# Correctness report — tbs_prte_optimal_transport **Verdict: NO ERROR FOUND** **Paper: MINOR ISSUES — no main result is shown false or weakened** Minor: - Theorem 2.3: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned - Proposition 3.1: condition added (bookkeeping or a pinned reading) - Theorem 3.4: condition added (bookkeeping or a pinned reading); ambiguous; the owner settled a reading - Theorem 3.7: condition added (bookkeeping or a pinned reading) - Theorem 3.8: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned - Proposition 4.3: condition added (bookkeeping or a pinned reading); ambiguous; the owner settled a reading - Proposition 4.4: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned - Theorem 4.7: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned - Proposition 4.8: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned - Theorem 4.10: condition added (bookkeeping or a pinned reading) - Lemma D.1: condition added (bookkeeping or a pinned reading) - Corollary O5.1: condition added (bookkeeping or a pinned reading); ambiguous; a reading was pinned - `ass:structure-consistency`: ambiguous; a reading was pinned - `ass:structure-threshold-crossing`: ambiguous; a reading was pinned - `ass:structure-uniform-resistance`: ambiguous; a reading was pinned - `ass:target-weight-identifiable`: ambiguous; a reading was pinned - `ass:continuity-p-on-components`: ambiguous; a reading was pinned - `ass:phi-smooth-differentiable`: ambiguous; a reading was pinned - `ass:phi-smooth-derivative`: ambiguous; a reading was pinned - `ass:overlap-policy`: ambiguous; a reading was pinned - `ass:1d-ot-probability-laws`: ambiguous; a reading was pinned - `ass:j-specialization-continuity`: ambiguous; a reading was pinned - `ass:dml-normality-first-stage`: ambiguous; a reading was pinned - `ass:dml-normality-score-clipping`: ambiguous; a reading was pinned - `ass:dml-normality-nuisance-rates`: ambiguous; a reading was pinned - `ass:dml-normality-binary`: ambiguous; a reading was pinned - `ass:continuous-learners-first-stage`: ambiguous; a reading was pinned - `ass:continuous-learners-outcome`: ambiguous; a reading was pinned - `ass:continuous-learners-boundary-grid-rate`: ambiguous; a reading was pinned - `ass:ot-formulation-untreated-compatibility`: ambiguous; a reading was pinned - `def:mte`: ambiguous; a reading was pinned - `def:ot-formulation`: ambiguous; a reading was pinned - `def:wasserstein-distance`: ambiguous; a reading was pinned - `def:range-regions`: ambiguous; a reading was pinned - `def:range-decomposition-cov`: ambiguous; a reading was pinned - `def:discrete-nuisance-components`: ambiguous; a reading was pinned - `def:j-specialization-boundaries`: ambiguous; a reading was pinned - `def:boundary-indicator-means`: ambiguous; a reading was pinned - `def:binary-score`: ambiguous; a reading was pinned - `def:dml-normality-binary-nuisances`: ambiguous; a reading was pinned - `def:boundary-treated-law`: ambiguous; a reading was pinned - `def:continuous-instrument`: ambiguous; a reading was pinned - `def:quantile-endpoint-convention`: ambiguous; a reading was pinned - `def:continuous-learners-selected-law`: ambiguous; a reading was pinned - `def:continuous-learners-clipped-grid`: ambiguous; a reading was pinned - `def:dml-level-recovery`: ambiguous; a reading was pinned - `def:dml-estimation-procedure`: ambiguous; a reading was pinned - `def:dml-continuous-outcome-algorithm`: ambiguous; a reading was pinned - `def:discrete-lower-bound-quantiles`: ambiguous; a reading was pinned - `def:continuous-lower-bound-quantiles`: ambiguous; a reading was pinned - `def:covariate-continuous-lower-bound-quantiles`: ambiguous; a reading was pinned - `def:dml-binary-rearrangement`: ambiguous; a reading was pinned - `def:covariate-continuous-lower-bound-rank-kernel`: ambiguous; a reading was pinned - `def:continuous-outcome-score`: ambiguous; a reading was pinned - `\mathcal{Y}`: notation ambiguous; a reading was pinned - `Y`: notation ambiguous; a reading was pinned - `\mathcal{Z}`: notation ambiguous; a reading was pinned - `\mathcal{X}`: notation ambiguous; a reading was pinned - `X`: notation ambiguous; a reading was pinned - `U`: notation ambiguous; a reading was pinned - `p`: notation ambiguous; a reading was pinned - `q`: notation ambiguous; a reading was pinned - `\phi`: notation ambiguous; a reading was pinned - `P`: notation ambiguous; a reading was pinned - `W^q`: notation ambiguous; a reading was pinned - `\omega`: notation ambiguous; a reading was pinned - `\theta_\omega`: notation ambiguous; a reading was pinned - `\theta_{\omega,w}`: notation ambiguous; a reading was pinned - `\hat{\underline{\theta}}_{\omega,1}`: notation ambiguous; a reading was pinned - `\pi_w`: notation ambiguous; a reading was pinned - `\tilde{\pi}`: notation ambiguous; a reading was pinned - `\tilde{\pi}_w`: notation ambiguous; a reading was pinned - `\mathcal{R}_w`: notation ambiguous; a reading was pinned - `H_{w,x}`: notation ambiguous; a reading was pinned - `H_w`: notation ambiguous; a reading was pinned - `K_{w,x}`: notation ambiguous; a reading was pinned - `\mu`: notation ambiguous; a reading was pinned - `\nu`: notation ambiguous; a reading was pinned - `c`: notation ambiguous; a reading was pinned - `\xi`: notation ambiguous; a reading was pinned - `K`: notation ambiguous; a reading was pinned - `p_i`: notation ambiguous; a reading was pinned - `I_i`: notation ambiguous; a reading was pinned - `\mu_{1,i}`: notation ambiguous; a reading was pinned - `Q_{Y,i}`: notation ambiguous; a reading was pinned - `Q_{\omega,i}`: notation ambiguous; a reading was pinned - `\underline{p}_k`: notation ambiguous; a reading was pinned - `\overline{p}_k`: notation ambiguous; a reading was pinned - `K_x`: notation ambiguous; a reading was pinned - `L_k`: notation ambiguous; a reading was pinned - `G_k`: notation ambiguous; a reading was pinned - `\mu_{1,k}`: notation ambiguous; a reading was pinned - `\mu_{1,k\mid x}`: notation ambiguous; a reading was pinned - `Q_{Y,k}`: notation ambiguous; a reading was pinned - `Q_{\omega,k}`: notation ambiguous; a reading was pinned - `Q_{Y,k\mid x}`: notation ambiguous; a reading was pinned - `Q_{\omega,k\mid x}`: notation ambiguous; a reading was pinned - `g_1^{\mathrm c}`: notation ambiguous; a reading was pinned - `\mathcal A`: notation ambiguous; a reading was pinned - `p_k`: notation ambiguous; a reading was pinned - `q_{j,k}`: notation ambiguous; a reading was pinned - `l_k`: notation ambiguous; a reading was pinned - `S_k`: notation ambiguous; a reading was pinned - `T_{j,k}`: notation ambiguous; a reading was pinned - `R_x`: notation ambiguous; a reading was pinned - `c_{\text{gap}}`: notation ambiguous; a reading was pinned - `c_\pi`: notation ambiguous; a reading was pinned - `c_h`: notation ambiguous; a reading was pinned - `\gamma_{\text{full},k}`: notation ambiguous; a reading was pinned - `\gamma_{j,k}`: notation ambiguous; a reading was pinned - `\gamma_K`: notation ambiguous; a reading was pinned - `\pi_k`: notation ambiguous; a reading was pinned - `\tau_{j,k}`: notation ambiguous; a reading was pinned - `a_{j,k}`: notation ambiguous; a reading was pinned - `\kappa_{j,k}`: notation ambiguous; a reading was pinned - `J_{\text{full},k}`: notation ambiguous; a reading was pinned - `J_{j,k}`: notation ambiguous; a reading was pinned - `\Delta_K`: notation ambiguous; a reading was pinned - `\nu_{j,k}`: notation ambiguous; a reading was pinned - `P_{1,k}`: notation ambiguous; a reading was pinned - `h_{j,k}^{-}`: notation ambiguous; a reading was pinned - `n`: notation ambiguous; a reading was pinned - `I_0`: notation ambiguous; a reading was pinned - `I_1`: notation ambiguous; a reading was pinned - `I_2`: notation ambiguous; a reading was pinned - `\eta`: notation ambiguous; a reading was pinned - `\hat\eta`: notation ambiguous; a reading was pinned - `\eta_0`: notation ambiguous; a reading was pinned - `\delta`: notation ambiguous; a reading was pinned - `\mathcal T`: notation ambiguous; a reading was pinned - `\psi`: notation ambiguous; a reading was pinned - `\hat\psi`: notation ambiguous; a reading was pinned - `J_{j,k}^{+}`: notation ambiguous; a reading was pinned - `J_{j,k}^{-}`: notation ambiguous; a reading was pinned - `J_{\text{full},k}^{+}`: notation ambiguous; a reading was pinned - `J_{\text{full},k}^{-}`: notation ambiguous; a reading was pinned - `M_{j,k}^{+}`: notation ambiguous; a reading was pinned - `M_{j,k}^{-}`: notation ambiguous; a reading was pinned - `R_{q,j,k}`: notation ambiguous; a reading was pinned - `R_{P,k}`: notation ambiguous; a reading was pinned - `R_{p,k}`: notation ambiguous; a reading was pinned - `I_{j,k}^{-}`: notation ambiguous; a reading was pinned - `D_{j,k}^{-}`: notation ambiguous; a reading was pinned - `E_{j,k}^{-}`: notation ambiguous; a reading was pinned - `\psi_{j,k}^J`: notation ambiguous; a reading was pinned - `\psi_{\text{full},k}^J`: notation ambiguous; a reading was pinned - `\psi_{j,k}^{\text{prod}}`: notation ambiguous; a reading was pinned - `\psi_{\text{full},k}^{\text{prod}}`: notation ambiguous; a reading was pinned - `\psi_{\Delta_K}`: notation ambiguous; a reading was pinned - `\psi_{j,k}^{h}`: notation ambiguous; a reading was pinned - `\underline{p}`: notation ambiguous; a reading was pinned - `\overline{p}`: notation ambiguous; a reading was pinned - `g_1`: notation ambiguous; a reading was pinned - `\hat p`: notation ambiguous; a reading was pinned - `\hat q`: notation ambiguous; a reading was pinned - `\hat P`: notation ambiguous; a reading was pinned - `\hat{\underline p}`: notation ambiguous; a reading was pinned - `\hat{\overline p}`: notation ambiguous; a reading was pinned - `\hat q^{\mathrm{tr}}`: notation ambiguous; a reading was pinned - `\hat g_1`: notation ambiguous; a reading was pinned - `\hat\mu_{1,\underline p\mid x}`: notation ambiguous; a reading was pinned - `\widehat Q_{1,\underline p\mid x}`: notation ambiguous; a reading was pinned - `\hat\pi_k`: notation ambiguous; a reading was pinned - `\hat\tau_{j,k}`: notation ambiguous; a reading was pinned - `r_{e,n}`: notation ambiguous; a reading was pinned - `r_{g,n}`: notation ambiguous; a reading was pinned - `r_{\mu,n}`: notation ambiguous; a reading was pinned - `r_{p,n}`: notation ambiguous; a reading was pinned - `r_{g,\mathrm{loc},n}`: notation ambiguous; a reading was pinned - `r_{Q,n}`: notation ambiguous; a reading was pinned - `\Lambda`: notation ambiguous; a reading was pinned - `\hat\Lambda`: notation ambiguous; a reading was pinned - `\delta_n`: notation ambiguous; a reading was pinned - `\varepsilon_n`: notation ambiguous; a reading was pinned - `M_g`: notation ambiguous; a reading was pinned - `M_Q`: notation ambiguous; a reading was pinned - `u_m`: notation ambiguous; a reading was pinned - `\tau_m`: notation ambiguous; a reading was pinned - `\mathcal F_n`: notation ambiguous; a reading was pinned - `\mathcal F_Q`: notation ambiguous; a reading was pinned - `\widetilde Q`: notation ambiguous; a reading was pinned - `\rho_\tau`: notation ambiguous; a reading was pinned - `\widehat C`: notation ambiguous; a reading was pinned - `\hat\tau_i`: notation ambiguous; a reading was pinned - `\hat\psi_1`: notation ambiguous; a reading was pinned - `\hat\psi_3`: notation ambiguous; a reading was pinned - `\rho_j`: notation ambiguous; a reading was pinned - `L_\phi`: notation ambiguous; a reading was pinned - `U_\ell`: notation ambiguous; a reading was pinned - `\hat U_\ell`: notation ambiguous; a reading was pinned - `\mathcal S_x`: notation ambiguous; a reading was pinned - `\Pi_x`: notation ambiguous; a reading was pinned - `\mu^{\mathrm{sel}}_{1,n\mid x}`: notation ambiguous; a reading was pinned - `Q^{\mathrm{sel}}_{1,n\mid x}`: notation ambiguous; a reading was pinned - `\widetilde Q_m^c`: notation ambiguous; a reading was pinned - `\mathcal J_{Q,n}`: notation ambiguous; a reading was pinned - `\pi`: notation ambiguous; a reading was pinned - `\mathcal B`: notation ambiguous; a reading was pinned - `\indep`: notation ambiguous; a reading was pinned - `\Gamma_{1,x}`: notation ambiguous; a reading was pinned - `B_Y`: notation ambiguous; a reading was pinned - `\bar p_2`: notation ambiguous; a reading was pinned - `\widehat S_k`: notation ambiguous; a reading was pinned - `\widehat T_{j,k}`: notation ambiguous; a reading was pinned - `\alpha_{p_k}`: notation ambiguous; a reading was pinned - `\alpha_{p_{k+1}}`: notation ambiguous; a reading was pinned - `IF_{J^+}`: notation ambiguous; a reading was pinned - `IF_{J^-}`: notation ambiguous; a reading was pinned - `IF_{M_{j,k}^+}`: notation ambiguous; a reading was pinned - `IF_{M_{j,k}^-}`: notation ambiguous; a reading was pinned - `\alpha_{P,k+1}^{(j,k)}`: notation ambiguous; a reading was pinned - `\alpha_{P,k}^{(j,k)}`: notation ambiguous; a reading was pinned - `\alpha_{p,k+1}^{(j,k)}`: notation ambiguous; a reading was pinned - `\alpha_{p,k}^{(j,k)}`: notation ambiguous; a reading was pinned - `m`: notation ambiguous; a reading was pinned - `\nu_{k\mid x}`: notation ambiguous; a reading was pinned - `T_x`: notation ambiguous; a reading was pinned - `\rho_{k\mid x}`: notation ambiguous; a reading was pinned - `F_{\omega,k\mid x}`: notation ambiguous; a reading was pinned - `R_{k,x}`: notation ambiguous; a reading was pinned - `S`: notation ambiguous; a reading was pinned - `\pi_{1,x}^\star`: notation ambiguous; a reading was pinned - `\mathcal E_n`: notation ambiguous; a reading was pinned - `r_{\mathrm{reg},n}`: notation ambiguous; a reading was pinned - `r_{\mathrm{wt},n}`: notation ambiguous; a reading was pinned - `r_{\mathrm{core},n}`: notation ambiguous; a reading was pinned 12 result(s) of the paper were examined. None was found wrong as printed. In Lean: 12 proved, 0 taken from the literature as cited, 0 not certified, 0 not formalized. ## What was wrong and what was done No error was found in the results examined. ## 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 2.3.** For $\rho\in\{\mu,\nu\}$ and $t\in(0,1)$, $Q_\rho(t)=\inf\{a\in\mathbb R:\rho((\! -\infty,a])\geqslant t\}$. - **Theorem 2.3.** For $00$ is the same constant used for $\pi_k$ in \cref{asp:overlap}. - **Theorem 4.7.** For a fresh $X$ with the population marginal law and independent of $I_1$, $\|\hat p(z,\cdot)-p(z,\cdot)\|_{L_2}^2=\mathbb E[(\hat p(z,X)-p(z,X))^2\mid I_1]$, and $n^{1/4}\max_{z\in\mathcal Z}\|\hat p(z,\cdot)-p(z,\cdot)\|_{L_2}\xrightarrow{P}0$ as $n\to\infty$. - **Theorem 4.7.** Pointwise in $x$, the fitted probability, propensity, share, and indicator-mean nuisances lie in $[0,1]$, $\hat\gamma_{\mathrm{full},k}(x)\in[-1,1]$, $\hat\nu_{j,k}(x),\hat J_{\mathrm{full},k}^{\pm}(x)\in[y_{\min},y_{\max}]$, and $\hat J_{j,k}^{\pm}(x)\in[\min\{0,y_{\min}\},\max\{0,y_{\max}\}]$, for each applicable fitted component; denominator clipping is applied further as stated. - **Theorem 4.7.** Componentwise $L_2$ errors integrate over the marginal law of $X$ under $\mathbb P_{\mathrm{obs}}$, holding each fitted function and any instrument argument $z$ fixed; $o_P(n^{-1/4})$ refers to convergence in probability over the observed sample and random partition as $n\to\infty$. - **Theorem 4.7.** $Y\in\{0,1\}$ almost surely. - **Theorem 4.7.** Two remaining labeled vertices are connected exactly when their averaged values differ by less than $c_{\mathrm{gap}}/2$. - **Theorem 4.7.** In this definition, $\psi_{\mathrm{full},k}^{\mathrm{prod}}$ and $\psi_{j,k}^{\mathrm{prod}}$ are the binary-outcome components of Section~\ref{app:scores_discrete}, and $\psi_{\Delta_K}$ is the common tail component of Section~\ref{app:scores_continuous}. - **Theorem 4.7.** The binary nuisance list supplements the separately fitted first-stage propensity, so $\eta(x)=\{p(z,x),p_k(x),P_{1,k}(x),\pi_k(x),\tau_{j,k}(x),a_{j,k}(z,x),\gamma_{\mathrm{full},k}(x),\gamma_{j,k}(x),\gamma_K(x)\}_{j,k}$, with $p_0=P_{1,0}=0$ fixed and no fitted $\pi_0$. - **Theorem 4.7.** The structural-zero assignment in Algorithm~\ref{alg:dml_continuous_outcome} is $\widehat S_0=\{z:\bar p_2(z)\leqslant\rho_n\}$, with deterministic $\rho_n\downarrow0$ satisfying $n^{1/4}\rho_n\to\infty$. - **Theorem 4.7.** The estimated baseline bins used to index $\widehat T_{j,k}$ include their left endpoints and exclude their right endpoints for $k0$ and $00\}|$, excluding the structural boundary $p_0=0$. In Theorem~\ref{thm:continuous_lower_bound}: $K$ is the number of ordered maximal closed intervals in $p(\mathcal Z)$ after merging overlaps, including degenerate singleton intervals and any interval containing zero. In Proposition~\ref{prop:discrete_reduction}, Proposition~\ref{prop:J_specialization}, Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: $K$ counts the distinct positive baseline propensity levels at each $x$ and does not depend on $x$; $k=0,\ldots,K-1$ indexes the identified intervals and $k=K$ indexes the residual tail. In Corollary~\ref{cor:continuous_primitive_rate}: $K(t)$ is the base one-dimensional smoothing kernel in $K_{h_n}(t)=K(t/h_n)/h_n$, rather than a propensity-level count. - $p_i$: $p_i$ denotes an ordered unique positive baseline propensity level for $1\leqslant i\leqslant K$, with structural integration boundaries $p_0=0$ and $p_{K+1}=1$; a structural boundary need not be an attained positive propensity level. In Theorem~\ref{thm:discrete_lower_bound}: The levels satisfy $0=p_00$ for every $x\in\mathcal X$ and $k=1,\ldots,K$. - $\tau_{j,k}$: $\tau_{j,k}(x)=\mathbb P_{\mathrm{obs}}(Z\in T_{j,k}\mid X=x)$ is a population conditional probability in $[0,1]$, with fitted policy-set nuisances indexed by nonempty $T_{j,k}$, $k=0,\ldots,K$ and $j=1,\ldots,l_k$; structural boundary sets are empty and have probability zero. In Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: $\tau_{j,k}(x)\geqslant c_\pi>0$ for every $x\in\mathcal X$ and every nonempty $T_{j,k}$. - $a_{j,k}$: $a_{j,k}(z,x)=\mathbb P_{\mathrm{obs}}(Z=z\mid Z\in T_{j,k},X=x)$ is the population assignment-share function for each nonempty policy set, indexed by $k=0,\ldots,K$ and $j=1,\ldots,l_k$, with $a_{j,k}(z,x)=0$ outside $T_{j,k}$ and $\sum_{z\in T_{j,k}}a_{j,k}(z,x)=1$ wherever the conditioning probability is positive. - $\kappa_{j,k}$: $\kappa_{j,k}(x)$ is the population relative threshold for a nonempty baseline interval, indexed by $k=0,\ldots,K-1$ and $j=0,\ldots,l_k+1$, with $\kappa_{0,k}(x)=0$ and $\kappa_{l_k+1,k}(x)=1$. - $J_{\text{full},k}$: $J_{\text{full},k}(x)$ is the population full-interval contribution $(p_{k+1}(x)-p_k(x))\mathbb E_{\mu_{1,k\mid x}}[Y]$ for $k=0,\ldots,K-1$. In Proposition~\ref{prop:J_specialization}, Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: In the binary-outcome case, $J_{\text{full},k}(x)=P_{1,k+1}(x)-P_{1,k}(x)$, where $P_{1,k}(x)=\mathbb E_{\text{obs}}[YW\mid Z\in S_k,X=x]$ for $k\geqslant1$ and $P_{1,0}(x)=0$. - $J_{j,k}$: $J_{j,k}(x)=(p_{k+1}(x)-p_k(x))\int_{\kappa_{j,k}(x)}^{\kappa_{j+1,k}(x)}Q_{Y,k\mid x}(u)\,\mathrm du$ is the population lower-bound subinterval contribution on a nondegenerate identified bin, with structural thresholds $\kappa_{0,k}=0$ and $\kappa_{l_k+1,k}=1$. In Proposition~\ref{prop:discrete_reduction}, Proposition~\ref{prop:J_specialization}, Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: The bound uses $J_{j-1,k}$ for $0\leq k\max\{q_{j-1,k}(x),p_{k+1}(x)-J_{\mathrm{full},k}(x)\}\big)=\indicator(h_{j,k}^{-}(x)>0)$ is the binary-score active-set indicator for $k=0,\ldots,K-1$ and $j=1,\ldots,l_k$, with $q_{0,k}(x)=p_k(x)$. - $D_{j,k}^{-}$: $D_{j,k}^{-}(x)=\indicator\!\big(q_{j-1,k}(x)p_{k+1}(x)-J_{\mathrm{full},k}(x)\big)$ is the binary-score dominance indicator for $k=0,\ldots,K-1$ and $j=1,\ldots,l_k$, with $q_{0,k}(x)=p_k(x)$; its strict inequality assigns zero at equality. - $\psi_{j,k}^J$: $\psi_{j,k}^{J}(O;\eta)$ is the continuous-outcome uncentered lower-bound score for $\mathbb E[J_{j,k}(X)]$, used for $0\leq k0$ is the fixed, sample-size-indexed localization window used in the fitted selection rule $\hat P\le\hat{\underline p}(X)+\delta_n$. In Corollary~\ref{cor:continuous_primitive_rate}: $\delta_n\to0$ as $n\to\infty$. - $\varepsilon_n$: $\varepsilon_n$ is the quantile-analysis trimming sequence satisfying $M_Q^{-1}\le\varepsilon_n<1/2$ and $\varepsilon_n\to0$; it restricts the learner analysis to intermediate quantiles without changing the estimator. - $M_g$: $M_g$ is a fixed positive integer tuning choice for the regression grid at each sample size, giving spacing $M_g^{-1}$ and $M_g+1$ grid points $u_m$, $m=0,\ldots,M_g$. - $M_Q$: $M_Q$ is the integer resolution of the quantile grid $\tau_m=m/M_Q$, $m=1,\ldots,M_Q-1$, and may vary with $n$. In Corollary~\ref{cor:continuous_primitive_rate}: $M_Q^{-1}\to0$, so the quantile grid resolution diverges with sample size. - $u_m$: $u_m=m/M_g$ for $m=0,\ldots,M_g$ is the fixed regression index grid spanning $[0,1]$, including both endpoints. - $\tau_m$: $\tau_m=m/M_Q$ for $m=1,\ldots,M_Q-1$ denotes the interior quantile grid; the interpolation endpoint levels $0$ and $1$ are separate from these indexed grid points. - $\mathcal F_n$: $\mathcal F_n$ is a user-chosen, potentially $n$-dependent class of real-valued functions of $x$, used for both the intercept function $g$ and the slope function $\beta$ in each local-linear fit. - $\mathcal F_Q$: $\mathcal F_Q$ is the chosen class of real-valued functions of $x$ over which the boundary conditional quantile check-function regression is minimized; with no covariates it is the class of constants. - $\widetilde Q$: $\widetilde Q$ is the raw interior-grid check-regression fit on the treated localized sample from $I_1$, whose selection rule is determined by the first stage on $I_0$, before clipping, rearrangement, and interpolation. - $\rho_\tau$: $\rho_\tau(u)=u(\tau-\indicator(u<0))$ is the check loss for $u\in\mathbb R$ at the quantile levels $\tau=\tau_m=m/M_Q$, $m=1,\ldots,M_Q-1$, used in the boundary quantile regression. - $\widehat C$: $\widehat C(\tau\mid x)=\int_0^\tau\widehat Q_{1,\underline p\mid x}(s)\,\mathrm ds$ for $\tau\in[0,1]$ and $x\in\mathcal X$, with grid values computed exactly by the trapezoid rule and off-grid evaluation implemented by linear interpolation of adjacent grid values. - $\hat\tau_i$: For $i\in I_2$ with $\hat{\underline p}(X_i)>0$, $\hat\tau_i=\min\{1,\hat q(Z_i,X_i)/\hat{\underline p}(X_i)\}\in[0,1]$ is the normalized boundary-integration cutoff; when $\hat{\underline p}(X_i)=0$ it is not evaluated, and when $\hat q(Z_i,X_i)\geqslant\hat{\underline p}(X_i)>0$ it equals one. - $\hat\psi_1$: $\hat\psi_1(O)$ denotes the estimated lower-bound boundary contribution $-\int_{\min\{\hat{\underline p}(X),\hat q(Z,X)\}}^{\hat{\underline p}(X)}\widehat Q_{1,\underline p\mid X}(v/\hat{\underline p}(X))\,\mathrm dv$, with value zero when $\hat{\underline p}(X)=0$. In Corollary~\ref{cor:continuous_primitive_rate}: For $\hat{\underline p}(X_i)>0$, the concrete implementation evaluates $-\hat{\underline p}(X_i)[\widehat C(1\mid X_i)-\widehat C(\hat\tau_i\mid X_i)]$, where $\hat\tau_i=\min\{1,\hat q(Z_i,X_i)/\hat{\underline p}(X_i)\}$; cumulative-integral grid values are computed by the trapezoid rule, and off-grid lookup uses linear interpolation between adjacent cumulative-integral grid values. For $\hat{\underline p}(X_i)=0$, it returns zero without evaluating $\hat\tau_i$. - $\hat\psi_3$: $\hat\psi_3(O)=y_{\min}(\hat q(Z,X)-\hat{\overline p}(X))_+$ is the fitted lower-bound tail score, evaluated at $O_i$ for $i\in I_2$ with the fitted score map held fixed conditional on $I_0,I_1$. - $\rho_j$: For $j\in\{0,1,2\}$, $\rho_j\in(0,1)$ is the limiting split proportion $|I_j|/n\to\rho_j$, with $\rho_0+\rho_1+\rho_2=1$. - $L_\phi$: $L_\phi<\infty$ is a uniform Lipschitz bound satisfying $|\phi(z,x,p')-\phi(z,x,p)|\leqslant L_\phi|p'-p|$ for every $(z,x,p,p')\in\mathcal Z\times\mathcal X\times[0,1]^2$. - $U_\ell$: For $\ell\in\{1,2\}$, $U_1=q^{\mathrm{tr}}(Z,X)$ and $U_2=P=p(Z,X)$ are the population evaluation indices, with $U_\ell\in\mathcal S_X$; their estimated counterparts are $\hat U_1=\hat q^{\mathrm{tr}}(Z,X)$ and $\hat U_2=\hat P$. - $\hat U_\ell$: For $\ell\in\{1,2\}$, $\hat U_1=\hat q^{\mathrm{tr}}(Z,X)$ and $\hat U_2=\hat P$ are the two target-evaluation indices, with evaluation over $(Z,X)$ conditional on $I_0,I_1$, and the population comparison for $\hat g_1(\hat U_\ell,X)$ is $g_1(\Pi_X(\hat U_\ell),X)$. - $\mathcal S_x$: $\mathcal S_x=[\underline p(x),\overline p(x)]$ is the population propensity interval at covariate value $x$, used as the domain of the population comparison after projection by $\Pi_x$. - $\Pi_x$: $\Pi_x:\mathbb R\to\mathcal S_x$ is projection onto the population support interval $\mathcal S_x=[\underline p(x),\overline p(x)]$, given by $\Pi_x(u)=\min\{\overline p(x),\max\{\underline p(x),u\}\}$. - $\mu^{\mathrm{sel}}_{1,n\mid x}$: $\mu^{\mathrm{sel}}_{1,n\mid x}$ is the conditional population outcome law under the fitted selection event $\{W=1,\hat p(Z,X)\le\hat{\underline p}(x)+\delta_n\}$ given $X=x$ and $I_0$, with the canonical boundary law used when that event has zero conditional probability. - $Q^{\mathrm{sel}}_{1,n\mid x}$: $Q^{\mathrm{sel}}_{1,n\mid x}:(0,1)\to\mathcal Y$ is the quantile function of the $I_0$-conditional selected law $\mu^{\mathrm{sel}}_{1,n\mid x}$, indexed by sample size $n$ and covariate value $x$, including its stipulated zero-selection version. - $\widetilde Q_m^c$: $\widetilde Q_m^c(x)$ is the clipped pre-rearrangement fit at $\tau_m=m/M_Q$, for $m=1,\ldots,M_Q-1$, obtained from the boundary quantile regression on $\mathcal I_{\delta,1}\subseteq I_1$ using the selection rule determined on $I_0$. - $\mathcal J_{Q,n}$: $\mathcal J_{Q,n}=\{m\in\{1,\ldots,M_Q-1\}:\varepsilon_n\le\tau_m\le1-\varepsilon_n\}$ is the interior grid subset used for analytical quantile trimming; it does not trim the estimator. - $\pi$: The meaning of $\pi$ is determined by its arguments: transport notation denotes a joint probability measure, whereas $\pi(z,x)$ denotes the conditional instrument assignment probability $\mathbb P(Z=z\mid X=x)$. In Theorem~\ref{thm:1d_ot}: $\pi$ ranges over $\Pi(\mu,\nu)\subseteq\mathcal P(\mathbb R^2)$. In Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: In instrument-specific probability expressions, $\pi(z,x)=\mathbb P_{\mathrm{obs}}(Z=z\mid X=x)$, rather than a transport coupling. - $\mathcal B$: The notation $\mathcal B$ denotes an OT state space when standing alone and the Borel $\sigma$-field operator when applied to an outcome domain. In Theorem~\ref{thm:1d_ot}: The second marginal state space is $\mathcal B=\mathbb R$. In Theorem~\ref{thm:covariate_continuous_lower_bound}: In the kernel definition, $A\in\mathcal B(\mathcal Y)$ means that $A$ is a Borel subset of $\mathcal Y$. - $\indep$: $\indep$ denotes independence, with $A\indep B\mid C$ denoting conditional independence given $C$, namely factorization of the conditional joint law into the two conditional marginal laws almost surely. - $\Gamma_{1,x}$: $\Gamma_{1,x}$ is the conditional treated feasible class defined for $\mathbb P_X$-almost every $x$, with its members understood through their Lebesgue-almost-everywhere kernel values in $u$. In Theorem~\ref{thm:covariate_continuous_lower_bound}: For $\mathbb P_X$-almost every $x$, the observational constraint equivalently holds as $\int_0^p\pi_{1,x}(\mathrm dy\mid u)\,\mathrm du=H_{1,x}(p,\mathrm dy)$ for every $p\in p(\mathcal Z,x)$, using the canonical continuous version of $H_{1,x}$. - $B_Y$: $B_Y$ is a fixed, finite, nonnegative constant determined by $y_{\min}$ and $y_{\max}$. In Theorem~\ref{thm:covariate_continuous_lower_bound}: $B_Y=\max\{|y_{\min}|,|y_{\max}|\}$ bounds the absolute magnitude of feasible outcomes. In Theorem~\ref{thm:dml_normality}, Corollary~\ref{cor:continuous_primitive_rate}: $B_Y=y_{\max}-y_{\min}$ is the outcome range, used to bound differences between values in $[y_{\min},y_{\max}]$. - $\bar p_2$: $\bar p_2(z)$ is the sample-dependent average of the clipped first-stage fit $\hat p(z,\cdot)$ trained on $I_1$ over the covariates indexed by $I_2$; conditional on $I_1$ it remains an empirical average over evaluation covariates, rather than the population marginal mean. - $\widehat S_k$: $\widehat S_k$ is the sample-dependent baseline level-set estimate obtained from the pre-recovery fits on $I_1$ and their averages over the covariates in $I_2$, with a threshold-defined structural-zero set and positively indexed baseline components ordered by their averages. - $\widehat T_{j,k}$: $\widehat T_{j,k}$ is the sample-dependent policy level-set estimate obtained from the pre-recovery fits on $I_1$ and their averages over the covariates in $I_2$; it depends on both of these inputs and is indexed by the recovered component ordering and baseline bins. - $\alpha_{p_k}$: $\alpha_{p_k}$ is the population lower-boundary representer for the particular component $J_{j,k}$, with the dependence on $j$ suppressed in its notation, for $k=1,\ldots,K-1$ and $j=0,\ldots,l_k-1$; there is no $p_0$ perturbation or correction. - $\alpha_{p_{k+1}}$: $\alpha_{p_{k+1}}$ denotes the upper interval-boundary representer for the continuous-outcome component $J_{j,k}$, with suppressed dependence on $j$, indexed by $k=0,\ldots,K-1$ and $j=0,\ldots,l_k-1$; it remains an upper-side representer at $k=0$. - $IF_{J^+}$: $IF_{J^+}$ denotes the plus-side observation-level residual component indexed implicitly by $k\in\{0,\ldots,K-1\}$ and $j\in\{0,\ldots,l_k-1\}$, evaluated with the nuisance functions supplied to the continuous-outcome score. - $IF_{J^-}$: $IF_{J^-}$ denotes the minus-side observation-level residual component indexed implicitly by $k\in\{0,\ldots,K-1\}$ and $j\in\{0,\ldots,l_k-1\}$, evaluated with the nuisance functions supplied to the continuous-outcome score, with its structural $k=0$ component identically zero. - $IF_{M_{j,k}^+}$: $IF_{M_{j,k}^+}$ is the observation-level residual indexed by $k=0,\ldots,K-1$ and boundary index $j=0,\ldots,l_k$. In Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: In the continuous-outcome part, the no-atom condition at $y_{\min}$ makes the true residual $IF_{M_{0,k}^+}$ vanish almost surely. - $IF_{M_{j,k}^-}$: $IF_{M_{j,k}^-}$ is the observation-level residual indexed by $k=0,\ldots,K-1$ and boundary index $j=0,\ldots,l_k$, with the structural zero convention at $k=0$. In Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: In the continuous-outcome part, the no-atom condition at $y_{\min}$ makes the true residual $IF_{M_{0,k}^-}$ vanish almost surely. - $\alpha_{P,k+1}^{(j,k)}$: $\alpha_{P,k+1}^{(j,k)}(x)$ is the coefficient of a perturbation of $P_{1,k+1}(x)$ in the binary component $h_{j,k}^{-}(x)$, indexed by $k=0,\ldots,K-1$ and $j=1,\ldots,l_k$, holding the propensity and alternative-policy levels and $P_{1,k}$ fixed. - $\alpha_{P,k}^{(j,k)}$: For $k\in\{0,\ldots,K-1\}$ and $j\in\{1,\ldots,l_k\}$, $\alpha_{P,k}^{(j,k)}(x)=-D_{j,k}^{-}(x)$ is the coefficient of the $P_{1,k}$ perturbation in the derivative of $h_{j,k}^{-}$, with the other nuisance arguments held fixed. - $\alpha_{p,k+1}^{(j,k)}$: $\alpha_{p,k+1}^{(j,k)}(x)$ is the direct upper-propensity coefficient for the binary score component $h_{j,k}^{-}$, indexed by $k=0,\ldots,K-1$ and $j=1,\ldots,l_k$, holding $P_{1,k}$, $P_{1,k+1}$, and the alternative-policy levels fixed. - $\alpha_{p,k}^{(j,k)}$: $\alpha_{p,k}^{(j,k)}(x)$ denotes the direct lower-propensity coefficient in the binary score for $h_{j,k}^{-}$, indexed by $k=0,\ldots,K-1$ and $j=1,\ldots,l_k$, holding the success regressions and alternative-policy levels fixed; at $k=0$ its residual correction is omitted because $p_0\equiv0$ is known. - $m$: The notation $m(X;\eta)$ denotes the binary lower-bound integrand, whereas bare $m$ in $u_m$ or $\tau_m$ denotes a grid index. In Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: In the binary-outcome case, $m(X;\eta)$ is the displayed integrand with nuisance argument $\eta$; evaluating at $\eta_0$ gives the population target integrand. In Corollary~\ref{cor:continuous_primitive_rate}: $m$ is an integer grid index: $m=0,\ldots,M_g$ for $u_m=m/M_g$ and $m=1,\ldots,M_Q-1$ for the fitted quantiles at $\tau_m=m/M_Q$. - $\nu_{k\mid x}$: $\nu_{k\mid x}$ is the normalized Lebesgue probability law on $G_k(x)$, equivalently the law of $U$ conditional on $X=x$ and $U\in G_k(x)$, for almost every $x$ and $k=0,\ldots,K_x-1$ with $|G_k(x)|>0$. - $T_x$: $T_x:[0,1]\to\mathbb R$, $T_x(u)=\omega(x,u)$, is the target-weight map with the covariate argument fixed at $x$. - $\rho_{k\mid x}$: $\rho_{k\mid x}=(T_x)_{\#}\mathrm{Unif}(G_k(x))$ is the population probability law of $\omega(x,U)$ for an auxiliary $U\sim\mathrm{Unif}(G_k(x))$, defined for fixed $x$ and $k=0,\ldots,K_x-1$ with $|G_k(x)|>0$. - $F_{\omega,k\mid x}$: $F_{\omega,k\mid x}$ is the population distribution function of the pushforward of $\nu_{k\mid x}$ under $u\mapsto\omega(x,u)$, using a jointly measurable version in $(x,v)$ for almost every $x$ and each nonempty gap indexed by $k=0,\ldots,K_x-1$. - $R_{k,x}$: $R_{k,x}$ is a measurable map of $(u,s)\in G_k(x)\times[0,1]$ for almost every $x$ and each $k\in\{0,\ldots,K_x-1\}$ with $|G_k(x)|>0$, using the left limit of $F_{\omega,k\mid x}$ to distribute each weight atom across its rank interval. - $S$: $S$ is an auxiliary $\mathrm{Unif}(0,1)$ draw independent of $U\sim\nu_{k\mid x}$ for each fixed $x$ and nonempty gap $G_k(x)$, used to randomize ties in $R_{k,x}(U,S)$ and integrated out in the attaining outcome kernel. - $\pi_{1,x}^\star$: $\pi_{1,x}^\star$ is a jointly measurable population probability kernel for $Y(1)$ given $(X,U)=(x,u)$, defined for almost every $x$ and pasted across the nonempty gaps, non-degenerate LIV intervals and trivial tail, with arbitrary probability-law values at region boundaries. - $\mathcal E_n$: $\mathcal E_n$ is the event of simultaneous exact recovery of all baseline and policy level sets with their population ordering and bin indices, including $\widehat S_0=S_0$, by the procedure using first-stage fits on $I_1$ and their averages over $\{X_i:i\in I_2\}$. - $r_{\mathrm{reg},n}$: $r_{\mathrm{reg},n}$ is the maximum population $L_2$ error of the fitted outcome regressions, propensity regressions, pooled policy levels, and boundary quantiles, with integration over the marginal law of $X$ while holding the fitted functions fixed. In Theorem~\ref{thm:dml_normality}: In the continuous-outcome proof, the errors are those of the oracle nuisance fits trained on $I_1$ using the population level sets; hence their magnitudes are fixed conditional on $I_1$ and satisfy $r_{\mathrm{reg},n}=o_P(n^{-1/4})$. - $r_{\mathrm{wt},n}$: $r_{\mathrm{wt},n}$ is the maximum of the componentwise $L_2(\mathbb P_{\mathrm{obs}}(\mathrm{d}x))$ errors of the fitted weights $\hat\pi_k,\hat\tau_{j,k},\hat a_{j,k}(z,\cdot),\hat\gamma_{j,k},\hat\gamma_{\mathrm{full},k},\hat\gamma_K$ relative to their population counterparts, over the finitely many fitted indices and instrument categories. - $r_{\mathrm{core},n}$: $r_{\mathrm{core},n}$ is the maximum of the $L_2(\mathbb P_X)$ norms of fitted-minus-population errors for the finitely many $P_{1,k}$, $p_k$, instrument-specific $p(z,\cdot)$, and pooled $q_{j,k}$ nuisances used by the binary oracle score, with the fitted $P_{1,k}$ taken after clipping and rearrangement and the structural zero boundaries retained. **Objections raised during the review and withdrawn.** The paper's owner showed that each misreads the paper; the printed statement stands under the reading given. - **Theorem 3.4.** A finite instrument domain $\mathcal Z$ carries the discrete topology, so the common-support condition $\mathrm{Supp}(Z\mid X=x)=\mathcal Z$ says that every instrument value has positive conditional probability given $X=x$. - **Proposition 4.3.** A finite instrument domain $\mathcal Z$ carries the discrete topology, so the common-support condition $\mathrm{Supp}(Z\mid X=x)=\mathcal Z$ says that every instrument value has positive conditional probability given $X=x$. ## 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 2.3 | proved in Lean, under 2 condition(s) made explicit | `Papers.TbsPrteOptimalTransport.productCost_transport_extrema` (`T1dOt.lean:22`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Proposition 3.1 | proved in Lean, under 4 condition(s) made explicit | `Papers.TbsPrteOptimalTransport.feasible_iff_structural` (`TSharpness.lean:25`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Theorem 3.4 | proved in Lean, under 14 condition(s) made explicit | `Papers.TbsPrteOptimalTransport.finite_instrument_sharp_bounds` (`TDiscreteLowerBound.lean:40`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Theorem 3.7 | proved in Lean, under 16 condition(s) made explicit | `Papers.TbsPrteOptimalTransport.general_instrument_sharp_bounds` (`TContinuousLowerBound.lean:29`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Theorem 3.8 | proved in Lean, under 13 condition(s) made explicit | `Papers.TbsPrteOptimalTransport.conditional_sharp_bounds` (`TCovariateLowerBound.lean:30`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Proposition 4.3 | proved in Lean, under 14 condition(s) made explicit | `Papers.TbsPrteOptimalTransport.discrete_reduction` (`TDiscreteReduction.lean:28`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Proposition 4.4 | proved in Lean, under 18 condition(s) made explicit | `Papers.TbsPrteOptimalTransport.J_specialization` (`TJSpecialization.lean:117`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Theorem 4.7 | proved in Lean, under 35 condition(s) made explicit | `Papers.TbsPrteOptimalTransport.dml_asymptotic_normality` (`TDmlNormality.lean:485`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Proposition 4.8 | proved in Lean, under 16 condition(s) made explicit | `Papers.TbsPrteOptimalTransport.continuous_reduction` (`TContinuousReduction.lean:30`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Theorem 4.10 | proved in Lean, under 16 condition(s) made explicit | `Papers.TbsPrteOptimalTransport.plugin_rate` (`TContinuousRate.lean:321`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Lemma D.1 | proved in Lean, under 29 condition(s) made explicit | `Papers.TbsPrteOptimalTransport.score_mean_and_orthogonality` (`Helpers/ScoreOrthogonality.lean:117`) | the standard ones only (propext, Classical.choice, Quot.sound) | | Corollary O5.1 | proved in Lean, under 24 condition(s) made explicit | `Papers.TbsPrteOptimalTransport.primitive_learner_rate` (`TContinuousPrimitiveRate.lean:48`) | the standard ones only (propext, Classical.choice, Quot.sound) | **Conditions made explicit during the formalization.** - For each upper : Bool, Integrable (connectedInt M zeroLaw upper (policyProp phi M.prop)) (zxLaw M). 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/draft.pdf` (130 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: `draft.tex` (source sha256 `7f4abb242c51b72bbd1ed2ce4f002f0690d0dfa649fd8452c1eb2ced13df86f7`) - stage completed: `V6` - results in scope: 12 | bucket | count | | --- | --- | | verified | 0 | | verified_with_bookkeeping | 12 | | conditional | 0 | | repaired | 0 | | 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. 1D Optimal Transport with Product Cost; see, e.g., \citep{villani2021topics, galichon2016optimal} `thm:1d-ot` | field | value | | --- | --- | | bucket | `verified_with_bookkeeping` | | source | `sections/preliminaries.tex:88-95` (`theorem`, label `thm:1d_ot`) | | statement | \label{thm:1d_ot} Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}$ with quantile functions $Q_\mu$ and $Q_\nu$. Suppose either that $\mu$ has bounded support and $\nu$ has a finite first moment, or vice versa, and let the cost be the simple product $c(a,b) = a \cdot b$. Then, the minimum and maximum of the expected cost over all valid couplings are completely characterized by the countermonotonic and comonotonic couplings, respectively: \begin{align*} \min_{\pi \in \Pi(\mu, \nu)} \int_{\mathbb{R}^2} a \cdot b \, \mathrm{d}\pi(a, b) &= \int_0^1 Q_\mu(t) Q_\nu(1-t) \, \mathrm{d}t, \\ \max_{\pi \in \Pi(\mu, \nu)} \int_{\mathbb{R}^2} a \cdot b \, \mathrm{d}\pi(a, b) &= \int_0^1 Q_\mu(t) Q_\nu(t) \, \mathrm{d}t. \end{align*} The minimum is attained by the countermonotonic coupling $(Q_\mu(\xi), Q_\nu(1-\xi))$, and the maximum by the comonotonic coupling $(Q_\mu(\xi), Q_\nu(\xi))$, where $\xi \sim \mathrm{Unif}(0,1)$. | | appended hypothesis list | ass:1d-ot-probability-laws, ass:1d-ot-moment-support, ass:1d-ot-product-cost | | Lean | `T1dOt.lean:22` — `Papers.TbsPrteOptimalTransport.productCost_transport_extrema` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity pinned · correctness ok | | added conditions | `ass:pin-20`, `ass:pin-63` | | cited results | — | | assumed substrate gates | — | | findings | [note, V1.5] pinned reading: For $00$ and $00$ is the same constant used for $\pi_k$ in \cref{asp:overlap}. — pins reading of ass:overlap-policy: Both clauses belong to the single quantified overlap assumption. The proof of \cref{thm:dml_normality} uses this assumption to bound the inverse-probability weights uniformly. - side condition added by the confirming referee `ass:pin-20`: For $\rho\in\{\mu,\nu\}$ and $t\in(0,1)$, $Q_\rho(t)=\inf\{a\in\mathbb R:\rho((\! -\infty,a])\geqslant t\}$. — pins reading of ass:1d-ot-probability-laws: The standard generalized-inverse convention for probability quantiles applies to the notation section and Theorem~\ref{thm:1d_ot}; the theorem evaluates these functions at uniform draws and inside Lebesgue integrals. - side condition added by the confirming referee `ass:pin-21`: For each fixed $k\in\{0,\dots,K-1\}$ and $j\in\{0,\dots,l_k-1\}$ to which part~(i) is applied, its continuity hypothesis is $\mu_{1,k\mid X}(\{\nu_{j,k}(X)\})=\mu_{1,k\mid X}(\{\nu_{j+1,k}(X)\})=0$ almost surely under the marginal law of $X$. — pins reading of ass:j-specialization-continuity: The proof of Proposition~\ref{prop:J_specialization}(i) fixes the band indices and uses continuity of the distribution function at the two boundaries to identify their probability levels and exclude boundary atoms. - side condition added by the confirming referee `ass:pin-22`: For a fresh $X$ with the population marginal law and independent of $I_1$, $\|\hat p(z,\cdot)-p(z,\cdot)\|_{L_2}^2=\mathbb E[(\hat p(z,X)-p(z,X))^2\mid I_1]$, and $n^{1/4}\max_{z\in\mathcal Z}\|\hat p(z,\cdot)-p(z,\cdot)\|_{L_2}\xrightarrow{P}0$ as $n\to\infty$. — pins reading of ass:dml-normality-first-stage: The exact-recovery argument in the proof of \cref{thm:dml_normality} compares $\mathbb E[\hat p(z,X)\mid I_1]$ with $\mathbb E[p(z,X)]$ using this norm, before evaluating the fit on independent covariates in $I_2$. - side condition added by the confirming referee `ass:pin-23`: Pointwise in $x$, the fitted probability, propensity, share, and indicator-mean nuisances lie in $[0,1]$, $\hat\gamma_{\mathrm{full},k}(x)\in[-1,1]$, $\hat\nu_{j,k}(x),\hat J_{\mathrm{full},k}^{\pm}(x)\in[y_{\min},y_{\max}]$, and $\hat J_{j,k}^{\pm}(x)\in[\min\{0,y_{\min}\},\max\{0,y_{\max}\}]$, for each applicable fitted component; denominator clipping is applied further as stated. — pins reading of ass:dml-normality-score-clipping: The nuisance definitions and estimation steps in \cref{app:procedure_continuous_outcome,app:procedure_binary_outcome} determine these ranges. The quantile remainder in (\ref{eq:pf_dml_bias}) explicitly uses both quantiles lying in $[y_{\min},y_{\max}]$. - side condition added by the confirming referee `ass:pin-24`: Componentwise $L_2$ errors integrate over the marginal law of $X$ under $\mathbb P_{\mathrm{obs}}$, holding each fitted function and any instrument argument $z$ fixed; $o_P(n^{-1/4})$ refers to convergence in probability over the observed sample and random partition as $n\to\infty$. — pins reading of ass:dml-normality-nuisance-rates: The sampling setup for Theorem~\ref{thm:dml_normality} specifies the random sample and partition; its supplied second-order remainder bound, equation~(\ref{eq:pf_dml_bias}), uses population covariate integrals of the fitted errors. - side condition added by the confirming referee `ass:pin-25`: $Y\in\{0,1\}$ almost surely. — pins reading of ass:dml-normality-binary: Proposition~\ref{prop:J_specialization}\textup{(ii)} explicitly states this coding before defining the binary subinterval contribution used by Theorem~\ref{thm:dml_normality}\textup{(ii)}. - side condition added by the confirming referee `ass:pin-29`: Every constraint in $\eqref{eq:ot_formulation}$ is imposed simultaneously for both $w\in\{0,1\}$. — pins reading of def:ot-formulation: The sentence immediately preceding equation $\eqref{eq:ot_formulation}$ explicitly says that every constraint is imposed for both treatment states. - side condition added by the confirming referee `ass:pin-3`: Two remaining labeled vertices are connected exactly when their averaged values differ by less than $c_{\mathrm{gap}}/2$. — pins reading of def:dml-estimation-procedure: Both Algorithm~\ref{alg:dml_continuous_outcome} and the Level Sets step in Section~\ref{app:procedure_continuous_outcome} explicitly use ``less than''. - side condition added by the confirming referee `ass:pin-32`: For $\mathbb P_X$-almost every $x$, the intervals $L_k(x)$, $1\leqslant k\leqslant K_x$, the gaps $G_k(x)$, $0\leqslant k0$ and $00$ and $00$ is the same constant used for $\pi_k$ in \cref{asp:overlap}. — pins reading of ass:overlap-policy: Both clauses belong to the single quantified overlap assumption. The proof of \cref{thm:dml_normality} uses this assumption to bound the inverse-probability weights uniformly. - side condition added by the confirming referee `ass:pin-20`: For $\rho\in\{\mu,\nu\}$ and $t\in(0,1)$, $Q_\rho(t)=\inf\{a\in\mathbb R:\rho((\! -\infty,a])\geqslant t\}$. — pins reading of ass:1d-ot-probability-laws: The standard generalized-inverse convention for probability quantiles applies to the notation section and Theorem~\ref{thm:1d_ot}; the theorem evaluates these functions at uniform draws and inside Lebesgue integrals. - side condition added by the confirming referee `ass:pin-21`: For each fixed $k\in\{0,\dots,K-1\}$ and $j\in\{0,\dots,l_k-1\}$ to which part~(i) is applied, its continuity hypothesis is $\mu_{1,k\mid X}(\{\nu_{j,k}(X)\})=\mu_{1,k\mid X}(\{\nu_{j+1,k}(X)\})=0$ almost surely under the marginal law of $X$. — pins reading of ass:j-specialization-continuity: The proof of Proposition~\ref{prop:J_specialization}(i) fixes the band indices and uses continuity of the distribution function at the two boundaries to identify their probability levels and exclude boundary atoms. - side condition added by the confirming referee `ass:pin-22`: For a fresh $X$ with the population marginal law and independent of $I_1$, $\|\hat p(z,\cdot)-p(z,\cdot)\|_{L_2}^2=\mathbb E[(\hat p(z,X)-p(z,X))^2\mid I_1]$, and $n^{1/4}\max_{z\in\mathcal Z}\|\hat p(z,\cdot)-p(z,\cdot)\|_{L_2}\xrightarrow{P}0$ as $n\to\infty$. — pins reading of ass:dml-normality-first-stage: The exact-recovery argument in the proof of \cref{thm:dml_normality} compares $\mathbb E[\hat p(z,X)\mid I_1]$ with $\mathbb E[p(z,X)]$ using this norm, before evaluating the fit on independent covariates in $I_2$. - side condition added by the confirming referee `ass:pin-23`: Pointwise in $x$, the fitted probability, propensity, share, and indicator-mean nuisances lie in $[0,1]$, $\hat\gamma_{\mathrm{full},k}(x)\in[-1,1]$, $\hat\nu_{j,k}(x),\hat J_{\mathrm{full},k}^{\pm}(x)\in[y_{\min},y_{\max}]$, and $\hat J_{j,k}^{\pm}(x)\in[\min\{0,y_{\min}\},\max\{0,y_{\max}\}]$, for each applicable fitted component; denominator clipping is applied further as stated. — pins reading of ass:dml-normality-score-clipping: The nuisance definitions and estimation steps in \cref{app:procedure_continuous_outcome,app:procedure_binary_outcome} determine these ranges. The quantile remainder in (\ref{eq:pf_dml_bias}) explicitly uses both quantiles lying in $[y_{\min},y_{\max}]$. - side condition added by the confirming referee `ass:pin-24`: Componentwise $L_2$ errors integrate over the marginal law of $X$ under $\mathbb P_{\mathrm{obs}}$, holding each fitted function and any instrument argument $z$ fixed; $o_P(n^{-1/4})$ refers to convergence in probability over the observed sample and random partition as $n\to\infty$. — pins reading of ass:dml-normality-nuisance-rates: The sampling setup for Theorem~\ref{thm:dml_normality} specifies the random sample and partition; its supplied second-order remainder bound, equation~(\ref{eq:pf_dml_bias}), uses population covariate integrals of the fitted errors. - side condition added by the confirming referee `ass:pin-25`: $Y\in\{0,1\}$ almost surely. — pins reading of ass:dml-normality-binary: Proposition~\ref{prop:J_specialization}\textup{(ii)} explicitly states this coding before defining the binary subinterval contribution used by Theorem~\ref{thm:dml_normality}\textup{(ii)}. - side condition added by the confirming referee `ass:pin-29`: Every constraint in $\eqref{eq:ot_formulation}$ is imposed simultaneously for both $w\in\{0,1\}$. — pins reading of def:ot-formulation: The sentence immediately preceding equation $\eqref{eq:ot_formulation}$ explicitly says that every constraint is imposed for both treatment states. - side condition added by the confirming referee `ass:pin-32`: For $\mathbb P_X$-almost every $x$, the intervals $L_k(x)$, $1\leqslant k\leqslant K_x$, the gaps $G_k(x)$, $0\leqslant k0$ and $00$ is the same constant used for $\pi_k$ in \cref{asp:overlap}. — pins reading of ass:overlap-policy: Both clauses belong to the single quantified overlap assumption. The proof of \cref{thm:dml_normality} uses this assumption to bound the inverse-probability weights uniformly. - clarification (`ass:pin-22`, used by `thm:dml-normality`): For a fresh $X$ with the population marginal law and independent of $I_1$, $\\|\hat p(z,\cdot)-p(z,\cdot)\\|_{L_2}^2=\mathbb E[(\hat p(z,X)-p(z,X))^2\mid I_1]$, and $n^{1/4}\max_{z\in\mathcal Z}\\|\hat p(z,\cdot)-p(z,\cdot)\\|_{L_2}\xrightarrow{P}0$ as $n\to\infty$. — pins reading of ass:dml-normality-first-stage: The exact-recovery argument in the proof of \cref{thm:dml_normality} compares $\mathbb E[\hat p(z,X)\mid I_1]$ with $\mathbb E[p(z,X)]$ using this norm, before evaluating the fit on independent covariates in $I_2$. - clarification (`ass:pin-23`, used by `thm:dml-normality`): Pointwise in $x$, the fitted probability, propensity, share, and indicator-mean nuisances lie in $[0,1]$, $\hat\gamma_{\mathrm{full},k}(x)\in[-1,1]$, $\hat\nu_{j,k}(x),\hat J_{\mathrm{full},k}^{\pm}(x)\in[y_{\min},y_{\max}]$, and $\hat J_{j,k}^{\pm}(x)\in[\min\{0,y_{\min}\},\max\{0,y_{\max}\}]$, for each applicable fitted component; denominator clipping is applied further as stated. — pins reading of ass:dml-normality-score-clipping: The nuisance definitions and estimation steps in \cref{app:procedure_continuous_outcome,app:procedure_binary_outcome} determine these ranges. The quantile remainder in (\ref{eq:pf_dml_bias}) explicitly uses both quantiles lying in $[y_{\min},y_{\max}]$. - clarification (`ass:pin-24`, used by `thm:dml-normality`): Componentwise $L_2$ errors integrate over the marginal law of $X$ under $\mathbb P_{\mathrm{obs}}$, holding each fitted function and any instrument argument $z$ fixed; $o_P(n^{-1/4})$ refers to convergence in probability over the observed sample and random partition as $n\to\infty$. — pins reading of ass:dml-normality-nuisance-rates: The sampling setup for Theorem~\ref{thm:dml_normality} specifies the random sample and partition; its supplied second-order remainder bound, equation~(\ref{eq:pf_dml_bias}), uses population covariate integrals of the fitted errors. - clarification (`ass:pin-25`, used by `thm:dml-normality`): $Y\in\{0,1\}$ almost surely. — pins reading of ass:dml-normality-binary: Proposition~\ref{prop:J_specialization}\textup{(ii)} explicitly states this coding before defining the binary subinterval contribution used by Theorem~\ref{thm:dml_normality}\textup{(ii)}. - clarification (`ass:pin-3`, used by `thm:dml-normality`): Two remaining labeled vertices are connected exactly when their averaged values differ by less than $c_{\mathrm{gap}}/2$. — pins reading of def:dml-estimation-procedure: Both Algorithm~\ref{alg:dml_continuous_outcome} and the Level Sets step in Section~\ref{app:procedure_continuous_outcome} explicitly use ``less than''. - clarification (`ass:pin-36`, used by `thm:dml-normality`): In this definition, $\psi_{\mathrm{full},k}^{\mathrm{prod}}$ and $\psi_{j,k}^{\mathrm{prod}}$ are the binary-outcome components of Section~\ref{app:scores_discrete}, and $\psi_{\Delta_K}$ is the common tail component of Section~\ref{app:scores_continuous}. — pins reading of def:binary-score: The binary nuisance-estimation subsection introduces this display and immediately directs the reader to Section~\ref{app:scores_discrete} for its component formulas. - clarification (`ass:pin-37`, used by `thm:dml-normality`): The binary nuisance list supplements the separately fitted first-stage propensity, so $\eta(x)=\{p(z,x),p_k(x),P_{1,k}(x),\pi_k(x),\tau_{j,k}(x),a_{j,k}(z,x),\gamma_{\mathrm{full},k}(x),\gamma_{j,k}(x),\gamma_K(x)\}_{j,k}$, with $p_0=P_{1,0}=0$ fixed and no fitted $\pi_0$. — pins reading of def:dml-normality-binary-nuisances: Theorem~\ref{thm:dml_normality} separately imposes the instrument-specific first-stage rate before specifying the remaining score nuisances. Section~\ref{app:procedure_binary_outcome} explicitly prints the full vector including $p(z,x)$ and states the structural-zero conventions. - clarification (`ass:pin-4`, used by `thm:dml-normality`): The structural-zero assignment in Algorithm~\ref{alg:dml_continuous_outcome} is $\widehat S_0=\{z:\bar p_2(z)\leqslant\rho_n\}$, with deterministic $\rho_n\downarrow0$ satisfying $n^{1/4}\rho_n\to\infty$. — pins reading of def:dml-continuous-outcome-algorithm: The Level Sets step in Section~\ref{app:procedure_continuous_outcome} explicitly defines this assignment and removes precisely its baseline vertices from the graph. - clarification (`ass:pin-43`, used by `thm:dml-normality`): The estimated baseline bins used to index $\widehat T_{j,k}$ include their left endpoints and exclude their right endpoints for $k0$ and $00$ is the same constant used for $\pi_k$ in \cref{asp:overlap}. (Both clauses belong to the single quantified overlap assumption. The proof of \cref{thm:dml_normality} uses this assumption to bound the inverse-probability weights uniformly.) - `ass:phi-smooth-derivative`: pinned reading: There exist finite constants $C,L\geqslant0$ such that, for all $z\in\mathcal Z$, $x\in\mathcal X$, and $p,p'\in[0,1]$, $\|\partial\phi/\partial p(z,x,p)\|\leqslant C$ and $\|\partial\phi/\partial p(z,x,p)-\partial\phi/\partial p(z,x,p')\|\leqslant L\|p-p'\|$. (The exact-recovery argument for \cref{thm:dml_normality} uses a uniformly bounded derivative, and (\ref{eq:pf_dml_bias}) uses its uniformly Lipschitz derivative for the quadratic policy remainder.) - `ass:phi-smooth-differentiable`: pinned reading: For every $z\in\mathcal Z$ and $x\in\mathcal X$, $p\mapsto\phi(z,x,p)$ is differentiable on $(0,1)$ and has finite right and left derivatives at $p=0$ and $p=1$, respectively; these are its endpoint values of $\partial\phi/\partial p$. (The policy map has domain $\mathcal Z\times\mathcal X\times[0,1]$. The estimation procedure clips $\hat p$ to this interval, and the policy Taylor remainder in (\ref{eq:pf_dml_bias}) uses differentiation within that domain.) - `ass:structure-consistency`: pinned reading: $Y=WY(1)+(1-W)Y(0)$ almost surely under the structural population law. (Assumption \ref{asp:structure} states consistency as a structural equality of random variables; the subsequent observational derivation states conditional identities almost everywhere, consistently with the standard almost-sure convention.) - `ass:structure-threshold-crossing`: pinned reading: $W=\indicator(U\leqslant p(Z,X))$ almost surely under the structural population law. (The threshold-crossing clause of Assumption \ref{asp:structure} is a structural equality of random variables. The derivation preceding equation \ref{eq:canonical_submeasures} uses it to establish observational identities almost everywhere.) - `ass:structure-uniform-resistance`: pinned reading: $\mathcal L(U\mid X=x)=\mathrm{Unif}(0,1)$ for $\mathbb P_X$-almost every $x$. (Assumption \ref{asp:structure} uses conditional-distribution notation without a pointwise covariate quantifier, while its common-support clause explicitly says every covariate value. The surrounding prose also recognizes that conditional probabilities are identified only almost surely.) - `ass:target-weight-identifiable`: pinned reading: Any two structural laws satisfying Structural Assumptions and inducing the same $\mathbb P_{obs}$ give the same $\omega(x,u)$ for $\mathbb P_{obs}(\mathrm dx)\,\mathrm du$-almost every $(x,u)$. (The paragraph following Structural Assumptions states that the observed law identifies conditional probabilities almost surely; equation (\ref{eq:omega_form}) constructs weights from such conditional expectations, and equation (\ref{eq:target_mte}) integrates against the covariate marginal and Lebesgue measure.) - `def:binary-score`: pinned reading: In this definition, $\psi_{\mathrm{full},k}^{\mathrm{prod}}$ and $\psi_{j,k}^{\mathrm{prod}}$ are the binary-outcome components of Section~\ref{app:scores_discrete}, and $\psi_{\Delta_K}$ is the common tail component of Section~\ref{app:scores_continuous}. (The binary nuisance-estimation subsection introduces this display and immediately directs the reader to Section~\ref{app:scores_discrete} for its component formulas.) - `def:boundary-indicator-means`: pinned reading: For $j=1,\ldots,l_k$, interpret $\nu_{j,k}(x)=Q_{Y,k\mid x}(\kappa_{j,k}(x))$ using $Q_{Y,k\mid x}(t)=\inf\{y\in\mathbb R:\mu_{1,k\mid x}(\mathcal Y\cap(-\infty,y])\geqslant t\}$ for $00$ and $00\}\|$, excluding the structural boundary $p_0=0$. In Theorem~\ref{thm:continuous_lower_bound}: $K$ is the number of ordered maximal closed intervals in $p(\mathcal Z)$ after merging overlaps, including degenerate singleton intervals and any interval containing zero. In Proposition~\ref{prop:discrete_reduction}, Proposition~\ref{prop:J_specialization}, Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: $K$ counts the distinct positive baseline propensity levels at each $x$ and does not depend on $x$; $k=0,\ldots,K-1$ indexes the identified intervals and $k=K$ indexes the residual tail. In Corollary~\ref{cor:continuous_primitive_rate}: $K(t)$ is the base one-dimensional smoothing kernel in $K_{h_n}(t)=K(t/h_n)/h_n$, rather than a propensity-level count. — Section~\ref{subsubsec:discrete_iv} defines the positive-level count; equation~\ref{eq:range_decomposition} defines the maximal-interval count; Assumption~\ref{asp:prop_gap}(i) fixes the finite estimation count across covariates; Section~\ref{app:procedure_continuous_instrument} defines $K_{h_n}(t)=K(t/h_n)/h_n$. - reading of the symbol `p_i`: $p_i$ denotes an ordered unique positive baseline propensity level for $1\leqslant i\leqslant K$, with structural integration boundaries $p_0=0$ and $p_{K+1}=1$; a structural boundary need not be an attained positive propensity level. In Theorem~\ref{thm:discrete_lower_bound}: The levels satisfy $0=p_00$ for every $x\in\mathcal X$ and $k=1,\ldots,K$. — The definition at flat.tex line 836, the structural-zero convention following Assumption~\ref{asp:overlap}, and the nuisance procedure at lines 1781--1787 specify these indices and omissions. - reading of the symbol `\tau_{j,k}`: $\tau_{j,k}(x)=\mathbb P_{\mathrm{obs}}(Z\in T_{j,k}\mid X=x)$ is a population conditional probability in $[0,1]$, with fitted policy-set nuisances indexed by nonempty $T_{j,k}$, $k=0,\ldots,K$ and $j=1,\ldots,l_k$; structural boundary sets are empty and have probability zero. In Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: $\tau_{j,k}(x)\geqslant c_\pi>0$ for every $x\in\mathcal X$ and every nonempty $T_{j,k}$. — The policy-set boundary conventions at flat.tex lines 737--745, the definition at line 836, and the fitting procedure at line 1787 specify the sets and indices. - reading of the symbol `a_{j,k}`: $a_{j,k}(z,x)=\mathbb P_{\mathrm{obs}}(Z=z\mid Z\in T_{j,k},X=x)$ is the population assignment-share function for each nonempty policy set, indexed by $k=0,\ldots,K$ and $j=1,\ldots,l_k$, with $a_{j,k}(z,x)=0$ outside $T_{j,k}$ and $\sum_{z\in T_{j,k}}a_{j,k}(z,x)=1$ wherever the conditioning probability is positive. — Flat.tex line 783 defines the two-argument conditional share; lines 1787--1791 give its simplex-valued estimation and pooled-policy identity. - reading of the symbol `\kappa_{j,k}`: $\kappa_{j,k}(x)$ is the population relative threshold for a nonempty baseline interval, indexed by $k=0,\ldots,K-1$ and $j=0,\ldots,l_k+1$, with $\kappa_{0,k}(x)=0$ and $\kappa_{l_k+1,k}(x)=1$. — Flat.tex lines 740--750 define the ordered policy levels and the integration-boundary conventions; lines 772--776 define their relative thresholds. Proposition~\ref{prop:J_specialization} uses only $j=0,\ldots,l_k$ as outcome-boundary indices. - reading of the symbol `J_{\text{full},k}`: $J_{\text{full},k}(x)$ is the population full-interval contribution $(p_{k+1}(x)-p_k(x))\mathbb E_{\mu_{1,k\mid x}}[Y]$ for $k=0,\ldots,K-1$. In Proposition~\ref{prop:J_specialization}, Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: In the binary-outcome case, $J_{\text{full},k}(x)=P_{1,k+1}(x)-P_{1,k}(x)$, where $P_{1,k}(x)=\mathbb E_{\text{obs}}[YW\mid Z\in S_k,X=x]$ for $k\geqslant1$ and $P_{1,0}(x)=0$. — The nuisance-component display immediately before Proposition~\ref{prop:discrete_reduction} defines the quantity; Proposition~\ref{prop:J_specialization}\textup{(ii)} gives its binary specialization and zero boundary convention. - reading of the symbol `J_{j,k}`: $J_{j,k}(x)=(p_{k+1}(x)-p_k(x))\int_{\kappa_{j,k}(x)}^{\kappa_{j+1,k}(x)}Q_{Y,k\mid x}(u)\,\mathrm du$ is the population lower-bound subinterval contribution on a nondegenerate identified bin, with structural thresholds $\kappa_{0,k}=0$ and $\kappa_{l_k+1,k}=1$. In Proposition~\ref{prop:discrete_reduction}, Proposition~\ref{prop:J_specialization}, Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: The bound uses $J_{j-1,k}$ for $0\leq k\max\{q_{j-1,k}(x),p_{k+1}(x)-J_{\mathrm{full},k}(x)\}\big)=\indicator(h_{j,k}^{-}(x)>0)$ is the binary-score active-set indicator for $k=0,\ldots,K-1$ and $j=1,\ldots,l_k$, with $q_{0,k}(x)=p_k(x)$. — Section~\ref{app:scores_discrete} gives both defining expressions; the finite-instrument decomposition specifies the sub-interval indices and structural lower boundary. - reading of the symbol `D_{j,k}^{-}`: $D_{j,k}^{-}(x)=\indicator\!\big(q_{j-1,k}(x)p_{k+1}(x)-J_{\mathrm{full},k}(x)\big)$ is the binary-score dominance indicator for $k=0,\ldots,K-1$ and $j=1,\ldots,l_k$, with $q_{0,k}(x)=p_k(x)$; its strict inequality assigns zero at equality. — Section~\ref{app:scores_discrete} defines the indicator; the finite-instrument decomposition specifies the sub-interval indices and structural lower boundary. - reading of the symbol `\psi_{j,k}^J`: $\psi_{j,k}^{J}(O;\eta)$ is the continuous-outcome uncentered lower-bound score for $\mathbb E[J_{j,k}(X)]$, used for $0\leq k0$ is the fixed, sample-size-indexed localization window used in the fitted selection rule $\hat P\le\hat{\underline p}(X)+\delta_n$. In Corollary~\ref{cor:continuous_primitive_rate}: $\delta_n\to0$ as $n\to\infty$. — Algorithm~\ref{alg:continuous_instrument} fixes the localization window and uses it to select treated observations; Corollary~\ref{cor:continuous_primitive_rate} explicitly requires $\delta_n\to0$. - reading of the symbol `\varepsilon_n`: $\varepsilon_n$ is the quantile-analysis trimming sequence satisfying $M_Q^{-1}\le\varepsilon_n<1/2$ and $\varepsilon_n\to0$; it restricts the learner analysis to intermediate quantiles without changing the estimator. — Condition (L3) states the inequalities and convergence; its explanatory paragraph states that trimming leaves the estimator unchanged. - reading of the symbol `M_g`: $M_g$ is a fixed positive integer tuning choice for the regression grid at each sample size, giving spacing $M_g^{-1}$ and $M_g+1$ grid points $u_m$, $m=0,\ldots,M_g$. — Flat.tex line 1893 and Algorithm~\ref{alg:continuous_instrument} specify the fixed grid $u_m=m/M_g$, $m=0,\ldots,M_g$. - reading of the symbol `M_Q`: $M_Q$ is the integer resolution of the quantile grid $\tau_m=m/M_Q$, $m=1,\ldots,M_Q-1$, and may vary with $n$. In Corollary~\ref{cor:continuous_primitive_rate}: $M_Q^{-1}\to0$, so the quantile grid resolution diverges with sample size. — Algorithm~\ref{alg:continuous_instrument} fixes the grid, and Corollary~\ref{cor:continuous_primitive_rate} requires $M_Q^{-1}\to0$. - reading of the symbol `u_m`: $u_m=m/M_g$ for $m=0,\ldots,M_g$ is the fixed regression index grid spanning $[0,1]$, including both endpoints. — Flat.tex line 1893 and the first step of Algorithm~\ref{alg:continuous_instrument} both give the same formula and index range. - reading of the symbol `\tau_m`: $\tau_m=m/M_Q$ for $m=1,\ldots,M_Q-1$ denotes the interior quantile grid; the interpolation endpoint levels $0$ and $1$ are separate from these indexed grid points. — The treated-boundary-law construction at line 1912 and Algorithm~\ref{alg:continuous_instrument} specify the index range and separately discuss endpoint values. - reading of the symbol `\mathcal F_n`: $\mathcal F_n$ is a user-chosen, potentially $n$-dependent class of real-valued functions of $x$, used for both the intercept function $g$ and the slope function $\beta$ in each local-linear fit. — The local-linear minimization and its following sentence in Section~\ref{app:procedure_continuous_instrument} specify that both $g$ and $\beta$ belong to $\mathcal F_n$. - reading of the symbol `\mathcal F_Q`: $\mathcal F_Q$ is the chosen class of real-valued functions of $x$ over which the boundary conditional quantile check-function regression is minimized; with no covariates it is the class of constants. — Equation~(\ref{eq:boundary_quantile_regression}) and the sentences immediately following it specify the optimization domain and the no-covariate convention. - reading of the symbol `\widetilde Q`: $\widetilde Q$ is the raw interior-grid check-regression fit on the treated localized sample from $I_1$, whose selection rule is determined by the first stage on $I_0$, before clipping, rearrangement, and interpolation. — Equation~\ref{eq:boundary_quantile_regression} gives the optimization; Algorithm~\ref{alg:continuous_instrument} specifies the grid and empty-sample branch. Line 1920 assigns interpolation and endpoint conventions to the resulting $\widehat Q_{1,\underline p\mid x}$. - reading of the symbol `\rho_\tau`: $\rho_\tau(u)=u(\tau-\indicator(u<0))$ is the check loss for $u\in\mathbb R$ at the quantile levels $\tau=\tau_m=m/M_Q$, $m=1,\ldots,M_Q-1$, used in the boundary quantile regression. — Equation~\ref{eq:boundary_quantile_regression} at flat.tex lines 1912--1918 specifies both the check loss and its interior quantile grid. - reading of the symbol `\widehat C`: $\widehat C(\tau\mid x)=\int_0^\tau\widehat Q_{1,\underline p\mid x}(s)\,\mathrm ds$ for $\tau\in[0,1]$ and $x\in\mathcal X$, with grid values computed exactly by the trapezoid rule and off-grid evaluation implemented by linear interpolation of adjacent grid values. — The definition and numerical evaluation convention are given at flat.tex lines 1931--1942 and in Algorithm~\ref{alg:continuous_instrument}. - reading of the symbol `\hat\tau_i`: For $i\in I_2$ with $\hat{\underline p}(X_i)>0$, $\hat\tau_i=\min\{1,\hat q(Z_i,X_i)/\hat{\underline p}(X_i)\}\in[0,1]$ is the normalized boundary-integration cutoff; when $\hat{\underline p}(X_i)=0$ it is not evaluated, and when $\hat q(Z_i,X_i)\geqslant\hat{\underline p}(X_i)>0$ it equals one. — Flat.tex lines 1930--1942 specify its positive-denominator domain, zero branch, and empty-region value. Algorithm~\ref{alg:continuous_instrument} defines it during evaluation on $I_2$. - reading of the symbol `\hat\psi_1`: $\hat\psi_1(O)$ denotes the estimated lower-bound boundary contribution $-\int_{\min\{\hat{\underline p}(X),\hat q(Z,X)\}}^{\hat{\underline p}(X)}\widehat Q_{1,\underline p\mid X}(v/\hat{\underline p}(X))\,\mathrm dv$, with value zero when $\hat{\underline p}(X)=0$. In Corollary~\ref{cor:continuous_primitive_rate}: For $\hat{\underline p}(X_i)>0$, the concrete implementation evaluates $-\hat{\underline p}(X_i)[\widehat C(1\mid X_i)-\widehat C(\hat\tau_i\mid X_i)]$, where $\hat\tau_i=\min\{1,\hat q(Z_i,X_i)/\hat{\underline p}(X_i)\}$; cumulative-integral grid values are computed by the trapezoid rule, and off-grid lookup uses linear interpolation between adjacent cumulative-integral grid values. For $\hat{\underline p}(X_i)=0$, it returns zero without evaluating $\hat\tau_i$. — The target-evaluation passage in Section~\ref{app:procedure_continuous_instrument} defines the integral and its zero branch, distinguishes $\hat\psi_1^{\,+}$, and specifies trapezoid evaluation and interpolation at flat.tex lines 1924--1942; Algorithm~\ref{alg:continuous_instrument} supplies the implementation. - reading of the symbol `\hat\psi_3`: $\hat\psi_3(O)=y_{\min}(\hat q(Z,X)-\hat{\overline p}(X))_+$ is the fitted lower-bound tail score, evaluated at $O_i$ for $i\in I_2$ with the fitted score map held fixed conditional on $I_0,I_1$. — Flat.tex lines 908--912 define evaluation on $I_2$; lines 1948--1954 and Algorithm~\ref{alg:continuous_instrument} give the score formula and its evaluation. - reading of the symbol `\rho_j`: For $j\in\{0,1,2\}$, $\rho_j\in(0,1)$ is the limiting split proportion $\|I_j\|/n\to\rho_j$, with $\rho_0+\rho_1+\rho_2=1$. — Flat.tex line 908 and line 1882 describe a partition into three samples with these positive limits; Corollary~\ref{cor:continuous_primitive_rate} repeats the index range at line 3014. - reading of the symbol `L_\phi`: $L_\phi<\infty$ is a uniform Lipschitz bound satisfying $\|\phi(z,x,p')-\phi(z,x,p)\|\leqslant L_\phi\|p'-p\|$ for every $(z,x,p,p')\in\mathcal Z\times\mathcal X\times[0,1]^2$. — Corollary~\ref{cor:continuous_primitive_rate}, flat.tex lines 3010--3013, states the bound and its full domain; its proof uses the same constant for the sup-norm policy error. - reading of the symbol `U_\ell`: For $\ell\in\{1,2\}$, $U_1=q^{\mathrm{tr}}(Z,X)$ and $U_2=P=p(Z,X)$ are the population evaluation indices, with $U_\ell\in\mathcal S_X$; their estimated counterparts are $\hat U_1=\hat q^{\mathrm{tr}}(Z,X)$ and $\hat U_2=\hat P$. — The definitions preceding Assumption~\ref{asp:continuous_learners} specify all four indices, and the proof of Corollary~\ref{cor:continuous_primitive_rate} explicitly uses $U_\ell\in\mathcal S_X$. - reading of the symbol `\hat U_\ell`: For $\ell\in\{1,2\}$, $\hat U_1=\hat q^{\mathrm{tr}}(Z,X)$ and $\hat U_2=\hat P$ are the two target-evaluation indices, with evaluation over $(Z,X)$ conditional on $I_0,I_1$, and the population comparison for $\hat g_1(\hat U_\ell,X)$ is $g_1(\Pi_X(\hat U_\ell),X)$. — The definitions at flat.tex lines 3039--3045 and condition (L2) at lines 3052--3061 specify the indices and conditional evaluation. Lines 3094--3097 explain that the policy index can have a different evaluation distribution and endpoint mass. - reading of the symbol `\mathcal S_x`: $\mathcal S_x=[\underline p(x),\overline p(x)]$ is the population propensity interval at covariate value $x$, used as the domain of the population comparison after projection by $\Pi_x$. — The definitions preceding Assumption~\ref{asp:continuous_learners} give $\mathcal S_x$ and $\Pi_x$; the subsequent explanation says projection keeps the population comparison inside its identified support. - reading of the symbol `\Pi_x`: $\Pi_x:\mathbb R\to\mathcal S_x$ is projection onto the population support interval $\mathcal S_x=[\underline p(x),\overline p(x)]$, given by $\Pi_x(u)=\min\{\overline p(x),\max\{\underline p(x),u\}\}$. — Lines 3039--3043 define $\mathcal S_x$ and $\Pi_x$ together; line 3094 explains that this projection keeps the population comparison inside its identified support. - reading of the symbol `\mu^{\mathrm{sel}}_{1,n\mid x}`: $\mu^{\mathrm{sel}}_{1,n\mid x}$ is the conditional population outcome law under the fitted selection event $\{W=1,\hat p(Z,X)\le\hat{\underline p}(x)+\delta_n\}$ given $X=x$ and $I_0$, with the canonical boundary law used when that event has zero conditional probability. — Lines 3063--3072 give the conditioning event and fallback explicitly; lines 3146--3155 treat this law, conditional on $I_0$, as a mixture of treated population laws. - reading of the symbol `Q^{\mathrm{sel}}_{1,n\mid x}`: $Q^{\mathrm{sel}}_{1,n\mid x}:(0,1)\to\mathcal Y$ is the quantile function of the $I_0$-conditional selected law $\mu^{\mathrm{sel}}_{1,n\mid x}$, indexed by sample size $n$ and covariate value $x$, including its stipulated zero-selection version. — Lines 3063--3073 define the selected law conditional on $I_0$, specify its zero-selection version, and introduce its quantile function; lines 3096 and 3219 identify this as the target of the localized quantile regression. Line 3169 only recalls monotonicity and boundedness and does not redefine the symbol. - reading of the symbol `\widetilde Q_m^c`: $\widetilde Q_m^c(x)$ is the clipped pre-rearrangement fit at $\tau_m=m/M_Q$, for $m=1,\ldots,M_Q-1$, obtained from the boundary quantile regression on $\mathcal I_{\delta,1}\subseteq I_1$ using the selection rule determined on $I_0$. — The construction and grid are specified at lines 1963--1975; lines 3074--3089 define the clipped fit and compare it to the selected-law quantile. - reading of the symbol `\mathcal J_{Q,n}`: $\mathcal J_{Q,n}=\{m\in\{1,\ldots,M_Q-1\}:\varepsilon_n\le\tau_m\le1-\varepsilon_n\}$ is the interior grid subset used for analytical quantile trimming; it does not trim the estimator. — Condition (L3) defines the subset, and the paragraph following Assumption~\ref{asp:continuous_learners} explicitly says that this analytical trimming leaves the estimator unchanged. - reading of the symbol `\pi`: The meaning of $\pi$ is determined by its arguments: transport notation denotes a joint probability measure, whereas $\pi(z,x)$ denotes the conditional instrument assignment probability $\mathbb P(Z=z\mid X=x)$. In Theorem~\ref{thm:1d_ot}: $\pi$ ranges over $\Pi(\mu,\nu)\subseteq\mathcal P(\mathbb R^2)$. In Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: In instrument-specific probability expressions, $\pi(z,x)=\mathbb P_{\mathrm{obs}}(Z=z\mid X=x)$, rather than a transport coupling. — The coupling definition at flat.tex line 233 fixes the transport meaning; the proof of Lemma~\ref{lem:orthogonality_discrete} explicitly uses $\mathbb E[\indicator(Z=z)\mid X]=\pi(z,X)$ at line 2859. - reading of the symbol `\mathcal B`: The notation $\mathcal B$ denotes an OT state space when standing alone and the Borel $\sigma$-field operator when applied to an outcome domain. In Theorem~\ref{thm:1d_ot}: The second marginal state space is $\mathcal B=\mathbb R$. In Theorem~\ref{thm:covariate_continuous_lower_bound}: In the kernel definition, $A\in\mathcal B(\mathcal Y)$ means that $A$ is a Borel subset of $\mathcal Y$. — The standalone state-space notation is introduced at line 233; Theorem~\ref{thm:1d_ot} places both marginals on $\mathbb R$. The kernel display at line 1742 uses $A\in\mathcal B(\mathcal Y)$, with the Borel convention specified at line 159. - reading of the symbol `\indep`: $\indep$ denotes independence, with $A\indep B\mid C$ denoting conditional independence given $C$, namely factorization of the conditional joint law into the two conditional marginal laws almost surely. — The notation paragraph at line 257 explicitly declares $\indep$ to mean (conditional) independence; Assumption~\ref{asp:structure} and the construction in the proof of Proposition~\ref{prop:sharpness} use conditional factorization given $X$. - reading of the symbol `\Gamma_{1,x}`: $\Gamma_{1,x}$ is the conditional treated feasible class defined for $\mathbb P_X$-almost every $x$, with its members understood through their Lebesgue-almost-everywhere kernel values in $u$. In Theorem~\ref{thm:covariate_continuous_lower_bound}: For $\mathbb P_X$-almost every $x$, the observational constraint equivalently holds as $\int_0^p\pi_{1,x}(\mathrm dy\mid u)\,\mathrm du=H_{1,x}(p,\mathrm dy)$ for every $p\in p(\mathcal Z,x)$, using the canonical continuous version of $H_{1,x}$. — The disintegration paragraph at lines 1632--1642 specifies the constraints and the almost-everywhere qualification in $x$; the paragraph preceding equation~(\ref{eq:pf_cts_obs}) extends the cumulative constraint by total-variation continuity. - reading of the symbol `B_Y`: $B_Y$ is a fixed, finite, nonnegative constant determined by $y_{\min}$ and $y_{\max}$. In Theorem~\ref{thm:covariate_continuous_lower_bound}: $B_Y=\max\{\|y_{\min}\|,\|y_{\max}\|\}$ bounds the absolute magnitude of feasible outcomes. In Theorem~\ref{thm:dml_normality}, Corollary~\ref{cor:continuous_primitive_rate}: $B_Y=y_{\max}-y_{\min}$ is the outcome range, used to bound differences between values in $[y_{\min},y_{\max}]$. — Line 1642 defines the absolute-magnitude bound; line 2204 explicitly redefines the notation as the outcome range. Lines 3136--3176 use the range bound for distances and quantile errors. - reading of the symbol `\bar p_2`: $\bar p_2(z)$ is the sample-dependent average of the clipped first-stage fit $\hat p(z,\cdot)$ trained on $I_1$ over the covariates indexed by $I_2$; conditional on $I_1$ it remains an empirical average over evaluation covariates, rather than the population marginal mean. — The formula at flat.tex lines 1762 and 1776 and the conditional concentration calculation at lines 2179--2191. - reading of the symbol `\widehat S_k`: $\widehat S_k$ is the sample-dependent baseline level-set estimate obtained from the pre-recovery fits on $I_1$ and their averages over the covariates in $I_2$, with a threshold-defined structural-zero set and positively indexed baseline components ordered by their averages. — The explicit recovery definition at flat.tex line 1780 and the algorithm at lines 1760--1764. - reading of the symbol `\widehat T_{j,k}`: $\widehat T_{j,k}$ is the sample-dependent policy level-set estimate obtained from the pre-recovery fits on $I_1$ and their averages over the covariates in $I_2$; it depends on both of these inputs and is indexed by the recovered component ordering and baseline bins. — The algorithm at flat.tex lines 1760--1764 and the explicit graph construction at line 1780. - reading of the symbol `\alpha_{p_k}`: $\alpha_{p_k}$ is the population lower-boundary representer for the particular component $J_{j,k}$, with the dependence on $j$ suppressed in its notation, for $k=1,\ldots,K-1$ and $j=0,\ldots,l_k-1$; there is no $p_0$ perturbation or correction. — The representer display and following explanation at flat.tex lines 2023--2031 give the component dependence and structural boundary convention; lines 2802--2818 verify the same formula and the absent standalone lower-endpoint term at $j=0$. - reading of the symbol `\alpha_{p_{k+1}}`: $\alpha_{p_{k+1}}$ denotes the upper interval-boundary representer for the continuous-outcome component $J_{j,k}$, with suppressed dependence on $j$, indexed by $k=0,\ldots,K-1$ and $j=0,\ldots,l_k-1$; it remains an upper-side representer at $k=0$. — Section~\ref{app:scores_continuous} gives the exact formula for each component, and the product-score construction uses $J_{j-1,k}$ for $j=1,\ldots,l_k$; only minus-side terms are omitted at $k=0$. - reading of the symbol `IF_{J^+}`: $IF_{J^+}$ denotes the plus-side observation-level residual component indexed implicitly by $k\in\{0,\ldots,K-1\}$ and $j\in\{0,\ldots,l_k-1\}$, evaluated with the nuisance functions supplied to the continuous-outcome score. — The defining display at flat.tex lines 2016--2020, the target aggregation at lines 2010--2014, and the conditional-expectation perturbation calculation at lines 2786--2792. - reading of the symbol `IF_{J^-}`: $IF_{J^-}$ denotes the minus-side observation-level residual component indexed implicitly by $k\in\{0,\ldots,K-1\}$ and $j\in\{0,\ldots,l_k-1\}$, evaluated with the nuisance functions supplied to the continuous-outcome score, with its structural $k=0$ component identically zero. — The influence-function residual display and following boundary convention at flat.tex lines 2016--2021, together with the score construction at lines 2042--2048 and the target aggregation at lines 2010--2014. - reading of the symbol `IF_{M_{j,k}^+}`: $IF_{M_{j,k}^+}$ is the observation-level residual indexed by $k=0,\ldots,K-1$ and boundary index $j=0,\ldots,l_k$. In Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: In the continuous-outcome part, the no-atom condition at $y_{\min}$ makes the true residual $IF_{M_{0,k}^+}$ vanish almost surely. — The defining display at flat.tex lines 2016--2019 supplies the residual; Proposition~\ref{prop:J_specialization} supplies the boundary indices, and the estimation paragraph at line 1810 supplies the lower-boundary convention. - reading of the symbol `IF_{M_{j,k}^-}`: $IF_{M_{j,k}^-}$ is the observation-level residual indexed by $k=0,\ldots,K-1$ and boundary index $j=0,\ldots,l_k$, with the structural zero convention at $k=0$. In Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: In the continuous-outcome part, the no-atom condition at $y_{\min}$ makes the true residual $IF_{M_{0,k}^-}$ vanish almost surely. — The residual display and structural-boundary paragraph at flat.tex lines 2016--2021 give its formula and zero convention; Proposition~\ref{prop:J_specialization} gives the boundary indices. - reading of the symbol `\alpha_{P,k+1}^{(j,k)}`: $\alpha_{P,k+1}^{(j,k)}(x)$ is the coefficient of a perturbation of $P_{1,k+1}(x)$ in the binary component $h_{j,k}^{-}(x)$, indexed by $k=0,\ldots,K-1$ and $j=1,\ldots,l_k$, holding the propensity and alternative-policy levels and $P_{1,k}$ fixed. — The pathwise-derivative paragraph of Section~\ref{app:scores_discrete} states that $h_{j,k}^{-}$ depends on the success regressions through $P_{1,k+1}-P_{1,k}$ and gives the coefficient $D_{j,k}^{-}$. - reading of the symbol `\alpha_{P,k}^{(j,k)}`: For $k\in\{0,\ldots,K-1\}$ and $j\in\{1,\ldots,l_k\}$, $\alpha_{P,k}^{(j,k)}(x)=-D_{j,k}^{-}(x)$ is the coefficient of the $P_{1,k}$ perturbation in the derivative of $h_{j,k}^{-}$, with the other nuisance arguments held fixed. — The binary score construction in Section~\ref{app:scores_discrete} gives the derivative coefficients; the proof of Lemma~\ref{lem:orthogonality_discrete}(ii) explicitly perturbs $P_{1,m}$ and identifies these coefficients. - reading of the symbol `\alpha_{p,k+1}^{(j,k)}`: $\alpha_{p,k+1}^{(j,k)}(x)$ is the direct upper-propensity coefficient for the binary score component $h_{j,k}^{-}$, indexed by $k=0,\ldots,K-1$ and $j=1,\ldots,l_k$, holding $P_{1,k}$, $P_{1,k+1}$, and the alternative-policy levels fixed. — Section~\ref{app:scores_discrete} gives $\alpha_{p,k+1}^{(j,k)}=-D_{j,k}^{-}$; the proof in Section~\ref{oa:orthogonality_binary} separates the integration-boundary channel from the alternative-policy channels. - reading of the symbol `\alpha_{p,k}^{(j,k)}`: $\alpha_{p,k}^{(j,k)}(x)$ denotes the direct lower-propensity coefficient in the binary score for $h_{j,k}^{-}$, indexed by $k=0,\ldots,K-1$ and $j=1,\ldots,l_k$, holding the success regressions and alternative-policy levels fixed; at $k=0$ its residual correction is omitted because $p_0\equiv0$ is known. — Section~\ref{app:scores_discrete} gives the piecewise coefficient, separates direct propensity derivatives from policy-level corrections, and sets $R_{p,0}\equiv0$ at the structural boundary. - reading of the symbol `m`: The notation $m(X;\eta)$ denotes the binary lower-bound integrand, whereas bare $m$ in $u_m$ or $\tau_m$ denotes a grid index. In Lemma~\ref{lem:orthogonality_discrete}, Theorem~\ref{thm:dml_normality}: In the binary-outcome case, $m(X;\eta)$ is the displayed integrand with nuisance argument $\eta$; evaluating at $\eta_0$ gives the population target integrand. In Corollary~\ref{cor:continuous_primitive_rate}: $m$ is an integer grid index: $m=0,\ldots,M_g$ for $u_m=m/M_g$ and $m=1,\ldots,M_Q-1$ for the fitted quantiles at $\tau_m=m/M_Q$. — The integrand is displayed at flat.tex lines 1870--1873; the two grid conventions appear at lines 1893 and 1912 and in Algorithm~\ref{alg:continuous_instrument}. - reading of the symbol `\nu_{k\mid x}`: $\nu_{k\mid x}$ is the normalized Lebesgue probability law on $G_k(x)$, equivalently the law of $U$ conditional on $X=x$ and $U\in G_k(x)$, for almost every $x$ and $k=0,\ldots,K_x-1$ with $\|G_k(x)\|>0$. — The gap argument around (\ref{eq:pf_cts_gap_constraint}) defines $\nu_{k\mid x}:=\mathrm{Unif}(G_k(x))$ after dividing by the positive gap length and identifies it as the gap $U$-marginal. - reading of the symbol `T_x`: $T_x:[0,1]\to\mathbb R$, $T_x(u)=\omega(x,u)$, is the target-weight map with the covariate argument fixed at $x$. — The proof of Theorem~\ref{thm:covariate_continuous_lower_bound} fixes a nonempty conditional gap and explicitly defines $T_x(u)=\omega(x,u)$ before pushing forward its resistance law. - reading of the symbol `\rho_{k\mid x}`: $\rho_{k\mid x}=(T_x)_{\#}\mathrm{Unif}(G_k(x))$ is the population probability law of $\omega(x,U)$ for an auxiliary $U\sim\mathrm{Unif}(G_k(x))$, defined for fixed $x$ and $k=0,\ldots,K_x-1$ with $\|G_k(x)\|>0$. — The conditional gap decomposition and the OT solution in the proof of Theorem~\ref{thm:covariate_continuous_lower_bound} specify the index range, omit empty gaps, and define the pushforward law. - reading of the symbol `F_{\omega,k\mid x}`: $F_{\omega,k\mid x}$ is the population distribution function of the pushforward of $\nu_{k\mid x}$ under $u\mapsto\omega(x,u)$, using a jointly measurable version in $(x,v)$ for almost every $x$ and each nonempty gap indexed by $k=0,\ldots,K_x-1$. — The sharpness construction defines $F_{\omega,k\mid x}$ by integration over $\nu_{k\mid x}$; the paragraph following (\ref{eq:pf_cts_rank_kernel}) establishes joint measurability. - reading of the symbol `R_{k,x}`: $R_{k,x}$ is a measurable map of $(u,s)\in G_k(x)\times[0,1]$ for almost every $x$ and each $k\in\{0,\ldots,K_x-1\}$ with $\|G_k(x)\|>0$, using the left limit of $F_{\omega,k\mid x}$ to distribute each weight atom across its rank interval. — The sharpness construction preceding (\ref{eq:pf_cts_rank_kernel}) defines the map on a nonempty gap; its subsequent measurability argument explains the measurable left limits. The construction explicitly accommodates atomic and nonmonotone weights. - reading of the symbol `S`: $S$ is an auxiliary $\mathrm{Unif}(0,1)$ draw independent of $U\sim\nu_{k\mid x}$ for each fixed $x$ and nonempty gap $G_k(x)$, used to randomize ties in $R_{k,x}(U,S)$ and integrated out in the attaining outcome kernel. — The sentence preceding (\ref{eq:pf_cts_rank_kernel}) explicitly specifies independent $U\sim\nu_{k\mid x}$ and $S\sim\mathrm{Unif}(0,1)$; the kernel display integrates over $s\in[0,1]$. - reading of the symbol `\pi_{1,x}^\star`: $\pi_{1,x}^\star$ is a jointly measurable population probability kernel for $Y(1)$ given $(X,U)=(x,u)$, defined for almost every $x$ and pasted across the nonempty gaps, non-degenerate LIV intervals and trivial tail, with arbitrary probability-law values at region boundaries. — Equation~(\ref{eq:pf_cts_rank_kernel}) and the following paragraph specify all pieces; the final paragraph of the proof of Theorem~\ref{thm:covariate_continuous_lower_bound} establishes joint measurability and compatibility. - reading of the symbol `\mathcal E_n`: $\mathcal E_n$ is the event of simultaneous exact recovery of all baseline and policy level sets with their population ordering and bin indices, including $\widehat S_0=S_0$, by the procedure using first-stage fits on $I_1$ and their averages over $\{X_i:i\in I_2\}$. — The exact-recovery paragraphs in the proof of Theorem~\ref{thm:dml_normality} establish structural-zero recovery and ordered graph-component recovery before defining $\mathcal E_n$; the binary-outcome proof uses the same event. - reading of the symbol `r_{\mathrm{reg},n}`: $r_{\mathrm{reg},n}$ is the maximum population $L_2$ error of the fitted outcome regressions, propensity regressions, pooled policy levels, and boundary quantiles, with integration over the marginal law of $X$ while holding the fitted functions fixed. In Theorem~\ref{thm:dml_normality}: In the continuous-outcome proof, the errors are those of the oracle nuisance fits trained on $I_1$ using the population level sets; hence their magnitudes are fixed conditional on $I_1$ and satisfy $r_{\mathrm{reg},n}=o_P(n^{-1/4})$. — The second-order remainder paragraph preceding (\ref{eq:pf_dml_bias}) defines the maximum and conditions on $I_1$; the preceding exact-recovery paragraph specifies that the subsequent calculation uses oracle nuisance fits. - reading of the symbol `r_{\mathrm{wt},n}`: $r_{\mathrm{wt},n}$ is the maximum of the componentwise $L_2(\mathbb P_{\mathrm{obs}}(\mathrm{d}x))$ errors of the fitted weights $\hat\pi_k,\hat\tau_{j,k},\hat a_{j,k}(z,\cdot),\hat\gamma_{j,k},\hat\gamma_{\mathrm{full},k},\hat\gamma_K$ relative to their population counterparts, over the finitely many fitted indices and instrument categories. — The definitions at flat.tex lines 2199 and 2267 use the same weight-error group in both outcome regimes; the surrounding remainder calculations integrate prediction errors over $X$ and condition on $I_1$ for the oracle estimator. - reading of the symbol `r_{\mathrm{core},n}`: $r_{\mathrm{core},n}$ is the maximum of the $L_2(\mathbb P_X)$ norms of fitted-minus-population errors for the finitely many $P_{1,k}$, $p_k$, instrument-specific $p(z,\cdot)$, and pooled $q_{j,k}$ nuisances used by the binary oracle score, with the fitted $P_{1,k}$ taken after clipping and rearrangement and the structural zero boundaries retained. — The proof of Theorem~\ref{thm:dml_normality}\textup{(ii)} first passes to the oracle estimator, then clips and rearranges the nonboundary treated-success regressions, integrates their squared errors over $x$, and defines the maximum nuisance error in its second-order remainder argument. - added condition (`ass:bk-3`, `thm:discrete-lower-bound`): For every nonempty conditional gap $G_k(x)$, extend $Q_{Y,k\mid x}$ to $[0,1]$ by $Q_{Y,k\mid x}(0):=y_{\min}$ and $Q_{Y,k\mid x}(1):=y_{\max}$. — Equation (\ref{eq:pf_cts_rank_kernel}) and its upper-bound analogue require feasible outcome-quantile values at exceptional randomized-rank endpoints. - added condition (`ass:bk-3`, `thm:continuous-lower-bound`): For every nonempty conditional gap $G_k(x)$, extend $Q_{Y,k\mid x}$ to $[0,1]$ by $Q_{Y,k\mid x}(0):=y_{\min}$ and $Q_{Y,k\mid x}(1):=y_{\max}$. — Equation (\ref{eq:pf_cts_rank_kernel}) and its upper-bound analogue require feasible outcome-quantile values at exceptional randomized-rank endpoints. - added condition (`ass:bk-3`, `thm:covariate-continuous-lower-bound`): For every nonempty conditional gap $G_k(x)$, extend $Q_{Y,k\mid x}$ to $[0,1]$ by $Q_{Y,k\mid x}(0):=y_{\min}$ and $Q_{Y,k\mid x}(1):=y_{\max}$. — Equation (\ref{eq:pf_cts_rank_kernel}) and its upper-bound analogue require feasible outcome-quantile values at exceptional randomized-rank endpoints. - added condition (`ass:bk-3`, `prop:discrete-reduction`): For every nonempty conditional gap $G_k(x)$, extend $Q_{Y,k\mid x}$ to $[0,1]$ by $Q_{Y,k\mid x}(0):=y_{\min}$ and $Q_{Y,k\mid x}(1):=y_{\max}$. — Equation (\ref{eq:pf_cts_rank_kernel}) and its upper-bound analogue require feasible outcome-quantile values at exceptional randomized-rank endpoints. - added condition (`ass:bk-3`, `prop:j-specialization`): For every nonempty conditional gap $G_k(x)$, extend $Q_{Y,k\mid x}$ to $[0,1]$ by $Q_{Y,k\mid x}(0):=y_{\min}$ and $Q_{Y,k\mid x}(1):=y_{\max}$. — Equation (\ref{eq:pf_cts_rank_kernel}) and its upper-bound analogue require feasible outcome-quantile values at exceptional randomized-rank endpoints. - added condition (`ass:bk-1`, `thm:dml-normality`): $K\geqslant1$. — The printed tail correction in $\psi_{\Delta_K}$ evaluates $1/\pi_K(X)$ at an observed positive-propensity level. - added condition (`ass:bk-2`, `thm:dml-normality`): $K\geqslant1$. — The printed tail score divides by $\pi_K(X)$; no denominator is defined at the structural index zero. - added condition (`ass:bk-3`, `thm:dml-normality`): For every nonempty conditional gap $G_k(x)$, extend $Q_{Y,k\mid x}$ to $[0,1]$ by $Q_{Y,k\mid x}(0):=y_{\min}$ and $Q_{Y,k\mid x}(1):=y_{\max}$. — Equation (\ref{eq:pf_cts_rank_kernel}) and its upper-bound analogue require feasible outcome-quantile values at exceptional randomized-rank endpoints. - added condition (`ass:bk-3`, `prop:continuous-reduction`): For every nonempty conditional gap $G_k(x)$, extend $Q_{Y,k\mid x}$ to $[0,1]$ by $Q_{Y,k\mid x}(0):=y_{\min}$ and $Q_{Y,k\mid x}(1):=y_{\max}$. — Equation (\ref{eq:pf_cts_rank_kernel}) and its upper-bound analogue require feasible outcome-quantile values at exceptional randomized-rank endpoints. - added condition (`ass:bk-3`, `thm:continuous-rate`): For every nonempty conditional gap $G_k(x)$, extend $Q_{Y,k\mid x}$ to $[0,1]$ by $Q_{Y,k\mid x}(0):=y_{\min}$ and $Q_{Y,k\mid x}(1):=y_{\max}$. — Equation (\ref{eq:pf_cts_rank_kernel}) and its upper-bound analogue require feasible outcome-quantile values at exceptional randomized-rank endpoints. - added condition (`ass:bk-1`, `lem:orthogonality-discrete`): $K\geqslant1$. — The printed tail correction in $\psi_{\Delta_K}$ evaluates $1/\pi_K(X)$ at an observed positive-propensity level. - added condition (`ass:bk-3`, `lem:orthogonality-discrete`): For every nonempty conditional gap $G_k(x)$, extend $Q_{Y,k\mid x}$ to $[0,1]$ by $Q_{Y,k\mid x}(0):=y_{\min}$ and $Q_{Y,k\mid x}(1):=y_{\max}$. — Equation (\ref{eq:pf_cts_rank_kernel}) and its upper-bound analogue require feasible outcome-quantile values at exceptional randomized-rank endpoints. - added condition (`ass:bk-3`, `thm:continuous-primitive-rate`): For every nonempty conditional gap $G_k(x)$, extend $Q_{Y,k\mid x}$ to $[0,1]$ by $Q_{Y,k\mid x}(0):=y_{\min}$ and $Q_{Y,k\mid x}(1):=y_{\max}$. — Equation (\ref{eq:pf_cts_rank_kernel}) and its upper-bound analogue require feasible outcome-quantile values at exceptional randomized-rank endpoints. ## Added conditions | id | where | class | condition | | --- | --- | --- | --- | | `ass:pin-20` | `thm:1d-ot` | V1.5 bookkeeping | For $\rho\in\{\mu,\nu\}$ and $t\in(0,1)$, $Q_\rho(t)=\inf\{a\in\mathbb R:\rho((\! -\infty,a])\geqslant t\}$. (pins reading of ass:1d-ot-probability-laws: The standard generalized-inverse convention for probability quantiles applies to the notation section and Theorem~\ref{thm:1d_ot}; the theorem evaluates these functions at uniform draws and inside Lebesgue integrals.) | | `ass:pin-63` | `thm:1d-ot` | V1.5 bookkeeping | For $00$ is the same constant used for $\pi_k$ in \cref{asp:overlap}. (pins reading of ass:overlap-policy: Both clauses belong to the single quantified overlap assumption. The proof of \cref{thm:dml_normality} uses this assumption to bound the inverse-probability weights uniformly.) | | `ass:pin-20` | `thm:dml-normality` | V1.5 bookkeeping | For $\rho\in\{\mu,\nu\}$ and $t\in(0,1)$, $Q_\rho(t)=\inf\{a\in\mathbb R:\rho((\! -\infty,a])\geqslant t\}$. (pins reading of ass:1d-ot-probability-laws: The standard generalized-inverse convention for probability quantiles applies to the notation section and Theorem~\ref{thm:1d_ot}; the theorem evaluates these functions at uniform draws and inside Lebesgue integrals.) | | `ass:pin-21` | `thm:dml-normality` | V1.5 bookkeeping | For each fixed $k\in\{0,\dots,K-1\}$ and $j\in\{0,\dots,l_k-1\}$ to which part~(i) is applied, its continuity hypothesis is $\mu_{1,k\mid X}(\{\nu_{j,k}(X)\})=\mu_{1,k\mid X}(\{\nu_{j+1,k}(X)\})=0$ almost surely under the marginal law of $X$. (pins reading of ass:j-specialization-continuity: The proof of Proposition~\ref{prop:J_specialization}(i) fixes the band indices and uses continuity of the distribution function at the two boundaries to identify their probability levels and exclude boundary atoms.) | | `ass:pin-22` | `thm:dml-normality` | V1.5 bookkeeping | For a fresh $X$ with the population marginal law and independent of $I_1$, $\\|\hat p(z,\cdot)-p(z,\cdot)\\|_{L_2}^2=\mathbb E[(\hat p(z,X)-p(z,X))^2\mid I_1]$, and $n^{1/4}\max_{z\in\mathcal Z}\\|\hat p(z,\cdot)-p(z,\cdot)\\|_{L_2}\xrightarrow{P}0$ as $n\to\infty$. (pins reading of ass:dml-normality-first-stage: The exact-recovery argument in the proof of \cref{thm:dml_normality} compares $\mathbb E[\hat p(z,X)\mid I_1]$ with $\mathbb E[p(z,X)]$ using this norm, before evaluating the fit on independent covariates in $I_2$.) | | `ass:pin-23` | `thm:dml-normality` | V1.5 bookkeeping | Pointwise in $x$, the fitted probability, propensity, share, and indicator-mean nuisances lie in $[0,1]$, $\hat\gamma_{\mathrm{full},k}(x)\in[-1,1]$, $\hat\nu_{j,k}(x),\hat J_{\mathrm{full},k}^{\pm}(x)\in[y_{\min},y_{\max}]$, and $\hat J_{j,k}^{\pm}(x)\in[\min\{0,y_{\min}\},\max\{0,y_{\max}\}]$, for each applicable fitted component; denominator clipping is applied further as stated. (pins reading of ass:dml-normality-score-clipping: The nuisance definitions and estimation steps in \cref{app:procedure_continuous_outcome,app:procedure_binary_outcome} determine these ranges. The quantile remainder in (\ref{eq:pf_dml_bias}) explicitly uses both quantiles lying in $[y_{\min},y_{\max}]$.) | | `ass:pin-24` | `thm:dml-normality` | V1.5 bookkeeping | Componentwise $L_2$ errors integrate over the marginal law of $X$ under $\mathbb P_{\mathrm{obs}}$, holding each fitted function and any instrument argument $z$ fixed; $o_P(n^{-1/4})$ refers to convergence in probability over the observed sample and random partition as $n\to\infty$. (pins reading of ass:dml-normality-nuisance-rates: The sampling setup for Theorem~\ref{thm:dml_normality} specifies the random sample and partition; its supplied second-order remainder bound, equation~(\ref{eq:pf_dml_bias}), uses population covariate integrals of the fitted errors.) | | `ass:pin-25` | `thm:dml-normality` | V1.5 bookkeeping | $Y\in\{0,1\}$ almost surely. (pins reading of ass:dml-normality-binary: Proposition~\ref{prop:J_specialization}\textup{(ii)} explicitly states this coding before defining the binary subinterval contribution used by Theorem~\ref{thm:dml_normality}\textup{(ii)}.) | | `ass:pin-29` | `thm:dml-normality` | V1.5 bookkeeping | Every constraint in $\eqref{eq:ot_formulation}$ is imposed simultaneously for both $w\in\{0,1\}$. (pins reading of def:ot-formulation: The sentence immediately preceding equation $\eqref{eq:ot_formulation}$ explicitly says that every constraint is imposed for both treatment states.) | | `ass:pin-3` | `thm:dml-normality` | V1.5 bookkeeping | Two remaining labeled vertices are connected exactly when their averaged values differ by less than $c_{\mathrm{gap}}/2$. (pins reading of def:dml-estimation-procedure: Both Algorithm~\ref{alg:dml_continuous_outcome} and the Level Sets step in Section~\ref{app:procedure_continuous_outcome} explicitly use ``less than''.) | | `ass:pin-32` | `thm:dml-normality` | V1.5 bookkeeping | For $\mathbb P_X$-almost every $x$, the intervals $L_k(x)$, $1\leqslant k\leqslant K_x$, the gaps $G_k(x)$, $0\leqslant k0$ and $00$ and $00$ is the same constant used for $\pi_k$ in \cref{asp:overlap}. (pins reading of ass:overlap-policy: Both clauses belong to the single quantified overlap assumption. The proof of \cref{thm:dml_normality} uses this assumption to bound the inverse-probability weights uniformly.) | | `ass:pin-20` | `lem:orthogonality-discrete` | V1.5 bookkeeping | For $\rho\in\{\mu,\nu\}$ and $t\in(0,1)$, $Q_\rho(t)=\inf\{a\in\mathbb R:\rho((\! -\infty,a])\geqslant t\}$. (pins reading of ass:1d-ot-probability-laws: The standard generalized-inverse convention for probability quantiles applies to the notation section and Theorem~\ref{thm:1d_ot}; the theorem evaluates these functions at uniform draws and inside Lebesgue integrals.) | | `ass:pin-21` | `lem:orthogonality-discrete` | V1.5 bookkeeping | For each fixed $k\in\{0,\dots,K-1\}$ and $j\in\{0,\dots,l_k-1\}$ to which part~(i) is applied, its continuity hypothesis is $\mu_{1,k\mid X}(\{\nu_{j,k}(X)\})=\mu_{1,k\mid X}(\{\nu_{j+1,k}(X)\})=0$ almost surely under the marginal law of $X$. (pins reading of ass:j-specialization-continuity: The proof of Proposition~\ref{prop:J_specialization}(i) fixes the band indices and uses continuity of the distribution function at the two boundaries to identify their probability levels and exclude boundary atoms.) | | `ass:pin-22` | `lem:orthogonality-discrete` | V1.5 bookkeeping | For a fresh $X$ with the population marginal law and independent of $I_1$, $\\|\hat p(z,\cdot)-p(z,\cdot)\\|_{L_2}^2=\mathbb E[(\hat p(z,X)-p(z,X))^2\mid I_1]$, and $n^{1/4}\max_{z\in\mathcal Z}\\|\hat p(z,\cdot)-p(z,\cdot)\\|_{L_2}\xrightarrow{P}0$ as $n\to\infty$. (pins reading of ass:dml-normality-first-stage: The exact-recovery argument in the proof of \cref{thm:dml_normality} compares $\mathbb E[\hat p(z,X)\mid I_1]$ with $\mathbb E[p(z,X)]$ using this norm, before evaluating the fit on independent covariates in $I_2$.) | | `ass:pin-23` | `lem:orthogonality-discrete` | V1.5 bookkeeping | Pointwise in $x$, the fitted probability, propensity, share, and indicator-mean nuisances lie in $[0,1]$, $\hat\gamma_{\mathrm{full},k}(x)\in[-1,1]$, $\hat\nu_{j,k}(x),\hat J_{\mathrm{full},k}^{\pm}(x)\in[y_{\min},y_{\max}]$, and $\hat J_{j,k}^{\pm}(x)\in[\min\{0,y_{\min}\},\max\{0,y_{\max}\}]$, for each applicable fitted component; denominator clipping is applied further as stated. (pins reading of ass:dml-normality-score-clipping: The nuisance definitions and estimation steps in \cref{app:procedure_continuous_outcome,app:procedure_binary_outcome} determine these ranges. The quantile remainder in (\ref{eq:pf_dml_bias}) explicitly uses both quantiles lying in $[y_{\min},y_{\max}]$.) | | `ass:pin-24` | `lem:orthogonality-discrete` | V1.5 bookkeeping | Componentwise $L_2$ errors integrate over the marginal law of $X$ under $\mathbb P_{\mathrm{obs}}$, holding each fitted function and any instrument argument $z$ fixed; $o_P(n^{-1/4})$ refers to convergence in probability over the observed sample and random partition as $n\to\infty$. (pins reading of ass:dml-normality-nuisance-rates: The sampling setup for Theorem~\ref{thm:dml_normality} specifies the random sample and partition; its supplied second-order remainder bound, equation~(\ref{eq:pf_dml_bias}), uses population covariate integrals of the fitted errors.) | | `ass:pin-25` | `lem:orthogonality-discrete` | V1.5 bookkeeping | $Y\in\{0,1\}$ almost surely. (pins reading of ass:dml-normality-binary: Proposition~\ref{prop:J_specialization}\textup{(ii)} explicitly states this coding before defining the binary subinterval contribution used by Theorem~\ref{thm:dml_normality}\textup{(ii)}.) | | `ass:pin-29` | `lem:orthogonality-discrete` | V1.5 bookkeeping | Every constraint in $\eqref{eq:ot_formulation}$ is imposed simultaneously for both $w\in\{0,1\}$. (pins reading of def:ot-formulation: The sentence immediately preceding equation $\eqref{eq:ot_formulation}$ explicitly says that every constraint is imposed for both treatment states.) | | `ass:pin-32` | `lem:orthogonality-discrete` | V1.5 bookkeeping | For $\mathbb P_X$-almost every $x$, the intervals $L_k(x)$, $1\leqslant k\leqslant K_x$, the gaps $G_k(x)$, $0\leqslant k0$ and $00$ is the same constant used for $\pi_k$ in \cref{asp:overlap}. (Both clauses belong to the single quantified overlap assumption. The proof of \cref{thm:dml_normality} uses this assumption to bound the inverse-probability weights uniformly.) - run-level `ass:1d-ot-probability-laws`: [note, V1.5] pinned reading: For $\rho\in\{\mu,\nu\}$ and $t\in(0,1)$, $Q_\rho(t)=\inf\{a\in\mathbb R:\rho((\! -\infty,a])\geqslant t\}$. (The standard generalized-inverse convention for probability quantiles applies to the notation section and Theorem~\ref{thm:1d_ot}; the theorem evaluates these functions at uniform draws and inside Lebesgue integrals.) - run-level `ass:j-specialization-continuity`: [note, V1.5] pinned reading: For each fixed $k\in\{0,\dots,K-1\}$ and $j\in\{0,\dots,l_k-1\}$ to which part~(i) is applied, its continuity hypothesis is $\mu_{1,k\mid X}(\{\nu_{j,k}(X)\})=\mu_{1,k\mid X}(\{\nu_{j+1,k}(X)\})=0$ almost surely under the marginal law of $X$. (The proof of Proposition~\ref{prop:J_specialization}(i) fixes the band indices and uses continuity of the distribution function at the two boundaries to identify their probability levels and exclude boundary atoms.) - run-level `ass:dml-normality-first-stage`: [note, V1.5] pinned reading: For a fresh $X$ with the population marginal law and independent of $I_1$, $\\|\hat p(z,\cdot)-p(z,\cdot)\\|_{L_2}^2=\mathbb E[(\hat p(z,X)-p(z,X))^2\mid I_1]$, and $n^{1/4}\max_{z\in\mathcal Z}\\|\hat p(z,\cdot)-p(z,\cdot)\\|_{L_2}\xrightarrow{P}0$ as $n\to\infty$. (The exact-recovery argument in the proof of \cref{thm:dml_normality} compares $\mathbb E[\hat p(z,X)\mid I_1]$ with $\mathbb E[p(z,X)]$ using this norm, before evaluating the fit on independent covariates in $I_2$.) - run-level `ass:dml-normality-score-clipping`: [note, V1.5] pinned reading: Pointwise in $x$, the fitted probability, propensity, share, and indicator-mean nuisances lie in $[0,1]$, $\hat\gamma_{\mathrm{full},k}(x)\in[-1,1]$, $\hat\nu_{j,k}(x),\hat J_{\mathrm{full},k}^{\pm}(x)\in[y_{\min},y_{\max}]$, and $\hat J_{j,k}^{\pm}(x)\in[\min\{0,y_{\min}\},\max\{0,y_{\max}\}]$, for each applicable fitted component; denominator clipping is applied further as stated. (The nuisance definitions and estimation steps in \cref{app:procedure_continuous_outcome,app:procedure_binary_outcome} determine these ranges. The quantile remainder in (\ref{eq:pf_dml_bias}) explicitly uses both quantiles lying in $[y_{\min},y_{\max}]$.) - run-level `ass:dml-normality-nuisance-rates`: [note, V1.5] pinned reading: Componentwise $L_2$ errors integrate over the marginal law of $X$ under $\mathbb P_{\mathrm{obs}}$, holding each fitted function and any instrument argument $z$ fixed; $o_P(n^{-1/4})$ refers to convergence in probability over the observed sample and random partition as $n\to\infty$. (The sampling setup for Theorem~\ref{thm:dml_normality} specifies the random sample and partition; its supplied second-order remainder bound, equation~(\ref{eq:pf_dml_bias}), uses population covariate integrals of the fitted errors.) - run-level `ass:dml-normality-binary`: [note, V1.5] pinned reading: $Y\in\{0,1\}$ almost surely. (Proposition~\ref{prop:J_specialization}\textup{(ii)} explicitly states this coding before defining the binary subinterval contribution used by Theorem~\ref{thm:dml_normality}\textup{(ii)}.) - run-level `ass:continuous-learners-first-stage`: [note, V1.5] pinned reading: $\\|\hat p-p\\|_\infty=\sup_{(z,x)\in\mathcal Z\times\mathcal X}\|\hat p(z,x)-p(z,x)\|$. (The prose introducing the Continuous-Instrument Learner Conditions requires uniform first-stage control because the fit determines conditional support extrema and shrinking boundary-sample membership.) - run-level `ass:continuous-learners-outcome`: [note, V1.5] pinned reading: In (L2), $O_P(r_{g,\mathrm{loc},n})$ describes stochastic order over the training observations and random partition, while $\mathbb E_{Z,X}[\cdot\mid I_0,I_1]$ averages an independent population draw of $(Z,X)$ conditional on the partition and the observations indexed by $I_0\cup I_1$. (The three-way sample-splitting description holds fitted functions fixed for evaluation; the prose following the learner conditions explicitly distinguishes training and target-evaluation laws. The notation section defines stochastic order as boundedness in probability.) - run-level `ass:continuous-learners-boundary-grid-rate`: [note, V1.5] pinned reading: In (L3), $O_P(r_{Q,n})$ describes stochastic order over the training observations and random partition, while $\mathbb E_X[\cdot\mid I_0,I_1]$ averages an independent population draw of $X$ conditional on the partition and the observations indexed by $I_0\cup I_1$. (The learner-condition preamble specifies training conditioning, and the three-way sample-splitting description fixes learned functions during target evaluation. The notation section defines stochastic order as boundedness in probability.) - run-level `ass:ot-formulation-untreated-compatibility`: [note, V1.5] pinned reading: The untreated compatibility equality holds for $\mathbb P_{Z,X}$-almost every $(z,x)$, where $\mathbb P_{Z,X}$ is the $(Z,X)$-marginal of $\mathbb P_{\mathrm{obs}}$. (Equation $\eqref{eq:ot_formulation}$ explicitly imposes the same compatibility constraints for $\mathbb P_{Z,X}$-almost every $(z,x)$.) - run-level `def:mte`: [note, V1.5] pinned reading: $\text{MTE}(x,u)$ denotes a jointly measurable version of $\mathbb E[Y(1)-Y(0)\mid X=x,U=u]$, determined only for $(\mathbb P_X\otimes\mathrm du)$-almost every $(x,u)$. (The structural assumption specifies $U\mid X\sim\mathrm{Unif}(0,1)$, and the target in equation~(\ref{eq:target_mte}) integrates this conditional mean against $\omega(X,u)$.) - run-level `def:ot-formulation`: [note, V1.5] pinned reading: Every constraint in $\eqref{eq:ot_formulation}$ is imposed simultaneously for both $w\in\{0,1\}$. (The sentence immediately preceding equation $\eqref{eq:ot_formulation}$ explicitly says that every constraint is imposed for both treatment states.) - run-level `def:wasserstein-distance`: [note, V1.5] pinned reading: For every probability measure $\mu$ on $\mathbb R$ and $00$ satisfies, for all $x\in\mathcal X$ and $v,v'\in R_x=\{p(z,x),q(z,x)\}_{z\in\mathcal Z}$, either $v'=v$ or $\|v-v'\|>c_{\text{gap}}$." (The recorded label omits the defining strict inequality and that separation covers both $p$ and $q$. The simulation choice $c_{\text{gap}}=0.04$ instantiates the same constant and does not redefine it.) - run-level `symbol c_\pi`: [note, V1.5] symbol table corrected: def "$c_\pi>0$ satisfies $\pi_k(x)\geqslant c_\pi$ for every $x\in\mathcal X$ and observed positive-propensity level $k\in\{1,\ldots,K\}$, and $\tau_{j,k}(x)\geqslant c_\pi$ for every nonempty alternative-policy level set $T_{j,k}$." (The vague lower-bound description omits the exact probabilities and excludes neither empty sets nor the structural zero index. The paper specifies both exclusions and uses the constant in denominator clipping.) - run-level `symbol c_h`: [note, V1.5] symbol table corrected: def "$c_h>0$ satisfies $\|p_{k+1}(X)-J_{\mathrm{full},k}(X)-q_{j,k}(X)\|\geqslant c_h$ almost surely for every $k=0,\ldots,K-1$ with $l_k\geqslant1$ and $j=0,\ldots,l_k$." (The recorded description omits the defining expression and its index restrictions. The separation concerns the binary lower-boundary branch arguments, including the structural lower boundary, rather than an unspecified outcome gap.) - run-level `symbol a_{j,k}`: [note, V1.5] symbol table corrected: space "$\mathcal Z\times\mathcal X\to[0,1]$" (The recorded domain omits the instrument argument $z$. The shares form a conditional probability vector within each policy set; they are distinct from their fitted simplex-valued estimates. Later pooling and error passages retain this meaning.) - run-level `symbol \kappa_{j,k}`: [note, V1.5] symbol table corrected: space "$\mathcal X\to[0,1]$" (The formula is correct, but the recorded space treats a covariate-dependent function as a scalar. The threshold indices include the integration endpoint $l_k+1$, whereas the displayed outcome-boundary quantiles stop at $l_k$.) - run-level `symbol I_0`: [note, V1.5] symbol table corrected: type "latent interval or sample index set", space "$2^{[0,1]}\cup 2^{\{1,\ldots,n\}}$", def "$I_0=(p_0,p_1]=(0,p_1]$ in the latent-interval decomposition; in sample splitting, $I_0\subseteq\{1,\ldots,n\}$ is a training index set in the partition $I_0,I_1,I_2$." (The recorded type and space cover only the latent interval. In the estimation results being read, $I_0$ instead denotes sample indices and, in conditioning notation, the corresponding training sample. SETTINGS already records this distinction correctly.) - run-level `symbol I_1`: [note, V1.5] symbol table corrected: type "latent interval or sample index set, according to context", space "$2^{[0,1]}\cup 2^{\{1,\ldots,n\}}$", def "$I_1=(p_1,p_2]$ in the latent-interval decomposition; in estimation, $I_1$ is a sample index set in the partition of the observed sample." (The recorded latent interval alone gives the wrong type and meaning in estimation. The paper reuses the notation for an auxiliary half-sample in finite-instrument estimation and for the nuisance sample in connected-instrument estimation.) - run-level `symbol I_2`: [note, V1.5] symbol table corrected: type "latent interval or main evaluation sample index set", space "$2^{[0,1]}$ in the latent decomposition; $2^{\{1,\ldots,n\}}$ in estimation", def "In the latent decomposition, $I_2=(p_2,p_3]$. In estimation, $I_2$ is the main evaluation index set in the sample partition." (The entry records only the latent interval, whereas all listed estimation results use sample indices. The finite-IV SETTINGS description of fits as functions only of training data applies to the oracle non-set nuisances: feasible remaining fits also depend on recovered sets constructed using all $I_2$ covariates, as the paper explicitly explains.) - run-level `symbol \delta`: [note, V1.5] symbol table corrected: type "context-dependent nuisance direction or boundary-localization window", space "The nuisance function space for $\delta$; real localization windows indexed by $n$ for $\delta_n$.", def "$\delta$ is a perturbation direction in the same function space as $\eta$; $\delta_n$ is the localization window used to form $\mathcal I_{\delta,1}=\{i\in I_1:W_i=1,\ \hat P_i\leqslant\hat{\underline p}(X_i)+\delta_n\}$." (The entry describes only a nuisance perturbation, but the continuous-instrument construction also uses the indexed notation for a fixed localization window. These meanings are separated by scope; the localization clause has balanced math delimiters.) - run-level `symbol \hat\sigma^2`: [note, V1.5] symbol table corrected: def "$\hat\sigma^2=\frac{1}{\|I_2\|}\sum_{i\in I_2}\bigl(\hat\psi(O_i;\hat\eta)-\hat{\underline\theta}_{\omega,1}\bigr)^2$." (The recorded description omits the displayed centering and normalization. The paper uses the evaluation-sample mean as the center and divides by $\|I_2\|$; subsequent occurrences retain this definition.) - run-level `symbol R_{q,j,k}`: [note, V1.5] symbol table corrected: def "For every nonempty alternative-policy level set, $R_{q,j,k}(O)=\frac{1}{\tau_{j,k}(X)}\sum_{z\in T_{j,k}}\frac{\partial\phi}{\partial p}(z,X,p(z,X))\indicator(Z=z)(W-p(z,X))$. Set $R_{q,0,k}=R_{q,l_k+1,k}\equiv0$ for the structural integration boundaries." (The pooled formula is recorded correctly for regular nonempty levels, but its structural-zero branches are missing. The normalization counts a tied policy value once, and the inference convention distinguishes it from the structural left boundary.) - run-level `symbol R_{P,k}`: [note, V1.5] symbol table corrected: def "For $k\geqslant1$, $R_{P,k}(O)=\frac{\indicator(Z\in S_k)}{\pi_k(X)}(YW-P_{1,k}(X))$. At the known structural boundary, set $R_{P,0}\equiv0$." (The recorded residual formula omits the zero-index branch used in the full-interval and binary fractional-interval scores. The paper defines this branch as zero rather than by conditioning on a possibly empty structural level set.) - run-level `symbol R_{p,k}`: [note, V1.5] symbol table corrected: def "For $k\geqslant1$, $R_{p,k}(O)=\frac{\indicator(Z\in S_k)}{\pi_k(X)}(W-p_k(X))$. At the known structural boundary, set $R_{p,0}\equiv0$." (The recorded formula is correct for positive baseline levels but omits the explicitly defined zero residual at index zero. That index denotes a structural boundary and need not have an observed level set.) - run-level `symbol \psi_{j,k}^J`: [note, V1.5] symbol table corrected: def "$\psi_{j,k}^{J}(O;\eta)=J_{j,k}(X)+IF_{J^{+}}(O)-IF_{J^{-}}(O)+\nu_{j,k}(X)\big(p_{k+1}(X)M_{j,k}^{+}(X)-p_k(X)M_{j,k}^{-}(X)-(q_{j,k}(X)-p_k(X))+IF_{M_{j,k}^{+}}(O)-IF_{M_{j,k}^{-}}(O)\big)-\nu_{j+1,k}(X)\big(p_{k+1}(X)M_{j+1,k}^{+}(X)-p_k(X)M_{j+1,k}^{-}(X)-(q_{j+1,k}(X)-p_k(X))+IF_{M_{j+1,k}^{+}}(O)-IF_{M_{j+1,k}^{-}}(O)\big)+\alpha_{p_k}(X)\frac{\indicator(Z\in S_k)}{\pi_k(X)}(W-p_k(X))+\alpha_{p_{k+1}}(X)\frac{\indicator(Z\in S_{k+1})}{\pi_{k+1}(X)}(W-p_{k+1}(X))-\nu_{j,k}(X)R_{q,j,k}(O)+\nu_{j+1,k}(X)R_{q,j+1,k}(O)$, omitting every minus-side term and the $p_k$ correction at $k=0$." (The label omits the outcome, quantile-moment, propensity, and pooled policy corrections. The paper fixes the lower-index conventions explicitly: the product score with index j uses this component with index j-1, and the structural boundary never requires division by pi_0.) - run-level `symbol \psi_{\text{full},k}^J`: [note, V1.5] symbol table corrected: def "For continuous outcomes, $\psi_{\text{full},k}^{J}(O;\eta)=J_{\text{full},k}(X)+\frac{\indicator(Z\in S_{k+1})W}{\pi_{k+1}(X)}(Y-J_{\text{full},k}^{+}(X))-\frac{\indicator(Z\in S_k)W}{\pi_k(X)}(Y-J_{\text{full},k}^{-}(X))+J_{\text{full},k}^{+}(X)\frac{\indicator(Z\in S_{k+1})}{\pi_{k+1}(X)}(W-p_{k+1}(X))-J_{\text{full},k}^{-}(X)\frac{\indicator(Z\in S_k)}{\pi_k(X)}(W-p_k(X))$, with every minus-side term omitted at $k=0$. For binary outcomes, $\psi_{\text{full},k}^{J}(O;\eta)=J_{\text{full},k}(X)+R_{P,k+1}(O)-R_{P,k}(O)$." (The label omits the complete score formula. The binary display is the specialized construction for the same full-interval target; both settings explicitly avoid division by the structural-boundary share.) - run-level `symbol \psi_{j,k}^{\text{prod}}`: [note, V1.5] symbol table corrected: def "For continuous outcomes, $\psi_{j,k}^{\text{prod}}(O;\eta)=\gamma_{j,k}(X)\psi_{j-1,k}^{J}(O;\eta)+J_{j-1,k}(X)\big(\indicator(Z\in\bigcup_{l=j}^{l_k}T_{l,k})-\gamma_{j,k}(X)\big)$; for binary outcomes, $\psi_{j,k}^{\text{prod}}(O;\eta)=\gamma_{j,k}(X)\psi_{j,k}^{h}(O;\eta)+h_{j,k}^{-}(X)\big(\indicator(Z\in\bigcup_{l=j}^{l_k}T_{l,k})-\gamma_{j,k}(X)\big)$." (The label omits the formula and the essential indexing offset: the continuous product with weight index j uses integral index j-1. Binary outcomes use its specialization h with index j; these are outcome-specific constructions rather than an unexplained later redefinition.) - run-level `symbol \psi_{\text{full},k}^{\text{prod}}`: [note, V1.5] symbol table corrected: def "$\psi_{\text{full},k}^{\text{prod}}(O;\eta)=\gamma_{\text{full},k}(X)\psi_{\text{full},k}^{J}(O;\eta)+J_{\text{full},k}(X)\big(\indicator(Z\in\bigcup_{m=k+1}^{K}\bigcup_{l=0}^{l_m}T_{l,m})-\indicator(Z\in\bigcup_{m=k+1}^{K}S_m)-\gamma_{\text{full},k}(X)\big)$." (The entry omits the product rule correction for the weighting regression. The product formula is common to both outcome settings; its component score has the setting-specific formulas recorded separately.) - run-level `symbol \psi_{\Delta_K}`: [note, V1.5] symbol table corrected: def "$\psi_{\Delta_K}(O;\eta)=y_{\min}\indicator\big(Z\in\bigcup_{j=1}^{l_K}T_{j,K}\big)(q(Z,X)-p_K(X))+y_{\min}\indicator\big(Z\in\bigcup_{j=1}^{l_K}T_{j,K}\big)\frac{\partial\phi}{\partial p}(Z,X,p(Z,X))(W-p(Z,X))-y_{\min}\gamma_K(X)\frac{\indicator(Z\in S_K)}{\pi_K(X)}(W-p_K(X))$." (The label omits the observed tail contribution and both propensity corrections. The paper specifies a common lower-bound score in both outcome settings; it is not simply the plug-in conditional tail expectation.) - run-level `symbol \psi_{j,k}^{h}`: [note, V1.5] symbol table corrected: def "$\psi_{j,k}^{h}(O;\eta)=h_{j,k}^{-}(X)+D_{j,k}^{-}(X)(R_{P,k+1}(O)-R_{P,k}(O))+\alpha_{p,k+1}^{(j,k)}(X)R_{p,k+1}(O)+\alpha_{p,k}^{(j,k)}(X)R_{p,k}(O)+I_{j,k}^{-}(X)R_{q,j,k}(O)-E_{j,k}^{-}(X)I_{j,k}^{-}(X)R_{q,j-1,k}(O)$." (The entry identifies the score only by a label and omits all residual corrections. Its formula and structural boundary conventions are explicit; this is an uncentered score, not a centered influence function.) - run-level `symbol \widehat Q_{1,\underline p\mid x}`: [note, V1.5] symbol table corrected: space "$[0,1]\to[y_{\min},y_{\max}]$", def "$\widehat Q_{1,\underline p\mid x}$ is the quantile function of the estimated treated boundary law $\hat\mu_{1,\underline p\mid x}$, estimating the quantile function $Q_{1,\underline p\mid x}$ of $\mu_{1,\underline p\mid x}$." (The vague fitted-version definition omits the estimated-law interpretation. Linear interpolation can take values outside a noninterval outcome domain, so the recorded codomain $\mathcal Y$ is too restrictive; the concrete regression construction applies only to the primitive-rate procedure.) - run-level `symbol r_{e,n}`: [note, V1.5] symbol table corrected: def "$r_{e,n}\to0$ satisfies $\mathbb E_{Z,X}[\|\hat\Lambda-\Lambda\|\mid I_0,I_1]=O_P(r_{e,n})$ for each $\Lambda\in\{\underline p(X),\min\{\underline p(X),q(Z,X)\},(q(Z,X)-\overline p(X))_+\}$, where $\hat\Lambda$ substitutes the estimated nuisances." (The recorded endpoint-and-tail label omits the three defining maps and their conditional losses. The later first-stage rate relation does not redefine the sequence.) - run-level `symbol r_{g,n}`: [note, V1.5] symbol table corrected: def "$r_{g,n}\to0$ satisfies $\mathbb E_{Z,X}[\|\hat g_1(\hat q^{\mathrm{tr}}(Z,X),X)-g_1(q^{\mathrm{tr}}(Z,X),X)\|+\|\hat g_1(\hat P,X)-g_1(P,X)\|\mid I_0,I_1]=O_P(r_{g,n})$." (The recorded label omits the defining loss and its estimated-versus-true evaluation arguments. The concrete learner supplies an additional rate relation, not a new meaning.) - run-level `symbol r_{\mu,n}`: [note, V1.5] symbol table corrected: def "$\mathbb E_X[\underline p(X)W_1(\hat\mu_{1,\underline p\mid X},\mu_{1,\underline p\mid X})\mid I_0,I_1]=O_P(r_{\mu,n})$, with $r_{\mu,n}\to0$." (The weighted boundary-law label omits the Wasserstein loss, conditioning, and canonical-law comparator. Its decomposition into learner, localization, trimming, and grid errors applies to the concrete construction; it is not the general definition.) - run-level `symbol r_{p,n}`: [note, V1.5] symbol table corrected: def "$\\|\hat p-p\\|_\infty=O_P(r_{p,n})$, with the supremum taken over $(z,x)\in\mathcal Z\times\mathcal X$." (The entry omits the explicit sup-norm condition and its domain. The later restriction $r_{p,n}=o(\delta_n)$ is a condition for the proposed series boundary-learner construction, not a redefinition of this rate.) - run-level `symbol r_{g,\mathrm{loc},n}`: [note, V1.5] symbol table corrected: def "$r_{g,\mathrm{loc},n}$ is the rate in $\max_{\ell\in\{1,2\}}\mathbb E_{Z,X}[\|\hat g_1(\hat U_\ell,X)-g_1(\Pi_X(\hat U_\ell),X)\|\mid I_0,I_1]=O_P(r_{g,\mathrm{loc},n})$, where $\hat U_1=\hat q^{\mathrm{tr}}(Z,X)$, $\hat U_2=\hat P$, and $\Pi_x(u)=\min\{\overline p(x),\max\{\underline p(x),u\}\}$." (A generated-design prediction-rate label does not specify the actual loss: prediction is required at both estimated evaluation arguments, including any endpoint mass, against the population regression at their projected values.) - run-level `symbol r_{Q,n}`: [note, V1.5] symbol table corrected: def "$r_{Q,n}$ is the rate in $\frac1{M_Q}\sum_{m\in\mathcal J_{Q,n}}\mathbb E_X[\underline p(X)\|\widetilde Q_m^c(X)-Q^{\mathrm{sel}}_{1,n\mid X}(\tau_m)\|\mid I_0,I_1]=O_P(r_{Q,n})$, where $\mathcal J_{Q,n}=\{m:\varepsilon_n\le\tau_m\le1-\varepsilon_n\}$, $\tau_m=m/M_Q$, and $\widetilde Q_m^c$ is the clipped grid fit before rearrangement." (The vague fit-rate label omits the weighted grid-average loss and its selected-law comparator. Localization, excluded quantile tails, and interpolation contribute separate errors and are not included in this learner rate.) - run-level `symbol \Lambda`: [note, V1.5] symbol table corrected: type "context-dependent population expression or logistic function", space "$[0,1]\cup\{f:\mathbb R\to(0,1)\}$", def "In the endpoint-rate notation, $\Lambda\in\{\underline p(X),\min\{\underline p(X),q(Z,X)\},(q(Z,X)-\overline p(X))_+\}$; in the simulation index $\xi(X)=\Lambda(X'\beta_1)$, $\Lambda$ is the logistic function." (The recorded endpoint definition is correct locally but omits the later logistic reuse. The two meanings are explicitly separated by context, so the reuse is resolvable.) - run-level `symbol \hat\Lambda`: [note, V1.5] symbol table corrected: def "$\hat\Lambda$ is respectively $\hat{\underline p}(X)$, $\min\{\hat{\underline p}(X),\hat q(Z,X)\}$, or $(\hat q(Z,X)-\hat{\overline p}(X))_+$, paired with the corresponding choice of $\Lambda$." (The generic plug-in label omits the three paired expressions. The displayed $[0,1]$ space is correct for their evaluated values; conditional evaluation does not turn those values into constants in $(Z,X)$.) - run-level `symbol \varepsilon_n`: [note, V1.5] symbol table corrected: space "$[M_Q^{-1},1/2)$", def "$M_Q^{-1}\le\varepsilon_n<1/2$, $\varepsilon_n\to0$, and $\varepsilon_n$ defines the interior range $[\varepsilon_n,1-\varepsilon_n]$." (The recorded positive range omits the explicit upper bound, grid-dependent lower bound, and vanishing requirement.) - run-level `symbol \hat\beta_m`: [note, V1.5] symbol table corrected: def "For $m=0,\ldots,M_g$, $(\hat g_1(u_m,\cdot),\hat\beta_m)\in\arg\min_{g,\beta\in\mathcal F_n}\sum_{i\in I_1}K_{h_n}(\hat P_i-u_m)[Y_iW_i-g(X_i)-\beta(X_i)(\hat P_i-u_m)]^2$." (The recorded slope label omits the defining weighted joint optimization, including the response, function class, kernel, and training indices. The corrected definition supplies these without requiring uniqueness of the minimizer.) - run-level `symbol \widetilde Q`: [note, V1.5] symbol table corrected: space "$\{f:\{m/M_Q:m=1,\ldots,M_Q-1\}\times\mathcal X\to\mathbb R\}$", def "For $m=1,\ldots,M_Q-1$, $\widetilde Q(\tau_m\mid\cdot)\in\arg\min_{f\in\mathcal F_Q}\sum_{i\in\mathcal I_{\delta,1}}\rho_{\tau_m}(Y_i-f(X_i))$, where $\rho_\tau(u)=u(\tau-\indicator(u<0))$ and $\mathcal I_{\delta,1}=\{i\in I_1:W_i=1,\ \hat P_i\le\hat{\underline p}(X_i)+\delta_n\}$. If $\mathcal I_{\delta,1}$ is empty, set $\widetilde Q(\tau_m\mid x)=y_{\min}$ for every $(m,x)$." (The entry incorrectly grants the raw fit a domain covering every $\tau\in(0,1)$ and omits its defining optimization and empty-sample branch. The paper defines it only on the interior grid; the processed quantile function receives interpolation and endpoint values.) - run-level `symbol \mu^{\mathrm{sel}}_{1,n\mid x}`: [note, V1.5] symbol table corrected: def "$\mu^{\mathrm{sel}}_{1,n\mid x}=\mathcal L(Y\mid W=1,\ \hat P\le\hat{\underline p}(x)+\delta_n,\ X=x,I_0)$. At any $x$ for which $\mathbb P(W=1,\hat P\le\hat{\underline p}(x)+\delta_n\mid X=x,I_0)=0$, set $\mu^{\mathrm{sel}}_{1,n\mid x}=\mu_{1,\underline p\mid x}$." (The recorded label does not specify the selection event or the conditioning that defines the target. The paper defines a training-conditional population law rather than the empirical law of the localized $I_1$ sample.) - run-level `symbol \pi`: [note, V1.5] symbol table corrected: type "context-dependent probability measure or conditional instrument assignment probability", space "$\mathcal P(\mathcal A\times\mathcal B)$ in transport notation; $[0,1]$ for $\pi(z,x)$", def "A coupling $\pi\in\Pi(\mu,\nu)$ is a joint probability measure with marginals $\mu$ and $\nu$; the notation $\pi(z,x)$ denotes $\mathbb P(Z=z\mid X=x)$." (The recorded coupling meaning is correct in the OT primer but cannot describe the conditional instrument assignment probabilities used in the discrete-score proof.) - run-level `symbol \mathcal B`: [note, V1.5] symbol table corrected: type "overloaded notation for an OT state space and the Borel sigma-field operator", def "$\mathcal B$ denotes the second OT marginal's state space; $\mathcal B(\mathcal Y)$ denotes the Borel $\sigma$-field of $\mathcal Y$." (The recorded state-space interpretation misses the Borel sigma-field usage in the conditional-kernel proof. These are distinct, contextually determined uses of the same notation.) - run-level `symbol \Gamma_1`: [note, V1.5] symbol table corrected: def "$\Gamma_1(\mathbb P_{\mathrm{obs}})$ is the set of probability measures on $\mathcal Y\times[0,1]\times\mathcal X$ satisfying $\int_0^{p(z,x)}\pi_1(\mathrm dy\mid u,x)\,\mathrm du=H_{1,x}(p(z,x),\mathrm dy)$ for $\mathbb P_{Z,X}$-almost every $(z,x)$ and $\pi_1(\mathrm du,\mathrm dx)=\mathrm du\,\mathbb P_{\mathrm{obs}}(\mathrm dx)$." (The recorded entry identifies the treated feasible class without stating what makes a measure feasible. Equations~(\ref{eq:pf_cts_constraint1}) and~(\ref{eq:pf_cts_constraint2}) supply its observational and marginal constraints; the corrected definition records both.) - run-level `symbol \Gamma_{1,x}`: [note, V1.5] symbol table corrected: space "$2^{\{\pi_{1,x}:[0,1]\to\mathcal P(\mathcal Y)\text{ a probability kernel}\}}$", def "$\Gamma_{1,x}$ is the set of probability kernels $\pi_{1,x}(\mathrm dy\mid u)$ on $\mathcal Y$ satisfying $\int_0^{p(z,x)}\pi_{1,x}(\mathrm dy\mid u)\,\mathrm du=H_{1,x}(p(z,x),\mathrm dy)$ for $\mathbb P_{\mathrm{obs}}(\mathrm dz\mid x)$-almost every $z$." (The recorded definition gives only a location and label, omitting the observational constraint; its space also includes arbitrary functions rather than probability kernels. The paper defines conditional feasibility almost everywhere in $x$ and uses continuity to impose its cumulative constraint throughout the conditional propensity support.) - run-level `symbol B_Y`: [note, V1.5] symbol table corrected: type "outcome bound constant with context-dependent definition", def "$B_Y$ denotes a fixed bound determined by the feasible outcome endpoints, with its precise definition depending on the proof context." (The entry records only the first absolute-magnitude definition. The later estimation proofs use the outcome diameter, which generally differs from that magnitude bound.) - run-level `symbol \widehat S_k`: [note, V1.5] symbol table corrected: def "Choose deterministic $\rho_n\downarrow0$ with $n^{1/4}\rho_n\to\infty$ and define $\widehat S_0=\{z:\bar p_2(z)\leqslant\rho_n\}$. Remove those baseline vertices and form the graph on remaining baseline vertices and all policy vertices, connecting values among $\{\bar p_2(z),\bar q_2(z):z\in\mathcal Z\}$ whose distance is less than $c_{\mathrm{gap}}/2$. The sets $\widehat S_k$, $k\geqslant1$, collect instruments whose remaining $(p,z)$ vertices share a connected component, ordered by component averages." (The entry omits both structural-zero thresholding and the connected-component rule. Recovery uses evaluation covariates, so these sets are not fixed conditional only on training data; the SETTINGS description of remaining fits overlooks this dependence. The proof's equality with population sets is an asymptotic recovery event.) - run-level `symbol \widehat T_{j,k}`: [note, V1.5] symbol table corrected: def "After removing baseline vertices belonging to $\widehat S_0$, form the undirected graph on the remaining baseline vertices and all policy vertices, connecting two vertices when their scalar values among $\{\bar p_2(z),\bar q_2(z):z\in\mathcal Z\}$ differ by less than $c_{\mathrm{gap}}/2$. The sets $\widehat T_{j,k}$ collect instruments whose $(q,z)$ vertices share a connected component, indexed by the baseline bin containing that component average, using half-open bins for $k0$ is the same constant used for $\pi_k$ in \cref{asp:overlap}. (Both clauses belong to the single quantified overlap assumption. The proof of \cref{thm:dml_normality} uses this assumption to bound the inverse-probability weights uniformly.) - note on `ass:phi-smooth-derivative`: pinned reading: There exist finite constants $C,L\geqslant0$ such that, for all $z\in\mathcal Z$, $x\in\mathcal X$, and $p,p'\in[0,1]$, $\|\partial\phi/\partial p(z,x,p)\|\leqslant C$ and $\|\partial\phi/\partial p(z,x,p)-\partial\phi/\partial p(z,x,p')\|\leqslant L\|p-p'\|$. (The exact-recovery argument for \cref{thm:dml_normality} uses a uniformly bounded derivative, and (\ref{eq:pf_dml_bias}) uses its uniformly Lipschitz derivative for the quadratic policy remainder.) - note on `ass:phi-smooth-differentiable`: pinned reading: For every $z\in\mathcal Z$ and $x\in\mathcal X$, $p\mapsto\phi(z,x,p)$ is differentiable on $(0,1)$ and has finite right and left derivatives at $p=0$ and $p=1$, respectively; these are its endpoint values of $\partial\phi/\partial p$. (The policy map has domain $\mathcal Z\times\mathcal X\times[0,1]$. The estimation procedure clips $\hat p$ to this interval, and the policy Taylor remainder in (\ref{eq:pf_dml_bias}) uses differentiation within that domain.) - note on `ass:structure-consistency`: pinned reading: $Y=WY(1)+(1-W)Y(0)$ almost surely under the structural population law. (Assumption \ref{asp:structure} states consistency as a structural equality of random variables; the subsequent observational derivation states conditional identities almost everywhere, consistently with the standard almost-sure convention.) - note on `ass:structure-threshold-crossing`: pinned reading: $W=\indicator(U\leqslant p(Z,X))$ almost surely under the structural population law. (The threshold-crossing clause of Assumption \ref{asp:structure} is a structural equality of random variables. The derivation preceding equation \ref{eq:canonical_submeasures} uses it to establish observational identities almost everywhere.) - note on `ass:structure-uniform-resistance`: pinned reading: $\mathcal L(U\mid X=x)=\mathrm{Unif}(0,1)$ for $\mathbb P_X$-almost every $x$. (Assumption \ref{asp:structure} uses conditional-distribution notation without a pointwise covariate quantifier, while its common-support clause explicitly says every covariate value. The surrounding prose also recognizes that conditional probabilities are identified only almost surely.) - note on `ass:target-weight-identifiable`: pinned reading: Any two structural laws satisfying Structural Assumptions and inducing the same $\mathbb P_{obs}$ give the same $\omega(x,u)$ for $\mathbb P_{obs}(\mathrm dx)\,\mathrm du$-almost every $(x,u)$. (The paragraph following Structural Assumptions states that the observed law identifies conditional probabilities almost surely; equation (\ref{eq:omega_form}) constructs weights from such conditional expectations, and equation (\ref{eq:target_mte}) integrates against the covariate marginal and Lebesgue measure.) - note on `def:binary-score`: pinned reading: In this definition, $\psi_{\mathrm{full},k}^{\mathrm{prod}}$ and $\psi_{j,k}^{\mathrm{prod}}$ are the binary-outcome components of Section~\ref{app:scores_discrete}, and $\psi_{\Delta_K}$ is the common tail component of Section~\ref{app:scores_continuous}. (The binary nuisance-estimation subsection introduces this display and immediately directs the reader to Section~\ref{app:scores_discrete} for its component formulas.) - note on `def:boundary-indicator-means`: pinned reading: For $j=1,\ldots,l_k$, interpret $\nu_{j,k}(x)=Q_{Y,k\mid x}(\kappa_{j,k}(x))$ using $Q_{Y,k\mid x}(t)=\inf\{y\in\mathbb R:\mu_{1,k\mid x}(\mathcal Y\cap(-\infty,y])\geqslant t\}$ for $00$ and $00$ satisfies, for all $x\in\mathcal X$ and $v,v'\in R_x=\{p(z,x),q(z,x)\}_{z\in\mathcal Z}$, either $v'=v$ or $\|v-v'\|>c_{\text{gap}}$." (The recorded label omits the defining strict inequality and that separation covers both $p$ and $q$. The simulation choice $c_{\text{gap}}=0.04$ instantiates the same constant and does not redefine it.) - note on `symbol c_\pi`: symbol table corrected: def "$c_\pi>0$ satisfies $\pi_k(x)\geqslant c_\pi$ for every $x\in\mathcal X$ and observed positive-propensity level $k\in\{1,\ldots,K\}$, and $\tau_{j,k}(x)\geqslant c_\pi$ for every nonempty alternative-policy level set $T_{j,k}$." (The vague lower-bound description omits the exact probabilities and excludes neither empty sets nor the structural zero index. The paper specifies both exclusions and uses the constant in denominator clipping.) - note on `symbol c_h`: symbol table corrected: def "$c_h>0$ satisfies $\|p_{k+1}(X)-J_{\mathrm{full},k}(X)-q_{j,k}(X)\|\geqslant c_h$ almost surely for every $k=0,\ldots,K-1$ with $l_k\geqslant1$ and $j=0,\ldots,l_k$." (The recorded description omits the defining expression and its index restrictions. The separation concerns the binary lower-boundary branch arguments, including the structural lower boundary, rather than an unspecified outcome gap.) - note on `symbol H_{w,x}`: symbol table corrected: space "$\operatorname{supp}(P\mid X=x)\to\mathcal M_+(\mathcal Y)$" (The definition already selects the correct canonical observed version, but the recorded domain $[0,1]$ wrongly supplies observed values outside the conditional propensity support. The paper fixes the arm index, support endpoints, submeasure masses, and notation without covariates.) - note on `symbol H_w`: symbol table corrected: space "$\operatorname{supp}(P)\to\mathcal M_+(\mathcal Y)$, with $H_1(0,\cdot)=0$ additionally specified at the structural boundary." (The recorded domain [0,1] suggests a uniquely identified observed map throughout unsupported gaps and tails. The paper identifies the canonical map only on propensity support, with the separately specified treated zero boundary; a compatible latent kernel supplies extensions elsewhere without making those extensions identified.) - note on `symbol I_0`: symbol table corrected: type "latent interval or sample index set", space "$2^{[0,1]}\cup 2^{\{1,\ldots,n\}}$", def "$I_0=(p_0,p_1]=(0,p_1]$ in the latent-interval decomposition; in sample splitting, $I_0\subseteq\{1,\ldots,n\}$ is a training index set in the partition $I_0,I_1,I_2$." (The recorded type and space cover only the latent interval. In the estimation results being read, $I_0$ instead denotes sample indices and, in conditioning notation, the corresponding training sample. SETTINGS already records this distinction correctly.) - note on `symbol I_1`: symbol table corrected: type "latent interval or sample index set, according to context", space "$2^{[0,1]}\cup 2^{\{1,\ldots,n\}}$", def "$I_1=(p_1,p_2]$ in the latent-interval decomposition; in estimation, $I_1$ is a sample index set in the partition of the observed sample." (The recorded latent interval alone gives the wrong type and meaning in estimation. The paper reuses the notation for an auxiliary half-sample in finite-instrument estimation and for the nuisance sample in connected-instrument estimation.) - note on `symbol I_2`: symbol table corrected: type "latent interval or main evaluation sample index set", space "$2^{[0,1]}$ in the latent decomposition; $2^{\{1,\ldots,n\}}$ in estimation", def "In the latent decomposition, $I_2=(p_2,p_3]$. In estimation, $I_2$ is the main evaluation index set in the sample partition." (The entry records only the latent interval, whereas all listed estimation results use sample indices. The finite-IV SETTINGS description of fits as functions only of training data applies to the oracle non-set nuisances: feasible remaining fits also depend on recovered sets constructed using all $I_2$ covariates, as the paper explicitly explains.) - note on `symbol IF_{J^-}`: symbol table corrected: def "$IF_{J^-}(O)=\pi_k(X)^{-1}\indicator(Z\in S_k)W\big(Y\indicator(\nu_{j,k}(X)