searches the deterministic finite grid $\Lambda_n=\{2\rho^k:k=0,\ldots,K_n\}$, where $1/2\leqslant\rho<1$ and $K_n=\lceil\log(2n^2)/\log(1/\rho)\rceil$; its selected value is denoted $\lambda_{\mathrm{dp}}$. In Proposition~\ref{prop:rkhs_qp}: $\lambda>0$ is a candidate penalty for the unrestricted quadratic program, before the numerical spectral restriction. In Lemma~\ref{eq:iter_trae_strong_propagation}: In each concentration application in the proof, $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ is a fixed candidate penalty.
- $r_\delta$: $r_\delta\in\tilde{\mathcal Q}$ is a perturbed right-hand side satisfying $\|r_\delta-r_0\|\leqslant\delta$.
- $\delta$: $\delta\geqslant0$ denotes the prescribed bound in $\|r_\delta-r_0\|\leqslant\delta$ in the classical inverse problem, and the radius of the population-norm restriction $\|f\|\leqslant\delta$ when used as the argument of $\mathfrak R_n(\delta;\mathcal F)$.
- $h_\lambda^\delta$: $h_\lambda^\delta$ is the minimizer over $h\in\tilde{\mathcal H}$ of $\|\mathcal T h-r_\delta\|^2+\lambda\|h\|^2$ for $0<\lambda<\infty$, with $h_\infty^\delta=0$ under the discrepancy rule.
- $\ell$: $\ell$ takes its meaning from its local binding: a prescribed residual-bracketing factor, an integer iteration index, or a real-valued loss function on $\mathbb R$. In Lemma~\ref{eq:iter_trae_strong_propagation}: In $M_{\leqslant j}=\max_{1\leqslant\ell\leqslant j}\{\lambda,\|f_\ell^*\|_\infty^2\}$ and the subsequent references to $f_\ell^*$, $\ell$ ranges over the integers $1,\ldots,j$. In \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:strong_convexity}: Put $M=\max\{1,B\}$. Here $\ell:\mathbb R\to\mathbb R$ is the globally Lipschitz loss $\ell(v)=M^2\bigl(\max\{-1,\min\{v,1\}\}\bigr)^2$, with Lipschitz constant $2M^2$; boundedness gives $\ell(h(X)/M)=h(X)^2$ for $h\in\mathcal H$. In Lemma~\ref{lemma:t_err_dp}: In the likelihood concentration argument, $\ell:\mathbb R\to\mathbb R$ is the fixed loss $\ell(v)=-\log\bigl(\max\{1+v,2^{-1/2}\}\bigr)$; it is $\sqrt2$-Lipschitz, satisfies $\ell(0)=0$, and obeys $\ell(f_g-1)=-\log f_g$. In Lemma~\ref{lemma:strong_convexity}: Put $M=\max\{1,B\}$. In the normalized squared-loss argument, $\ell:\mathbb R\to\mathbb R$ is the loss $\ell(u)=M^2\bigl(\max\{-1,\min\{u,1\}\}\bigr)^2$, which is $2M^2$-Lipschitz and satisfies $\ell(h(X)/M)=h(X)^2$ for $h\in\mathcal H$. In \cite[Localized Concentration (source label lem:local-concen)]{li2024regularized}, used in the proof of Lemma~\ref{lemma:t_err_dp}: In the likelihood concentration argument, $\ell:\mathbb R\to\mathbb R$ is the fixed loss $\ell(v)=-\log\bigl(\max\{1+v,2^{-1/2}\}\bigr)$; it is $\sqrt2$-Lipschitz, satisfies $\ell(0)=0$, and obeys $\ell(f_g-1)=-\log f_g$.
- $\xi$: $\xi\in(0,1)$ is a prescribed failure-probability level, with fixed numerical rescalings absorbed into the rate bounds and the level relabelled after intersecting finitely many event families. In Corollary~\ref{cor:normality}: The confidence level used in each primal and dual tuning rule is $\xi_n/2$, where $\xi_n\downarrow0$ is deterministic; the simultaneous nuisance bounds consequently have failure probability at most $\xi_n$.
- $\rho$: In the discrepancy grid, $\rho\in[1/2,1)$ is the fixed search factor, independent of $n$; the notation $\rho$ in the experimental designs of Section~\ref{sec:experiments} instead denotes their instrument-strength parameter, equal to the correlation of $(X,Z)$ in the linear Gaussian design.
- $\hat h_\lambda$: $\hat h_\lambda$ denotes the candidate solution at penalty $\lambda$, with the discrepancy search using the family indexed by $\lambda\in\Lambda_n$ and the candidate construction specified by the estimation method. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}: $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$, where $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Lemma~\ref{lemma:err_l_dp}, Lemma~\ref{lemma:err_r_dp}, Lemma~\ref{lemma:str_dp}, Corollary~\ref{cor:rdiv_discrepancy_bound}, Lemma~\ref{lemma:rdiv_accepted_bounds}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $\hat h_\lambda$ is a measurable selected minimizer over $h\in\mathcal H$ of $\mathbb E_{n,\mathrm{fit}}[(Y-\hat{\mathcal T}h(Z))^2]+\lambda\mathbb E_{n,\mathrm{fit}}[h(X)^2]$. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}: The candidate supplied to the search is $\hat h_{t,\lambda}$, with $\hat h_{0,\lambda}=0$ and measurable selected minimizers $\hat h_{j,\lambda}\in\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\|h-\hat h_{j-1,\lambda}\|_{2,n}^2\}$ for $j=1,\ldots,t$, using the RDIV fitting loss. In Corollary~\ref{cor:normality}: The primal candidate is the TRAE measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $\mathcal H$; the corresponding dual construction exchanges the primal and dual spaces and operators. For the selected primal fit $\hat h$, take jointly measurable fitted evaluations representing the fitted function for almost every nuisance-training and tuning realization and agreeing almost surely with $\hat h(X(W_i))$ simultaneously for $i=1,\ldots,n$ in the evaluation sample. In Proposition~\ref{prop:rkhs_qp}: $\widehat h_\lambda$ minimizes $L_{N,\eta}(h)+\lambda\mathbb E_N[h(X)^2]$ over the full RKHS $\mathcal H$, taking the minimum-RKHS-norm representative among functions with the same fitting-sample evaluations; it has the form $\widehat h_\lambda(\cdot)=\sum_{j=1}^N\widehat\alpha_{\lambda,j}k_{\mathcal H}(X_j,\cdot)$ with coefficients solving \eqref{eq:rkhs_qp}. In Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}: The candidate supplied to the search is $\hat h_{t,\lambda}$, with $\hat h_{0,\lambda}=0$ and measurable selected minimizers $\hat h_{j,\lambda}\in\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\|h-\hat h_{j-1,\lambda}\|_{2,n}^2\}$ for $j=1,\ldots,t$, using the TRAE adversarial loss.
- $D_n$: $D_n$ is the nonnegative discrepancy functional supplied to the search, evaluated at the candidate for each $\lambda\in\Lambda_n$, with its functional definition specified by the estimation method. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:err_l_dp}, Corollary~\ref{cor:rdiv_discrepancy_bound}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: $D_n(h)=\mathbb E_{n,\mathrm{cal}}[(\hat r(Z)-\hat{\mathcal T}h(Z))^2]$, the projected calibration residual; the same functional is evaluated at $\hat h_{t,\lambda}$ for iterative candidates. In Corollary~\ref{cor:normality}: For each nuisance problem, the discrepancy is its empirical adversarial loss: $D_n=L_n$ for the primal problem, with the corresponding loss after exchanging the primal and dual spaces and operators for the dual problem. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{lemma:trae_accepted_bounds}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}: $D_n(h)=L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$, with the same loss evaluated at the final iterate for iterative candidates.
- $A_n$: $A_n:\Lambda_n\to[0,\infty)$ supplies the noise level in the acceptance threshold $\tfrac32A_n(\lambda)$, with prescribed statistical-error bounds in the theoretical constructions and data-dependent noise-floor estimates in the experimental construction. In Theorem~\ref{thm:common_dp}: $A_n$ is prescribed, and is required to be nonincreasing only over $\Lambda_n\cap[\lambda^\circ,2]$. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}: $A_n=A_n^{\mathrm R}$, where $A_n^{\mathrm R}(\lambda)=\delta_n^2/\lambda+\nu_n^2$ for $\lambda\in\Lambda_n$. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}: $A_n=A_{n,t}^{\mathrm R}$, where $A_{n,t}^{\mathrm R}(\lambda)=s_{n,t}^{\mathrm R}/\lambda+\nu_n^2$ and $s_{n,t}^{\mathrm R}=a_t\delta_n^2$. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Corollary~\ref{cor:normality}: $A_n=A_n^{\mathrm T}$, where $A_n^{\mathrm T}(\lambda)=\delta_n^2$ for $\lambda\in\Lambda_n$, using the TRAE accuracy scale for the relevant problem. In Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}: $A_n=A_{n,t}^{\mathrm T}$, where $A_{n,t}^{\mathrm T}(\lambda)=\delta_n^2+\min\{\lambda,s_{n,t}^{\mathrm T}/\lambda\}$ and $s_{n,t}^{\mathrm T}=b_t\delta_n^2$; on the oracle segment $[\lambda_{\mathrm{IT}}^\circ,2]$ this equals $\delta_n^2+s_{n,t}^{\mathrm T}/\lambda$.
- $\mathcal S_n$: $\mathcal S_n\subseteq(0,2]$ is the prescribed stability domain for the generic comparisons, which are required at grid points in $\Lambda_n\cap\mathcal S_n$. In Theorem~\ref{thm:deepiv_rate}, Theorem~\ref{thm:tikhonv_rate}: $\mathcal S_n=(0,2]$. In Theorem~\ref{thm:iter_rdiv_rate}: $\mathcal S_n=[\lambda_{\mathrm{IR}}^\circ,2]$ in the application of the generic comparison argument. In Theorem~\ref{thm:iter_trae_rate}: $\mathcal S_n=[\lambda_{\mathrm{IT}}^\circ,2]$ in the application of the generic comparison argument.
- $\mathcal E_n$: $\mathcal E_n$ is a common event of probability at least $1-\xi$ on which the relevant comparisons hold simultaneously over their stated function and grid domains. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}: $\mathcal E_n$ is the common TRAE concentration event formed using $\eta_n=\xi/(K_n+1)$ and a union bound over $\Lambda_n$; it includes the uniform loss comparison, the localized squared-norm comparison, and $L_n(h_0)\leqslant C\delta_n^2$.
- $\gamma$: $\gamma$ is locally bound to its construction, with $\gamma\in[0,1]$ denoting the scalar multiplier in $\operatorname{star}(\mathcal F)=\{\gamma f:\gamma\in[0,1],f\in\mathcal F\}$. In Theorem~\ref{thm:common_dp}: $\gamma\in(0,1]$ is a fixed exponent in the strong-error comparison and the resulting oracle bound. In Theorem~\ref{thm:deepiv_rate}, Theorem~\ref{thm:tikhonv_rate}: When applying the oracle guarantee, the exponent is $\gamma=\min\{\beta,1\}/(1+\min\{\beta,1\})$. In Theorem~\ref{thm:iter_rdiv_rate}, Theorem~\ref{thm:iter_trae_rate}: When applying the oracle guarantee, the exponent is $\gamma=\beta/(\beta+1)$. In Lemma~\ref{lemma:iter_bias}: In the spectral bound, $\gamma>0$ is an arbitrary real exponent in $\sup_{0\leqslant u\leqslant1}u^\gamma\{\lambda/(u+\lambda)\}^{2j}\leqslant C_{\gamma,j}\lambda^{\min\{\gamma,2j\}}$; it is then specialized to $\beta$ and $\beta+1$. In Proposition~\ref{prop:rkhs_qp}: $\gamma\in\mathbb R^N$ is the critic coefficient vector in $f_\gamma(\cdot)=\sum_{j=1}^N\gamma_j k_{\mathcal F}(Z_j,\cdot)$, over which the finite-dimensional critic objective is maximized.
- $\lambda^\circ$: $\lambda^\circ\in\Lambda_n$ is any oracle comparison point satisfying $\Lambda_n\cap[\lambda^\circ,2]\subseteq\mathcal S_n$, nonincreasing $A_n$ on that grid segment, and $B_n(\lambda^\circ)\leqslant\tfrac12A_n(\lambda^\circ)$; no uniqueness or computable selection rule is required.
- $\lambda_{\mathrm{dp}}$: $\lambda_{\mathrm{dp}}=2\rho^{k_*}$, where $k_*=\min\{k\in\{0,\ldots,K_n\}:D_n(\hat h_{2\rho^k})\leqslant\tfrac32A_n(2\rho^k)\}$ if this set is nonempty, and $k_*=K_n$ otherwise, using the candidates, discrepancy, and noise level supplied to the rule. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}: The rule uses the one-step RDIV candidates $\hat h_\lambda$, the projected calibration discrepancy, and $A_n^{\mathrm R}(\lambda)=\delta_n^2/\lambda+\nu_n^2$. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}: The rule uses the one-step TRAE candidates, $D_n=L_n$, and $A_n^{\mathrm T}(\lambda)=\delta_n^2$. In Corollary~\ref{cor:normality}: The primal and dual TRAE problems are tuned separately at confidence levels $\xi_n/2$ using a common deterministic accuracy bound $\delta_n$; the dual output is $\lambda_{\mathrm{dp}}^{\text{dual}}$. In Proposition~\ref{prop:exist_lambda_iter_rdiv}: The rule uses the terminal iterative RDIV candidates $\hat h_{t,\lambda}$, the projected calibration discrepancy, and $A_{n,t}^{\mathrm R}(\lambda)=s_{n,t}^{\mathrm R}/\lambda+\nu_n^2$; its selected parameter is denoted $\lambda_{\mathrm{dp},t}^{\mathrm R}$. In Proposition~\ref{prop:exist_lambda_iter_trae}: The rule uses the terminal iterative TRAE candidates and $A_{n,t}^{\mathrm T}(\lambda)=\delta_n^2+\min\{\lambda,s_{n,t}^{\mathrm T}/\lambda\}$.
- $g_0$: $g_0(\cdot\mid z)\in\mathcal G$ is the true conditional density of $X$ given $Z=z$ with respect to $\mu$, normalized by $\int g_0(x\mid z)\,\mu(dx)=1$ for $P_Z$-almost every $z$, with $g_0(x\mid z)\geqslant c_0$ on the common conditional-density support.
- $\hat r$: $\hat r$ is the pilot regression estimate of $r_0(z)=\mathbb E[Y\mid Z=z]$ fitted from $Y$ on $Z$ using $I_{\mathrm{pre}}$.
- $I_{\mathrm{pre}}$: In conditioning expressions, $I_{\mathrm{pre}}$ denotes the observed preliminary sample, so conditioning on $I_{\mathrm{pre}}$ means conditioning on $\sigma(W_i:i\in I_{\mathrm{pre}})$.
- $I_{\mathrm{fit}}$: $I_{\mathrm{fit}}$ is the fitting block of the three-way RDIV split, with $|I_{\mathrm{fit}}|=n$, and $\mathbb E_n=\mathbb E_{n,\mathrm{fit}}$ in the RDIV analysis.
- $I_{\mathrm{cal}}$: $I_{\mathrm{cal}}$ is the calibration block of the three-way RDIV split, with $|I_{\mathrm{cal}}|=n$.
- $L_n$: $L_n$ denotes the empirical fitting loss on $\mathcal H$, with $\mathbb E_n$ taken over the fitting observations of the applicable procedure. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:err_l_dp}, Lemma~\ref{lemma:err_r_dp}, Lemma~\ref{lemma:str_dp}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:iter_bias}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $L_n(h)=\mathbb E_{n,\mathrm{fit}}[(Y-\hat{\mathcal T}h(Z))^2]$; the iterative procedure retains this loss and changes the penalty center. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Corollary~\ref{cor:normality}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}: $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$, with $\mathbb E_n$ the TRAE fitting-sample average; the iterative procedure retains this loss and changes the penalty center. In \cite[Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:iter_rdiv_stability}, Lemma (Recursive RDIV stability): $L_n(h)=\mathbb E_{n,\mathrm{fit}}[(Y-\hat{\mathcal T}h(Z))^2]$; changing the penalty center in a concentration comparison does not change this fitting loss. In \cite[Lemma 9 (lem:concentration)]{bennett2023source}, used in the proof of Lemma~\ref{eq:iter_trae_strong_propagation}: $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$, where $\mathbb E_n$ is the TRAE fitting-sample average; changing the penalty center in a concentration comparison does not change this fitting loss.
- $L$: $L$ denotes the method-specific population loss functional, defined using population expectations. In Lemma~\ref{lemma:t_err_dp}, Lemma~\ref{lemma:err_l_dp}, Lemma~\ref{lemma:err_r_dp}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $L(h)=\mathbb E[(Y-\mathcal T h(Z))^2]=\sigma^2_{Y\mid Z}+\|\mathcal T(h-h_0)\|^2$, where $\sigma^2_{Y\mid Z}=\mathbb E[(Y-r_0(Z))^2]$. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Corollary~\ref{cor:normality}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}: $L(h)=\sup_{f\in\mathcal F}\mathbb E[2m(W;f)-2h(X)f(Z)-f(Z)^2]$; under the stated closeness assumption, $L(h)=\|\mathcal T(h-h_0)\|^2$ for $h\in\mathcal H$. In \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:strong_convexity}: $L(h)=\sup_{f\in\mathcal F}\mathbb E[2m(W;f)-2h(X)f(Z)-f(Z)^2]$; under the stated closeness assumption, $L(h)=\|\mathcal T(h-h_0)\|^2$ for $h\in\mathcal H$. In \cite[Lemma 9 (lem:concentration)]{bennett2023source}, used in the proof of Lemma~\ref{eq:iter_trae_strong_propagation}: $L(h)=\sup_{f\in\mathcal F}\mathbb E[2m(W;f)-2h(X)f(Z)-f(Z)^2]$; under the stated closeness assumption, $L(h)=\|\mathcal T(h-h_0)\|^2$ for $h\in\mathcal H$.
- $B$: $B$ is a fixed finite nonnegative uniform envelope bound for the quantities specified in the applicable boundedness assumptions. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Corollary~\ref{cor:normality}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:strong_convexity}: $|m(W;f)|$, $|h(X)|$, and $|f(Z)|$ are almost surely bounded by $B$, uniformly over $h\in\mathcal H$ and $f\in\mathcal F$; for a dual fit the corresponding bound uses the exchanged primal and dual objects. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:t_err_dp}, Lemma~\ref{lemma:err_l_dp}, Corollary~\ref{cor:rdiv_discrepancy_bound}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $|Y|\leqslant B$ almost surely, $|h(x)|\leqslant B$ for every $h\in\mathcal H$ on the common support of the conditional densities, and $|\hat r(z)|\leqslant B$. In \cite[Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:iter_rdiv_stability}, Lemma (Recursive RDIV stability): $|Y|\leqslant B$ almost surely, $|h(x)|\leqslant B$ for every $h\in\mathcal H$ on the common support of the conditional densities, and $|\hat r(z)|\leqslant B$. In \cite[Lemma 9 (lem:concentration)]{bennett2023source}, used in the proof of Lemma~\ref{eq:iter_trae_strong_propagation}: $|m(W;f)|$, $|h(X)|$, and $|f(Z)|$ are almost surely bounded by $B$, uniformly over $h\in\mathcal H$ and $f\in\mathcal F$.
- $c_0$: $c_0>0$ is a fixed finite constant, independent of sample size, such that $g_0(x\mid z)\geqslant c_0$ on the common conditional-density support.
- $C_G$: $C_G>0$ is a fixed finite constant, independent of sample size, such that $\sup_{g\in\mathcal G}\|g\|_\infty\leqslant C_G$ on the common conditional-density support.
- $\mathcal T_g$: For every $g\in\mathcal G$, $\mathcal T_g$ maps $h\in\mathcal H$ to the $P_Z$-square-integrable function $(\mathcal T_g h)(z)=\int h(x)g(x\mid z)\,\mu(dx)$, with $z$ understood $P_Z$-almost everywhere.
- $\mathcal G_0$: $\mathcal G_0$ is the measurable class $\{(x,z)\mapsto\sqrt{\{g(x\mid z)+g_0(x\mid z)\}/\{2g_0(x\mid z)\}}:g\in\mathcal G\}$ on the common conditional-density support, with input distribution the true joint law of $(X,Z)$ and positive denominator there.
- $\eta_n$: $\eta_n=\xi/(K_n+1)$ is the deterministic grid confidence allocation at the prescribed failure level $\xi$. In Corollary~\ref{cor:normality}: For each of the primal and dual tuning rules, the prescribed failure level is $\xi_n/2$, so $\eta_n=\xi_n/\{2(K_n+1)\}$, where $\xi_n\downarrow0$ is deterministic.
- $\delta_{n,r}$: $\delta_{n,r}=\delta_{n,r}(\xi)\geqslant0$ is a prescribed deterministic, $\beta$-independent accuracy bound such that $\|\hat r-r_0\|_{L_2(Z)}\leqslant\delta_{n,r}$ with probability at least $1-\xi$ over the preliminary sample. In Lemma~\ref{lemma:t_err_dp}: The pilot-regression accuracy bound is evaluated at failure probability $\xi/2$ and relabelled $\delta_{n,r}$, so here $\delta_{n,r}$ denotes $\delta_{n,r}(\xi/2)$ and satisfies $\|\hat r-r_0\|_{L_2(Z)}\leqslant\delta_{n,r}$ with probability at least $1-\xi/2$ over the preliminary sample.
- $\delta_{n,\mathcal G}$: $\delta_{n,\mathcal G}$ is a prescribed deterministic, $\beta$-independent upper bound on the critical radius of $\mathcal G_0$ under the joint input distribution of $(X,Z)$, evaluated at the failure probability allocated to the conditional-density fit and satisfying $\delta_{n,\mathcal G}\geqslant C\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. In Lemma~\ref{lemma:t_err_dp}: The conditional-density bound is evaluated at failure probability $\xi/2$ and relabelled $\delta_{n,\mathcal G}$, so here $\delta_{n,\mathcal G}$ denotes $\delta_{n,\mathcal G}(\xi/2)$.
- $\delta_{n,\mathcal H}$: $\delta_{n,\mathcal H}=\delta_{n,\mathcal H}(\eta_n)$ is a prescribed deterministic common upper bound on the critical radii of $\mathcal H$ under the distribution of $X$ and of $\mathcal T_g\mathcal H$ under the distribution of $Z$, uniformly for $g\in\mathcal G$, with $\eta_n=\xi/(K_n+1)$ and $\delta_{n,\mathcal H}\geqslant C\sqrt{\{\log\log(en)+\log(1/\eta_n)\}/n}$. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: The iterative concentration comparisons allocate failure probability $\eta_{n,t}=\xi/\{t(K_n+1)\}$ over iteration--grid pairs; the resulting confidence adjustment is absorbed into $\delta_n$ for fixed $t$.
- $\delta_n$: $\delta_n$ is a prescribed deterministic, method-specific statistical-error bound varying with $n$, calibrated before fitting by fixed multipliers independent of $n$, $\lambda$, and the unknown source parameter. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: The iteration--grid confidence allocation is $\eta_{n,t}=\xi/\{t(K_n+1)\}$; the resulting $\log t$ contribution is absorbed into a constant adjustment of $\delta_n$ for fixed $t$. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:strong_convexity}: $\delta_n$ is the TRAE scale, unrelated to the RDIV scales, satisfying $\delta_n\geqslant\max\{\tilde\delta_n(\eta_n),n^{-1/2}\}$ and $\delta_n=o(1)$, with $\eta_n=\xi/(K_n+1)$. In Corollary~\ref{cor:normality}: $\delta_n$ is a common deterministic TRAE bound for both primal and dual problems, each tuned at confidence level $\xi_n/2$ for a deterministic $\xi_n\downarrow0$, and additionally satisfies $\delta_n=o(n^{-\alpha_{\mathrm{dr}}})$, where $\alpha_{\mathrm{dr}}=(1+\min\{\beta_m,1\})/(2+4\min\{\beta_m,1\})$ and $\beta_m=\max\{\beta_h,\beta_q\}$. In Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}: $\delta_n$ is the TRAE scale satisfying $\delta_n\geqslant\max\{\tilde\delta_n(\eta_{n,t}),n^{-1/2}\}$ and $\delta_n=o(1)$, with $\eta_{n,t}=\xi/\{t(K_n+1)\}$. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:t_err_dp}, Lemma~\ref{lemma:err_l_dp}, Lemma~\ref{lemma:err_r_dp}, Lemma~\ref{lemma:str_dp}, Corollary~\ref{cor:rdiv_discrepancy_bound}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $\delta_n$ is the RDIV fitting scale satisfying $\delta_n\geqslant\delta_{n,\mathcal G}+\delta_{n,\mathcal H}+\delta_{n,\xi}$ and $\delta_n+\nu_n=o(1)$.
- $\nu_n$: $\nu_n$ is a prescribed deterministic RDIV projected-discrepancy accuracy bound, fixed before fitting the candidates, with $\nu_n\geqslant\delta_{n,\mathcal G}+\delta_{n,r}+\delta_{n,\xi}$ and $\delta_n+\nu_n=o(1)$; its calibration multipliers are independent of $n$, $\lambda$, and the unknown source parameter. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: The bounds use the iteration--grid confidence allocation $\eta_{n,t}=\xi/\{t(K_n+1)\}$; the resulting $\log t$ contribution is absorbed into a constant adjustment of $\nu_n$ for the prescribed fixed $t$. In \cite[Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:iter_rdiv_stability}, Lemma (Recursive RDIV stability): The bounds use the iteration--grid confidence allocation $\eta_{n,t}=\xi/\{t(K_n+1)\}$; the resulting $\log t$ contribution is absorbed into a constant adjustment of $\nu_n$ for the prescribed fixed $t$.
- $\tilde\delta_n$: $\tilde\delta_n(\xi)$ is a prescribed common critical-radius upper bound indexed by sample size $n$ and failure probability $\xi\in(0,1)$ for the five star-shaped classes specified in Assumption~\ref{asp:critical_radius}.
- $\alpha$: $\alpha$ is local notation whose scalar-exponent and RKHS-coefficient meanings are distinct. In Proposition~\ref{prop:rkhs_qp}: $\alpha\in\mathbb R^N$ is the optimization variable representing $h_\alpha(\cdot)=\sum_{j=1}^N\alpha_j k_{\mathcal H}(X_j,\cdot)$; its fitting-sample evaluation vector is $K_{\mathcal H}\alpha$. In Theorem~\ref{thm:deepiv_rate}, Lemma~\ref{lemma:variational_source}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:strong_convexity}, Lemma~\ref{lemma:trae_accepted_bounds}: $\alpha=\min\{\beta,1\}/(1+\min\{\beta,1\})=(p-1)/p$ is the scalar interpolation exponent.
- $\lambda_{\mathrm R}^\circ$: Whenever the defining set is nonempty, $\lambda_{\mathrm R}^\circ$ is the deterministic oracle point $2\rho^{k^\circ}$, where $k^\circ=\min\{k\in\{0,\ldots,K_n\}:2\rho^k\leqslant c_{\mathrm R}\max\{\delta_n^{2/(p+1)},\nu_n^{2/p}\}\}$ and $c_{\mathrm R}>0$ is a sufficiently small fixed constant; this point exists for all sufficiently large $n$.
- $\lambda_{\mathrm T}^\circ$: For all sufficiently large $n$, $\lambda_{\mathrm T}^\circ$ is the deterministic oracle point $2\rho^{k^\circ}$, where $k^\circ=\min\{k\in\{0,\ldots,K_n\}:2\rho^k\leqslant c_{\mathrm T}\delta_n^{2/p}\}$, $c_{\mathrm T}>0$ is a sufficiently small fixed constant, and $\delta_n$ is the TRAE scale in \eqref{eq:delta_trae}.
- $t$: $t\geqslant1$ is a prescribed integer iteration count held fixed as $n$ varies and chosen independently of the data and the source parameter $\beta$.
- $a_t$: $a_t\geqslant1$ is a deterministic calibrated RDIV propagation constant depending only on the prescribed $t$ and fixed population quantities, independent of $n$ and $\lambda$.
- $\lambda_{\mathrm{IR}}^\circ$: Whenever the defining set is nonempty, $\lambda_{\mathrm{IR}}^\circ$ is the deterministic oracle point $2\rho^{k^\circ}$, where $k^\circ=\min\{k\in\{0,\ldots,K_n\}:2\rho^k\leqslant c_{\mathrm{IR}}\max\{(s_{n,t}^{\mathrm R})^{1/(q_t+1)},\nu_n^{2/q_t}\}\}$ and $c_{\mathrm{IR}}>0$ is a sufficiently small fixed constant; this point exists for all sufficiently large $n$.
- $b_t$: $b_t\geqslant2$ is a calibrated deterministic propagation constant depending only on the prescribed iteration count $t$ and the fixed boundedness and mean-square-continuity constants, independently of $n$, $\beta$, and $\|w_0\|$. In Lemma~\ref{eq:iter_trae_strong_propagation}: $b_t$ is the terminal member of a nondecreasing sequence with $b_0=0$ and $b_j\geqslant2$ for $j=1,\ldots,t$; in the proof one may choose $b_j=C(1+b_{j-1})$ for a sufficiently large fixed $C\geqslant2$.
- $\lambda_{\mathrm{IT}}^\circ$: For all sufficiently large $n$, $\lambda_{\mathrm{IT}}^\circ$ is the deterministic oracle point $2\rho^{k^\circ}$, where $k^\circ=\min\{k\in\{0,\ldots,K_n\}:2\rho^k\leqslant c_{\mathrm{IT}}(s_{n,t}^{\mathrm T})^{1/(q_t+1)}\}$ and $c_{\mathrm{IT}}>0$ is a sufficiently small fixed constant.
- $h_{j,\lambda}^*$: $h_{j,\lambda}^*$ is the population minimizer over $\tilde{\mathcal H}$ for positive finite $\lambda$, initialized by $h_{0,\lambda}^*=0$ and recursively defined for positive integer $j$; the iterative procedures use $j=1,\ldots,t$. In Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}: $h_{j,\lambda}^*\in\mathcal H$ for $j=1,\ldots,t$ and $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$, for every $n$ for which the oracle point is defined. In Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: $h_{j,\lambda}^*\in\mathcal H$ for $j=1,\ldots,t$ and $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$, for every $n$ for which the oracle point is defined.
- $\hat h_{j,\lambda}$: $\hat h_{j,\lambda}$ denotes a selected measurable empirical minimizer in the recursion initialized by $\hat h_{0,\lambda}=0$, with $j=1,\ldots,t$ and the empirical penalty norm evaluated on the fitting sample. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: The loss in the recursion is $L_n(h)=\mathbb E_{n,\mathrm{fit}}[(Y-\hat{\mathcal T}h(Z))^2]$. In Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}: The loss in the recursion is the adversarial loss $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$. In Lemma~\ref{lemma:iter_rdiv_stability}, Lemma~\ref{lemma:iter_trae_accepted_stability}: The empirical penalty center remains $\hat h_{j-1,\lambda}$; the fixed-center concentration comparisons use population iterates as centers. In \cite[Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:iter_rdiv_stability}, Lemma (Recursive RDIV stability): The empirical iterates use $L_n(h)=\mathbb E_{n,\mathrm{fit}}[(Y-\hat{\mathcal T}h(Z))^2]$ and penalty center $\hat h_{j-1,\lambda}$; the fixed-center concentration comparisons use the deterministic population iterates as comparators and centers. In \cite[Lemma 9 (lem:concentration)]{bennett2023source}, used in the proof of Lemma~\ref{eq:iter_trae_strong_propagation}: The empirical iterates use $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$ and penalty center $\hat h_{j-1,\lambda}$; the fixed-center concentration comparisons use the deterministic population iterates as comparators and centers.
- $a_j$: $\{a_j\}_{j=0}^t$ is a deterministic RDIV propagation sequence initialized at $a_0=0$, with $a_j\geqslant1$ for $1\leqslant j\leqslant t$; each positive-index term depends only on $j$ and fixed population quantities and is independent of $n$ and $\lambda$.
- $b_j$: $\{b_j\}_{j=0}^t$ is a nondecreasing deterministic propagation sequence with $b_0=0$ and $b_j\geqslant2$ for $1\leqslant j\leqslant t$, each positive-index term depending only on $j$ and the fixed boundedness and mean-square-continuity constants, independently of $\beta$ and $\|w_0\|$.
- $\mathcal E_n^{\mathrm R}$: $\mathcal E_n^{\mathrm R}$ is the intersection of the preliminary, fitting, and calibration events under the paper's confidence convention, with probability at least $1-\xi$, on which the fitting comparisons hold simultaneously for every $h\in\mathcal H$ and $\lambda\in\Lambda_n$, and the projected calibration comparisons hold simultaneously for the fitted candidates $\hat h_\lambda$, $\lambda\in\Lambda_n$.
- $\mathcal E_{n,t}^{\mathrm{IR}}$: $\mathcal E_{n,t}^{\mathrm{IR}}$ is the common preliminary, recursive fitting, and final calibration event constructed with allocation $\eta_{n,t}=\xi/\{t(K_n+1)\}$, of probability at least $1-\xi$, on which the recursive RDIV stability comparisons hold simultaneously for $j=1,\ldots,t$ and $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$, and the projected calibration comparison holds for the final candidates on that same grid segment.
- $\mathcal E_{n,t}^{\mathrm{IT}}$: $\mathcal E_{n,t}^{\mathrm{IT}}$ is the common iterative TRAE concentration event obtained by intersecting the requisite iteration--grid concentration events with failure-probability allocation $\eta_{n,t}=\xi/\{t(K_n+1)\}$, with probability at least $1-\xi$ under the paper's confidence convention. In Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}: The concentration comparisons are simultaneous over the oracle-relevant grid segment $\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$, including the recursive comparisons for $j=1,\ldots,t$.
- $b$: $b=\min\{\beta,1\}\in(0,1]$ is the truncated source exponent.
- $r_\lambda$: $r_\lambda(u)=\lambda/(u+\lambda)$ for finite $\lambda>0$ and $u\geqslant0$, and $r_\lambda(\mathcal A)$ denotes this function applied to $\mathcal A$ by spectral functional calculus.
- $H^2$: $H^2(g_1,g_2)=\int(\sqrt{g_1(x)}-\sqrt{g_2(x)})^2\,\mu(dx)$ is defined for probability densities $g_1,g_2$ with respect to the common measure $\mu$ on $\mathcal X$; for conditional densities it is evaluated at each fixed $z$, with any averaging over $Z$ written separately.
- $\beta_h$: $\beta_h>0$ is a source exponent satisfying $h_0=(\mathcal T^*\mathcal T)^{\beta_h/2}w_h$ for some $w_h\in\tilde{\mathcal H}$; it need not be the largest exponent for which such a representation exists.
- $\beta_q$: $\beta_q>0$ is a source exponent satisfying $q_0=(\mathcal T\mathcal T^*)^{\beta_q/2}w_q$ for some $w_q\in\tilde{\mathcal Q}$; it need not be the largest exponent for which such a representation exists.
- $\boldsymbol y$: $\boldsymbol y$ denotes the vector of outcomes on the fitting block, on the outcome scale used by the fit. In Proposition~\ref{prop:rkhs_qp}: $\boldsymbol y=(Y_1,\ldots,Y_N)^\top$, with $N=n_1$ the fitting-sample size.

**Proofs completed** (a step of the printed argument could not be checked as it stands, so it was completed; no error was found).

- Lemma C.2

## 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 3.1 | proved in Lean, under 1 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.common_dp` (`TCommonDp.lean:24`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Proposition 3.5 | proved in Lean, given 2 cited result(s), under 5 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.exist_lambda_deepiv` (`TExistLambdaDeepiv.lean:65`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Theorem 3.6 | proved in Lean, given 2 cited result(s), under 8 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.deepiv_rate` (`TDeepivRate.lean:64`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Proposition 3.8 | proved in Lean, given 4 cited result(s), under 7 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.exist_lambda_iter_rdiv` (`TExistLambdaIterRdiv.lean:196`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Theorem 3.9 | proved in Lean, given 4 cited result(s), under 6 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.iter_rdiv_rate` (`TIterRdivRate.lean:56`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Proposition 3.13 | proved in Lean, given 2 cited result(s), under 5 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.exist_lambda` (`TExistLambda.lean:122`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Theorem 3.14 | proved in Lean, given 2 cited result(s), under 7 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.tikhonov_rate` (`TTikhonvRate.lean:45`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Proposition 3.17 | proved in Lean, given 2 cited result(s), under 5 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.exist_lambda_iter_trae` (`TExistLambdaIterTrae.lean:118`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Theorem 3.18 | proved in Lean, given 4 cited result(s), under 10 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.iter_trae_rate` (`TIterTraeRate.lean:392`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Corollary 3.19 | proved in Lean (the corrected statement), given 2 cited result(s), under 7 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.dr_asymptotic_normality` (`TNormality.lean:41`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma A.1 | proved in Lean | `Papers.TsAdaptiveEstimationInference.variational_source_uniform` (`Helpers/SpectralRegularization.lean:544`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma B.1 | proved in Lean, given 1 cited result(s), under 2 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.t_err_dp` (`Helpers/RDIV.lean:85`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma B.2 | proved in Lean, given 2 cited result(s), under 3 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.err_l_dp` (`Helpers/RDIV.lean:354`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma B.3 | proved in Lean, given 2 cited result(s), under 3 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.err_r_dp` (`Helpers/RDIV.lean:605`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma B.4 | proved in Lean, given 2 cited result(s), under 3 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.str_dp` (`Helpers/RDIV.lean:705`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Corollary B.5 | proved in Lean, given 2 cited result(s), under 4 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.rdiv_discrepancy_bound` (`TRdivDiscrepancyBound.lean:411`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma B.6 | proved in Lean, given 2 cited result(s), under 4 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.rdiv_accepted_bounds` (`Helpers/RDIV.lean:833`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma B.7 | proved in Lean | `Papers.TsAdaptiveEstimationInference.iter_bias` (`Helpers/IterBias.lean:382`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma B.8 | taken from the literature as cited; not proved here | — | — |
| Corollary B.9 | proved in Lean, given 4 cited result(s), under 4 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.iter_rdiv_discrepancy_bound` (`TIterRdivDiscrepancyBound.lean:98`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma B.10 | proved in Lean, given 4 cited result(s), under 3 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.iter_rdiv_accepted_bounds` (`Helpers/IterRDIV.lean:329`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma C.1 | proved in Lean, given 1 cited result(s), under 2 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.concentrate_l` (`Helpers/TRAELossEvents.lean:436`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma C.2 | proved in Lean, given 2 cited result(s), under 3 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.strong_convexity` (`Helpers/TRAE.lean:233`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Corollary C.3 | proved in Lean, given 2 cited result(s), under 4 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.trae_discrepancy_bound` (`TTraeDiscrepancyBound.lean:35`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma C.4 | proved in Lean, given 2 cited result(s), under 3 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.trae_accepted_bounds` (`Helpers/TRAE.lean:284`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma C.2 | proved in Lean, given 3 cited result(s), under 4 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.iter_trae_strong_propagation` (`Helpers/IterTRAEPropagation.lean:38`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Corollary C.6 | proved in Lean, given 3 cited result(s), under 5 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.iter_trae_discrepancy_bound` (`TIterTraeDiscrepancyBound.lean:53`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Lemma C.7 | proved in Lean, given 3 cited result(s), under 5 condition(s) made explicit | `Papers.TsAdaptiveEstimationInference.iter_trae_accepted_stability` (`Helpers/IterTRAEStability.lean:282`) | the standard ones only (propext, Classical.choice, Quot.sound) |
| Proposition D.1 | proved in Lean | `Papers.TsAdaptiveEstimationInference.rkhs_qp` (`TRkhsQp.lean:19`) | the standard ones only (propext, Classical.choice, Quot.sound) |

**Taken from the literature, not proved here.**

- foster2023orthogonal, Lemma 14 — checked against source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt

  What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.LipschitzVectorLocalizedConcentration` (`Helpers/RDIVSetup.lean:896`)

  ```lean
  noncomputable def LipschitzVectorLocalizedConcentration
      (V : IVWorld 𝒲 𝒳 𝒵) (P : Measure Ω)
      (sample : AnalysisSample V P) (split : SampleSplit) : Sort 0 :=
    IsProbabilityMeasure P →
    ∀ (mu : Measure 𝒳) (G : Set (𝒳 → 𝒵 → ℝ)) (g₀ : 𝒳 → 𝒵 → ℝ)
      (gHat : ℕ → Ω → 𝒳 → 𝒵 → ℝ) (support : Set (𝒳 × 𝒵))
      (B c₀ CG ρ : ℝ),
      RDIVSetting V mu g₀ → RangeCondition V → H0MemHypothesisClass V →
      DensityRealizable V mu G g₀ → DensityBounds G g₀ support c₀ CG →
      OutcomeBounded V B → HypothesisClassBounded V support B →
      MLEMaximizerSelected V P sample split G gHat →
      RDIVClassesMeasurable V P sample split mu G g₀ gHat →
      RDIVFittingSupremumVersion V P sample split mu gHat ρ →
      ∃ C > 0, ∀ (n : ℕ) (ζ r lam : ℝ), 0 < n →
        ζ ∈ Set.Ioo (0 : ℝ) 1 → lam ∈ regGrid ρ n → lam ∈ Set.Ioc (0 : ℝ) 2 →
        confidenceFloor n ζ ≤ r →
        IsCriticalRadiusBound (V.law.map V.xOf) (primalVersion V '' V.H) n r →
        (∀ g ∈ G, IsCriticalRadiusBound (V.law.map V.zOf)
          (densityOperator mu g '' (primalVersion V '' V.H)) n r) →
        ∀ᵐ preliminary ∂P,
          1 - ζ ≤ P.real {ω | ∀ h ∈ V.H,
            |(conditionalRisk V mu gHat n preliminary lam h -
                conditionalRisk V mu gHat n preliminary lam (minNormSolution V)) -
              (empiricalRisk V P sample split mu (fun _ _ => gHat n preliminary) n ω lam h -
                empiricalRisk V P sample split mu (fun _ _ => gHat n preliminary) n ω lam
                  (minNormSolution V))| ≤
            C * (r * inputL2Norm (V.law.map V.zOf)
              (densityOperator mu (gHat n preliminary)
                (primalVersion V h - primalVersion V (minNormSolution V))) +
              r * ‖h - minNormSolution V‖ + r ^ 2)}
  ```

  The passage of the proof:

  > For each fixed $\lambda\in\Lambda_n$, apply the localized concentration inequality in \citet[Lemma~14]{foster2023orthogonal} to the two-coordinate class
  > \[
  > h\longmapsto\{\hat{\mathcal T}h(Z),h(X)\}
  > \]
  > at the fixed comparator $h_0$ and to the loss
  > $\phi_\lambda(y,u,v)=(y-u)^2+\lambda v^2$. This loss is uniformly Lipschitz on the bounded range because $\lambda\leqslant2$. The critical-radius bounds in \cref{asp:cr_deepiv} give
  > \[
  > \left|\{\widehat R^\lambda(h)-\widehat R^\lambda(h_0)\}
  > -\{R_n^\lambda(h)-R_n^\lambda(h_0)\}\right|
  > \leqslant C\left[\delta_{n,\mathcal H}
  > \{\|\hat{\mathcal T}(h-h_0)\|+\|h-h_0\|\}
  > +\delta_{n,\mathcal H}^2\right].
  > \]
- li2024regularized, Localized Concentration (source label lem:local-concen) — checked against source supplied by the user at discovery/cited_sources/li2024regularized.txt

  What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.LiLocalizedConcentration` (`Helpers/RDIVSetup.lean:308`)

  ```lean
  noncomputable def LiLocalizedConcentration : Sort 0 :=
    ∀ (mu : Measure 𝒳) (G : Set (𝒳 → 𝒵 → ℝ)) (g₀ : 𝒳 → 𝒵 → ℝ)
      (support : Set (𝒳 × 𝒵)) (c₀ CG : ℝ),
      RDIVSetting V mu g₀ → DensityRealizable V mu G g₀ → DensityBounds G g₀ support c₀ CG →
      LocalizedProcessesMeasurable (V.law.map (fun w => (V.xOf w, V.zOf w)))
        (likelihoodTransformClass G g₀) →
      ∀ (δG : ℕ → ℝ → ℝ), DensityCriticalRadiusBound V G g₀ δG →
      (∃ c > 0, ∀ n : ℕ, 0 < n → ∀ ξ ∈ Set.Ioo (0 : ℝ) 1,
        c * confidenceFloor n ξ ≤ δG n ξ) →
      ∃ A > 0, ∀ n : ℕ, 0 < n → ∀ ξ ∈ Set.Ioo (0 : ℝ) 1,
        ∃ E : Set (Fin n → 𝒳 × 𝒵), MeasurableSet E ∧
          1 - ξ / 2 ≤ (Measure.pi (fun _ : Fin n =>
            V.law.map (fun w => (V.xOf w, V.zOf w)))).real E ∧
          ∀ x ∈ E, ∀ g ∈ G,
            let f := fun y : 𝒳 × 𝒵 =>
              Real.sqrt ((g y.1 y.2 + g₀ y.1 y.2) / (2 * g₀ y.1 y.2))
            |(n : ℝ)⁻¹ * (∑ i, -Real.log (f (x i))) -
              (∫ y, -Real.log (f y) ∂V.law.map (fun w => (V.xOf w, V.zOf w)))| ≤
              A * (δG n (ξ / 2) *
                inputL2Norm (V.law.map (fun w => (V.xOf w, V.zOf w)))
                  (f - fun _ => 1) + δG n (ξ / 2) ^ 2)
  ```

  The passage of the proof:

  > We establish the density bound directly using the Localized Concentration lemma of \citet{li2024regularized}. Write $P=\mathbb E_{X,Z}$ and $P_n=\mathbb E_{n,\mathrm{pre}}$, and define
  > \[
  > f_g(x,z)=\sqrt{\frac{g(x\mid z)+g_0(x\mid z)}{2g_0(x\mid z)}},\qquad
  > \mathcal U=\{f_g-1:g\in\mathcal G\}.
  > \]
  > Realizability gives $f_{g_0}=1$, so $0\in\mathcal U$. The density bounds give
  > \[
  > 2^{-1/2}\leqslant f_g\leqslant M,\qquad
  > M=\sqrt{\frac{C_G/c_0+1}{2}}.
  > \]
  > Thus $\mathcal U$ is uniformly bounded by a fixed constant. Define the globally Lipschitz loss
  > \[
  > \ell(v)=-\log\bigl(\max\{1+v,2^{-1/2}\}\bigr).
  > \]
  > The maximum map is $1$-Lipschitz, and the derivative of $-\log t$ has absolute value at most $\sqrt2$ for $t\geqslant2^{-1/2}$. Hence $\ell$ is $\sqrt2$-Lipschitz, $\ell(0)=0$, and $\ell(f_g-1)=-\log f_g$.
  > 
  > We verify the localized complexity condition for this application. Put $\mathcal S=\operatorname{star}(\mathcal G_0-\mathcal G_0)$ and choose a deterministic critical radius $d\leqslant\delta_{n,\mathcal G}$ as supplied by Assumption~\ref{asp:cr_deepiv}. By the critical-radius definition, $\mathfrak R_n(d;\mathcal S)\leqslant d^2$. For $s\geqslant d$, star-shapedness implies that $(d/s)v\in\mathcal S$ and $\|(d/s)v\|_{L_2(X,Z)}\leqslant d$ whenever $v\in\mathcal S$ and $\|v\|_{L_2(X,Z)}\leqslant s$. Scaling inside the Rademacher supremum therefore gives
  > \[
  > \mathfrak R_n(s;\mathcal S)
  > \leqslant\frac{s}{d}\mathfrak R_n(d;\mathcal S)
  > \leqslant sd.
  > \]
  > Since $1\in\mathcal G_0$, we have $\operatorname{star}(\mathcal U)\subseteq\mathcal S$. Taking $s=\delta_{n,\mathcal G}$ yields
  > \[
  > \mathfrak R_n\bigl(\delta_{n,\mathcal G};\operatorname{star}(\mathcal U)\bigr)
  > \leqslant\delta_{n,\mathcal G}d
  > \leqslant\delta_{n,\mathcal G}^2.
  > \]
  > Apply the cited localized concentration lemma with input $(X,Z)$, one coordinate, class $\mathcal U$, fixed comparator $0$, loss $\ell$, and failure probability $\xi/2$. The measurability and logarithmic radius floor are supplied by Assumption~\ref{asp:cr_deepiv}, with the confidence evaluation specified in the proof. Consequently, for a fixed constant $A>0$, with probability at least $1-\xi/2$,
  > \[
  > \left|(P_n-P)(-\log f_g)\right|
  > \leqslant A\left\{\delta_{n,\mathcal G}\|f_g-1\|_{L_2(X,Z)}+\delta_{n,\mathcal G}^2\right\},
  > \qquad g\in\mathcal G.
  > \]
- foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling — checked against source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt

  What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.IterRDIVFixedCenterLocalizedConcentration` (`Helpers/IterRDIVConcentration.lean:128`)

  ```lean
  noncomputable def IterRDIVFixedCenterLocalizedConcentration : Sort 0 :=
    ∀ (mu : Measure 𝒳) (G : Set (𝒳 → 𝒵 → ℝ)) (g₀ : 𝒳 → 𝒵 → ℝ)
      (support : Set (𝒳 × 𝒵)) (B c₀ CG : ℝ),
      RDIVSetting V mu g₀ → DensityRealizable V mu G g₀ →
      DensityBounds G g₀ support c₀ CG → OutcomeBounded V B →
      HypothesisClassBounded V support B →
      ∃ C > 0, ∃ c₁ > 0, ∃ c₂ > 0,
      ∀ (P : Measure Ω) (_hProbability : IsProbabilityMeasure P)
        (sample : AnalysisSample V P) (split : SampleSplit)
        (gHat : ℕ → Ω → 𝒳 → 𝒵 → ℝ) (ρ : ℝ),
      MLEMaximizerSelected V P sample split G gHat →
      RDIVClassesMeasurable V P sample split mu G g₀ gHat →
      IterRDIVFixedCenterSupremumVersion V P sample split mu gHat ρ →
      ∀ (n t j : ℕ) (lam r : ℝ) (sourceRadius : ℝ)
        (_hSourceScope : gridValue ρ (terminalIndex ρ n) ≤ sourceRadius ∧
          oraclePoint ρ n sourceRadius ≤ lam), 0 < n → 1 ≤ j → j ≤ t →
      lam ∈ regGrid ρ n → lam ∈ Set.Ioc (0 : ℝ) 2 →
      populationIterate V lam j ∈ V.H →
      (populationIterate V lam (j - 1) ∈ V.H ∨ j = 1) →
      0 < r → 1 < 2 * c₂ * n →
      8 * Real.log (41 * Real.log (2 * c₂ * n)) / (c₂ * n) ≤ r ^ 2 →
      iterRdivAnchoredRadius (V.law.map V.xOf) (primalVersion V '' V.H)
        (primalVersion V (populationIterate V lam j)) n r →
      (∀ g ∈ G, iterRdivAnchoredRadius (V.law.map V.zOf)
        (densityOperator mu g '' (primalVersion V '' V.H))
        (densityOperator mu g (primalVersion V (populationIterate V lam j))) n r) →
      ∀ᵐ a ∂P, ∃ E : Set (Fin n → 𝒲), MeasurableSet E ∧
        1 - 2 * c₁ * Real.exp (-c₂ * n * r ^ 2 / 8) ≤
          (Measure.pi (fun _ : Fin n => V.law)).real E ∧
        ∀ ws ∈ E, ∀ h ∈ V.H,
          let increment := iterRdivFixedCenterLoss V mu (gHat n a) lam j h -
            iterRdivFixedCenterLoss V mu (gHat n a) lam j (populationIterate V lam j)
          |(n : ℝ)⁻¹ * (∑ i, increment (ws i)) - ∫ w, increment w ∂V.law| ≤
            C * r * (inputL2Norm (V.law.map V.zOf)
              (densityOperator mu (gHat n a) (primalVersion V h -
                primalVersion V (populationIterate V lam j))) +
              ‖h - populationIterate V lam j‖ + r)
  ```

  The passage of the proof:

  > The fitting loss is Lipschitz in the two coordinates
  > $\{\hat{\mathcal T}h(Z),h(X)\}$, uniformly over these bounded centers. This is the fixed-comparator fitting application of \citet[Lemma~14]{foster2023orthogonal}, with comparator $h_{j,\lambda}^*$ and penalty center $h_{j-1,\lambda}^*$.
- foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling — checked against source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt

  What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.IterRDIVSquaredLossLocalizedConcentration` (`Helpers/IterRDIVConcentration.lean:187`)

  ```lean
  noncomputable def IterRDIVSquaredLossLocalizedConcentration : Sort 0 :=
    ∀ (mu : Measure 𝒳) (G : Set (𝒳 → 𝒵 → ℝ)) (g₀ : 𝒳 → 𝒵 → ℝ)
      (support : Set (𝒳 × 𝒵)) (B c₀ CG : ℝ),
      RDIVSetting V mu g₀ → DensityRealizable V mu G g₀ →
      DensityBounds G g₀ support c₀ CG → HypothesisClassBounded V support B →
      ∃ C > 0, ∃ c₁ > 0, ∃ c₂ > 0,
      ∀ (P : Measure Ω) (_hProbability : IsProbabilityMeasure P)
        (sample : AnalysisSample V P) (split : SampleSplit)
        (gHat : ℕ → Ω → 𝒳 → 𝒵 → ℝ) (ρ : ℝ),
      RDIVClassesMeasurable V P sample split mu G g₀ gHat →
      IterRDIVFixedCenterSupremumVersion V P sample split mu gHat ρ →
      ∀ (n t k : ℕ) (lam r : ℝ) (sourceRadius : ℝ)
        (_hSourceScope : gridValue ρ (terminalIndex ρ n) ≤ sourceRadius ∧
          oraclePoint ρ n sourceRadius ≤ lam), 0 < n → 1 ≤ k → k ≤ t →
      lam ∈ regGrid ρ n → lam ∈ Set.Ioc (0 : ℝ) 2 →
      populationIterate V lam k ∈ V.H →
      0 < r → 1 < 2 * c₂ * n →
      4 * Real.log (41 * Real.log (2 * c₂ * n)) / (c₂ * n) ≤ r ^ 2 →
      iterRdivAnchoredRadius (V.law.map V.xOf) (primalVersion V '' V.H)
        (primalVersion V (populationIterate V lam k)) n r →
      ∀ᵐ a ∂P, ∃ E : Set (Fin n → 𝒲), MeasurableSet E ∧
        1 - 2 * c₁ * Real.exp (-c₂ * n * r ^ 2 / 4) ≤
          (Measure.pi (fun _ : Fin n => V.law)).real E ∧
        ∀ ws ∈ E, ∀ h ∈ V.H,
          |(n : ℝ)⁻¹ * (∑ i, (primalVersion V h (V.xOf (ws i)) -
            primalVersion V (populationIterate V lam k) (V.xOf (ws i))) ^ 2) -
            ‖h - populationIterate V lam k‖ ^ 2| ≤
            C * r * (‖h - populationIterate V lam k‖ + r)
  ```

  The passage of the proof:

  > Apply the localized squared-loss comparison to
  > $(h-h_{k,\lambda}^*)^2$, whose comparator and center are fixed. Here the localized squared-loss comparison is the scalar application of \citet[Lemma~14]{foster2023orthogonal} at the fixed comparator $h_{k,\lambda}^*$.
- bennett2023source, Lemma 9 — checked against source supplied by the user at discovery/cited_sources/bennett2023source.txt

  What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.TRAELocalizedConcentration` (`Helpers/TRAESetup.lean:845`)

  ```lean
  noncomputable def TRAELocalizedConcentration : Sort 0 :=
    ∀ (B : ℝ) (tildeδ : ℕ → ℝ → ℝ),
      RangeCondition V → ClosenessCondition V → MeanSquareContinuity V →
      TRAEBoundedness V B → TRAECommonNullSetBoundedness V B → TRAECriticalRadiusBound V tildeδ →
      ∃ C > 0, ∀ (δ : ℕ → ℝ) (allocation : ℕ → ℝ),
      TRAEScaleCalibration δ tildeδ allocation →
      ∀ (n : ℕ), 0 < n → allocation n ∈ Set.Ioo (0 : ℝ) 1 →
      ∃ E : Set (Fin n → 𝒲), MeasurableSet E ∧
        1 - allocation n ≤ (Measure.pi (fun _ : Fin n => V.law)).real E ∧
        ∀ x ∈ E, ∀ h ∈ V.H, ∀ f ∈ V.F,
          let u := fun w => ![V.m f w,
            primalVersion V h (V.xOf w) * dualVersion V f (V.zOf w),
            dualVersion V f (V.zOf w)]
          |(n : ℝ)⁻¹ * (∑ i, traeClippedLoss B (x i) (u (x i))) -
            (∫ w, traeClippedLoss B w (u w) ∂V.law)| ≤
            C * (δ n * (∑ i : Fin 3, inputL2Norm V.law (fun w => u w i)) + δ n ^ 2)
  ```

  The passage of the proof:

  > The localized concentration bound used in \citet[Lemma~9]{bennett2023source}
  > gives
  > \begin{align*}
  > L( h) &=\sup_{f\in \mathcal{F}}\mathbb{E}\left[ 2( m( W;f) -f( Z) h( X)) -f^{2}( Z)\right] \\
  > & =\mathbb{E}\left[ 2( m( W;f_{h}) -f_{h}( Z) h( X)) -f_{h}^{2}( Z)\right]\\
  >  & \leqslant \mathbb{E}_{n}\left[ 2( m( W;f_{h}) -f_{h}( Z) h( X)) -f_{h}^{2}( Z)\right] +O\left( \delta _{n}^{2} +\delta _{n} \| f_{h} \| \right)\\
  >  & \leqslant \sup_{f\in \mathcal{F}}\mathbb{E}_{n}\left[ 2( m( W;f) -f( Z) h( X)) -f^{2}( Z)\right] +O\left( \delta _{n}^{2} +\delta _{n} \| f_{h} \| \right)\\
  >  & \leqslant L_{n}( h) +O\left( \delta _{n}^{2} +\delta _{n} \| f_{h} \| \right) .
  > \end{align*}
  > Since $\|f_h\|^2=L(h)$, Young's inequality gives
  > \[
  > L( h) \leqslant 2L_{n}( h) +O\left( \delta _{n}^{2}\right) .
  > \]
  > Similarly, 
  > \begin{align*}
  > L_{n}( h)&  =\sup_{f\in \mathcal{F}}\mathbb{E}_{n}\left[ 2( m( W;f) -f( Z) h( X)) -f^{2}( Z)\right] \\
  >  & \leqslant \sup_{f\in\mathcal F}\left\{\mathbb E\left[2(m(W;f)-f(Z)h(X))-f(Z)^2\right]+C\left(\delta_n^2+\delta_n\|f\|\right)\right\}\\
  >  & \leqslant \sup_{f\in\widetilde{\mathcal Q}}\mathbb{E}\left[ 2( m( W;f) -f( Z) h( X)) -\frac{4}{5} f^{2}( Z)\right] +O\left( \delta _{n}^{2}\right)\\
  >  & =\frac{5}{4}\|\mathcal T(h_0-h)\|^2+O\left( \delta _{n}^{2}\right)\\
  >  & =\frac{5}{4} L( h) +O\left( \delta _{n}^{2}\right) .
  > \end{align*}
- foster2023orthogonal, Lemma 14 — checked against source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt

  What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.ScalarLipschitzLocalizedConcentration` (`Helpers/TRAESetup.lean:937`)

  ```lean
  noncomputable def ScalarLipschitzLocalizedConcentration
      (V : IVWorld 𝒲 𝒳 𝒵) (P : Measure Ω)
      (sample : AnalysisSample V P) (split : SampleSplit) : Sort 0 :=
    IsProbabilityMeasure P →
    ∀ (B : ℝ) (tildeδ : ℕ → ℝ → ℝ),
    TRAEBoundedness V B → TRAECommonNullSetBoundedness V B → TRAECriticalRadiusBound V tildeδ →
    ∃ C > 0, ∀ (δ allocation : ℕ → ℝ),
    H0MemHypothesisClass V → TRAEScaleCalibration δ tildeδ allocation →
    ∀ n : ℕ, 0 < n → allocation n ∈ Set.Ioo (0 : ℝ) 1 →
      ∃ E : Set Ω, MeasurableSet E ∧ 1 - allocation n ≤ P.real E ∧
        ∀ ω ∈ E, ∀ h ∈ V.H,
          |(primalEmpSq V P sample split n ω h -
              primalEmpSq V P sample split n ω (minNormSolution V)) -
            (‖h‖ ^ 2 - ‖minNormSolution V‖ ^ 2)| ≤
              C * (δ n * ‖h - minNormSolution V‖ + δ n ^ 2)
  ```

  The passage of the proof:

  > The localized squared-norm inequality
  > in \citet[Lemma~14]{foster2023orthogonal} gives, uniformly over
  > $h\in\mathcal H$,
  > \[
  > \left|\{\|h\|_{2,n}^2-\|h_0\|_{2,n}^2\}
  > -\{\|h\|^2-\|h_0\|^2\}\right|
  > \leqslant C\{\delta_n\|h-h_0\|+\delta_n^2\}.
  > \]
- bennett2023source, Lemma 9 (lem:concentration) — checked against source supplied by the user at discovery/cited_sources/bennett2023source.txt

  What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.IterTRAEAnchoredLocalizedConcentration` (`Helpers/IterTRAEConcentration.lean:92`)

  ```lean
  noncomputable def IterTRAEAnchoredLocalizedConcentration : Sort 0 :=
    ∀ B cfloor : ℝ, TRAEBoundedness V B → TRAECommonNullSetBoundedness V B → 0 < cfloor →
    ∃ C > 0, ∀ n : ℕ, 3 ≤ n → ∀ ζ ∈ Set.Ioo (0 : ℝ) 1,
    ∀ r : ℝ, cfloor * Real.sqrt
      ((3 * Real.log (Real.log n) + Real.log (1 / ζ)) / n) ≤ r →
    0 < r →
    (∀ F ∈ ({starClass (momentImageClass V),
        starClass (productObservationClass V), starClass (criticObservationClass V)} :
          Set (Set (𝒲 → ℝ))),
      IsCriticalRadiusBound V.law F n r) →
    ∀ (ρ c b β : ℝ) (δ : ℕ → ℝ) (t : ℕ),
    GridDomain ρ → 0 < c → 2 ≤ b → 0 < β → 1 ≤ t →
    let radius := c * (b * δ n ^ 2) ^ (1 / (qualificationExponent β t + 1))
    ∀ anchor : 𝒲 → Fin 3 → ℝ,
    (anchor = 0 ∨ ∃ j : ℕ, 1 ≤ j ∧ j ≤ t ∧
      ∃ lam ∈ regGrid ρ n,
        (∃ k ≤ terminalIndex ρ n, gridValue ρ k ≤ radius) ∧
        oraclePoint ρ n radius ≤ lam ∧ lam ≤ 2 ∧ 0 < lam ∧
        populationIterate V lam j ∈ V.H ∧
        projCondExpOp V (minNormSolution V - populationIterate V lam j) ∈ V.F ∧
        anchor = iterTraePopulationAnchor V j lam) →
    anchor ∈ traeLossClass V →
    AELocalizedSupremaMeasurableAt V.law (traeLossClass V) anchor (traeClippedLoss B) n →
    ∃ E : Set (Fin n → 𝒲), MeasurableSet E ∧
      1 - ζ ≤ (Measure.pi (fun _ : Fin n => V.law)).real E ∧
      ∀ x ∈ E, ∀ u ∈ traeLossClass V,
        |(n : ℝ)⁻¹ * (∑ i, (traeClippedLoss B (x i) (u (x i)) -
            traeClippedLoss B (x i) (anchor (x i)))) -
          ∫ w, (traeClippedLoss B w (u w) - traeClippedLoss B w (anchor w)) ∂V.law| ≤
        C * (r * (∑ i : Fin 3, inputL2Norm V.law (fun w => u w i - anchor w i)) + r ^ 2)
  ```

  The passage of the proof:

  > Apply the localized concentration inequality of \citet[Lemma~9]{bennett2023source} to the product and moment-image classes in \cref{asp:critical_radius}, and apply its bounded Lipschitz version to the squared test functions.
- bennett2023source, Lemma 9 (lem:concentration) — checked against source supplied by the user at discovery/cited_sources/bennett2023source.txt

  What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.IterTRAEFixedCenterSquaredLossConcentration` (`Helpers/IterTRAEConcentration.lean:214`)

  ```lean
  noncomputable def IterTRAEFixedCenterSquaredLossConcentration : Sort 0 :=
    ∀ B cfloor : ℝ, TRAEBoundedness V B → TRAECommonNullSetBoundedness V B → 0 < cfloor →
    ∃ C > 0, ∀ n : ℕ, 3 ≤ n → ∀ ζ ∈ Set.Ioo (0 : ℝ) 1,
    ∀ r : ℝ, cfloor * Real.sqrt
      ((Real.log (Real.log n) + Real.log (1 / ζ)) / n) ≤ r → 0 < r →
    ∀ (F : Set (𝒲 → ℝ)) (anchor center : 𝒲 → ℝ),
    IterTRAEScalarSourceApplication V n F anchor center →
    anchor ∈ F →
    (∀ᵐ w ∂V.law, ∀ f ∈ F, |f w| ≤ 2 * max 1 B) →
    (∀ w (u v : Fin 1 → ℝ),
      |iterTraeSourceSquare B center w u - iterTraeSourceSquare B center w v| ≤
        |u 0 - v 0|) →
    expectedRademacher V.law {f ∈ starClass F | eLpNorm f 2 V.law ≤ ENNReal.ofReal r} n ≤
      ENNReal.ofReal (r ^ 2) →
    AELocalizedSupremaMeasurableAt V.law (iterTraeScalarClass F)
      (fun w => fun _ : Fin 1 => anchor w) (iterTraeSourceSquare B center) n →
    ∃ E : Set (Fin n → 𝒲), MeasurableSet E ∧
      1 - ζ ≤ (Measure.pi (fun _ : Fin n => V.law)).real E ∧
      ∀ x ∈ E, ∀ f ∈ F,
        |(n : ℝ)⁻¹ * (∑ i,
            (iterTraeSourceSquare B center (x i) (fun _ => f (x i)) -
             iterTraeSourceSquare B center (x i) (fun _ => anchor (x i)))) -
          ∫ w, (iterTraeSourceSquare B center w (fun _ => f w) -
            iterTraeSourceSquare B center w (fun _ => anchor w)) ∂V.law| ≤
        C * (r * inputL2Norm V.law (fun w => f w - anchor w) + r ^ 2)
  ```

  The passage of the proof:

  > For the fixed-center squared-loss comparisons, apply the same localized concentration inequality of \citet[Lemma~9]{bennett2023source} to the bounded clipped losses corresponding to $(h-h_{j-1,\lambda}^*)^2$ at comparator $h_{j,\lambda}^*$ and $(h-h_{k,\lambda}^*)^2$ at comparator $h_{k,\lambda}^*$, for the fixed iteration--grid pairs under consideration.

**Conditions made explicit during the formalization.**

- Explicitly threaded the already approved IterRDIVFixedCenterSupremumVersion condition into both iterative selection theorem signatures.
- Threaded the existing IterRDIVFixedCenterSupremumVersion condition through both iterative RDIV selection theorems.

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` (96 pages). `bundle/MATCH.md` sets each result beside its Lean declaration; `bundle/lean/` holds the Lean code.

The mechanical checks leave nothing open.

# Technical appendix

The records behind the report above, by the run's own identifiers.

- paper: `main.tex` (source sha256 `6bd532e9261f88ffcf0c7f74572dda0bc2c38292dbf4950d2259b0e5601c2e35`)
- stage completed: `V6`
- results in scope: 29

| bucket | count |
| --- | --- |
| verified | 3 |
| verified_with_bookkeeping | 1 |
| conditional | 22 |
| repaired | 3 |
| 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. Oracle guarantee for the discrepancy rule `thm:common-dp`

| field | value |
| --- | --- |
| bucket | `verified_with_bookkeeping` |
| source | `main.tex:260-285` (`theorem`, label `thm:common_dp`) |
| statement | \label{thm:common_dp} Let $A_n:\Lambda_n\to[0,\infty)$, and let $\mathcal S_n\subseteq(0,2]$ be a stability region. Fix constants $C_A,C_V,C_b>0$ and exponents $\gamma\in(0,1]$ and $q>1$, and write $B_n(\lambda)=C_b\lambda^q$. Suppose that, on an event $\mathcal E_n$ with probability at least $1-\xi$, the discrepancy bound \begin{equation}\label{eq:common_discrepancy_bound} D_n(\hat h_\lambda)\leqslant A_n(\lambda)+B_n(\lambda), \end{equation} holds simultaneously over $\lambda\in\Lambda_n\cap\mathcal S_n$, and every $\lambda\in\Lambda_n\cap\mathcal S_n$ satisfies \begin{equation}\label{eq:common_accepted_bounds} \begin{aligned} \\|\mathcal T(\hat h_\lambda-h_0)\\|^2 &\leqslant C_A\{D_n(\hat h_\lambda)+A_n(\lambda)\},\\ \\|\hat h_\lambda-h_0\\|^2 &\leqslant C_V\left[\frac{D_n(\hat h_\lambda)+A_n(\lambda)}{\lambda}+\{D_n(\hat h_\lambda)+A_n(\lambda)\}^\gamma\right], \end{aligned} \end{equation} Suppose that some grid point $\lambda^\circ$ satisfies $\Lambda_n\cap[\lambda^\circ,2]\subseteq\mathcal S_n$, that $A_n$ is nonincreasing over the grid points in $[\lambda^\circ,2]$, and that \begin{equation}\label{eq:common_oracle_crossing} B_n(\lambda^\circ)\leqslant\tfrac12A_n(\lambda^\circ). \end{equation} Then \cref{alg:discrepancy} accepts a candidate before exhausting the grid, and returns a parameter satisfying $\lambda_{\mathrm{dp}}\geqslant\lambda^\circ$ with \begin{align}\label{eq:common_rate} \\|\mathcal T(\hat h_{\lambda_{\mathrm{dp}}}-h_0)\\|^2 &\leqslant O\{A_n(\lambda^\circ)\},\\ \\|\hat h_{\lambda_{\mathrm{dp}}}-h_0\\|^2 &\leqslant O\{A_n(\lambda^\circ)/\lambda^\circ+A_n(\lambda^\circ)^\gamma\}. \end{align} |
| appended hypothesis list | ass:common-dp-noise-domain, ass:common-dp-stability-domain, ass:common-dp-parameters, ass:common-dp-event-comparisons, ass:common-dp-stable-segment, ass:common-dp-monotonicity, ass:common-dp-oracle-crossing |
| Lean | `TCommonDp.lean:24` — `Papers.TsAdaptiveEstimationInference.common_dp` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity pinned · correctness ok |
| added conditions | `ass:pin-142` |
| cited results | — |
| assumed substrate gates | — |
| findings | [note, V1.5] pinned reading: On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. (The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.) |

### 2. prop:exist_lambda_deepiv `prop:exist-lambda-deepiv`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `main.tex:425-427` (`proposition`, label `prop:exist_lambda_deepiv`) |
| statement | \label{prop:exist_lambda_deepiv} Suppose \cref{asp:range,asp:source_condition,asp:closeness_deepiv,asp:re_cd,asp:cr_deepiv} hold. For every $n\geqslant N_{\mathrm R}(B)$, \cref{alg:discrepancy}, with the projected statistic \eqref{eq:rdiv_projected_loss} and noise level \eqref{eq:rdiv_common_scales}, accepts a candidate after at most $K_n+1=O(\log n)$ regularized fits, with probability at least $1-\xi$. If $h_0=0$, the initial parameter is accepted. In general, the returned parameter either is the initial parameter or is a noninitial grid point satisfying \eqref{eq:discrepancy_principle}. |
| Lean | `TExistLambdaDeepiv.lean:65` — `Papers.TsAdaptiveEstimationInference.exist_lambda_deepiv` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity pinned · correctness ok |
| added conditions | `ass:pin-153`, `ass:pin-155`, `ass:pin-172`, `ass:pin-174`, `ass:pin-fix-4d0cd055` |
| cited results | `lem:rdiv-localized-concentration-lemma-14` (foster2023orthogonal, Lemma 14); `lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)) |
| assumed substrate gates | — |
| findings | [note, V1.5] pinned reading: For each fixed $n\geqslant N_{\mathrm R}(B)$, all probabilistic conclusions, including initial acceptance when $h_0=0$, hold on the same event $\mathcal E_n^{\mathrm R}$ of probability at least $1-\xi$. (The proof of Proposition~\ref{prop:exist_lambda_deepiv} explicitly works throughout on $\mathcal E_n^{\mathrm R}$ and derives initial acceptance there.) |

### 3. thm:deepiv_rate `thm:deepiv-rate`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `main.tex:429-448` (`theorem`, label `thm:deepiv_rate`) |
| statement | \label{thm:deepiv_rate} Under the assumptions of \cref{prop:exist_lambda_deepiv}, let $\lambda_{\mathrm{dp}}$ be its output and write \[ p=\min\{\beta+1,2\}, \qquad \alpha=\frac{\min\{\beta,1\}}{1+\min\{\beta,1\}}, \qquad \varepsilon_{n,\beta}=\delta_n^{2p/(p+1)}+\nu_n^2. \] For every $n\geqslant N_{\mathrm R}(B)$, with probability at least $1-\xi$, \begin{align*} \\|\mathcal T(\hat h_{\lambda_{\mathrm{dp}}}-h_0)\\|^2 \leqslant O(\varepsilon_{n,\beta}), \quad \\|\hat h_{\lambda_{\mathrm{dp}}}-h_0\\|^2 \leqslant O\!\left( \delta_n^{2(p-1)/(p+1)} +\varepsilon_{n,\beta}^{\alpha}\right) \leqslant O(\varepsilon_{n,\beta}^\alpha). \end{align*} If $h_0=0$, the squared strong and weak errors are both $O(\delta_n^2)$. |
| Lean | `TDeepivRate.lean:64` — `Papers.TsAdaptiveEstimationInference.deepiv_rate` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity pinned · correctness ok |
| added conditions | `ass:pin-142`, `ass:pin-153`, `ass:pin-155`, `ass:pin-172`, `ass:pin-174`, `ass:pin-190`, `ass:pin-196`, `ass:pin-fix-4d0cd055` |
| cited results | `lem:rdiv-localized-concentration-lemma-14` (foster2023orthogonal, Lemma 14); `lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)) |
| assumed substrate gates | — |
| findings | [note, V1.5] pinned reading: For each fixed $n\geqslant N_{\mathrm R}(B)$, the displayed error bounds and, when $h_0=0$, both sharper $O(\delta_n^2)$ bounds hold jointly with probability at least $1-\xi$ over the preliminary, fitting, and calibration samples. (The proof obtains the zero-signal bounds from the probabilistic initial-acceptance conclusion of Proposition~\ref{prop:exist_lambda_deepiv} and Lemma~\ref{lemma:str_dp}; the general case uses the common RDIV event and the stated confidence convention.) |

### 4. prop:exist_lambda_iter_rdiv `prop:exist-lambda-iter-rdiv`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `main.tex:527-529` (`proposition`, label `prop:exist_lambda_iter_rdiv`) |
| statement | \label{prop:exist_lambda_iter_rdiv} Suppose the assumptions of \cref{prop:exist_lambda_deepiv} hold. For every $n\geqslant N_{\mathrm R}(B)$, \cref{alg:discrepancy}, applied to $\hat h_{t,\lambda}$ with the noise level \eqref{eq:iter_rdiv_common_scales}, accepts a candidate with probability at least $1-\xi$. It solves at most $t(K_n+1)=O(t\log n)$ regularized least-squares problems. If $h_0=0$, the initial parameter is accepted. Otherwise, the output is either the initial parameter or a noninitial point satisfying \eqref{eq:discrepancy_principle} with $D_n(\hat h_{t,\lambda})$ and $A_{n,t}^{\mathrm R}(\lambda)$. |
| Lean | `TExistLambdaIterRdiv.lean:196` — `Papers.TsAdaptiveEstimationInference.exist_lambda_iter_rdiv` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity pinned · correctness ok |
| added conditions | `ass:pin-153`, `ass:pin-155`, `ass:pin-187`, `ass:pin-205`, `ass:pin-fix-4d0cd055`, `ass:filler:27:prop:exist-lambda-iter-rdiv:14:ass:bk-owner-4`, `ass:filler:27:prop:exist-lambda-iter-rdiv:14:ass:bk-owner-4:1` |
| cited results | `lem:iter-rdiv-fixed-center-concentration-lemma-14` (foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling); `lem:iter-rdiv-squared-loss-concentration-lemma-14` (foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling); `lem:rdiv-localized-concentration-lemma-14` (foster2023orthogonal, Lemma 14); `lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)) |
| assumed substrate gates | — |
| findings | [note, V1.5] pinned reading: For each fixed $n\geqslant N_{\mathrm R}(B)$ and prescribed $t$, there is an event of probability at least $1-\xi$ on which the rule accepts a candidate and satisfies the stated output alternatives, including acceptance of $\lambda_0=2$ when $h_0=0$. (The proof of Proposition~\ref{prop:exist_lambda_iter_rdiv} explicitly places initial acceptance on the corresponding concentration event.) |

### 5. Adaptive iterative RDIV `thm:iter-rdiv-rate`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `main.tex:531-553` (`theorem`, label `thm:iter_rdiv_rate`) |
| statement | \label{thm:iter_rdiv_rate} Under the assumptions of \cref{prop:exist_lambda_iter_rdiv,asp:iter_realizability_rdiv}, let $\lambda_{\mathrm{dp},t}^{\mathrm R}$ denote the selected parameter and set \[ \varepsilon_{n,t,\beta}^{\mathrm R} =(s_{n,t}^{\mathrm R})^{q_t/(q_t+1)}+\nu_n^2. \] For every fixed $t$ and every $n\geqslant N_{\mathrm R}(B)$, with probability at least $1-\xi$, \begin{align*} \\|\mathcal T(\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm R}}-h_0)\\|^2 \leqslant O_t(\varepsilon_{n,t,\beta}^{\mathrm R}), \quad \\|\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm R}}-h_0\\|^2 \leqslant O_t\!\left( (s_{n,t}^{\mathrm R})^{(q_t-1)/(q_t+1)} +(\varepsilon_{n,t,\beta}^{\mathrm R})^{\beta/(\beta+1)} \right). \end{align*} Here $O_t(\cdot)$ allows constants that depend on the prescribed iteration count and fixed population quantities. In particular, $a_t$ is independent of $n$. If $\nu_n=O(\delta_n)$ and $2t\geqslant\beta+1$, the bounds reduce to \begin{align*} \\|\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm R}}-h_0\\|^2 =O_t\!\left(\delta_n^{2\beta/(\beta+2)}\right), \quad \\|\mathcal T(\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm R}}-h_0)\\|^2 =O_t\!\left(\delta_n^{2(\beta+1)/(\beta+2)}\right). \end{align*} |
| Lean | `TIterRdivRate.lean:56` — `Papers.TsAdaptiveEstimationInference.iter_rdiv_rate` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-142`, `ass:pin-153`, `ass:pin-155`, `ass:pin-187`, `ass:pin-205`, `ass:pin-fix-4d0cd055` |
| cited results | `lem:iter-rdiv-fixed-center-concentration-lemma-14` (foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling); `lem:iter-rdiv-squared-loss-concentration-lemma-14` (foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling); `lem:rdiv-localized-concentration-lemma-14` (foster2023orthogonal, Lemma 14); `lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)) |
| assumed substrate gates | — |
| findings | — |

### 6. prop:exist_lambda `prop:exist-lambda`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `main.tex:646-657` (`proposition`, label `prop:exist_lambda`) |
| statement | \label{prop:exist_lambda} Suppose that Assumptions \ref{asp:range}, \ref{asp:source_condition}, \ref{asp:closeness_deepiv}, \ref{asp:closeness}, \ref{asp:mean_sq_continuity}, and \ref{asp:critical_radius} hold. Run \cref{alg:discrepancy} with inputs $\lambda_0 = 2$, a fixed $1/2\leqslant\rho<1$, and noise level $A_n^{\mathrm T}(\lambda)=\delta^2_n$, where $\delta_n$ satisfies \eqref{eq:delta_trae}. For every $n\geqslant N_{\mathrm T}(B)$, the algorithm accepts a candidate after at most $K_n+1=O(\log n)$ optimization problems, with probability at least $1-\xi$. The returned parameter either is $\lambda_0$ through the initial-acceptance branch or is a noninitial grid point satisfying \eqref{eq:discrepancy_principle}. |
| Lean | `TExistLambda.lean:122` — `Papers.TsAdaptiveEstimationInference.exist_lambda` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity pinned · correctness ok |
| added conditions | `ass:pin-156`, `ass:pin-160`, `ass:pin-193`, `ass:pin-206`, `ass:pin-fix-958aeb7a` |
| cited results | `lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9); `lem:trae-squared-norm-lemma-14` (foster2023orthogonal, Lemma 14) |
| assumed substrate gates | — |
| findings | [note, V1.5] pinned reading: For each fixed $n\geqslant N_{\mathrm T}(B)$, acceptance within the stated number of optimization problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_n$, with probability at least $1-\xi$ over the fitting sample. (The proof begins by restricting throughout to $\mathcal E_n$, establishes terminal acceptance there, and derives both return branches on that same event.) |

### 7. thm:tikhonv_rate `thm:tikhonv-rate`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `main.tex:660-667` (`theorem`, label `thm:tikhonv_rate`) |
| statement | \label{thm:tikhonv_rate} Under the assumptions of \cref{prop:exist_lambda}, let $ {\lambda_{\mathrm{dp}}} $ be chosen by \cref{alg:discrepancy}. For every $n\geqslant N_{\mathrm T}(B)$, with probability at least $ 1-\xi $, \[ \\| \hat{h}_{\lambda_{\mathrm{dp}}} -h_{0} \\| ^{2} \leqslant O\left( \delta _{n}^{2\min\{\beta ,1\} /( 1+\min\{\beta ,1\})}\right) ,\qquad \\| \mathcal{T}(\hat{h}_{\lambda_{\mathrm{dp}}} -h_{0}) \\| ^{2} \leqslant O\left( \delta _{n}^{2}\right) , \] where $\delta_n$ is defined in (\ref{eq:delta_trae}). |
| Lean | `TTikhonvRate.lean:45` — `Papers.TsAdaptiveEstimationInference.tikhonov_rate` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity pinned · correctness ok |
| added conditions | `ass:pin-142`, `ass:pin-156`, `ass:pin-160`, `ass:pin-193`, `ass:pin-194`, `ass:pin-206`, `ass:pin-fix-958aeb7a` |
| cited results | `lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9); `lem:trae-squared-norm-lemma-14` (foster2023orthogonal, Lemma 14) |
| assumed substrate gates | — |
| findings | [note, V1.5] pinned reading: With $\beta$ and $\xi$ fixed as $n\to\infty$, for each $n\geqslant N_{\mathrm T}(B)$ the displayed strong- and weak-error bounds hold jointly on a fitting-sample event of probability at least $1-\xi$. (The proof of Theorem~\ref{thm:tikhonv_rate} works on $\mathcal E_n$, which the preceding confidence allocation constructs separately for each sample size.) |

### 8. prop:exist_lambda_iter_trae `prop:exist-lambda-iter-trae`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `main.tex:737-739` (`proposition`, label `prop:exist_lambda_iter_trae`) |
| statement | \label{prop:exist_lambda_iter_trae} Suppose the assumptions of \cref{prop:exist_lambda} hold. For every $n\geqslant N_{\mathrm{IT}}(B)$, \cref{alg:discrepancy}, applied to $\hat h_{t,\lambda}$ with the noise level \eqref{eq:iter_trae_scale}, accepts a candidate after at most $t(K_n+1)=O(t\log n)$ minimax problems, with probability at least $1-\xi$. The returned parameter is either the initial point or a noninitial point satisfying \eqref{eq:discrepancy_principle} with the iterative loss and threshold. |
| Lean | `TExistLambdaIterTrae.lean:118` — `Papers.TsAdaptiveEstimationInference.exist_lambda_iter_trae` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity pinned · correctness ok |
| added conditions | `ass:pin-156`, `ass:pin-160`, `ass:pin-195`, `ass:pin-206`, `ass:pin-fix-958aeb7a` |
| cited results | `lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9); `lem:trae-squared-norm-lemma-14` (foster2023orthogonal, Lemma 14) |
| assumed substrate gates | — |
| findings | [note, V1.5] pinned reading: For each fixed $n\geqslant N_{\mathrm{IT}}(B)$ and prescribed $t$, acceptance within the stated number of minimax problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_{n,t}^{\mathrm{IT}}$, with probability at least $1-\xi$ over the fitting sample. (The proof works throughout on $\mathcal E_{n,t}^{\mathrm{IT}}$ and derives terminal acceptance and the first-accepted-index alternative there.) |

### 9. Adaptive iterative TRAE `thm:iter-trae-rate`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `main.tex:741-765` (`theorem`, label `thm:iter_trae_rate`) |
| statement | \label{thm:iter_trae_rate} Under the assumptions of \cref{prop:exist_lambda_iter_trae,asp:iter_realizability_trae,asp:iter_mean_sq_continuity}, let $\lambda_{\mathrm{dp},t}^{\mathrm T}$ denote the selected parameter, with $s_{n,t}^{\mathrm T}$ and $q_t$ defined above. For every fixed $t$ and every $n\geqslant N_{\mathrm{IT}}(B)$, with probability at least $1-\xi$, \begin{align*} \\|\mathcal T(\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm T}}-h_0)\\|^2 \leqslant O_t\!\left( (s_{n,t}^{\mathrm T})^{q_t/(q_t+1)} \right), \quad \\|\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm T}}-h_0\\|^2 \leqslant O_t\!\left( (s_{n,t}^{\mathrm T})^{(q_t-1)/(q_t+1)} \right). \end{align*} Because $t$ is fixed, $s_{n,t}^{\mathrm T}=b_t\delta_n^2=O_t(\delta_n^2)$, so $b_t$ affects only the constants hidden by $O_t(\cdot)$. If $2t\geqslant\beta+1$, these bounds become \begin{align*} \\|\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm T}}-h_0\\|^2 =O_t\!\left(\delta_n^{2\beta/(\beta+2)}\right), \quad \\|\mathcal T(\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm T}}-h_0)\\|^2 =O_t\!\left(\delta_n^{2(\beta+1)/(\beta+2)}\right). \end{align*} |
| Lean | `TIterTraeRate.lean:392` — `Papers.TsAdaptiveEstimationInference.iter_trae_rate` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity pinned · correctness ok |
| added conditions | `ass:bk-12`, `ass:pin-142`, `ass:pin-156`, `ass:pin-160`, `ass:pin-188`, `ass:pin-195`, `ass:pin-203`, `ass:pin-204`, `ass:pin-206`, `ass:pin-fix-958aeb7a` |
| cited results | `lem:iter-trae-anchored-concentration-lemma-9` (bennett2023source, Lemma 9 (lem:concentration)); `lem:iter-trae-fixed-center-squared-loss-lemma-9` (bennett2023source, Lemma 9 (lem:concentration)); `lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9); `lem:trae-squared-norm-lemma-14` (foster2023orthogonal, Lemma 14) |
| assumed substrate gates | — |
| findings | [note, V1.5] pinned reading: The constants hidden by $O_t(\cdot)$ may depend on the prescribed $t$, fixed population quantities including $\beta$ and $\\|w_0\\|$, and the fixed grid ratio $\rho$, but are independent of $n$ and the candidate $\lambda$. (The text following the bounds in Theorem~\ref{thm:iter_rdiv_rate} explicitly allows dependence on the prescribed iteration count and fixed population quantities; Lemma~\ref{lemma:iter_trae_accepted_stability} supplies the comparison constants used here under the same convention.) |

### 10. cor:normality `thm:normality`

| field | value |
| --- | --- |
| bucket | `repaired` |
| source | `main.tex:791-807` (`corollary`, label `cor:normality`) |
| statement | \label{cor:normality} Suppose that the assumptions of \cref{thm:tikhonv_rate} hold for both $h_0$ and $q_0$ with (potentially different) source conditions $\beta_h,\beta_q>0$. Choose a deterministic sequence $\xi_n\downarrow0$, and run the primal and dual tuning rules at confidence levels $\xi_n/2$. Let a common deterministic sequence $\delta_n$ satisfy \eqref{eq:delta_trae} for both problems at these confidence levels. Assume that, for some $\gamma_h,\gamma_q>0$, the score maps satisfy \begin{align*} \mathbb{E}\left[\left(\tilde{m}( W;h_{1}) -\tilde{m}( W;h_{2})\right)^{2}\right] & \leqslant C\\| h_{1} -h_{2} \\| ^{\gamma_h },\\ \mathbb{E}\left[\left(m( W;q_{1}) -m( W;q_{2})\right)^{2}\right] & \leqslant C\\| q_{1} -q_{2} \\| ^{\gamma_q } \end{align*} for all $h_1,h_2\in\mathcal H$ and $q_1,q_2\in\mathcal Q$. Let $\beta_m=\max\{\beta_h,\beta_q\}$ and assume that $\delta_n$ satisfies \[ \delta _{n} =o\left( n^{-\alpha _{\mathrm{dr}}}\right) ,\qquad \alpha _{\mathrm{dr}} :=\frac{1+\min\{\beta _{m} ,1\}}{2+4\min\{\beta _{m} ,1\}} . \] Let $\hat h,\hat q$ be the adaptive adversarial Tikhonov estimators, trained and tuned independently of the evaluation sample used in $\hat\theta(\hat h,\hat q)$. Finally, assume that $0<\sigma_0^2:=\mathbb E[\psi_0(W)^2]<\infty$. Then, \[ \sqrt{n}(\hat{\theta }(\hat{h},\hat{q}) -\theta _{0}) =\frac{1}{\sqrt{n}}\sum _{i=1}^{n} \psi _{0}( W_i) +o_{p}( 1) , \] where $\psi _{0}( W) =\tilde{m}( W;h_{0}) +m( W;q_{0}) -q_{0}( Z) h_{0}( X) -\theta _{0}$. Consequently, $\sqrt n(\hat\theta(\hat h,\hat q)-\theta_0)\rightsquigarrow N(0,\sigma_0^2)$. |
| appended hypothesis list | ass:normality-source-parameters, ass:normality-confidence, ass:normality-common-scale, ass:normality-score-continuity, ass:normality-rate, ass:normality-independent-evaluation, ass:normality-variance |
| Lean | `TNormality.lean:41` — `Papers.TsAdaptiveEstimationInference.dr_asymptotic_normality` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-142`, `ass:pin-156`, `ass:pin-160`, `ass:pin-193`, `ass:pin-194`, `ass:pin-206`, `ass:pin-fix-958aeb7a` |
| cited results | `lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9); `lem:trae-squared-norm-lemma-14` (foster2023orthogonal, Lemma 14) |
| assumed substrate gates | — |
| findings | [refuted, V1.5] [F3 reviewer] The evaluated random score in Theorem~\ref{cor:normality} needs a coherent jointly measurable version. The printed statement (main.tex:791--807) and its conditional Chebyshev argument (appendices.tex:1106--1123) do not state this well-definedness convention. IndependentEvaluationSample currently supplies jointly measurable versions of the nuisance evaluations only, not of the fitted score maps. Sectionwise measurability and the displayed population $L_2$ continuity bounds do not by themselves supply evaluation coherence. Add only the following measurability bookkeeping to the theorem and corrected paper: writing $S(w;h,q)=\tilde m(w;h)+m(w;q)-h(X(w))q(Z(w))$ and $d_n(w)=S(w;\hat h,\hat q)-S(w;h_0,q_0)$, for each $n$ the fitted score difference admits a version jointly measurable in the full nuisance-training/tuning information and a fresh observation, agreeing with $d_n$ for almost every training realization and almost every fresh observation, and agreeing almost surely at every observation in the finite evaluation block. Preserve the existing independent-evaluation and nuisance-version requirements. This is only well-definedness of the evaluated random score and its conditional expectation, not a supplied probability bound, convergence claim or conditional Chebyshev conclusion. Do not add countability, separability, compactness, a uniform empirical-process condition, or joint measurability of every score on the entire hypothesis space. Keep every rate, source, confidence and variance condition and the full asymptotic conclusion. The fixer must write this minimal convention explicitly into the corrected theorem; V1.5 must review it before Lean uses it. No substantive weakening of a main result is authorized. witness: The evaluated random score in Theorem~\ref{cor:normality} needs a coherent jointly measurable version. The printed statement (main.tex:791--807) and its conditional Chebyshev argument (appendices.tex:1106--1123) do not state this well-definedness convention. IndependentEvaluationSample currently supplies jointly measurable versions of the nuisance evaluations only, not of the fitted score maps. Sectionwise measurability and the displayed population $L_2$ continuity bounds do not by themselves supply evaluation coherence. Add only the following measurability bookkeeping to the theorem and corrected paper: writing $S(w;h,q)=\tilde m(w;h)+m(w;q)-h(X(w))q(Z(w))$ and $d_n(w)=S(w;\hat h,\hat q)-S(w;h_0,q_0)$, for each $n$ the fitted score difference admits a version jointly measurable in the full nuisance-training/tuning information and a fresh observation, agreeing with $d_n$ for almost every training realization and almost every fresh observation, and agreeing almost surely at every observation in the finite evaluation block. Preserve the existing independent-evaluation and nuisance-version requirements. This is only well-definedness of the evaluated random score and its conditional expectation, not a supplied probability bound, convergence claim or conditional Chebyshev conclusion. Do not add countability, separability, compactness, a uniform empirical-process condition, or joint measurability of every score on the entire hypothesis space. Keep every rate, source, confidence and variance condition and the full asymptotic conclusion. The fixer must write this minimal convention explicitly into the corrected theorem; V1.5 must review it before Lean uses it. No substantive weakening of a main result is authorized. minimal patch (confirmed): corrected statement: Suppose that the assumptions of \cref{thm:tikhonv_rate} hold for both $h_0$ and $q_0$ with (potentially different) source conditions $\beta_h,\beta_q>0$. Choose a deterministic sequence $\xi_n\downarrow0$, and run the primal and dual tuning rules at confidence levels $\xi_n/2$. Let a common deterministic sequence $\delta_n$ satisfy \eqref{eq:delta_trae} for both problems at these confidence levels. Assume that, for some $\gamma_h,\gamma_q>0$, the score maps satisfy \begin{align*} \mathbb{E}\left[\left(\tilde{m}( W;h_{1}) -\tilde{m}( W;h_{2})\right)^{2}\right] & \leqslant C\\| h_{1} -h_{2} \\| ^{\gamma_h },\\ \mathbb{E}\left[\left(m( W;q_{1}) -m( W;q_{2})\right)^{2}\right] & \leqslant C\\| q_{1} -q_{2} \\| ^{\gamma_q } \end{align*} for all $h_1,h_2\in\mathcal H$ and $q_1,q_2\in\mathcal Q$. Let $\beta_m=\max\{\beta_h,\beta_q\}$ and assume that $\delta_n$ satisfies \[ \delta _{n} =o\left( n^{-\alpha _{\mathrm{dr}}}\right) ,\qquad \alpha _{\mathrm{dr}} :=\frac{1+\min\{\beta _{m} ,1\}}{2+4\min\{\beta _{m} ,1\}} . \] Let $\hat h,\hat q$ be the adaptive adversarial Tikhonov estimators, trained and tuned independently of the evaluation sample used in $\hat\theta(\hat h,\hat q)$. For each $n$, write $\mathcal G_n$ for the full nuisance-training and tuning sigma-field. Take jointly measurable versions of the fitted nuisance evaluations, representing the fitted functions for almost every training realization and agreeing almost surely with their evaluations at every observation in the finite evaluation sample. Write \[ S(w;h,q)=\tilde m(w;h)+m(w;q)-h(X(w))q(Z(w)),\qquad  d_n(w)=S(w;\hat h,\hat q)-S(w;h_0,q_0). \] Suppose that, for each $n$, $d_n$ admits a version jointly measurable in $\mathcal G_n$ and a fresh observation, agreeing with $d_n$ for almost every training realization and almost every fresh observation, and agreeing almost surely with $d_n(W_i)$ simultaneously for $i=1,\ldots,n$ in the evaluation sample. Finally, assume that $0<\sigma_0^2:=\mathbb E[\psi_0(W)^2]<\infty$. Then, \[ \sqrt{n}(\hat{\theta }(\hat{h},\hat{q}) -\theta _{0}) =\frac{1}{\sqrt{n}}\sum _{i=1}^{n} \psi _{0}( W_i) +o_{p}( 1) , \] where $\psi _{0}( W) =\tilde{m}( W;h_{0}) +m( W;q_{0}) -q_{0}( Z) h_{0}( X) -\theta _{0}$. Consequently, $\sqrt n(\hat\theta(\hat h,\hat q)-\theta_0)\rightsquigarrow N(0,\sigma_0^2)$. · proof change: Apply \cref{thm:tikhonv_rate} to both nuisance estimators at the prescribed confidence levels $\xi_n/2$. Their rate bounds then hold on a common event with probability at least $1-\xi_n\to1$. Because $\beta_h,\beta_q>0$ and $\delta_n\to0$, these bounds give \begin{equation}\label{eq:conv_hq} \\| \hat{h} -h_{0} \\| =o_{p}( 1) ,\qquad \\| \hat{q} -q_{0}\\| =o_{p}( 1) . \end{equation} The remaining bias condition is \begin{equation}\label{eq:square_root} \sqrt{n}\mathbb{E}[(\hat{q}( Z) -q_{0}( Z))(\hat{h}( X) -h_{0}( X))] =o_{p}( 1) . \end{equation} Suppose first that $\beta_m=\beta_h\geqslant\beta_q$. By the adjoint identity and the Cauchy--Schwarz inequality, \begin{align*} \left\|\mathbb{E}[(\hat{q}( Z) -q_{0}( Z))(\hat{h}( X) -h_{0}( X))]\right\| & \leqslant \\|\hat h-h_0\\|\,\\|\mathcal T^*(\hat q-q_0)\\|\\  & =O_{p}\left( \delta _{n}^{\min\{\beta _{m} ,1\} /( 1+\min\{\beta _{m} ,1\})} \cdot \delta _{n}\right)\\  & =O_{p}\left( \delta _{n}^{1+\min\{\beta _{m} ,1\} /( 1+\min\{\beta _{m} ,1\})}\right) . \end{align*} If $\beta_m=\beta_q\geqslant\beta_h$, the symmetric bound \[ \left\|\mathbb{E}[(\hat{q}( Z) -q_{0}( Z))(\hat{h}( X) -h_{0}( X))]\right\| \leqslant \\|\hat q-q_0\\|\,\\|\mathcal T(\hat h-h_0)\\| \] gives the same rate. By assumption, $\delta _{n} =o\left( n^{-( 1+\min\{\beta _{m} ,1\}) /( 2+4\min\{\beta _{m} ,1\})}\right)$, and hence \[ \left\|\mathbb{E}[(\hat{q}( Z) -q_{0}( Z))(\hat{h}( X) -h_{0}( X))]\right\| =o_{p}\left( n^{-1/2}\right), \] which proves \eqref{eq:square_root}.  Use the jointly measurable version of $d_n$ stipulated in Corollary~\ref{cor:normality}. Its agreement with the fitted score difference for almost every training realization and almost every fresh observation identifies its conditional population integrals with those of the score difference. Its almost sure agreement at every evaluation observation identifies its evaluation average with the difference of the two evaluated score averages. Conditional on the nuisance-training data, the evaluation observations are independent and identically distributed. Boundedness of the nuisance classes and the score continuity assumptions imply \[ \mathbb E[d_n(W)^2\mid\text{training}] \leqslant C\left\{ \\|\hat h-h_0\\|^{\gamma_h}+\\|\hat q-q_0\\|^{\gamma_q} +\\|\hat h-h_0\\|^2+\\|\hat q-q_0\\|^2 \right\}=o_p(1). \] Conditional Chebyshev's inequality therefore gives $\sqrt n(\mathbb E_{n,\mathrm{eval}}-\mathbb E)d_n=o_p(1)$, where $\mathbb E d_n$ integrates a fresh observation conditional on the training data. The primal and dual moment equations give \[ \mathbb E d_n =-\mathbb E[(\hat h(X)-h_0(X))(\hat q(Z)-q_0(Z))]. \] Combining this identity with \eqref{eq:square_root} yields \[ \sqrt n\{\hat\theta(\hat h,\hat q)-\theta_0\} =\frac1{\sqrt n}\sum_{i=1}^n\psi_0(W_i)+o_p(1). \] The moment equations also imply $\mathbb E\psi_0(W)=0$. The classical central limit theorem applies because $0<\mathbb E[\psi_0(W)^2]<\infty$, proving the stated normal limit. |

### 11. Variational source inequality `lem:variational-source`

| field | value |
| --- | --- |
| bucket | `verified` |
| source | `appendices.tex:38-59` (`lemma`, label `lemma:variational_source`) |
| statement | \label{lemma:variational_source} Let $\mathcal A=\mathcal T^*\mathcal T$, $b=\min\{\beta,1\}$, $p=1+b$, and $\alpha=b/(1+b)$. Under \cref{asp:source_condition}, every $u\in\widetilde{\mathcal H}$ satisfies \[ \|\langle h_0,u\rangle\| \leqslant \\|w_0\\|\,\\|\mathcal Tu\\|^b\\|u\\|^{1-b}. \] Consequently, for every fixed $\epsilon\in(0,1)$ and $\lambda\in(0,2]$, \begin{align*} 2\lambda\|\langle h_0,u\rangle\| &\leqslant \epsilon\{\\|\mathcal Tu\\|^2+\lambda\\|u\\|^2\} +C_\epsilon\\|w_0\\|^2\lambda^p,\\ 2\|\langle h_0,u\rangle\| &\leqslant \epsilon\\|u\\|^2 +C_\epsilon\\|w_0\\|^{2/(1+b)} \\|\mathcal Tu\\|^{2b/(1+b)}. \end{align*} |
| Lean | `Helpers/SpectralRegularization.lean:544` — `Papers.TsAdaptiveEstimationInference.variational_source_uniform` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | — |
| cited results | — |
| assumed substrate gates | — |
| findings | — |

### 12. Preliminary RDIV nuisance and operator bounds `lem:t-err-dp`

| field | value |
| --- | --- |
| bucket | `repaired` |
| source | `appendices.tex:102-124` (`lemma`, label `lemma:t_err_dp`) |
| statement | \label{lemma:t_err_dp} Under \cref{asp:re_cd,asp:cr_deepiv}, with probability at least $1-\xi$, \begin{align} \mathbb E_Z\!\left[H^2\{\hat g(\cdot\mid Z),g_0(\cdot\mid Z)\}\right] &\leqslant C\delta_{n,\mathcal G}^2, \label{eq:rdiv_mle_hellinger}\\ \\|\hat r-r_0\\|_{L_2(Z)}&\leqslant \delta_{n,r},\nonumber\\ \sup_{h\in\mathcal H} \\|(\hat{\mathcal T}-\mathcal T)h\\|_{L_2(Z)} &\leqslant C\delta_{n,\mathcal G},\label{eq:rdiv_op_global}\\ \\|(\hat{\mathcal T}-\mathcal T)(h-h')\\|_{L_1(Z)} &\leqslant C\delta_{n,\mathcal G}\\|h-h'\\|, \qquad h,h'\in\mathcal H.\label{eq:rdiv_op_relative} \end{align} Moreover, if \[ \widehat L(h):=\mathbb E[(Y-\hat{\mathcal T}h(Z))^2\mid I_{\mathrm{pre}}], \] then, uniformly over $h,h'\in\mathcal H$, \begin{equation}\label{eq:rdiv_first_stage_loss} \|\{\widehat L(h)-\widehat L(h')\}-\{L(h)-L(h')\}\| \leqslant C\delta_{n,\mathcal G}\\|h-h'\\|. \end{equation} |
| Lean | `Helpers/RDIV.lean:85` — `Papers.TsAdaptiveEstimationInference.t_err_dp` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-153`, `ass:pin-fix-4d0cd055` |
| cited results | `lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)) |
| assumed substrate gates | — |
| findings | [refuted, V1.5] the step its proof takes from [li2024regularized, Theorem 24] (lem:conditional-mle-theorem-24) does not hold as the proof uses it: (a) The cited Theorem 24 explicitly imports the source-condition hypothesis of its referenced RDIV theorem. Lemma~\ref{lemma:t_err_dp} assumes only Assumptions~\ref{asp:re_cd} and \ref{asp:cr_deepiv}; these do not imply a source condition. Thus the citation, as stated, has an unestablished hypothesis. (b) The supplied theorem states an unsigned exponential confidence formula, and its proof explicitly gives $1-\exp(n\delta_n^2)$. Neither supplies the asserted probability $1-\xi/2$. The supplied localized concentration lemma has a usable confidence statement, but obtaining the density-only bound from it requires an explicit specialization rather than the printed direct invocation of Theorem 24. witness: (a) Let $X,Z\in[0,1]$, let $\phi_j$ be the mean-zero orthonormal Rademacher functions, and put $\sigma_j=2^{-j^2-2}$ and $g_0(x\mid z)=1+\sum_{j\geqslant1}\sigma_j\phi_j(x)\phi_j(z)$, with $Z$ uniform. This is a bounded conditional density bounded away from zero. Take the closed primal and dual spaces generated by the constants and the $\phi_j$, set $h_0=\sum_{j\geqslant1}j^{-2}\phi_j$, $r_0=\mathcal T h_0$, $Y=r_0(Z)$, $\hat r=r_0$, $\mathcal H=\{c h_0:1/2\leqslant c\leqslant3/2\}$, and $\mathcal G=\{g_0,1\}$. The host's density, boundedness, measurability, pilot, and critical-radius conditions can all be satisfied. However, for every $\beta>0$, $\sum_{j\geqslant1}j^{-4}\sigma_j^{-2\beta}=\infty$, so the source hypothesis required by the cited theorem fails. (b) The source proof's reported probability $1-\exp(n\delta_n^2)$ is nonpositive, hence gives no positive confidence guarantee; replacing it by $1-\xi/2$ does not follow from that formula. minimal patch (confirmed): proof change: Allocate failure probability $\xi/2$ to the conditional-density estimator and $\xi/2$ to the pilot regression. As elsewhere, evaluate the corresponding deterministic accuracy bounds at these confidence levels and relabel them $\delta_{n,\mathcal G}$ and $\delta_{n,r}$. We establish the density bound directly using the Localized Concentration lemma of \citet{li2024regularized}. Write $P=\mathbb E_{X,Z}$ and $P_n=\mathbb E_{n,\mathrm{pre}}$, and define \[ f_g(x,z)=\sqrt{\frac{g(x\mid z)+g_0(x\mid z)}{2g_0(x\mid z)}},\qquad \mathcal U=\{f_g-1:g\in\mathcal G\}. \] Realizability gives $f_{g_0}=1$, so $0\in\mathcal U$. The density bounds give \[ 2^{-1/2}\leqslant f_g\leqslant M,\qquad M=\sqrt{\frac{C_G/c_0+1}{2}}. \] Thus $\mathcal U$ is uniformly bounded by a fixed constant. Define the globally Lipschitz loss \[ \ell(v)=-\log\bigl(\max\{1+v,2^{-1/2}\}\bigr). \] The maximum map is $1$-Lipschitz, and the derivative of $-\log t$ has absolute value at most $\sqrt2$ for $t\geqslant2^{-1/2}$. Hence $\ell$ is $\sqrt2$-Lipschitz, $\ell(0)=0$, and $\ell(f_g-1)=-\log f_g$.  We verify the localized complexity condition for this application. Put $\mathcal S=\operatorname{star}(\mathcal G_0-\mathcal G_0)$ and choose a deterministic critical radius $d\leqslant\delta_{n,\mathcal G}$ as supplied by Assumption~\ref{asp:cr_deepiv}. By the critical-radius definition, $\mathfrak R_n(d;\mathcal S)\leqslant d^2$. For $s\geqslant d$, star-shapedness implies that $(d/s)v\in\mathcal S$ and $\\|(d/s)v\\|_{L_2(X,Z)}\leqslant d$ whenever $v\in\mathcal S$ and $\\|v\\|_{L_2(X,Z)}\leqslant s$. Scaling inside the Rademacher supremum therefore gives \[ \mathfrak R_n(s;\mathcal S) \leqslant\frac{s}{d}\mathfrak R_n(d;\mathcal S) \leqslant sd. \] Since $1\in\mathcal G_0$, we have $\operatorname{star}(\mathcal U)\subseteq\mathcal S$. Taking $s=\delta_{n,\mathcal G}$ yields \[ \mathfrak R_n\bigl(\delta_{n,\mathcal G};\operatorname{star}(\mathcal U)\bigr) \leqslant\delta_{n,\mathcal G}d \leqslant\delta_{n,\mathcal G}^2. \] Apply the cited localized concentration lemma with input $(X,Z)$, one coordinate, class $\mathcal U$, fixed comparator $0$, loss $\ell$, and failure probability $\xi/2$. The measurability and logarithmic radius floor are supplied by Assumption~\ref{asp:cr_deepiv}, with the confidence evaluation specified in the proof. Consequently, for a fixed constant $A>0$, with probability at least $1-\xi/2$, \[ \left\|(P_n-P)(-\log f_g)\right\| \leqslant A\left\{\delta_{n,\mathcal G}\\|f_g-1\\|_{L_2(X,Z)}+\delta_{n,\mathcal G}^2\right\}, \qquad g\in\mathcal G. \]  To use empirical optimality, note that $P_n\log g_0(X\mid Z)$ is finite by the density bounds. Since $\hat g$ maximizes the empirical log likelihood and all candidate densities are bounded above, $\hat g(X_i\mid Z_i)>0$ at every preliminary observation, almost surely. For $a>0$, the arithmetic-geometric mean inequality gives $(a+1)/2\geqslant\sqrt a$, and hence \[ -\log\sqrt{\frac{a+1}{2}}\leqslant-\frac14\log a. \] Taking $a=\hat g(X_i\mid Z_i)/g_0(X_i\mid Z_i)$ and averaging gives \[ P_n(-\log f_{\hat g}) \leqslant-\frac14P_n\log\frac{\hat g(X\mid Z)}{g_0(X\mid Z)} \leqslant0. \] For every $g\in\mathcal G$, conditional density normalization gives \[ P f_g^2 =\mathbb E_Z\int\frac{g(x\mid Z)+g_0(x\mid Z)}2\,\mu(dx)=1. \] Since $\log t\leqslant t-1$ for $t>0$, it follows that \[ P(-\log f_g)\geqslant1-Pf_g =\frac12P(f_g-1)^2. \] On the concentration event, put $b=\\|f_{\hat g}-1\\|_{L_2(X,Z)}$. Combining these inequalities and using Young's inequality gives \[ \frac{b^2}{2} \leqslant P(-\log f_{\hat g}) \leqslant A\delta_{n,\mathcal G}b+A\delta_{n,\mathcal G}^2 \leqslant\frac{b^2}{4}+(A^2+A)\delta_{n,\mathcal G}^2. \] Thus $b^2\leqslant4(A^2+A)\delta_{n,\mathcal G}^2$.  Finally, for $t\geqslant0$ and $f=\sqrt{(t^2+1)/2}$, we have $f\leqslant t+1$ and \[ \|t-1\|=\frac{2(f+1)}{t+1}\|f-1\|\leqslant4\|f-1\|. \] Apply this pointwise with $t=\sqrt{\hat g/g_0}$ and integrate under the true joint distribution to obtain \[ \begin{aligned} \mathbb E_Z\!\left[H^2\{\hat g(\cdot\mid Z),g_0(\cdot\mid Z)\}\right] &=P\left(\sqrt{\frac{\hat g(X\mid Z)}{g_0(X\mid Z)}}-1\right)^2\\ &\leqslant16b^2 \leqslant64(A^2+A)\delta_{n,\mathcal G}^2. \end{aligned} \] This proves \eqref{eq:rdiv_mle_hellinger} with probability at least $1-\xi/2$. The nuisance-rate assumption gives $\\|\hat r-r_0\\|_{L_2(Z)}\leqslant\delta_{n,r}$ with probability at least $1-\xi/2$. Therefore, without requiring independence between $\hat g$ and $\hat r$, the union bound shows that both inequalities hold simultaneously with probability at least $1-\xi$. We work on this intersection for the rest of the proof. Here \[ H^2(g_1,g_2)=\int\{\sqrt{g_1(x)}-\sqrt{g_2(x)}\}^2\,\mu(dx). \]  Let $P_X$ be the true marginal distribution of $X$ and let $\hat P_X$ be the marginal induced by $Z\sim P_Z$ and $X\mid Z\sim\hat g(\cdot\mid Z)$. The density bounds in \cref{asp:re_cd} give $\hat g(x\mid z)/g_0(x\mid z)\leqslant C_G/c_0$ and hence \[ \frac{d\hat P_X}{dP_X}\leqslant \frac{C_G}{c_0}. \] The total-variation--Hellinger inequality and $\\|h\\|_\infty\leqslant B$ give \[ \\|(\hat{\mathcal T}-\mathcal T)h\\|^2 \leqslant B^2\mathbb E_Z \left[\left\{\int\|\hat g(x\mid Z)-g_0(x\mid Z)\|\,\mu(dx)\right\}^2\right] \leqslant C B^2\delta_{n,\mathcal G}^2, \] which proves \eqref{eq:rdiv_op_global}. For $u=h-h'$, factor $\hat g/g_0-1=(\sqrt{\hat g/g_0}-1)(\sqrt{\hat g/g_0}+1)$. Cauchy--Schwarz and the conditional density-ratio bound yield \begin{align*} \\|(\hat{\mathcal T}-\mathcal T)u\\|_1 &\leqslant \mathbb E\left[\|u(X)\| \left\|\sqrt{\frac{\hat g(X\mid Z)}{g_0(X\mid Z)}}-1\right\| \left\{\sqrt{\frac{\hat g(X\mid Z)}{g_0(X\mid Z)}}+1\right\}\right]\\ &\leqslant C\\|u\\| \mathbb E_Z\!\left[H^2\{\hat g(\cdot\mid Z),g_0(\cdot\mid Z)\}\right]^{1/2} \leqslant C\delta_{n,\mathcal G}\\|u\\|, \end{align*} which proves \eqref{eq:rdiv_op_relative}. \citet{li2024regularized} use the same operator argument in their MLE analysis.  Finally, write $e_h=(\hat{\mathcal T}-\mathcal T)h$. Pointwise, \begin{align*} &(Y-\hat{\mathcal T}h)^2-(Y-\hat{\mathcal T}h')^2 -\{(Y-\mathcal T h)^2-(Y-\mathcal T h')^2\}\\ &\quad=-(\hat{\mathcal T}-\mathcal T)u (2Y-\hat{\mathcal T}h-\hat{\mathcal T}h') +\mathcal T u(e_h+e_{h'}). \end{align*} The expectation of the first term is at most $4B\\|(\hat{\mathcal T}-\mathcal T)u\\|_1$; the second is at most $\\|\mathcal T u\\|(\\|e_h\\|+\\|e_{h'}\\|)$. Apply \eqref{eq:rdiv_op_global}, \eqref{eq:rdiv_op_relative}, and conditional Jensen to obtain \eqref{eq:rdiv_first_stage_loss}. |

### 13. Uniform RDIV empirical comparisons `lem:err-l-dp`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:183-196` (`lemma`, label `lemma:err_l_dp`) |
| statement | \label{lemma:err_l_dp} Under \cref{asp:range,asp:closeness_deepiv,asp:re_cd,asp:cr_deepiv}, there is an event $\mathcal E_n^{\mathrm R}$ with probability at least $1-\xi$ on which, simultaneously for every $\lambda\in\Lambda_n$ and $h\in\mathcal H$, \begin{align} &\left\|\{\widehat R^\lambda(h)-\widehat R^\lambda(h_0)\} -\{R_n^\lambda(h)-R_n^\lambda(h_0)\}\right\| \leqslant C\{\delta_{n,\mathcal H}\\|h-h_0\\| +\delta_{n,\mathcal H}^2\},\label{eq:rdiv_fit_comparison}\\ &\left\|D_n(\hat h_\lambda) -\\|\mathcal T(\hat h_\lambda-h_0)\\|^2\right\| \leqslant C\nu_n\\|\mathcal T(\hat h_\lambda-h_0)\\|+C\nu_n^2, \label{eq:rdiv_projected_comparison} \end{align} where $\widehat R^\lambda(h)=\widehat L(h)+\lambda\\|h\\|^2$. |
| Lean | `Helpers/RDIV.lean:354` — `Papers.TsAdaptiveEstimationInference.err_l_dp` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-153`, `ass:pin-155`, `ass:pin-fix-4d0cd055` |
| cited results | `lem:rdiv-localized-concentration-lemma-14` (foster2023orthogonal, Lemma 14); `lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)) |
| assumed substrate gates | — |
| findings | — |

### 14. RDIV population--empirical risk comparison `lem:err-r-dp`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:237-246` (`lemma`, label `lemma:err_r_dp`) |
| statement | \label{lemma:err_r_dp} On $\mathcal E_n^{\mathrm R}$, uniformly over $h\in\mathcal H$ and $\lambda\in\Lambda_n$, \[ \left\| R^\lambda(h)-R^\lambda(h_0) -\{R_n^\lambda(h)-R_n^\lambda(h_0)\} \right\| \leqslant C\{\delta_n\\|h-h_0\\|+\delta_n^2\}. \] |
| Lean | `Helpers/RDIV.lean:605` — `Papers.TsAdaptiveEstimationInference.err_r_dp` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-153`, `ass:pin-155`, `ass:pin-fix-4d0cd055` |
| cited results | `lem:rdiv-localized-concentration-lemma-14` (foster2023orthogonal, Lemma 14); `lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)) |
| assumed substrate gates | — |
| findings | — |

### 15. Direct RDIV control `lem:str-dp`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:252-261` (`lemma`, label `lemma:str_dp`) |
| statement | \label{lemma:str_dp} On $\mathcal E_n^{\mathrm R}$, for every $\lambda\in\Lambda_n$, \[ \\|\mathcal T(\hat h_\lambda-h_0)\\|^2 +\lambda\\|\hat h_\lambda-h_0\\|^2 \leqslant C\left\{\frac{\delta_n^2}{\lambda} +\\|w_0\\|^2\lambda^p\right\}, \qquad p=\min\{\beta+1,2\}. \] |
| Lean | `Helpers/RDIV.lean:705` — `Papers.TsAdaptiveEstimationInference.str_dp` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-153`, `ass:pin-155`, `ass:pin-fix-4d0cd055` |
| cited results | `lem:rdiv-localized-concentration-lemma-14` (foster2023orthogonal, Lemma 14); `lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)) |
| assumed substrate gates | — |
| findings | — |

### 16. RDIV discrepancy bound `thm:rdiv-discrepancy-bound`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:278-280` (`corollary`, label `cor:rdiv_discrepancy_bound`) |
| statement | \label{cor:rdiv_discrepancy_bound} Under the assumptions of \cref{thm:deepiv_rate}, the calibrated RDIV scales satisfy \eqref{eq:rdiv_discrepancy_bound} simultaneously for every $\lambda\in\Lambda_n$ on $\mathcal E_n^{\mathrm R}$, for a fixed constant $C_{\mathrm R}>0$. |
| Lean | `TRdivDiscrepancyBound.lean:411` — `Papers.TsAdaptiveEstimationInference.rdiv_discrepancy_bound` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity pinned · correctness ok |
| added conditions | `ass:pin-153`, `ass:pin-155`, `ass:pin-174`, `ass:pin-fix-4d0cd055` |
| cited results | `lem:rdiv-localized-concentration-lemma-14` (foster2023orthogonal, Lemma 14); `lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)) |
| assumed substrate gates | — |
| findings | [note, V1.5] pinned reading: The assumptions imported from Theorem~\ref{thm:deepiv_rate} are the general hypotheses of Proposition~\ref{prop:exist_lambda_deepiv}, without imposing $h_0=0$. (Theorem~\ref{thm:deepiv_rate} imports the proposition's assumptions and treats $h_0=0$ only in a separate conditional conclusion. The present proof uses Lemmas~\ref{lemma:err_l_dp} and~\ref{lemma:str_dp} for general $h_0$.) |

### 17. RDIV error bounds `lem:rdiv-accepted-bounds`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:312-324` (`lemma`, label `lemma:rdiv_accepted_bounds`) |
| statement | \label{lemma:rdiv_accepted_bounds} Under the assumptions of \cref{thm:deepiv_rate}, for every $\lambda\in\Lambda_n$, on $\mathcal E_n^{\mathrm R}$, \begin{align*} \\|\mathcal T(\hat h_\lambda-h_0)\\|^2 &\leqslant C \{D_n(\hat h_\lambda)+A_n^{\mathrm R}(\lambda)\},\\ \\|\hat h_\lambda-h_0\\|^2 &\leqslant C\left[V_n^{\mathrm R}(\lambda) +\{D_n(\hat h_\lambda)+A_n^{\mathrm R}(\lambda)\}^\alpha\right], \end{align*} where the noise level and stability bound are defined in \eqref{eq:rdiv_common_scales} and $\alpha=\min\{\beta,1\}/\{1+\min\{\beta,1\}\}$. |
| Lean | `Helpers/RDIV.lean:833` — `Papers.TsAdaptiveEstimationInference.rdiv_accepted_bounds` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity pinned · correctness ok |
| added conditions | `ass:pin-153`, `ass:pin-155`, `ass:pin-196`, `ass:pin-fix-4d0cd055` |
| cited results | `lem:rdiv-localized-concentration-lemma-14` (foster2023orthogonal, Lemma 14); `lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)) |
| assumed substrate gates | — |
| findings | [note, V1.5] pinned reading: The assumptions of \cref{thm:deepiv_rate} mean its standing hypotheses inherited from \cref{prop:exist_lambda_deepiv}; the bounds here hold for arbitrary $h_0$, without imposing $h_0=0$. (The zero condition introduces only the last conclusion of \cref{thm:deepiv_rate}; the proof of \cref{lemma:rdiv_accepted_bounds} treats arbitrary $h_0$ using the variational source inequality.) |

### 18. Iterative Tikhonov bias `lem:iter-bias`

| field | value |
| --- | --- |
| bucket | `verified` |
| source | `appendices.tex:394-414` (`lemma`, label `lemma:iter_bias`) |
| statement | \label{lemma:iter_bias} Let $\mathcal A=\mathcal T^*\mathcal T$ and $r_\lambda(u)=\lambda/(u+\lambda)$. For every $j\geqslant1$, \begin{align} h_0-h_{j,\lambda}^*&=r_\lambda(\mathcal A)^j h_0 =\mathcal A^{\beta/2}r_\lambda(\mathcal A)^j w_0, \label{eq:iter_bias_identity}\\ \\|h_{j,\lambda}^*-h_0\\|^2 &\leqslant C\\|w_0\\|^2\lambda^{\min\{\beta,2j\}}, \label{eq:iter_strong_bias}\\ \\|\mathcal T(h_{j,\lambda}^*-h_0)\\|^2 &\leqslant C\\|w_0\\|^2\lambda^{\min\{\beta+1,2j\}}. \label{eq:iter_weak_bias} \end{align} Moreover, if $z_{j,\lambda}=r_\lambda(\mathcal A)^jw_0$, then $\\|z_{j,\lambda}\\|\leqslant\\|w_0\\|$ and \begin{equation}\label{eq:iter_bias_interpolation} \\|h_{j,\lambda}^*-h_0\\|^2 \leqslant \\|w_0\\|^{2/(\beta+1)} \\|\mathcal T(h_{j,\lambda}^*-h_0)\\|^{2\beta/(\beta+1)}. \end{equation} |
| Lean | `Helpers/IterBias.lean:382` — `Papers.TsAdaptiveEstimationInference.iter_bias` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | — |
| cited results | — |
| assumed substrate gates | — |
| findings | — |

### 19. Recursive RDIV stability `lem:iter-rdiv-squared-loss-concentration-lemma-14`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:429-449` (`lemma`, label `lemma:iter_rdiv_stability`) |
| statement | Apply the localized squared-loss comparison to $(h-h_{k,\lambda}^*)^2$, whose comparator and center are fixed. Here the localized squared-loss comparison is the scalar application of \citet[Lemma~14]{foster2023orthogonal} at the fixed comparator $h_{k,\lambda}^*$. |
| Lean | no declaration anchored |
| crosswalk verdict | — |
| axioms | — |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-153`, `ass:pin-155` |
| cited results | `lem:iter-rdiv-squared-loss-concentration-lemma-14` (foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling) |
| assumed substrate gates | — |
| findings | [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |

### 20. Iterative RDIV discrepancy bound `thm:iter-rdiv-discrepancy-bound`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:507-516` (`corollary`, label `cor:iter_rdiv_discrepancy_bound`) |
| statement | \label{cor:iter_rdiv_discrepancy_bound} Under the assumptions of \cref{thm:iter_rdiv_rate}, on $\mathcal E_{n,t}^{\mathrm{IR}}$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$, \[ D_n(\hat h_{t,\lambda}) \leqslant \frac{a_t\delta_n^2}{\lambda}+\nu_n^2 +C_{\mathrm{IR}}\\|w_0\\|^2\lambda^{q_t}, \qquad q_t=\min\{\beta+1,2t\}. \] |
| Lean | `TIterRdivDiscrepancyBound.lean:98` — `Papers.TsAdaptiveEstimationInference.iter_rdiv_discrepancy_bound` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-153`, `ass:pin-155`, `ass:pin-187`, `ass:pin-fix-4d0cd055` |
| cited results | `lem:iter-rdiv-fixed-center-concentration-lemma-14` (foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling); `lem:iter-rdiv-squared-loss-concentration-lemma-14` (foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling); `lem:rdiv-localized-concentration-lemma-14` (foster2023orthogonal, Lemma 14); `lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)) |
| assumed substrate gates | — |
| findings | — |

### 21. Iterative RDIV error bounds `lem:iter-rdiv-accepted-bounds`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:528-543` (`lemma`, label `lemma:iter_rdiv_accepted_bounds`) |
| statement | \label{lemma:iter_rdiv_accepted_bounds} Under the assumptions of \cref{thm:iter_rdiv_rate}, for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$, on $\mathcal E_{n,t}^{\mathrm{IR}}$, \begin{align*} \\|\mathcal T(\hat h_{t,\lambda}-h_0)\\|^2 &\leqslant C_t \{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm R}(\lambda)\},\\ \\|\hat h_{t,\lambda}-h_0\\|^2 &\leqslant C_t\left[ V_{n,t}^{\mathrm R}(\lambda) +\{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm R}(\lambda)\}^{\beta/(\beta+1)} \right], \end{align*} where the noise level and stability bound are defined in \eqref{eq:iter_rdiv_common_scales}. |
| Lean | `Helpers/IterRDIV.lean:329` — `Papers.TsAdaptiveEstimationInference.iter_rdiv_accepted_bounds` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-153`, `ass:pin-155`, `ass:pin-fix-4d0cd055` |
| cited results | `lem:iter-rdiv-fixed-center-concentration-lemma-14` (foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling); `lem:iter-rdiv-squared-loss-concentration-lemma-14` (foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling); `lem:rdiv-localized-concentration-lemma-14` (foster2023orthogonal, Lemma 14); `lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)) |
| assumed substrate gates | — |
| findings | — |

### 22. Uniform TRAE loss comparison `lem:concentrate-l`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:624-629` (`lemma`, label `lemma:concentrate_l`) |
| statement | \label{lemma:concentrate_l} Under \cref{asp:closeness,asp:mean_sq_continuity,asp:critical_radius}, on $\mathcal E_n$, the following inequalities hold simultaneously for all $h\in\mathcal H$: \[ \frac{1}{2} L( h) -O\left( \delta _{n}^{2}\right) \leqslant L_{n}( h) \leqslant \frac{5}{4} L( h) +O\left( \delta _{n}^{2}\right) . \] |
| Lean | `Helpers/TRAELossEvents.lean:436` — `Papers.TsAdaptiveEstimationInference.concentrate_l` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-206`, `ass:pin-fix-958aeb7a` |
| cited results | `lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9) |
| assumed substrate gates | — |
| findings | — |

### 23. Direct TRAE control `lem:strong-convexity`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:656-673` (`lemma`, label `lemma:strong_convexity`) |
| statement | \label{lemma:strong_convexity} Under the assumptions of \cref{thm:tikhonv_rate}, on $\mathcal E_n$, the following bound holds simultaneously for all $\lambda\in\Lambda_n$: \[ L_n(\hat h_\lambda)+\frac{\lambda}{2} \\|\hat h_\lambda-h_0\\|^2 \leqslant C\{\delta_n^2+\\|w_0\\|^2\lambda^p\}, \qquad p=\min\{\beta+1,2\}. \] Moreover, for all $\lambda\in\Lambda_n$, \[ \\|\hat h_\lambda-h_0\\|^2 \leqslant C\left[\frac{\delta_n^2}{\lambda} +\{L_n(\hat h_\lambda)+\delta_n^2\}^{\alpha}\right], \qquad \alpha=\frac{\min\{\beta,1\}}{1+\min\{\beta,1\}}. \] |
| Lean | `Helpers/TRAE.lean:233` — `Papers.TsAdaptiveEstimationInference.strong_convexity` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-156`, `ass:pin-206`, `ass:pin-fix-958aeb7a` |
| cited results | `lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9); `lem:trae-squared-norm-lemma-14` (foster2023orthogonal, Lemma 14) |
| assumed substrate gates | — |
| findings | — |

### 24. TRAE discrepancy bound `thm:trae-discrepancy-bound`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:722-724` (`corollary`, label `cor:trae_discrepancy_bound`) |
| statement | \label{cor:trae_discrepancy_bound} Under the assumptions of \cref{thm:tikhonv_rate}, the calibrated TRAE scale satisfies \eqref{eq:trae_discrepancy_bound} simultaneously for every $\lambda\in\Lambda_n$ on $\mathcal E_n$, for a fixed constant $C_{\mathrm T}>0$. |
| Lean | `TTraeDiscrepancyBound.lean:35` — `Papers.TsAdaptiveEstimationInference.trae_discrepancy_bound` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-156`, `ass:pin-160`, `ass:pin-206`, `ass:pin-fix-958aeb7a` |
| cited results | `lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9); `lem:trae-squared-norm-lemma-14` (foster2023orthogonal, Lemma 14) |
| assumed substrate gates | — |
| findings | — |

### 25. TRAE error bounds `lem:trae-accepted-bounds`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:739-751` (`lemma`, label `lemma:trae_accepted_bounds`) |
| statement | \label{lemma:trae_accepted_bounds} Suppose that the assumptions of \cref{thm:tikhonv_rate} hold. For every $\lambda\in\Lambda_n$, on $\mathcal E_n$, \begin{align*} \\|\mathcal T(\hat h_\lambda-h_0)\\|^2 &\leqslant C \{L_n(\hat h_\lambda)+\delta_n^2\},\\ \\|\hat h_\lambda-h_0\\|^2 &\leqslant C\left[V_n^{\mathrm T}(\lambda) +\{L_n(\hat h_\lambda)+\delta_n^2\}^\alpha\right], \end{align*} where the noise level and stability bound are defined in \eqref{eq:trae_common_scales} and $\alpha=\min\{\beta,1\}/\{1+\min\{\beta,1\}\}$. |
| Lean | `Helpers/TRAE.lean:284` — `Papers.TsAdaptiveEstimationInference.trae_accepted_bounds` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:pin-156`, `ass:pin-206`, `ass:pin-fix-958aeb7a` |
| cited results | `lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9); `lem:trae-squared-norm-lemma-14` (foster2023orthogonal, Lemma 14) |
| assumed substrate gates | — |
| findings | — |

### 26. Iterative TRAE propagation `lem:iter-trae-strong-propagation`

| field | value |
| --- | --- |
| bucket | `repaired` |
| source | `appendices.tex:807-822` (`lemma`, label `eq:iter_trae_strong_propagation`) |
| statement | Under the assumptions of \cref{thm:iter_trae_rate}, there is a nondecreasing deterministic sequence with $b_0=0$ and $b_j\geqslant2$, $j=1,\ldots,t$, where each $b_j$ depends only on $j$ and the prescribed boundedness and mean-square-continuity constants, independently of $\beta$ and $\\|w_0\\|$. This sequence may be calibrated so that, on $\mathcal E_{n,t}^{\mathrm{IT}}$, simultaneously for $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$, \begin{align} \\|\hat h_{j,\lambda}-h_{j,\lambda}^*\\|^2 &\leqslant C\frac{b_j\delta_n^2}{\lambda^2}, \qquad j=1,\ldots,t, \label{eq:iter_trae_strong_propagation}\\ L(\hat h_{t,\lambda}) &\leqslant C\left[ \delta_n^2+\min\left\{\lambda, \frac{b_t\delta_n^2}{\lambda}\right\} +\\|w_0\\|^2\lambda^{q_t} \right]. \label{eq:iter_trae_weak_propagation} \end{align} |
| Lean | `Helpers/IterTRAEPropagation.lean:38` — `Papers.TsAdaptiveEstimationInference.iter_trae_strong_propagation` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness bookkeeping |
| added conditions | `ass:bk-12`, `ass:pin-156`, `ass:pin-206`, `ass:pin-fix-958aeb7a` |
| cited results | `lem:iter-trae-anchored-concentration-lemma-9` (bennett2023source, Lemma 9 (lem:concentration)); `lem:iter-trae-fixed-center-squared-loss-lemma-9` (bennett2023source, Lemma 9 (lem:concentration)); `lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9) |
| assumed substrate gates | — |
| findings | [refuted, V1.5] the step its proof takes from [bennett2023source, Lemma 11] (lem:iter-trae-excess-loss-lemma-11) does not hold as the proof uses it: (a) The citation's hypotheses are not established: Lemma 11 assumes the conditions of the source's iterative theorem, which inherits condition (c), $\mathbb E[h_0(X)-h(X)\mid Z=\cdot]\in\mathcal F$. The host assumes only $\mathcal T(h_0-h)\in\mathcal F$, allowing a proper projected test space. (b) The source's standing assumption ``Realizability and Closedness'' requires convexity of $\mathcal H$, whereas the host allows general hypothesis classes. These are defects of the citation application; the random penalty center itself is permitted by the source. Also: the step its proof takes from [bennett2023source, Lemma 10] (lem:iter-trae-excess-loss-lemma-10) does not hold as the proof uses it: (a) Lemma 10 assumes the conditions of the source's one-step theorem, including condition (c), $\mathbb E[h_0(X)-h(X)\mid Z=\cdot]\in\mathcal F$. The host's projected closedness assumption does not establish this condition. (b) The source's standing ``Realizability and Closedness'' assumption requires convexity of $\mathcal H$, which the host does not impose. Substituting the random centered penalty is supported by the source, but it does not resolve these missing hypotheses. witness: (a) Let $V,\varepsilon$ be independent Rademacher variables, $X=Z=V$, $Y=1/2+\varepsilon/4$, $\tilde{\mathcal H}=\operatorname{span}\{1,x\}$, $\tilde{\mathcal Q}=\operatorname{span}\{1\}$, and $m(W;f)=Yf(Z)$. Take $\mathcal H=\{x\mapsto a+bx:0\leqslant a\leqslant1/2,\ b\in\{-1/4,0,1/4\}\}$ and $\mathcal F=\{z\mapsto d:\|d\|\leqslant1\}$. Then $\mathcal T(a+bx)=a$, $h_0=1/2$, and every population iterate $h_{j,\lambda}^*=\{1-(\lambda/(1+\lambda))^j\}/2$ belongs to $\mathcal H$. The host's boundedness, source, projected closedness, continuity, complexity, and measurable-minimum requirements hold. However, for $h(x)=x/4$, the unprojected residual $1/2-z/4$ is not in $\mathcal F$, violating the source's condition (c). (b) In the same instance, $0,x/4\in\mathcal H$ but $x/8\notin\mathcal H$, violating the source's standing convexity hypothesis. minimal patch (confirmed): proof change: We adapt the iterative argument of \citet[Theorem~6]{bennett2023source} to the finite grid and give the recursive comparison explicitly. Fix $j$ and $\lambda$, and set \[ e_j=\\|\hat h_{j,\lambda}-h_{j,\lambda}^*\\|, \qquad u_j=\\|\mathcal T(\hat h_{j,\lambda}-h_{j,\lambda}^*)\\|, \qquad f_j^*=\mathcal T(h_0-h_{j,\lambda}^*). \] Also define \[ M_{\leqslant j} =\max_{1\leqslant\ell\leqslant j} \left\{\lambda,\\|f_\ell^*\\|_\infty^2\right\}. \] By \cref{asp:iter_realizability_trae,asp:closeness}, $f_\ell^*\in\mathcal F$. The boundedness part of \cref{asp:critical_radius} therefore gives $M_{\leqslant j}\leqslant\max\{2,B^2\}$ for every grid point under consideration.  To track the random penalty center, write \begin{align*} I_{n,j} &=\\|h_{j,\lambda}^*-\hat h_{j-1,\lambda}\\|_{2,n}^2 -\\|\hat h_{j,\lambda}-\hat h_{j-1,\lambda}\\|_{2,n}^2,\\ I_j &=\\|h_{j,\lambda}^*-h_{j-1,\lambda}^*\\|^2 -\\|\hat h_{j,\lambda}-h_{j-1,\lambda}^*\\|^2. \end{align*} We derive the excess-loss comparison directly. Write $g=h_{j,\lambda}^*$, $k=f_j^*$, $\bar f=\mathcal T(h_0-\hat h_{j,\lambda})$, and \[ \ell(h,f)=2m(W;f)-2h(X)f(Z)-f(Z)^2. \] By \cref{asp:closeness,asp:iter_realizability_trae}, $k,\bar f\in\mathcal F$. For every $h\in\mathcal H$ and $f\in\mathcal F$, the Riesz identity in Section~\ref{sec:pre} and the definition of the projected operator give \[ \mathbb E[\ell(h,f)] =2\langle\mathcal T(h_0-h),f\rangle-\\|f\\|^2 =\\|\mathcal T(h_0-h)\\|^2-\\|f-\mathcal T(h_0-h)\\|^2. \] Indeed, $f\in\widetilde{\mathcal Q}$ implies $\mathbb E[h(X)f(Z)]=\langle\mathcal T h,f\rangle$, and $\mathbb E[m(W;f)]=\langle r_0,f\rangle=\langle\mathcal T h_0,f\rangle$.  Apply the localized concentration inequality of \citet[Lemma~9]{bennett2023source} to the product and moment-image classes in \cref{asp:critical_radius}, and apply its bounded Lipschitz version to the squared test functions. With the deterministic anchors $g$ and $k$, these inequalities give, uniformly over $h\in\mathcal H$ and $f\in\mathcal F$, \[ \begin{split} &\left\|(\mathbb E_n-\mathbb E)[\ell(h,f)-\ell(g,k)]\right\|\\ &\quad\leqslant C\delta_n\left\{ \bigl(\mathbb E[m(W;f-k)^2]\bigr)^{1/2} +\\|h(X)f(Z)-g(X)k(Z)\\|_{L_2(W)} +\\|f-k\\|\right\}+C\delta_n^2\\ &\quad\leqslant C\left\{\delta_n\\|h-g\\|\\|k\\|_\infty +\delta_n\\|f-k\\|+\delta_n^2\right\}. \end{split} \] For the second inequality, \cref{asp:iter_mean_sq_continuity} bounds the first norm by $C\\|f-k\\|$, while \[ h(X)f(Z)-g(X)k(Z) =h(X)(f-k)(Z)+(h-g)(X)k(Z) \] implies that the product norm is at most $B\\|f-k\\|+\\|h-g\\|\\|k\\|_\infty$. The square map is $2B$-Lipschitz on the range of the test functions. Also, $h_0\in\mathcal H$ and \cref{asp:closeness} imply $0\in\mathcal F$. Applying the same concentration inequality with zero moment, product, and test-function anchors, and using \cref{asp:mean_sq_continuity}, gives \[ \left\|(\mathbb E_n-\mathbb E)\ell(h,f)\right\| \leqslant C\{\delta_n\\|f\\|+\delta_n^2\}, \qquad h\in\mathcal H,\ f\in\mathcal F. \] Allocate fixed fractions of $\eta_{n,t}$ to these finitely many concentration inequalities and to the comparison of Lemma~\ref{lemma:concentrate_l} used below, absorbing the resulting fixed constants into the calibrated scale.  Taking $h=g$ in the increment bound and using the population identity yields \[ \begin{split} \mathbb E_n[\ell(g,f)-\ell(g,k)] &\leqslant-\\|f-k\\|^2+C\delta_n\\|f-k\\|+C\delta_n^2\\ &\leqslant C\delta_n^2. \end{split} \] Taking the supremum over $f\in\mathcal F$ therefore proves \[ 0\leqslant L_n(g)-\mathbb E_n\ell(g,k)\leqslant C\delta_n^2, \] without requiring an empirical maximizing test function. Next, $\\|\bar f-k\\|=u_j$, so the increment bound at $(\hat h_{j,\lambda},\bar f)$ gives \[ \begin{split} L(\hat h_{j,\lambda})-L(g) &\leqslant\mathbb E_n[\ell(\hat h_{j,\lambda},\bar f)-\ell(g,k)] +C\{\delta_ne_j\\|k\\|_\infty+\delta_nu_j+\delta_n^2\}\\ &\leqslant L_n(\hat h_{j,\lambda})-L_n(g) +C\{\delta_ne_j\\|k\\|_\infty+\delta_nu_j+\delta_n^2\}. \end{split} \] Empirical optimality relative to $g\in\mathcal H$ gives $L_n(\hat h_{j,\lambda})-L_n(g)\leqslant\lambda I_{n,j}$, including when the penalty center is random. Consequently, \begin{equation*} L(\hat h_{j,\lambda})-L(h_{j,\lambda}^*) \leqslant \lambda I_{n,j} +C\left\{ \delta_ne_j\\|f_j^*\\|_\infty +\delta_nu_j+\delta_n^2 \right\}. \end{equation*} This comparison uses the localized bounds for $\operatorname{star}(\mathcal H\cdotp\mathcal F)$, $\operatorname{star}(m\circ\mathcal F)$, and $\operatorname{star}(\mathcal F)$ from \cref{asp:critical_radius}, together with \cref{asp:iter_mean_sq_continuity}. In particular, that bound controls $\mathbb E[m(W;f_{\hat h_{j,\lambda}}-f_j^*)^2]$ by $Cu_j^2$, where $f_h=\mathcal T(h_0-h)$. The population normal equation for $h_{j,\lambda}^*$ gives the exact quadratic lower bound \begin{equation}\label{eq:iter_trae_population_curvature} \lambda e_j^2+u_j^2 \leqslant L(\hat h_{j,\lambda})-L(h_{j,\lambda}^*)-\lambda I_j. \end{equation} The quadratic identity for \eqref{eq:iter_population}, together with $L(h)=\\|\mathcal T(h-h_0)\\|^2$ on $\mathcal H$, proves this bound.  Expanding the squared penalties separates the center error as $2\langle\hat h_{j,\lambda}-h_{j,\lambda}^*, \hat h_{j-1,\lambda}-h_{j-1,\lambda}^*\rangle_n$. For the fixed-center squared-loss comparisons, apply the same localized concentration inequality of \citet[Lemma~9]{bennett2023source} to the bounded clipped losses corresponding to $(h-h_{j-1,\lambda}^*)^2$ at comparator $h_{j,\lambda}^*$ and $(h-h_{k,\lambda}^*)^2$ at comparator $h_{k,\lambda}^*$, for the fixed iteration--grid pairs under consideration. The fixed-population-center squared-loss measurability condition in Assumption~\ref{asp:critical_radius} supplies measurable localized empirical and Rademacher suprema for these applications.  Here is the scalar specialization and the penalty expansion. Put $M=\max\{1,B\}$ and $\mathcal U=\mathcal H/M$. For each deterministic center $c=h_{j-1,\lambda}^*$ or $c=h_{k,\lambda}^*$, use the loss \[ \ell_c(v;X)=\{M\max\{-1,\min\{v,1\}\}-c(X)\}^2. \] On the common full-probability set, $\|c(X)\|\leqslant M$, including $c=h_{0,\lambda}^*=0$, so this loss is $4M^2$-Lipschitz in $v$ and agrees with $(h(X)-c(X))^2$ at $v=h(X)/M$. For a comparator $g\in\mathcal H$, the inclusion $\operatorname{star}(\mathcal H-g)\subseteq\mathcal S:=\operatorname{star}(\mathcal H-\mathcal H)$ supplies the anchored complexity. Indeed, if $d\leqslant\delta_n$ is a critical radius for $\mathcal S$, then $\mathfrak R_n(d;\mathcal S)\leqslant d^2$ because $0\in\mathcal S$ and $\mathcal S\subseteq\operatorname{star}(\mathcal S-\mathcal S)$. Star scaling gives $\mathfrak R_n(s;\mathcal S)\leqslant sd$ for $s\geqslant d$. Hence, for $s\geqslant\delta_n$, \[ \mathfrak R_n(s;\operatorname{star}(\mathcal U-g/M)) \leqslant M^{-1}\mathfrak R_n(Ms;\mathcal S) \leqslant s\delta_n. \] The measurable squared-loss processes in Assumption~\ref{asp:critical_radius} are exactly the loss-increment processes for the stated centers and comparators, after this normalization. Lemma~9 therefore gives, on the allocated concentration event, \[ \left\|(\mathbb E_n-\mathbb E) \bigl[\{h(X)-h_{j-1,\lambda}^*(X)\}^2 -\{h_{j,\lambda}^*(X)-h_{j-1,\lambda}^*(X)\}^2\bigr]\right\| \leqslant C\{\delta_n\\|h-h_{j,\lambda}^*\\|+\delta_n^2\}, \] and, for $k=1,\ldots,t$, \[ \left\|\\|h-h_{k,\lambda}^*\\|_{2,n}^2-\\|h-h_{k,\lambda}^*\\|^2\right\| \leqslant C\{\delta_n\\|h-h_{k,\lambda}^*\\|+\delta_n^2\}. \] Allocate fixed fractions of $\eta_{n,t}$ to these scalar comparisons. Their constants depend only on the fixed class bounds.  Write $\Delta_j=\hat h_{j,\lambda}-h_{j,\lambda}^*$ and $\Delta_0=0$. Expansion gives \[ \begin{split} I_{n,j}-I_j ={}&(\mathbb E_n-\mathbb E) \bigl[\{h_{j,\lambda}^*-h_{j-1,\lambda}^*\}^2 -\{\hat h_{j,\lambda}-h_{j-1,\lambda}^*\}^2\bigr] +2\mathbb E_n[\Delta_j(X)\Delta_{j-1}(X)]. \end{split} \] The first term is at most $C(\delta_ne_j+\delta_n^2)$. The second scalar comparison and $2\delta_ne_k\leqslant e_k^2+\delta_n^2$ give $\\|\Delta_k\\|_{2,n}\leqslant C(e_k+\delta_n)$ for $k\geqslant1$; this bound also holds for $k=0$ because $\Delta_0=0$. Empirical Cauchy--Schwarz thus bounds the last term by \[ 2\\|\Delta_j\\|_{2,n}\\|\Delta_{j-1}\\|_{2,n} \leqslant 2Ce_je_{j-1}+C\delta_n(e_j+e_{j-1})+C\delta_n^2. \] Combining these bounds yields \begin{equation}\label{eq:iter_trae_center_comparison} I_{n,j}-I_j \leqslant C\delta_n(e_j+e_{j-1})+C\delta_n^2+2Ce_je_{j-1}. \end{equation} Combining \eqref{eq:iter_trae_biasless}--\eqref{eq:iter_trae_center_comparison} and applying Young's inequality gives \[ e_j^2\leqslant C\frac{\delta_n^2M_{\leqslant j}}{\lambda^2}+Ce_{j-1}^2. \] All terms proportional to $\delta_n^2$, including those multiplied by $\lambda\leqslant2$, are covered by $\delta_n^2M_{\leqslant j}/\lambda$ before division by $\lambda$. Since $e_0=0$ and $M_{\leqslant j}$ is uniformly bounded, enlarge the fixed constant $C\geqslant2$ if necessary, set $b_0=0$, and recursively choose $b_j=C(1+b_{j-1})$. Induction gives \eqref{eq:iter_trae_strong_propagation}. Thus each $b_j$ depends only on $j$ and the fixed constants.  For the weak bound, put $v_j=\\|\mathcal T(h_{j,\lambda}^*-h_0)\\|=\\|k\\|$. The absolute concentration inequality established in this proof, evaluated at the feasible residual $\bar f$, gives \[ L(\hat h_{j,\lambda}) \leqslant L_n(\hat h_{j,\lambda}) +C\{\delta_n\\|\bar f\\|+\delta_n^2\}. \] The assumptions of Lemma~\ref{lemma:concentrate_l} are inherited from the assumptions of \cref{thm:iter_trae_rate}. Its upper comparison, applied to $g=h_{j,\lambda}^*\in\mathcal H$ with the allocated confidence level, gives \[ L_n(g)\leqslant\frac54L(g)+C\delta_n^2 =\frac54v_j^2+C\delta_n^2. \] Empirical optimality, $L(g)=v_j^2$, and $\\|\bar f\\|\leqslant v_j+u_j$ therefore imply \[ \begin{split} L(\hat h_{j,\lambda})-L(g) &\leqslant\lambda I_{n,j}+\frac14v_j^2 +C\delta_n(v_j+u_j)+C\delta_n^2\\ &\leqslant\lambda I_{n,j}+C\{v_j^2+\delta_nu_j+\delta_n^2\}, \end{split} \] where the last inequality uses $\delta_nv_j\leqslant(v_j^2+\delta_n^2)/2$. Combining this with \eqref{eq:iter_trae_population_curvature} gives \[ \lambda e_j^2+u_j^2 \leqslant C\{v_j^2+\delta_nu_j+\delta_n^2\} +\lambda(I_{n,j}-I_j). \] Substituting \eqref{eq:iter_trae_center_comparison} and absorbing the linear terms gives \[ \frac{\lambda}{2}e_j^2+\frac12u_j^2 \leqslant C\left(v_j^2+\delta_n^2+\lambda e_{j-1}^2\right). \] At the final iteration, boundedness of $\mathcal H$ and \eqref{eq:iter_trae_strong_propagation} give the two simultaneous bounds \[ \lambda e_{t-1}^2\leqslant C\lambda, \qquad \lambda e_{t-1}^2 \leqslant C\frac{b_t\delta_n^2}{\lambda}. \] Taking their minimum, applying the squared triangle inequality, and using the weak-bias bound \eqref{eq:iter_weak_bias} prove \eqref{eq:iter_trae_weak_propagation}. The argument used only $\lambda\in(0,2]$, with the upper endpoint absorbed into fixed constants. Each comparison is applied to the fixed iteration--grid pairs in the oracle-relevant segment. The definition of $\eta_{n,t}$ and a union bound give the simultaneous event. |

### 27. Iterative TRAE discrepancy bound `thm:iter-trae-discrepancy-bound`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:924-926` (`corollary`, label `cor:iter_trae_discrepancy_bound`) |
| statement | \label{cor:iter_trae_discrepancy_bound} Under the assumptions of \cref{thm:iter_trae_rate}, the calibrated iterative TRAE scales satisfy \eqref{eq:iter_trae_discrepancy_bound} simultaneously for all $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ on $\mathcal E_{n,t}^{\mathrm{IT}}$. |
| Lean | `TIterTraeDiscrepancyBound.lean:53` — `Papers.TsAdaptiveEstimationInference.iter_trae_discrepancy_bound` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | `ass:bk-12`, `ass:pin-156`, `ass:pin-188`, `ass:pin-206`, `ass:pin-fix-958aeb7a` |
| cited results | `lem:iter-trae-anchored-concentration-lemma-9` (bennett2023source, Lemma 9 (lem:concentration)); `lem:iter-trae-fixed-center-squared-loss-lemma-9` (bennett2023source, Lemma 9 (lem:concentration)); `lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9) |
| assumed substrate gates | — |
| findings | — |

### 28. Iterative TRAE error bounds `lem:iter-trae-accepted-stability`

| field | value |
| --- | --- |
| bucket | `conditional` |
| source | `appendices.tex:941-966` (`lemma`, label `lemma:iter_trae_accepted_stability`) |
| statement | \label{lemma:iter_trae_accepted_stability} Let $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ satisfy $\lambda\geqslant\sqrt{s_{n,t}^{\mathrm T}}$. On $\mathcal E_{n,t}^{\mathrm{IT}}$, \begin{align} &\\|\mathcal T(\hat h_{t,\lambda}-h_0)\\|^2 \leqslant C_t \{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm T}(\lambda)\}, \label{eq:iter_trae_accepted_total}\\ &\\|\hat h_{t,\lambda}-h_0\\|^2 \leqslant C_t\left[ \frac{1}{\lambda}\{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm T}(\lambda)\} +\{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm T}(\lambda)\}^{\beta/(\beta+1)} \right], \label{eq:iter_trae_accepted_total_strong}\\ &\lambda\\|\hat h_{t,\lambda}-h_{t,\lambda}^*\\|^2 +\\|\mathcal T(\hat h_{t,\lambda}-h_{t,\lambda}^*)\\|^2 \leqslant C_t \{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm T}(\lambda)\}, \label{eq:iter_trae_accepted_variance}\\ &\\|\mathcal T(h_{t,\lambda}^*-h_0)\\|^2 \leqslant C_t \{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm T}(\lambda)\}, \label{eq:iter_trae_accepted_bias}\\ &\\|\hat h_{t,\lambda}-h_{t,\lambda}^*\\|^2 \leqslant \frac{C_t}{\lambda} \{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm T}(\lambda)\}. \label{eq:iter_trae_accepted_strong} \end{align} |
| appended hypothesis list | ass:iter-trae-accepted-stability-branch |
| Lean | `Helpers/IterTRAEStability.lean:282` — `Papers.TsAdaptiveEstimationInference.iter_trae_accepted_stability` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity pinned · correctness ok |
| added conditions | `ass:bk-12`, `ass:pin-156`, `ass:pin-203`, `ass:pin-206`, `ass:pin-fix-958aeb7a` |
| cited results | `lem:iter-trae-anchored-concentration-lemma-9` (bennett2023source, Lemma 9 (lem:concentration)); `lem:iter-trae-fixed-center-squared-loss-lemma-9` (bennett2023source, Lemma 9 (lem:concentration)); `lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9) |
| assumed substrate gates | — |
| findings | [note, V1.5] pinned reading: The constant $C_t$ may depend on the prescribed $t$ and fixed population quantities, including $\beta$ and $\\|w_0\\|$, but is independent of $n$, $\lambda$, and the realized fitting observations. (The proof applies Lemma~\ref{lemma:iter_bias}, whose interpolation inequality contributes $\\|w_0\\|^{2/(\beta+1)}$. The iterative-rate convention also explicitly permits constants depending on the prescribed iteration count and fixed population quantities.) |

### 29. RKHS quadratic reduction `prop:rkhs-qp`

| field | value |
| --- | --- |
| bucket | `verified` |
| source | `appendices.tex:1297-1312` (`proposition`, label `prop:rkhs_qp`) |
| statement | \label{prop:rkhs_qp} Let $\mathcal H$ and $\mathcal F$ be the RKHSs generated by $k_{\mathcal H}$ and $k_{\mathcal F}$, and let $K_{\mathcal H}$, $K_{\mathcal F}$, and $P_\eta$ be defined as in \cref{sec:rkhs_computation}. Among functions with the same fitting-sample evaluations, take the representative with minimum RKHS norm. Then the minimizer of \[ L_{N,\eta}(h)+\lambda\mathbb E_N[h(X)^2] \] has the form $\widehat h_\lambda(\cdot)=\sum_{j=1}^N\widehat\alpha_{\lambda,j}k_{\mathcal H}(X_j,\cdot)$, where $\widehat\alpha_\lambda$ solves \eqref{eq:rkhs_qp}. The minimum-Euclidean-norm coefficient vector is \begin{equation} \label{eq:rkhs_closed_form} \widehat\alpha_\lambda =\left(K_{\mathcal H}P_\eta K_{\mathcal H} +\lambda K_{\mathcal H}^2\right)^\dagger K_{\mathcal H}P_\eta\boldsymbol y, \end{equation} where $A^\dagger$ denotes the Moore--Penrose pseudoinverse of $A$. |
| appended hypothesis list | ass:rkhs-qp-spaces, ass:rkhs-qp-representative |
| Lean | `TRkhsQp.lean:19` — `Papers.TsAdaptiveEstimationInference.rkhs_qp` |
| crosswalk verdict | `equivalent` |
| axioms | clean — `propext`, `Classical.choice`, `Quot.sound` |
| V1.5 review | faithfulness ok · ambiguity ok · correctness ok |
| added conditions | — |
| cited results | — |
| assumed substrate gates | — |
| findings | — |

## Errors by root cause

3 independent error(s) — 3 repaired by a minimal patch, 0 not repaired, 0 unfixable and left out of Lean; 0 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_results list covers the headline oracle guarantee, one-step and iterative rates, and doubly robust inference. The carried repair to thm:normality preserves its contribution, and the propagation proof repair remains consistent with the downstream discrepancy and rate results.

- **`thm:normality`** (`main.tex:791-807`): [F3 reviewer] The evaluated random score in Theorem~\ref{cor:normality} needs a coherent jointly measurable version. The printed statement (main.tex:791--807) and its conditional Chebyshev argument (appendices.tex:1106--1123) do not state this well-definedness convention. IndependentEvaluationSample currently supplies jointly measurable versions of the nuisance evaluations only, not of the fitted score maps. Sectionwise measurability and the displayed population $L_2$ continuity bounds do not by themselves supply evaluation coherence. Add only the following measurability bookkeeping to the theorem and corrected paper: writing $S(w;h,q)=\tilde m(w;h)+m(w;q)-h(X(w))q(Z(w))$ and $d_n(w)=S(w;\hat h,\hat q)-S(w;h_0,q_0)$, for each $n$ the fitted score difference admits a version jointly measurable in the full nuisance-training/tuning information and a fresh observation, agreeing with $d_n$ for almost every training realization and almost every fresh observation, and agreeing almost surely at every observation in the finite evaluation block. Preserve the existing independent-evaluation and nuisance-version requirements. This is only well-definedness of the evaluated random score and its conditional expectation, not a supplied probability bound, convergence claim or conditional Chebyshev conclusion. Do not add countability, separability, compactness, a uniform empirical-process condition, or joint measurability of every score on the entire hypothesis space. Keep every rate, source, confidence and variance condition and the full asymptotic conclusion. The fixer must write this minimal convention explicitly into the corrected theorem; V1.5 must review it before Lean uses it. No substantive weakening of a main result is authorized.
  - witness: The evaluated random score in Theorem~\ref{cor:normality} needs a coherent jointly measurable version. The printed statement (main.tex:791--807) and its conditional Chebyshev argument (appendices.tex:1106--1123) do not state this well-definedness convention. IndependentEvaluationSample currently supplies jointly measurable versions of the nuisance evaluations only, not of the fitted score maps. Sectionwise measurability and the displayed population $L_2$ continuity bounds do not by themselves supply evaluation coherence. Add only the following measurability bookkeeping to the theorem and corrected paper: writing $S(w;h,q)=\tilde m(w;h)+m(w;q)-h(X(w))q(Z(w))$ and $d_n(w)=S(w;\hat h,\hat q)-S(w;h_0,q_0)$, for each $n$ the fitted score difference admits a version jointly measurable in the full nuisance-training/tuning information and a fresh observation, agreeing with $d_n$ for almost every training realization and almost every fresh observation, and agreeing almost surely at every observation in the finite evaluation block. Preserve the existing independent-evaluation and nuisance-version requirements. This is only well-definedness of the evaluated random score and its conditional expectation, not a supplied probability bound, convergence claim or conditional Chebyshev conclusion. Do not add countability, separability, compactness, a uniform empirical-process condition, or joint measurability of every score on the entire hypothesis space. Keep every rate, source, confidence and variance condition and the full asymptotic conclusion. The fixer must write this minimal convention explicitly into the corrected theorem; V1.5 must review it before Lean uses it. No substantive weakening of a main result is authorized.
  - Claim: Corollary~\ref{cor:normality} asserts asymptotic linearity and a normal limit for the independently evaluated doubly robust estimator.
  - What is wrong: ``For completeness, define the uncentered score \vsDisplay{S(W;h,q)=\tilde m(W;h)+m(W;q)-h(X)q(Z), \qquad d_n(W)=S(W;\hat h,\hat q)-S(W;h_0,q_0).\mbox{''}} Defining the fitted score difference sectionwise does not ensure a jointly measurable version coherent with both its conditional population integrals and its evaluations on the finite evaluation sample. Counterexample: Let $U$ and $V$ be independent uniform random variables on $[0,1]$, realized on the uncompleted Borel product probability space, with $U$ representing training information and $V$ a fresh observation. Let $A\subset[0,1]$ be a non-Borel set. Consider the sectionwise representatives $g(u,v)=\mathbf 1_A(u)\mathbf 1_{\{0\}}(v)$ and $\bar g(u,v)=0$, which represent the same zero population-$L_2$ function for every fixed $u$. Every fixed-training section of $g$ is measurable and has zero population-$L_2$ norm, but $g(U,V)$ is not measurable on this product space because its nonzero set is $A\times\{0\}$; choosing the coherent measurable version $\bar g$ resolves this version ambiguity.
  - Proposed fix: State explicitly the jointly measurable nuisance-evaluation versions and the coherent jointly measurable version of the fitted score difference, including agreement for almost every training realization and fresh observation and almost sure agreement throughout the finite evaluation sample; use that version in the proof, preserving the full asymptotic linearity and normal limit of Corollary~\ref{cor:normality}.
  - repair confirmed by an independent referee pass (see Corrected version); the corrected statement is proved in Lean
  - corrected statement: Suppose that the assumptions of \cref{thm:tikhonv_rate} hold for both $h_0$ and $q_0$ with (potentially different) source conditions $\beta_h,\beta_q>0$. Choose a deterministic sequence $\xi_n\downarrow0$, and run the primal and dual tuning rules at confidence levels $\xi_n/2$. Let a common deterministic sequence $\delta_n$ satisfy \eqref{eq:delta_trae} for both problems at these confidence levels. Assume that, for some $\gamma_h,\gamma_q>0$, the score maps satisfy \begin{align*} \mathbb{E}\left[\left(\tilde{m}( W;h_{1}) -\tilde{m}( W;h_{2})\right)^{2}\right] & \leqslant C\| h_{1} -h_{2} \| ^{\gamma_h },\\ \mathbb{E}\left[\left(m( W;q_{1}) -m( W;q_{2})\right)^{2}\right] & \leqslant C\| q_{1} -q_{2} \| ^{\gamma_q } \end{align*} for all $h_1,h_2\in\mathcal H$ and $q_1,q_2\in\mathcal Q$. Let $\beta_m=\max\{\beta_h,\beta_q\}$ and assume that $\delta_n$ satisfies \[ \delta _{n} =o\left( n^{-\alpha _{\mathrm{dr}}}\right) ,\qquad \alpha _{\mathrm{dr}} :=\frac{1+\min\{\beta _{m} ,1\}}{2+4\min\{\beta _{m} ,1\}} . \] Let $\hat h,\hat q$ be the adaptive adversarial Tikhonov estimators, trained and tuned independently of the evaluation sample used in $\hat\theta(\hat h,\hat q)$. For each $n$, write $\mathcal G_n$ for the full nuisance-training and tuning sigma-field. Take jointly measurable versions of the fitted nuisance evaluations, representing the fitted functions for almost every training realization and agreeing almost surely with their evaluations at every observation in the finite evaluation sample. Write \[ S(w;h,q)=\tilde m(w;h)+m(w;q)-h(X(w))q(Z(w)),\qquad  d_n(w)=S(w;\hat h,\hat q)-S(w;h_0,q_0). \] Suppose that, for each $n$, $d_n$ admits a version jointly measurable in $\mathcal G_n$ and a fresh observation, agreeing with $d_n$ for almost every training realization and almost every fresh observation, and agreeing almost surely with $d_n(W_i)$ simultaneously for $i=1,\ldots,n$ in the evaluation sample. Finally, assume that $0<\sigma_0^2:=\mathbb E[\psi_0(W)^2]<\infty$. Then, \[ \sqrt{n}(\hat{\theta }(\hat{h},\hat{q}) -\theta _{0}) =\frac{1}{\sqrt{n}}\sum _{i=1}^{n} \psi _{0}( W_i) +o_{p}( 1) , \] where $\psi _{0}( W) =\tilde{m}( W;h_{0}) +m( W;q_{0}) -q_{0}( Z) h_{0}( X) -\theta _{0}$. Consequently, $\sqrt n(\hat\theta(\hat h,\hat q)-\theta_0)\rightsquigarrow N(0,\sigma_0^2)$.
- **`lem:t-err-dp`** (`appendices.tex:102-124`): the step its proof takes from [li2024regularized, Theorem 24] (lem:conditional-mle-theorem-24) does not hold as the proof uses it: (a) The cited Theorem 24 explicitly imports the source-condition hypothesis of its referenced RDIV theorem. Lemma~\ref{lemma:t_err_dp} assumes only Assumptions~\ref{asp:re_cd} and \ref{asp:cr_deepiv}; these do not imply a source condition. Thus the citation, as stated, has an unestablished hypothesis. (b) The supplied theorem states an unsigned exponential confidence formula, and its proof explicitly gives $1-\exp(n\delta_n^2)$. Neither supplies the asserted probability $1-\xi/2$. The supplied localized concentration lemma has a usable confidence statement, but obtaining the density-only bound from it requires an explicit specialization rather than the printed direct invocation of Theorem 24.
  - witness: (a) Let $X,Z\in[0,1]$, let $\phi_j$ be the mean-zero orthonormal Rademacher functions, and put $\sigma_j=2^{-j^2-2}$ and $g_0(x\mid z)=1+\sum_{j\geqslant1}\sigma_j\phi_j(x)\phi_j(z)$, with $Z$ uniform. This is a bounded conditional density bounded away from zero. Take the closed primal and dual spaces generated by the constants and the $\phi_j$, set $h_0=\sum_{j\geqslant1}j^{-2}\phi_j$, $r_0=\mathcal T h_0$, $Y=r_0(Z)$, $\hat r=r_0$, $\mathcal H=\{c h_0:1/2\leqslant c\leqslant3/2\}$, and $\mathcal G=\{g_0,1\}$. The host's density, boundedness, measurability, pilot, and critical-radius conditions can all be satisfied. However, for every $\beta>0$, $\sum_{j\geqslant1}j^{-4}\sigma_j^{-2\beta}=\infty$, so the source hypothesis required by the cited theorem fails. (b) The source proof's reported probability $1-\exp(n\delta_n^2)$ is nonpositive, hence gives no positive confidence guarantee; replacing it by $1-\xi/2$ does not follow from that formula.
  - Claim: Lemma~\ref{lemma:t_err_dp} asserts the simultaneous density, nuisance, operator, and first-stage loss bounds with probability at least $1-\xi$ under Assumptions~\ref{asp:re_cd} and~\ref{asp:cr_deepiv}.
  - What is wrong: (a) ``\dots applied with $g_0\in\mathcal G$ and the critical radius of $\mathcal G_0$, gives \eqref{eq:rdiv_mle_hellinger} with probability at least $1-\xi/2$.'' This step uses the conditional-MLE bound of \citet[Theorem~24]{li2024regularized}, which the paper assumes only for the density lower bound and the conditions of the source's referenced RDIV theorem, including its source condition, but applies it to a model satisfying only Assumptions~\ref{asp:re_cd} and~\ref{asp:cr_deepiv}. (b) In the step quoted in (a): The cited theorem states an exponential confidence formula without a negative exponent, and its proof gives $1-\exp(n\delta_n^2)$, which is nonpositive and does not imply the asserted probability $1-\xi/2$. Counterexample: Let $\mu$ be Lebesgue measure on $[0,1]$, let $Z$ be uniform on $[0,1]$, and let $\phi_j$ be the mean-zero orthonormal Rademacher functions. Set $\sigma_j=2^{-j^2-2}$ and $g_0(x\mid z)=1+\sum_{j\geqslant1}\sigma_j\phi_j(x)\phi_j(z)$; both marginals are uniform, and $1/2\leqslant g_0\leqslant2$. Take $\tilde{\mathcal H}=\overline{\operatorname{span}}\{1,\phi_j:j\geqslant1\}\subseteq L_2(X)$ and $\tilde{\mathcal Q}=L_2(Z)$, so that $\mathcal T\phi_j=\sigma_j\phi_j$ and $\mathcal T^*\mathcal T\phi_j=\sigma_j^2\phi_j$. Set $h_0=\sum_{j\geqslant1}j^{-2}\phi_j$, $r_0=\mathcal T h_0$, $Y=r_0(Z)$, and $\hat r=r_0$. Take $\mathcal H=\{c h_0:1/2\leqslant c\leqslant3/2\}$, $\mathcal G=\{g_0,1\}$, $B=3$, $c_0=1/2$, and $C_G=2$. Choose $\delta_{n,r}=0$ and deterministic, $\beta$-independent critical-radius upper bounds satisfying the logarithmic floors in Assumption~\ref{asp:cr_deepiv}; these exist because the density class is finite and the primal and density-induced classes are bounded scalar intervals. Fix $n\geqslant1$ and $\xi\in(0,1)$. These objects satisfy Assumptions~\ref{asp:re_cd} and~\ref{asp:cr_deepiv}, but $\sum_{j\geqslant1}j^{-4}\sigma_j^{-2\beta}=\infty$ for every $\beta>0$, so the cited source condition fails; moreover, $1-\exp(n\delta_{n,\mathcal G}^2)\leqslant0<1-\xi/2$, so the reported confidence formula supplies no asserted positive probability guarantee even in this model.
  - Proposed fix: Replace the invocation of Theorem~24 by an explicit density-only specialization of the source's Localized Concentration lemma to the centered likelihood-transform class, followed by empirical likelihood optimality and a direct Hellinger comparison. This changes only the proof, adds no source condition, and protects Theorems~\ref{thm:deepiv_rate} and~\ref{thm:iter_rdiv_rate} and Corollaries~\ref{cor:rdiv_discrepancy_bound} and~\ref{cor:iter_rdiv_discrepancy_bound} at full strength.
  - repair confirmed by an independent referee pass (see Corrected version); the corrected statement is proved in Lean
- **`lem:iter-trae-strong-propagation`** (`appendices.tex:807-822`): the step its proof takes from [bennett2023source, Lemma 11] (lem:iter-trae-excess-loss-lemma-11) does not hold as the proof uses it: (a) The citation's hypotheses are not established: Lemma 11 assumes the conditions of the source's iterative theorem, which inherits condition (c), $\mathbb E[h_0(X)-h(X)\mid Z=\cdot]\in\mathcal F$. The host assumes only $\mathcal T(h_0-h)\in\mathcal F$, allowing a proper projected test space. (b) The source's standing assumption ``Realizability and Closedness'' requires convexity of $\mathcal H$, whereas the host allows general hypothesis classes. These are defects of the citation application; the random penalty center itself is permitted by the source. Also: the step its proof takes from [bennett2023source, Lemma 10] (lem:iter-trae-excess-loss-lemma-10) does not hold as the proof uses it: (a) Lemma 10 assumes the conditions of the source's one-step theorem, including condition (c), $\mathbb E[h_0(X)-h(X)\mid Z=\cdot]\in\mathcal F$. The host's projected closedness assumption does not establish this condition. (b) The source's standing ``Realizability and Closedness'' assumption requires convexity of $\mathcal H$, which the host does not impose. Substituting the random centered penalty is supported by the source, but it does not resolve these missing hypotheses.
  - witness: (a) Let $V,\varepsilon$ be independent Rademacher variables, $X=Z=V$, $Y=1/2+\varepsilon/4$, $\tilde{\mathcal H}=\operatorname{span}\{1,x\}$, $\tilde{\mathcal Q}=\operatorname{span}\{1\}$, and $m(W;f)=Yf(Z)$. Take $\mathcal H=\{x\mapsto a+bx:0\leqslant a\leqslant1/2,\ b\in\{-1/4,0,1/4\}\}$ and $\mathcal F=\{z\mapsto d:|d|\leqslant1\}$. Then $\mathcal T(a+bx)=a$, $h_0=1/2$, and every population iterate $h_{j,\lambda}^*=\{1-(\lambda/(1+\lambda))^j\}/2$ belongs to $\mathcal H$. The host's boundedness, source, projected closedness, continuity, complexity, and measurable-minimum requirements hold. However, for $h(x)=x/4$, the unprojected residual $1/2-z/4$ is not in $\mathcal F$, violating the source's condition (c). (b) In the same instance, $0,x/4\in\mathcal H$ but $x/8\notin\mathcal H$, violating the source's standing convexity hypothesis.
  - Claim: The simultaneous propagation bounds assert strong-error control for every fitted iteration and a final weak-loss bound under projected closeness and general hypothesis classes.
  - What is wrong: (a) ``\dots applied at this fixed iteration--grid pair, gives \vsDisplay{L(\hat h_{j,\lambda})-L(h_{j,\lambda}^*) \leqslant \lambda I_{n,j} +C\left\{ \delta_ne_j\|f_j^*\|_\infty +\delta_nu_j+\delta_n^2 \right\}.\mbox{''}} This step uses the excess-loss comparison of \citet[Lemma~11]{bennett2023source}, which the paper assumes only for the source's iterative-theorem conditions, including $\mathbb E[h_0(X)-h(X)\mid Z=\cdot]\in\mathcal F$ for every $h\in\mathcal H$, but applies it to a projected operator for which only $\mathcal T(h_0-h)\in\mathcal F$ is assumed. (b) The step quoted in (a) uses the excess-loss comparison of \citet[Lemma~11]{bennett2023source}, which the paper assumes only for the source's standing Realizability and Closedness assumption, which requires convexity of $\mathcal H$, but applies it to the general hypothesis class $\mathcal H$, which need not be convex. (c) ``For the weak bound, the ordinary excess-loss comparison \citep[Lemma~10]{bennett2023source} retains the population bias.'' This step uses the excess-loss comparison of \citet[Lemma~10]{bennett2023source}, which the paper assumes only for the source's one-step-theorem conditions, including $\mathbb E[h_0(X)-h(X)\mid Z=\cdot]\in\mathcal F$ for every $h\in\mathcal H$, but applies it to a projected operator for which only $\mathcal T(h_0-h)\in\mathcal F$ is assumed. (d) The step quoted in (c) uses the excess-loss comparison of \citet[Lemma~10]{bennett2023source}, which the paper assumes only for the source's standing Realizability and Closedness assumption, which requires convexity of $\mathcal H$, but applies it to the general hypothesis class $\mathcal H$, which need not be convex. (e) ``The localized squared-norm comparison for $\operatorname{star}(\mathcal H-\mathcal H)$ then yields \vsDisplay{I_{n,j}-I_j \leqslant C\delta_n(e_j+e_{j-1})+C\delta_n^2+2Ce_je_{j-1}.\mbox{''}} The localized concentration application for squares centered at $h_{j-1,\lambda}^*$ or $h_{k,\lambda}^*$ requires measurable localized loss suprema, which do not follow from measurability for unshifted squares centered at $h_0$ or for the specified zero-comparator adversarial process. Counterexample: For the citation-hypothesis obstruction, let $V$ and $\varepsilon$ be independent Rademacher variables, and set $X=Z=V$, $Y=1/2+\varepsilon/4$, and $W=(Y,X,Z)$. Take $\widetilde{\mathcal H}=\operatorname{span}\{1,x\}$, $\widetilde{\mathcal Q}=\operatorname{span}\{1\}$, and $m(W;q)=Yq(Z)$. Let $\mathcal H=\{x\mapsto a+bx:0\leqslant a\leqslant1/2,\ b\in\{-1/4,0,1/4\}\}$ and $\mathcal F=\{z\mapsto d:|d|\leqslant1\}$. Take $\beta=1$ and $w_0=h_0=1/2$; then $\mathcal T(a+bx)=a$ and $h_{j,\lambda}^*=\{1-(\lambda/(1+\lambda))^j\}/2\in\mathcal H$ for every $j\geqslant1$ and $\lambda>0$. Use independent copies of $W$, fix $t\geqslant1$ and $\xi\in(0,1)$, choose sufficiently large deterministic critical-radius bounds and calibrated scales of order $\sqrt{\{\log\log(en)+\log(t(K_n+1)/\xi)+1\}/n}$, and select measurable empirical minimizers over the compact class $\mathcal H$. For a separate instance of the squared-loss measurability obstruction, let $A\subset(0,1/2)$ be non-Lebesgue-measurable and let $W=X=Z$ have the completed uniform law on $[0,1]$. Take $\widetilde{\mathcal H}=L_2(X)$, $\widetilde{\mathcal Q}=\operatorname{span}\{1\}$, $\mathcal T h=\mathbb E[h(X)]$, $r_0=h_0=w_0=1$, $\beta=1$, $m(W;f)=f$, and $\mathcal F$ equal to the constant functions with values in $[0,1]$. Using the supplied representatives, take $\mathcal H=\{x\mapsto c:0\leqslant c\leqslant1\}\cup\{1-\mathbf1_{\{a\}}:a\in[0,1]\}\cup\{1-\tfrac12\mathbf1_{\{a\}}:a\in A\}$, set $t=2$ and $\rho=1/2$, and choose the constant empirical minimizers $\hat h_{j,\lambda}=h_{j,\lambda}^*=1-\{\lambda/(1+\lambda)\}^j$. Use $B=1$ and deterministic critical-radius bounds of order $\sqrt{\{\log\log(en)+\log(t(K_n+1)/\xi)+1\}/n}$; the localized linear Rademacher suprema are measurable with bounds $O(r/\sqrt n+1/n)$, and the optional singleton increments are dominated by the singleton increments indexed by $[0,1]$ for both the stipulated unshifted squared-loss and zero-comparator adversarial suprema. In the first instance, for $h(x)=x/4$ the unprojected residual $1/2-z/4$ is outside $\mathcal F$, and $0,x/4\in\mathcal H$ while their midpoint $x/8\notin\mathcal H$, so both excess-loss citations lack their required hypotheses; in the separate uniform-law instance, for $\lambda=1$ and sufficiently large $n$, the localized centered squared-loss supremum at $h_{1,1}^*=1/2$ with radius $1/2$ equals $(4n)^{-1}\mathbf1_{\{\exists i:X_i\in A\}}$ off the measurable null set of sample ties and is not measurable under the fitting-sample law.
  - Proposed fix: Replace the two excess-loss citations by direct localized-concentration arguments using the existing Riesz identity in Section~\ref{sec:pre}. Add to Assumption~\ref{asp:critical_radius} measurability of the localized empirical and Rademacher fixed-population-center squared-loss suprema used in the penalty comparisons, and apply Lemma~9 explicitly to their normalized clipped losses. Expand the empirical penalties and use empirical Cauchy--Schwarz to obtain the center comparison. This regularity condition is inherited by all the iterative TRAE results through Assumption~\ref{asp:critical_radius}, protecting Theorem~\ref{thm:iter_trae_rate} and Corollary~\ref{cor:iter_trae_discrepancy_bound} at full strength.
  - repair confirmed by an independent referee pass (see Corrected version); the corrected statement is proved in Lean

## Corrected version

26 of 29 result(s) differ from the version you provided — 3 repaired, 1 with the proof completed (no error), 22 only with a side condition or reading made explicit; the others are as printed. The Lean formalization is of the corrected version.

### `thm:common-dp` (`main.tex:260-285`)

- as printed: \label{thm:common_dp} Let $A_n:\Lambda_n\to[0,\infty)$, and let $\mathcal S_n\subseteq(0,2]$ be a stability region. Fix constants $C_A,C_V,C_b>0$ and exponents $\gamma\in(0,1]$ and $q>1$, and write $B_n(\lambda)=C_b\lambda^q$. Suppose that, on an event $\mathcal E_n$ with probability at least $1-\xi$, the discrepancy bound \begin{equation}\label{eq:common_discrepancy_bound} D_n(\hat h_\lambda)\leqslant A_n(\lambda)+B_n(\lambda), \end{equation} holds simultaneously over $\lambda\in\Lambda_n\cap\mathcal S_n$, and every $\lambda\in\Lambda_n\cap\mathcal S_n$ satisfies \begin{equation}\label{eq:common_accepted_bounds} \begin{aligned} \|\mathcal T(\hat h_\lambda-h_0)\|^2 &\leqslant C_A\{D_n(\hat h_\lambda)+A_n(\lambda)\},\\ \|\hat h_\lambda-h_0\|^2 &\leqslant C_V\left[\frac{D_n(\hat h_\lambda)+A_n(\lambda)}{\lambda}+\{D_n(\hat h_\lambda)+A_n(\lambda)\}^\gamma\right], \end{aligned} \end{equation} Suppose that some grid point $\lambda^\circ$ satisfies $\Lambda_n\cap[\lambda^\circ,2]\subseteq\mathcal S_n$, that $A_n$ is nonincreasing over the grid points in $[\lambda^\circ,2]$, and that \begin{equation}\label{eq:common_oracle_crossing} B_n(\lambda^\circ)\leqslant\tfrac12A_n(\lambda^\circ). \end{equation} Then \cref{alg:discrepancy} accepts a candidate before exhausting the grid, and returns a parameter satisfying $\lambda_{\mathrm{dp}}\geqslant\lambda^\circ$ with \begin{align}\label{eq:common_rate} \|\mathcal T(\hat h_{\lambda_{\mathrm{dp}}}-h_0)\|^2 &\leqslant O\{A_n(\lambda^\circ)\},\\ \|\hat h_{\lambda_{\mathrm{dp}}}-h_0\|^2 &\leqslant O\{A_n(\lambda^\circ)/\lambda^\circ+A_n(\lambda^\circ)^\gamma\}. \end{align}
- 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-142`: On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. — pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.

### `prop:exist-lambda-deepiv` (`main.tex:425-427`)

- as printed: \label{prop:exist_lambda_deepiv} Suppose \cref{asp:range,asp:source_condition,asp:closeness_deepiv,asp:re_cd,asp:cr_deepiv} hold. For every $n\geqslant N_{\mathrm R}(B)$, \cref{alg:discrepancy}, with the projected statistic \eqref{eq:rdiv_projected_loss} and noise level \eqref{eq:rdiv_common_scales}, accepts a candidate after at most $K_n+1=O(\log n)$ regularized fits, with probability at least $1-\xi$. If $h_0=0$, the initial parameter is accepted. In general, the returned parameter either is the initial parameter or is a noninitial grid point satisfying \eqref{eq:discrepancy_principle}.
- 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-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-155`: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.
- side condition added by the confirming referee `ass:pin-172`: For each fixed $n\geqslant N_{\mathrm R}(B)$, all probabilistic conclusions, including initial acceptance when $h_0=0$, hold on the same event $\mathcal E_n^{\mathrm R}$ of probability at least $1-\xi$. — pins reading of prop:exist-lambda-deepiv: The proof of Proposition~\ref{prop:exist_lambda_deepiv} explicitly works throughout on $\mathcal E_n^{\mathrm R}$ and derives initial acceptance there.
- side condition added by the confirming referee `ass:pin-174`: The assumptions imported from Theorem~\ref{thm:deepiv_rate} are the general hypotheses of Proposition~\ref{prop:exist_lambda_deepiv}, without imposing $h_0=0$. — pins reading of thm:rdiv-discrepancy-bound: Theorem~\ref{thm:deepiv_rate} imports the proposition's assumptions and treats $h_0=0$ only in a separate conditional conclusion. The present proof uses Lemmas~\ref{lemma:err_l_dp} and~\ref{lemma:str_dp} for general $h_0$.
- side condition added by the confirming referee `ass:pin-fix-4d0cd055`: For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair

### `thm:deepiv-rate` (`main.tex:429-448`)

- as printed: \label{thm:deepiv_rate} Under the assumptions of \cref{prop:exist_lambda_deepiv}, let $\lambda_{\mathrm{dp}}$ be its output and write \[ p=\min\{\beta+1,2\}, \qquad \alpha=\frac{\min\{\beta,1\}}{1+\min\{\beta,1\}}, \qquad \varepsilon_{n,\beta}=\delta_n^{2p/(p+1)}+\nu_n^2. \] For every $n\geqslant N_{\mathrm R}(B)$, with probability at least $1-\xi$, \begin{align*} \|\mathcal T(\hat h_{\lambda_{\mathrm{dp}}}-h_0)\|^2 \leqslant O(\varepsilon_{n,\beta}), \quad \|\hat h_{\lambda_{\mathrm{dp}}}-h_0\|^2 \leqslant O\!\left( \delta_n^{2(p-1)/(p+1)} +\varepsilon_{n,\beta}^{\alpha}\right) \leqslant O(\varepsilon_{n,\beta}^\alpha). \end{align*} If $h_0=0$, the squared strong and weak errors are both $O(\delta_n^2)$.
- 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-142`: On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. — pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.
- side condition added by the confirming referee `ass:pin-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-155`: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.
- side condition added by the confirming referee `ass:pin-172`: For each fixed $n\geqslant N_{\mathrm R}(B)$, all probabilistic conclusions, including initial acceptance when $h_0=0$, hold on the same event $\mathcal E_n^{\mathrm R}$ of probability at least $1-\xi$. — pins reading of prop:exist-lambda-deepiv: The proof of Proposition~\ref{prop:exist_lambda_deepiv} explicitly works throughout on $\mathcal E_n^{\mathrm R}$ and derives initial acceptance there.
- side condition added by the confirming referee `ass:pin-174`: The assumptions imported from Theorem~\ref{thm:deepiv_rate} are the general hypotheses of Proposition~\ref{prop:exist_lambda_deepiv}, without imposing $h_0=0$. — pins reading of thm:rdiv-discrepancy-bound: Theorem~\ref{thm:deepiv_rate} imports the proposition's assumptions and treats $h_0=0$ only in a separate conditional conclusion. The present proof uses Lemmas~\ref{lemma:err_l_dp} and~\ref{lemma:str_dp} for general $h_0$.
- side condition added by the confirming referee `ass:pin-190`: For each fixed $n\geqslant N_{\mathrm R}(B)$, the displayed error bounds and, when $h_0=0$, both sharper $O(\delta_n^2)$ bounds hold jointly with probability at least $1-\xi$ over the preliminary, fitting, and calibration samples. — pins reading of thm:deepiv-rate: The proof obtains the zero-signal bounds from the probabilistic initial-acceptance conclusion of Proposition~\ref{prop:exist_lambda_deepiv} and Lemma~\ref{lemma:str_dp}; the general case uses the common RDIV event and the stated confidence convention.
- side condition added by the confirming referee `ass:pin-196`: The assumptions of \cref{thm:deepiv_rate} mean its standing hypotheses inherited from \cref{prop:exist_lambda_deepiv}; the bounds here hold for arbitrary $h_0$, without imposing $h_0=0$. — pins reading of lem:rdiv-accepted-bounds: The zero condition introduces only the last conclusion of \cref{thm:deepiv_rate}; the proof of \cref{lemma:rdiv_accepted_bounds} treats arbitrary $h_0$ using the variational source inequality.
- side condition added by the confirming referee `ass:pin-fix-4d0cd055`: For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair

### `prop:exist-lambda-iter-rdiv` (`main.tex:527-529`)

- as printed: \label{prop:exist_lambda_iter_rdiv} Suppose the assumptions of \cref{prop:exist_lambda_deepiv} hold. For every $n\geqslant N_{\mathrm R}(B)$, \cref{alg:discrepancy}, applied to $\hat h_{t,\lambda}$ with the noise level \eqref{eq:iter_rdiv_common_scales}, accepts a candidate with probability at least $1-\xi$. It solves at most $t(K_n+1)=O(t\log n)$ regularized least-squares problems. If $h_0=0$, the initial parameter is accepted. Otherwise, the output is either the initial parameter or a noninitial point satisfying \eqref{eq:discrepancy_principle} with $D_n(\hat h_{t,\lambda})$ and $A_{n,t}^{\mathrm R}(\lambda)$.
- 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-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-155`: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.
- side condition added by the confirming referee `ass:pin-187`: The displayed discrepancy inequality holds on $\mathcal E_{n,t}^{\mathrm{IR}}$, with $\mathbb P(\mathcal E_{n,t}^{\mathrm{IR}})\geqslant1-\xi$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$. — pins reading of def:iter-rdiv-discrepancy-bound: The sentence introducing the display cites Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, which expressly states simultaneity on this event; Lemma~\ref{lemma:iter_rdiv_stability} specifies its probability.
- side condition added by the confirming referee `ass:pin-205`: For each fixed $n\geqslant N_{\mathrm R}(B)$ and prescribed $t$, there is an event of probability at least $1-\xi$ on which the rule accepts a candidate and satisfies the stated output alternatives, including acceptance of $\lambda_0=2$ when $h_0=0$. — pins reading of prop:exist-lambda-iter-rdiv: The proof of Proposition~\ref{prop:exist_lambda_iter_rdiv} explicitly places initial acceptance on the corresponding concentration event.
- side condition added by the confirming referee `ass:pin-fix-4d0cd055`: For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair

### `thm:iter-rdiv-rate` (`main.tex:531-553`)

- as printed: \label{thm:iter_rdiv_rate} Under the assumptions of \cref{prop:exist_lambda_iter_rdiv,asp:iter_realizability_rdiv}, let $\lambda_{\mathrm{dp},t}^{\mathrm R}$ denote the selected parameter and set \[ \varepsilon_{n,t,\beta}^{\mathrm R} =(s_{n,t}^{\mathrm R})^{q_t/(q_t+1)}+\nu_n^2. \] For every fixed $t$ and every $n\geqslant N_{\mathrm R}(B)$, with probability at least $1-\xi$, \begin{align*} \|\mathcal T(\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm R}}-h_0)\|^2 \leqslant O_t(\varepsilon_{n,t,\beta}^{\mathrm R}), \quad \|\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm R}}-h_0\|^2 \leqslant O_t\!\left( (s_{n,t}^{\mathrm R})^{(q_t-1)/(q_t+1)} +(\varepsilon_{n,t,\beta}^{\mathrm R})^{\beta/(\beta+1)} \right). \end{align*} Here $O_t(\cdot)$ allows constants that depend on the prescribed iteration count and fixed population quantities. In particular, $a_t$ is independent of $n$. If $\nu_n=O(\delta_n)$ and $2t\geqslant\beta+1$, the bounds reduce to \begin{align*} \|\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm R}}-h_0\|^2 =O_t\!\left(\delta_n^{2\beta/(\beta+2)}\right), \quad \|\mathcal T(\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm R}}-h_0)\|^2 =O_t\!\left(\delta_n^{2(\beta+1)/(\beta+2)}\right). \end{align*}
- 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-142`: On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. — pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.
- side condition added by the confirming referee `ass:pin-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-155`: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.
- side condition added by the confirming referee `ass:pin-187`: The displayed discrepancy inequality holds on $\mathcal E_{n,t}^{\mathrm{IR}}$, with $\mathbb P(\mathcal E_{n,t}^{\mathrm{IR}})\geqslant1-\xi$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$. — pins reading of def:iter-rdiv-discrepancy-bound: The sentence introducing the display cites Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, which expressly states simultaneity on this event; Lemma~\ref{lemma:iter_rdiv_stability} specifies its probability.
- side condition added by the confirming referee `ass:pin-205`: For each fixed $n\geqslant N_{\mathrm R}(B)$ and prescribed $t$, there is an event of probability at least $1-\xi$ on which the rule accepts a candidate and satisfies the stated output alternatives, including acceptance of $\lambda_0=2$ when $h_0=0$. — pins reading of prop:exist-lambda-iter-rdiv: The proof of Proposition~\ref{prop:exist_lambda_iter_rdiv} explicitly places initial acceptance on the corresponding concentration event.
- side condition added by the confirming referee `ass:pin-fix-4d0cd055`: For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair

### `prop:exist-lambda` (`main.tex:646-657`)

- as printed: \label{prop:exist_lambda} Suppose that Assumptions \ref{asp:range}, \ref{asp:source_condition}, \ref{asp:closeness_deepiv}, \ref{asp:closeness}, \ref{asp:mean_sq_continuity}, and \ref{asp:critical_radius} hold. Run \cref{alg:discrepancy} with inputs $\lambda_0 = 2$, a fixed $1/2\leqslant\rho<1$, and noise level $A_n^{\mathrm T}(\lambda)=\delta^2_n$, where $\delta_n$ satisfies \eqref{eq:delta_trae}. For every $n\geqslant N_{\mathrm T}(B)$, the algorithm accepts a candidate after at most $K_n+1=O(\log n)$ optimization problems, with probability at least $1-\xi$. The returned parameter either is $\lambda_0$ through the initial-acceptance branch or is a noninitial grid point satisfying \eqref{eq:discrepancy_principle}.
- 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-156`: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.
- side condition added by the confirming referee `ass:pin-160`: For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. — pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.
- side condition added by the confirming referee `ass:pin-193`: For each fixed $n\geqslant N_{\mathrm T}(B)$, acceptance within the stated number of optimization problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_n$, with probability at least $1-\xi$ over the fitting sample. — pins reading of prop:exist-lambda: The proof begins by restricting throughout to $\mathcal E_n$, establishes terminal acceptance there, and derives both return branches on that same event.
- side condition added by the confirming referee `ass:pin-206`: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- side condition added by the confirming referee `ass:pin-fix-958aeb7a`: The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair

### `thm:tikhonv-rate` (`main.tex:660-667`)

- as printed: \label{thm:tikhonv_rate} Under the assumptions of \cref{prop:exist_lambda}, let $ {\lambda_{\mathrm{dp}}} $ be chosen by \cref{alg:discrepancy}. For every $n\geqslant N_{\mathrm T}(B)$, with probability at least $ 1-\xi $, \[ \| \hat{h}_{\lambda_{\mathrm{dp}}} -h_{0} \| ^{2} \leqslant O\left( \delta _{n}^{2\min\{\beta ,1\} /( 1+\min\{\beta ,1\})}\right) ,\qquad \| \mathcal{T}(\hat{h}_{\lambda_{\mathrm{dp}}} -h_{0}) \| ^{2} \leqslant O\left( \delta _{n}^{2}\right) , \] where $\delta_n$ is defined in (\ref{eq:delta_trae}).
- 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-142`: On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. — pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.
- side condition added by the confirming referee `ass:pin-156`: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.
- side condition added by the confirming referee `ass:pin-160`: For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. — pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.
- side condition added by the confirming referee `ass:pin-193`: For each fixed $n\geqslant N_{\mathrm T}(B)$, acceptance within the stated number of optimization problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_n$, with probability at least $1-\xi$ over the fitting sample. — pins reading of prop:exist-lambda: The proof begins by restricting throughout to $\mathcal E_n$, establishes terminal acceptance there, and derives both return branches on that same event.
- side condition added by the confirming referee `ass:pin-194`: With $\beta$ and $\xi$ fixed as $n\to\infty$, for each $n\geqslant N_{\mathrm T}(B)$ the displayed strong- and weak-error bounds hold jointly on a fitting-sample event of probability at least $1-\xi$. — pins reading of thm:tikhonv-rate: The proof of Theorem~\ref{thm:tikhonv_rate} works on $\mathcal E_n$, which the preceding confidence allocation constructs separately for each sample size.
- side condition added by the confirming referee `ass:pin-206`: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- side condition added by the confirming referee `ass:pin-fix-958aeb7a`: The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair

### `prop:exist-lambda-iter-trae` (`main.tex:737-739`)

- as printed: \label{prop:exist_lambda_iter_trae} Suppose the assumptions of \cref{prop:exist_lambda} hold. For every $n\geqslant N_{\mathrm{IT}}(B)$, \cref{alg:discrepancy}, applied to $\hat h_{t,\lambda}$ with the noise level \eqref{eq:iter_trae_scale}, accepts a candidate after at most $t(K_n+1)=O(t\log n)$ minimax problems, with probability at least $1-\xi$. The returned parameter is either the initial point or a noninitial point satisfying \eqref{eq:discrepancy_principle} with the iterative loss and threshold.
- 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-156`: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.
- side condition added by the confirming referee `ass:pin-160`: For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. — pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.
- side condition added by the confirming referee `ass:pin-195`: For each fixed $n\geqslant N_{\mathrm{IT}}(B)$ and prescribed $t$, acceptance within the stated number of minimax problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_{n,t}^{\mathrm{IT}}$, with probability at least $1-\xi$ over the fitting sample. — pins reading of prop:exist-lambda-iter-trae: The proof works throughout on $\mathcal E_{n,t}^{\mathrm{IT}}$ and derives terminal acceptance and the first-accepted-index alternative there.
- side condition added by the confirming referee `ass:pin-206`: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- side condition added by the confirming referee `ass:pin-fix-958aeb7a`: The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair

### `thm:iter-trae-rate` (`main.tex:741-765`)

- as printed: \label{thm:iter_trae_rate} Under the assumptions of \cref{prop:exist_lambda_iter_trae,asp:iter_realizability_trae,asp:iter_mean_sq_continuity}, let $\lambda_{\mathrm{dp},t}^{\mathrm T}$ denote the selected parameter, with $s_{n,t}^{\mathrm T}$ and $q_t$ defined above. For every fixed $t$ and every $n\geqslant N_{\mathrm{IT}}(B)$, with probability at least $1-\xi$, \begin{align*} \|\mathcal T(\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm T}}-h_0)\|^2 \leqslant O_t\!\left( (s_{n,t}^{\mathrm T})^{q_t/(q_t+1)} \right), \quad \|\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm T}}-h_0\|^2 \leqslant O_t\!\left( (s_{n,t}^{\mathrm T})^{(q_t-1)/(q_t+1)} \right). \end{align*} Because $t$ is fixed, $s_{n,t}^{\mathrm T}=b_t\delta_n^2=O_t(\delta_n^2)$, so $b_t$ affects only the constants hidden by $O_t(\cdot)$. If $2t\geqslant\beta+1$, these bounds become \begin{align*} \|\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm T}}-h_0\|^2 =O_t\!\left(\delta_n^{2\beta/(\beta+2)}\right), \quad \|\mathcal T(\hat h_{t,\lambda_{\mathrm{dp},t}^{\mathrm T}}-h_0)\|^2 =O_t\!\left(\delta_n^{2(\beta+1)/(\beta+2)}\right). \end{align*}
- 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-12`: With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\|u-u_{j,\lambda}^*\|_{L_2(W;\mathbb R^3)}\leqslant r}|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}|$ and $\sup_{u\in\mathcal U,\,\|u-u_{j,\lambda}^*\|_{L_2(W;\mathbb R^3)}\leqslant r}|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$. — The localized excess-loss concentration uses this nonzero deterministic anchor; the stipulated zero-anchor and fixed-center squared-loss measurability do not supply these suprema.
- side condition added by the confirming referee `ass:pin-142`: On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. — pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.
- side condition added by the confirming referee `ass:pin-156`: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.
- side condition added by the confirming referee `ass:pin-160`: For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. — pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.
- side condition added by the confirming referee `ass:pin-188`: Under the assumptions of Theorem~\ref{thm:iter_trae_rate}, the displayed inequality holds simultaneously for all $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ on $\mathcal E_{n,t}^{\mathrm{IT}}$, where $\mathbb P(\mathcal E_{n,t}^{\mathrm{IT}})\geqslant1-\xi$. — pins reading of def:iter-trae-discrepancy-bound: The sentence immediately preceding the display invokes Corollary~\ref{cor:iter_trae_discrepancy_bound}, whose statement explicitly places this simultaneous bound on the common concentration event.
- side condition added by the confirming referee `ass:pin-195`: For each fixed $n\geqslant N_{\mathrm{IT}}(B)$ and prescribed $t$, acceptance within the stated number of minimax problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_{n,t}^{\mathrm{IT}}$, with probability at least $1-\xi$ over the fitting sample. — pins reading of prop:exist-lambda-iter-trae: The proof works throughout on $\mathcal E_{n,t}^{\mathrm{IT}}$ and derives terminal acceptance and the first-accepted-index alternative there.
- side condition added by the confirming referee `ass:pin-203`: The constant $C_t$ may depend on the prescribed $t$ and fixed population quantities, including $\beta$ and $\|w_0\|$, but is independent of $n$, $\lambda$, and the realized fitting observations. — pins reading of lem:iter-trae-accepted-stability: The proof applies Lemma~\ref{lemma:iter_bias}, whose interpolation inequality contributes $\|w_0\|^{2/(\beta+1)}$. The iterative-rate convention also explicitly permits constants depending on the prescribed iteration count and fixed population quantities.
- side condition added by the confirming referee `ass:pin-204`: The constants hidden by $O_t(\cdot)$ may depend on the prescribed $t$, fixed population quantities including $\beta$ and $\|w_0\|$, and the fixed grid ratio $\rho$, but are independent of $n$ and the candidate $\lambda$. — pins reading of thm:iter-trae-rate: The text following the bounds in Theorem~\ref{thm:iter_rdiv_rate} explicitly allows dependence on the prescribed iteration count and fixed population quantities; Lemma~\ref{lemma:iter_trae_accepted_stability} supplies the comparison constants used here under the same convention.
- side condition added by the confirming referee `ass:pin-206`: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- side condition added by the confirming referee `ass:pin-fix-958aeb7a`: The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair

### `thm:normality` (`main.tex:791-807`)

- as printed: \label{cor:normality} Suppose that the assumptions of \cref{thm:tikhonv_rate} hold for both $h_0$ and $q_0$ with (potentially different) source conditions $\beta_h,\beta_q>0$. Choose a deterministic sequence $\xi_n\downarrow0$, and run the primal and dual tuning rules at confidence levels $\xi_n/2$. Let a common deterministic sequence $\delta_n$ satisfy \eqref{eq:delta_trae} for both problems at these confidence levels. Assume that, for some $\gamma_h,\gamma_q>0$, the score maps satisfy \begin{align*} \mathbb{E}\left[\left(\tilde{m}( W;h_{1}) -\tilde{m}( W;h_{2})\right)^{2}\right] & \leqslant C\| h_{1} -h_{2} \| ^{\gamma_h },\\ \mathbb{E}\left[\left(m( W;q_{1}) -m( W;q_{2})\right)^{2}\right] & \leqslant C\| q_{1} -q_{2} \| ^{\gamma_q } \end{align*} for all $h_1,h_2\in\mathcal H$ and $q_1,q_2\in\mathcal Q$. Let $\beta_m=\max\{\beta_h,\beta_q\}$ and assume that $\delta_n$ satisfies \[ \delta _{n} =o\left( n^{-\alpha _{\mathrm{dr}}}\right) ,\qquad \alpha _{\mathrm{dr}} :=\frac{1+\min\{\beta _{m} ,1\}}{2+4\min\{\beta _{m} ,1\}} . \] Let $\hat h,\hat q$ be the adaptive adversarial Tikhonov estimators, trained and tuned independently of the evaluation sample used in $\hat\theta(\hat h,\hat q)$. Finally, assume that $0<\sigma_0^2:=\mathbb E[\psi_0(W)^2]<\infty$. Then, \[ \sqrt{n}(\hat{\theta }(\hat{h},\hat{q}) -\theta _{0}) =\frac{1}{\sqrt{n}}\sum _{i=1}^{n} \psi _{0}( W_i) +o_{p}( 1) , \] where $\psi _{0}( W) =\tilde{m}( W;h_{0}) +m( W;q_{0}) -q_{0}( Z) h_{0}( X) -\theta _{0}$. Consequently, $\sqrt n(\hat\theta(\hat h,\hat q)-\theta_0)\rightsquigarrow N(0,\sigma_0^2)$.
- corrected: Suppose that the assumptions of \cref{thm:tikhonv_rate} hold for both $h_0$ and $q_0$ with (potentially different) source conditions $\beta_h,\beta_q>0$. Choose a deterministic sequence $\xi_n\downarrow0$, and run the primal and dual tuning rules at confidence levels $\xi_n/2$. Let a common deterministic sequence $\delta_n$ satisfy \eqref{eq:delta_trae} for both problems at these confidence levels. Assume that, for some $\gamma_h,\gamma_q>0$, the score maps satisfy \begin{align*} \mathbb{E}\left[\left(\tilde{m}( W;h_{1}) -\tilde{m}( W;h_{2})\right)^{2}\right] & \leqslant C\| h_{1} -h_{2} \| ^{\gamma_h },\\ \mathbb{E}\left[\left(m( W;q_{1}) -m( W;q_{2})\right)^{2}\right] & \leqslant C\| q_{1} -q_{2} \| ^{\gamma_q } \end{align*} for all $h_1,h_2\in\mathcal H$ and $q_1,q_2\in\mathcal Q$. Let $\beta_m=\max\{\beta_h,\beta_q\}$ and assume that $\delta_n$ satisfies \[ \delta _{n} =o\left( n^{-\alpha _{\mathrm{dr}}}\right) ,\qquad \alpha _{\mathrm{dr}} :=\frac{1+\min\{\beta _{m} ,1\}}{2+4\min\{\beta _{m} ,1\}} . \] Let $\hat h,\hat q$ be the adaptive adversarial Tikhonov estimators, trained and tuned independently of the evaluation sample used in $\hat\theta(\hat h,\hat q)$. For each $n$, write $\mathcal G_n$ for the full nuisance-training and tuning sigma-field. Take jointly measurable versions of the fitted nuisance evaluations, representing the fitted functions for almost every training realization and agreeing almost surely with their evaluations at every observation in the finite evaluation sample. Write \[ S(w;h,q)=\tilde m(w;h)+m(w;q)-h(X(w))q(Z(w)),\qquad d_n(w)=S(w;\hat h,\hat q)-S(w;h_0,q_0). \] Suppose that, for each $n$, $d_n$ admits a version jointly measurable in $\mathcal G_n$ and a fresh observation, agreeing with $d_n$ for almost every training realization and almost every fresh observation, and agreeing almost surely with $d_n(W_i)$ simultaneously for $i=1,\ldots,n$ in the evaluation sample. Finally, assume that $0<\sigma_0^2:=\mathbb E[\psi_0(W)^2]<\infty$. Then, \[ \sqrt{n}(\hat{\theta }(\hat{h},\hat{q}) -\theta _{0}) =\frac{1}{\sqrt{n}}\sum _{i=1}^{n} \psi _{0}( W_i) +o_{p}( 1) , \] where $\psi _{0}( W) =\tilde{m}( W;h_{0}) +m( W;q_{0}) -q_{0}( Z) h_{0}( X) -\theta _{0}$. Consequently, $\sqrt n(\hat\theta(\hat h,\hat q)-\theta_0)\rightsquigarrow N(0,\sigma_0^2)$.
- side condition added by the confirming referee `ass:pin-142`: On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. — pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.
- side condition added by the confirming referee `ass:pin-156`: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.
- side condition added by the confirming referee `ass:pin-160`: For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. — pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.
- side condition added by the confirming referee `ass:pin-193`: For each fixed $n\geqslant N_{\mathrm T}(B)$, acceptance within the stated number of optimization problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_n$, with probability at least $1-\xi$ over the fitting sample. — pins reading of prop:exist-lambda: The proof begins by restricting throughout to $\mathcal E_n$, establishes terminal acceptance there, and derives both return branches on that same event.
- side condition added by the confirming referee `ass:pin-194`: With $\beta$ and $\xi$ fixed as $n\to\infty$, for each $n\geqslant N_{\mathrm T}(B)$ the displayed strong- and weak-error bounds hold jointly on a fitting-sample event of probability at least $1-\xi$. — pins reading of thm:tikhonv-rate: The proof of Theorem~\ref{thm:tikhonv_rate} works on $\mathcal E_n$, which the preceding confidence allocation constructs separately for each sample size.
- side condition added by the confirming referee `ass:pin-206`: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- side condition added by the confirming referee `ass:pin-fix-958aeb7a`: The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair
- proof change: the printed proof, revised step by step] Apply \cref{thm:tikhonv_rate} to both nuisance estimators at the prescribed confidence levels $\xi_n/2$. Their rate bounds then hold on a common event with probability at least $1-\xi_n\to1$. Because $\beta_h,\beta_q>0$ and $\delta_n\to0$, these bounds give \begin{equation}\label{eq:conv_hq} \| \hat{h} -h_{0} \| =o_{p}( 1) ,\qquad \| \hat{q} -q_{0}\| =o_{p}( 1) . \end{equation} The remaining bias condition is \begin{equation}\label{eq:square_root} \sqrt{n}\mathbb{E}[(\hat{q}( Z) -q_{0}( Z))(\hat{h}( X) -h_{0}( X))] =o_{p}( 1) . \end{equation} Suppose first that $\beta_m=\beta_h\geqslant\beta_q$. By the adjoint identity and the Cauchy--Schwarz inequality, \begin{align*} \left|\mathbb{E}[(\hat{q}( Z) -q_{0}( Z))(\hat{h}( X) -h_{0}( X))]\right| & \leqslant \|\hat h-h_0\|\,\|\mathcal T^*(\hat q-q_0)\|\\  & =O_{p}\left( \delta _{n}^{\min\{\beta _{m} ,1\} /( 1+\min\{\beta _{m} ,1\})} \cdot \delta _{n}\right)\\  & =O_{p}\left( \delta _{n}^{1+\min\{\beta _{m} ,1\} /( 1+\min\{\beta _{m} ,1\})}\right) . \end{align*} If $\beta_m=\beta_q\geqslant\beta_h$, the symmetric bound \[ \left|\mathbb{E}[(\hat{q}( Z) -q_{0}( Z))(\hat{h}( X) -h_{0}( X))]\right| \leqslant \|\hat q-q_0\|\,\|\mathcal T(\hat h-h_0)\| \] gives the same rate. By assumption, $\delta _{n} =o\left( n^{-( 1+\min\{\beta _{m} ,1\}) /( 2+4\min\{\beta _{m} ,1\})}\right)$, and hence \[ \left|\mathbb{E}[(\hat{q}( Z) -q_{0}( Z))(\hat{h}( X) -h_{0}( X))]\right| =o_{p}\left( n^{-1/2}\right), \] which proves \eqref{eq:square_root}.  Use the jointly measurable version of $d_n$ stipulated in Corollary~\ref{cor:normality}. Its agreement with the fitted score difference for almost every training realization and almost every fresh observation identifies its conditional population integrals with those of the score difference. Its almost sure agreement at every evaluation observation identifies its evaluation average with the difference of the two evaluated score averages. Conditional on the nuisance-training data, the evaluation observations are independent and identically distributed. Boundedness of the nuisance classes and the score continuity assumptions imply \[ \mathbb E[d_n(W)^2\mid\text{training}] \leqslant C\left\{ \|\hat h-h_0\|^{\gamma_h}+\|\hat q-q_0\|^{\gamma_q} +\|\hat h-h_0\|^2+\|\hat q-q_0\|^2 \right\}=o_p(1). \] Conditional Chebyshev's inequality therefore gives $\sqrt n(\mathbb E_{n,\mathrm{eval}}-\mathbb E)d_n=o_p(1)$, where $\mathbb E d_n$ integrates a fresh observation conditional on the training data. The primal and dual moment equations give \[ \mathbb E d_n =-\mathbb E[(\hat h(X)-h_0(X))(\hat q(Z)-q_0(Z))]. \] Combining this identity with \eqref{eq:square_root} yields \[ \sqrt n\{\hat\theta(\hat h,\hat q)-\theta_0\} =\frac1{\sqrt n}\sum_{i=1}^n\psi_0(W_i)+o_p(1). \] The moment equations also imply $\mathbb E\psi_0(W)=0$. The classical central limit theorem applies because $0<\mathbb E[\psi_0(W)^2]<\infty$, proving the stated normal limit.
- why: [F3 reviewer] The evaluated random score in Theorem~\ref{cor:normality} needs a coherent jointly measurable version. The printed statement (main.tex:791--807) and its conditional Chebyshev argument (appendices.tex:1106--1123) do not state this well-definedness convention. IndependentEvaluationSample currently supplies jointly measurable versions of the nuisance evaluations only, not of the fitted score maps. Sectionwise measurability and the displayed population $L_2$ continuity bounds do not by themselves supply evaluation coherence. Add only the following measurability bookkeeping to the theorem and corrected paper: writing $S(w;h,q)=\tilde m(w;h)+m(w;q)-h(X(w))q(Z(w))$ and $d_n(w)=S(w;\hat h,\hat q)-S(w;h_0,q_0)$, for each $n$ the fitted score difference admits a version jointly measurable in the full nuisance-training/tuning information and a fresh observation, agreeing with $d_n$ for almost every training realization and almost every fresh observation, and agreeing almost surely at every observation in the finite evaluation block. Preserve the existing independent-evaluation and nuisance-version requirements. This is only well-definedness of the evaluated random score and its conditional expectation, not a supplied probability bound, convergence claim or conditional Chebyshev conclusion. Do not add countability, separability, compactness, a uniform empirical-process condition, or joint measurability of every score on the entire hypothesis space. Keep every rate, source, confidence and variance condition and the full asymptotic conclusion. The fixer must write this minimal convention explicitly into the corrected theorem; V1.5 must review it before Lean uses it. No substantive weakening of a main result is authorized.

### `lem:t-err-dp` (`appendices.tex:102-124`)

- as printed: \label{lemma:t_err_dp} Under \cref{asp:re_cd,asp:cr_deepiv}, with probability at least $1-\xi$, \begin{align} \mathbb E_Z\!\left[H^2\{\hat g(\cdot\mid Z),g_0(\cdot\mid Z)\}\right] &\leqslant C\delta_{n,\mathcal G}^2, \label{eq:rdiv_mle_hellinger}\\ \|\hat r-r_0\|_{L_2(Z)}&\leqslant \delta_{n,r},\nonumber\\ \sup_{h\in\mathcal H} \|(\hat{\mathcal T}-\mathcal T)h\|_{L_2(Z)} &\leqslant C\delta_{n,\mathcal G},\label{eq:rdiv_op_global}\\ \|(\hat{\mathcal T}-\mathcal T)(h-h')\|_{L_1(Z)} &\leqslant C\delta_{n,\mathcal G}\|h-h'\|, \qquad h,h'\in\mathcal H.\label{eq:rdiv_op_relative} \end{align} Moreover, if \[ \widehat L(h):=\mathbb E[(Y-\hat{\mathcal T}h(Z))^2\mid I_{\mathrm{pre}}], \] then, uniformly over $h,h'\in\mathcal H$, \begin{equation}\label{eq:rdiv_first_stage_loss} |\{\widehat L(h)-\widehat L(h')\}-\{L(h)-L(h')\}| \leqslant C\delta_{n,\mathcal G}\|h-h'\|. \end{equation}
- corrected: statement unchanged
- no longer uses `lem:conditional-mle-theorem-24`
- side condition added by the confirming referee `ass:pin-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-fix-4d0cd055`: For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair
- proof change: the printed proof, revised step by step] Allocate failure probability $\xi/2$ to the conditional-density estimator and $\xi/2$ to the pilot regression. As elsewhere, evaluate the corresponding deterministic accuracy bounds at these confidence levels and relabel them $\delta_{n,\mathcal G}$ and $\delta_{n,r}$. We establish the density bound directly using the Localized Concentration lemma of \citet{li2024regularized}. Write $P=\mathbb E_{X,Z}$ and $P_n=\mathbb E_{n,\mathrm{pre}}$, and define \[ f_g(x,z)=\sqrt{\frac{g(x\mid z)+g_0(x\mid z)}{2g_0(x\mid z)}},\qquad \mathcal U=\{f_g-1:g\in\mathcal G\}. \] Realizability gives $f_{g_0}=1$, so $0\in\mathcal U$. The density bounds give \[ 2^{-1/2}\leqslant f_g\leqslant M,\qquad M=\sqrt{\frac{C_G/c_0+1}{2}}. \] Thus $\mathcal U$ is uniformly bounded by a fixed constant. Define the globally Lipschitz loss \[ \ell(v)=-\log\bigl(\max\{1+v,2^{-1/2}\}\bigr). \] The maximum map is $1$-Lipschitz, and the derivative of $-\log t$ has absolute value at most $\sqrt2$ for $t\geqslant2^{-1/2}$. Hence $\ell$ is $\sqrt2$-Lipschitz, $\ell(0)=0$, and $\ell(f_g-1)=-\log f_g$.  We verify the localized complexity condition for this application. Put $\mathcal S=\operatorname{star}(\mathcal G_0-\mathcal G_0)$ and choose a deterministic critical radius $d\leqslant\delta_{n,\mathcal G}$ as supplied by Assumption~\ref{asp:cr_deepiv}. By the critical-radius definition, $\mathfrak R_n(d;\mathcal S)\leqslant d^2$. For $s\geqslant d$, star-shapedness implies that $(d/s)v\in\mathcal S$ and $\|(d/s)v\|_{L_2(X,Z)}\leqslant d$ whenever $v\in\mathcal S$ and $\|v\|_{L_2(X,Z)}\leqslant s$. Scaling inside the Rademacher supremum therefore gives \[ \mathfrak R_n(s;\mathcal S) \leqslant\frac{s}{d}\mathfrak R_n(d;\mathcal S) \leqslant sd. \] Since $1\in\mathcal G_0$, we have $\operatorname{star}(\mathcal U)\subseteq\mathcal S$. Taking $s=\delta_{n,\mathcal G}$ yields \[ \mathfrak R_n\bigl(\delta_{n,\mathcal G};\operatorname{star}(\mathcal U)\bigr) \leqslant\delta_{n,\mathcal G}d \leqslant\delta_{n,\mathcal G}^2. \] Apply the cited localized concentration lemma with input $(X,Z)$, one coordinate, class $\mathcal U$, fixed comparator $0$, loss $\ell$, and failure probability $\xi/2$. The measurability and logarithmic radius floor are supplied by Assumption~\ref{asp:cr_deepiv}, with the confidence evaluation specified in the proof. Consequently, for a fixed constant $A>0$, with probability at least $1-\xi/2$, \[ \left|(P_n-P)(-\log f_g)\right| \leqslant A\left\{\delta_{n,\mathcal G}\|f_g-1\|_{L_2(X,Z)}+\delta_{n,\mathcal G}^2\right\}, \qquad g\in\mathcal G. \]  To use empirical optimality, note that $P_n\log g_0(X\mid Z)$ is finite by the density bounds. Since $\hat g$ maximizes the empirical log likelihood and all candidate densities are bounded above, $\hat g(X_i\mid Z_i)>0$ at every preliminary observation, almost surely. For $a>0$, the arithmetic-geometric mean inequality gives $(a+1)/2\geqslant\sqrt a$, and hence \[ -\log\sqrt{\frac{a+1}{2}}\leqslant-\frac14\log a. \] Taking $a=\hat g(X_i\mid Z_i)/g_0(X_i\mid Z_i)$ and averaging gives \[ P_n(-\log f_{\hat g}) \leqslant-\frac14P_n\log\frac{\hat g(X\mid Z)}{g_0(X\mid Z)} \leqslant0. \] For every $g\in\mathcal G$, conditional density normalization gives \[ P f_g^2 =\mathbb E_Z\int\frac{g(x\mid Z)+g_0(x\mid Z)}2\,\mu(dx)=1. \] Since $\log t\leqslant t-1$ for $t>0$, it follows that \[ P(-\log f_g)\geqslant1-Pf_g =\frac12P(f_g-1)^2. \] On the concentration event, put $b=\|f_{\hat g}-1\|_{L_2(X,Z)}$. Combining these inequalities and using Young's inequality gives \[ \frac{b^2}{2} \leqslant P(-\log f_{\hat g}) \leqslant A\delta_{n,\mathcal G}b+A\delta_{n,\mathcal G}^2 \leqslant\frac{b^2}{4}+(A^2+A)\delta_{n,\mathcal G}^2. \] Thus $b^2\leqslant4(A^2+A)\delta_{n,\mathcal G}^2$.  Finally, for $t\geqslant0$ and $f=\sqrt{(t^2+1)/2}$, we have $f\leqslant t+1$ and \[ |t-1|=\frac{2(f+1)}{t+1}|f-1|\leqslant4|f-1|. \] Apply this pointwise with $t=\sqrt{\hat g/g_0}$ and integrate under the true joint distribution to obtain \[ \begin{aligned} \mathbb E_Z\!\left[H^2\{\hat g(\cdot\mid Z),g_0(\cdot\mid Z)\}\right] &=P\left(\sqrt{\frac{\hat g(X\mid Z)}{g_0(X\mid Z)}}-1\right)^2\\ &\leqslant16b^2 \leqslant64(A^2+A)\delta_{n,\mathcal G}^2. \end{aligned} \] This proves \eqref{eq:rdiv_mle_hellinger} with probability at least $1-\xi/2$. The nuisance-rate assumption gives $\|\hat r-r_0\|_{L_2(Z)}\leqslant\delta_{n,r}$ with probability at least $1-\xi/2$. Therefore, without requiring independence between $\hat g$ and $\hat r$, the union bound shows that both inequalities hold simultaneously with probability at least $1-\xi$. We work on this intersection for the rest of the proof. Here \[ H^2(g_1,g_2)=\int\{\sqrt{g_1(x)}-\sqrt{g_2(x)}\}^2\,\mu(dx). \]  Let $P_X$ be the true marginal distribution of $X$ and let $\hat P_X$ be the marginal induced by $Z\sim P_Z$ and $X\mid Z\sim\hat g(\cdot\mid Z)$. The density bounds in \cref{asp:re_cd} give $\hat g(x\mid z)/g_0(x\mid z)\leqslant C_G/c_0$ and hence \[ \frac{d\hat P_X}{dP_X}\leqslant \frac{C_G}{c_0}. \] The total-variation--Hellinger inequality and $\|h\|_\infty\leqslant B$ give \[ \|(\hat{\mathcal T}-\mathcal T)h\|^2 \leqslant B^2\mathbb E_Z \left[\left\{\int|\hat g(x\mid Z)-g_0(x\mid Z)|\,\mu(dx)\right\}^2\right] \leqslant C B^2\delta_{n,\mathcal G}^2, \] which proves \eqref{eq:rdiv_op_global}. For $u=h-h'$, factor $\hat g/g_0-1=(\sqrt{\hat g/g_0}-1)(\sqrt{\hat g/g_0}+1)$. Cauchy--Schwarz and the conditional density-ratio bound yield \begin{align*} \|(\hat{\mathcal T}-\mathcal T)u\|_1 &\leqslant \mathbb E\left[|u(X)| \left|\sqrt{\frac{\hat g(X\mid Z)}{g_0(X\mid Z)}}-1\right| \left\{\sqrt{\frac{\hat g(X\mid Z)}{g_0(X\mid Z)}}+1\right\}\right]\\ &\leqslant C\|u\| \mathbb E_Z\!\left[H^2\{\hat g(\cdot\mid Z),g_0(\cdot\mid Z)\}\right]^{1/2} \leqslant C\delta_{n,\mathcal G}\|u\|, \end{align*} which proves \eqref{eq:rdiv_op_relative}. \citet{li2024regularized} use the same operator argument in their MLE analysis.  Finally, write $e_h=(\hat{\mathcal T}-\mathcal T)h$. Pointwise, \begin{align*} &(Y-\hat{\mathcal T}h)^2-(Y-\hat{\mathcal T}h')^2 -\{(Y-\mathcal T h)^2-(Y-\mathcal T h')^2\}\\ &\quad=-(\hat{\mathcal T}-\mathcal T)u (2Y-\hat{\mathcal T}h-\hat{\mathcal T}h') +\mathcal T u(e_h+e_{h'}). \end{align*} The expectation of the first term is at most $4B\|(\hat{\mathcal T}-\mathcal T)u\|_1$; the second is at most $\|\mathcal T u\|(\|e_h\|+\|e_{h'}\|)$. Apply \eqref{eq:rdiv_op_global}, \eqref{eq:rdiv_op_relative}, and conditional Jensen to obtain \eqref{eq:rdiv_first_stage_loss}.
- why: the step its proof takes from [li2024regularized, Theorem 24] (lem:conditional-mle-theorem-24) does not hold as the proof uses it: (a) The cited Theorem 24 explicitly imports the source-condition hypothesis of its referenced RDIV theorem. Lemma~\ref{lemma:t_err_dp} assumes only Assumptions~\ref{asp:re_cd} and \ref{asp:cr_deepiv}; these do not imply a source condition. Thus the citation, as stated, has an unestablished hypothesis. (b) The supplied theorem states an unsigned exponential confidence formula, and its proof explicitly gives $1-\exp(n\delta_n^2)$. Neither supplies the asserted probability $1-\xi/2$. The supplied localized concentration lemma has a usable confidence statement, but obtaining the density-only bound from it requires an explicit specialization rather than the printed direct invocation of Theorem 24.

### `lem:err-l-dp` (`appendices.tex:183-196`)

- as printed: \label{lemma:err_l_dp} Under \cref{asp:range,asp:closeness_deepiv,asp:re_cd,asp:cr_deepiv}, there is an event $\mathcal E_n^{\mathrm R}$ with probability at least $1-\xi$ on which, simultaneously for every $\lambda\in\Lambda_n$ and $h\in\mathcal H$, \begin{align} &\left|\{\widehat R^\lambda(h)-\widehat R^\lambda(h_0)\} -\{R_n^\lambda(h)-R_n^\lambda(h_0)\}\right| \leqslant C\{\delta_{n,\mathcal H}\|h-h_0\| +\delta_{n,\mathcal H}^2\},\label{eq:rdiv_fit_comparison}\\ &\left|D_n(\hat h_\lambda) -\|\mathcal T(\hat h_\lambda-h_0)\|^2\right| \leqslant C\nu_n\|\mathcal T(\hat h_\lambda-h_0)\|+C\nu_n^2, \label{eq:rdiv_projected_comparison} \end{align} where $\widehat R^\lambda(h)=\widehat L(h)+\lambda\|h\|^2$.
- 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-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-155`: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.
- side condition added by the confirming referee `ass:pin-fix-4d0cd055`: For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair

### `lem:err-r-dp` (`appendices.tex:237-246`)

- as printed: \label{lemma:err_r_dp} On $\mathcal E_n^{\mathrm R}$, uniformly over $h\in\mathcal H$ and $\lambda\in\Lambda_n$, \[ \left| R^\lambda(h)-R^\lambda(h_0) -\{R_n^\lambda(h)-R_n^\lambda(h_0)\} \right| \leqslant C\{\delta_n\|h-h_0\|+\delta_n^2\}. \]
- 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-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-155`: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.
- side condition added by the confirming referee `ass:pin-fix-4d0cd055`: For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair

### `lem:str-dp` (`appendices.tex:252-261`)

- as printed: \label{lemma:str_dp} On $\mathcal E_n^{\mathrm R}$, for every $\lambda\in\Lambda_n$, \[ \|\mathcal T(\hat h_\lambda-h_0)\|^2 +\lambda\|\hat h_\lambda-h_0\|^2 \leqslant C\left\{\frac{\delta_n^2}{\lambda} +\|w_0\|^2\lambda^p\right\}, \qquad p=\min\{\beta+1,2\}. \]
- 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-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-155`: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.
- side condition added by the confirming referee `ass:pin-fix-4d0cd055`: For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair

### `thm:rdiv-discrepancy-bound` (`appendices.tex:278-280`)

- as printed: \label{cor:rdiv_discrepancy_bound} Under the assumptions of \cref{thm:deepiv_rate}, the calibrated RDIV scales satisfy \eqref{eq:rdiv_discrepancy_bound} simultaneously for every $\lambda\in\Lambda_n$ on $\mathcal E_n^{\mathrm R}$, for a fixed constant $C_{\mathrm R}>0$.
- 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-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-155`: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.
- side condition added by the confirming referee `ass:pin-174`: The assumptions imported from Theorem~\ref{thm:deepiv_rate} are the general hypotheses of Proposition~\ref{prop:exist_lambda_deepiv}, without imposing $h_0=0$. — pins reading of thm:rdiv-discrepancy-bound: Theorem~\ref{thm:deepiv_rate} imports the proposition's assumptions and treats $h_0=0$ only in a separate conditional conclusion. The present proof uses Lemmas~\ref{lemma:err_l_dp} and~\ref{lemma:str_dp} for general $h_0$.
- side condition added by the confirming referee `ass:pin-fix-4d0cd055`: For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair

### `lem:rdiv-accepted-bounds` (`appendices.tex:312-324`)

- as printed: \label{lemma:rdiv_accepted_bounds} Under the assumptions of \cref{thm:deepiv_rate}, for every $\lambda\in\Lambda_n$, on $\mathcal E_n^{\mathrm R}$, \begin{align*} \|\mathcal T(\hat h_\lambda-h_0)\|^2 &\leqslant C \{D_n(\hat h_\lambda)+A_n^{\mathrm R}(\lambda)\},\\ \|\hat h_\lambda-h_0\|^2 &\leqslant C\left[V_n^{\mathrm R}(\lambda) +\{D_n(\hat h_\lambda)+A_n^{\mathrm R}(\lambda)\}^\alpha\right], \end{align*} where the noise level and stability bound are defined in \eqref{eq:rdiv_common_scales} and $\alpha=\min\{\beta,1\}/\{1+\min\{\beta,1\}\}$.
- 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-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-155`: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.
- side condition added by the confirming referee `ass:pin-196`: The assumptions of \cref{thm:deepiv_rate} mean its standing hypotheses inherited from \cref{prop:exist_lambda_deepiv}; the bounds here hold for arbitrary $h_0$, without imposing $h_0=0$. — pins reading of lem:rdiv-accepted-bounds: The zero condition introduces only the last conclusion of \cref{thm:deepiv_rate}; the proof of \cref{lemma:rdiv_accepted_bounds} treats arbitrary $h_0$ using the variational source inequality.
- side condition added by the confirming referee `ass:pin-fix-4d0cd055`: For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair

### `lem:iter-rdiv-squared-loss-concentration-lemma-14` (`appendices.tex:429-449`)

- as printed: Apply the localized squared-loss comparison to $(h-h_{k,\lambda}^*)^2$, whose comparator and center are fixed. Here the localized squared-loss comparison is the scalar application of \citet[Lemma~14]{foster2023orthogonal} at the fixed comparator $h_{k,\lambda}^*$.
- 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-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-155`: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.

### `thm:iter-rdiv-discrepancy-bound` (`appendices.tex:507-516`)

- as printed: \label{cor:iter_rdiv_discrepancy_bound} Under the assumptions of \cref{thm:iter_rdiv_rate}, on $\mathcal E_{n,t}^{\mathrm{IR}}$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$, \[ D_n(\hat h_{t,\lambda}) \leqslant \frac{a_t\delta_n^2}{\lambda}+\nu_n^2 +C_{\mathrm{IR}}\|w_0\|^2\lambda^{q_t}, \qquad q_t=\min\{\beta+1,2t\}. \]
- 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-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-155`: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.
- side condition added by the confirming referee `ass:pin-187`: The displayed discrepancy inequality holds on $\mathcal E_{n,t}^{\mathrm{IR}}$, with $\mathbb P(\mathcal E_{n,t}^{\mathrm{IR}})\geqslant1-\xi$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$. — pins reading of def:iter-rdiv-discrepancy-bound: The sentence introducing the display cites Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, which expressly states simultaneity on this event; Lemma~\ref{lemma:iter_rdiv_stability} specifies its probability.
- side condition added by the confirming referee `ass:pin-fix-4d0cd055`: For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair

### `lem:iter-rdiv-accepted-bounds` (`appendices.tex:528-543`)

- as printed: \label{lemma:iter_rdiv_accepted_bounds} Under the assumptions of \cref{thm:iter_rdiv_rate}, for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$, on $\mathcal E_{n,t}^{\mathrm{IR}}$, \begin{align*} \|\mathcal T(\hat h_{t,\lambda}-h_0)\|^2 &\leqslant C_t \{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm R}(\lambda)\},\\ \|\hat h_{t,\lambda}-h_0\|^2 &\leqslant C_t\left[ V_{n,t}^{\mathrm R}(\lambda) +\{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm R}(\lambda)\}^{\beta/(\beta+1)} \right], \end{align*} where the noise level and stability bound are defined in \eqref{eq:iter_rdiv_common_scales}.
- 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-153`: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- side condition added by the confirming referee `ass:pin-155`: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.
- side condition added by the confirming referee `ass:pin-fix-4d0cd055`: For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair

### `lem:concentrate-l` (`appendices.tex:624-629`)

- as printed: \label{lemma:concentrate_l} Under \cref{asp:closeness,asp:mean_sq_continuity,asp:critical_radius}, on $\mathcal E_n$, the following inequalities hold simultaneously for all $h\in\mathcal H$: \[ \frac{1}{2} L( h) -O\left( \delta _{n}^{2}\right) \leqslant L_{n}( h) \leqslant \frac{5}{4} L( h) +O\left( \delta _{n}^{2}\right) . \]
- 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:pin-206`: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- side condition added by the confirming referee `ass:pin-fix-958aeb7a`: The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair

### `lem:strong-convexity` (`appendices.tex:656-673`)

- as printed: \label{lemma:strong_convexity} Under the assumptions of \cref{thm:tikhonv_rate}, on $\mathcal E_n$, the following bound holds simultaneously for all $\lambda\in\Lambda_n$: \[ L_n(\hat h_\lambda)+\frac{\lambda}{2} \|\hat h_\lambda-h_0\|^2 \leqslant C\{\delta_n^2+\|w_0\|^2\lambda^p\}, \qquad p=\min\{\beta+1,2\}. \] Moreover, for all $\lambda\in\Lambda_n$, \[ \|\hat h_\lambda-h_0\|^2 \leqslant C\left[\frac{\delta_n^2}{\lambda} +\{L_n(\hat h_\lambda)+\delta_n^2\}^{\alpha}\right], \qquad \alpha=\frac{\min\{\beta,1\}}{1+\min\{\beta,1\}}. \]
- corrected: statement unchanged
- no error: the proof as printed could not be checked, so its steps were completed (below)
- side condition added by the confirming referee `ass:pin-156`: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.
- side condition added by the confirming referee `ass:pin-206`: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- side condition added by the confirming referee `ass:pin-fix-958aeb7a`: The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair
- proof change: the printed proof, revised step by step] Write $e_\lambda=\hat h_\lambda-h_0$. The localized squared-norm inequality in \citet[Lemma~14]{foster2023orthogonal}, applied using the squared-loss measurability condition in Assumption~\ref{asp:critical_radius}, gives, uniformly over $h\in\mathcal H$, \[ \left|\{\|h\|_{2,n}^2-\|h_0\|_{2,n}^2\} -\{\|h\|^2-\|h_0\|^2\}\right| \leqslant C\{\delta_n\|h-h_0\|+\delta_n^2\}. \] \footnote{Put $M=\max\{1,B\}$, $\mathcal U=\{h/M:h\in\mathcal H\}$, and $u_0=h_0/M$. Assumption~\ref{asp:closeness_deepiv} gives $u_0\in\mathcal U$, and boundedness gives $|u(X)|\leqslant1$ uniformly over $u\in\mathcal U$ on the common full-probability set. The loss \[ \ell(v)=M^2\bigl(\max\{-1,\min\{v,1\}\}\bigr)^2 \] is $L$-Lipschitz with $L=2M^2$, since clipping is $1$-Lipschitz and $|a^2-b^2|\leqslant2|a-b|$ for $a,b\in[-1,1]$. Moreover, $\ell(h(X)/M)=h(X)^2$.  Set $\mathcal S=\operatorname{star}(\mathcal H-\mathcal H)$. By Assumption~\ref{asp:critical_radius} and \eqref{eq:delta_trae}, there is a deterministic critical radius $d\leqslant\delta_n$ for $\mathcal S$. Since $0\in\mathcal S$, we have $\mathcal S\subseteq\operatorname{star}(\mathcal S-\mathcal S)$, so the critical-radius definition implies $\mathfrak R_n(d;\mathcal S)\leqslant d^2$. For $s\geqslant d$, scaling by $d/s$ preserves membership in $\mathcal S$ and gives \[ \mathfrak R_n(s;\mathcal S) \leqslant\frac{s}{d}\mathfrak R_n(d;\mathcal S) \leqslant sd. \] Also, $h_0\in\mathcal H$ implies $\operatorname{star}(\mathcal H-h_0)\subseteq\mathcal S$. Thus, for every $s\geqslant\delta_n$, \[ \begin{split} \mathfrak R_n\bigl(s;\operatorname{star}(\mathcal U-u_0)\bigr) &=M^{-1}\mathfrak R_n\bigl(Ms;\operatorname{star}(\mathcal H-h_0)\bigr)\\ &\leqslant M^{-1}\mathfrak R_n(Ms;\mathcal S) \leqslant sd\leqslant s\delta_n. \end{split} \] In particular, every $r\geqslant\delta_n$ satisfies the anchored fixed-point inequality for the normalized class.  For $R>0$, define \[ Z_n(R)=\sup_{\substack{u\in\mathcal U\\\|u-u_0\|\leqslant R}} \left|(\mathbb E_n-\mathbb E)[\ell(u(X))-\ell(u_0(X))]\right|. \] Its empirical and Rademacher suprema are precisely the squared-loss suprema in Assumption~\ref{asp:critical_radius}, with localization radius $MR$, and hence are measurable. The localized concentration inequality used in the proof of \citet[Lemma~14]{foster2023orthogonal}, applied to this measurable loss process and the anchored fixed point, gives universal constants $c_1,c_2>0$ such that, for $R\geqslant r\geqslant\delta_n$ and $v>0$, \[ \mathbb P\{Z_n(R)\geqslant16LRr+v\} \leqslant c_1\exp\left\{-\frac{c_2nv^2}{L^2R^2+2Lv}\right\}. \] Taking $v=LRr$ and using $r\leqslant R$ gives \[ \mathbb P\{Z_n(R)\geqslant17LRr\} \leqslant c_1\exp\{-c_2nr^2/3\}. \] Choose $r=a\delta_n$ with a fixed sufficiently large $a\geqslant1$. If $r\geqslant2$, set $J=0$; otherwise set $J=\lceil\log(2/r)/\log2\rceil$. Put $R_j=2^jr$ for $j=0,\ldots,J$. A union bound shows that the measurable event \[ Z_n(R_j)<17LR_jr,\qquad j=0,\ldots,J, \] has failure probability at most $(J+1)c_1\exp\{-c_2nr^2/3\}$. Since $r\geqslant\delta_n\geqslant n^{-1/2}$, there is a universal constant $C_0$ such that $J+1\leqslant C_0\log(en)$. Equation~\eqref{eq:delta_trae} gives $n\delta_n^2\geqslant1$ and a fixed positive lower bound for \[ \frac{n\delta_n^2}{\log\log(en)+\log(1/\eta_n)}. \] Consequently, a fixed sufficiently large $a$ makes this failure probability at most $\eta_n/4$.  On this event, let $u\in\mathcal U$ and write $b=\|u-u_0\|$. If $b\leqslant r$, the loss increment is bounded by $Z_n(r)<17Lr^2$. If $b>r$, boundedness gives $b\leqslant2$, so there is an index $j\in\{1,\ldots,J\}$ with $R_{j-1}<b\leqslant R_j$. Then $R_j<2b$, and the loss increment is bounded by $Z_n(R_j)<34Lrb$. Thus, in both cases, \[ \left|(\mathbb E_n-\mathbb E)[\ell(u(X))-\ell(u_0(X))]\right| \leqslant34Lr(b+r). \] Substitution of $u=h/M$, $L=2M^2$, and $r=a\delta_n$ bounds the squared-norm difference by \[ 68M^2a\delta_n\{\|h-h_0\|/M+a\delta_n\} \leqslant C\{\delta_n\|h-h_0\|+\delta_n^2\}, \] uniformly over $h\in\mathcal H$. The constant depends only on the fixed boundedness and calibration constants. Allocating $\eta_n/4$ to this comparison at each grid point uses at most $(K_n+1)\eta_n/4=\xi/4$ of the total failure probability, supplying the squared-norm comparison included in $\mathcal E_n$.} The event $\mathcal E_n$ includes this bound and $L_n(h_0)\leqslant C\delta_n^2$. Empirical optimality relative to $h_0\in\mathcal H$ therefore gives \begin{equation}\label{eq:trae_empirical_optimality} L_n(\hat h_\lambda) +\lambda\{\|e_\lambda\|^2+2\langle h_0,e_\lambda\rangle\} \leqslant C\{\delta_n^2+\lambda\delta_n\|e_\lambda\|\}. \end{equation}  By \cref{lemma:concentrate_l}, $L(\hat h_\lambda)\leqslant2L_n(\hat h_\lambda)+C\delta_n^2$. Apply the first consequence of \cref{lemma:variational_source} to the inner-product term in \eqref{eq:trae_empirical_optimality}. Young's inequality also gives $C\lambda\delta_n\|e_\lambda\| \leqslant\lambda\|e_\lambda\|^2/8+C\delta_n^2$. Absorbing the resulting empirical-loss and squared-norm terms proves the first bound.  For the second bound, the lower comparison in \cref{lemma:concentrate_l} gives \[ L(\hat h_\lambda)\leqslant2L_n(\hat h_\lambda)+C\delta_n^2 \leqslant C \{L_n(\hat h_\lambda)+\delta_n^2\}. \] By \cref{asp:closeness_deepiv,asp:closeness}, taking $h=h_0$ gives $0=\mathcal T(h_0-h_0)\in\mathcal F$. Hence $L_n(\hat h_\lambda)\geqslant0$. Drop this term from \eqref{eq:trae_empirical_optimality}, divide by $\lambda$, and apply Young's inequality to obtain \[ \|e_\lambda\|^2 \leqslant C\frac{\delta_n^2}{\lambda} +C|\langle h_0,e_\lambda\rangle|. \] The second consequence of \cref{lemma:variational_source} and the weak bound above complete the proof.

### `thm:trae-discrepancy-bound` (`appendices.tex:722-724`)

- as printed: \label{cor:trae_discrepancy_bound} Under the assumptions of \cref{thm:tikhonv_rate}, the calibrated TRAE scale satisfies \eqref{eq:trae_discrepancy_bound} simultaneously for every $\lambda\in\Lambda_n$ on $\mathcal E_n$, for a fixed constant $C_{\mathrm T}>0$.
- 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-156`: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.
- side condition added by the confirming referee `ass:pin-160`: For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. — pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.
- side condition added by the confirming referee `ass:pin-206`: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- side condition added by the confirming referee `ass:pin-fix-958aeb7a`: The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair

### `lem:trae-accepted-bounds` (`appendices.tex:739-751`)

- as printed: \label{lemma:trae_accepted_bounds} Suppose that the assumptions of \cref{thm:tikhonv_rate} hold. For every $\lambda\in\Lambda_n$, on $\mathcal E_n$, \begin{align*} \|\mathcal T(\hat h_\lambda-h_0)\|^2 &\leqslant C \{L_n(\hat h_\lambda)+\delta_n^2\},\\ \|\hat h_\lambda-h_0\|^2 &\leqslant C\left[V_n^{\mathrm T}(\lambda) +\{L_n(\hat h_\lambda)+\delta_n^2\}^\alpha\right], \end{align*} where the noise level and stability bound are defined in \eqref{eq:trae_common_scales} and $\alpha=\min\{\beta,1\}/\{1+\min\{\beta,1\}\}$.
- 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-156`: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.
- side condition added by the confirming referee `ass:pin-206`: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- side condition added by the confirming referee `ass:pin-fix-958aeb7a`: The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair

### `lem:iter-trae-strong-propagation` (`appendices.tex:807-822`)

- as printed: Under the assumptions of \cref{thm:iter_trae_rate}, there is a nondecreasing deterministic sequence with $b_0=0$ and $b_j\geqslant2$, $j=1,\ldots,t$, where each $b_j$ depends only on $j$ and the prescribed boundedness and mean-square-continuity constants, independently of $\beta$ and $\|w_0\|$. This sequence may be calibrated so that, on $\mathcal E_{n,t}^{\mathrm{IT}}$, simultaneously for $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$, \begin{align} \|\hat h_{j,\lambda}-h_{j,\lambda}^*\|^2 &\leqslant C\frac{b_j\delta_n^2}{\lambda^2}, \qquad j=1,\ldots,t, \label{eq:iter_trae_strong_propagation}\\ L(\hat h_{t,\lambda}) &\leqslant C\left[ \delta_n^2+\min\left\{\lambda, \frac{b_t\delta_n^2}{\lambda}\right\} +\|w_0\|^2\lambda^{q_t} \right]. \label{eq:iter_trae_weak_propagation} \end{align}
- corrected: statement unchanged
- no longer uses `lem:iter-trae-excess-loss-lemma-11`
- no longer uses `lem:iter-trae-excess-loss-lemma-10`
- side condition added by the confirming referee `ass:bk-12`: With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\|u-u_{j,\lambda}^*\|_{L_2(W;\mathbb R^3)}\leqslant r}|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}|$ and $\sup_{u\in\mathcal U,\,\|u-u_{j,\lambda}^*\|_{L_2(W;\mathbb R^3)}\leqslant r}|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$. — The localized excess-loss concentration uses this nonzero deterministic anchor; the stipulated zero-anchor and fixed-center squared-loss measurability do not supply these suprema.
- side condition added by the confirming referee `ass:pin-156`: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.
- side condition added by the confirming referee `ass:pin-206`: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- side condition added by the confirming referee `ass:pin-fix-958aeb7a`: The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair
- proof change: the printed proof, revised step by step] We adapt the iterative argument of \citet[Theorem~6]{bennett2023source} to the finite grid and give the recursive comparison explicitly. Fix $j$ and $\lambda$, and set \[ e_j=\|\hat h_{j,\lambda}-h_{j,\lambda}^*\|, \qquad u_j=\|\mathcal T(\hat h_{j,\lambda}-h_{j,\lambda}^*)\|, \qquad f_j^*=\mathcal T(h_0-h_{j,\lambda}^*). \] Also define \[ M_{\leqslant j} =\max_{1\leqslant\ell\leqslant j} \left\{\lambda,\|f_\ell^*\|_\infty^2\right\}. \] By \cref{asp:iter_realizability_trae,asp:closeness}, $f_\ell^*\in\mathcal F$. The boundedness part of \cref{asp:critical_radius} therefore gives $M_{\leqslant j}\leqslant\max\{2,B^2\}$ for every grid point under consideration.  To track the random penalty center, write \begin{align*} I_{n,j} &=\|h_{j,\lambda}^*-\hat h_{j-1,\lambda}\|_{2,n}^2 -\|\hat h_{j,\lambda}-\hat h_{j-1,\lambda}\|_{2,n}^2,\\ I_j &=\|h_{j,\lambda}^*-h_{j-1,\lambda}^*\|^2 -\|\hat h_{j,\lambda}-h_{j-1,\lambda}^*\|^2. \end{align*} We derive the excess-loss comparison directly. Write $g=h_{j,\lambda}^*$, $k=f_j^*$, $\bar f=\mathcal T(h_0-\hat h_{j,\lambda})$, and \[ \ell(h,f)=2m(W;f)-2h(X)f(Z)-f(Z)^2. \] By \cref{asp:closeness,asp:iter_realizability_trae}, $k,\bar f\in\mathcal F$. For every $h\in\mathcal H$ and $f\in\mathcal F$, the Riesz identity in Section~\ref{sec:pre} and the definition of the projected operator give \[ \mathbb E[\ell(h,f)] =2\langle\mathcal T(h_0-h),f\rangle-\|f\|^2 =\|\mathcal T(h_0-h)\|^2-\|f-\mathcal T(h_0-h)\|^2. \] Indeed, $f\in\widetilde{\mathcal Q}$ implies $\mathbb E[h(X)f(Z)]=\langle\mathcal T h,f\rangle$, and $\mathbb E[m(W;f)]=\langle r_0,f\rangle=\langle\mathcal T h_0,f\rangle$.  Apply the localized concentration inequality of \citet[Lemma~9]{bennett2023source} to the product and moment-image classes in \cref{asp:critical_radius}, and apply its bounded Lipschitz version to the squared test functions. With the deterministic anchors $g$ and $k$, these inequalities give, uniformly over $h\in\mathcal H$ and $f\in\mathcal F$, \[ \begin{split} &\left|(\mathbb E_n-\mathbb E)[\ell(h,f)-\ell(g,k)]\right|\\ &\quad\leqslant C\delta_n\left\{ \bigl(\mathbb E[m(W;f-k)^2]\bigr)^{1/2} +\|h(X)f(Z)-g(X)k(Z)\|_{L_2(W)} +\|f-k\|\right\}+C\delta_n^2\\ &\quad\leqslant C\left\{\delta_n\|h-g\|\|k\|_\infty +\delta_n\|f-k\|+\delta_n^2\right\}. \end{split} \] For the second inequality, \cref{asp:iter_mean_sq_continuity} bounds the first norm by $C\|f-k\|$, while \[ h(X)f(Z)-g(X)k(Z) =h(X)(f-k)(Z)+(h-g)(X)k(Z) \] implies that the product norm is at most $B\|f-k\|+\|h-g\|\|k\|_\infty$. The square map is $2B$-Lipschitz on the range of the test functions. Also, $h_0\in\mathcal H$ and \cref{asp:closeness} imply $0\in\mathcal F$. Applying the same concentration inequality with zero moment, product, and test-function anchors, and using \cref{asp:mean_sq_continuity}, gives \[ \left|(\mathbb E_n-\mathbb E)\ell(h,f)\right| \leqslant C\{\delta_n\|f\|+\delta_n^2\}, \qquad h\in\mathcal H,\ f\in\mathcal F. \] Allocate fixed fractions of $\eta_{n,t}$ to these finitely many concentration inequalities and to the comparison of Lemma~\ref{lemma:concentrate_l} used below, absorbing the resulting fixed constants into the calibrated scale.  Taking $h=g$ in the increment bound and using the population identity yields \[ \begin{split} \mathbb E_n[\ell(g,f)-\ell(g,k)] &\leqslant-\|f-k\|^2+C\delta_n\|f-k\|+C\delta_n^2\\ &\leqslant C\delta_n^2. \end{split} \] Taking the supremum over $f\in\mathcal F$ therefore proves \[ 0\leqslant L_n(g)-\mathbb E_n\ell(g,k)\leqslant C\delta_n^2, \] without requiring an empirical maximizing test function. Next, $\|\bar f-k\|=u_j$, so the increment bound at $(\hat h_{j,\lambda},\bar f)$ gives \[ \begin{split} L(\hat h_{j,\lambda})-L(g) &\leqslant\mathbb E_n[\ell(\hat h_{j,\lambda},\bar f)-\ell(g,k)] +C\{\delta_ne_j\|k\|_\infty+\delta_nu_j+\delta_n^2\}\\ &\leqslant L_n(\hat h_{j,\lambda})-L_n(g) +C\{\delta_ne_j\|k\|_\infty+\delta_nu_j+\delta_n^2\}. \end{split} \] Empirical optimality relative to $g\in\mathcal H$ gives $L_n(\hat h_{j,\lambda})-L_n(g)\leqslant\lambda I_{n,j}$, including when the penalty center is random. Consequently, \begin{equation*} L(\hat h_{j,\lambda})-L(h_{j,\lambda}^*) \leqslant \lambda I_{n,j} +C\left\{ \delta_ne_j\|f_j^*\|_\infty +\delta_nu_j+\delta_n^2 \right\}. \end{equation*} This comparison uses the localized bounds for $\operatorname{star}(\mathcal H\cdotp\mathcal F)$, $\operatorname{star}(m\circ\mathcal F)$, and $\operatorname{star}(\mathcal F)$ from \cref{asp:critical_radius}, together with \cref{asp:iter_mean_sq_continuity}. In particular, that bound controls $\mathbb E[m(W;f_{\hat h_{j,\lambda}}-f_j^*)^2]$ by $Cu_j^2$, where $f_h=\mathcal T(h_0-h)$. The population normal equation for $h_{j,\lambda}^*$ gives the exact quadratic lower bound \begin{equation}\label{eq:iter_trae_population_curvature} \lambda e_j^2+u_j^2 \leqslant L(\hat h_{j,\lambda})-L(h_{j,\lambda}^*)-\lambda I_j. \end{equation} The quadratic identity for \eqref{eq:iter_population}, together with $L(h)=\|\mathcal T(h-h_0)\|^2$ on $\mathcal H$, proves this bound.  Expanding the squared penalties separates the center error as $2\langle\hat h_{j,\lambda}-h_{j,\lambda}^*, \hat h_{j-1,\lambda}-h_{j-1,\lambda}^*\rangle_n$. For the fixed-center squared-loss comparisons, apply the same localized concentration inequality of \citet[Lemma~9]{bennett2023source} to the bounded clipped losses corresponding to $(h-h_{j-1,\lambda}^*)^2$ at comparator $h_{j,\lambda}^*$ and $(h-h_{k,\lambda}^*)^2$ at comparator $h_{k,\lambda}^*$, for the fixed iteration--grid pairs under consideration. The fixed-population-center squared-loss measurability condition in Assumption~\ref{asp:critical_radius} supplies measurable localized empirical and Rademacher suprema for these applications.  Here is the scalar specialization and the penalty expansion. Put $M=\max\{1,B\}$ and $\mathcal U=\mathcal H/M$. For each deterministic center $c=h_{j-1,\lambda}^*$ or $c=h_{k,\lambda}^*$, use the loss \[ \ell_c(v;X)=\{M\max\{-1,\min\{v,1\}\}-c(X)\}^2. \] On the common full-probability set, $|c(X)|\leqslant M$, including $c=h_{0,\lambda}^*=0$, so this loss is $4M^2$-Lipschitz in $v$ and agrees with $(h(X)-c(X))^2$ at $v=h(X)/M$. For a comparator $g\in\mathcal H$, the inclusion $\operatorname{star}(\mathcal H-g)\subseteq\mathcal S:=\operatorname{star}(\mathcal H-\mathcal H)$ supplies the anchored complexity. Indeed, if $d\leqslant\delta_n$ is a critical radius for $\mathcal S$, then $\mathfrak R_n(d;\mathcal S)\leqslant d^2$ because $0\in\mathcal S$ and $\mathcal S\subseteq\operatorname{star}(\mathcal S-\mathcal S)$. Star scaling gives $\mathfrak R_n(s;\mathcal S)\leqslant sd$ for $s\geqslant d$. Hence, for $s\geqslant\delta_n$, \[ \mathfrak R_n(s;\operatorname{star}(\mathcal U-g/M)) \leqslant M^{-1}\mathfrak R_n(Ms;\mathcal S) \leqslant s\delta_n. \] The measurable squared-loss processes in Assumption~\ref{asp:critical_radius} are exactly the loss-increment processes for the stated centers and comparators, after this normalization. Lemma~9 therefore gives, on the allocated concentration event, \[ \left|(\mathbb E_n-\mathbb E) \bigl[\{h(X)-h_{j-1,\lambda}^*(X)\}^2 -\{h_{j,\lambda}^*(X)-h_{j-1,\lambda}^*(X)\}^2\bigr]\right| \leqslant C\{\delta_n\|h-h_{j,\lambda}^*\|+\delta_n^2\}, \] and, for $k=1,\ldots,t$, \[ \left|\|h-h_{k,\lambda}^*\|_{2,n}^2-\|h-h_{k,\lambda}^*\|^2\right| \leqslant C\{\delta_n\|h-h_{k,\lambda}^*\|+\delta_n^2\}. \] Allocate fixed fractions of $\eta_{n,t}$ to these scalar comparisons. Their constants depend only on the fixed class bounds.  Write $\Delta_j=\hat h_{j,\lambda}-h_{j,\lambda}^*$ and $\Delta_0=0$. Expansion gives \[ \begin{split} I_{n,j}-I_j ={}&(\mathbb E_n-\mathbb E) \bigl[\{h_{j,\lambda}^*-h_{j-1,\lambda}^*\}^2 -\{\hat h_{j,\lambda}-h_{j-1,\lambda}^*\}^2\bigr] +2\mathbb E_n[\Delta_j(X)\Delta_{j-1}(X)]. \end{split} \] The first term is at most $C(\delta_ne_j+\delta_n^2)$. The second scalar comparison and $2\delta_ne_k\leqslant e_k^2+\delta_n^2$ give $\|\Delta_k\|_{2,n}\leqslant C(e_k+\delta_n)$ for $k\geqslant1$; this bound also holds for $k=0$ because $\Delta_0=0$. Empirical Cauchy--Schwarz thus bounds the last term by \[ 2\|\Delta_j\|_{2,n}\|\Delta_{j-1}\|_{2,n} \leqslant 2Ce_je_{j-1}+C\delta_n(e_j+e_{j-1})+C\delta_n^2. \] Combining these bounds yields \begin{equation}\label{eq:iter_trae_center_comparison} I_{n,j}-I_j \leqslant C\delta_n(e_j+e_{j-1})+C\delta_n^2+2Ce_je_{j-1}. \end{equation} Combining \eqref{eq:iter_trae_biasless}--\eqref{eq:iter_trae_center_comparison} and applying Young's inequality gives \[ e_j^2\leqslant C\frac{\delta_n^2M_{\leqslant j}}{\lambda^2}+Ce_{j-1}^2. \] All terms proportional to $\delta_n^2$, including those multiplied by $\lambda\leqslant2$, are covered by $\delta_n^2M_{\leqslant j}/\lambda$ before division by $\lambda$. Since $e_0=0$ and $M_{\leqslant j}$ is uniformly bounded, enlarge the fixed constant $C\geqslant2$ if necessary, set $b_0=0$, and recursively choose $b_j=C(1+b_{j-1})$. Induction gives \eqref{eq:iter_trae_strong_propagation}. Thus each $b_j$ depends only on $j$ and the fixed constants.  For the weak bound, put $v_j=\|\mathcal T(h_{j,\lambda}^*-h_0)\|=\|k\|$. The absolute concentration inequality established in this proof, evaluated at the feasible residual $\bar f$, gives \[ L(\hat h_{j,\lambda}) \leqslant L_n(\hat h_{j,\lambda}) +C\{\delta_n\|\bar f\|+\delta_n^2\}. \] The assumptions of Lemma~\ref{lemma:concentrate_l} are inherited from the assumptions of \cref{thm:iter_trae_rate}. Its upper comparison, applied to $g=h_{j,\lambda}^*\in\mathcal H$ with the allocated confidence level, gives \[ L_n(g)\leqslant\frac54L(g)+C\delta_n^2 =\frac54v_j^2+C\delta_n^2. \] Empirical optimality, $L(g)=v_j^2$, and $\|\bar f\|\leqslant v_j+u_j$ therefore imply \[ \begin{split} L(\hat h_{j,\lambda})-L(g) &\leqslant\lambda I_{n,j}+\frac14v_j^2 +C\delta_n(v_j+u_j)+C\delta_n^2\\ &\leqslant\lambda I_{n,j}+C\{v_j^2+\delta_nu_j+\delta_n^2\}, \end{split} \] where the last inequality uses $\delta_nv_j\leqslant(v_j^2+\delta_n^2)/2$. Combining this with \eqref{eq:iter_trae_population_curvature} gives \[ \lambda e_j^2+u_j^2 \leqslant C\{v_j^2+\delta_nu_j+\delta_n^2\} +\lambda(I_{n,j}-I_j). \] Substituting \eqref{eq:iter_trae_center_comparison} and absorbing the linear terms gives \[ \frac{\lambda}{2}e_j^2+\frac12u_j^2 \leqslant C\left(v_j^2+\delta_n^2+\lambda e_{j-1}^2\right). \] At the final iteration, boundedness of $\mathcal H$ and \eqref{eq:iter_trae_strong_propagation} give the two simultaneous bounds \[ \lambda e_{t-1}^2\leqslant C\lambda, \qquad \lambda e_{t-1}^2 \leqslant C\frac{b_t\delta_n^2}{\lambda}. \] Taking their minimum, applying the squared triangle inequality, and using the weak-bias bound \eqref{eq:iter_weak_bias} prove \eqref{eq:iter_trae_weak_propagation}. The argument used only $\lambda\in(0,2]$, with the upper endpoint absorbed into fixed constants. Each comparison is applied to the fixed iteration--grid pairs in the oracle-relevant segment. The definition of $\eta_{n,t}$ and a union bound give the simultaneous event.
- why: the step its proof takes from [bennett2023source, Lemma 11] (lem:iter-trae-excess-loss-lemma-11) does not hold as the proof uses it: (a) The citation's hypotheses are not established: Lemma 11 assumes the conditions of the source's iterative theorem, which inherits condition (c), $\mathbb E[h_0(X)-h(X)\mid Z=\cdot]\in\mathcal F$. The host assumes only $\mathcal T(h_0-h)\in\mathcal F$, allowing a proper projected test space. (b) The source's standing assumption ``Realizability and Closedness'' requires convexity of $\mathcal H$, whereas the host allows general hypothesis classes. These are defects of the citation application; the random penalty center itself is permitted by the source. Also: the step its proof takes from [bennett2023source, Lemma 10] (lem:iter-trae-excess-loss-lemma-10) does not hold as the proof uses it: (a) Lemma 10 assumes the conditions of the source's one-step theorem, including condition (c), $\mathbb E[h_0(X)-h(X)\mid Z=\cdot]\in\mathcal F$. The host's projected closedness assumption does not establish this condition. (b) The source's standing ``Realizability and Closedness'' assumption requires convexity of $\mathcal H$, which the host does not impose. Substituting the random centered penalty is supported by the source, but it does not resolve these missing hypotheses.

### `thm:iter-trae-discrepancy-bound` (`appendices.tex:924-926`)

- as printed: \label{cor:iter_trae_discrepancy_bound} Under the assumptions of \cref{thm:iter_trae_rate}, the calibrated iterative TRAE scales satisfy \eqref{eq:iter_trae_discrepancy_bound} simultaneously for all $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ on $\mathcal E_{n,t}^{\mathrm{IT}}$.
- 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-12`: With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\|u-u_{j,\lambda}^*\|_{L_2(W;\mathbb R^3)}\leqslant r}|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}|$ and $\sup_{u\in\mathcal U,\,\|u-u_{j,\lambda}^*\|_{L_2(W;\mathbb R^3)}\leqslant r}|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$. — The localized excess-loss concentration uses this nonzero deterministic anchor; the stipulated zero-anchor and fixed-center squared-loss measurability do not supply these suprema.
- side condition added by the confirming referee `ass:pin-156`: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.
- side condition added by the confirming referee `ass:pin-188`: Under the assumptions of Theorem~\ref{thm:iter_trae_rate}, the displayed inequality holds simultaneously for all $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ on $\mathcal E_{n,t}^{\mathrm{IT}}$, where $\mathbb P(\mathcal E_{n,t}^{\mathrm{IT}})\geqslant1-\xi$. — pins reading of def:iter-trae-discrepancy-bound: The sentence immediately preceding the display invokes Corollary~\ref{cor:iter_trae_discrepancy_bound}, whose statement explicitly places this simultaneous bound on the common concentration event.
- side condition added by the confirming referee `ass:pin-206`: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- side condition added by the confirming referee `ass:pin-fix-958aeb7a`: The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair

### `lem:iter-trae-accepted-stability` (`appendices.tex:941-966`)

- as printed: \label{lemma:iter_trae_accepted_stability} Let $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ satisfy $\lambda\geqslant\sqrt{s_{n,t}^{\mathrm T}}$. On $\mathcal E_{n,t}^{\mathrm{IT}}$, \begin{align} &\|\mathcal T(\hat h_{t,\lambda}-h_0)\|^2 \leqslant C_t \{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm T}(\lambda)\}, \label{eq:iter_trae_accepted_total}\\ &\|\hat h_{t,\lambda}-h_0\|^2 \leqslant C_t\left[ \frac{1}{\lambda}\{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm T}(\lambda)\} +\{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm T}(\lambda)\}^{\beta/(\beta+1)} \right], \label{eq:iter_trae_accepted_total_strong}\\ &\lambda\|\hat h_{t,\lambda}-h_{t,\lambda}^*\|^2 +\|\mathcal T(\hat h_{t,\lambda}-h_{t,\lambda}^*)\|^2 \leqslant C_t \{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm T}(\lambda)\}, \label{eq:iter_trae_accepted_variance}\\ &\|\mathcal T(h_{t,\lambda}^*-h_0)\|^2 \leqslant C_t \{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm T}(\lambda)\}, \label{eq:iter_trae_accepted_bias}\\ &\|\hat h_{t,\lambda}-h_{t,\lambda}^*\|^2 \leqslant \frac{C_t}{\lambda} \{D_n(\hat h_{t,\lambda})+A_{n,t}^{\mathrm T}(\lambda)\}. \label{eq:iter_trae_accepted_strong} \end{align}
- 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-12`: With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\|u-u_{j,\lambda}^*\|_{L_2(W;\mathbb R^3)}\leqslant r}|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}|$ and $\sup_{u\in\mathcal U,\,\|u-u_{j,\lambda}^*\|_{L_2(W;\mathbb R^3)}\leqslant r}|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$. — The localized excess-loss concentration uses this nonzero deterministic anchor; the stipulated zero-anchor and fixed-center squared-loss measurability do not supply these suprema.
- side condition added by the confirming referee `ass:pin-156`: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.
- side condition added by the confirming referee `ass:pin-203`: The constant $C_t$ may depend on the prescribed $t$ and fixed population quantities, including $\beta$ and $\|w_0\|$, but is independent of $n$, $\lambda$, and the realized fitting observations. — pins reading of lem:iter-trae-accepted-stability: The proof applies Lemma~\ref{lemma:iter_bias}, whose interpolation inequality contributes $\|w_0\|^{2/(\beta+1)}$. The iterative-rate convention also explicitly permits constants depending on the prescribed iteration count and fixed population quantities.
- side condition added by the confirming referee `ass:pin-206`: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\|f\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- side condition added by the confirming referee `ass:pin-fix-958aeb7a`: The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair

## Suggestions to the authors

Not errors: readings the paper determines but does not state, and side conditions the proofs use silently.

- clarification (`ass:pin-142`, used by `thm:common-dp`): On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. — pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.
- clarification (`ass:pin-153`, used by `prop:exist-lambda-deepiv`): For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. — pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.
- clarification (`ass:pin-155`, used by `prop:exist-lambda-deepiv`): For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.
- clarification (`ass:pin-172`, used by `prop:exist-lambda-deepiv`): For each fixed $n\geqslant N_{\mathrm R}(B)$, all probabilistic conclusions, including initial acceptance when $h_0=0$, hold on the same event $\mathcal E_n^{\mathrm R}$ of probability at least $1-\xi$. — pins reading of prop:exist-lambda-deepiv: The proof of Proposition~\ref{prop:exist_lambda_deepiv} explicitly works throughout on $\mathcal E_n^{\mathrm R}$ and derives initial acceptance there.
- clarification (`ass:pin-174`, used by `prop:exist-lambda-deepiv`): The assumptions imported from Theorem~\ref{thm:deepiv_rate} are the general hypotheses of Proposition~\ref{prop:exist_lambda_deepiv}, without imposing $h_0=0$. — pins reading of thm:rdiv-discrepancy-bound: Theorem~\ref{thm:deepiv_rate} imports the proposition's assumptions and treats $h_0=0$ only in a separate conditional conclusion. The present proof uses Lemmas~\ref{lemma:err_l_dp} and~\ref{lemma:str_dp} for general $h_0$.
- clarification (`ass:pin-fix-4d0cd055`, used by `prop:exist-lambda-deepiv`): For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. — pins reading of def:rdiv-preliminary: stated by the repair
- clarification (`ass:pin-190`, used by `thm:deepiv-rate`): For each fixed $n\geqslant N_{\mathrm R}(B)$, the displayed error bounds and, when $h_0=0$, both sharper $O(\delta_n^2)$ bounds hold jointly with probability at least $1-\xi$ over the preliminary, fitting, and calibration samples. — pins reading of thm:deepiv-rate: The proof obtains the zero-signal bounds from the probabilistic initial-acceptance conclusion of Proposition~\ref{prop:exist_lambda_deepiv} and Lemma~\ref{lemma:str_dp}; the general case uses the common RDIV event and the stated confidence convention.
- clarification (`ass:pin-196`, used by `thm:deepiv-rate`): The assumptions of \cref{thm:deepiv_rate} mean its standing hypotheses inherited from \cref{prop:exist_lambda_deepiv}; the bounds here hold for arbitrary $h_0$, without imposing $h_0=0$. — pins reading of lem:rdiv-accepted-bounds: The zero condition introduces only the last conclusion of \cref{thm:deepiv_rate}; the proof of \cref{lemma:rdiv_accepted_bounds} treats arbitrary $h_0$ using the variational source inequality.
- clarification (`ass:pin-187`, used by `prop:exist-lambda-iter-rdiv`): The displayed discrepancy inequality holds on $\mathcal E_{n,t}^{\mathrm{IR}}$, with $\mathbb P(\mathcal E_{n,t}^{\mathrm{IR}})\geqslant1-\xi$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$. — pins reading of def:iter-rdiv-discrepancy-bound: The sentence introducing the display cites Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, which expressly states simultaneity on this event; Lemma~\ref{lemma:iter_rdiv_stability} specifies its probability.
- clarification (`ass:pin-205`, used by `prop:exist-lambda-iter-rdiv`): For each fixed $n\geqslant N_{\mathrm R}(B)$ and prescribed $t$, there is an event of probability at least $1-\xi$ on which the rule accepts a candidate and satisfies the stated output alternatives, including acceptance of $\lambda_0=2$ when $h_0=0$. — pins reading of prop:exist-lambda-iter-rdiv: The proof of Proposition~\ref{prop:exist_lambda_iter_rdiv} explicitly places initial acceptance on the corresponding concentration event.
- clarification (`ass:pin-156`, used by `prop:exist-lambda`): The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. — pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.
- clarification (`ass:pin-160`, used by `prop:exist-lambda`): For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. — pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.
- clarification (`ass:pin-193`, used by `prop:exist-lambda`): For each fixed $n\geqslant N_{\mathrm T}(B)$, acceptance within the stated number of optimization problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_n$, with probability at least $1-\xi$ over the fitting sample. — pins reading of prop:exist-lambda: The proof begins by restricting throughout to $\mathcal E_n$, establishes terminal acceptance there, and derives both return branches on that same event.
- clarification (`ass:pin-206`, used by `prop:exist-lambda`): There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. — pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.
- clarification (`ass:pin-fix-958aeb7a`, used by `prop:exist-lambda`): The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. — pins reading of ass:critical-radius-complexity: stated by the repair
- clarification (`ass:pin-194`, used by `thm:tikhonv-rate`): With $\beta$ and $\xi$ fixed as $n\to\infty$, for each $n\geqslant N_{\mathrm T}(B)$ the displayed strong- and weak-error bounds hold jointly on a fitting-sample event of probability at least $1-\xi$. — pins reading of thm:tikhonv-rate: The proof of Theorem~\ref{thm:tikhonv_rate} works on $\mathcal E_n$, which the preceding confidence allocation constructs separately for each sample size.
- clarification (`ass:pin-195`, used by `prop:exist-lambda-iter-trae`): For each fixed $n\geqslant N_{\mathrm{IT}}(B)$ and prescribed $t$, acceptance within the stated number of minimax problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_{n,t}^{\mathrm{IT}}$, with probability at least $1-\xi$ over the fitting sample. — pins reading of prop:exist-lambda-iter-trae: The proof works throughout on $\mathcal E_{n,t}^{\mathrm{IT}}$ and derives terminal acceptance and the first-accepted-index alternative there.
- clarification (`ass:pin-188`, used by `thm:iter-trae-rate`): Under the assumptions of Theorem~\ref{thm:iter_trae_rate}, the displayed inequality holds simultaneously for all $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ on $\mathcal E_{n,t}^{\mathrm{IT}}$, where $\mathbb P(\mathcal E_{n,t}^{\mathrm{IT}})\geqslant1-\xi$. — pins reading of def:iter-trae-discrepancy-bound: The sentence immediately preceding the display invokes Corollary~\ref{cor:iter_trae_discrepancy_bound}, whose statement explicitly places this simultaneous bound on the common concentration event.
- clarification (`ass:pin-203`, used by `thm:iter-trae-rate`): The constant $C_t$ may depend on the prescribed $t$ and fixed population quantities, including $\beta$ and $\\|w_0\\|$, but is independent of $n$, $\lambda$, and the realized fitting observations. — pins reading of lem:iter-trae-accepted-stability: The proof applies Lemma~\ref{lemma:iter_bias}, whose interpolation inequality contributes $\\|w_0\\|^{2/(\beta+1)}$. The iterative-rate convention also explicitly permits constants depending on the prescribed iteration count and fixed population quantities.
- clarification (`ass:pin-204`, used by `thm:iter-trae-rate`): The constants hidden by $O_t(\cdot)$ may depend on the prescribed $t$, fixed population quantities including $\beta$ and $\\|w_0\\|$, and the fixed grid ratio $\rho$, but are independent of $n$ and the candidate $\lambda$. — pins reading of thm:iter-trae-rate: The text following the bounds in Theorem~\ref{thm:iter_rdiv_rate} explicitly allows dependence on the prescribed iteration count and fixed population quantities; Lemma~\ref{lemma:iter_trae_accepted_stability} supplies the comparison constants used here under the same convention.
- `ass:cr-deepiv-measurability`: pinned reading: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.)
- `ass:mean-sq-continuity`: pinned reading: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.)
- `ass:rdiv-minima`: pinned reading: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.)
- `ass:trae-minima`: pinned reading: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.)
- `def:iter-rdiv-discrepancy-bound`: pinned reading: The displayed discrepancy inequality holds on $\mathcal E_{n,t}^{\mathrm{IR}}$, with $\mathbb P(\mathcal E_{n,t}^{\mathrm{IR}})\geqslant1-\xi$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$. (The sentence introducing the display cites Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, which expressly states simultaneity on this event; Lemma~\ref{lemma:iter_rdiv_stability} specifies its probability.)
- `def:iter-trae-discrepancy-bound`: pinned reading: Under the assumptions of Theorem~\ref{thm:iter_trae_rate}, the displayed inequality holds simultaneously for all $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ on $\mathcal E_{n,t}^{\mathrm{IT}}$, where $\mathbb P(\mathcal E_{n,t}^{\mathrm{IT}})\geqslant1-\xi$. (The sentence immediately preceding the display invokes Corollary~\ref{cor:iter_trae_discrepancy_bound}, whose statement explicitly places this simultaneous bound on the common concentration event.)
- `def:trae-discrepancy-bound`: pinned reading: For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. (The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.)
- `lem:iter-trae-accepted-stability`: pinned reading: The constant $C_t$ may depend on the prescribed $t$ and fixed population quantities, including $\beta$ and $\\|w_0\\|$, but is independent of $n$, $\lambda$, and the realized fitting observations. (The proof applies Lemma~\ref{lemma:iter_bias}, whose interpolation inequality contributes $\\|w_0\\|^{2/(\beta+1)}$. The iterative-rate convention also explicitly permits constants depending on the prescribed iteration count and fixed population quantities.)
- `lem:rdiv-accepted-bounds`: pinned reading: The assumptions of \cref{thm:deepiv_rate} mean its standing hypotheses inherited from \cref{prop:exist_lambda_deepiv}; the bounds here hold for arbitrary $h_0$, without imposing $h_0=0$. (The zero condition introduces only the last conclusion of \cref{thm:deepiv_rate}; the proof of \cref{lemma:rdiv_accepted_bounds} treats arbitrary $h_0$ using the variational source inequality.)
- `prop:exist-lambda`: pinned reading: For each fixed $n\geqslant N_{\mathrm T}(B)$, acceptance within the stated number of optimization problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_n$, with probability at least $1-\xi$ over the fitting sample. (The proof begins by restricting throughout to $\mathcal E_n$, establishes terminal acceptance there, and derives both return branches on that same event.)
- `prop:exist-lambda-deepiv`: pinned reading: For each fixed $n\geqslant N_{\mathrm R}(B)$, all probabilistic conclusions, including initial acceptance when $h_0=0$, hold on the same event $\mathcal E_n^{\mathrm R}$ of probability at least $1-\xi$. (The proof of Proposition~\ref{prop:exist_lambda_deepiv} explicitly works throughout on $\mathcal E_n^{\mathrm R}$ and derives initial acceptance there.)
- `prop:exist-lambda-iter-rdiv`: pinned reading: For each fixed $n\geqslant N_{\mathrm R}(B)$ and prescribed $t$, there is an event of probability at least $1-\xi$ on which the rule accepts a candidate and satisfies the stated output alternatives, including acceptance of $\lambda_0=2$ when $h_0=0$. (The proof of Proposition~\ref{prop:exist_lambda_iter_rdiv} explicitly places initial acceptance on the corresponding concentration event.)
- `prop:exist-lambda-iter-trae`: pinned reading: For each fixed $n\geqslant N_{\mathrm{IT}}(B)$ and prescribed $t$, acceptance within the stated number of minimax problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_{n,t}^{\mathrm{IT}}$, with probability at least $1-\xi$ over the fitting sample. (The proof works throughout on $\mathcal E_{n,t}^{\mathrm{IT}}$ and derives terminal acceptance and the first-accepted-index alternative there.)
- `thm:common-dp`: pinned reading: On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. (The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.)
- `thm:deepiv-rate`: pinned reading: For each fixed $n\geqslant N_{\mathrm R}(B)$, the displayed error bounds and, when $h_0=0$, both sharper $O(\delta_n^2)$ bounds hold jointly with probability at least $1-\xi$ over the preliminary, fitting, and calibration samples. (The proof obtains the zero-signal bounds from the probabilistic initial-acceptance conclusion of Proposition~\ref{prop:exist_lambda_deepiv} and Lemma~\ref{lemma:str_dp}; the general case uses the common RDIV event and the stated confidence convention.)
- `thm:iter-trae-rate`: pinned reading: The constants hidden by $O_t(\cdot)$ may depend on the prescribed $t$, fixed population quantities including $\beta$ and $\\|w_0\\|$, and the fixed grid ratio $\rho$, but are independent of $n$ and the candidate $\lambda$. (The text following the bounds in Theorem~\ref{thm:iter_rdiv_rate} explicitly allows dependence on the prescribed iteration count and fixed population quantities; Lemma~\ref{lemma:iter_trae_accepted_stability} supplies the comparison constants used here under the same convention.)
- `thm:rdiv-discrepancy-bound`: pinned reading: The assumptions imported from Theorem~\ref{thm:deepiv_rate} are the general hypotheses of Proposition~\ref{prop:exist_lambda_deepiv}, without imposing $h_0=0$. (Theorem~\ref{thm:deepiv_rate} imports the proposition's assumptions and treats $h_0=0$ only in a separate conditional conclusion. The present proof uses Lemmas~\ref{lemma:err_l_dp} and~\ref{lemma:str_dp} for general $h_0$.)
- `thm:tikhonv-rate`: pinned reading: With $\beta$ and $\xi$ fixed as $n\to\infty$, for each $n\geqslant N_{\mathrm T}(B)$ the displayed strong- and weak-error bounds hold jointly on a fitting-sample event of probability at least $1-\xi$. (The proof of Theorem~\ref{thm:tikhonv_rate} works on $\mathcal E_n$, which the preceding confidence allocation constructs separately for each sample size.)
- reading of the symbol `r_0`: $r_0\in\tilde{\mathcal Q}$ is the Riesz representer of the continuous linear functional $q\mapsto\mathbb E[m(W;q)]$, satisfying $\mathbb E[m(W;q)]=\mathbb E[r_0(Z)q(Z)]$ for every $q\in\tilde{\mathcal Q}$. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:t_err_dp}, Lemma~\ref{lemma:err_l_dp}, Lemma~\ref{lemma:err_r_dp}, Lemma~\ref{lemma:str_dp}, Corollary~\ref{cor:rdiv_discrepancy_bound}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $\tilde{\mathcal Q}=L_2(Z)$ and $r_0(z)=\mathbb E[Y\mid Z=z]$. — Section~\ref{sec:rdiv} specializes the Riesz representer to the conditional outcome mean. The enclosing iterative RDIV results use the same specialization.
- reading of the symbol `q`: The letter $q$ is locally reused for function-valued arguments in $\tilde{\mathcal Q}$ and for a deterministic scalar bias exponent in $(1,\infty)$; these uses denote distinct objects. In Theorem~\ref{thm:common_dp}: In $B_n(\lambda)=C_b\lambda^q$, $q>1$ is a fixed scalar exponent. In Theorem~\ref{thm:deepiv_rate}, Theorem~\ref{thm:tikhonv_rate}: In the application of the generic discrepancy guarantee, the scalar exponent is $q=p=\min\{\beta+1,2\}$. In Theorem~\ref{thm:iter_rdiv_rate}, Theorem~\ref{thm:iter_trae_rate}: In the proof's scalar abbreviation and application of the generic discrepancy guarantee, $q=q_t=\min\{\beta+1,2t\}$. In Corollary~\ref{cor:normality}: In the dual minimization, $q\in\mathcal Q$ is the candidate function of $Z$; in $\hat\theta(h,q)$ it is the dual nuisance argument. — The Riesz identity in Section~\ref{sec:pre} quantifies over $q\in\tilde{\mathcal Q}$. Theorem~\ref{thm:common_dp} introduces the scalar exponent. The proofs of Theorems~\ref{thm:deepiv_rate} and~\ref{thm:tikhonv_rate} set $q=p$, while the proofs of Theorems~\ref{thm:iter_rdiv_rate} and~\ref{thm:iter_trae_rate} set $q=q_t$. The dual minimization preceding Corollary~\ref{cor:normality} ranges over $q\in\mathcal Q$.
- reading of the symbol `Y`: $Y$ is the scalar outcome, with $r_0(z)=\mathbb E[Y\mid Z=z]$ in the IV setting. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:t_err_dp}, Lemma~\ref{lemma:err_l_dp}, Lemma~\ref{lemma:err_r_dp}, Lemma~\ref{lemma:str_dp}, Corollary~\ref{cor:rdiv_discrepancy_bound}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $\|Y\|\leqslant B$ almost surely for a fixed $B<\infty$. — Assumption~\ref{asp:cr_deepiv} bounds the outcome, and the iterative RDIV results retain that assumption.
- reading of the symbol `\mathcal H`: $\mathcal H$ denotes the primal hypothesis class of functions of $X$. In Proposition~\ref{prop:rkhs_qp}: $\mathcal H$ is the full RKHS generated by $k_{\mathcal H}$, before restriction to the retained numerical spectral span; its RKHS norm specifies the minimum-norm representative among functions with the same fitting-sample evaluations. — Section~\ref{sec:pre} introduces $\mathcal H\subset\tilde{\mathcal H}$ as a general function class; Proposition~\ref{prop:rkhs_qp} expressly specializes the notation to the RKHS generated by $k_{\mathcal H}$.
- reading of the symbol `\mathcal F`: $\mathcal F$ is a function class on the input domain used in the surrounding complexity or loss definition. In Corollary~\ref{cor:normality}: For the primal application, $\mathcal F\subset\tilde{\mathcal Q}$ is the adversarial test class; for the dual application the corresponding test class is $\mathcal S\subseteq\tilde{\mathcal H}$, with the primal and dual spaces exchanged. In Proposition~\ref{prop:rkhs_qp}: $\mathcal F$ is the full RKHS generated by $k_{\mathcal F}$, and its empirical supremum includes the penalty $-\eta\\|f\\|_{\mathcal F}^2$ with $\eta>0$. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:strong_convexity}: $\mathcal F\subset\tilde{\mathcal Q}$ is the test class in $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$ and its population counterpart; it contains $\mathcal T(h_0-h)$ for every $h\in\mathcal H$ under \cref{asp:closeness}. In \cite[Lemma 9 (lem:concentration)]{bennett2023source}, used in the proof of Lemma~\ref{eq:iter_trae_strong_propagation}: $\mathcal F\subset\tilde{\mathcal Q}$ is the test class in $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$ and its population counterpart; it contains $\mathcal T(h_0-h)$ for every $h\in\mathcal H$ under \cref{asp:closeness}. — The generic complexity notation allows an arbitrary function class. Section~\ref{sec:trae} specifies the adversarial test class, and Assumption~\ref{asp:closeness} supplies the residual inclusion used in the iterative TRAE comparisons.
- reading of the symbol `\mathcal S`: $\mathcal S$ is a fixed population function class whose local definition specifies its role and whether its functions have argument $X$ or $(X,Z)$. In Corollary~\ref{cor:normality}: $\mathcal S$ is the dual test class appearing in $L_n^{\mathrm{dual}}(q)=\sup_{s\in\mathcal S}\mathbb E_n[2\tilde m(W;s)-2q(Z)s(X)-s(X)^2]$. In Lemma~\ref{lemma:strong_convexity}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:strong_convexity}: $\mathcal S=\operatorname{star}(\mathcal H-\mathcal H)$ is the star-shaped difference class used to verify the anchored critical-radius condition. In Lemma~\ref{lemma:t_err_dp}: In the likelihood concentration argument, $\mathcal S=\operatorname{star}(\mathcal G_0-\mathcal G_0)$ is the fixed star-shaped difference class of likelihood-transform functions on $(X,Z)$, with norm $\\|\cdot\\|_{L_2(X,Z)}$. In \cite[Localized Concentration (source label lem:local-concen)]{li2024regularized}, used in the proof of Lemma~\ref{lemma:t_err_dp}: In the likelihood concentration argument, $\mathcal S=\operatorname{star}(\mathcal G_0-\mathcal G_0)$ is the fixed star-shaped difference class of likelihood-transform functions on $(X,Z)$, with norm $\\|\cdot\\|_{L_2(X,Z)}$. — Lemma lem:rdiv-mle-localized-concentration-li2024 explicitly sets $\mathcal S=\operatorname{star}(\mathcal G_0-\mathcal G_0)$ and uses the population $L_2(X,Z)$ norm.
- reading of the symbol `a_0`: The notation $a_0$ is local to either the dual inverse problem or the scalar recursion for $(a_j)_{j\geqslant0}$. In Corollary~\ref{cor:normality}: $a_0\in\tilde{\mathcal H}$ is the Riesz representer satisfying $\mathbb E[\tilde m(W;h)]=\mathbb E[a_0(X)h(X)]$ for every $h\in\tilde{\mathcal H}$, and is the right-hand side of $\mathcal T^*q_0=a_0$. In Lemma~\ref{lemma:iter_rdiv_stability}: In the proof's recursion $a_j=C(1+a_{j-1})$, $a_0=0$ is a scalar initial value, distinct from the Riesz representer in the dual inverse problem. — The preliminaries at line 150 and the dual formulation at lines 775--777 define the Riesz representer; the proof of Lemma~\ref{lemma:iter_rdiv_stability} at lines 1764--1765 separately initializes the scalar recursion.
- reading of the symbol `\hat\theta`: $\hat\theta$ denotes an empirical estimate of $\theta_0$, with its nuisance arguments displayed when the doubly robust score functional is intended. In Corollary~\ref{cor:normality}: $\hat\theta(h,q)=\mathbb E_{n,\mathrm{eval}}[\tilde m(W;h)+m(W;q)-h(X)q(Z)]$, and the estimator in the conclusion is its evaluation at $(h,q)=(\hat h,\hat q)$; here $q\in\mathcal Q$ is a nuisance-function argument. — The inference subsection defines $\hat\theta(h,q)$ by the evaluation-sample score average. The experimental evaluation paragraph explicitly assigns the unindexed symbol to a held-out plug-in average.
- reading of the symbol `\hat h`: $\hat h$ denotes an estimated primal solution, with $\hat h_\lambda$ denoting a candidate at penalty $\lambda$, $\hat h_{j,\lambda}$ denoting an empirical iterate, and $\hat h_{\lambda_{\mathrm{dp}}}$ or $\hat h_{t,\lambda_{\mathrm{dp}}}$ denoting the corresponding discrepancy-selected estimate. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: $\hat h_{0,\lambda}=0$ and $\hat h_{j,\lambda}=\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\\|h-\hat h_{j-1,\lambda}\\|_{2,n}^2\}$ for $j=1,\ldots,t$, with the RDIV squared loss; the candidate used for discrepancy selection is $\hat h_{t,\lambda}$. In Corollary~\ref{cor:normality}: $\hat h=\hat h_{\lambda_{\mathrm{dp}}}$ is the one-step adaptive adversarial Tikhonov estimate, with the primal tuning rule run at confidence level $\xi_n/2$ and with the common deterministic sequence $\delta_n$ specified for both nuisance problems. Its evaluations use jointly measurable versions relative to the full nuisance-training and tuning sigma-field $\mathcal G_n$ and the observation, representing the fitted function for almost every training realization and agreeing almost surely with its evaluations at every observation in the finite evaluation sample. In Proposition~\ref{prop:exist_lambda_deepiv}: $\hat h_\lambda=\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\mathbb E_n[h(X)^2]\}$ with $L_n(h)=\mathbb E_{n,\mathrm{fit}}[(Y-(\hat{\mathcal T}h)(Z))^2]$. In Proposition~\ref{prop:exist_lambda}: $\hat h_\lambda=\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\mathbb E_n[h(X)^2]\}$, where $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$. In Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}: $\hat h_{0,\lambda}=0$ and $\hat h_{j,\lambda}=\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\\|h-\hat h_{j-1,\lambda}\\|_{2,n}^2\}$ for $j=1,\ldots,t$, with the adversarial loss; the candidate used for discrepancy selection is $\hat h_{t,\lambda}$. — Corollary~\ref{cor:normality} specifies jointly measurable versions of fitted nuisance evaluations, representing the fitted functions for almost every training realization and agreeing almost surely with their evaluations at every observation in the finite evaluation sample.
- reading of the symbol `\hat q`: $\hat q$ denotes the dual empirical adversarial Tikhonov fit at the discrepancy-selected dual penalty. In Corollary~\ref{cor:normality}: The dual discrepancy rule is run at confidence level $\xi_n/2$, using the common deterministic sequence $\delta_n$ that satisfies \eqref{eq:delta_trae} for both nuisance problems. Take jointly measurable fitted evaluations of $\hat q$ representing the fitted function for almost every nuisance-training and tuning realization and agreeing almost surely with $\hat q(Z(W_i))$ simultaneously for $i=1,\ldots,n$ in the evaluation sample. — The dual-estimation displays define the empirical loss and penalized minimizer. Corollary~\ref{cor:normality} specifies the dual confidence level and accuracy sequence and requires jointly measurable versions of the fitted nuisance evaluations.
- reading of the symbol `\beta`: $\beta>0$ is a source exponent satisfying $h_0=(\mathcal T^*\mathcal T)^{\beta/2}w_0$ for some $w_0\in\widetilde{\mathcal H}$. In Corollary~\ref{cor:normality}: The two applications use potentially different source exponents: $h_0=(\mathcal T^*\mathcal T)^{\beta_h/2}w_h$ and $q_0=(\mathcal T\mathcal T^*)^{\beta_q/2}w_q$, with source elements in their respective Hilbert spaces; $\beta_m=\max\{\beta_h,\beta_q\}$. — Assumption~\ref{asp:source_condition} explicitly specifies the power $\beta/2$. Corollary~\ref{cor:normality} applies the primal and dual source conditions with potentially different exponents $\beta_h$ and $\beta_q$.
- reading of the symbol `w_0`: $w_0\in\tilde{\mathcal H}$ is a chosen source witness satisfying $h_0=(\mathcal T^*\mathcal T)^{\beta/2}w_0$, with no uniqueness or unit-norm requirement. In Proposition~\ref{prop:exist_lambda_deepiv}, Proposition~\ref{prop:exist_lambda_iter_rdiv}: When $h_0=0$, take $w_0=0$ in the initial-acceptance argument. — Assumption~\ref{asp:source_condition} specifies existence of a source witness. The proofs of Propositions~\ref{prop:exist_lambda_deepiv} and~\ref{prop:exist_lambda_iter_rdiv} explicitly choose the zero witness for the zero solution.
- reading of the symbol `\lambda`: $\lambda$ is the scalar regularization parameter. In the classical Hilbert-space construction \eqref{eq:tikh_sol} and Definition~\ref{def:dp}, finite penalties satisfy $0<\lambda<\infty$, with $\lambda=\infty$ denoting the zero-estimator convention. The value $\lambda=0$ is admitted only for the separate finite-dimensional computational convention of Section~\ref{sec:rkhs_computation}, which selects a minimum-Euclidean-norm solution on the retained spectral subspace; it does not define an unregularized solution of \eqref{eq:tikh_sol} for arbitrary $r_\delta$. In Theorem~\ref{thm:common_dp}, Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Corollary~\ref{cor:normality}: The tuning rule searches the deterministic finite grid $\Lambda_n=\{2\rho^k:k=0,\ldots,K_n\}$, where $1/2\leqslant\rho<1$ and $K_n=\lceil\log(2n^2)/\log(1/\rho)\rceil$; its selected value is denoted $\lambda_{\mathrm{dp}}$. In Proposition~\ref{prop:rkhs_qp}: $\lambda>0$ is a candidate penalty for the unrestricted quadratic program, before the numerical spectral restriction. In Lemma~\ref{eq:iter_trae_strong_propagation}: In each concentration application in the proof, $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ is a fixed candidate penalty. — The proof of Lemma~\ref{eq:iter_trae_strong_propagation} applies its comparisons to fixed iteration--grid pairs in the oracle-relevant segment. Section~\ref{sec:rkhs_computation} retains the separate computational convention at $\lambda=0$.
- reading of the symbol `r_\delta`: $r_\delta\in\tilde{\mathcal Q}$ is a perturbed right-hand side satisfying $\\|r_\delta-r_0\\|\leqslant\delta$. — The opening paragraph on classical linear inverse problems explicitly imposes $\\|r_\delta-r_0\\|\leqslant\delta$.
- reading of the symbol `\delta`: $\delta\geqslant0$ denotes the prescribed bound in $\\|r_\delta-r_0\\|\leqslant\delta$ in the classical inverse problem, and the radius of the population-norm restriction $\\|f\\|\leqslant\delta$ when used as the argument of $\mathfrak R_n(\delta;\mathcal F)$. — The classical inverse-problem paragraph supplies the perturbation bound; the notation paragraph defines $\mathfrak R_n(\delta;\mathcal F)$ by restricting to $\\|f\\|\leqslant\delta$ and states that this is the population norm.
- reading of the symbol `h_\lambda^\delta`: $h_\lambda^\delta$ is the minimizer over $h\in\tilde{\mathcal H}$ of $\\|\mathcal T h-r_\delta\\|^2+\lambda\\|h\\|^2$ for $0<\lambda<\infty$, with $h_\infty^\delta=0$ under the discrepancy rule. — Equation~\ref{eq:tikh_sol} defines the minimizer, and Definition~\ref{def:dp} identifies the choice $\lambda=\infty$ with $h=0$.
- reading of the symbol `\ell`: $\ell$ takes its meaning from its local binding: a prescribed residual-bracketing factor, an integer iteration index, or a real-valued loss function on $\mathbb R$. In Lemma~\ref{eq:iter_trae_strong_propagation}: In $M_{\leqslant j}=\max_{1\leqslant\ell\leqslant j}\{\lambda,\\|f_\ell^*\\|_\infty^2\}$ and the subsequent references to $f_\ell^*$, $\ell$ ranges over the integers $1,\ldots,j$. In \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:strong_convexity}: Put $M=\max\{1,B\}$. Here $\ell:\mathbb R\to\mathbb R$ is the globally Lipschitz loss $\ell(v)=M^2\bigl(\max\{-1,\min\{v,1\}\}\bigr)^2$, with Lipschitz constant $2M^2$; boundedness gives $\ell(h(X)/M)=h(X)^2$ for $h\in\mathcal H$. In Lemma~\ref{lemma:t_err_dp}: In the likelihood concentration argument, $\ell:\mathbb R\to\mathbb R$ is the fixed loss $\ell(v)=-\log\bigl(\max\{1+v,2^{-1/2}\}\bigr)$; it is $\sqrt2$-Lipschitz, satisfies $\ell(0)=0$, and obeys $\ell(f_g-1)=-\log f_g$. In Lemma~\ref{lemma:strong_convexity}: Put $M=\max\{1,B\}$. In the normalized squared-loss argument, $\ell:\mathbb R\to\mathbb R$ is the loss $\ell(u)=M^2\bigl(\max\{-1,\min\{u,1\}\}\bigr)^2$, which is $2M^2$-Lipschitz and satisfies $\ell(h(X)/M)=h(X)^2$ for $h\in\mathcal H$. In \cite[Localized Concentration (source label lem:local-concen)]{li2024regularized}, used in the proof of Lemma~\ref{lemma:t_err_dp}: In the likelihood concentration argument, $\ell:\mathbb R\to\mathbb R$ is the fixed loss $\ell(v)=-\log\bigl(\max\{1+v,2^{-1/2}\}\bigr)$; it is $\sqrt2$-Lipschitz, satisfies $\ell(0)=0$, and obeys $\ell(f_g-1)=-\log f_g$. — Lemma lem:rdiv-mle-localized-concentration-li2024 explicitly defines the clipped logarithmic loss and verifies its Lipschitz constant and its identity on the likelihood-transform class.
- reading of the symbol `\xi`: $\xi\in(0,1)$ is a prescribed failure-probability level, with fixed numerical rescalings absorbed into the rate bounds and the level relabelled after intersecting finitely many event families. In Corollary~\ref{cor:normality}: The confidence level used in each primal and dual tuning rule is $\xi_n/2$, where $\xi_n\downarrow0$ is deterministic; the simultaneous nuisance bounds consequently have failure probability at most $\xi_n$. — The confidence convention immediately preceding Theorem~\ref{thm:common_dp} explicitly permits numerical rescaling and relabelling. Corollary~\ref{cor:normality} prescribes a deterministic sequence and splits its level between the two tuning rules.
- reading of the symbol `\rho`: In the discrepancy grid, $\rho\in[1/2,1)$ is the fixed search factor, independent of $n$; the notation $\rho$ in the experimental designs of Section~\ref{sec:experiments} instead denotes their instrument-strength parameter, equal to the correlation of $(X,Z)$ in the linear Gaussian design. — The paragraph defining the grid fixes the search factor independently of $n$, and Algorithm~\ref{alg:discrepancy} uses it to generate successive penalties. Section~\ref{sec:experiment_design} separately specifies $X=\rho Z+\sqrt{1-\rho^2}\,V$ with independent standard normal $Z,V$.
- reading of the symbol `\hat h_\lambda`: $\hat h_\lambda$ denotes the candidate solution at penalty $\lambda$, with the discrepancy search using the family indexed by $\lambda\in\Lambda_n$ and the candidate construction specified by the estimation method. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}: $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$, where $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Lemma~\ref{lemma:err_l_dp}, Lemma~\ref{lemma:err_r_dp}, Lemma~\ref{lemma:str_dp}, Corollary~\ref{cor:rdiv_discrepancy_bound}, Lemma~\ref{lemma:rdiv_accepted_bounds}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $\hat h_\lambda$ is a measurable selected minimizer over $h\in\mathcal H$ of $\mathbb E_{n,\mathrm{fit}}[(Y-\hat{\mathcal T}h(Z))^2]+\lambda\mathbb E_{n,\mathrm{fit}}[h(X)^2]$. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}: The candidate supplied to the search is $\hat h_{t,\lambda}$, with $\hat h_{0,\lambda}=0$ and measurable selected minimizers $\hat h_{j,\lambda}\in\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\\|h-\hat h_{j-1,\lambda}\\|_{2,n}^2\}$ for $j=1,\ldots,t$, using the RDIV fitting loss. In Corollary~\ref{cor:normality}: The primal candidate is the TRAE measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $\mathcal H$; the corresponding dual construction exchanges the primal and dual spaces and operators. For the selected primal fit $\hat h$, take jointly measurable fitted evaluations representing the fitted function for almost every nuisance-training and tuning realization and agreeing almost surely with $\hat h(X(W_i))$ simultaneously for $i=1,\ldots,n$ in the evaluation sample. In Proposition~\ref{prop:rkhs_qp}: $\widehat h_\lambda$ minimizes $L_{N,\eta}(h)+\lambda\mathbb E_N[h(X)^2]$ over the full RKHS $\mathcal H$, taking the minimum-RKHS-norm representative among functions with the same fitting-sample evaluations; it has the form $\widehat h_\lambda(\cdot)=\sum_{j=1}^N\widehat\alpha_{\lambda,j}k_{\mathcal H}(X_j,\cdot)$ with coefficients solving \eqref{eq:rkhs_qp}. In Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}: The candidate supplied to the search is $\hat h_{t,\lambda}$, with $\hat h_{0,\lambda}=0$ and measurable selected minimizers $\hat h_{j,\lambda}\in\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\\|h-\hat h_{j-1,\lambda}\\|_{2,n}^2\}$ for $j=1,\ldots,t$, using the TRAE adversarial loss. — Corollary~\ref{cor:normality} requires jointly measurable versions of the fitted nuisance evaluations, representing the fitted functions for almost every training realization and agreeing almost surely with their evaluations throughout the finite evaluation sample.
- reading of the symbol `D_n`: $D_n$ is the nonnegative discrepancy functional supplied to the search, evaluated at the candidate for each $\lambda\in\Lambda_n$, with its functional definition specified by the estimation method. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:err_l_dp}, Corollary~\ref{cor:rdiv_discrepancy_bound}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: $D_n(h)=\mathbb E_{n,\mathrm{cal}}[(\hat r(Z)-\hat{\mathcal T}h(Z))^2]$, the projected calibration residual; the same functional is evaluated at $\hat h_{t,\lambda}$ for iterative candidates. In Corollary~\ref{cor:normality}: For each nuisance problem, the discrepancy is its empirical adversarial loss: $D_n=L_n$ for the primal problem, with the corresponding loss after exchanging the primal and dual spaces and operators for the dual problem. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{lemma:trae_accepted_bounds}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}: $D_n(h)=L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$, with the same loss evaluated at the final iterate for iterative candidates. — Equation \eqref{eq:rdiv_projected_loss} retains the projected calibration residual, while the paragraph introducing \eqref{eq:trae_common_scales} sets $D_n=L_n$. The iterative procedures retain their respective discrepancies.
- reading of the symbol `A_n`: $A_n:\Lambda_n\to[0,\infty)$ supplies the noise level in the acceptance threshold $\tfrac32A_n(\lambda)$, with prescribed statistical-error bounds in the theoretical constructions and data-dependent noise-floor estimates in the experimental construction. In Theorem~\ref{thm:common_dp}: $A_n$ is prescribed, and is required to be nonincreasing only over $\Lambda_n\cap[\lambda^\circ,2]$. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}: $A_n=A_n^{\mathrm R}$, where $A_n^{\mathrm R}(\lambda)=\delta_n^2/\lambda+\nu_n^2$ for $\lambda\in\Lambda_n$. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}: $A_n=A_{n,t}^{\mathrm R}$, where $A_{n,t}^{\mathrm R}(\lambda)=s_{n,t}^{\mathrm R}/\lambda+\nu_n^2$ and $s_{n,t}^{\mathrm R}=a_t\delta_n^2$. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Corollary~\ref{cor:normality}: $A_n=A_n^{\mathrm T}$, where $A_n^{\mathrm T}(\lambda)=\delta_n^2$ for $\lambda\in\Lambda_n$, using the TRAE accuracy scale for the relevant problem. In Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}: $A_n=A_{n,t}^{\mathrm T}$, where $A_{n,t}^{\mathrm T}(\lambda)=\delta_n^2+\min\{\lambda,s_{n,t}^{\mathrm T}/\lambda\}$ and $s_{n,t}^{\mathrm T}=b_t\delta_n^2$; on the oracle segment $[\lambda_{\mathrm{IT}}^\circ,2]$ this equals $\delta_n^2+s_{n,t}^{\mathrm T}/\lambda$. — Equations~\eqref{eq:rdiv_common_scales}, \eqref{eq:iter_rdiv_common_scales}, \eqref{eq:trae_common_scales}, and \eqref{eq:iter_trae_scale} retain the method-specific definitions; the affected scopes must be restricted to the results still using $A_n$.
- reading of the symbol `\mathcal S_n`: $\mathcal S_n\subseteq(0,2]$ is the prescribed stability domain for the generic comparisons, which are required at grid points in $\Lambda_n\cap\mathcal S_n$. In Theorem~\ref{thm:deepiv_rate}, Theorem~\ref{thm:tikhonv_rate}: $\mathcal S_n=(0,2]$. In Theorem~\ref{thm:iter_rdiv_rate}: $\mathcal S_n=[\lambda_{\mathrm{IR}}^\circ,2]$ in the application of the generic comparison argument. In Theorem~\ref{thm:iter_trae_rate}: $\mathcal S_n=[\lambda_{\mathrm{IT}}^\circ,2]$ in the application of the generic comparison argument. — Theorem~\ref{thm:common_dp} defines the domain; the proofs of Theorems~\ref{thm:deepiv_rate}, \ref{thm:iter_rdiv_rate}, \ref{thm:tikhonv_rate}, and \ref{thm:iter_trae_rate} specify the respective domains.
- reading of the symbol `\mathcal E_n`: $\mathcal E_n$ is a common event of probability at least $1-\xi$ on which the relevant comparisons hold simultaneously over their stated function and grid domains. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}: $\mathcal E_n$ is the common TRAE concentration event formed using $\eta_n=\xi/(K_n+1)$ and a union bound over $\Lambda_n$; it includes the uniform loss comparison, the localized squared-norm comparison, and $L_n(h_0)\leqslant C\delta_n^2$. — Theorem~\ref{thm:common_dp} retains the generic event, and the paragraph after \eqref{eq:delta_trae} retains the common TRAE event. The separately named RDIV and iterative events no longer supply scopes for this symbol.
- reading of the symbol `\gamma`: $\gamma$ is locally bound to its construction, with $\gamma\in[0,1]$ denoting the scalar multiplier in $\operatorname{star}(\mathcal F)=\{\gamma f:\gamma\in[0,1],f\in\mathcal F\}$. In Theorem~\ref{thm:common_dp}: $\gamma\in(0,1]$ is a fixed exponent in the strong-error comparison and the resulting oracle bound. In Theorem~\ref{thm:deepiv_rate}, Theorem~\ref{thm:tikhonv_rate}: When applying the oracle guarantee, the exponent is $\gamma=\min\{\beta,1\}/(1+\min\{\beta,1\})$. In Theorem~\ref{thm:iter_rdiv_rate}, Theorem~\ref{thm:iter_trae_rate}: When applying the oracle guarantee, the exponent is $\gamma=\beta/(\beta+1)$. In Lemma~\ref{lemma:iter_bias}: In the spectral bound, $\gamma>0$ is an arbitrary real exponent in $\sup_{0\leqslant u\leqslant1}u^\gamma\{\lambda/(u+\lambda)\}^{2j}\leqslant C_{\gamma,j}\lambda^{\min\{\gamma,2j\}}$; it is then specialized to $\beta$ and $\beta+1$. In Proposition~\ref{prop:rkhs_qp}: $\gamma\in\mathbb R^N$ is the critic coefficient vector in $f_\gamma(\cdot)=\sum_{j=1}^N\gamma_j k_{\mathcal F}(Z_j,\cdot)$, over which the finite-dimensional critic objective is maximized. — The notation paragraph defines the star-hull multiplier. Theorem~\ref{thm:common_dp} introduces a fixed exponent; the proofs of Theorems~\ref{thm:deepiv_rate}, \ref{thm:tikhonv_rate}, \ref{thm:iter_rdiv_rate}, and~\ref{thm:iter_trae_rate} specify its substitutions. The proof of Lemma~\ref{lemma:iter_bias} quantifies over positive spectral exponents. The proof of Proposition~\ref{prop:rkhs_qp} introduces the critic coefficient vector.
- reading of the symbol `\lambda^\circ`: $\lambda^\circ\in\Lambda_n$ is any oracle comparison point satisfying $\Lambda_n\cap[\lambda^\circ,2]\subseteq\mathcal S_n$, nonincreasing $A_n$ on that grid segment, and $B_n(\lambda^\circ)\leqslant\tfrac12A_n(\lambda^\circ)$; no uniqueness or computable selection rule is required. — Theorem~\ref{thm:common_dp} assumes that some grid point satisfies exactly these three conditions.
- reading of the symbol `\lambda_{\mathrm{dp}}`: $\lambda_{\mathrm{dp}}=2\rho^{k_*}$, where $k_*=\min\{k\in\{0,\ldots,K_n\}:D_n(\hat h_{2\rho^k})\leqslant\tfrac32A_n(2\rho^k)\}$ if this set is nonempty, and $k_*=K_n$ otherwise, using the candidates, discrepancy, and noise level supplied to the rule. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}: The rule uses the one-step RDIV candidates $\hat h_\lambda$, the projected calibration discrepancy, and $A_n^{\mathrm R}(\lambda)=\delta_n^2/\lambda+\nu_n^2$. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}: The rule uses the one-step TRAE candidates, $D_n=L_n$, and $A_n^{\mathrm T}(\lambda)=\delta_n^2$. In Corollary~\ref{cor:normality}: The primal and dual TRAE problems are tuned separately at confidence levels $\xi_n/2$ using a common deterministic accuracy bound $\delta_n$; the dual output is $\lambda_{\mathrm{dp}}^{\text{dual}}$. In Proposition~\ref{prop:exist_lambda_iter_rdiv}: The rule uses the terminal iterative RDIV candidates $\hat h_{t,\lambda}$, the projected calibration discrepancy, and $A_{n,t}^{\mathrm R}(\lambda)=s_{n,t}^{\mathrm R}/\lambda+\nu_n^2$; its selected parameter is denoted $\lambda_{\mathrm{dp},t}^{\mathrm R}$. In Proposition~\ref{prop:exist_lambda_iter_trae}: The rule uses the terminal iterative TRAE candidates and $A_{n,t}^{\mathrm T}(\lambda)=\delta_n^2+\min\{\lambda,s_{n,t}^{\mathrm T}/\lambda\}$. — Algorithm~\ref{alg:discrepancy} retains the same selection rule. Propositions~\ref{prop:exist_lambda_iter_rdiv} and~\ref{prop:exist_lambda_iter_trae} retain its iterative inputs; the former scope groups now include results no longer listed as using this symbol.
- reading of the symbol `g_0`: $g_0(\cdot\mid z)\in\mathcal G$ is the true conditional density of $X$ given $Z=z$ with respect to $\mu$, normalized by $\int g_0(x\mid z)\,\mu(dx)=1$ for $P_Z$-almost every $z$, with $g_0(x\mid z)\geqslant c_0$ on the common conditional-density support. — The RDIV conditional-density construction and Assumption~\ref{asp:re_cd} specify the true density, realizability, dominating measure, and lower bound; Assumption~\ref{asp:cr_deepiv} identifies the common support.
- reading of the symbol `\hat r`: $\hat r$ is the pilot regression estimate of $r_0(z)=\mathbb E[Y\mid Z=z]$ fitted from $Y$ on $Z$ using $I_{\mathrm{pre}}$. — The RDIV setup at flat.tex lines 299--305 explicitly fits the regression on $I_{\mathrm{pre}}$; line 324 explains its independence from calibration. The linear-smoother formula at line 2447 belongs to the experimental implementation.
- reading of the symbol `I_{\mathrm{pre}}`: In conditioning expressions, $I_{\mathrm{pre}}$ denotes the observed preliminary sample, so conditioning on $I_{\mathrm{pre}}$ means conditioning on $\sigma(W_i:i\in I_{\mathrm{pre}})$. — Section~\ref{sec:rdiv} introduces the preliminary sample and fits the nuisances on it; the proof of Lemma~\ref{lemma:err_l_dp} explicitly conditions on it to fix $\hat{\mathcal T}$.
- reading of the symbol `I_{\mathrm{fit}}`: $I_{\mathrm{fit}}$ is the fitting block of the three-way RDIV split, with $\|I_{\mathrm{fit}}\|=n$, and $\mathbb E_n=\mathbb E_{n,\mathrm{fit}}$ in the RDIV analysis. — The RDIV setup at flat.tex line 299 specifies the block sizes and the empirical-average convention; the convention is repeated at lines 1363--1366.
- reading of the symbol `I_{\mathrm{cal}}`: $I_{\mathrm{cal}}$ is the calibration block of the three-way RDIV split, with $\|I_{\mathrm{cal}}\|=n$. — The RDIV setup at flat.tex line 299 specifies three independent samples, each of size $n$, and the projected discrepancy uses the empirical average on the calibration block.
- reading of the symbol `L_n`: $L_n$ denotes the empirical fitting loss on $\mathcal H$, with $\mathbb E_n$ taken over the fitting observations of the applicable procedure. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:err_l_dp}, Lemma~\ref{lemma:err_r_dp}, Lemma~\ref{lemma:str_dp}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:iter_bias}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $L_n(h)=\mathbb E_{n,\mathrm{fit}}[(Y-\hat{\mathcal T}h(Z))^2]$; the iterative procedure retains this loss and changes the penalty center. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Corollary~\ref{cor:normality}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}: $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$, with $\mathbb E_n$ the TRAE fitting-sample average; the iterative procedure retains this loss and changes the penalty center. In \cite[Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:iter_rdiv_stability}, Lemma (Recursive RDIV stability): $L_n(h)=\mathbb E_{n,\mathrm{fit}}[(Y-\hat{\mathcal T}h(Z))^2]$; changing the penalty center in a concentration comparison does not change this fitting loss. In \cite[Lemma 9 (lem:concentration)]{bennett2023source}, used in the proof of Lemma~\ref{eq:iter_trae_strong_propagation}: $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$, where $\mathbb E_n$ is the TRAE fitting-sample average; changing the penalty center in a concentration comparison does not change this fitting loss. — The RDIV fitting definition and the loss definitions in Section~\ref{sec:trae} specify the two meanings. The iterative definitions retain these losses, including in the fixed-center comparisons in the proofs of Lemma~\ref{lemma:iter_rdiv_stability}, Lemma~\ref{eq:iter_trae_strong_propagation}, and Lemma~\ref{lemma:iter_trae_accepted_stability}.
- reading of the symbol `L`: $L$ denotes the method-specific population loss functional, defined using population expectations. In Lemma~\ref{lemma:t_err_dp}, Lemma~\ref{lemma:err_l_dp}, Lemma~\ref{lemma:err_r_dp}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $L(h)=\mathbb E[(Y-\mathcal T h(Z))^2]=\sigma^2_{Y\mid Z}+\\|\mathcal T(h-h_0)\\|^2$, where $\sigma^2_{Y\mid Z}=\mathbb E[(Y-r_0(Z))^2]$. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Corollary~\ref{cor:normality}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}: $L(h)=\sup_{f\in\mathcal F}\mathbb E[2m(W;f)-2h(X)f(Z)-f(Z)^2]$; under the stated closeness assumption, $L(h)=\\|\mathcal T(h-h_0)\\|^2$ for $h\in\mathcal H$. In \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:strong_convexity}: $L(h)=\sup_{f\in\mathcal F}\mathbb E[2m(W;f)-2h(X)f(Z)-f(Z)^2]$; under the stated closeness assumption, $L(h)=\\|\mathcal T(h-h_0)\\|^2$ for $h\in\mathcal H$. In \cite[Lemma 9 (lem:concentration)]{bennett2023source}, used in the proof of Lemma~\ref{eq:iter_trae_strong_propagation}: $L(h)=\sup_{f\in\mathcal F}\mathbb E[2m(W;f)-2h(X)f(Z)-f(Z)^2]$; under the stated closeness assumption, $L(h)=\\|\mathcal T(h-h_0)\\|^2$ for $h\in\mathcal H$. — Section~\ref{sec:trae} defines the adversarial population loss. Its weak-error identity under Assumption~\ref{asp:closeness} is retained in the iterative comparisons in the proofs of Lemma~\ref{eq:iter_trae_strong_propagation} and Lemma~\ref{lemma:iter_trae_accepted_stability}.
- reading of the symbol `B`: $B$ is a fixed finite nonnegative uniform envelope bound for the quantities specified in the applicable boundedness assumptions. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Corollary~\ref{cor:normality}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:strong_convexity}: $\|m(W;f)\|$, $\|h(X)\|$, and $\|f(Z)\|$ are almost surely bounded by $B$, uniformly over $h\in\mathcal H$ and $f\in\mathcal F$; for a dual fit the corresponding bound uses the exchanged primal and dual objects. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:t_err_dp}, Lemma~\ref{lemma:err_l_dp}, Corollary~\ref{cor:rdiv_discrepancy_bound}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $\|Y\|\leqslant B$ almost surely, $\|h(x)\|\leqslant B$ for every $h\in\mathcal H$ on the common support of the conditional densities, and $\|\hat r(z)\|\leqslant B$. In \cite[Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:iter_rdiv_stability}, Lemma (Recursive RDIV stability): $\|Y\|\leqslant B$ almost surely, $\|h(x)\|\leqslant B$ for every $h\in\mathcal H$ on the common support of the conditional densities, and $\|\hat r(z)\|\leqslant B$. In \cite[Lemma 9 (lem:concentration)]{bennett2023source}, used in the proof of Lemma~\ref{eq:iter_trae_strong_propagation}: $\|m(W;f)\|$, $\|h(X)\|$, and $\|f(Z)\|$ are almost surely bounded by $B$, uniformly over $h\in\mathcal H$ and $f\in\mathcal F$. — Assumption~\ref{asp:cr_deepiv} supplies the RDIV envelopes. Assumption~\ref{asp:critical_radius} supplies the TRAE envelopes used in the iterative concentration comparisons.
- reading of the symbol `c_0`: $c_0>0$ is a fixed finite constant, independent of sample size, such that $g_0(x\mid z)\geqslant c_0$ on the common conditional-density support. — Assumption~\ref{asp:re_cd} gives the fixed positive lower bound; Assumption~\ref{asp:cr_deepiv} identifies the common support, and the proof of Lemma~\ref{lemma:t_err_dp} uses the bound in density ratios.
- reading of the symbol `C_G`: $C_G>0$ is a fixed finite constant, independent of sample size, such that $\sup_{g\in\mathcal G}\\|g\\|_\infty\leqslant C_G$ on the common conditional-density support. — Assumption~\ref{asp:re_cd} states the fixed constant and uniform density bound; the density-ratio argument in the proof of Lemma~\ref{lemma:t_err_dp} applies it to $\hat g/g_0$.
- reading of the symbol `\mathcal T_g`: For every $g\in\mathcal G$, $\mathcal T_g$ maps $h\in\mathcal H$ to the $P_Z$-square-integrable function $(\mathcal T_g h)(z)=\int h(x)g(x\mid z)\,\mu(dx)$, with $z$ understood $P_Z$-almost everywhere. — Assumption~\ref{asp:cr_deepiv} introduces the operator for $g\in\mathcal G$ and defines its image class under the respective input distribution; the notation paragraph specifies the population norm for functions of $Z$.
- reading of the symbol `\mathcal G_0`: $\mathcal G_0$ is the measurable class $\{(x,z)\mapsto\sqrt{\{g(x\mid z)+g_0(x\mid z)\}/\{2g_0(x\mid z)\}}:g\in\mathcal G\}$ on the common conditional-density support, with input distribution the true joint law of $(X,Z)$ and positive denominator there. — Assumptions~\ref{asp:re_cd} and~\ref{asp:cr_deepiv} give positivity, the common support, measurability, and the displayed class; the conditional-MLE application in the proof of Lemma~\ref{lemma:t_err_dp} uses its critical radius.
- reading of the symbol `\eta_n`: $\eta_n=\xi/(K_n+1)$ is the deterministic grid confidence allocation at the prescribed failure level $\xi$. In Corollary~\ref{cor:normality}: For each of the primal and dual tuning rules, the prescribed failure level is $\xi_n/2$, so $\eta_n=\xi_n/\{2(K_n+1)\}$, where $\xi_n\downarrow0$ is deterministic. — Assumption~\ref{asp:cr_deepiv} and the paragraph introducing \eqref{eq:delta_trae} define the allocation; Corollary~\ref{cor:normality} prescribes confidence levels $\xi_n/2$ for both tuning rules.
- reading of the symbol `\delta_{n,r}`: $\delta_{n,r}=\delta_{n,r}(\xi)\geqslant0$ is a prescribed deterministic, $\beta$-independent accuracy bound such that $\\|\hat r-r_0\\|_{L_2(Z)}\leqslant\delta_{n,r}$ with probability at least $1-\xi$ over the preliminary sample. In Lemma~\ref{lemma:t_err_dp}: The pilot-regression accuracy bound is evaluated at failure probability $\xi/2$ and relabelled $\delta_{n,r}$, so here $\delta_{n,r}$ denotes $\delta_{n,r}(\xi/2)$ and satisfies $\\|\hat r-r_0\\|_{L_2(Z)}\leqslant\delta_{n,r}$ with probability at least $1-\xi/2$ over the preliminary sample. — The proof of Lemma~\ref{lemma:t_err_dp} allocates failure probability $\xi/2$ to the pilot regression and explicitly evaluates and relabels the corresponding deterministic accuracy bound as $\delta_{n,r}$.
- reading of the symbol `\delta_{n,\mathcal G}`: $\delta_{n,\mathcal G}$ is a prescribed deterministic, $\beta$-independent upper bound on the critical radius of $\mathcal G_0$ under the joint input distribution of $(X,Z)$, evaluated at the failure probability allocated to the conditional-density fit and satisfying $\delta_{n,\mathcal G}\geqslant C\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. In Lemma~\ref{lemma:t_err_dp}: The conditional-density bound is evaluated at failure probability $\xi/2$ and relabelled $\delta_{n,\mathcal G}$, so here $\delta_{n,\mathcal G}$ denotes $\delta_{n,\mathcal G}(\xi/2)$. — Assumption~\ref{asp:cr_deepiv} defines the radius bound and its confidence-dependent floor. The proof of Lemma~\ref{lemma:t_err_dp} says: Allocate failure probability $\xi/2$ to the conditional-density estimator and $\xi/2$ to the pilot regression. It then explicitly evaluates and relabels the corresponding deterministic accuracy bounds.
- reading of the symbol `\delta_{n,\mathcal H}`: $\delta_{n,\mathcal H}=\delta_{n,\mathcal H}(\eta_n)$ is a prescribed deterministic common upper bound on the critical radii of $\mathcal H$ under the distribution of $X$ and of $\mathcal T_g\mathcal H$ under the distribution of $Z$, uniformly for $g\in\mathcal G$, with $\eta_n=\xi/(K_n+1)$ and $\delta_{n,\mathcal H}\geqslant C\sqrt{\{\log\log(en)+\log(1/\eta_n)\}/n}$. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: The iterative concentration comparisons allocate failure probability $\eta_{n,t}=\xi/\{t(K_n+1)\}$ over iteration--grid pairs; the resulting confidence adjustment is absorbed into $\delta_n$ for fixed $t$. — Assumption~\ref{asp:cr_deepiv} still defines the common radius bound. The iterative confidence convention and the proof of Lemma~\ref{lemma:iter_rdiv_stability} retain the allocation over iteration--grid pairs, already recorded in the surviving iterative scope.
- reading of the symbol `\delta_n`: $\delta_n$ is a prescribed deterministic, method-specific statistical-error bound varying with $n$, calibrated before fitting by fixed multipliers independent of $n$, $\lambda$, and the unknown source parameter. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: The iteration--grid confidence allocation is $\eta_{n,t}=\xi/\{t(K_n+1)\}$; the resulting $\log t$ contribution is absorbed into a constant adjustment of $\delta_n$ for fixed $t$. In Proposition~\ref{prop:exist_lambda}, Theorem~\ref{thm:tikhonv_rate}, Lemma~\ref{lemma:concentrate_l}, Lemma~\ref{lemma:strong_convexity}, Corollary~\ref{cor:trae_discrepancy_bound}, Lemma~\ref{lemma:trae_accepted_bounds}, \cite[Lemma 9]{bennett2023source}, used in the proof of Lemma~\ref{lemma:concentrate_l}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:strong_convexity}: $\delta_n$ is the TRAE scale, unrelated to the RDIV scales, satisfying $\delta_n\geqslant\max\{\tilde\delta_n(\eta_n),n^{-1/2}\}$ and $\delta_n=o(1)$, with $\eta_n=\xi/(K_n+1)$. In Corollary~\ref{cor:normality}: $\delta_n$ is a common deterministic TRAE bound for both primal and dual problems, each tuned at confidence level $\xi_n/2$ for a deterministic $\xi_n\downarrow0$, and additionally satisfies $\delta_n=o(n^{-\alpha_{\mathrm{dr}}})$, where $\alpha_{\mathrm{dr}}=(1+\min\{\beta_m,1\})/(2+4\min\{\beta_m,1\})$ and $\beta_m=\max\{\beta_h,\beta_q\}$. In Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}: $\delta_n$ is the TRAE scale satisfying $\delta_n\geqslant\max\{\tilde\delta_n(\eta_{n,t}),n^{-1/2}\}$ and $\delta_n=o(1)$, with $\eta_{n,t}=\xi/\{t(K_n+1)\}$. In Proposition~\ref{prop:exist_lambda_deepiv}, Theorem~\ref{thm:deepiv_rate}, Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:t_err_dp}, Lemma~\ref{lemma:err_l_dp}, Lemma~\ref{lemma:err_r_dp}, Lemma~\ref{lemma:str_dp}, Corollary~\ref{cor:rdiv_discrepancy_bound}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}, \cite[Lemma 14]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:err_l_dp}: $\delta_n$ is the RDIV fitting scale satisfying $\delta_n\geqslant\delta_{n,\mathcal G}+\delta_{n,\mathcal H}+\delta_{n,\xi}$ and $\delta_n+\nu_n=o(1)$. — Equation~\eqref{eq:delta_deepiv} defines the RDIV scale, equation~\eqref{eq:delta_trae} defines the TRAE scale, and the iterative confidence conventions remain in the enclosing iterative arguments. The existing scoped clauses for those arguments already record these conventions.
- reading of the symbol `\nu_n`: $\nu_n$ is a prescribed deterministic RDIV projected-discrepancy accuracy bound, fixed before fitting the candidates, with $\nu_n\geqslant\delta_{n,\mathcal G}+\delta_{n,r}+\delta_{n,\xi}$ and $\delta_n+\nu_n=o(1)$; its calibration multipliers are independent of $n$, $\lambda$, and the unknown source parameter. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: The bounds use the iteration--grid confidence allocation $\eta_{n,t}=\xi/\{t(K_n+1)\}$; the resulting $\log t$ contribution is absorbed into a constant adjustment of $\nu_n$ for the prescribed fixed $t$. In \cite[Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:iter_rdiv_stability}, Lemma (Recursive RDIV stability): The bounds use the iteration--grid confidence allocation $\eta_{n,t}=\xi/\{t(K_n+1)\}$; the resulting $\log t$ contribution is absorbed into a constant adjustment of $\nu_n$ for the prescribed fixed $t$. — The paragraph following the definition of $\eta_{n,t}$ retains $\delta_n$ and $\nu_n$ with a fixed-$t$ confidence adjustment. The proof of Lemma~\ref{lemma:iter_rdiv_stability} applies the fixed-comparator and squared-loss comparisons on the iteration--grid event with allocation $\eta_{n,t}$.
- reading of the symbol `\tilde\delta_n`: $\tilde\delta_n(\xi)$ is a prescribed common critical-radius upper bound indexed by sample size $n$ and failure probability $\xi\in(0,1)$ for the five star-shaped classes specified in Assumption~\ref{asp:critical_radius}. — Assumption~\ref{asp:critical_radius} lists the five classes, displays the confidence argument and lower order bound, and equation \eqref{eq:delta_trae} evaluates it at $\eta_n=\xi/(K_n+1)$.
- reading of the symbol `\alpha`: $\alpha$ is local notation whose scalar-exponent and RKHS-coefficient meanings are distinct. In Proposition~\ref{prop:rkhs_qp}: $\alpha\in\mathbb R^N$ is the optimization variable representing $h_\alpha(\cdot)=\sum_{j=1}^N\alpha_j k_{\mathcal H}(X_j,\cdot)$; its fitting-sample evaluation vector is $K_{\mathcal H}\alpha$. In Theorem~\ref{thm:deepiv_rate}, Lemma~\ref{lemma:variational_source}, Lemma~\ref{lemma:rdiv_accepted_bounds}, Lemma~\ref{lemma:strong_convexity}, Lemma~\ref{lemma:trae_accepted_bounds}: $\alpha=\min\{\beta,1\}/(1+\min\{\beta,1\})=(p-1)/p$ is the scalar interpolation exponent. — Theorem~\ref{thm:deepiv_rate} and Lemma~\ref{lemma:variational_source} define the scalar exponent. Corollary~\ref{cor:normality} instead uses the distinct notation $\alpha_{\mathrm{dr}}$.
- reading of the symbol `\lambda_{\mathrm R}^\circ`: Whenever the defining set is nonempty, $\lambda_{\mathrm R}^\circ$ is the deterministic oracle point $2\rho^{k^\circ}$, where $k^\circ=\min\{k\in\{0,\ldots,K_n\}:2\rho^k\leqslant c_{\mathrm R}\max\{\delta_n^{2/(p+1)},\nu_n^{2/p}\}\}$ and $c_{\mathrm R}>0$ is a sufficiently small fixed constant; this point exists for all sufficiently large $n$. — The definition preceding \eqref{eq:rdiv_oracle_scales} supplies the threshold. The proof of Theorem~\ref{thm:deepiv_rate} explicitly states that the point exists for all sufficiently large $n$, and Algorithm~\ref{alg:discrepancy} fixes the decreasing grid order.
- reading of the symbol `\lambda_{\mathrm T}^\circ`: For all sufficiently large $n$, $\lambda_{\mathrm T}^\circ$ is the deterministic oracle point $2\rho^{k^\circ}$, where $k^\circ=\min\{k\in\{0,\ldots,K_n\}:2\rho^k\leqslant c_{\mathrm T}\delta_n^{2/p}\}$, $c_{\mathrm T}>0$ is a sufficiently small fixed constant, and $\delta_n$ is the TRAE scale in \eqref{eq:delta_trae}. — The definition preceding \eqref{eq:trae_oracle_scales} states 'For all sufficiently large $n$'; the convention following Assumption~\ref{asp:critical_radius} fixes the TRAE meaning of $\delta_n$, and Algorithm~\ref{alg:discrepancy} fixes the grid order.
- reading of the symbol `t`: $t\geqslant1$ is a prescribed integer iteration count held fixed as $n$ varies and chosen independently of the data and the source parameter $\beta$. — The iterative constructions prescribe the iteration count, and Theorems~\ref{thm:iter_rdiv_rate} and~\ref{thm:iter_trae_rate} quantify over fixed $t$. The proof of Lemma~\ref{lemma:t_err_dp} now invokes the conditional-MLE bound directly and contains no derivative of $-\log t$.
- reading of the symbol `a_t`: $a_t\geqslant1$ is a deterministic calibrated RDIV propagation constant depending only on the prescribed $t$ and fixed population quantities, independent of $n$ and $\lambda$. — Lines 484--485 specify determinism and dependence only on $t$ and fixed population quantities; the noise-calibration paragraph absorbs propagation constants, and Theorem~\ref{thm:iter_rdiv_rate} explicitly states independence of $n$.
- reading of the symbol `\lambda_{\mathrm{IR}}^\circ`: Whenever the defining set is nonempty, $\lambda_{\mathrm{IR}}^\circ$ is the deterministic oracle point $2\rho^{k^\circ}$, where $k^\circ=\min\{k\in\{0,\ldots,K_n\}:2\rho^k\leqslant c_{\mathrm{IR}}\max\{(s_{n,t}^{\mathrm R})^{1/(q_t+1)},\nu_n^{2/q_t}\}\}$ and $c_{\mathrm{IR}}>0$ is a sufficiently small fixed constant; this point exists for all sufficiently large $n$. — The definition preceding \eqref{eq:iter_rdiv_oracle_scales} specifies the threshold, and Assumption~\ref{asp:iter_realizability_rdiv} explicitly says 'For every $n$ for which $\lambda_{\mathrm{IR}}^\circ$ is defined'; Algorithm~\ref{alg:discrepancy} specifies the decreasing grid order.
- reading of the symbol `b_t`: $b_t\geqslant2$ is a calibrated deterministic propagation constant depending only on the prescribed iteration count $t$ and the fixed boundedness and mean-square-continuity constants, independently of $n$, $\beta$, and $\\|w_0\\|$. In Lemma~\ref{eq:iter_trae_strong_propagation}: $b_t$ is the terminal member of a nondecreasing sequence with $b_0=0$ and $b_j\geqslant2$ for $j=1,\ldots,t$; in the proof one may choose $b_j=C(1+b_{j-1})$ for a sufficiently large fixed $C\geqslant2$. — The paragraph preceding equation~\eqref{eq:iter_trae_scale} specifies its dependencies. Lemma~\ref{eq:iter_trae_strong_propagation} supplies the calibrated sequence, and its proof constructs that sequence recursively.
- reading of the symbol `\lambda_{\mathrm{IT}}^\circ`: For all sufficiently large $n$, $\lambda_{\mathrm{IT}}^\circ$ is the deterministic oracle point $2\rho^{k^\circ}$, where $k^\circ=\min\{k\in\{0,\ldots,K_n\}:2\rho^k\leqslant c_{\mathrm{IT}}(s_{n,t}^{\mathrm T})^{1/(q_t+1)}\}$ and $c_{\mathrm{IT}}>0$ is a sufficiently small fixed constant. — The definition preceding \eqref{eq:iter_trae_oracle_scales} explicitly restricts the construction to all sufficiently large $n$; Algorithm~\ref{alg:discrepancy} orders the grid from $k=0$ to $K_n$.
- reading of the symbol `h_{j,\lambda}^*`: $h_{j,\lambda}^*$ is the population minimizer over $\tilde{\mathcal H}$ for positive finite $\lambda$, initialized by $h_{0,\lambda}^*=0$ and recursively defined for positive integer $j$; the iterative procedures use $j=1,\ldots,t$. In Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}: $h_{j,\lambda}^*\in\mathcal H$ for $j=1,\ldots,t$ and $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$, for every $n$ for which the oracle point is defined. In Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: $h_{j,\lambda}^*\in\mathcal H$ for $j=1,\ldots,t$ and $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$, for every $n$ for which the oracle point is defined. — Assumptions~\ref{asp:iter_realizability_rdiv} and~\ref{asp:iter_realizability_trae} specify membership of the population iterates on the respective oracle grid segments. The population definition and the remaining stability arguments retain the same comparators and penalty centers.
- reading of the symbol `\hat h_{j,\lambda}`: $\hat h_{j,\lambda}$ denotes a selected measurable empirical minimizer in the recursion initialized by $\hat h_{0,\lambda}=0$, with $j=1,\ldots,t$ and the empirical penalty norm evaluated on the fitting sample. In Proposition~\ref{prop:exist_lambda_iter_rdiv}, Theorem~\ref{thm:iter_rdiv_rate}, Lemma~\ref{lemma:iter_rdiv_stability}, Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, Lemma~\ref{lemma:iter_rdiv_accepted_bounds}: The loss in the recursion is $L_n(h)=\mathbb E_{n,\mathrm{fit}}[(Y-\hat{\mathcal T}h(Z))^2]$. In Proposition~\ref{prop:exist_lambda_iter_trae}, Theorem~\ref{thm:iter_trae_rate}, Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}: The loss in the recursion is the adversarial loss $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$. In Lemma~\ref{lemma:iter_rdiv_stability}, Lemma~\ref{lemma:iter_trae_accepted_stability}: The empirical penalty center remains $\hat h_{j-1,\lambda}$; the fixed-center concentration comparisons use population iterates as centers. In \cite[Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling]{foster2023orthogonal}, used in the proof of Lemma~\ref{lemma:iter_rdiv_stability}, Lemma (Recursive RDIV stability): The empirical iterates use $L_n(h)=\mathbb E_{n,\mathrm{fit}}[(Y-\hat{\mathcal T}h(Z))^2]$ and penalty center $\hat h_{j-1,\lambda}$; the fixed-center concentration comparisons use the deterministic population iterates as comparators and centers. In \cite[Lemma 9 (lem:concentration)]{bennett2023source}, used in the proof of Lemma~\ref{eq:iter_trae_strong_propagation}: The empirical iterates use $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$ and penalty center $\hat h_{j-1,\lambda}$; the fixed-center concentration comparisons use the deterministic population iterates as comparators and centers. — The recursions in \eqref{eq:iter_rdiv_estimator} and the iterative TRAE definition specify the losses and empirical centers. The fixed-center comparisons in the proofs of Lemma~\ref{lemma:iter_rdiv_stability} and Lemma~\ref{lemma:iter_trae_accepted_stability} use population centers.
- reading of the symbol `a_j`: $\{a_j\}_{j=0}^t$ is a deterministic RDIV propagation sequence initialized at $a_0=0$, with $a_j\geqslant1$ for $1\leqslant j\leqslant t$; each positive-index term depends only on $j$ and fixed population quantities and is independent of $n$ and $\lambda$. — Lemma~\ref{lemma:iter_rdiv_stability} specifies the positive-index constants and their dependencies; its proof initializes $a_0=0$ and chooses $a_j=C(1+a_{j-1})$ for fixed $C>1$.
- reading of the symbol `b_j`: $\{b_j\}_{j=0}^t$ is a nondecreasing deterministic propagation sequence with $b_0=0$ and $b_j\geqslant2$ for $1\leqslant j\leqslant t$, each positive-index term depending only on $j$ and the fixed boundedness and mean-square-continuity constants, independently of $\beta$ and $\\|w_0\\|$. — The statement of Lemma~\ref{eq:iter_trae_strong_propagation} gives the index range, monotonicity, and dependencies; its proof chooses $b_j=C(1+b_{j-1})$ with a fixed $C\geqslant2$.
- reading of the symbol `\mathcal E_n^{\mathrm R}`: $\mathcal E_n^{\mathrm R}$ is the intersection of the preliminary, fitting, and calibration events under the paper's confidence convention, with probability at least $1-\xi$, on which the fitting comparisons hold simultaneously for every $h\in\mathcal H$ and $\lambda\in\Lambda_n$, and the projected calibration comparisons hold simultaneously for the fitted candidates $\hat h_\lambda$, $\lambda\in\Lambda_n$. — Lemma~\ref{lemma:err_l_dp} states the simultaneous fitting and projected calibration comparisons. Its proof conditions first on the preliminary sample for fitting and then on preliminary and fitting samples for calibration, and concludes by combining all three event families.
- reading of the symbol `\mathcal E_{n,t}^{\mathrm{IR}}`: $\mathcal E_{n,t}^{\mathrm{IR}}$ is the common preliminary, recursive fitting, and final calibration event constructed with allocation $\eta_{n,t}=\xi/\{t(K_n+1)\}$, of probability at least $1-\xi$, on which the recursive RDIV stability comparisons hold simultaneously for $j=1,\ldots,t$ and $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$, and the projected calibration comparison holds for the final candidates on that same grid segment. — Lemma~\ref{lemma:iter_rdiv_stability} states the probability, index ranges, and common calibration comparison. Its proof unions the recursive comparisons over iteration--grid pairs and intersects their event with the final calibration event under the confidence allocation.
- reading of the symbol `\mathcal E_{n,t}^{\mathrm{IT}}`: $\mathcal E_{n,t}^{\mathrm{IT}}$ is the common iterative TRAE concentration event obtained by intersecting the requisite iteration--grid concentration events with failure-probability allocation $\eta_{n,t}=\xi/\{t(K_n+1)\}$, with probability at least $1-\xi$ under the paper's confidence convention. In Lemma~\ref{eq:iter_trae_strong_propagation}, Corollary~\ref{cor:iter_trae_discrepancy_bound}, Lemma~\ref{lemma:iter_trae_accepted_stability}: The concentration comparisons are simultaneous over the oracle-relevant grid segment $\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$, including the recursive comparisons for $j=1,\ldots,t$. — The defining sentence at line 2070 specifies the common event and allocation. The proof of Lemma~\ref{eq:iter_trae_strong_propagation} applies the comparisons at fixed iteration--grid pairs and concludes with a union bound; the confidence convention preceding Theorem~\ref{thm:common_dp} permits fixed numerical rescalings.
- reading of the symbol `b`: $b=\min\{\beta,1\}\in(0,1]$ is the truncated source exponent. — Lemma~\ref{lemma:variational_source} defines $b=\min\{\beta,1\}$. The current squared-norm concentration argument no longer introduces a normalized-class distance called $b$.
- reading of the symbol `r_\lambda`: $r_\lambda(u)=\lambda/(u+\lambda)$ for finite $\lambda>0$ and $u\geqslant0$, and $r_\lambda(\mathcal A)$ denotes this function applied to $\mathcal A$ by spectral functional calculus. — Lemma~\ref{lemma:iter_bias} defines the scalar filter and immediately uses its operator functional calculus in $r_\lambda(\mathcal A)^j$; its proof uses the spectrum of $\mathcal A$ in $[0,1]$.
- reading of the symbol `H^2`: $H^2(g_1,g_2)=\int(\sqrt{g_1(x)}-\sqrt{g_2(x)})^2\,\mu(dx)$ is defined for probability densities $g_1,g_2$ with respect to the common measure $\mu$ on $\mathcal X$; for conditional densities it is evaluated at each fixed $z$, with any averaging over $Z$ written separately. — The proof of Lemma~\ref{lemma:t_err_dp} gives the integral definition, and its statement writes $\mathbb E_Z[H^2\{\hat g(\cdot\mid Z),g_0(\cdot\mid Z)\}]$.
- reading of the symbol `\beta_h`: $\beta_h>0$ is a source exponent satisfying $h_0=(\mathcal T^*\mathcal T)^{\beta_h/2}w_h$ for some $w_h\in\tilde{\mathcal H}$; it need not be the largest exponent for which such a representation exists. — Assumption~\ref{asp:source_condition} defines a source exponent by existence of a source element, and Corollary~\ref{cor:normality} applies that condition to $h_0$ with exponent $\beta_h$.
- reading of the symbol `\beta_q`: $\beta_q>0$ is a source exponent satisfying $q_0=(\mathcal T\mathcal T^*)^{\beta_q/2}w_q$ for some $w_q\in\tilde{\mathcal Q}$; it need not be the largest exponent for which such a representation exists. — The dual equation is $\mathcal T^*q_0=a_0$; Corollary~\ref{cor:normality} applies Assumption~\ref{asp:source_condition} to this dual problem with exponent $\beta_q$.
- reading of the symbol `\boldsymbol y`: $\boldsymbol y$ denotes the vector of outcomes on the fitting block, on the outcome scale used by the fit. In Proposition~\ref{prop:rkhs_qp}: $\boldsymbol y=(Y_1,\ldots,Y_N)^\top$, with $N=n_1$ the fitting-sample size. — Section~\ref{sec:rkhs_computation} defines $\boldsymbol y=(Y_1,\ldots,Y_N)^\top$; the later RDIV computation explicitly uses standardized outcomes.
- added condition (`ass:bk-12`, `thm:iter-trae-rate`): With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}\|$ and $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}\|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$. — The localized excess-loss concentration uses this nonzero deterministic anchor; the stipulated zero-anchor and fixed-center squared-loss measurability do not supply these suprema.
- added condition (`ass:bk-12`, `lem:iter-trae-strong-propagation`): With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}\|$ and $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}\|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$. — The localized excess-loss concentration uses this nonzero deterministic anchor; the stipulated zero-anchor and fixed-center squared-loss measurability do not supply these suprema.
- added condition (`ass:bk-12`, `thm:iter-trae-discrepancy-bound`): With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}\|$ and $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}\|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$. — The localized excess-loss concentration uses this nonzero deterministic anchor; the stipulated zero-anchor and fixed-center squared-loss measurability do not supply these suprema.
- added condition (`ass:bk-12`, `lem:iter-trae-accepted-stability`): With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}\|$ and $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}\|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$. — The localized excess-loss concentration uses this nonzero deterministic anchor; the stipulated zero-anchor and fixed-center squared-loss measurability do not supply these suprema.

## Added conditions

| id | where | class | condition |
| --- | --- | --- | --- |
| `ass:pin-142` | `thm:common-dp` | V1.5 bookkeeping | On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. (pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.) |
| `ass:pin-153` | `prop:exist-lambda-deepiv` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-155` | `prop:exist-lambda-deepiv` | V1.5 bookkeeping | For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.) |
| `ass:pin-172` | `prop:exist-lambda-deepiv` | V1.5 bookkeeping | For each fixed $n\geqslant N_{\mathrm R}(B)$, all probabilistic conclusions, including initial acceptance when $h_0=0$, hold on the same event $\mathcal E_n^{\mathrm R}$ of probability at least $1-\xi$. (pins reading of prop:exist-lambda-deepiv: The proof of Proposition~\ref{prop:exist_lambda_deepiv} explicitly works throughout on $\mathcal E_n^{\mathrm R}$ and derives initial acceptance there.) |
| `ass:pin-174` | `prop:exist-lambda-deepiv` | V1.5 bookkeeping | The assumptions imported from Theorem~\ref{thm:deepiv_rate} are the general hypotheses of Proposition~\ref{prop:exist_lambda_deepiv}, without imposing $h_0=0$. (pins reading of thm:rdiv-discrepancy-bound: Theorem~\ref{thm:deepiv_rate} imports the proposition's assumptions and treats $h_0=0$ only in a separate conditional conclusion. The present proof uses Lemmas~\ref{lemma:err_l_dp} and~\ref{lemma:str_dp} for general $h_0$.) |
| `ass:pin-fix-4d0cd055` | `prop:exist-lambda-deepiv` | V1.5 bookkeeping | For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. (pins reading of def:rdiv-preliminary: stated by the repair) |
| `ass:pin-142` | `thm:deepiv-rate` | V1.5 bookkeeping | On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. (pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.) |
| `ass:pin-153` | `thm:deepiv-rate` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-155` | `thm:deepiv-rate` | V1.5 bookkeeping | For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.) |
| `ass:pin-172` | `thm:deepiv-rate` | V1.5 bookkeeping | For each fixed $n\geqslant N_{\mathrm R}(B)$, all probabilistic conclusions, including initial acceptance when $h_0=0$, hold on the same event $\mathcal E_n^{\mathrm R}$ of probability at least $1-\xi$. (pins reading of prop:exist-lambda-deepiv: The proof of Proposition~\ref{prop:exist_lambda_deepiv} explicitly works throughout on $\mathcal E_n^{\mathrm R}$ and derives initial acceptance there.) |
| `ass:pin-174` | `thm:deepiv-rate` | V1.5 bookkeeping | The assumptions imported from Theorem~\ref{thm:deepiv_rate} are the general hypotheses of Proposition~\ref{prop:exist_lambda_deepiv}, without imposing $h_0=0$. (pins reading of thm:rdiv-discrepancy-bound: Theorem~\ref{thm:deepiv_rate} imports the proposition's assumptions and treats $h_0=0$ only in a separate conditional conclusion. The present proof uses Lemmas~\ref{lemma:err_l_dp} and~\ref{lemma:str_dp} for general $h_0$.) |
| `ass:pin-190` | `thm:deepiv-rate` | V1.5 bookkeeping | For each fixed $n\geqslant N_{\mathrm R}(B)$, the displayed error bounds and, when $h_0=0$, both sharper $O(\delta_n^2)$ bounds hold jointly with probability at least $1-\xi$ over the preliminary, fitting, and calibration samples. (pins reading of thm:deepiv-rate: The proof obtains the zero-signal bounds from the probabilistic initial-acceptance conclusion of Proposition~\ref{prop:exist_lambda_deepiv} and Lemma~\ref{lemma:str_dp}; the general case uses the common RDIV event and the stated confidence convention.) |
| `ass:pin-196` | `thm:deepiv-rate` | V1.5 bookkeeping | The assumptions of \cref{thm:deepiv_rate} mean its standing hypotheses inherited from \cref{prop:exist_lambda_deepiv}; the bounds here hold for arbitrary $h_0$, without imposing $h_0=0$. (pins reading of lem:rdiv-accepted-bounds: The zero condition introduces only the last conclusion of \cref{thm:deepiv_rate}; the proof of \cref{lemma:rdiv_accepted_bounds} treats arbitrary $h_0$ using the variational source inequality.) |
| `ass:pin-fix-4d0cd055` | `thm:deepiv-rate` | V1.5 bookkeeping | For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. (pins reading of def:rdiv-preliminary: stated by the repair) |
| `ass:pin-153` | `prop:exist-lambda-iter-rdiv` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-155` | `prop:exist-lambda-iter-rdiv` | V1.5 bookkeeping | For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.) |
| `ass:pin-187` | `prop:exist-lambda-iter-rdiv` | V1.5 bookkeeping | The displayed discrepancy inequality holds on $\mathcal E_{n,t}^{\mathrm{IR}}$, with $\mathbb P(\mathcal E_{n,t}^{\mathrm{IR}})\geqslant1-\xi$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$. (pins reading of def:iter-rdiv-discrepancy-bound: The sentence introducing the display cites Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, which expressly states simultaneity on this event; Lemma~\ref{lemma:iter_rdiv_stability} specifies its probability.) |
| `ass:pin-205` | `prop:exist-lambda-iter-rdiv` | V1.5 bookkeeping | For each fixed $n\geqslant N_{\mathrm R}(B)$ and prescribed $t$, there is an event of probability at least $1-\xi$ on which the rule accepts a candidate and satisfies the stated output alternatives, including acceptance of $\lambda_0=2$ when $h_0=0$. (pins reading of prop:exist-lambda-iter-rdiv: The proof of Proposition~\ref{prop:exist_lambda_iter_rdiv} explicitly places initial acceptance on the corresponding concentration event.) |
| `ass:pin-fix-4d0cd055` | `prop:exist-lambda-iter-rdiv` | V1.5 bookkeeping | For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. (pins reading of def:rdiv-preliminary: stated by the repair) |
| `ass:pin-142` | `thm:iter-rdiv-rate` | V1.5 bookkeeping | On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. (pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.) |
| `ass:pin-153` | `thm:iter-rdiv-rate` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-155` | `thm:iter-rdiv-rate` | V1.5 bookkeeping | For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.) |
| `ass:pin-187` | `thm:iter-rdiv-rate` | V1.5 bookkeeping | The displayed discrepancy inequality holds on $\mathcal E_{n,t}^{\mathrm{IR}}$, with $\mathbb P(\mathcal E_{n,t}^{\mathrm{IR}})\geqslant1-\xi$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$. (pins reading of def:iter-rdiv-discrepancy-bound: The sentence introducing the display cites Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, which expressly states simultaneity on this event; Lemma~\ref{lemma:iter_rdiv_stability} specifies its probability.) |
| `ass:pin-205` | `thm:iter-rdiv-rate` | V1.5 bookkeeping | For each fixed $n\geqslant N_{\mathrm R}(B)$ and prescribed $t$, there is an event of probability at least $1-\xi$ on which the rule accepts a candidate and satisfies the stated output alternatives, including acceptance of $\lambda_0=2$ when $h_0=0$. (pins reading of prop:exist-lambda-iter-rdiv: The proof of Proposition~\ref{prop:exist_lambda_iter_rdiv} explicitly places initial acceptance on the corresponding concentration event.) |
| `ass:pin-fix-4d0cd055` | `thm:iter-rdiv-rate` | V1.5 bookkeeping | For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. (pins reading of def:rdiv-preliminary: stated by the repair) |
| `ass:pin-156` | `prop:exist-lambda` | V1.5 bookkeeping | The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.) |
| `ass:pin-160` | `prop:exist-lambda` | V1.5 bookkeeping | For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. (pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.) |
| `ass:pin-193` | `prop:exist-lambda` | V1.5 bookkeeping | For each fixed $n\geqslant N_{\mathrm T}(B)$, acceptance within the stated number of optimization problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_n$, with probability at least $1-\xi$ over the fitting sample. (pins reading of prop:exist-lambda: The proof begins by restricting throughout to $\mathcal E_n$, establishes terminal acceptance there, and derives both return branches on that same event.) |
| `ass:pin-206` | `prop:exist-lambda` | V1.5 bookkeeping | There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.) |
| `ass:pin-fix-958aeb7a` | `prop:exist-lambda` | V1.5 bookkeeping | The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. (pins reading of ass:critical-radius-complexity: stated by the repair) |
| `ass:pin-142` | `thm:tikhonv-rate` | V1.5 bookkeeping | On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. (pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.) |
| `ass:pin-156` | `thm:tikhonv-rate` | V1.5 bookkeeping | The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.) |
| `ass:pin-160` | `thm:tikhonv-rate` | V1.5 bookkeeping | For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. (pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.) |
| `ass:pin-193` | `thm:tikhonv-rate` | V1.5 bookkeeping | For each fixed $n\geqslant N_{\mathrm T}(B)$, acceptance within the stated number of optimization problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_n$, with probability at least $1-\xi$ over the fitting sample. (pins reading of prop:exist-lambda: The proof begins by restricting throughout to $\mathcal E_n$, establishes terminal acceptance there, and derives both return branches on that same event.) |
| `ass:pin-194` | `thm:tikhonv-rate` | V1.5 bookkeeping | With $\beta$ and $\xi$ fixed as $n\to\infty$, for each $n\geqslant N_{\mathrm T}(B)$ the displayed strong- and weak-error bounds hold jointly on a fitting-sample event of probability at least $1-\xi$. (pins reading of thm:tikhonv-rate: The proof of Theorem~\ref{thm:tikhonv_rate} works on $\mathcal E_n$, which the preceding confidence allocation constructs separately for each sample size.) |
| `ass:pin-206` | `thm:tikhonv-rate` | V1.5 bookkeeping | There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.) |
| `ass:pin-fix-958aeb7a` | `thm:tikhonv-rate` | V1.5 bookkeeping | The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. (pins reading of ass:critical-radius-complexity: stated by the repair) |
| `ass:pin-156` | `prop:exist-lambda-iter-trae` | V1.5 bookkeeping | The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.) |
| `ass:pin-160` | `prop:exist-lambda-iter-trae` | V1.5 bookkeeping | For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. (pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.) |
| `ass:pin-195` | `prop:exist-lambda-iter-trae` | V1.5 bookkeeping | For each fixed $n\geqslant N_{\mathrm{IT}}(B)$ and prescribed $t$, acceptance within the stated number of minimax problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_{n,t}^{\mathrm{IT}}$, with probability at least $1-\xi$ over the fitting sample. (pins reading of prop:exist-lambda-iter-trae: The proof works throughout on $\mathcal E_{n,t}^{\mathrm{IT}}$ and derives terminal acceptance and the first-accepted-index alternative there.) |
| `ass:pin-206` | `prop:exist-lambda-iter-trae` | V1.5 bookkeeping | There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.) |
| `ass:pin-fix-958aeb7a` | `prop:exist-lambda-iter-trae` | V1.5 bookkeeping | The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. (pins reading of ass:critical-radius-complexity: stated by the repair) |
| `ass:bk-12` | `thm:iter-trae-rate` | V1.5 bookkeeping | With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}\|$ and $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}\|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$. (The localized excess-loss concentration uses this nonzero deterministic anchor; the stipulated zero-anchor and fixed-center squared-loss measurability do not supply these suprema.) |
| `ass:pin-142` | `thm:iter-trae-rate` | V1.5 bookkeeping | On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. (pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.) |
| `ass:pin-156` | `thm:iter-trae-rate` | V1.5 bookkeeping | The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.) |
| `ass:pin-160` | `thm:iter-trae-rate` | V1.5 bookkeeping | For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. (pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.) |
| `ass:pin-188` | `thm:iter-trae-rate` | V1.5 bookkeeping | Under the assumptions of Theorem~\ref{thm:iter_trae_rate}, the displayed inequality holds simultaneously for all $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ on $\mathcal E_{n,t}^{\mathrm{IT}}$, where $\mathbb P(\mathcal E_{n,t}^{\mathrm{IT}})\geqslant1-\xi$. (pins reading of def:iter-trae-discrepancy-bound: The sentence immediately preceding the display invokes Corollary~\ref{cor:iter_trae_discrepancy_bound}, whose statement explicitly places this simultaneous bound on the common concentration event.) |
| `ass:pin-195` | `thm:iter-trae-rate` | V1.5 bookkeeping | For each fixed $n\geqslant N_{\mathrm{IT}}(B)$ and prescribed $t$, acceptance within the stated number of minimax problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_{n,t}^{\mathrm{IT}}$, with probability at least $1-\xi$ over the fitting sample. (pins reading of prop:exist-lambda-iter-trae: The proof works throughout on $\mathcal E_{n,t}^{\mathrm{IT}}$ and derives terminal acceptance and the first-accepted-index alternative there.) |
| `ass:pin-203` | `thm:iter-trae-rate` | V1.5 bookkeeping | The constant $C_t$ may depend on the prescribed $t$ and fixed population quantities, including $\beta$ and $\\|w_0\\|$, but is independent of $n$, $\lambda$, and the realized fitting observations. (pins reading of lem:iter-trae-accepted-stability: The proof applies Lemma~\ref{lemma:iter_bias}, whose interpolation inequality contributes $\\|w_0\\|^{2/(\beta+1)}$. The iterative-rate convention also explicitly permits constants depending on the prescribed iteration count and fixed population quantities.) |
| `ass:pin-204` | `thm:iter-trae-rate` | V1.5 bookkeeping | The constants hidden by $O_t(\cdot)$ may depend on the prescribed $t$, fixed population quantities including $\beta$ and $\\|w_0\\|$, and the fixed grid ratio $\rho$, but are independent of $n$ and the candidate $\lambda$. (pins reading of thm:iter-trae-rate: The text following the bounds in Theorem~\ref{thm:iter_rdiv_rate} explicitly allows dependence on the prescribed iteration count and fixed population quantities; Lemma~\ref{lemma:iter_trae_accepted_stability} supplies the comparison constants used here under the same convention.) |
| `ass:pin-206` | `thm:iter-trae-rate` | V1.5 bookkeeping | There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.) |
| `ass:pin-fix-958aeb7a` | `thm:iter-trae-rate` | V1.5 bookkeeping | The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. (pins reading of ass:critical-radius-complexity: stated by the repair) |
| `ass:pin-142` | `thm:normality` | V1.5 bookkeeping | On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. (pins reading of thm:common-dp: The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.) |
| `ass:pin-156` | `thm:normality` | V1.5 bookkeeping | The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.) |
| `ass:pin-160` | `thm:normality` | V1.5 bookkeeping | For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. (pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.) |
| `ass:pin-193` | `thm:normality` | V1.5 bookkeeping | For each fixed $n\geqslant N_{\mathrm T}(B)$, acceptance within the stated number of optimization problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_n$, with probability at least $1-\xi$ over the fitting sample. (pins reading of prop:exist-lambda: The proof begins by restricting throughout to $\mathcal E_n$, establishes terminal acceptance there, and derives both return branches on that same event.) |
| `ass:pin-194` | `thm:normality` | V1.5 bookkeeping | With $\beta$ and $\xi$ fixed as $n\to\infty$, for each $n\geqslant N_{\mathrm T}(B)$ the displayed strong- and weak-error bounds hold jointly on a fitting-sample event of probability at least $1-\xi$. (pins reading of thm:tikhonv-rate: The proof of Theorem~\ref{thm:tikhonv_rate} works on $\mathcal E_n$, which the preceding confidence allocation constructs separately for each sample size.) |
| `ass:pin-206` | `thm:normality` | V1.5 bookkeeping | There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.) |
| `ass:pin-fix-958aeb7a` | `thm:normality` | V1.5 bookkeeping | The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. (pins reading of ass:critical-radius-complexity: stated by the repair) |
| `ass:pin-153` | `lem:t-err-dp` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-fix-4d0cd055` | `lem:t-err-dp` | V1.5 bookkeeping | For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. (pins reading of def:rdiv-preliminary: stated by the repair) |
| `ass:pin-153` | `lem:err-l-dp` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-155` | `lem:err-l-dp` | V1.5 bookkeeping | For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.) |
| `ass:pin-fix-4d0cd055` | `lem:err-l-dp` | V1.5 bookkeeping | For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. (pins reading of def:rdiv-preliminary: stated by the repair) |
| `ass:pin-153` | `lem:err-r-dp` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-155` | `lem:err-r-dp` | V1.5 bookkeeping | For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.) |
| `ass:pin-fix-4d0cd055` | `lem:err-r-dp` | V1.5 bookkeeping | For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. (pins reading of def:rdiv-preliminary: stated by the repair) |
| `ass:pin-153` | `lem:str-dp` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-155` | `lem:str-dp` | V1.5 bookkeeping | For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.) |
| `ass:pin-fix-4d0cd055` | `lem:str-dp` | V1.5 bookkeeping | For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. (pins reading of def:rdiv-preliminary: stated by the repair) |
| `ass:pin-153` | `thm:rdiv-discrepancy-bound` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-155` | `thm:rdiv-discrepancy-bound` | V1.5 bookkeeping | For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.) |
| `ass:pin-174` | `thm:rdiv-discrepancy-bound` | V1.5 bookkeeping | The assumptions imported from Theorem~\ref{thm:deepiv_rate} are the general hypotheses of Proposition~\ref{prop:exist_lambda_deepiv}, without imposing $h_0=0$. (pins reading of thm:rdiv-discrepancy-bound: Theorem~\ref{thm:deepiv_rate} imports the proposition's assumptions and treats $h_0=0$ only in a separate conditional conclusion. The present proof uses Lemmas~\ref{lemma:err_l_dp} and~\ref{lemma:str_dp} for general $h_0$.) |
| `ass:pin-fix-4d0cd055` | `thm:rdiv-discrepancy-bound` | V1.5 bookkeeping | For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. (pins reading of def:rdiv-preliminary: stated by the repair) |
| `ass:pin-153` | `lem:rdiv-accepted-bounds` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-155` | `lem:rdiv-accepted-bounds` | V1.5 bookkeeping | For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.) |
| `ass:pin-196` | `lem:rdiv-accepted-bounds` | V1.5 bookkeeping | The assumptions of \cref{thm:deepiv_rate} mean its standing hypotheses inherited from \cref{prop:exist_lambda_deepiv}; the bounds here hold for arbitrary $h_0$, without imposing $h_0=0$. (pins reading of lem:rdiv-accepted-bounds: The zero condition introduces only the last conclusion of \cref{thm:deepiv_rate}; the proof of \cref{lemma:rdiv_accepted_bounds} treats arbitrary $h_0$ using the variational source inequality.) |
| `ass:pin-fix-4d0cd055` | `lem:rdiv-accepted-bounds` | V1.5 bookkeeping | For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. (pins reading of def:rdiv-preliminary: stated by the repair) |
| `ass:pin-153` | `lem:iter-rdiv-squared-loss-concentration-lemma-14` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-155` | `lem:iter-rdiv-squared-loss-concentration-lemma-14` | V1.5 bookkeeping | For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.) |
| `ass:pin-153` | `thm:iter-rdiv-discrepancy-bound` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-155` | `thm:iter-rdiv-discrepancy-bound` | V1.5 bookkeeping | For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.) |
| `ass:pin-187` | `thm:iter-rdiv-discrepancy-bound` | V1.5 bookkeeping | The displayed discrepancy inequality holds on $\mathcal E_{n,t}^{\mathrm{IR}}$, with $\mathbb P(\mathcal E_{n,t}^{\mathrm{IR}})\geqslant1-\xi$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$. (pins reading of def:iter-rdiv-discrepancy-bound: The sentence introducing the display cites Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, which expressly states simultaneity on this event; Lemma~\ref{lemma:iter_rdiv_stability} specifies its probability.) |
| `ass:pin-fix-4d0cd055` | `thm:iter-rdiv-discrepancy-bound` | V1.5 bookkeeping | For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. (pins reading of def:rdiv-preliminary: stated by the repair) |
| `ass:pin-153` | `lem:iter-rdiv-accepted-bounds` | V1.5 bookkeeping | For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (pins reading of ass:cr-deepiv-measurability: Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.) |
| `ass:pin-155` | `lem:iter-rdiv-accepted-bounds` | V1.5 bookkeeping | For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:rdiv-minima: The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.) |
| `ass:pin-fix-4d0cd055` | `lem:iter-rdiv-accepted-bounds` | V1.5 bookkeeping | For every preliminary sample size $n\geqslant1$, the empirical log likelihood $g\mapsto\mathbb E_{n,\mathrm{pre}}[\log g(X\mid Z)]$ attains its maximum over $\mathcal G$ almost surely, and $\hat g\in\mathcal G$ is a measurable maximizer selected using only $I_{\mathrm{pre}}$. (pins reading of def:rdiv-preliminary: stated by the repair) |
| `ass:pin-206` | `lem:concentrate-l` | V1.5 bookkeeping | There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.) |
| `ass:pin-fix-958aeb7a` | `lem:concentrate-l` | V1.5 bookkeeping | The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. (pins reading of ass:critical-radius-complexity: stated by the repair) |
| `ass:pin-156` | `lem:strong-convexity` | V1.5 bookkeeping | The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.) |
| `ass:pin-206` | `lem:strong-convexity` | V1.5 bookkeeping | There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.) |
| `ass:pin-fix-958aeb7a` | `lem:strong-convexity` | V1.5 bookkeeping | The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. (pins reading of ass:critical-radius-complexity: stated by the repair) |
| `ass:pin-156` | `thm:trae-discrepancy-bound` | V1.5 bookkeeping | The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.) |
| `ass:pin-160` | `thm:trae-discrepancy-bound` | V1.5 bookkeeping | For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. (pins reading of def:trae-discrepancy-bound: The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.) |
| `ass:pin-206` | `thm:trae-discrepancy-bound` | V1.5 bookkeeping | There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.) |
| `ass:pin-fix-958aeb7a` | `thm:trae-discrepancy-bound` | V1.5 bookkeeping | The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. (pins reading of ass:critical-radius-complexity: stated by the repair) |
| `ass:pin-156` | `lem:trae-accepted-bounds` | V1.5 bookkeeping | The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.) |
| `ass:pin-206` | `lem:trae-accepted-bounds` | V1.5 bookkeeping | There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.) |
| `ass:pin-fix-958aeb7a` | `lem:trae-accepted-bounds` | V1.5 bookkeeping | The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. (pins reading of ass:critical-radius-complexity: stated by the repair) |
| `ass:bk-12` | `lem:iter-trae-strong-propagation` | V1.5 bookkeeping | With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}\|$ and $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}\|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$. (The localized excess-loss concentration uses this nonzero deterministic anchor; the stipulated zero-anchor and fixed-center squared-loss measurability do not supply these suprema.) |
| `ass:pin-156` | `lem:iter-trae-strong-propagation` | V1.5 bookkeeping | The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.) |
| `ass:pin-206` | `lem:iter-trae-strong-propagation` | V1.5 bookkeeping | There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.) |
| `ass:pin-fix-958aeb7a` | `lem:iter-trae-strong-propagation` | V1.5 bookkeeping | The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. (pins reading of ass:critical-radius-complexity: stated by the repair) |
| `ass:bk-12` | `thm:iter-trae-discrepancy-bound` | V1.5 bookkeeping | With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}\|$ and $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}\|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$. (The localized excess-loss concentration uses this nonzero deterministic anchor; the stipulated zero-anchor and fixed-center squared-loss measurability do not supply these suprema.) |
| `ass:pin-156` | `thm:iter-trae-discrepancy-bound` | V1.5 bookkeeping | The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.) |
| `ass:pin-188` | `thm:iter-trae-discrepancy-bound` | V1.5 bookkeeping | Under the assumptions of Theorem~\ref{thm:iter_trae_rate}, the displayed inequality holds simultaneously for all $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ on $\mathcal E_{n,t}^{\mathrm{IT}}$, where $\mathbb P(\mathcal E_{n,t}^{\mathrm{IT}})\geqslant1-\xi$. (pins reading of def:iter-trae-discrepancy-bound: The sentence immediately preceding the display invokes Corollary~\ref{cor:iter_trae_discrepancy_bound}, whose statement explicitly places this simultaneous bound on the common concentration event.) |
| `ass:pin-206` | `thm:iter-trae-discrepancy-bound` | V1.5 bookkeeping | There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.) |
| `ass:pin-fix-958aeb7a` | `thm:iter-trae-discrepancy-bound` | V1.5 bookkeeping | The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. (pins reading of ass:critical-radius-complexity: stated by the repair) |
| `ass:bk-12` | `lem:iter-trae-accepted-stability` | V1.5 bookkeeping | With $\mathcal U$ and $\ell_B$ as in Assumption~\ref{asp:critical_radius}, for every fixed iteration--grid pair $(j,\lambda)$ under consideration and every $r>0$, both $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|(\mathbb E_n-\mathbb E)\{\ell_B(u(W))-\ell_B(u_{j,\lambda}^*(W))\}\|$ and $\sup_{u\in\mathcal U,\,\\|u-u_{j,\lambda}^*\\|_{L_2(W;\mathbb R^3)}\leqslant r}\|n^{-1}\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(u_{j,\lambda}^*(W_i))\}\|$ are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, where $f_j^*=\mathcal T(h_0-h_{j,\lambda}^*)$ and $u_{j,\lambda}^*(W)=(m(W;f_j^*),h_{j,\lambda}^*(X)f_j^*(Z),f_j^*(Z))$. (The localized excess-loss concentration uses this nonzero deterministic anchor; the stipulated zero-anchor and fixed-center squared-loss measurability do not supply these suprema.) |
| `ass:pin-156` | `lem:iter-trae-accepted-stability` | V1.5 bookkeeping | The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (pins reading of ass:trae-minima: The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.) |
| `ass:pin-203` | `lem:iter-trae-accepted-stability` | V1.5 bookkeeping | The constant $C_t$ may depend on the prescribed $t$ and fixed population quantities, including $\beta$ and $\\|w_0\\|$, but is independent of $n$, $\lambda$, and the realized fitting observations. (pins reading of lem:iter-trae-accepted-stability: The proof applies Lemma~\ref{lemma:iter_bias}, whose interpolation inequality contributes $\\|w_0\\|^{2/(\beta+1)}$. The iterative-rate convention also explicitly permits constants depending on the prescribed iteration count and fixed population quantities.) |
| `ass:pin-206` | `lem:iter-trae-accepted-stability` | V1.5 bookkeeping | There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (pins reading of ass:mean-sq-continuity: Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.) |
| `ass:pin-fix-958aeb7a` | `lem:iter-trae-accepted-stability` | V1.5 bookkeeping | The lower bound denoted by $\Omega$ means that there exists a constant $c>0$, independent of $n$ and $\xi$, such that for every integer $n\geqslant1$ and every $\xi\in(0,1)$, $\tilde\delta_n(\xi)\geqslant c\sqrt{\{\log\log(en)+\log(1/\xi)\}/n}$. (pins reading of ass:critical-radius-complexity: stated by the repair) |
| `ass:filler:27:prop:exist-lambda-iter-rdiv:14:ass:bk-owner-4` | `prop:exist-lambda-iter-rdiv` | faithful-refinement | Explicitly threaded the already approved IterRDIVFixedCenterSupremumVersion condition into both iterative selection theorem signatures. |
| `ass:filler:27:prop:exist-lambda-iter-rdiv:14:ass:bk-owner-4:1` | `prop:exist-lambda-iter-rdiv` | faithful-refinement | Threaded the existing IterRDIVFixedCenterSupremumVersion condition through both iterative RDIV selection theorems. |

## Conditions assumed

### Cited results

| id | used by | citation | locator | checked against |
| --- | --- | --- | --- | --- |
| `lem:rdiv-localized-concentration-lemma-14` | `prop:exist-lambda-deepiv` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-mle-localized-concentration-li2024` | `prop:exist-lambda-deepiv` | li2024regularized | Localized Concentration (source label lem:local-concen) | source supplied by the user at discovery/cited_sources/li2024regularized.txt |
| `lem:rdiv-localized-concentration-lemma-14` | `thm:deepiv-rate` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-mle-localized-concentration-li2024` | `thm:deepiv-rate` | li2024regularized | Localized Concentration (source label lem:local-concen) | source supplied by the user at discovery/cited_sources/li2024regularized.txt |
| `lem:iter-rdiv-fixed-center-concentration-lemma-14` | `prop:exist-lambda-iter-rdiv` | foster2023orthogonal | Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:iter-rdiv-squared-loss-concentration-lemma-14` | `prop:exist-lambda-iter-rdiv` | foster2023orthogonal | Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-localized-concentration-lemma-14` | `prop:exist-lambda-iter-rdiv` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-mle-localized-concentration-li2024` | `prop:exist-lambda-iter-rdiv` | li2024regularized | Localized Concentration (source label lem:local-concen) | source supplied by the user at discovery/cited_sources/li2024regularized.txt |
| `lem:iter-rdiv-fixed-center-concentration-lemma-14` | `thm:iter-rdiv-rate` | foster2023orthogonal | Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:iter-rdiv-squared-loss-concentration-lemma-14` | `thm:iter-rdiv-rate` | foster2023orthogonal | Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-localized-concentration-lemma-14` | `thm:iter-rdiv-rate` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-mle-localized-concentration-li2024` | `thm:iter-rdiv-rate` | li2024regularized | Localized Concentration (source label lem:local-concen) | source supplied by the user at discovery/cited_sources/li2024regularized.txt |
| `lem:trae-localized-concentration-lemma-9` | `prop:exist-lambda` | bennett2023source | Lemma 9 | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-squared-norm-lemma-14` | `prop:exist-lambda` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:trae-localized-concentration-lemma-9` | `thm:tikhonv-rate` | bennett2023source | Lemma 9 | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-squared-norm-lemma-14` | `thm:tikhonv-rate` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:trae-localized-concentration-lemma-9` | `prop:exist-lambda-iter-trae` | bennett2023source | Lemma 9 | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-squared-norm-lemma-14` | `prop:exist-lambda-iter-trae` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:iter-trae-anchored-concentration-lemma-9` | `thm:iter-trae-rate` | bennett2023source | Lemma 9 (lem:concentration) | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:iter-trae-fixed-center-squared-loss-lemma-9` | `thm:iter-trae-rate` | bennett2023source | Lemma 9 (lem:concentration) | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-localized-concentration-lemma-9` | `thm:iter-trae-rate` | bennett2023source | Lemma 9 | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-squared-norm-lemma-14` | `thm:iter-trae-rate` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:trae-localized-concentration-lemma-9` | `thm:normality` | bennett2023source | Lemma 9 | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-squared-norm-lemma-14` | `thm:normality` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-mle-localized-concentration-li2024` | `lem:t-err-dp` | li2024regularized | Localized Concentration (source label lem:local-concen) | source supplied by the user at discovery/cited_sources/li2024regularized.txt |
| `lem:rdiv-localized-concentration-lemma-14` | `lem:err-l-dp` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-mle-localized-concentration-li2024` | `lem:err-l-dp` | li2024regularized | Localized Concentration (source label lem:local-concen) | source supplied by the user at discovery/cited_sources/li2024regularized.txt |
| `lem:rdiv-localized-concentration-lemma-14` | `lem:err-r-dp` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-mle-localized-concentration-li2024` | `lem:err-r-dp` | li2024regularized | Localized Concentration (source label lem:local-concen) | source supplied by the user at discovery/cited_sources/li2024regularized.txt |
| `lem:rdiv-localized-concentration-lemma-14` | `lem:str-dp` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-mle-localized-concentration-li2024` | `lem:str-dp` | li2024regularized | Localized Concentration (source label lem:local-concen) | source supplied by the user at discovery/cited_sources/li2024regularized.txt |
| `lem:rdiv-localized-concentration-lemma-14` | `thm:rdiv-discrepancy-bound` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-mle-localized-concentration-li2024` | `thm:rdiv-discrepancy-bound` | li2024regularized | Localized Concentration (source label lem:local-concen) | source supplied by the user at discovery/cited_sources/li2024regularized.txt |
| `lem:rdiv-localized-concentration-lemma-14` | `lem:rdiv-accepted-bounds` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-mle-localized-concentration-li2024` | `lem:rdiv-accepted-bounds` | li2024regularized | Localized Concentration (source label lem:local-concen) | source supplied by the user at discovery/cited_sources/li2024regularized.txt |
| `lem:iter-rdiv-squared-loss-concentration-lemma-14` | `lem:iter-rdiv-squared-loss-concentration-lemma-14` | foster2023orthogonal | Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:iter-rdiv-fixed-center-concentration-lemma-14` | `thm:iter-rdiv-discrepancy-bound` | foster2023orthogonal | Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:iter-rdiv-squared-loss-concentration-lemma-14` | `thm:iter-rdiv-discrepancy-bound` | foster2023orthogonal | Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-localized-concentration-lemma-14` | `thm:iter-rdiv-discrepancy-bound` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-mle-localized-concentration-li2024` | `thm:iter-rdiv-discrepancy-bound` | li2024regularized | Localized Concentration (source label lem:local-concen) | source supplied by the user at discovery/cited_sources/li2024regularized.txt |
| `lem:iter-rdiv-fixed-center-concentration-lemma-14` | `lem:iter-rdiv-accepted-bounds` | foster2023orthogonal | Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:iter-rdiv-squared-loss-concentration-lemma-14` | `lem:iter-rdiv-accepted-bounds` | foster2023orthogonal | Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-localized-concentration-lemma-14` | `lem:iter-rdiv-accepted-bounds` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:rdiv-mle-localized-concentration-li2024` | `lem:iter-rdiv-accepted-bounds` | li2024regularized | Localized Concentration (source label lem:local-concen) | source supplied by the user at discovery/cited_sources/li2024regularized.txt |
| `lem:trae-localized-concentration-lemma-9` | `lem:concentrate-l` | bennett2023source | Lemma 9 | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-localized-concentration-lemma-9` | `lem:strong-convexity` | bennett2023source | Lemma 9 | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-squared-norm-lemma-14` | `lem:strong-convexity` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:trae-localized-concentration-lemma-9` | `thm:trae-discrepancy-bound` | bennett2023source | Lemma 9 | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-squared-norm-lemma-14` | `thm:trae-discrepancy-bound` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:trae-localized-concentration-lemma-9` | `lem:trae-accepted-bounds` | bennett2023source | Lemma 9 | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-squared-norm-lemma-14` | `lem:trae-accepted-bounds` | foster2023orthogonal | Lemma 14 | source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt |
| `lem:iter-trae-anchored-concentration-lemma-9` | `lem:iter-trae-strong-propagation` | bennett2023source | Lemma 9 (lem:concentration) | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:iter-trae-fixed-center-squared-loss-lemma-9` | `lem:iter-trae-strong-propagation` | bennett2023source | Lemma 9 (lem:concentration) | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-localized-concentration-lemma-9` | `lem:iter-trae-strong-propagation` | bennett2023source | Lemma 9 | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:iter-trae-anchored-concentration-lemma-9` | `thm:iter-trae-discrepancy-bound` | bennett2023source | Lemma 9 (lem:concentration) | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:iter-trae-fixed-center-squared-loss-lemma-9` | `thm:iter-trae-discrepancy-bound` | bennett2023source | Lemma 9 (lem:concentration) | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-localized-concentration-lemma-9` | `thm:iter-trae-discrepancy-bound` | bennett2023source | Lemma 9 | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:iter-trae-anchored-concentration-lemma-9` | `lem:iter-trae-accepted-stability` | bennett2023source | Lemma 9 (lem:concentration) | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:iter-trae-fixed-center-squared-loss-lemma-9` | `lem:iter-trae-accepted-stability` | bennett2023source | Lemma 9 (lem:concentration) | source supplied by the user at discovery/cited_sources/bennett2023source.txt |
| `lem:trae-localized-concentration-lemma-9` | `lem:iter-trae-accepted-stability` | bennett2023source | Lemma 9 | source supplied by the user at discovery/cited_sources/bennett2023source.txt |

`lem:rdiv-localized-concentration-lemma-14` (foster2023orthogonal, Lemma 14), invoked in the proof of `lem:err-l-dp`:

What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.LipschitzVectorLocalizedConcentration` (`Helpers/RDIVSetup.lean:896`)

```lean
noncomputable def LipschitzVectorLocalizedConcentration
    (V : IVWorld 𝒲 𝒳 𝒵) (P : Measure Ω)
    (sample : AnalysisSample V P) (split : SampleSplit) : Sort 0 :=
  IsProbabilityMeasure P →
  ∀ (mu : Measure 𝒳) (G : Set (𝒳 → 𝒵 → ℝ)) (g₀ : 𝒳 → 𝒵 → ℝ)
    (gHat : ℕ → Ω → 𝒳 → 𝒵 → ℝ) (support : Set (𝒳 × 𝒵))
    (B c₀ CG ρ : ℝ),
    RDIVSetting V mu g₀ → RangeCondition V → H0MemHypothesisClass V →
    DensityRealizable V mu G g₀ → DensityBounds G g₀ support c₀ CG →
    OutcomeBounded V B → HypothesisClassBounded V support B →
    MLEMaximizerSelected V P sample split G gHat →
    RDIVClassesMeasurable V P sample split mu G g₀ gHat →
    RDIVFittingSupremumVersion V P sample split mu gHat ρ →
    ∃ C > 0, ∀ (n : ℕ) (ζ r lam : ℝ), 0 < n →
      ζ ∈ Set.Ioo (0 : ℝ) 1 → lam ∈ regGrid ρ n → lam ∈ Set.Ioc (0 : ℝ) 2 →
      confidenceFloor n ζ ≤ r →
      IsCriticalRadiusBound (V.law.map V.xOf) (primalVersion V '' V.H) n r →
      (∀ g ∈ G, IsCriticalRadiusBound (V.law.map V.zOf)
        (densityOperator mu g '' (primalVersion V '' V.H)) n r) →
      ∀ᵐ preliminary ∂P,
        1 - ζ ≤ P.real {ω | ∀ h ∈ V.H,
          |(conditionalRisk V mu gHat n preliminary lam h -
              conditionalRisk V mu gHat n preliminary lam (minNormSolution V)) -
            (empiricalRisk V P sample split mu (fun _ _ => gHat n preliminary) n ω lam h -
              empiricalRisk V P sample split mu (fun _ _ => gHat n preliminary) n ω lam
                (minNormSolution V))| ≤
          C * (r * inputL2Norm (V.law.map V.zOf)
            (densityOperator mu (gHat n preliminary)
              (primalVersion V h - primalVersion V (minNormSolution V))) +
            r * ‖h - minNormSolution V‖ + r ^ 2)}
```

The passage of the proof:

> For each fixed $\lambda\in\Lambda_n$, apply the localized concentration inequality in \citet[Lemma~14]{foster2023orthogonal} to the two-coordinate class
> \[
> h\longmapsto\{\hat{\mathcal T}h(Z),h(X)\}
> \]
> at the fixed comparator $h_0$ and to the loss
> $\phi_\lambda(y,u,v)=(y-u)^2+\lambda v^2$. This loss is uniformly Lipschitz on the bounded range because $\lambda\leqslant2$. The critical-radius bounds in \cref{asp:cr_deepiv} give
> \[
> \left|\{\widehat R^\lambda(h)-\widehat R^\lambda(h_0)\}
> -\{R_n^\lambda(h)-R_n^\lambda(h_0)\}\right|
> \leqslant C\left[\delta_{n,\mathcal H}
> \{\|\hat{\mathcal T}(h-h_0)\|+\|h-h_0\|\}
> +\delta_{n,\mathcal H}^2\right].
> \]

`lem:rdiv-mle-localized-concentration-li2024` (li2024regularized, Localized Concentration (source label lem:local-concen)), invoked in the proof of `lem:t-err-dp`:

What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.LiLocalizedConcentration` (`Helpers/RDIVSetup.lean:308`)

```lean
noncomputable def LiLocalizedConcentration : Sort 0 :=
  ∀ (mu : Measure 𝒳) (G : Set (𝒳 → 𝒵 → ℝ)) (g₀ : 𝒳 → 𝒵 → ℝ)
    (support : Set (𝒳 × 𝒵)) (c₀ CG : ℝ),
    RDIVSetting V mu g₀ → DensityRealizable V mu G g₀ → DensityBounds G g₀ support c₀ CG →
    LocalizedProcessesMeasurable (V.law.map (fun w => (V.xOf w, V.zOf w)))
      (likelihoodTransformClass G g₀) →
    ∀ (δG : ℕ → ℝ → ℝ), DensityCriticalRadiusBound V G g₀ δG →
    (∃ c > 0, ∀ n : ℕ, 0 < n → ∀ ξ ∈ Set.Ioo (0 : ℝ) 1,
      c * confidenceFloor n ξ ≤ δG n ξ) →
    ∃ A > 0, ∀ n : ℕ, 0 < n → ∀ ξ ∈ Set.Ioo (0 : ℝ) 1,
      ∃ E : Set (Fin n → 𝒳 × 𝒵), MeasurableSet E ∧
        1 - ξ / 2 ≤ (Measure.pi (fun _ : Fin n =>
          V.law.map (fun w => (V.xOf w, V.zOf w)))).real E ∧
        ∀ x ∈ E, ∀ g ∈ G,
          let f := fun y : 𝒳 × 𝒵 =>
            Real.sqrt ((g y.1 y.2 + g₀ y.1 y.2) / (2 * g₀ y.1 y.2))
          |(n : ℝ)⁻¹ * (∑ i, -Real.log (f (x i))) -
            (∫ y, -Real.log (f y) ∂V.law.map (fun w => (V.xOf w, V.zOf w)))| ≤
            A * (δG n (ξ / 2) *
              inputL2Norm (V.law.map (fun w => (V.xOf w, V.zOf w)))
                (f - fun _ => 1) + δG n (ξ / 2) ^ 2)
```

The passage of the proof:

> We establish the density bound directly using the Localized Concentration lemma of \citet{li2024regularized}. Write $P=\mathbb E_{X,Z}$ and $P_n=\mathbb E_{n,\mathrm{pre}}$, and define
> \[
> f_g(x,z)=\sqrt{\frac{g(x\mid z)+g_0(x\mid z)}{2g_0(x\mid z)}},\qquad
> \mathcal U=\{f_g-1:g\in\mathcal G\}.
> \]
> Realizability gives $f_{g_0}=1$, so $0\in\mathcal U$. The density bounds give
> \[
> 2^{-1/2}\leqslant f_g\leqslant M,\qquad
> M=\sqrt{\frac{C_G/c_0+1}{2}}.
> \]
> Thus $\mathcal U$ is uniformly bounded by a fixed constant. Define the globally Lipschitz loss
> \[
> \ell(v)=-\log\bigl(\max\{1+v,2^{-1/2}\}\bigr).
> \]
> The maximum map is $1$-Lipschitz, and the derivative of $-\log t$ has absolute value at most $\sqrt2$ for $t\geqslant2^{-1/2}$. Hence $\ell$ is $\sqrt2$-Lipschitz, $\ell(0)=0$, and $\ell(f_g-1)=-\log f_g$.
> 
> We verify the localized complexity condition for this application. Put $\mathcal S=\operatorname{star}(\mathcal G_0-\mathcal G_0)$ and choose a deterministic critical radius $d\leqslant\delta_{n,\mathcal G}$ as supplied by Assumption~\ref{asp:cr_deepiv}. By the critical-radius definition, $\mathfrak R_n(d;\mathcal S)\leqslant d^2$. For $s\geqslant d$, star-shapedness implies that $(d/s)v\in\mathcal S$ and $\|(d/s)v\|_{L_2(X,Z)}\leqslant d$ whenever $v\in\mathcal S$ and $\|v\|_{L_2(X,Z)}\leqslant s$. Scaling inside the Rademacher supremum therefore gives
> \[
> \mathfrak R_n(s;\mathcal S)
> \leqslant\frac{s}{d}\mathfrak R_n(d;\mathcal S)
> \leqslant sd.
> \]
> Since $1\in\mathcal G_0$, we have $\operatorname{star}(\mathcal U)\subseteq\mathcal S$. Taking $s=\delta_{n,\mathcal G}$ yields
> \[
> \mathfrak R_n\bigl(\delta_{n,\mathcal G};\operatorname{star}(\mathcal U)\bigr)
> \leqslant\delta_{n,\mathcal G}d
> \leqslant\delta_{n,\mathcal G}^2.
> \]
> Apply the cited localized concentration lemma with input $(X,Z)$, one coordinate, class $\mathcal U$, fixed comparator $0$, loss $\ell$, and failure probability $\xi/2$. The measurability and logarithmic radius floor are supplied by Assumption~\ref{asp:cr_deepiv}, with the confidence evaluation specified in the proof. Consequently, for a fixed constant $A>0$, with probability at least $1-\xi/2$,
> \[
> \left|(P_n-P)(-\log f_g)\right|
> \leqslant A\left\{\delta_{n,\mathcal G}\|f_g-1\|_{L_2(X,Z)}+\delta_{n,\mathcal G}^2\right\},
> \qquad g\in\mathcal G.
> \]

`lem:iter-rdiv-fixed-center-concentration-lemma-14` (foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling), invoked in the proof of `lem:iter-rdiv-stability`:

What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.IterRDIVFixedCenterLocalizedConcentration` (`Helpers/IterRDIVConcentration.lean:128`)

```lean
noncomputable def IterRDIVFixedCenterLocalizedConcentration : Sort 0 :=
  ∀ (mu : Measure 𝒳) (G : Set (𝒳 → 𝒵 → ℝ)) (g₀ : 𝒳 → 𝒵 → ℝ)
    (support : Set (𝒳 × 𝒵)) (B c₀ CG : ℝ),
    RDIVSetting V mu g₀ → DensityRealizable V mu G g₀ →
    DensityBounds G g₀ support c₀ CG → OutcomeBounded V B →
    HypothesisClassBounded V support B →
    ∃ C > 0, ∃ c₁ > 0, ∃ c₂ > 0,
    ∀ (P : Measure Ω) (_hProbability : IsProbabilityMeasure P)
      (sample : AnalysisSample V P) (split : SampleSplit)
      (gHat : ℕ → Ω → 𝒳 → 𝒵 → ℝ) (ρ : ℝ),
    MLEMaximizerSelected V P sample split G gHat →
    RDIVClassesMeasurable V P sample split mu G g₀ gHat →
    IterRDIVFixedCenterSupremumVersion V P sample split mu gHat ρ →
    ∀ (n t j : ℕ) (lam r : ℝ) (sourceRadius : ℝ)
      (_hSourceScope : gridValue ρ (terminalIndex ρ n) ≤ sourceRadius ∧
        oraclePoint ρ n sourceRadius ≤ lam), 0 < n → 1 ≤ j → j ≤ t →
    lam ∈ regGrid ρ n → lam ∈ Set.Ioc (0 : ℝ) 2 →
    populationIterate V lam j ∈ V.H →
    (populationIterate V lam (j - 1) ∈ V.H ∨ j = 1) →
    0 < r → 1 < 2 * c₂ * n →
    8 * Real.log (41 * Real.log (2 * c₂ * n)) / (c₂ * n) ≤ r ^ 2 →
    iterRdivAnchoredRadius (V.law.map V.xOf) (primalVersion V '' V.H)
      (primalVersion V (populationIterate V lam j)) n r →
    (∀ g ∈ G, iterRdivAnchoredRadius (V.law.map V.zOf)
      (densityOperator mu g '' (primalVersion V '' V.H))
      (densityOperator mu g (primalVersion V (populationIterate V lam j))) n r) →
    ∀ᵐ a ∂P, ∃ E : Set (Fin n → 𝒲), MeasurableSet E ∧
      1 - 2 * c₁ * Real.exp (-c₂ * n * r ^ 2 / 8) ≤
        (Measure.pi (fun _ : Fin n => V.law)).real E ∧
      ∀ ws ∈ E, ∀ h ∈ V.H,
        let increment := iterRdivFixedCenterLoss V mu (gHat n a) lam j h -
          iterRdivFixedCenterLoss V mu (gHat n a) lam j (populationIterate V lam j)
        |(n : ℝ)⁻¹ * (∑ i, increment (ws i)) - ∫ w, increment w ∂V.law| ≤
          C * r * (inputL2Norm (V.law.map V.zOf)
            (densityOperator mu (gHat n a) (primalVersion V h -
              primalVersion V (populationIterate V lam j))) +
            ‖h - populationIterate V lam j‖ + r)
```

The passage of the proof:

> The fitting loss is Lipschitz in the two coordinates
> $\{\hat{\mathcal T}h(Z),h(X)\}$, uniformly over these bounded centers. This is the fixed-comparator fitting application of \citet[Lemma~14]{foster2023orthogonal}, with comparator $h_{j,\lambda}^*$ and penalty center $h_{j-1,\lambda}^*$.

`lem:iter-rdiv-squared-loss-concentration-lemma-14` (foster2023orthogonal, Lemma 14 (lem:lipschitz_wainwright); signed-tail evidence: lem:prob_localized and lem:peeling), invoked in the proof of `lem:iter-rdiv-stability`:

What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.IterRDIVSquaredLossLocalizedConcentration` (`Helpers/IterRDIVConcentration.lean:187`)

```lean
noncomputable def IterRDIVSquaredLossLocalizedConcentration : Sort 0 :=
  ∀ (mu : Measure 𝒳) (G : Set (𝒳 → 𝒵 → ℝ)) (g₀ : 𝒳 → 𝒵 → ℝ)
    (support : Set (𝒳 × 𝒵)) (B c₀ CG : ℝ),
    RDIVSetting V mu g₀ → DensityRealizable V mu G g₀ →
    DensityBounds G g₀ support c₀ CG → HypothesisClassBounded V support B →
    ∃ C > 0, ∃ c₁ > 0, ∃ c₂ > 0,
    ∀ (P : Measure Ω) (_hProbability : IsProbabilityMeasure P)
      (sample : AnalysisSample V P) (split : SampleSplit)
      (gHat : ℕ → Ω → 𝒳 → 𝒵 → ℝ) (ρ : ℝ),
    RDIVClassesMeasurable V P sample split mu G g₀ gHat →
    IterRDIVFixedCenterSupremumVersion V P sample split mu gHat ρ →
    ∀ (n t k : ℕ) (lam r : ℝ) (sourceRadius : ℝ)
      (_hSourceScope : gridValue ρ (terminalIndex ρ n) ≤ sourceRadius ∧
        oraclePoint ρ n sourceRadius ≤ lam), 0 < n → 1 ≤ k → k ≤ t →
    lam ∈ regGrid ρ n → lam ∈ Set.Ioc (0 : ℝ) 2 →
    populationIterate V lam k ∈ V.H →
    0 < r → 1 < 2 * c₂ * n →
    4 * Real.log (41 * Real.log (2 * c₂ * n)) / (c₂ * n) ≤ r ^ 2 →
    iterRdivAnchoredRadius (V.law.map V.xOf) (primalVersion V '' V.H)
      (primalVersion V (populationIterate V lam k)) n r →
    ∀ᵐ a ∂P, ∃ E : Set (Fin n → 𝒲), MeasurableSet E ∧
      1 - 2 * c₁ * Real.exp (-c₂ * n * r ^ 2 / 4) ≤
        (Measure.pi (fun _ : Fin n => V.law)).real E ∧
      ∀ ws ∈ E, ∀ h ∈ V.H,
        |(n : ℝ)⁻¹ * (∑ i, (primalVersion V h (V.xOf (ws i)) -
          primalVersion V (populationIterate V lam k) (V.xOf (ws i))) ^ 2) -
          ‖h - populationIterate V lam k‖ ^ 2| ≤
          C * r * (‖h - populationIterate V lam k‖ + r)
```

The passage of the proof:

> Apply the localized squared-loss comparison to
> $(h-h_{k,\lambda}^*)^2$, whose comparator and center are fixed. Here the localized squared-loss comparison is the scalar application of \citet[Lemma~14]{foster2023orthogonal} at the fixed comparator $h_{k,\lambda}^*$.

`lem:trae-localized-concentration-lemma-9` (bennett2023source, Lemma 9), invoked in the proof of `lem:concentrate-l`:

What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.TRAELocalizedConcentration` (`Helpers/TRAESetup.lean:845`)

```lean
noncomputable def TRAELocalizedConcentration : Sort 0 :=
  ∀ (B : ℝ) (tildeδ : ℕ → ℝ → ℝ),
    RangeCondition V → ClosenessCondition V → MeanSquareContinuity V →
    TRAEBoundedness V B → TRAECommonNullSetBoundedness V B → TRAECriticalRadiusBound V tildeδ →
    ∃ C > 0, ∀ (δ : ℕ → ℝ) (allocation : ℕ → ℝ),
    TRAEScaleCalibration δ tildeδ allocation →
    ∀ (n : ℕ), 0 < n → allocation n ∈ Set.Ioo (0 : ℝ) 1 →
    ∃ E : Set (Fin n → 𝒲), MeasurableSet E ∧
      1 - allocation n ≤ (Measure.pi (fun _ : Fin n => V.law)).real E ∧
      ∀ x ∈ E, ∀ h ∈ V.H, ∀ f ∈ V.F,
        let u := fun w => ![V.m f w,
          primalVersion V h (V.xOf w) * dualVersion V f (V.zOf w),
          dualVersion V f (V.zOf w)]
        |(n : ℝ)⁻¹ * (∑ i, traeClippedLoss B (x i) (u (x i))) -
          (∫ w, traeClippedLoss B w (u w) ∂V.law)| ≤
          C * (δ n * (∑ i : Fin 3, inputL2Norm V.law (fun w => u w i)) + δ n ^ 2)
```

The passage of the proof:

> The localized concentration bound used in \citet[Lemma~9]{bennett2023source}
> gives
> \begin{align*}
> L( h) &=\sup_{f\in \mathcal{F}}\mathbb{E}\left[ 2( m( W;f) -f( Z) h( X)) -f^{2}( Z)\right] \\
> & =\mathbb{E}\left[ 2( m( W;f_{h}) -f_{h}( Z) h( X)) -f_{h}^{2}( Z)\right]\\
>  & \leqslant \mathbb{E}_{n}\left[ 2( m( W;f_{h}) -f_{h}( Z) h( X)) -f_{h}^{2}( Z)\right] +O\left( \delta _{n}^{2} +\delta _{n} \| f_{h} \| \right)\\
>  & \leqslant \sup_{f\in \mathcal{F}}\mathbb{E}_{n}\left[ 2( m( W;f) -f( Z) h( X)) -f^{2}( Z)\right] +O\left( \delta _{n}^{2} +\delta _{n} \| f_{h} \| \right)\\
>  & \leqslant L_{n}( h) +O\left( \delta _{n}^{2} +\delta _{n} \| f_{h} \| \right) .
> \end{align*}
> Since $\|f_h\|^2=L(h)$, Young's inequality gives
> \[
> L( h) \leqslant 2L_{n}( h) +O\left( \delta _{n}^{2}\right) .
> \]
> Similarly, 
> \begin{align*}
> L_{n}( h)&  =\sup_{f\in \mathcal{F}}\mathbb{E}_{n}\left[ 2( m( W;f) -f( Z) h( X)) -f^{2}( Z)\right] \\
>  & \leqslant \sup_{f\in\mathcal F}\left\{\mathbb E\left[2(m(W;f)-f(Z)h(X))-f(Z)^2\right]+C\left(\delta_n^2+\delta_n\|f\|\right)\right\}\\
>  & \leqslant \sup_{f\in\widetilde{\mathcal Q}}\mathbb{E}\left[ 2( m( W;f) -f( Z) h( X)) -\frac{4}{5} f^{2}( Z)\right] +O\left( \delta _{n}^{2}\right)\\
>  & =\frac{5}{4}\|\mathcal T(h_0-h)\|^2+O\left( \delta _{n}^{2}\right)\\
>  & =\frac{5}{4} L( h) +O\left( \delta _{n}^{2}\right) .
> \end{align*}

`lem:trae-squared-norm-lemma-14` (foster2023orthogonal, Lemma 14), invoked in the proof of `lem:strong-convexity`:

What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.ScalarLipschitzLocalizedConcentration` (`Helpers/TRAESetup.lean:937`)

```lean
noncomputable def ScalarLipschitzLocalizedConcentration
    (V : IVWorld 𝒲 𝒳 𝒵) (P : Measure Ω)
    (sample : AnalysisSample V P) (split : SampleSplit) : Sort 0 :=
  IsProbabilityMeasure P →
  ∀ (B : ℝ) (tildeδ : ℕ → ℝ → ℝ),
  TRAEBoundedness V B → TRAECommonNullSetBoundedness V B → TRAECriticalRadiusBound V tildeδ →
  ∃ C > 0, ∀ (δ allocation : ℕ → ℝ),
  H0MemHypothesisClass V → TRAEScaleCalibration δ tildeδ allocation →
  ∀ n : ℕ, 0 < n → allocation n ∈ Set.Ioo (0 : ℝ) 1 →
    ∃ E : Set Ω, MeasurableSet E ∧ 1 - allocation n ≤ P.real E ∧
      ∀ ω ∈ E, ∀ h ∈ V.H,
        |(primalEmpSq V P sample split n ω h -
            primalEmpSq V P sample split n ω (minNormSolution V)) -
          (‖h‖ ^ 2 - ‖minNormSolution V‖ ^ 2)| ≤
            C * (δ n * ‖h - minNormSolution V‖ + δ n ^ 2)
```

The passage of the proof:

> The localized squared-norm inequality
> in \citet[Lemma~14]{foster2023orthogonal} gives, uniformly over
> $h\in\mathcal H$,
> \[
> \left|\{\|h\|_{2,n}^2-\|h_0\|_{2,n}^2\}
> -\{\|h\|^2-\|h_0\|^2\}\right|
> \leqslant C\{\delta_n\|h-h_0\|+\delta_n^2\}.
> \]

`lem:iter-trae-anchored-concentration-lemma-9` (bennett2023source, Lemma 9 (lem:concentration)), invoked in the proof of `lem:iter-trae-strong-propagation`:

What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.IterTRAEAnchoredLocalizedConcentration` (`Helpers/IterTRAEConcentration.lean:92`)

```lean
noncomputable def IterTRAEAnchoredLocalizedConcentration : Sort 0 :=
  ∀ B cfloor : ℝ, TRAEBoundedness V B → TRAECommonNullSetBoundedness V B → 0 < cfloor →
  ∃ C > 0, ∀ n : ℕ, 3 ≤ n → ∀ ζ ∈ Set.Ioo (0 : ℝ) 1,
  ∀ r : ℝ, cfloor * Real.sqrt
    ((3 * Real.log (Real.log n) + Real.log (1 / ζ)) / n) ≤ r →
  0 < r →
  (∀ F ∈ ({starClass (momentImageClass V),
      starClass (productObservationClass V), starClass (criticObservationClass V)} :
        Set (Set (𝒲 → ℝ))),
    IsCriticalRadiusBound V.law F n r) →
  ∀ (ρ c b β : ℝ) (δ : ℕ → ℝ) (t : ℕ),
  GridDomain ρ → 0 < c → 2 ≤ b → 0 < β → 1 ≤ t →
  let radius := c * (b * δ n ^ 2) ^ (1 / (qualificationExponent β t + 1))
  ∀ anchor : 𝒲 → Fin 3 → ℝ,
  (anchor = 0 ∨ ∃ j : ℕ, 1 ≤ j ∧ j ≤ t ∧
    ∃ lam ∈ regGrid ρ n,
      (∃ k ≤ terminalIndex ρ n, gridValue ρ k ≤ radius) ∧
      oraclePoint ρ n radius ≤ lam ∧ lam ≤ 2 ∧ 0 < lam ∧
      populationIterate V lam j ∈ V.H ∧
      projCondExpOp V (minNormSolution V - populationIterate V lam j) ∈ V.F ∧
      anchor = iterTraePopulationAnchor V j lam) →
  anchor ∈ traeLossClass V →
  AELocalizedSupremaMeasurableAt V.law (traeLossClass V) anchor (traeClippedLoss B) n →
  ∃ E : Set (Fin n → 𝒲), MeasurableSet E ∧
    1 - ζ ≤ (Measure.pi (fun _ : Fin n => V.law)).real E ∧
    ∀ x ∈ E, ∀ u ∈ traeLossClass V,
      |(n : ℝ)⁻¹ * (∑ i, (traeClippedLoss B (x i) (u (x i)) -
          traeClippedLoss B (x i) (anchor (x i)))) -
        ∫ w, (traeClippedLoss B w (u w) - traeClippedLoss B w (anchor w)) ∂V.law| ≤
      C * (r * (∑ i : Fin 3, inputL2Norm V.law (fun w => u w i - anchor w i)) + r ^ 2)
```

The passage of the proof:

> Apply the localized concentration inequality of \citet[Lemma~9]{bennett2023source} to the product and moment-image classes in \cref{asp:critical_radius}, and apply its bounded Lipschitz version to the squared test functions.

`lem:iter-trae-fixed-center-squared-loss-lemma-9` (bennett2023source, Lemma 9 (lem:concentration)), invoked in the proof of `lem:iter-trae-strong-propagation`:

What Lean assumes for it: `Papers.TsAdaptiveEstimationInference.IterTRAEFixedCenterSquaredLossConcentration` (`Helpers/IterTRAEConcentration.lean:214`)

```lean
noncomputable def IterTRAEFixedCenterSquaredLossConcentration : Sort 0 :=
  ∀ B cfloor : ℝ, TRAEBoundedness V B → TRAECommonNullSetBoundedness V B → 0 < cfloor →
  ∃ C > 0, ∀ n : ℕ, 3 ≤ n → ∀ ζ ∈ Set.Ioo (0 : ℝ) 1,
  ∀ r : ℝ, cfloor * Real.sqrt
    ((Real.log (Real.log n) + Real.log (1 / ζ)) / n) ≤ r → 0 < r →
  ∀ (F : Set (𝒲 → ℝ)) (anchor center : 𝒲 → ℝ),
  IterTRAEScalarSourceApplication V n F anchor center →
  anchor ∈ F →
  (∀ᵐ w ∂V.law, ∀ f ∈ F, |f w| ≤ 2 * max 1 B) →
  (∀ w (u v : Fin 1 → ℝ),
    |iterTraeSourceSquare B center w u - iterTraeSourceSquare B center w v| ≤
      |u 0 - v 0|) →
  expectedRademacher V.law {f ∈ starClass F | eLpNorm f 2 V.law ≤ ENNReal.ofReal r} n ≤
    ENNReal.ofReal (r ^ 2) →
  AELocalizedSupremaMeasurableAt V.law (iterTraeScalarClass F)
    (fun w => fun _ : Fin 1 => anchor w) (iterTraeSourceSquare B center) n →
  ∃ E : Set (Fin n → 𝒲), MeasurableSet E ∧
    1 - ζ ≤ (Measure.pi (fun _ : Fin n => V.law)).real E ∧
    ∀ x ∈ E, ∀ f ∈ F,
      |(n : ℝ)⁻¹ * (∑ i,
          (iterTraeSourceSquare B center (x i) (fun _ => f (x i)) -
           iterTraeSourceSquare B center (x i) (fun _ => anchor (x i)))) -
        ∫ w, (iterTraeSourceSquare B center w (fun _ => f w) -
          iterTraeSourceSquare B center w (fun _ => anchor w)) ∂V.law| ≤
      C * (r * inputL2Norm V.law (fun w => f w - anchor w) + r ^ 2)
```

The passage of the proof:

> For the fixed-center squared-loss comparisons, apply the same localized concentration inequality of \citet[Lemma~9]{bennett2023source} to the bounded clipped losses corresponding to $(h-h_{j-1,\lambda}^*)^2$ at comparator $h_{j,\lambda}^*$ and $(h-h_{k,\lambda}^*)^2$ at comparator $h_{k,\lambda}^*$, for the fixed iteration--grid pairs under consideration.

### Deferred substrate gates

None: every substrate this run needed was built and proved.

## Unresolved / notes

- run-level `ass:cr-deepiv-measurability`: [note, V1.5] pinned reading: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.)
- run-level `ass:mean-sq-continuity`: [note, V1.5] pinned reading: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.)
- run-level `ass:rdiv-minima`: [note, V1.5] pinned reading: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.)
- run-level `ass:trae-minima`: [note, V1.5] pinned reading: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.)
- run-level `def:iter-rdiv-discrepancy-bound`: [note, V1.5] pinned reading: The displayed discrepancy inequality holds on $\mathcal E_{n,t}^{\mathrm{IR}}$, with $\mathbb P(\mathcal E_{n,t}^{\mathrm{IR}})\geqslant1-\xi$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$. (The sentence introducing the display cites Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, which expressly states simultaneity on this event; Lemma~\ref{lemma:iter_rdiv_stability} specifies its probability.)
- run-level `def:trae-discrepancy-bound`: [note, V1.5] pinned reading: For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. (The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.)
- run-level `def:iter-trae-discrepancy-bound`: [note, V1.5] pinned reading: Under the assumptions of Theorem~\ref{thm:iter_trae_rate}, the displayed inequality holds simultaneously for all $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ on $\mathcal E_{n,t}^{\mathrm{IT}}$, where $\mathbb P(\mathcal E_{n,t}^{\mathrm{IT}})\geqslant1-\xi$. (The sentence immediately preceding the display invokes Corollary~\ref{cor:iter_trae_discrepancy_bound}, whose statement explicitly places this simultaneous bound on the common concentration event.)
- run-level `symbol q`: [note, V1.5] symbol table corrected: type "test function or scalar bias exponent, according to local context", space "$\tilde{\mathcal Q}\cup(1,\infty)$", def "The letter $q$ denotes a function in the moment and dual-estimation formulas, and is separately reused for the scalar exponent $q>1$ in $B_n(\lambda)=C_b\lambda^q$." (The table records only a test function, but the generic discrepancy theorem and four rate proofs also use $q$ as a scalar exponent. The paper explicitly distinguishes these local uses, and the dual optimization restricts its function-valued candidate to $\mathcal Q$.)
- run-level `symbol \mathcal F`: [note, V1.5] symbol table corrected: type "function class", space "A class of real-valued functions on the input domain under consideration.", def "$\mathcal F$ denotes the function class under consideration; its domain and specialization are supplied by the surrounding definition." (The test-class-only entry misses the generic complexity notation and full-RKHS specialization. The existing correction remains valid; the additional clause identifies the adversarial specialization in the separately registered iterative TRAE concentration steps.)
- run-level `symbol \mathcal S`: [note, V1.5] symbol table corrected: type "dual test class or locally defined star-shaped difference class", space "$\mathcal S\subseteq L_2(X)\ \text{or}\ \mathcal S\subseteq L_2(X,Z)$", def "In the dual empirical adversarial loss, $\mathcal S\subseteq\widetilde{\mathcal H}$ is the test class. In the normalized squared-loss arguments, $\mathcal S=\operatorname{star}(\mathcal H-\mathcal H)$. In the likelihood concentration argument, $\mathcal S=\operatorname{star}(\mathcal G_0-\mathcal G_0)$." (The existing local class definitions remain valid. The likelihood-transform difference-class reading must also be attached directly to the registered likelihood concentration lemma.)
- run-level `symbol \hat q`: [note, V1.5] symbol table corrected: def "$\hat q=\arg\min_{q\in\mathcal Q}\{L_n^{\text{dual}}(q)+\lambda_{\mathrm{dp}}^{\text{dual}}\mathbb E_n[q(Z)^2]\}$, where $L_n^{\text{dual}}(q)=\sup_{s\in\mathcal S}\mathbb E_n[2\tilde m(W;s)-2q(Z)s(X)-s(X)^2]$ and $\lambda_{\mathrm{dp}}^{\text{dual}}$ is selected with the discrepancy principle." (The recorded label omits the defining dual optimization, so the existing definition fix remains necessary. The inference scope additionally specifies jointly measurable versions of the selected dual evaluations.)
- run-level `symbol \lambda`: [note, V1.5] symbol table corrected: space "$[0,\infty]$" (The computational zero-penalty convention still requires the existing space correction. The fixed-candidate qualification now belongs to the propagation proof; the other readings remain valid.)
- run-level `symbol \ell`: [note, V1.5] symbol table corrected: type "locally bound residual-bracketing constant, iteration index, or loss function", space "$(1,\infty)\cup\mathbb N\cup\{f:\mathbb R\to\mathbb R\}$", def "In the classical discrepancy principle, $\ell\in(1,\infty)$ is the prescribed residual-bracketing factor. In $M_{\leqslant j}=\max_{1\leqslant\ell\leqslant j}\{\lambda,\\|f_\ell^*\\|_\infty^2\}$, it is an integer index ranging from $1$ to $j$. In the normalized squared-loss argument, $\ell(v)=M^2\bigl(\max\{-1,\min\{v,1\}\}\bigr)^2$, where $M=\max\{1,B\}$. In the likelihood concentration argument, $\ell(v)=-\log\bigl(\max\{1+v,2^{-1/2}\}\bigr)$." (The existing meanings remain valid. The logarithmic-loss reading must also apply directly to the registered likelihood concentration lemma, which does not inherit a scope attached only to its consuming lemma.)
- run-level `symbol A_n`: [note, V1.5] symbol table corrected: def "$A_n(\lambda)$ is the noise level supplied to the discrepancy rule, whose acceptance threshold is $\tfrac32A_n(\lambda)$." (The recorded prescribed-bound definition excludes the experimental noise-floor meaning. The existing general reading and theoretical formulas remain valid; four clauses require retagging.)
- run-level `symbol \gamma`: [note, V1.5] symbol table corrected: type "locally bound scalar parameter or critic coefficient vector", space "$\mathbb R\cup\bigcup_{N\geqslant1}\mathbb R^N$", def "In $\operatorname{star}(\mathcal F)=\{\gamma f:\gamma\in[0,1],f\in\mathcal F\}$, $\gamma$ is the scalar multiplier; elsewhere it is a locally specified positive exponent or the coefficient vector in $f_\gamma(\cdot)=\sum_{j=1}^N\gamma_j k_{\mathcal F}(Z_j,\cdot)$." (The recorded star-hull meaning is correct locally but omits distinct uses that change the symbol's type and domain. The paper also uses a fixed oracle exponent, an arbitrary positive spectral exponent, and an unrestricted RKHS critic coefficient vector.)
- run-level `symbol L_n`: [note, V1.5] symbol table corrected: def "For RDIV, $L_n(h)=\mathbb E_{n,\mathrm{fit}}[(Y-\hat{\mathcal T}h(Z))^2]$. For TRAE, $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$, where $\mathbb E_n$ averages over its fitting sample." (The table records only the RDIV squared loss, whereas TRAE reuses the symbol for an adversarial supremum. The existing fix remains valid; the additional scopes supply these meanings to the separately registered iterative concentration steps.)
- run-level `symbol L`: [note, V1.5] symbol table corrected: type "method-specific population loss", def "RDIV uses $L(h)=\mathbb E[(Y-\mathcal T h(Z))^2]=\sigma^2_{Y\mid Z}+\\|\mathcal T(h-h_0)\\|^2$, and TRAE uses $L(h)=\sup_{f\in\mathcal F}\mathbb E[2m(W;f)-2h(X)f(Z)-f(Z)^2]$." (The table gives only the RDIV population squared loss. The existing method-specific correction remains valid; the additional clause specifies the adversarial meaning in the separately registered iterative TRAE concentration steps.)
- run-level `symbol \delta_n`: [note, V1.5] symbol table corrected: def "A prescribed deterministic calibrated statistical-error bound: for RDIV, $\delta_n\geqslant\delta_{n,\mathcal G}+\delta_{n,\mathcal H}+\delta_{n,\xi}$ with $\delta_n+\nu_n=o(1)$; for TRAE, $\delta_n\geqslant\max\{\tilde\delta_n(\eta_n),n^{-1/2}\}$ with $\delta_n=o(1)$. Iterative procedures use $\eta_{n,t}=\xi/\{t(K_n+1)\}$ in their concentration bounds. For inference it is a common deterministic bound satisfying the TRAE requirements for both primal and dual problems at confidence levels $\xi_n/2$." (The original entry covers only RDIV, whereas the symbol also denotes the distinct TRAE scale and the common inference bound. The existing correction and surviving scopes remain valid; only obsolete cited-step scopes are removed.)
- run-level `symbol \tilde\delta_n`: [note, V1.5] symbol table corrected: type "confidence-indexed critical-radius upper-bound sequence", space "$(0,1)\to[0,\infty)$", def "$\tilde\delta_n(\xi)=\Omega\!\left(\sqrt{(\log\log(en)+\log(1/\xi))/n}\right)$ is a common upper bound on the critical radii of $\operatorname{star}(\mathcal H\cdotp\mathcal F)$, $\operatorname{star}(m\circ\mathcal F)$, $\operatorname{star}(\mathcal F)$, $\operatorname{star}(\mathcal H)$, and $\operatorname{star}(\mathcal H-\mathcal H)$." (The recorded scalar space suppresses the confidence argument, and the vague definition omits the five classes and the stipulated lower order bound. In particular, calibration uses its value at the allocated failure probability.)
- run-level `symbol \alpha`: [note, V1.5] symbol table corrected: type "locally defined interpolation exponent or RKHS coefficient vector", space "$(0,1/2]\cup\mathbb R^N$", def "The notation $\alpha$ denotes the interpolation exponent $\min\{\beta,1\}/(1+\min\{\beta,1\})$ in the source and error bounds, and a coefficient vector in $\mathbb R^N$ in the RKHS quadratic program, with $h_\alpha(x)=\sum_{j=1}^N\alpha_j k_{\mathcal H}(X_j,x)$." (The scalar-only entry omits the RKHS coefficient-vector meaning. Both meanings and the existing RKHS scope remain valid; only the obsolete normality tag is removed from the scalar scope.)
- run-level `symbol \hat h_{j,\lambda}`: [note, V1.5] symbol table corrected: def "Initialize $\hat h_{0,\lambda}=0$ and, for $j=1,\ldots,t$, define $\hat h_{j,\lambda}=\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\\|h-\hat h_{j-1,\lambda}\\|_{2,n}^2\}$, where $L_n$ is the fitting loss of the respective RDIV or TRAE procedure." (The recorded RDIV-only description omits the TRAE recursion. The existing correction remains valid; the additional clauses specify the method and distinguish the empirical recursion from the population centers used in the separately registered concentration comparisons.)
- run-level `symbol a_j`: [note, V1.5] symbol table corrected: space "$[0,\infty)$, with $a_0=0$ and $a_j\geqslant1$ for $j=1,\ldots,t$" (The recorded space excludes the explicitly used initialization $a_0=0$. The paper fixes the positive-index constants uniformly over sample size and penalty. Proposition~\ref{prop:rkhs_qp} uses coefficient symbols $\alpha_j$, not the RDIV constants $a_j$, so it introduces no second meaning.)
- run-level `symbol \lambda_{\mathrm{dp},t}^{\mathrm R}`: [note, V1.5] symbol table corrected: def "The output of Algorithm~\ref{alg:discrepancy} applied to $\hat h_{t,\lambda}$ with the projected calibration statistic in equation~\eqref{eq:rdiv_projected_loss} and noise level $A_{n,t}^{\mathrm R}(\lambda)=s_{n,t}^{\mathrm R}/\lambda+\nu_n^2$: return $2\rho^{k^*}$ for the smallest $k^*\in\{0,\ldots,K_n\}$ satisfying $D_n(\hat h_{t,2\rho^{k^*}})\leqslant\tfrac32 A_{n,t}^{\mathrm R}(2\rho^{k^*})$; if no such index exists, return $2\rho^{K_n}$." (The recorded label omits the selection rule. The parameter is the first accepted grid point using the projected RDIV discrepancy and iterative RDIV threshold, with terminal fallback.)
- run-level `symbol \lambda_{\mathrm{dp},t}^{\mathrm T}`: [note, V1.5] symbol table corrected: def "The output of Algorithm~\ref{alg:discrepancy} applied to $\hat h_{t,\lambda}$ with $D_n=L_n$ and noise level $A_{n,t}^{\mathrm T}(\lambda)=\delta_n^2+\min\{\lambda,s_{n,t}^{\mathrm T}/\lambda\}$: return $2\rho^{k^*}$ for the smallest $k^*\in\{0,\ldots,K_n\}$ satisfying $L_n(\hat h_{t,2\rho^{k^*}})\leqslant\tfrac32 A_{n,t}^{\mathrm T}(2\rho^{k^*})$; if no such index exists, return $2\rho^{K_n}$." (The recorded label does not specify which parameter is selected. The definition requires the first accepted point in decreasing grid order, with the iterative TRAE threshold and terminal fallback.)
- run-level `symbol \gamma_h`: [note, V1.5] symbol table corrected: def "An exponent $\gamma_h>0$ such that $\mathbb E[(\tilde m(W;h_1)-\tilde m(W;h_2))^2]\leqslant C\\|h_1-h_2\\|^{\gamma_h}$ for all $h_1,h_2\in\mathcal H$, where the norm is the population $L_2(X)$ norm and $C$ is a uniform constant." (The label omits the precise score-continuity inequality. The coefficient vector $\widehat\gamma_h$ in the RKHS proof is a distinct symbol and does not redefine $\gamma_h$.)
- run-level `symbol \gamma_q`: [note, V1.5] symbol table corrected: def "An exponent $\gamma_q>0$ such that $\mathbb E[(m(W;q_1)-m(W;q_2))^2]\leqslant C\\|q_1-q_2\\|^{\gamma_q}$ for all $q_1,q_2\in\mathcal Q$, where the norm is the population $L_2(Z)$ norm and $C$ is a uniform constant." (The label alone omits the defining inequality. The exponent applies to the norm on the right, while the score difference on the left is squared.)
- run-level `symbol \xi_n`: [note, V1.5] symbol table corrected: space "$\{(\xi_n)_{n\geqslant1}:\xi_n>0,\ \xi_n\downarrow0\}$" (The deterministic sequence decreasing to zero is correctly described, but the recorded space requires every term to be strictly below one. The paper explicitly allows $\xi_n=1/\log(en)$, whose first term equals one; the tuning rules use confidence levels $\xi_n/2$.)
- run-level `symbol P_\eta`: [note, V1.5] symbol table corrected: type "ridge-stabilized critic smoothing matrix" (The displayed formula is correct, but the matrix is generally not an idempotent projection: its eigenvalues are $d/(d+N\eta)$ for eigenvalues $d$ of $K_{\mathcal F}$. Proposition~\ref{prop:rkhs_qp} refers back to the same definition.)
- run-level `symbol \widehat\alpha_\lambda`: [note, V1.5] symbol table corrected: def "$\widehat\alpha_\lambda=(K_{\mathcal H}P_\eta K_{\mathcal H}+\lambda K_{\mathcal H}^{2})^{\dagger}K_{\mathcal H}P_\eta\boldsymbol y$, where $A^\dagger$ denotes the Moore--Penrose pseudoinverse of $A$." (The recorded entry identifies the coefficient vector by a program without recording its defining formula. The paper specifies the minimum-Euclidean-norm vector explicitly; the subsequent computation on a retained spectral span does not replace the unrestricted vector in Proposition~\ref{prop:rkhs_qp}.)
- run-level `lem:rdiv-localized-concentration-lemma-14`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt
- run-level `lem:trae-localized-concentration-lemma-9`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/bennett2023source.txt
- run-level `lem:trae-squared-norm-lemma-14`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt
- run-level `lem:iter-trae-anchored-concentration-lemma-9`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/bennett2023source.txt
- run-level `lem:iter-trae-fixed-center-squared-loss-lemma-9`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/bennett2023source.txt
- run-level `lem:rdiv-mle-localized-concentration-li2024`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/li2024regularized.txt
- run-level `lem:iter-rdiv-fixed-center-concentration-lemma-14`: [note, V1.5] cited source checked: source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt
- run-level `ass:critical-radius-boundedness`: [refuted, V1.5] [F3 reviewer] The localized moment/product/critic loss processes needed to instantiate Bennett Lemma 9 are not covered by ass:critical-radius-boundedness: its accepted measurability clause covers only localized squared-primal losses. The measurable empirical loss and selected minimizer (main.tex:572) likewise do not establish these localized centered and Rademacher suprema. The specific local loss is ell_B(u)=2*u_1-2*u_2-clip_B(u_3)^2 over the product of star(m o F), star(H*F), and star(F), with zero comparator and vector-L2 localization. Require only measurability of its centered and signed localized suprema for each positive n and radius, including the exchanged dual instance where used. This is empirical-process well-definedness bookkeeping; it supplies no deviation inequality or rate. Preserve the existing squared-loss condition and all earlier repairs. Do not add countability, separability, convexity, compactness, arbitrary-Lipschitz-loss regularity or a supplied TRAE concentration conclusion. The minimal explicit condition must enter Assumption asp:critical_radius and the corrected paper through V1.5, and all instantiation and probability bounds remain local. See Helpers/TRAESetup.lean:595-610 and Helpers/TRAEDeviations.lean:29-47. The original filler named the Lean helper traeLocalizedDeviationBoth rather than a patchable core node; the orchestrator corrects only that routing error. This hypothesis of the paper is used by prop:exist-lambda, thm:tikhonv-rate, prop:exist-lambda-iter-trae, thm:iter-trae-rate, thm:normality, lem:concentrate-l, lem:strong-convexity, thm:trae-discrepancy-bound, lem:trae-accepted-bounds, lem:iter-trae-strong-propagation, thm:iter-trae-discrepancy-bound, lem:iter-trae-accepted-stability, lem:trae-localized-concentration-lemma-9, lem:trae-squared-norm-lemma-14. 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: The localized moment/product/critic loss processes needed to instantiate Bennett Lemma 9 are not covered by ass:critical-radius-boundedness: its accepted measurability clause covers only localized squared-primal losses. The measurable empirical loss and selected minimizer (main.tex:572) likewise do not establish these localized centered and Rademacher suprema. The specific local loss is ell_B(u)=2*u_1-2*u_2-clip_B(u_3)^2 over the product of star(m o F), star(H*F), and star(F), with zero comparator and vector-L2 localization. Require only measurability of its centered and signed localized suprema for each positive n and radius, including the exchanged dual instance where used. This is empirical-process well-definedness bookkeeping; it supplies no deviation inequality or rate. Preserve the existing squared-loss condition and all earlier repairs. Do not add countability, separability, convexity, compactness, arbitrary-Lipschitz-loss regularity or a supplied TRAE concentration conclusion. The minimal explicit condition must enter Assumption asp:critical_radius and the corrected paper through V1.5, and all instantiation and probability bounds remain local. See Helpers/TRAESetup.lean:595-610 and Helpers/TRAEDeviations.lean:29-47. The original filler named the Lean helper traeLocalizedDeviationBoth rather than a patchable core node; the orchestrator corrects only that routing error. minimal patch (confirmed): corrected statement: For a fixed $B<\infty$, suppose that $\|m(W;f)\|$, $\|h(X)\|$, and $\|f(Z)\|$ are a.s. bounded by $B$, uniformly over $ h\in \mathcal{H} ,f\in \mathcal{F}$. Suppose in addition that, for every integer $n\geqslant1$ and every $r>0$, the localized squared-loss suprema \[ \sup_{\substack{h\in\mathcal H\\ \\|h-h_0\\|\leqslant r}} \left\|(\mathbb E_n-\mathbb E)[h(X)^2-h_0(X)^2]\right\| \quad\text{and}\quad \sup_{\substack{h\in\mathcal H\\ \\|h-h_0\\|\leqslant r}} \left\|\frac1n\sum_{i=1}^n\epsilon_i\{h(X_i)^2-h_0(X_i)^2\}\right\| \] are measurable random variables under the fitting-sample law and its product with the independent Rademacher law, respectively. Regard the functions in $\mathcal H\cdotp\mathcal F$ and $\mathcal F$ as functions of $W$ through $(X,Z)$ and $Z$, respectively, and put \[ \mathcal U=\operatorname{star}(m\circ\mathcal F) \times\operatorname{star}(\mathcal H\cdotp\mathcal F) \times\operatorname{star}(\mathcal F), \qquad \\|u\\|_{L_2(W;\mathbb R^3)}^2 =\mathbb E\!\left[\sum_{j=1}^3u_j(W)^2\right]. \] Define the loss with zero comparator by \[ \operatorname{clip}_B(a)=\max\{-B,\min\{a,B\}\}, \qquad \ell_B(u)=2u_1-2u_2-\operatorname{clip}_B(u_3)^2, \qquad \ell_B(0)=0. \] Suppose also that, for every integer $n\geqslant1$ and every $r>0$, the localized loss suprema \[ \sup_{\substack{u\in\mathcal U\\ \\|u\\|_{L_2(W;\mathbb R^3)}\leqslant r}} \left\|(\mathbb E_n-\mathbb E)[\ell_B(u(W))-\ell_B(0)]\right\| \quad\text{and}\quad \sup_{\substack{u\in\mathcal U\\ \\|u\\|_{L_2(W;\mathbb R^3)}\leqslant r}} \left\|\frac1n\sum_{i=1}^n\epsilon_i\{\ell_B(u(W_i))-\ell_B(0)\}\right\| \] are measurable random variables under the fitting-sample law and its product with the independent Rademacher law, respectively. Whenever the primal and dual roles are exchanged, suppose that these same two localized loss suprema for the exchanged problem are measurable under its fitting-sample law and its product with the independent Rademacher law, respectively. For iterative TRAE, suppose in addition that the following fixed-population-center squared-loss processes are measurable. For every integer $n\geqslant1$, prescribed integer $t\geqslant1$, grid point $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ for which $\lambda_{\mathrm{IT}}^\circ$ is defined, radius $r>0$, and $j=1,\ldots,t$, put \[ a_{j,\lambda}(h;X) =\{h(X)-h_{j-1,\lambda}^*(X)\}^2 -\{h_{j,\lambda}^*(X)-h_{j-1,\lambda}^*(X)\}^2, \qquad d_{j,\lambda}(h;X)=\{h(X)-h_{j,\lambda}^*(X)\}^2, \] where the population iterates are defined by \eqref{eq:iter_population}. For each $v\in\{a_{j,\lambda},d_{j,\lambda}\}$, suppose that both \[ \sup_{\substack{h\in\mathcal H\\\\|h-h_{j,\lambda}^*\\|\leqslant r}} \left\|(\mathbb E_n-\mathbb E)v(h;X)\right\| \quad\text{and}\quad \sup_{\substack{h\in\mathcal H\\\\|h-h_{j,\lambda}^*\\|\leqslant r}} \left\|\frac1n\sum_{i=1}^n\epsilon_i v(h;X_i)\right\| \] are measurable under the fitting-sample law and its product with the independent Rademacher law, respectively, using the same function representatives as the empirical penalties. The zero-iterate comparison uses $\hat h_{0,\lambda}=h_{0,\lambda}^*=0$. · proof change: The proof as printed proves the repaired statement; no step changes.
- run-level `lem:trae-squared-norm-lemma-14`: [refuted, V1.5] [F2.5 reviewer] [statement-wrong] The carrier assumes class-level squared-loss process measurability absent from the frozen TRAE setting; if this regularity is necessary, it requires an explicit referee-approved condition rather than a carrier-only premise. witness: [statement-wrong] The carrier assumes class-level squared-loss process measurability absent from the frozen TRAE setting; if this regularity is necessary, it requires an explicit referee-approved condition rather than a carrier-only premise. minimal patch (confirmed): proof change: We use the squared-loss measurability condition in Assumption~\ref{asp:critical_radius}. Put $M=\max\{1,B\}$, $\mathcal U=\{h/M:h\in\mathcal H\}$, and $u_0=h_0/M$. Assumption~\ref{asp:closeness_deepiv} gives $u_0\in\mathcal U$, and the boundedness condition gives $\|u(X)\|\leqslant1$ uniformly over $u\in\mathcal U$. Define the globally Lipschitz loss \[ \ell(v)=M^2\bigl(\max\{-1,\min\{v,1\}\}\bigr)^2. \] The clipping map is $1$-Lipschitz, and $\|a^2-b^2\|\leqslant2\|a-b\|$ for $a,b\in[-1,1]$. Hence $\ell$ is $L$-Lipschitz with $L=2M^2$, and $\ell(h(X)/M)=h(X)^2$.  We verify the anchored complexity condition. Set $\mathcal S=\operatorname{star}(\mathcal H-\mathcal H)$. By Assumption~\ref{asp:critical_radius} and \eqref{eq:delta_trae}, there is a deterministic critical radius $d\leqslant\delta_n$ for $\mathcal S$. Since $0\in\mathcal S$, we have $\mathcal S\subset\operatorname{star}(\mathcal S-\mathcal S)$, and the critical-radius definition implies \[ \mathfrak R_n(d;\mathcal S)\leqslant d^2. \] For $s\geqslant d$, star-shapedness shows that $(d/s)g\in\mathcal S$ and $\\|(d/s)g\\|\leqslant d$ whenever $g\in\mathcal S$ and $\\|g\\|\leqslant s$. Applying this scaling inside the Rademacher supremum gives \[ \mathfrak R_n(s;\mathcal S) \leqslant\frac{s}{d}\mathfrak R_n(d;\mathcal S) \leqslant sd. \] Also, $h_0\in\mathcal H$ implies $\operatorname{star}(\mathcal H-h_0)\subset\mathcal S$. Therefore, for every $s\geqslant\delta_n$, \[ \begin{split} \mathfrak R_n\bigl(s;\operatorname{star}(\mathcal U-u_0)\bigr) &=M^{-1}\mathfrak R_n\bigl(Ms;\operatorname{star}(\mathcal H-h_0)\bigr)\\ &\leqslant M^{-1}\mathfrak R_n(Ms;\mathcal S) \leqslant sd\leqslant s\delta_n. \end{split} \] In particular, every $r\geqslant\delta_n$ satisfies the anchored fixed-point inequality for the normalized class.  For $R>0$, define \[ Z_n(R)=\sup_{\substack{u\in\mathcal U\\\\|u-u_0\\|\leqslant R}} \left\|(\mathbb E_n-\mathbb E)[\ell(u(X))-\ell(u_0(X))]\right\|. \] Its empirical and Rademacher suprema are precisely the squared-loss suprema in Assumption~\ref{asp:critical_radius}, with localization radius $MR$, and thus are measurable. The localized concentration inequality used in the proof of \citet[Lemma~14]{foster2023orthogonal}, applied to this measurable loss process and the anchored fixed point just verified, gives universal constants $c_1,c_2>0$ such that, for $R\geqslant r\geqslant\delta_n$ and $v>0$, \[ \mathbb P\{Z_n(R)\geqslant16LRr+v\} \leqslant c_1\exp\left\{-\frac{c_2nv^2}{L^2R^2+2Lv}\right\}. \] Taking $v=LRr$ and using $r\leqslant R$ yields \[ \mathbb P\{Z_n(R)\geqslant17LRr\} \leqslant c_1\exp\{-c_2nr^2/3\}. \]  Choose $r=a\delta_n$ with a fixed sufficiently large $a\geqslant1$. If $r\geqslant2$, set $J=0$. If $r<2$, set $J=\lceil\log(2/r)/\log2\rceil$. For $j=0,\ldots,J$, put $R_j=2^jr$. A union bound shows that the measurable event \[ Z_n(R_j)<17LR_jr,\qquad j=0,\ldots,J, \] has failure probability at most $(J+1)c_1\exp\{-c_2nr^2/3\}$. Since $r\geqslant\delta_n\geqslant n^{-1/2}$, there is a universal constant $C_0$ such that $J+1\leqslant C_0\log(en)$. Moreover, \eqref{eq:delta_trae} gives $n\delta_n^2\geqslant1$ and a fixed positive lower bound for $n\delta_n^2/\{\log\log(en)+\log(1/\eta_n)\}$. Consequently, a fixed sufficiently large $a$ makes this failure probability at most $\eta_n/4$.  On this event, let $u\in\mathcal U$ and write $b=\\|u-u_0\\|$. If $b\leqslant r$, then the loss increment is bounded by $Z_n(r)<17Lr^2$. If $b>r$, boundedness gives $b\leqslant2$, so there is an index $j\in\{1,\ldots,J\}$ with $R_{j-1}<b\leqslant R_j$. For that index, $R_j<2b$, and the loss increment is bounded by $Z_n(R_j)<34Lrb$. Thus, in both cases, \[ \left\|(\mathbb E_n-\mathbb E)[\ell(u(X))-\ell(u_0(X))]\right\| \leqslant34Lr(b+r). \] Substituting $u=h/M$, $L=2M^2$, and $r=a\delta_n$ proves, uniformly over $h\in\mathcal H$, \[ \begin{split} \left\|\{\\|h\\|_{2,n}^2-\\|h_0\\|_{2,n}^2\} -\{\\|h\\|^2-\\|h_0\\|^2\}\right\| &\leqslant68M^2a\delta_n\{\\|h-h_0\\|/M+a\delta_n\}\\ &\leqslant C\{\delta_n\\|h-h_0\\|+\delta_n^2\}, \end{split} \] where $C$ depends only on the fixed boundedness and calibration constants. Allocating $\eta_n/4$ to this comparison at each grid point uses at most $(K_n+1)\eta_n/4=\xi/4$ of the total failure probability. This supplies the squared-norm comparison included in $\mathcal E_n$.
- run-level `ass:critical-radius-complexity`: [note, V1.5] clarified — the reading the statement means is now stated: [F2.5 reviewer] [statement-wrong] The Ω clause leaves its n-onset and ξ-uniformity unspecified; eventual bounds for fixed ξ, eventual uniform bounds, and Lean's uniform bound for every positive n impose different assumptions. This hypothesis of the paper is used by prop:exist-lambda, thm:tikhonv-rate, prop:exist-lambda-iter-trae, thm:iter-trae-rate, thm:normality, lem:concentrate-l, lem:strong-convexity, thm:trae-discrepancy-bound, lem:trae-accepted-bounds, lem:iter-trae-strong-propagation, thm:iter-trae-discrepancy-bound, lem:iter-trae-accepted-stability, lem:trae-localized-concentration-lemma-9, lem:trae-squared-norm-lemma-14, lem:iter-trae-excess-loss-lemma-11, lem:iter-trae-excess-loss-lemma-10. 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 proof as printed proves the repaired statement; no step changes.
- note on `ass:cr-deepiv-measurability`: pinned reading: For $\mathcal H$ and, for each fixed $g\in\mathcal G$, $\mathcal T_g\mathcal H$, the empirical-process suprema used in the critical-radius analysis are measurable under their respective input distributions. (Assumption \ref{asp:cr_deepiv} places the measurability sentence immediately before the critical-radius requirements for these classes; the notation section defines the relevant Rademacher complexities using expectations of suprema.)
- note on `ass:critical-radius-complexity`: clarified — the reading the statement means is now stated: [F2.5 reviewer] [statement-wrong] The Ω clause leaves its n-onset and ξ-uniformity unspecified; eventual bounds for fixed ξ, eventual uniform bounds, and Lean's uniform bound for every positive n impose different assumptions. This hypothesis of the paper is used by prop:exist-lambda, thm:tikhonv-rate, prop:exist-lambda-iter-trae, thm:iter-trae-rate, thm:normality, lem:concentrate-l, lem:strong-convexity, thm:trae-discrepancy-bound, lem:trae-accepted-bounds, lem:iter-trae-strong-propagation, thm:iter-trae-discrepancy-bound, lem:iter-trae-accepted-stability, lem:trae-localized-concentration-lemma-9, lem:trae-squared-norm-lemma-14, lem:iter-trae-excess-loss-lemma-11, lem:iter-trae-excess-loss-lemma-10. 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:mean-sq-continuity`: pinned reading: There exists a fixed $C<\infty$, independent of $f$, $n$, and $\lambda$, such that $\mathbb E[m(W;f)^2]\leqslant C\\|f\\|^2$ for every $f\in\mathcal F$. (Assumption~\ref{asp:mean_sq_continuity} names mean-square continuity; the calibration convention in discovery/flat.tex, line 292, expressly treats its constant as fixed and independent of $n$ and $\lambda$.)
- note on `ass:rdiv-minima`: pinned reading: For every $\lambda\in\Lambda_n$, the empirical RDIV minimum is attained and $\hat h_\lambda$ is a measurable selected minimizer of $L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (The assumption immediately follows the RDIV estimator's defining minimization in the Adaptive Regularized DeepIV Estimator subsection, and the binding definition of $\hat h_\lambda$ specifies a measurable selected minimizer.)
- note on `ass:trae-minima`: pinned reading: The empirical loss $L_n$ is measurable, and the asserted attainment and measurable selection concern the minimum of $h\mapsto L_n(h)+\lambda\mathbb E_n[h(X)^2]$ over $h\in\mathcal H$. (The sentence immediately follows the displayed definition of the TRAE estimator as the minimizer of this penalized objective.)
- note on `def:iter-rdiv-discrepancy-bound`: pinned reading: The displayed discrepancy inequality holds on $\mathcal E_{n,t}^{\mathrm{IR}}$, with $\mathbb P(\mathcal E_{n,t}^{\mathrm{IR}})\geqslant1-\xi$, simultaneously for every $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IR}}^\circ,2]$. (The sentence introducing the display cites Corollary~\ref{cor:iter_rdiv_discrepancy_bound}, which expressly states simultaneity on this event; Lemma~\ref{lemma:iter_rdiv_stability} specifies its probability.)
- note on `def:iter-trae-discrepancy-bound`: pinned reading: Under the assumptions of Theorem~\ref{thm:iter_trae_rate}, the displayed inequality holds simultaneously for all $\lambda\in\Lambda_n\cap[\lambda_{\mathrm{IT}}^\circ,2]$ on $\mathcal E_{n,t}^{\mathrm{IT}}$, where $\mathbb P(\mathcal E_{n,t}^{\mathrm{IT}})\geqslant1-\xi$. (The sentence immediately preceding the display invokes Corollary~\ref{cor:iter_trae_discrepancy_bound}, whose statement explicitly places this simultaneous bound on the common concentration event.)
- note on `def:trae-discrepancy-bound`: pinned reading: For a fixed constant $C_{\mathrm T}>0$, both displayed inequalities hold on $\mathcal E_n$, simultaneously for all $\lambda\in\Lambda_n$, under the stated calibration convention. (The sentence introducing equation~\ref{eq:trae_discrepancy_bound} expressly attributes it to Corollary~\ref{cor:trae_discrepancy_bound} and says that it holds on $\mathcal E_n$.)
- note on `lem:iter-rdiv-fixed-center-concentration-lemma-14`: cited source checked: source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt
- note on `lem:iter-rdiv-squared-loss-concentration-lemma-14`: cited source checked: source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt
- note on `lem:iter-trae-accepted-stability`: pinned reading: The constant $C_t$ may depend on the prescribed $t$ and fixed population quantities, including $\beta$ and $\\|w_0\\|$, but is independent of $n$, $\lambda$, and the realized fitting observations. (The proof applies Lemma~\ref{lemma:iter_bias}, whose interpolation inequality contributes $\\|w_0\\|^{2/(\beta+1)}$. The iterative-rate convention also explicitly permits constants depending on the prescribed iteration count and fixed population quantities.)
- note on `lem:iter-trae-anchored-concentration-lemma-9`: cited source checked: source supplied by the user at discovery/cited_sources/bennett2023source.txt
- note on `lem:iter-trae-fixed-center-squared-loss-lemma-9`: cited source checked: source supplied by the user at discovery/cited_sources/bennett2023source.txt
- note on `lem:rdiv-accepted-bounds`: pinned reading: The assumptions of \cref{thm:deepiv_rate} mean its standing hypotheses inherited from \cref{prop:exist_lambda_deepiv}; the bounds here hold for arbitrary $h_0$, without imposing $h_0=0$. (The zero condition introduces only the last conclusion of \cref{thm:deepiv_rate}; the proof of \cref{lemma:rdiv_accepted_bounds} treats arbitrary $h_0$ using the variational source inequality.)
- note on `lem:rdiv-localized-concentration-lemma-14`: cited source checked: source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt
- note on `lem:rdiv-mle-localized-concentration-li2024`: cited source checked: source supplied by the user at discovery/cited_sources/li2024regularized.txt
- note on `lem:trae-localized-concentration-lemma-9`: cited source checked: source supplied by the user at discovery/cited_sources/bennett2023source.txt
- note on `lem:trae-squared-norm-lemma-14`: cited source checked: source supplied by the user at discovery/cited_sources/foster2023orthogonal.txt
- note on `prop:exist-lambda`: pinned reading: For each fixed $n\geqslant N_{\mathrm T}(B)$, acceptance within the stated number of optimization problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_n$, with probability at least $1-\xi$ over the fitting sample. (The proof begins by restricting throughout to $\mathcal E_n$, establishes terminal acceptance there, and derives both return branches on that same event.)
- note on `prop:exist-lambda-deepiv`: pinned reading: For each fixed $n\geqslant N_{\mathrm R}(B)$, all probabilistic conclusions, including initial acceptance when $h_0=0$, hold on the same event $\mathcal E_n^{\mathrm R}$ of probability at least $1-\xi$. (The proof of Proposition~\ref{prop:exist_lambda_deepiv} explicitly works throughout on $\mathcal E_n^{\mathrm R}$ and derives initial acceptance there.)
- note on `prop:exist-lambda-iter-rdiv`: pinned reading: For each fixed $n\geqslant N_{\mathrm R}(B)$ and prescribed $t$, there is an event of probability at least $1-\xi$ on which the rule accepts a candidate and satisfies the stated output alternatives, including acceptance of $\lambda_0=2$ when $h_0=0$. (The proof of Proposition~\ref{prop:exist_lambda_iter_rdiv} explicitly places initial acceptance on the corresponding concentration event.)
- note on `prop:exist-lambda-iter-trae`: pinned reading: For each fixed $n\geqslant N_{\mathrm{IT}}(B)$ and prescribed $t$, acceptance within the stated number of minimax problems and the stated alternative for the returned parameter hold jointly on $\mathcal E_{n,t}^{\mathrm{IT}}$, with probability at least $1-\xi$ over the fitting sample. (The proof works throughout on $\mathcal E_{n,t}^{\mathrm{IT}}$ and derives terminal acceptance and the first-accepted-index alternative there.)
- note on `symbol \alpha`: symbol table corrected: type "locally defined interpolation exponent or RKHS coefficient vector", space "$(0,1/2]\cup\mathbb R^N$", def "The notation $\alpha$ denotes the interpolation exponent $\min\{\beta,1\}/(1+\min\{\beta,1\})$ in the source and error bounds, and a coefficient vector in $\mathbb R^N$ in the RKHS quadratic program, with $h_\alpha(x)=\sum_{j=1}^N\alpha_j k_{\mathcal H}(X_j,x)$." (The scalar-only entry omits the RKHS coefficient-vector meaning. Both meanings and the existing RKHS scope remain valid; only the obsolete normality tag is removed from the scalar scope.)
- note on `symbol \delta_n`: symbol table corrected: def "A prescribed deterministic calibrated statistical-error bound: for RDIV, $\delta_n\geqslant\delta_{n,\mathcal G}+\delta_{n,\mathcal H}+\delta_{n,\xi}$ with $\delta_n+\nu_n=o(1)$; for TRAE, $\delta_n\geqslant\max\{\tilde\delta_n(\eta_n),n^{-1/2}\}$ with $\delta_n=o(1)$. Iterative procedures use $\eta_{n,t}=\xi/\{t(K_n+1)\}$ in their concentration bounds. For inference it is a common deterministic bound satisfying the TRAE requirements for both primal and dual problems at confidence levels $\xi_n/2$." (The original entry covers only RDIV, whereas the symbol also denotes the distinct TRAE scale and the common inference bound. The existing correction and surviving scopes remain valid; only obsolete cited-step scopes are removed.)
- note on `symbol \ell`: symbol table corrected: type "locally bound residual-bracketing constant, iteration index, or loss function", space "$(1,\infty)\cup\mathbb N\cup\{f:\mathbb R\to\mathbb R\}$", def "In the classical discrepancy principle, $\ell\in(1,\infty)$ is the prescribed residual-bracketing factor. In $M_{\leqslant j}=\max_{1\leqslant\ell\leqslant j}\{\lambda,\\|f_\ell^*\\|_\infty^2\}$, it is an integer index ranging from $1$ to $j$. In the normalized squared-loss argument, $\ell(v)=M^2\bigl(\max\{-1,\min\{v,1\}\}\bigr)^2$, where $M=\max\{1,B\}$. In the likelihood concentration argument, $\ell(v)=-\log\bigl(\max\{1+v,2^{-1/2}\}\bigr)$." (The existing meanings remain valid. The logarithmic-loss reading must also apply directly to the registered likelihood concentration lemma, which does not inherit a scope attached only to its consuming lemma.)
- note on `symbol \gamma`: symbol table corrected: type "locally bound scalar parameter or critic coefficient vector", space "$\mathbb R\cup\bigcup_{N\geqslant1}\mathbb R^N$", def "In $\operatorname{star}(\mathcal F)=\{\gamma f:\gamma\in[0,1],f\in\mathcal F\}$, $\gamma$ is the scalar multiplier; elsewhere it is a locally specified positive exponent or the coefficient vector in $f_\gamma(\cdot)=\sum_{j=1}^N\gamma_j k_{\mathcal F}(Z_j,\cdot)$." (The recorded star-hull meaning is correct locally but omits distinct uses that change the symbol's type and domain. The paper also uses a fixed oracle exponent, an arbitrary positive spectral exponent, and an unrestricted RKHS critic coefficient vector.)
- note on `symbol \gamma_h`: symbol table corrected: def "An exponent $\gamma_h>0$ such that $\mathbb E[(\tilde m(W;h_1)-\tilde m(W;h_2))^2]\leqslant C\\|h_1-h_2\\|^{\gamma_h}$ for all $h_1,h_2\in\mathcal H$, where the norm is the population $L_2(X)$ norm and $C$ is a uniform constant." (The label omits the precise score-continuity inequality. The coefficient vector $\widehat\gamma_h$ in the RKHS proof is a distinct symbol and does not redefine $\gamma_h$.)
- note on `symbol \gamma_q`: symbol table corrected: def "An exponent $\gamma_q>0$ such that $\mathbb E[(m(W;q_1)-m(W;q_2))^2]\leqslant C\\|q_1-q_2\\|^{\gamma_q}$ for all $q_1,q_2\in\mathcal Q$, where the norm is the population $L_2(Z)$ norm and $C$ is a uniform constant." (The label alone omits the defining inequality. The exponent applies to the norm on the right, while the score difference on the left is squared.)
- note on `symbol \hat h_{j,\lambda}`: symbol table corrected: def "Initialize $\hat h_{0,\lambda}=0$ and, for $j=1,\ldots,t$, define $\hat h_{j,\lambda}=\arg\min_{h\in\mathcal H}\{L_n(h)+\lambda\\|h-\hat h_{j-1,\lambda}\\|_{2,n}^2\}$, where $L_n$ is the fitting loss of the respective RDIV or TRAE procedure." (The recorded RDIV-only description omits the TRAE recursion. The existing correction remains valid; the additional clauses specify the method and distinguish the empirical recursion from the population centers used in the separately registered concentration comparisons.)
- note on `symbol \hat q`: symbol table corrected: def "$\hat q=\arg\min_{q\in\mathcal Q}\{L_n^{\text{dual}}(q)+\lambda_{\mathrm{dp}}^{\text{dual}}\mathbb E_n[q(Z)^2]\}$, where $L_n^{\text{dual}}(q)=\sup_{s\in\mathcal S}\mathbb E_n[2\tilde m(W;s)-2q(Z)s(X)-s(X)^2]$ and $\lambda_{\mathrm{dp}}^{\text{dual}}$ is selected with the discrepancy principle." (The recorded label omits the defining dual optimization, so the existing definition fix remains necessary. The inference scope additionally specifies jointly measurable versions of the selected dual evaluations.)
- note on `symbol \lambda`: symbol table corrected: space "$[0,\infty]$" (The computational zero-penalty convention still requires the existing space correction. The fixed-candidate qualification now belongs to the propagation proof; the other readings remain valid.)
- note on `symbol \lambda_{\mathrm{dp},t}^{\mathrm R}`: symbol table corrected: def "The output of Algorithm~\ref{alg:discrepancy} applied to $\hat h_{t,\lambda}$ with the projected calibration statistic in equation~\eqref{eq:rdiv_projected_loss} and noise level $A_{n,t}^{\mathrm R}(\lambda)=s_{n,t}^{\mathrm R}/\lambda+\nu_n^2$: return $2\rho^{k^*}$ for the smallest $k^*\in\{0,\ldots,K_n\}$ satisfying $D_n(\hat h_{t,2\rho^{k^*}})\leqslant\tfrac32 A_{n,t}^{\mathrm R}(2\rho^{k^*})$; if no such index exists, return $2\rho^{K_n}$." (The recorded label omits the selection rule. The parameter is the first accepted grid point using the projected RDIV discrepancy and iterative RDIV threshold, with terminal fallback.)
- note on `symbol \lambda_{\mathrm{dp},t}^{\mathrm T}`: symbol table corrected: def "The output of Algorithm~\ref{alg:discrepancy} applied to $\hat h_{t,\lambda}$ with $D_n=L_n$ and noise level $A_{n,t}^{\mathrm T}(\lambda)=\delta_n^2+\min\{\lambda,s_{n,t}^{\mathrm T}/\lambda\}$: return $2\rho^{k^*}$ for the smallest $k^*\in\{0,\ldots,K_n\}$ satisfying $L_n(\hat h_{t,2\rho^{k^*}})\leqslant\tfrac32 A_{n,t}^{\mathrm T}(2\rho^{k^*})$; if no such index exists, return $2\rho^{K_n}$." (The recorded label does not specify which parameter is selected. The definition requires the first accepted point in decreasing grid order, with the iterative TRAE threshold and terminal fallback.)
- note on `symbol \mathcal F`: symbol table corrected: type "function class", space "A class of real-valued functions on the input domain under consideration.", def "$\mathcal F$ denotes the function class under consideration; its domain and specialization are supplied by the surrounding definition." (The test-class-only entry misses the generic complexity notation and full-RKHS specialization. The existing correction remains valid; the additional clause identifies the adversarial specialization in the separately registered iterative TRAE concentration steps.)
- note on `symbol \mathcal S`: symbol table corrected: type "dual test class or locally defined star-shaped difference class", space "$\mathcal S\subseteq L_2(X)\ \text{or}\ \mathcal S\subseteq L_2(X,Z)$", def "In the dual empirical adversarial loss, $\mathcal S\subseteq\widetilde{\mathcal H}$ is the test class. In the normalized squared-loss arguments, $\mathcal S=\operatorname{star}(\mathcal H-\mathcal H)$. In the likelihood concentration argument, $\mathcal S=\operatorname{star}(\mathcal G_0-\mathcal G_0)$." (The existing local class definitions remain valid. The likelihood-transform difference-class reading must also be attached directly to the registered likelihood concentration lemma.)
- note on `symbol \tilde\delta_n`: symbol table corrected: type "confidence-indexed critical-radius upper-bound sequence", space "$(0,1)\to[0,\infty)$", def "$\tilde\delta_n(\xi)=\Omega\!\left(\sqrt{(\log\log(en)+\log(1/\xi))/n}\right)$ is a common upper bound on the critical radii of $\operatorname{star}(\mathcal H\cdotp\mathcal F)$, $\operatorname{star}(m\circ\mathcal F)$, $\operatorname{star}(\mathcal F)$, $\operatorname{star}(\mathcal H)$, and $\operatorname{star}(\mathcal H-\mathcal H)$." (The recorded scalar space suppresses the confidence argument, and the vague definition omits the five classes and the stipulated lower order bound. In particular, calibration uses its value at the allocated failure probability.)
- note on `symbol \widehat\alpha_\lambda`: symbol table corrected: def "$\widehat\alpha_\lambda=(K_{\mathcal H}P_\eta K_{\mathcal H}+\lambda K_{\mathcal H}^{2})^{\dagger}K_{\mathcal H}P_\eta\boldsymbol y$, where $A^\dagger$ denotes the Moore--Penrose pseudoinverse of $A$." (The recorded entry identifies the coefficient vector by a program without recording its defining formula. The paper specifies the minimum-Euclidean-norm vector explicitly; the subsequent computation on a retained spectral span does not replace the unrestricted vector in Proposition~\ref{prop:rkhs_qp}.)
- note on `symbol \xi_n`: symbol table corrected: space "$\{(\xi_n)_{n\geqslant1}:\xi_n>0,\ \xi_n\downarrow0\}$" (The deterministic sequence decreasing to zero is correctly described, but the recorded space requires every term to be strictly below one. The paper explicitly allows $\xi_n=1/\log(en)$, whose first term equals one; the tuning rules use confidence levels $\xi_n/2$.)
- note on `symbol a_j`: symbol table corrected: space "$[0,\infty)$, with $a_0=0$ and $a_j\geqslant1$ for $j=1,\ldots,t$" (The recorded space excludes the explicitly used initialization $a_0=0$. The paper fixes the positive-index constants uniformly over sample size and penalty. Proposition~\ref{prop:rkhs_qp} uses coefficient symbols $\alpha_j$, not the RDIV constants $a_j$, so it introduces no second meaning.)
- note on `symbol A_n`: symbol table corrected: def "$A_n(\lambda)$ is the noise level supplied to the discrepancy rule, whose acceptance threshold is $\tfrac32A_n(\lambda)$." (The recorded prescribed-bound definition excludes the experimental noise-floor meaning. The existing general reading and theoretical formulas remain valid; four clauses require retagging.)
- note on `symbol L`: symbol table corrected: type "method-specific population loss", def "RDIV uses $L(h)=\mathbb E[(Y-\mathcal T h(Z))^2]=\sigma^2_{Y\mid Z}+\\|\mathcal T(h-h_0)\\|^2$, and TRAE uses $L(h)=\sup_{f\in\mathcal F}\mathbb E[2m(W;f)-2h(X)f(Z)-f(Z)^2]$." (The table gives only the RDIV population squared loss. The existing method-specific correction remains valid; the additional clause specifies the adversarial meaning in the separately registered iterative TRAE concentration steps.)
- note on `symbol L_n`: symbol table corrected: def "For RDIV, $L_n(h)=\mathbb E_{n,\mathrm{fit}}[(Y-\hat{\mathcal T}h(Z))^2]$. For TRAE, $L_n(h)=\sup_{f\in\mathcal F}\mathbb E_n[2m(W;f)-2h(X)f(Z)-f(Z)^2]$, where $\mathbb E_n$ averages over its fitting sample." (The table records only the RDIV squared loss, whereas TRAE reuses the symbol for an adversarial supremum. The existing fix remains valid; the additional scopes supply these meanings to the separately registered iterative concentration steps.)
- note on `symbol P_\eta`: symbol table corrected: type "ridge-stabilized critic smoothing matrix" (The displayed formula is correct, but the matrix is generally not an idempotent projection: its eigenvalues are $d/(d+N\eta)$ for eigenvalues $d$ of $K_{\mathcal F}$. Proposition~\ref{prop:rkhs_qp} refers back to the same definition.)
- note on `symbol q`: symbol table corrected: type "test function or scalar bias exponent, according to local context", space "$\tilde{\mathcal Q}\cup(1,\infty)$", def "The letter $q$ denotes a function in the moment and dual-estimation formulas, and is separately reused for the scalar exponent $q>1$ in $B_n(\lambda)=C_b\lambda^q$." (The table records only a test function, but the generic discrepancy theorem and four rate proofs also use $q$ as a scalar exponent. The paper explicitly distinguishes these local uses, and the dual optimization restricts its function-valued candidate to $\mathcal Q$.)
- note on `thm:common-dp`: pinned reading: On $\mathcal E_n$, the algorithm terminates through its acceptance branch at an index $k\leqslant k^\circ$, where $\lambda^\circ=2\rho^{k^\circ}$ and $k^\circ\leqslant K_n$; acceptance at $k=K_n$ is allowed, and both error bounds hold on this event. (The prose following the algorithm restricts acceptance guarantees to the stated events. The proof of Theorem~\ref{thm:common_dp} establishes acceptance at the oracle point and explicitly identifies the consequence as avoiding fallback.)
- note on `thm:deepiv-rate`: pinned reading: For each fixed $n\geqslant N_{\mathrm R}(B)$, the displayed error bounds and, when $h_0=0$, both sharper $O(\delta_n^2)$ bounds hold jointly with probability at least $1-\xi$ over the preliminary, fitting, and calibration samples. (The proof obtains the zero-signal bounds from the probabilistic initial-acceptance conclusion of Proposition~\ref{prop:exist_lambda_deepiv} and Lemma~\ref{lemma:str_dp}; the general case uses the common RDIV event and the stated confidence convention.)
- note on `thm:iter-trae-rate`: pinned reading: The constants hidden by $O_t(\cdot)$ may depend on the prescribed $t$, fixed population quantities including $\beta$ and $\\|w_0\\|$, and the fixed grid ratio $\rho$, but are independent of $n$ and the candidate $\lambda$. (The text following the bounds in Theorem~\ref{thm:iter_rdiv_rate} explicitly allows dependence on the prescribed iteration count and fixed population quantities; Lemma~\ref{lemma:iter_trae_accepted_stability} supplies the comparison constants used here under the same convention.)
- note on `thm:rdiv-discrepancy-bound`: pinned reading: The assumptions imported from Theorem~\ref{thm:deepiv_rate} are the general hypotheses of Proposition~\ref{prop:exist_lambda_deepiv}, without imposing $h_0=0$. (Theorem~\ref{thm:deepiv_rate} imports the proposition's assumptions and treats $h_0=0$ only in a separate conditional conclusion. The present proof uses Lemmas~\ref{lemma:err_l_dp} and~\ref{lemma:str_dp} for general $h_0$.)
- note on `thm:tikhonv-rate`: pinned reading: With $\beta$ and $\xi$ fixed as $n\to\infty$, for each $n\geqslant N_{\mathrm T}(B)$ the displayed strong- and weak-error bounds hold jointly on a fitting-sample event of probability at least $1-\xi$. (The proof of Theorem~\ref{thm:tikhonv_rate} works on $\mathcal E_n$, which the preceding confidence allocation constructs separately for each sample size.)

### V1 lint warnings

- thm:deepiv-rate: thm:deepiv_rate references \ref{prop:exist_lambda_deepiv} (prop:exist_lambda_deepiv) but `depends_on` lacks ass:exist-lambda-deepiv-zero-case; V1.5 must confirm whether the paper assumes it or merely mentions it
- thm:deepiv-rate: thm:deepiv_rate references \ref{thm:common_dp} (thm:common_dp) but `depends_on` lacks ass:common-dp-noise-domain, ass:common-dp-stability-domain, ass:common-dp-parameters, ass:common-dp-event-comparisons, ass:common-dp-stable-segment, ass:common-dp-monotonicity, ass:common-dp-oracle-crossing, def:common-dp-bias; V1.5 must confirm whether the paper assumes it or merely mentions it
- thm:deepiv-rate: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:deepiv-rate: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:deepiv-rate: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:deepiv-rate: depends on ass:re-cd-realizability which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:deepiv-rate: depends on ass:re-cd-density-bounds which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:deepiv-rate: depends on ass:re-cd-nuisance-rate which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:deepiv-rate: depends on ass:cr-deepiv-outcome which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:deepiv-rate: depends on ass:cr-deepiv-hypotheses which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:deepiv-rate: depends on ass:cr-deepiv-pilot which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:deepiv-rate: depends on ass:cr-deepiv-measurability which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:deepiv-rate: depends on ass:cr-deepiv-density-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:deepiv-rate: depends on ass:cr-deepiv-hypothesis-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:deepiv-rate: depends on ass:cr-deepiv-radius-floor which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: prop:exist_lambda_iter_rdiv references \ref{prop:exist_lambda_deepiv} (prop:exist_lambda_deepiv) but `depends_on` lacks ass:exist-lambda-deepiv-zero-case, prop:exist-lambda-deepiv; V1.5 must confirm whether the paper assumes it or merely mentions it
- prop:exist-lambda-iter-rdiv: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: depends on ass:re-cd-realizability which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: depends on ass:re-cd-density-bounds which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: depends on ass:re-cd-nuisance-rate which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: depends on ass:cr-deepiv-outcome which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: depends on ass:cr-deepiv-hypotheses which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: depends on ass:cr-deepiv-pilot which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: depends on ass:cr-deepiv-measurability which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: depends on ass:cr-deepiv-density-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: depends on ass:cr-deepiv-hypothesis-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-rdiv: depends on ass:cr-deepiv-radius-floor which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: thm:iter_rdiv_rate references \ref{prop:exist_lambda_iter_rdiv} (prop:exist_lambda_iter_rdiv) but `depends_on` lacks ass:exist-lambda-iter-rdiv-zero-case; V1.5 must confirm whether the paper assumes it or merely mentions it
- thm:iter-rdiv-rate: thm:iter_rdiv_rate references \ref{thm:common_dp} (thm:common_dp) but `depends_on` lacks ass:common-dp-noise-domain, ass:common-dp-stability-domain, ass:common-dp-parameters, ass:common-dp-event-comparisons, ass:common-dp-stable-segment, ass:common-dp-monotonicity, ass:common-dp-oracle-crossing, def:common-dp-bias; V1.5 must confirm whether the paper assumes it or merely mentions it
- thm:iter-rdiv-rate: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: depends on ass:re-cd-realizability which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: depends on ass:re-cd-density-bounds which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: depends on ass:re-cd-nuisance-rate which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: depends on ass:cr-deepiv-outcome which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: depends on ass:cr-deepiv-hypotheses which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: depends on ass:cr-deepiv-pilot which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: depends on ass:cr-deepiv-measurability which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: depends on ass:cr-deepiv-density-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: depends on ass:cr-deepiv-hypothesis-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-rate: depends on ass:cr-deepiv-radius-floor which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:tikhonv-rate: thm:tikhonv_rate references \ref{thm:common_dp} (thm:common_dp) but `depends_on` lacks ass:common-dp-noise-domain, ass:common-dp-stability-domain, ass:common-dp-parameters, ass:common-dp-event-comparisons, ass:common-dp-stable-segment, ass:common-dp-monotonicity, ass:common-dp-oracle-crossing, def:common-dp-bias; V1.5 must confirm whether the paper assumes it or merely mentions it
- thm:tikhonv-rate: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:tikhonv-rate: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:tikhonv-rate: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:tikhonv-rate: depends on ass:closeness which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:tikhonv-rate: depends on ass:mean-sq-continuity which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:tikhonv-rate: depends on ass:critical-radius-boundedness which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:tikhonv-rate: depends on ass:critical-radius-complexity which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-trae: prop:exist_lambda_iter_trae references \ref{prop:exist_lambda} (prop:exist_lambda) but `depends_on` lacks prop:exist-lambda; V1.5 must confirm whether the paper assumes it or merely mentions it
- prop:exist-lambda-iter-trae: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-trae: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-trae: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-trae: depends on ass:closeness which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-trae: depends on ass:mean-sq-continuity which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-trae: depends on ass:critical-radius-boundedness which the environment never references; V1.5 must confirm the paper actually assumes it
- prop:exist-lambda-iter-trae: depends on ass:critical-radius-complexity which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-rate: thm:iter_trae_rate references \ref{lemma:iter_trae_accepted_stability} (lemma:iter_trae_accepted_stability) but `depends_on` lacks ass:iter-trae-accepted-stability-branch; V1.5 must confirm whether the paper assumes it or merely mentions it
- thm:iter-trae-rate: thm:iter_trae_rate references \ref{thm:common_dp} (thm:common_dp) but `depends_on` lacks ass:common-dp-noise-domain, ass:common-dp-stability-domain, ass:common-dp-parameters, ass:common-dp-event-comparisons, ass:common-dp-stable-segment, ass:common-dp-monotonicity, ass:common-dp-oracle-crossing, def:common-dp-bias; V1.5 must confirm whether the paper assumes it or merely mentions it
- thm:iter-trae-rate: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-rate: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-rate: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-rate: depends on ass:closeness which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-rate: depends on ass:mean-sq-continuity which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-rate: depends on ass:critical-radius-boundedness which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-rate: depends on ass:critical-radius-complexity which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:normality: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:normality: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:normality: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:normality: depends on ass:closeness which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:normality: depends on ass:mean-sq-continuity which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:normality: depends on ass:critical-radius-boundedness which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:normality: depends on ass:critical-radius-complexity which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:str-dp: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:str-dp: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: cor:rdiv_discrepancy_bound references \ref{thm:deepiv_rate} (thm:deepiv_rate) but `depends_on` lacks ass:deepiv-rate-zero-case, thm:deepiv-rate; V1.5 must confirm whether the paper assumes it or merely mentions it
- thm:rdiv-discrepancy-bound: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: depends on ass:re-cd-realizability which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: depends on ass:re-cd-density-bounds which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: depends on ass:re-cd-nuisance-rate which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: depends on ass:cr-deepiv-outcome which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: depends on ass:cr-deepiv-hypotheses which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: depends on ass:cr-deepiv-pilot which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: depends on ass:cr-deepiv-measurability which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: depends on ass:cr-deepiv-density-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: depends on ass:cr-deepiv-hypothesis-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:rdiv-discrepancy-bound: depends on ass:cr-deepiv-radius-floor which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: lemma:rdiv_accepted_bounds references \ref{thm:deepiv_rate} (thm:deepiv_rate) but `depends_on` lacks ass:deepiv-rate-zero-case, thm:deepiv-rate; V1.5 must confirm whether the paper assumes it or merely mentions it
- lem:rdiv-accepted-bounds: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: depends on ass:re-cd-realizability which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: depends on ass:re-cd-density-bounds which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: depends on ass:re-cd-nuisance-rate which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: depends on ass:cr-deepiv-outcome which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: depends on ass:cr-deepiv-hypotheses which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: depends on ass:cr-deepiv-pilot which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: depends on ass:cr-deepiv-measurability which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: depends on ass:cr-deepiv-density-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: depends on ass:cr-deepiv-hypothesis-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:rdiv-accepted-bounds: depends on ass:cr-deepiv-radius-floor which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-bias: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-bias: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: lemma:iter_rdiv_stability references \ref{thm:iter_rdiv_rate} (thm:iter_rdiv_rate) but `depends_on` lacks ass:iter-rdiv-rate-comparable-noise, ass:iter-rdiv-rate-qualification, thm:iter-rdiv-rate; V1.5 must confirm whether the paper assumes it or merely mentions it
- lem:iter-rdiv-stability: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:re-cd-realizability which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:re-cd-density-bounds which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:re-cd-nuisance-rate which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:cr-deepiv-outcome which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:cr-deepiv-hypotheses which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:cr-deepiv-pilot which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:cr-deepiv-measurability which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:cr-deepiv-density-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:cr-deepiv-hypothesis-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:cr-deepiv-radius-floor which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-stability: depends on ass:iter-realizability-rdiv which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: cor:iter_rdiv_discrepancy_bound references \ref{thm:iter_rdiv_rate} (thm:iter_rdiv_rate) but `depends_on` lacks ass:iter-rdiv-rate-comparable-noise, ass:iter-rdiv-rate-qualification, thm:iter-rdiv-rate; V1.5 must confirm whether the paper assumes it or merely mentions it
- thm:iter-rdiv-discrepancy-bound: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:re-cd-realizability which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:re-cd-density-bounds which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:re-cd-nuisance-rate which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:cr-deepiv-outcome which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:cr-deepiv-hypotheses which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:cr-deepiv-pilot which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:cr-deepiv-measurability which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:cr-deepiv-density-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:cr-deepiv-hypothesis-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:cr-deepiv-radius-floor which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-rdiv-discrepancy-bound: depends on ass:iter-realizability-rdiv which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: lemma:iter_rdiv_accepted_bounds references \ref{thm:iter_rdiv_rate} (thm:iter_rdiv_rate) but `depends_on` lacks ass:iter-rdiv-rate-comparable-noise, ass:iter-rdiv-rate-qualification, thm:iter-rdiv-rate; V1.5 must confirm whether the paper assumes it or merely mentions it
- lem:iter-rdiv-accepted-bounds: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:re-cd-realizability which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:re-cd-density-bounds which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:re-cd-nuisance-rate which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:cr-deepiv-outcome which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:cr-deepiv-hypotheses which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:cr-deepiv-pilot which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:cr-deepiv-measurability which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:cr-deepiv-density-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:cr-deepiv-hypothesis-radius which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:cr-deepiv-radius-floor which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-rdiv-accepted-bounds: depends on ass:iter-realizability-rdiv which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:strong-convexity: lemma:strong_convexity references \ref{thm:tikhonv_rate} (thm:tikhonv_rate) but `depends_on` lacks thm:tikhonv-rate; V1.5 must confirm whether the paper assumes it or merely mentions it
- lem:strong-convexity: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:strong-convexity: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:strong-convexity: depends on ass:mean-sq-continuity which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:strong-convexity: depends on ass:critical-radius-boundedness which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:strong-convexity: depends on ass:critical-radius-complexity which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:trae-discrepancy-bound: cor:trae_discrepancy_bound references \ref{thm:tikhonv_rate} (thm:tikhonv_rate) but `depends_on` lacks thm:tikhonv-rate; V1.5 must confirm whether the paper assumes it or merely mentions it
- thm:trae-discrepancy-bound: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:trae-discrepancy-bound: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:trae-discrepancy-bound: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:trae-discrepancy-bound: depends on ass:closeness which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:trae-discrepancy-bound: depends on ass:mean-sq-continuity which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:trae-discrepancy-bound: depends on ass:critical-radius-boundedness which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:trae-discrepancy-bound: depends on ass:critical-radius-complexity which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:trae-accepted-bounds: lemma:trae_accepted_bounds references \ref{thm:tikhonv_rate} (thm:tikhonv_rate) but `depends_on` lacks thm:tikhonv-rate; V1.5 must confirm whether the paper assumes it or merely mentions it
- lem:trae-accepted-bounds: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:trae-accepted-bounds: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:trae-accepted-bounds: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:trae-accepted-bounds: depends on ass:closeness which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:trae-accepted-bounds: depends on ass:mean-sq-continuity which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:trae-accepted-bounds: depends on ass:critical-radius-boundedness which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:trae-accepted-bounds: depends on ass:critical-radius-complexity which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-strong-propagation: eq:iter_trae_strong_propagation references \ref{thm:iter_trae_rate} (thm:iter_trae_rate) but `depends_on` lacks ass:iter-trae-rate-qualification, thm:iter-trae-rate; V1.5 must confirm whether the paper assumes it or merely mentions it
- lem:iter-trae-strong-propagation: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-strong-propagation: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-strong-propagation: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-strong-propagation: depends on ass:mean-sq-continuity which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-discrepancy-bound: cor:iter_trae_discrepancy_bound references \ref{thm:iter_trae_rate} (thm:iter_trae_rate) but `depends_on` lacks ass:iter-trae-rate-qualification, thm:iter-trae-rate; V1.5 must confirm whether the paper assumes it or merely mentions it
- thm:iter-trae-discrepancy-bound: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-discrepancy-bound: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-discrepancy-bound: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-discrepancy-bound: depends on ass:closeness which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-discrepancy-bound: depends on ass:mean-sq-continuity which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-discrepancy-bound: depends on ass:critical-radius-boundedness which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-discrepancy-bound: depends on ass:critical-radius-complexity which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-discrepancy-bound: depends on ass:iter-realizability-trae which the environment never references; V1.5 must confirm the paper actually assumes it
- thm:iter-trae-discrepancy-bound: depends on ass:iter-mean-sq-continuity which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-accepted-stability: depends on ass:range which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-accepted-stability: depends on ass:source-condition which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-accepted-stability: depends on ass:closeness-deepiv which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-accepted-stability: depends on ass:closeness which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-accepted-stability: depends on ass:mean-sq-continuity which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-accepted-stability: depends on ass:critical-radius-boundedness which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-accepted-stability: depends on ass:critical-radius-complexity which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-accepted-stability: depends on ass:iter-realizability-trae which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-accepted-stability: depends on ass:iter-mean-sq-continuity which the environment never references; V1.5 must confirm the paper actually assumes it
- lem:iter-trae-strong-propagation: the proof of eq:iter_trae_strong_propagation cites [bennett2023source, Theorem 6] (eq:iter_trae_strong_propagation/proof/1/1); 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 proof explicitly adapts the iterative argument and carries out the recursive comparisons and induction itself.). paper: "We adapt the iterative argument of \citet[Theorem~6]{bennett2023source} to the finite grid and give the…"

### Parser warnings

- smile.sty: \usepackage{algorithmic} loads algorithmic.sty, which the paper's source ships; the flattener does not read a package, so the agents and the page do not see its definitions — it defines 62 command(s), 5 of them used in the extracted results (\STATE, \IF, \FOR, \ENDIF, \ENDFOR); inline algorithmic.sty into the main file's preamble in place of the \usepackage (between \makeatletter and \makeatother) in the run's source copy

## Lean evidence

- build of the run barrel: **ok**
- source scan (sorry / admit / axiom): clean

| declaration | axioms |
| --- | --- |
| `Papers.TsAdaptiveEstimationInference.common_dp` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.exist_lambda_deepiv` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.deepiv_rate` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.exist_lambda_iter_rdiv` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.iter_rdiv_rate` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.exist_lambda` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.tikhonov_rate` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.exist_lambda_iter_trae` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.iter_trae_rate` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.dr_asymptotic_normality` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.variational_source_uniform` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.t_err_dp` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.err_l_dp` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.err_r_dp` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.str_dp` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.rdiv_discrepancy_bound` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.rdiv_accepted_bounds` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.iter_bias` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.iter_rdiv_discrepancy_bound` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.iter_rdiv_accepted_bounds` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.concentrate_l` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.strong_convexity` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.trae_discrepancy_bound` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.trae_accepted_bounds` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.iter_trae_strong_propagation` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.iter_trae_discrepancy_bound` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.iter_trae_accepted_stability` | clean — `propext`, `Classical.choice`, `Quot.sound` |
| `Papers.TsAdaptiveEstimationInference.rkhs_qp` | clean — `propext`, `Classical.choice`, `Quot.sound` |

## Inputs

| artifact | status |
| --- | --- |
| `discovery/v1_scope.json` | present (42 environment(s)) |
| `discovery/v1_5_review.json` | present (verdict PASS) |
| `formalization/crosswalk_full.json` | present (853 object(s)) |
| `graph.json` | present (865 node(s)) |
| Lean evidence | collected (build ok, 0 scan issue(s), 28 declaration(s)) |

## Annotated paper

Annotated paper: `annotated/main.tex` (171 annotation(s), 0 unlocated; index `annotated/ANNOTATIONS.md`); PDF: `annotated/main.pdf`