# Correctness report — tbs_extreme_effects **Verdict: ERRORS FOUND — REPAIRED BY MINIMAL PATCHES** 7 result(s) of the paper were examined. 5 were found wrong as printed: 5 corrected (5 by a weaker statement), 0 not repaired. In Lean: 7 proved, 0 taken from the literature as cited, 0 not certified, 0 not formalized. ## What was wrong and what was done ### Proposition 3.3 — corrected; the corrected statement is weaker - **As printed.** \label{prop:id}Suppose that $U$ is multivariate regularly varying and Assumption~\ref{assump:exo}, \ref{assump:iid}, \ref{assump:f_homo} hold, we have \begin{align*} \theta^{\tmop{NETE}} = \lim_{t \rightarrow \infty} \mathbb{E}[g (X, 1, U /\|U\|) - g (X, 0, U /\|U\|) \mid \|U\|> t] \cdot \lim_{t \rightarrow \infty} \mathbb{E}[\|U\|^{\alpha} / t^{\alpha} \mid \|U\|> t] . \end{align*} - **What is wrong.** ``\dots \[ \begin{array}{@{}c@{}}\displaystyle \lim_{t \rightarrow \infty} \mathbb{E} \left[ \frac{Y (1) - Y (0)}{t^{\alpha}} \mid \|U\|> t \right] \\ \displaystyle = \lim_{t \rightarrow \infty} \mathbb{E} \left[ \frac{f (X, 1, U) - f (X, 0, U)}{t^{\alpha}} \mid \|U\|> t \right] \\ \displaystyle = \lim_{t \rightarrow \infty} \mathbb{E} \left[ \frac{f (X, 1, U) - f (X, 0, U)}{\|U\|^{\alpha}} \cdot \left( \frac{\|U\|}{t} \right)^{\alpha} \mid \|U\|> t \right] \\ \displaystyle = \lim_{t \rightarrow \infty} \mathbb{E} \left[ (g (X, 1, U / \| U \|) - g (X, 0, U / \| U \|) + 2 e (t)) \cdot \left( \frac{\|U\|}{t} \right)^{\alpha} \mid \|U\|> t \right]\end{array}\mbox{''} \] This step uses the conditional-mean identification supplied by exogeneity and consistency, which the paper assumes only for conditioning on $X$, but applies it to conditioning on $(X,U)$ and then on $\{\|U\|>t\}$. In the step quoted in (a): The homogeneity inequality supplies a signed remainder bounded at the realized radius by $2e(\|U\|)$, rather than the displayed equality with the positive term $2e(t)$. ``\dots \[ \frac{\int_1^{\infty} \alpha P (\| U \| > {rt}) r^{\alpha - 1} \tmop{dr}}{P (\| U \| > t)} \leqslant 2\int^{\infty}_1 \alpha r^{\alpha -\beta + \epsilon - 1} \tmop{dr} < \infty .\mbox{''} \] The choice of a positive $\epsilon$ satisfying $\alpha-\beta+\epsilon<0$ requires $\alpha<\beta$, which the proposition does not assume. ``We next argue that $\lim_{t \rightarrow \infty} \mathbb{E} [(\| U \| / t)^{\alpha} \mid \| U \| > t] = \alpha / (\beta - \alpha)$.'' The radial moment limit is $\beta/(\beta-\alpha)$ when $\alpha<\beta$, rather than $\alpha/(\beta-\alpha)$. ``We conclude that $h (U)$ is uniformly integrable.'' A uniform bound on first absolute moments does not imply uniform integrability. - **Counterexample.** Let $X\sim\tmop{Unif}([0,1])$, $D\sim\tmop{Ber}(1/2)$, and $V\sim\tmop{Unif}([0,1])$ be mutually independent. Let $W$ be independent of $(X,D,V)$ with $\mathbb P(W>w)=w^{-3}$ for $w\geq1$, and set $U=W(V,1-V)$ and $\alpha=1$. Define $T(w)=(1-w^{-3})^{-1/3}$, $Y(1)=DW+(1-D)T(W)$, $Y(0)=0$, and $Y=Y(D)$. Take independent copies for the observations, and take $g(x,1,s)=1$, $g(x,0,s)=0$, and $e(r)=0$. The variables $T(W)$ and $W$ have the same distribution, so $(Y(1),Y(0))\perp D\mid X$, while $U\perp(X,D)$ and exact homogeneity holds. The displayed product is $3/2$, whereas $\mathbb E[T(W)\mid W>t]\to1$ and $\mathbb E[W\mid W>t]=3t/2$ give $\theta^{\tmop{NETE}}=3/4$. - **Correction.** Suppose that $U$ is multivariate regularly varying and Assumption~\ref{assump:exo}, \ref{assump:iid}, \ref{assump:f_homo} hold. Suppose in addition that $\alpha<\beta$, $0t]<\infty$ for both treatments and all sufficiently large $t$. Then we have \begin{align*} \theta^{\tmop{NETE}} &= \lim_{t\rightarrow\infty}\mathbb E[g(X,1,U/\|U\|)-g(X,0,U/\|U\|)\mid\|U\|>t]\cdot\lim_{t\rightarrow\infty}\mathbb E[\|U\|^{\alpha}/t^{\alpha}\mid\|U\|>t]. \end{align*} - **Proof.** Write $R_U=\|U\|$, $S_U=U/\|U\|$, and $\bar e(t)=\sup_{r\geq t}e(r)$; the last quantity is finite for sufficiently large $t$ and tends to zero. Independence of $U$ from $(X,D)$ gives $\mathbb P(D=1\mid X,U)=p(X)$. Positivity, consistency, and conditional mean exogeneity given $(X,U)$ therefore give, for both treatments, \[ f(X,d,U)=\mathbb E[Y(d)\mid X,U,D=d]=\mathbb E[Y(d)\mid X,U]. \] The assumed absolute tail means justify taking conditional expectations on $\{R_U>t\}$. Homogeneity at the realized radius yields \[ \frac{f(X,1,U)-f(X,0,U)}{R_U^{\alpha}}=g(X,1,S_U)-g(X,0,S_U)+\varepsilon(X,U),\qquad |\varepsilon(X,U)|\leq2\bar e(t)\quad\hbox{on }\{R_U>t\}. \] Consequently, \[ \left|\mathbb … (the corrected paper gives it in full) - **What the correction gives up.** Identification for distributions that satisfy exogeneity given $X$ but violate conditional mean exogeneity given $(X,U)$, distributions with a degenerate true propensity, the range $\alpha\geq\beta$, and counterfactuals without finite absolute tail means. (The paper's owner accepted the weaker statement.) - **Formal check.** proved in Lean (the corrected statement), under 7 condition(s) made explicit — `Papers.TbsExtremeEffects.nete_identification` (`TId.lean:18`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Theorem 3.5 — corrected; the corrected statement is weaker - **As printed.** \label{thm:rate}Suppose that Assumption \ref{assump:exo}, \ref{assump:overlap}, \ref{assump:iid}, \ref{assump:f_homo}, \ref{assump:mvary} hold, $\alpha < \beta$, where $\alpha$ and $\beta$ are defined in \cref{assump:f_homo} and \cref{assump:mvary} respectively. Furthermore, for any fixed $t$, with probability at least $1 - \delta$, \begin{align*} |p (X) - \hat{p} (X) | \leqslant R_p (n, \delta) & , | \hat{\alpha}_{n} - \alpha| \leqslant R_\alpha(n,\delta),\\ |\mathbb{E}[Y/\|U\|^{\alpha} \mid X, D, U /\|U\|, \|U\|> t] &- \hat{g} (X, D, U /\|U\|) | \leqslant R_g (n_t, \delta), \end{align*} where $n_t = \sum_{i = 1}^{n/2} I (\|U_i \|> t)$ and $R_p, R_g, R_\alpha$ are estimation errors that are monotonically decreasing with respect to sample size. Then, with probability at least $1 - \delta, \delta \in (0, 1 / 2)$, we have \begin{align} \left| {{\widehat{\theta }_{n, t}^{\text{DR} }} } - \theta^{NETE} \right| & \leqslant O (\sqrt{R_p (n/2, \delta)R_g ({n_t}, \delta)} + t^{\beta / 2} n^{-1 / 2}+ \log (1 / \delta) n^{- 1 / (2 +\beta)} \notag\\ & \quad\quad \quad \quad + t^{- \min \{1, \beta\}} + t^{- \beta s / (1 - 2 s)} + \log(t)R_\alpha(n,\delta)+ e (t)).\label{eq:rate_dr} \end{align} and \begin{align} \left| {{\widehat{\theta }_{n, t}^{\text{IPW}}} } - \theta^{NETE} \right| & \leqslant O (R_p (n/2, \delta) +t^{\beta / 2} n^{- 1 / 2} + \log (1 / \delta) n^{- 1 / (2 + \beta)} \notag\\ &\quad\quad \quad \quad + t^{- \min \{1, \beta\}} + t^{- \beta s / (1 - 2 s)}+ \log(t)R_\alpha(n,\delta) + e (t)). \label{eq:rate_ipw} \end{align} - **What is wrong.** ``\dots \[ \begin{array}{@{}c@{}}\displaystyle | \hat{\theta}_{n, t}^{\tmop{DR}} - \theta^{\tmop{NETE}} | = | \hat{\eta}_{n, t}^{\tmop{DR}} \cdot \hat{\mu}_n - \eta \cdot \mu | \\ \displaystyle \leqslant | \mu | \cdot | \hat{\eta}_{n, t}^{\tmop{DR}} - \eta | + | \hat{\eta}_{n, t}^{\tmop{DR}} | \cdot | \hat{\mu}_n - \mu |.\end{array}\mbox{''} \] This step uses identification of the potential-outcome conditional means by $f(X,d,U)$, which the paper assumes only for the conditional mean exogeneity condition $\mathbb E[Y(d)\mid X,U,D]=\mathbb E[Y(d)\mid X,U]$ for both treatments, but applies it to potential outcomes satisfying only $(Y(1),Y(0))\perp D\mid X$ and noise satisfying $U\perp(X,D)$. ``\dots \[ \begin{array}{@{}c@{}}\displaystyle | \hat{\theta}_{n, t}^{\tmop{DR}} - \theta^{\tmop{NETE}} | \leqslant O ( \sqrt{R_p (n/2, \delta) R_g (nt^{- \beta}, \delta)} + t^{\beta / 2}n^{- 1 / 2} + \log (1 / \delta) n^{- 1 / (2 + \beta)} \\ \displaystyle \quad\quad\quad\quad + t^{- \min \{1,\beta\}} + t^{- \beta s / (1 - 2 s)} + \log (t) R_{\alpha} (n, \delta) + e(t)) .\end{array}\mbox{''} \] The sampling term $t^{\beta/2}n^{-1/2}$ requires control of normalized outcome fluctuations, whereas the assumptions allow integrable stable noise whose sample averages converge more slowly. ``\dots \[ | \hat{\mu}_n - \mu | = \left| \frac{1}{1 - \hat{\alpha}_{n} \hat{\gamma}_n} - \frac{1}{1 - \alpha \gamma} \right| = O (| \hat{\alpha}_{n} - \alpha | + | \hat{\gamma}_n - \gamma |) = O (\log (1 / \delta) n^{- 1 / (2 + \beta)} + R_{\alpha} (n, \delta))\mbox{''} \] The reciprocal difference contains the factor $|1-\hat\alpha_n\hat\gamma_n|^{-1}$, which boundedness of $\hat\gamma_n$ and sample-size monotonicity of $R_\alpha$ do not control. In the step quoted in (b): The exponent perturbation contains $t^a-1+Kat^a$, with $a=R_\alpha(n,\varepsilon)$, and cannot be bounded by a constant multiple of $a\log t$ when $a\log t$ is unbounded. In the step quoted in (b): Homogeneity is used at radii exceeding $t$, so its contribution is controlled by $\bar e(t)=\sup_{r\geq t}e(r)$ rather than by $e(t)$. In the step quoted in (b): The angular bias can have order $t^{-1}\log t$ when $\beta=1$, whereas the displayed bound retains only $t^{-1}$. ``By Lemma \ref{lemma:mu_rate} and \ref{lemma:eta_rate}, with high probability, $ \hat{\gamma}_n $ and $| \hat{\eta}_{n,t}^{\tmop{DR}} |$ is bounded.'' A nuisance event with failure probability $\delta$ leaves no allowance for sampling or radial-estimation failures, and sample-size monotonicity does not compare nuisance bounds at different confidence tolerances. In the step quoted in (c): Lemma~\ref{lemma:mu_rate} retains an adaptation logarithm, an additional endpoint logarithm when $\beta=1$, and exponent $1/3$ when $0<\beta<1$, none of which is supplied by the displayed conversion to $\log(1/\delta)n^{-1/(2+\beta)}$. - **Counterexample.** Take $m=d=d_z=d_u=2$, $A=I_2$, and independent Pareto type-II components $Z_1,Z_2$ with index $\beta=2$, and set $U=AZ$. Choose $l=1/2$, $u=2$, $\sigma=1/2$, $s=2/5$, $\xi=1/2$, and $\zeta=1$, so that the nominal component density satisfies the tail-model conditions. Let $X=0$, let $r_0$ be a median of $\|U\|$, and set $B=I\{\|U\|>r_0\}$. Let $V$ be an independent fair Bernoulli variable, let $\varepsilon$ be independent standard Gaussian noise, and define $D=B\mathbin{\oplus}V$, $Y(d)=2d+V+\varepsilon$, and $Y=Y(D)$. Take independent copies for the observations, set $\alpha=\hat\alpha_n=0$, $g(x,d,u)=1+d$, $e(r)=I\{r\leq r_0\}$, $p=\hat p=1/2$, $c=1/4$, and $\hat g(x,d,u)=1+d$. For each tolerance $\varepsilon_0\in(0,1)$, use $R_p(q,\varepsilon_0)=(q+1)^{-10}$, $R_\alpha(q,\varepsilon_0)=(q+1)^{-10}$, and $R_g(q,\varepsilon_0)=2+(q+1)^{-10}$, which satisfy the joint uniform nuisance inequalities. Fix $\delta=1/4$ and take $t=n^{1/4}$ along even sample sizes. Here $D$ is independent of both $U$ and the potential-outcome pair, but $\theta^{\tmop{NETE}}=2$ while both estimators converge in probability to $1$ and both displayed error bounds tend to zero. - **Rests on.** Lemma A.3, Lemma A.2, Lemma A.1, Proposition 3.3, also found wrong as printed. - **Correction.** Suppose that Assumption \ref{assump:exo}, \ref{assump:overlap}, \ref{assump:iid}, \ref{assump:f_homo}, \ref{assump:mvary} hold, $\alpha < \beta$, where $\alpha$ and $\beta$ are defined in \cref{assump:f_homo} and \cref{assump:mvary} respectively. Suppose in addition that the conditions of Proposition~\ref{prop:id} hold. Suppose that the fitted propensity, regression and exponent are measurable with respect to the training split and independent auxiliary data, denoted together by $\mathcal T$, and that $\mathbb E[|Y/\|U\|^{\hat\alpha_n}|\mid\mathcal T,\|U\|>t]<\infty$ on each joint nuisance event. Furthermore, for any fixed $t$, with probability at least $1 - \delta$, \begin{align*} |p (X) - \hat{p} (X) | \leqslant R_p (n, \delta) & , | \hat{\alpha}_{n} - \alpha| \leqslant R_\alpha(n,\delta),\\ |\mathbb{E}[Y/\|U\|^{\alpha} \mid X, D, U /\|U\|, \|U\|> t] &- \hat{g} (X, D, U /\|U\|) | \leqslant R_g (n_t, \delta), \end{align*} where $n_t = \sum_{i = 1}^{n/2} I (\|U_i \|> t)$ and $R_p, R_g, R_\alpha$ are estimation errors that are monotonically decreasing with respect to sample size. Fix $\delta\in(0,1/2)$ and a deterministic threshold $t$. Run the adaptive Hill selection in Algorithm~\ref{alg:nete} at tolerance $\delta/6$, and write its full-input output as $\hat\gamma_n=\widetilde\gamma_n$. Put $\pi_t=\mathbb P(\|U\|>t)$, $M=|\mathcal I|$, $\bar e(t)=\sup_{r\geq t}e(r)$, and $G_t=C+\bar e(t)$, where $C$ bounds $|g|$ on its compact domain. Use zero for each first-factor estimate when $M=0$. For a generic evaluation observation put $S_U=U/\|U\|$ and define \[ \begin{aligned} \psi^{\mathrm{DR}}={}&\hat g(X,1,S_U)-\hat g(X,0,S_U)\\ &+\frac{D-\hat p(X)}{\hat p(X)(1-\hat p(X))}\left(\frac{Y}{\|U\|^{\hat\alpha_n}}-\hat g(X,D,S_U)\right),\\ \psi^{\mathrm{IPW}}={}&\frac{Y}{\|U\|^{\hat\alpha_n}}\left(\frac{D}{\hat p(X)}-\frac{1-D}{1-\hat p(X)}\right). \end{aligned} \] For each $q\in\{\mathrm{DR},\mathrm{IPW}\}$, choose $B_q>0$ using only $\mathcal T$, and put \[ \begin{aligned} v_t^q(B_q)&=\mathbb E[(\psi^q)^2I\{|\psi^q|\leq B_q\}\mid\mathcal T,\|U\|>t],\\ b_t^q(B_q)&=\mathbb E[|\psi^q|I\{|\psi^q|>B_q\}\mid\mathcal T,\|U\|>t],\\ \mathcal S_t^q&=\sqrt{\frac{12v_t^q(B_q)}{\delta\max\{M,1\}}}+\left(1+\frac{12}{\delta}\right)b_t^q(B_q). \end{aligned} \] Fix $a_0\in(0,\beta)$ and put \[ \begin{aligned} a&=R_\alpha(n,\delta/12),\qquad K=\frac{4}{\beta-a_0},\\ \mathcal A_t&=2LC_W\left((\log t)^k t^{-\min\{1,\beta\}}+t^{-\beta s/(1-2s)}\right),\\ \mathcal E_{\mathrm{DR}}={}&\mathcal S_t^{\mathrm{DR}}+\frac2c R_p(n/2,\delta/12)R_g(n_t,\delta/12)\\ &+2\bar e(t)+\mathcal A_t+\frac{2G_t}{c}(t^a-1+Kat^a),\\ \mathcal E_{\mathrm{IPW}}={}&\mathcal S_t^{\mathrm{IPW}}+\frac{2G_t}{c}R_p(n/2,\delta/12)\\ &+2\bar e(t)+\mathcal A_t+\frac{2G_t}{c}(t^a-1+Kat^a). \end{aligned} \] Suppose moreover, as in Lemma~\ref{lemma:eta_rate}, that the angular Wasserstein term satisfies the bound that Proposition 3.1 of \citet{zhang2023wasserstein} gives for this Pareto-type model, which is taken from that work and not proved here: there are constants $C_W>0$, $k\geq0$ and $t_W\geq1$ such that \[ W_1(\mathcal L_{U/\|U\|}^t,\mathcal L_{U/\|U\|})\leq C_W(\log t)^k t^{-\min\{1,\beta\}}+C_W t^{-\beta s/(1-2s)}\qquad(t\geq t_W). \] These are the constants $C_W$ and $k$ in the angular term $\mathcal A_t$ above. Let \[ b_n=C_{\delta/6} \begin{cases} \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/3},&0<\beta<1,\\ \left(\dfrac{\log n\,\log((48/\delta)\log n)}{n}\right)^{1/3},&\beta=1,\\ \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/(2+\beta)},&\beta>1, \end{cases} \] where $C_{\delta/6}$ is the constant supplied by Lemma~\ref{lemma:mu_rate}. Define \[ \begin{aligned} \gamma&=1/\beta,\qquad h=1-\alpha\gamma>0,\\ Q_n&=a(\gamma+b_n)+|\alpha|b_n,\qquad d_n=|1-\hat\alpha_n\hat\gamma_n|,\\ \mathcal H_n&= \begin{cases} Q_n/(h d_n),&d_n>0,\\ 1/h,&d_n=0. \end{cases} \end{aligned} \] Use Algorithm~\ref{alg:nete}'s convention $\hat\mu_n=0$ when $d_n=0$. For sufficiently large $n$ and $t\geq t_W$, if $a\leq a_0$ and $n\pi_t\geq16\log(12/\delta)$, then, with probability at least $1-\delta$, we have $M\geq n\pi_t/4$ and \[ \left|\widehat\theta_{n,t}^{\mathrm{DR}}-\theta^{\tmop{NETE}}\right| \leq |\hat\mu_n|\mathcal E_{\mathrm{DR}}+|\eta|\mathcal H_n, \] and \[ \left|\widehat\theta_{n,t}^{\mathrm{IPW}}-\theta^{\tmop{NETE}}\right| \leq |\hat\mu_n|\mathcal E_{\mathrm{IPW}}+|\eta|\mathcal H_n, \] where $\eta=\lim_{r\rightarrow\infty}\mathbb E[g(X,1,U/\|U\|)-g(X,0,U/\|U\|)\mid\|U\|>r]$. If moreover, for a constant $V<\infty$ independent of $n,t,\delta$ and every $q\in\{\mathrm{DR},\mathrm{IPW}\}$, $\mathbb E[(\psi^q)^2\mid\mathcal T,\|U\|>t]\leq V$ on the joint nuisance event, $t\geq\exp(1)$, $a\log t\leq1$ and $Q_n\leq h/2$, then on the same joint success event of probability at least $1-\delta$ we also have \[ \begin{aligned} |\widehat\theta_{n,t}^{\mathrm{DR}}-\theta^{NETE}|\leq C\biggl(&\sqrt{\frac{V}{\delta n\pi_t}}+R_p(n/2,\delta/12)R_g(n_t,\delta/12)\\ &+\bar e(t)+\mathcal A_t+R_\alpha(n,\delta/12)\log t+b_n\biggr),\\ |\widehat\theta_{n,t}^{\mathrm{IPW}}-\theta^{NETE}|\leq C\biggl(&\sqrt{\frac{V}{\delta n\pi_t}}+R_p(n/2,\delta/12)\\ &+\bar e(t)+\mathcal A_t+R_\alpha(n,\delta/12)\log t+b_n\biggr), \end{aligned} \] where $C$ does not depend on $n,t,\delta$, and $\pi_t\asymp t^{-\beta}$ makes the first term of order $\sqrt{V/\delta}\,t^{\beta/2}n^{-1/2}$. - **Proof.** Proposition~\ref{prop:id} applies because its identification and integrability conditions are assumed. The limiting angular contrast is $\eta$, and the limiting radial moment is \[ \mu=\int_1^\infty r^\alpha\beta r^{-\beta-1}\,\tmop{dr}=\frac{\beta}{\beta-\alpha}=\frac1h. \] The integral converges because $\alpha<\beta$. Consequently $\theta^{\tmop{NETE}}=\eta\mu$. Since $|g|\leq C$, we also have $|\eta|\leq2C$. For either estimator, the product identity gives \[ \widehat\theta_{n,t}^{\mathrm{estimator}}-\theta^{\tmop{NETE}} =\hat\mu_n(\hat\eta_{n,t}^{\mathrm{estimator}}-\eta)+\eta(\hat\mu_n-\mu). \] Let $E_\alpha$ be the joint nuisance event at tolerance $\delta/12$. The nuisance … (the corrected paper gives it in full) - **What the correction gives up.** The unconditional claim under covariate-only exogeneity is lost, including cases where treatment and potential outcomes become dependent after conditioning on $U$. Fitted nuisances using evaluation observations and fitted-exponent pseudo-outcomes without finite conditional absolute means are excluded. The suggested first-factor bound requires $R_\alpha(n,\delta/12)\leq a_0<\beta$ and enough expected exceedances for its probability guarantee. A universal square-root sampling rate without outcome-tail control, a uniformly bounded reciprocal perturbation near its pole, the linear exponent term without control of $R_\alpha(n,\delta)\log t$, the pointwise homogeneity term $e(t)$, and the logarithm-free angular and radial rates are lost. The nuisance bounds are evaluated at the smaller confidence tolerance $\delta/12$. The appended square-root sampling and linear exponent bounds apply only to scores satisfying the stated conditional second-moment bound, thresholds $t\geq\exp(1)$ with $R_\alpha(n,\delta/12)\log t\leq1$, and perturbations satisfying $Q_n\leq h/2$; the general product bounds still retain the reciprocal denominator and full exponent perturbation outside those additional conditions. (The paper's owner accepted the weaker statement.) - **Formal check.** proved in Lean (the corrected statement), under 25 condition(s) made explicit — `Papers.TbsExtremeEffects.nete_rate` (`TRate.lean:96`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Lemma A.1 — corrected; the corrected statement is weaker - **As printed.** \label{lemma:mu_rate}Under Assumption \ref{assump:iid}, \ref{assump:mvary} with probability at least $1 - \delta$, for sufficiently large $n$, we have \begin{align*} | \gamma - \hat{\gamma}_n | & \leqslant O \left( \left( \frac{\log (2 / \delta)}{n} \right)^{1 / (2 + \beta)} \right), \end{align*} where $\gamma = 1 / \beta$ is the EVI of $U$. - **What is wrong.** ``\dots choosing $k$ to be \[ k = \max \left\{ k \in \{l_n, \cdots, n\} \text{and} \ \forall i \in \{l_n, \cdots, n\}, | \hat{\gamma}(i) - \hat{\gamma}(k) | \leqslant \frac{\hat{\gamma}(i) r_n (\delta)}{\sqrt{i}} \right\} \] where $\hat{\gamma}(i) = \frac{1}{i} \sum_{j=1}^i \log \frac{\|U_{(j)}\|}{\|U_{(i+1)}\|}$ and $r_n (\delta)$ scales like $\sqrt{\log ((2 / \delta) \log (n))}$.'' The comparison at $i=n$ requires rank $n+1$, although Algorithm~\ref{alg:nete} orders at most $n/2$ evaluation exceedances. In the step quoted in (a): The common-denominator Hill statistic in this step differs from Algorithm~\ref{alg:nete}'s adjacent-ratio statistic, which telescopes to $k^{-1}\log(\|U_{(1)}\|/\|U_{(k+1)}\|)$. In the step quoted in (a): Comparisons against every candidate can leave the admissible set empty, even when all candidate Hill statistics are available. In the step quoted in (a): The lower rank and tolerance are not calibrated, so their stated orders alone do not establish the adaptive deviation bound. ``By Assumption~\ref{assump:mvary}, we know that \[ g (z) = c \alpha^m \prod_{i = 1}^m (1 + z_i)^{- \alpha - 1}, \|z\|_1 > \zeta k^{(1 - 2 s) / \alpha} .\mbox{''} \] The component density uses the outcome exponent $\alpha$ instead of the tail index $\beta$. ``Let $\tilde{Z}_1, \cdots, \tilde{Z}_m \sim \alpha c^{1 / m} (1 + z)^{- \alpha - 1}, z \geqslant c^{1 / (m \alpha)} - 1$.'' The shifted component distributions alter the nominal product law, so exact proportionality of the original component density does not identify their radial convolution with the radial law of $U$. ``\dots we have that when $\|U\|_1 > \max_i \{a_i \} \zeta k^{(1 - 2 s) / \alpha}$, \[ f (t) = C \beta t^{- \beta - 1} (1 + D (1 - 1 / \beta) t^{- 1} + o (t^{- 1})) .\mbox{''} \] The constant-coefficient density expansion omits a logarithmic term at $\beta=1$. In the step quoted in (g): The stated survival coefficient $D$ corresponds to density coefficient $D(1+1/\beta)$ rather than $D(1-1/\beta)$. ``\dots we get \[ \eta (t) = \frac{1}{\beta (1 - DC^{- 1 / \beta} t^{- 1 / \beta} + o (t^{1 / \beta}))} - \frac{1}{\beta} = O (t^{- 1 / \beta}) .\mbox{''} \] At $\beta=1$ the von Mises function can have order $t^{-1}\log t$, which is not $O(t^{-1})$. In the step quoted in (i): For $\beta>1$ the survival expansion gives von Mises coefficient $-DC^{-1/\beta}/\beta^2$, with the opposite sign to the displayed substitution. ``\dots with probability at least $1 - \delta$, we have \[ | \gamma - \hat{\gamma}_n | \leqslant O \left( \left( \frac{\log (2 / \delta)}{n} \right)^{1 / (2 + \beta)} \right),\mbox{''} \] The cited adaptive deviation bound retains $\log((2/\delta)\log n)$, which cannot be replaced by $\log(2/\delta)$ for fixed $\delta$. - **Counterexample.** Take $m=d=d_z=d_u=2$, $A=I_2$, and independent $Z_1,Z_2$ with density $(1+z)^{-2}$ on $z\geq0$. Set $U=AZ$, $\beta=1$, $l=1/4$, $u=2$, $\sigma=1/8$, $s=1/4$, $\xi=1/2$, and $\zeta=1$. Let $X=0$, let $D\sim\tmop{Ber}(1/2)$ be independent of $U$, set $Y=0$, and take independent copies for the observations. Choose even $n$, $t=1$, and $l_n=\lceil100\log n\rceil$ for sufficiently large $n$. Let $R=\|U\|_1$, whose survival is $\bar F_R(r)=2/(r+2)+2\log(r+1)/(r+2)^2$ and satisfies $\mathbb P(R>r)\leq4/r$. The comparison at $i=n$ requests an unavailable rank $n+1$; even after unavailable ranks are suppressed, with probability at least $1-4/n$ every computable adjacent-ratio candidate with $k\geq l_n$ satisfies $k^{-1}\log(\|U_{(1)}\|/\|U_{(k+1)}\|)\leq2\log n/l_n\leq1/50$, so any selected output has error from $\gamma=1$ at least $49/50$, whereas the claimed bound tends to zero. - **Correction.** Under Assumption \ref{assump:iid}, \ref{assump:mvary}, using the standing convention $\|\cdot\|=\|\cdot\|_1$, let $R_i=\|U_i\|_1$ for all $i\in\{1,\ldots,n\}$ and order these full-input radii as $R_{[1]}\geq\cdots\geq R_{[n]}$. For $n\geq2$, define \[ H_n(k)=\frac1k\sum_{j=1}^k\log\frac{R_{[j]}}{R_{[k+1]}},\qquad 1\leq k\leq n-1. \] For $\delta\in(0,1)$, set \[ \begin{aligned} L_n&=\min\{n-1,\lceil100\log n\rceil\},&J_n&=1+\lceil\log n/\log2\rceil,\\ q_n(\delta)&=\log(4J_n/\delta),&c_n(\delta)&=2\sqrt{q_n(\delta)}+\frac{q_n(\delta)}{\sqrt{L_n}},\\ \rho_n(\delta)&=16c_n(\delta). \end{aligned} \] Let \[ \begin{aligned} \widetilde k_n&=\max\bigl\{k\in\{L_n,\ldots,n-1\}:\\ &\qquad |H_n(i)-H_n(k)|\leq\frac{H_n(i)\rho_n(\delta)}{\sqrt i}\\ &\qquad \text{for every }i\in\{L_n,\ldots,k\}\bigr\},\\ \widetilde\gamma_n&=H_n(\widetilde k_n). \end{aligned} \] For each fixed $\delta\in(0,1)$ and every sufficiently large $n$, with probability at least $1-\delta$, we have \[ |\gamma-\widetilde\gamma_n|\leq C_\delta \begin{cases} \left(\dfrac{\log((8/\delta)\log n)}{n}\right)^{1/3},&0<\beta<1,\\ \left(\dfrac{\log n\,\log((8/\delta)\log n)}{n}\right)^{1/3},&\beta=1,\\ \left(\dfrac{\log((8/\delta)\log n)}{n}\right)^{1/(2+\beta)},&\beta>1, \end{cases} \] where $C_\delta$ may depend on the fixed distribution and on $\delta$. where $\gamma = 1 / \beta$ is the EVI of $U$. - **Proof.** We adopt the non-asymptotic analysis of the adaptive Hill estimator for EVI in {\citep{boucheron2015tail}}. We give the exponential-spacing argument and the confidence calibration explicitly. Write $R=\|U\|_1$. Fix a member of $M_K$, put $a_i=\|A_{\cdot i}\|_1$, and let $Z_i^0$ be independent nominal Pareto type-II components of index $\beta$. The component density is a positive constant $c_0$ times the nominal product density whenever $\|z\|_1>\zeta K^{(1-2s)/\beta}$. Since \[ R=\sum_{i=1}^m a_iZ_i,\qquad l\leq a_i\leq u,\qquad R\leq u\|Z\|_1, \] the survival and density of $R$ beyond $u\zeta K^{(1-2s)/\beta}$ are $c_0$ times those of $T=\sum_i a_iZ_i^0$. This transfer uses the original … (the corrected paper gives it in full) - **What the correction gives up.** The guarantee for the printed evaluation-exceedance adjacent-ratio estimate and the selector comparing every $i\in\{l_n,\ldots,n\}$ is replaced by a guarantee for a different full-input Hill estimate with prefix selection. The logarithm-free radial bound is lost, and the radial exponent $1/(2+\beta)$ is replaced by $1/3$ for $0<\beta<1$. (An intermediate result: its strongest true version was adopted.) - **Formal check.** proved in Lean (the corrected statement), under 4 condition(s) made explicit — `Papers.TbsExtremeEffects.adaptive_hill_rate` (`Helpers/HillRate.lean:40`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Lemma A.2 — corrected; the corrected statement is weaker - **As printed.** \label{lemma:eta_rate}Undet the assumption of Theorem \ref{thm:rate}, with probability at least $1 - \delta$, we have \begin{align*} | \hat{\eta}_{n, t}^{\tmop{DR}} - \eta | & \leqslant O (\sqrt{R_p (n/2, \delta) R_g (t^{- \beta} n, \delta)} + t^{\beta / 2} n^{- 1 / 2} + t^{- \min \{1, \beta\}} + t^{- \beta s / (1 - 2 s)} + e (t) + \log (t) R_{\alpha} (n,\delta)) . \end{align*} - **What is wrong.** ``\dots by standard DML theory {\citep{foster2023orthogonal}}, we have \[ | \hat{\eta}_{n, t}^{\tmop{DR}} - \eta^t | \leqslant O \left( \sqrt{R_p (n/2, \delta) R_g (n_t, \delta)} + n_t^{- 1 / 2} \right),\mbox{''} \] Square-root concentration requires control of the normalized DR score's variance, which the assumptions do not supply, and its evaluation average has $|\mathcal I|$ summands rather than the training count $n_t$. ``\dots \[ \begin{array}{@{}c@{}}\displaystyle | \eta^t - \eta | = \left| \mathbb{E} \left[ \frac{f (X, 1, U) - f (X, 0, U)}{\|U\|^{\hat{\alpha}_{n}}} \mid \|U\|> t \right] -\mathbb{E}[g (X, 1, U /\|U\|) - g (X, 0, U /\|U\|)] \right| \\ \displaystyle \leqslant \left| \mathbb{E} \left[ \frac{f (X, 1, U) - f (X, 0, U)}{\|U\|^{\alpha}} - g (X, 1, U /\|U\|) - g (X, 0, U /\|U\|) \mid \|U\|> t \right] \right| \\ \displaystyle \quad \quad + |\mathbb{E}[g (X, 1, U /\|U\|) - g (X, 0, U /\|U\|)] -\mathbb{E}[g (X, 1, U /\|U\|) - g (X, 0, U /\|U\|) \mid \|U\|> t] | \\ \displaystyle \qquad + \left| \mathbb{E} \left[ \frac{f (X, 1, U) - f (X, 0, U)}{\|U\|^{\alpha}} \left( 1 - {\| U \|^{\alpha - \hat{\alpha}_{n}}} \right) \right] \right| \\ \displaystyle \leqslant 2 e (t) + |\mathbb{E}[g (X, 1, U /\|U\|) - g (X, 0, U /\|U\|)] -\mathbb{E}[g (X, 1, U /\|U\|) - g (X, 0, U /\|U\|) \mid \|U\|> t] |, \\ \displaystyle \qquad + \left| \mathbb{E} \left[ \frac{f (X, 1, U) - f (X, 0, U)}{\|U\|^{\alpha}} \left( 1 - {\| U \|^{\alpha - \hat{\alpha}_{n}}} \right) \right] \mid \|U\|> t \right|\end{array}\mbox{''} \] Homogeneity over exceedances bounds the remainder by $2\mathbb E[e(\|U\|)\mid\|U\|>t]$ or $2\sup_{r\geq t}e(r)$, and neither quantity is bounded by $2e(t)$ for an arbitrary error function tending to zero. ``\dots we have \[ \begin{array}{@{}c@{}}\displaystyle | \eta^t - \eta | \leqslant 2 e (t) + LW_1 (\mathcal{L}_{U /\|U\|}^t, \mathcal{L}_{U /\|U\|}) + O (\log (t) R_{\alpha} (n, \delta)) \\ \displaystyle \leqslant 2 e (t) + O (t^{- \min \{1, \beta\}} + t^{- \beta s / (1 - 2 s)} + \log (t) R_{\alpha} (n, \delta)),\end{array}\mbox{''} \] At the admissible value $\beta=1$, the angular bias can have order $t^{-1}\log t$, exceeding the displayed $O(t^{-1})$ term. ``\dots \[ \begin{array}{@{}c@{}}\displaystyle \left| \mathbb{E} \left[ \frac{f (X, 1, U) - f (X, 0, U)}{\|U\|^{\alpha}} \left( 1 - {\| U \|^{\alpha - \hat{\alpha}_{n}}} \right) \right] \mid \|U\|> t \right| \leqslant C\mathbb{E} \left[ \left| 1 - {\| U\|^{\alpha - \hat{\alpha}_{n}}} \right| \mid \| U \| > t \right] \\ \displaystyle = O (\log (t) R_{\alpha} (n, \delta)) .\end{array}\mbox{''} \] A uniformly linear exponent-error bound requires control of $R_\alpha(n,\delta)\log t$, whereas $R_\alpha(n,\delta)\to0$ alone permits $t^{R_\alpha(n,\delta)}$ to grow faster than the displayed term. ``and we have \[ | \hat{\eta}_{n, t}^{\tmop{DR}} - \eta^t | \leqslant O \left( \sqrt{R_p (n/2, \delta) R_g (nt^{- \beta}, \delta)} + t^{\beta / 2} n^{- 1 / 2} \right),\mbox{''} \] Monotonicity only gives a bound at a genuine lower bound for $n_t$ and does not permit replacing $R_g(n_t,\delta)$ by $R_g(nt^{-\beta},\delta)$ across arbitrary discontinuous decreasing error functions. ``\dots we get \[ | \hat{\eta}_n^{t} - \eta | \leqslant O (\sqrt{R_p (n/2, \delta) R_g (t^{- \beta} n, \delta)} + t^{\beta / 2} n^{- 1 / 2} + t^{- \min \{1, \beta\}} + t^{- \beta s / (1 - 2 s)} + e (t) + \log (t) R_{\alpha} (n,\delta)) .\mbox{''} \] Combining nuisance, count, and evaluation guarantees with separate failure probabilities does not give total failure probability $\delta$ at the same nuisance-bound arguments. ``By Bernstein's inequality, with probability at least $1 - \delta$, \[ n_t - n\mathbb{P}(\|U\|> t)/2 \geqslant O \left( \log (1 / \delta) + \sqrt{n\mathbb{P}(\|U\|> t) \log (1 / \delta)} \right)\mbox{''} \] A Bernstein lower-tail bound subtracts its deviation term from the expected count, rather than asserting the displayed positive deviation. In the step quoted in (b): The triangle inequality requires subtracting the whole contrast $g(X,1,U/\|U\|)-g(X,0,U/\|U\|)$ and retaining tail conditioning in the exponent-error expectation. ``\dots where $\mathcal{L}_{U /\|U\|}^t$ is the distribution of $U /\|U\|$ conditioning on $\|U\|> t$ and $\mathcal{L}_{U /\|U\|}$ is its limiting distribution as $t \rightarrow \infty$.'' The treatment contrast has Lipschitz constant at most $2L$, so its Wasserstein bound requires $2L$ rather than $L$. - **Counterexample.** Let $X\sim\tmop{Unif}([0,1])$ and $D\sim\tmop{Ber}(1/2)$ be independent, and let $U=(Z_1,Z_2)$ have independent Pareto type-II components with index $\beta=3$. Take $m=d=d_z=d_u=2$, $A=I_2$, $l=1/4$, $u=2$, $\sigma=1/8$, $s=1/4$, $\xi=1/2$, and $\zeta=1$, and use independent observations. Let $\epsilon$ be independent and symmetric with characteristic function $\mathbb E[\exp(iv\epsilon)]=\exp(-|v|^{3/2})$, and define $Y(d)=\|U\|(d+\epsilon)$ and $Y=Y(D)$. Set $\alpha=\hat\alpha_n=1$, $\hat p=1/2$, $\hat g(x,d,s)=d$, $g(x,d,s)=d$, and $e(t)=0$. Use the positive decreasing bounds $R_p(n,\delta)=R_\alpha(n,\delta)=(1+n)^{-10}$ and $R_g(q,\delta)=(1+q)^{-10}$, which cover these oracle errors surely. Take $t=n^{1/5}$ and any fixed confidence tolerance in $(0,1)$. Here $\eta=1$ and, conditional on $M=|\mathcal I|$, the estimator error has distribution $2M^{-1/3}\epsilon$; since $M$ has order $n^{2/5}$, its scale is $n^{-2/15}$ while the entire printed bound has order $n^{-1/5}$, so its coverage tends to zero. - **Rests on.** Proposition 3.3, also found wrong as printed. - **Correction.** Under the assumption of Theorem \ref{thm:rate}, suppose in addition that the conditions of Proposition~\ref{prop:id} hold. Suppose that the fitted propensity, regression, and exponent are measurable with respect to the training split and independent auxiliary data. Suppose that the nuisance inequalities of Theorem~\ref{thm:rate} hold jointly with probability at least $1-\varepsilon$, uniformly almost surely over the covariate and direction support and both treatments, for each tolerance $\varepsilon$. Suppose that $0t)$ and $M=|\mathcal I|$, and set $\hat\eta_{n,t}^{\tmop{DR}}=0$ when $M=0$. Let $\bar e(t)=\sup_{r\geq t}e(r)$, let $C$ bound $|g|$ on its compact domain, and put $G_t=C+\bar e(t)$. For a generic evaluation observation define \[ \psi=\hat g(X,1,U/\|U\|)-\hat g(X,0,U/\|U\|)+\frac{D-\hat p(X)}{\hat p(X)(1-\hat p(X))}\left(\frac{Y}{\|U\|^{\hat\alpha_n}}-\hat g(X,D,U/\|U\|)\right). \] For any positive truncation level $B$ chosen using only the training split and independent auxiliary data, define \[ \begin{aligned} v_t(B)&=\mathbb E[\psi^2 I\{|\psi|\leq B\}\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t],\\ a_t(B)&=\mathbb E[|\psi|I\{|\psi|>B\}\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t]. \end{aligned} \] Fix $a_0\in(0,\beta)$ and write $a=R_\alpha(n,\delta/4)$. For sufficiently large $t\geq1$, if $a\leq a_0$ and $n\pi_t\geq16\log(4/\delta)$, then with probability at least $1-\delta$ we have $M\geq n\pi_t/4$ and \begin{align*} |\hat\eta_{n,t}^{\tmop{DR}}-\eta|\leq{}&\sqrt{\frac{4v_t(B)}{\delta M}}+\left(1+\frac4\delta\right)a_t(B)+\frac2c R_p(n/2,\delta/4)R_g(n_t,\delta/4)\\ &+2\bar e(t)+2L W_1(\mathcal L_{U/\|U\|}^t,\mathcal L_{U/\|U\|})+\frac{2G_t}{c}\left(t^a-1+K a t^a\right), \end{align*} where $K$ depends on the radial law and $a_0$. Suppose moreover that the angular Wasserstein term satisfies the bound that Proposition 3.1 of \citet{zhang2023wasserstein} gives for this Pareto-type model, which is taken from that work and not proved here: there are constants $C_W>0$, $k\geq0$ and $t_W\geq1$ such that \[ W_1(\mathcal L_{U/\|U\|}^t,\mathcal L_{U/\|U\|})\leq C_W(\log t)^k t^{-\min\{1,\beta\}}+C_W t^{-\beta s/(1-2s)}\qquad(t\geq t_W). \] Then for $t\geq t_W$ the angular term $2L W_1(\mathcal L_{U/\|U\|}^t,\mathcal L_{U/\|U\|})$ of the display is at most $2LC_W(\log t)^k t^{-\min\{1,\beta\}}+2LC_W t^{-\beta s/(1-2s)}$, with the logarithmic factor retained. When the conditional score variance $V_t$ is finite, the sampling term may instead be taken to be $\sqrt{4V_t/(\delta M)}$. - **Proof.** Condition on the training split and independent auxiliary data, denoted together by $\mathcal T$. Write $R_U=\|U\|$, $S_U=U/\|U\|$, and \[ m_d^t(x,s)=\mathbb E[Y/R_U^{\alpha}\mid X=x,D=d,S_U=s,R_U>t],\qquad \eta_\alpha^t=\mathbb E[(f(X,1,U)-f(X,0,U))/R_U^{\alpha}\mid R_U>t]. \] Independence of $U$ from $(X,D)$ gives $\mathbb E[D\mid X,S_U,R_U>t]=p(X)$. Let $\psi_\alpha$ be the score $\psi$ with $\hat\alpha_n$ replaced by $\alpha$, keeping the same fitted functions. Direct conditional expectation over treatment and outcome gives \begin{align*} \mathbb E[\psi_\alpha\mid\mathcal T,R_U>t]-\eta_\alpha^t=\mathbb E\left[(p(X)-\hat p(X))\left\{\frac{m_1^t(X,S_U)-\hat g(X,1,S_U)}{\hat … (the corrected paper gives it in full) - **What the correction gives up.** A finite universal square-root rate for all integrable scores, bounds using only $e(t)$, a logarithm-free angular bias at $\beta=1$, uniform linear dependence on $R_\alpha\log t$, replacement of the random training count inside arbitrary decreasing $R_g$, and confidence $1-\delta$ with unchanged nuisance arguments. The suggested conditional evaluation argument also excludes fits that use the evaluation observations and fits whose nuisance guarantees hold only at a random evaluation point; its nonvacuous bound requires an integrable fitted-exponent pseudo-outcome and an exponent-error bound smaller than $\beta$. Cases in which conditional mean exogeneity given $(X,U)$ fails, or in which either potential outcome has an infinite absolute tail mean, are also excluded. The rate of the angular Wasserstein term is taken from Proposition 3.1 of \citet{zhang2023wasserstein} as an explicit assumption, with its poly-logarithmic factor, and is not proved. (An intermediate result: its strongest true version was adopted.) - **Formal check.** proved in Lean (the corrected statement), under 19 condition(s) made explicit — `Papers.TbsExtremeEffects.eta_dr_rate` (`Helpers/EtaDR.lean:420`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Lemma A.3 — corrected; the corrected statement is weaker - **As printed.** \label{lemma:eta_ipw} Under the assumption of Theorem \ref{thm:rate}, with probability at least $1 - \delta$, we have \begin{align*} | \hat{\eta}_{n, t}^{\text{IPW}} - \eta | & \leqslant O (R_p (n/2, \delta) + t^{\beta / 2} n^{- 1 / 2} + t^{- \min \{1, \beta\}} + t^{- \beta s / (1 - 2 s)}) + e (t) . \end{align*} - **What is wrong.** ``\dots we have \[ | \hat{\eta}_{n, t}^{\text{IPW}} - \eta^t | \leqslant O (R_p (n/2, \delta) + n_t^{- 1 / 2}) = O (R_p (n/2, \delta) + t^{\beta / 2} n^{- 1 / 2} + \log (t) R_{\alpha} (n,\delta)) .\mbox{''} \] The conclusion omits the exponent perturbation $\mathbb E[\|U\|^{\alpha-\hat\alpha_n}-1\mid\|U\|>t]$ present in this step, and its proposed linear order requires control of $\log(t)R_\alpha(n,\delta)$. The step quoted in (a) uses the sampling bound of Theorem 1 of \cite{su2023estimated}, which the paper assumes only for outcomes satisfying that theorem's moment bounds and its specified logistic propensity-estimation framework, but applies it to the normalized IPW scores under only the assumptions of Theorem~\ref{thm:rate}. ``Term $| \eta^t - \eta |$ can be bounded in the same way as in the proof of Lemma \ref{lemma:eta_rate}.'' The homogeneity error on exceedances is controlled by $\mathbb E[e(\|U\|)\mid\|U\|>t]$, which need not be bounded by $e(t)$. In the step quoted in (c): The angular bias may have order $t^{-1}\log t$ when $\beta=1$, rather than the asserted $t^{-1}$. ``The rest of the proof is similar.'' The concluding sentence derives neither the missing exponent term nor a joint event with probability at least $1-\delta$. - **Counterexample.** Take $m=d=d_z=d_u=2$, $A=I_2$, independent nominal Pareto type-II components with $\beta=2$, and $l=1/2$, $u=2$, $\sigma=1/2$, $s=2/5$, $\xi=1/2$, and $\zeta=1$. Let $X\sim\tmop{Unif}([0,1])$ and $D\sim\tmop{Ber}(1/2)$ be independent of the components and each other, use independent copies for observations, and take $p=\hat p=1/2$ and $c=1/4$. Set $Y(d)=d$, $Y=Y(D)$, $\alpha=0$, $g(x,d,s)=\hat g(x,d,s)=d$, $e=0$, and the deterministic training-measurable fit $\hat\alpha_n=1/\log n$. Choose $R_\alpha(n,\delta)=1/\log n$, $R_p(n,\delta)=R_g(n,\delta)=n^{-10}$, $t=n^{1/6}$, and $\delta=1/4$. The IPW average converges in probability to $\exp(-1/6)$ while $\eta=1$ and the entire printed bound tends to zero. - **Rests on.** Lemma A.2, Proposition 3.3, also found wrong as printed. - **Correction.** Under the assumption of Theorem \ref{thm:rate}, suppose in addition that the conditions of Proposition~\ref{prop:id} hold. Suppose in addition that the conditions of Lemma~\ref{lemma:eta_rate} hold. Use its training split and independent auxiliary data, denoted together by $\mathcal T$, its $\pi_t=\mathbb P(\|U\|>t)$, $M=|\mathcal I|$, $\bar e(t)=\sup_{r\geq t}e(r)$, and $G_t=C+\bar e(t)$, where $C$ bounds $|g|$ on its compact domain. Set $\hat\eta_{n,t}^{\text{IPW}}=0$ if $M=0$. For a generic evaluation observation define \[ \phi=\frac{Y}{\|U\|^{\hat\alpha_n}}\left(\frac{D}{\hat p(X)}-\frac{1-D}{1-\hat p(X)}\right).\] For any $B>0$ chosen using only $\mathcal T$, put \[\begin{aligned} v_t^{\text{IPW}}(B)&=\mathbb E[\phi^2 I\{|\phi|\leq B\}\mid\mathcal T,\|U\|>t],\\ b_t^{\text{IPW}}(B)&=\mathbb E[|\phi|I\{|\phi|>B\}\mid\mathcal T,\|U\|>t].\end{aligned}\] Fix $a_0\in(0,\beta)$ and put $a=R_\alpha(n,\delta/4)$. For sufficiently large $t\geq1$, if $a\leq a_0$ and $n\pi_t\geq16\log(4/\delta)$, then with probability at least $1-\delta$ we have $M\geq n\pi_t/4$ and \[\begin{aligned} |\hat\eta_{n,t}^{\text{IPW}}-\eta|\leq{}&\sqrt{\frac{4v_t^{\text{IPW}}(B)}{\delta M}}+\left(1+\frac4\delta\right)b_t^{\text{IPW}}(B)+\frac{2G_t}{c}R_p(n/2,\delta/4)\\ &+2\bar e(t)+2L W_1(\mathcal L_{U/\|U\|}^t,\mathcal L_{U/\|U\|})+\frac{2G_t}{c}(t^a-1+K a t^a),\end{aligned}\] where $K$ depends on the radial law and $a_0$. Under the angular assumption of Lemma~\ref{lemma:eta_rate}, taken from Proposition 3.1 of \citet{zhang2023wasserstein}, the angular Wasserstein term has the bound stated there for $t\geq t_W$, with its logarithmic factor retained. When $V_t^{\text{IPW}}=\tmop{Var}(\phi\mid\mathcal T,\|U\|>t)$ is finite, the sampling term may instead be taken to be $\sqrt{4V_t^{\text{IPW}}/(\delta M)}$. - **Proof.** Write $R_U=\|U\|$ and $S_U=U/\|U\|$, and define \[\begin{aligned} m_d^t(x,s)&=\mathbb E[Y/R_U^\alpha\mid X=x,D=d,S_U=s,R_U>t],\\ \eta_\alpha^t&=\mathbb E[(f(X,1,U)-f(X,0,U))/R_U^\alpha\mid R_U>t].\end{aligned}\] Independence of $U$ from $(X,D)$ gives $\mathbb P(D=1\mid X,S_U,R_U>t)=p(X)$. Let $\phi_\alpha$ be $\phi$ with $\hat\alpha_n$ replaced by $\alpha$. Conditional expectation over treatment and outcome gives \[\begin{aligned} &\mathbb E[\phi_\alpha\mid\mathcal T,R_U>t]-\eta_\alpha^t\\ &\quad=\mathbb E\left[(p(X)-\hat p(X))\left\{\frac{m_1^t(X,S_U)}{\hat p(X)}+\frac{m_0^t(X,S_U)}{1-\hat p(X)}\right\}\mid\mathcal T,R_U>t\right].\end{aligned}\] Indeed, the treatment coefficients after … (the corrected paper gives it in full) - **What the correction gives up.** The universal square-root sampling rate for integrable scores of infinite variance, the omission of exponent error, the pointwise approximation term $e(t)$, the logarithm-free angular bound at $\beta=1$, nonuniform nuisance guarantees, fits using evaluation observations, degenerate true propensities, fitted-exponent pseudo-outcomes without finite absolute tail means, exponent-error bounds not below a fixed $a_0<\beta$, and thresholds with insufficient expected exceedance counts. Cases in which conditional mean exogeneity given $(X,U)$ fails, or in which either potential outcome has an infinite absolute tail mean, are also excluded. (An intermediate result: its strongest true version was adopted.) - **Formal check.** proved in Lean (the corrected statement), under 21 condition(s) made explicit — `Papers.TbsExtremeEffects.eta_ipw_rate` (`Helpers/EtaIPW.lean:411`); axioms: the standard ones only (propext, Classical.choice, Quot.sound). ### Restated to follow a correction above No error of their own: these results use a corrected result and are restated accordingly. - **Corollary 3.6.** Under the assumptions of Theorem~\ref{thm:rate}, further suppose that \begin{align*} R_p(n,\delta)&=\Theta(\log(1/\delta)n^{-1/2}),\qquad R_g(n,\delta)=\Theta(\log(1/\delta)n^{-1/2}),\\ R_\alpha(n,\delta)&=\Theta(\log(1/\delta)n^{-c_\alpha}) \end{align*} for some $c_\alpha>0$. Suppose in addition that the conditions of Proposition~\ref{prop:id} hold. Let $\mathcal T$ consist of the training split and independent auxiliary data. Suppose that $\hat p$, $\hat g$ and $\hat\alpha_n$ are $\mathcal T$-measurable and are the same functions and exponent for every input threshold. Write $R_U=\|U\|$, $S_U=U/\|U\|$, $\gamma=1/\beta$, $\pi_t=\mathbb P(R_U>t)$, and \[ m_d^t(x,v)=\mathbb E[Y/R_U^\alpha\mid X=x,D=d,S_U=v,R_U>t]. \] In each case below, let $\kappa$ be the exponent function specified in that case. Use fixed constants $0t)$ is the training exceedance count. On this same event suppose that the fitted-exponent pseudo-outcome has finite conditional absolute mean for every $t\in I_n$ and that, for a constant $V<\infty$ independent of $n$ and $t$, \[ \mathbb E[\psi^2\mid\mathcal T,R_U>t]\leq V\qquad(t\in I_n), \] where \[ \psi=\hat g(X,1,S_U)-\hat g(X,0,S_U)+\frac{D-\hat p(X)}{\hat p(X)(1-\hat p(X))}\left(\frac{Y}{R_U^{\hat\alpha_n}}-\hat g(X,D,S_U)\right). \] Set the first-factor estimate to zero if there are no evaluation exceedances. Run the full-input Hill selection of Algorithm~\ref{alg:nete} at tolerance $\delta/6$, and put \[ b_n=C_{\delta/6}\begin{cases} \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/3},&0<\beta<1,\\ \left(\dfrac{\log n\,\log((48/\delta)\log n)}{n}\right)^{1/3},&\beta=1,\\ \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/(2+\beta)},&\beta>1, \end{cases} \] with the constant supplied by Lemma~\ref{lemma:mu_rate}. Define \[ \bar e(t)=\sup_{r\geq t}e(r). \] Then, for each fixed $\delta\in(0,1/2)$ and every sufficiently large even $n$, the following conclusions hold. Here $k\geq0$ is the exponent of the logarithm in the angular assumption of Theorem~\ref{thm:rate}, taken from Proposition 3.1 of \citet{zhang2023wasserstein}; under that assumption the angular term at the threshold $t_n$ is of the order shown, up to this power of $\log n$. \begin{enumerate} \item If $s\in(0,1/(2+\max\{1,\beta\}))$, take $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ and put $\kappa(v)=(1-2s)v$. With probability at least $1-\delta$, we have \[ \begin{aligned} |\widehat\theta_{n,t_n}^{\text{DR}}-\theta^{NETE}|=O\bigl(&\sqrt{V/\delta}\,n^{-s}+\log^2(6/\delta)n^{-1/2-s}\\ &+\bar e(t_n)+(\log n)^k n^{-s}+n^{-c_\alpha}\log n\log(6/\delta)+|\alpha|b_n\bigr). \end{aligned} \] \item If $s\in[1/(2+\max\{1,\beta\}),1/2)$, take $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ and put $\kappa(v)=v/(1+2\min\{1,v\})$. Write $q=1/(2+\max\{\beta,1\})$. With probability at least $1-\delta$, we have \[ \begin{aligned} |\widehat\theta_{n,t_n}^{\text{DR}}-\theta^{NETE}|=O\bigl(&\sqrt{V/\delta}\,n^{-q}+\log^2(6/\delta)n^{-1/2-q}\\ &+\bar e(t_n)+(\log n)^k n^{-q}+n^{-c_\alpha}\log n\log(6/\delta)+|\alpha|b_n\bigr). \end{aligned} \] \end{enumerate} - **Corollary A.4.** Under the assumptions of Theorem~\ref{thm:rate}, further suppose that \begin{align*} R_p(n,\delta)&=\Theta(\log(1/\delta)n^{-1/2}),\qquad R_g(n,\delta)=\Theta(\log(1/\delta)n^{-1/2}),\\ R_\alpha(n,\delta)&=\Theta(\log(1/\delta)n^{-c_\alpha}) \end{align*} for some $c_\alpha>0$. Suppose in addition that the conditions of Proposition~\ref{prop:id} hold. Let $\mathcal T$ consist of the training split and independent auxiliary data. Suppose that $\hat p$, $\hat g$ and $\hat\alpha_n$ are $\mathcal T$-measurable and are the same functions and exponent for every input threshold. Write $R_U=\|U\|$, $S_U=U/\|U\|$, $\gamma=1/\beta$, $\pi_t=\mathbb P(R_U>t)$, and \[ m_d^t(x,v)=\mathbb E[Y/R_U^\alpha\mid X=x,D=d,S_U=v,R_U>t]. \] In each case below, let $\kappa$ be the exponent function specified in that case. Use fixed constants $0t)$ is the training exceedance count. On this same event suppose that the fitted-exponent pseudo-outcome has finite conditional absolute mean for every $t\in I_n$ and that, for a constant $V<\infty$ independent of $n$ and $t$, \[ \mathbb E[\phi^2\mid\mathcal T,R_U>t]\leq V\qquad(t\in I_n),\qquad \phi=\frac{Y}{R_U^{\hat\alpha_n}}\left(\frac{D}{\hat p(X)}-\frac{1-D}{1-\hat p(X)}\right). \] Set the first-factor estimate to zero if there are no evaluation exceedances. Run the full-input Hill selection of Algorithm~\ref{alg:nete} at tolerance $\delta/6$, and put \[ b_n=C_{\delta/6}\begin{cases} \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/3},&0<\beta<1,\\ \left(\dfrac{\log n\,\log((48/\delta)\log n)}{n}\right)^{1/3},&\beta=1,\\ \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/(2+\beta)},&\beta>1, \end{cases} \] with the constant supplied by Lemma~\ref{lemma:mu_rate}. Define \[ \bar e(t)=\sup_{r\geq t}e(r). \] Then, for each fixed $\delta\in(0,1/2)$ and every sufficiently large even $n$, the following conclusions hold. Here $k\geq0$ is the exponent of the logarithm in the angular assumption of Theorem~\ref{thm:rate}, taken from Proposition 3.1 of \citet{zhang2023wasserstein}; under that assumption the angular term at the threshold $t_n$ is of the order shown, up to this power of $\log n$. \begin{enumerate} \item If $s\in(0,1/(2+\max\{1,\beta\}))$, take $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ and put $\kappa(v)=(1-2s)v$. With probability at least $1-\delta$, we have \[ \begin{aligned} |\widehat\theta_{n,t_n}^{\text{IPW}}-\theta^{NETE}|=O\bigl(&\sqrt{V/\delta}\,n^{-s}+\log(6/\delta)n^{-1/2}\\ &+\bar e(t_n)+(\log n)^k n^{-s}+n^{-c_\alpha}\log n\log(6/\delta)+|\alpha|b_n\bigr). \end{aligned} \] \item If $s\in[1/(2+\max\{1,\beta\}),1/2)$, take $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ and put $\kappa(v)=v/(1+2\min\{1,v\})$. Write $q=1/(2+\max\{\beta,1\})$. With probability at least $1-\delta$, we have \[ \begin{aligned} |\widehat\theta_{n,t_n}^{\text{IPW}}-\theta^{NETE}|=O\bigl(&\sqrt{V/\delta}\,n^{-q}+\log(6/\delta)n^{-1/2}\\ &+\bar e(t_n)+(\log n)^k n^{-q}+n^{-c_\alpha}\log n\log(6/\delta)+|\alpha|b_n\bigr). \end{aligned} \] \end{enumerate} ## 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.** - **Proposition 3.3.** $U\perp(X,D)$. - **Proposition 3.3.** $\|X\|_1\leqslant R$ almost surely. - **Proposition 3.3.** For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\|U\|/b_n,U/\|U\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. - **Proposition 3.3.** $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. - **Proposition 3.3.** The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\|x\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\|u\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. - **Proposition 3.3.** $Y(D)=Y$ almost surely throughout the paper. - **Proposition 3.3.** The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\|u\|_1=1$. - **Theorem 3.5.** For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\|z\|_1>\zeta k^{(1-2s)/\beta}$. - **Theorem 3.5.** The existential cutoff $t_W$ is chosen sufficiently large for Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} and so that $\sup_{r\geq t_W}e(r)\leq1$ and $\mathbb P(\|U\|>tr)/\mathbb P(\|U\|>t)\leq4r^{-\beta}$ for every $t\geq t_W$ and $r\geq1$. - **Theorem 3.5.** For each fixed $\mathcal T$-measurable choice $B>0$, the displayed probability bound uses the base conditions of Lemma~\ref{lemma:eta_rate}; its angular specialization additionally uses that lemma's angular assumption and $t\geq t_W$, and its variance alternative additionally requires $V_t^{\text{IPW}}<\infty$. - **Theorem 3.5.** For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. - **Theorem 3.5.** The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. - **Theorem 3.5.** The equal split in Algorithm~\ref{alg:nete} is defined for positive even sample sizes $n$, with $|\mathcal D_1|=|\mathcal D_2|=n/2$. - **Theorem 3.5.** The component dimension in the density of Assumption~\ref{assump:mvary} is $m=d_z$. - **Theorem 3.5.** For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments. - **Theorem 3.5.** Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$. - **Theorem 3.5.** The fitted-exponent pseudo-outcome has finite conditional absolute mean under the evaluation tail law: $\mathbb E[|Y|/\|U\|^{\hat\alpha_n}\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t]<\infty$ almost surely on each joint nuisance event. - **Theorem 3.5.** The homogeneity inequality holds for both treatments and every positive radius, on the positive unit sphere for the $\ell_1$ norm. - **Theorem 3.5.** The equal split uses $n\in2\mathbb N$. - **Theorem 3.5.** The symbol $V_t$ denotes $\tmop{Var}(\psi\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t)$. - **Theorem 3.5.** The sample-size domain of $R_g(q,\varepsilon)$ includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. - **Theorem 3.5.** For each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that, for every integer $n\geq n_0$, the bound has probability at least $1-\delta$ under the law of the sample of size $n$. - **Theorem 3.5.** The count $n_t=\sum_{i=1}^{n/2}I(\|U_i\|>t)$ is the training-split exceedance count, with the training observations indexed first after the random split. - **Theorem 3.5.** The sampling term comprises both $\sqrt{4v_t^{\text{IPW}}(B)/(\delta M)}$ and $(1+4/\delta)b_t^{\text{IPW}}(B)$. - **Theorem 3.5.** The target $\eta$ denotes $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. - **Corollary 3.6.** For each fixed $\delta\in(0,1/2)$, the probability guarantee quantifies separately over every sufficiently large even $n$. - **Corollary 3.6.** The constants $c_-$ and $c_+$ in $t_n=\Theta(n^{\kappa(\hat\gamma_n)})$ are deterministic and independent of $n$ and the observed sample. - **Corollary A.4.** For each fixed $\delta\in(0,1/2)$, the probability guarantee quantifies separately over every sufficiently large even $n$. - **Corollary A.4.** The constants $c_-$ and $c_+$ in $t_n=\Theta(n^{\kappa(\hat\gamma_n)})$ are deterministic and independent of $n$ and the observed sample. **Notation.** - $X$: $X\in\mathcal{X}$ denotes the covariate in the potential-outcome model. In Proposition~\ref{prop:id}, Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $\|X\|_1\leqslant R$, so the covariate takes values in $B_R$. In Lemma~\ref{lemma:eta_rate}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: In $\eta=\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, $\mathbb E_X$ integrates with respect to the population marginal law of $X$, rather than an empirical covariate distribution. - $D$: $D\in\{0,1\}$ denotes the treatment indicator. In Lemma~\ref{lemma:mu_rate}: In the proof's expansions $f(t)=C\beta t^{-\beta-1}(1+D(1-1/\beta)t^{-1}+o(t^{-1}))$ and $\bar F(t)=Ct^{-\beta}(1+Dt^{-1}+o(t^{-1}))$, $D$ denotes a fixed expansion coefficient rather than the treatment indicator. - $Y$: $Y$ is the observed outcome satisfying consistency $Y=Y(D)$, where $Y(d)$ denotes the potential outcome when treatment is set to $d$. - $Y(d)$: $Y(d)$ denotes the potential outcome under intervention $d\in\{0,1\}$, with consistency $Y(D)=Y$ throughout. In Proposition~\ref{prop:id}: For both $d\in\{0,1\}$, $\mathbb E[Y(d)\mid X,U,D]=\mathbb E[Y(d)\mid X,U]$ almost surely, and $\mathbb E[|Y(d)|\mid\|U\|>t]<\infty$ for all sufficiently large $t$. - $U$: $U$ denotes the extreme-noise vector in $\mathbb R_+^d$, with $\|U\|=\|U\|_1$ throughout the treatment-effect analysis. In Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}: Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$, with $U=AZ$ and $m=d_z$ in the component density. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_ipw}: Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$, with $U=AZ$ and $m=d_z$ in the component density. - $d$: $d$ denotes the dimension in $U\in\mathbb R_+^d$ and in sphere notation; when used as the treatment argument in $Y(d)$, $f(x,d,u)$, or $g(x,d,u)$, it is a locally bound treatment value in $\{0,1\}$. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $m\geq d\geq2$. - $t$: $t>0$ is the level defining the strict exceedance event $\{\|U\|_1>t\}$, except when it is the argument of a scalar function in a tail calculation. In Proposition~\ref{prop:id}: $t$ is a deterministic limit parameter tending to infinity. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: The finite-threshold analysis uses a fixed deterministic $t$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The selected threshold is $t=t_n$, with $c_-n^{\kappa(\hat\gamma_n)}\leq t_n\leq c_+n^{\kappa(\hat\gamma_n)}$, where $0t)$. In Lemma~\ref{lemma:eta_ipw}: The bound concerns sufficiently large $t\geq1$, subject to $n\pi_t\geq16\log(4/\delta)$, where $\pi_t=\mathbb P(\|U\|>t)$. In Theorem~\ref{thm:rate}: The main bounds concern deterministic $t\geq t_W$, subject to $n\pi_t\geq16\log(12/\delta)$, where $\pi_t=\mathbb P(\|U\|>t)$; the additional simplified bounds also require $t\geq\exp(1)$ and $a\log t\leq1$, where $a=R_\alpha(n,\delta/12)$. - $\alpha$: $\alpha\in\mathbb R$ is the fixed population homogeneity exponent governing the normalization $f(x,d,tu)/t^\alpha$ and the definition of $\theta^{\tmop{NETE}}$; it is distinct from the supplied approximation $\hat\alpha_n$. In Proposition~\ref{prop:id}, Theorem~\ref{thm:rate}, Corollary~\ref{cor:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate_ipw}: $\alpha<\beta$. - $\beta$: $\beta\in(0,\infty)$ is the population regular-variation index of the extreme-noise law, with radial limiting measure satisfying $\nu_\beta([r,\infty))=r^{-\beta}$ for $r>0$. - $R$: $R$ is the fixed covariate bound in $\|X\|_1\leqslant R$, except where explicitly rebound as a local radial variable. In Lemma~\ref{lemma:mu_rate}: In the proof, $R=\|U\|_1$ is the generic radial random variable; $f_R$, $F_R$, and $\bar F_R$ refer to its density, distribution function, and survival function. This local $R$ is distinct from the covariate bound. - $f$: $f$ denotes a population function with its domain and meaning determined by its local definition, rather than an estimated regression function. In Proposition~\ref{prop:id}, Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: In the extreme-effect model and its decompositions, $f$ is the real-valued population conditional-mean function satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$, whose pointwise values $f(x,d,u)$ are those appearing in the asymptotic homogeneous property. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: In the extreme-effect model and its decompositions, $f$ is the real-valued population conditional-mean function satisfying $f(X,D,U)=\mathbb E[Y\mid X,D,U]$, whose pointwise values $f(x,d,u)$ appear in the asymptotic homogeneous property. - $g$: $g(x,d,u)$ is the population asymptotic normalized conditional-mean function, given by $g(x,d,u)=\lim_{t\to\infty}f(x,d,tu)/t^\alpha$ on its covariate, treatment, and direction domain. In Proposition~\ref{prop:id}, Lemma~\ref{lemma:eta_rate}: The homogeneity inequality $|f(x,d,tu)/t^\alpha-g(x,d,u)|\leqslant e(t)$ holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\|u\|_1=1$; $g$ is $L$-Lipschitz on this domain. In Lemma~\ref{lemma:eta_ipw}: The homogeneity inequality $|f(x,d,tu)/t^\alpha-g(x,d,u)|\leqslant e(t)$ holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\|u\|_1=1$; $g$ is $L$-Lipschitz on this domain. - $e$: $e(t)$ is a deterministic, nonnegative uniform approximation-error bound in the asymptotic homogeneous property, allowing $e(t)=0$ and tending to zero as $t\rightarrow\infty$. In Proposition~\ref{prop:id}, Lemma~\ref{lemma:eta_rate}: The bound $|f(x,d,ru)/r^\alpha-g(x,d,u)|\leq e(r)$ holds for both $d\in\{0,1\}$ and every $r>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\|u\|_1=1$. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The bound $|f(x,d,ru)/r^\alpha-g(x,d,u)|\leq e(r)$ holds for both $d\in\{0,1\}$ and every $r>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\|u\|_1=1$. In Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: The approximation-error envelope used in the bound is $\bar e(t)=\sup_{r\geq t}e(r)$, and $G_t=C+\bar e(t)$; no equality $\bar e(t)=e(t)$ is assumed. In Theorem~\ref{thm:rate}: The approximation-error envelope used in the bound is $\bar e(t)=\sup_{r\geq t}e(r)$, and $G_t=C+\bar e(t)$; no equality $\bar e(t)=e(t)$ is assumed. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The approximation-error envelope used in the bound is $\bar e(t)=\sup_{r\geq t}e(r)$, evaluated at the selected threshold $t_n$; no equality $\bar e(t)=e(t)$ is assumed. - $c$: $c$ denotes a deterministic positive constant whose identity and additional restrictions are supplied by its local definition. In Proposition~\ref{prop:id}, Theorem~\ref{thm:rate}, Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: In the regular-variation definition, $c>0$ is the mass multiplier in $c\nu_\beta\times S$, independently of any overlap constant. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: In the overlap assumption, $c\in(0,1/2)$ bounds the estimated propensity uniformly by $c\leq\hat p(x)\leq1-c$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: In the regular-variation definition, $c>0$ is the mass multiplier in $c\nu_\beta\times S$, independently of the overlap constant; in the overlap assumption and the error bounds containing $1/c$, $c\in(0,1/2)$ satisfies $c\leq\hat p(x)\leq1-c$ for every $x\in\mathcal X$. The constants $c_1,c_2,c_\alpha$ are separate symbols. - $p$: $p(x)=\mathbb P(D=1\mid X=x)$ is the population propensity function on $\mathcal X$, distinct from its fitted estimate $\hat p(x)$. In Lemma~\ref{lemma:eta_ipw}: $00$, the locally chosen $p>1$ with $p\alpha<\beta$ is a fixed scalar moment exponent in $|G(S_U)(R_U/t)^\alpha|^p$ and $(R_U/t)^{p\alpha}$. - $\hat{p}$: $\hat p$ is the sample-dependent estimator of the propensity function fitted using the first half $\mathcal D_1$ of the split sample by regression of $D$ on $X$, with $\hat p(x)\in[c,1-c]$ for every $x\in\mathcal X$; the fitted function is fixed conditional on the training split and all randomness used in fitting. In Lemma~\ref{lemma:eta_rate}: $\hat p$ is measurable with respect to $\mathcal D_1$ and independent auxiliary data; its error inequality holds jointly with the regression and exponent inequalities with probability at least $1-\varepsilon$, uniformly almost surely over the covariate support, for each tolerance $\varepsilon$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $\hat p$ is measurable with respect to the training split and independent auxiliary data $\mathcal T$, and is the same fitted function for every input threshold. With probability at least $1-\delta/6$, jointly for every $t\in I_n=[\ell_n,u_n]$, its inequality $|p(x)-\hat p(x)|\leq R_p(n/2,\delta/6)$ holds together with the regression and exponent inequalities, simultaneously almost surely over the covariate and conditional direction support and both treatments. The additional estimation-rate condition is $R_p(n,\delta)=\Theta(\log(1/\delta)n^{-1/2})$. In Theorem~\ref{thm:rate}: $\hat p$ is measurable with respect to the training split and independent auxiliary data $\mathcal T$; for each fixed $t$ and $\varepsilon\in(0,1)$, its inequality $|p(X)-\hat p(X)|\leq R_p(n,\varepsilon)$ holds jointly with the regression and exponent inequalities on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments. In Lemma~\ref{lemma:eta_ipw}: $\hat p$ is measurable with respect to the training split and independent auxiliary data $\mathcal T$; its error inequality holds jointly with the regression and exponent inequalities with probability at least $1-\varepsilon$, uniformly almost surely over the covariate support, for each tolerance $\varepsilon$. - $i$: $i$ is a locally bound integer index whose range is supplied by the expression in which it occurs. In Lemma~\ref{lemma:mu_rate}: In $R_i=\|U_i\|_1$, $i\in\{1,\ldots,n\}$ indexes the full input sample; in the adaptive comparison, $i\in\{L_n,\ldots,k\}$ and the compared statistic is $H_n(i)=i^{-1}\sum_{j=1}^{i}\log(R_{[j]}/R_{[i+1]})$. In Theorem~\ref{thm:rate}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: In $n_t=\sum_{i=1}^{n/2}I(\|U_i\|>t)$, $i\in\{1,\ldots,n/2\}$ indexes the training observations, indexed first after the random split. - $X_i$: $X_i\in\mathcal{X}$ is the covariate coordinate of the $i$th sampled tuple $(X_i,D_i,Y_i,U_i)$, indexed by $i=1,\ldots,N$ in the model and $i=1,\ldots,n$ in the estimation notation. In Proposition~\ref{prop:id}, Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $X_i$ takes values in $B_R$, so $\|X_i\|_1\leq R$, under the bounded-covariate assumption. In Theorem~\ref{thm:rate}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: After the random equal split, $X_i$ is the covariate coordinate of a training observation for $1\leq i\leq n/2$ and of an evaluation observation for $n/2t)$, $U_i$ belongs to the training split, whose observations are indexed first after the random split. In Lemma~\ref{lemma:mu_rate}: $R_i=\|U_i\|_1$ for $i\in\{1,\ldots,n\}$ uses every full-input observation, without restriction to a split or exceedance set. - $S$: As a spectral measure, $S$ is the population probability measure on $\mathbb{S}^+=\{u\in\mathbb{R}_+^d:\|u\|=1\}$ in the regular-variation limit, equivalently the limiting law of $U/\|U\|$ conditional on $\|U\|>t$ as $t\rightarrow\infty$, with the $\ell_1$ norm used after the notation convention. - $b_n$: $b_n$ is a deterministic positive sequence on the index domain of its local definition. In Proposition~\ref{prop:id}, Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: In the inherited regular-variation definition, $b_n$, $n\geq1$, is chosen for the law of $U$ and the chosen norm, tends to infinity, and satisfies $n\mathbb P((\|U\|/b_n,U/\|U\|)\in\cdot)\overset{w}{\rightarrow}c\nu_\beta\times S$. In Theorem~\ref{thm:rate}: In $Q_n$ and the radial estimation bound, $b_n=C_{\delta/6}\begin{cases}(\log((48/\delta)\log n)/n)^{1/3},&0<\beta<1,\\(\log n\,\log((48/\delta)\log n)/n)^{1/3},&\beta=1,\\(\log((48/\delta)\log n)/n)^{1/(2+\beta)},&\beta>1.\end{cases}$ For fixed $\delta$ and the fixed noise law this is a deterministic error-bound sequence tending to zero, defined for sufficiently large $n$. The inherited regular-variation definition separately uses a normalizing sequence tending to infinity. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: In the radial estimation bound and the term $|\alpha|b_n$, $b_n=C_{\delta/6}\begin{cases}(\log((48/\delta)\log n)/n)^{1/3},&0<\beta<1,\\(\log n\,\log((48/\delta)\log n)/n)^{1/3},&\beta=1,\\(\log((48/\delta)\log n)/n)^{1/(2+\beta)},&\beta>1.\end{cases}$ For each fixed $\delta\in(0,1/2)$ and fixed noise law, this is a deterministic error-bound sequence tending to zero, used for sufficiently large even $n$. The inherited regular-variation definition separately uses a normalizing sequence tending to infinity. - $Z$: $Z$ denotes the random variable specified by its local definition: an illustrative regularly varying scalar, a nonnegative component vector in $U=AZ$, or a scalar truncated evaluation score. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $Z\in\mathbb{R}_+^{d_z}$ is the random component vector with $\mathcal{L}(Z)\in\tilde M_k$ for a positive integer $k$ in the population representation $U=AZ$; the approximation condition does not assert independence of its components. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:mu_rate}: $Z\in\mathbb R_+^{d_z}$ is the random component vector with $\mathcal L(Z)\in\tilde M_k$ for a positive integer $k$ in the population representation $U=AZ$; the approximation condition does not assert independence of its components. In Lemma~\ref{lemma:eta_rate}: In the tail-model assumption, $Z\in\mathbb R_+^{d_z}$ is the component vector with $\mathcal L(Z)\in\tilde M_k$ for a positive integer $k$, without an assertion of component independence. In the proof's conditional sampling argument, the local definition $Z=\psi I\{|\psi|\leq B\}$ denotes a scalar truncated evaluation score under the law conditional on $\mathcal T$ and $R_U>t$; $Z_i$ are its evaluation realizations. In Lemma~\ref{lemma:eta_ipw}: In the tail-model assumption, $Z\in\mathbb R_+^{d_z}$ is the component vector with $\mathcal L(Z)\in\tilde M_k$ for a positive integer $k$, without an assertion of component independence. In the proof's conditional sampling argument, the local definition $Z=\phi I\{|\phi|\leq B\}$ denotes a scalar truncated evaluation score under the law conditional on $\mathcal T$ and $R_U>t$; $Z_i$ are its evaluation realizations. - $A$: In the representation $U=AZ$, $A$ is a nonnegative $d_u\times d_z$ matrix. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: $A\in\mathcal{A}$ is a fixed matrix specifying the population representation $U=AZ$, rather than an estimated matrix or a matrix redrawn for each observation. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $A\in\mathcal A$ is a fixed matrix specifying the population representation $U=AZ$, rather than an estimated matrix or a matrix redrawn for each observation. - $\tilde M_k$: $\tilde M_k$ is a fixed population class of component laws indexed by $k\in\{1,2,\ldots\}$, with fixed $\beta>0$, $00$; its joint-density conditions do not assert independence of the components. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: $m=d_z$ in the defining density conditions, so the nominal density is $\beta^{d_z}\prod_{i=1}^{d_z}(1+z_i)^{-(\beta+1)}$ and every component product ranges over $i\in\{1,\ldots,d_z\}$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $m=d_z$ in the defining density conditions, so the nominal density is $\beta^{d_z}\prod_{i=1}^{d_z}(1+z_i)^{-(\beta+1)}$ and every component product ranges over $i\in\{1,\ldots,d_z\}$. - $\mathcal{A}$: $\mathcal{A}$ is the fixed population matrix class, with column indices $i\in\{1,\ldots,d_z\}$ and fixed constants $00$. - $M_k$: $M_k$ is a fixed population distribution class indexed by $k\in\{1,2,\ldots\}$, rather than by the sample size or a selected Hill order statistic. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: $M_k=\{\mathcal L(U):U=AZ,\ A\in\mathcal A,\ \mathcal L(Z)\in\tilde M_k\}$; the class concerns the noise law, with $m=d_z$ in the component density. - $M$: $M$ denotes the object specified by its local definition, either a class of probability laws or an observed exceedance count. In Lemma~\ref{lemma:mu_rate}: $M=\bigcup_{k=1}^{\infty}M_k$ is a class of noise laws, with $M_k=\{\mathcal L(U):U=AZ,\ A\in\mathcal A,\ \mathcal L(Z)\in\tilde M_k\}$ and $m=d_z$ in the component density. In Lemma~\ref{lemma:eta_rate}: In the score bound and the zero-count convention, $M=|\mathcal I|$ is the realized second-half evaluation exceedance count. The tail-model assumption separately concerns the noise law $\mathcal L(U)$, with $m=d_z$ in its component density. In Theorem~\ref{thm:rate}: In the sampling terms, count bound and zero-count convention, $M=|\mathcal I|$ is the realized second-half evaluation exceedance count. In the inherited tail-model definition, $M=\bigcup_{k=1}^{\infty}M_k$ separately denotes the class of noise laws, with component dimension $m=d_z$. In Lemma~\ref{lemma:eta_ipw}: In the score bound and zero-count convention, $M=|\mathcal I|$ is the realized second-half evaluation exceedance count. The tail-model assumption separately concerns the noise law $\mathcal L(U)$, with $m=d_z$ in its component density. - $s$: In the tail model, $s\in(0,1/2)$ is a fixed model constant governing both the density error $\xi k^{-s}$ and the exact-tail threshold $\zeta k^{(1-2s)/\beta}$. In Lemma~\ref{lemma:eta_rate}: In $m_d^t(x,s)$, $S_U=s$, $s\mapsto\mathbb E_X[g(X,1,s)-g(X,0,s)]$, and $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, $s\in\mathbb S^+=\{u\in\mathbb R_+^d:\|u\|_1=1\}$ is an angular argument or bound integration variable. In $t^{-\beta s/(1-2s)}$, $s$ retains its scalar tail-model meaning. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $s$ is the scalar tail-model constant in the threshold exponent and the two item restrictions: the first item assumes $00$ is a fixed model constant imposing $JA=\sqrt{\det(A^\mathsf{T}A)}\geq\sigma$ on matrices in $\mathcal{A}$. - $\xi$: $\xi\in(0,1)$ is a fixed model constant bounding the relative density approximation error by $\xi k^{-s}$. - $\zeta$: $\zeta>0$ is a fixed constant in the tail-model class: for each class index $k$, the exact proportional-density region is $\|z\|_1>\zeta k^{(1-2s)/\beta}$. - $n$: $n$ is the size of the full input dataset $\mathcal{D}=\{(X_i,D_i,Y_i,U_i)\}_{i=1}^n$, before its split into two equal halves; when used as the argument of an estimation-error function, it denotes that function's sample-size argument. In Lemma~\ref{lemma:mu_rate}: $n$ is the integer size of the full input sample whose radii are ordered, with $n\geq2$ for the definition of $H_n(k)$; for each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that the deviation bound holds with probability at least $1-\delta$ under the sample-size-$n$ law for every integer $n\geq n_0$. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The full-input sample size $n$ is a positive even integer, with $|\mathcal D_1|=|\mathcal D_2|=n/2$; this restriction does not restrict a generic sample-size argument of an estimation-error function to even integers. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: For each fixed $\delta\in(0,1/2)$, the probability guarantee holds separately under the sample-size-$n$ law for every sufficiently large even $n$. - $\delta$: $\delta$ is a fixed confidence tolerance in $(0,1)$, used in probability guarantees of the form $1-\delta$ and in the adaptive Hill tuning rule. In Lemma~\ref{lemma:mu_rate}: For each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that, for every integer $n\geq n_0$, the bound has probability at least $1-\delta$ under the law of the sample of size $n$; $n_0$ and $C_\delta$ may depend on $\delta$. In Theorem~\ref{thm:rate}: $\delta\in(0,1/2)$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: For each fixed $\delta\in(0,1/2)$, the probability guarantee holds separately under the sample-size-$n$ law for every sufficiently large even $n$; the sample-size requirement may depend on $\delta$. The full-input Hill selection uses tolerance $\delta/6$. - $n_t$: $n_t$ is the observed number of threshold exceedances in the designated half of the sample, with the half and strictness of the threshold determined by its local definition. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $n_t=\sum_{i=1}^{n/2}I(\|U_i\|>t)$ is the realized training-split exceedance count for positive even $n$, taking values in $\{0,\ldots,n/2\}$; it is distinct from the evaluation count $|\mathcal I|$ and from its expectation $n\mathbb P(\|U\|>t)/2$. - $R_p$: $R_p(n,\delta)$ is a deterministic positive error-bound function, monotonically decreasing in its sample-size argument, used in the high-probability inequality $|p(X)-\hat p(X)|\leqslant R_p(n,\delta)$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $R_p(n,\varepsilon)=\Theta(\log(1/\varepsilon)n^{-1/2})$. On one joint nuisance event of probability at least $1-\delta/6$, the three nuisance inequalities hold simultaneously for every $t\in I_n$, almost surely over the covariate and conditional direction support and both treatments; the propensity inequality is $|p(x)-\hat p(x)|\leqslant R_p(n/2,\delta/6)$, using the same fitted propensity for every input threshold. In Theorem~\ref{thm:rate}: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, conditional direction support and both treatments. The displayed propensity inequality uses $R_p(n,\varepsilon)$; the training-split propensity bound entering the final error bounds uses $R_p(n/2,\varepsilon)$. In Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: For each tolerance $\varepsilon$, the nuisance inequalities hold jointly with probability at least $1-\varepsilon$, uniformly almost surely over the covariate and direction support and both treatments. The training-split propensity bound used in the proof is $|p(x)-\hat p(x)|\leqslant R_p(n/2,\varepsilon)$. - $R_g$: $R_g(q,\delta)$ is a nonnegative regression-error bound function, nonincreasing in its sample-size argument $q$, for estimating the conditional mean of $Y/\|U\|^\alpha$ given $(X,D,U/\|U\|)$ and $\|U\|>t$; evaluation at $q=n_t$ uses the observed tail count. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $R_g(n,\delta)=\Theta(\log(1/\delta)n^{-1/2})$. On one joint nuisance event of probability at least $1-\delta/6$, simultaneously for every $t\in I_n$, $|m_d^t(x,v)-\hat g(x,d,v)|\leqslant R_g(n_t,\delta/6)$ almost surely over the covariate and conditional direction support and both treatments, where $m_d^t(x,v)=\mathbb E[Y/\|U\|^\alpha\mid X=x,D=d,U/\|U\|=v,\|U\|>t]$ and $n_t=\sum_{i=1}^{n/2}I(\|U_i\|>t)$ is the training-split exceedance count. In Lemma~\ref{lemma:eta_rate}: For each tolerance $\varepsilon\in(0,1)$, the bound $R_g(n_t,\varepsilon)$ applies uniformly almost surely over the covariate and direction support and both treatments, on the joint nuisance event of probability at least $1-\varepsilon$; its sample-size argument $n_t$ is the training-split exceedance count, distinct from $M=|\mathcal I|$. For positive even $n$, its domain includes every $q\in\{0,1,\ldots,n/2\}$, including $q=0$. In Theorem~\ref{thm:rate}: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the regression-error bound $R_g(n_t,\varepsilon)$ holds together with the propensity and exponent inequalities on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, conditional direction support, and both treatments; $n_t=\sum_{i=1}^{n/2}I(\|U_i\|>t)$ is the training-split exceedance count, with training observations indexed first after the random split. In Lemma~\ref{lemma:eta_ipw}: For each tolerance $\varepsilon$, $R_g(n_t,\varepsilon)$ applies uniformly almost surely over the covariate and direction support and both treatments on the joint nuisance event of probability at least $1-\varepsilon$; $n_t$ is the training-split exceedance count, distinct from $M=|\mathcal I|$. Its sample-size domain includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. - $R_\alpha$: $R_\alpha(n,\delta)$ is a deterministic positive error-bound function, monotonically decreasing in its sample-size argument, used in the high-probability inequality $|\hat\alpha_n-\alpha|\leqslant R_\alpha(n,\delta)$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $R_\alpha(n,\delta)=\Theta(\log(1/\delta)n^{-c_\alpha})$ for some $c_\alpha>0$. - $\widehat{\theta}_{n,t}^{\text{DR}}$: $\widehat\theta_{n,t}^{\text{DR}}$ is the sample estimator formed by multiplying the split-sample normalized DR first-factor estimate by $\hat\mu_n=(1-\hat\alpha_n\hat\gamma_n)^{-1}$, wherever that denominator and the first-factor estimate are defined. In Corollary~\ref{cor:rate}: The estimator is evaluated at $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ in the first item and $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ in the second item, with deterministic positive order constants independent of $n$ and the sample. The fitted $\hat p$, $\hat g$, and $\hat\alpha_n$ are measurable with respect to the training split and independent auxiliary data $\mathcal T$ and are the same functions and exponent for every input threshold. The Hill estimate $\hat\gamma_n$ uses all $n$ input radii at tolerance $\delta/6$. Use $\hat\eta_{n,t}^{\mathrm{DR}}=0$ when there are no evaluation exceedances and $\hat\mu_n=0$ when $1-\hat\alpha_n\hat\gamma_n=0$. In Theorem~\ref{thm:rate}: The threshold $t$ is deterministic, the fitted propensity, regression, and exponent are measurable with respect to the training split and independent auxiliary data $\mathcal T$, and $\hat\gamma_n=\widetilde\gamma_n$ is computed from all $n$ input radii at tolerance $\delta/6$; use $\hat\eta_{n,t}^{\mathrm{DR}}=0$ when $M=0$ and $\hat\mu_n=0$ when $1-\hat\alpha_n\hat\gamma_n=0$. - $\widehat{\theta}_{n,t}^{\text{IPW}}$: $\widehat\theta_{n,t}^{\text{IPW}}$ is the split-sample product $\hat\eta_{n,t}^{\mathrm{IPW}}\hat\mu_n$, using $\hat p$ fitted on $\mathcal D_1$, the supplied exponent $\hat\alpha_n$, and the normalized IPW score averaged over evaluation exceedances in $\mathcal D_2$, with $\hat\mu_n=(1-\hat\alpha_n\hat\gamma_n)^{-1}$ wherever the average and reciprocal are defined. In Corollary~\ref{cor:rate_ipw}: The estimator is evaluated at $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ in the first item and $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ in the second item, with deterministic positive order constants independent of $n$ and the sample. The fitted $\hat p$, $\hat g$, and $\hat\alpha_n$ are measurable with respect to the training split and independent auxiliary data $\mathcal T$ and are the same functions and exponent for every input threshold. The Hill estimate $\hat\gamma_n$ uses all $n$ input radii at tolerance $\delta/6$. Use $\hat\eta_{n,t}^{\mathrm{IPW}}=0$ when there are no evaluation exceedances and $\hat\mu_n=0$ when $1-\hat\alpha_n\hat\gamma_n=0$. In Theorem~\ref{thm:rate}: The threshold $t$ is deterministic, the fitted propensity, regression, and exponent are measurable with respect to the training split and independent auxiliary data $\mathcal T$, and $\hat\gamma_n=\widetilde\gamma_n$ is computed from all $n$ input radii at tolerance $\delta/6$; use $\hat\eta_{n,t}^{\mathrm{IPW}}=0$ when $M=0$ and $\hat\mu_n=0$ when $1-\hat\alpha_n\hat\gamma_n=0$. - $\hat{\eta}_{n,t}^{\tmop{DR}}$: $\hat\eta_{n,t}^{\tmop{DR}}$ is the average normalized doubly robust score over exceedances in the second half $\mathcal{D}_2$, using $\hat p$ and $\hat g$ fitted on the first half $\mathcal{D}_1$ and the supplied exponent $\hat\alpha_n$, wherever $|\mathcal{I}|>0$. In Lemma~\ref{lemma:eta_rate}: Set $\hat\eta_{n,t}^{\tmop{DR}}=0$ when $M=|\mathcal I|=0$; conditional on $\mathcal D_1$ and the independent auxiliary data, the fitted $\hat p$, $\hat g$, and $\hat\alpha_n$ are fixed, and for $M>0$ the estimator averages the displayed score $\psi$ over the evaluation exceedances. In Corollary~\ref{cor:rate}: The estimator is evaluated at $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ in the first item and $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ in the second item, with deterministic positive order constants independent of $n$ and the sample. Set $\hat\eta_{n,t}^{\tmop{DR}}=0$ when there are no evaluation exceedances. The fitted $\hat p$, $\hat g$, and $\hat\alpha_n$ are $\mathcal T$-measurable, where $\mathcal T$ consists of the training split and independent auxiliary data, and are the same functions and exponent for every input threshold; they are held fixed in the score moments conditional on $\mathcal T$ and $\|U\|>t$. In Theorem~\ref{thm:rate}: Set $\hat\eta_{n,t}^{\tmop{DR}}=0$ when $M=|\mathcal I|=0$; conditional on the training split and independent auxiliary data $\mathcal T$, the fitted $\hat p$, $\hat g$, and $\hat\alpha_n$ are fixed, and for $M>0$ the estimator averages $\psi^{\mathrm{DR}}$ over the evaluation exceedances. In Lemma~\ref{lemma:eta_ipw}: Set $\hat\eta_{n,t}^{\tmop{DR}}=0$ when $M=|\mathcal I|=0$; conditional on the training split and independent auxiliary data $\mathcal T$, the fitted $\hat p$, $\hat g$, and $\hat\alpha_n$ are fixed, and for $M>0$ the estimator averages the normalized doubly robust score over the evaluation exceedances. - $\hat{\eta}_{n,t}^{\text{IPW}}$: $\hat\eta_{n,t}^{\mathrm{IPW}}$ is the sample average over the threshold exceedances in the second half $\mathcal{D}_2$ of the split sample, using $\hat p$ fitted on the first half $\mathcal{D}_1$ and the supplied exponent $\hat\alpha_n$, wherever $|\mathcal{I}|>0$. In Corollary~\ref{cor:rate_ipw}: The estimator is evaluated at $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ in the first item and $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ in the second item, with deterministic positive order constants independent of $n$ and the sample. Set $\hat\eta_{n,t}^{\mathrm{IPW}}=0$ when there are no evaluation exceedances. The fitted propensity and exponent are $\mathcal T$-measurable, where $\mathcal T$ consists of the training split and independent auxiliary data, and are the same function and exponent for every input threshold; they are held fixed in the score moments conditional on $\mathcal T$ and $\|U\|>t$. In Lemma~\ref{lemma:eta_ipw}: $\hat\eta_{n,t}^{\mathrm{IPW}}=0$ when $M=|\mathcal I|=0$. The fitted propensity and exponent are measurable with respect to $\mathcal T$, the training split together with independent auxiliary data, and are held fixed in the score moments conditional on $\mathcal T$ and $\|U\|>t$. In Theorem~\ref{thm:rate}: $\hat\eta_{n,t}^{\mathrm{IPW}}=0$ when $M=|\mathcal I|=0$. The fitted propensity and exponent are measurable with respect to $\mathcal T$, the training split together with independent auxiliary data, and are held fixed in the score moments conditional on $\mathcal T$ and $\|U\|>t$; the threshold $t$ is deterministic. - $\eta$: $\eta$ is a population quantity whose scalar or functional meaning is determined by its local use. In Lemma~\ref{lemma:eta_rate}: $\eta=\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\mathrm{d}s)$, where $S$ is the limiting spectral probability measure. In Theorem~\ref{thm:rate}: $\eta=\lim_{t\to\infty}\mathbb E[g(X,1,U/\|U\|)-g(X,0,U/\|U\|)\mid\|U\|>t]$ is the scalar population first-factor target. In Lemma~\ref{lemma:eta_ipw}: $\eta=\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. In Lemma~\ref{lemma:mu_rate}: $\eta(v)=vQ'(v)/Q(v)-\gamma=\bar F_R(Q(v))/(Q(v)f_R(Q(v)))-1/\beta$ is the deterministic von Mises function on the differentiable tail of $Q(v)=F_R^{-1}(1-1/v)$, where $R=\|U\|_1$, $F_R$ is its population distribution function, $\bar F_R$ its survival function, $f_R$ its density, and $\gamma=1/\beta$. - $\mathcal{D}$: $\mathcal D=\{(X_i,D_i,Y_i,U_i)\}_{i=1}^n$ denotes the indexed input dataset of $n$ observations. In Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: The sample size $n$ is a positive even integer, and the equal split satisfies $|\mathcal D_1|=|\mathcal D_2|=n/2$. In Theorem~\ref{thm:rate}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The input size $n$ is a positive even integer; after the random equal split, indices $1,\ldots,n/2$ designate the training observations and indices $n/2+1,\ldots,n$ designate the evaluation observations. - $\mathcal{D}_1$: $\mathcal D_1$ is the first half of the random equal split of the indexed dataset $\mathcal D$, containing $n/2$ observations and supplying the observations for nuisance regression. In Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: The sample size $n$ is a positive even integer, and $|\mathcal D_1|=|\mathcal D_2|=n/2$. In Theorem~\ref{thm:rate}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The input size $n$ is a positive even integer, and after the random split the training observations are indexed first, so $\mathcal D_1=\{(X_i,D_i,Y_i,U_i)\}_{i=1}^{n/2}$ and $n_t=\sum_{i=1}^{n/2}I(\|U_i\|>t)$ counts its exceedances. - $\mathcal{D}_2$: $\mathcal D_2$ is the second equal part of the split dataset, used for evaluation, with the equal split defined for positive even sample sizes $n$ and $|\mathcal D_1|=|\mathcal D_2|=n/2$. - $\hat{\alpha}_n$: $\hat{\alpha}_n\in\mathbb{R}$ is the supplied approximate value of the outcome scaling index $\alpha$, obtainable from prior knowledge or heuristics. In Lemma~\ref{lemma:eta_rate}: $\hat{\alpha}_n$ is measurable with respect to the training split and independent auxiliary data; for each tolerance $\varepsilon$, its bound $|\hat{\alpha}_n-\alpha|\leq R_\alpha(n,\varepsilon)$ holds jointly with the other nuisance inequalities with probability at least $1-\varepsilon$. The bound uses $a=R_\alpha(n,\delta/4)\leq a_0$ for fixed $a_0\in(0,\beta)$. In Theorem~\ref{thm:rate}: $\hat\alpha_n$ is measurable with respect to $\mathcal T$, the training split and independent auxiliary data. For every fixed $t$ and tolerance $\varepsilon\in(0,1)$, $|\hat\alpha_n-\alpha|\leq R_\alpha(n,\varepsilon)$ holds on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, jointly with the other nuisance inequalities, simultaneously almost surely over the covariate support, conditional direction support and both treatments. The bound uses $a=R_\alpha(n,\delta/12)\leq a_0$ for fixed $a_0\in(0,\beta)$. In Lemma~\ref{lemma:eta_ipw}: $\hat{\alpha}_n$ is measurable with respect to the training split and independent auxiliary data; for every tolerance $\varepsilon$, its inequality $|\hat{\alpha}_n-\alpha|\leq R_\alpha(n,\varepsilon)$ holds jointly with the other nuisance inequalities with probability at least $1-\varepsilon$. The exponent-error quantity is $a=R_\alpha(n,\delta/4)\leq a_0$, where $a_0\in(0,\beta)$ is fixed. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $\hat\alpha_n$ is $\mathcal T$-measurable, where $\mathcal T$ consists of the training split and independent auxiliary data, and is the same exponent for every input threshold. Its rate satisfies $R_\alpha(n,\delta)=\Theta(\log(1/\delta)n^{-c_\alpha})$ for some fixed $c_\alpha>0$. With probability at least $1-\delta/6$, the inequality $|\hat\alpha_n-\alpha|\leq R_\alpha(n,\delta/6)$ holds jointly with the propensity and regression inequalities for every $t\in I_n=[(c_-/2)n^{\kappa(\gamma)},2c_+n^{\kappa(\gamma)}]$, simultaneously almost surely over the covariate and conditional direction support and both treatments. - $\hat{g}$: $\hat g(x,d,s)$ is the fitted regression of $Y/\|U\|^{\hat\alpha_n}$ on $(X,D,U/\|U\|)$ using $\mathcal D_1$, with $s\in\mathbb S^+$ for the paper's $\ell_1$ norm. In Lemma~\ref{lemma:eta_rate}: $\hat g$ is measurable with respect to $\mathcal D_1$ and independent auxiliary data and is held fixed in the conditional evaluation-score expectations; its nuisance inequality holds uniformly almost surely over the covariate and direction support and both treatments on the stipulated joint nuisance events. In Theorem~\ref{thm:rate}: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, $\hat g$ is measurable with respect to the training split and independent auxiliary data, denoted together by $\mathcal T$, and is held fixed in the conditional evaluation-score expectations; its nuisance comparison is with $\mathbb E[Y/\|U\|^{\alpha}\mid X,D,U/\|U\|,\|U\|>t]$, with bound $R_g(n_t,\varepsilon)$, although the fitted response uses $\hat\alpha_n$. This inequality holds simultaneously almost surely over the covariate support, conditional direction support and both treatments on one $\mathcal T$-measurable joint nuisance event of probability at least $1-\varepsilon$. In Lemma~\ref{lemma:eta_ipw}: $\hat g$ is measurable with respect to the training split and independent auxiliary data, denoted together by $\mathcal T$; its nuisance inequality holds uniformly almost surely over the covariate and direction support and both treatments on the stipulated joint nuisance events. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $\hat g$ is measurable with respect to the training split and independent auxiliary data, denoted together by $\mathcal T$, and is the same fitted function for every input threshold; with probability at least $1-\delta/6$, jointly for every $t\in I_n$, $|m_d^t(x,v)-\hat g(x,d,v)|\leq R_g(n_t,\delta/6)$ simultaneously almost surely over the covariate and conditional direction support and both treatments, where $m_d^t(x,v)=\mathbb E[Y/\|U\|^\alpha\mid X=x,D=d,U/\|U\|=v,\|U\|>t]$ and $n_t$ is the training exceedance count. - $\mathcal{I}$: $\mathcal I=\{i\in\{1,\ldots,n\}: (X_i,D_i,Y_i,U_i)\in\mathcal D_2,\ \|U_i\|>t\}$ is the evaluation exceedance index set at the threshold supplied to the estimator. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: At the selected threshold $t_n$, $\mathcal I=\{i\in\{1,\ldots,n\}:(X_i,D_i,Y_i,U_i)\in\mathcal D_2,\ \|U_i\|>t_n\}$. - $\hat{\gamma}_n$: $\hat\gamma_n$ denotes a sample EVI estimate computed from ordered $\ell_1$ radii; the displayed definition in Algorithm~\ref{alg:nete} is $\hat\gamma_n=k^{-1}\sum_{j=1}^k\log(\|U_{(j)}\|/\|U_{(j+1)}\|)$ on the evaluation exceedances, with the displayed adaptive choice of $k$. In Theorem~\ref{thm:rate}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: The full-input convention is $\hat\gamma_n=\widetilde\gamma_n=H_n(\widetilde k_n)$ at tolerance $\delta/6$. Here $R_{[1]}\geq\cdots\geq R_{[n]}$ order all input radii, $H_n(k)=k^{-1}\sum_{j=1}^k\log(R_{[j]}/R_{[k+1]})$ for $1\leq k\leq n-1$, and $\widetilde k_n$ is the maximum $k\in\{L_n,\ldots,n-1\}$ satisfying $|H_n(i)-H_n(k)|\leq H_n(i)\rho_n(\delta/6)/\sqrt i$ for every $i\in\{L_n,\ldots,k\}$, using the displayed tuning definitions for $L_n$ and $\rho_n$. - $t_n$: $t_n$ denotes the selected threshold sequence, depending on $\hat\gamma_n$, with $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ for $s\in(0,1/(2+\max\{1,\beta\}))$ and $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ for $s\in[1/(2+\max\{1,\beta\}),1/2)$; the $t$ in the corresponding estimator denotes this selected threshold. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The threshold order means $c_-n^{\kappa(\hat\gamma_n)}\leq t_n\leq c_+n^{\kappa(\hat\gamma_n)}$, with deterministic constants $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. Identification `prop:id` | field | value | | --- | --- | | bucket | `repaired` | | source | `Arxiv_template.tex:234-243` (`proposition`, label `prop:id`) | | statement | \label{prop:id}Suppose that $U$ is multivariate regularly varying and Assumption~\ref{assump:exo}, \ref{assump:iid}, \ref{assump:f_homo} hold, we have \begin{align*} \theta^{\tmop{NETE}} = \lim_{t \rightarrow \infty} \mathbb{E}[g (X, 1, U /\\|U\\|) - g (X, 0, U /\\|U\\|) \mid \\|U\\|> t] \cdot \lim_{t \rightarrow \infty} \mathbb{E}[\\|U\\|^{\alpha} / t^{\alpha} \mid \\|U\\|> t] . \end{align*} | | appended hypothesis list | ass:prop-id-regular-variation, ass:exogeneity, ass:iid-sampling, ass:iid-regular-variation, ass:iid-independence, ass:f-homo-bounded-x, ass:f-homo-asymptotic | | Lean | `TId.lean:18` — `Papers.TbsExtremeEffects.nete_identification` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-33`, `ass:pin-34`, `ass:pin-38`, `ass:pin-39`, `ass:pin-43`, `ass:pin-7`, `ass:pin-fix-8cd5c960` | | cited results | — | | assumed substrate gates | — | | findings | [refuted, V1.5] The core carries the printed hypotheses and identification setting faithfully. The homogeneity display could quantify over one fixed treatment argument and radius, or uniformly over both treatment arguments and all positive radii; its sphere could also mean the usual Euclidean sphere or the positive unit sphere for the paper's norm. The proof and global norm convention determine the latter, uniform reading. The substantive error is the proof's first equality: $(Y(1),Y(0))\perp D\mid X$ and $U\perp(X,D)$ do not imply conditional exogeneity given $(X,U)$. The conclusion fails even with overlap, exact homogeneity, finite outcome means, and $\alpha<\beta$. In that moment range, the remaining radial-angular passage can be closed using Potter bounds, a higher radial moment for uniform integrability, and $\sup_{r\geq t}e(r)\to0$; bounded first moments alone do not establish uniform integrability. The preliminary radial-moment sentence also drops the additive one, although the subsequent calculation supplies the correct value $\beta/(\beta-\alpha)$. witness: Take independent $X\sim\text{Unif}([0,1])$, $D\sim\text{Ber}(1/2)$, $V\sim\text{Unif}([0,1])$, and $W$ with $\mathbb P(W>w)=w^{-3}$ for $w\geq1$. Set $U=W(V,1-V)$, $\alpha=1$, and $T(w)=(1-w^{-3})^{-1/3}$. Define $Y(1)=DW+(1-D)T(W)$, $Y(0)=0$, and $Y=Y(D)$. Since $W^{-3}$ is uniform, $T(W)$ has the same law as $W$. Thus $(Y(1),Y(0))\perp D\mid X$, and $U\perp(X,D)$; independent copies satisfy the sampling assumption. Nevertheless, $f(x,1,u)=\\|u\\|$, $f(x,0,u)=0$, so exact homogeneity holds with $g(x,1,s)=1$, $g(x,0,s)=0$, and $e(t)=0$. The printed right-hand side is $3/2$. But $\mathbb E[W\mid W>t]=3t/2$ and $\mathbb E[T(W)\mid W>t]\to1$, giving $\theta^{\tmop{NETE}}=3/4$. The false step replaces the conditional counterfactual mean by the observed conditional mean after conditioning on $U$; neither stated independence licenses that replacement. minimal patch (confirmed): corrected statement: Suppose that $U$ is multivariate regularly varying and Assumption~\ref{assump:exo}, \ref{assump:iid}, \ref{assump:f_homo} hold. Suppose in addition that $\alpha<\beta$, $0t]<\infty$ for both treatments and all sufficiently large $t$. Then we have \begin{align*} \theta^{\tmop{NETE}} &= \lim_{t\rightarrow\infty}\mathbb E[g(X,1,U/\\|U\\|)-g(X,0,U/\\|U\\|)\mid\\|U\\|>t]\cdot\lim_{t\rightarrow\infty}\mathbb E[\\|U\\|^{\alpha}/t^{\alpha}\mid\\|U\\|>t]. \end{align*} · proof change: Write $R_U=\\|U\\|$, $S_U=U/\\|U\\|$, and $\bar e(t)=\sup_{r\geq t}e(r)$; the last quantity is finite for sufficiently large $t$ and tends to zero. Independence of $U$ from $(X,D)$ gives $\mathbb P(D=1\mid X,U)=p(X)$. Positivity, consistency, and conditional mean exogeneity given $(X,U)$ therefore give, for both treatments, \[ f(X,d,U)=\mathbb E[Y(d)\mid X,U,D=d]=\mathbb E[Y(d)\mid X,U]. \] The assumed absolute tail means justify taking conditional expectations on $\{R_U>t\}$. Homogeneity at the realized radius yields \[ \frac{f(X,1,U)-f(X,0,U)}{R_U^{\alpha}}=g(X,1,S_U)-g(X,0,S_U)+\varepsilon(X,U),\qquad \|\varepsilon(X,U)\|\leq2\bar e(t)\quad\hbox{on }\{R_U>t\}. \] Consequently, \[ \left\|\mathbb E\left[\frac{Y(1)-Y(0)}{t^{\alpha}}\mid R_U>t\right]-\mathbb E\left[(g(X,1,S_U)-g(X,0,S_U))(R_U/t)^{\alpha}\mid R_U>t\right]\right\|\leq2\bar e(t)\mathbb E[(R_U/t)^{\alpha}\mid R_U>t]. \] We next establish that the radial moment tends to $\beta/(\beta-\alpha)$. For every real $a<\beta$, the identity $x^a=1+\int_1^x a r^{a-1}\,\tmop{dr}$, together with absolute integrability of the integral, gives \[ \mathbb E[(R_U/t)^a\mid R_U>t]=1+\int_1^{\infty}a r^{a-1}\frac{\mathbb P(R_U>tr)}{\mathbb P(R_U>t)}\,\tmop{dr}. \] For $a=0$ the integral is zero; for $a<0$ its absolute integrability follows from $\int_1^{\infty}\|a\|r^{a-1}\,\tmop{dr}=1$. Note that $\\| U \\|$ is also regularly varying. By Potter's theorem \cite[Theorem 1.56]{bingham1989regular}, for any $\epsilon >0$ and sufficiently large $t$, we have \begin{align*} \frac{P (\\| U \\| > {rt})}{P (\\| U \\| > t)} & \leqslant 2 r^{-\beta + \epsilon} . \end{align*} For $0t]\longrightarrow1+\int_1^{\infty}a r^{a-\beta-1}\,\tmop{dr}=\frac{\beta}{\beta-a}. \] In particular, the radial moments with $a=\alpha$ are bounded, and the preceding homogeneity error tends to zero. Define \[ G(s)=\mathbb E_X[g(X,1,s)-g(X,0,s)],\qquad h_t(r,s)=G(s)r^{\alpha}. \] Independence of $X$ and $U$ shows that the remaining conditional expectation is $\mathbb E[h_t(R_U/t,S_U)\mid R_U>t]$. Note that by Assumption \ref{assump:f_homo}, $g$ is a continuous function on a compact set and thus is bounded by some constant $C > 0$. Thus $\|G(s)\|\leq2C$, and $G$ is continuous. If $\alpha\leq0$, then $\|h_t(r,s)\|\leq2C$ for $r\geq1$. If $\alpha>0$, choose $p>1$ with $p\alpha<\beta$. The radial moment calculation gives \[ \sup_{t\geq t_0}\mathbb E[\|G(S_U)(R_U/t)^{\alpha}\|^p\mid R_U>t]\leq(2C)^p\sup_{t\geq t_0}\mathbb E[(R_U/t)^{p\alpha}\mid R_U>t]<\infty. \] This establishes uniform integrability in both cases. Definition~\ref{def:mvarying} gives conditional weak convergence of $(R_U/t,S_U)$ to the product of a Pareto radial law with survival $r^{-\beta}$ for $r\geq1$ and the spectral probability measure $S$. For a bounded truncation of $G(s)r^{\alpha}$, weak convergence gives convergence of expectations. Uniform integrability makes the truncation errors tend to zero uniformly, so \[ \lim_{t\rightarrow\infty}\mathbb E[G(S_U)(R_U/t)^{\alpha}\mid R_U>t]=\left(\int G(s)\,S(\tmop{ds})\right)\frac{\beta}{\beta-\alpha}. \] Weak convergence of the angular marginal and bounded continuity of $G$ also give \[ \lim_{t\rightarrow\infty}\mathbb E[g(X,1,S_U)-g(X,0,S_U)\mid R_U>t]=\int G(s)\,S(\tmop{ds}). \] Combining these two limits with the vanishing homogeneity error proves the stated product formula. · contribution narrowed (accepted by the owner): Identification for distributions that satisfy exogeneity given $X$ but violate conditional mean exogeneity given $(X,U)$, distributions with a degenerate true propensity, the range $\alpha\geq\beta$, and counterfactuals without finite absolute tail means. ／ [note, F4] [statement-wrong] 'g being L-Lipschitz on this domain' admits two formal readings. (A) Per treatment: for each d, \|g(x,d,u) − g(x',d,u')\| ≤ L(‖x−x'‖_1 + ‖u−u'‖_1) on B_R × S^{d−1}. (B) Joint in (x,d,u): \|g(x,d,u) − g(x',d',u')\| ≤ L(‖x−x'‖_1 + \|d−d'\| + ‖u−u'‖_1), which additionally forces \|g(x,1,u) − g(x,0,u)\| ≤ L. LipschitzG realizes (B). NL edit to pin it: say either 'for each d, g(·,d,·) is L-Lipschitz on B_R × S^{d−1}' or 'g is L-Lipschitz jointly in (x,d,u), with d in {0,1} as a real coordinate'. (rewind rejected by the orchestrator on prop:id: The author settled the reading (session, 2026-10-09): in Assumption~\ref{assump:f_homo}, ``$g$ is $L$-Lipschitz'' is per treatment. For each $d\in\{0,1\}$, $g(\cdot,d,\cdot)$ is $L$-Lipschitz on $B_R\times S^{d-1}$ under the $\ell_1$ norm: $\|g(x,d,u)-g(x^\prime,d,u^\prime)\|\le L(\\|x-x^\prime\\|_1+\\|u-u^\prime\\|_1)$. Nothing is assumed across the two treatments; in particular $\|g(x,1,u)-g(x,0,u)\|\le L$ is not assumed. The Lean statement is being changed to this reading.) | ### 2. `thm:rate` | field | value | | --- | --- | | bucket | `repaired` | | source | `Arxiv_template.tex:330-357` (`theorem`, label `thm:rate`) | | statement | \label{thm:rate}Suppose that Assumption \ref{assump:exo}, \ref{assump:overlap}, \ref{assump:iid}, \ref{assump:f_homo}, \ref{assump:mvary} hold, $\alpha < \beta$, where $\alpha$ and $\beta$ are defined in \cref{assump:f_homo} and \cref{assump:mvary} respectively. Furthermore, for any fixed $t$, with probability at least $1 - \delta$, \begin{align*} \|p (X) - \hat{p} (X) \| \leqslant R_p (n, \delta) & , \| \hat{\alpha}_{n} - \alpha\| \leqslant R_\alpha(n,\delta),\\ \|\mathbb{E}[Y/\\|U\\|^{\alpha} \mid X, D, U /\\|U\\|, \\|U\\|> t] &- \hat{g} (X, D, U /\\|U\\|) \| \leqslant R_g (n_t, \delta), \end{align*} where $n_t = \sum_{i = 1}^{n/2} I (\\|U_i \\|> t)$ and $R_p, R_g, R_\alpha$ are estimation errors that are monotonically decreasing with respect to sample size. Then, with probability at least $1 - \delta, \delta \in (0, 1 / 2)$, we have \begin{align} \left\| {{\widehat{\theta }_{n, t}^{\text{DR} }} } - \theta^{NETE} \right\| & \leqslant O (\sqrt{R_p (n/2, \delta)R_g ({n_t}, \delta)} + t^{\beta / 2} n^{-1 / 2}+ \log (1 / \delta) n^{- 1 / (2 +\beta)} \notag\\ & \quad\quad \quad \quad + t^{- \min \{1, \beta\}} + t^{- \beta s / (1 - 2 s)} + \log(t)R_\alpha(n,\delta)+ e (t)).\label{eq:rate_dr} \end{align} and \begin{align} \left\| {{\widehat{\theta }_{n, t}^{\text{IPW}}} } - \theta^{NETE} \right\| & \leqslant O (R_p (n/2, \delta) +t^{\beta / 2} n^{- 1 / 2} + \log (1 / \delta) n^{- 1 / (2 + \beta)} \notag\\ &\quad\quad \quad \quad + t^{- \min \{1, \beta\}} + t^{- \beta s / (1 - 2 s)}+ \log(t)R_\alpha(n,\delta) + e (t)). \label{eq:rate_ipw} \end{align} | | appended hypothesis list | ass:exogeneity, ass:overlap, ass:iid-sampling, ass:iid-regular-variation, ass:iid-independence, ass:f-homo-bounded-x, ass:f-homo-asymptotic, ass:mvary-density, ass:mvary-matrix, ass:mvary-constants, ass:rate-alpha-less-beta, ass:rate-nuisance-errors | | Lean | `TRate.lean:96` — `Papers.TbsExtremeEffects.nete_rate` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity pinned · correctness ok | | added conditions | `ass:pin-33`, `ass:pin-34`, `ass:pin-36`, `ass:pin-38`, `ass:pin-39`, `ass:pin-41`, `ass:pin-42`, `ass:pin-43`, `ass:pin-44`, `ass:pin-45`, `ass:pin-7`, `ass:pin-fix-0b814294`, `ass:pin-fix-10c2ec7b`, `ass:pin-fix-6857bb4d`, `ass:pin-fix-8964f8dd`, `ass:pin-fix-8a129280`, `ass:pin-fix-8cd5c960`, `ass:pin-fix-8ef6fd29`, `ass:pin-fix-9567640d`, `ass:pin-fix-b3a62d30`, `ass:pin-fix-c3605ca3`, `ass:pin-fix-c4d18fdb`, `ass:pin-fix-c90a58f3`, `ass:pin-fix-ebdb71b8`, `ass:pin-fix-f18f3355` | | cited results | — | | assumed substrate gates | — | | findings | [note, V1.5] the corrected statement stands as one statement; the referee would state it as a general statement with corollaries: Keep the general score-dependent product bounds as the main statement and separate the finite-variance specialization, which additionally requires the second-moment bound, $t\geq\exp(1)$, $a\log t\leq1$, and $Q_n\leq h/2$. ／ [note, V1.5] pinned reading: The existential cutoff $t_W$ is chosen sufficiently large for Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} and so that $\sup_{r\geq t_W}e(r)\leq1$ and $\mathbb P(\\|U\\|>tr)/\mathbb P(\\|U\\|>t)\leq4r^{-\beta}$ for every $t\geq t_W$ and $r\geq1$. (The proof of Theorem~\ref{thm:rate} invokes the first-factor lemmas' sufficiently-large-threshold conclusions, uses the radial moment inequality with $K=4/(\beta-a_0)$, and bounds $G_t$ by $C+1$. The stated Pareto-type radial law and $e(t)\to0$ permit this enlargement of the existential cutoff.) ／ [refuted, V1.5] The nuisance inequalities could concern only random evaluation arguments or hold simultaneously over their support; the proof's applications of the first-factor lemmas determine the latter reading. Under that stronger reading, the following defects remain. (a) The opening identification equality is false: exogeneity conditional on covariates and independence of the noise from treatment do not imply exogeneity conditional on the noise. (b) The asserted sampling rate has no hypothesis controlling normalized outcome fluctuations. (c) The radial-moment inversion is not uniformly Lipschitz near its pole, and monotone estimation-error bounds need not approach zero. (d) The exponent contribution cannot generally be replaced by $\log(t)R_\alpha(n,\delta)$: its reproduced form contains $t^a-1+Kat^a$. (e) The homogeneity contribution requires $\bar e(t)$, rather than the point value $e(t)$. (f) The angular rate omits a necessary logarithm when $\beta=1$. (g) A nuisance event with failure probability $\delta$ leaves no probability budget for evaluation error; sample-size monotonicity does not compare error bounds at different confidence tolerances. (h) The proof also absorbs the logarithmic factors and the different regimes of Lemma~\ref{lemma:mu_rate} into a rate that this lemma does not supply. These are statement failures in (a)--(g), with the false proof steps identified below; (h) is an additional invalid conversion of the available dependency bound. witness: Common setup: take $m=d=d_z=d_u=2$, $A=I_2$, independent Pareto type-II components $Z_1,Z_2$ with index $\beta$, and $U=Z$. Choose $l=1/2$, $u=2$, $\sigma=1/2$, $s=2/5$, $\xi=1/2$, and $\zeta=1$. The nominal density belongs to the stated model class. Take independent bounded covariates, and use positive, nonincreasing nuisance bounds; exact errors can be bounded by $q^{-10}$ or $(q+1)^{-10}$. Except in (f), use $\beta=2$. Each example has a nonzero NETE. (a) Let $r_0$ be a radial median, $B=I\{\\|U\\|>r_0\}$, and let $V$ be an independent fair Bernoulli variable. Set $D=B\mathbin{\oplus}V$, with $\oplus$ denoting exclusive-or. Let $\varepsilon$ be independent standard Gaussian noise and set $Y(d)=2d+V+\varepsilon$. Because $B$ and $V$ are independent fair Bernoulli variables, $D$ is independent of both $U$ and $V$. Consequently the printed exogeneity and consistency assumptions hold. For $\\|U\\|>r_0$, however, the observed conditional mean is $f(x,d,U)=1+d$. Thus $\alpha=0$, $g(x,d,u)=1+d$, and one may take $e(r)=I\{r\le r_0\}$. The true treatment contrast is always $2$, so $\theta^{\tmop{NETE}}=2$, whereas $\eta\mu=1$. With $\hat p=1/2$, $\hat g=1+d$, and $\hat\alpha_n=0$, both estimators converge to $1$ along $t=n^{1/4}$. The regression-error inequality holds uniformly with, for example, $R_g(q,\delta)=2+(q+1)^{-10}$, while the displayed right sides approach zero. The false step is the opening equality in the theorem's proof. (b) Instead take independent $D\sim\text{Ber}(1/2)$ and $Y(d)=d+\varepsilon$, where $\varepsilon$ is symmetric stable noise with characteristic function $\mathbb E[\exp(\mathrm{i}v\varepsilon)]=\exp(-\|v\|^{3/2})$. Its absolute first moment is finite. Here $\alpha=0$, $g=d$, $e=0$, and $\theta^{\tmop{NETE}}=1$. Use exact nuisances and $\hat\alpha_n=0$, so $\hat\mu_n=1$. Conditional on $M=\|\mathcal I\|>0$, the DR error has exactly the distribution $2M^{-1/3}\varepsilon$. At $t=n^{1/4}$, $M/n^{1/2}\to1$ in probability. The error therefore has scale $n^{-1/6}$, whereas the claimed right side is $O(n^{-1/4})$. The IPW estimator has the same stable term and an additional smaller binomial term. Both claimed probability bounds fail. The false step is substituting $t^{\beta/2}n^{-1/2}$ for the unrestricted sampling contribution of the first-factor lemmas. (c) Take $Y(d)=d\\|U\\|$, $\alpha=1$, $g=d$, $e=0$, and $\hat\alpha_n=2-n^{-1}$. The exponent inequality holds with the decreasing bound $R_\alpha(n,\delta)=1+n^{-1}$. Bounded fitted regressions satisfy the regression inequality with a constant bound. For any fixed $t>2$, the IPW first factor converges to the positive number $\mathbb E[\\|U\\|^{-1}\mid\\|U\\|>t]$. The full-input Hill estimator in the algorithm converges to $1/2$. Hence $1-(2-n^{-1})\hat\gamma_n\to0$ in probability and $\|\hat\mu_n\|\to\infty$ in probability. Its denominator is nonzero almost surely at each sample size. The IPW error diverges although the asserted right side stays bounded. The exact inversion is $\|\hat\mu_n-\mu\|=\|\hat\gamma_n(\hat\alpha_n-\alpha)+\alpha(\hat\gamma_n-\gamma)\|/(\|1-\hat\alpha_n\hat\gamma_n\|\,\|1-\alpha\gamma\|)$; the proof drops its uncontrolled first denominator. (d) Take $Y(d)=d$, $\alpha=0$, $g=d$, $e=0$, and $\hat\alpha_n=-a_n$, where $a_n=(\log(n+2))^{-1/2}$. Use exact propensity and a bounded fitted regression, whose error is bounded by $R_g(q,\delta)=2+(q+1)^{-10}$. Set $R_\alpha(n,\delta)=a_n$ and $t=n^{1/4}$. The IPW first factor divided by $t^{a_n}$ converges to $1$, and $\hat\mu_n=(1+a_n\hat\gamma_n)^{-1}\to1$. Thus the estimator grows like $\exp(a_n\log n/4)$, while the claimed exponent contribution is only $a_n\log n/4$. The approximation $t^a-1=O(a\log t)$ is false without a bound on $a\log t$. (e) Define $b(r)=1/\log r$ on the disjoint intervals $[2\cdot4^j,3\cdot4^j]$, $j\ge1$, and zero elsewhere. Let $Y(d)=d(1+b(\\|U\\|))$, $\alpha=0$, $g=d$, and $e(r)=b(r)$. Use $\hat p=1/2$, $\hat g=d$, $\hat\alpha_n=0$, and $R_g(q,\delta)=1+(q+1)^{-10}$. These nuisance inequalities hold uniformly. At $t_j=4^j$ and $n_j=t_j^4$, $e(t_j)=0$, but regular variation gives a fixed positive conditional probability of $\\|U\\|\in[2t_j,3t_j]$. Consequently $\mathbb E[b(\\|U\\|)\mid\\|U\\|>t_j]\ge c_0/\log t_j$ for some $c_0>0$. Both estimators have this bias and sampling error $O_{\mathbb P}(t_j^{-1})$. Their target is $1$, while the claimed right sides are $O(t_j^{-1})$. The false step replaces an error integrated over radii exceeding $t$ by its value at $t$. (f) Use $\beta=1$, put $h(u)=u_1u_2$ on the positive unit sphere, and take $Y(d)=d(1+h(U/\\|U\\|))$. Then $\alpha=0$, $g(x,d,u)=d(1+h(u))$, $e=0$, and the spectral measure is supported on the coordinate axes, giving $\theta^{\tmop{NETE}}=1$. With exact nuisances, the DR estimator averages $1+h(U/\\|U\\|)$. Integrating the nominal density over $Z_1\in[t,2t]$ and $Z_2\in[1,t]$ gives $\mathbb E[h(U/\\|U\\|)I\{\\|U\\|>t\}]\ge c_1t^{-2}\log t$, while $\mathbb P(\\|U\\|>t)=O(t^{-1})$. Its positive bias is therefore at least $c_2t^{-1}\log t$. At $t=n^{1/3}$, bounded-score sampling fluctuations are $O_{\mathbb P}(n^{-1/3})$, but the bias exceeds every constant multiple of the claimed $n^{-1/3}$ rate with probability approaching one. The IPW estimator has the same bias. The false step drops the logarithm explicitly retained in the angular bounds of the first-factor lemmas. (g) Take $Y(d)=d+\varepsilon$ with independent standard Gaussian noise, $\alpha=0$, and exact supplied exponent. Let a training-covariate event have probability $\delta_0=1/4$, independently of evaluation observations. Off that event set $\hat p=1/2$, $\hat g(x,1,u)=1$, and $\hat g(x,0,u)=0$. On it set $\hat p=1/4$, $\hat g(x,1,u)=2$, and $\hat g(x,0,u)=0$. For each tolerance $\varepsilon$, choose $R_p(q,\varepsilon)=q^{-10}+\tfrac14 I\{\varepsilon<1/4\}$ and $R_g(q,\varepsilon)=(q+1)^{-10}+I\{\varepsilon<1/4\}$. The stipulated nuisance guarantees hold jointly and uniformly, with bounds decreasing in sample size. At $\delta=\delta_0$ and $t=n^{1/4}$, the claimed right side is $O(n^{-1/4})$. On the bad training event the DR estimator converges to zero instead of its target $1$. On the good event, $n^{1/4}(\widehat\theta_{n,t}^{\text{DR}}-1)$ converges to a Gaussian with variance $4$. For every finite bound constant, the limiting failure probability is therefore strictly greater than $\delta_0$: it includes the full bad-event probability and a positive Gaussian tail on the good event. Replacing confidence tolerances by smaller ones changes the nuisance bounds and cannot be justified by their sample-size monotonicity. (h) For fixed $\delta$, the radial bound supplied by Lemma~\ref{lemma:mu_rate} has order $(\log((8/\delta)\log n)/n)^{1/(2+\beta)}$ when $\beta>1$, $(\log n\,\log((8/\delta)\log n)/n)^{1/3}$ when $\beta=1$, and $(\log((8/\delta)\log n)/n)^{1/3}$ when $\beta<1$. Each divided by the proof's asserted $\log(1/\delta)n^{-1/(2+\beta)}$ tends to infinity. Thus the displayed conversion cannot be obtained by absorbing constants; the reproduced product bound must retain the available radial regimes and factors. minimal patch (confirmed): corrected statement: Suppose that Assumption \ref{assump:exo}, \ref{assump:overlap}, \ref{assump:iid}, \ref{assump:f_homo}, \ref{assump:mvary} hold, $\alpha < \beta$, where $\alpha$ and $\beta$ are defined in \cref{assump:f_homo} and \cref{assump:mvary} respectively. Suppose in addition that the conditions of Proposition~\ref{prop:id} hold. Suppose that the fitted propensity, regression and exponent are measurable with respect to the training split and independent auxiliary data, denoted together by $\mathcal T$, and that $\mathbb E[\|Y/\\|U\\|^{\hat\alpha_n}\|\mid\mathcal T,\\|U\\|>t]<\infty$ on each joint nuisance event. Furthermore, for any fixed $t$, with probability at least $1 - \delta$, \begin{align*} \|p (X) - \hat{p} (X) \| \leqslant R_p (n, \delta) & , \| \hat{\alpha}_{n} - \alpha\| \leqslant R_\alpha(n,\delta),\\ \|\mathbb{E}[Y/\\|U\\|^{\alpha} \mid X, D, U /\\|U\\|, \\|U\\|> t] &- \hat{g} (X, D, U /\\|U\\|) \| \leqslant R_g (n_t, \delta), \end{align*} where $n_t = \sum_{i = 1}^{n/2} I (\\|U_i \\|> t)$ and $R_p, R_g, R_\alpha$ are estimation errors that are monotonically decreasing with respect to sample size. Fix $\delta\in(0,1/2)$ and a deterministic threshold $t$. Run the adaptive Hill selection in Algorithm~\ref{alg:nete} at tolerance $\delta/6$, and write its full-input output as $\hat\gamma_n=\widetilde\gamma_n$. Put $\pi_t=\mathbb P(\\|U\\|>t)$, $M=\|\mathcal I\|$, $\bar e(t)=\sup_{r\geq t}e(r)$, and $G_t=C+\bar e(t)$, where $C$ bounds $\|g\|$ on its compact domain. Use zero for each first-factor estimate when $M=0$. For a generic evaluation observation put $S_U=U/\\|U\\|$ and define \[ \begin{aligned} \psi^{\mathrm{DR}}={}&\hat g(X,1,S_U)-\hat g(X,0,S_U)\\ &+\frac{D-\hat p(X)}{\hat p(X)(1-\hat p(X))}\left(\frac{Y}{\\|U\\|^{\hat\alpha_n}}-\hat g(X,D,S_U)\right),\\ \psi^{\mathrm{IPW}}={}&\frac{Y}{\\|U\\|^{\hat\alpha_n}}\left(\frac{D}{\hat p(X)}-\frac{1-D}{1-\hat p(X)}\right). \end{aligned} \] For each $q\in\{\mathrm{DR},\mathrm{IPW}\}$, choose $B_q>0$ using only $\mathcal T$, and put \[ \begin{aligned} v_t^q(B_q)&=\mathbb E[(\psi^q)^2I\{\|\psi^q\|\leq B_q\}\mid\mathcal T,\\|U\\|>t],\\ b_t^q(B_q)&=\mathbb E[\|\psi^q\|I\{\|\psi^q\|>B_q\}\mid\mathcal T,\\|U\\|>t],\\ \mathcal S_t^q&=\sqrt{\frac{12v_t^q(B_q)}{\delta\max\{M,1\}}}+\left(1+\frac{12}{\delta}\right)b_t^q(B_q). \end{aligned} \] Fix $a_0\in(0,\beta)$ and put \[ \begin{aligned} a&=R_\alpha(n,\delta/12),\qquad K=\frac{4}{\beta-a_0},\\ \mathcal A_t&=2LC_W\left((\log t)^k t^{-\min\{1,\beta\}}+t^{-\beta s/(1-2s)}\right),\\ \mathcal E_{\mathrm{DR}}={}&\mathcal S_t^{\mathrm{DR}}+\frac2c R_p(n/2,\delta/12)R_g(n_t,\delta/12)\\ &+2\bar e(t)+\mathcal A_t+\frac{2G_t}{c}(t^a-1+Kat^a),\\ \mathcal E_{\mathrm{IPW}}={}&\mathcal S_t^{\mathrm{IPW}}+\frac{2G_t}{c}R_p(n/2,\delta/12)\\ &+2\bar e(t)+\mathcal A_t+\frac{2G_t}{c}(t^a-1+Kat^a). \end{aligned} \] Suppose moreover, as in Lemma~\ref{lemma:eta_rate}, that the angular Wasserstein term satisfies the bound that Proposition 3.1 of \citet{zhang2023wasserstein} gives for this Pareto-type model, which is taken from that work and not proved here: there are constants $C_W>0$, $k\geq0$ and $t_W\geq1$ such that \[ W_1(\mathcal L_{U/\\|U\\|}^t,\mathcal L_{U/\\|U\\|})\leq C_W(\log t)^k t^{-\min\{1,\beta\}}+C_W t^{-\beta s/(1-2s)}\qquad(t\geq t_W). \] These are the constants $C_W$ and $k$ in the angular term $\mathcal A_t$ above. Let \[ b_n=C_{\delta/6} \begin{cases} \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/3},&0<\beta<1,\\ \left(\dfrac{\log n\,\log((48/\delta)\log n)}{n}\right)^{1/3},&\beta=1,\\ \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/(2+\beta)},&\beta>1, \end{cases} \] where $C_{\delta/6}$ is the constant supplied by Lemma~\ref{lemma:mu_rate}. Define \[ \begin{aligned} \gamma&=1/\beta,\qquad h=1-\alpha\gamma>0,\\ Q_n&=a(\gamma+b_n)+\|\alpha\|b_n,\qquad d_n=\|1-\hat\alpha_n\hat\gamma_n\|,\\ \mathcal H_n&= \begin{cases} Q_n/(h d_n),&d_n>0,\\ 1/h,&d_n=0. \end{cases} \end{aligned} \] Use Algorithm~\ref{alg:nete}'s convention $\hat\mu_n=0$ when $d_n=0$. For sufficiently large $n$ and $t\geq t_W$, if $a\leq a_0$ and $n\pi_t\geq16\log(12/\delta)$, then, with probability at least $1-\delta$, we have $M\geq n\pi_t/4$ and \[ \left\|\widehat\theta_{n,t}^{\mathrm{DR}}-\theta^{\tmop{NETE}}\right\| \leq \|\hat\mu_n\|\mathcal E_{\mathrm{DR}}+\|\eta\|\mathcal H_n, \] and \[ \left\|\widehat\theta_{n,t}^{\mathrm{IPW}}-\theta^{\tmop{NETE}}\right\| \leq \|\hat\mu_n\|\mathcal E_{\mathrm{IPW}}+\|\eta\|\mathcal H_n, \] where $\eta=\lim_{r\rightarrow\infty}\mathbb E[g(X,1,U/\\|U\\|)-g(X,0,U/\\|U\\|)\mid\\|U\\|>r]$. If moreover, for a constant $V<\infty$ independent of $n,t,\delta$ and every $q\in\{\mathrm{DR},\mathrm{IPW}\}$, $\mathbb E[(\psi^q)^2\mid\mathcal T,\\|U\\|>t]\leq V$ on the joint nuisance event, $t\geq\exp(1)$, $a\log t\leq1$ and $Q_n\leq h/2$, then on the same joint success event of probability at least $1-\delta$ we also have \[ \begin{aligned} \|\widehat\theta_{n,t}^{\mathrm{DR}}-\theta^{NETE}\|\leq C\biggl(&\sqrt{\frac{V}{\delta n\pi_t}}+R_p(n/2,\delta/12)R_g(n_t,\delta/12)\\ &+\bar e(t)+\mathcal A_t+R_\alpha(n,\delta/12)\log t+b_n\biggr),\\ \|\widehat\theta_{n,t}^{\mathrm{IPW}}-\theta^{NETE}\|\leq C\biggl(&\sqrt{\frac{V}{\delta n\pi_t}}+R_p(n/2,\delta/12)\\ &+\bar e(t)+\mathcal A_t+R_\alpha(n,\delta/12)\log t+b_n\biggr), \end{aligned} \] where $C$ does not depend on $n,t,\delta$, and $\pi_t\asymp t^{-\beta}$ makes the first term of order $\sqrt{V/\delta}\,t^{\beta/2}n^{-1/2}$. · proof change: Proposition~\ref{prop:id} applies because its identification and integrability conditions are assumed. The limiting angular contrast is $\eta$, and the limiting radial moment is \[ \mu=\int_1^\infty r^\alpha\beta r^{-\beta-1}\,\tmop{dr}=\frac{\beta}{\beta-\alpha}=\frac1h. \] The integral converges because $\alpha<\beta$. Consequently $\theta^{\tmop{NETE}}=\eta\mu$. Since $\|g\|\leq C$, we also have $\|\eta\|\leq2C$. For either estimator, the product identity gives \[ \widehat\theta_{n,t}^{\mathrm{estimator}}-\theta^{\tmop{NETE}} =\hat\mu_n(\hat\eta_{n,t}^{\mathrm{estimator}}-\eta)+\eta(\hat\mu_n-\mu). \] Let $E_\alpha$ be the joint nuisance event at tolerance $\delta/12$. The nuisance hypothesis gives $\mathbb P(E_\alpha^c)\leq\delta/12$, and on this event $\|\hat\alpha_n-\alpha\|\leq R_\alpha(n,\delta/12)=a$. Apply Lemma~\ref{lemma:eta_rate} at confidence tolerance $\delta/3$, using truncation level $B_{\mathrm{DR}}$. Its internal nuisance tolerance is $(\delta/3)/4=\delta/12$, its exponent bound is $a$, and its count requirement is $n\pi_t\geq16\log(12/\delta)$. It supplies an event $E_{\mathrm{DR}}$ with failure probability at most $\delta/3$ on which $M\geq n\pi_t/4$ and $\|\hat\eta_{n,t}^{\mathrm{DR}}-\eta\|\leq\mathcal E_{\mathrm{DR}}$. Apply Lemma~\ref{lemma:eta_ipw} at tolerance $\delta/3$ with $B_{\mathrm{IPW}}$; its sampling coefficients are $12/\delta$ and its nuisance and exponent arguments are also $\delta/12$. It supplies an event $E_{\mathrm{IPW}}$ with failure probability at most $\delta/3$ on which $M\geq n\pi_t/4$ and $\|\hat\eta_{n,t}^{\mathrm{IPW}}-\eta\|\leq\mathcal E_{\mathrm{IPW}}$. The use of $R_p(n/2,\delta/12)$ is permitted by its sample-size monotonicity, since $R_p(n,\delta/12)\leq R_p(n/2,\delta/12)$; no comparison between confidence tolerances is used. Lemma~\ref{lemma:mu_rate}, applied to the full-input Hill selection run at tolerance $\delta/6$, supplies an event $E_\gamma$ with failure probability at most $\delta/6$ on which $\|\hat\gamma_n-\gamma\|\leq b_n$. Thus \[ E=E_\alpha\cap E_{\mathrm{DR}}\cap E_{\mathrm{IPW}}\cap E_\gamma \] satisfies \[ \mathbb P(E^c)\leq\frac{\delta}{12}+\frac{\delta}{3}+\frac{\delta}{3}+\frac{\delta}{6}=\frac{11\delta}{12}\leq\delta. \] This intersection explicitly contains the exponent inequality as well as both first-factor inequalities and the radial inequality. Independence between these events is unnecessary. On $E$, we have $\|\hat\gamma_n\|\leq\gamma+b_n$ and hence \[ \|\hat\alpha_n\hat\gamma_n-\alpha\gamma\| =\|\hat\gamma_n(\hat\alpha_n-\alpha)+\alpha(\hat\gamma_n-\gamma)\| \leq a(\gamma+b_n)+\|\alpha\|b_n=Q_n. \] If $d_n>0$, exact reciprocal subtraction gives \[ \|\hat\mu_n-\mu\| =\frac{\|\hat\alpha_n\hat\gamma_n-\alpha\gamma\|}{h d_n} \leq\frac{Q_n}{h d_n}=\mathcal H_n. \] If $d_n=0$, Algorithm~\ref{alg:nete} sets $\hat\mu_n=0$, so \[ \|\hat\mu_n-\mu\|=1/h=\mathcal H_n. \] Thus $\|\hat\mu_n-\mu\|\leq\mathcal H_n$ throughout $E$, without any uniform Lipschitz assertion near the pole. On $E$, the DR product identity and the triangle inequality give \[ \begin{aligned} \|\widehat\theta_{n,t}^{\mathrm{DR}}-\theta^{\tmop{NETE}}\| &\leq\|\hat\mu_n\|\,\|\hat\eta_{n,t}^{\mathrm{DR}}-\eta\|+\|\eta\|\,\|\hat\mu_n-\mu\|\\ &\leq\|\hat\mu_n\|\mathcal E_{\mathrm{DR}}+\|\eta\|\mathcal H_n. \end{aligned} \] Here Lemma~\ref{lemma:eta_rate} supplies the score-dependent sampling term, the product $R_p(n/2,\delta/12)R_g(n_t,\delta/12)$, the tail envelope $2\bar e(t)$, the angular Wasserstein term and the full exponent perturbation. Its exponent calculation uses $\|\hat\alpha_n-\alpha\|\leq a$ on $E_\alpha$ and gives $\mathbb E[\\|U\\|^a-1\mid\\|U\\|>t]\leq t^a-1+Kat^a$ for $K=4/(\beta-a_0)$. No linearization of this expression, replacement of $\bar e(t)$ by $e(t)$, or suppression of angular or radial logarithms is made. The count inequality also holds on $E$. For IPW, the product identity gives \[ \widehat\theta_{n,t}^{\mathrm{IPW}}-\theta^{\tmop{NETE}} =\hat\mu_n(\hat\eta_{n,t}^{\mathrm{IPW}}-\eta)+\eta(\hat\mu_n-\mu). \] On $E$, Lemma~\ref{lemma:eta_ipw} gives $\|\hat\eta_{n,t}^{\mathrm{IPW}}-\eta\|\leq\mathcal E_{\mathrm{IPW}}$, and the reciprocal calculation gives $\|\hat\mu_n-\mu\|\leq\mathcal H_n$. Therefore \[ \begin{aligned} \|\widehat\theta_{n,t}^{\mathrm{IPW}}-\theta^{\tmop{NETE}}\| &\leq\|\hat\mu_n\|\,\|\hat\eta_{n,t}^{\mathrm{IPW}}-\eta\|+\|\eta\|\,\|\hat\mu_n-\mu\|\\ &\leq\|\hat\mu_n\|\mathcal E_{\mathrm{IPW}}+\|\eta\|\mathcal H_n. \end{aligned} \] Both bounds and $M\geq n\pi_t/4$ thus hold together with probability at least $1-\delta$. For the additional assertion, we choose the success events in the preceding proof using the constructions in Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw}. Let $E_M=\{M\geq n\pi_t/4\}$. Its failure probability is at most $\exp(-n\pi_t/16)\leq\delta/12$. For each $q\in\{\mathrm{DR},\mathrm{IPW}\}$ put $Z^q=\psi^q I\{\|\psi^q\|\leq B_q\}$ and $W^q=\psi^q I\{\|\psi^q\|>B_q\}$. Conditional on $\mathcal T$ and positive $M$, Chebyshev's and Markov's inequalities give two events $E_{Z,q}$ and $E_{W,q}$, each with conditional failure probability at most $\delta/12$, on which \[ \left\|M^{-1}\sum_{i\in\mathcal I}Z_i^q-\mathbb E[Z^q\mid\mathcal T,\\|U\\|>t]\right\|\leq\sqrt{\frac{12v_t^q(B_q)}{\delta M}},\qquad M^{-1}\sum_{i\in\mathcal I}\|W_i^q\|\leq\frac{12b_t^q(B_q)}\delta. \] If $b_t^q(B_q)=0$, the tail variables vanish almost surely. Take $E_{\mathrm{DR}}=E_\alpha\cap E_M\cap E_{Z,\mathrm{DR}}\cap E_{W,\mathrm{DR}}$ and $E_{\mathrm{IPW}}=E_\alpha\cap E_M\cap E_{Z,\mathrm{IPW}}\cap E_{W,\mathrm{IPW}}$. These events give exactly the first-factor conclusions used in the preceding proof, by the two cited lemmas, and each has failure probability at most $\delta/3$. Since their nuisance and count events are shared, their intersection with $E_\gamma$ actually has failure probability at most \[ \delta/12+\delta/12+4\delta/12+\delta/6=8\delta/12. \] On $E_\alpha$, the additional second-moment assumption gives $\tmop{Var}(\psi^q\mid\mathcal T,\\|U\\|>t)\leq V$. Conditional Chebyshev supplies an event $F_q$ whose failure on $E_\alpha\cap E_M$ has probability at most $\delta/12$ and on which \[ \left\|\hat\eta_{n,t}^q-\mathbb E[\psi^q\mid\mathcal T,\\|U\\|>t]\right\|\leq\sqrt{\frac{12V}{\delta M}}\leq\sqrt{\frac{48V}{\delta n\pi_t}}. \] The assertion follows directly if $V=0$, because the centered scores then vanish almost surely. Integrating the conditional failure bounds and using a union bound shows that $E_*=E\cap F_{\mathrm{DR}}\cap F_{\mathrm{IPW}}$ has failure probability at most $10\delta/12\leq\delta$. Thus both general bounds and both finite-variance bounds hold on this one event; it contains the exponent inequality explicitly. The finite-variance alternatives in Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} give the same conditional-mean bounds on $E_\alpha$, with this sampling term. Write $z=a\log t\in[0,1]$. Convexity of the exponential gives $\exp(z)-1\leq(\exp(1)-1)z$; also $t^a\leq\exp(1)$ and $a\leq a\log t$. Therefore \[ t^a-1+Kat^a\leq(\exp(1)-1+K\exp(1))a\log t. \] For sufficiently large $t$, $\bar e(t)\leq1$, so $G_t$ is bounded by the fixed constant $C+1$. On $E_*$, $d_n\geq h-Q_n\geq h/2>0$, and consequently \[ \|\hat\mu_n\|\leq2/h,\qquad \mathcal H_n\leq2Q_n/h^2, \qquad Q_n\leq\gamma a+(a_0+\|\alpha\|)b_n\leq C_1(a\log t+b_n). \] Here $C_1$ depends only on the fixed model and $a_0$. Substituting the DR mean bound and the finite-variance sampling bound in the DR product identity gives the additional DR display. Substituting the IPW mean bound and its finite-variance sampling bound in the IPW product identity gives the additional IPW display. The factors $2/h$, $\|\eta\|$, $2/c$, the bounded $G_t$ and the displayed exponent constant can all be absorbed into a constant independent of $n,t,\delta$. Finally the radial tail in the stated model is $\pi_t\asymp t^{-\beta}$, which gives the asserted order of the sampling term. The definition of $b_n$ and every nuisance tolerance remain exactly those displayed in the statement. · contribution narrowed (accepted by the owner): The unconditional claim under covariate-only exogeneity is lost, including cases where treatment and potential outcomes become dependent after conditioning on $U$. Fitted nuisances using evaluation observations and fitted-exponent pseudo-outcomes without finite conditional absolute means are excluded. The suggested first-factor bound requires $R_\alpha(n,\delta/12)\leq a_0<\beta$ and enough expected exceedances for its probability guarantee. A universal square-root sampling rate without outcome-tail control, a uniformly bounded reciprocal perturbation near its pole, the linear exponent term without control of $R_\alpha(n,\delta)\log t$, the pointwise homogeneity term $e(t)$, and the logarithm-free angular and radial rates are lost. The nuisance bounds are evaluated at the smaller confidence tolerance $\delta/12$. The appended square-root sampling and linear exponent bounds apply only to scores satisfying the stated conditional second-moment bound, thresholds $t\geq\exp(1)$ with $R_\alpha(n,\delta/12)\log t\leq1$, and perturbations satisfying $Q_n\leq h/2$; the general product bounds still retain the reciprocal denominator and full exponent perturbation outside those additional conditions. | | inherits refutation from | `lem:eta-ipw`, `lem:eta-rate`, `lem:mu-rate`, `prop:id` | ### 3. cor:rate `thm:cor-rate` | field | value | | --- | --- | | bucket | `verified_with_bookkeeping` | | source | `Arxiv_template.tex:363-386` (`corollary`, label `cor:rate`) | | statement | \label{cor:rate} Under the assumptions of Theorem~\ref{thm:rate}, further suppose that \begin{align*} R_p (n, \delta) = \Theta (\log (1 / \delta) n^{- 1 / 2}) , R_g (n, \delta) = \Theta (\log (1 / \delta) n^{- 1 / 2}), R_\alpha (n, \delta) = \Theta( \log(1/\delta) n^{-c_\alpha}), \end{align*} for some $c_\alpha > 0 $, the following conclusions hold. \begin{enumerate} \item If $s \in (0, 1 / (2 + \max \{1, \beta\}))$, takes $t_n = \Theta (n^{ (1 - 2 s) \hat{\gamma}_{n}})$, with probability at least $1 - \delta$, we have \begin{align*} \| {{\widehat{\theta }_{n, t}^{\text{DR} }} } - \theta^{NETE} \| & = O (e (t_n) + n^{- s} \log (1 / \delta) + n^{-c_\alpha} \log(n)\log (1 / \delta)) . \end{align*} \item If $s \in [1 / (2 + \max \{1, \beta\}), 1 / 2)$, \ takes $t = \Theta (n^{(\hat{\gamma}_{n} / (1 + 2 \min \{1, \hat{\gamma}_{n} \})})$, with probability at least $1 - \delta$, we have \begin{align*} \| {{\widehat{\theta }_{n, t}^{\text{DR} }} } - \theta^{NETE} \| & = O (e (t_n) + n^{- 1 / (2 + \max \{\beta, 1\})} \log (1 / \delta) + n^{-c_\alpha} \log(n)\log (1 / \delta)) . \end{align*} \end{enumerate} | | appended hypothesis list | ass:cor-rate-errors, ass:consistency, ass:exogeneity, ass:overlap, ass:iid-sampling, ass:iid-regular-variation, ass:iid-independence, ass:f-homo-bounded-x, ass:f-homo-asymptotic, ass:mvary-density, ass:mvary-matrix, ass:mvary-constants, ass:rate-alpha-less-beta, ass:rate-nuisance-errors | | Lean | `TCorRate.lean:27` — `Papers.TbsExtremeEffects.nete_dr_rate_adaptive_threshold` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-33`, `ass:pin-34`, `ass:pin-36`, `ass:pin-38`, `ass:pin-39`, `ass:pin-41`, `ass:pin-42`, `ass:pin-43`, `ass:pin-44`, `ass:pin-45`, `ass:pin-7`, `ass:pin-fix-0b814294`, `ass:pin-fix-10c2ec7b`, `ass:pin-fix-27ee3ba1`, `ass:pin-fix-6857bb4d`, `ass:pin-fix-8964f8dd`, `ass:pin-fix-8a129280`, `ass:pin-fix-8cd5c960`, `ass:pin-fix-8ef6fd29`, `ass:pin-fix-9567640d`, `ass:pin-fix-b3a62d30`, `ass:pin-fix-c3605ca3`, `ass:pin-fix-c4d18fdb`, `ass:pin-fix-c90a58f3`, `ass:pin-fix-d303828c`, `ass:pin-fix-ebdb71b8`, `ass:pin-fix-f18f3355` | | cited results | — | | assumed substrate gates | — | | findings | [note, V1.5] the corrected statement stands as one statement; the referee would state it as a general statement with corollaries: Separate the two rates into propositions under their respective ranges of $s$ and threshold rules; since neither is a general claim governing the other, the parent should be an umbrella. | ### 4. lemma:mu_rate `lem:mu-rate` | field | value | | --- | --- | | bucket | `repaired` | | source | `Arxiv_template.tex:607-616` (`lemma`, label `lemma:mu_rate`) | | statement | \label{lemma:mu_rate}Under Assumption \ref{assump:iid}, \ref{assump:mvary} with probability at least $1 - \delta$, for sufficiently large $n$, we have \begin{align*} \| \gamma - \hat{\gamma}_n \| & \leqslant O \left( \left( \frac{\log (2 / \delta)}{n} \right)^{1 / (2 + \beta)} \right), \end{align*} where $\gamma = 1 / \beta$ is the EVI of $U$. | | appended hypothesis list | ass:iid-sampling, ass:iid-regular-variation, ass:iid-independence, ass:mvary-density, ass:mvary-matrix, ass:mvary-constants | | Lean | `Helpers/HillRate.lean:40` — `Papers.TbsExtremeEffects.adaptive_hill_rate` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-33`, `ass:pin-36`, `ass:pin-38`, `ass:pin-fix-c4d18fdb` | | cited results | — | | assumed substrate gates | — | | findings | [refuted, V1.5] The core faithfully retains the assumptions and algorithm reference. (a) The unresolved readings are the outcome exponent versus a local tail-index meaning of $\alpha$, and the algorithm's adjacent-ratio statistic versus the proof's common-denominator Hill statistic. The input collection, admissible ranks, and calibration of $l_n$ and $r_n(\delta)$ are not reconciled. (b) Algorithm~\ref{alg:nete} compares candidates through $n$, although its evaluation collection contains at most $n/2$ observations; moreover, its displayed statistic telescopes and lacks the weights required for a Hill estimator. (c) The proof's comparison against every candidate index is not the cited adaptive selection rule and can leave the admissible set empty even after removing unavailable ranks. (d) The asserted second-order tail expansion, and hence $\eta(t)=O(t^{-1/\beta})$, fail at the admissible value $\beta=1$, where a logarithmic term is present. (e) Two additional proof slips can be closed locally: the proportional component tail should be transferred using the original product Pareto law, rather than the shifted component law; and differentiating the displayed survival expansion gives the density coefficient $D(1+1/\beta)$, not $D(1-1/\beta)$. These corrections do not resolve (b)--(d). (f) I also could not close the final removal of the adaptation factor: [Boucheron and Thomas, Corollary 3.12](https://arxiv.org/pdf/1503.05077) retains $\log((2/\delta)\log n)$, whereas the printed rate retains only $\log(2/\delta)$. witness: (b) Take independent Pareto type-II components $Z_1,Z_2$ with $\beta=1$, $U=Z$, $A=I_2$, $m=d=d_z=d_u=2$, $l=1/4$, $u=2$, $\sigma=1/8$, $s=1/4$, $\xi=1/2$, and $\zeta=1$, with independent observations and fixed threshold $t=1$. This exact product law satisfies every stated tail-model restriction. Nevertheless, $M=\|\mathcal I\|\leq n/2$, so the comparison at $i=n$ requires an unavailable order statistic. Independently, the algorithm's display equals $k^{-1}\log(\\|U_{(1)}\\|/\\|U_{(k+1)}\\|)$, whereas the proof's statistic equals $k^{-1}\sum_{j=1}^k j\log(\\|U_{(j)}\\|/\\|U_{(j+1)}\\|)$. They are different random variables. For the permitted choice $l_n=\lceil(\log n)^2\rceil$, every computable algorithmic candidate satisfies $0\leq\hat\gamma_k\leq\log(\max_i\\|U_i\\|/t)/l_n=O_{\mathbb P}(1/\log n)$, incompatible with $\gamma=1$. (c) On the same radial law, even using the proof's full-sample Hill statistic and deleting the unavailable endpoint, choose $l_n=\lceil(\log n)^2\rceil$ and $r_n(\delta)$ of the stated order. Then $\hat\gamma(l_n)\to1$ in probability, while $\hat\gamma(n-1)=\tfrac12\log n+O_{\mathbb P}(1)$ because the minimum radius is of order $n^{-1/2}$. Both comparison tolerances tend to zero. No candidate can satisfy comparisons with both indices, so the printed maximum is empty with probability tending to one. (d) Direct convolution on this exact product law gives $\bar F(t)=2/(t+2)+2\log(t+1)/(t+2)^2=2t^{-1}+(2\log t-4)t^{-2}+o(t^{-2})$. Thus the displayed constant-coefficient second-order expansion is false. Its tail quantile satisfies $U(q)=2q+\log(2q)-2+o(1)$, and the von Mises function has leading term $-\log q/(2q)$, which is not $O(q^{-1})$. This failure persists even under the favorable interpretation that the density calculation's $\alpha$ means $\beta$. minimal patch (confirmed): corrected statement: Under Assumption \ref{assump:iid}, \ref{assump:mvary}, using the standing convention $\\|\cdot\\|=\\|\cdot\\|_1$, let $R_i=\\|U_i\\|_1$ for all $i\in\{1,\ldots,n\}$ and order these full-input radii as $R_{[1]}\geq\cdots\geq R_{[n]}$. For $n\geq2$, define \[ H_n(k)=\frac1k\sum_{j=1}^k\log\frac{R_{[j]}}{R_{[k+1]}},\qquad 1\leq k\leq n-1. \] For $\delta\in(0,1)$, set \[ \begin{aligned} L_n&=\min\{n-1,\lceil100\log n\rceil\},&J_n&=1+\lceil\log n/\log2\rceil,\\ q_n(\delta)&=\log(4J_n/\delta),&c_n(\delta)&=2\sqrt{q_n(\delta)}+\frac{q_n(\delta)}{\sqrt{L_n}},\\ \rho_n(\delta)&=16c_n(\delta). \end{aligned} \] Let \[ \begin{aligned} \widetilde k_n&=\max\bigl\{k\in\{L_n,\ldots,n-1\}:\\ &\qquad \|H_n(i)-H_n(k)\|\leq\frac{H_n(i)\rho_n(\delta)}{\sqrt i}\\ &\qquad \text{for every }i\in\{L_n,\ldots,k\}\bigr\},\\ \widetilde\gamma_n&=H_n(\widetilde k_n). \end{aligned} \] For each fixed $\delta\in(0,1)$ and every sufficiently large $n$, with probability at least $1-\delta$, we have \[ \|\gamma-\widetilde\gamma_n\|\leq C_\delta \begin{cases} \left(\dfrac{\log((8/\delta)\log n)}{n}\right)^{1/3},&0<\beta<1,\\ \left(\dfrac{\log n\,\log((8/\delta)\log n)}{n}\right)^{1/3},&\beta=1,\\ \left(\dfrac{\log((8/\delta)\log n)}{n}\right)^{1/(2+\beta)},&\beta>1, \end{cases} \] where $C_\delta$ may depend on the fixed distribution and on $\delta$. where $\gamma = 1 / \beta$ is the EVI of $U$. · proof change: We adopt the non-asymptotic analysis of the adaptive Hill estimator for EVI in {\citep{boucheron2015tail}}. We give the exponential-spacing argument and the confidence calibration explicitly. Write $R=\\|U\\|_1$. Fix a member of $M_K$, put $a_i=\\|A_{\cdot i}\\|_1$, and let $Z_i^0$ be independent nominal Pareto type-II components of index $\beta$. The component density is a positive constant $c_0$ times the nominal product density whenever $\\|z\\|_1>\zeta K^{(1-2s)/\beta}$. Since \[ R=\sum_{i=1}^m a_iZ_i,\qquad l\leq a_i\leq u,\qquad R\leq u\\|Z\\|_1, \] the survival and density of $R$ beyond $u\zeta K^{(1-2s)/\beta}$ are $c_0$ times those of $T=\sum_i a_iZ_i^0$. This transfer uses the original nominal components without shifting their supports. Put $W_i=a_iZ_i^0$, $V_i=\sum_{j\ne i}W_j$, and $f_i(x)=\beta a_i^\beta(a_i+x)^{-\beta-1}$ for $x\geq0$, extended by zero for $x<0$. Partitioning the convolution integral according to the largest summand, whose ties have probability zero, gives \[ f_T(r)=\sum_i\mathbb E\left[f_i(r-V_i)I\{\max_{j\ne i}W_j\leq r-V_i\}\right]. \] On this event $r-V_i\geq r/m$. If $V_i\leq r/2$, the event holds and the mean value theorem gives $\|f_i(r-V_i)-f_i(r)\|\leq C r^{-\beta-2}V_i$. On the part where $V_i>r/2$, the indicated density is at most $C r^{-\beta-1}$. A union bound using the tails of the $m-1$ summands gives $\mathbb P(V_i>x)\leq Cx^{-\beta}$ for $x\geq1$. Integrating this bound and using $\mathbb E[V_iI\{V_i\leq r/2\}]\leq\mathbb E[\min\{V_i,r/2\}]$ gives \[ \mathbb E[V_iI\{V_i\leq r/2\}]\leq \begin{cases}Cr^{1-\beta},&\beta<1,\\ C\log r,&\beta=1,\\ C,&\beta>1.\end{cases} \] Consequently \[ \left\|f_T(r)-\sum_i f_i(r)\right\| \leq C\sum_i\left(r^{-\beta-2}\mathbb E[V_iI\{V_i\leq r/2\}]+r^{-\beta-1}\mathbb P(V_i>r/2)\right). \] Since $\sum_i f_i(r)=\beta\sum_i a_i^\beta r^{-\beta-1}(1+O(r^{-1}))$, set \[ b_\beta(r)=\begin{cases}r^{-\beta},&\beta<1,\\r^{-1}\log r,&\beta=1,\\r^{-1},&\beta>1,\end{cases}\qquad C_0=c_0\sum_i a_i^\beta>0. \] The actual radial density and survival therefore satisfy, for sufficiently large $r$, \[ f_R(r)=\beta C_0r^{-\beta-1}(1+O(b_\beta(r))),\qquad \bar F_R(r)=C_0r^{-\beta}(1+O(b_\beta(r))). \] The second estimate follows by integrating the first error from $r$ to infinity: the integrals of $r^{-2\beta-1}$, $r^{-3}\log r$, and $r^{-\beta-2}$ have orders $r^{-2\beta}$, $r^{-2}\log r$, and $r^{-\beta-1}$, respectively. Thus no constant-coefficient expansion at $\beta=1$ is required. When $\beta>1$ and a survival expansion with coefficient $D$ is used, differentiation gives density coefficient $D(1+1/\beta)$; the density and survival envelopes just proved avoid differentiating an unspecified remainder. Let $Q(v)=F_R^{-1}(1-1/v)$ be the radial tail quantile. Its differentiable tail has $Q(v)\sim(C_0v)^{1/\beta}$ and \[ \eta(v)=\frac{vQ'(v)}{Q(v)}-\gamma =\frac{\bar F_R(Q(v))}{Q(v)f_R(Q(v))}-\frac1\beta =O(b_\beta(Q(v))). \] Choose a fixed $v_0>1$ in that tail and define $\bar\eta(v)=\sup_{w\geq\max\{v,v_0\}}\|\eta(w)\|$ for every $v>0$. For sufficiently large $v$ this gives \[ \bar\eta(v)\leq \begin{cases}Cv^{-1},&\beta<1,\\Cv^{-1}\log v,&\beta=1,\\Cv^{-1/\beta},&\beta>1.\end{cases} \] For $y\geq z\geq\log v_0$, integrating the quantile derivative gives \[ \|\log Q(e^y)-\log Q(e^z)-\gamma(y-z)\|\leq\bar\eta(e^z)(y-z). \] We next prove the simultaneous stochastic bound used for selection. Let $Y_{(1)}\geq\cdots\geq Y_{(n)}$ be decreasing order statistics of independent standard exponential variables, with $Y_{(n+1)}=0$. Their joint density is $n!\exp(-\sum_j y_j)$ on the ordered region. The change of variables $E_j=j(Y_{(j)}-Y_{(j+1)})$ has Jacobian $1/n!$ and $\sum_j y_j=\sum_j E_j$, so the $E_j$ are independent standard exponential variables. The quantile transform represents the ordered radii as $Q(e^{Y_{(j)}})$. Define $B_i=\sum_{j=1}^i(E_j-1)$ and $A_i=i^{-1}\sum_{j=1}^iE_j$ for this argument only. For $0<\lambda<1$, \[ \log\mathbb E e^{\lambda(E_j-1)}=-\lambda-\log(1-\lambda)\leq\frac{\lambda^2}{2(1-\lambda)},\qquad \log\mathbb E e^{-\lambda(E_j-1)}=\lambda-\log(1+\lambda)\leq\frac{\lambda^2}{2(1-\lambda)}. \] Hence $\exp(\pm\lambda B_i-i\lambda^2/(2(1-\lambda)))$ are nonnegative supermartingales with initial value one. The maximal supermartingale inequality implies \[ \mathbb P\left(\max_{i\leq m}\|B_i\|\geq\sqrt{2mx}+x\right)\leq2e^{-x}. \] Indeed, each one-sided probability is at most $\exp(-\lambda a+m\lambda^2/(2(1-\lambda)))$ for $a=\sqrt{2mx}+x$; taking $\lambda=\sqrt{2x/m}/(1+\sqrt{2x/m})$ makes this bound $e^{-x}$. Apply this inequality at the ends of the dyadic intervals $2^a\leq i<2^{a+1}$, using $m=\min\{n,2^{a+1}\}$ and $x=q_n(\delta)$. There are at most $J_n$ such intervals. On each interval $\sqrt{2m}/\sqrt i\leq2$. The union bound therefore supplies an event $E$ of probability at least $1-\delta/2$ on which \[ \sqrt i\,\|A_i-1\|\leq2\sqrt{q_n(\delta)}+q_n(\delta)/\sqrt i\leq c_n(\delta),\qquad L_n\leq i\leq n. \] For fixed $\delta$, $c_n(\delta)=O_\delta(\sqrt{\log\log n})$ and $c_n(\delta)/\sqrt{L_n}\to0$. Write $c_n=c_n(\delta)$ and consider the deterministic index \[ k_* =\max\left\{k\in\{L_n,\ldots,n-1\}:\ \frac{n}{2k}\geq v_0,\quad \sqrt k\,\bar\eta\left(\frac{n}{2k}\right)\leq\gamma c_n\right\}. \] The choices below show that this set is nonempty for all sufficiently large $n$. Put $T_*=e^{Y_{(k_*+1)}}$. The number of exponential observations exceeding $\log(n/(2k_*))$ has binomial law with mean $2k_*$. Its multiplicative lower-tail bound gives \[ \mathbb P\left(T_*<\frac{n}{2k_*}\right)\leq\exp(-k_*/4)\leq\delta/2 \] for sufficiently large $n$, because $k_*\geq L_n$. Indeed, if $B\sim\tmop{Bin}(n,p)$ and $\mu=np$, Markov's inequality gives \[ \mathbb P(B\leq\mu/2)\leq e^{\mu\log2/2}\mathbb E[2^{-B}] =e^{\mu\log2/2}(1-p/2)^n \leq e^{-\mu(1-\log2)/2}\leq e^{-\mu/8}, \] using $\log2\leq3/4$; here $p=2k_*/n$ and $\mu=2k_*$. Let $E_*$ be the complementary event. On $E_*$ all spacings involved in $H_n(i)$ for $i\leq k_*$ lie above $\log v_0$. Since $H_n(i)=i^{-1}\sum_{j=1}^i j\log(R_{[j]}/R_{[j+1]})$ by telescoping, the quantile inequality thus gives \[ \|H_n(i)-\gamma A_i\|\leq\bar\eta(T_*)A_i\leq\frac{\gamma c_n}{\sqrt{k_*}}A_i,\qquad L_n\leq i\leq k_*. \] On $E\cap E_*$, for large enough $n$, $c_n/\sqrt{L_n}\leq1/6$. It follows that \[ \|H_n(i)-\gamma\|\leq\frac{\gamma c_n}{\sqrt i}+\frac{\gamma c_n}{\sqrt{k_*}}\left(1+\frac{c_n}{\sqrt i}\right) \leq\frac{3\gamma c_n}{\sqrt i},\qquad \gamma/2\leq H_n(i)\leq3\gamma/2. \] For every $L_n\leq i\leq k\leq k_*$, \[ \|H_n(i)-H_n(k)\|\leq\frac{6\gamma c_n}{\sqrt i} \leq\frac{H_n(i)\rho_n(\delta)}{\sqrt i}. \] Thus $k_*$ qualifies for the displayed prefix rule, and its selected index $\widetilde k_n$ is at least $k_*$. The set in that rule is nonempty almost surely because $k=L_n$ only compares with itself. The comparison at $i=k_*$ now yields \[ \|\widetilde\gamma_n-\gamma\|\leq\|H_n(k_*)-\gamma\|+\frac{H_n(k_*)\rho_n(\delta)}{\sqrt{k_*}} \leq\frac{27\gamma c_n}{\sqrt{k_*}}. \] The two events used have total failure probability at most $\delta$; independence between them is unnecessary. To obtain the displayed rates, when $\beta\ne1$ put $p=\min\{1,1/\beta\}$ and choose a sufficiently small fixed $a_*>0$. Let \[ k_0=\left\lfloor a_* n^{2p/(1+2p)}c_n^{2/(1+2p)}\right\rfloor. \] For all sufficiently large $n$, $k_0\geq L_n$, $k_01$ it is $1/(2+\beta)$. When $\beta=1$, take \[ k_0=\left\lfloor a_*n^{2/3}\left(\frac{c_n}{\log n}\right)^{2/3}\right\rfloor. \] Again $k_0\geq L_n$ and $k_0t_j]=\Theta((\log t_j)^{-1})$. Taking even $n_j$ of order $t_j^{10}$ makes sampling error negligible; the printed right-hand side is $O(t_j^{-1})$, whereas the estimator's bias has order $(\log t_j)^{-1}$. The step bounding the homogeneity contribution by $2e(t)$ is false because the assumption bounds the error at each realized radius separately. (c) Set $\beta=1$, $\alpha=1/2$, $s=2/5$, $A=I_2$, and $Y(d)=d\\|U\\|^{1/2}[1+\min\{U_1,U_2\}/\\|U\\|]$. Then $e(t)=0$, $g(x,1,s)=1+\min\{s_1,s_2\}$, $g(x,0,s)=0$, and the limiting spectral measure is supported on the two coordinate axes, giving $\eta=1$. Oracle nuisances make the score exactly $1+\min\{U_1,U_2\}/\\|U\\|$. Direct integration of the product density gives $\mathbb E[\min\{U_1,U_2\}/\\|U\\|\mid\\|U\\|>t]=\Theta((\log t)/t)$. For example, integrating over $1\leq Z_1\leq t/2$ and $t\leq Z_2\leq2t$ gives a lower bound of this order, since $\mathbb P(\\|U\\|>t)\sim2/t$. With $n$ of order $t^5$, sampling error is $O(t^{-2})$ and the printed bound is $O(t^{-1})$. The actual angular bias therefore violates the conclusion; $\beta=1$ is an interior value of the stated tail-index domain. (d) Set $\beta=3$, $\alpha=1$, $s=1/4$, $A=I_2$, and $Y(d)=d\\|U\\|$. Let $\kappa_n=(\log(n+\exp(1)))^{-1/2}$, $\hat\alpha_n=1-\kappa_n$, and $R_\alpha(n,\delta)=\kappa_n$. Use $\hat p=1/2$ and $\hat g(x,d,s)=d$, obtainable by regression over that singleton class; the displayed nuisance assumption concerns the true-exponent regression and is satisfied exactly. The score is $1+2D(\\|U\\|^{\kappa_n}-1)$, with tail expectation asymptotic to $3t^{\kappa_n}/(3-\kappa_n)$. At $t=n^{1/5}$, its bias grows as $\exp(\sqrt{\log n}/5)$, whereas the printed exponent term is only $O(\sqrt{\log n})$ and sampling fluctuations are negligible relative to the bias. Here $R_\alpha\to0$ and the relevant radial moments are finite. The false step linearizes $\\|U\\|^{\alpha-\hat\alpha_n}$ despite an unbounded product $R_\alpha\log t$. (e) Set $\beta=3$, $\alpha=1$, $s=1/4$, $A=I_2/2$, and $Y(d)=d\\|U\\|$. Let $q_j=2^{4^j}$ and define a bounded nonincreasing function $b(q)=q_j^{-1/4}$ on $4q_{j-1}t)\sim t^{-3}/4$. Thus $b(n_t)=q_j^{-1/4}$ with probability tending to one, and the conditional DR mean is $1+b(n_t)/6$. But $n_jt_j^{-3}=8q_j$, where $b(8q_j)=q_{j+1}^{-1/4}=q_j^{-1}$. Every term in the claimed bound is $O(q_j^{-1/2})$, while the actual bias is $q_j^{-1/4}/6$. This disproves the replacement of the argument of $R_g$ using monotonicity alone. (f) Finally, take $\beta=3$, $\alpha=1$, $s=1/4$, $A=I_2$, and $Y(d)=\\|U\\|(d+\epsilon)$ with independent standard normal $\epsilon$. Fix $\delta=1/10$. Let $B$ be the event that the first training observation has covariate below $1/10$. On $B^c$, fit the oracle nuisances; on $B$, fit $\hat p=3/5$, $\hat g(x,1,s)=2$, and $\hat g(x,0,s)=0$. At this confidence tolerance, take the small positive decreasing nuisance bounds above: their joint guarantee holds with probability exactly $9/10$. For smaller tolerances, bounds $1/10+(1+n)^{-10}$ and $1+(1+q)^{-10}$ cover the bad fit surely, so the assumption holds for every tolerance. At $t=n^{1/5}$, the bad-fit estimator converges to $1+1/6$, while on $B^c$ the scaled error $n^{1/5}(\hat\eta_{n,t}^{\tmop{DR}}-1)$ converges to a nondegenerate normal law. The printed right-hand side is a fixed multiple of $n^{-1/5}$. For every finite such multiple, limiting coverage is strictly below $9/10$: the bad branch fails, and the good branch has positive Gaussian failure probability. Consequently the same-$\delta$ confidence conclusion does not follow from the stated nuisance guarantee. minimal patch (confirmed): corrected statement: Under the assumption of Theorem \ref{thm:rate}, suppose in addition that the conditions of Proposition~\ref{prop:id} hold. Suppose that the fitted propensity, regression, and exponent are measurable with respect to the training split and independent auxiliary data. Suppose that the nuisance inequalities of Theorem~\ref{thm:rate} hold jointly with probability at least $1-\varepsilon$, uniformly almost surely over the covariate and direction support and both treatments, for each tolerance $\varepsilon$. Suppose that $0t)$ and $M=\|\mathcal I\|$, and set $\hat\eta_{n,t}^{\tmop{DR}}=0$ when $M=0$. Let $\bar e(t)=\sup_{r\geq t}e(r)$, let $C$ bound $\|g\|$ on its compact domain, and put $G_t=C+\bar e(t)$. For a generic evaluation observation define \[ \psi=\hat g(X,1,U/\\|U\\|)-\hat g(X,0,U/\\|U\\|)+\frac{D-\hat p(X)}{\hat p(X)(1-\hat p(X))}\left(\frac{Y}{\\|U\\|^{\hat\alpha_n}}-\hat g(X,D,U/\\|U\\|)\right). \] For any positive truncation level $B$ chosen using only the training split and independent auxiliary data, define \[ \begin{aligned} v_t(B)&=\mathbb E[\psi^2 I\{\|\psi\|\leq B\}\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t],\\ a_t(B)&=\mathbb E[\|\psi\|I\{\|\psi\|>B\}\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t]. \end{aligned} \] Fix $a_0\in(0,\beta)$ and write $a=R_\alpha(n,\delta/4)$. For sufficiently large $t\geq1$, if $a\leq a_0$ and $n\pi_t\geq16\log(4/\delta)$, then with probability at least $1-\delta$ we have $M\geq n\pi_t/4$ and \begin{align*} \|\hat\eta_{n,t}^{\tmop{DR}}-\eta\|\leq{}&\sqrt{\frac{4v_t(B)}{\delta M}}+\left(1+\frac4\delta\right)a_t(B)+\frac2c R_p(n/2,\delta/4)R_g(n_t,\delta/4)\\ &+2\bar e(t)+2L W_1(\mathcal L_{U/\\|U\\|}^t,\mathcal L_{U/\\|U\\|})+\frac{2G_t}{c}\left(t^a-1+K a t^a\right), \end{align*} where $K$ depends on the radial law and $a_0$. Suppose moreover that the angular Wasserstein term satisfies the bound that Proposition 3.1 of \citet{zhang2023wasserstein} gives for this Pareto-type model, which is taken from that work and not proved here: there are constants $C_W>0$, $k\geq0$ and $t_W\geq1$ such that \[ W_1(\mathcal L_{U/\\|U\\|}^t,\mathcal L_{U/\\|U\\|})\leq C_W(\log t)^k t^{-\min\{1,\beta\}}+C_W t^{-\beta s/(1-2s)}\qquad(t\geq t_W). \] Then for $t\geq t_W$ the angular term $2L W_1(\mathcal L_{U/\\|U\\|}^t,\mathcal L_{U/\\|U\\|})$ of the display is at most $2LC_W(\log t)^k t^{-\min\{1,\beta\}}+2LC_W t^{-\beta s/(1-2s)}$, with the logarithmic factor retained. When the conditional score variance $V_t$ is finite, the sampling term may instead be taken to be $\sqrt{4V_t/(\delta M)}$. · proof change: Condition on the training split and independent auxiliary data, denoted together by $\mathcal T$. Write $R_U=\\|U\\|$, $S_U=U/\\|U\\|$, and \[ m_d^t(x,s)=\mathbb E[Y/R_U^{\alpha}\mid X=x,D=d,S_U=s,R_U>t],\qquad \eta_\alpha^t=\mathbb E[(f(X,1,U)-f(X,0,U))/R_U^{\alpha}\mid R_U>t]. \] Independence of $U$ from $(X,D)$ gives $\mathbb E[D\mid X,S_U,R_U>t]=p(X)$. Let $\psi_\alpha$ be the score $\psi$ with $\hat\alpha_n$ replaced by $\alpha$, keeping the same fitted functions. Direct conditional expectation over treatment and outcome gives \begin{align*} \mathbb E[\psi_\alpha\mid\mathcal T,R_U>t]-\eta_\alpha^t=\mathbb E\left[(p(X)-\hat p(X))\left\{\frac{m_1^t(X,S_U)-\hat g(X,1,S_U)}{\hat p(X)}+\frac{m_0^t(X,S_U)-\hat g(X,0,S_U)}{1-\hat p(X)}\right\}\mid\mathcal T,R_U>t\right]. \end{align*} Indeed, the coefficients of $m_1^t-\hat g_1$ and $m_0^t-\hat g_0$ are respectively $p/\hat p-1$ and $1-(1-p)/(1-\hat p)$, which equal $(p-\hat p)/\hat p$ and $(p-\hat p)/(1-\hat p)$. On the joint nuisance event at tolerance $\delta/4$, the uniform bounds and estimated-propensity overlap therefore imply \[ \|\mathbb E[\psi_\alpha\mid\mathcal T,R_U>t]-\eta_\alpha^t\|\leq\frac2c R_p(n/2,\delta/4)R_g(n_t,\delta/4). \] where $n_t = \sum_{i = 1}^{n/2} I (\\|U_i \\|> t)$ is a random variable. Keep this training count as the argument of $R_g$; no regularity of that error function beyond its stated monotonicity is needed. Homogeneity at each realized radius gives \[ \|\eta_\alpha^t-\mathbb E[g(X,1,S_U)-g(X,0,S_U)\mid R_U>t]\|\leq2\bar e(t). \] The function $s\mapsto\mathbb E_X[g(X,1,s)-g(X,0,s)]$ is $2L$-Lipschitz. Independence of $X$ and $U$ and the definition of the limiting angular target therefore give \[ \|\eta_\alpha^t-\eta\|\leq2\bar e(t)+2L W_1(\mathcal L_{U/\\|U\\|}^t,\mathcal L_{U/\\|U\\|}). \] Proposition 3.1 of \cite{zhang2023wasserstein}, with Wasserstein order one and tail index $\beta$, supplies the stated angular bound with its logarithmic factors. Next, the fitted functions cancel from $\psi-\psi_\alpha$. Taking conditional expectation over the outcome and treatment leaves \[ \mathbb E\left[\left\{\frac{p(X)}{\hat p(X)}\frac{f(X,1,U)}{R_U^{\alpha}}-\frac{1-p(X)}{1-\hat p(X)}\frac{f(X,0,U)}{R_U^{\alpha}}\right\}\left(R_U^{\alpha-\hat\alpha_n}-1\right)\mid\mathcal T,R_U>t\right]. \] On the nuisance event, $\|\alpha-\hat\alpha_n\|\leq a$. Since $R_U\geq t\geq1$, we have $\|R_U^{\alpha-\hat\alpha_n}-1\|\leq R_U^a-1$. Homogeneity also gives $\|f(X,d,U)/R_U^{\alpha}\|\leq G_t$, so \[ \|\mathbb E[\psi-\psi_\alpha\mid\mathcal T,R_U>t]\|\leq\frac{2G_t}{c}\mathbb E[R_U^a-1\mid R_U>t]. \] Choose $\epsilon=(\beta-a_0)/2>0$. Potter's bound gives, uniformly for $r\geq1$ and sufficiently large $t$, \[ \frac{\mathbb P(R_U>tr)}{\pi_t}\leq2r^{-\beta+\epsilon}. \] Integrating the radial tail, for $0\leq a\leq a_0$, yields \begin{align*} \mathbb E[R_U^a-1\mid R_U>t]&=t^a-1+t^a\int_1^{\infty}a r^{a-1}\frac{\mathbb P(R_U>tr)}{\pi_t}\,\tmop{dr}\\ &\leq t^a-1+\frac{2a t^a}{\beta-\epsilon-a}\leq t^a-1+K a t^a. \end{align*} This also covers $a=0$, when both sides vanish. Combining the three mean-error bounds controls $\|\mathbb E[\psi\mid\mathcal T,R_U>t]-\eta\|$ by the nonsampling terms in the statement. For evaluation, $M$ is binomial with parameters $n/2$ and $\pi_t$, independent of the training split and auxiliary data. The binomial lower-tail bound gives \[ \mathbb P(MB\}$. Chebyshev's inequality gives \[ \mathbb P\left(\left\|\frac1M\sum Z_i-\mathbb E[Z\mid\mathcal T,R_U>t]\right\|>\sqrt{\frac{4v_t(B)}{\delta M}}\ \middle\|\ \mathcal T,M\right)\leq\delta/4. \] Markov's inequality gives \[ \mathbb P\left(\frac1M\sum\|W_i\|>\frac{4a_t(B)}\delta\ \middle\|\ \mathcal T,M\right)\leq\delta/4; \] if $a_t(B)=0$, the tail variables vanish almost surely and this assertion holds directly. On the intersection of these events, the triangle inequality gives \[ \left\|\frac1M\sum\psi_i-\mathbb E[\psi\mid\mathcal T,R_U>t]\right\|\leq\sqrt{\frac{4v_t(B)}{\delta M}}+\left(1+\frac4\delta\right)a_t(B). \] The joint nuisance event fails with probability at most $\delta/4$. Adding its failure probability, the count failure probability, and the two conditional sampling failure probabilities gives total failure probability at most $\delta$. The estimator is the average of these scores on the resulting event, which proves the bound. If $V_t<\infty$, apply Chebyshev directly to the untruncated score average with failure probability $\delta/4$; this gives the stated replacement sampling term, with total failure probability still at most $\delta$. · narrowed: A finite universal square-root rate for all integrable scores, bounds using only $e(t)$, a logarithm-free angular bias at $\beta=1$, uniform linear dependence on $R_\alpha\log t$, replacement of the random training count inside arbitrary decreasing $R_g$, and confidence $1-\delta$ with unchanged nuisance arguments. The suggested conditional evaluation argument also excludes fits that use the evaluation observations and fits whose nuisance guarantees hold only at a random evaluation point; its nonvacuous bound requires an integrable fitted-exponent pseudo-outcome and an exponent-error bound smaller than $\beta$. Cases in which conditional mean exogeneity given $(X,U)$ fails, or in which either potential outcome has an infinite absolute tail mean, are also excluded. The rate of the angular Wasserstein term is taken from Proposition 3.1 of \citet{zhang2023wasserstein} as an explicit assumption, with its poly-logarithmic factor, and is not proved. | | inherits refutation from | `prop:id` | ### 6. lemma:eta_ipw `lem:eta-ipw` | field | value | | --- | --- | | bucket | `repaired` | | source | `Arxiv_template.tex:787-795` (`lemma`, label `lemma:eta_ipw`) | | statement | \label{lemma:eta_ipw} Under the assumption of Theorem \ref{thm:rate}, with probability at least $1 - \delta$, we have \begin{align*} \| \hat{\eta}_{n, t}^{\text{IPW}} - \eta \| & \leqslant O (R_p (n/2, \delta) + t^{\beta / 2} n^{- 1 / 2} + t^{- \min \{1, \beta\}} + t^{- \beta s / (1 - 2 s)}) + e (t) . \end{align*} | | appended hypothesis list | ass:exogeneity, ass:overlap, ass:iid-sampling, ass:iid-regular-variation, ass:iid-independence, ass:f-homo-bounded-x, ass:f-homo-asymptotic, ass:mvary-density, ass:mvary-matrix, ass:mvary-constants, ass:rate-alpha-less-beta, ass:rate-nuisance-errors | | Lean | `Helpers/EtaIPW.lean:411` — `Papers.TbsExtremeEffects.eta_ipw_rate` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity pinned · correctness ok | | added conditions | `ass:pin-33`, `ass:pin-34`, `ass:pin-36`, `ass:pin-38`, `ass:pin-39`, `ass:pin-42`, `ass:pin-43`, `ass:pin-44`, `ass:pin-45`, `ass:pin-7`, `ass:pin-fix-0b814294`, `ass:pin-fix-10c2ec7b`, `ass:pin-fix-8964f8dd`, `ass:pin-fix-8a129280`, `ass:pin-fix-8cd5c960`, `ass:pin-fix-8ef6fd29`, `ass:pin-fix-9567640d`, `ass:pin-fix-b3a62d30`, `ass:pin-fix-c3605ca3`, `ass:pin-fix-ebdb71b8`, `ass:pin-fix-f18f3355` | | cited results | — | | assumed substrate gates | — | | findings | [note, V1.5] the corrected statement stands as one statement; the referee would state it as a general statement with corollaries: Keep the truncated-score IPW bound as the main statement, and separate its angular-bound specialization and finite-variance sampling alternative into blocks under their respective additional conditions. ／ [note, V1.5] pinned reading: For each fixed $\mathcal T$-measurable choice $B>0$, the displayed probability bound uses the base conditions of Lemma~\ref{lemma:eta_rate}; its angular specialization additionally uses that lemma's angular assumption and $t\geq t_W$, and its variance alternative additionally requires $V_t^{\text{IPW}}<\infty$. (The statement explicitly says ``Under the angular assumption'' and ``When'' the variance is finite; the amended proof applies conditional sampling inequalities to the chosen $B$.) ／ [refuted, V1.5] The lint warnings are dismissed: ``Under the assumption of Theorem~\ref{thm:rate}'' imports its hypotheses, not its conclusions. The core correctly wires each listed assumption directly, and no dependency on the theorem's conclusion is required. The model-class readings $\mathcal L(X)$ versus $\mathcal L(U)$ and unspecified $m$ versus $m=d_z$ remain unresolved. Reproducing the IPW conditional mean gives propensity error, exponent perturbation, radial approximation bias, angular bias, and sampling fluctuation. (a) The displayed conclusion drops the exponent term retained by its own proof. (b) The cited sampling bound requires outcome-tail conditions absent here; [Su et al., Theorem 1](https://arxiv.org/html/2303.17102v1) assumes outcome moment bounds and a specified logistic estimation framework. (c) Referencing Lemma~\ref{lemma:eta_rate} does not permit replacing its radial error envelope by $e(t)$. (d) Its angular bound retains a logarithm at $\beta=1$, which the displayed conclusion omits. witness: For every example take $m=d=d_z=d_u=2$, $A=I_2$, independent nominal Pareto type-II components, $l=1/2$, $u=2$, $\sigma=1/2$, $s=2/5$, $\xi=1/2$, $\zeta=1$, independent $X\sim\text{Unif}([0,1])$, independent $D\sim\text{Ber}(1/2)$, and $p=\hat p=1/2$. Use i.i.d. observations and training-measurable nuisance fits; singleton regression classes realize the specified deterministic fits. Fix $\delta=1/4$. All potential outcomes below are independent of treatment. (a) Take $\beta=2$, $\alpha=0$, $Y(d)=d$, $g=\hat g=d$, $e=0$, and $\hat\alpha_n=1/\log n$. The nuisance inequalities hold with $R_\alpha(n,\delta)=1/\log n$ and arbitrarily small propensity and regression bounds. At deterministic $t=n^{1/6}$, the IPW average converges in probability to $\exp(-1/6)$, since its mean is $\mathbb E[\\|U\\|^{-1/\log n}\mid\\|U\\|>n^{1/6}]$. But $\eta=1$ and every term in the printed bound tends to zero. The omitted term in the conditional mean is $\mathbb E[\\|U\\|^{\alpha-\hat\alpha_n}-1\mid\\|U\\|>t]$. (b) Take $\beta=2$, $\alpha=\hat\alpha_n=0$, $Y(d)=d+E$, $g=\hat g=d$, and $e=0$, with independent centered symmetric stable noise $E$ of index $6/5$. All conditional absolute means are finite and nuisance errors are exact. The IPW score is $2D+2(2D-1)E$. Conditional on $M=\|\mathcal I\|$, its stable fluctuation has scale $M^{-1/6}$, not $M^{-1/2}$. At $t=n^{1/6}$ it has scale $n^{-1/9}$ while the printed bound is of order $n^{-1/6}$; failure probability tends to one. (c) With $\beta=2$ and $\alpha=\hat\alpha_n=0$, put $t_j=2^{j^2}$, $b(r)=\sum_{j\geq2}j^{-1}I\{2t_j\leq r\leq3t_j\}$, $Y(d)=d(1+b(\\|U\\|))$, $g=\hat g=d$, and $e=b$. The regression inequality holds with $R_g=1$. At $t_j$, $e(t_j)=0$ but $\mathbb E[b(\\|U\\|)\mid\\|U\\|>t_j]$ is at least a positive constant times $1/j$. With sample sizes of order $t_j^{10}$, sampling error is negligible and the printed right-hand side is of order $t_j^{-1}$. Thus the claimed approximation-bias step is false. (d) Take $\beta=1$, $\alpha=\hat\alpha_n=0$, $Y(d)=d(1+\min\{U_1,U_2\}/\\|U\\|)$, exact regression, and $e=0$. The limiting angular contrast is $1$. Restricting integration to $U_1\in[t,2t]$, $U_2\in[1,t/2]$ gives $\mathbb E[\min\{U_1,U_2\}/\\|U\\|\mid\\|U\\|>t]\geq C(\log t)/t$ for some $C>0$. With sample sizes of order $t^6$, the displayed bound is only of order $t^{-1}$. minimal patch (confirmed): corrected statement: Under the assumption of Theorem \ref{thm:rate}, suppose in addition that the conditions of Proposition~\ref{prop:id} hold. Suppose in addition that the conditions of Lemma~\ref{lemma:eta_rate} hold. Use its training split and independent auxiliary data, denoted together by $\mathcal T$, its $\pi_t=\mathbb P(\\|U\\|>t)$, $M=\|\mathcal I\|$, $\bar e(t)=\sup_{r\geq t}e(r)$, and $G_t=C+\bar e(t)$, where $C$ bounds $\|g\|$ on its compact domain. Set $\hat\eta_{n,t}^{\text{IPW}}=0$ if $M=0$. For a generic evaluation observation define \[ \phi=\frac{Y}{\\|U\\|^{\hat\alpha_n}}\left(\frac{D}{\hat p(X)}-\frac{1-D}{1-\hat p(X)}\right).\] For any $B>0$ chosen using only $\mathcal T$, put \[\begin{aligned} v_t^{\text{IPW}}(B)&=\mathbb E[\phi^2 I\{\|\phi\|\leq B\}\mid\mathcal T,\\|U\\|>t],\\ b_t^{\text{IPW}}(B)&=\mathbb E[\|\phi\|I\{\|\phi\|>B\}\mid\mathcal T,\\|U\\|>t].\end{aligned}\] Fix $a_0\in(0,\beta)$ and put $a=R_\alpha(n,\delta/4)$. For sufficiently large $t\geq1$, if $a\leq a_0$ and $n\pi_t\geq16\log(4/\delta)$, then with probability at least $1-\delta$ we have $M\geq n\pi_t/4$ and \[\begin{aligned} \|\hat\eta_{n,t}^{\text{IPW}}-\eta\|\leq{}&\sqrt{\frac{4v_t^{\text{IPW}}(B)}{\delta M}}+\left(1+\frac4\delta\right)b_t^{\text{IPW}}(B)+\frac{2G_t}{c}R_p(n/2,\delta/4)\\ &+2\bar e(t)+2L W_1(\mathcal L_{U/\\|U\\|}^t,\mathcal L_{U/\\|U\\|})+\frac{2G_t}{c}(t^a-1+K a t^a),\end{aligned}\] where $K$ depends on the radial law and $a_0$. Under the angular assumption of Lemma~\ref{lemma:eta_rate}, taken from Proposition 3.1 of \citet{zhang2023wasserstein}, the angular Wasserstein term has the bound stated there for $t\geq t_W$, with its logarithmic factor retained. When $V_t^{\text{IPW}}=\tmop{Var}(\phi\mid\mathcal T,\\|U\\|>t)$ is finite, the sampling term may instead be taken to be $\sqrt{4V_t^{\text{IPW}}/(\delta M)}$. · proof change: Write $R_U=\\|U\\|$ and $S_U=U/\\|U\\|$, and define \[\begin{aligned} m_d^t(x,s)&=\mathbb E[Y/R_U^\alpha\mid X=x,D=d,S_U=s,R_U>t],\\ \eta_\alpha^t&=\mathbb E[(f(X,1,U)-f(X,0,U))/R_U^\alpha\mid R_U>t].\end{aligned}\] Independence of $U$ from $(X,D)$ gives $\mathbb P(D=1\mid X,S_U,R_U>t)=p(X)$. Let $\phi_\alpha$ be $\phi$ with $\hat\alpha_n$ replaced by $\alpha$. Conditional expectation over treatment and outcome gives \[\begin{aligned} &\mathbb E[\phi_\alpha\mid\mathcal T,R_U>t]-\eta_\alpha^t\\ &\quad=\mathbb E\left[(p(X)-\hat p(X))\left\{\frac{m_1^t(X,S_U)}{\hat p(X)}+\frac{m_0^t(X,S_U)}{1-\hat p(X)}\right\}\mid\mathcal T,R_U>t\right].\end{aligned}\] Indeed, the treatment coefficients after subtraction are $p/\hat p-1=(p-\hat p)/\hat p$ and $1-(1-p)/(1-\hat p)=(p-\hat p)/(1-\hat p)$. Homogeneity gives $\|m_d^t\|\leq G_t$. On the joint nuisance event at tolerance $\delta/4$, estimated-propensity overlap therefore gives \[ \|\mathbb E[\phi_\alpha\mid\mathcal T,R_U>t]-\eta_\alpha^t\|\leq\frac{2G_t}{c}R_p(n/2,\delta/4).\] Taking conditional expectations over the outcome in the exponent difference gives \[\begin{aligned} &\mathbb E[\phi-\phi_\alpha\mid\mathcal T,R_U>t]\\ &=\mathbb E\left[\left\{\frac{p(X)f(X,1,U)}{\hat p(X)R_U^\alpha}-\frac{(1-p(X))f(X,0,U)}{(1-\hat p(X))R_U^\alpha}\right\}(R_U^{\alpha-\hat\alpha_n}-1)\mid\mathcal T,R_U>t\right].\end{aligned}\] Since $R_U>t\geq1$ and $\|\alpha-\hat\alpha_n\|\leq a$, the absolute value is at most $(2G_t/c)\mathbb E[R_U^a-1\mid R_U>t]$. Put $\epsilon=(\beta-a_0)/2$. Potter's bound gives $\mathbb P(R_U>tr)/\pi_t\leq2r^{-\beta+\epsilon}$ for all $r\geq1$ and sufficiently large $t$. Integration gives \[\begin{aligned} \mathbb E[R_U^a-1\mid R_U>t]&=t^a-1+t^a\int_1^\infty a r^{a-1}\frac{\mathbb P(R_U>tr)}{\pi_t}\,\tmop{dr}\\ &\leq t^a-1+\frac{2a t^a}{\beta-\epsilon-a}\leq t^a-1+K a t^a,\end{aligned}\] with $K=4/(\beta-a_0)$; for $a=0$ both sides vanish. The triangle inequality now supplies the decomposition into sampling fluctuation, propensity error, exponent error, and $\|\eta_\alpha^t-\eta\|$. Homogeneity at the realized radius gives \[ \|\eta_\alpha^t-\mathbb E[g(X,1,S_U)-g(X,0,S_U)\mid R_U>t]\|\leq2\bar e(t).\] The function $s\mapsto\mathbb E_X[g(X,1,s)-g(X,0,s)]$ is $2L$-Lipschitz. Independence of $X$ and $U$ and the Wasserstein coupling inequality therefore give \[ \|\eta_\alpha^t-\eta\|\leq2\bar e(t)+2L W_1(\mathcal L_{U/\\|U\\|}^t,\mathcal L_{U/\\|U\\|}).\] Lemma~\ref{lemma:eta_rate} supplies the angular bound under the identical model assumptions, including its logarithmic factors. Conditional on $\mathcal T$ and positive $M$, the evaluation scores are independent with the conditional exceedance law. Put $Z=\phi I\{\|\phi\|\leq B\}$ and $W=\phi I\{\|\phi\|>B\}$. Chebyshev's and Markov's inequalities give, respectively, conditional failure probabilities at most $\delta/4$ for \[\begin{aligned} \left\|M^{-1}\sum Z_i-\mathbb E[Z\mid\mathcal T,R_U>t]\right\|&\leq\sqrt{4v_t^{\text{IPW}}(B)/(\delta M)},\\ M^{-1}\sum\|W_i\|&\leq4b_t^{\text{IPW}}(B)/\delta.\end{aligned}\] If $b_t^{\text{IPW}}(B)=0$, the second inequality holds almost surely because $W=0$ almost surely. Combining them with $\|\mathbb E[W\mid\mathcal T,R_U>t]\|\leq b_t^{\text{IPW}}(B)$ bounds the sampling fluctuation by the first two terms in the statement. The count has law $M\sim\tmop{Bin}(n/2,\pi_t)$, so \[ \mathbb P(M 0 $, the following conclusions hold. \begin{enumerate} \item If $s \in (0, 1 / (2 + \max \{1, \beta\}))$, takes $t_n = \Theta (n^{ (1 - 2 s) \hat{\gamma}_{n}})$, with probability at least $1 - \delta$, we have \begin{align*} \| {{\widehat{\theta }_{n, t}^{\text{IPW} }} } - \theta^{NETE} \| & = O (e (t_n) + n^{- s} \log (1 / \delta) + n^{-c_\alpha} \log(n)\log (1 / \delta)) . \end{align*} \item If $s \in [1 / (2 + \max \{1, \beta\}), 1 / 2)$, \ takes $t = \Theta (n^{(\hat{\gamma}_{n} / (1 + 2 \min \{1, \hat{\gamma}_{n} \})})$, with probability at least $1 - \delta$, we have \begin{align*} \| {{\widehat{\theta }_{n, t}^{\text{IPW} }} } - \theta^{NETE} \| & = O (e (t_n) + n^{- 1 / (2 + \max \{\beta, 1\})} \log (1 / \delta) + n^{-c_\alpha} \log(n)\log (1 / \delta)) . \end{align*} \end{enumerate} | | appended hypothesis list | ass:cor-rate-ipw-errors, ass:consistency, ass:exogeneity, ass:overlap, ass:iid-sampling, ass:iid-regular-variation, ass:iid-independence, ass:f-homo-bounded-x, ass:f-homo-asymptotic, ass:mvary-density, ass:mvary-matrix, ass:mvary-constants, ass:rate-alpha-less-beta, ass:rate-nuisance-errors | | Lean | `TCorRateIpw.lean:30` — `Papers.TbsExtremeEffects.nete_ipw_rate_adaptive_threshold` | | crosswalk verdict | `equivalent` | | axioms | clean — `propext`, `Classical.choice`, `Quot.sound` | | V1.5 review | faithfulness ok · ambiguity ok · correctness ok | | added conditions | `ass:pin-33`, `ass:pin-34`, `ass:pin-36`, `ass:pin-38`, `ass:pin-39`, `ass:pin-41`, `ass:pin-42`, `ass:pin-43`, `ass:pin-44`, `ass:pin-45`, `ass:pin-7`, `ass:pin-fix-0b814294`, `ass:pin-fix-10c2ec7b`, `ass:pin-fix-6857bb4d`, `ass:pin-fix-8964f8dd`, `ass:pin-fix-8a129280`, `ass:pin-fix-8cd5c960`, `ass:pin-fix-8ef6fd29`, `ass:pin-fix-9567640d`, `ass:pin-fix-b3a62d30`, `ass:pin-fix-c3605ca3`, `ass:pin-fix-c4d18fdb`, `ass:pin-fix-c90a58f3`, `ass:pin-fix-ebdb71b8`, `ass:pin-fix-f18f3355`, `ass:pin-fix-fc4621e4`, `ass:pin-fix-fdafcd87` | | cited results | — | | assumed substrate gates | — | | findings | [note, V1.5] the corrected statement stands as one statement; the referee would state it as a general statement with corollaries: Separate the two rates into propositions under their respective ranges of $s$ and threshold rules; since neither is a general claim governing the other, the parent should be an umbrella. | ## Errors by root cause 2 independent error(s) — 2 repaired by a minimal patch, 0 not repaired, 0 unfixable and left out of Lean; 3 result(s) inherit one. ### Errors found — repaired The result as printed is false (the witness shows why). The smallest change that keeps the paper's key contribution was confirmed by an independent referee pass; the F stages formalize it in place of the printed statement. Key contribution, as judged by the confirming referee: The key-result list includes prop:id, thm:rate, thm:cor-rate, and thm:cor-rate-ipw, covering identification, finite-sample bounds, and both adaptive-rate results. Their repairs preserve the NETE estimand and the contribution at the owner-accepted scope, consistently retaining the additional conditions and angular, radial, homogeneity, and exponent-error terms. - **`prop:id`** (`Arxiv_template.tex:234-243`): The core carries the printed hypotheses and identification setting faithfully. The homogeneity display could quantify over one fixed treatment argument and radius, or uniformly over both treatment arguments and all positive radii; its sphere could also mean the usual Euclidean sphere or the positive unit sphere for the paper's norm. The proof and global norm convention determine the latter, uniform reading. The substantive error is the proof's first equality: $(Y(1),Y(0))\perp D\mid X$ and $U\perp(X,D)$ do not imply conditional exogeneity given $(X,U)$. The conclusion fails even with overlap, exact homogeneity, finite outcome means, and $\alpha<\beta$. In that moment range, the remaining radial-angular passage can be closed using Potter bounds, a higher radial moment for uniform integrability, and $\sup_{r\geq t}e(r)\to0$; bounded first moments alone do not establish uniform integrability. The preliminary radial-moment sentence also drops the additive one, although the subsequent calculation supplies the correct value $\beta/(\beta-\alpha)$. - witness: Take independent $X\sim\text{Unif}([0,1])$, $D\sim\text{Ber}(1/2)$, $V\sim\text{Unif}([0,1])$, and $W$ with $\mathbb P(W>w)=w^{-3}$ for $w\geq1$. Set $U=W(V,1-V)$, $\alpha=1$, and $T(w)=(1-w^{-3})^{-1/3}$. Define $Y(1)=DW+(1-D)T(W)$, $Y(0)=0$, and $Y=Y(D)$. Since $W^{-3}$ is uniform, $T(W)$ has the same law as $W$. Thus $(Y(1),Y(0))\perp D\mid X$, and $U\perp(X,D)$; independent copies satisfy the sampling assumption. Nevertheless, $f(x,1,u)=\|u\|$, $f(x,0,u)=0$, so exact homogeneity holds with $g(x,1,s)=1$, $g(x,0,s)=0$, and $e(t)=0$. The printed right-hand side is $3/2$. But $\mathbb E[W\mid W>t]=3t/2$ and $\mathbb E[T(W)\mid W>t]\to1$, giving $\theta^{\tmop{NETE}}=3/4$. The false step replaces the conditional counterfactual mean by the observed conditional mean after conditioning on $U$; neither stated independence licenses that replacement. - Claim: Proposition~\ref{prop:id} identifies $\theta^{\tmop{NETE}}$ from the observed conditional mean under exogeneity given $X$ and independence of $U$ from $(X,D)$. - What is wrong: (a) ``\dots \vsDisplay{\begin{array}{@{}c@{}}\displaystyle \lim_{t \rightarrow \infty} \mathbb{E} \left[ \frac{Y (1) - Y (0)}{t^{\alpha}} \mid \|U\|> t \right] \\ \displaystyle = \lim_{t \rightarrow \infty} \mathbb{E} \left[ \frac{f (X, 1, U) - f (X, 0, U)}{t^{\alpha}} \mid \|U\|> t \right] \\ \displaystyle = \lim_{t \rightarrow \infty} \mathbb{E} \left[ \frac{f (X, 1, U) - f (X, 0, U)}{\|U\|^{\alpha}} \cdot \left( \frac{\|U\|}{t} \right)^{\alpha} \mid \|U\|> t \right] \\ \displaystyle = \lim_{t \rightarrow \infty} \mathbb{E} \left[ (g (X, 1, U / \| U \|) - g (X, 0, U / \| U \|) + 2 e (t)) \cdot \left( \frac{\|U\|}{t} \right)^{\alpha} \mid \|U\|> t \right]\end{array}\mbox{''}} This step uses the conditional-mean identification supplied by exogeneity and consistency, which the paper assumes only for conditioning on $X$, but applies it to conditioning on $(X,U)$ and then on $\{\|U\|>t\}$. (b) In the step quoted in (a): The homogeneity inequality supplies a signed remainder bounded at the realized radius by $2e(\|U\|)$, rather than the displayed equality with the positive term $2e(t)$. (c) ``\dots \vsDisplay{\frac{\int_1^{\infty} \alpha P (\| U \| > {rt}) r^{\alpha - 1} \tmop{dr}}{P (\| U \| > t)} \leqslant 2\int^{\infty}_1 \alpha r^{\alpha -\beta + \epsilon - 1} \tmop{dr} < \infty .\mbox{''}} The choice of a positive $\epsilon$ satisfying $\alpha-\beta+\epsilon<0$ requires $\alpha<\beta$, which the proposition does not assume. (d) ``We next argue that $\lim_{t \rightarrow \infty} \mathbb{E} [(\| U \| / t)^{\alpha} \mid \| U \| > t] = \alpha / (\beta - \alpha)$.'' The radial moment limit is $\beta/(\beta-\alpha)$ when $\alpha<\beta$, rather than $\alpha/(\beta-\alpha)$. (e) ``We conclude that $h (U)$ is uniformly integrable.'' A uniform bound on first absolute moments does not imply uniform integrability. Counterexample: Let $X\sim\tmop{Unif}([0,1])$, $D\sim\tmop{Ber}(1/2)$, and $V\sim\tmop{Unif}([0,1])$ be mutually independent. Let $W$ be independent of $(X,D,V)$ with $\mathbb P(W>w)=w^{-3}$ for $w\geq1$, and set $U=W(V,1-V)$ and $\alpha=1$. Define $T(w)=(1-w^{-3})^{-1/3}$, $Y(1)=DW+(1-D)T(W)$, $Y(0)=0$, and $Y=Y(D)$. Take independent copies for the observations, and take $g(x,1,s)=1$, $g(x,0,s)=0$, and $e(r)=0$. The variables $T(W)$ and $W$ have the same distribution, so $(Y(1),Y(0))\perp D\mid X$, while $U\perp(X,D)$ and exact homogeneity holds. The displayed product is $3/2$, whereas $\mathbb E[T(W)\mid W>t]\to1$ and $\mathbb E[W\mid W>t]=3t/2$ give $\theta^{\tmop{NETE}}=3/4$. - Proposed fix: The suggested version retains the identification formula but adds conditional mean exogeneity given $(X,U)$, true propensity positivity, $\alpha<\beta$, and finite absolute counterfactual tail means; its proof uses the tail supremum of $e$ and a higher radial moment. No admissible repair preserves the identification claim of Proposition~\ref{prop:id} at its printed scope. - Gives up: Identification for distributions that satisfy exogeneity given $X$ but violate conditional mean exogeneity given $(X,U)$, distributions with a degenerate true propensity, the range $\alpha\geq\beta$, and counterfactuals without finite absolute tail means. - repair confirmed by an independent referee pass (see Corrected version); the corrected statement is proved in Lean - corrected statement: Suppose that $U$ is multivariate regularly varying and Assumption~\ref{assump:exo}, \ref{assump:iid}, \ref{assump:f_homo} hold. Suppose in addition that $\alpha<\beta$, $0t]<\infty$ for both treatments and all sufficiently large $t$. Then we have \begin{align*} \theta^{\tmop{NETE}} &= \lim_{t\rightarrow\infty}\mathbb E[g(X,1,U/\|U\|)-g(X,0,U/\|U\|)\mid\|U\|>t]\cdot\lim_{t\rightarrow\infty}\mathbb E[\|U\|^{\alpha}/t^{\alpha}\mid\|U\|>t]. \end{align*} - contribution narrowed: Identification for distributions that satisfy exogeneity given $X$ but violate conditional mean exogeneity given $(X,U)$, distributions with a degenerate true propensity, the range $\alpha\geq\beta$, and counterfactuals without finite absolute tail means. (the owner accepted the weaker version) - inherited by: `thm:rate`, `lem:eta-rate`, `lem:eta-ipw` - **`lem:mu-rate`** (`Arxiv_template.tex:607-616`): The core faithfully retains the assumptions and algorithm reference. (a) The unresolved readings are the outcome exponent versus a local tail-index meaning of $\alpha$, and the algorithm's adjacent-ratio statistic versus the proof's common-denominator Hill statistic. The input collection, admissible ranks, and calibration of $l_n$ and $r_n(\delta)$ are not reconciled. (b) Algorithm~\ref{alg:nete} compares candidates through $n$, although its evaluation collection contains at most $n/2$ observations; moreover, its displayed statistic telescopes and lacks the weights required for a Hill estimator. (c) The proof's comparison against every candidate index is not the cited adaptive selection rule and can leave the admissible set empty even after removing unavailable ranks. (d) The asserted second-order tail expansion, and hence $\eta(t)=O(t^{-1/\beta})$, fail at the admissible value $\beta=1$, where a logarithmic term is present. (e) Two additional proof slips can be closed locally: the proportional component tail should be transferred using the original product Pareto law, rather than the shifted component law; and differentiating the displayed survival expansion gives the density coefficient $D(1+1/\beta)$, not $D(1-1/\beta)$. These corrections do not resolve (b)--(d). (f) I also could not close the final removal of the adaptation factor: [Boucheron and Thomas, Corollary 3.12](https://arxiv.org/pdf/1503.05077) retains $\log((2/\delta)\log n)$, whereas the printed rate retains only $\log(2/\delta)$. - witness: (b) Take independent Pareto type-II components $Z_1,Z_2$ with $\beta=1$, $U=Z$, $A=I_2$, $m=d=d_z=d_u=2$, $l=1/4$, $u=2$, $\sigma=1/8$, $s=1/4$, $\xi=1/2$, and $\zeta=1$, with independent observations and fixed threshold $t=1$. This exact product law satisfies every stated tail-model restriction. Nevertheless, $M=|\mathcal I|\leq n/2$, so the comparison at $i=n$ requires an unavailable order statistic. Independently, the algorithm's display equals $k^{-1}\log(\|U_{(1)}\|/\|U_{(k+1)}\|)$, whereas the proof's statistic equals $k^{-1}\sum_{j=1}^k j\log(\|U_{(j)}\|/\|U_{(j+1)}\|)$. They are different random variables. For the permitted choice $l_n=\lceil(\log n)^2\rceil$, every computable algorithmic candidate satisfies $0\leq\hat\gamma_k\leq\log(\max_i\|U_i\|/t)/l_n=O_{\mathbb P}(1/\log n)$, incompatible with $\gamma=1$. (c) On the same radial law, even using the proof's full-sample Hill statistic and deleting the unavailable endpoint, choose $l_n=\lceil(\log n)^2\rceil$ and $r_n(\delta)$ of the stated order. Then $\hat\gamma(l_n)\to1$ in probability, while $\hat\gamma(n-1)=\tfrac12\log n+O_{\mathbb P}(1)$ because the minimum radius is of order $n^{-1/2}$. Both comparison tolerances tend to zero. No candidate can satisfy comparisons with both indices, so the printed maximum is empty with probability tending to one. (d) Direct convolution on this exact product law gives $\bar F(t)=2/(t+2)+2\log(t+1)/(t+2)^2=2t^{-1}+(2\log t-4)t^{-2}+o(t^{-2})$. Thus the displayed constant-coefficient second-order expansion is false. Its tail quantile satisfies $U(q)=2q+\log(2q)-2+o(1)$, and the von Mises function has leading term $-\log q/(2q)$, which is not $O(q^{-1})$. This failure persists even under the favorable interpretation that the density calculation's $\alpha$ means $\beta$. - Claim: Lemma~\ref{lemma:mu_rate} asserts that the radial estimator in Algorithm~\ref{alg:nete} has error of order $(\log(2/\delta)/n)^{1/(2+\beta)}$ under the stated Pareto-type model. - What is wrong: (a) ``\dots choosing $k$ to be \vsDisplay{k = \max \left\{ k \in \{l_n, \cdots, n\} \text{and} \ \forall i \in \{l_n, \cdots, n\}, | \hat{\gamma}(i) - \hat{\gamma}(k) | \leqslant \frac{\hat{\gamma}(i) r_n (\delta)}{\sqrt{i}} \right\}} where $\hat{\gamma}(i) = \frac{1}{i} \sum_{j=1}^i \log \frac{\|U_{(j)}\|}{\|U_{(i+1)}\|}$ and $r_n (\delta)$ scales like $\sqrt{\log ((2 / \delta) \log (n))}$.'' The comparison at $i=n$ requires rank $n+1$, although Algorithm~\ref{alg:nete} orders at most $n/2$ evaluation exceedances. (b) In the step quoted in (a): The common-denominator Hill statistic in this step differs from Algorithm~\ref{alg:nete}'s adjacent-ratio statistic, which telescopes to $k^{-1}\log(\|U_{(1)}\|/\|U_{(k+1)}\|)$. (c) In the step quoted in (a): Comparisons against every candidate can leave the admissible set empty, even when all candidate Hill statistics are available. (d) In the step quoted in (a): The lower rank and tolerance are not calibrated, so their stated orders alone do not establish the adaptive deviation bound. (e) ``By Assumption~\ref{assump:mvary}, we know that \vsDisplay{g (z) = c \alpha^m \prod_{i = 1}^m (1 + z_i)^{- \alpha - 1}, \|z\|_1 > \zeta k^{(1 - 2 s) / \alpha} .\mbox{''}} The component density uses the outcome exponent $\alpha$ instead of the tail index $\beta$. (f) ``Let $\tilde{Z}_1, \cdots, \tilde{Z}_m \sim \alpha c^{1 / m} (1 + z)^{- \alpha - 1}, z \geqslant c^{1 / (m \alpha)} - 1$.'' The shifted component distributions alter the nominal product law, so exact proportionality of the original component density does not identify their radial convolution with the radial law of $U$. (g) ``\dots we have that when $\|U\|_1 > \max_i \{a_i \} \zeta k^{(1 - 2 s) / \alpha}$, \vsDisplay{f (t) = C \beta t^{- \beta - 1} (1 + D (1 - 1 / \beta) t^{- 1} + o (t^{- 1})) .\mbox{''}} The constant-coefficient density expansion omits a logarithmic term at $\beta=1$. (h) In the step quoted in (g): The stated survival coefficient $D$ corresponds to density coefficient $D(1+1/\beta)$ rather than $D(1-1/\beta)$. (i) ``\dots we get \vsDisplay{\eta (t) = \frac{1}{\beta (1 - DC^{- 1 / \beta} t^{- 1 / \beta} + o (t^{1 / \beta}))} - \frac{1}{\beta} = O (t^{- 1 / \beta}) .\mbox{''}} At $\beta=1$ the von Mises function can have order $t^{-1}\log t$, which is not $O(t^{-1})$. (j) In the step quoted in (i): For $\beta>1$ the survival expansion gives von Mises coefficient $-DC^{-1/\beta}/\beta^2$, with the opposite sign to the displayed substitution. (k) ``\dots with probability at least $1 - \delta$, we have \vsDisplay{| \gamma - \hat{\gamma}_n | \leqslant O \left( \left( \frac{\log (2 / \delta)}{n} \right)^{1 / (2 + \beta)} \right),\mbox{''}} The cited adaptive deviation bound retains $\log((2/\delta)\log n)$, which cannot be replaced by $\log(2/\delta)$ for fixed $\delta$. Counterexample: Take $m=d=d_z=d_u=2$, $A=I_2$, and independent $Z_1,Z_2$ with density $(1+z)^{-2}$ on $z\geq0$. Set $U=AZ$, $\beta=1$, $l=1/4$, $u=2$, $\sigma=1/8$, $s=1/4$, $\xi=1/2$, and $\zeta=1$. Let $X=0$, let $D\sim\tmop{Ber}(1/2)$ be independent of $U$, set $Y=0$, and take independent copies for the observations. Choose even $n$, $t=1$, and $l_n=\lceil100\log n\rceil$ for sufficiently large $n$. Let $R=\|U\|_1$, whose survival is $\bar F_R(r)=2/(r+2)+2\log(r+1)/(r+2)^2$ and satisfies $\mathbb P(R>r)\leq4/r$. The comparison at $i=n$ requests an unavailable rank $n+1$; even after unavailable ranks are suppressed, with probability at least $1-4/n$ every computable adjacent-ratio candidate with $k\geq l_n$ satisfies $k^{-1}\log(\|U_{(1)}\|/\|U_{(k+1)}\|)\leq2\log n/l_n\leq1/50$, so any selected output has error from $\gamma=1$ at least $49/50$, whereas the claimed bound tends to zero. - Proposed fix: The suggested statement replaces the evaluation-exceedance adjacent-ratio estimate by the separately defined full-input Hill estimate $\widetilde\gamma_n$, with ranks through $n-1$, calibrated tolerance, and prefix comparisons. Its proof uses the original product Pareto law and the standing $\ell_1$ norm, and its bound retains the adaptation logarithm and the endpoint logarithm at $\beta=1$. Theorem~\ref{thm:rate} and Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} can use this radial bound only for NETE estimators and thresholds formed with $\widetilde\gamma_n$; it supplies no vanishing radial bound for their printed evaluation-exceedance adjacent-ratio estimate. - Gives up: The guarantee for the printed evaluation-exceedance adjacent-ratio estimate and the selector comparing every $i\in\{l_n,\ldots,n\}$ is replaced by a guarantee for a different full-input Hill estimate with prefix selection. The logarithm-free radial bound is lost, and the radial exponent $1/(2+\beta)$ is replaced by $1/3$ for $0<\beta<1$. - repair confirmed by an independent referee pass (see Corrected version); the corrected statement is proved in Lean - corrected statement: Under Assumption \ref{assump:iid}, \ref{assump:mvary}, using the standing convention $\|\cdot\|=\|\cdot\|_1$, let $R_i=\|U_i\|_1$ for all $i\in\{1,\ldots,n\}$ and order these full-input radii as $R_{[1]}\geq\cdots\geq R_{[n]}$. For $n\geq2$, define \[ H_n(k)=\frac1k\sum_{j=1}^k\log\frac{R_{[j]}}{R_{[k+1]}},\qquad 1\leq k\leq n-1. \] For $\delta\in(0,1)$, set \[ \begin{aligned} L_n&=\min\{n-1,\lceil100\log n\rceil\},&J_n&=1+\lceil\log n/\log2\rceil,\\ q_n(\delta)&=\log(4J_n/\delta),&c_n(\delta)&=2\sqrt{q_n(\delta)}+\frac{q_n(\delta)}{\sqrt{L_n}},\\ \rho_n(\delta)&=16c_n(\delta). \end{aligned} \] Let \[ \begin{aligned} \widetilde k_n&=\max\bigl\{k\in\{L_n,\ldots,n-1\}:\\ &\qquad |H_n(i)-H_n(k)|\leq\frac{H_n(i)\rho_n(\delta)}{\sqrt i}\\ &\qquad \text{for every }i\in\{L_n,\ldots,k\}\bigr\},\\ \widetilde\gamma_n&=H_n(\widetilde k_n). \end{aligned} \] For each fixed $\delta\in(0,1)$ and every sufficiently large $n$, with probability at least $1-\delta$, we have \[ |\gamma-\widetilde\gamma_n|\leq C_\delta \begin{cases} \left(\dfrac{\log((8/\delta)\log n)}{n}\right)^{1/3},&0<\beta<1,\\ \left(\dfrac{\log n\,\log((8/\delta)\log n)}{n}\right)^{1/3},&\beta=1,\\ \left(\dfrac{\log((8/\delta)\log n)}{n}\right)^{1/(2+\beta)},&\beta>1, \end{cases} \] where $C_\delta$ may depend on the fixed distribution and on $\delta$. where $\gamma = 1 / \beta$ is the EVI of $U$. - narrowed: The guarantee for the printed evaluation-exceedance adjacent-ratio estimate and the selector comparing every $i\in\{l_n,\ldots,n\}$ is replaced by a guarantee for a different full-input Hill estimate with prefix selection. The logarithm-free radial bound is lost, and the radial exponent $1/(2+\beta)$ is replaced by $1/3$ for $0<\beta<1$. (an intermediate result: its strongest true version was applied automatically) - inherited by: `thm:rate` ## Corrected version 7 of 7 result(s) differ from the version you provided — 7 repaired, 0 with the proof written out in full (no error), 0 only with a side condition or reading made explicit; the others are as printed. The Lean formalization is of the corrected version. ### `prop:id` (`Arxiv_template.tex:234-243`) - as printed: \label{prop:id}Suppose that $U$ is multivariate regularly varying and Assumption~\ref{assump:exo}, \ref{assump:iid}, \ref{assump:f_homo} hold, we have \begin{align*} \theta^{\tmop{NETE}} = \lim_{t \rightarrow \infty} \mathbb{E}[g (X, 1, U /\|U\|) - g (X, 0, U /\|U\|) \mid \|U\|> t] \cdot \lim_{t \rightarrow \infty} \mathbb{E}[\|U\|^{\alpha} / t^{\alpha} \mid \|U\|> t] . \end{align*} - corrected: Suppose that $U$ is multivariate regularly varying and Assumption~\ref{assump:exo}, \ref{assump:iid}, \ref{assump:f_homo} hold. Suppose in addition that $\alpha<\beta$, $0t]<\infty$ for both treatments and all sufficiently large $t$. Then we have \begin{align*} \theta^{\tmop{NETE}} &= \lim_{t\rightarrow\infty}\mathbb E[g(X,1,U/\|U\|)-g(X,0,U/\|U\|)\mid\|U\|>t]\cdot\lim_{t\rightarrow\infty}\mathbb E[\|U\|^{\alpha}/t^{\alpha}\mid\|U\|>t]. \end{align*} - side condition added by the confirming referee `ass:pin-33`: $U\perp(X,D)$. — pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly. - side condition added by the confirming referee `ass:pin-34`: $\|X\|_1\leqslant R$ almost surely. — pins reading of ass:f-homo-bounded-x: Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm. - side condition added by the confirming referee `ass:pin-38`: For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\|U\|/b_n,U/\|U\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. — pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation. - side condition added by the confirming referee `ass:pin-39`: $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. — pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely. - side condition added by the confirming referee `ass:pin-43`: The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\|x\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\|u\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. — pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere. - side condition added by the confirming referee `ass:pin-7`: $Y(D)=Y$ almost surely throughout the paper. — pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely. - side condition added by the confirming referee `ass:pin-fix-8cd5c960`: The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\|u\|_1=1$. — pins reading of prop:id: stated by the repair - proof change: the printed proof, revised step by step] Write $R_U=\|U\|$, $S_U=U/\|U\|$, and $\bar e(t)=\sup_{r\geq t}e(r)$; the last quantity is finite for sufficiently large $t$ and tends to zero. Independence of $U$ from $(X,D)$ gives $\mathbb P(D=1\mid X,U)=p(X)$. Positivity, consistency, and conditional mean exogeneity given $(X,U)$ therefore give, for both treatments, \[ f(X,d,U)=\mathbb E[Y(d)\mid X,U,D=d]=\mathbb E[Y(d)\mid X,U]. \] The assumed absolute tail means justify taking conditional expectations on $\{R_U>t\}$. Homogeneity at the realized radius yields \[ \frac{f(X,1,U)-f(X,0,U)}{R_U^{\alpha}}=g(X,1,S_U)-g(X,0,S_U)+\varepsilon(X,U),\qquad |\varepsilon(X,U)|\leq2\bar e(t)\quad\hbox{on }\{R_U>t\}. \] Consequently, \[ \left|\mathbb E\left[\frac{Y(1)-Y(0)}{t^{\alpha}}\mid R_U>t\right]-\mathbb E\left[(g(X,1,S_U)-g(X,0,S_U))(R_U/t)^{\alpha}\mid R_U>t\right]\right|\leq2\bar e(t)\mathbb E[(R_U/t)^{\alpha}\mid R_U>t]. \] We next establish that the radial moment tends to $\beta/(\beta-\alpha)$. For every real $a<\beta$, the identity $x^a=1+\int_1^x a r^{a-1}\,\tmop{dr}$, together with absolute integrability of the integral, gives \[ \mathbb E[(R_U/t)^a\mid R_U>t]=1+\int_1^{\infty}a r^{a-1}\frac{\mathbb P(R_U>tr)}{\mathbb P(R_U>t)}\,\tmop{dr}. \] For $a=0$ the integral is zero; for $a<0$ its absolute integrability follows from $\int_1^{\infty}|a|r^{a-1}\,\tmop{dr}=1$. Note that $\| U \|$ is also regularly varying. By Potter's theorem \cite[Theorem 1.56]{bingham1989regular}, for any $\epsilon >0$ and sufficiently large $t$, we have \begin{align*} \frac{P (\| U \| > {rt})}{P (\| U \| > t)} & \leqslant 2 r^{-\beta + \epsilon} . \end{align*} For $0t]\longrightarrow1+\int_1^{\infty}a r^{a-\beta-1}\,\tmop{dr}=\frac{\beta}{\beta-a}. \] In particular, the radial moments with $a=\alpha$ are bounded, and the preceding homogeneity error tends to zero. Define \[ G(s)=\mathbb E_X[g(X,1,s)-g(X,0,s)],\qquad h_t(r,s)=G(s)r^{\alpha}. \] Independence of $X$ and $U$ shows that the remaining conditional expectation is $\mathbb E[h_t(R_U/t,S_U)\mid R_U>t]$. Note that by Assumption \ref{assump:f_homo}, $g$ is a continuous function on a compact set and thus is bounded by some constant $C > 0$. Thus $|G(s)|\leq2C$, and $G$ is continuous. If $\alpha\leq0$, then $|h_t(r,s)|\leq2C$ for $r\geq1$. If $\alpha>0$, choose $p>1$ with $p\alpha<\beta$. The radial moment calculation gives \[ \sup_{t\geq t_0}\mathbb E[|G(S_U)(R_U/t)^{\alpha}|^p\mid R_U>t]\leq(2C)^p\sup_{t\geq t_0}\mathbb E[(R_U/t)^{p\alpha}\mid R_U>t]<\infty. \] This establishes uniform integrability in both cases. Definition~\ref{def:mvarying} gives conditional weak convergence of $(R_U/t,S_U)$ to the product of a Pareto radial law with survival $r^{-\beta}$ for $r\geq1$ and the spectral probability measure $S$. For a bounded truncation of $G(s)r^{\alpha}$, weak convergence gives convergence of expectations. Uniform integrability makes the truncation errors tend to zero uniformly, so \[ \lim_{t\rightarrow\infty}\mathbb E[G(S_U)(R_U/t)^{\alpha}\mid R_U>t]=\left(\int G(s)\,S(\tmop{ds})\right)\frac{\beta}{\beta-\alpha}. \] Weak convergence of the angular marginal and bounded continuity of $G$ also give \[ \lim_{t\rightarrow\infty}\mathbb E[g(X,1,S_U)-g(X,0,S_U)\mid R_U>t]=\int G(s)\,S(\tmop{ds}). \] Combining these two limits with the vanishing homogeneity error proves the stated product formula. - why: The core carries the printed hypotheses and identification setting faithfully. The homogeneity display could quantify over one fixed treatment argument and radius, or uniformly over both treatment arguments and all positive radii; its sphere could also mean the usual Euclidean sphere or the positive unit sphere for the paper's norm. The proof and global norm convention determine the latter, uniform reading. The substantive error is the proof's first equality: $(Y(1),Y(0))\perp D\mid X$ and $U\perp(X,D)$ do not imply conditional exogeneity given $(X,U)$. The conclusion fails even with overlap, exact homogeneity, finite outcome means, and $\alpha<\beta$. In that moment range, the remaining radial-angular passage can be closed using Potter bounds, a higher radial moment for uniform integrability, and $\sup_{r\geq t}e(r)\to0$; bounded first moments alone do not establish uniform integrability. The preliminary radial-moment sentence also drops the additive one, although the subsequent calculation supplies the correct value $\beta/(\beta-\alpha)$. ### `thm:rate` (`Arxiv_template.tex:330-357`) - as printed: \label{thm:rate}Suppose that Assumption \ref{assump:exo}, \ref{assump:overlap}, \ref{assump:iid}, \ref{assump:f_homo}, \ref{assump:mvary} hold, $\alpha < \beta$, where $\alpha$ and $\beta$ are defined in \cref{assump:f_homo} and \cref{assump:mvary} respectively. Furthermore, for any fixed $t$, with probability at least $1 - \delta$, \begin{align*} |p (X) - \hat{p} (X) | \leqslant R_p (n, \delta) & , | \hat{\alpha}_{n} - \alpha| \leqslant R_\alpha(n,\delta),\\ |\mathbb{E}[Y/\|U\|^{\alpha} \mid X, D, U /\|U\|, \|U\|> t] &- \hat{g} (X, D, U /\|U\|) | \leqslant R_g (n_t, \delta), \end{align*} where $n_t = \sum_{i = 1}^{n/2} I (\|U_i \|> t)$ and $R_p, R_g, R_\alpha$ are estimation errors that are monotonically decreasing with respect to sample size. Then, with probability at least $1 - \delta, \delta \in (0, 1 / 2)$, we have \begin{align} \left| {{\widehat{\theta }_{n, t}^{\text{DR} }} } - \theta^{NETE} \right| & \leqslant O (\sqrt{R_p (n/2, \delta)R_g ({n_t}, \delta)} + t^{\beta / 2} n^{-1 / 2}+ \log (1 / \delta) n^{- 1 / (2 +\beta)} \notag\\ & \quad\quad \quad \quad + t^{- \min \{1, \beta\}} + t^{- \beta s / (1 - 2 s)} + \log(t)R_\alpha(n,\delta)+ e (t)).\label{eq:rate_dr} \end{align} and \begin{align} \left| {{\widehat{\theta }_{n, t}^{\text{IPW}}} } - \theta^{NETE} \right| & \leqslant O (R_p (n/2, \delta) +t^{\beta / 2} n^{- 1 / 2} + \log (1 / \delta) n^{- 1 / (2 + \beta)} \notag\\ &\quad\quad \quad \quad + t^{- \min \{1, \beta\}} + t^{- \beta s / (1 - 2 s)}+ \log(t)R_\alpha(n,\delta) + e (t)). \label{eq:rate_ipw} \end{align} - corrected: Suppose that Assumption \ref{assump:exo}, \ref{assump:overlap}, \ref{assump:iid}, \ref{assump:f_homo}, \ref{assump:mvary} hold, $\alpha < \beta$, where $\alpha$ and $\beta$ are defined in \cref{assump:f_homo} and \cref{assump:mvary} respectively. Suppose in addition that the conditions of Proposition~\ref{prop:id} hold. Suppose that the fitted propensity, regression and exponent are measurable with respect to the training split and independent auxiliary data, denoted together by $\mathcal T$, and that $\mathbb E[|Y/\|U\|^{\hat\alpha_n}|\mid\mathcal T,\|U\|>t]<\infty$ on each joint nuisance event. Furthermore, for any fixed $t$, with probability at least $1 - \delta$, \begin{align*} |p (X) - \hat{p} (X) | \leqslant R_p (n, \delta) & , | \hat{\alpha}_{n} - \alpha| \leqslant R_\alpha(n,\delta),\\ |\mathbb{E}[Y/\|U\|^{\alpha} \mid X, D, U /\|U\|, \|U\|> t] &- \hat{g} (X, D, U /\|U\|) | \leqslant R_g (n_t, \delta), \end{align*} where $n_t = \sum_{i = 1}^{n/2} I (\|U_i \|> t)$ and $R_p, R_g, R_\alpha$ are estimation errors that are monotonically decreasing with respect to sample size. Fix $\delta\in(0,1/2)$ and a deterministic threshold $t$. Run the adaptive Hill selection in Algorithm~\ref{alg:nete} at tolerance $\delta/6$, and write its full-input output as $\hat\gamma_n=\widetilde\gamma_n$. Put $\pi_t=\mathbb P(\|U\|>t)$, $M=|\mathcal I|$, $\bar e(t)=\sup_{r\geq t}e(r)$, and $G_t=C+\bar e(t)$, where $C$ bounds $|g|$ on its compact domain. Use zero for each first-factor estimate when $M=0$. For a generic evaluation observation put $S_U=U/\|U\|$ and define \[ \begin{aligned} \psi^{\mathrm{DR}}={}&\hat g(X,1,S_U)-\hat g(X,0,S_U)\\ &+\frac{D-\hat p(X)}{\hat p(X)(1-\hat p(X))}\left(\frac{Y}{\|U\|^{\hat\alpha_n}}-\hat g(X,D,S_U)\right),\\ \psi^{\mathrm{IPW}}={}&\frac{Y}{\|U\|^{\hat\alpha_n}}\left(\frac{D}{\hat p(X)}-\frac{1-D}{1-\hat p(X)}\right). \end{aligned} \] For each $q\in\{\mathrm{DR},\mathrm{IPW}\}$, choose $B_q>0$ using only $\mathcal T$, and put \[ \begin{aligned} v_t^q(B_q)&=\mathbb E[(\psi^q)^2I\{|\psi^q|\leq B_q\}\mid\mathcal T,\|U\|>t],\\ b_t^q(B_q)&=\mathbb E[|\psi^q|I\{|\psi^q|>B_q\}\mid\mathcal T,\|U\|>t],\\ \mathcal S_t^q&=\sqrt{\frac{12v_t^q(B_q)}{\delta\max\{M,1\}}}+\left(1+\frac{12}{\delta}\right)b_t^q(B_q). \end{aligned} \] Fix $a_0\in(0,\beta)$ and put \[ \begin{aligned} a&=R_\alpha(n,\delta/12),\qquad K=\frac{4}{\beta-a_0},\\ \mathcal A_t&=2LC_W\left((\log t)^k t^{-\min\{1,\beta\}}+t^{-\beta s/(1-2s)}\right),\\ \mathcal E_{\mathrm{DR}}={}&\mathcal S_t^{\mathrm{DR}}+\frac2c R_p(n/2,\delta/12)R_g(n_t,\delta/12)\\ &+2\bar e(t)+\mathcal A_t+\frac{2G_t}{c}(t^a-1+Kat^a),\\ \mathcal E_{\mathrm{IPW}}={}&\mathcal S_t^{\mathrm{IPW}}+\frac{2G_t}{c}R_p(n/2,\delta/12)\\ &+2\bar e(t)+\mathcal A_t+\frac{2G_t}{c}(t^a-1+Kat^a). \end{aligned} \] Suppose moreover, as in Lemma~\ref{lemma:eta_rate}, that the angular Wasserstein term satisfies the bound that Proposition 3.1 of \citet{zhang2023wasserstein} gives for this Pareto-type model, which is taken from that work and not proved here: there are constants $C_W>0$, $k\geq0$ and $t_W\geq1$ such that \[ W_1(\mathcal L_{U/\|U\|}^t,\mathcal L_{U/\|U\|})\leq C_W(\log t)^k t^{-\min\{1,\beta\}}+C_W t^{-\beta s/(1-2s)}\qquad(t\geq t_W). \] These are the constants $C_W$ and $k$ in the angular term $\mathcal A_t$ above. Let \[ b_n=C_{\delta/6} \begin{cases} \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/3},&0<\beta<1,\\ \left(\dfrac{\log n\,\log((48/\delta)\log n)}{n}\right)^{1/3},&\beta=1,\\ \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/(2+\beta)},&\beta>1, \end{cases} \] where $C_{\delta/6}$ is the constant supplied by Lemma~\ref{lemma:mu_rate}. Define \[ \begin{aligned} \gamma&=1/\beta,\qquad h=1-\alpha\gamma>0,\\ Q_n&=a(\gamma+b_n)+|\alpha|b_n,\qquad d_n=|1-\hat\alpha_n\hat\gamma_n|,\\ \mathcal H_n&= \begin{cases} Q_n/(h d_n),&d_n>0,\\ 1/h,&d_n=0. \end{cases} \end{aligned} \] Use Algorithm~\ref{alg:nete}'s convention $\hat\mu_n=0$ when $d_n=0$. For sufficiently large $n$ and $t\geq t_W$, if $a\leq a_0$ and $n\pi_t\geq16\log(12/\delta)$, then, with probability at least $1-\delta$, we have $M\geq n\pi_t/4$ and \[ \left|\widehat\theta_{n,t}^{\mathrm{DR}}-\theta^{\tmop{NETE}}\right| \leq |\hat\mu_n|\mathcal E_{\mathrm{DR}}+|\eta|\mathcal H_n, \] and \[ \left|\widehat\theta_{n,t}^{\mathrm{IPW}}-\theta^{\tmop{NETE}}\right| \leq |\hat\mu_n|\mathcal E_{\mathrm{IPW}}+|\eta|\mathcal H_n, \] where $\eta=\lim_{r\rightarrow\infty}\mathbb E[g(X,1,U/\|U\|)-g(X,0,U/\|U\|)\mid\|U\|>r]$. If moreover, for a constant $V<\infty$ independent of $n,t,\delta$ and every $q\in\{\mathrm{DR},\mathrm{IPW}\}$, $\mathbb E[(\psi^q)^2\mid\mathcal T,\|U\|>t]\leq V$ on the joint nuisance event, $t\geq\exp(1)$, $a\log t\leq1$ and $Q_n\leq h/2$, then on the same joint success event of probability at least $1-\delta$ we also have \[ \begin{aligned} |\widehat\theta_{n,t}^{\mathrm{DR}}-\theta^{NETE}|\leq C\biggl(&\sqrt{\frac{V}{\delta n\pi_t}}+R_p(n/2,\delta/12)R_g(n_t,\delta/12)\\ &+\bar e(t)+\mathcal A_t+R_\alpha(n,\delta/12)\log t+b_n\biggr),\\ |\widehat\theta_{n,t}^{\mathrm{IPW}}-\theta^{NETE}|\leq C\biggl(&\sqrt{\frac{V}{\delta n\pi_t}}+R_p(n/2,\delta/12)\\ &+\bar e(t)+\mathcal A_t+R_\alpha(n,\delta/12)\log t+b_n\biggr), \end{aligned} \] where $C$ does not depend on $n,t,\delta$, and $\pi_t\asymp t^{-\beta}$ makes the first term of order $\sqrt{V/\delta}\,t^{\beta/2}n^{-1/2}$. - side condition added by the confirming referee `ass:pin-33`: $U\perp(X,D)$. — pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly. - side condition added by the confirming referee `ass:pin-34`: $\|X\|_1\leqslant R$ almost surely. — pins reading of ass:f-homo-bounded-x: Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm. - side condition added by the confirming referee `ass:pin-36`: For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\|z\|_1>\zeta k^{(1-2s)/\beta}$. — pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$. - side condition added by the confirming referee `ass:pin-38`: For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\|U\|/b_n,U/\|U\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. — pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation. - side condition added by the confirming referee `ass:pin-39`: $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. — pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely. - side condition added by the confirming referee `ass:pin-41`: The existential cutoff $t_W$ is chosen sufficiently large for Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} and so that $\sup_{r\geq t_W}e(r)\leq1$ and $\mathbb P(\|U\|>tr)/\mathbb P(\|U\|>t)\leq4r^{-\beta}$ for every $t\geq t_W$ and $r\geq1$. — pins reading of thm:rate: The proof of Theorem~\ref{thm:rate} invokes the first-factor lemmas' sufficiently-large-threshold conclusions, uses the radial moment inequality with $K=4/(\beta-a_0)$, and bounds $G_t$ by $C+1$. The stated Pareto-type radial law and $e(t)\to0$ permit this enlargement of the existential cutoff. - side condition added by the confirming referee `ass:pin-42`: For each fixed $\mathcal T$-measurable choice $B>0$, the displayed probability bound uses the base conditions of Lemma~\ref{lemma:eta_rate}; its angular specialization additionally uses that lemma's angular assumption and $t\geq t_W$, and its variance alternative additionally requires $V_t^{\text{IPW}}<\infty$. — pins reading of lem:eta-ipw: The statement explicitly says ``Under the angular assumption'' and ``When'' the variance is finite; the amended proof applies conditional sampling inequalities to the chosen $B$. - side condition added by the confirming referee `ass:pin-43`: The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\|x\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\|u\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. — pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere. - side condition added by the confirming referee `ass:pin-44`: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. — pins reading of ass:rate-nuisance-errors: Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation. - side condition added by the confirming referee `ass:pin-45`: The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. — pins reading of def:nete-algorithm: Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention. - side condition added by the confirming referee `ass:pin-7`: $Y(D)=Y$ almost surely throughout the paper. — pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely. - side condition added by the confirming referee `ass:pin-fix-0b814294`: The equal split in Algorithm~\ref{alg:nete} is defined for positive even sample sizes $n$, with $|\mathcal D_1|=|\mathcal D_2|=n/2$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-10c2ec7b`: The component dimension in the density of Assumption~\ref{assump:mvary} is $m=d_z$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-6857bb4d`: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments. — pins reading of thm:rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8964f8dd`: Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8a129280`: The fitted-exponent pseudo-outcome has finite conditional absolute mean under the evaluation tail law: $\mathbb E[|Y|/\|U\|^{\hat\alpha_n}\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t]<\infty$ almost surely on each joint nuisance event. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8cd5c960`: The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\|u\|_1=1$. — pins reading of prop:id: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8ef6fd29`: The homogeneity inequality holds for both treatments and every positive radius, on the positive unit sphere for the $\ell_1$ norm. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9567640d`: The equal split uses $n\in2\mathbb N$. — pins reading of def:nete-algorithm: stated by the repair - side condition added by the confirming referee `ass:pin-fix-b3a62d30`: The symbol $V_t$ denotes $\tmop{Var}(\psi\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t)$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c3605ca3`: The sample-size domain of $R_g(q,\varepsilon)$ includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c4d18fdb`: For each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that, for every integer $n\geq n_0$, the bound has probability at least $1-\delta$ under the law of the sample of size $n$. — pins reading of lem:mu-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c90a58f3`: The count $n_t=\sum_{i=1}^{n/2}I(\|U_i\|>t)$ is the training-split exceedance count, with the training observations indexed first after the random split. — pins reading of thm:rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ebdb71b8`: The sampling term comprises both $\sqrt{4v_t^{\text{IPW}}(B)/(\delta M)}$ and $(1+4/\delta)b_t^{\text{IPW}}(B)$. — pins reading of lem:eta-ipw: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f18f3355`: The target $\eta$ denotes $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. — pins reading of lem:eta-rate: stated by the repair - proof change: the printed proof, revised step by step] Proposition~\ref{prop:id} applies because its identification and integrability conditions are assumed. The limiting angular contrast is $\eta$, and the limiting radial moment is \[ \mu=\int_1^\infty r^\alpha\beta r^{-\beta-1}\,\tmop{dr}=\frac{\beta}{\beta-\alpha}=\frac1h. \] The integral converges because $\alpha<\beta$. Consequently $\theta^{\tmop{NETE}}=\eta\mu$. Since $|g|\leq C$, we also have $|\eta|\leq2C$. For either estimator, the product identity gives \[ \widehat\theta_{n,t}^{\mathrm{estimator}}-\theta^{\tmop{NETE}} =\hat\mu_n(\hat\eta_{n,t}^{\mathrm{estimator}}-\eta)+\eta(\hat\mu_n-\mu). \] Let $E_\alpha$ be the joint nuisance event at tolerance $\delta/12$. The nuisance hypothesis gives $\mathbb P(E_\alpha^c)\leq\delta/12$, and on this event $|\hat\alpha_n-\alpha|\leq R_\alpha(n,\delta/12)=a$. Apply Lemma~\ref{lemma:eta_rate} at confidence tolerance $\delta/3$, using truncation level $B_{\mathrm{DR}}$. Its internal nuisance tolerance is $(\delta/3)/4=\delta/12$, its exponent bound is $a$, and its count requirement is $n\pi_t\geq16\log(12/\delta)$. It supplies an event $E_{\mathrm{DR}}$ with failure probability at most $\delta/3$ on which $M\geq n\pi_t/4$ and $|\hat\eta_{n,t}^{\mathrm{DR}}-\eta|\leq\mathcal E_{\mathrm{DR}}$. Apply Lemma~\ref{lemma:eta_ipw} at tolerance $\delta/3$ with $B_{\mathrm{IPW}}$; its sampling coefficients are $12/\delta$ and its nuisance and exponent arguments are also $\delta/12$. It supplies an event $E_{\mathrm{IPW}}$ with failure probability at most $\delta/3$ on which $M\geq n\pi_t/4$ and $|\hat\eta_{n,t}^{\mathrm{IPW}}-\eta|\leq\mathcal E_{\mathrm{IPW}}$. The use of $R_p(n/2,\delta/12)$ is permitted by its sample-size monotonicity, since $R_p(n,\delta/12)\leq R_p(n/2,\delta/12)$; no comparison between confidence tolerances is used. Lemma~\ref{lemma:mu_rate}, applied to the full-input Hill selection run at tolerance $\delta/6$, supplies an event $E_\gamma$ with failure probability at most $\delta/6$ on which $|\hat\gamma_n-\gamma|\leq b_n$. Thus \[ E=E_\alpha\cap E_{\mathrm{DR}}\cap E_{\mathrm{IPW}}\cap E_\gamma \] satisfies \[ \mathbb P(E^c)\leq\frac{\delta}{12}+\frac{\delta}{3}+\frac{\delta}{3}+\frac{\delta}{6}=\frac{11\delta}{12}\leq\delta. \] This intersection explicitly contains the exponent inequality as well as both first-factor inequalities and the radial inequality. Independence between these events is unnecessary. On $E$, we have $|\hat\gamma_n|\leq\gamma+b_n$ and hence \[ |\hat\alpha_n\hat\gamma_n-\alpha\gamma| =|\hat\gamma_n(\hat\alpha_n-\alpha)+\alpha(\hat\gamma_n-\gamma)| \leq a(\gamma+b_n)+|\alpha|b_n=Q_n. \] If $d_n>0$, exact reciprocal subtraction gives \[ |\hat\mu_n-\mu| =\frac{|\hat\alpha_n\hat\gamma_n-\alpha\gamma|}{h d_n} \leq\frac{Q_n}{h d_n}=\mathcal H_n. \] If $d_n=0$, Algorithm~\ref{alg:nete} sets $\hat\mu_n=0$, so \[ |\hat\mu_n-\mu|=1/h=\mathcal H_n. \] Thus $|\hat\mu_n-\mu|\leq\mathcal H_n$ throughout $E$, without any uniform Lipschitz assertion near the pole. On $E$, the DR product identity and the triangle inequality give \[ \begin{aligned} |\widehat\theta_{n,t}^{\mathrm{DR}}-\theta^{\tmop{NETE}}| &\leq|\hat\mu_n|\,|\hat\eta_{n,t}^{\mathrm{DR}}-\eta|+|\eta|\,|\hat\mu_n-\mu|\\ &\leq|\hat\mu_n|\mathcal E_{\mathrm{DR}}+|\eta|\mathcal H_n. \end{aligned} \] Here Lemma~\ref{lemma:eta_rate} supplies the score-dependent sampling term, the product $R_p(n/2,\delta/12)R_g(n_t,\delta/12)$, the tail envelope $2\bar e(t)$, the angular Wasserstein term and the full exponent perturbation. Its exponent calculation uses $|\hat\alpha_n-\alpha|\leq a$ on $E_\alpha$ and gives $\mathbb E[\|U\|^a-1\mid\|U\|>t]\leq t^a-1+Kat^a$ for $K=4/(\beta-a_0)$. No linearization of this expression, replacement of $\bar e(t)$ by $e(t)$, or suppression of angular or radial logarithms is made. The count inequality also holds on $E$. For IPW, the product identity gives \[ \widehat\theta_{n,t}^{\mathrm{IPW}}-\theta^{\tmop{NETE}} =\hat\mu_n(\hat\eta_{n,t}^{\mathrm{IPW}}-\eta)+\eta(\hat\mu_n-\mu). \] On $E$, Lemma~\ref{lemma:eta_ipw} gives $|\hat\eta_{n,t}^{\mathrm{IPW}}-\eta|\leq\mathcal E_{\mathrm{IPW}}$, and the reciprocal calculation gives $|\hat\mu_n-\mu|\leq\mathcal H_n$. Therefore \[ \begin{aligned} |\widehat\theta_{n,t}^{\mathrm{IPW}}-\theta^{\tmop{NETE}}| &\leq|\hat\mu_n|\,|\hat\eta_{n,t}^{\mathrm{IPW}}-\eta|+|\eta|\,|\hat\mu_n-\mu|\\ &\leq|\hat\mu_n|\mathcal E_{\mathrm{IPW}}+|\eta|\mathcal H_n. \end{aligned} \] Both bounds and $M\geq n\pi_t/4$ thus hold together with probability at least $1-\delta$. For the additional assertion, we choose the success events in the preceding proof using the constructions in Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw}. Let $E_M=\{M\geq n\pi_t/4\}$. Its failure probability is at most $\exp(-n\pi_t/16)\leq\delta/12$. For each $q\in\{\mathrm{DR},\mathrm{IPW}\}$ put $Z^q=\psi^q I\{|\psi^q|\leq B_q\}$ and $W^q=\psi^q I\{|\psi^q|>B_q\}$. Conditional on $\mathcal T$ and positive $M$, Chebyshev's and Markov's inequalities give two events $E_{Z,q}$ and $E_{W,q}$, each with conditional failure probability at most $\delta/12$, on which \[ \left|M^{-1}\sum_{i\in\mathcal I}Z_i^q-\mathbb E[Z^q\mid\mathcal T,\|U\|>t]\right|\leq\sqrt{\frac{12v_t^q(B_q)}{\delta M}},\qquad M^{-1}\sum_{i\in\mathcal I}|W_i^q|\leq\frac{12b_t^q(B_q)}\delta. \] If $b_t^q(B_q)=0$, the tail variables vanish almost surely. Take $E_{\mathrm{DR}}=E_\alpha\cap E_M\cap E_{Z,\mathrm{DR}}\cap E_{W,\mathrm{DR}}$ and $E_{\mathrm{IPW}}=E_\alpha\cap E_M\cap E_{Z,\mathrm{IPW}}\cap E_{W,\mathrm{IPW}}$. These events give exactly the first-factor conclusions used in the preceding proof, by the two cited lemmas, and each has failure probability at most $\delta/3$. Since their nuisance and count events are shared, their intersection with $E_\gamma$ actually has failure probability at most \[ \delta/12+\delta/12+4\delta/12+\delta/6=8\delta/12. \] On $E_\alpha$, the additional second-moment assumption gives $\tmop{Var}(\psi^q\mid\mathcal T,\|U\|>t)\leq V$. Conditional Chebyshev supplies an event $F_q$ whose failure on $E_\alpha\cap E_M$ has probability at most $\delta/12$ and on which \[ \left|\hat\eta_{n,t}^q-\mathbb E[\psi^q\mid\mathcal T,\|U\|>t]\right|\leq\sqrt{\frac{12V}{\delta M}}\leq\sqrt{\frac{48V}{\delta n\pi_t}}. \] The assertion follows directly if $V=0$, because the centered scores then vanish almost surely. Integrating the conditional failure bounds and using a union bound shows that $E_*=E\cap F_{\mathrm{DR}}\cap F_{\mathrm{IPW}}$ has failure probability at most $10\delta/12\leq\delta$. Thus both general bounds and both finite-variance bounds hold on this one event; it contains the exponent inequality explicitly. The finite-variance alternatives in Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} give the same conditional-mean bounds on $E_\alpha$, with this sampling term. Write $z=a\log t\in[0,1]$. Convexity of the exponential gives $\exp(z)-1\leq(\exp(1)-1)z$; also $t^a\leq\exp(1)$ and $a\leq a\log t$. Therefore \[ t^a-1+Kat^a\leq(\exp(1)-1+K\exp(1))a\log t. \] For sufficiently large $t$, $\bar e(t)\leq1$, so $G_t$ is bounded by the fixed constant $C+1$. On $E_*$, $d_n\geq h-Q_n\geq h/2>0$, and consequently \[ |\hat\mu_n|\leq2/h,\qquad \mathcal H_n\leq2Q_n/h^2, \qquad Q_n\leq\gamma a+(a_0+|\alpha|)b_n\leq C_1(a\log t+b_n). \] Here $C_1$ depends only on the fixed model and $a_0$. Substituting the DR mean bound and the finite-variance sampling bound in the DR product identity gives the additional DR display. Substituting the IPW mean bound and its finite-variance sampling bound in the IPW product identity gives the additional IPW display. The factors $2/h$, $|\eta|$, $2/c$, the bounded $G_t$ and the displayed exponent constant can all be absorbed into a constant independent of $n,t,\delta$. Finally the radial tail in the stated model is $\pi_t\asymp t^{-\beta}$, which gives the asserted order of the sampling term. The definition of $b_n$ and every nuisance tolerance remain exactly those displayed in the statement. - why: The nuisance inequalities could concern only random evaluation arguments or hold simultaneously over their support; the proof's applications of the first-factor lemmas determine the latter reading. Under that stronger reading, the following defects remain. (a) The opening identification equality is false: exogeneity conditional on covariates and independence of the noise from treatment do not imply exogeneity conditional on the noise. (b) The asserted sampling rate has no hypothesis controlling normalized outcome fluctuations. (c) The radial-moment inversion is not uniformly Lipschitz near its pole, and monotone estimation-error bounds need not approach zero. (d) The exponent contribution cannot generally be replaced by $\log(t)R_\alpha(n,\delta)$: its reproduced form contains $t^a-1+Kat^a$. (e) The homogeneity contribution requires $\bar e(t)$, rather than the point value $e(t)$. (f) The angular rate omits a necessary logarithm when $\beta=1$. (g) A nuisance event with failure probability $\delta$ leaves no probability budget for evaluation error; sample-size monotonicity does not compare error bounds at different confidence tolerances. (h) The proof also absorbs the logarithmic factors and the different regimes of Lemma~\ref{lemma:mu_rate} into a rate that this lemma does not supply. These are statement failures in (a)--(g), with the false proof steps identified below; (h) is an additional invalid conversion of the available dependency bound. ### `thm:cor-rate` (`Arxiv_template.tex:363-386`) - as printed: \label{cor:rate} Under the assumptions of Theorem~\ref{thm:rate}, further suppose that \begin{align*} R_p (n, \delta) = \Theta (\log (1 / \delta) n^{- 1 / 2}) , R_g (n, \delta) = \Theta (\log (1 / \delta) n^{- 1 / 2}), R_\alpha (n, \delta) = \Theta( \log(1/\delta) n^{-c_\alpha}), \end{align*} for some $c_\alpha > 0 $, the following conclusions hold. \begin{enumerate} \item If $s \in (0, 1 / (2 + \max \{1, \beta\}))$, takes $t_n = \Theta (n^{ (1 - 2 s) \hat{\gamma}_{n}})$, with probability at least $1 - \delta$, we have \begin{align*} | {{\widehat{\theta }_{n, t}^{\text{DR} }} } - \theta^{NETE} | & = O (e (t_n) + n^{- s} \log (1 / \delta) + n^{-c_\alpha} \log(n)\log (1 / \delta)) . \end{align*} \item If $s \in [1 / (2 + \max \{1, \beta\}), 1 / 2)$, \ takes $t = \Theta (n^{(\hat{\gamma}_{n} / (1 + 2 \min \{1, \hat{\gamma}_{n} \})})$, with probability at least $1 - \delta$, we have \begin{align*} | {{\widehat{\theta }_{n, t}^{\text{DR} }} } - \theta^{NETE} | & = O (e (t_n) + n^{- 1 / (2 + \max \{\beta, 1\})} \log (1 / \delta) + n^{-c_\alpha} \log(n)\log (1 / \delta)) . \end{align*} \end{enumerate} - corrected: Under the assumptions of Theorem~\ref{thm:rate}, further suppose that \begin{align*} R_p(n,\delta)&=\Theta(\log(1/\delta)n^{-1/2}),\qquad R_g(n,\delta)=\Theta(\log(1/\delta)n^{-1/2}),\\ R_\alpha(n,\delta)&=\Theta(\log(1/\delta)n^{-c_\alpha}) \end{align*} for some $c_\alpha>0$. Suppose in addition that the conditions of Proposition~\ref{prop:id} hold. Let $\mathcal T$ consist of the training split and independent auxiliary data. Suppose that $\hat p$, $\hat g$ and $\hat\alpha_n$ are $\mathcal T$-measurable and are the same functions and exponent for every input threshold. Write $R_U=\|U\|$, $S_U=U/\|U\|$, $\gamma=1/\beta$, $\pi_t=\mathbb P(R_U>t)$, and \[ m_d^t(x,v)=\mathbb E[Y/R_U^\alpha\mid X=x,D=d,S_U=v,R_U>t]. \] In each case below, let $\kappa$ be the exponent function specified in that case. Use fixed constants $0t)$ is the training exceedance count. On this same event suppose that the fitted-exponent pseudo-outcome has finite conditional absolute mean for every $t\in I_n$ and that, for a constant $V<\infty$ independent of $n$ and $t$, \[ \mathbb E[\psi^2\mid\mathcal T,R_U>t]\leq V\qquad(t\in I_n), \] where \[ \psi=\hat g(X,1,S_U)-\hat g(X,0,S_U)+\frac{D-\hat p(X)}{\hat p(X)(1-\hat p(X))}\left(\frac{Y}{R_U^{\hat\alpha_n}}-\hat g(X,D,S_U)\right). \] Set the first-factor estimate to zero if there are no evaluation exceedances. Run the full-input Hill selection of Algorithm~\ref{alg:nete} at tolerance $\delta/6$, and put \[ b_n=C_{\delta/6}\begin{cases} \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/3},&0<\beta<1,\\ \left(\dfrac{\log n\,\log((48/\delta)\log n)}{n}\right)^{1/3},&\beta=1,\\ \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/(2+\beta)},&\beta>1, \end{cases} \] with the constant supplied by Lemma~\ref{lemma:mu_rate}. Define \[ \bar e(t)=\sup_{r\geq t}e(r). \] Then, for each fixed $\delta\in(0,1/2)$ and every sufficiently large even $n$, the following conclusions hold. Here $k\geq0$ is the exponent of the logarithm in the angular assumption of Theorem~\ref{thm:rate}, taken from Proposition 3.1 of \citet{zhang2023wasserstein}; under that assumption the angular term at the threshold $t_n$ is of the order shown, up to this power of $\log n$. \begin{enumerate} \item If $s\in(0,1/(2+\max\{1,\beta\}))$, take $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ and put $\kappa(v)=(1-2s)v$. With probability at least $1-\delta$, we have \[ \begin{aligned} |\widehat\theta_{n,t_n}^{\text{DR}}-\theta^{NETE}|=O\bigl(&\sqrt{V/\delta}\,n^{-s}+\log^2(6/\delta)n^{-1/2-s}\\ &+\bar e(t_n)+(\log n)^k n^{-s}+n^{-c_\alpha}\log n\log(6/\delta)+|\alpha|b_n\bigr). \end{aligned} \] \item If $s\in[1/(2+\max\{1,\beta\}),1/2)$, take $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ and put $\kappa(v)=v/(1+2\min\{1,v\})$. Write $q=1/(2+\max\{\beta,1\})$. With probability at least $1-\delta$, we have \[ \begin{aligned} |\widehat\theta_{n,t_n}^{\text{DR}}-\theta^{NETE}|=O\bigl(&\sqrt{V/\delta}\,n^{-q}+\log^2(6/\delta)n^{-1/2-q}\\ &+\bar e(t_n)+(\log n)^k n^{-q}+n^{-c_\alpha}\log n\log(6/\delta)+|\alpha|b_n\bigr). \end{aligned} \] \end{enumerate} - no longer uses `ass:cor-rate-errors` - side condition added by the confirming referee `ass:pin-33`: $U\perp(X,D)$. — pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly. - side condition added by the confirming referee `ass:pin-34`: $\|X\|_1\leqslant R$ almost surely. — pins reading of ass:f-homo-bounded-x: Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm. - side condition added by the confirming referee `ass:pin-36`: For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\|z\|_1>\zeta k^{(1-2s)/\beta}$. — pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$. - side condition added by the confirming referee `ass:pin-38`: For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\|U\|/b_n,U/\|U\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. — pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation. - side condition added by the confirming referee `ass:pin-39`: $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. — pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely. - side condition added by the confirming referee `ass:pin-41`: The existential cutoff $t_W$ is chosen sufficiently large for Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} and so that $\sup_{r\geq t_W}e(r)\leq1$ and $\mathbb P(\|U\|>tr)/\mathbb P(\|U\|>t)\leq4r^{-\beta}$ for every $t\geq t_W$ and $r\geq1$. — pins reading of thm:rate: The proof of Theorem~\ref{thm:rate} invokes the first-factor lemmas' sufficiently-large-threshold conclusions, uses the radial moment inequality with $K=4/(\beta-a_0)$, and bounds $G_t$ by $C+1$. The stated Pareto-type radial law and $e(t)\to0$ permit this enlargement of the existential cutoff. - side condition added by the confirming referee `ass:pin-42`: For each fixed $\mathcal T$-measurable choice $B>0$, the displayed probability bound uses the base conditions of Lemma~\ref{lemma:eta_rate}; its angular specialization additionally uses that lemma's angular assumption and $t\geq t_W$, and its variance alternative additionally requires $V_t^{\text{IPW}}<\infty$. — pins reading of lem:eta-ipw: The statement explicitly says ``Under the angular assumption'' and ``When'' the variance is finite; the amended proof applies conditional sampling inequalities to the chosen $B$. - side condition added by the confirming referee `ass:pin-43`: The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\|x\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\|u\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. — pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere. - side condition added by the confirming referee `ass:pin-44`: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. — pins reading of ass:rate-nuisance-errors: Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation. - side condition added by the confirming referee `ass:pin-45`: The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. — pins reading of def:nete-algorithm: Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention. - side condition added by the confirming referee `ass:pin-7`: $Y(D)=Y$ almost surely throughout the paper. — pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely. - side condition added by the confirming referee `ass:pin-fix-0b814294`: The equal split in Algorithm~\ref{alg:nete} is defined for positive even sample sizes $n$, with $|\mathcal D_1|=|\mathcal D_2|=n/2$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-10c2ec7b`: The component dimension in the density of Assumption~\ref{assump:mvary} is $m=d_z$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-27ee3ba1`: For each fixed $\delta\in(0,1/2)$, the probability guarantee quantifies separately over every sufficiently large even $n$. — pins reading of thm:cor-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-6857bb4d`: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments. — pins reading of thm:rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8964f8dd`: Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8a129280`: The fitted-exponent pseudo-outcome has finite conditional absolute mean under the evaluation tail law: $\mathbb E[|Y|/\|U\|^{\hat\alpha_n}\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t]<\infty$ almost surely on each joint nuisance event. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8cd5c960`: The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\|u\|_1=1$. — pins reading of prop:id: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8ef6fd29`: The homogeneity inequality holds for both treatments and every positive radius, on the positive unit sphere for the $\ell_1$ norm. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9567640d`: The equal split uses $n\in2\mathbb N$. — pins reading of def:nete-algorithm: stated by the repair - side condition added by the confirming referee `ass:pin-fix-b3a62d30`: The symbol $V_t$ denotes $\tmop{Var}(\psi\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t)$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c3605ca3`: The sample-size domain of $R_g(q,\varepsilon)$ includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c4d18fdb`: For each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that, for every integer $n\geq n_0$, the bound has probability at least $1-\delta$ under the law of the sample of size $n$. — pins reading of lem:mu-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c90a58f3`: The count $n_t=\sum_{i=1}^{n/2}I(\|U_i\|>t)$ is the training-split exceedance count, with the training observations indexed first after the random split. — pins reading of thm:rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-d303828c`: The constants $c_-$ and $c_+$ in $t_n=\Theta(n^{\kappa(\hat\gamma_n)})$ are deterministic and independent of $n$ and the observed sample. — pins reading of thm:cor-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ebdb71b8`: The sampling term comprises both $\sqrt{4v_t^{\text{IPW}}(B)/(\delta M)}$ and $(1+4/\delta)b_t^{\text{IPW}}(B)$. — pins reading of lem:eta-ipw: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f18f3355`: The target $\eta$ denotes $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. — pins reading of lem:eta-rate: stated by the repair - proof change: the printed proof, revised step by step] Fix $\delta\in(0,1/2)$ and an even $n$. Let $E_T$ be the joint uniform nuisance and moment event in the statement. Let $E_\gamma$ be the event supplied by Lemma~\ref{lemma:mu_rate} at tolerance $\delta/6$. Each event has failure probability at most $\delta/6$. On $E_\gamma$, $|\hat\gamma_n-\gamma|\leq b_n$. All three displayed definitions of $b_n$ satisfy $b_n\log n\to0$ for fixed $\delta$. The function $v\mapsto(1-2s)v$ is Lipschitz with constant at most $1$. The function $v\mapsto v/(1+2\min\{1,v\})$ is continuous at $v=1$ and has derivative $1/(1+2v)^2$ for $v<1$ and $1/3$ for $v>1$, so it too is Lipschitz with constant at most $1$. Consequently \[ \frac{n^{\kappa(\hat\gamma_n)}}{n^{\kappa(\gamma)}}=\exp\{(\kappa(\hat\gamma_n)-\kappa(\gamma))\log n\}\in[1/2,2] \] on $E_\gamma$ for sufficiently large $n$, and hence $t_n\in I_n$. In particular, the correct radial identity is \[ n^{\beta\hat\gamma_n}=n\exp\{\beta(\hat\gamma_n-\gamma)\log n\},\qquad n^{\beta(\hat\gamma_n-\gamma)}=1+O(b_n\log n) \] on this event. Put $m=n/2$ and $r=(1-\beta\kappa(\gamma))/2$. In the first case $\beta\kappa(\gamma)=1-2s$, so $r=s$. In the second case, with $a_* =\min\{1,\beta\}$, we have $\kappa(\gamma)=1/(\beta+2a_*)$ and $r=a_*/(\beta+2a_*)=q$. Thus $r>0$ in both cases. The Pareto-type model gives $\pi_t\asymp t^{-\beta}$ for large $t$, and therefore \[ \pi_{\ell_n}\asymp\pi_{u_n}\asymp n^{-1+2r},\qquad m\pi_{u_n}\asymp n^{2r}. \] The training and evaluation exceedance counts at the deterministic threshold $u_n$ are each binomial with parameters $m,\pi_{u_n}$. For either count $N_{u_n}$, the binomial lower-tail inequality gives \[ \mathbb P(N_{u_n}t)-P_T[\psi I(R_U>t)]\},\qquad C(t)=\sum_{i=1}^{m}\{I(R_i>t)-\pi_t\}. \] For completeness, an independent conditional copy $Z'_1,\ldots,Z'_m$ and Jensen's inequality yield \[ \mathbb E[\sup_{t\in I_n}|A(t)|^2\mid\mathcal T]\leq\mathbb E\left[\sup_{t\in I_n}\left|\sum_{i=1}^{m}\{\psi(Z_i)I(R_i>t)-\psi(Z'_i)I(R'_i>t)\}\right|^2\middle|\mathcal T\right]. \] Introduce independent symmetric signs $\varepsilon_i$. Swapping each independent pair leaves its law invariant, so the right-hand side equals the same expectation with each difference multiplied by $\varepsilon_i$. The inequality $|a-b|^2\leq2|a|^2+2|b|^2$ then bounds it by \[ 4\mathbb E\left[\sup_{t\in I_n}\left|\sum_{i=1}^{m}\varepsilon_i\psi(Z_i)I(R_i>t)\right|^2\middle|\mathcal T\right]. \] Conditional on the observations, order those with $R_i>\ell_n$ by decreasing radius. The signed sums indexed by $t\in I_n$ are among the partial sums of independent mean-zero variables in this order. The square-integrable martingale maximal inequality bounds their expected squared maximum by $4\sum_i\psi(Z_i)^2I(R_i>\ell_n)$. Thus, on $E_T$, \[ \mathbb E[\sup_{t\in I_n}|A(t)|^2\mid\mathcal T]\leq16mV\pi_{\ell_n}. \] The same argument with $\psi=1$ gives $\mathbb E[\sup_{t\in I_n}|C(t)|^2\mid\mathcal T]\leq16m\pi_{\ell_n}$. Markov's inequality therefore supplies two conditional events, each with failure probability at most $\delta/6$, on which \[ \sup_{t\in I_n}|A(t)|\leq\sqrt{96mV\pi_{\ell_n}/\delta},\qquad \sup_{t\in I_n}|C(t)|\leq\sqrt{96m\pi_{\ell_n}/\delta}. \] These conditional failure bounds also hold unconditionally on $E_T$ by integration. Suprema of these threshold processes are measurable because they can be taken over a countable dense set together with the band endpoints. Write $\mu_t=P_T[\psi\mid R_U>t]$ and $M_t=\sum_i I(R_i>t)$. On $E_T$, Cauchy--Schwarz gives $|\mu_t|\leq\sqrt V$. Since \[ \sum_i\psi(Z_i)I(R_i>t)-M_t\mu_t=A(t)-\mu_t C(t), \] the two maximal bounds and the evaluation-count bound imply, simultaneously for every $t\in I_n$, \[ |\hat\eta_{n,t}^{\tmop{DR}}-\mu_t|\leq4\sqrt{\frac{96V\pi_{\ell_n}}{\delta m\pi_{u_n}^2}}=O(\sqrt{V/\delta}\,n^{-r}). \] This inequality holds at $t_n$ on $E_\gamma$. It remains to bound the conditional score mean. Put $a=R_\alpha(n,\delta/6)$, choose $a_0\in(0,\beta)$, let $C_g$ bound $|g|$ on its compact domain, and set $G_t=C_g+\bar e(t)$. For sufficiently large $n$, $a\leq a_0$ and $G_t$ is bounded uniformly on $I_n$. Let $\psi_\alpha$ denote the score with the exponent replaced by $\alpha$, and let \[ \eta_\alpha^t=\mathbb E[(f(X,1,U)-f(X,0,U))/R_U^\alpha\mid R_U>t]. \] Independence of $U$ from $(X,D)$ and conditional expectation over treatment give \[ P_T[\psi_\alpha\mid R_U>t]-\eta_\alpha^t=\mathbb E\left[(\hat p-p)\left\{\frac{\hat g(X,1,S_U)-m_1^t(X,S_U)}{\hat p}+\frac{\hat g(X,0,S_U)-m_0^t(X,S_U)}{1-\hat p}\right\}\middle|\mathcal T,R_U>t\right]. \] Estimated-propensity overlap and the uniform nuisance event bound its absolute value by $(2/c)R_p(n/2,\delta/6)R_g(n_t,\delta/6)$. The training-count bound and the assumed rates make this $O(\log^2(6/\delta)n^{-1/2-r})$ uniformly on $I_n$. The exponent calculation in Lemma~\ref{lemma:eta_rate} is a conditional-mean calculation: taking the conditional outcome mean first and using homogeneity bounds the exponent difference by \[ \frac{2G_t}{c}\mathbb E[R_U^a-1\mid R_U>t]\leq\frac{2G_t}{c}(t^a-1+Kat^a),\qquad K=4/(\beta-a_0). \] This population inequality holds for every sufficiently large $t$, so it is available throughout $I_n$. Since $a=O(\log(6/\delta)n^{-c_\alpha})$ and $\log t=O(\log n)$ on this band, $a\log t\to0$. The inequalities $\exp(x)-1\leq2x$ for sufficiently small $x\geq0$ and $t^a\leq2$ show that this term is $O(\log(6/\delta)n^{-c_\alpha}\log n)$. Homogeneity at every radius above $t$ gives an error at most $2\bar e(t)$. The function $v\mapsto\mathbb E_X[g(X,1,v)-g(X,0,v)]$ is $2L$-Lipschitz. Independence of $X$ and $U$ and the Wasserstein coupling inequality therefore give \[ |\eta_\alpha^t-\eta|\leq2\bar e(t)+\mathcal A_t, \] where $\eta$ is the limiting angular contrast. Lemma~\ref{lemma:eta_rate} supplies the bound for $\mathcal A_t$; we retain that term without suppressing any logarithm. Finally Proposition~\ref{prop:id} gives $\theta^{\tmop{NETE}}=\eta/h$, where $h=1-\alpha\gamma>0$. On $E_T\cap E_\gamma$, \[ |\hat\alpha_n\hat\gamma_n-\alpha\gamma|\leq a(\gamma+b_n)+|\alpha|b_n=:Q_n\to0. \] For sufficiently large $n$, $Q_n\leq h/2$, whence $|\hat\mu_n|\leq2/h$ and exact reciprocal subtraction gives $|\hat\mu_n-1/h|\leq2Q_n/h^2$. The product identity \[ \widehat\theta_{n,t_n}^{\text{DR}}-\theta^{\tmop{NETE}}=\hat\mu_n(\hat\eta_{n,t_n}^{\tmop{DR}}-\eta)+\eta(\hat\mu_n-1/h) \] now proves the stated bound with $r=s$ in the first case and $r=q$ in the second. The term $a$ in $Q_n$ is absorbed by $a\log n$, while $|\alpha|b_n$ is retained. There are six failure events: the uniform nuisance event, the Hill event, the two count events, and the two maximal-process events. Each contributes at most $\delta/6$, so their union has probability at most $\delta$. No sequence-wide probability assertion is used. ### `lem:mu-rate` (`Arxiv_template.tex:607-616`) - as printed: \label{lemma:mu_rate}Under Assumption \ref{assump:iid}, \ref{assump:mvary} with probability at least $1 - \delta$, for sufficiently large $n$, we have \begin{align*} | \gamma - \hat{\gamma}_n | & \leqslant O \left( \left( \frac{\log (2 / \delta)}{n} \right)^{1 / (2 + \beta)} \right), \end{align*} where $\gamma = 1 / \beta$ is the EVI of $U$. - corrected: Under Assumption \ref{assump:iid}, \ref{assump:mvary}, using the standing convention $\|\cdot\|=\|\cdot\|_1$, let $R_i=\|U_i\|_1$ for all $i\in\{1,\ldots,n\}$ and order these full-input radii as $R_{[1]}\geq\cdots\geq R_{[n]}$. For $n\geq2$, define \[ H_n(k)=\frac1k\sum_{j=1}^k\log\frac{R_{[j]}}{R_{[k+1]}},\qquad 1\leq k\leq n-1. \] For $\delta\in(0,1)$, set \[ \begin{aligned} L_n&=\min\{n-1,\lceil100\log n\rceil\},&J_n&=1+\lceil\log n/\log2\rceil,\\ q_n(\delta)&=\log(4J_n/\delta),&c_n(\delta)&=2\sqrt{q_n(\delta)}+\frac{q_n(\delta)}{\sqrt{L_n}},\\ \rho_n(\delta)&=16c_n(\delta). \end{aligned} \] Let \[ \begin{aligned} \widetilde k_n&=\max\bigl\{k\in\{L_n,\ldots,n-1\}:\\ &\qquad |H_n(i)-H_n(k)|\leq\frac{H_n(i)\rho_n(\delta)}{\sqrt i}\\ &\qquad \text{for every }i\in\{L_n,\ldots,k\}\bigr\},\\ \widetilde\gamma_n&=H_n(\widetilde k_n). \end{aligned} \] For each fixed $\delta\in(0,1)$ and every sufficiently large $n$, with probability at least $1-\delta$, we have \[ |\gamma-\widetilde\gamma_n|\leq C_\delta \begin{cases} \left(\dfrac{\log((8/\delta)\log n)}{n}\right)^{1/3},&0<\beta<1,\\ \left(\dfrac{\log n\,\log((8/\delta)\log n)}{n}\right)^{1/3},&\beta=1,\\ \left(\dfrac{\log((8/\delta)\log n)}{n}\right)^{1/(2+\beta)},&\beta>1, \end{cases} \] where $C_\delta$ may depend on the fixed distribution and on $\delta$. where $\gamma = 1 / \beta$ is the EVI of $U$. - side condition added by the confirming referee `ass:pin-33`: $U\perp(X,D)$. — pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly. - side condition added by the confirming referee `ass:pin-36`: For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\|z\|_1>\zeta k^{(1-2s)/\beta}$. — pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$. - side condition added by the confirming referee `ass:pin-38`: For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\|U\|/b_n,U/\|U\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. — pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation. - side condition added by the confirming referee `ass:pin-fix-c4d18fdb`: For each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that, for every integer $n\geq n_0$, the bound has probability at least $1-\delta$ under the law of the sample of size $n$. — pins reading of lem:mu-rate: stated by the repair - proof change: the printed proof, revised step by step] We adopt the non-asymptotic analysis of the adaptive Hill estimator for EVI in {\citep{boucheron2015tail}}. We give the exponential-spacing argument and the confidence calibration explicitly. Write $R=\|U\|_1$. Fix a member of $M_K$, put $a_i=\|A_{\cdot i}\|_1$, and let $Z_i^0$ be independent nominal Pareto type-II components of index $\beta$. The component density is a positive constant $c_0$ times the nominal product density whenever $\|z\|_1>\zeta K^{(1-2s)/\beta}$. Since \[ R=\sum_{i=1}^m a_iZ_i,\qquad l\leq a_i\leq u,\qquad R\leq u\|Z\|_1, \] the survival and density of $R$ beyond $u\zeta K^{(1-2s)/\beta}$ are $c_0$ times those of $T=\sum_i a_iZ_i^0$. This transfer uses the original nominal components without shifting their supports. Put $W_i=a_iZ_i^0$, $V_i=\sum_{j\ne i}W_j$, and $f_i(x)=\beta a_i^\beta(a_i+x)^{-\beta-1}$ for $x\geq0$, extended by zero for $x<0$. Partitioning the convolution integral according to the largest summand, whose ties have probability zero, gives \[ f_T(r)=\sum_i\mathbb E\left[f_i(r-V_i)I\{\max_{j\ne i}W_j\leq r-V_i\}\right]. \] On this event $r-V_i\geq r/m$. If $V_i\leq r/2$, the event holds and the mean value theorem gives $|f_i(r-V_i)-f_i(r)|\leq C r^{-\beta-2}V_i$. On the part where $V_i>r/2$, the indicated density is at most $C r^{-\beta-1}$. A union bound using the tails of the $m-1$ summands gives $\mathbb P(V_i>x)\leq Cx^{-\beta}$ for $x\geq1$. Integrating this bound and using $\mathbb E[V_iI\{V_i\leq r/2\}]\leq\mathbb E[\min\{V_i,r/2\}]$ gives \[ \mathbb E[V_iI\{V_i\leq r/2\}]\leq \begin{cases}Cr^{1-\beta},&\beta<1,\\ C\log r,&\beta=1,\\ C,&\beta>1.\end{cases} \] Consequently \[ \left|f_T(r)-\sum_i f_i(r)\right| \leq C\sum_i\left(r^{-\beta-2}\mathbb E[V_iI\{V_i\leq r/2\}]+r^{-\beta-1}\mathbb P(V_i>r/2)\right). \] Since $\sum_i f_i(r)=\beta\sum_i a_i^\beta r^{-\beta-1}(1+O(r^{-1}))$, set \[ b_\beta(r)=\begin{cases}r^{-\beta},&\beta<1,\\r^{-1}\log r,&\beta=1,\\r^{-1},&\beta>1,\end{cases}\qquad C_0=c_0\sum_i a_i^\beta>0. \] The actual radial density and survival therefore satisfy, for sufficiently large $r$, \[ f_R(r)=\beta C_0r^{-\beta-1}(1+O(b_\beta(r))),\qquad \bar F_R(r)=C_0r^{-\beta}(1+O(b_\beta(r))). \] The second estimate follows by integrating the first error from $r$ to infinity: the integrals of $r^{-2\beta-1}$, $r^{-3}\log r$, and $r^{-\beta-2}$ have orders $r^{-2\beta}$, $r^{-2}\log r$, and $r^{-\beta-1}$, respectively. Thus no constant-coefficient expansion at $\beta=1$ is required. When $\beta>1$ and a survival expansion with coefficient $D$ is used, differentiation gives density coefficient $D(1+1/\beta)$; the density and survival envelopes just proved avoid differentiating an unspecified remainder. Let $Q(v)=F_R^{-1}(1-1/v)$ be the radial tail quantile. Its differentiable tail has $Q(v)\sim(C_0v)^{1/\beta}$ and \[ \eta(v)=\frac{vQ'(v)}{Q(v)}-\gamma =\frac{\bar F_R(Q(v))}{Q(v)f_R(Q(v))}-\frac1\beta =O(b_\beta(Q(v))). \] Choose a fixed $v_0>1$ in that tail and define $\bar\eta(v)=\sup_{w\geq\max\{v,v_0\}}|\eta(w)|$ for every $v>0$. For sufficiently large $v$ this gives \[ \bar\eta(v)\leq \begin{cases}Cv^{-1},&\beta<1,\\Cv^{-1}\log v,&\beta=1,\\Cv^{-1/\beta},&\beta>1.\end{cases} \] For $y\geq z\geq\log v_0$, integrating the quantile derivative gives \[ |\log Q(e^y)-\log Q(e^z)-\gamma(y-z)|\leq\bar\eta(e^z)(y-z). \] We next prove the simultaneous stochastic bound used for selection. Let $Y_{(1)}\geq\cdots\geq Y_{(n)}$ be decreasing order statistics of independent standard exponential variables, with $Y_{(n+1)}=0$. Their joint density is $n!\exp(-\sum_j y_j)$ on the ordered region. The change of variables $E_j=j(Y_{(j)}-Y_{(j+1)})$ has Jacobian $1/n!$ and $\sum_j y_j=\sum_j E_j$, so the $E_j$ are independent standard exponential variables. The quantile transform represents the ordered radii as $Q(e^{Y_{(j)}})$. Define $B_i=\sum_{j=1}^i(E_j-1)$ and $A_i=i^{-1}\sum_{j=1}^iE_j$ for this argument only. For $0<\lambda<1$, \[ \log\mathbb E e^{\lambda(E_j-1)}=-\lambda-\log(1-\lambda)\leq\frac{\lambda^2}{2(1-\lambda)},\qquad \log\mathbb E e^{-\lambda(E_j-1)}=\lambda-\log(1+\lambda)\leq\frac{\lambda^2}{2(1-\lambda)}. \] Hence $\exp(\pm\lambda B_i-i\lambda^2/(2(1-\lambda)))$ are nonnegative supermartingales with initial value one. The maximal supermartingale inequality implies \[ \mathbb P\left(\max_{i\leq m}|B_i|\geq\sqrt{2mx}+x\right)\leq2e^{-x}. \] Indeed, each one-sided probability is at most $\exp(-\lambda a+m\lambda^2/(2(1-\lambda)))$ for $a=\sqrt{2mx}+x$; taking $\lambda=\sqrt{2x/m}/(1+\sqrt{2x/m})$ makes this bound $e^{-x}$. Apply this inequality at the ends of the dyadic intervals $2^a\leq i<2^{a+1}$, using $m=\min\{n,2^{a+1}\}$ and $x=q_n(\delta)$. There are at most $J_n$ such intervals. On each interval $\sqrt{2m}/\sqrt i\leq2$. The union bound therefore supplies an event $E$ of probability at least $1-\delta/2$ on which \[ \sqrt i\,|A_i-1|\leq2\sqrt{q_n(\delta)}+q_n(\delta)/\sqrt i\leq c_n(\delta),\qquad L_n\leq i\leq n. \] For fixed $\delta$, $c_n(\delta)=O_\delta(\sqrt{\log\log n})$ and $c_n(\delta)/\sqrt{L_n}\to0$. Write $c_n=c_n(\delta)$ and consider the deterministic index \[ k_* =\max\left\{k\in\{L_n,\ldots,n-1\}:\ \frac{n}{2k}\geq v_0,\quad \sqrt k\,\bar\eta\left(\frac{n}{2k}\right)\leq\gamma c_n\right\}. \] The choices below show that this set is nonempty for all sufficiently large $n$. Put $T_*=e^{Y_{(k_*+1)}}$. The number of exponential observations exceeding $\log(n/(2k_*))$ has binomial law with mean $2k_*$. Its multiplicative lower-tail bound gives \[ \mathbb P\left(T_*<\frac{n}{2k_*}\right)\leq\exp(-k_*/4)\leq\delta/2 \] for sufficiently large $n$, because $k_*\geq L_n$. Indeed, if $B\sim\tmop{Bin}(n,p)$ and $\mu=np$, Markov's inequality gives \[ \mathbb P(B\leq\mu/2)\leq e^{\mu\log2/2}\mathbb E[2^{-B}] =e^{\mu\log2/2}(1-p/2)^n \leq e^{-\mu(1-\log2)/2}\leq e^{-\mu/8}, \] using $\log2\leq3/4$; here $p=2k_*/n$ and $\mu=2k_*$. Let $E_*$ be the complementary event. On $E_*$ all spacings involved in $H_n(i)$ for $i\leq k_*$ lie above $\log v_0$. Since $H_n(i)=i^{-1}\sum_{j=1}^i j\log(R_{[j]}/R_{[j+1]})$ by telescoping, the quantile inequality thus gives \[ |H_n(i)-\gamma A_i|\leq\bar\eta(T_*)A_i\leq\frac{\gamma c_n}{\sqrt{k_*}}A_i,\qquad L_n\leq i\leq k_*. \] On $E\cap E_*$, for large enough $n$, $c_n/\sqrt{L_n}\leq1/6$. It follows that \[ |H_n(i)-\gamma|\leq\frac{\gamma c_n}{\sqrt i}+\frac{\gamma c_n}{\sqrt{k_*}}\left(1+\frac{c_n}{\sqrt i}\right) \leq\frac{3\gamma c_n}{\sqrt i},\qquad \gamma/2\leq H_n(i)\leq3\gamma/2. \] For every $L_n\leq i\leq k\leq k_*$, \[ |H_n(i)-H_n(k)|\leq\frac{6\gamma c_n}{\sqrt i} \leq\frac{H_n(i)\rho_n(\delta)}{\sqrt i}. \] Thus $k_*$ qualifies for the displayed prefix rule, and its selected index $\widetilde k_n$ is at least $k_*$. The set in that rule is nonempty almost surely because $k=L_n$ only compares with itself. The comparison at $i=k_*$ now yields \[ |\widetilde\gamma_n-\gamma|\leq|H_n(k_*)-\gamma|+\frac{H_n(k_*)\rho_n(\delta)}{\sqrt{k_*}} \leq\frac{27\gamma c_n}{\sqrt{k_*}}. \] The two events used have total failure probability at most $\delta$; independence between them is unnecessary. To obtain the displayed rates, when $\beta\ne1$ put $p=\min\{1,1/\beta\}$ and choose a sufficiently small fixed $a_*>0$. Let \[ k_0=\left\lfloor a_* n^{2p/(1+2p)}c_n^{2/(1+2p)}\right\rfloor. \] For all sufficiently large $n$, $k_0\geq L_n$, $k_01$ it is $1/(2+\beta)$. When $\beta=1$, take \[ k_0=\left\lfloor a_*n^{2/3}\left(\frac{c_n}{\log n}\right)^{2/3}\right\rfloor. \] Again $k_0\geq L_n$ and $k_0t)$ and $M=|\mathcal I|$, and set $\hat\eta_{n,t}^{\tmop{DR}}=0$ when $M=0$. Let $\bar e(t)=\sup_{r\geq t}e(r)$, let $C$ bound $|g|$ on its compact domain, and put $G_t=C+\bar e(t)$. For a generic evaluation observation define \[ \psi=\hat g(X,1,U/\|U\|)-\hat g(X,0,U/\|U\|)+\frac{D-\hat p(X)}{\hat p(X)(1-\hat p(X))}\left(\frac{Y}{\|U\|^{\hat\alpha_n}}-\hat g(X,D,U/\|U\|)\right). \] For any positive truncation level $B$ chosen using only the training split and independent auxiliary data, define \[ \begin{aligned} v_t(B)&=\mathbb E[\psi^2 I\{|\psi|\leq B\}\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t],\\ a_t(B)&=\mathbb E[|\psi|I\{|\psi|>B\}\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t]. \end{aligned} \] Fix $a_0\in(0,\beta)$ and write $a=R_\alpha(n,\delta/4)$. For sufficiently large $t\geq1$, if $a\leq a_0$ and $n\pi_t\geq16\log(4/\delta)$, then with probability at least $1-\delta$ we have $M\geq n\pi_t/4$ and \begin{align*} |\hat\eta_{n,t}^{\tmop{DR}}-\eta|\leq{}&\sqrt{\frac{4v_t(B)}{\delta M}}+\left(1+\frac4\delta\right)a_t(B)+\frac2c R_p(n/2,\delta/4)R_g(n_t,\delta/4)\\ &+2\bar e(t)+2L W_1(\mathcal L_{U/\|U\|}^t,\mathcal L_{U/\|U\|})+\frac{2G_t}{c}\left(t^a-1+K a t^a\right), \end{align*} where $K$ depends on the radial law and $a_0$. Suppose moreover that the angular Wasserstein term satisfies the bound that Proposition 3.1 of \citet{zhang2023wasserstein} gives for this Pareto-type model, which is taken from that work and not proved here: there are constants $C_W>0$, $k\geq0$ and $t_W\geq1$ such that \[ W_1(\mathcal L_{U/\|U\|}^t,\mathcal L_{U/\|U\|})\leq C_W(\log t)^k t^{-\min\{1,\beta\}}+C_W t^{-\beta s/(1-2s)}\qquad(t\geq t_W). \] Then for $t\geq t_W$ the angular term $2L W_1(\mathcal L_{U/\|U\|}^t,\mathcal L_{U/\|U\|})$ of the display is at most $2LC_W(\log t)^k t^{-\min\{1,\beta\}}+2LC_W t^{-\beta s/(1-2s)}$, with the logarithmic factor retained. When the conditional score variance $V_t$ is finite, the sampling term may instead be taken to be $\sqrt{4V_t/(\delta M)}$. - side condition added by the confirming referee `ass:pin-33`: $U\perp(X,D)$. — pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly. - side condition added by the confirming referee `ass:pin-34`: $\|X\|_1\leqslant R$ almost surely. — pins reading of ass:f-homo-bounded-x: Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm. - side condition added by the confirming referee `ass:pin-36`: For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\|z\|_1>\zeta k^{(1-2s)/\beta}$. — pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$. - side condition added by the confirming referee `ass:pin-38`: For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\|U\|/b_n,U/\|U\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. — pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation. - side condition added by the confirming referee `ass:pin-39`: $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. — pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely. - side condition added by the confirming referee `ass:pin-43`: The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\|x\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\|u\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. — pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere. - side condition added by the confirming referee `ass:pin-44`: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. — pins reading of ass:rate-nuisance-errors: Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation. - side condition added by the confirming referee `ass:pin-45`: The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. — pins reading of def:nete-algorithm: Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention. - side condition added by the confirming referee `ass:pin-7`: $Y(D)=Y$ almost surely throughout the paper. — pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely. - side condition added by the confirming referee `ass:pin-fix-0b814294`: The equal split in Algorithm~\ref{alg:nete} is defined for positive even sample sizes $n$, with $|\mathcal D_1|=|\mathcal D_2|=n/2$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-10c2ec7b`: The component dimension in the density of Assumption~\ref{assump:mvary} is $m=d_z$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8964f8dd`: Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8a129280`: The fitted-exponent pseudo-outcome has finite conditional absolute mean under the evaluation tail law: $\mathbb E[|Y|/\|U\|^{\hat\alpha_n}\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t]<\infty$ almost surely on each joint nuisance event. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8cd5c960`: The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\|u\|_1=1$. — pins reading of prop:id: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8ef6fd29`: The homogeneity inequality holds for both treatments and every positive radius, on the positive unit sphere for the $\ell_1$ norm. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9567640d`: The equal split uses $n\in2\mathbb N$. — pins reading of def:nete-algorithm: stated by the repair - side condition added by the confirming referee `ass:pin-fix-b3a62d30`: The symbol $V_t$ denotes $\tmop{Var}(\psi\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t)$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c3605ca3`: The sample-size domain of $R_g(q,\varepsilon)$ includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f18f3355`: The target $\eta$ denotes $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. — pins reading of lem:eta-rate: stated by the repair - proof change: the printed proof, revised step by step] Condition on the training split and independent auxiliary data, denoted together by $\mathcal T$. Write $R_U=\|U\|$, $S_U=U/\|U\|$, and \[ m_d^t(x,s)=\mathbb E[Y/R_U^{\alpha}\mid X=x,D=d,S_U=s,R_U>t],\qquad \eta_\alpha^t=\mathbb E[(f(X,1,U)-f(X,0,U))/R_U^{\alpha}\mid R_U>t]. \] Independence of $U$ from $(X,D)$ gives $\mathbb E[D\mid X,S_U,R_U>t]=p(X)$. Let $\psi_\alpha$ be the score $\psi$ with $\hat\alpha_n$ replaced by $\alpha$, keeping the same fitted functions. Direct conditional expectation over treatment and outcome gives \begin{align*} \mathbb E[\psi_\alpha\mid\mathcal T,R_U>t]-\eta_\alpha^t=\mathbb E\left[(p(X)-\hat p(X))\left\{\frac{m_1^t(X,S_U)-\hat g(X,1,S_U)}{\hat p(X)}+\frac{m_0^t(X,S_U)-\hat g(X,0,S_U)}{1-\hat p(X)}\right\}\mid\mathcal T,R_U>t\right]. \end{align*} Indeed, the coefficients of $m_1^t-\hat g_1$ and $m_0^t-\hat g_0$ are respectively $p/\hat p-1$ and $1-(1-p)/(1-\hat p)$, which equal $(p-\hat p)/\hat p$ and $(p-\hat p)/(1-\hat p)$. On the joint nuisance event at tolerance $\delta/4$, the uniform bounds and estimated-propensity overlap therefore imply \[ |\mathbb E[\psi_\alpha\mid\mathcal T,R_U>t]-\eta_\alpha^t|\leq\frac2c R_p(n/2,\delta/4)R_g(n_t,\delta/4). \] where $n_t = \sum_{i = 1}^{n/2} I (\|U_i \|> t)$ is a random variable. Keep this training count as the argument of $R_g$; no regularity of that error function beyond its stated monotonicity is needed. Homogeneity at each realized radius gives \[ |\eta_\alpha^t-\mathbb E[g(X,1,S_U)-g(X,0,S_U)\mid R_U>t]|\leq2\bar e(t). \] The function $s\mapsto\mathbb E_X[g(X,1,s)-g(X,0,s)]$ is $2L$-Lipschitz. Independence of $X$ and $U$ and the definition of the limiting angular target therefore give \[ |\eta_\alpha^t-\eta|\leq2\bar e(t)+2L W_1(\mathcal L_{U/\|U\|}^t,\mathcal L_{U/\|U\|}). \] Proposition 3.1 of \cite{zhang2023wasserstein}, with Wasserstein order one and tail index $\beta$, supplies the stated angular bound with its logarithmic factors. Next, the fitted functions cancel from $\psi-\psi_\alpha$. Taking conditional expectation over the outcome and treatment leaves \[ \mathbb E\left[\left\{\frac{p(X)}{\hat p(X)}\frac{f(X,1,U)}{R_U^{\alpha}}-\frac{1-p(X)}{1-\hat p(X)}\frac{f(X,0,U)}{R_U^{\alpha}}\right\}\left(R_U^{\alpha-\hat\alpha_n}-1\right)\mid\mathcal T,R_U>t\right]. \] On the nuisance event, $|\alpha-\hat\alpha_n|\leq a$. Since $R_U\geq t\geq1$, we have $|R_U^{\alpha-\hat\alpha_n}-1|\leq R_U^a-1$. Homogeneity also gives $|f(X,d,U)/R_U^{\alpha}|\leq G_t$, so \[ |\mathbb E[\psi-\psi_\alpha\mid\mathcal T,R_U>t]|\leq\frac{2G_t}{c}\mathbb E[R_U^a-1\mid R_U>t]. \] Choose $\epsilon=(\beta-a_0)/2>0$. Potter's bound gives, uniformly for $r\geq1$ and sufficiently large $t$, \[ \frac{\mathbb P(R_U>tr)}{\pi_t}\leq2r^{-\beta+\epsilon}. \] Integrating the radial tail, for $0\leq a\leq a_0$, yields \begin{align*} \mathbb E[R_U^a-1\mid R_U>t]&=t^a-1+t^a\int_1^{\infty}a r^{a-1}\frac{\mathbb P(R_U>tr)}{\pi_t}\,\tmop{dr}\\ &\leq t^a-1+\frac{2a t^a}{\beta-\epsilon-a}\leq t^a-1+K a t^a. \end{align*} This also covers $a=0$, when both sides vanish. Combining the three mean-error bounds controls $|\mathbb E[\psi\mid\mathcal T,R_U>t]-\eta|$ by the nonsampling terms in the statement. For evaluation, $M$ is binomial with parameters $n/2$ and $\pi_t$, independent of the training split and auxiliary data. The binomial lower-tail bound gives \[ \mathbb P(MB\}$. Chebyshev's inequality gives \[ \mathbb P\left(\left|\frac1M\sum Z_i-\mathbb E[Z\mid\mathcal T,R_U>t]\right|>\sqrt{\frac{4v_t(B)}{\delta M}}\ \middle|\ \mathcal T,M\right)\leq\delta/4. \] Markov's inequality gives \[ \mathbb P\left(\frac1M\sum|W_i|>\frac{4a_t(B)}\delta\ \middle|\ \mathcal T,M\right)\leq\delta/4; \] if $a_t(B)=0$, the tail variables vanish almost surely and this assertion holds directly. On the intersection of these events, the triangle inequality gives \[ \left|\frac1M\sum\psi_i-\mathbb E[\psi\mid\mathcal T,R_U>t]\right|\leq\sqrt{\frac{4v_t(B)}{\delta M}}+\left(1+\frac4\delta\right)a_t(B). \] The joint nuisance event fails with probability at most $\delta/4$. Adding its failure probability, the count failure probability, and the two conditional sampling failure probabilities gives total failure probability at most $\delta$. The estimator is the average of these scores on the resulting event, which proves the bound. If $V_t<\infty$, apply Chebyshev directly to the untruncated score average with failure probability $\delta/4$; this gives the stated replacement sampling term, with total failure probability still at most $\delta$. - why: The reference to Theorem~\ref{thm:rate} imports its hypotheses, not its conclusion, so the lint observation about an absent theorem edge is dismissed. Exogeneity, overlap, the three i.i.d. blocks, and the three Pareto-model blocks are all inherited; the other eight lint observations are dismissed. The setting faithfully distinguishes training and evaluation counts. Ambiguity remains between the displayed covariate-law class and the noise-law interpretation, and between equal and distinct component dimensions. I use the binding limiting angular meaning of $\eta$. Reproduction yields six independent failures: (a) the sampling step asserts a square-root rate without controlling the variance of the normalized score; (b) the homogeneity remainder over exceedances is controlled by an average of $e(\|U\|)$ or its tail supremum, not by $e(t)$; (c) at $\beta=1$, the angular bias can be $\Theta(t^{-1}\log t)$, so the logarithm omitted from [Zhang et al., Proposition 3.1](https://arxiv.org/pdf/2312.09862) is material; (d) exponent error is not uniformly linear in $R_\alpha(n,\delta)\log t$ without a smallness restriction; (e) monotonicity does not permit replacing a random training count by $nt^{-\beta}$ inside an arbitrary $R_g$; and (f) separate nuisance and evaluation guarantees do not retain failure probability $\delta$ with the same nuisance-bound arguments. The evaluation sampling term uses $|\mathcal I|$, rather than the training $n_t$. Both counts can concentrate, but this does not repair (e). Correcting the proof's triangle-inequality signs, conditioning notation, Lipschitz factor, and Bernstein lower-tail sign leaves all six counterexamples intact. ### `lem:eta-ipw` (`Arxiv_template.tex:787-795`) - as printed: \label{lemma:eta_ipw} Under the assumption of Theorem \ref{thm:rate}, with probability at least $1 - \delta$, we have \begin{align*} | \hat{\eta}_{n, t}^{\text{IPW}} - \eta | & \leqslant O (R_p (n/2, \delta) + t^{\beta / 2} n^{- 1 / 2} + t^{- \min \{1, \beta\}} + t^{- \beta s / (1 - 2 s)}) + e (t) . \end{align*} - corrected: Under the assumption of Theorem \ref{thm:rate}, suppose in addition that the conditions of Proposition~\ref{prop:id} hold. Suppose in addition that the conditions of Lemma~\ref{lemma:eta_rate} hold. Use its training split and independent auxiliary data, denoted together by $\mathcal T$, its $\pi_t=\mathbb P(\|U\|>t)$, $M=|\mathcal I|$, $\bar e(t)=\sup_{r\geq t}e(r)$, and $G_t=C+\bar e(t)$, where $C$ bounds $|g|$ on its compact domain. Set $\hat\eta_{n,t}^{\text{IPW}}=0$ if $M=0$. For a generic evaluation observation define \[ \phi=\frac{Y}{\|U\|^{\hat\alpha_n}}\left(\frac{D}{\hat p(X)}-\frac{1-D}{1-\hat p(X)}\right).\] For any $B>0$ chosen using only $\mathcal T$, put \[\begin{aligned} v_t^{\text{IPW}}(B)&=\mathbb E[\phi^2 I\{|\phi|\leq B\}\mid\mathcal T,\|U\|>t],\\ b_t^{\text{IPW}}(B)&=\mathbb E[|\phi|I\{|\phi|>B\}\mid\mathcal T,\|U\|>t].\end{aligned}\] Fix $a_0\in(0,\beta)$ and put $a=R_\alpha(n,\delta/4)$. For sufficiently large $t\geq1$, if $a\leq a_0$ and $n\pi_t\geq16\log(4/\delta)$, then with probability at least $1-\delta$ we have $M\geq n\pi_t/4$ and \[\begin{aligned} |\hat\eta_{n,t}^{\text{IPW}}-\eta|\leq{}&\sqrt{\frac{4v_t^{\text{IPW}}(B)}{\delta M}}+\left(1+\frac4\delta\right)b_t^{\text{IPW}}(B)+\frac{2G_t}{c}R_p(n/2,\delta/4)\\ &+2\bar e(t)+2L W_1(\mathcal L_{U/\|U\|}^t,\mathcal L_{U/\|U\|})+\frac{2G_t}{c}(t^a-1+K a t^a),\end{aligned}\] where $K$ depends on the radial law and $a_0$. Under the angular assumption of Lemma~\ref{lemma:eta_rate}, taken from Proposition 3.1 of \citet{zhang2023wasserstein}, the angular Wasserstein term has the bound stated there for $t\geq t_W$, with its logarithmic factor retained. When $V_t^{\text{IPW}}=\tmop{Var}(\phi\mid\mathcal T,\|U\|>t)$ is finite, the sampling term may instead be taken to be $\sqrt{4V_t^{\text{IPW}}/(\delta M)}$. - side condition added by the confirming referee `ass:pin-33`: $U\perp(X,D)$. — pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly. - side condition added by the confirming referee `ass:pin-34`: $\|X\|_1\leqslant R$ almost surely. — pins reading of ass:f-homo-bounded-x: Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm. - side condition added by the confirming referee `ass:pin-36`: For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\|z\|_1>\zeta k^{(1-2s)/\beta}$. — pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$. - side condition added by the confirming referee `ass:pin-38`: For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\|U\|/b_n,U/\|U\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. — pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation. - side condition added by the confirming referee `ass:pin-39`: $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. — pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely. - side condition added by the confirming referee `ass:pin-42`: For each fixed $\mathcal T$-measurable choice $B>0$, the displayed probability bound uses the base conditions of Lemma~\ref{lemma:eta_rate}; its angular specialization additionally uses that lemma's angular assumption and $t\geq t_W$, and its variance alternative additionally requires $V_t^{\text{IPW}}<\infty$. — pins reading of lem:eta-ipw: The statement explicitly says ``Under the angular assumption'' and ``When'' the variance is finite; the amended proof applies conditional sampling inequalities to the chosen $B$. - side condition added by the confirming referee `ass:pin-43`: The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\|x\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\|u\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. — pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere. - side condition added by the confirming referee `ass:pin-44`: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. — pins reading of ass:rate-nuisance-errors: Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation. - side condition added by the confirming referee `ass:pin-45`: The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. — pins reading of def:nete-algorithm: Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention. - side condition added by the confirming referee `ass:pin-7`: $Y(D)=Y$ almost surely throughout the paper. — pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely. - side condition added by the confirming referee `ass:pin-fix-0b814294`: The equal split in Algorithm~\ref{alg:nete} is defined for positive even sample sizes $n$, with $|\mathcal D_1|=|\mathcal D_2|=n/2$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-10c2ec7b`: The component dimension in the density of Assumption~\ref{assump:mvary} is $m=d_z$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8964f8dd`: Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8a129280`: The fitted-exponent pseudo-outcome has finite conditional absolute mean under the evaluation tail law: $\mathbb E[|Y|/\|U\|^{\hat\alpha_n}\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t]<\infty$ almost surely on each joint nuisance event. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8cd5c960`: The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\|u\|_1=1$. — pins reading of prop:id: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8ef6fd29`: The homogeneity inequality holds for both treatments and every positive radius, on the positive unit sphere for the $\ell_1$ norm. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9567640d`: The equal split uses $n\in2\mathbb N$. — pins reading of def:nete-algorithm: stated by the repair - side condition added by the confirming referee `ass:pin-fix-b3a62d30`: The symbol $V_t$ denotes $\tmop{Var}(\psi\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t)$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c3605ca3`: The sample-size domain of $R_g(q,\varepsilon)$ includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ebdb71b8`: The sampling term comprises both $\sqrt{4v_t^{\text{IPW}}(B)/(\delta M)}$ and $(1+4/\delta)b_t^{\text{IPW}}(B)$. — pins reading of lem:eta-ipw: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f18f3355`: The target $\eta$ denotes $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. — pins reading of lem:eta-rate: stated by the repair - proof change: the printed proof, revised step by step] Write $R_U=\|U\|$ and $S_U=U/\|U\|$, and define \[\begin{aligned} m_d^t(x,s)&=\mathbb E[Y/R_U^\alpha\mid X=x,D=d,S_U=s,R_U>t],\\ \eta_\alpha^t&=\mathbb E[(f(X,1,U)-f(X,0,U))/R_U^\alpha\mid R_U>t].\end{aligned}\] Independence of $U$ from $(X,D)$ gives $\mathbb P(D=1\mid X,S_U,R_U>t)=p(X)$. Let $\phi_\alpha$ be $\phi$ with $\hat\alpha_n$ replaced by $\alpha$. Conditional expectation over treatment and outcome gives \[\begin{aligned} &\mathbb E[\phi_\alpha\mid\mathcal T,R_U>t]-\eta_\alpha^t\\ &\quad=\mathbb E\left[(p(X)-\hat p(X))\left\{\frac{m_1^t(X,S_U)}{\hat p(X)}+\frac{m_0^t(X,S_U)}{1-\hat p(X)}\right\}\mid\mathcal T,R_U>t\right].\end{aligned}\] Indeed, the treatment coefficients after subtraction are $p/\hat p-1=(p-\hat p)/\hat p$ and $1-(1-p)/(1-\hat p)=(p-\hat p)/(1-\hat p)$. Homogeneity gives $|m_d^t|\leq G_t$. On the joint nuisance event at tolerance $\delta/4$, estimated-propensity overlap therefore gives \[ |\mathbb E[\phi_\alpha\mid\mathcal T,R_U>t]-\eta_\alpha^t|\leq\frac{2G_t}{c}R_p(n/2,\delta/4).\] Taking conditional expectations over the outcome in the exponent difference gives \[\begin{aligned} &\mathbb E[\phi-\phi_\alpha\mid\mathcal T,R_U>t]\\ &=\mathbb E\left[\left\{\frac{p(X)f(X,1,U)}{\hat p(X)R_U^\alpha}-\frac{(1-p(X))f(X,0,U)}{(1-\hat p(X))R_U^\alpha}\right\}(R_U^{\alpha-\hat\alpha_n}-1)\mid\mathcal T,R_U>t\right].\end{aligned}\] Since $R_U>t\geq1$ and $|\alpha-\hat\alpha_n|\leq a$, the absolute value is at most $(2G_t/c)\mathbb E[R_U^a-1\mid R_U>t]$. Put $\epsilon=(\beta-a_0)/2$. Potter's bound gives $\mathbb P(R_U>tr)/\pi_t\leq2r^{-\beta+\epsilon}$ for all $r\geq1$ and sufficiently large $t$. Integration gives \[\begin{aligned} \mathbb E[R_U^a-1\mid R_U>t]&=t^a-1+t^a\int_1^\infty a r^{a-1}\frac{\mathbb P(R_U>tr)}{\pi_t}\,\tmop{dr}\\ &\leq t^a-1+\frac{2a t^a}{\beta-\epsilon-a}\leq t^a-1+K a t^a,\end{aligned}\] with $K=4/(\beta-a_0)$; for $a=0$ both sides vanish. The triangle inequality now supplies the decomposition into sampling fluctuation, propensity error, exponent error, and $|\eta_\alpha^t-\eta|$. Homogeneity at the realized radius gives \[ |\eta_\alpha^t-\mathbb E[g(X,1,S_U)-g(X,0,S_U)\mid R_U>t]|\leq2\bar e(t).\] The function $s\mapsto\mathbb E_X[g(X,1,s)-g(X,0,s)]$ is $2L$-Lipschitz. Independence of $X$ and $U$ and the Wasserstein coupling inequality therefore give \[ |\eta_\alpha^t-\eta|\leq2\bar e(t)+2L W_1(\mathcal L_{U/\|U\|}^t,\mathcal L_{U/\|U\|}).\] Lemma~\ref{lemma:eta_rate} supplies the angular bound under the identical model assumptions, including its logarithmic factors. Conditional on $\mathcal T$ and positive $M$, the evaluation scores are independent with the conditional exceedance law. Put $Z=\phi I\{|\phi|\leq B\}$ and $W=\phi I\{|\phi|>B\}$. Chebyshev's and Markov's inequalities give, respectively, conditional failure probabilities at most $\delta/4$ for \[\begin{aligned} \left|M^{-1}\sum Z_i-\mathbb E[Z\mid\mathcal T,R_U>t]\right|&\leq\sqrt{4v_t^{\text{IPW}}(B)/(\delta M)},\\ M^{-1}\sum|W_i|&\leq4b_t^{\text{IPW}}(B)/\delta.\end{aligned}\] If $b_t^{\text{IPW}}(B)=0$, the second inequality holds almost surely because $W=0$ almost surely. Combining them with $|\mathbb E[W\mid\mathcal T,R_U>t]|\leq b_t^{\text{IPW}}(B)$ bounds the sampling fluctuation by the first two terms in the statement. The count has law $M\sim\tmop{Bin}(n/2,\pi_t)$, so \[ \mathbb P(M 0 $, the following conclusions hold. \begin{enumerate} \item If $s \in (0, 1 / (2 + \max \{1, \beta\}))$, takes $t_n = \Theta (n^{ (1 - 2 s) \hat{\gamma}_{n}})$, with probability at least $1 - \delta$, we have \begin{align*} | {{\widehat{\theta }_{n, t}^{\text{IPW} }} } - \theta^{NETE} | & = O (e (t_n) + n^{- s} \log (1 / \delta) + n^{-c_\alpha} \log(n)\log (1 / \delta)) . \end{align*} \item If $s \in [1 / (2 + \max \{1, \beta\}), 1 / 2)$, \ takes $t = \Theta (n^{(\hat{\gamma}_{n} / (1 + 2 \min \{1, \hat{\gamma}_{n} \})})$, with probability at least $1 - \delta$, we have \begin{align*} | {{\widehat{\theta }_{n, t}^{\text{IPW} }} } - \theta^{NETE} | & = O (e (t_n) + n^{- 1 / (2 + \max \{\beta, 1\})} \log (1 / \delta) + n^{-c_\alpha} \log(n)\log (1 / \delta)) . \end{align*} \end{enumerate} - corrected: Under the assumptions of Theorem~\ref{thm:rate}, further suppose that \begin{align*} R_p(n,\delta)&=\Theta(\log(1/\delta)n^{-1/2}),\qquad R_g(n,\delta)=\Theta(\log(1/\delta)n^{-1/2}),\\ R_\alpha(n,\delta)&=\Theta(\log(1/\delta)n^{-c_\alpha}) \end{align*} for some $c_\alpha>0$. Suppose in addition that the conditions of Proposition~\ref{prop:id} hold. Let $\mathcal T$ consist of the training split and independent auxiliary data. Suppose that $\hat p$, $\hat g$ and $\hat\alpha_n$ are $\mathcal T$-measurable and are the same functions and exponent for every input threshold. Write $R_U=\|U\|$, $S_U=U/\|U\|$, $\gamma=1/\beta$, $\pi_t=\mathbb P(R_U>t)$, and \[ m_d^t(x,v)=\mathbb E[Y/R_U^\alpha\mid X=x,D=d,S_U=v,R_U>t]. \] In each case below, let $\kappa$ be the exponent function specified in that case. Use fixed constants $0t)$ is the training exceedance count. On this same event suppose that the fitted-exponent pseudo-outcome has finite conditional absolute mean for every $t\in I_n$ and that, for a constant $V<\infty$ independent of $n$ and $t$, \[ \mathbb E[\phi^2\mid\mathcal T,R_U>t]\leq V\qquad(t\in I_n),\qquad \phi=\frac{Y}{R_U^{\hat\alpha_n}}\left(\frac{D}{\hat p(X)}-\frac{1-D}{1-\hat p(X)}\right). \] Set the first-factor estimate to zero if there are no evaluation exceedances. Run the full-input Hill selection of Algorithm~\ref{alg:nete} at tolerance $\delta/6$, and put \[ b_n=C_{\delta/6}\begin{cases} \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/3},&0<\beta<1,\\ \left(\dfrac{\log n\,\log((48/\delta)\log n)}{n}\right)^{1/3},&\beta=1,\\ \left(\dfrac{\log((48/\delta)\log n)}{n}\right)^{1/(2+\beta)},&\beta>1, \end{cases} \] with the constant supplied by Lemma~\ref{lemma:mu_rate}. Define \[ \bar e(t)=\sup_{r\geq t}e(r). \] Then, for each fixed $\delta\in(0,1/2)$ and every sufficiently large even $n$, the following conclusions hold. Here $k\geq0$ is the exponent of the logarithm in the angular assumption of Theorem~\ref{thm:rate}, taken from Proposition 3.1 of \citet{zhang2023wasserstein}; under that assumption the angular term at the threshold $t_n$ is of the order shown, up to this power of $\log n$. \begin{enumerate} \item If $s\in(0,1/(2+\max\{1,\beta\}))$, take $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ and put $\kappa(v)=(1-2s)v$. With probability at least $1-\delta$, we have \[ \begin{aligned} |\widehat\theta_{n,t_n}^{\text{IPW}}-\theta^{NETE}|=O\bigl(&\sqrt{V/\delta}\,n^{-s}+\log(6/\delta)n^{-1/2}\\ &+\bar e(t_n)+(\log n)^k n^{-s}+n^{-c_\alpha}\log n\log(6/\delta)+|\alpha|b_n\bigr). \end{aligned} \] \item If $s\in[1/(2+\max\{1,\beta\}),1/2)$, take $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ and put $\kappa(v)=v/(1+2\min\{1,v\})$. Write $q=1/(2+\max\{\beta,1\})$. With probability at least $1-\delta$, we have \[ \begin{aligned} |\widehat\theta_{n,t_n}^{\text{IPW}}-\theta^{NETE}|=O\bigl(&\sqrt{V/\delta}\,n^{-q}+\log(6/\delta)n^{-1/2}\\ &+\bar e(t_n)+(\log n)^k n^{-q}+n^{-c_\alpha}\log n\log(6/\delta)+|\alpha|b_n\bigr). \end{aligned} \] \end{enumerate} - no longer uses `ass:cor-rate-ipw-errors` - side condition added by the confirming referee `ass:pin-33`: $U\perp(X,D)$. — pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly. - side condition added by the confirming referee `ass:pin-34`: $\|X\|_1\leqslant R$ almost surely. — pins reading of ass:f-homo-bounded-x: Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm. - side condition added by the confirming referee `ass:pin-36`: For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\|z\|_1>\zeta k^{(1-2s)/\beta}$. — pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$. - side condition added by the confirming referee `ass:pin-38`: For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\|U\|/b_n,U/\|U\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. — pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation. - side condition added by the confirming referee `ass:pin-39`: $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. — pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely. - side condition added by the confirming referee `ass:pin-41`: The existential cutoff $t_W$ is chosen sufficiently large for Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} and so that $\sup_{r\geq t_W}e(r)\leq1$ and $\mathbb P(\|U\|>tr)/\mathbb P(\|U\|>t)\leq4r^{-\beta}$ for every $t\geq t_W$ and $r\geq1$. — pins reading of thm:rate: The proof of Theorem~\ref{thm:rate} invokes the first-factor lemmas' sufficiently-large-threshold conclusions, uses the radial moment inequality with $K=4/(\beta-a_0)$, and bounds $G_t$ by $C+1$. The stated Pareto-type radial law and $e(t)\to0$ permit this enlargement of the existential cutoff. - side condition added by the confirming referee `ass:pin-42`: For each fixed $\mathcal T$-measurable choice $B>0$, the displayed probability bound uses the base conditions of Lemma~\ref{lemma:eta_rate}; its angular specialization additionally uses that lemma's angular assumption and $t\geq t_W$, and its variance alternative additionally requires $V_t^{\text{IPW}}<\infty$. — pins reading of lem:eta-ipw: The statement explicitly says ``Under the angular assumption'' and ``When'' the variance is finite; the amended proof applies conditional sampling inequalities to the chosen $B$. - side condition added by the confirming referee `ass:pin-43`: The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\|x\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\|u\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. — pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere. - side condition added by the confirming referee `ass:pin-44`: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. — pins reading of ass:rate-nuisance-errors: Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation. - side condition added by the confirming referee `ass:pin-45`: The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. — pins reading of def:nete-algorithm: Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention. - side condition added by the confirming referee `ass:pin-7`: $Y(D)=Y$ almost surely throughout the paper. — pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely. - side condition added by the confirming referee `ass:pin-fix-0b814294`: The equal split in Algorithm~\ref{alg:nete} is defined for positive even sample sizes $n$, with $|\mathcal D_1|=|\mathcal D_2|=n/2$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-10c2ec7b`: The component dimension in the density of Assumption~\ref{assump:mvary} is $m=d_z$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-6857bb4d`: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments. — pins reading of thm:rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8964f8dd`: Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8a129280`: The fitted-exponent pseudo-outcome has finite conditional absolute mean under the evaluation tail law: $\mathbb E[|Y|/\|U\|^{\hat\alpha_n}\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t]<\infty$ almost surely on each joint nuisance event. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8cd5c960`: The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\|u\|_1=1$. — pins reading of prop:id: stated by the repair - side condition added by the confirming referee `ass:pin-fix-8ef6fd29`: The homogeneity inequality holds for both treatments and every positive radius, on the positive unit sphere for the $\ell_1$ norm. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-9567640d`: The equal split uses $n\in2\mathbb N$. — pins reading of def:nete-algorithm: stated by the repair - side condition added by the confirming referee `ass:pin-fix-b3a62d30`: The symbol $V_t$ denotes $\tmop{Var}(\psi\mid\mathcal D_1,\hbox{ auxiliary data},\|U\|>t)$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c3605ca3`: The sample-size domain of $R_g(q,\varepsilon)$ includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c4d18fdb`: For each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that, for every integer $n\geq n_0$, the bound has probability at least $1-\delta$ under the law of the sample of size $n$. — pins reading of lem:mu-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-c90a58f3`: The count $n_t=\sum_{i=1}^{n/2}I(\|U_i\|>t)$ is the training-split exceedance count, with the training observations indexed first after the random split. — pins reading of thm:rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-ebdb71b8`: The sampling term comprises both $\sqrt{4v_t^{\text{IPW}}(B)/(\delta M)}$ and $(1+4/\delta)b_t^{\text{IPW}}(B)$. — pins reading of lem:eta-ipw: stated by the repair - side condition added by the confirming referee `ass:pin-fix-f18f3355`: The target $\eta$ denotes $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. — pins reading of lem:eta-rate: stated by the repair - side condition added by the confirming referee `ass:pin-fix-fc4621e4`: For each fixed $\delta\in(0,1/2)$, the probability guarantee quantifies separately over every sufficiently large even $n$. — pins reading of thm:cor-rate-ipw: stated by the repair - side condition added by the confirming referee `ass:pin-fix-fdafcd87`: The constants $c_-$ and $c_+$ in $t_n=\Theta(n^{\kappa(\hat\gamma_n)})$ are deterministic and independent of $n$ and the observed sample. — pins reading of thm:cor-rate-ipw: stated by the repair - proof change: Fix $\delta\in(0,1/2)$ and an even $n$. Let $E_T$ be the uniform nuisance and moment event from the statement. Its failure probability is at most $\delta/6$. Lemma~\ref{lemma:mu_rate}, applied to the full-input Hill selection at tolerance $\delta/6$, supplies an event $E_\gamma$ with failure probability at most $\delta/6$ on which $|\hat\gamma_n-\gamma|\leq b_n$. Each displayed radial regime gives $b_n\log n\to0$ for fixed $\delta$. The first exponent function has derivative $1-2s$. For the second exponent function the derivative is $1/(1+2v)^2$ when $v<1$ and $1/3$ when $v>1$, and the function is continuous at $v=1$. Both exponent functions are therefore Lipschitz with constant at most $1$. On $E_\gamma$, \[ \frac{n^{\kappa(\hat\gamma_n)}}{n^{\kappa(\gamma)}}=\exp\{(\kappa(\hat\gamma_n)-\kappa(\gamma))\log n\}\in[1/2,2] \] for sufficiently large $n$. The threshold order consequently implies $t_n\in I_n$. Put $m=n/2$ and $r=(1-\beta\kappa(\gamma))/2$. In the first case $\beta\kappa(\gamma)=1-2s$, so $r=s$. In the second case, put $a_*=\min\{1,\beta\}$. Then $\kappa(\gamma)=1/(\beta+2a_*)$ and $r=a_*/(\beta+2a_*)=q$. The Pareto-type model gives $\pi_t\asymp t^{-\beta}$, whence \[ \pi_{\ell_n}\asymp\pi_{u_n}\asymp n^{-1+2r},\qquad m\pi_{u_n}\asymp n^{2r}. \] Both $r=s$ and $r=q$ are positive. The training and evaluation counts at $u_n$ are binomial with parameters $m,\pi_{u_n}$. For either count $N_{u_n}$, \[ \mathbb P(N_{u_n}t)-P_T[\phi I(R_U>t)]\},\qquad C(t)=\sum_{i=1}^{m}\{I(R_i>t)-\pi_t\}. \] Introduce an independent conditional copy $Z'_1,\ldots,Z'_m$. Jensen's inequality first bounds the conditional expected squared supremum of $A$ by the expected squared supremum of the difference of the two sample sums. Independent symmetric signs $\varepsilon_i$ may multiply these differences because swapping either member of each independent pair preserves its law. Applying $|a-b|^2\leq2|a|^2+2|b|^2$ to the signed difference gives \[ \mathbb E[\sup_{t\in I_n}|A(t)|^2\mid\mathcal T]\leq4\mathbb E\left[\sup_{t\in I_n}\left|\sum_{i=1}^{m}\varepsilon_i\phi(Z_i)I(R_i>t)\right|^2\middle|\mathcal T\right]. \] Conditional on the observations, order those with $R_i>\ell_n$ by decreasing radius. Each sum indexed by $t\in I_n$ is a partial sum of the independent mean-zero signed variables in that order. The square-integrable martingale maximal inequality bounds its expected squared maximum by $4\sum_i\phi(Z_i)^2I(R_i>\ell_n)$. Hence, on $E_T$, \[ \mathbb E[\sup_{t\in I_n}|A(t)|^2\mid\mathcal T]\leq16mV\pi_{\ell_n}. \] Repeating these inequalities with the constant score $1$ gives $\mathbb E[\sup_{t\in I_n}|C(t)|^2\mid\mathcal T]\leq16m\pi_{\ell_n}$. Markov's inequality supplies two events, each with conditional failure probability at most $\delta/6$, on which \[ \sup_{t\in I_n}|A(t)|\leq\sqrt{96mV\pi_{\ell_n}/\delta},\qquad \sup_{t\in I_n}|C(t)|\leq\sqrt{96m\pi_{\ell_n}/\delta}. \] Integration over $\mathcal T$ preserves these failure bounds on $E_T$. The suprema are measurable by taking a countable dense set of thresholds and the band endpoints. Let $\mu_t=P_T[\phi\mid R_U>t]$ and $M_t=\sum_i I(R_i>t)$. On $E_T$, $|\mu_t|\leq\sqrt V$ by Cauchy--Schwarz. The identity \[ \sum_i\phi(Z_i)I(R_i>t)-M_t\mu_t=A(t)-\mu_t C(t) \] and the evaluation-count bound imply, simultaneously for every $t\in I_n$, \[ |\hat\eta_{n,t}^{\text{IPW}}-\mu_t|\leq4\sqrt{\frac{96V\pi_{\ell_n}}{\delta m\pi_{u_n}^2}}=O(\sqrt{V/\delta}\,n^{-r}). \] It is therefore valid at the selected $t_n$ on $E_\gamma$. To bound the conditional mean, put $a=R_\alpha(n,\delta/6)$, fix $a_0\in(0,\beta)$, and let $G_t=C_g+\bar e(t)$, where $C_g$ bounds $|g|$. For large $n$, $a\leq a_0$ and $G_t$ is uniformly bounded on $I_n$. Let $\phi_\alpha$ be the score with exponent $\alpha$ and put \[ \eta_\alpha^t=\mathbb E[(f(X,1,U)-f(X,0,U))/R_U^\alpha\mid R_U>t]. \] Independence of $U$ from $(X,D)$ gives the treatment probability $p(X)$ after conditioning on direction and exceedance. Conditional expectation over treatment and outcome therefore yields \[ P_T[\phi_\alpha\mid R_U>t]-\eta_\alpha^t=\mathbb E\left[(p-\hat p)\left\{\frac{m_1^t(X,S_U)}{\hat p}+\frac{m_0^t(X,S_U)}{1-\hat p}\right\}\middle|\mathcal T,R_U>t\right]. \] Homogeneity gives $|m_d^t|\leq G_t$, so overlap and the uniform propensity inequality bound this difference by $(2G_t/c)R_p(n/2,\delta/6)=O(\log(6/\delta)n^{-1/2})$ throughout $I_n$. Taking the conditional outcome mean in the exponent difference gives \[ \begin{aligned} P_T[\phi-\phi_\alpha\mid R_U>t]=\mathbb E\biggl[\biggl\{&\frac{p(X)f(X,1,U)}{\hat p(X)R_U^\alpha}\\ &-\frac{(1-p(X))f(X,0,U)}{(1-\hat p(X))R_U^\alpha}\biggr\}(R_U^{\alpha-\hat\alpha_n}-1)\biggm|\mathcal T,R_U>t\biggr]. \end{aligned} \] Its absolute value is at most $(2G_t/c)\mathbb E[R_U^a-1\mid R_U>t]$. The radial integration bound in Lemma~\ref{lemma:eta_ipw} gives, uniformly for sufficiently large $t$, \[ \mathbb E[R_U^a-1\mid R_U>t]\leq t^a-1+Kat^a,\qquad K=4/(\beta-a_0). \] On $I_n$, $\log t=O(\log n)$ and $a\log t\to0$. Thus $t^a\leq2$ and $t^a-1\leq2a\log t$ for large $n$, and the exponent contribution is $O(\log(6/\delta)n^{-c_\alpha}\log n)$. Homogeneity at the realized radius bounds the difference between $\eta_\alpha^t$ and the conditional angular contrast by $2\bar e(t)$. The function $v\mapsto\mathbb E_X[g(X,1,v)-g(X,0,v)]$ is $2L$-Lipschitz. Independence of $X$ and $U$ and the Wasserstein coupling inequality then give \[ |\eta_\alpha^t-\eta|\leq2\bar e(t)+\mathcal A_t. \] Lemma~\ref{lemma:eta_rate} supplies the angular bound, including its logarithmic factors, and the displayed conclusion retains $\mathcal A_{t_n}$ itself. By Proposition~\ref{prop:id}, $\theta^{\tmop{NETE}}=\eta/h$ with $h=1-\alpha\gamma>0$. On $E_T\cap E_\gamma$, \[ |\hat\alpha_n\hat\gamma_n-\alpha\gamma|\leq a(\gamma+b_n)+|\alpha|b_n=:Q_n\to0. \] For sufficiently large $n$, $Q_n\leq h/2$. Hence $|1-\hat\alpha_n\hat\gamma_n|\geq h/2$, $|\hat\mu_n|\leq2/h$, and exact reciprocal subtraction gives $|\hat\mu_n-1/h|\leq2Q_n/h^2$. Finally, \[ \widehat\theta_{n,t_n}^{\text{IPW}}-\theta^{\tmop{NETE}}=\hat\mu_n(\hat\eta_{n,t_n}^{\text{IPW}}-\eta)+\eta(\hat\mu_n-1/h). \] Combining the sampling, propensity, homogeneity, angular and exponent bounds proves the claimed display with $r=s$ in the first case and $r=q$ in the second. The contribution of $a$ from $Q_n$ is absorbed by $a\log n$, and $|\alpha|b_n$ is retained. The uniform nuisance event, Hill event, two count events and two maximal-process events have total failure probability at most $6(\delta/6)=\delta$. This establishes a separate guarantee for each sufficiently large even $n$. ## Findings withdrawn by the owner Not errors: the paper's owner showed each finding misreads the paper. The printed statement stands, under the owner's reading (or side condition), which the formalization adopts. - `thm:cor-rate` (raised at F2.5): the owner's reading (`ass:pin-owner-1`): In the rate assumptions $R_p(n,\delta)=\Theta(\log(1/\delta)n^{-1/2})$, $R_g(n,\delta)=\Theta(\log(1/\delta)n^{-1/2})$ and $R_\alpha(n,\delta)=\Theta(\log(1/\delta)n^{-c_\alpha})$, the dependence on the tolerance $\delta$ is the displayed factor $\log(1/\delta)$: the implied constants and the sample size from which the bounds hold do not depend on $\delta$. — the withdrawn finding: [statement-wrong] The nuisance Θ rates leave confidence uniformity unspecified: fixed-tolerance ∀ε ∃cε,Cε,n0ε versus uniform ∃c,C,n0 ∀ε; RateTheta selects the uniform reading. Edit the frozen rate clause to state the intended dependence explicitly. - `thm:cor-rate-ipw` (raised at F2.5): the owner's reading (`ass:pin-owner-2`): In the rate assumptions $R_p(n,\delta)=\Theta(\log(1/\delta)n^{-1/2})$, $R_g(n,\delta)=\Theta(\log(1/\delta)n^{-1/2})$ and $R_\alpha(n,\delta)=\Theta(\log(1/\delta)n^{-c_\alpha})$, the dependence on the tolerance $\delta$ is the displayed factor $\log(1/\delta)$: the implied constants and the sample size from which the bounds hold do not depend on $\delta$. — the withdrawn finding: [statement-wrong] The three Theta assumptions leave open whether their constants and sufficiently-large-n onset depend on tolerance; Lean selects uniform constants and onset over all tolerances. - `def:nete-algorithm` (raised at F2.5): the owner's reading (`ass:pin-owner-3`): In Algorithm~\ref{alg:nete} the outputs are given a value on every sample by convention: when the index set $\mathcal I$ is empty the averages defining $\hat\eta_{n,t}^{\mathrm{IPW}}$ and $\hat\eta_{n,t}^{\mathrm{DR}}$ are taken to be $0$, and a Hill statistic whose reference radius $\\|U_{(k+1)}\\|$ is $0$ is taken to be $0$. These conventions concern events the results exclude: the bounds are stated on events where $\|\mathcal I\|\geq1$, and the radii are positive almost surely. — the withdrawn finding: [statement-wrong] The frozen algorithm leaves empty evaluation averages and zero Hill radial denominators undefined on admissible data, whereas Lean selects totalized values. - `ass:overlap` (raised at F2.5): the owner's reading (`ass:pin-owner-4`): In Assumption~\ref{assump:overlap} the overlap constant $c\in(0,1/2)$ is one deterministic constant: it does not depend on the sample size $n$, on the sample, or on the threshold $t$. — the withdrawn finding: [statement-wrong] The frozen overlap condition does not distinguish a deterministic overlap constant for each fixed sample size from one deterministic constant shared by all sample sizes; Lean selects the latter. ## Suggestions to the authors Not errors: readings the paper determines but does not state, and side conditions the proofs use silently. - clarification (`ass:pin-33`, used by `prop:id`): $U\perp(X,D)$. — pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly. - clarification (`ass:pin-34`, used by `prop:id`): $\\|X\\|_1\leqslant R$ almost surely. — pins reading of ass:f-homo-bounded-x: Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm. - clarification (`ass:pin-38`, used by `prop:id`): For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\\|U\\|/b_n,U/\\|U\\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. — pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation. - clarification (`ass:pin-39`, used by `prop:id`): $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. — pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely. - clarification (`ass:pin-43`, used by `prop:id`): The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\\|x\\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\\|u\\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. — pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere. - clarification (`ass:pin-7`, used by `prop:id`): $Y(D)=Y$ almost surely throughout the paper. — pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely. - clarification (`ass:pin-fix-8cd5c960`, used by `prop:id`): The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\\|u\\|_1=1$. — pins reading of prop:id: stated by the repair - clarification (`ass:pin-36`, used by `thm:rate`): For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\\|z\\|_1>\zeta k^{(1-2s)/\beta}$. — pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$. - clarification (`ass:pin-41`, used by `thm:rate`): The existential cutoff $t_W$ is chosen sufficiently large for Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} and so that $\sup_{r\geq t_W}e(r)\leq1$ and $\mathbb P(\\|U\\|>tr)/\mathbb P(\\|U\\|>t)\leq4r^{-\beta}$ for every $t\geq t_W$ and $r\geq1$. — pins reading of thm:rate: The proof of Theorem~\ref{thm:rate} invokes the first-factor lemmas' sufficiently-large-threshold conclusions, uses the radial moment inequality with $K=4/(\beta-a_0)$, and bounds $G_t$ by $C+1$. The stated Pareto-type radial law and $e(t)\to0$ permit this enlargement of the existential cutoff. - clarification (`ass:pin-42`, used by `thm:rate`): For each fixed $\mathcal T$-measurable choice $B>0$, the displayed probability bound uses the base conditions of Lemma~\ref{lemma:eta_rate}; its angular specialization additionally uses that lemma's angular assumption and $t\geq t_W$, and its variance alternative additionally requires $V_t^{\text{IPW}}<\infty$. — pins reading of lem:eta-ipw: The statement explicitly says ``Under the angular assumption'' and ``When'' the variance is finite; the amended proof applies conditional sampling inequalities to the chosen $B$. - clarification (`ass:pin-44`, used by `thm:rate`): For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. — pins reading of ass:rate-nuisance-errors: Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation. - clarification (`ass:pin-45`, used by `thm:rate`): The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. — pins reading of def:nete-algorithm: Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention. - clarification (`ass:pin-fix-0b814294`, used by `thm:rate`): The equal split in Algorithm~\ref{alg:nete} is defined for positive even sample sizes $n$, with $\|\mathcal D_1\|=\|\mathcal D_2\|=n/2$. — pins reading of lem:eta-rate: stated by the repair - clarification (`ass:pin-fix-10c2ec7b`, used by `thm:rate`): The component dimension in the density of Assumption~\ref{assump:mvary} is $m=d_z$. — pins reading of lem:eta-rate: stated by the repair - clarification (`ass:pin-fix-6857bb4d`, used by `thm:rate`): For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments. — pins reading of thm:rate: stated by the repair - clarification (`ass:pin-fix-8964f8dd`, used by `thm:rate`): Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$. — pins reading of lem:eta-rate: stated by the repair - clarification (`ass:pin-fix-8a129280`, used by `thm:rate`): The fitted-exponent pseudo-outcome has finite conditional absolute mean under the evaluation tail law: $\mathbb E[\|Y\|/\\|U\\|^{\hat\alpha_n}\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t]<\infty$ almost surely on each joint nuisance event. — pins reading of lem:eta-rate: stated by the repair - clarification (`ass:pin-fix-8ef6fd29`, used by `thm:rate`): The homogeneity inequality holds for both treatments and every positive radius, on the positive unit sphere for the $\ell_1$ norm. — pins reading of lem:eta-rate: stated by the repair - clarification (`ass:pin-fix-9567640d`, used by `thm:rate`): The equal split uses $n\in2\mathbb N$. — pins reading of def:nete-algorithm: stated by the repair - clarification (`ass:pin-fix-b3a62d30`, used by `thm:rate`): The symbol $V_t$ denotes $\tmop{Var}(\psi\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t)$. — pins reading of lem:eta-rate: stated by the repair - clarification (`ass:pin-fix-c3605ca3`, used by `thm:rate`): The sample-size domain of $R_g(q,\varepsilon)$ includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. — pins reading of lem:eta-rate: stated by the repair - clarification (`ass:pin-fix-c4d18fdb`, used by `thm:rate`): For each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that, for every integer $n\geq n_0$, the bound has probability at least $1-\delta$ under the law of the sample of size $n$. — pins reading of lem:mu-rate: stated by the repair - clarification (`ass:pin-fix-c90a58f3`, used by `thm:rate`): The count $n_t=\sum_{i=1}^{n/2}I(\\|U_i\\|>t)$ is the training-split exceedance count, with the training observations indexed first after the random split. — pins reading of thm:rate: stated by the repair - clarification (`ass:pin-fix-ebdb71b8`, used by `thm:rate`): The sampling term comprises both $\sqrt{4v_t^{\text{IPW}}(B)/(\delta M)}$ and $(1+4/\delta)b_t^{\text{IPW}}(B)$. — pins reading of lem:eta-ipw: stated by the repair - clarification (`ass:pin-fix-f18f3355`, used by `thm:rate`): The target $\eta$ denotes $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. — pins reading of lem:eta-rate: stated by the repair - clarification (`ass:pin-fix-27ee3ba1`, used by `thm:cor-rate`): For each fixed $\delta\in(0,1/2)$, the probability guarantee quantifies separately over every sufficiently large even $n$. — pins reading of thm:cor-rate: stated by the repair - clarification (`ass:pin-fix-d303828c`, used by `thm:cor-rate`): The constants $c_-$ and $c_+$ in $t_n=\Theta(n^{\kappa(\hat\gamma_n)})$ are deterministic and independent of $n$ and the observed sample. — pins reading of thm:cor-rate: stated by the repair - clarification (`ass:pin-fix-fc4621e4`, used by `thm:cor-rate-ipw`): For each fixed $\delta\in(0,1/2)$, the probability guarantee quantifies separately over every sufficiently large even $n$. — pins reading of thm:cor-rate-ipw: stated by the repair - clarification (`ass:pin-fix-fdafcd87`, used by `thm:cor-rate-ipw`): The constants $c_-$ and $c_+$ in $t_n=\Theta(n^{\kappa(\hat\gamma_n)})$ are deterministic and independent of $n$ and the observed sample. — pins reading of thm:cor-rate-ipw: stated by the repair - `ass:consistency`: pinned reading: $Y(D)=Y$ almost surely throughout the paper. (The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely.) - `ass:f-homo-asymptotic`: pinned reading: The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\\|x\\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\\|u\\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. (Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere.) - `ass:f-homo-bounded-x`: pinned reading: $\\|X\\|_1\leqslant R$ almost surely. (Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm.) - `ass:iid-independence`: pinned reading: $U\perp(X,D)$. (The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly.) - `ass:mvary-density`: pinned reading: For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\\|z\\|_1>\zeta k^{(1-2s)/\beta}$. (Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$.) - `ass:rate-nuisance-errors`: pinned reading: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. (Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation.) - `def:conditional-mean`: pinned reading: $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. (Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely.) - `def:nete-algorithm`: pinned reading: The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. (Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention.) - `def:regular-variation`: pinned reading: For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\\|U\\|/b_n,U/\\|U\\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. (Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation.) - reading of the symbol `X`: $X\in\mathcal{X}$ denotes the covariate in the potential-outcome model. In Proposition~\ref{prop:id}, Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $\\|X\\|_1\leqslant R$, so the covariate takes values in $B_R$. In Lemma~\ref{lemma:eta_rate}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: In $\eta=\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, $\mathbb E_X$ integrates with respect to the population marginal law of $X$, rather than an empirical covariate distribution. — Assumption~\ref{assump:f_homo} gives bounded support, inherited by Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw}. The binding target definitions for Lemma~\ref{lemma:eta_rate} and both corollaries specify integration against the population marginal law of $X$. - reading of the symbol `D`: $D\in\{0,1\}$ denotes the treatment indicator. In Lemma~\ref{lemma:mu_rate}: In the proof's expansions $f(t)=C\beta t^{-\beta-1}(1+D(1-1/\beta)t^{-1}+o(t^{-1}))$ and $\bar F(t)=Ct^{-\beta}(1+Dt^{-1}+o(t^{-1}))$, $D$ denotes a fixed expansion coefficient rather than the treatment indicator. — Section~\ref{sec:prelin} introduces $D$ as binary treatment. In the proof of Lemma~\ref{lemma:mu_rate}, equation~\ref{eq:density_f} and the following survival expansion instead use $D$ as an asymptotic coefficient; there is no explicit redefining sentence. - reading of the symbol `Y`: $Y$ is the observed outcome satisfying consistency $Y=Y(D)$, where $Y(d)$ denotes the potential outcome when treatment is set to $d$. — The potential-outcome setup in Section~\ref{sec:prelin} explicitly assumes $Y(D)=Y$ throughout the paper; the extreme-data setup retains that definition. - reading of the symbol `Y(d)`: $Y(d)$ denotes the potential outcome under intervention $d\in\{0,1\}$, with consistency $Y(D)=Y$ throughout. In Proposition~\ref{prop:id}: For both $d\in\{0,1\}$, $\mathbb E[Y(d)\mid X,U,D]=\mathbb E[Y(d)\mid X,U]$ almost surely, and $\mathbb E[\|Y(d)\|\mid\\|U\\|>t]<\infty$ for all sufficiently large $t$. — The potential-outcome paragraph in Section~\ref{sec:prelin} defines the binary intervention and consistency. Proposition~\ref{prop:id} additionally imposes conditional-mean exogeneity given $(X,U)$ and conditional absolute integrability in sufficiently high tails. - reading of the symbol `U`: $U$ denotes the extreme-noise vector in $\mathbb R_+^d$, with $\\|U\\|=\\|U\\|_1$ throughout the treatment-effect analysis. In Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}: Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$, with $U=AZ$ and $m=d_z$ in the component density. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_ipw}: Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$, with $U=AZ$ and $m=d_z$ in the component density. — Assumption~\ref{assump:iid} specifies the extreme-noise vector. The current proof of Lemma~\ref{lemma:mu_rate} in discovery/core.json writes the scalar radial tail quantile as $Q(v)=F_R^{-1}(1-1/v)$ and no longer uses $U(t)$ or its former expansion. - reading of the symbol `d`: $d$ denotes the dimension in $U\in\mathbb R_+^d$ and in sphere notation; when used as the treatment argument in $Y(d)$, $f(x,d,u)$, or $g(x,d,u)$, it is a locally bound treatment value in $\{0,1\}$. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $m\geq d\geq2$. — Assumption~\ref{assump:mvary} imposes $m\geq d\geq2$. Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} inherit this assumption through Theorem~\ref{thm:rate}. - reading of the symbol `t`: $t>0$ is the level defining the strict exceedance event $\{\\|U\\|_1>t\}$, except when it is the argument of a scalar function in a tail calculation. In Proposition~\ref{prop:id}: $t$ is a deterministic limit parameter tending to infinity. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: The finite-threshold analysis uses a fixed deterministic $t$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The selected threshold is $t=t_n$, with $c_-n^{\kappa(\hat\gamma_n)}\leq t_n\leq c_+n^{\kappa(\hat\gamma_n)}$, where $0t)$. In Lemma~\ref{lemma:eta_ipw}: The bound concerns sufficiently large $t\geq1$, subject to $n\pi_t\geq16\log(4/\delta)$, where $\pi_t=\mathbb P(\\|U\\|>t)$. In Theorem~\ref{thm:rate}: The main bounds concern deterministic $t\geq t_W$, subject to $n\pi_t\geq16\log(12/\delta)$, where $\pi_t=\mathbb P(\\|U\\|>t)$; the additional simplified bounds also require $t\geq\exp(1)$ and $a\log t\leq1$, where $a=R_\alpha(n,\delta/12)$. — Theorem~\ref{thm:rate} states its main bounds for sufficiently large $n$ and $t\geq t_W$, and its additional simplified bounds require $t\geq\exp(1)$ and $a\log t\leq1$. - reading of the symbol `\alpha`: $\alpha\in\mathbb R$ is the fixed population homogeneity exponent governing the normalization $f(x,d,tu)/t^\alpha$ and the definition of $\theta^{\tmop{NETE}}$; it is distinct from the supplied approximation $\hat\alpha_n$. In Proposition~\ref{prop:id}, Theorem~\ref{thm:rate}, Corollary~\ref{cor:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate_ipw}: $\alpha<\beta$. — Assumption~\ref{assump:f_homo} introduces the exponent through its homogeneity inequality, and equation~\ref{eq:nete} calls it a known index from prior knowledge. Proposition~\ref{prop:id} and Theorem~\ref{thm:rate} impose $\alpha<\beta$; the remaining listed results inherit that restriction. The positive constant in the synthetic example specializes the same exponent. - reading of the symbol `\beta`: $\beta\in(0,\infty)$ is the population regular-variation index of the extreme-noise law, with radial limiting measure satisfying $\nu_\beta([r,\infty))=r^{-\beta}$ for $r>0$. — Definition~\ref{def:mvarying} defines the index through the radial limiting measure; Assumption~\ref{assump:mvary} uses the same $\beta$ in its Pareto-type model. - reading of the symbol `R`: $R$ is the fixed covariate bound in $\\|X\\|_1\leqslant R$, except where explicitly rebound as a local radial variable. In Lemma~\ref{lemma:mu_rate}: In the proof, $R=\\|U\\|_1$ is the generic radial random variable; $f_R$, $F_R$, and $\bar F_R$ refer to its density, distribution function, and survival function. This local $R$ is distinct from the covariate bound. — Assumption~\ref{assump:f_homo} introduces the covariate bound. The replacement proof of Lemma~\ref{lemma:mu_rate} in discovery/core.json explicitly says, "Write $R=\\|U\\|_1$." - reading of the symbol `f`: $f$ denotes a population function with its domain and meaning determined by its local definition, rather than an estimated regression function. In Proposition~\ref{prop:id}, Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: In the extreme-effect model and its decompositions, $f$ is the real-valued population conditional-mean function satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$, whose pointwise values $f(x,d,u)$ are those appearing in the asymptotic homogeneous property. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: In the extreme-effect model and its decompositions, $f$ is the real-valued population conditional-mean function satisfying $f(X,D,U)=\mathbb E[Y\mid X,D,U]$, whose pointwise values $f(x,d,u)$ appear in the asymptotic homogeneous property. — The Pareto type-II definition supplies the univariate density. Assumption~\ref{assump:f_homo} defines $f(X,D,U)=\mathbb E[Y\mid X,D,U]$, and the printed tail-index calculation separately defines a weighted-sum density. Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} inherit the conditional-mean definition through Theorem~\ref{thm:rate}; their additional definitions of $m_d^t(x,v)$ specify threshold-conditional normalized means and do not redefine $f$. - reading of the symbol `g`: $g(x,d,u)$ is the population asymptotic normalized conditional-mean function, given by $g(x,d,u)=\lim_{t\to\infty}f(x,d,tu)/t^\alpha$ on its covariate, treatment, and direction domain. In Proposition~\ref{prop:id}, Lemma~\ref{lemma:eta_rate}: The homogeneity inequality $\|f(x,d,tu)/t^\alpha-g(x,d,u)\|\leqslant e(t)$ holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\\|u\\|_1=1$; $g$ is $L$-Lipschitz on this domain. In Lemma~\ref{lemma:eta_ipw}: The homogeneity inequality $\|f(x,d,tu)/t^\alpha-g(x,d,u)\|\leqslant e(t)$ holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\\|u\\|_1=1$; $g$ is $L$-Lipschitz on this domain. — Assumption~\ref{assump:f_homo} and the existing scoped readings still determine the conditional-mean meaning. The replacement proof of Lemma~\ref{lemma:mu_rate}, recorded in discovery/core.json, uses nominal component densities without introducing a density named $g$. - reading of the symbol `e`: $e(t)$ is a deterministic, nonnegative uniform approximation-error bound in the asymptotic homogeneous property, allowing $e(t)=0$ and tending to zero as $t\rightarrow\infty$. In Proposition~\ref{prop:id}, Lemma~\ref{lemma:eta_rate}: The bound $\|f(x,d,ru)/r^\alpha-g(x,d,u)\|\leq e(r)$ holds for both $d\in\{0,1\}$ and every $r>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\\|u\\|_1=1$. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The bound $\|f(x,d,ru)/r^\alpha-g(x,d,u)\|\leq e(r)$ holds for both $d\in\{0,1\}$ and every $r>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\\|u\\|_1=1$. In Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: The approximation-error envelope used in the bound is $\bar e(t)=\sup_{r\geq t}e(r)$, and $G_t=C+\bar e(t)$; no equality $\bar e(t)=e(t)$ is assumed. In Theorem~\ref{thm:rate}: The approximation-error envelope used in the bound is $\bar e(t)=\sup_{r\geq t}e(r)$, and $G_t=C+\bar e(t)$; no equality $\bar e(t)=e(t)$ is assumed. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The approximation-error envelope used in the bound is $\bar e(t)=\sup_{r\geq t}e(r)$, evaluated at the selected threshold $t_n$; no equality $\bar e(t)=e(t)$ is assumed. — Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} explicitly define $\bar e(t)=\sup_{r\geq t}e(r)$ and use $\bar e(t_n)$ in their bounds. Assumption~\ref{assump:f_homo} assumes convergence to zero without monotonicity; the inherited positive-sphere homogeneity interpretation remains unchanged. - reading of the symbol `c`: $c$ denotes a deterministic positive constant whose identity and additional restrictions are supplied by its local definition. In Proposition~\ref{prop:id}, Theorem~\ref{thm:rate}, Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: In the regular-variation definition, $c>0$ is the mass multiplier in $c\nu_\beta\times S$, independently of any overlap constant. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: In the overlap assumption, $c\in(0,1/2)$ bounds the estimated propensity uniformly by $c\leq\hat p(x)\leq1-c$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: In the regular-variation definition, $c>0$ is the mass multiplier in $c\nu_\beta\times S$, independently of the overlap constant; in the overlap assumption and the error bounds containing $1/c$, $c\in(0,1/2)$ satisfies $c\leq\hat p(x)\leq1-c$ for every $x\in\mathcal X$. The constants $c_1,c_2,c_\alpha$ are separate symbols. — Definition~\ref{def:mvarying} and Assumption~\ref{assump:overlap} define distinct local constants. The error bounds in Theorem~\ref{thm:rate} and Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} use the overlap constant in their factors $1/c$. Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} inherit both definitions through Theorem~\ref{thm:rate}; their displayed conclusions do not explicitly contain $1/c$. The printed tail-density calculation locally uses $c$ as a proportionality constant. - reading of the symbol `p`: $p(x)=\mathbb P(D=1\mid X=x)$ is the population propensity function on $\mathcal X$, distinct from its fitted estimate $\hat p(x)$. In Lemma~\ref{lemma:eta_ipw}: $00$, the locally chosen $p>1$ with $p\alpha<\beta$ is a fixed scalar moment exponent in $\|G(S_U)(R_U/t)^\alpha\|^p$ and $(R_U/t)^{p\alpha}$. — The proof of Proposition~\ref{prop:id} says: If $\alpha>0$, choose $p>1$ with $p\alpha<\beta$. Its statement retains population propensity positivity, as does Lemma~\ref{lemma:eta_rate}. - reading of the symbol `\hat{p}`: $\hat p$ is the sample-dependent estimator of the propensity function fitted using the first half $\mathcal D_1$ of the split sample by regression of $D$ on $X$, with $\hat p(x)\in[c,1-c]$ for every $x\in\mathcal X$; the fitted function is fixed conditional on the training split and all randomness used in fitting. In Lemma~\ref{lemma:eta_rate}: $\hat p$ is measurable with respect to $\mathcal D_1$ and independent auxiliary data; its error inequality holds jointly with the regression and exponent inequalities with probability at least $1-\varepsilon$, uniformly almost surely over the covariate support, for each tolerance $\varepsilon$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $\hat p$ is measurable with respect to the training split and independent auxiliary data $\mathcal T$, and is the same fitted function for every input threshold. With probability at least $1-\delta/6$, jointly for every $t\in I_n=[\ell_n,u_n]$, its inequality $\|p(x)-\hat p(x)\|\leq R_p(n/2,\delta/6)$ holds together with the regression and exponent inequalities, simultaneously almost surely over the covariate and conditional direction support and both treatments. The additional estimation-rate condition is $R_p(n,\delta)=\Theta(\log(1/\delta)n^{-1/2})$. In Theorem~\ref{thm:rate}: $\hat p$ is measurable with respect to the training split and independent auxiliary data $\mathcal T$; for each fixed $t$ and $\varepsilon\in(0,1)$, its inequality $\|p(X)-\hat p(X)\|\leq R_p(n,\varepsilon)$ holds jointly with the regression and exponent inequalities on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments. In Lemma~\ref{lemma:eta_ipw}: $\hat p$ is measurable with respect to the training split and independent auxiliary data $\mathcal T$; its error inequality holds jointly with the regression and exponent inequalities with probability at least $1-\varepsilon$, uniformly almost surely over the covariate support, for each tolerance $\varepsilon$. — Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} specify $\mathcal T$-measurability, the same fitted function for every threshold, and a joint nuisance event uniform over the deterministic threshold band. - reading of the symbol `i`: $i$ is a locally bound integer index whose range is supplied by the expression in which it occurs. In Lemma~\ref{lemma:mu_rate}: In $R_i=\\|U_i\\|_1$, $i\in\{1,\ldots,n\}$ indexes the full input sample; in the adaptive comparison, $i\in\{L_n,\ldots,k\}$ and the compared statistic is $H_n(i)=i^{-1}\sum_{j=1}^{i}\log(R_{[j]}/R_{[i+1]})$. In Theorem~\ref{thm:rate}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: In $n_t=\sum_{i=1}^{n/2}I(\\|U_i\\|>t)$, $i\in\{1,\ldots,n/2\}$ indexes the training observations, indexed first after the random split. — Theorem~\ref{thm:rate} explicitly identifies the indices in $n_t$ as training observations indexed first after the random split; its corollaries and Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} inherit the training-split convention. - reading of the symbol `X_i`: $X_i\in\mathcal{X}$ is the covariate coordinate of the $i$th sampled tuple $(X_i,D_i,Y_i,U_i)$, indexed by $i=1,\ldots,N$ in the model and $i=1,\ldots,n$ in the estimation notation. In Proposition~\ref{prop:id}, Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $X_i$ takes values in $B_R$, so $\\|X_i\\|_1\leq R$, under the bounded-covariate assumption. In Theorem~\ref{thm:rate}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: After the random equal split, $X_i$ is the covariate coordinate of a training observation for $1\leq i\leq n/2$ and of an evaluation observation for $n/2t)$, $U_i$ belongs to the training split, whose observations are indexed first after the random split. In Lemma~\ref{lemma:mu_rate}: $R_i=\\|U_i\\|_1$ for $i\in\{1,\ldots,n\}$ uses every full-input observation, without restriction to a split or exceedance set. — Assumption~\ref{assump:iid} uses observation-indexed noise vectors. Theorem~\ref{thm:rate} explicitly indexes training observations first after splitting; Lemma~\ref{lemma:mu_rate} instead defines radii from every full-input observation. - reading of the symbol `S`: As a spectral measure, $S$ is the population probability measure on $\mathbb{S}^+=\{u\in\mathbb{R}_+^d:\\|u\\|=1\}$ in the regular-variation limit, equivalently the limiting law of $U/\\|U\\|$ conditional on $\\|U\\|>t$ as $t\rightarrow\infty$, with the $\ell_1$ norm used after the notation convention. — Definition~\ref{def:mvarying} defines the measure, and the identification proof describes the limiting conditional radial-angular law. The paragraph following (\ref{eq:semi_syn}) separately defines the experimental scalar $S$. - reading of the symbol `b_n`: $b_n$ is a deterministic positive sequence on the index domain of its local definition. In Proposition~\ref{prop:id}, Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: In the inherited regular-variation definition, $b_n$, $n\geq1$, is chosen for the law of $U$ and the chosen norm, tends to infinity, and satisfies $n\mathbb P((\\|U\\|/b_n,U/\\|U\\|)\in\cdot)\overset{w}{\rightarrow}c\nu_\beta\times S$. In Theorem~\ref{thm:rate}: In $Q_n$ and the radial estimation bound, $b_n=C_{\delta/6}\begin{cases}(\log((48/\delta)\log n)/n)^{1/3},&0<\beta<1,\\(\log n\,\log((48/\delta)\log n)/n)^{1/3},&\beta=1,\\(\log((48/\delta)\log n)/n)^{1/(2+\beta)},&\beta>1.\end{cases}$ For fixed $\delta$ and the fixed noise law this is a deterministic error-bound sequence tending to zero, defined for sufficiently large $n$. The inherited regular-variation definition separately uses a normalizing sequence tending to infinity. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: In the radial estimation bound and the term $\|\alpha\|b_n$, $b_n=C_{\delta/6}\begin{cases}(\log((48/\delta)\log n)/n)^{1/3},&0<\beta<1,\\(\log n\,\log((48/\delta)\log n)/n)^{1/3},&\beta=1,\\(\log((48/\delta)\log n)/n)^{1/(2+\beta)},&\beta>1.\end{cases}$ For each fixed $\delta\in(0,1/2)$ and fixed noise law, this is a deterministic error-bound sequence tending to zero, used for sufficiently large even $n$. The inherited regular-variation definition separately uses a normalizing sequence tending to infinity. — Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} explicitly define $b_n$ using the constant supplied by Lemma~\ref{lemma:mu_rate}, and use it in the term $\|\alpha\|b_n$. - reading of the symbol `Z`: $Z$ denotes the random variable specified by its local definition: an illustrative regularly varying scalar, a nonnegative component vector in $U=AZ$, or a scalar truncated evaluation score. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $Z\in\mathbb{R}_+^{d_z}$ is the random component vector with $\mathcal{L}(Z)\in\tilde M_k$ for a positive integer $k$ in the population representation $U=AZ$; the approximation condition does not assert independence of its components. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:mu_rate}: $Z\in\mathbb R_+^{d_z}$ is the random component vector with $\mathcal L(Z)\in\tilde M_k$ for a positive integer $k$ in the population representation $U=AZ$; the approximation condition does not assert independence of its components. In Lemma~\ref{lemma:eta_rate}: In the tail-model assumption, $Z\in\mathbb R_+^{d_z}$ is the component vector with $\mathcal L(Z)\in\tilde M_k$ for a positive integer $k$, without an assertion of component independence. In the proof's conditional sampling argument, the local definition $Z=\psi I\{\|\psi\|\leq B\}$ denotes a scalar truncated evaluation score under the law conditional on $\mathcal T$ and $R_U>t$; $Z_i$ are its evaluation realizations. In Lemma~\ref{lemma:eta_ipw}: In the tail-model assumption, $Z\in\mathbb R_+^{d_z}$ is the component vector with $\mathcal L(Z)\in\tilde M_k$ for a positive integer $k$, without an assertion of component independence. In the proof's conditional sampling argument, the local definition $Z=\phi I\{\|\phi\|\leq B\}$ denotes a scalar truncated evaluation score under the law conditional on $\mathcal T$ and $R_U>t$; $Z_i$ are its evaluation realizations. — Assumption~\ref{assump:mvary} defines the component vector. The proofs of Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} additionally define $Z=\psi I\{\|\psi\|\leq B\}$ and $Z=\phi I\{\|\phi\|\leq B\}$, respectively, in their conditional sampling arguments. - reading of the symbol `A`: In the representation $U=AZ$, $A$ is a nonnegative $d_u\times d_z$ matrix. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: $A\in\mathcal{A}$ is a fixed matrix specifying the population representation $U=AZ$, rather than an estimated matrix or a matrix redrawn for each observation. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $A\in\mathcal A$ is a fixed matrix specifying the population representation $U=AZ$, rather than an estimated matrix or a matrix redrawn for each observation. — Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} inherit Assumption~\ref{assump:mvary} through Theorem~\ref{thm:rate}; their printed threshold rules do not introduce $A_n$. - reading of the symbol `\tilde M_k`: $\tilde M_k$ is a fixed population class of component laws indexed by $k\in\{1,2,\ldots\}$, with fixed $\beta>0$, $00$; its joint-density conditions do not assert independence of the components. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: $m=d_z$ in the defining density conditions, so the nominal density is $\beta^{d_z}\prod_{i=1}^{d_z}(1+z_i)^{-(\beta+1)}$ and every component product ranges over $i\in\{1,\ldots,d_z\}$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $m=d_z$ in the defining density conditions, so the nominal density is $\beta^{d_z}\prod_{i=1}^{d_z}(1+z_i)^{-(\beta+1)}$ and every component product ranges over $i\in\{1,\ldots,d_z\}$. — The defining display is in Assumption~\ref{assump:mvary}. Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} incorporate the component-dimension convention through Lemmas~\ref{lemma:mu_rate} and~\ref{lemma:eta_rate}. - reading of the symbol `\mathcal{A}`: $\mathcal{A}$ is the fixed population matrix class, with column indices $i\in\{1,\ldots,d_z\}$ and fixed constants $00$. — The matrix-class display in Assumption~\ref{assump:mvary} defines a subset of the matrix space, rather than a matrix or a probability measure on matrices. - reading of the symbol `M_k`: $M_k$ is a fixed population distribution class indexed by $k\in\{1,2,\ldots\}$, rather than by the sample size or a selected Hill order statistic. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:mu_rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: $M_k=\{\mathcal L(U):U=AZ,\ A\in\mathcal A,\ \mathcal L(Z)\in\tilde M_k\}$; the class concerns the noise law, with $m=d_z$ in the component density. — Assumption~\ref{assump:mvary} defines $M=\bigcup_{k=1}^{\infty}M_k$. Lemmas~\ref{lemma:mu_rate} and~\ref{lemma:eta_rate} specify the noise law $\mathcal L(U)$, and Theorem~\ref{thm:rate} and Lemma~\ref{lemma:eta_ipw} incorporate those conditions. - reading of the symbol `M`: $M$ denotes the object specified by its local definition, either a class of probability laws or an observed exceedance count. In Lemma~\ref{lemma:mu_rate}: $M=\bigcup_{k=1}^{\infty}M_k$ is a class of noise laws, with $M_k=\{\mathcal L(U):U=AZ,\ A\in\mathcal A,\ \mathcal L(Z)\in\tilde M_k\}$ and $m=d_z$ in the component density. In Lemma~\ref{lemma:eta_rate}: In the score bound and the zero-count convention, $M=\|\mathcal I\|$ is the realized second-half evaluation exceedance count. The tail-model assumption separately concerns the noise law $\mathcal L(U)$, with $m=d_z$ in its component density. In Theorem~\ref{thm:rate}: In the sampling terms, count bound and zero-count convention, $M=\|\mathcal I\|$ is the realized second-half evaluation exceedance count. In the inherited tail-model definition, $M=\bigcup_{k=1}^{\infty}M_k$ separately denotes the class of noise laws, with component dimension $m=d_z$. In Lemma~\ref{lemma:eta_ipw}: In the score bound and zero-count convention, $M=\|\mathcal I\|$ is the realized second-half evaluation exceedance count. The tail-model assumption separately concerns the noise law $\mathcal L(U)$, with $m=d_z$ in its component density. — Theorem~\ref{thm:rate} now explicitly puts $M=\|\mathcal I\|$ and uses it in the sampling terms and zero-count convention, while retaining Assumption~\ref{assump:mvary}. - reading of the symbol `s`: In the tail model, $s\in(0,1/2)$ is a fixed model constant governing both the density error $\xi k^{-s}$ and the exact-tail threshold $\zeta k^{(1-2s)/\beta}$. In Lemma~\ref{lemma:eta_rate}: In $m_d^t(x,s)$, $S_U=s$, $s\mapsto\mathbb E_X[g(X,1,s)-g(X,0,s)]$, and $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, $s\in\mathbb S^+=\{u\in\mathbb R_+^d:\\|u\\|_1=1\}$ is an angular argument or bound integration variable. In $t^{-\beta s/(1-2s)}$, $s$ retains its scalar tail-model meaning. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $s$ is the scalar tail-model constant in the threshold exponent and the two item restrictions: the first item assumes $00$ is a fixed model constant imposing $JA=\sqrt{\det(A^\mathsf{T}A)}\geq\sigma$ on matrices in $\mathcal{A}$. — Assumption~\ref{assump:mvary} defines $JA$, imposes its lower bound, and explicitly lists $\sigma$ among the constants. - reading of the symbol `\xi`: $\xi\in(0,1)$ is a fixed model constant bounding the relative density approximation error by $\xi k^{-s}$. — Assumption~\ref{assump:mvary} gives the density inequality and explicitly calls $\xi$ a constant. - reading of the symbol `\zeta`: $\zeta>0$ is a fixed constant in the tail-model class: for each class index $k$, the exact proportional-density region is $\\|z\\|_1>\zeta k^{(1-2s)/\beta}$. — Assumption~\ref{assump:mvary} defines the region in $\tilde M_k$ and explicitly lists $\zeta>0$ among its constants. - reading of the symbol `n`: $n$ is the size of the full input dataset $\mathcal{D}=\{(X_i,D_i,Y_i,U_i)\}_{i=1}^n$, before its split into two equal halves; when used as the argument of an estimation-error function, it denotes that function's sample-size argument. In Lemma~\ref{lemma:mu_rate}: $n$ is the integer size of the full input sample whose radii are ordered, with $n\geq2$ for the definition of $H_n(k)$; for each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that the deviation bound holds with probability at least $1-\delta$ under the sample-size-$n$ law for every integer $n\geq n_0$. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The full-input sample size $n$ is a positive even integer, with $\|\mathcal D_1\|=\|\mathcal D_2\|=n/2$; this restriction does not restrict a generic sample-size argument of an estimation-error function to even integers. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: For each fixed $\delta\in(0,1/2)$, the probability guarantee holds separately under the sample-size-$n$ law for every sufficiently large even $n$. — Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} quantify separately over every sufficiently large even $n$ for each fixed $\delta\in(0,1/2)$. The equal-split convention remains in force. - reading of the symbol `\delta`: $\delta$ is a fixed confidence tolerance in $(0,1)$, used in probability guarantees of the form $1-\delta$ and in the adaptive Hill tuning rule. In Lemma~\ref{lemma:mu_rate}: For each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that, for every integer $n\geq n_0$, the bound has probability at least $1-\delta$ under the law of the sample of size $n$; $n_0$ and $C_\delta$ may depend on $\delta$. In Theorem~\ref{thm:rate}: $\delta\in(0,1/2)$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: For each fixed $\delta\in(0,1/2)$, the probability guarantee holds separately under the sample-size-$n$ law for every sufficiently large even $n$; the sample-size requirement may depend on $\delta$. The full-input Hill selection uses tolerance $\delta/6$. — Theorem~\ref{thm:rate} fixes $\delta\in(0,1/2)$. Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} give their guarantees separately for each fixed $\delta\in(0,1/2)$ and every sufficiently large even $n$, and run Hill selection at tolerance $\delta/6$. - reading of the symbol `n_t`: $n_t$ is the observed number of threshold exceedances in the designated half of the sample, with the half and strictness of the threshold determined by its local definition. In Theorem~\ref{thm:rate}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $n_t=\sum_{i=1}^{n/2}I(\\|U_i\\|>t)$ is the realized training-split exceedance count for positive even $n$, taking values in $\{0,\ldots,n/2\}$; it is distinct from the evaluation count $\|\mathcal I\|$ and from its expectation $n\mathbb P(\\|U\\|>t)/2$. — Theorem~\ref{thm:rate} defines the first-half count with strict exceedances; Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} use its nuisance conditions. Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} use the same count in their simultaneous threshold-dependent regression bounds. - reading of the symbol `R_p`: $R_p(n,\delta)$ is a deterministic positive error-bound function, monotonically decreasing in its sample-size argument, used in the high-probability inequality $\|p(X)-\hat p(X)\|\leqslant R_p(n,\delta)$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $R_p(n,\varepsilon)=\Theta(\log(1/\varepsilon)n^{-1/2})$. On one joint nuisance event of probability at least $1-\delta/6$, the three nuisance inequalities hold simultaneously for every $t\in I_n$, almost surely over the covariate and conditional direction support and both treatments; the propensity inequality is $\|p(x)-\hat p(x)\|\leqslant R_p(n/2,\delta/6)$, using the same fitted propensity for every input threshold. In Theorem~\ref{thm:rate}: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, conditional direction support and both treatments. The displayed propensity inequality uses $R_p(n,\varepsilon)$; the training-split propensity bound entering the final error bounds uses $R_p(n/2,\varepsilon)$. In Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: For each tolerance $\varepsilon$, the nuisance inequalities hold jointly with probability at least $1-\varepsilon$, uniformly almost surely over the covariate and direction support and both treatments. The training-split propensity bound used in the proof is $\|p(x)-\hat p(x)\|\leqslant R_p(n/2,\varepsilon)$. — Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} explicitly require the three nuisance bounds jointly for every $t\in I_n$, with probability at least $1-\delta/6$, and use $R_p(n/2,\delta/6)$ for the fitted training-split propensity. - reading of the symbol `R_g`: $R_g(q,\delta)$ is a nonnegative regression-error bound function, nonincreasing in its sample-size argument $q$, for estimating the conditional mean of $Y/\\|U\\|^\alpha$ given $(X,D,U/\\|U\\|)$ and $\\|U\\|>t$; evaluation at $q=n_t$ uses the observed tail count. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $R_g(n,\delta)=\Theta(\log(1/\delta)n^{-1/2})$. On one joint nuisance event of probability at least $1-\delta/6$, simultaneously for every $t\in I_n$, $\|m_d^t(x,v)-\hat g(x,d,v)\|\leqslant R_g(n_t,\delta/6)$ almost surely over the covariate and conditional direction support and both treatments, where $m_d^t(x,v)=\mathbb E[Y/\\|U\\|^\alpha\mid X=x,D=d,U/\\|U\\|=v,\\|U\\|>t]$ and $n_t=\sum_{i=1}^{n/2}I(\\|U_i\\|>t)$ is the training-split exceedance count. In Lemma~\ref{lemma:eta_rate}: For each tolerance $\varepsilon\in(0,1)$, the bound $R_g(n_t,\varepsilon)$ applies uniformly almost surely over the covariate and direction support and both treatments, on the joint nuisance event of probability at least $1-\varepsilon$; its sample-size argument $n_t$ is the training-split exceedance count, distinct from $M=\|\mathcal I\|$. For positive even $n$, its domain includes every $q\in\{0,1,\ldots,n/2\}$, including $q=0$. In Theorem~\ref{thm:rate}: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the regression-error bound $R_g(n_t,\varepsilon)$ holds together with the propensity and exponent inequalities on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, conditional direction support, and both treatments; $n_t=\sum_{i=1}^{n/2}I(\\|U_i\\|>t)$ is the training-split exceedance count, with training observations indexed first after the random split. In Lemma~\ref{lemma:eta_ipw}: For each tolerance $\varepsilon$, $R_g(n_t,\varepsilon)$ applies uniformly almost surely over the covariate and direction support and both treatments on the joint nuisance event of probability at least $1-\varepsilon$; $n_t$ is the training-split exceedance count, distinct from $M=\|\mathcal I\|$. Its sample-size domain includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. — Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} specify a joint nuisance event of probability at least $1-\delta/6$ for every threshold in the deterministic band $I_n$, with the regression bound indexed by the training exceedance count. - reading of the symbol `R_\alpha`: $R_\alpha(n,\delta)$ is a deterministic positive error-bound function, monotonically decreasing in its sample-size argument, used in the high-probability inequality $\|\hat\alpha_n-\alpha\|\leqslant R_\alpha(n,\delta)$. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $R_\alpha(n,\delta)=\Theta(\log(1/\delta)n^{-c_\alpha})$ for some $c_\alpha>0$. — Theorem~\ref{thm:rate} gives the defining error inequality and states monotonicity in sample size. Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} additionally prescribe its rate. - reading of the symbol `\widehat{\theta}_{n,t}^{\text{DR}}`: $\widehat\theta_{n,t}^{\text{DR}}$ is the sample estimator formed by multiplying the split-sample normalized DR first-factor estimate by $\hat\mu_n=(1-\hat\alpha_n\hat\gamma_n)^{-1}$, wherever that denominator and the first-factor estimate are defined. In Corollary~\ref{cor:rate}: The estimator is evaluated at $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ in the first item and $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ in the second item, with deterministic positive order constants independent of $n$ and the sample. The fitted $\hat p$, $\hat g$, and $\hat\alpha_n$ are measurable with respect to the training split and independent auxiliary data $\mathcal T$ and are the same functions and exponent for every input threshold. The Hill estimate $\hat\gamma_n$ uses all $n$ input radii at tolerance $\delta/6$. Use $\hat\eta_{n,t}^{\mathrm{DR}}=0$ when there are no evaluation exceedances and $\hat\mu_n=0$ when $1-\hat\alpha_n\hat\gamma_n=0$. In Theorem~\ref{thm:rate}: The threshold $t$ is deterministic, the fitted propensity, regression, and exponent are measurable with respect to the training split and independent auxiliary data $\mathcal T$, and $\hat\gamma_n=\widetilde\gamma_n$ is computed from all $n$ input radii at tolerance $\delta/6$; use $\hat\eta_{n,t}^{\mathrm{DR}}=0$ when $M=0$ and $\hat\mu_n=0$ when $1-\hat\alpha_n\hat\gamma_n=0$. — Corollary~\ref{cor:rate} specifies fitted nuisances shared across thresholds, full-input Hill selection at tolerance $\delta/6$, and zero first-factor estimates for empty evaluation tails; it inherits the reciprocal convention of Theorem~\ref{thm:rate}. - reading of the symbol `\widehat{\theta}_{n,t}^{\text{IPW}}`: $\widehat\theta_{n,t}^{\text{IPW}}$ is the split-sample product $\hat\eta_{n,t}^{\mathrm{IPW}}\hat\mu_n$, using $\hat p$ fitted on $\mathcal D_1$, the supplied exponent $\hat\alpha_n$, and the normalized IPW score averaged over evaluation exceedances in $\mathcal D_2$, with $\hat\mu_n=(1-\hat\alpha_n\hat\gamma_n)^{-1}$ wherever the average and reciprocal are defined. In Corollary~\ref{cor:rate_ipw}: The estimator is evaluated at $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ in the first item and $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ in the second item, with deterministic positive order constants independent of $n$ and the sample. The fitted $\hat p$, $\hat g$, and $\hat\alpha_n$ are measurable with respect to the training split and independent auxiliary data $\mathcal T$ and are the same functions and exponent for every input threshold. The Hill estimate $\hat\gamma_n$ uses all $n$ input radii at tolerance $\delta/6$. Use $\hat\eta_{n,t}^{\mathrm{IPW}}=0$ when there are no evaluation exceedances and $\hat\mu_n=0$ when $1-\hat\alpha_n\hat\gamma_n=0$. In Theorem~\ref{thm:rate}: The threshold $t$ is deterministic, the fitted propensity, regression, and exponent are measurable with respect to the training split and independent auxiliary data $\mathcal T$, and $\hat\gamma_n=\widetilde\gamma_n$ is computed from all $n$ input radii at tolerance $\delta/6$; use $\hat\eta_{n,t}^{\mathrm{IPW}}=0$ when $M=0$ and $\hat\mu_n=0$ when $1-\hat\alpha_n\hat\gamma_n=0$. — Corollary~\ref{cor:rate_ipw} specifies fitted nuisances shared across thresholds, full-input Hill selection at tolerance $\delta/6$, and zero first-factor estimates for empty evaluation tails; it inherits the reciprocal convention of Theorem~\ref{thm:rate}. - reading of the symbol `\hat{\eta}_{n,t}^{\tmop{DR}}`: $\hat\eta_{n,t}^{\tmop{DR}}$ is the average normalized doubly robust score over exceedances in the second half $\mathcal{D}_2$, using $\hat p$ and $\hat g$ fitted on the first half $\mathcal{D}_1$ and the supplied exponent $\hat\alpha_n$, wherever $\|\mathcal{I}\|>0$. In Lemma~\ref{lemma:eta_rate}: Set $\hat\eta_{n,t}^{\tmop{DR}}=0$ when $M=\|\mathcal I\|=0$; conditional on $\mathcal D_1$ and the independent auxiliary data, the fitted $\hat p$, $\hat g$, and $\hat\alpha_n$ are fixed, and for $M>0$ the estimator averages the displayed score $\psi$ over the evaluation exceedances. In Corollary~\ref{cor:rate}: The estimator is evaluated at $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ in the first item and $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ in the second item, with deterministic positive order constants independent of $n$ and the sample. Set $\hat\eta_{n,t}^{\tmop{DR}}=0$ when there are no evaluation exceedances. The fitted $\hat p$, $\hat g$, and $\hat\alpha_n$ are $\mathcal T$-measurable, where $\mathcal T$ consists of the training split and independent auxiliary data, and are the same functions and exponent for every input threshold; they are held fixed in the score moments conditional on $\mathcal T$ and $\\|U\\|>t$. In Theorem~\ref{thm:rate}: Set $\hat\eta_{n,t}^{\tmop{DR}}=0$ when $M=\|\mathcal I\|=0$; conditional on the training split and independent auxiliary data $\mathcal T$, the fitted $\hat p$, $\hat g$, and $\hat\alpha_n$ are fixed, and for $M>0$ the estimator averages $\psi^{\mathrm{DR}}$ over the evaluation exceedances. In Lemma~\ref{lemma:eta_ipw}: Set $\hat\eta_{n,t}^{\tmop{DR}}=0$ when $M=\|\mathcal I\|=0$; conditional on the training split and independent auxiliary data $\mathcal T$, the fitted $\hat p$, $\hat g$, and $\hat\alpha_n$ are fixed, and for $M>0$ the estimator averages the normalized doubly robust score over the evaluation exceedances. — Corollary~\ref{cor:rate} specifies training-measurable fitted functions and exponent shared across thresholds, zero for an empty evaluation tail, and deterministic constants in the adaptive threshold order. - reading of the symbol `\hat{\eta}_{n,t}^{\text{IPW}}`: $\hat\eta_{n,t}^{\mathrm{IPW}}$ is the sample average over the threshold exceedances in the second half $\mathcal{D}_2$ of the split sample, using $\hat p$ fitted on the first half $\mathcal{D}_1$ and the supplied exponent $\hat\alpha_n$, wherever $\|\mathcal{I}\|>0$. In Corollary~\ref{cor:rate_ipw}: The estimator is evaluated at $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ in the first item and $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ in the second item, with deterministic positive order constants independent of $n$ and the sample. Set $\hat\eta_{n,t}^{\mathrm{IPW}}=0$ when there are no evaluation exceedances. The fitted propensity and exponent are $\mathcal T$-measurable, where $\mathcal T$ consists of the training split and independent auxiliary data, and are the same function and exponent for every input threshold; they are held fixed in the score moments conditional on $\mathcal T$ and $\\|U\\|>t$. In Lemma~\ref{lemma:eta_ipw}: $\hat\eta_{n,t}^{\mathrm{IPW}}=0$ when $M=\|\mathcal I\|=0$. The fitted propensity and exponent are measurable with respect to $\mathcal T$, the training split together with independent auxiliary data, and are held fixed in the score moments conditional on $\mathcal T$ and $\\|U\\|>t$. In Theorem~\ref{thm:rate}: $\hat\eta_{n,t}^{\mathrm{IPW}}=0$ when $M=\|\mathcal I\|=0$. The fitted propensity and exponent are measurable with respect to $\mathcal T$, the training split together with independent auxiliary data, and are held fixed in the score moments conditional on $\mathcal T$ and $\\|U\\|>t$; the threshold $t$ is deterministic. — Corollary~\ref{cor:rate_ipw} specifies training-measurable fitted functions and exponent shared across thresholds, zero for an empty evaluation tail, and deterministic constants in the adaptive threshold order. - reading of the symbol `\eta`: $\eta$ is a population quantity whose scalar or functional meaning is determined by its local use. In Lemma~\ref{lemma:eta_rate}: $\eta=\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\mathrm{d}s)$, where $S$ is the limiting spectral probability measure. In Theorem~\ref{thm:rate}: $\eta=\lim_{t\to\infty}\mathbb E[g(X,1,U/\\|U\\|)-g(X,0,U/\\|U\\|)\mid\\|U\\|>t]$ is the scalar population first-factor target. In Lemma~\ref{lemma:eta_ipw}: $\eta=\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. In Lemma~\ref{lemma:mu_rate}: $\eta(v)=vQ'(v)/Q(v)-\gamma=\bar F_R(Q(v))/(Q(v)f_R(Q(v)))-1/\beta$ is the deterministic von Mises function on the differentiable tail of $Q(v)=F_R^{-1}(1-1/v)$, where $R=\\|U\\|_1$, $F_R$ is its population distribution function, $\bar F_R$ its survival function, $f_R$ its density, and $\gamma=1/\beta$. — The proof of Lemma~\ref{lemma:mu_rate} in discovery/core.json defines $Q(v)=F_R^{-1}(1-1/v)$ and displays the von Mises function in terms of $Q$, the radial survival function, and the radial density. - reading of the symbol `\mathcal{D}`: $\mathcal D=\{(X_i,D_i,Y_i,U_i)\}_{i=1}^n$ denotes the indexed input dataset of $n$ observations. In Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: The sample size $n$ is a positive even integer, and the equal split satisfies $\|\mathcal D_1\|=\|\mathcal D_2\|=n/2$. In Theorem~\ref{thm:rate}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The input size $n$ is a positive even integer; after the random equal split, indices $1,\ldots,n/2$ designate the training observations and indices $n/2+1,\ldots,n$ designate the evaluation observations. — Algorithm~\ref{alg:nete} specifies the indexed input and equal split. Theorem~\ref{thm:rate} specifies training-first indexing after that split, inherited by Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw}. - reading of the symbol `\mathcal{D}_1`: $\mathcal D_1$ is the first half of the random equal split of the indexed dataset $\mathcal D$, containing $n/2$ observations and supplying the observations for nuisance regression. In Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: The sample size $n$ is a positive even integer, and $\|\mathcal D_1\|=\|\mathcal D_2\|=n/2$. In Theorem~\ref{thm:rate}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The input size $n$ is a positive even integer, and after the random split the training observations are indexed first, so $\mathcal D_1=\{(X_i,D_i,Y_i,U_i)\}_{i=1}^{n/2}$ and $n_t=\sum_{i=1}^{n/2}I(\\|U_i\\|>t)$ counts its exceedances. — The binding reading of Theorem~\ref{thm:rate} specifies that the training observations are indexed first after the random split; Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} inherit its assumptions. Algorithm~\ref{alg:nete} uses equal halves. - reading of the symbol `\mathcal{D}_2`: $\mathcal D_2$ is the second equal part of the split dataset, used for evaluation, with the equal split defined for positive even sample sizes $n$ and $\|\mathcal D_1\|=\|\mathcal D_2\|=n/2$. — Algorithm~\ref{alg:nete} defines the equal split and uses its second part for evaluation; the binding reading of Lemma~\ref{lemma:eta_rate} explicitly specifies positive even $n$ and both split cardinalities. - reading of the symbol `\hat{\alpha}_n`: $\hat{\alpha}_n\in\mathbb{R}$ is the supplied approximate value of the outcome scaling index $\alpha$, obtainable from prior knowledge or heuristics. In Lemma~\ref{lemma:eta_rate}: $\hat{\alpha}_n$ is measurable with respect to the training split and independent auxiliary data; for each tolerance $\varepsilon$, its bound $\|\hat{\alpha}_n-\alpha\|\leq R_\alpha(n,\varepsilon)$ holds jointly with the other nuisance inequalities with probability at least $1-\varepsilon$. The bound uses $a=R_\alpha(n,\delta/4)\leq a_0$ for fixed $a_0\in(0,\beta)$. In Theorem~\ref{thm:rate}: $\hat\alpha_n$ is measurable with respect to $\mathcal T$, the training split and independent auxiliary data. For every fixed $t$ and tolerance $\varepsilon\in(0,1)$, $\|\hat\alpha_n-\alpha\|\leq R_\alpha(n,\varepsilon)$ holds on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, jointly with the other nuisance inequalities, simultaneously almost surely over the covariate support, conditional direction support and both treatments. The bound uses $a=R_\alpha(n,\delta/12)\leq a_0$ for fixed $a_0\in(0,\beta)$. In Lemma~\ref{lemma:eta_ipw}: $\hat{\alpha}_n$ is measurable with respect to the training split and independent auxiliary data; for every tolerance $\varepsilon$, its inequality $\|\hat{\alpha}_n-\alpha\|\leq R_\alpha(n,\varepsilon)$ holds jointly with the other nuisance inequalities with probability at least $1-\varepsilon$. The exponent-error quantity is $a=R_\alpha(n,\delta/4)\leq a_0$, where $a_0\in(0,\beta)$ is fixed. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $\hat\alpha_n$ is $\mathcal T$-measurable, where $\mathcal T$ consists of the training split and independent auxiliary data, and is the same exponent for every input threshold. Its rate satisfies $R_\alpha(n,\delta)=\Theta(\log(1/\delta)n^{-c_\alpha})$ for some fixed $c_\alpha>0$. With probability at least $1-\delta/6$, the inequality $\|\hat\alpha_n-\alpha\|\leq R_\alpha(n,\delta/6)$ holds jointly with the propensity and regression inequalities for every $t\in I_n=[(c_-/2)n^{\kappa(\gamma)},2c_+n^{\kappa(\gamma)}]$, simultaneously almost surely over the covariate and conditional direction support and both treatments. — Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} require $\mathcal T$-measurability, the same exponent for every input threshold, and a joint nuisance event of probability at least $1-\delta/6$ uniformly over their deterministic threshold band. - reading of the symbol `\hat{g}`: $\hat g(x,d,s)$ is the fitted regression of $Y/\\|U\\|^{\hat\alpha_n}$ on $(X,D,U/\\|U\\|)$ using $\mathcal D_1$, with $s\in\mathbb S^+$ for the paper's $\ell_1$ norm. In Lemma~\ref{lemma:eta_rate}: $\hat g$ is measurable with respect to $\mathcal D_1$ and independent auxiliary data and is held fixed in the conditional evaluation-score expectations; its nuisance inequality holds uniformly almost surely over the covariate and direction support and both treatments on the stipulated joint nuisance events. In Theorem~\ref{thm:rate}: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, $\hat g$ is measurable with respect to the training split and independent auxiliary data, denoted together by $\mathcal T$, and is held fixed in the conditional evaluation-score expectations; its nuisance comparison is with $\mathbb E[Y/\\|U\\|^{\alpha}\mid X,D,U/\\|U\\|,\\|U\\|>t]$, with bound $R_g(n_t,\varepsilon)$, although the fitted response uses $\hat\alpha_n$. This inequality holds simultaneously almost surely over the covariate support, conditional direction support and both treatments on one $\mathcal T$-measurable joint nuisance event of probability at least $1-\varepsilon$. In Lemma~\ref{lemma:eta_ipw}: $\hat g$ is measurable with respect to the training split and independent auxiliary data, denoted together by $\mathcal T$; its nuisance inequality holds uniformly almost surely over the covariate and direction support and both treatments on the stipulated joint nuisance events. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: $\hat g$ is measurable with respect to the training split and independent auxiliary data, denoted together by $\mathcal T$, and is the same fitted function for every input threshold; with probability at least $1-\delta/6$, jointly for every $t\in I_n$, $\|m_d^t(x,v)-\hat g(x,d,v)\|\leq R_g(n_t,\delta/6)$ simultaneously almost surely over the covariate and conditional direction support and both treatments, where $m_d^t(x,v)=\mathbb E[Y/\\|U\\|^\alpha\mid X=x,D=d,U/\\|U\\|=v,\\|U\\|>t]$ and $n_t$ is the training exceedance count. — Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} require the same $\mathcal T$-measurable fitted functions for every input threshold and specify a joint nuisance event over every $t\in I_n$ with probability at least $1-\delta/6$. - reading of the symbol `\mathcal{I}`: $\mathcal I=\{i\in\{1,\ldots,n\}: (X_i,D_i,Y_i,U_i)\in\mathcal D_2,\ \\|U_i\\|>t\}$ is the evaluation exceedance index set at the threshold supplied to the estimator. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: At the selected threshold $t_n$, $\mathcal I=\{i\in\{1,\ldots,n\}:(X_i,D_i,Y_i,U_i)\in\mathcal D_2,\ \\|U_i\\|>t_n\}$. — Algorithm~\ref{alg:nete} defines the evaluation exceedance set. Corollaries~\ref{cor:rate} and~\ref{cor:rate_ipw} substitute their selected threshold into the estimator but do not introduce $M_{t_n}$ or specify the previously recorded full-sample threshold construction. - reading of the symbol `\hat{\gamma}_n`: $\hat\gamma_n$ denotes a sample EVI estimate computed from ordered $\ell_1$ radii; the displayed definition in Algorithm~\ref{alg:nete} is $\hat\gamma_n=k^{-1}\sum_{j=1}^k\log(\\|U_{(j)}\\|/\\|U_{(j+1)}\\|)$ on the evaluation exceedances, with the displayed adaptive choice of $k$. In Theorem~\ref{thm:rate}, Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}, Lemma~\ref{lemma:eta_rate}, Lemma~\ref{lemma:eta_ipw}: The full-input convention is $\hat\gamma_n=\widetilde\gamma_n=H_n(\widetilde k_n)$ at tolerance $\delta/6$. Here $R_{[1]}\geq\cdots\geq R_{[n]}$ order all input radii, $H_n(k)=k^{-1}\sum_{j=1}^k\log(R_{[j]}/R_{[k+1]})$ for $1\leq k\leq n-1$, and $\widetilde k_n$ is the maximum $k\in\{L_n,\ldots,n-1\}$ satisfying $\|H_n(i)-H_n(k)\|\leq H_n(i)\rho_n(\delta/6)/\sqrt i$ for every $i\in\{L_n,\ldots,k\}$, using the displayed tuning definitions for $L_n$ and $\rho_n$. — Equation~\ref{eq:mu} prints the evaluation-exceedance formula with adjacent order statistics. Theorem~\ref{thm:rate} instead explicitly identifies its full-input output with $\widetilde\gamma_n$ of Lemma~\ref{lemma:mu_rate}; the two corollaries and first-factor lemmas inherit the theorem's assumptions. - reading of the symbol `t_n`: $t_n$ denotes the selected threshold sequence, depending on $\hat\gamma_n$, with $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ for $s\in(0,1/(2+\max\{1,\beta\}))$ and $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ for $s\in[1/(2+\max\{1,\beta\}),1/2)$; the $t$ in the corresponding estimator denotes this selected threshold. In Corollary~\ref{cor:rate}, Corollary~\ref{cor:rate_ipw}: The threshold order means $c_-n^{\kappa(\hat\gamma_n)}\leq t_n\leq c_+n^{\kappa(\hat\gamma_n)}$, with deterministic constants $00$ such that, for every $a>0$, the polar measures $n\tmop{P}((\\|U\\|/b_n,U/\\|U\\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. (pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation.) | | `ass:pin-39` | `prop:id` | V1.5 bookkeeping | $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. (pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely.) | | `ass:pin-43` | `prop:id` | V1.5 bookkeeping | The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\\|x\\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\\|u\\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. (pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere.) | | `ass:pin-7` | `prop:id` | V1.5 bookkeeping | $Y(D)=Y$ almost surely throughout the paper. (pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely.) | | `ass:pin-fix-8cd5c960` | `prop:id` | V1.5 bookkeeping | The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\\|u\\|_1=1$. (pins reading of prop:id: stated by the repair) | | `ass:pin-33` | `thm:rate` | V1.5 bookkeeping | $U\perp(X,D)$. (pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly.) | | `ass:pin-34` | `thm:rate` | V1.5 bookkeeping | $\\|X\\|_1\leqslant R$ almost surely. (pins reading of ass:f-homo-bounded-x: Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm.) | | `ass:pin-36` | `thm:rate` | V1.5 bookkeeping | For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\\|z\\|_1>\zeta k^{(1-2s)/\beta}$. (pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$.) | | `ass:pin-38` | `thm:rate` | V1.5 bookkeeping | For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\\|U\\|/b_n,U/\\|U\\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. (pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation.) | | `ass:pin-39` | `thm:rate` | V1.5 bookkeeping | $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. (pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely.) | | `ass:pin-41` | `thm:rate` | V1.5 bookkeeping | The existential cutoff $t_W$ is chosen sufficiently large for Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} and so that $\sup_{r\geq t_W}e(r)\leq1$ and $\mathbb P(\\|U\\|>tr)/\mathbb P(\\|U\\|>t)\leq4r^{-\beta}$ for every $t\geq t_W$ and $r\geq1$. (pins reading of thm:rate: The proof of Theorem~\ref{thm:rate} invokes the first-factor lemmas' sufficiently-large-threshold conclusions, uses the radial moment inequality with $K=4/(\beta-a_0)$, and bounds $G_t$ by $C+1$. The stated Pareto-type radial law and $e(t)\to0$ permit this enlargement of the existential cutoff.) | | `ass:pin-42` | `thm:rate` | V1.5 bookkeeping | For each fixed $\mathcal T$-measurable choice $B>0$, the displayed probability bound uses the base conditions of Lemma~\ref{lemma:eta_rate}; its angular specialization additionally uses that lemma's angular assumption and $t\geq t_W$, and its variance alternative additionally requires $V_t^{\text{IPW}}<\infty$. (pins reading of lem:eta-ipw: The statement explicitly says ``Under the angular assumption'' and ``When'' the variance is finite; the amended proof applies conditional sampling inequalities to the chosen $B$.) | | `ass:pin-43` | `thm:rate` | V1.5 bookkeeping | The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\\|x\\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\\|u\\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. (pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere.) | | `ass:pin-44` | `thm:rate` | V1.5 bookkeeping | For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. (pins reading of ass:rate-nuisance-errors: Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation.) | | `ass:pin-45` | `thm:rate` | V1.5 bookkeeping | The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. (pins reading of def:nete-algorithm: Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention.) | | `ass:pin-7` | `thm:rate` | V1.5 bookkeeping | $Y(D)=Y$ almost surely throughout the paper. (pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely.) | | `ass:pin-fix-0b814294` | `thm:rate` | V1.5 bookkeeping | The equal split in Algorithm~\ref{alg:nete} is defined for positive even sample sizes $n$, with $\|\mathcal D_1\|=\|\mathcal D_2\|=n/2$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-10c2ec7b` | `thm:rate` | V1.5 bookkeeping | The component dimension in the density of Assumption~\ref{assump:mvary} is $m=d_z$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-6857bb4d` | `thm:rate` | V1.5 bookkeeping | For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments. (pins reading of thm:rate: stated by the repair) | | `ass:pin-fix-8964f8dd` | `thm:rate` | V1.5 bookkeeping | Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-8a129280` | `thm:rate` | V1.5 bookkeeping | The fitted-exponent pseudo-outcome has finite conditional absolute mean under the evaluation tail law: $\mathbb E[\|Y\|/\\|U\\|^{\hat\alpha_n}\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t]<\infty$ almost surely on each joint nuisance event. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-8cd5c960` | `thm:rate` | V1.5 bookkeeping | The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\\|u\\|_1=1$. (pins reading of prop:id: stated by the repair) | | `ass:pin-fix-8ef6fd29` | `thm:rate` | V1.5 bookkeeping | The homogeneity inequality holds for both treatments and every positive radius, on the positive unit sphere for the $\ell_1$ norm. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-9567640d` | `thm:rate` | V1.5 bookkeeping | The equal split uses $n\in2\mathbb N$. (pins reading of def:nete-algorithm: stated by the repair) | | `ass:pin-fix-b3a62d30` | `thm:rate` | V1.5 bookkeeping | The symbol $V_t$ denotes $\tmop{Var}(\psi\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t)$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-c3605ca3` | `thm:rate` | V1.5 bookkeeping | The sample-size domain of $R_g(q,\varepsilon)$ includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-c4d18fdb` | `thm:rate` | V1.5 bookkeeping | For each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that, for every integer $n\geq n_0$, the bound has probability at least $1-\delta$ under the law of the sample of size $n$. (pins reading of lem:mu-rate: stated by the repair) | | `ass:pin-fix-c90a58f3` | `thm:rate` | V1.5 bookkeeping | The count $n_t=\sum_{i=1}^{n/2}I(\\|U_i\\|>t)$ is the training-split exceedance count, with the training observations indexed first after the random split. (pins reading of thm:rate: stated by the repair) | | `ass:pin-fix-ebdb71b8` | `thm:rate` | V1.5 bookkeeping | The sampling term comprises both $\sqrt{4v_t^{\text{IPW}}(B)/(\delta M)}$ and $(1+4/\delta)b_t^{\text{IPW}}(B)$. (pins reading of lem:eta-ipw: stated by the repair) | | `ass:pin-fix-f18f3355` | `thm:rate` | V1.5 bookkeeping | The target $\eta$ denotes $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-33` | `thm:cor-rate` | V1.5 bookkeeping | $U\perp(X,D)$. (pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly.) | | `ass:pin-34` | `thm:cor-rate` | V1.5 bookkeeping | $\\|X\\|_1\leqslant R$ almost surely. (pins reading of ass:f-homo-bounded-x: Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm.) | | `ass:pin-36` | `thm:cor-rate` | V1.5 bookkeeping | For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\\|z\\|_1>\zeta k^{(1-2s)/\beta}$. (pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$.) | | `ass:pin-38` | `thm:cor-rate` | V1.5 bookkeeping | For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\\|U\\|/b_n,U/\\|U\\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. (pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation.) | | `ass:pin-39` | `thm:cor-rate` | V1.5 bookkeeping | $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. (pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely.) | | `ass:pin-41` | `thm:cor-rate` | V1.5 bookkeeping | The existential cutoff $t_W$ is chosen sufficiently large for Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} and so that $\sup_{r\geq t_W}e(r)\leq1$ and $\mathbb P(\\|U\\|>tr)/\mathbb P(\\|U\\|>t)\leq4r^{-\beta}$ for every $t\geq t_W$ and $r\geq1$. (pins reading of thm:rate: The proof of Theorem~\ref{thm:rate} invokes the first-factor lemmas' sufficiently-large-threshold conclusions, uses the radial moment inequality with $K=4/(\beta-a_0)$, and bounds $G_t$ by $C+1$. The stated Pareto-type radial law and $e(t)\to0$ permit this enlargement of the existential cutoff.) | | `ass:pin-42` | `thm:cor-rate` | V1.5 bookkeeping | For each fixed $\mathcal T$-measurable choice $B>0$, the displayed probability bound uses the base conditions of Lemma~\ref{lemma:eta_rate}; its angular specialization additionally uses that lemma's angular assumption and $t\geq t_W$, and its variance alternative additionally requires $V_t^{\text{IPW}}<\infty$. (pins reading of lem:eta-ipw: The statement explicitly says ``Under the angular assumption'' and ``When'' the variance is finite; the amended proof applies conditional sampling inequalities to the chosen $B$.) | | `ass:pin-43` | `thm:cor-rate` | V1.5 bookkeeping | The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\\|x\\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\\|u\\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. (pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere.) | | `ass:pin-44` | `thm:cor-rate` | V1.5 bookkeeping | For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. (pins reading of ass:rate-nuisance-errors: Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation.) | | `ass:pin-45` | `thm:cor-rate` | V1.5 bookkeeping | The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. (pins reading of def:nete-algorithm: Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention.) | | `ass:pin-7` | `thm:cor-rate` | V1.5 bookkeeping | $Y(D)=Y$ almost surely throughout the paper. (pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely.) | | `ass:pin-fix-0b814294` | `thm:cor-rate` | V1.5 bookkeeping | The equal split in Algorithm~\ref{alg:nete} is defined for positive even sample sizes $n$, with $\|\mathcal D_1\|=\|\mathcal D_2\|=n/2$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-10c2ec7b` | `thm:cor-rate` | V1.5 bookkeeping | The component dimension in the density of Assumption~\ref{assump:mvary} is $m=d_z$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-27ee3ba1` | `thm:cor-rate` | V1.5 bookkeeping | For each fixed $\delta\in(0,1/2)$, the probability guarantee quantifies separately over every sufficiently large even $n$. (pins reading of thm:cor-rate: stated by the repair) | | `ass:pin-fix-6857bb4d` | `thm:cor-rate` | V1.5 bookkeeping | For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments. (pins reading of thm:rate: stated by the repair) | | `ass:pin-fix-8964f8dd` | `thm:cor-rate` | V1.5 bookkeeping | Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-8a129280` | `thm:cor-rate` | V1.5 bookkeeping | The fitted-exponent pseudo-outcome has finite conditional absolute mean under the evaluation tail law: $\mathbb E[\|Y\|/\\|U\\|^{\hat\alpha_n}\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t]<\infty$ almost surely on each joint nuisance event. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-8cd5c960` | `thm:cor-rate` | V1.5 bookkeeping | The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\\|u\\|_1=1$. (pins reading of prop:id: stated by the repair) | | `ass:pin-fix-8ef6fd29` | `thm:cor-rate` | V1.5 bookkeeping | The homogeneity inequality holds for both treatments and every positive radius, on the positive unit sphere for the $\ell_1$ norm. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-9567640d` | `thm:cor-rate` | V1.5 bookkeeping | The equal split uses $n\in2\mathbb N$. (pins reading of def:nete-algorithm: stated by the repair) | | `ass:pin-fix-b3a62d30` | `thm:cor-rate` | V1.5 bookkeeping | The symbol $V_t$ denotes $\tmop{Var}(\psi\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t)$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-c3605ca3` | `thm:cor-rate` | V1.5 bookkeeping | The sample-size domain of $R_g(q,\varepsilon)$ includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-c4d18fdb` | `thm:cor-rate` | V1.5 bookkeeping | For each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that, for every integer $n\geq n_0$, the bound has probability at least $1-\delta$ under the law of the sample of size $n$. (pins reading of lem:mu-rate: stated by the repair) | | `ass:pin-fix-c90a58f3` | `thm:cor-rate` | V1.5 bookkeeping | The count $n_t=\sum_{i=1}^{n/2}I(\\|U_i\\|>t)$ is the training-split exceedance count, with the training observations indexed first after the random split. (pins reading of thm:rate: stated by the repair) | | `ass:pin-fix-d303828c` | `thm:cor-rate` | V1.5 bookkeeping | The constants $c_-$ and $c_+$ in $t_n=\Theta(n^{\kappa(\hat\gamma_n)})$ are deterministic and independent of $n$ and the observed sample. (pins reading of thm:cor-rate: stated by the repair) | | `ass:pin-fix-ebdb71b8` | `thm:cor-rate` | V1.5 bookkeeping | The sampling term comprises both $\sqrt{4v_t^{\text{IPW}}(B)/(\delta M)}$ and $(1+4/\delta)b_t^{\text{IPW}}(B)$. (pins reading of lem:eta-ipw: stated by the repair) | | `ass:pin-fix-f18f3355` | `thm:cor-rate` | V1.5 bookkeeping | The target $\eta$ denotes $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-33` | `lem:mu-rate` | V1.5 bookkeeping | $U\perp(X,D)$. (pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly.) | | `ass:pin-36` | `lem:mu-rate` | V1.5 bookkeeping | For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\\|z\\|_1>\zeta k^{(1-2s)/\beta}$. (pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$.) | | `ass:pin-38` | `lem:mu-rate` | V1.5 bookkeeping | For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\\|U\\|/b_n,U/\\|U\\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. (pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation.) | | `ass:pin-fix-c4d18fdb` | `lem:mu-rate` | V1.5 bookkeeping | For each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that, for every integer $n\geq n_0$, the bound has probability at least $1-\delta$ under the law of the sample of size $n$. (pins reading of lem:mu-rate: stated by the repair) | | `ass:pin-33` | `lem:eta-rate` | V1.5 bookkeeping | $U\perp(X,D)$. (pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly.) | | `ass:pin-34` | `lem:eta-rate` | V1.5 bookkeeping | $\\|X\\|_1\leqslant R$ almost surely. (pins reading of ass:f-homo-bounded-x: Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm.) | | `ass:pin-36` | `lem:eta-rate` | V1.5 bookkeeping | For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\\|z\\|_1>\zeta k^{(1-2s)/\beta}$. (pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$.) | | `ass:pin-38` | `lem:eta-rate` | V1.5 bookkeeping | For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\\|U\\|/b_n,U/\\|U\\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. (pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation.) | | `ass:pin-39` | `lem:eta-rate` | V1.5 bookkeeping | $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. (pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely.) | | `ass:pin-43` | `lem:eta-rate` | V1.5 bookkeeping | The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\\|x\\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\\|u\\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. (pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere.) | | `ass:pin-44` | `lem:eta-rate` | V1.5 bookkeeping | For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. (pins reading of ass:rate-nuisance-errors: Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation.) | | `ass:pin-45` | `lem:eta-rate` | V1.5 bookkeeping | The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. (pins reading of def:nete-algorithm: Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention.) | | `ass:pin-7` | `lem:eta-rate` | V1.5 bookkeeping | $Y(D)=Y$ almost surely throughout the paper. (pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely.) | | `ass:pin-fix-0b814294` | `lem:eta-rate` | V1.5 bookkeeping | The equal split in Algorithm~\ref{alg:nete} is defined for positive even sample sizes $n$, with $\|\mathcal D_1\|=\|\mathcal D_2\|=n/2$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-10c2ec7b` | `lem:eta-rate` | V1.5 bookkeeping | The component dimension in the density of Assumption~\ref{assump:mvary} is $m=d_z$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-8964f8dd` | `lem:eta-rate` | V1.5 bookkeeping | Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-8a129280` | `lem:eta-rate` | V1.5 bookkeeping | The fitted-exponent pseudo-outcome has finite conditional absolute mean under the evaluation tail law: $\mathbb E[\|Y\|/\\|U\\|^{\hat\alpha_n}\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t]<\infty$ almost surely on each joint nuisance event. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-8cd5c960` | `lem:eta-rate` | V1.5 bookkeeping | The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\\|u\\|_1=1$. (pins reading of prop:id: stated by the repair) | | `ass:pin-fix-8ef6fd29` | `lem:eta-rate` | V1.5 bookkeeping | The homogeneity inequality holds for both treatments and every positive radius, on the positive unit sphere for the $\ell_1$ norm. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-9567640d` | `lem:eta-rate` | V1.5 bookkeeping | The equal split uses $n\in2\mathbb N$. (pins reading of def:nete-algorithm: stated by the repair) | | `ass:pin-fix-b3a62d30` | `lem:eta-rate` | V1.5 bookkeeping | The symbol $V_t$ denotes $\tmop{Var}(\psi\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t)$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-c3605ca3` | `lem:eta-rate` | V1.5 bookkeeping | The sample-size domain of $R_g(q,\varepsilon)$ includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-f18f3355` | `lem:eta-rate` | V1.5 bookkeeping | The target $\eta$ denotes $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-33` | `lem:eta-ipw` | V1.5 bookkeeping | $U\perp(X,D)$. (pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly.) | | `ass:pin-34` | `lem:eta-ipw` | V1.5 bookkeeping | $\\|X\\|_1\leqslant R$ almost surely. (pins reading of ass:f-homo-bounded-x: Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm.) | | `ass:pin-36` | `lem:eta-ipw` | V1.5 bookkeeping | For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\\|z\\|_1>\zeta k^{(1-2s)/\beta}$. (pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$.) | | `ass:pin-38` | `lem:eta-ipw` | V1.5 bookkeeping | For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\\|U\\|/b_n,U/\\|U\\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. (pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation.) | | `ass:pin-39` | `lem:eta-ipw` | V1.5 bookkeeping | $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. (pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely.) | | `ass:pin-42` | `lem:eta-ipw` | V1.5 bookkeeping | For each fixed $\mathcal T$-measurable choice $B>0$, the displayed probability bound uses the base conditions of Lemma~\ref{lemma:eta_rate}; its angular specialization additionally uses that lemma's angular assumption and $t\geq t_W$, and its variance alternative additionally requires $V_t^{\text{IPW}}<\infty$. (pins reading of lem:eta-ipw: The statement explicitly says ``Under the angular assumption'' and ``When'' the variance is finite; the amended proof applies conditional sampling inequalities to the chosen $B$.) | | `ass:pin-43` | `lem:eta-ipw` | V1.5 bookkeeping | The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\\|x\\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\\|u\\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. (pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere.) | | `ass:pin-44` | `lem:eta-ipw` | V1.5 bookkeeping | For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. (pins reading of ass:rate-nuisance-errors: Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation.) | | `ass:pin-45` | `lem:eta-ipw` | V1.5 bookkeeping | The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. (pins reading of def:nete-algorithm: Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention.) | | `ass:pin-7` | `lem:eta-ipw` | V1.5 bookkeeping | $Y(D)=Y$ almost surely throughout the paper. (pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely.) | | `ass:pin-fix-0b814294` | `lem:eta-ipw` | V1.5 bookkeeping | The equal split in Algorithm~\ref{alg:nete} is defined for positive even sample sizes $n$, with $\|\mathcal D_1\|=\|\mathcal D_2\|=n/2$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-10c2ec7b` | `lem:eta-ipw` | V1.5 bookkeeping | The component dimension in the density of Assumption~\ref{assump:mvary} is $m=d_z$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-8964f8dd` | `lem:eta-ipw` | V1.5 bookkeeping | Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-8a129280` | `lem:eta-ipw` | V1.5 bookkeeping | The fitted-exponent pseudo-outcome has finite conditional absolute mean under the evaluation tail law: $\mathbb E[\|Y\|/\\|U\\|^{\hat\alpha_n}\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t]<\infty$ almost surely on each joint nuisance event. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-8cd5c960` | `lem:eta-ipw` | V1.5 bookkeeping | The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\\|u\\|_1=1$. (pins reading of prop:id: stated by the repair) | | `ass:pin-fix-8ef6fd29` | `lem:eta-ipw` | V1.5 bookkeeping | The homogeneity inequality holds for both treatments and every positive radius, on the positive unit sphere for the $\ell_1$ norm. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-9567640d` | `lem:eta-ipw` | V1.5 bookkeeping | The equal split uses $n\in2\mathbb N$. (pins reading of def:nete-algorithm: stated by the repair) | | `ass:pin-fix-b3a62d30` | `lem:eta-ipw` | V1.5 bookkeeping | The symbol $V_t$ denotes $\tmop{Var}(\psi\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t)$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-c3605ca3` | `lem:eta-ipw` | V1.5 bookkeeping | The sample-size domain of $R_g(q,\varepsilon)$ includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-ebdb71b8` | `lem:eta-ipw` | V1.5 bookkeeping | The sampling term comprises both $\sqrt{4v_t^{\text{IPW}}(B)/(\delta M)}$ and $(1+4/\delta)b_t^{\text{IPW}}(B)$. (pins reading of lem:eta-ipw: stated by the repair) | | `ass:pin-fix-f18f3355` | `lem:eta-ipw` | V1.5 bookkeeping | The target $\eta$ denotes $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-33` | `thm:cor-rate-ipw` | V1.5 bookkeeping | $U\perp(X,D)$. (pins reading of ass:iid-independence: The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly.) | | `ass:pin-34` | `thm:cor-rate-ipw` | V1.5 bookkeeping | $\\|X\\|_1\leqslant R$ almost surely. (pins reading of ass:f-homo-bounded-x: Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm.) | | `ass:pin-36` | `thm:cor-rate-ipw` | V1.5 bookkeeping | For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\\|z\\|_1>\zeta k^{(1-2s)/\beta}$. (pins reading of ass:mvary-density: Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$.) | | `ass:pin-38` | `thm:cor-rate-ipw` | V1.5 bookkeeping | For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\\|U\\|/b_n,U/\\|U\\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. (pins reading of def:regular-variation: Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation.) | | `ass:pin-39` | `thm:cor-rate-ipw` | V1.5 bookkeeping | $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. (pins reading of def:conditional-mean: Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely.) | | `ass:pin-41` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The existential cutoff $t_W$ is chosen sufficiently large for Lemmas~\ref{lemma:eta_rate} and~\ref{lemma:eta_ipw} and so that $\sup_{r\geq t_W}e(r)\leq1$ and $\mathbb P(\\|U\\|>tr)/\mathbb P(\\|U\\|>t)\leq4r^{-\beta}$ for every $t\geq t_W$ and $r\geq1$. (pins reading of thm:rate: The proof of Theorem~\ref{thm:rate} invokes the first-factor lemmas' sufficiently-large-threshold conclusions, uses the radial moment inequality with $K=4/(\beta-a_0)$, and bounds $G_t$ by $C+1$. The stated Pareto-type radial law and $e(t)\to0$ permit this enlargement of the existential cutoff.) | | `ass:pin-42` | `thm:cor-rate-ipw` | V1.5 bookkeeping | For each fixed $\mathcal T$-measurable choice $B>0$, the displayed probability bound uses the base conditions of Lemma~\ref{lemma:eta_rate}; its angular specialization additionally uses that lemma's angular assumption and $t\geq t_W$, and its variance alternative additionally requires $V_t^{\text{IPW}}<\infty$. (pins reading of lem:eta-ipw: The statement explicitly says ``Under the angular assumption'' and ``When'' the variance is finite; the amended proof applies conditional sampling inequalities to the chosen $B$.) | | `ass:pin-43` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\\|x\\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\\|u\\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. (pins reading of ass:f-homo-asymptotic: Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere.) | | `ass:pin-44` | `thm:cor-rate-ipw` | V1.5 bookkeeping | For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. (pins reading of ass:rate-nuisance-errors: Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation.) | | `ass:pin-45` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. (pins reading of def:nete-algorithm: Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention.) | | `ass:pin-7` | `thm:cor-rate-ipw` | V1.5 bookkeeping | $Y(D)=Y$ almost surely throughout the paper. (pins reading of ass:consistency: The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely.) | | `ass:pin-fix-0b814294` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The equal split in Algorithm~\ref{alg:nete} is defined for positive even sample sizes $n$, with $\|\mathcal D_1\|=\|\mathcal D_2\|=n/2$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-10c2ec7b` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The component dimension in the density of Assumption~\ref{assump:mvary} is $m=d_z$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-6857bb4d` | `thm:cor-rate-ipw` | V1.5 bookkeeping | For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments. (pins reading of thm:rate: stated by the repair) | | `ass:pin-fix-8964f8dd` | `thm:cor-rate-ipw` | V1.5 bookkeeping | Assumption~\ref{assump:mvary} specifies the noise law $\mathcal L(U)$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-8a129280` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The fitted-exponent pseudo-outcome has finite conditional absolute mean under the evaluation tail law: $\mathbb E[\|Y\|/\\|U\\|^{\hat\alpha_n}\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t]<\infty$ almost surely on each joint nuisance event. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-8cd5c960` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The homogeneity inequality holds for every $d\in\{0,1\}$ and every radius $t>0$, uniformly over $x\in B_R$ and $u\in\mathbb R_+^d$ with $\\|u\\|_1=1$. (pins reading of prop:id: stated by the repair) | | `ass:pin-fix-8ef6fd29` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The homogeneity inequality holds for both treatments and every positive radius, on the positive unit sphere for the $\ell_1$ norm. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-9567640d` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The equal split uses $n\in2\mathbb N$. (pins reading of def:nete-algorithm: stated by the repair) | | `ass:pin-fix-b3a62d30` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The symbol $V_t$ denotes $\tmop{Var}(\psi\mid\mathcal D_1,\hbox{ auxiliary data},\\|U\\|>t)$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-c3605ca3` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The sample-size domain of $R_g(q,\varepsilon)$ includes every possible training exceedance count $q\in\{0,1,\ldots,n/2\}$, including $q=0$. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-c4d18fdb` | `thm:cor-rate-ipw` | V1.5 bookkeeping | For each fixed $\delta\in(0,1)$ there is a deterministic $n_0$ such that, for every integer $n\geq n_0$, the bound has probability at least $1-\delta$ under the law of the sample of size $n$. (pins reading of lem:mu-rate: stated by the repair) | | `ass:pin-fix-c90a58f3` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The count $n_t=\sum_{i=1}^{n/2}I(\\|U_i\\|>t)$ is the training-split exceedance count, with the training observations indexed first after the random split. (pins reading of thm:rate: stated by the repair) | | `ass:pin-fix-ebdb71b8` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The sampling term comprises both $\sqrt{4v_t^{\text{IPW}}(B)/(\delta M)}$ and $(1+4/\delta)b_t^{\text{IPW}}(B)$. (pins reading of lem:eta-ipw: stated by the repair) | | `ass:pin-fix-f18f3355` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The target $\eta$ denotes $\int\mathbb E_X[g(X,1,s)-g(X,0,s)]\,S(\tmop{ds})$, where $S$ is the limiting spectral probability measure. (pins reading of lem:eta-rate: stated by the repair) | | `ass:pin-fix-fc4621e4` | `thm:cor-rate-ipw` | V1.5 bookkeeping | For each fixed $\delta\in(0,1/2)$, the probability guarantee quantifies separately over every sufficiently large even $n$. (pins reading of thm:cor-rate-ipw: stated by the repair) | | `ass:pin-fix-fdafcd87` | `thm:cor-rate-ipw` | V1.5 bookkeeping | The constants $c_-$ and $c_+$ in $t_n=\Theta(n^{\kappa(\hat\gamma_n)})$ are deterministic and independent of $n$ and the observed sample. (pins reading of thm:cor-rate-ipw: stated by the repair) | ## Conditions assumed ### Cited results None: no result rests on a theorem imported from the literature. ### Deferred substrate gates None: every substrate this run needed was built and proved. ## Unresolved / notes - run-level `ass:consistency`: [note, V1.5] pinned reading: $Y(D)=Y$ almost surely throughout the paper. (The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely.) - run-level `ass:f-homo-bounded-x`: [note, V1.5] pinned reading: $\\|X\\|_1\leqslant R$ almost surely. (Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm.) - run-level `ass:f-homo-asymptotic`: [note, V1.5] pinned reading: The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\\|x\\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\\|u\\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. (Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere.) - run-level `ass:rate-nuisance-errors`: [note, V1.5] pinned reading: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. (Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation.) - run-level `def:nete-algorithm`: [note, V1.5] pinned reading: The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. (Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention.) - run-level `ass:iid-independence`: [note, V1.5] pinned reading: $U\perp(X,D)$. (The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly.) - run-level `ass:mvary-density`: [note, V1.5] pinned reading: For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\\|z\\|_1>\zeta k^{(1-2s)/\beta}$. (Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$.) - run-level `def:regular-variation`: [note, V1.5] pinned reading: For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\\|U\\|/b_n,U/\\|U\\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. (Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation.) - run-level `def:conditional-mean`: [note, V1.5] pinned reading: $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. (Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely.) - run-level `symbol \theta^{\tmop{NETE}}`: [note, V1.5] symbol table corrected: def "$\theta^{\tmop{NETE}}=\lim_{t\rightarrow\infty}\mathbb E\left[\frac{Y(1)-Y(0)}{t^\alpha}\mid\\|U\\|>t\right]$." (The recorded name omits the defining tail-conditional limit and its scaling. Equation~\ref{eq:nete} defines the population estimand; Proposition~\ref{prop:id} gives an identification formula for that same quantity.) - run-level `symbol f`: [note, V1.5] symbol table corrected: type "locally defined population function", space "A locally specified function space: $(0,\infty)\to[0,\infty)$ for densities, and $\mathcal X\times\{0,1\}\times\mathbb R_+^d\to\mathbb R$ for the conditional mean.", def "$f(x)=\beta(1+x)^{-(\beta+1)}$, $x>0$, in the Pareto type-II definition; $f(X,D,U)=\mathbb E[Y\mid X,D,U]$ in the extreme-effect model; $f(t)$ is the density of $\sum_{i=1}^m a_i\tilde z_i$ in the printed tail-index calculation." (The recorded univariate positive-function space does not cover the real-valued conditional mean or distinguish all local density uses. The existing fixes and scoped readings remain applicable, but the evidence asserting that the corollaries do not introduce $m_d^t$ is now false.) - run-level `symbol g`: [note, V1.5] symbol table corrected: type "asymptotic normalized conditional-mean function", def "Assumption~\ref{assump:f_homo} specifies an $L$-Lipschitz function $g(x,d,u)$ satisfying $\|f(x,d,tu)/t^\alpha-g(x,d,u)\|\leqslant e(t)$ for $x\in B_R$ and $u\in S^{d-1}$, where $f(X,D,U)=\mathbb E[Y\mid X,D,U]$ and $e(t)\to0$." (The conditional-mean fix and readings remain applicable. The replacement Hill proof removes the former local density definition, so the redefinition sentence on file no longer applies.) - run-level `symbol e`: [note, V1.5] symbol table corrected: space "$(0,\infty)\to[0,\infty)$", def "$e$ satisfies $\lim_{t\rightarrow\infty}e(t)=0$ and $\|f(x,d,tu)/t^\alpha-g(x,d,u)\|\leq e(t)$ for all $x\in B_R$ and $u\in S^{d-1}$, for the binary treatment values $d$." (The original entry omits the uniform approximation inequality and excludes the explicitly allowed zero error. Retain the general reading and every existing scoped clause, explicitly including the homogeneity clause shared by the theorem, IPW lemma, and both corollaries; add the corollaries' use of the tail envelope.) - run-level `symbol c`: [note, V1.5] symbol table corrected: type "locally defined positive constant", space "$(0,\infty)$", def "$c$ is locally defined: the overlap constant satisfies $c\in(0,1/2)$ and $\hat p(x)\in[c,1-c]$ for every $x\in\mathcal X$; the regular-variation constant satisfies $c>0$ in the limit $c\nu_\beta\times S$; in the printed tail-density calculation it denotes a density proportionality constant." (The recorded overlap-only entry misses the distinct regular-variation and density constants. The existing local readings remain applicable; only the evidence claiming explicit overlap factors in the corollary bounds is outdated.) - run-level `symbol i`: [note, V1.5] symbol table corrected: type "locally bound index", space "$\mathbb{N}$", def "$i$ is an index with the range specified by its local sum, product, quantifier, or indexed dataset." (The local-index repair still applies. The training-count indices now additionally have an explicit interpretation after splitting; the existing full-input and adaptive-comparison clause remains valid.) - run-level `symbol b_n`: [note, V1.5] symbol table corrected: type "locally defined regular-variation normalizing sequence or EVI error-bound sequence", def "$b_n\to\infty$ is the normalizing sequence in the regular-variation definition; its local definition as an EVI error-bound sequence instead applies where explicitly given." (The corollaries now explicitly use the deterministic EVI error-bound sequence, which tends to zero. Their scopes must distinguish this sequence from the diverging regular-variation normalizer.) - run-level `symbol \nu_{\beta}`: [note, V1.5] symbol table corrected: space "The nonnegative Radon measures on $(0,\infty]$." (The tail identity is correct, but $\nu_\beta$ is not a probability measure: $\nu_\beta([r,\infty))=r^{-\beta}$ exceeds one for $0\zeta k^{(1-2s)/\beta}$." (The entry describes a class of laws but gives the space of a single law and omits the defining density restrictions. The existing repair is retained, with the dimension convention additionally recorded for both corollaries.) - run-level `symbol \mathcal{A}`: [note, V1.5] symbol table corrected: space "$\{B:B\subseteq\mathbb{R}_+^{d_u\times d_z}\}$", def "$\mathcal{A}=\{A\in\mathbb{R}_+^{d_u\times d_z}:l\leq\min_i\\|A_{\cdot i}\\|_1\leq\max_i\\|A_{\cdot i}\\|_1\leq u\ \text{and}\ JA\geq\sigma\}$, where $JA=\sqrt{\det(A^\mathsf{T}A)}$." (The defining constraints and the general reading on file remain valid. The value space must be the collection of subsets of the matrix space; the notation $\mathcal{P}$ used elsewhere for probability measures would misclassify this deterministic matrix class.) - run-level `symbol M_k`: [note, V1.5] symbol table corrected: space "$\{B:B\subseteq\mathcal{P}(\mathbb{R}_+^{d_u})\}$" (The recorded construction is appropriate, but a class of probability laws is a subset of the probability-measure space, not one probability measure. The supplied statements settle the printed $\mathcal L(X)$ notation as $\mathcal L(U)$ and distinguish the model index $k$ from the adaptive Hill index.) - run-level `symbol M`: [note, V1.5] symbol table corrected: type "locally defined model class or evaluation exceedance count", space "Classes of probability laws, or nonnegative integer counts, according to the local definition.", def "$M=\bigcup_{k=1}^{\infty}M_k$ in the tail-model definition; the local definition $M=\|\mathcal I\|$ instead denotes the evaluation exceedance count." (The existing general distinction remains valid, but the theorem now locally defines the evaluation count as well as inheriting the model class.) - run-level `symbol R_p`: [note, V1.5] symbol table corrected: type "propensity estimation error-bound function", space "$\mathbb N\times(0,1)\to\mathbb R^+$", def "$R_p(n,\delta)$ bounds $\|p(X)-\hat p(X)\|$ with probability at least $1-\delta$ and is monotonically decreasing with respect to sample size." (The recorded scalar entry omits the sample-size and tolerance arguments of the propensity-error bound. The former corollary clause saying that no simultaneous threshold event is specified is now false and is replaced; the general reading and other scopes on file remain valid.) - run-level `symbol R_g`: [note, V1.5] symbol table corrected: type "regression estimation-error bound function", space "$\mathbb Z_{\geqslant0}\times(0,1)\to\mathbb R^+$", def "$R_g(n_t,\delta)$ bounds $\|\mathbb E[Y/\\|U\\|^\alpha\mid X,D,U/\\|U\\|,\\|U\\|>t]-\hat g(X,D,U/\\|U\\|)\|$ with probability at least $1-\delta$ for any fixed $t$, and is monotonically decreasing with respect to sample size." (The recorded scalar space and vague estimation-error definition omit the bound's sample-size and tolerance arguments and its conditional-mean target. The corollary scope now also includes simultaneous bounds over $I_n$; the general reading and other scopes on file remain valid.) - run-level `symbol R_\alpha`: [note, V1.5] symbol table corrected: type "scaling-index estimation error-bound function", space "$\mathbb N\times(0,1)\to\mathbb R^+$", def "$R_\alpha(n,\delta)$ bounds $\|\hat\alpha_n-\alpha\|$ with probability at least $1-\delta$ and is monotonically decreasing with respect to sample size." (The recorded scalar space and vague error label omit that this is a function of sample size and confidence tolerance bounding an estimation error, not the realized error itself. Its polynomial rate is specific to the corollaries.) - run-level `symbol \widehat{\theta}_{n,t}^{\text{DR}}`: [note, V1.5] symbol table corrected: def "$\widehat\theta_{n,t}^{\text{DR}}=\hat\eta_{n,t}^{\mathrm{DR}}\hat\mu_n=\hat\eta_{n,t}^{\mathrm{DR}}/(1-\hat\alpha_n\hat\gamma_n)$, wherever the first-factor estimate and reciprocal are defined." (The defining product remains necessary. Its adaptive-threshold scope now also specifies shared fitted nuisances, full-input Hill selection, deterministic threshold-order constants, and the zero conventions.) - run-level `symbol \widehat{\theta}_{n,t}^{\text{IPW}}`: [note, V1.5] symbol table corrected: def "$\widehat\theta_{n,t}^{\text{IPW}}=\hat\eta_{n,t}^{\mathrm{IPW}}\hat\mu_n$, where $\hat\eta_{n,t}^{\mathrm{IPW}}=\frac{1}{\|\mathcal I\|}\sum_{i\in\mathcal I}\frac{Y_i}{\\|U_i\\|^{\hat\alpha_n}}\left(\frac{D_i}{\hat p(X_i)}-\frac{1-D_i}{1-\hat p(X_i)}\right)$, $\hat\mu_n=(1-\hat\alpha_n\hat\gamma_n)^{-1}$, and $\mathcal I=\{i:\\|U_i\\|>t,\ (X_i,D_i,Y_i,U_i)\in\mathcal D_2\}$, wherever the average and reciprocal are defined." (The defining product remains necessary. Its adaptive-threshold scope now also specifies shared fitted nuisances, full-input Hill selection, deterministic threshold-order constants, and the zero conventions.) - run-level `symbol \hat{\eta}_{n,t}^{\tmop{DR}}`: [note, V1.5] symbol table corrected: def "$\hat\eta_{n,t}^{\tmop{DR}}=\frac{1}{\|\mathcal I\|}\sum_{i\in\mathcal I}\left[\hat g(X_i,1,S_i)-\hat g(X_i,0,S_i)+\frac{D_i-\hat p(X_i)}{\hat p(X_i)(1-\hat p(X_i))}\left(Y_i/\\|U_i\\|^{\hat\alpha_n}-\hat g(X_i,D_i,S_i)\right)\right]$, where $S_i=U_i/\\|U_i\\|$ and $\mathcal I=\{i:\\|U_i\\|>t,\ (X_i,D_i,Y_i,U_i)\in\mathcal D_2\}$, whenever $\|\mathcal I\|>0$." (The recorded label omits the defining normalized DR average. The adaptive-threshold reading additionally requires the stated empty-tail convention and fitted nuisances shared across all thresholds.) - run-level `symbol \hat{\eta}_{n,t}^{\text{IPW}}`: [note, V1.5] symbol table corrected: def "$\hat\eta_{n,t}^{\mathrm{IPW}}=\frac{1}{\|\mathcal I\|}\sum_{i\in\mathcal I}\frac{Y_i}{\\|U_i\\|^{\hat\alpha_n}}\left(\frac{D_i}{\hat p(X_i)}-\frac{1-D_i}{1-\hat p(X_i)}\right)$, where $\mathcal I=\{i:\\|U_i\\|>t,\ (X_i,D_i,Y_i,U_i)\in\mathcal D_2\}$, whenever $\|\mathcal I\|>0$." (The recorded label omits the defining normalized IPW average. The adaptive-threshold reading additionally requires the stated empty-tail convention and fitted propensity and exponent shared across all thresholds.) - run-level `symbol \eta`: [note, V1.5] symbol table corrected: type "locally designated population quantity", space "$\mathbb R$ or a real-valued function, according to its local use", def "$\eta$ denotes the population first factor when compared with $\hat\eta_{n,t}^{\mathrm{DR}}$ or $\hat\eta_{n,t}^{\mathrm{IPW}}$; $\eta(t)$ denotes the von Mises function in the tail-index analysis." (The scalar first-factor readings remain valid. The functional reading previously used $U(t)$ and the printed density identity; the current proof instead defines the von Mises function using $Q(v)$ and the radial survival-to-density ratio.) - run-level `symbol \mathcal{D}`: [note, V1.5] symbol table corrected: space "$\left(\mathcal X\times\{0,1\}\times\mathbb R\times\mathbb R_+^d\right)^n$", def "$\mathcal D=\{(X_i,D_i,Y_i,U_i)\}_{i=1}^n$ is the indexed dataset supplied to Algorithm~\ref{alg:nete}." (The recorded space is a particular dataset rather than its value space. Training-first indexing now explicitly determines how the input coordinates are used.) - run-level `symbol \mathcal{D}_1`: [note, V1.5] symbol table corrected: space "$\left(\mathcal X\times\{0,1\}\times\mathbb R\times\mathbb R_+^d\right)^{n/2}$" (The recorded space is the dataset itself rather than a space of training samples. The new reading fixes which indexed observations form the training half in the theorem and its corollaries.) - run-level `symbol \mathcal{D}_2`: [note, V1.5] symbol table corrected: space "$\{E\subseteq\mathcal D:\|E\|=n/2\}$" (The recorded space treats the split as an element of the dataset rather than a subset of size $n/2$. The binding convention also settles the sample-size domain of the equal split.) - run-level `symbol \hat{g}`: [note, V1.5] symbol table corrected: space "$\mathcal{X}\times\{0,1\}\times\mathbb{S}^+\to\mathbb{R}$" (The direction domain remains the positive unit sphere for the paper's $\ell_1$ norm. The previous corollary clause specified separate fixed-threshold nuisance events; the current texts require one common fitted function and simultaneous bounds throughout $I_n$. The general reading and other scoped clauses remain valid.) - run-level `symbol \mathcal{I}`: [note, V1.5] symbol table corrected: space "$\mathcal{P}(\{1,\ldots,n\})$" (The symbol takes values in a power set, rather than in the displayed realized index set. The general reading remains valid; the earlier corollary clause contains a count notation and full-sample convention absent from the current text.) - run-level `symbol \hat{\gamma}_n`: [note, V1.5] symbol table corrected: def "$\hat\gamma_n$ is the sample estimate of $\gamma$ produced by adaptive order-statistic selection, with its input radii and formula specified by the applicable estimator convention." (The table globally specifies evaluation exceedances, whereas the theorem specifies all input radii and the common-denominator Hill formula. These conventions must be separated; the adjacent-denominator formula printed in the algorithm cannot silently be replaced everywhere.) - run-level `symbol t_n`: [note, V1.5] symbol table corrected: def "The selected threshold has $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ when $s\in(0,1/(2+\max\{1,\beta\}))$, and $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ when $s\in[1/(2+\max\{1,\beta\}),1/2)$." (The recorded vague definition omits the two threshold orders. The existing general reading remains valid, but the definition's former distinction between $t$ and $t_n$ is obsolete, and the deterministic order constants are now explicit.) - run-level `symbol S_i`: [note, V1.5] symbol table corrected: space "$\mathbb S^+=\{s\in\mathbb R_+^d:\\|s\\|_1=1\}$" (The stored definition and finite-threshold direction reading remain valid, with the positive $\ell_1$ sphere replacing the recorded unrestricted sphere. The stored evidence that no listed result uses this symbol is no longer applicable.) - run-level `symbol l_n`: [note, V1.5] symbol table corrected: def "The deterministic lower candidate index, defined in Algorithm~\ref{alg:nete} by $l_n=\min\{n-1,\lceil100\log n\rceil\}$ for $n\geq2$; the experimental implementation chooses $l_n=30$." (The vague endpoint definition omits the algorithm's explicit calibration. The general deterministic-endpoint reading remains valid, but the evidence claiming that theoretical calibration is unspecified no longer holds.) - run-level `symbol k`: [note, V1.5] symbol table corrected: type "context-dependent Hill index, model-class index, or angular logarithmic exponent", space "$\mathbb N$ for Hill and model-class indices; $[0,\infty)$ for the angular logarithmic exponent", def "In adaptive Hill estimation, $k$ is a candidate index or the data-dependent selected index; Algorithm~\ref{alg:nete} selects $k=\max\{q\in\{l_n,\ldots,n-1\}:\|\hat\gamma(i)-\hat\gamma(q)\|\leq\hat\gamma(i)r_n(\delta)/\sqrt i\ \text{for every }i\in\{l_n,\ldots,q\}\}$. In $M=\bigcup_{k=1}^{\infty}M_k$, $k$ is the positive integer model-class index. In the angular Wasserstein bound, $k\geq0$ is the constant exponent in $(\log t)^k$." (The recorded integer Hill index cannot also represent the fixed real exponent $k\geq0$ introduced in the angular bounds. These uses and the model-class index are distinct local meanings. The existing Hill clauses remain unchanged, explicitly retained alongside the additional angular clause.) - run-level `symbol r_n`: [note, V1.5] symbol table corrected: space "$(0,1)\to(0,\infty)$ for each admissible integer sample size $n$", def "The deterministic tolerance $r_n(\delta)$ in the adaptive Hill selection rule." (The function-space correction on file still applies. The algorithm now determines the tolerance exactly, so its calibration must be added to the binding reading.) - run-level `symbol \hat{\gamma}_i`: [note, V1.5] symbol table corrected: space "$\mathbb R_{\geq0}$", def "$\hat\gamma_i=\hat\gamma(i)=\frac1i\sum_{j=1}^i\log\frac{\\|U_{(j)}\\|}{\\|U_{(i+1)}\\|}$, where the full-input radii are ordered decreasingly, $1\leq i\leq n-1$, and the reference radius is positive." (The vague estimate label omits the Hill formula, and the statistic can equal zero. The algorithm now settles the full-input ordering and index range generally, rather than only in the scopes on file.) - run-level `symbol \widehat{\theta}_{n,t}^{\mathrm{estimator}}`: [note, V1.5] symbol table corrected: def "$\widehat{\theta}_{n,t}^{\mathrm{estimator}}=\hat\eta_{n,t}^{\mathrm{estimator}}\cdot\hat\mu_n$, where $\hat\mu_n=1/(1-\hat\alpha_n\hat\gamma_n)$ when its denominator is nonzero and $\hat\mu_n=0$ otherwise; $\hat\eta_{n,t}^{\mathrm{estimator}}$ is given by Equation~\ref{eq:eta_ipw} or Equation~\ref{eq:eta_dr}, according to the supplied option." (The recorded label omits the defining product. The definition on file also omits the algorithm's now explicit zero-denominator convention; the general reading must include its full-input radial calculation.) - note on `ass:consistency`: pinned reading: $Y(D)=Y$ almost surely throughout the paper. (The Causal Inference paragraph states consistency for random variables in the potential-outcome framework; equality of random variables in this framework is understood almost surely.) - note on `ass:f-homo-asymptotic`: pinned reading: The homogeneity inequality holds for every $d\in\{0,1\}$ and every $t>0$, uniformly over $x\in B_R=\{x:\\|x\\|_1\leqslant R\}$ and $u\in S^{d-1}=\{u\in\mathbb R_+^d:\\|u\\|_1=1\}$, with $g$ being $L$-Lipschitz on this domain under the $\ell_1$ norm convention. (Assumption~\ref{assump:f_homo} defines the approximation using the conditional mean on the nonnegative noise domain. The Notations paragraph fixes the norm; Proposition~\ref{prop:id} uses both treatment values, and its added homogeneity condition explicitly specifies every positive radius and the positive unit sphere.) - note on `ass:f-homo-bounded-x`: pinned reading: $\\|X\\|_1\leqslant R$ almost surely. (Assumption~\ref{assump:f_homo} bounds the random covariate, and the Notations paragraph explicitly identifies both norm symbols with the $\ell_1$-norm.) - note on `ass:iid-independence`: pinned reading: $U\perp(X,D)$. (The opening model description calls $U$ an independent extreme-noise vector, and Assumption~\ref{assump:iid} states that it is independent of $X,D$. Standard multivariate independence treats the listed variables jointly.) - note on `ass:mvary-density`: pinned reading: For each component law in $\tilde M_k$, there exists one constant $C_k>0$, independent of $z$, such that $h(z)=C_k\prod_{i=1}^{m}(1+z_i)^{-(\beta+1)}$ whenever $\\|z\\|_1>\zeta k^{(1-2s)/\beta}$. (Assumption~\ref{assump:mvary} uses $\varpropto$ on the stated tail region and separately bounds the relative error from the normalized nominal density. Proportionality fixes a constant multiplier, without requiring that multiplier to equal $\beta^m$.) - note on `ass:rate-nuisance-errors`: pinned reading: For each fixed $t$ and tolerance $\varepsilon\in(0,1)$, the three nuisance inequalities, with $\delta=\varepsilon$, hold on one $\mathcal T$-measurable event of probability at least $1-\varepsilon$, simultaneously almost surely over the covariate support, the conditional direction support and both treatments, where $\mathcal T$ consists of the training split and independent auxiliary data. (Algorithm~\ref{alg:nete} fits nuisances on the training split before evaluation; Theorem~\ref{thm:rate}'s stated nuisance reading explicitly specifies the joint training-measurable event and simultaneous support interpretation.) - note on `def:conditional-mean`: pinned reading: $f$ is a fixed measurable version satisfying $f(X,D,U)=\mathbb{E}[Y\mid X,D,U]$ almost surely, and this same version is used in Assumption~\ref{assump:f_homo}. (Assumption~\ref{assump:f_homo} first defines $f$ through conditional expectation and then immediately imposes a pointwise homogeneity property on that function. The standard meaning of conditional expectation fixes the equality almost surely.) - note on `def:nete-algorithm`: pinned reading: The training indices are chosen uniformly from the subsets of $\{1,\ldots,n\}$ of size $n/2$, independently of the sampled observations; the remaining indices form $\mathcal D_2$. (Algorithm~\ref{alg:nete} specifies a random equal split, and the following estimation paragraph uses the first half for fitting and the second half for evaluation, invoking the standard independent sample-splitting convention.) - note on `def:regular-variation`: pinned reading: For each norm, there exist $S$, $b_n\rightarrow\infty$, and $c>0$ such that, for every $a>0$, the polar measures $n\tmop{P}((\\|U\\|/b_n,U/\\|U\\|)\in\cdot)$ restricted to $[a,\infty]\times\mathbb{S}^+$ converge weakly as finite measures to the corresponding restriction of $c\nu_\beta\times S$. (Definition~\ref{def:mvarying} specifies $\nu_\beta([r,\infty))=r^{-\beta}$, which is finite away from zero and infinite over the entire radial space. The immediately following discussion identifies the intended Pareto tail limit, fixing the standard tail-measure interpretation.) - note on `lem:eta-ipw`: the corrected statement stands as one statement; the referee would state it as a general statement with corollaries: Keep the truncated-score IPW bound as the main statement, and separate its angular-bound specialization and finite-variance sampling alternative into blocks under their respective additional conditions. - note on `lem:eta-rate`: the corrected statement stands as one statement; the referee would state it as a general statement with corollaries: Keep the general truncated-score DR bound as the main statement; separate the angular-rate specialization under the additional Wasserstein hypothesis and the sampling specialization under finite conditional score variance into blocks. - note on `prop:id`: [statement-wrong] 'g being L-Lipschitz on this domain' admits two formal readings. (A) Per treatment: for each d, \|g(x,d,u) − g(x',d,u')\| ≤ L(‖x−x'‖_1 + ‖u−u'‖_1) on B_R × S^{d−1}. (B) Joint in (x,d,u): \|g(x,d,u) − g(x',d',u')\| ≤ L(‖x−x'‖_1 + \|d−d'\| + ‖u−u'‖_1), which additionally forces \|g(x,1,u) − g(x,0,u)\| ≤ L. LipschitzG realizes (B). NL edit to pin it: say either 'for each d, g(·,d,·) is L-Lipschitz on B_R × S^{d−1}' or 'g is L-Lipschitz jointly in (x,d,u), with d in {0,1} as a real coordinate'. (rewind rejected by the orchestrator on prop:id: The author settled the reading (session, 2026-10-09): in Assumption~\ref{assump:f_homo}, ``$g$ is $L$-Lipschitz'' is per treatment. For each $d\in\{0,1\}$, $g(\cdot,d,\cdot)$ is $L$-Lipschitz on $B_R\times S^{d-1}$ under the $\ell_1$ norm: $\|g(x,d,u)-g(x^\prime,d,u^\prime)\|\le L(\\|x-x^\prime\\|_1+\\|u-u^\prime\\|_1)$. Nothing is assumed across the two treatments; in particular $\|g(x,1,u)-g(x,0,u)\|\le L$ is not assumed. The Lean statement is being changed to this reading.) - note on `symbol \eta`: symbol table corrected: type "locally designated population quantity", space "$\mathbb R$ or a real-valued function, according to its local use", def "$\eta$ denotes the population first factor when compared with $\hat\eta_{n,t}^{\mathrm{DR}}$ or $\hat\eta_{n,t}^{\mathrm{IPW}}$; $\eta(t)$ denotes the von Mises function in the tail-index analysis." (The scalar first-factor readings remain valid. The functional reading previously used $U(t)$ and the printed density identity; the current proof instead defines the von Mises function using $Q(v)$ and the radial survival-to-density ratio.) - note on `symbol \hat{\eta}_{n,t}^{\text{IPW}}`: symbol table corrected: def "$\hat\eta_{n,t}^{\mathrm{IPW}}=\frac{1}{\|\mathcal I\|}\sum_{i\in\mathcal I}\frac{Y_i}{\\|U_i\\|^{\hat\alpha_n}}\left(\frac{D_i}{\hat p(X_i)}-\frac{1-D_i}{1-\hat p(X_i)}\right)$, where $\mathcal I=\{i:\\|U_i\\|>t,\ (X_i,D_i,Y_i,U_i)\in\mathcal D_2\}$, whenever $\|\mathcal I\|>0$." (The recorded label omits the defining normalized IPW average. The adaptive-threshold reading additionally requires the stated empty-tail convention and fitted propensity and exponent shared across all thresholds.) - note on `symbol \hat{\eta}_{n,t}^{\tmop{DR}}`: symbol table corrected: def "$\hat\eta_{n,t}^{\tmop{DR}}=\frac{1}{\|\mathcal I\|}\sum_{i\in\mathcal I}\left[\hat g(X_i,1,S_i)-\hat g(X_i,0,S_i)+\frac{D_i-\hat p(X_i)}{\hat p(X_i)(1-\hat p(X_i))}\left(Y_i/\\|U_i\\|^{\hat\alpha_n}-\hat g(X_i,D_i,S_i)\right)\right]$, where $S_i=U_i/\\|U_i\\|$ and $\mathcal I=\{i:\\|U_i\\|>t,\ (X_i,D_i,Y_i,U_i)\in\mathcal D_2\}$, whenever $\|\mathcal I\|>0$." (The recorded label omits the defining normalized DR average. The adaptive-threshold reading additionally requires the stated empty-tail convention and fitted nuisances shared across all thresholds.) - note on `symbol \hat{\gamma}_i`: symbol table corrected: space "$\mathbb R_{\geq0}$", def "$\hat\gamma_i=\hat\gamma(i)=\frac1i\sum_{j=1}^i\log\frac{\\|U_{(j)}\\|}{\\|U_{(i+1)}\\|}$, where the full-input radii are ordered decreasingly, $1\leq i\leq n-1$, and the reference radius is positive." (The vague estimate label omits the Hill formula, and the statistic can equal zero. The algorithm now settles the full-input ordering and index range generally, rather than only in the scopes on file.) - note on `symbol \hat{\gamma}_n`: symbol table corrected: def "$\hat\gamma_n$ is the sample estimate of $\gamma$ produced by adaptive order-statistic selection, with its input radii and formula specified by the applicable estimator convention." (The table globally specifies evaluation exceedances, whereas the theorem specifies all input radii and the common-denominator Hill formula. These conventions must be separated; the adjacent-denominator formula printed in the algorithm cannot silently be replaced everywhere.) - note on `symbol \hat{g}`: symbol table corrected: space "$\mathcal{X}\times\{0,1\}\times\mathbb{S}^+\to\mathbb{R}$" (The direction domain remains the positive unit sphere for the paper's $\ell_1$ norm. The previous corollary clause specified separate fixed-threshold nuisance events; the current texts require one common fitted function and simultaneous bounds throughout $I_n$. The general reading and other scoped clauses remain valid.) - note on `symbol \mathcal{A}`: symbol table corrected: space "$\{B:B\subseteq\mathbb{R}_+^{d_u\times d_z}\}$", def "$\mathcal{A}=\{A\in\mathbb{R}_+^{d_u\times d_z}:l\leq\min_i\\|A_{\cdot i}\\|_1\leq\max_i\\|A_{\cdot i}\\|_1\leq u\ \text{and}\ JA\geq\sigma\}$, where $JA=\sqrt{\det(A^\mathsf{T}A)}$." (The defining constraints and the general reading on file remain valid. The value space must be the collection of subsets of the matrix space; the notation $\mathcal{P}$ used elsewhere for probability measures would misclassify this deterministic matrix class.) - note on `symbol \mathcal{D}`: symbol table corrected: space "$\left(\mathcal X\times\{0,1\}\times\mathbb R\times\mathbb R_+^d\right)^n$", def "$\mathcal D=\{(X_i,D_i,Y_i,U_i)\}_{i=1}^n$ is the indexed dataset supplied to Algorithm~\ref{alg:nete}." (The recorded space is a particular dataset rather than its value space. Training-first indexing now explicitly determines how the input coordinates are used.) - note on `symbol \mathcal{D}_1`: symbol table corrected: space "$\left(\mathcal X\times\{0,1\}\times\mathbb R\times\mathbb R_+^d\right)^{n/2}$" (The recorded space is the dataset itself rather than a space of training samples. The new reading fixes which indexed observations form the training half in the theorem and its corollaries.) - note on `symbol \mathcal{D}_2`: symbol table corrected: space "$\{E\subseteq\mathcal D:\|E\|=n/2\}$" (The recorded space treats the split as an element of the dataset rather than a subset of size $n/2$. The binding convention also settles the sample-size domain of the equal split.) - note on `symbol \mathcal{I}`: symbol table corrected: space "$\mathcal{P}(\{1,\ldots,n\})$" (The symbol takes values in a power set, rather than in the displayed realized index set. The general reading remains valid; the earlier corollary clause contains a count notation and full-sample convention absent from the current text.) - note on `symbol \nu_{\beta}`: symbol table corrected: space "The nonnegative Radon measures on $(0,\infty]$." (The tail identity is correct, but $\nu_\beta$ is not a probability measure: $\nu_\beta([r,\infty))=r^{-\beta}$ exceeds one for $0t\right]$." (The recorded name omits the defining tail-conditional limit and its scaling. Equation~\ref{eq:nete} defines the population estimand; Proposition~\ref{prop:id} gives an identification formula for that same quantity.) - note on `symbol \tilde M_k`: symbol table corrected: space "$\{B:B\subseteq\mathcal P(\mathbb R_+^{d_z})\}$", def "$\tilde M_k$ consists of laws $\mathcal L(Z)$ admitting a Lebesgue density $h$ on $\mathbb R_+^{d_z}$ such that $\left\|\frac{h(z)-\beta^m\prod_{i=1}^m(1+z_i)^{-(\beta+1)}}{\beta^m\prod_{i=1}^{d_z}(1+z_i)^{-(\beta+1)}}\right\|\leq\xi k^{-s}$ for every $z$, and $h(z)\varpropto\prod_{i=1}^m(1+z_i)^{-(\beta+1)}$ if $\\|z\\|_1>\zeta k^{(1-2s)/\beta}$." (The entry describes a class of laws but gives the space of a single law and omits the defining density restrictions. The existing repair is retained, with the dimension convention additionally recorded for both corollaries.) - note on `symbol \widehat{\theta}_{n,t}^{\mathrm{estimator}}`: symbol table corrected: def "$\widehat{\theta}_{n,t}^{\mathrm{estimator}}=\hat\eta_{n,t}^{\mathrm{estimator}}\cdot\hat\mu_n$, where $\hat\mu_n=1/(1-\hat\alpha_n\hat\gamma_n)$ when its denominator is nonzero and $\hat\mu_n=0$ otherwise; $\hat\eta_{n,t}^{\mathrm{estimator}}$ is given by Equation~\ref{eq:eta_ipw} or Equation~\ref{eq:eta_dr}, according to the supplied option." (The recorded label omits the defining product. The definition on file also omits the algorithm's now explicit zero-denominator convention; the general reading must include its full-input radial calculation.) - note on `symbol \widehat{\theta}_{n,t}^{\text{DR}}`: symbol table corrected: def "$\widehat\theta_{n,t}^{\text{DR}}=\hat\eta_{n,t}^{\mathrm{DR}}\hat\mu_n=\hat\eta_{n,t}^{\mathrm{DR}}/(1-\hat\alpha_n\hat\gamma_n)$, wherever the first-factor estimate and reciprocal are defined." (The defining product remains necessary. Its adaptive-threshold scope now also specifies shared fitted nuisances, full-input Hill selection, deterministic threshold-order constants, and the zero conventions.) - note on `symbol \widehat{\theta}_{n,t}^{\text{IPW}}`: symbol table corrected: def "$\widehat\theta_{n,t}^{\text{IPW}}=\hat\eta_{n,t}^{\mathrm{IPW}}\hat\mu_n$, where $\hat\eta_{n,t}^{\mathrm{IPW}}=\frac{1}{\|\mathcal I\|}\sum_{i\in\mathcal I}\frac{Y_i}{\\|U_i\\|^{\hat\alpha_n}}\left(\frac{D_i}{\hat p(X_i)}-\frac{1-D_i}{1-\hat p(X_i)}\right)$, $\hat\mu_n=(1-\hat\alpha_n\hat\gamma_n)^{-1}$, and $\mathcal I=\{i:\\|U_i\\|>t,\ (X_i,D_i,Y_i,U_i)\in\mathcal D_2\}$, wherever the average and reciprocal are defined." (The defining product remains necessary. Its adaptive-threshold scope now also specifies shared fitted nuisances, full-input Hill selection, deterministic threshold-order constants, and the zero conventions.) - note on `symbol b_n`: symbol table corrected: type "locally defined regular-variation normalizing sequence or EVI error-bound sequence", def "$b_n\to\infty$ is the normalizing sequence in the regular-variation definition; its local definition as an EVI error-bound sequence instead applies where explicitly given." (The corollaries now explicitly use the deterministic EVI error-bound sequence, which tends to zero. Their scopes must distinguish this sequence from the diverging regular-variation normalizer.) - note on `symbol c`: symbol table corrected: type "locally defined positive constant", space "$(0,\infty)$", def "$c$ is locally defined: the overlap constant satisfies $c\in(0,1/2)$ and $\hat p(x)\in[c,1-c]$ for every $x\in\mathcal X$; the regular-variation constant satisfies $c>0$ in the limit $c\nu_\beta\times S$; in the printed tail-density calculation it denotes a density proportionality constant." (The recorded overlap-only entry misses the distinct regular-variation and density constants. The existing local readings remain applicable; only the evidence claiming explicit overlap factors in the corollary bounds is outdated.) - note on `symbol e`: symbol table corrected: space "$(0,\infty)\to[0,\infty)$", def "$e$ satisfies $\lim_{t\rightarrow\infty}e(t)=0$ and $\|f(x,d,tu)/t^\alpha-g(x,d,u)\|\leq e(t)$ for all $x\in B_R$ and $u\in S^{d-1}$, for the binary treatment values $d$." (The original entry omits the uniform approximation inequality and excludes the explicitly allowed zero error. Retain the general reading and every existing scoped clause, explicitly including the homogeneity clause shared by the theorem, IPW lemma, and both corollaries; add the corollaries' use of the tail envelope.) - note on `symbol f`: symbol table corrected: type "locally defined population function", space "A locally specified function space: $(0,\infty)\to[0,\infty)$ for densities, and $\mathcal X\times\{0,1\}\times\mathbb R_+^d\to\mathbb R$ for the conditional mean.", def "$f(x)=\beta(1+x)^{-(\beta+1)}$, $x>0$, in the Pareto type-II definition; $f(X,D,U)=\mathbb E[Y\mid X,D,U]$ in the extreme-effect model; $f(t)$ is the density of $\sum_{i=1}^m a_i\tilde z_i$ in the printed tail-index calculation." (The recorded univariate positive-function space does not cover the real-valued conditional mean or distinguish all local density uses. The existing fixes and scoped readings remain applicable, but the evidence asserting that the corollaries do not introduce $m_d^t$ is now false.) - note on `symbol g`: symbol table corrected: type "asymptotic normalized conditional-mean function", def "Assumption~\ref{assump:f_homo} specifies an $L$-Lipschitz function $g(x,d,u)$ satisfying $\|f(x,d,tu)/t^\alpha-g(x,d,u)\|\leqslant e(t)$ for $x\in B_R$ and $u\in S^{d-1}$, where $f(X,D,U)=\mathbb E[Y\mid X,D,U]$ and $e(t)\to0$." (The conditional-mean fix and readings remain applicable. The replacement Hill proof removes the former local density definition, so the redefinition sentence on file no longer applies.) - note on `symbol i`: symbol table corrected: type "locally bound index", space "$\mathbb{N}$", def "$i$ is an index with the range specified by its local sum, product, quantifier, or indexed dataset." (The local-index repair still applies. The training-count indices now additionally have an explicit interpretation after splitting; the existing full-input and adaptive-comparison clause remains valid.) - note on `symbol k`: symbol table corrected: type "context-dependent Hill index, model-class index, or angular logarithmic exponent", space "$\mathbb N$ for Hill and model-class indices; $[0,\infty)$ for the angular logarithmic exponent", def "In adaptive Hill estimation, $k$ is a candidate index or the data-dependent selected index; Algorithm~\ref{alg:nete} selects $k=\max\{q\in\{l_n,\ldots,n-1\}:\|\hat\gamma(i)-\hat\gamma(q)\|\leq\hat\gamma(i)r_n(\delta)/\sqrt i\ \text{for every }i\in\{l_n,\ldots,q\}\}$. In $M=\bigcup_{k=1}^{\infty}M_k$, $k$ is the positive integer model-class index. In the angular Wasserstein bound, $k\geq0$ is the constant exponent in $(\log t)^k$." (The recorded integer Hill index cannot also represent the fixed real exponent $k\geq0$ introduced in the angular bounds. These uses and the model-class index are distinct local meanings. The existing Hill clauses remain unchanged, explicitly retained alongside the additional angular clause.) - note on `symbol l_n`: symbol table corrected: def "The deterministic lower candidate index, defined in Algorithm~\ref{alg:nete} by $l_n=\min\{n-1,\lceil100\log n\rceil\}$ for $n\geq2$; the experimental implementation chooses $l_n=30$." (The vague endpoint definition omits the algorithm's explicit calibration. The general deterministic-endpoint reading remains valid, but the evidence claiming that theoretical calibration is unspecified no longer holds.) - note on `symbol M`: symbol table corrected: type "locally defined model class or evaluation exceedance count", space "Classes of probability laws, or nonnegative integer counts, according to the local definition.", def "$M=\bigcup_{k=1}^{\infty}M_k$ in the tail-model definition; the local definition $M=\|\mathcal I\|$ instead denotes the evaluation exceedance count." (The existing general distinction remains valid, but the theorem now locally defines the evaluation count as well as inheriting the model class.) - note on `symbol M_k`: symbol table corrected: space "$\{B:B\subseteq\mathcal{P}(\mathbb{R}_+^{d_u})\}$" (The recorded construction is appropriate, but a class of probability laws is a subset of the probability-measure space, not one probability measure. The supplied statements settle the printed $\mathcal L(X)$ notation as $\mathcal L(U)$ and distinguish the model index $k$ from the adaptive Hill index.) - note on `symbol R_\alpha`: symbol table corrected: type "scaling-index estimation error-bound function", space "$\mathbb N\times(0,1)\to\mathbb R^+$", def "$R_\alpha(n,\delta)$ bounds $\|\hat\alpha_n-\alpha\|$ with probability at least $1-\delta$ and is monotonically decreasing with respect to sample size." (The recorded scalar space and vague error label omit that this is a function of sample size and confidence tolerance bounding an estimation error, not the realized error itself. Its polynomial rate is specific to the corollaries.) - note on `symbol R_g`: symbol table corrected: type "regression estimation-error bound function", space "$\mathbb Z_{\geqslant0}\times(0,1)\to\mathbb R^+$", def "$R_g(n_t,\delta)$ bounds $\|\mathbb E[Y/\\|U\\|^\alpha\mid X,D,U/\\|U\\|,\\|U\\|>t]-\hat g(X,D,U/\\|U\\|)\|$ with probability at least $1-\delta$ for any fixed $t$, and is monotonically decreasing with respect to sample size." (The recorded scalar space and vague estimation-error definition omit the bound's sample-size and tolerance arguments and its conditional-mean target. The corollary scope now also includes simultaneous bounds over $I_n$; the general reading and other scopes on file remain valid.) - note on `symbol r_n`: symbol table corrected: space "$(0,1)\to(0,\infty)$ for each admissible integer sample size $n$", def "The deterministic tolerance $r_n(\delta)$ in the adaptive Hill selection rule." (The function-space correction on file still applies. The algorithm now determines the tolerance exactly, so its calibration must be added to the binding reading.) - note on `symbol R_p`: symbol table corrected: type "propensity estimation error-bound function", space "$\mathbb N\times(0,1)\to\mathbb R^+$", def "$R_p(n,\delta)$ bounds $\|p(X)-\hat p(X)\|$ with probability at least $1-\delta$ and is monotonically decreasing with respect to sample size." (The recorded scalar entry omits the sample-size and tolerance arguments of the propensity-error bound. The former corollary clause saying that no simultaneous threshold event is specified is now false and is replaced; the general reading and other scopes on file remain valid.) - note on `symbol S_i`: symbol table corrected: space "$\mathbb S^+=\{s\in\mathbb R_+^d:\\|s\\|_1=1\}$" (The stored definition and finite-threshold direction reading remain valid, with the positive $\ell_1$ sphere replacing the recorded unrestricted sphere. The stored evidence that no listed result uses this symbol is no longer applicable.) - note on `symbol t_n`: symbol table corrected: def "The selected threshold has $t_n=\Theta(n^{(1-2s)\hat\gamma_n})$ when $s\in(0,1/(2+\max\{1,\beta\}))$, and $t_n=\Theta(n^{\hat\gamma_n/(1+2\min\{1,\hat\gamma_n\})})$ when $s\in[1/(2+\max\{1,\beta\}),1/2)$." (The recorded vague definition omits the two threshold orders. The existing general reading remains valid, but the definition's former distinction between $t$ and $t_n$ is obsolete, and the deterministic order constants are now explicit.) - note on `symbol Z`: symbol table corrected: type "locally defined scalar random variable or component vector", space "$\mathbb R$ or $\mathbb R_+^{d_z}$, according to the local definition.", def "$Z$ denotes the regularly varying scalar in the illustrative tail expectation or the component vector in $U=AZ$; in the local score-truncation definitions it denotes $Z=\psi I\{\|\psi\|\leq B\}$ or $Z=\phi I\{\|\phi\|\leq B\}$." (The component-vector entry does not cover the scalar truncated scores newly defined in the two proofs, which may be negative. Their local definitions must be distinguished from the population component vector; the existing corollary clause remains valid.) - note on `thm:cor-rate`: the corrected statement stands as one statement; the referee would state it as a general statement with corollaries: Separate the two rates into propositions under their respective ranges of $s$ and threshold rules; since neither is a general claim governing the other, the parent should be an umbrella. - note on `thm:cor-rate-ipw`: the corrected statement stands as one statement; the referee would state it as a general statement with corollaries: Separate the two rates into propositions under their respective ranges of $s$ and threshold rules; since neither is a general claim governing the other, the parent should be an umbrella. - note on `thm:rate`: the corrected statement stands as one statement; the referee would state it as a general statement with corollaries: Keep the general score-dependent product bounds as the main statement and separate the finite-variance specialization, which additionally requires the second-moment bound, $t\geq\exp(1)$, $a\log t\leq1$, and $Q_n\leq h/2$. ### V1 lint warnings - thm:cor-rate: depends on ass:exogeneity which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate: depends on ass:overlap which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate: depends on ass:iid-sampling which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate: depends on ass:iid-regular-variation which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate: depends on ass:iid-independence which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate: depends on ass:f-homo-bounded-x which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate: depends on ass:f-homo-asymptotic which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate: depends on ass:mvary-density which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate: depends on ass:mvary-matrix which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate: depends on ass:mvary-constants which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-rate: lemma:eta_rate references \ref{thm:rate} (thm:rate) but `depends_on` lacks thm:rate; V1.5 must confirm whether the paper assumes it or merely mentions it - lem:eta-rate: depends on ass:exogeneity which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-rate: depends on ass:overlap which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-rate: depends on ass:iid-sampling which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-rate: depends on ass:iid-regular-variation which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-rate: depends on ass:iid-independence which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-rate: depends on ass:mvary-density which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-rate: depends on ass:mvary-matrix which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-rate: depends on ass:mvary-constants which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-ipw: lemma:eta_ipw references \ref{thm:rate} (thm:rate) but `depends_on` lacks thm:rate; V1.5 must confirm whether the paper assumes it or merely mentions it - lem:eta-ipw: depends on ass:exogeneity which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-ipw: depends on ass:overlap which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-ipw: depends on ass:iid-sampling which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-ipw: depends on ass:iid-regular-variation which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-ipw: depends on ass:iid-independence which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-ipw: depends on ass:f-homo-bounded-x which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-ipw: depends on ass:f-homo-asymptotic which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-ipw: depends on ass:mvary-density which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-ipw: depends on ass:mvary-matrix which the environment never references; V1.5 must confirm the paper actually assumes it - lem:eta-ipw: depends on ass:mvary-constants which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate-ipw: depends on ass:exogeneity which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate-ipw: depends on ass:overlap which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate-ipw: depends on ass:iid-sampling which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate-ipw: depends on ass:iid-regular-variation which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate-ipw: depends on ass:iid-independence which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate-ipw: depends on ass:f-homo-bounded-x which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate-ipw: depends on ass:f-homo-asymptotic which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate-ipw: depends on ass:mvary-density which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate-ipw: depends on ass:mvary-matrix which the environment never references; V1.5 must confirm the paper actually assumes it - thm:cor-rate-ipw: depends on ass:mvary-constants which the environment never references; V1.5 must confirm the paper actually assumes it ### Parser warnings - Arxiv_template.tex:523: untitled \begin{proof} follows two proof-less results (remark, remark); not bound — say which it proves (hand-fix) - segment: proof at flat lines 704–782 bound to prop:id — Appendix proof derives the identification formula. ## Lean evidence - build of the run barrel: **ok** - source scan (sorry / admit / axiom): clean | declaration | axioms | | --- | --- | | `Papers.TbsExtremeEffects.nete_identification` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsExtremeEffects.nete_rate` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsExtremeEffects.nete_dr_rate_adaptive_threshold` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsExtremeEffects.adaptive_hill_rate` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsExtremeEffects.eta_dr_rate` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsExtremeEffects.eta_ipw_rate` | clean — `propext`, `Classical.choice`, `Quot.sound` | | `Papers.TbsExtremeEffects.nete_ipw_rate_adaptive_threshold` | clean — `propext`, `Classical.choice`, `Quot.sound` | ## Inputs | artifact | status | | --- | --- | | `discovery/v1_scope.json` | present (14 environment(s)) | | `discovery/v1_5_review.json` | present (verdict PASS) | | `formalization/crosswalk_full.json` | present (959 object(s)) | | `graph.json` | present (962 node(s)) | | Lean evidence | collected (build ok, 0 scan issue(s), 7 declaration(s)) | ## Annotated paper Annotated paper: `annotated/Arxiv_template.tex` (113 annotation(s), 0 unlocated; index `annotated/ANNOTATIONS.md`); PDF: `annotated/Arxiv_template.pdf`