Contour Instruments for Partially Linear Models

An observation is the triple $O=(X,T,Y)$, where $X$ takes values in a measurable covariate space $\mathcal X$ and $T,Y\in\mathbb R$. A model over a parameter record consists of a probability law $P$ for $O$ under which $T$ and $Y$ are integrable, a real coefficient $\theta_0$, measurable regression functions $g_0,q_0$ on the covariate space with $$g_0(X)=E_P[T\mid X],\qquad q_0(X)=E_P[Y\mid X]$$ $P$-almost surely, and deterministic measurable supplied code sequences $\widehat g=(\widehat g_n)_{n\in\mathbb N}$ and $\widehat q=(\widehat q_n)_{n\in\mathbb N}$ for $g_0$ and $q_0$. The treatment and outcome innovations are $$\eta=T-g_0(X),\qquad \xi=Y-q_0(X)-\theta_0\eta,$$ and the clipped supplied codes at index $n$ are $$\bar g_n(x)=\min\{\max\{\widehat g_n(x),-C_g\},C_g\},\qquad \bar q_n(x)=\min\{\max\{\widehat q_n(x),-C_q\},C_q\}.$$ The covariate marginal of $P$ is written $P_X$. A model is written $m$, and when the law has to be displayed the coefficient is written $\theta_0(P)$.

In the setting of Definition P-19, Definition P-18, the $k$th cumulant of the treatment innovation $\eta$ satisfies $$|\kappa_k(\eta)|\ge\delta.$$

The estimator $\widehat\theta_{\mathrm{spec},n}$ is the packaged adaptive contour estimator on the public domain where the current supplied treatment-regression code underlying $\bar g_n$ is measurable. It takes as inputs $O_1,\ldots,O_n$, the supplied treatment-regression code sequence through its current clipped values $\bar g_n$, the parameter record $p$, and fixed certified records $\mathfrak p_{\star}$ and $\mathfrak c_{\theta,\star}$ representing $k,\delta,\psi_{\eta},R_0,R_1$, and $C_{\theta}$, with $$R_0=A_k\left(\frac{\psi_{\eta}^2}{\delta}\right)^{(k-2)^{-1}}, \qquad R_1=R_0+1.$$ Here $p$ contains $n\ge1$, $r,k\in\mathbb N$ with $k=r+1$ and $r\ge2$, $s\in[0,\infty]$ with $r\le s$, $\gamma\in(1/2,1)$, positive constants $C_{\theta},C_g,C_q,\psi_{\eta},\psi_{\xi},\delta,\sigma$, and nonnegative nonincreasing sequences $\varepsilon_{1n},\varepsilon_{2n}:\mathbb N\to\mathbb R$ on positive indices.

1. Define the deterministic folds $$I_0=\{i:i<\lfloor n/2\rfloor\}, \qquad I_1=\{i:i\ge\lfloor n/2\rfloor\}.$$ For each real observation coordinate $x$ and integer $m\ge0$, define the floor-dyadic interval $$\mathcal I_m(x)= \left[ 2^{-(m+1)}\lfloor2^{m+1} x\rfloor, 2^{-(m+1)}\{\lfloor2^{m+1} x\rfloor+1\} \right].$$
2. Form the residualized treatment values $$Z_{n,i}=T_i-\bar g_n(X_i), \qquad i=1,\ldots,n.$$ For each fold $a\in\{0,1\}$, derivative order $p$, and complex $z$, define $$\widehat F_{a,n}^{(p)}(z) = \bigl(\max\{|I_a|,1\}\bigr)^{-1}\sum_{i\in I_a}Z_{n,i}^{p}e^{zZ_{n,i}}, \qquad p\in\{0,1,2\},$$ and $$\widehat G_{a,n}^{(p)}(z) = \bigl(\max\{|I_a|,1\}\bigr)^{-1}\sum_{i\in I_a}Y_iZ_{n,i}^{p}e^{zZ_{n,i}}, \qquad p\in\{0,1\}.$$ Thus an empty fold contributes the empty sum and is normalized by one.
3. For represented observation data, define $$\widetilde{\mathfrak s} = \bigl((\mathfrak t_i,\mathfrak y_i,\mathfrak g_i)_{i=1}^n\bigr),$$ where $\mathfrak t_i$, $\mathfrak y_i$, and $\mathfrak g_i$ are certified-real records naming $T_i$, $Y_i$, and $\bar g_n(X_i)$. The full represented-data input is $$\mathfrak s_{\star} = \bigl((\mathfrak t_i,\mathfrak y_i,\mathfrak g_i)_{i=1}^n, \mathfrak p_{\star},\mathfrak c_{\theta,\star}\bigr).$$ For $i=1,\ldots,n$, put $$\mathfrak z_i=\mathfrak t_i-\mathfrak g_i.$$
4. Invoke the contour bank determined by $\mathfrak p_{\star}$. For the $j$th circle, define $$C_j=\{z_j(t):0\le t\le1\}, \qquad z_j(t)=\rho_je^{2\pi it}.$$ At error tolerance one, refine $\mathfrak z_i$, $\mathfrak y_i$, and $\mathfrak\rho_j$ using their own executable moduli, and let $U_{Z,i}$, $U_{Y,i}$, and $U_{\rho,j}$ be the largest absolute endpoint of the corresponding rational intervals. For a canonical observation name this is the interval $\mathcal I_{\mu(1)}$ with the convention above; for $\mathfrak z_i$ it is the certified subtraction interval obtained from $\mathfrak t_i$ and $\mathfrak g_i$, whose modulus uses the two half-error input moduli. Using $e^x\le3^{\lceil x\rceil}$ for $x\ge0$ and $\pi<4$, form rational upper bounds for the empirical derivative sums on each $C_j$.
5. For each $j$, compute a rational bracket of width at most $a_{\star}/64$ enclosing $$\inf_{z\in C_j}|\widehat F_{0,n}(z)|.$$ Let $m_j$ be the lower endpoint of this bracket. When $m_j>0$, define $$W_{0j}(t)= \rho_je^{2\pi it} \frac{\widehat F'_{0,n}(z_j(t))} {\widehat F_{0,n}(z_j(t))}.$$ Use the rational Lipschitz bound $$|W'_{0j}(t)|\le 64\left[ \rho_j\frac{\sup|\widehat F'_{0,n}|}{m_j} +\rho_j^2\left\{ \frac{\sup|\widehat F''_{0,n}|}{m_j} +\frac{\sup|\widehat F'_{0,n}|^2}{m_j^2} \right\}\right].$$
6. Compute a rectangle enclosure, with coordinate widths strictly below $1/4$, for $$\int_0^1W_{0j}(t)\,dt = \frac{1}{2\pi i}\oint_{C_j} \frac{\widehat F'_{0,n}(z)} {\widehat F_{0,n}(z)}\,dz.$$ Define $$N_j= \frac{1}{2\pi i}\oint_{C_j} \frac{\widehat F'_{0,n}(z)} {\widehat F_{0,n}(z)}\,dz$$ when the real coordinate contains a unique nonnegative integer and the imaginary coordinate contains zero. A circle is admissible when $$m_j\ge a_{\star}/2 \qquad\text{and}\qquad N_j\ge1.$$ Let $$j_{\star} = \min\left\{ j: j\in\arg\max_{\ell:\,m_{\ell}\ge a_{\star}/2,\ N_{\ell}\ge1}m_{\ell} \right\}.$$ If the admissible set is empty, set $$\widehat\theta_{\mathrm{spec},n}=0.$$
7. On the evaluation fold for the selected circle $C_{j_{\star}}$, compute a lower endpoint $$m_{j_{\star}}^{(1)} \le \inf_{z\in C_{j_{\star}}}|\widehat F_{1,n}(z)|.$$ If $$m_{j_{\star}}^{(1)}<a_{\star}/4,$$ set $$\widehat\theta_{\mathrm{spec},n}=0.$$
8. Otherwise define the evaluation-fold contour integrand $$V_{j_{\star}}(t)= \rho_{j_{\star}}e^{2\pi it} \frac{\widehat G_{1,n}(z_{j_{\star}}(t))} {\widehat F_{1,n}(z_{j_{\star}}(t))}.$$ Use the rational Lipschitz bound $$|V'_{j_{\star}}(t)|\le 64\left[ \rho_{j_{\star}}\frac{\sup|\widehat G_{1,n}|}{m_{j_{\star}}^{(1)}} +\rho_{j_{\star}}^2\left\{ \frac{\sup|\widehat G'_{1,n}|}{m_{j_{\star}}^{(1)}} +\frac{\sup|\widehat G_{1,n}|\sup|\widehat F'_{1,n}|}{(m_{j_{\star}}^{(1)})^2} \right\}\right].$$ For each rational tolerance $e>0$ and rational Lipschitz bound $L$, the schedule spends one third of the tolerance on discretization and uses $$N_{\mathrm{mesh}}=\left\lceil\frac{3L}{e}\right\rceil+1,$$ so that $L/N_{\mathrm{mesh}}\le e/3$.
9. Enclose the unnormalized contour integral $$\int_0^1V_{j_{\star}}(t)\,dt = \frac{1}{2\pi i}\oint_{C_{j_{\star}}} \frac{\widehat G_{1,n}(z)} {\widehat F_{1,n}(z)}\,dz.$$ Divide the resulting rectangle by $N_{j_{\star}}$ to enclose $$\frac{1}{N_{j_{\star}}2\pi i} \oint_{C_{j_{\star}}} \frac{\widehat G_{1,n}(z)} {\widehat F_{1,n}(z)}\,dz$$ with coordinate radius at most $1/n$. Take $y\in\mathbb Q$ to be the real midpoint of that enclosure, so that $y$ approximates the normalized contour integral to within $1/n$: $$\left|\,y-\operatorname{Re}\left(\frac{1}{N_{j_{\star}}2\pi i} \oint_{C_{j_{\star}}}\frac{\widehat G_{1,n}(z)}{\widehat F_{1,n}(z)}\,dz\right)\right|\le\frac1n.$$ Only rational quantities are ever formed, so the estimator returns this certified rational approximation rather than the integral itself; the $1/n$ tolerance is negligible beside the statistical error.
10. Output $$\widehat\theta_{\mathrm{spec},n} = \operatorname{clip}_{C_{\theta}}(y) = \frac{|y+C_{\theta}|-|y-C_{\theta}|}{2} = \Pi_{[-C_{\theta},C_{\theta}]}(y).$$

Fix $r\ge 2$ and positive constants $\delta,C_{\theta},C_g,C_q,\psi_{\eta},\psi_{\xi}$. Then there are constants $c>0$ and $C>0$, depending only on these fixed primitive constants, such that the following statements hold. For every measurable covariate space, let $p_0$ be a parameter record whose primitive entries $r,\delta,C_{\theta},C_g,C_q,\psi_{\eta},\psi_{\xi}$ equal the fixed constants above, and fix one experiment-wide record consisting of the certified contour-bank input and the represented range input for $p_0$. For every parameter record $p$ with the same primitive entries as $p_0$, and for every base model satisfying the usual measurability, conditional-mean, and integrability requirements, write $n$ for the sample size of $p$. Let $\widehat\theta_{\mathrm{spec},n}$ be the total Borel statistic of Definition P-13, constructed from the transported fixed records and the base treatment-code sequence. Define the fixed-code non-Gaussian class $$\mathcal P_{\mathrm{NG},n}(p;\mathrm{base}) = \{P:\ P\in\mathcal P_{\mathrm{NG},n}\text{ of }\text{def:non-gaussian-class},\ \bar g_n(P)=\bar g_n(\mathrm{base}),\ \bar q_n(P)=\bar q_n(\mathrm{base})\}.$$ Define the fixed-code JMS ACE class $$\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}}(p;\mathrm{base}) = \{P:\ P\in\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}}\text{ of }\text{def:jms-ace-class},\ \bar g_n(P)=\bar g_n(\mathrm{base}),\ \bar q_n(P)=\bar q_n(\mathrm{base})\}.$$

• (Measurable statistic.) The statistic $\widehat\theta_{\mathrm{spec},n}$ is measurable.
• (Represented execution.) For every compiled bounded spectral adapter satisfying the full canonical build-and-compilation specification for $p$, the transported fixed records, and the base treatment-code sequence, the represented-data execution realizes the same $\widehat\theta_{\mathrm{spec},n}$ as in Definition P-13.
• (Class relations.) Every law in $\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}}$ belongs to $\mathcal P_{\mathrm{NG},n}$. If $s=r$, then $\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}}$ coincides with the $L^s(P_X)$-restricted subclass $\mathcal P_{\mathrm{NG},n}^{\mathrm{ACE}}$ of Definition P-10.
• (Non-Gaussian fixed-code risk.) If the sample law is the i.i.d. product law of Assumption A-1, $\mathcal P_{\mathrm{NG},n}(p;\mathrm{base})$ is nonempty, $n\ge 2$, and $$\varepsilon_{1,n}\le \{4R_1\exp(2C_gR_1)\}^{-1},$$ where $R_1$ is the search radius from the contour-bank construction in Definition P-13, then $$\frac{c}{n} \le \mathfrak R_n^{\mathrm{NG}} \le \sup_{P\in\mathcal P_{\mathrm{NG},n}(p;\mathrm{base})} E_P\!\left[(\widehat\theta_{\mathrm{spec},n}-\theta_0(P))^2\right] \le \frac{C}{n},$$ and $$\sup_{P\in\mathcal P_{\mathrm{NG},n}(p;\mathrm{base})} Q_{P,1-\gamma}\!\left(|\widehat\theta_{\mathrm{spec},n}-\theta_0(P)|\right) \le \sqrt{\frac{C}{\gamma n}},$$ with generalized lower quantiles as in Definition P-15.
• (ACE fixed-code risk.) If $\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}}(p;\mathrm{base})$ is nonempty, $n\ge 2$, and $$\varepsilon_{1,n}\le \{4R_1\exp(2C_gR_1)\}^{-1},$$ then $$\frac{c}{n} \le \inf_{\widetilde\theta} \sup_{P\in\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}}(p;\mathrm{base})} E_P\!\left[(\widetilde\theta-\theta_0(P))^2\right] \le \sup_{P\in\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}}(p;\mathrm{base})} E_P\!\left[(\widehat\theta_{\mathrm{spec},n}-\theta_0(P))^2\right] \le \frac{C}{n},$$ where the infimum is over measurable real-valued estimators of $O_1,\ldots,O_n$. Moreover, $$\sup_{P\in\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}}(p;\mathrm{base})} Q_{P,1-\gamma}\!\left(|\widehat\theta_{\mathrm{spec},n}-\theta_0(P)|\right) \le \sqrt{\frac{C}{\gamma n}}.$$

Fix an ACE order $r\ge2$ and positive constants $\delta,C_{\theta},C_g,C_q,\psi_{\eta},\psi_{\xi}$. There is a constant $C_{\mathrm{spec}}>0$, depending only on these fixed constants, such that the following statements hold. Let $\gamma\in(1/2,1)$, let $C_{\gamma}>0$ be the constant in the published order-$r$ ACE generalized-quantile guarantee, and write $\widehat\theta_{\mathrm{ACE},r,n}$ for the estimator supplied by that published ACE handle. Let $p_0$ be a parameter record whose $r,\delta,C_{\theta},C_g,C_q,\psi_{\eta},\psi_{\xi}$, and $\gamma$ coordinates equal the displayed constants, and fix the certified primitive bank and range records associated with $p_0$. For any parameter record $p$ sharing the same fixed experiment constants as $p_0$ and the same probability level $\gamma$, and for any supplied code sequences whose clipped versions are $\bar g_n$ and $\bar q_n$, the following hold, under the usual measurability and integrability conditions.

• Class relations. At the sample size $n$ of $p$, $$\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}} \subseteq \mathcal P_{\mathrm{NG},n}, \qquad \mathcal P_{\mathrm{NG},n}^{\mathrm{ACE}} \subseteq \mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}}.$$ If the comparison exponent of $p$ satisfies $s=r$, then the common clipped-code convention gives $$\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}} =\mathcal P_{\mathrm{NG},n}^{\mathrm{ACE}}.$$ Here the classes and the comparison subclass are those of Definition P-3, Definition P-1, Definition P-10.
• ACE upper guarantee. If a law $P\in\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}}$ uses the clipped supplied treatment and outcome codes $\bar g_n,\bar q_n$, if the JMS eligibility condition $\mathsf E_n^{\mathrm{JMS}}$ of Definition P-7, Definition P-17 holds, and if $\varepsilon_{1,n}>0$ and $\varepsilon_{2,n}\ge0$, then $$Q_{P,1-\gamma} \bigl(|\widehat\theta_{\mathrm{ACE},r,n}-\theta_0(P)|\bigr) \le B_{\mathrm{ACE},n,\gamma},$$ where $Q_{P,1-\gamma}$ is the generalized quantile in Definition P-15 and $$\begin{aligned} B_{\mathrm{ACE},n,\gamma} &= C_{\gamma} r!16^r\delta^{-1} \Big[ \varepsilon_{1,n}^r\varepsilon_{2,n} + C_{\theta}\varepsilon_{1,n}^{r+1} \\ &\qquad\qquad +64(C_g+\psi_{\eta})^r \{r^2(C_g+\psi_{\eta})+\psi_{\xi}+C_{\theta}\psi_{\eta}\} (\gamma n)^{-1/2} \Big]. \end{aligned}$$
• Spectral upper guarantee. If $\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}}$ contains at least one law using the clipped supplied codes, $n\ge2$, $\mathsf E_n^{\mathrm{JMS}}$ holds, $\varepsilon_{1,n}>0$, and $$\varepsilon_{1,n} \le \{4R_1\exp(2C_gR_1)\}^{-1},$$ where $R_1$ is the outer contour radius of Definition P-12 determined by the parameter record in force, then the certified contour statistic $\widehat\theta_{\mathrm{spec},n}$ of Definition P-13, built from the fixed primitive records, satisfies $$\sup_{P\in\mathcal P_{\mathrm{ACE},n}^{\mathrm{JMS}}} Q_{P,1-\gamma} \bigl(|\widehat\theta_{\mathrm{spec},n}-\theta_0(P)|\bigr) \le \sqrt{\frac{C_{\mathrm{spec}}}{\gamma n}},$$ where the supremum ranges over laws in the published ACE class whose clipped supplied treatment and outcome codes are $\bar g_n$ and $\bar q_n$.
• Upper-guarantee separation along sequences. For any sequence $(p_n)$ of parameter records whose sample size at index $n$ is $n$, with the same fixed experiment constants as $p_0$, the same radius sequences $\varepsilon_{1,\cdot},\varepsilon_{2,\cdot}$, and $p_n.\gamma=\gamma$, suppose that eventually $n\ge2$, $\mathsf E_n^{\mathrm{JMS}}$ holds, $\varepsilon_{1,n}>0$, $\varepsilon_{2,n}\ge0$, $$\varepsilon_{1,n} \le \{4R_1\exp(2C_gR_1)\}^{-1},$$ and the published ACE class contains a law using the clipped supplied codes. If $$\frac{ \varepsilon_{1,n}^r\varepsilon_{2,n} + C_{\theta}\varepsilon_{1,n}^{r+1} }{n^{-1/2}} \longrightarrow \infty,$$ then $$\frac{ \sqrt{C_{\mathrm{spec}}/(\gamma n)} }{ B_{\mathrm{ACE},n,\gamma} } \longrightarrow 0 .$$

Let $(\delta_n)_{n\in\mathbb N}$ be a real sequence with $\delta_n>0$ for every $n$, nonincreasing in $n$, and $\delta_n\to0$. Fix $C_{\theta},C_g,C_q,\psi_{\xi}\in\mathbb R$. The following two assertions hold.

• (Gaussian--Rademacher path.) There is a constant $C>0$, depending only on $C_{\theta},C_g,C_q,\psi_{\xi}$, such that the following holds. Let $p$ be any parameter record whose constants $C_{\theta},C_g,C_q,\psi_{\xi}$ are the ones just fixed and whose orders are $k=4$ and $r=3$. For any $a\in(0,1]$ and any law $P$ in a model $m$, assume that the treatment noise satisfies $$\eta\sim \sqrt{1-a^2}\,G_{\circ}+aS,$$ where $G_{\circ}\sim N(0,1)$, $S$ is symmetric Rademacher, and $G_{\circ}\perp S$. Assume also Assumption A-1, Assumption A-2, Assumption A-3, Assumption A-4, Assumption A-5, Assumption A-6, Assumption A-9. At the sample size $n$ of $p$, assume the supplied clipped treatment code $\bar g_n$ satisfies $$x\mapsto |\bar g_n(x)-g_0(x)|\in L^1(P_X), \qquad \int |\bar g_n(x)-g_0(x)|\,dP_X(x)\le \varepsilon_{1,n}.$$ Then the conclusion of Lemma L-5 — the explicit mean-squared-error bound along the Gaussian--Rademacher path — holds with constant $C$ for $p,m,a$.
• (Local ACE oracle envelope.) For every published ACE handle, every $\gamma\in\mathbb R$, and every $C_{\gamma}\in\mathbb R$, if the handle satisfies the conclusion of Theorem~5.4 of \citet{JinMackeySyrgkanis2025} at $(\gamma,C_{\gamma})$, then $C_{\gamma}>0$ and the following local ACE statement holds. For every parameter record $p$ whose probability level is $\gamma$, writing $n$ for its sample size, if $\varepsilon_{1,n}>0$ and $\varepsilon_{2,n}>0$, then for every deterministic treatment code $\widehat g$, deterministic outcome code $\widehat q$, and every model, whenever $$\bar g_n=\Pi_{[-C_g,C_g]}\widehat g_n, \qquad \bar q_n=\Pi_{[-C_q,C_q]}\widehat q_n,$$ and, writing $r$ and $k$ for the orders of $p$ and $\Delta=\delta_n$, the local ACE class conditions $$1\le n, \qquad |\kappa_k(\eta)|\ge \Delta,$$ hold together with Assumption A-2, Assumption A-3, Assumption A-4, Assumption A-5, Assumption A-6, Assumption A-9, the treatment-noise exponential envelope $$o\mapsto \exp\{\eta(o)^2/\psi_{\eta}^2\}\in L^1(P), \qquad \int \exp\{\eta(o)^2/\psi_{\eta}^2\}\,dP(o)\le 2,$$ and the nuisance-radius bounds $$\|\bar g_n-g_0\|_{L^r(P_X)}\le \varepsilon_{1,n}, \qquad \|\bar q_n-q_0\|_{L^r(P_X)}\le \varepsilon_{2,n},$$ and whenever the local JMS eligibility quantities $$a_{1,n}=2\log\!\left(6(C_g+\psi_{\eta})\varepsilon_{1,n}^{-1}\right), \qquad b_{1,n}=\log(\gamma n/9),$$ $$a_2=4(C_g+\psi_{\eta}), \qquad b_{2,n}(\Delta)= \frac{200\min\{1,C_{\theta}\}\Delta}{\max\{\varepsilon_{1,n},\varepsilon_{2,n},(\gamma n)^{-1/2}(\psi_{\xi}+C_{\theta}\psi_{\eta})\}}$$ satisfy $$a_{1,n}\ne0, \qquad 0<\frac{b_{1,n}}{a_{1,n}}, \qquad 0<a_2b_{2,n}(\Delta), \qquad a_2\log\{a_2b_{2,n}(\Delta)\}\ne0,$$ and $$r\le \min\left\{ \frac{b_{1,n}}{a_{1,n}}-a_{1,n}^{-1}\log\!\left(\frac{b_{1,n}}{a_{1,n}}\right), \frac{b_{2,n}(\Delta)}{a_2\log\{a_2b_{2,n}(\Delta)\}} \right\},$$ the published order-$r$ ACE estimator obeys $$Q_{P,1-\gamma} \!\left(\left|\widehat\theta_{\mathrm{ACE},r,n}-\theta_0(P)\right|\right) \le B_{\mathrm{ACE},n,\gamma}(\delta_n;C_{\gamma}),$$ where $$B_{\mathrm{ACE},n,\gamma}(\Delta;C_{\gamma}) = C_{\gamma}\,r!\,16^r\,\Delta^{-1} \left( \varepsilon_{1,n}^r\varepsilon_{2,n} +C_{\theta}\varepsilon_{1,n}^{r+1} +64(C_g+\psi_{\eta})^r \bigl(r^2(C_g+\psi_{\eta})+\psi_{\xi}+C_{\theta}\psi_{\eta}\bigr) (\gamma n)^{-1/2} \right).$$

A natural next question concerns triangular treatment-noise laws satisfying $$|\kappa_k(\eta_n)|\ge\delta_n \qquad\text{with}\qquad \delta_n\downarrow0$$ under the same supplied code sequences. The target is a data-driven selector among ordinary DML, finite-order ACE, and global-contour procedures that attains the sharp minimax mean-squared-error rate as a function of $$(n,\varepsilon_{1,n},\varepsilon_{2,n},\delta_n),$$ supports uniformly valid inference across these regimes, and admits a matching local minimax lower bound. Future work could determine the sharp-rate functional, the corresponding uniform-inference criterion, and the local lower-bound construction for this frontier.