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Uniform Expected Risk for Distance-Based Boundary Regression Designs

Abstract

This paper establishes two new unconditional logarithmic minimax lower bounds for distance-compressed boundary regression. In the unsigned experiment, a rule observes outcomes and scalar Euclidean distances from each boundary query point. Over the compact nonparametric law class $P_{NP} (L, q)$ in Definition 9, with $q \geq 1$ and $L \geq 4$ , both the CTY common-map risk in Definition 12 and the larger point-indexed outer risk in Definition 14 have minimax expected boundary sup-loss bounded below on the scale $a_{n} = (\frac{lo g n}{n})^{1/4} .$ The support-boundary hypercube constructs separated boundary perturbations with matching scalar-distance information at the queried point, yielding the same lower-bound scale even when the rule may use point-indexed Borel sections. Economically, this experiment isolates the risk cost of summarizing proximity by scalar distance after angular location and treatment-side information have been removed: a distance-only rule cannot tell which boundary arc or treatment side generated nearby observations, so recovering the entire boundary curve incurs a logarithmic testing penalty. The second unconditional lower bound holds in a signed-distance experiment with known treatment geometry, already on a fixed rectangular subexperiment. Conditional on the three CTY-style analytic inputs collected in Definition 33, the stabilized local-polynomial estimator and that lower bound characterize the signed-distance expected outer-risk rate on the same scale for every fixed polynomial order $p$ , moment exponent $ν \geq 2$ , and envelope $L \geq L_{0} (p)$ . Thus the unsigned contribution is lower-bound sharpness, while the signed two-sided frontier is explicitly conditional on $AI_{p, ν, L}$ .

Introduction

Regression discontinuity designs use discontinuities in treatment assignment to identify local causal effects near assignment thresholds. The classical scalar-cutoff design begins with the threshold assignment framework of Thistlethwaite et al. (1960) and the continuity-based identification analysis of Hahn et al. (2001); modern econometric treatments develop local comparisons, bandwidth selection, bias correction, and inference around that threshold (Imbens et al., 2008; Lee et al., 2010; Imbens et al., 2012; Calonico et al., 2014; Calonico et al., 2020; Armstrong et al., 2020; Kolesár et al., 2018; Imbens et al., 2019). Boundary and geographic RD designs extend the same logic to bivariate or multivariate scores, where treatment changes across a curve or interface rather than at a scalar cutoff (Keele et al., 2015; Keele et al., 2015; Keele et al., 2016; Papay et al., 2011; Reardon et al., 2012; Wong et al., 2013; Choi et al., 2018; Díaz et al., 2023; Sawada et al., 2024; Merlano, 2025; Kendall et al., 2026).

Distance-to-boundary reductions are central in this setting. They turn a two-dimensional local problem into a one-dimensional running-coordinate problem and are used both as empirical summaries of proximity to the boundary and as coordinates for local-polynomial fitting. Cattaneo et al. (2026) develop distance-based identification and estimation tools for boundary discontinuity designs, including local-polynomial methods for boundary treatment-effect curves. This paper studies the minimax expected-risk consequences of two precise distance experiments: an unsigned experiment in which angular and side information is compressed away, and a known-geometry signed experiment in which the rule retains the treatment side induced by the boundary.

The first experiment is an unsigned distance-compressed boundary regression problem. The law class $P_{NP} (L, q)$ in Definition 9 consists of compact bivariate random-design regression laws with bounded continuous design density, Hölder regression smoothness of order $q$ , and uniformly nondegenerate conditional variance. At a boundary query point $x$ , the data available to the rule are $V_{n, P} (x) = ((Y_{i}, ∥ X_{i} - x ∥_{2}) : 1 \leq i \leq n),$ as defined in Definition 10. The target is the boundary regression function $x \mapsto μ_{P} (x)$ , and the loss is expected supremum error over $bd (X_{P})$ .

For this unsigned experiment, Theorem 1 establishes that, for every integer $q \geq 1$ and every $L \geq 4$ , the common-map risk $R_{n}^{NP}$ in Definition 12 is bounded below on the scale $a_{n} = (lo g n / n)^{1/4}$ . Theorem 2 gives the same lower scale for the larger point-indexed class in Definition 13, evaluated with the outer-expectation risk in Definition 14. Thus the logarithmic lower bound is tied to the unsigned distance information set itself: even law-independent Borel sections chosen separately at each boundary point face the same order of uniform expected loss over $P_{NP} (L, q)$ .

The proof of the unsigned converse combines a direct-product testing inequality with a support-boundary packing. Lemma 1 converts many small coordinatewise overlaps into a lower bound on the probability of at least one decoding error. Lemma 2 builds many separated boundary points on a square support and assigns an independent binary regression perturbation to each local cell. The perturbations are smooth enough to remain in $P_{NP} (L, q)$ , their boundary values differ by order $a_{n}$ , and the unsigned distance laws at each coordinate have Kullback–Leibler divergence of logarithmic size. The support-boundary packing yields separated boundary signals of order $a_{n}$ and logarithmic compressed-distance KL control.

The second experiment is a known-geometry signed-distance boundary RD problem. The law class $P_{12} (p, ν, L)$ in Definition 22 is the displayed Euclidean-distance, uniform-kernel specialization of the CTY Assumptions 1–2 boundary design, with rectangular support, bounded continuous density, smooth potential-outcome regressions, continuous variances, a $2 + ν$ moment envelope, a rectifiable interior interface, two-sided local mass, and a population Gram lower bound. The rule observes signed-distance data $U_{n, P}^{\pm} (x)$ from Definition 21 and may depend on the known geometry $G_{P}$ from Definition 20. The target is the interface treatment-effect curve $τ_{P} (x) = μ_{1, P} (x) - μ_{0, P} (x)$ in Definition 23.

For the signed-distance experiment, Theorem 3 proves a lower bound on the signed-distance minimax risk $R_{n}^{12, \pm} (p, ν, L)$ in Definition 26. For every fixed order $p$ , there is an envelope threshold $L_{0} (p) \geq 48$ such that, for every $ν \geq 2$ and $L \geq L_{0} (p)$ , the risk is bounded below on the $a_{n}$ scale. The same lower bound is already present in the fixed rectangular subexperiment in Definition 27. The hard family in Lemma 6 places independent local perturbations along the bottom edge of a known treated rectangle and keeps all vertices inside the same known-geometry design.

The signed-distance upper bound uses an explicit local-polynomial rule. Definition 28 defines a degree- $p$ estimator that winsorizes outcomes at $B (h) = h^{- 1/3}$ , inverts the empirical Gram matrix only on a uniform conditioning event, and clips the resulting treatment-effect estimate to $[- 2 L, 2 L]$ . Under the distance-identification, first-order bias, and expected maximal inputs collected in Definition 33, Proposition 1 shows that this estimator with bandwidth $h_{n} = a_{n}$ has expected outer risk bounded by a constant multiple of $a_{n}$ uniformly over $P \in P_{12} (p, ν, L)$ .

Under $AI_{p, ν, L}$ , the signed-distance upper bound combines with the lower bound to give the conditional frontier in Theorem 4. For every fixed $p$ , $ν \geq 2$ , and $L \geq L_{0} (p)$ , the signed-distance minimax risk satisfies $0 < c \leq n \to \infty lim inf \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}} \leq n \to \infty lim sup \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}} \leq C < \infty.$ The same theorem records that the winsorized, Gram-stabilized signed-distance local-polynomial estimator attains the upper side of this bound under the same three analytic inputs. A short reproducibility note records the checked scope for the anchored mathematical statements in Section D.

The comparison between the two experiments is organized by law class, information set, and risk. In the unsigned problem, scalar Euclidean distances from each query point create a boundary-indexed compression in which many angularly separated alternatives remain hard to distinguish, producing unconditional lower bounds. In the signed problem, known boundary geometry and treatment side supply the one-dimensional coordinate used by local-polynomial RD methods; its fixed-geometry lower bound is unconditional, while its two-sided rate characterization uses the displayed analytic inputs. The common normalization $a_{n}$ therefore records a shared logarithmic lower scale, with a conditional upper side for the signed experiment.

The remainder of the paper proceeds as follows. Section 3 defines the compact nonparametric unsigned distance experiment, its decision classes, and its minimax risks. Section 4 defines the known-geometry signed-distance RD experiment, the A1/A2 law class, the treatment-effect target, and the stabilized local-polynomial estimator. Section 5 states the unsigned lower bounds, the signed-distance lower bound, the conditional upper bound, and the matched signed-distance frontier. Section 6 discusses the interpretation for boundary RD designs. Sections A, B, C, D, and E contain the direct-product and packing arguments, the signed-distance empirical-process ingredients, the fixed-geometry signed hypercube, the reproducibility note, and deferred proofs.

Regression discontinuity designs originate in the threshold assignment framework of Thistlethwaite et al. (1960) and the identification analysis of Hahn et al. (2001), with modern econometric treatments emphasizing local comparisons, bandwidth choice, and inferential refinements (Imbens et al., 2008; Lee et al., 2010; Imbens et al., 2012). Geographic and boundary designs extend this logic to assignment regions in space, where the score is vector-valued and the estimand is indexed by an interface rather than a scalar cutoff (Keele et al., 2015; Keele et al., 2015; Keele et al., 2016; Papay et al., 2011; Reardon et al., 2012; Wong et al., 2013; Choi et al., 2018; Díaz et al., 2023; Sawada et al., 2024; Merlano, 2025). Within that literature, Cattaneo et al. (2026) develop distance-based identification and estimation tools for boundary regression designs. Their analysis establishes identification from distance-to-boundary reductions, sequence-scoped bias expansions, and probability-order stochastic control for local-polynomial procedures under their maintained design conditions.

Theorem 6 of Cattaneo et al. (2026) studies the unsigned minimax risk $T_{n} in f P \in P_{NP} (L, q) sup E_{P} [x \in bd (X_{P}) sup ∣ T_{n} (U_{n} (x)) - μ_{P} (x) ∣], U_{n} (x) = ((Y_{i}, ∥ X_{i} - x ∥_{2}) : i \leq n),$ where one law-independent measurable map $T_{n}$ is used at every boundary point. Their theorem proves a positive lower bound after multiplication by $n^{1/4}$ . The printed conjecture immediately following that theorem replaces $n^{1/4}$ by $(n / lo g n)^{1/4}$ , matching the scale of their uniform upper bound. The unsigned common-map result here proves that conjectured strengthening without changing the law class, transformed data, target, decision class, or expected supremum loss. The point-indexed result changes only the decision rule: a law-independent Borel section may now be selected separately for every query point, while the transformed data and regression target remain the same; outer expectation supplies the measurable envelope for the boundary supremum. It proves the same logarithmic lower scale for this larger class.

The signed result concerns a different econometric target and information set. Motivated by CTY’s signed-distance identification and local-polynomial analysis, it studies the boundary treatment-effect curve when the assignment geometry and treatment side are available to the rule. The paper proves a new minimax lower bound already on one fixed known rectangular geometry. Its expected-risk upper bound is stated under the displayed identification, bias, and expected maximal hypotheses. Table 1 records these three deliverables in parallel.

Nearest CTY benchmarks and the exact change in target, information, decision class, risk, or conditioning status.
Deliverable	Nearest CTY result	Exact increment in this paper
Unsigned CTY decision problem	Theorem 6 of Cattaneo et al. (2026): $P_{NP} (L, q)$ , data $U_{n} (x) = ((Y_{i}, ∥ X_{i} - x ∥_{2}) : i \leq n)$ , one common map $T_{n}$ , boundary regression target, and expected boundary sup-loss; lower scale $n^{- 1/4}$ .	Keeps the law class, information, target, decision class, and risk fixed and proves the conjectured lower scale $a_{n}$ .
Point-indexed unsigned rules	CTY’s lower-bound theorem uses one law-independent common map across query points.	Allows a separate law-independent Borel section for each query point, uses outer expectation for the boundary supremum, and still proves the $a_{n}$ lower scale.
Signed known-geometry treatment effects	Theorems 1–2 and Supplemental SA-8.3–SA-8.4 provide the nearest identification, bias, and stochastic-process ingredients for signed-distance local-polynomial analysis.	Changes the target to the boundary treatment-effect curve and the information to signed distance with known geometry; proves an unconditional $a_{n}$ lower bound on a fixed rectangular subexperiment and, under $AI_{p, ν, L}$ , a stabilized expected outer-risk upper bound at the same scale.

This paper studies the uniform expected-risk consequences of those distance reductions. In the unsigned experiment, the relevant object is the compact bivariate nonparametric regression class $P_{NP}$ $(L, q)$ in Definition 9, where a rule observes outcomes and unsigned Euclidean distances as in Definition 10. The decision problem uses the common-map class $T_{n}^{NP}$ in Definition 11 and its completed-expectation risk $R_{n}^{NP}$ in Definition 12. The resulting converse is a same-class statement: over the law class and information set just described, the minimax expected boundary sup-loss has the logarithmic scale $a_{n}$ introduced in Definition 8. The point-indexed extension uses the sectionwise-Borel outer-risk setup in Definitions 13 and 14, thereby characterizing the same compression barrier for rules that may choose a law-independent Borel section separately at each boundary point.

This focus differs from the usual role of distance in geographic RD applications. Empirical distance reductions often serve as low-dimensional summaries of proximity to a boundary or as practical coordinates for local fitting (Keele et al., 2015; Keele et al., 2015; Keele et al., 2016; Kendall et al., 2026). Here the distance reduction is part of the statistical experiment: the rule receives the outcome and the scalar unsigned distance from the query point, and the risk is the expected supremum over boundary points. The lower bound therefore speaks to information loss under the compressed observation scheme, rather than to specification choice in a geographic RD regression. Related diagnostic and design tools, including manipulation tests and geographic balance analyses (McCrary, 2008; Cattaneo et al., 2022; Cattaneo et al., 2019; Cattaneo et al., 2023; Cattaneo et al., 2025), address complementary features of RD credibility; the present comparison is organized around the target, the available data transformation, and the uniform loss criterion.

The signed-distance part connects to local-polynomial boundary estimation. Local-polynomial methods provide the standard approximation architecture for nonparametric regression and RD estimation (Fan et al., 1996; Ruppert et al., 1994; Calonico et al., 2014; Calonico et al., 2020; Imbens et al., 2019). Robust bias correction and optimal inference results refine confidence interval construction and coverage for scalar or boundary-local targets (Calonico et al., 2014; Calonico et al., 2020; Armstrong et al., 2020; Kolesár et al., 2018). The known-geometry signed-distance experiment fixes the CTY Assumptions 1--2 potential-outcome boundary design through $P_{12}$ $(p, ν, L)$ in Definition 22, observes signed-distance data in Definition 21, and evaluates treatment-effect rules in Definitions 25 and 26. Assuming the CTY distance-identification, uniform first-order bias, and expected maximal inputs collected as $AI_{p, ν, L}$ in Definition 33, the paper combines the signed-distance lower bound with the winsorized, Gram-stabilized local-polynomial upper bound to obtain the two-sided expected-risk rate.

The minimax structure draws on classical nonparametric lower-bound and empirical-process ideas. Packing and modulus arguments underlie the rate calculations in nonparametric estimation (Stone, 1982; Tsybakov, 2009), while boundary and set-indexed empirical-process tools support the stochastic controls used in local smoothing problems (Mammen et al., 1995; Cuevas et al., 2004; Vapnik et al., 1971; Pollard, 1984; van der Vaart et al., 1996). The lower-bound construction in this paper adapts those ideas to boundary-indexed distance compression: many separated boundary locations carry independent binary perturbations, while scalar distance observations constrain the coordinatewise information available to any admissible rule. The empirical-process component enters the signed-distance upper result through the stated analytic inputs and the bounded-envelope maximal inequality used for the winsorized score. Together, these ingredients place the paper between geographic RD methodology and minimax nonparametric theory: the estimand is a boundary treatment-effect or boundary regression curve, the information set is a distance-compressed experiment, and the criterion is uniform expected loss over the displayed law classes.

Setup: Distance-Compressed Boundary Regression

This section fixes the statistical experiment used for the unsigned lower bounds. The object of interest is a boundary-indexed regression function under a compact bivariate random-design law, and the information available at a boundary point is the outcome together with the scalar Euclidean distance from that point. The normalization used throughout the converse arguments is introduced first.

Definition 1 [synth_25] (Law operator

L

For a random element $Z$ taking values in a measurable space $(E, E)$ under a probability law $P$ , write $L_{P} (Z) = P \circ Z^{- 1}$ for the distribution of $Z$ ; when $P$ is clear, write $L (Z)$ . For an event $A$ with $P (A) > 0$ , write $L_{P} (Z ∣ A)$ for the conditional distribution satisfying $L_{P} (Z ∣ A) (B) = P (Z \in B, A) / P (A)$ for every $B \in E$ .

Definition 2 [synth_28] (Smallest quadratic-form eigenvalue

λ_{m i n}

For a real symmetric matrix $Ψ$ , we write $λ_{m i n} (Ψ) = in f_{∥ v ∥_{2} = 1} v^{⊤} Ψ v$ , equivalently $λ_{m i n} (Ψ) \geq c$ means $c ∥ v ∥_{2}^{2} \leq v^{⊤} Ψ v$ for every vector $v$ .

Definition 3 [synth_43] (Closed Euclidean ball

B_{2} (x, r)

For $x \in R^{2}$ and $r \geq 0$ , we write $B_{2} (x, r)$ for the closed Euclidean ball ${z \in R^{2} : ∥ z - x ∥_{2} \leq r}$ .

Definition 4 [synth_36] (Euclidean

q

-Hölder ball

H^{q} (X_{P}, L)

We say $g \in H^{q} (X_{P}, L)$ when $g : R^{2} \to R$ is $C^{q - 1}$ on $X_{P}$ , satisfies $∥ D^{j} g (x)∥ \leq L$ for every $0 \leq j \leq q - 1$ and every $x \in X_{P}$ , and its $(q - 1)$ st Fréchet derivative is $L$ -Lipschitz on $X_{P}$ : $∥ D^{q - 1} g (x) - D^{q - 1} g (y)∥ \leq L ∥ x - y ∥_{2}$ for all $x, y \in X_{P}$ .

Definition 5 [synth_72] (Faithful random-design law space

P

We say $P \in P$ when $P$ is a probability law of $(Y, X) \in R \times R^{2}$ with square-integrable outcome, whose carried objects $X_{P}$ , $f_{P}$ , $μ_{P}$ , and $σ_{P}^{2}$ are tied to the law as follows: $X_{P}$ is the topological support of the $X$ -marginal, the $X$ -marginal has Lebesgue density $f_{P} 1_{X_{P}}$ , $μ_{P}$ is an $X$ -marginal-almost-everywhere version of $E_{P} [Y ∣ X = x]$ , and $σ_{P}^{2}$ is an $X$ -marginal-almost-everywhere version of $Var_{P} (Y ∣ X = x)$ .

Definition 6 [synth_138] (Bivariate design support

X

For a law $P$ of $(Y, X)$ on $R \times R^{2}$ , we say $X$ is the bivariate design support when $X = supp (L_{P} (X))$ , the topological support of the $X$ -marginal under $P$ .

Definition 7 [synth_29] (Total variation distance

∥ Q_{j, 0} - Q_{j, 1} ∥_{TV}

For the two probability laws $Q_{j, 0}$ and $Q_{j, 1}$ on the common measurable sample space of $S_{j}$ , define their total variation distance by $∥ Q_{j, 0} - Q_{j, 1} ∥_{TV} = A sup ∣ Q_{j, 0} (A) - Q_{j, 1} (A) ∣,$ where the supremum ranges over all measurable sets $A$ in that sample space.

These preliminary conventions give the basic probabilistic and geometric vocabulary. In particular, the law-space convention ties the support, density, regression, and conditional variance to the same underlying law, while the Hölder and ball notation provide the local smoothness and Euclidean localization language used in the packing arguments. Total variation enters later through the direct-product comparison of coordinatewise compressed experiments.

Definition 8 [def:frontier-rates] (Frontier rate

a_{n}

The frontier normalization is $a_{n} = (\frac{lo g n}{n})^{1/4} .$

⊢ Lean

The scale in Definition 8 is the common normalization for the unsigned boundary lower bounds. Its logarithmic factor reflects the uniform boundary supremum loss rather than a pointwise loss criterion.

The nonparametric class now fixes the bivariate random-design model. It combines compact support, a bounded continuous design density, Hölder regression smoothness, and uniformly nondegenerate conditional variance.

Definition 9 [def:cty-nonparametric-class] (Nonparametric law class

P_{NP} (L, q)

For $q \in N$ and $L \in R$ , the class $P_{NP} (L, q)$ consists of all random-design regression laws $P$ satisfying the following conditions.

(Parameter regime.) $1 \leq q$ and $4 \leq L$ .
(Sampling.) $P$ is a probability law for one observation $(Y, X) \in R \times R^{2}$ , $Y$ is square-integrable under $P$ , and for every $n$ , $(Y_{i}, X_{i})_{i = 1}^{n}$ are sampled independently and identically from $P$ .
(Design.) The $X$ -marginal has Lebesgue density $f_{P} 1_{X_{P}}$ , where $X_{P}$ is the exact topological support of the $X$ -marginal. Write $f_{P}$ for this density. It is continuous on $X_{P}$ , $X_{P}$ is compact and satisfies $X_{P} \subseteq [- L, L]^{2}$ , and $L^{- 1} \leq f_{P} (x) \leq L$ for every $x \in X_{P}$ . The boundary $bd (X_{P})$ admits a Lipschitz parametrization by $[0, 1]$ .
(Regression.) The function $μ_{P}$ is an $X$ -marginal-almost-everywhere version of $E_{P} [Y ∣ X = x]$ and belongs to $H^{q} (X_{P}, L)$ .
(Variance.) The function $σ_{P}^{2}$ is an $X$ -marginal-almost-everywhere version of $Var_{P} (Y ∣ X = x)$ , is continuous on $X_{P}$ , and satisfies $L^{- 1} \leq σ_{P}^{2} (x) \leq L$ for every $x \in X_{P}$ .

Equivalently, $P_{NP} (L, q) = {P : P satisfies the conditions above} .$

⊢ Lean

The class in Definition 9 is a compact-support version of a random-design nonparametric regression model. The density bounds keep local sample sizes comparable across admissible laws, the Lipschitz boundary condition gives a one-dimensional interface over which a supremum can be evaluated, and the Hölder condition calibrates how quickly regression values may change near separated boundary locations. These conditions are standard in nonparametric minimax analysis, with compactness and smoothness playing the same organizational role as in classical regression lower bounds (Stone, 1982; Tsybakov, 2009).

At a boundary query point, the unsigned experiment compresses each covariate to its distance from that point. The rule retains the outcome and the scalar distance, so all angular information around the query point is absorbed into this one-dimensional summary.

Definition 10 [def:cty-distance-data] (Distance data

V_{n, P} (x)

For $x \in bd (X_{P})$ , the distance data are $V_{n, P} (x) = ((Y_{i}, ∥ X_{i} - x ∥_{2}) : 1 \leq i \leq n) .$

⊢ Lean

The observation in Definition 10 is indexed by the boundary point because the same sample is re-expressed relative to each query location. This formulation matches the distance-to-boundary reductions studied by Cattaneo et al. (2026), while the present risk criterion evaluates the resulting boundary-indexed regression rule uniformly over $bd (X_{P})$ .

The first decision class uses one law-independent map for all query points. This common-map restriction captures procedures that process the distance data by a single measurable rule once the sample size is fixed.

Definition 11 [def:cty-distance-decision-class] (Common-map class

T_{n}^{NP}

For a represented rule, write $T_{n}$ for its selected law-independent measurable map. Define the sample-size-indexed class by $T_{n}^{NP} = {μ_{n} : there exists a measurable map T_{n} : (R \times [0, \infty))^{n} \to R selected independently of P, μ_{n} (x) = T_{n} (V_{n, P} (x)) for every x \in bd (X_{P})} .$

⊢ Lean

Definition 12 [def:cty-distance-risk] (Common-map risk

R_{n}^{NP}

The common-map minimax risk is $R_{n}^{NP} = μ_{n} \in T_{n}^{NP} in f P \in P_{NP} (L, q) sup E_{P} [x \in bd (X_{P}) sup ∣ μ_{n} (x) - μ_{P} (x) ∣] .$

⊢ Lean

Together, Definitions 11 and 12 define the completed-expectation minimax problem for common-map rules. The expectation is the ordinary expectation under the product law generated by $P$ , and the loss is the supremum error along the support boundary. The infimum therefore ranges over estimators that share the same measurable distance-data transformation across all admissible laws and all boundary points.

For comparison, the point-indexed decision class lets the measurable section depend on the query point while remaining law independent. This keeps the unsigned distance information set fixed and enlarges the admissible class by allowing a separate Borel rule at each $x$ .

Definition 13 [def:point-indexed-distance-decision-class] (Point-indexed class

T_{n}^{PI}

The point-indexed decision class is $T_{n}^{PI} = ⎩ ⎨ ⎧ μ_{n}^{PI} : there exists a family {T_{n, x} : x \in R^{2}} selected independently of P, T_{n, x} : (R \times [0, \infty))^{n} \to R is Borel measurable for every fixed x, μ_{n, P}^{PI} (x) = T_{n, x} (V_{n, P} (x)) for every P \in P_{NP} (L, q) and every x \in bd (X_{P}) ⎭ ⎬ ⎫ .$ This class consists exactly of such law-independent fixed- $x$ Borel sections.

⊢ Lean

Definition 14 [def:point-indexed-distance-risk] (Point-indexed risk

R_{n}^{PI}

For an extended nonnegative function $Z$ on the sample space, define the outer expectation by $E_{P}^{*} [Z] = in f {E_{P} [W] : W is extended nonnegative and measurable, and W \geq Z pointwise} .$ The point-indexed distance risk is $R_{n}^{PI} = μ_{n}^{PI} \in T_{n}^{PI} in f P \in P_{NP} (L, q) sup E_{P}^{*} [x \in bd (X_{P}) sup μ_{n, P}^{PI} (x) - μ_{P} (x)] .$

⊢ Lean

The outer expectation in Definition 14 follows the standard empirical-process convention for suprema that may require measurable majorants (van der Vaart et al., 1996). With this convention, Definitions 13 and 14 define a sectionwise Borel problem that preserves the same distance-compressed observations as Definition 10. The main lower bounds compare these two unsigned risks after the common notation and signed-distance setup have been fixed.

Setup: Known-Geometry Signed-Distance Designs

The second experiment keeps the boundary geometry available to the rule and records which side of the interface each nearby observation occupies. This is the setting in which local-polynomial methods use distance as a one-dimensional coordinate while preserving the sign induced by treatment assignment, as in boundary regression designs developed by Cattaneo et al. (2026). We first fix the assignment, interface, basis, and coordinate conventions that underlie the signed-distance reduction.

Definition 15 [synth_4] (Assignment regions

A_{t, P}

We say $A_{t, P}$ is the arm- $t$ assignment region of $P$ , for $t \in {0, 1}$ , when $A_{0, P}$ and $A_{1, P}$ are Borel subsets of $X_{P}$ satisfying $A_{0, P} \cup A_{1, P} = X_{P}$ and $A_{0, P} \cap A_{1, P} = \emptyset$ , with deterministic treatment assignment $T = 1 {X \in A_{1, P}}$ .

Definition 16 [synth_35] (Known treatment interface

B

We say $B$ is the known treatment interface for the fixed hypercube geometry when it is the boundary of the treated rectangle, $B = \partial ([- 1, 1] \times [0, 2]) .$

Definition 17 [synth_71] (Degree-

p

Polynomial Basis

r_{p}

For an integer $p \geq 0$ and $u \in R$ , define $r_{p} (u) = (1, u, \dots, u^{p})^{⊤} \in R^{p + 1}$ .

Definition 18 [synth_24] (Score-coordinate projection

score

We say $score$ is the score-coordinate projection when, for every potential-outcome observation $w = (y_{0}, y_{1}, x) \in R \times R \times R^{2}$ , $score (w) = x$ .

The partition in Definition 15 makes treatment deterministic given location, so the side of the boundary is a feature of the design rather than an additional random assignment variable. The fixed interface in Definition 16 is the rectangular geometry used in the lower-bound construction, while Definition 17 supplies the signed-distance polynomial basis used by the local fits. The projection in Definition 18 records that the score coordinate is the Euclidean covariate.

The known-geometry object packages the support, assignment regions, interface, Euclidean metric, and smoothing weight. The canonical smoothing weight is the one-dimensional uniform kernel $K_{□}$ , which selects observations whose signed distance lies within the bandwidth window.

Definition 19 [def:cty-uniform-kernel] (Uniform kernel

K_{□}

The fixed uniform kernel is $K_{□} (u) = 1 {∣ u ∣ \leq 1}, u \in R .$

⊢ Lean

Definition 20 [def:cty-known-geometry] (Known geometry

G_{P}

Define $A$ to be the class of admissible known geometries $G = (X, A_{0}, A_{1}, B, d, K_{□})$ , where $A_{0}, A_{1}$ are a Borel partition of $X$ , $B = bd (A_{0}) \cap bd (A_{1}) \subset int (X), d (x, z) = ∥ x - z ∥_{2} .$ For a member law $P$ , write $A_{0, P}, A_{1, P}$ for its assignment regions and write $B_{P} = bd (A_{0, P}) \cap bd (A_{1, P})$ . Define its known design object by $G_{P} = (X_{P}, A_{0, P}, A_{1, P}, B_{P}, d_{P}, K_{□}) \in A .$

⊢ Lean

Definition 21 [def:cty-known-geometry-signed-distance-data] (Signed-distance data

U_{n, P}^{\pm} (x)

For $x \in B_{P}$ , define the signed distance $D_{P, x}^{\pm} = {1 (X \in A_{1, P}) - 1 (X \in A_{0, P})} ∥ X - x ∥_{2}$ and the signed-distance sample $U_{n, P}^{\pm} (x) = ((Y_{i}, D_{P, x, i}^{\pm}) : 1 \leq i \leq n) .$

⊢ Lean

The known design object $G_{P}$ in Definition 20 gives the rule the relevant boundary and side labels. For an interface point $x$ , the signed distance $D_{P, x}^{\pm}$ in Definition 21 retains the Euclidean distance magnitude and assigns its sign according to the treatment region. Thus the signed-distance sample $U_{n, P}^{\pm} (x)$ provides the one-dimensional running coordinate used by local-polynomial RD estimators (Fan et al., 1996; Ruppert et al., 1994; Calonico et al., 2014; Calonico et al., 2020; Imbens et al., 2019).

The law class below is the displayed Euclidean, uniform-kernel specialization of the CTY Assumptions 1–2 boundary design. Its clauses have four statistical roles. The support, assignment regions, interface, metric, and signed coordinate define the experiment and information available to a rule. The density, regression smoothness, conditional variance, and moment bounds provide uniform design and outcome envelopes. The population-Gram, arm-mass, distance-slice, and VC conditions keep local-polynomial fitting uniformly conditioned along the interface. The separate inputs $AI_{p, ν, L}$ , introduced in Definition 33, supply identification, first-order approximation, and expected uniform stochastic control for the conditional upper bound.

Definition 22 [def:cty-a1-a2-class] (A1/A2 law class

P_{12} (p, ν, L)

For integers $p \geq 0$ , real numbers $ν \geq 2$ , and $L \geq 4$ , let $P_{12} (p, ν, L)$ be the set of laws $P$ satisfying the following conditions.

Write $Y_{i} (0), Y_{i} (1)$ for the sampled potential outcomes of observation $i$ , and write $H^{1} (B_{P})$ for the one-dimensional Hausdorff measure of the common interface.

(Sampling and consistency.) For every $n$ , the vectors $(Y_{i} (0), Y_{i} (1), X_{i})$ , $1 \leq i \leq n$ , are independently and identically distributed, $T = 1 {X \in A_{1, P}}, Y = T Y (1) + (1 - T) Y (0) .$
(Rectangular support and density.) The support has the form $X_{P} = [\underline{x}_{P}, \overline{x}_{P}]^{2} \subseteq [- L, L]^{2} .$ The variable $X$ has a Lebesgue density $f_{P}$ , continuous on $X_{P}$ , such that $L^{- 1} \leq f_{P} (x) \leq L, x \in X_{P} .$
(Smooth potential-outcome regressions.) For each $t \in {0, 1}$ , the regression $μ_{t, P} (x) = E_{P} [Y (t) ∣ X = x]$ has a $C^{p + 1}$ extension to an open neighborhood of $X_{P}$ , and $0 \leq ∣ α ∣ \leq p max x \in X_{P} sup ∣ \partial^{α} μ_{t, P} (x) ∣ + 0 \leq ∣ α ∣ \leq p max x \neq = z \in X_{P} sup \frac{∣ \partial ^{α} μ _{t, P} ( x ) - \partial ^{α} μ _{t, P} ( z ) ∣}{∥ x - z ∥ _{2}} \leq L .$
(Conditional variances.) For each $t \in {0, 1}$ , the conditional variance $σ_{t, P}^{2} (x) = Var_{P} (Y (t) ∣ X = x)$ is continuous and satisfies $L^{- 1} \leq σ_{t, P}^{2} (x) \leq L, x \in X_{P} .$
(Moment envelope.) For each $t \in {0, 1}$ and every $x \in X_{P}$ , $E_{P} [∣ Y (t) ∣^{2 + ν} ∣ X = x] \leq L .$
(Assignment geometry.) The sets $A_{0, P}$ and $A_{1, P}$ form a Borel partition of $X_{P}$ , and their known common interface $B_{P} \subset int (X_{P})$ is a compact rectifiable curve satisfying $L^{- 1} \leq H^{1} (B_{P}) \leq L .$
(Metric, kernel, and VC bound.) The known metric and kernel are $d_{P} (x, z) = ∥ x - z ∥_{2}, K_{□} .$ The collection ${{z \in X_{P} : ∥ z - x ∥_{2} \leq r} : x \in X_{P}, r \geq 0}$ is a VC class with VC index at most four.
(Population Gram lower bound.) Put $r_{p} (u) = (1, u, \dots, u^{p})^{⊤}, I_{1} = [0, \infty), I_{0} = (- \infty, 0),$ and $Ψ_{t, P, x} (h) = E_{P} [\frac{1}{h ^{2}} 1 {D_{P, x}^{\pm} \in I_{t}} K_{□} (D_{P, x}^{\pm} / h) r_{p} (D_{P, x}^{\pm} / h) r_{p} (D_{P, x}^{\pm} / h)^{⊤}] .$ For every $0 < h \leq L^{- 1}$ , every $x \in B_{P}$ , and both $t \in {0, 1}$ , $λ_{m i n} (Ψ_{t, P, x} (h)) \geq L^{- 1} .$
(Small-bandwidth arm mass.) For every $0 < h \leq L^{- 1}$ , every $x \in B_{P}$ , and both $t \in {0, 1}$ , $\frac{1}{h ^{2}} \int_{A_{t, P}} K_{□} ({(2 t - 1) ∥ z - x ∥_{2}} / h) d z \geq L^{- 1} .$
(Distance-slice density.) For every $x \in B_{P}$ , both $t \in {0, 1}$ , and every $0 < s \leq L^{- 1}$ , the one-dimensional Hausdorff integral of $f_{P}$ over ${z \in A_{t, P} : ∥ z - x ∥_{2} = s}$ is finite and strictly positive.

This class is the displayed $L$ -uniformized Euclidean-distance, uniform-kernel specialization of the two baseline smoothness and design conditions, augmented by the displayed envelope restrictions and quantitative small-bandwidth thresholds.

⊢ Lean

The assumptions in Definition 22 fall into three groups. First, the support, assignment partition, rectifiable interface, signed coordinate, and distance-slice conditions define the design and the information observed by the signed-distance procedure. Second, the density, moments, smoothness, variance, arm-mass, population-Gram, and VC conditions provide uniform outcome, conditioning, and complexity envelopes for local fitting. Third, $AI_{p, ν, L}$ in Definition 33 collects the identification, bias, and expected maximal inputs invoked specifically by the conditional upper bound. The unconditional signed lower bound uses the fixed-geometry subexperiment inside the first two groups; the stabilized estimator and conditional frontier use all three groups.

The potential outcomes $Y (t)$ and deterministic treatment $T$ in Definition 22 give the boundary target its causal RD interpretation. The arm-specific regressions $μ_{t, P}$ support degree- $p$ signed-distance approximation, while the arm-specific variances $σ_{t, P}^{2}$ and $2 + ν$ moment envelope keep the outcome scale uniform. Geometrically, the class includes compact rectifiable interfaces of controlled length inside rectangular supports and requires two-sided local mass and stable signed-distance Gram matrices at every interface point. These quantitative conditions also cover the corners of the fixed treated rectangle: Lemma 6 verifies the Gram and arm-mass thresholds there rather than relying on a smooth-boundary shortcut.

The treatment-effect target is the jump between the two smooth potential-outcome regression surfaces at the interface. The population local-polynomial coefficient records the signed-distance normal equations that the estimator approximates.

Definition 23 [synth_46] (Interface treatment-effect curve

τ_{P} (x)

For a law $P \in P_{12} (p, ν, L)$ and an interface point $x \in B_{P}$ , define the treatment-effect curve by $τ_{P} (x) = μ_{1, P} (x) - μ_{0, P} (x)$ , where $μ_{t, P} (x) = E_{P} [Y (t) ∣ X = x]$ is the continuous conditional-mean version for arm $t \in {0, 1}$ .

Definition 24 [synth_128] (Population local-polynomial coefficient

β_{t, P, x}^{*} (h)

We say $β_{t, P, x}^{*} (h)$ is the unwinsorized population degree- $p$ signed-distance local-polynomial coefficient when $β_{t, P, x}^{*} (h) \in R^{p + 1}$ satisfies the normal equations $Ψ_{t, P, x} (h) β_{t, P, x}^{*} (h) = E_{P} [\frac{1}{h ^{2}} 1 {D_{P, x}^{\pm} \in I_{t}} K_{□} (D_{P, x}^{\pm} / h) r_{p} (D_{P, x}^{\pm} / h) Y] .$

The curve $τ_{P}$ in Definition 23 is evaluated uniformly over $B_{P}$ . The coefficient in Definition 24 is a population projection onto the signed-distance polynomial basis; its intercept difference is the population analogue of the treatment-effect estimator, and its approximation error is the first-order bias component isolated later.

Rules in the signed-distance experiment may depend on the known geometry and the query point through law-independent Borel sections. The risk then uses the outer expectation already introduced for point-indexed suprema in Definition 14.

Definition 25 [def:cty-a1-a2-point-indexed-decision-class] (Known-geometry rules

T_{n}^{12, \pm}

Let $T_{n}^{12, \pm}$ be the class of rules for which there is one collection ${T_{n, G, x} : G is an admissible known geometry, x \in R^{2}}$ selected independently of the unknown member law, such that every fixed $(G, x)$ section is a Borel map from $(R \times R)^{n}$ to $R$ and, for every $P \in P_{12} (p, ν, L)$ , every $p \geq 0$ , and every $x \in B_{P}$ , $τ_{n, P} (x) = T_{n, G_{P}, x} (U_{n, P}^{\pm} (x)) .$ The imposed regularity is fixed- $(G, x)$ Borel measurability of each section.

⊢ Lean

Definition 26 [def:cty-a1-a2-outer-risk] (Signed-distance minimax risk

R_{n}^{12, \pm} (p, ν, L)

Using the outer expectation $E_{P}^{*}$ defined in Definition 14, for $p \geq 0$ the signed-distance minimax risk is $R_{n}^{12, \pm} (p, ν, L) = τ_{n} \in T_{n}^{12, \pm} in f P \in P_{12} (p, ν, L) sup E_{P}^{*} [x \in B_{P} sup ∣ τ_{n, P} (x) - τ_{P} (x) ∣] .$

⊢ Lean

Definition 27 [synth_126] (Fixed-geometry signed-distance outer risk

R_{n}^{12, \pm, fix} (p, ν, L)

We define $R_{n}^{12, \pm, fix} (p, ν, L)$ as the signed-distance outer minimax risk with the law supremum restricted to the fixed hard geometry: $R_{n}^{12, \pm, fix} (p, ν, L) = τ_{n} \in T_{n}^{12, \pm} in f P \in P_{12} (p, ν, L) : X_{P} = [- 3, 3]^{2}, A_{1, P} = [- 1, 1] \times [0, 2], A_{0, P} = X_{P} ∖ A_{1, P}, B_{P} = \partial A_{1, P}, d_{P} (x, z) = ∥ x - z ∥_{2}, and the known kernel is K_{□} sup E_{P}^{*} [x \in B_{P} sup ∣ τ_{n, P} (x) - τ_{P} (x) ∣] .$

The decision class $T_{n}^{12, \pm}$ in Definition 25 gives procedures access to the known geometry while keeping the section family fixed before the member law is chosen. The signed-distance risk $R_{n}^{12, \pm}$ in Definition 26 measures the expected worst interface error for treatment-effect estimation. The fixed-geometry risk in Definition 27 is the same criterion restricted to the rectangular hard family, which is the family used for the signed-distance converse in Theorem 3.

It remains to name the estimator used for the upper bound. The construction follows the standard local-polynomial template, with three stabilizing features: outcomes are winsorized at a bandwidth-dependent level, the empirical Gram matrix is inverted only when it has a uniform quadratic-form lower bound, and the final treatment-effect estimate is clipped to the natural envelope scale.

Definition 28 [def:cty-stabilized-local-polynomial-estimator] (Stabilized local-polynomial estimator

τ_{n, h}^{12, LP}

For $B \in R$ , define the winsorization map by $ψ_{B} (y) = sign (y) min {∣ y ∣, B} .$ For each $t \in {0, 1}$ , query point $x$ , and bandwidth $h$ , define $Ψ_{t, x} (h) = \frac{1}{n h ^{2}} i = 1 \sum n 1 {D_{P, x, i}^{\pm} \in I_{t}} K_{□} (D_{P, x, i}^{\pm} / h) r_{p} (D_{P, x, i}^{\pm} / h) r_{p} (D_{P, x, i}^{\pm} / h)^{⊤} .$ With $B (h) = h^{- 1/3}$ , define $m_{t, x}^{B} (h) = \frac{1}{n h ^{2}} i = 1 \sum n 1 {D_{P, x, i}^{\pm} \in I_{t}} K_{□} (D_{P, x, i}^{\pm} / h) r_{p} (D_{P, x, i}^{\pm} / h) ψ_{B (h)} (Y_{i}) .$ The stabilized coefficient is $β_{t, x}^{B} (h) = {Ψ_{t, x} (h)^{- 1} m_{t, x}^{B} (h), 0, if \forall v \in R^{p + 1}, (2 L)^{- 1} \sum_{j = 0}^{p} v_{j}^{2} \leq v^{⊤} Ψ_{t, x} (h) v, otherwise.$ With $e_{1} = (1, 0, \dots, 0)^{⊤}$ , the clipped signed-distance local-polynomial treatment-effect estimator is $τ_{n, h}^{12, LP} (x) = max {- 2 L, min {2 L, e_{1}^{⊤} (β_{1, x}^{B} (h) - β_{0, x}^{B} (h))}} .$

⊢ Lean

The winsorization map $ψ_{B}$ in Definition 28 converts the moment envelope into a bounded-score component at bandwidth $h$ . The empirical Gram matrix estimates the population Gram $Ψ_{t, P, x} (h)$ , and the fallback rule keeps the coefficient map stable on samples where the empirical design is ill conditioned. The resulting estimator $τ_{n, h}^{12, LP}$ is the concrete rule used in the signed-distance upper result.

The following three deviation objects separate approximation, design, and score variation. They are stated here because the main theorem collects analytic inputs in terms of these quantities before applying the estimator in Definition 28.

Definition 29 [synth_39] (Uniform first-order bias ratio

BiasRatio_{p, ν^{'}, L} (h_{n})

For $h_{n} > 0$ , define the uniform first-order bias ratio by $BiasRatio_{p, ν^{'}, L} (h_{n}) = P \in P_{12} (p, ν^{'}, L) sup x \in B_{P} sup \frac{e _{1}^{⊤} { β _{1, P, x}^{*} ( h _{n} ) - β _{0, P, x}^{*} ( h _{n} )} - τ _{P} ( x )}{h _{n}} .$

Definition 30 [synth_62] (Uniform empirical Gram deviation

GramDev_{n, p, P} (h_{n})

For a law $P \in P_{12} (p, ν, L)$ and bandwidth $h_{n} > 0$ , we write $GramDev_{n, p, P} (h_{n})$ for the extended nonnegative sample-space random variable $GramDev_{n, p, P} (h_{n}) = t \in {0, 1} sup x \in B_{P} sup 0 \leq j, k \leq p max Ψ_{t, x} (h_{n})_{j k} - Ψ_{t, P, x} (h_{n})_{j k} .$

Definition 31 [synth_63] (Uniform raw-score deviation

ScoreDev_{n, p, P} (h_{n})

Define $ScoreDev_{n, p, P} (h_{n})$ as the sample-dependent uniform centered local-polynomial score deviation $ScoreDev_{n, p, P} (h_{n}) = t \in {0, 1} sup x \in B_{P} sup \frac{1}{n h _{n}^{2}} i = 1 \sum n 1 {D_{P, x, i}^{\pm} \in I_{t}} K_{□} (D_{P, x, i}^{\pm} / h_{n}) r_{p} (D_{P, x, i}^{\pm} / h_{n}) Y_{i} - E_{P} [\frac{1}{h _{n}^{2}} 1 {D_{P, x}^{\pm} \in I_{t}} K_{□} (D_{P, x}^{\pm} / h_{n}) r_{p} (D_{P, x}^{\pm} / h_{n}) Y],$ with the norm taken in $R^{p + 1}$ .

Definition 32 [synth_131] (Winsorized score

score_{t, x, h, j}^{B}

We say $score_{t, x, h, j}^{B}$ is the winsorized local-polynomial residual score when, for $P \in P_{12} (p, ν, L)$ , sample size $n$ , arm $t \in {0, 1}$ , interface point $x \in B_{P}$ , bandwidth $h > 0$ , winsorization level $B > 0$ , and coordinate $j \in {0, \dots, p}$ , it is the $P^{n}$ -random variable $score_{t, x, h, j}^{B} = \frac{1}{n h ^{2}} i = 1 \sum n 1 {D_{P, x, i}^{\pm} \in I_{t}} K_{□} (\frac{D _{P, x, i}^{\pm}}{h}) r_{p} (\frac{D _{P, x, i}^{\pm}}{h})_{j} ψ_{B} (Y_{i}) - r_{p} (\frac{D _{P, x, i}^{\pm}}{h})^{⊤} β_{t, P, x}^{*} (h),$ with $β_{t, P, x}^{*} (h)$ the unwinsorized population local-polynomial coefficient at $(t, x, h)$ .

The bias ratio in Definition 29 measures the uniform first-order approximation error of the population signed-distance fit. The Gram deviation in Definition 30 tracks the uniform empirical conditioning of the local design, and the raw-score deviation in Definition 31 records the centered stochastic part before winsorization. The winsorized score in Definition 32 is the bounded-envelope score component used by the maximal inequality in the upper-bound appendix. These objects place the later risk result in the usual bias, design-stability, and empirical-process decomposition for local-polynomial RD estimators (Fan et al., 1996; Ruppert et al., 1994; Calonico et al., 2014; Calonico et al., 2020; van der Vaart et al., 1996).

Main Results

The results now put the two distance-compressed experiments on a common risk scale. The unsigned problem uses the nonparametric regression class and distance data introduced in Definitions 9 and 10; the signed problem uses the known-geometry law class, signed-distance data, and treatment-effect risk in Definitions 22, 21, and 26. Throughout the section, the normalizing sequence is the rate $a_{n}$ in Definition 8.

The first result is the same-class converse for the unsigned common-map experiment. It evaluates exactly the decision class in Definition 11 under the completed-expectation risk in Definition 12.

Theorem 1 [thm:cty-same-class-log-converse] (Same-Class Log Converse).

For every integer $q \geq 1$ and every $L \geq 4$ , there exists a constant $c_{CTY} = c_{CTY} (q, L) > 0$ such that, for the common-map minimax risk $R_{n}^{NP}$ in Definition 12 and $a_{n} = (lo g n / n)^{1/4}$ , $n \to \infty lim inf (\frac{n}{lo g n})^{1/4} R_{n}^{NP} = n \to \infty lim inf \frac{R _{n}^{NP}}{a _{n}} \geq c_{CTY} .$

⊢ Lean

The theorem establishes that a single law-independent distance rule incurs expected boundary sup-loss at least of logarithmic order $a_{n}$ over $P_{NP} (L, q)$ . The packing calculation behind the rate is easiest to read from the separated boundary construction used in the appendix. If the signal amplitude is $Δ$ , then the construction places $M ≍ Δ^{- 1/ q}$ active locations along the boundary and uses cells with area scale $w^{2} ≍ Δ^{2/ q}$ . Unsigned distance compression leaves each coordinate with information of order $n Δ^{4}$ , so the many-coordinate testing barrier is calibrated by $n Δ^{4} ≍ lo g M, hence Δ ≍ (\frac{lo g n}{n})^{1/4} .$ The smoothness order $q$ affects the packing constants and the geometric calibration of $w$ , while the exponent is governed by the fourth-order distance-compression information calculation. In the signed-distance local-polynomial results below, the polynomial order $p$ likewise affects constants and regularity thresholds while the displayed rate normalization remains $a_{n}$ .

The next result carries the same logarithmic converse to the point-indexed unsigned class. The point-indexed class in Definition 13 permits a separate Borel section at each query point, and the risk is evaluated with outer expectation as in Definition 14.

Theorem 2 [thm:point-indexed-distance-log-converse] (Logarithmic Distance Converse).

For every $q \in N$ and $L \in R$ satisfying $q \geq 1$ and $L \geq 4$ , there exists a constant $c_{PI} = c_{PI} (q, L) > 0$ such that, with $R_{n}^{PI}$ the point-indexed outer-expectation minimax risk in Definition 14 and $a_{n} = (lo g n / n)^{1/4}$ , $n \to \infty lim inf (\frac{n}{lo g n})^{1/4} R_{n}^{PI} = n \to \infty lim inf \frac{R _{n}^{PI}}{a _{n}} \geq c_{PI} .$

⊢ Lean

Thus the lower-bound scale is a property of the unsigned distance information itself, even when the rule may choose its measurable section point by point. Outer expectation in this statement is the same measurable-majorant criterion introduced in Definition 14, which is the standard way to state uniform risks over potentially nonseparable boundary index sets (van der Vaart et al., 1996).

The signed-distance lower bound uses the known-geometry experiment in Definitions 25 and 26. It is stated for all local-polynomial orders after an envelope threshold that depends on the order.

Theorem 3 [thm:cty-a1-a2-point-indexed-log-converse-all-orders] (Point-Indexed Log Converse).

For every nonnegative integer $p$ , there exists a real number $L_{0} = L_{0} (p)$ such that:

(Envelope threshold.) $L_{0} \geq 48$ .
(Moment exponent.) $ν \geq 2$ .
(Uniform envelope.) $L \geq L_{0}$ .
(Risk definition.) $R_{n}^{12, \pm} (p, ν, L)$ is the signed-distance minimax risk in Definition 26.
(Fixed-geometry risk.) $R_{n}^{12, \pm, fix} (p, ν, L)$ is the infimum over the same known-geometry point-indexed decision rules of the same worst-case outer-expected interface sup-loss, with the law supremum further restricted to laws in $P_{12} (p, ν, L)$ satisfying the fixed hard-geometry condition.

For every $ν$ and $L$ satisfying these conditions, there exists a constant $c = c (p, ν, L) > 0$ such that, writing $a_{n} = (lo g n / n)^{1/4}$ , $n \to \infty lim inf a_{n}^{- 1} R_{n}^{12, \pm} (p, ν, L) = n \to \infty lim inf \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}},$ and $c \leq n \to \infty lim inf \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}}, c \leq n \to \infty lim inf \frac{R _{n}^{12, \pm, fix} ( p , ν , L )}{a _{n}} .$

⊢ Lean

Theorem 3 establishes the same logarithmic lower scale for the known-geometry signed-distance minimax risk. The second inequality records that the lower bound is already present inside the fixed rectangular geometry from Definition 27. Consequently, the converse is calibrated by a concrete Euclidean boundary experiment rather than by variation over arbitrary interfaces.

The upper result for the signed-distance experiment combines the estimator in Definition 28 with three analytic inputs. These inputs are distance identification, uniform first-order bias control, and expected maximal inequalities; every signed-distance upper or matched-rate statement below is conditioned on this collection.

Definition 33 [synth_139] (Analytic inputs for the signed-distance upper bound).

Define $AI_{p, ν, L}$ as the following three signed-distance analytic inputs used by the expected-risk upper bound.

(CTY distance identification.) For every A1/A2 law $P$ and metric $d$ , the support is rectangular, the density is continuous and positive on the support, both potential outcomes are integrable, both arm regressions have $C^{p + 1}$ extensions, the boundary is a positive-length rectifiable curve, $d$ is a metric uniformly equivalent to Euclidean distance on the support, and every sufficiently small positive armwise distance slice at every boundary point has finite positive density mass. Under these conditions there is one source-coherent selected conditional law of the observed outcome given signed distance, with armwise mean versions $θ_{1}, θ_{0}$ whose one-sided limits at zero satisfy $τ_{P} (x) = u ↓ 0 lim θ_{1} (u) - u ↑ 0 lim θ_{0} (u) .$
(Uniform first-order bias.) There is $C_{b} > 0$ , depending only on $p, L$ , such that for every $ν^{'} \geq 2$ and every positive antitone deterministic sequence $h_{n} \to 0$ with $n h_{n}^{2} \to \infty$ , $n \to \infty lim sup BiasRatio_{p, ν^{'}, L} (h_{n}) \leq C_{b} .$
(Expected maximal bounds.) There is $C_{m} > 0$ such that, along every positive deterministic $h_{n} \to 0$ satisfying $\frac{n ^{(1 + ν) / (2 + ν)} h _{n}^{2}}{lo g ( h _{n}^{- 1} )} \to \infty,$ all sufficiently large $n$ and every $P \in P_{12} (p, ν, L)$ satisfy $E_{P}^{*} [GramDev_{n, p, P} (h_{n})] \leq C_{m} \frac{lo g ( h _{n}^{- 1} )}{n h _{n}^{2}}$ and $E_{P}^{*} [ScoreDev_{n, p, P} (h_{n})] \leq C_{m} {\frac{lo g ( h _{n}^{- 1} )}{n h _{n}^{2}} + \frac{lo g ( h _{n}^{- 1} )}{n ^{(1 + ν) / (2 + ν)} h _{n}^{2}}} .$

The three components of $AI_{p, ν, L}$ are statistical assumptions with distinct roles. Identification connects the one-sided signed-distance conditional-mean limits to the boundary treatment-effect curve (Cattaneo et al., 2026). Approximation bounds the population local-polynomial intercept error uniformly at first order in the bandwidth (Fan et al., 1996; Ruppert et al., 1994). Stochastic control bounds the expected uniform Gram and score deviations over arms, boundary points, and laws (Vapnik et al., 1971; Pollard, 1984; van der Vaart et al., 1996). The next proposition is conditional on all three displayed inputs and derives the expected-risk rate for the stabilized estimator from them.

Proposition 1 [prop:cty-a1-a2-winsorized-expected-outer-upper] (Expected Outer Upper Bound).

Let $p \geq 0$ be an integer, let $ν \geq 2$ , and let $L \geq 4$ . Assume:

(Distance identification.) For every A1/A2 law $P$ , every distance map $d$ , and every $x \in B_{P}$ satisfying the CTY identification assumptions at order $p$ , there are signed-distance conditional-mean versions $θ_{1}, θ_{0} : R \to R$ and one-sided limits $θ_{1} (0 +)$ and $θ_{0} (0 -)$ such that $τ_{P} (x) = θ_{1} (0 +) - θ_{0} (0 -) .$
(Uniform first-order bias.) There is a constant $C_{b} > 0$ , depending on $p$ and $L$ , such that for every $ν^{'} \geq 2$ and every positive antitone deterministic bandwidth sequence $h_{n}$ with $h_{n} \to 0$ and $n h_{n}^{2} \to \infty$ , $n \to \infty lim sup BiasRatio_{p, ν^{'}, L} (h_{n}) \leq C_{b} .$
(Expected maximal bounds.) There is a constant $C_{m} > 0$ such that, for every positive deterministic bandwidth sequence $h_{n}$ with $h_{n} \to 0$ and $\frac{n ^{(1 + ν) / (2 + ν)} h _{n}^{2}}{lo g ( h _{n}^{- 1} )} \to \infty,$ the following bounds hold for all sufficiently large $n$ , uniformly over $P \in P_{12} (p, ν, L)$ : $E_{P}^{*} [GramDev_{n, p, P} (h_{n})] \leq C_{m} \frac{lo g ( h _{n}^{- 1} )}{n h _{n}^{2}},$ and $E_{P}^{*} [ScoreDev_{n, p, P} (h_{n})] \leq C_{m} {\frac{lo g ( h _{n}^{- 1} )}{n h _{n}^{2}} + \frac{lo g ( h _{n}^{- 1} )}{n ^{(1 + ν) / (2 + ν)} h _{n}^{2}}} .$

With $a_{n} = (lo g n / n)^{1/4}$ as in Definition 8, take the bandwidth $h_{n} = a_{n}$ and the winsorization level $B_{n} = a_{n}^{- 1/3}$ in the winsorized, Gram-stabilized signed-distance local-polynomial estimator clipped to $[- 2 L, 2 L]$ . Then there are constants $C > 0$ and $N$ such that, for every $n \geq N$ , $P \in P_{12} (p, ν, L) sup E_{P}^{*} [x \in B_{P} sup τ_{n, h_{n}}^{12, LP} (x) - τ_{P} (x)] \leq C a_{n} .$

⊢ Lean

The proposition gives a concrete signed-distance procedure with expected outer risk bounded by a constant multiple of $a_{n}$ . The bandwidth choice balances the first-order signed-distance bias, of order $h_{n}$ , with the uniform stochastic scale ${lo g (h_{n}^{- 1}) / (n h_{n}^{2})}^{1/2}$ . At $h_{n} = a_{n}$ , both terms have order $a_{n}$ , and the additional heavy-tail term in the assumed maximal bound is controlled under the displayed moment condition. The winsorized-score lemma used in the appendix supplies the bounded-envelope score component needed for this decomposition. The full proposition additionally uses the assumed uniform Gram control and the assumed raw-score, law-uniform, heavy-tail maximal bound with exponent $n^{(1 + ν) / (2 + ν)}$ , so the displayed expected-risk conclusion is tied to the full collection $AI_{p, ν, L}$ in Definition 33.

Conditional signed-distance frontier.

Under the analytic inputs in Definition 33, the signed-distance lower and upper statements combine to yield the matched outer-expected rate over $P_{12} (p, ν, L)$ .

Theorem 4 [thm:cty-a1-a2-winsorized-matched-frontier] (Matched Frontier Rate).

For every integer $p \geq 0$ , there exists a constant $L_{0} \geq 48$ such that, for every $ν \geq 2$ and every $L \geq L_{0}$ , the following conditions imply the frontier conclusion below:

(Distance identification.) For every A1/A2 law $P$ , every map $d$ satisfying the CTY identification assumptions for $p$ and $P$ , and every $x \in B_{P}$ , there are selected source-coherent signed-distance conditional-mean versions $θ_{1}, θ_{0} : R \to R$ whose right and left limits at zero exist and identify $τ_{P} (x) = u ↓ 0 lim θ_{1} (u) - u ↑ 0 lim θ_{0} (u) .$
(Uniform first-order bias.) There is a constant depending only on $p$ and $L$ such that, uniformly over every $ν^{'} \geq 2$ and every positive antitone deterministic bandwidth sequence $h_{n} \to 0$ with $n h_{n}^{2} \to \infty$ , the finite limsup of the normalized uniform first-order bias ratio is bounded by that constant.
(Expected maximal bounds.) For this $p, ν, L$ , there is a constant $C_{max} > 0$ such that, along every positive bandwidth sequence $h_{n} \to 0$ with $\frac{n ^{(1 + ν) / (2 + ν)} h _{n}^{2}}{lo g ( h _{n}^{- 1} )} \to \infty,$ for all sufficiently large $n$ and every $P \in P_{12} (p, ν, L)$ , the expected outer Gram-deviation supremum satisfies $E_{P}^{*} [t \in {0, 1} sup x \in B_{P} sup j, k sup Ψ_{t, x} (h_{n})_{j k} - Ψ_{t, P, x} (h_{n})_{j k}] \leq C_{max} \frac{lo g ( h _{n}^{- 1} )}{n h _{n}^{2}},$ and the expected outer centered raw-score supremum satisfies $E_{P}^{*} [t \in {0, 1} sup x \in B_{P} sup ∥ m_{t, x} (h_{n}) - m_{t, P, x} (h_{n}) ∥] \leq C_{max} (\frac{lo g ( h _{n}^{- 1} )}{n h _{n}^{2}} + \frac{lo g ( h _{n}^{- 1} )}{n ^{(1 + ν) / (2 + ν)} h _{n}^{2}}) .$

Then there exist constants $c$ and $C$ with $0 < c \leq C$ such that, for the signed-distance minimax risk $R_{n}^{12, \pm} (p, ν, L)$ of Definition 26 and $a_{n} = (lo g n / n)^{1/4}$ , $c \leq n \to \infty lim inf \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}} \leq n \to \infty lim sup \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}} \leq C .$ Moreover, the explicit winsorized, Gram-stabilized degree- $p$ signed-distance local-polynomial estimator with bandwidth $h_{n} = a_{n}$ and winsorization level $B_{n} = a_{n}^{- 1/3}$ , clipped to $[- 2 L, 2 L]$ , has uniform outer risk over $P \in P_{12} (p, ν, L)$ satisfying $n \to \infty lim sup \frac{sup _{P \in P_{12} (p, ν, L)} E _{P}^{*} [ sup _{x \in B_{P}} τ _{n, a_{n}}^{12, LP} ( x ) - τ _{P} ( x ) ]}{a _{n}} \leq C .$

⊢ Lean

Under the analytic inputs in Definition 33, Theorem 4 characterizes the signed-distance known-geometry minimax risk up to constants under the same envelope threshold as the lower bound. The lower inequality is supplied by Theorem 3, while the upper inequality is attained by the explicit winsorized, Gram-stabilized local-polynomial estimator from Definition 28. The result places the signed-distance experiment on the logarithmic expected outer-risk scale $a_{n}$ for every fixed polynomial order $p$ , moment exponent $ν \geq 2$ , and envelope $L$ above the stated order-dependent threshold, with the upper side governed by the three analytic inputs.

Discussion and Extensions

The preceding results separate two roles of distance in boundary regression designs. For the unsigned experiment, Theorems 1 and 2 establish logarithmic lower bounds for boundary sup-loss when the rule observes only outcomes and scalar Euclidean distances from the query point. The common-map statement applies to the decision class and completed-expectation risk in Definitions 11 and 12, while the point-indexed statement applies to the sectionwise Borel class and outer-expectation risk in Definitions 13 and 14. These conclusions identify an information barrier created by unsigned distance compression over the compact bivariate law class in Definition 9.

The signed-distance experiment has a different information set. There the rule observes signed distances generated by the known geometry in Definitions 20 and 21 and is evaluated over the A1/A2 law class and risk in Definitions 22 and 26. Assuming the analytic inputs collected in Definition 33, Proposition 1 and Theorem 4 combine the signed-distance lower bound with the stabilized local-polynomial upper argument to obtain a two-sided expected outer-risk rate. The role of Cattaneo et al. (2026) is therefore most direct in the signed design: their distance-identification framework supplies the population interpretation of the one-dimensional signed-distance coordinate, while the conditional result here packages the displayed inputs with the new lower bound and stabilized estimator.

The common normalization $a_{n} = (lo g n / n)^{1/4}$ in Definition 8 indexes different delivered objects across the two experiments. The unsigned experiment has unconditional common-map and point-indexed lower bounds: many separated boundary perturbations remain difficult because scalar distance hides their angular locations and treatment sides. The signed experiment has an unconditional fixed-geometry lower bound and, under the three inputs in Definition 33, a conditional upper bound attained by a procedure that uses the treatment side, known boundary geometry, and local-polynomial structure. Accordingly, the shared normalization compares lower scales; a two-sided rate conclusion is available here for the signed experiment conditional on $AI_{p, ν, L}$ .

The signed-distance result is stated for the known Euclidean geometry in Definition 20 and the one-dimensional uniform kernel in Definition 19. The law class in Definition 22 imposes rectangular support, bounded continuous density, smooth potential-outcome regressions, continuous variances, a $2 + ν$ moment envelope, rectifiable interior interface, two-sided small-bandwidth mass, and a population Gram lower bound. These conditions describe the boundary designs for which the signed-distance local-polynomial estimator has uniformly controlled bias, stable local design matrices, and expected stochastic fluctuations. They also align with the geographic RD setting in which boundary location and treatment side are part of the design information (Keele et al., 2015; Keele et al., 2015; Keele et al., 2016; Kendall et al., 2026).

The Euclidean and envelope restrictions also clarify the intended empirical interpretation. Geographic RD applications often use distance to a boundary as a practical running coordinate while retaining design knowledge about assignment regions and local geography (Papay et al., 2011; Reardon et al., 2012; Wong et al., 2013; Choi et al., 2018; Díaz et al., 2023; Sawada et al., 2024; Merlano, 2025). The signed-distance analysis formalizes that information set through the known-geometry class, whereas the unsigned analysis formalizes a more compressed experiment in which angular and side information are absent from the observation available at a query point. The results therefore organize comparisons by the data transformation and target risk, rather than by a universal ranking of distance-based empirical strategies.

Open questions.

The unsigned results in Theorems 1 and 2 are lower bounds only and establish no matching unsigned upper bound here. The signed-distance matched rate in Theorem 4 is conditional on $AI_{p, ν, L}$ , so a fully self-contained derivation of the signed-distance upper ingredients within the same displayed law class remains an open direction. Further extensions include non-Euclidean metrics, kernels beyond $K_{□}$ , alternative interface regularity classes, and risk criteria adapted to inference rather than uniform expected loss.

Appendices

Direct-Product and Support-Boundary Lower Bound

This appendix records the finite-experiment ingredients used to derive the unsigned logarithmic lower bounds. The argument has two components. The first is an overlap product inequality for many binary coordinates after each coordinate has been compressed. The second is a support-boundary hypercube inside the nonparametric class of Definition 9, constructed so that the boundary regression values differ at many separated points while the corresponding unsigned distance experiments remain close coordinate by coordinate. These ingredients yield the common-map converse in Theorem 1; after the strict inclusion statement below, the same construction yields the point-indexed converse in Theorem 2.

The direct-product step is stated abstractly because the same form is also useful for the signed-distance lower experiment. It converts many small coordinatewise overlaps into a lower bound on the probability of at least one decoding error.

Lemma 1 [lem:coordinatewise-overlap-direct-product] (Coordinatewise overlap product).

Fix an integer $M \geq 1$ . Assume the following conditions.

(Spaces.) The spaces carrying $Z_{0}, Z_{1}, \dots, Z_{M}, S_{1}, \dots, S_{M}$ are standard Borel.
(Binary coordinates.) The variables $Ω_{1}, \dots, Ω_{M}$ are independent and uniformly distributed on ${0, 1}$ .
(Conditional product structure.) Conditionally on $(Ω_{1}, \dots, Ω_{M})$ , the variables $Z_{0}, Z_{1}, \dots, Z_{M}$ are independent, the law of $Z_{j}$ depends only on $Ω_{j}$ for $1 \leq j \leq M$ , and the law of $Z_{0}$ is independent of $(Ω_{1}, \dots, Ω_{M})$ .
(Compressed coordinate decoding.) For each $j$ , $S_{j} = c_{j} (Z_{j})$ for a measurable map $c_{j}$ . The decoder $Ω_{j}$ is a measurable ${0, 1}$ -valued function of $(S_{j}, (Z_{1}, \dots, Z_{M}), Z_{0})$ and is unchanged when its raw coordinate vector is altered only in coordinate $j$ while $(S_{j}, Z_{- j}, Z_{0})$ is fixed. Define $Q_{j, b} = L (S_{j} ∣ Ω_{j} = b), ρ_{j} = 1 - ∥ Q_{j, 0} - Q_{j, 1} ∥_{TV} .$

Then every such decentralized decoder satisfies $Pr {Ω_{j} = Ω_{j} for every j} \leq \frac{1}{2} {1 + j = 1 \prod M (1 - ρ_{j})} .$ Moreover, for every real $κ < 1$ , if $KL (Q_{j, 0}, Q_{j, 1}) \leq max {κ lo g M, 0} for all j,$ then $Pr {at least one decoding error} \geq \frac{1}{2} [1 - exp {- M^{1 - κ} /2}] .$

Proof of Lemma 1.

Write $S_{j}$ for the measurable space carrying $S_{j}$ , and let $g_{j}$ be the measurable map representing the $j$ th decoder, so that $Ω_{j} = g_{j} (S_{j}, (Z_{1}, \dots, Z_{M}), Z_{0}),$ and so that the invariance hypothesis reads: for every $s \in S_{j}$ , every $z_{0}$ , and all raw vectors $z, z^{'}$ with $z_{k} = z_{k}^{'}$ for every $k \neq = j$ , $g_{j} (s, z, z_{0}) = g_{j} (s, z^{'}, z_{0}) .$ The two laws compared at coordinate $j$ are the compressed conditional laws $Q_{j, b} = L (S_{j} ∣ Ω_{j} = b)$ of Definition 1, and $ρ_{j} = 1 - ∥ Q_{j, 0} - Q_{j, 1} ∥_{TV}$ with the total variation distance of Definition 7. Since $0 \leq ∥ Q_{j, 0} - Q_{j, 1} ∥_{TV} \leq 1$ , we have $0 \leq ρ_{j} \leq 1$ for every $j$ .

Step 1: a coordinatewise coupling whose compressions agree with probability $ρ_{j}$ . The raw bit-conditional law at coordinate $j$ is $L (Z_{j} ∣ Ω_{j} = b)$ , and $Q_{j, b}$ is its image under $c_{j}$ . Let $K_{j}$ be the common part of $Q_{j, 0}$ and $Q_{j, 1}$ , that is, the measure on $S_{j}$ with density $\frac{d K _{j}}{d ( Q _{j, 0} + Q _{j, 1} )} = min {\frac{d Q _{j, 0}}{d ( Q _{j, 0} + Q _{j, 1} )}, \frac{d Q _{j, 1}}{d ( Q _{j, 0} + Q _{j, 1} )}}$ with respect to $Q_{j, 0} + Q_{j, 1}$ . Its total mass is $K_{j} (S_{j}) = 1 - ∥ Q_{j, 0} - Q_{j, 1} ∥_{TV} = ρ_{j},$ because $min {a, b} = {a + b - ∣ a - b ∣} /2$ , both densities integrate to one, and half the $L^{1}$ distance between them is the total variation distance.

Each residual $Q_{j, 0} - K_{j}$ and $Q_{j, 1} - K_{j}$ then has total mass $1 - ρ_{j}$ . When $ρ_{j} < 1$ , the maximal coupling of $Q_{j, 0}$ and $Q_{j, 1}$ is the law on $S_{j} \times S_{j}$ given by $(image of K_{j} under s \mapsto (s, s)) + (1 - ρ_{j})^{- 1} {(Q_{j, 0} - K_{j}) \times (Q_{j, 1} - K_{j})},$ and when $ρ_{j} = 1$ it is the image of $Q_{j, 0}$ under $s \mapsto (s, s)$ . In both cases its two marginals are $Q_{j, 0}$ and $Q_{j, 1}$ , and its diagonal carries mass at least $K_{j} (S_{j}) = ρ_{j}$ .

Because all the spaces are standard Borel, the regular conditional distributions of $Z_{j}$ given $c_{j} (Z_{j})$ exist under both raw bit-conditional laws; feeding the two coordinates of the maximal coupling into them produces a law $C_{j}$ on pairs of raw coordinate blocks, whose coordinates we write $(Z_{j}^{0}, Z_{j}^{1})$ , such that $L (Z_{j}^{0}) = L (Z_{j} ∣ Ω_{j} = 0), L (Z_{j}^{1}) = L (Z_{j} ∣ Ω_{j} = 1), C_{j} {c_{j} (Z_{j}^{0}) = c_{j} (Z_{j}^{1})} \geq ρ_{j} .$ The first two equalities hold because the lift preserves each marginal, and the third because the image of $C_{j}$ under $(z, z^{'}) \mapsto (c_{j} (z), c_{j} (z^{'}))$ is exactly the maximal coupling above, whose diagonal has mass at least $ρ_{j}$ .

Step 2: the coupled experiment reproduces the original one at every vertex. Let $C$ be the product of the coordinatewise couplings, that is, the law of the array $(Z_{j}^{0}, Z_{j}^{1})_{1 \leq j \leq M}$ determined by $C {(Z_{1}^{0}, Z_{1}^{1}) \in A_{1}, \dots, (Z_{M}^{0}, Z_{M}^{1}) \in A_{M}} = j = 1 \prod M C_{j} (A_{j}),$ and let $R = L (Z_{0})$ , which by the conditional product structure does not depend on $(Ω_{1}, \dots, Ω_{M})$ . Write $R \times C$ for the law under which $Z_{0}$ has law $R$ and is independent of the array. For a vertex $ω \in {0, 1}^{M}$ define the selected raw vector by $Z_{j} (ω) = Z_{j}^{ω_{j}}, 1 \leq j \leq M .$ Since the two marginals of $C_{j}$ are the two raw bit-conditional laws at coordinate $j$ , the selection map pushes $C$ forward to the product over $j$ of $L (Z_{j} ∣ Ω_{j} = ω_{j})$ , so that $L (Z_{0}, (Z_{1} (ω), \dots, Z_{M} (ω))) = L (Z_{0}, (Z_{1}, \dots, Z_{M}) ∣ Ω = ω),$ which is precisely the conditional product law assumed in the statement.

Consequently, writing $G (ω) = {g_{j} (c_{j} (Z_{j} (ω)), (Z_{1} (ω), \dots, Z_{M} (ω)), Z_{0}) = ω_{j} for every 1 \leq j \leq M}$ for the simultaneous-correctness event at the vertex $ω$ , the uniform hypercube prior gives $Pr {Ω_{j} = Ω_{j} for every j} = 2^{- M} ω \in {0, 1}^{M} \sum (R \times C) (G (ω)) .$

Step 3: the total-disagreement event. Let $N = {c_{j} (Z_{j}^{0}) \neq = c_{j} (Z_{j}^{1}) for every 1 \leq j \leq M}$ be the event that every coupled coordinate has unequal compressed values. It is a product event across coordinates, so Step 1 gives $C (N) = j = 1 \prod M C_{j} {c_{j} (Z_{j}^{0}) \neq = c_{j} (Z_{j}^{1})} \leq j = 1 \prod M ∥ Q_{j, 0} - Q_{j, 1} ∥_{TV} = j = 1 \prod M (1 - ρ_{j}),$ each factor being the complement of an event of probability at least $ρ_{j}$ .

Step 4: a pointwise bound on the number of correctly decoded vertices. Fix one realization of $Z_{0}$ and of the coupled array $(Z_{j}^{0}, Z_{j}^{1})_{j \leq M}$ . We claim $ω \in {0, 1}^{M} \sum 1 {G (ω)} \leq \frac{2 ^{M}}{2} (1 + 1_{N}) .$ On $N$ this is immediate: each of the $2^{M}$ indicators is at most one, and the right-hand side equals $2^{M}$ .

Off $N$ there is a coordinate $j$ with $c_{j} (Z_{j}^{0}) = c_{j} (Z_{j}^{1})$ . Let $ω^{(j)}$ be $ω$ with coordinate $j$ flipped. Selection at a coordinate $k \neq = j$ reads the same bit under $ω$ and $ω^{(j)}$ , while at coordinate $j$ the two selected blocks have equal compressions by the choice of $j$ ; hence $Z_{k} (ω) = Z_{k} (ω^{(j)}) (k \neq = j), c_{j} (Z_{j} (ω)) = c_{j} (Z_{j} (ω^{(j)})) .$ The invariance hypothesis applied to these two raw vectors therefore gives $g_{j} (c_{j} (Z_{j} (ω)), (Z_{k} (ω))_{k \leq M}, Z_{0}) = g_{j} (c_{j} (Z_{j} (ω^{(j)})), (Z_{k} (ω^{(j)}))_{k \leq M}, Z_{0}),$ so this single value cannot equal both $ω_{j}$ and $ω_{j}^{(j)} = 1 - ω_{j}$ , and consequently $1 {G (ω)} + 1 {G (ω^{(j)})} \leq 1 for every ω \in {0, 1}^{M} .$

Summing this over all $ω$ and using that $ω \mapsto ω^{(j)}$ is an involution of ${0, 1}^{M}$ , hence a bijection under which the sum is unchanged, $2 ω \sum 1 {G (ω)} = ω \sum (1 {G (ω)} + 1 {G (ω^{(j)})}) \leq ω \sum 1 = 2^{M},$ that is, $\sum_{ω} 1 {G (ω)} \leq 2^{M} /2$ , which is the claim off $N$ .

Step 5: the simultaneous success bound. Integrating the pointwise bound of Step 4 against $R \times C$ and using Step 3, $ω \in {0, 1}^{M} \sum (R \times C) (G (ω)) = \int ω \sum 1 {G (ω)} d (R \times C) \leq \frac{2 ^{M}}{2} {1 + C (N)} \leq \frac{2 ^{M}}{2} {1 + j = 1 \prod M (1 - ρ_{j})} .$ Dividing by $2^{M}$ and inserting the identity of Step 2 gives the first assertion, $Pr {Ω_{j} = Ω_{j} for every j} \leq \frac{1}{2} {1 + j = 1 \prod M (1 - ρ_{j})} .$

Step 6: calibrating the product by the Kullback–Leibler budget. Fix a real $κ < 1$ and assume $KL (Q_{j, 0}, Q_{j, 1}) \leq max {κ lo g M, 0} for every j .$ Since $M \geq 1$ we have $lo g M \geq 0$ , and we split on the sign of $κ$ .

If $κ \geq 0$ , then $κ lo g M \geq 0$ , so $max {κ lo g M, 0} = κ lo g M$ and the budget reads $KL (Q_{j, 0}, Q_{j, 1}) \leq κ lo g M$ . The Bretagnolle–Huber overlap bound applied to the pair $(Q_{j, 0}, Q_{j, 1})$ states $\frac{1}{2} exp {- KL (Q_{j, 0}, Q_{j, 1})} \leq 1 - ∥ Q_{j, 0} - Q_{j, 1} ∥_{TV} = ρ_{j},$ so monotonicity of $t \mapsto exp (- t)$ together with the budget gives $ρ_{j} \geq \frac{1}{2} exp {- κ lo g M} for every 1 \leq j \leq M .$

Every factor $1 - ρ_{j}$ is nonnegative, so the elementary inequality $1 - t \leq exp (- t)$ applied coordinatewise and then the common floor just obtained yield $j = 1 \prod M (1 - ρ_{j}) \leq exp {- j = 1 \sum M ρ_{j}} \leq exp {- M \cdot \frac{1}{2} exp (- κ lo g M)} = exp {- M^{1 - κ} /2},$ the final equality because $M exp (- κ lo g M) = M^{1 - κ}$ .

If $κ < 0$ , then $κ lo g M \leq 0$ , so $max {κ lo g M, 0} = 0$ and the budget forces $KL (Q_{j, 0}, Q_{j, 1}) = 0$ , hence $Q_{j, 0} = Q_{j, 1}$ and $∥ Q_{j, 0} - Q_{j, 1} ∥_{TV} = 0$ , that is, $ρ_{j} = 1$ for every $j$ . Since $M \geq 1$ , the product below has at least one factor, so $j = 1 \prod M (1 - ρ_{j}) = 0 \leq exp {- M^{1 - κ} /2} .$

Step 7: passing to the error probability. Write $p = \prod_{j = 1}^{M} (1 - ρ_{j})$ and $e = exp {- M^{1 - κ} /2}$ . Then $0 \leq p$ , because every factor is a total variation distance; $e \leq 1$ , because $M^{1 - κ} \geq 0$ ; and $p \leq e$ by Step 6. Hence $\frac{1}{2} (1 - e) + \frac{1}{2} (1 + p) = 1 + \frac{p - e}{2} \leq 1,$ so subtracting the success bound of Step 5 from one gives $Pr {at least one decoding error} = 1 - Pr {Ω_{j} = Ω_{j} for every j} \geq \frac{1}{2} [1 - exp {- M^{1 - κ} /2}] .$

Finally, along any sequence $M_{n} \to \infty$ with $κ < 1$ fixed, $M_{n}^{1 - κ} \to \infty$ and hence $exp {- M_{n}^{1 - κ} /2} \to 0$ , so the displayed lower bound is $1/2 - o (1)$ .

∎

The invariance requirement in Lemma 1 matches the distance-compressed experiment: when a rule estimates the value attached to coordinate $j$ , the information about the $j$ th perturbation that remains after compression is summarized by $S_{j}$ , while the remaining raw coordinates and the background block may be held fixed. The total-variation overlap coefficient measures how often the compressed coordinate laws can be coupled to agree. The Kullback–Leibler consequence gives the logarithmic calibration used in the boundary packing, in the same testing spirit as standard minimax lower-bound arguments (Stone, 1982; Tsybakov, 2009).

The next statement supplies the geometric and probabilistic certificate for the unsigned law class. It places many separated locations on the boundary of a square support and assigns one binary regression perturbation to each local half-disc.

Lemma 2 [lem:cty-support-boundary-angular-packing] (Angular Packing Certificate).

For every integer $q \geq 1$ and every envelope $L \geq 4$ , there exist constants $c_{0}, c_{1}, c_{w}^{-}, c_{w}^{+}, c_{rad}, α > 0, α < \frac{1}{8},$ such that, for all sufficiently large $n$ , writing $a_{n} = (lo g n / n)^{1/4}$ , there are an integer $M_{n}$ , a radius $w_{n}$ , a mass $m_{n}$ , boundary points $x_{1}, \dots, x_{M_{n}}$ , laws $P_{ω}$ indexed by $ω \in {0, 1}^{M_{n}}$ , and values $u_{j, 0}, u_{j, 1}$ such that:

(Packing size and scale.) $c_{0} a_{n}^{- 1/ q} \leq M_{n}, c_{w}^{-} a_{n}^{1/ q} \leq w_{n} \leq c_{w}^{+} a_{n}^{1/ q} .$
(Boundary packing.) With $S = [- 1/2, 1/2]^{2}$ , each $x_{j}$ lies on $bd (S)$ , and for $i \neq = j$ , $∥ x_{i} - x_{j} ∥_{2} \geq c_{w}^{-} a_{n}^{1/ q} .$ The cells $C_{j} = B_{2} (x_{j}, w_{n}) \cap S$ are pairwise disjoint.
(Model class and support.) For every $ω \in {0, 1}^{M_{n}}$ , the law $P_{ω}$ belongs to $P_{NP} (L, q)$ and has covariate support $S$ .
(Cell masses.) For every $ω$ and every $j$ , $P_{ω} (X \in C_{j}) = m_{n} .$
(Local dependence on one bit.) If $ω_{j} = ω_{j}^{'}$ , then the restricted laws of $(Y, X)$ on the cell $C_{j}$ agree under $P_{ω}$ and $P_{ω^{'}}$ . The restricted laws of $(Y, X)$ on $S ∖ ⋃_{j = 1}^{M_{n}} C_{j}$ are the same for all $ω$ .
(Boundary signal.) For every $ω$ and $j$ , $μ_{P_{ω}} (x_{j}) = u_{j, ω_{j}}, ∣ u_{j, 1} - u_{j, 0} ∣ \geq c_{1} a_{n} .$
(Radial invariance.) Let $ω^{(j)}$ be $ω$ with coordinate $j$ flipped. For every $ω$ and $j$ , the laws of $∥ X - x_{j} ∥_{2}$ under $P_{ω}$ and $P_{ω^{(j)}}$ agree. Moreover, the corresponding one-point distance laws agree after restriction to radii $z_{2} \geq c_{rad} a_{n}$ .
(Compressed-distance KL bound.) For every $ω$ and $j$ , $KL (L_{P_{ω}} {V_{n, P_{ω}} (x_{j})}, L_{P_{ω^{(j)}}} {V_{n, P_{ω^{(j)}}} (x_{j})}) \leq α lo g M_{n} .$

⊢ Lean

Proof of Lemma 2.

Fix $q \geq 1$ and $L \geq 4$ , and write $S = [- 1/2, 1/2]^{2}$ .

Step 1: the fixed profiles. Let $φ : R^{2} \to [0, 1]$ be the normalized radial bump, that is, the smooth radially symmetric function with $φ (0) = 1, 0 \leq φ \leq 1, φ (z) = 0 whenever ∥ z ∥_{2} \geq 1,$ and let $Ψ : R \to [0, 1]$ be the smooth transition function, which vanishes on $(- \infty, 0]$ and equals one on $[1, \infty)$ . For each integer $j \geq 0$ , fix a nonnegative derivative envelope $b_{j}$ satisfying $∥ D^{j} φ (z)∥ \leq b_{j} for every z \in R^{2} .$ These envelopes are finite fixed constants attached to the bump profile.

Step 2: the constants. Define the smoothness-dependent derivative scale and the one-observation divergence constant by $D_{q} = 1 + q + 1310720 + j = 0 \sum q b_{j}, C_{KL} = 1310720,$ and set $c_{0} = \frac{1}{24}, c_{1} = \frac{1}{1024 D _{q}}, c_{w}^{-} = c_{w}^{+} = 1, c_{rad} = \frac{1}{8}, α = \frac{1}{16} .$ Since every $b_{j} \geq 0$ , the scale $D_{q}$ is finite and satisfies $D_{q} \geq 1, D_{q} \geq q, D_{q} \geq C_{KL} .$ Hence all six constants are positive and $α = 1/16 < 1/8$ .

The three displayed inequalities give the divergence budget used below: $256 q C_{KL} \leq 256 D_{q}^{2} \leq (1024 D_{q})^{4},$ the first step because $q \leq D_{q}$ and $C_{KL} \leq D_{q}$ , the second because $D_{q} \geq 1$ .

Step 3: the eventual regime. With $a_{n} = (lo g n / n)^{1/4}$ as in Definition 8, set $w_{n} = a_{n}^{1/ q}, δ_{n} = \frac{a _{n}}{1024 D _{q}}, M_{n} = ⌊ \frac{1}{12 w _{n}} ⌋ .$

Because $a_{n} \to 0$ , for all sufficiently large $n$ we have $n \geq 2$ , $a_{n} \leq 1$ , and $0 < w_{n} \leq \frac{1}{24}, 0 < δ_{n} \leq \frac{1}{8} .$

Moreover, since $n a_{n}^{4} = lo g n$ , $n C_{KL} δ_{n}^{4} = C_{KL} \frac{lo g n}{( 1024 D _{q} ) ^{4}} \leq \frac{lo g n}{256 q} by the budget of Step 2,$ while $M_{n} \geq 1/ (24 w_{n}) = \frac{1}{24} a_{n}^{- 1/ q}$ gives $lo g M_{n} \geq \frac{1}{4 q} lo g n - o (lo g n)$ ; hence, enlarging $n$ once more, $n C_{KL} δ_{n}^{4} \leq α lo g M_{n} .$

Finally, for every $r \geq c_{rad} a_{n} = a_{n} /8$ , $2 \cdot (8 δ_{n}) \leq \frac{r}{8},$ because $D_{q} \geq 1$ makes $16 δ_{n} = a_{n} / (64 D_{q}) \leq a_{n} /64 \leq r /8$ .

Fix any $n$ in this eventual regime; all remaining claims are verified at that $n$ .

Step 4: the packing points and cells. Take the equispaced grid on the middle half of the lower edge of $S$ , $x_{j} = (- \frac{1}{4} + \frac{j}{2 ( M _{n} + 1 )}, - \frac{1}{2}), j = 1, \dots, M_{n},$ and let $C_{j} = B_{2} (x_{j}, w_{n}) \cap S$ as in the statement, with $B_{2}$ the closed Euclidean ball of Definition 3. Each $x_{j}$ has second coordinate $- 1/2$ and first coordinate of modulus strictly below $1/4$ , so $x_{j} \in bd (S)$ ; and since $w_{n} \leq 1/24$ , each $C_{j}$ is the closed upper half-disc ${z \in B_{2} (x_{j}, w_{n}) : z^{(2)} \geq - 1/2}$ , where $z = (z^{(1)}, z^{(2)})$ .

From the definition of $M_{n}$ , $M_{n} \leq 1/ (12 w_{n})$ . The small-radius bound $w_{n} \leq 1/24$ also gives $1 \leq 1/ (12 w_{n})$ , and therefore $M_{n} + 1 \leq \frac{1}{12 w _{n}} + \frac{1}{12 w _{n}} = \frac{1}{6 w _{n}} .$ Thus the grid spacing obeys $3 w_{n} \leq \frac{1}{2 ( M _{n} + 1 )},$ so that for $i \neq = j$ $∥ x_{i} - x_{j} ∥_{2} \geq 3 w_{n} \geq w_{n} = c_{w}^{-} a_{n}^{1/ q},$ and the closed cells $C_{i}, C_{j}$ , of radius $w_{n}$ around centers at distance at least $3 w_{n}$ , are disjoint.

The radius bounds $c_{w}^{-} a_{n}^{1/ q} \leq w_{n} \leq c_{w}^{+} a_{n}^{1/ q}$ hold with equality since $c_{w}^{\pm} = 1$ . For the packing size, $w_{n} \leq 1/24$ forces $1/ (12 w_{n}) \geq 2$ , and $⌊ u ⌋ \geq u /2$ for $u \geq 2$ , so $M_{n} \geq \frac{1}{24 w _{n}} = \frac{1}{24} a_{n}^{- 1/ q} = c_{0} a_{n}^{- 1/ q} .$

Step 5: the laws $P_{ω}$ . Fix $ω \in {0, 1}^{M_{n}}$ . Define the regression profile on $R^{2}$ by $μ_{ω} (x) = \frac{1}{2} + \frac{x ^{(1)}}{8} + j = 1 \sum M_{n} ω_{j} δ_{n} φ (\frac{x - x _{j}}{w _{n}}),$

and the design density on $S$ by $f_{ω} (x) = 1 + j = 1 \sum M_{n} ω_{j} t_{n} (∥ x - x_{j} ∥_{2}) κ_{j} (x), κ_{j} (x) = ⎩ ⎨ ⎧ \frac{x ^{(1)} - x _{j}^{(1)}}{∥ x - x _{j} ∥ _{2}}, 0, x \neq = x_{j}, x = x_{j},$ where the radial tilt is $t_{n} (r) = - \frac{2 δ _{n} φ ( \frac{r}{w _{n}} e _{\circ} ) χ _{n} ( r )}{max { r /8 , 8 δ _{n} }}, χ_{n} (r) = Ψ (\frac{r}{64 δ _{n}} - 1),$ and $e_{\circ}$ is a unit vector of $R^{2}$ (the bump is radial, so the choice is immaterial).

Let $P_{ω}$ be the joint law on observation coordinates obtained as follows. Its $X$ -marginal is the measure with Lebesgue density $f_{ω} 1_{S}$ , and at score $x$ the outcome kernel is the probability measure $μ_{ω} (x) N (1, 1) + {1 - μ_{ω} (x)} N (0, 1) .$ Equivalently, $P_{ω}$ is the image under $(x, y) \mapsto (y, x)$ of the product-kernel measure generated by this $X$ -marginal and this outcome kernel. The declared regression and variance profiles of the resulting law are $μ_{P_{ω}} (x) = μ_{ω} (x), σ_{P_{ω}}^{2} (x) = 1 + μ_{ω} (x) {1 - μ_{ω} (x)} .$ They are $X$ -marginal-almost-everywhere versions of the conditional mean and variance in the sense of Definition 9.

Since $∣ x^{(1)} ∣ \leq 1/2$ on $S$ , the bumps have disjoint supports $B_{2} (x_{j}, w_{n})$ , and $δ_{n} \leq 1/8$ , $\frac{1}{4} \leq \frac{1}{2} - \frac{1}{16} \leq μ_{ω} (x) \leq \frac{1}{2} + \frac{1}{16} + δ_{n} \leq \frac{3}{4} for x \in S,$ so the mixture weight is legitimate and the clipping of $μ_{ω}$ to $[0, 1]$ needed off $S$ , and to $[1/4, 3/4]$ used in Step 10, are both silent on $S$ .

Step 6: model class, support, and cell masses. The angular tilt obeys $∣ t_{n} (r) κ_{j} (x) ∣ \leq 1/4$ uniformly, and the bumps attached to distinct centers have disjoint supports, so $\frac{3}{4} \leq f_{ω} (x) \leq \frac{5}{4} (x \in S),$ whence $L^{- 1} \leq f_{ω} \leq L$ for $L \geq 4$ ; $f_{ω}$ is continuous on $S$ , and $S$ is compact with $S \subseteq [- L, L]^{2}$ and with $bd (S)$ Lipschitz-parametrized by $[0, 1]$ . The conditional variance $1 + μ_{ω} (1 - μ_{ω})$ is continuous and lies in $[1, 5/4] \subseteq [L^{- 1}, L]$ . Finally, for $0 \leq j \leq q$ , $δ_{n} w_{n}^{- j} b_{j} = (a_{n} w_{n}^{- j}) \cdot \frac{b _{j}}{1024 D _{q}} = w_{n}^{q - j} \cdot \frac{b _{j}}{1024 D _{q}} \leq 1,$ using $w_{n}^{q} = a_{n}$ , $w_{n} \leq 1$ and $b_{j} \leq D_{q}$ ; since the $j$ th derivative of the localized bump $δ_{n} φ ((\cdot - x_{j}) / w_{n})$ is bounded by $δ_{n} w_{n}^{- j} b_{j}$ , these bounds give $∥ D^{j} μ_{ω} ∥ \leq L$ on $S$ for $0 \leq j \leq q - 1$ together with $L$ -Lipschitz continuity of $D^{q - 1} μ_{ω}$ , that is $μ_{ω} \in H^{q} (S, L)$ in the sense of Definition 4. Consequently $P_{ω} \in P_{NP} (L, q)$ as in Definition 9, with $X_{P_{ω}} = S$ .

For the cell masses, the only $ω$ -dependent part of $f_{ω}$ on $C_{j}$ is $ω_{j} t_{n} (∥ x - x_{j} ∥_{2}) κ_{j} (x)$ , and reflecting $C_{j}$ in the vertical line through $x_{j}$ changes the sign of $κ_{j}$ while fixing $∥ x - x_{j} ∥_{2}$ ; hence that term integrates to zero over $C_{j}$ and $P_{ω} (X \in C_{j}) = Leb (C_{j}) = m_{n},$ where $m_{n} = Leb (C_{1})$ is common to all $j$ because the cells are translates of each other along the lower edge.

Step 7: locality in one bit. The functions $φ ((\cdot - x_{i}) / w_{n})$ and $t_{n} (∥ \cdot - x_{i} ∥_{2}) κ_{i} (\cdot)$ vanish outside $B_{2} (x_{i}, w_{n})$ , and the cells are pairwise disjoint. Therefore, on $C_{j}$ both $f_{ω}$ and $μ_{ω}$ depend on $ω$ only through $ω_{j}$ , so $ω_{j} = ω_{j}^{'} ⟹ P_{ω}_{{X \in C_{j}}} = P_{ω^{'}}_{{X \in C_{j}}},$ and on $S ∖ ⋃_{j} C_{j}$ we have $f_{ω} \equiv 1$ and $μ_{ω} (x) \equiv \frac{1}{2} + \frac{x ^{(1)}}{8}$ , so $P_{ω}_{{X \in S ∖ ⋃_{j} C_{j}}} = P_{ω^{'}}_{{X \in S ∖ ⋃_{j} C_{j}}} for all ω, ω^{'} .$

Step 8: boundary signal. Put $u_{j, 0} = \frac{1}{2} + \frac{x _{j}^{(1)}}{8}, u_{j, 1} = u_{j, 0} + δ_{n} .$ At $x_{j}$ every bump attached to another center vanishes, because $∥ x_{j} - x_{i} ∥_{2} \geq 3 w_{n} > w_{n}$ for $i \neq = j$ , while $φ (0) = 1$ ; hence $μ_{P_{ω}} (x_{j}) = u_{j, ω_{j}}, ∣ u_{j, 1} - u_{j, 0} ∣ = δ_{n} = c_{1} a_{n} .$

Step 9: radial invariance. Let $ω^{(j)}$ be $ω$ with coordinate $j$ flipped. For every measurable $A \subseteq [0, \infty)$ , the reflection of Step 6 applied to the annular slice ${x : ∥ x - x_{j} ∥_{2} \in A}$ again cancels the $ω_{j}$ -dependent density term, so $L_{P_{ω}} (∥ X - x_{j} ∥_{2}) = L_{P_{ω^{(j)}}} (∥ X - x_{j} ∥_{2}) .$

For the tail statement, fix $r \geq c_{rad} a_{n}$ . The last display of Step 3 gives $16 δ_{n} \leq r /8$ , so $r / (64 δ_{n}) - 1 \geq 1$ and $r /8 \geq 8 δ_{n}$ ; therefore the cutoff is fully active, $χ_{n} (r) = 1, t_{n} (r) = - \frac{16 δ _{n} φ ( \frac{r}{w _{n}} e _{\circ} )}{r} .$

Let $A \subseteq [c_{rad} a_{n}, \infty)$ be measurable and let $D_{j, A} = {x \in B_{2} (x_{j}, w_{n}) : x^{(2)} > - \frac{1}{2}, ∥ x - x_{j} ∥_{2} \in A}$ be the part of the cell $C_{j}$ on which the two vertices differ. Assume $ω_{j} = 1$ , so that $ω_{j}^{(j)} = 0$ ; the reverse case is symmetric. On $D_{j, A}$ the difference of the two outcome-weighted densities is $μ_{ω} f_{ω} - μ_{ω^{(j)}} f_{ω^{(j)}} = δ_{n} φ (\frac{r}{w _{n}} e_{\circ}) + {\frac{1}{2} + \frac{x ^{(1)}}{8} + δ_{n} φ (\frac{r}{w _{n}} e_{\circ})} t_{n} (r) κ_{j} (x), r = ∥ x - x_{j} ∥_{2},$ because on the cell all other bumps and angular corrections vanish.

Splitting the brace into the radius-only part $\frac{1}{2} + \frac{x _{j}^{(1)}}{8} + δ_{n} φ (\frac{r}{w _{n}} e_{\circ})$ and the remainder $\frac{1}{8} (x^{(1)} - x_{j}^{(1)})$ , the first part contributes $\int_{D_{j, A}} g (r) κ_{j} = 0$ by the same horizontal reflection, and the second contributes $\int_{D_{j, A}} \frac{x ^{(1)} - x _{j}^{(1)}}{8} t_{n} (r) κ_{j} (x) d x = \int_{D_{j, A}} \frac{r}{8} t_{n} (r) κ_{j} (x)^{2} d x = - \int_{D_{j, A}} 2 δ_{n} φ (\frac{r}{w _{n}} e_{\circ}) κ_{j} (x)^{2} d x .$

Since the average of $κ_{j}^{2}$ over each half-circle ${∥ x - x_{j} ∥_{2} = r, x^{(2)} > - 1/2}$ equals $1/2$ , the two remaining terms cancel: $\int_{D_{j, A}} (μ_{ω} f_{ω} - μ_{ω^{(j)}} f_{ω^{(j)}}) d x = 0.$

Together with the equality of radial slice masses this says that, for every measurable $A \subseteq [c_{rad} a_{n}, \infty)$ , the two experiments assign the same mass and the same conditional-mean mass to the slice ${∥ X - x_{j} ∥_{2} \in A}$ ; since the outcome mixture is determined by its conditional mean, the restricted one-point distance laws agree: $L_{P_{ω}} (Y, ∥ X - x_{j} ∥_{2})_{{z_{2} \geq c_{rad} a_{n}}} = L_{P_{ω^{(j)}}} (Y, ∥ X - x_{j} ∥_{2})_{{z_{2} \geq c_{rad} a_{n}}} .$

Step 10: the compressed-distance divergence. Below the fully active radius the same cancellation is replaced by a localized setwise bound. For every measurable $A \subseteq [0, \infty)$ , $\int_{{∥ X - x_{j} ∥_{2} \in A}} μ_{ω} d P_{ω}^{X} - \int_{{∥ X - x_{j} ∥_{2} \in A}} μ_{ω^{(j)}} d P_{ω^{(j)}}^{X} \leq δ_{n} P_{ω} (∥ X - x_{j} ∥_{2} \in A \cap [0, 128 δ_{n})),$ where $P^{X}$ denotes the covariate marginal: outside the radii $r < 2 \cdot (8 δ_{n}) / (1/8) = 128 δ_{n}$ the cutoff is fully active and Step 9 gives exact cancellation, while on those radii the uncancelled increment is bounded by the bump amplitude per unit radial mass.

Moreover, since $f_{ω} \leq 5/4$ and a disc of radius $R$ has area $π R^{2} \leq 4 R^{2}$ , $P_{ω} (∥ X - x_{j} ∥_{2} < R) \leq 5 R^{2} (R \geq 0) .$

The complete radial laws of Step 9 provide a common marginal law for $R = ∥ X - x_{j} ∥_{2}$ under the two adjacent vertices. Disintegrating the one-observation laws with respect to this common radial marginal gives measurable radial success parameters, clipped to $[1/4, 3/4]$ and equal almost everywhere to the corresponding conditional Bernoulli weights, such that the preceding setwise bound implies $∣ g_{ω} (r) - g_{ω^{(j)}} (r) ∣ \leq 2 δ_{n} 1 {r < 128 δ_{n}} for the common radial marginal-almost every r .$ Here the factor $2 δ_{n}$ is the envelope used for the almost-everywhere Radon–Nikodym comparison, and the middle-half clipping is silent by Step 5. For two Bernoulli-plus-Gaussian mixtures with success parameters in $[1/4, 3/4]$ , the divergence at a fixed radius is at most four times the squared difference of the success parameters. Averaging over the common radial marginal therefore gives $KL (L_{P_{ω}} (Y, ∥ X - x_{j} ∥_{2}), L_{P_{ω^{(j)}}} (Y, ∥ X - x_{j} ∥_{2})) \leq 4 (2 δ_{n})^{2} P_{ω} (∥ X - x_{j} ∥_{2} < 128 δ_{n}) \leq 4 (2 δ_{n})^{2} \cdot 5 (128 δ_{n})^{2} = C_{KL} δ_{n}^{4},$ using $4 \cdot 4 \cdot 5 \cdot 12 8^{2} = 1310720 = C_{KL}$ .

The distance data $V_{n, P} (x_{j})$ of Definition 10 are $n$ independent copies of the one-observation pair $(Y, ∥ X - x_{j} ∥_{2})$ , so the divergence tensorizes; combining with the eventual budget of Step 3, $KL (L_{P_{ω}} {V_{n, P_{ω}} (x_{j})}, L_{P_{ω^{(j)}}} {V_{n, P_{ω^{(j)}}} (x_{j})}) \leq n C_{KL} δ_{n}^{4} \leq α lo g M_{n} .$

Steps 4–10 supply every displayed clause with the constants of Step 2, for all sufficiently large $n$ .

∎

Lemma 2 is the concrete bridge between smooth boundary regression and the abstract product experiment. The Hölder restriction determines the relation between the signal height $a_{n}$ and the cell radius $w_{n}$ , while the boundary packing supplies $M_{n}$ separated query points. Local dependence on one bit makes each cell a coordinate of the hypercube, and the radial invariance clause expresses the angular information loss created by unsigned distance observations. The compressed-distance Kullback–Leibler bound then puts those coordinates in the range of Lemma 1. This is the finite-sample testing structure that underlies the rate calculation summarized in the main text.

For common-map rules, the decoder associated with a proposed estimator applies the same measurable map to each query point. If the estimator were uniformly accurate on all $x_{j}$ ’s, the signs of the boundary signals in Lemma 2 would be decoded correctly across all coordinates. Combining the signal separation with the overlap lower bound gives the positive lower scale in Theorem 1. The argument uses the law class and completed-expectation criterion exactly as defined in Definitions 9, 11, and 12.

The point-indexed class enlarges the set of admissible rules by allowing the Borel section to vary with the query point. The following inclusion statement records that relationship before applying the same hypercube construction to the outer-expectation risk.

Lemma 3 [lem:common-map-strict-in-point-indexed] (Strict Common-Map Inclusion).

For parameters satisfying

(Sample size.) $n \geq 1$ .
(Smoothness index.) $q \geq 1$ .
(Envelope.) $L \geq 4$ .

the common-map decision class and the point-indexed decision class of Definitions 11 and 13 satisfy $T_{n}^{NP} ⊊ T_{n}^{PI} .$

⊢ Lean

Proof of Lemma 3.

Throughout, a point of $R^{2}$ is written $x = (x^{(1)}, x^{(2)})$ , and a sample point is written $w = ((y_{i}, z_{i}) : 1 \leq i \leq n) \in (R \times R^{2})^{n}$ , so that the distance data of Definition 10 evaluated at $w$ and at a query point $x$ are $V_{n, P} (x)_{w} = ((y_{i}, ∥ z_{i} - x ∥_{2}) : 1 \leq i \leq n) .$

Inclusion. Let $μ_{n} \in T_{n}^{NP}$ and let $T_{n} : (R \times [0, \infty))^{n} \to R$ be a measurable map, selected independently of $P$ , representing it as in Definition 11. Define a point-indexed family by $T_{n, x} = T_{n} for every x \in R^{2} .$ Each section is the single Borel map $T_{n}$ , hence Borel measurable for every fixed $x$ , and the family is selected independently of $P$ because $T_{n}$ is. Moreover, for every $P \in P_{NP} (L, q)$ , every sample point, and every $x \in bd (X_{P})$ , $μ_{n} (x) = T_{n} (V_{n, P} (x)) = T_{n, x} (V_{n, P} (x)),$ so $μ_{n} \in T_{n}^{PI}$ in the sense of Definition 13. Therefore $T_{n}^{NP} \subseteq T_{n}^{PI} .$

A point-indexed rule outside the common-map class. Let the rule return the first coordinate of the query point, ignoring the data: $μ_{n}^{PI} (x) = x^{(1)}, generated by the sections T_{n, x} (v) = x^{(1)} for every v \in (R \times [0, \infty))^{n} .$ For each fixed $x$ the section $T_{n, x}$ is constant in $v$ , hence Borel, and the family ${T_{n, x}}$ does not depend on the member law. Thus $μ_{n}^{PI} \in T_{n}^{PI}$ .

Suppose, for contradiction, that this same rule also belongs to $T_{n}^{NP}$ , with common measurable map $T_{n}$ . Apply Lemma 2 at the given $q \geq 1$ and $L \geq 4$ , at any sample size beyond the threshold it provides, and take the vertex $ω = (0, \dots, 0)$ ; its model-class and support clause supplies a law $P \in P_{NP} (L, q), X_{P} = S = [- 1/2, 1/2]^{2} .$

Set $x_{-} = (- 1/4, - 1/2), x_{+} = (1/4, - 1/2), z = (0, - 1/2) .$ Each of $x_{-}$ and $x_{+}$ has second coordinate $- 1/2$ and first coordinate of modulus $1/4 < 1/2$ , so it is a non-corner point of the lower edge of $S$ and therefore $x_{-} \in bd (X_{P}), x_{+} \in bd (X_{P}) .$

Take the deterministic sample point $w$ whose every observation equals $(0, z)$ , that is $y_{i} = 0$ and $z_{i} = z$ for all $i \leq n$ . Since $z$ , $x_{-}$ and $x_{+}$ share their second coordinate, the Euclidean distances reduce to horizontal ones: $∥ z - x_{-} ∥_{2} = ∣ 0 - (- 1/4) ∣ = \frac{1}{4} = ∣ 0 - 1/4 ∣ = ∥ z - x_{+} ∥_{2} .$

Consequently the compressed samples at the two query points coincide: $V_{n, P} (x_{-})_{w} = ((0, 1/4) : 1 \leq i \leq n) = V_{n, P} (x_{+})_{w} .$

Both $x_{-}$ and $x_{+}$ lie on $bd (X_{P})$ , so the assumed common-map representation applies at each of them and, with the displayed equality of distance data, gives $x_{-}^{(1)} = T_{n} (V_{n, P} (x_{-})_{w}) = T_{n} (V_{n, P} (x_{+})_{w}) = x_{+}^{(1)} .$ But $x_{-}^{(1)} = - 1/4$ and $x_{+}^{(1)} = 1/4$ , a contradiction.

Hence $μ_{n}^{PI} \in T_{n}^{PI} ∖ T_{n}^{NP}$ , so the inclusion proved first is proper: $T_{n}^{NP} ⊊ T_{n}^{PI} .$

∎

The strict inclusion in Lemma 3 places the two unsigned risks in the expected order of admissible decision classes. For the converse, the relevant fact is that each point-indexed section remains law independent and receives the same unsigned distance data $V_{n, P} (x)$ from Definition 10. Thus the coordinatewise testing reduction induced by Lemma 2 continues to apply after replacing a single common map by fixed- $x$ Borel sections. The outer expectation in Definition 14 supplies the measurable-majorant convention for the boundary supremum, and the resulting lower bound is the point-indexed logarithmic converse in Theorem 2.

Signed-Distance Geometry, Maximal Inequalities, and Upper Bound

This appendix collects the empirical-process ingredients used by the signed-distance upper bound in Proposition 1. The objects are those of the known-geometry design in Definitions 20, 21, 22, and 28: local neighborhoods are Euclidean balls generated by the uniform weight $K_{□}$ , the outcome moment envelope is handled through winsorization, and the stochastic term is measured by an outer-expected supremum over arms and interface points. The two statements below organize these roles before the estimator bound is assembled in Proposition 1.

The first ingredient is geometric. Because the local windows are closed Euclidean balls, the relevant index class has finite VC complexity, giving the covering control used by the maximal inequality (Vapnik et al., 1971; Pollard, 1984; van der Vaart et al., 1996).

Lemma 4 [lem:euclidean-balls-vc] (Planar Balls VC Bound).

For every finite set $S$ of score points in $R^{2}$ , if $∣ S ∣ \geq 4$ , then $S$ is not shattered by the class of closed Euclidean balls ${B_{2} (x, r) : x \in R^{2}, r \geq 0} .$ Equivalently, no such $S$ has the property that every subset of $S$ can be written as $S \cap B_{2} (x, r)$ for some center $x \in R^{2}$ and radius $r \geq 0$ .

⊢ Lean

Proof of Lemma 4.

Suppose, for contradiction, that a finite set $S \subset R^{2}$ with $∣ S ∣ \geq 4$ is shattered by closed Euclidean balls, that is, for every subset $T \subseteq S$ there are a center $c \in R^{2}$ and a radius $r \geq 0$ with $T = S \cap B_{2} (c, r) = {z \in S : ∥ z - c ∥_{2} \leq r} .$ Since $∣ S ∣ \geq 4$ we may fix four pairwise distinct points $z_{1}, z_{2}, z_{3}, z_{4} \in S$ .

Writing $z_{i} = (z_{i, 1}, z_{i, 2})$ , put $Φ (z_{i}) = (1, z_{i, 1}, z_{i, 2}, ∥ z_{i} ∥_{2}^{2}) \in R^{4}, i \in {1, 2, 3, 4} .$

An algebraic identity. Let $ℓ = (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}) \in R^{4}$ be arbitrary coefficients and let $w \in R^{4}$ be the corresponding combination, $w = i = 1 \sum 4 ℓ_{i} Φ (z_{i}) .$ For a center $c = (c_{1}, c_{2}) \in R^{2}$ and a radius $r \geq 0$ , expanding the squared distance gives, for each $i$ , $∥ z_{i} - c ∥_{2}^{2} - r^{2} = ∥ z_{i} ∥_{2}^{2} - 2 c_{1} z_{i, 1} - 2 c_{2} z_{i, 2} + (∥ c ∥_{2}^{2} - r^{2}) \cdot 1,$ which is one and the same linear combination of the four entries of $Φ (z_{i})$ , with coefficients not depending on $i$ . Multiplying by $ℓ_{i}$ and summing therefore replaces those entries by the entries of $w$ : $i = 1 \sum 4 ℓ_{i} {∥ z_{i} - c ∥_{2}^{2} - r^{2}} = w_{4} - 2 c_{1} w_{2} - 2 c_{2} w_{3} + (∥ c ∥_{2}^{2} - r^{2}) w_{1} .$

A sign contradiction. We next record the contradiction used in both cases below. Suppose the coefficients $ℓ$ are not all zero, that their combination $w$ satisfies $w_{1} = 0$ , and that $i = 1 \sum 4 ℓ_{i} {∥ z_{i} - c ∥_{2}^{2} - r^{2}} \geq 0 for every c \in R^{2} and every r \geq 0.$ Reading off the first entry of $w$ and using $Φ (z_{i})_{1} = 1$ for every $i$ , $i = 1 \sum 4 ℓ_{i} = w_{1} = 0,$

so, some $ℓ_{i}$ being nonzero, at least one coefficient is strictly negative; fix an index $i_{-}$ with $ℓ_{i_{-}} < 0$ .

Apply the shattering hypothesis to the subset $T = {z_{i} : 1 \leq i \leq 4, ℓ_{i} > 0} \subseteq S,$ obtaining a center $c$ and a radius $r \geq 0$ with $T = S \cap B_{2} (c, r)$ . Because $z_{1}, \dots, z_{4}$ are distinct, $z_{i} \in T$ holds exactly when $ℓ_{i} > 0$ . Hence every term of the weighted sum is nonpositive, $ℓ_{i} {∥ z_{i} - c ∥_{2}^{2} - r^{2}} \leq 0, i \in {1, 2, 3, 4} :$ if $ℓ_{i} > 0$ then $z_{i} \in T \subseteq B_{2} (c, r)$ , so $∥ z_{i} - c ∥_{2} \leq r$ , the bracket is nonpositive, and the product is nonpositive; if $ℓ_{i} \leq 0$ then $z_{i} \in / T$ , so $∥ z_{i} - c ∥_{2} > r$ , the bracket is strictly positive, and again the product is nonpositive.

At $i_{-}$ the second alternative occurs with a strictly negative coefficient, so $ℓ_{i_{-}} {∥ z_{i_{-}} - c ∥_{2}^{2} - r^{2}} < 0,$

and summing the four terms gives $i = 1 \sum 4 ℓ_{i} {∥ z_{i} - c ∥_{2}^{2} - r^{2}} < 0,$ which contradicts the assumed nonnegativity at this $(c, r)$ .

Case 1: $Φ (z_{1}), \dots, Φ (z_{4})$ are linearly independent. Four linearly independent vectors in $R^{4}$ form a basis, so the fourth standard basis vector has an expansion $e_{4} = (0, 0, 0, 1) = i = 1 \sum 4 ℓ_{i} Φ (z_{i})$ with coefficients $ℓ$ that are not all zero, since $e_{4} \neq = 0$ . Here $w = e_{4}$ , so $w_{1} = w_{2} = w_{3} = 0$ and $w_{4} = 1$ , and the algebraic identity gives, for every $c \in R^{2}$ and every $r \geq 0$ , $i = 1 \sum 4 ℓ_{i} {∥ z_{i} - c ∥_{2}^{2} - r^{2}} = 1 \geq 0.$ The sign contradiction applies.

Case 2: $Φ (z_{1}), \dots, Φ (z_{4})$ are linearly dependent. Choose coefficients $ℓ$ , not all zero, with $i = 1 \sum 4 ℓ_{i} Φ (z_{i}) = 0.$ Here $w = 0$ , so in particular $w_{1} = 0$ , and the algebraic identity gives, for every $c \in R^{2}$ and every $r \geq 0$ , $i = 1 \sum 4 ℓ_{i} {∥ z_{i} - c ∥_{2}^{2} - r^{2}} = 0 \geq 0.$ The sign contradiction applies again.

Both cases are impossible, so no finite $S$ with $∣ S ∣ \geq 4$ is shattered by closed Euclidean balls.

∎

The bound in Lemma 4 is used with the ball notation introduced in Definition 3. In the signed-distance design, the event selected by the uniform kernel is a side-restricted Euclidean neighborhood of the query point. Finite VC complexity therefore converts the continuum of possible interface centers into an empirical-process class with logarithmic entropy, matching the $lo g (h^{- 1})$ factors in the risk calculation.

The second auxiliary statement supplies the expected maximal bound for the centered winsorized local-polynomial score. It combines the VC geometry of Lemma 4 with the bounded envelope created by winsorizing the outcome at level $B$ , using the outer-expectation convention already built into Definition 26.

Lemma 5 [lem:cty-winsorized-score-maximal-bound] (Winsorized Score Maximal Bound).

Fix a nonnegative integer $p$ and a real number $L$ . There is a constant $C = C (p, L) > 0$ such that the following holds for every real $ν$ .

(Law class.) The law $P$ belongs to the A1/A2 class $P_{12} (p, ν, L)$ of Definition 22.
(Sample size.) The sample size satisfies $n \geq 1$ .
(Bandwidth and clipping.) The bandwidth and winsorization level satisfy $0 < h \leq L^{- 1}$ and $B \geq 1$ .

Let $P^{n}$ denote the $n$ -fold i.i.d. law of the potential-outcome observations. For each $t \in {0, 1}$ and $x \in B_{P}$ , let the winsorized centered local-polynomial score be the vector whose $j$ th coordinate is $score_{t, x, h, j}^{B} - E_{P^{n}} [score_{t, x, h, j}^{B}],$ where the empirical score uses the uniform kernel, the polynomial basis $r_{p}$ , bandwidth $h$ , and the clipped outcome transform $ψ_{B}$ , centered around the corresponding population score. Then $E_{P^{n}}^{*} [t \in {0, 1}, x \in B_{P} sup score_{t, x, h}^{B} - E_{P^{n}} [score_{t, x, h}^{B}]] \leq C {\frac{lo g ( B / h )}{n h ^{2}} + \frac{B lo g ( B / h )}{n h ^{2}}} .$

⊢ Lean

Proof of Lemma 5.

Throughout, $Z = (Y (0), Y (1), X)$ denotes one potential-outcome observation, $Y = T Y (1) + (1 - T) Y (0)$ its observed outcome, and for $u \in R^{p + 1}$ we write $∥ u ∥ = max_{0 \leq j \leq p} ∣ u_{j} ∣$ for the coordinate maximum, which is the norm appearing in the displayed conclusion. All constants below are fixed before the moment exponent $ν$ , the law $P$ , the sample size $n$ , the bandwidth $h$ , and the winsorization level $B$ .

Step 1: a coefficient radius depending only on $p$ and $L$ . Set $R = 1 + ∣ L (p + 1) \cdot 16 L (2 + L) ∣,$ which is positive. We claim that for every $P \in P_{12} (p, ν, L)$ , every $t \in {0, 1}$ , every $x \in B_{P}$ , every $0 < h \leq L^{- 1}$ and every $0 \leq j \leq p$ , $β_{t, P, x}^{*} (h)_{j} \leq R .$ Write $m_{t, P, x} (h) = E_{P} [\frac{1}{h ^{2}} 1 {D_{P, x}^{\pm} \in I_{t}} K_{□} (D_{P, x}^{\pm} / h) r_{p} (D_{P, x}^{\pm} / h) Y]$ for the right-hand side of the population normal equations in Definition 24. Its coordinates obey $0 \leq j \leq p max ∣ m_{t, P, x} (h)_{j} ∣ \leq 16 L (2 + L) .$ Indeed, the uniform kernel forces $∣ D_{P, x}^{\pm} ∣ \leq h$ , hence $X \in B_{2} (x, h)$ and $∣ r_{p} (D_{P, x}^{\pm} / h)_{j} ∣ \leq 1$ , so the integrand is dominated pointwise by $h^{- 2} 1 {X \in B_{2} (x, h)} {1 + 2 Y (0)^{2} + 2 Y (1)^{2}}$ . The density envelope $f_{P} \leq L$ of Definition 22 and the area $π h^{2} \leq 4 h^{2}$ of a planar disk give $P {X \in B_{2} (x, h)} \leq 4 L h^{2} .$ The moment envelope in Definition 22 gives, since $ν \geq 2$ , the conditional second-moment estimate $E_{P} [Y (t)^{2} ∣ X = x] \leq 1 + L, x \in X_{P},$ using $y^{2} \leq 1 + ∣ y ∣^{2 + ν}$ . Localizing this bound to $B_{2} (x, h)$ gives $E_{P} [1 {X \in B_{2} (x, h)} Y (t)^{2}] \leq (1 + L) 4 L h^{2}$ for each arm, so the displayed coordinate bound follows from $h^{- 2} {4 L h^{2} + 4 (1 + L) 4 L h^{2}} = 20 L + 16 L^{2} \leq 16 L (2 + L) .$

The population Gram lower bound of Definition 22 and the normal equations of Definition 24, written with $β^{*} = β_{t, P, x}^{*} (h)$ , then give $L^{- 1} ∥ β^{*} ∥^{2} \leq L^{- 1} k = 0 \sum p (β_{k}^{*})^{2} \leq (β^{*})^{⊤} Ψ_{t, P, x} (h) β^{*} = (β^{*})^{⊤} m_{t, P, x} (h) \leq (p + 1) ∥ β^{*} ∥ \cdot 16 L (2 + L),$ the first step because $∥ β^{*} ∥^{2}$ is one of the summands. Hence $∥ β^{*} ∥ \leq L (p + 1) \cdot 16 L (2 + L) \leq R$ , which is the claim.

Step 2: the bounded score class. For $b \in R^{p + 1}$ let $Π_{R} b$ be its componentwise clipping, $(Π_{R} b)_{k} = max {- R, min {R, b_{k}}}, 0 \leq k \leq p,$ and let $I_{P, h} = {ι = (q, t, x, b, j) : q \in [h, 2 h), t \in {0, 1}, x \in B_{P}, b \in R^{p + 1}, 0 \leq j \leq p} .$ For $ι = (q, t, x, b, j) \in I_{P, h}$ define the scalar function of one observation $g_{ι} (Z) = 1 {D_{P, x}^{\pm} \in I_{t}} K_{□} (\frac{D _{P, x}^{\pm}}{q}) r_{p} (\frac{D _{P, x}^{\pm}}{h})_{j} ⎩ ⎨ ⎧ ψ_{B} (Y) - r_{p} (\frac{D _{P, x}^{\pm}}{h})^{⊤} Π_{R} b ⎭ ⎬ ⎫ .$ Only the kernel window uses the enlarged bandwidth $q$ ; the polynomial arguments keep $h$ , so the target score is the member with $q = h$ . The half-open enlargement $[h, 2 h)$ supplies the support slack used in Step 6, and clipping the coefficient gives a constant-envelope class.

Step 3: the score is a member of the class. Fix $t \in {0, 1}$ , $x \in B_{P}$ and $0 \leq j \leq p$ , and put $ι = (h, t, x, β_{t, P, x}^{*} (h), j) \in I_{P, h}$ . By Step 1 the clipping is inactive at this index, $Π_{R} β_{t, P, x}^{*} (h) = β_{t, P, x}^{*} (h)$ , so comparing with the definition of the winsorized score in Definition 32, $score_{t, x, h, j}^{B} = \frac{1}{h ^{2}} \cdot \frac{1}{n} i = 1 \sum n g_{ι} (Z_{i}) .$

The envelope bound of Step 5 makes $g_{ι}$ integrable, so averaging the previous display over the independent and identically distributed observations gives $E_{P^{n}} [score_{t, x, h, j}^{B}] = \frac{1}{h ^{2}} \int g_{ι} d P .$

Step 4: reduction to one scalar empirical process. For each fixed sample the coordinate maximum $∥ \cdot ∥$ is attained at some $j$ , and every pair $(t, x)$ with $x \in B_{P}$ together with that $j$ produces an index of $I_{P, h}$ by Step 3. Hence, pointwise on the sample space, $t \in {0, 1}, x \in B_{P} sup score_{t, x, h}^{B} - E_{P^{n}} [score_{t, x, h}^{B}] \leq \frac{1}{h ^{2}} ι \in I_{P, h} sup \frac{1}{n} i = 1 \sum n g_{ι} (Z_{i}) - \int g_{ι} d P .$

Outer expectation is monotone and pulls out the positive finite constant $h^{- 2}$ , so $E_{P^{n}}^{*} [t \in {0, 1}, x \in B_{P} sup score_{t, x, h}^{B} - E_{P^{n}} [score_{t, x, h}^{B}]] \leq \frac{1}{h ^{2}} E_{P^{n}}^{*} [ι \in I_{P, h} sup \frac{1}{n} i = 1 \sum n g_{ι} (Z_{i}) - \int g_{ι} d P] .$

Step 5: envelope, radius and covering witnesses. Put $Λ_{env} = (2 \cdot 4^{p} + 2) {1 + (p + 1) R}, Λ_{rad} = 1 + 3 \cdot 4^{p} {2 (1 + L) + {(p + 1) 2^{p} R}^{2}} 16 L,$ $C_{0} = 1 + max {Λ_{rad}, Λ_{env}} > 0.$ There are constants $A \geq exp (1)$ and $v \geq 1$ , depending only on $p$ and $R$ and therefore only on $p$ and $L$ , such that for every real $ν$ , every $P \in P_{12} (p, ν, L)$ and all $0 < h < B$ with $B \geq 1$ , the class ${g_{ι} : ι \in I_{P, h}}$ satisfies the following three properties.

(Envelope.) For every $ι \in I_{P, h}$ and every observation $z$ , $∣ g_{ι} (z) ∣ \leq C_{0} B .$ On the kernel window $∣ D_{P, x}^{\pm} ∣ \leq q < 2 h$ , so $∣ r_{p} (D_{P, x}^{\pm} / h)_{k} ∣ \leq 2^{p}$ for every $k$ . Together with $∣ ψ_{B} (Y) ∣ \leq B$ and $∣ (Π_{R} b)_{k} ∣ \leq R$ , this gives $∣ g_{ι} (z) ∣ \leq (2 \cdot 4^{p} + 2) {B + (p + 1) R} \leq Λ_{env} B \leq C_{0} B,$ where the middle inequality uses $B \geq 1$ .
(Population radius.) For every $ι \in I_{P, h}$ , $(\int g_{ι}^{2} d P)^{1/2} \leq C_{0} h .$ On the almost-sure support of $X$ , the squared separable score is bounded pointwise by $3 \cdot 4^{p} 1 {X \in B_{2} (x, 2 h)} {Y (0)^{2} + Y (1)^{2} + {(p + 1) 2^{p} R}^{2}} .$ The kernel window gives the radius $2 h$ because $q < 2 h$ . The probability of this disk is at most $16 L h^{2}$ , while the two localized arm-square integrals are each bounded by $(1 + L) 16 L h^{2}$ , using the conditional second-moment estimate from Step 1. Hence $\int g_{ι}^{2} d P$ is bounded by $3 \cdot 4^{p} {2 (1 + L) + {(p + 1) 2^{p} R}^{2}} 16 L h^{2} .$ Taking square roots and using $u \leq 1 + u$ for $u \geq 0$ gives the displayed $Λ_{rad} h \leq C_{0} h$ bound. The winsorization level is absorbed through $∣ ψ_{B} (Y) ∣ \leq ∣ Y ∣$ , so this radius is uniform in $B$ .
(Polynomial covering.) For every non-empty finite list $z_{1}, \dots, z_{m}$ of observations and every $0 < ε \leq 1$ there is a finite set $C \subseteq I_{P, h}$ with $κ \in C min (\frac{1}{m} i = 1 \sum m {g_{ι} (z_{i}) - g_{κ} (z_{i})}^{2})^{1/2} < ε C_{0} B for every ι \in I_{P, h},$ and with $∣ C ∣ \leq (A / ε)^{v}$ . The certificate is obtained as follows. First, the fixed-dimensional radial residual-polynomial class with bounded response $∣ ψ_{B} (Y) ∣ \leq B$ , bounded arm weights, and coefficient box $[- R, R]^{p + 1}$ has polynomial empirical $L^{2}$ covers with envelope $B + (p + 1) R$ ; this is applied with radial ranges $[0, 2]$ for the positive and negative arm pieces and $[0, 0]$ for the zero-distance and outside-support pieces. Second, Lemma 4 bounds the shattering of the planar closed-ball class, so the classifier family $1 {∥ X - x ∥_{2} \leq q}$ indexed by the center-radius parameter has Vapnik–Chervonenkis dimension at most three; the standard passage from a finite Vapnik–Chervonenkis dimension to empirical $L^{2}$ covering numbers (Pollard, 1984; van der Vaart et al., 1996) then supplies a fixed polynomial cover for that classifier family. Product closure combines each radial piece with the ball classifier for the positive and negative pieces, and finite-sum closure combines the four pieces $positive arm, negative arm, zero-distance arm, outside support .$ On the negative arm the factors $(- 1)^{k}$ are absorbed into the clipped coefficient vector, and the zero-distance and outside-support terms are degree-zero radial residual terms. A pullback then ties the ball center, radial center, arm, coefficient vector and coordinate to the same index $ι$ . These finitely many product, sum, envelope-enlargement and pullback operations preserve polynomial empirical $L^{2}$ entropy, changing only the constants; after the final envelope enlargement from $(2 \cdot 4^{p} + 2) {B + (p + 1) R}$ to $C_{0} B$ , the witnesses are the displayed $A$ and $v$ .

The three witnesses $A$ , $v$ , $C_{0}$ are chosen before $ν$ , $P$ , $h$ and $B$ , which is what makes the final constant uniform in the moment exponent.

The parameter-regime clause of $P_{12} (p, ν, L)$ gives $L \geq 4$ , so $h \leq L^{- 1} < 1 \leq B$ and $h < B$ ; the certificate therefore applies at the bandwidth and clipping level of the statement.

Step 6: countable reduction. Equip $I_{P, h}$ with the product topology inherited from $[h, 2 h) \subset R$ , the discrete arm and coordinate sets, the relative topology on $B_{P} \subset R^{2}$ , and the Euclidean topology on $R^{p + 1}$ . Choose a fixed countable dense sequence $(κ_{m})_{m \geq 1}$ in this space. Let $S = {z : X (z) \in X_{P}, X (z) \in / B_{P}} .$ The set $S$ has $P$ -probability one: $X \in X_{P}$ almost surely, and $P {X \in B_{P}} = 0$ because the interface has finite one-dimensional Hausdorff measure by Definition 22 and hence vanishing planar Lebesgue measure under the displayed density.

Fix $ι = (q, t, x, b, j) \in I_{P, h}$ . For $ℓ \geq 1$ set $δ_{ℓ} = \frac{2 h - q}{16 ℓ} > 0$ and consider the target point $(q + 4 δ_{ℓ}, t, x, b, j) \in I_{P, h}$ . By density of $(κ_{m})$ , choose $m_{ℓ}$ so that $κ_{m_{ℓ}} = (q_{ℓ}, t_{ℓ}, x_{ℓ}, b_{ℓ}, j_{ℓ})$ satisfies $∥ x_{ℓ} - x ∥_{2} < δ_{ℓ}, ∣ q_{ℓ} - (q + 4 δ_{ℓ}) ∣ < δ_{ℓ}, ∣ b_{ℓ} (k) - b (k) ∣ < δ_{ℓ} (0 \leq k \leq p),$ with $t_{ℓ} = t$ and $j_{ℓ} = j$ . Then $κ_{m_{ℓ}} \to ι$ , and the one-sided construction gives $q + ∥ x_{ℓ} - x ∥_{2} < q_{ℓ} for every ℓ .$ Thus the approximating support bandwidth converges to the target bandwidth while staying above it by enough to cover the center perturbation. For every $z \in S$ , this slack preserves the closed-kernel decision when $∣ D_{P, x}^{\pm} (z) ∣ \leq q$ , including the endpoint case $∣ D_{P, x}^{\pm} (z) ∣ = q$ ; when $∣ D_{P, x}^{\pm} (z) ∣ > q$ , ordinary continuity of distance and of $q_{ℓ}$ makes the approximating kernel decision eventually agree with the target one. Since $z \in X_{P} ∖ B_{P}$ , the assignment partition in Definition 22 fixes the signed-arm indicator along the approximating boundary centers, and continuity handles the polynomial and clipped-coefficient factors. Therefore $g_{κ_{m_{ℓ}}} (z) ⟶ g_{ι} (z), z \in S .$ The class is dominated by the integrable envelope $z \mapsto 2^{p} (∣ Y (0) ∣ + ∣ Y (1) ∣ + (p + 1) 2^{p} R),$ so dominated convergence also gives convergence of the corresponding population integrals. Consequently, for every $n$ , almost surely under $P^{n}$ , $ι \in I_{P, h} sup \frac{1}{n} i = 1 \sum n g_{ι} (Z_{i}) - \int g_{ι} d P = m \geq 1 sup \frac{1}{n} i = 1 \sum n g_{κ_{m}} (Z_{i}) - \int g_{κ_{m}} d P .$

Step 7: the variance-adaptive maximal inequality. We use the following countable-reduction form of the variance-adaptive VC maximal inequality. Let $μ$ be a probability law and ${G_{ι} : ι \in I}$ a real-valued class. Suppose that $0 < σ < U$ , $A \geq exp (1)$ , $v \geq 1$ , every $G_{ι}$ is measurable, $∣ G_{ι} ∣ \leq U$ , $(\int G_{ι}^{2} d μ)^{1/2} \leq σ$ , every countable subfamily has empirical $L^{2}$ covers of size at most $(A / ε)^{v}$ at radius $ε U$ for $0 < ε \leq 1$ , and a countable subfamily realizes the centered empirical supremum almost surely under every finite product law. Then, with $ℓ_{μ} = lo g (max {exp (1), \frac{A U}{σ}}),$ one has, for every $n \geq 1$ , $E_{μ^{n}}^{*} [ι \in I sup \frac{1}{n} i = 1 \sum n G_{ι} (W_{i}) - \int G_{ι} d μ] \leq c_{VC} {σ \frac{v ℓ _{μ}}{n} + \frac{v U ℓ _{μ}}{n}}, c_{VC} = 16384.$ Indeed, choose a reducing sequence $(G_{m})_{m \geq 1}$ . The envelope makes the countable supremum measurable, integrable, and bounded by $2 U$ , so the outer expectation of the continuum supremum is bounded by the ordinary expectation of the countable supremum. Countable-class symmetrization bounds that expectation by twice the Rademacher complexity. The Rademacher bound applies Dudley chaining to the empirical $L^{2}$ polynomial covers, uses the population $L^{2}$ radius $σ$ , and controls the empirical radius by symmetrizing the squared bounded class; the resulting quadratic inequality yields the displayed leading $σ$ -term and second-order $U$ -term, with the numerical constant rounded to $16384$ .

Apply this inequality with $μ = P$ , $G_{ι} = g_{ι}$ , $U = C_{0} B$ , $σ = C_{0} h$ , and with the entropy and countability witnesses from Steps 5 and 6. Writing $ℓ = lo g (max {exp (1), \frac{A C _{0} B}{C _{0} h}}),$ we obtain, for every $n \geq 1$ , $E_{P^{n}}^{*} [ι \in I_{P, h} sup \frac{1}{n} i = 1 \sum n g_{ι} (Z_{i}) - \int g_{ι} d P] \leq c_{VC} {C_{0} h \frac{v ℓ}{n} + \frac{v C _{0} B ℓ}{n}}, c_{VC} = 16384.$

Step 8: calibrating the logarithm. Since $L \geq 4$ , $h \leq L^{- 1}$ and $B \geq 1$ , $4 h \leq L h \leq 1 \leq B, so \frac{B}{h} \geq 4 and lo g (B / h) \geq 1.$ Also $A \geq exp (1)$ gives $lo g A \geq 1$ , so $A (B / h) \geq exp (1)$ and the maximum in $ℓ$ is attained at the second entry. Putting $k = 1 + lo g A$ , $ℓ = lo g A + lo g (B / h) \leq k lo g (B / h),$ because ${lo g A - 1} {lo g (B / h) - 1} \geq 0$ .

Step 9: conclusion. Abbreviate $Γ_{1} = \frac{lo g ( B / h )}{n h ^{2}}, Γ_{2} = \frac{B lo g ( B / h )}{n h ^{2}},$ both nonnegative. Multiplying the two terms of Step 7 by $h^{- 2}$ and using Step 8, $\frac{1}{h ^{2}} C_{0} h \frac{v ℓ}{n} \leq C_{0} v k \frac{1}{h} \frac{lo g ( B / h )}{n} = C_{0} v k Γ_{1}, \frac{1}{h ^{2}} \cdot \frac{v C _{0} B ℓ}{n} \leq C_{0} v k Γ_{2} .$ Therefore, chaining Steps 4, 6, 7 and the two displays above and setting $C = 1 + c_{VC} C_{0} {v k + v k},$ we obtain $E_{P^{n}}^{*} [t \in {0, 1}, x \in B_{P} sup score_{t, x, h}^{B} - E_{P^{n}} [score_{t, x, h}^{B}]] \leq c_{VC} C_{0} {v k + v k} (Γ_{1} + Γ_{2}) \leq C {\frac{lo g ( B / h )}{n h ^{2}} + \frac{B lo g ( B / h )}{n h ^{2}}} .$ The constant $C$ is positive and, since $R$ , $C_{0}$ , $A$ and $v$ were fixed from $p$ and $L$ alone, depends only on $p$ and $L$ .

∎

The two terms in Lemma 5 have the usual variance and envelope forms for bounded empirical processes indexed by VC-type sets (Pollard, 1984; van der Vaart et al., 1996). The scaling $h^{- 2}$ reflects the two-dimensional local window around an interface point, while the supremum over $t \in {0, 1}$ and $x \in B_{P}$ matches the signed-distance risk criterion in Definition 26. Together with the Gram lower bound in Definition 22, the statement controls the stochastic perturbation of the intercepts in the stabilized estimator.

These ingredients enter the upper-bound calculation as follows. The population part is separated into the first-order local-polynomial approximation term and the winsorization term, the latter controlled by the $2 + ν$ conditional moment envelope in Definition 22, which bounds the winsorization remainder by a multiple of $B^{- 3}$ . The sample part is decomposed into empirical Gram fluctuation and centered score fluctuation, with the score controlled by Lemma 5 and the Gram term handled by the expected maximal condition stated in Proposition 1. The stabilization rule in Definition 28 converts these uniform matrix and score bounds into a uniform intercept bound for each arm, and clipping keeps the treatment-effect estimate on the envelope scale.

At the bandwidth $h_{n} = a_{n}$ from Definition 8 and winsorization level $B_{n} = a_{n}^{- 1/3}$ , the first-order bias, winsorization bias, and stochastic terms in Proposition 1 are all bounded by a constant multiple of $a_{n}$ for sufficiently large $n$ . Thus the auxiliary results in this appendix support the expected outer-risk upper bound for the explicit signed-distance local-polynomial estimator, completing the upper side of the matched-rate comparison stated in Theorem 4.

Signed-Distance Hypercube and Matched Rate

This appendix records the hard signed-distance subexperiment used for the lower side of the matched-rate statement. The construction fixes the known rectangular geometry from Definition 27 and places many separated local perturbations along the bottom edge of the treatment boundary. Each perturbation changes the treatment-effect target at one boundary point by a calibrated amount, while the corresponding one-observation signed-distance laws remain close enough for the overlap direct-product argument in Lemma 1 to apply.

Lemma 6 [lem:cty-a1-a2-rectangle-angular-hypercube-all-orders] (Rectangle Angular Hypercube).

For every integer $p \geq 0$ , there is a constant $L_{0} = L_{0} (p) \geq 48$ such that, for every $ν \geq 2$ and every $L \geq L_{0}$ , there are constants $c, A, C, C_{0} > 0$ , with $2 C < A$ when $p = 0$ , and a constant $δ_{0} > 0$ such that the following construction is available for every $0 < Δ \leq δ_{0}$ . Writing $q = p + 1$ , there exist an integer $M$ , constants $w, ρ > 0$ , points $x_{1}, \dots, x_{M}$ , laws $P_{ω}$ indexed by $ω \in {0, 1}^{M}$ , and probability laws $Q_{j, b}$ for $j = 1, \dots, M$ and $b \in {0, 1}$ , satisfying:

(Size and calibration.) $M \geq c Δ^{- 1/ q}, w = A Δ^{1/ q}, ρ = \frac{π w ^{2}}{36} .$
(Common geometry.) The support is $X = [- 3, 3]^{2}$ , the treated region is $A_{1} = [- 1, 1] \times [0, 2]$ , the control region is $A_{0} = X ∖ A_{1}$ , the boundary is $B = \partial A_{1}$ , the metric is Euclidean distance, and the kernel is $K_{□}$ . The points $x_{j}$ lie on the middle of the bottom edge, and the disks $S_{j} = B_{2} (x_{j}, w)$ are contained in $X$ , pairwise separated by distance at least $2 w$ , and pairwise disjoint.
(Class membership.) For every $ω \in {0, 1}^{M}$ , the law $P_{ω}$ belongs to $P_{12} (p, ν, L)$ with the common geometry above.
(Bit structure.) For every $ω$ and $j$ , $P_{ω} {z : score (z) \in S_{j}} = ρ .$ The restriction of $P_{ω}$ to $S_{j}$ depends on $ω$ only through $ω_{j}$ , and the restriction of $P_{ω}$ outside $⋃_{j = 1}^{M} S_{j}$ is common across all $ω$ .
(Signed-observation laws.) Conditional on $score (Z) \in S_{j}$ , the one-observation signed-distance law at $x_{j}$ under $P_{ω}$ is $Q_{j, ω_{j}}$ .
(Signal separation.) If two vertices agree in coordinate $j$ , then their treatment-effect targets at $x_{j}$ agree. Flipping coordinate $j$ changes the treatment-effect target at $x_{j}$ by exactly $Δ$ : $τ_{P_{ω}} (x_{j}) - τ_{P_{ω^{(j)}}} (x_{j}) = Δ.$
(Local comparison laws.) For each $j$ , the laws $Q_{j, 0}$ and $Q_{j, 1}$ have the same signed-distance marginal, agree outside the event $0 < D_{P, x_{j}}^{\pm} < 2 C Δ$ , and obey the two Kullback–Leibler bounds $KL (Q_{j, 0}, Q_{j, 1}) \leq C_{0} \frac{Δ ^{4}}{w ^{2}}, KL (Q_{j, 1}, Q_{j, 0}) \leq C_{0} \frac{Δ ^{4}}{w ^{2}} .$

⊢ Lean

Proof of Lemma 6.

Throughout, a point of $R^{2}$ is written $x = (x^{(1)}, x^{(2)})$ , and we abbreviate $q = p + 1$ . The arm sign attached to the side intervals $I_{1} = [0, \infty)$ , $I_{0} = (- \infty, 0)$ of Definition 22 is written $(2 t - 1)$ , as in the arm-mass clause there.

Two fixed smooth profiles. Fix once and for all a radially symmetric function $φ : R^{2} \to [0, 1]$ of class $C^{\infty}$ with $φ (0) = 1, φ (z) = 0 whenever ∥ z ∥_{2} \geq 1,$ let $φ_{r} : [0, \infty) \to [0, 1]$ be its radial profile, so that $φ (z) = φ_{r} (∥ z ∥_{2})$ , and put $b_{k} = z \in R^{2} sup ∥ D^{k} φ (z)∥ < \infty, k \geq 0.$

Fix also a nondecreasing $C^{\infty}$ function $χ : R \to [0, 1]$ with $χ (s) = 0$ for $s \leq 0$ and $χ (s) = 1$ for $s \geq 1$ .

The constants. Set $A = 257 + (1 + q + 1310720 + k = 0 \sum q b_{k}), c = \frac{1}{24 A}, C = 128, C_{0} = 262144.$ All four are positive, $A > 256$ and $A \geq 1$ because the bracket is positive, and, since every $b_{k}$ is nonnegative and appears in the bracketed sum, $b_{k} \leq A for every 0 \leq k \leq q .$

Because $2 C = 256 < A$ , the required order-zero separation $2 C < A$ holds; it in fact holds for every $p$ .

The small-separation threshold. The map $Δ \mapsto A Δ^{1/ q}$ is continuous at $0$ with value $0$ , so there is $δ_{0} > 0$ such that every $0 < Δ \leq δ_{0}$ satisfies $Δ \leq \frac{1}{16}, Δ A \leq 1, w := A Δ^{1/ q} \leq \frac{1}{24} .$

The envelope threshold. The signed radial polynomial energy is coercive: there is $c_{p} > 0$ such that, for both $t \in {0, 1}$ and every $v = (v_{0}, \dots, v_{p})^{⊤} \in R^{p + 1}$ , $c_{p} ∥ v ∥_{2}^{2} \leq \int_{0}^{1} {v^{⊤} r_{p} ((2 t - 1) u)}^{2} u d u,$ with $r_{p}$ the basis of Definition 17.

Put $γ_{p} = \frac{1}{48} \cdot \frac{π}{2} \cdot c_{p} > 0, L_{0} = max {48, γ_{p}^{- 1}},$ so that $L_{0} \geq 48$ .

Fix $ν \geq 2$ and $L \geq L_{0}$ ; in particular $L \geq 48$ . Fix also $0 < Δ \leq δ_{0}$ .

Common geometry and grid. Take the fixed known geometry of Definition 20, $X = [- 3, 3]^{2}, A_{1} = [- 1, 1] \times [0, 2], A_{0} = X ∖ A_{1}, B = \partial A_{1}, d (x, z) = ∥ x - z ∥_{2},$ with the uniform kernel $K_{□}$ of Definition 19; the interface $B$ is the one displayed in Definition 16. With $w = A Δ^{1/ q}$ put $M = ⌊ \frac{1}{12 w} ⌋, x_{j} = (- \frac{1}{4} + \frac{j}{2 ( M + 1 )}, 0), j = 1, \dots, M, S_{j} = B_{2} (x_{j}, w),$ the balls being those of Definition 3.

Each $x_{j}$ has second coordinate $0$ and first coordinate of modulus at most $1/4$ , so it lies on the middle of the bottom edge ${(s, 0) : ∣ s ∣ \leq 1/2}$ of the treated rectangle, hence on $B$ .

Since $w \leq 1/24$ gives $1/ (12 w) \geq 2$ and therefore $M \geq 2$ and $M \geq 1/ (24 w)$ , $M \geq \frac{1}{24 w} = \frac{1}{24 A} Δ^{- 1/ q} = c Δ^{- 1/ q} .$

The centers are equispaced with gap $1/ {2 (M + 1)}$ , so for $i \neq = j$ $∥ x_{i} - x_{j} ∥_{2} = \frac{∣ i - j ∣}{2 ( M + 1 )} \geq \frac{1}{2 ( M + 1 )} \geq 4 w \geq 2 w,$ using $2 (M + 1) \leq 3 M \leq 1/ (4 w)$ , which is where $M \geq 2$ and $M \leq 1/ (12 w)$ are used.

Consequently the disks $S_{j}$ are pairwise disjoint, and each is contained in $X$ because $∣ x_{j}^{(1)} ∣ \leq 1/4$ and $w \leq 1/24$ .

The vertex laws. For $ω \in {0, 1}^{M}$ let $P_{ω}$ be the law with the following two ingredients. First, $X$ has Lebesgue density $f_{P_{ω}}$ supported on $X$ , given there by $f_{P_{ω}} (x) = \frac{1}{36} {1 + j = 1 \sum M ω_{j} t_{Δ} (∥ x - x_{j} ∥_{2}) η_{j} (x)},$ where the radial tilt and the horizontal direction cosine at $x_{j}$ are $t_{Δ} (r) = - \frac{2Δ φ _{r} ( r / w ) χ ( \frac{r}{128Δ} - 1 )}{max { r /16 , 8Δ }}, η_{j} (x) = \frac{x ^{(1)} - x _{j}^{(1)}}{∥ x - x _{j} ∥ _{2}} (x \neq = x_{j}), η_{j} (x_{j}) = 0.$

Second, given $X = x$ the two potential outcomes are conditionally independent Bernoulli variables whose means are the arm regressions of Definition 22, $μ_{0, P_{ω}} (x) = \frac{1}{2}, μ_{1, P_{ω}} (x) = \frac{1}{2} + \frac{x ^{(1)}}{16} + j = 1 \sum M ω_{j} Δ φ (\frac{x - x _{j}}{w}),$ both truncated to $[0, 1]$ off $X$ , with $T = 1 {X \in A_{1}}$ and $Y = T Y (1) + (1 - T) Y (0)$ .

Because the disks are $4 w$ -separated while $φ$ vanishes outside the unit ball, at most one summand is nonzero at any $x$ ; together with $∣ x^{(1)} ∣/16 \leq 3/16$ and $Δ \leq 1/16$ this gives $\frac{1}{4} \leq μ_{1, P_{ω}} (x) \leq \frac{3}{4}, x \in X,$ so the truncation is inactive on $X$ .

For the same reason the angular sum has modulus at most $1/4$ , whence $\frac{1}{48} \leq f_{P_{ω}} (x) \leq \frac{5}{144}, x \in X .$

Class membership. We check the clauses of Definition 22 in turn, using $L \geq 48$ throughout.

The support is the square $X = [- 3, 3]^{2} \subseteq [- L, L]^{2}$ , and $f_{P_{ω}}$ is continuous on $X$ with $L^{- 1} \leq \frac{1}{48} \leq f_{P_{ω}} (x) \leq \frac{5}{144} \leq L, x \in X .$

The control regression is constant, and the treatment regression is the restriction to $X$ of the globally $C^{\infty}$ function $g_{ω} (x) = \frac{1}{2} + \frac{x ^{(1)}}{16} + j = 1 \sum M ω_{j} Δ φ (\frac{x - x _{j}}{w}),$ which satisfies the displayed envelope $0 \leq ∣ α ∣ \leq p max x \in X sup ∣ \partial^{α} g_{ω} (x) ∣ + 0 \leq ∣ α ∣ \leq p max x \neq = z \in X sup \frac{∣ \partial ^{α} g _{ω} ( x ) - \partial ^{α} g _{ω} ( z ) ∣}{∥ x - z ∥ _{2}} \leq 48 \leq L .$

The bump terms are controlled because the bandwidth is calibrated to the amplitude: for every $0 \leq k \leq q$ the $k$ th derivative of one bump is bounded by $Δ w^{- k} b_{k}$ , and $Δ w^{- k} b_{k} = Δ^{1 - k / q} A^{- k} b_{k} \leq 1,$ since $Δ \leq 1$ and $k \leq q$ give $Δ^{1 - k / q} \leq 1$ , while $b_{k} \leq A \leq A^{k}$ for $k \geq 1$ ; at $k = 0$ the bound is $Δ b_{0} \leq Δ A \leq 1$ . The affine part $1/2 + x^{(1)} /16$ contributes at most $1/2 + 3/16$ to the derivative maximum, because $∣ x^{(1)} ∣/16 \leq 3/16$ on $X$ and its only nonvanishing derivative is the constant $1/16$ , and at most $1/16$ to the Lipschitz maximum; and at most one bump is active at a time.

The conditional variances are the Bernoulli variances of the two means, so $σ_{t, P_{ω}}^{2} (x) = μ_{t, P_{ω}} (x) {1 - μ_{t, P_{ω}} (x)} \in [\frac{3}{16}, \frac{1}{4}] \subseteq [L^{- 1}, L], x \in X, t \in {0, 1},$ and they are continuous on $X$ .

Each $Y (t)$ takes values in ${0, 1}$ , so the moment envelope holds with room to spare: $E_{P_{ω}} [∣ Y (t) ∣^{2 + ν} ∣ X = x] \leq 1 \leq L, x \in X .$

The sets $A_{0}, A_{1}$ are a Borel partition of $X$ with $bd (A_{0}) \cap bd (A_{1}) = \partial A_{1} = B \subset int (X)$ , and $B$ is a compact rectifiable curve, being the boundary of a $2 \times 2$ square. Its size obeys $2 \leq H^{1} (B) \leq 16, hence L^{- 1} \leq H^{1} (B) \leq L,$ the lower bound because $B$ contains the segment from $(- 1, 0)$ to $(1, 0)$ and the upper bound from a Lipschitz parametrization of the perimeter.

The metric is Euclidean and the kernel is $K_{□}$ ; the traces of closed Euclidean balls on $X$ form a VC class of index at most four by Lemma 4.

For the population Gram matrix, at every $x \in B$ , both $t \in {0, 1}$ , every $0 < h \leq L^{- 1}$ and every $v \in R^{p + 1}$ , $\frac{1}{48} \cdot \frac{π}{2} \int_{0}^{1} {v^{⊤} r_{p} ((2 t - 1) u)}^{2} u d u \leq v^{⊤} Ψ_{t, P_{ω}, x} (h) v,$ because the density is at least $1/48$ and, in polar coordinates around $x$ , the arm- $t$ part of the window $B_{2} (x, h)$ contains a quarter-disk on which the uniform kernel equals one, namely a sector of opening $π /2$ with one sign choice per coordinate determined by where $x$ sits on $B$ . A quarter, not a half, is what is available at every interface point: at the four corners of $A_{1}$ the arm- $t$ part of a small window is exactly a quarter-disk, and $h \leq L^{- 1} \leq 1$ keeps that sector inside $X$ . Its angular measure $π /2$ is the second factor in the display. Combining this with the coercivity constant, $γ_{p} ∥ v ∥_{2}^{2} \leq v^{⊤} Ψ_{t, P_{ω}, x} (h) v$ , and $L \geq γ_{p}^{- 1}$ gives the class threshold $λ_{m i n} (Ψ_{t, P_{ω}, x} (h)) \geq L^{- 1} .$

For the small-bandwidth arm mass, at every $x \in B$ , both $t$ , and every $0 < h \leq L^{- 1}$ , $\frac{1}{h ^{2}} \int_{A_{t}} K_{□} ({(2 t - 1) ∥ z - x ∥_{2}} / h) d z = \frac{1}{h ^{2}} \int_{A_{t} \cap B_{2} (x, h)} d z \geq \frac{1}{8} \geq L^{- 1},$ since every point of $\partial A_{1}$ , corners included, retains at least an eighth of the disk area in each arm.

Finally, for every $x \in B$ , both $t$ , and every $0 < s \leq L^{- 1}$ , the one-dimensional Hausdorff integral of $f_{P_{ω}}$ over ${z \in A_{t} : ∥ z - x ∥_{2} = s}$ is finite and strictly positive, because that slice is a nondegenerate circular arc and the density lies in $[1/48, 5/144]$ .

Hence $P_{ω} \in P_{12} (p, ν, L)$ , with the common geometry displayed above.

Bit structure. Put $ρ = π w^{2} /36 > 0$ . Point reflection through $x_{j}$ maps the disk $S_{j}$ onto itself, preserves $∥ x - x_{j} ∥_{2}$ and negates $η_{j}$ , so the angular correction integrates to zero over $S_{j}$ ; the remaining background density is the constant $1/36$ . With the score projection of Definition 18, $P_{ω} {z : score (z) \in S_{j}} = \frac{1}{36} \int_{S_{j}} d z = \frac{π w ^{2}}{36} = ρ .$

Each bump and each angular correction indexed by $k$ vanishes off $S_{k}$ , and the disks are disjoint; therefore the restriction of $P_{ω}$ to ${z : score (z) \in S_{j}}$ depends on $ω$ only through $ω_{j}$ , and the restriction of $P_{ω}$ to the complement of $⋃_{j} S_{j}$ is the same for every $ω$ .

Conditional signed-observation laws. Every $z \in S_{j}$ has $∣ z^{(1)} ∣ \leq 1/4 + w \leq 1$ and $∣ z^{(2)} ∣ \leq w \leq 2$ , so $z \in A_{1}$ exactly when $z^{(2)} \geq 0$ . Since $x_{j}^{(2)} = 0$ , the signed distance of Definition 21 therefore reduces on $S_{j}$ to the vertically signed radius, $D_{P_{ω}, x_{j}}^{\pm} = {∥ X - x_{j} ∥_{2}, - ∥ X - x_{j} ∥_{2}, X^{(2)} \geq 0, X^{(2)} < 0.$

Using the conditional-law notation of Definition 1, define $Q_{j, b} = L_{P_{ω}} ((Y, D_{P_{ω}, x_{j}}^{\pm}) ∣ X \in S_{j}) for any ω with ω_{j} = b,$ which is unambiguous by the one-bit locality just proved and is a probability law. Equivalently, the image of $P_{ω}$ restricted to ${X \in S_{j}}$ under $(Y, D_{P_{ω}, x_{j}}^{\pm})$ equals $ρ Q_{j, ω_{j}}$ .

Signal separation. On $X$ the treatment-effect curve of Definition 23 is $τ_{P_{ω}} (x) = μ_{1, P_{ω}} (x) - μ_{0, P_{ω}} (x) = \frac{x ^{(1)}}{16} + j = 1 \sum M ω_{j} Δ φ (\frac{x - x _{j}}{w}),$ and at $x = x_{j}$ only the $j$ th bump is active, with $φ (0) = 1$ , so $τ_{P_{ω}} (x_{j}) = \frac{x _{j}^{(1)}}{16} + ω_{j} Δ.$

Hence two vertices agreeing in coordinate $j$ have the same target at $x_{j}$ , and flipping coordinate $j$ moves that target by exactly the amplitude, $τ_{P_{ω}} (x_{j}) - τ_{P_{ω^{(j)}}} (x_{j}) = Δ.$

Common signed-distance marginal. The same reflection argument applies slice by slice: for every Borel $A \subseteq R$ , the set ${z \in S_{j} : D_{P_{ω}, x_{j}}^{\pm} (z) \in A}$ is invariant under reflection through $x_{j}$ , so the angular correction contributes nothing to its mass and $Q_{j, 0} (R \times A) = \frac{1}{36 ρ} \int_{{z \in S_{j} : D_{P_{ω}, x_{j}}^{\pm} (z) \in A}} d z = Q_{j, 1} (R \times A),$ which is the asserted equality of signed-distance marginals.

Localized comparison of the two bit laws. Under $Q_{j, b}$ the outcome is Bernoulli given the signed distance, with success parameter in $[1/4, 3/4]$ by the middle-half bound above. The two success parameters differ only at short positive radii: for every Borel $B \subseteq R$ , $\int_{{u \in B}} y d Q_{j, 1} (y, u) - \int_{{u \in B}} y d Q_{j, 0} (y, u) \leq Δ Q_{j, 0} (R \times (B \cap (0, 2 C Δ))) .$

Indeed, enabling the $j$ th bit adds $Δ φ_{r} (r / w)$ to the conditional mean on the upper half of the radial slice at radius $r$ and simultaneously tilts the design density by $t_{Δ} (r) η_{j}$ . For $r \geq 2 C Δ = 256Δ$ the cutoff is fully active, $χ (r / (128Δ) - 1) = 1$ and $max {r /16, 8Δ} = r /16$ , so $t_{Δ} (r) = - 32Δ φ_{r} (r / w) / r$ ; since the affine part of the treatment regression contributes the factor $(x^{(1)} - x_{j}^{(1)}) /16 = r η_{j} (x) /16$ , the tilt contributes $- 2Δ φ_{r} (r / w) η_{j} (x)^{2}$ per unit slice mass, and the mean of $η_{j}^{2}$ over the upper half circle of radius $r$ is $1/2$ , which cancels the added bump exactly. For $0 < r < 256Δ$ the two conditional means differ by at most $Δ$ , and on the control side, where the signed distance is negative, both laws carry the common mean $1/2$ .

Consequently the restrictions of the two laws to the complement of the short positive window agree: writing $E = {(y, u) : 0 < u < 2 C Δ}$ , $Q_{j, 0} (\cdot \cap E^{c}) = Q_{j, 1} (\cdot \cap E^{c}),$ which is the asserted agreement outside $0 < D_{P, x_{j}}^{\pm} < 2 C Δ$ .

The two divergence bounds. Two Bernoulli outcome laws over a common statistic, with success parameters in $[1/4, 3/4]$ and setwise mean-mass discrepancy at most $Δ$ localized to $(0, 2 C Δ)$ , satisfy $KL (Q_{j, 0}, Q_{j, 1}) \leq 4 Δ^{2} Q_{j, 0} (R \times (0, 2 C Δ)),$ and the same bound with the two laws interchanged, the marginals being common.

Because ${z \in S_{j} : 0 < D_{P_{ω}, x_{j}}^{\pm} (z) < R} \subseteq B_{2} (x_{j}, R)$ , the localized mass obeys $Q_{j, 0} (R \times (0, R)) = \frac{1}{36 ρ} \int_{{z \in S_{j} : 0 < D_{P_{ω}, x_{j}}^{\pm} (z) < R}} d z \leq \frac{π R ^{2}}{36 ρ} .$

Taking $R = 2 C Δ = 256Δ$ and substituting $ρ = π w^{2} /36$ , $4 Δ^{2} \cdot \frac{π ( 256Δ ) ^{2}}{36 ρ} = \frac{4 \cdot 65536 Δ ^{4}}{w ^{2}} = C_{0} \frac{Δ ^{4}}{w ^{2}},$ so both directed divergences are at most $C_{0} Δ^{4} / w^{2}$ .

Collecting the displayed size and calibration identities, the common geometry, class membership, the bit structure, the conditional signed-observation laws, the exact signal separation, and the three local comparison facts gives every clause of the statement for the constants $c, A, C = 128, C_{0} = 262144$ and the threshold $δ_{0}$ .

∎

The constants in Lemma 6 separate geometry from signal size. The radius $w = A Δ^{1/ q}$ gives room for a $C^{p + 1}$ perturbation of height $Δ$ , and the packing size $M ≳ Δ^{- 1/ q}$ supplies enough boundary coordinates to generate the logarithmic uniform-risk scale. The common rectangular geometry places every $P_{ω}$ inside the same fixed-known-geometry subexperiment, while the bit structure makes each disk a coordinate of the hypercube. The local comparison clause supplies the testing input: after $n$ observations, the effective information in a coordinate is calibrated by the cell probability $ρ$ and the Kullback–Leibler bounds.

To obtain Theorem 3, set the hypercube signal to the risk scale $Δ ≍ (lo g n / n)^{1/4}$ . Since $q = p + 1$ , the calibration in Lemma 6 gives $M ≳ Δ^{- 1/ q}$ , $w = A Δ^{1/ q}$ , and $ρ = π w^{2} /36$ . The comparison laws then satisfy $n ρ KL (Q_{j, 0}, Q_{j, 1}) ≲ n w^{2} \frac{Δ ^{4}}{w ^{2}} = n Δ^{4} ≍ lo g n,$ with the same order for the reverse divergence. This calibration keeps the coordinatewise overlap controlled while the maximum over the packed boundary points contributes the logarithmic factor. Applying Lemma 1 to the signed-distance observations $U_{n, P_{ω}}^{\pm} (x_{j})$ converts the overlap into a lower bound for recovering the hypercube coordinates, and the signal-separation clause converts that testing loss into the interface sup-loss in Definition 26. Because all vertices lie in the fixed rectangular subexperiment, the conclusion applies both to $R_{n}^{12, \pm, fix} (p, ν, L)$ from Definition 27 and to the full risk $R_{n}^{12, \pm} (p, ν, L)$ .

The matched statement follows by combining the lower and upper components already stated in the main text. Theorem 3 gives the positive lower constant for the signed-distance risk after the envelope threshold $L_{0} (p)$ is met. Proposition 1 gives a uniform outer-expected upper bound of order $a_{n}$ for the explicit winsorized, Gram-stabilized local-polynomial estimator with bandwidth $h_{n} = a_{n}$ and winsorization level $B_{n} = a_{n}^{- 1/3}$ . Therefore Theorem 4 characterizes the displayed known-geometry signed-distance minimax risk on the normalization $a_{n} = (lo g n / n)^{1/4}$ , with the estimator in Definition 28 attaining the upper side of the rate over $P_{12} (p, ν, L)$ .

Reproducibility Note

The displayed definitions and theorem implications have been machine checked, including the two unsigned converses, the signed fixed-geometry lower bound, and the assembly of the conditional signed upper and frontier results. In Proposition 1 and Theorem 4, distance identification, uniform first-order bias, and expected maximal control enter as hypotheses.

Proofs of the main results

Proof of Proposition 1.

Throughout, $Z = (Y (0), Y (1), X)$ denotes one potential-outcome observation and $Y = T Y (1) + (1 - T) Y (0)$ its observed outcome, as in Definition 22; $Z_{1}, \dots, Z_{n}$ is the sample and $P^{n}$ its law. For $u \in R^{p + 1}$ we write $∥ u ∥ = 0 \leq j \leq p max ∣ u_{j} ∣, ∥ u ∥_{2} = (j = 0 \sum p u_{j}^{2})^{1/2},$ so that $∥ u ∥_{2} \leq (p + 1) ∥ u ∥$ , the norm $∥ \cdot ∥$ being the one appearing in Lemma 5 and in Definition 30. Fix $p \geq 0$ , $ν \geq 2$ and $L \geq 4$ , and abbreviate the bandwidth and clipping schedules of the statement by $h_{n} = a_{n} = (\frac{lo g n}{n})^{1/4}, B_{n} = a_{n}^{- 1/3},$ as in Definition 8 and Definition 28. We write $Ψ_{t, x} (h)$ , $m_{t, x}^{B} (h)$ , $β_{t, x}^{B} (h)$ and $τ_{n, h}^{12, LP} (x)$ for the objects of Definition 28, $Ψ_{t, P, x} (h)$ for the population Gram of Definition 22, $β_{t, P, x}^{*} (h)$ for the population coefficient of Definition 24, and $τ_{P}$ for the target of Definition 23.

The first input of Definition 33 fixes what is being estimated: it is what makes $τ_{P} (x)$ the boundary contrast approached by the two one-sided signed-distance conditional-mean limits, and the bias input of the same collection is stated relative to that same target through Definition 29. Everything below is quantitative and uses the bias and expected-maximal inputs, Lemma 5, and the envelopes of Definition 22.

Step 0: the constants. Let $C_{b} > 0$ be the constant supplied by the uniform first-order bias input of the statement and put $\overset{ˉ}{C}_{b} = C_{b} + 1 > 0.$ Let $C_{m} > 0$ be the constant supplied by the expected maximal bounds of the statement, and let $C_{sc} = C_{sc} (p, L) > 0$ be the constant of Lemma 5. Define $C = \overset{ˉ}{C}_{b} + 32 L^{3} (p + 1) + 16 L (p + 1) C_{sc} + 8 L^{2} (p + 1) C_{m} > 0.$ All four summands are positive because $L \geq 4 > 0$ . This is the constant of the conclusion; the threshold $N$ is fixed at the end of Step 2.

Step 1: the four bandwidth calibrations. Let $n \geq 3$ and write $h = h_{n}$ , $B = B_{n}$ . We claim $B^{- 3} \leq h, \frac{lo g ( B / h )}{n h ^{2}} \leq 2 h, \frac{B lo g ( B / h )}{n h ^{2}} \leq 2 h, \frac{lo g ( h ^{- 1} )}{n h ^{2}} \leq h .$ First, $lo g n < n$ for every $n \geq 1$ , so $lo g n / n < 1$ and hence $0 < h < 1$ and $B = h^{- 1/3} \geq 1$ . Raising the definition of $a_{n}$ to the fourth power gives the identity that drives every calibration, $h^{4} = \frac{lo g n}{n}, equivalently \frac{lo g n}{n h ^{2}} = h^{2} .$

The first claim is the exponent computation $B^{- 3} = (h^{- 1/3})^{- 3} = h$ .

For the remaining three, take logarithms in the definition of $h$ : $lo g (h^{- 1}) = \frac{lo g n - lo g lo g n}{4}, lo g (B / h) = lo g (h^{- 4/3}) = \frac{4}{3} lo g (h^{- 1}),$ and, since $n \geq 3$ forces $lo g lo g n > 0$ , both are nonnegative and satisfy $lo g (h^{- 1}) \leq lo g n, lo g (B / h) \leq \frac{lo g n}{3} \leq 2 lo g n .$ Dividing by $n h^{2}$ and using the displayed identity, $\frac{lo g ( B / h )}{n h ^{2}} \leq \frac{2 lo g n}{n h ^{2}} = 2 h^{2}, \frac{lo g ( h ^{- 1} )}{n h ^{2}} \leq \frac{lo g n}{n h ^{2}} = h^{2} .$ Taking square roots gives the second claim, because $2 h^{2} \leq (2 h)^{2}$ , and the fourth claim directly. For the third, factor out $B$ and use the previous display together with $B h = h^{- 1/3} h = h^{2/3} \leq 1,$ which holds because $0 < h < 1$ ; thus $B lo g (B / h) / (n h^{2}) \leq B \cdot 2 h^{2} = 2 h (B h) \leq 2 h$ .

Step 2: activating the two assumed inputs along the frontier bandwidth. The bias input of the statement is asserted along antitone bandwidth sequences, and $n \mapsto a_{n}$ is antitone only from $n = 3$ onward. Set therefore $h_{n}^{♯} = a_{m a x {n, 3}}, n \in N .$ Then $h_{n}^{♯} > 0$ for every $n$ , the sequence $h^{♯}$ is antitone, $h_{n}^{♯} \to 0$ , and $n (h_{n}^{♯})^{2} = n lo g n \to \infty (n \geq 3),$ so the bias input applies to $h^{♯}$ with $ν^{'} = ν$ and yields $n \to \infty lim sup BiasRatio_{p, ν, L} (h_{n}^{♯}) \leq C_{b} < \overset{ˉ}{C}_{b} .$ Since $h_{n}^{♯} = a_{n}$ for every $n \geq 3$ , there is $N_{1}$ such that $BiasRatio_{p, ν, L} (a_{n}) \leq \overset{ˉ}{C}_{b}, n \geq N_{1} .$

The expected maximal bounds of the statement are asserted along positive deterministic sequences tending to zero whose heavy-tail regime diverges. At $h_{n} = a_{n}$ , using $lo g (a_{n}^{- 1}) = (lo g n - lo g lo g n) /4$ and $a_{n}^{2} = lo g n / n$ , $\frac{n ^{(1 + ν) / (2 + ν)} a _{n}^{2}}{lo g ( a _{n}^{- 1} )} = \frac{4 n ^{(1 + ν) / (2 + ν) - 1/2} ( lo g n ) ^{1/2}}{lo g n - lo g lo g n} ⟶ \infty,$ because $(1 + ν) / (2 + ν) - 1/2 = ν / {2 (2 + ν)} > 0$ for $ν \geq 2$ , so the numerator’s polynomial factor dominates the logarithm. Hence there is $N_{2}$ such that, for every $n \geq N_{2}$ and every $P \in P_{12} (p, ν, L)$ , $E_{P}^{*} [GramDev_{n, p, P} (a_{n})] \leq C_{m} \frac{lo g ( a _{n}^{- 1} )}{n a _{n}^{2}},$ with $GramDev$ as in Definition 30. Only this first half of the expected maximal bounds is used; the stochastic control of the score is supplied instead by Lemma 5, whose bounded envelope is created by the winsorization built into Definition 28.

Finally $a_{n} \to 0$ , so there is $N_{3}$ with $a_{n} \leq L^{- 1}$ for $n \geq N_{3}$ , which is the small-bandwidth regime in which the population Gram floor, the arm-mass threshold and Lemma 5 are all available. Put $N = max {3, N_{1}, N_{2}, N_{3}}$ and fix $n \geq N$ and $P \in P_{12} (p, ν, L)$ for the rest of the proof. Write $h = a_{n}$ and $B = B_{n} = h^{- 1/3}$ ; by Step 1, $1 \leq B$ and $0 < h \leq L^{- 1}$ .

Step 3: a small Gram deviation activates the stabilization guard. Put $δ = \frac{1}{2 L ( p + 1 )} > 0.$ We claim that on the event ${GramDev_{n, p, P} (h) \leq δ}$ the guard of Definition 28 holds simultaneously at every arm $t \in {0, 1}$ and every interface point $x \in B_{P}$ ; that is, $(2 L)^{- 1} j = 0 \sum p v_{j}^{2} \leq v^{⊤} Ψ_{t, x} (h) v, v \in R^{p + 1} .$ Indeed, $GramDev_{n, p, P} (h) \leq δ$ means exactly that every entry satisfies $∣ Ψ_{t, x} (h)_{j k} - Ψ_{t, P, x} (h)_{j k} ∣ \leq δ$ , and for any symmetric perturbation with entries bounded by $δ$ , $v^{⊤} {Ψ_{t, x} (h) - Ψ_{t, P, x} (h)} v \leq j, k \sum ∣ v_{j} ∣ δ ∣ v_{k} ∣ \leq δ j, k \sum \frac{v _{j}^{2} + v _{k}^{2}}{2} = (p + 1) δ j = 0 \sum p v_{j}^{2} = (2 L)^{- 1} j = 0 \sum p v_{j}^{2},$ the middle step by $2∣ v_{j} ∣∣ v_{k} ∣ \leq v_{j}^{2} + v_{k}^{2}$ .

The population Gram lower bound of Definition 22, valid because $0 < h \leq L^{- 1}$ and $x \in B_{P}$ , gives $v^{⊤} Ψ_{t, P, x} (h) v \geq L^{- 1} \sum_{j} v_{j}^{2} = 2 (2 L)^{- 1} \sum_{j} v_{j}^{2}$ . Subtracting the previous display leaves $v^{⊤} Ψ_{t, x} (h) v \geq (2 L)^{- 1} \sum_{j} v_{j}^{2}$ , which is the guard. This is where the factor $2$ in the guard threshold $(2 L)^{- 1}$ is spent: it is exactly the room left by the population floor $L^{- 1}$ after the empirical perturbation is absorbed.

Step 4: the guarded coefficient error is the inverted score. Suppose the guard holds at $(t, x)$ . Then $Ψ_{t, x} (h)$ is invertible: if $Ψ_{t, x} (h) v = 0$ then the guard forces $\sum_{j} v_{j}^{2} \leq 0$ , so $v = 0$ .

Moreover, for every $ζ \in R^{p + 1}$ , $Ψ_{t, x} (h)^{- 1} ζ_{2} \leq 2 L ∥ ζ ∥_{2} .$ Indeed, applying the guard to the vector $Ψ_{t, x} (h)^{- 1} ζ$ and then the Cauchy–Schwarz inequality, $(2 L)^{- 1} Ψ_{t, x} (h)^{- 1} ζ_{2}^{2} \leq {Ψ_{t, x} (h)^{- 1} ζ}^{⊤} Ψ_{t, x} (h) {Ψ_{t, x} (h)^{- 1} ζ} = {Ψ_{t, x} (h)^{- 1} ζ}^{⊤} ζ \leq Ψ_{t, x} (h)^{- 1} ζ_{2} ∥ ζ ∥_{2},$ and dividing by $∥ Ψ_{t, x} (h)^{- 1} ζ ∥_{2}$ when it is nonzero gives the claim; when it is zero the claim is trivial.

On the guarded branch $β_{t, x}^{B} (h) = Ψ_{t, x} (h)^{- 1} m_{t, x}^{B} (h)$ , so subtracting $β_{t, P, x}^{*} (h)$ and using the definition of the winsorized score in Definition 32, $β_{t, x}^{B} (h) - β_{t, P, x}^{*} (h) = Ψ_{t, x} (h)^{- 1} {m_{t, x}^{B} (h) - Ψ_{t, x} (h) β_{t, P, x}^{*} (h)} = Ψ_{t, x} (h)^{- 1} score_{t, x, h}^{B} .$ Selecting the intercept, bounding a coordinate by the Euclidean norm, and converting norms through $∥ u ∥_{2} \leq (p + 1) ∥ u ∥$ , $e_{1}^{⊤} {β_{t, x}^{B} (h) - β_{t, P, x}^{*} (h)} \leq Ψ_{t, x} (h)^{- 1} score_{t, x, h}^{B}_{2} \leq 2 L score_{t, x, h}^{B}_{2} \leq 2 L (p + 1) score_{t, x, h}^{B} .$

Step 5: the mean of the winsorized score is of order $B^{- 3}$ . Write $WScoreDev_{n, p, P} (h, B) = t \in {0, 1} sup x \in B_{P} sup score_{t, x, h}^{B} - E_{P^{n}} [score_{t, x, h}^{B}]$ for the centered supremum appearing in Lemma 5, so that for every $t$ and every $x \in B_{P}$ , $score_{t, x, h}^{B} \leq WScoreDev_{n, p, P} (h, B) + E_{P^{n}} [score_{t, x, h}^{B}] .$ We claim the deterministic bound $E_{P^{n}} [score_{t, x, h}^{B}] \leq 8 L^{2} B^{- 3}, t \in {0, 1}, x \in B_{P} .$ Fix $t$ , $x \in B_{P}$ , and a coordinate $0 \leq j \leq p$ , and write the single-observation weight $ξ_{j} (Z) = 1 {D_{P, x}^{\pm} \in I_{t}} K_{□} (D_{P, x}^{\pm} / h) r_{p} (D_{P, x}^{\pm} / h)_{j},$ with $D_{P, x}^{\pm}$ the signed distance of Definition 21, $K_{□}$ the uniform kernel of Definition 19, and $r_{p}$ the basis of Definition 17. Since $K_{□}$ vanishes off ${∣ D_{P, x}^{\pm} ∣ \leq h}$ and $∣ r_{p} (u)_{j} ∣ \leq 1$ for $∣ u ∣ \leq 1$ , $∣ ξ_{j} (Z) ∣ \leq 1 {X \in B_{2} (x, h)},$ with $B_{2} (x, h)$ the ball of Definition 3; the equivalence $∣ D_{P, x}^{\pm} ∣ \leq h ⟺ X \in B_{2} (x, h)$ is immediate from the definition of the signed distance, whose magnitude is $∥ X - x ∥_{2}$ . By Definition 32 the $j$ th coordinate of the winsorized score is the sample average $h^{- 2} n^{- 1} \sum_{i \leq n} ξ_{j} (Z_{i}) {ψ_{B} (Y_{i}) - r_{p} (D_{P, x, i}^{\pm} / h)^{⊤} β_{t, P, x}^{*} (h)}$ , so the observations being independent and identically distributed and the summand integrable, $E_{P^{n}} [score_{t, x, h, j}^{B}] = \frac{1}{h ^{2}} \int ξ_{j} (Z) {ψ_{B} (Y) - r_{p} (D_{P, x}^{\pm} / h)^{⊤} β_{t, P, x}^{*} (h)} d P .$

Split the integrand by inserting and removing the unwinsorized outcome, $ξ_{j} (Z) {ψ_{B} (Y) - r_{p}^{⊤} β^{*}} = population residual ξ_{j} (Z) {Y - r_{p}^{⊤} β^{*}} + winsorization remainder ξ_{j} (Z) {ψ_{B} (Y) - Y} .$ The first term integrates to zero: by the normal equations of Definition 24, $\frac{1}{h ^{2}} \int ξ_{j} (Z) {Y - r_{p} (D_{P, x}^{\pm} / h)^{⊤} β_{t, P, x}^{*} (h)} d P = E_{P} [\frac{1}{h ^{2}} 1 {D_{P, x}^{\pm} \in I_{t}} K_{□} (D_{P, x}^{\pm} / h) r_{p} (D_{P, x}^{\pm} / h) Y]_{j} - {Ψ_{t, P, x} (h) β_{t, P, x}^{*} (h)}_{j} = 0.$

For the second term, the moment envelope of Definition 22 is converted into a deterministic $B^{- 3}$ envelope pointwise in the outcome. For $B \geq 1$ and $ν \geq 2$ , every real $y$ satisfies $∣ y - ψ_{B} (y) ∣ \leq ∣ y ∣^{2 + ν} B^{- 3} .$ If $∣ y ∣ \leq B$ the left side vanishes; otherwise $∣ y ∣ > B \geq 1$ , so $∣ y - ψ_{B} (y) ∣ = ∣ y ∣ - B \leq ∣ y ∣ \leq ∣ y ∣^{4} B^{- 3} \leq ∣ y ∣^{2 + ν} B^{- 3}$ , using $∣ y ∣^{3} \geq B^{3}$ and then $∣ y ∣ \geq 1$ with $2 + ν \geq 4$ .

Integrating this against the selected conditional law of $Y (t^{'})$ given $X = z$ and applying the moment envelope $E_{P} [∣ Y (t^{'}) ∣^{2 + ν} ∣ X = z] \leq L$ of Definition 22 gives, for each arm $t^{'} \in {0, 1}$ and each $z \in X_{P}$ , $E_{P} [∣ Y (t^{'}) - ψ_{B} (Y (t^{'})) ∣ X = z] \leq L B^{- 3} .$ Note that it is this absolute conditional remainder, and not merely the difference of the two conditional means, that the argument requires, because the remainder is multiplied by the random weight $ξ_{j}$ before it is integrated.

Combining the weight bound, the consistency relation $Y = T Y (1) + (1 - T) Y (0)$ of Definition 22, which makes $∣ ψ_{B} (Y) - Y ∣$ at most the sum of the two armwise remainders, and the density envelope $f_{P} \leq L$ together with the planar area bound $π h^{2} \leq 4 h^{2}$ , which give $P {X \in B_{2} (x, h)} \leq 4 L h^{2},$

we obtain $\int ∣ ξ_{j} (Z) {ψ_{B} (Y) - Y} ∣ d P \leq t^{'} \in {0, 1} \sum E_{P} [1 {X \in B_{2} (x, h)} Y (t^{'}) - ψ_{B} (Y (t^{'}))] \leq 2 \cdot L B^{- 3} \cdot 4 L h^{2} = 8 L^{2} B^{- 3} h^{2} .$

Multiplying by $h^{- 2}$ and taking the maximum over $j$ proves the claim, since the bound is free of $j$ , $t$ and $x$ .

Step 6: the pointwise interface loss. We claim that for every sample, $x \in B_{P} sup τ_{n, h}^{12, LP} (x) - τ_{P} (x) \leq {\overset{ˉ}{C}_{b} h + 4 L (p + 1) \cdot 8 L^{2} B^{- 3}} + 4 L (p + 1) WScoreDev_{n, p, P} (h, B) + 8 L^{2} (p + 1) GramDev_{n, p, P} (h) .$ Two preliminary facts hold at every $x \in B_{P}$ . First, $B_{P} \subset int (X_{P})$ , and the derivative envelope of Definition 22 bounds each arm regression by $∣ μ_{t, P} (x) ∣ \leq L$ , so $∣ τ_{P} (x) ∣ \leq 2 L, that is τ_{P} (x) \in [- 2 L, 2 L] .$

Consequently, since projecting onto an interval containing $τ_{P} (x)$ cannot increase the distance to $τ_{P} (x)$ , the clipping in Definition 28 is harmless: $τ_{n, h}^{12, LP} (x) - τ_{P} (x) \leq e_{1}^{⊤} {β_{1, x}^{B} (h) - β_{0, x}^{B} (h)} - τ_{P} (x) .$ Second, $n \geq N_{1}$ and the definition of the bias ratio in Definition 29 give the deterministic first-order bias bound $e_{1}^{⊤} {β_{1, P, x}^{*} (h) - β_{0, P, x}^{*} (h)} - τ_{P} (x) \leq \overset{ˉ}{C}_{b} h, x \in B_{P} .$ Now split on the design.

Case 1: $GramDev_{n, p, P} (h) \leq δ$ . By Step 3 the guard holds at both arms and every $x \in B_{P}$ , so Step 4 applies at both arms. Inserting the two population coefficients and using the triangle inequality, $e_{1}^{⊤} {β_{1, x}^{B} (h) - β_{0, x}^{B} (h)} - τ_{P} (x) \leq t \in {0, 1} \sum e_{1}^{⊤} {β_{t, x}^{B} (h) - β_{t, P, x}^{*} (h)} + \overset{ˉ}{C}_{b} h .$ Bounding each summand by Step 4 and then each score by Step 5, $t \in {0, 1} \sum e_{1}^{⊤} {β_{t, x}^{B} (h) - β_{t, P, x}^{*} (h)} \leq 2 \cdot 2 L (p + 1) {WScoreDev_{n, p, P} (h, B) + 8 L^{2} B^{- 3}},$ which is $4 L (p + 1) \cdot 8 L^{2} B^{- 3} + 4 L (p + 1) WScoreDev_{n, p, P} (h, B) .$ The right-hand side no longer depends on $x$ , so taking the supremum over $x \in B_{P}$ and adding the nonnegative Gram term gives the claim in this case.

Case 2: $GramDev_{n, p, P} (h) > δ$ . Here the clipping alone controls the loss: $∣ τ_{n, h}^{12, LP} (x) ∣ \leq 2 L$ by construction and $∣ τ_{P} (x) ∣ \leq 2 L$ , so $x \in B_{P} sup τ_{n, h}^{12, LP} (x) - τ_{P} (x) \leq 4 L .$ On the other hand the case hypothesis and the value of $δ$ give $8 L^{2} (p + 1) GramDev_{n, p, P} (h) \geq 8 L^{2} (p + 1) \cdot \frac{1}{2 L ( p + 1 )} = 4 L,$ so the Gram term alone already dominates the loss, and the claim holds because the two remaining terms are nonnegative. This is the calibration that fixes the coefficient $8 L^{2} (p + 1)$ : it is the smallest multiple of $δ^{- 1}$ that pays for the trivial envelope $4 L$ on the ill-conditioned event.

Step 7: integration. Take outer expectations under $P^{n}$ in Step 6. The outer expectation of Definition 14 is monotone, is subadditive on sums, returns a constant on a constant because $P^{n}$ is a probability measure, and satisfies $E_{P}^{*} [c W] \leq c E_{P}^{*} [W]$ for every finite constant $c > 0$ . Applying these four properties in turn to the three summands of Step 6, $E_{P}^{*} [x \in B_{P} sup τ_{n, h}^{12, LP} (x) - τ_{P} (x)] \leq \overset{ˉ}{C}_{b} h + 32 L^{3} (p + 1) B^{- 3} + 4 L (p + 1) E_{P}^{*} [WScoreDev_{n, p, P} (h, B)] + 8 L^{2} (p + 1) E_{P}^{*} [GramDev_{n, p, P} (h)],$ where $4 L (p + 1) \cdot 8 L^{2} = 32 L^{3} (p + 1)$ . The measurability, integrability and extended-arithmetic bookkeeping needed to pass from the pointwise inequality to this display is routine and is suppressed.

Step 8: substituting the two maximal bounds. The three terms are now evaluated at $h = a_{n}$ and $B = B_{n} = a_{n}^{- 1/3}$ using Step 1.

The deterministic term. By the first calibration $B_{n}^{- 3} = a_{n}$ , so $\overset{ˉ}{C}_{b} a_{n} + 32 L^{3} (p + 1) B_{n}^{- 3} = {\overset{ˉ}{C}_{b} + 32 L^{3} (p + 1)} a_{n} .$

The score term. Since $n \geq 1$ , $0 < a_{n} \leq L^{- 1}$ and $B_{n} \geq 1$ , Lemma 5 applies to $P \in P_{12} (p, ν, L)$ at this bandwidth and clipping level and supplies $E_{P}^{*} [WScoreDev_{n, p, P} (a_{n}, B_{n})] \leq C_{sc} {\frac{lo g ( B _{n} / a _{n} )}{n a _{n}^{2}} + \frac{B _{n} lo g ( B _{n} / a _{n} )}{n a _{n}^{2}}} \leq C_{sc} {2 a_{n} + 2 a_{n}} = 4 C_{sc} a_{n},$ the last inequality by the second and third calibrations of Step 1. Hence $4 L (p + 1) E_{P}^{*} [WScoreDev_{n, p, P} (a_{n}, B_{n})] \leq 16 L (p + 1) C_{sc} a_{n} .$ This is the step at which the winsorization level is paid for twice over, and in opposite directions: raising $B_{n}$ shrinks the deterministic term $32 L^{3} (p + 1) B_{n}^{- 3}$ but inflates the envelope term $B_{n} lo g (B_{n} / a_{n}) / (n a_{n}^{2})$ . The choice $B_{n} = a_{n}^{- 1/3}$ is exactly the one that places both on the scale $a_{n}$ .

The Gram term. By Step 2 and the fourth calibration of Step 1, $8 L^{2} (p + 1) E_{P}^{*} [GramDev_{n, p, P} (a_{n})] \leq 8 L^{2} (p + 1) C_{m} \frac{lo g ( a _{n}^{- 1} )}{n a _{n}^{2}} \leq 8 L^{2} (p + 1) C_{m} a_{n} .$

Adding the three displays and recalling the definition of $C$ in Step 0, $E_{P}^{*} [x \in B_{P} sup τ_{n, a_{n}}^{12, LP} (x) - τ_{P} (x)] \leq {\overset{ˉ}{C}_{b} + 32 L^{3} (p + 1) + 16 L (p + 1) C_{sc} + 8 L^{2} (p + 1) C_{m}} a_{n} = C a_{n} .$ Every quantity in $C$ was fixed in Step 0, before $P$ and $n$ , so the bound is uniform over the law class, and taking the supremum over $P \in P_{12} (p, ν, L)$ gives $P \in P_{12} (p, ν, L) sup E_{P}^{*} [x \in B_{P} sup τ_{n, a_{n}}^{12, LP} (x) - τ_{P} (x)] \leq C a_{n}, n \geq N,$ which is the assertion, with $C > 0$ and $N$ as constructed.

∎

Proof of Theorem 3.

Step 1 (constants). Fix $p \geq 0$ and put $q = p + 1$ . Let $L_{0} = L_{0} (p)$ be the envelope threshold supplied by Lemma 6, so $L_{0} \geq 48$ . Fix $ν \geq 2$ and $L \geq L_{0}$ , and let $c_{□}, A, C, C_{0}, δ_{0}$ be the positive constants that lemma attaches to $(p, ν, L)$ . Set $z = 128 q C_{0}, γ = min {1, z^{- 1}}, d = \frac{1}{2} {1 - exp (- 1/2)}, c = \frac{γ d}{8} .$ Then $γ > 0$ , $d > 0$ and $c > 0$ . Since $0 < γ \leq 1$ we have $γ^{4} \leq γ$ , and $γ \leq z^{- 1}$ therefore gives the budget inequality $128 q C_{0} γ^{4} \leq 128 q C_{0} γ = z γ \leq 1.$

Step 2 (the logarithmic packing budget). We first record, as an eventual statement in $n$ , the calibration that will put the hypercube of Lemma 6 inside the range of Lemma 1. For $n \geq 2$ the rate $a_{n} = (lo g n / n)^{1/4}$ of Definition 8 satisfies $n a_{n}^{4} = lo g n$ , so the budget inequality above gives $C_{0} n (γ a_{n})^{4} = C_{0} γ^{4} n a_{n}^{4} = C_{0} γ^{4} lo g n \leq \frac{lo g n}{128 q} .$ Also $lo g a_{n} = \frac{1}{4} (lo g lo g n - lo g n)$ , so every integer $M$ with $M \geq c_{□} (γ a_{n})^{- 1/ q}$ obeys $lo g M \geq lo g c_{□} - \frac{1}{q} lo g γ - \frac{1}{4 q} (lo g lo g n - lo g n) .$ For all sufficiently large $n$ we have $lo g lo g n \leq \frac{1}{2} lo g n$ , whence $- \frac{1}{4 q} (lo g lo g n - lo g n) \geq \frac{1}{8 q} lo g n$ ; and, since $lo g n \to \infty$ , also $lo g c_{□} - \frac{1}{q} lo g γ \geq - \frac{1}{32 q} lo g n$ . For such $n$ the last display yields $lo g M \geq - \frac{1}{32 q} lo g n + \frac{1}{8 q} lo g n \geq \frac{1}{32 q} lo g n .$ Combining the two estimates, for all sufficiently large $n$ and every integer $M \geq c_{□} (γ a_{n})^{- 1/ q}$ , $C_{0} n (γ a_{n})^{4} \leq \frac{lo g n}{128 q} = \frac{1}{4} \cdot \frac{lo g n}{32 q} \leq \frac{1}{4} lo g M .$

The de-Poissonization remainder $exp {- n (1 - lo g 2)}$ decays exponentially in $n$ , while $a_{n}$ decays only polynomially, so $exp {- n (1 - lo g 2)} = o (a_{n})$ .

Since $a_{n} \to 0$ , we may therefore fix $N \geq 2$ such that, for every $n \geq N$ , $γ a_{n} \leq δ_{0}, 2 L exp {- n (1 - lo g 2)} \leq c a_{n},$ and the displayed logarithmic packing budget holds.

Step 3 (the hard family at scale $Δ$ ). Fix $n \geq N$ and a known-geometry point-indexed rule $τ_{n} \in T_{n}^{12, \pm}$ , and choose a family of Borel sections ${T_{n, G, x}}$ representing it as in Definition 25. Put $Δ = γ a_{n}$ , so $0 < Δ \leq δ_{0}$ . Applying Lemma 6 at separation $Δ$ produces an integer $M$ , a radius $w = A Δ^{1/ q}$ , the disk mass $ρ = π w^{2} /36 > 0$ , centers $x_{1}, \dots, x_{M}$ on the middle of the bottom edge with pairwise disjoint disks $S_{j} = B_{2} (x_{j}, w) \subseteq [- 3, 3]^{2}$ , laws $P_{ω} \in P_{12} (p, ν, L)$ indexed by $ω \in {0, 1}^{M}$ all carrying the same fixed rectangular geometry $X = [- 3, 3]^{2}, A_{1} = [- 1, 1] \times [0, 2], A_{0} = X ∖ A_{1}, B = \partial A_{1},$ and conditional signed-distance laws $Q_{j, b}$ with $M \geq c_{□} Δ^{- 1/ q}$ . In particular $M \geq 1$ , because $c_{□} Δ^{- 1/ q} > 0$ .

Step 4 (per-coordinate divergence budget). The Kullback–Leibler clause of Lemma 6, the identity $ρ = π w^{2} /36$ , and $π \leq 4$ give, for every $j$ , $KL (Q_{j, 0}, Q_{j, 1}) \cdot 2 n ρ \leq \frac{C _{0} Δ ^{4}}{w ^{2}} \cdot 2 n \frac{π w ^{2}}{36} = C_{0} n Δ^{4} \cdot \frac{π}{18} \leq C_{0} n Δ^{4} \leq \frac{1}{4} lo g M,$ where the last inequality uses the logarithmic packing budget above and $M \geq c_{□} (γ a_{n})^{- 1/ q}$ .

Step 5 (target values at the centers). For $j \leq M$ and $b \in {0, 1}$ let $ϵ (j, b) \in {0, 1}^{M}$ be the vertex with $j$ th coordinate $b$ and all other coordinates $0$ , and put $v_{j, b} = τ_{P_{ϵ (j, b)}} (x_{j}) .$ Because the treatment-effect target at $x_{j}$ depends on the vertex only through its $j$ th coordinate, and because flipping that coordinate moves the target at $x_{j}$ by exactly $Δ$ (Lemma 6), $τ_{P_{ω}} (x_{j}) = v_{j, ω_{j}} for every ω, and ∣ v_{j, 1} - v_{j, 0} ∣ = Δ.$

Step 6 (the Poissonized experiment). For a probability law $Q$ on a standard Borel space and $λ \geq 0$ , write $Π_{λ} (Q)$ for the law of the finite marked configuration obtained by drawing $N \sim Pois (λ)$ points independently from $Q$ , attaching to each point an independent mark uniform on $(0, 1]$ , and listing the points in increasing order of their marks; the marks are independent of the points and almost surely distinct, so they only fix a canonical ordering. For a configuration $s$ write $∣ s ∣$ for its number of points and, when $∣ s ∣ \geq n$ , write $(Y_{1}, X_{1}), \dots, (Y_{n}, X_{n})$ for its first $n$ points. Let $G_{0}$ be the common known geometry of the hypercube, which is the same tuple for every vertex because all $P_{ω}$ share $X, A_{0}, A_{1}, B$ , the Euclidean metric and $K_{□}$ . Define the Poissonized value of the rule at a point $x$ by $τ_{n}^{Π} (x; s) = {T_{n, G_{0}, x} ((Y_{i}, D_{x, i}^{\pm}) : 1 \leq i \leq n), 0, ∣ s ∣ \geq n, ∣ s ∣ < n, D_{x, i}^{\pm} = {1 (X_{i} \in A_{1}) - 1 (X_{i} \in A_{0})} ∥ X_{i} - x ∥_{2},$ and the Poissonized maximum loss at a vertex $ω$ by $Λ_{ω} (s) = 1 \leq j \leq M max τ_{n}^{Π} (x_{j}; s) - v_{j, ω_{j}} .$ Both are measurable, each section $T_{n, G_{0}, x_{j}}$ being Borel.

Step 7 (independent blocks and their compressions). Split a configuration drawn from $Π_{2 n} (P_{ω})$ according to the partition of $[- 3, 3]^{2}$ into the pairwise disjoint disks $S_{1}, \dots, S_{M}$ and their complement. Because thinning a marked Poisson configuration by a measurable partition produces independent marked Poisson configurations, the blocks $Z_{j} = {points with score \in S_{j}} (1 \leq j \leq M), Z_{0} = {points with score \in / k = 1 ⋃ M S_{k}}$ are independent, $Z_{j} \sim Π_{2 n ρ} (P_{ω} (\cdot ∣ score \in S_{j}))$ , and $Z_{0}$ is the corresponding marked Poisson configuration built from the normalized law of $P_{ω}$ off $⋃_{k} S_{k}$ at the complementary intensity. By the bit-structure clause of Lemma 6 the law of $Z_{j}$ depends on $ω$ only through $ω_{j}$ , each disk having the same mass $ρ$ , while the law of $Z_{0}$ is common across all vertices.

Compress the $j$ th block by replacing each of its points by its signed observation at $x_{j}$ under the common geometry, keeping its mark; the compressed blocks $S_{1}, \dots, S_{M}$ are the compressed coordinates required by Lemma 1: $S_{j} = c_{j} (Z_{j}), c_{j} : (Y, X, mark) ⟼ (Y, {1 (X \in A_{1}) - 1 (X \in A_{0})} ∥ X - x_{j} ∥_{2}, mark) .$ By the signed-observation clause of Lemma 6, the point law of the compressed $j$ th block at bit $b$ is exactly $Q_{j, b}$ , so $L (S_{j} ∣ Ω_{j} = b) = Π_{2 n ρ} (Q_{j, b}), b \in {0, 1} .$

Poisson tensorization gives $KL$ equal to $2 n ρ KL (Q_{j, 0}, Q_{j, 1})$ for the two unordered marked configurations, and mark-ordering is a measurable map, which cannot increase divergence; hence, with the per-coordinate divergence bound above, $KL (L (S_{j} ∣ Ω_{j} = 0), L (S_{j} ∣ Ω_{j} = 1)) \leq 2 n ρ KL (Q_{j, 0}, Q_{j, 1}) \leq \frac{1}{4} lo g M .$

Step 8 (direct-product testing bound). Let $Ω_{1}, \dots, Ω_{M}$ be independent uniform bits, so that the vertex $Ω = (Ω_{1}, \dots, Ω_{M})$ is uniform on ${0, 1}^{M}$ . The block product experiment first draws the common block and the cell blocks conditionally independently according to the bit-conditional laws above, and then superposes those blocks; at vertex $ω$ , the superposed configuration has the canonical marked-Poisson law $Π_{2 n} (P_{ω})$ . Attach to coordinate $j$ the midpoint decoder $Ω_{j} = 1 {∣ τ_{n}^{Π} (x_{j}; \cdot) - v_{j, 1} ∣ \leq ∣ τ_{n}^{Π} (x_{j}; \cdot) - v_{j, 0} ∣} .$ The configuration feeding $τ_{n}^{Π} (x_{j}; \cdot)$ is obtained by superposing the compressed blocks $c_{j} (Z_{k})$ , $k \neq = j$ , the compressed ancillary block $c_{j} (Z_{0})$ , and the already compressed block $S_{j}$ ; the raw $j$ th block therefore enters only through $S_{j}$ . Hence $Ω_{j}$ is a measurable function of $(S_{j}, (Z_{1}, \dots, Z_{M}), Z_{0})$ that is unchanged when the raw coordinate vector is altered only in coordinate $j$ while $(S_{j}, Z_{- j}, Z_{0})$ is held fixed.

All hypotheses of Lemma 1 are thus met, and its Kullback–Leibler consequence with $κ = 1/4$ , fed by the compressed-coordinate divergence bound, gives for the block experiment $Pr {at least one decoding error} \geq \frac{1}{2} {1 - exp (- M^{3/4} /2)} \geq \frac{1}{2} {1 - exp (- 1/2)} = d,$ the middle inequality because $M \geq 1$ forces $M^{3/4} \geq 1$ .

Step 9 (from decoding error to loss, and averaging). If the midpoint decoder is wrong at coordinate $j$ under the block product experiment, then the estimate is at least as close to the wrong value as to the true one, so the separation $∣ v_{j, 1} - v_{j, 0} ∣ = Δ$ forces $Ω_{j} \neq = ω_{j} ⟹ τ_{n}^{Π} (x_{j}; s) - v_{j, ω_{j}} \geq \frac{Δ}{2} .$

After superposition, this is a statement under the canonical marked-Poisson law $Π_{2 n} (P_{ω})$ . Consequently, for every vertex $ω$ , $\frac{Δ}{2} Π_{2 n} (P_{ω}) Pr {\exists j : Ω_{j} \neq = ω_{j}} \leq E_{Π_{2 n} (P_{ω})} [Λ_{ω}] .$ Averaging this inequality over the $2^{M}$ vertices and bounding the average of the right-hand side by its maximum selects a vertex $ω$ with $\frac{Δ}{2} \cdot 2^{- M} ω^{'} \in {0, 1}^{M} \sum Π_{2 n} (P_{ω^{'}}) Pr {\exists j : Ω_{j} \neq = ω_{j}^{'}} \leq E_{Π_{2 n} (P_{ω})} [Λ_{ω}] .$ The left factor is the average block-product decoding-error probability bounded below in the direct-product display, transported through the superposition map; and $(Δ/2) d = (γ a_{n} /2) d = 4 c a_{n}$ by the definition of $c$ . Therefore $4 c a_{n} \leq E_{Π_{2 n} (P_{ω})} [Λ_{ω}] .$

Step 10 (de-Poissonization). Write $P = P_{ω}$ . The hypercube construction gives $P \in P_{12} (p, ν, L)$ and the fixed rectangular geometry, so $G_{P} = G_{0}$ and each center lies on $B_{P} = \partial ([- 1, 1] \times [0, 2])$ . Define the finite packing loss of the rule on a sample $W$ of size $n$ by $ℓ_{n, P}^{pack} (W) = 1 \leq j \leq M max τ_{n, P} (x_{j}) - τ_{P} (x_{j}),$ which is measurable in $W$ because each section $T_{n, G_{P}, x_{j}}$ is Borel and $W \mapsto U_{n, P}^{\pm} (x_{j})$ is measurable.

On the event ${∣ s ∣ \geq n}$ the Poissonized value at $x_{j}$ is the value of the rule on the retained prefix, because $G_{0} = G_{P}$ makes the two signed observations at $x_{j}$ coincide, and $τ_{P} (x_{j}) = v_{j, ω_{j}}$ by the target identity above; hence $Λ_{ω} (s) = ℓ_{n, P}^{pack} ((Y_{i}, X_{i})_{i \leq n}) whenever ∣ s ∣ \geq n .$

On the complementary event the Poissonized value is $0$ , so $Λ_{ω} (s) \leq max_{j} ∣ v_{j, ω_{j}} ∣$ ; and since the two potential-outcome regressions of Definition 22 satisfy $sup_{x \in X_{P}} ∣ μ_{t, P} (x) ∣ \leq L$ , the target $τ_{P} = μ_{1, P} - μ_{0, P}$ of Definition 23 obeys $∣ v_{j, ω_{j}} ∣ = ∣ τ_{P} (x_{j}) ∣ \leq 2 L .$

Conditionally on ${∣ s ∣ \geq n}$ the first $n$ points of the configuration are i.i.d. from $P$ , and $Pr {Pois (2 n) < n} \leq exp {- n (1 - lo g 2)}$ , so splitting the Poissonized expectation over the two events gives $E_{Π_{2 n} (P)} [Λ_{ω}] \leq \int ℓ_{n, P}^{pack} d P^{n} + 2 L exp {- n (1 - lo g 2)} .$

By the choice of $N$ the second term is at most $c a_{n}$ . Combining this upper bound with the preceding Poissonized lower bound and weakening $4 c a_{n}$ to $2 c a_{n}$ , $2 c a_{n} \leq \int ℓ_{n, P}^{pack} d P^{n} + c a_{n} .$ Cancelling the finite common term $c a_{n}$ yields $c a_{n} \leq \int ℓ_{n, P}^{pack} d P^{n} .$

Step 11 (passage to the two risks). Each $x_{j}$ lies on $B_{P}$ , so pointwise in the sample $ℓ_{n, P}^{pack} \leq x \in B_{P} sup τ_{n, P} (x) - τ_{P} (x) .$ The finite packing loss is measurable, so its integral is at most the outer expectation of the boundary supremum. Hence $c a_{n} \leq E_{P}^{*} [x \in B_{P} sup τ_{n, P} (x) - τ_{P} (x)] .$

Since $P \in P_{12} (p, ν, L)$ , taking the supremum over the law class and then the infimum over $τ_{n} \in T_{n}^{12, \pm}$ gives, by Definition 26, $c a_{n} \leq R_{n}^{12, \pm} (p, ν, L) for every n \geq N .$ Since $P$ also satisfies the fixed hard-geometry condition, the same two steps taken inside the restricted supremum of Definition 27 give $c a_{n} \leq R_{n}^{12, \pm, fix} (p, ν, L) for every n \geq N .$

Step 12 (normalization and liminf). For $n \geq 2$ we have $a_{n} = (lo g n / n)^{1/4} > 0$ , hence $a_{n}^{- 1} R_{n}^{12, \pm} (p, ν, L) = \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}} .$ The two sequences are eventually equal, so their lower limits agree, which is the first asserted identity.

Dividing the two eventual risk inequalities by $a_{n} > 0$ gives, for every $n \geq max {N, 2}$ , $c \leq \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}}, c \leq \frac{R _{n}^{12, \pm, fix} ( p , ν , L )}{a _{n}},$ and the order property of $lim inf$ under an eventual lower bound yields $c \leq n \to \infty lim inf \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}}, c \leq n \to \infty lim inf \frac{R _{n}^{12, \pm, fix} ( p , ν , L )}{a _{n}} .$ This proves the theorem.

∎

Proof of Theorem 1.

Write $a_{n} = (lo g n / n)^{1/4}$ .

By Theorem 2, applied with the same $q$ and $L$ , choose $c_{PI} > 0$ such that $c_{PI} \leq n \to \infty lim inf \frac{R _{n}^{PI}}{a _{n}} .$ Set $c_{CTY} = c_{PI}$ .
The two normalizations of the common-map risk agree eventually. Indeed, for all sufficiently large $n$ , $a_{n} > 0$ and $(\frac{n}{lo g n})^{1/4} R_{n}^{NP} = \frac{R _{n}^{NP}}{a _{n}},$ because $a_{n} = (lo g n / n)^{1/4}$ is the reciprocal of $(n / lo g n)^{1/4}$ . Taking liminfs of these eventually equal sequences gives $n \to \infty lim inf (\frac{n}{lo g n})^{1/4} R_{n}^{NP} = n \to \infty lim inf \frac{R _{n}^{NP}}{a _{n}} .$
For each $n$ , the common-map risk dominates the point-indexed risk: $R_{n}^{PI} \leq R_{n}^{NP} .$ Two facts combine to give this.

First, a common map in Definition 11 is a point-indexed rule in Definition 13: take the same Borel section $T_{n, x} = T_{n}$ at every boundary point, as recorded by Lemma 3. Hence the infimum defining $R_{n}^{PI}$ runs over a superset of the rules defining $R_{n}^{NP}$ .

Second, for a common-map rule the two expectation conventions coincide term by term. Fix $P \in P_{NP} (L, q)$ and a common map $T_{n}$ , and write the boundary loss at the sample point $w$ as $Λ_{P} (w) = x \in bd (X_{P}) sup T_{n} (V_{n, P} (x)_{w}) - μ_{P} (x) .$ The map $(x, w) ⟼ T_{n} (V_{n, P} (x)_{w}) - μ_{P} (x), (x, w) \in bd (X_{P}) \times (R \times R^{2})^{n},$ is jointly Borel: $T_{n}$ is measurable, $(x, w) \mapsto V_{n, P} (x) ∣_{w}$ is continuous in $x$ and Borel in $w$ , and $μ_{P}$ is continuous on the compact set $X_{P} \supseteq bd (X_{P})$ because it lies in the Hölder ball of Definition 4.

Consequently, for every level $a$ , ${w : Λ_{P} (w) > a} = {w : \exists x \in bd (X_{P}), T_{n} (V_{n, P} (x)_{w}) - μ_{P} (x) > a}$ is the projection onto the sample coordinate of a Borel subset of the product of the Polish space $bd (X_{P})$ with the sample space, hence an analytic set.

Analytic sets are universally measurable, so every superlevel set of $Λ_{P}$ is measurable for the completion of the product law generated by $P$ ; therefore $Λ_{P}$ is measurable on that completed space and its completed integral coincides with its outer expectation, $\int Λ_{P} d \overline{P^{n}} = E_{P}^{*} [x \in bd (X_{P}) sup μ_{n} (x) - μ_{P} (x)] .$

Taking the supremum over $P \in P_{NP} (L, q)$ in the last display and then the infimum over the larger point-indexed class in Definition 14 bounds $R_{n}^{PI}$ above by the common-map risk in Definition 12, which is the displayed inequality. Dividing by the same nonnegative normalizing factor preserves this order: $\frac{R _{n}^{PI}}{a _{n}} \leq \frac{R _{n}^{NP}}{a _{n}} .$
Applying monotonicity of liminf to the preceding pointwise normalized-risk inequality gives $n \to \infty lim inf \frac{R _{n}^{PI}}{a _{n}} \leq n \to \infty lim inf \frac{R _{n}^{NP}}{a _{n}} .$ Combining this with the choice of $c_{CTY}$ in the first step yields $c_{CTY} \leq n \to \infty lim inf \frac{R _{n}^{NP}}{a _{n}},$ and the equality of normalizations from the second step gives the displayed conclusion.

∎

Proof of Theorem 2.

Fix $q \geq 1$ and $L \geq 4$ , and write $a_{n} = (lo g n / n)^{1/4}$ .

A finite packing lower bound, uniform over point-indexed rules. Let $c_{0}, c_{1}, c_{w}^{-}, c_{w}^{+}, c_{rad}, α > 0$ with $α < 1/8$ be the constants of Lemma 2, and put $d = \frac{1}{2} (1 - e^{- 1/2}) > 0, c = \frac{c _{1} d}{4} > 0.$ Take $n$ beyond the threshold of that lemma, and let $M_{n}$ , $w_{n}$ , $m_{n}$ , the boundary points $x_{1}, \dots, x_{M_{n}}$ , the laws $P_{ω}$ , and the values $u_{j, 0}, u_{j, 1}$ be the objects it supplies, with $S = [- 1/2, 1/2]^{2}$ the common covariate support. Two of its clauses are used at once: $M_{n} \geq c_{0} a_{n}^{- 1/ q} > 0$ , so $M_{n} \geq 1$ ; and $m_{n} > 0$ , because $C_{j}$ contains a relatively open neighbourhood of the support point $x_{j}$ inside $S$ , which therefore has positive covariate mass, and that mass is $m_{n}$ .

Fix a point-indexed rule $μ_{n, P}^{PI} \in T_{n}^{PI}$ and law-independent Borel sections ${T_{n, x}}$ representing it as in Definition 13.

The Poissonized product experiment. Put independent uniform binary coordinates $Ω_{1}, \dots, Ω_{M_{n}}$ on ${0, 1}$ . Conditionally on $Ω = ω$ , let $N$ be Poisson with mean $2 n$ , independent of an i.i.d. stream of observations from $P_{ω}$ , and attach to each observation an independent mark uniform on $(0, 1]$ ; list the resulting $N$ marked observations in increasing mark order. Split this configuration along the partition of $S$ into the cells $C_{1}, \dots, C_{M_{n}}$ and their complement, and write $Z_{j} = the sub-configuration with covariate in C_{j} (1 \leq j \leq M_{n}), Z_{0} = the sub-configuration with covariate in S ∖ k ⋃ C_{k} .$ Splitting a Poisson configuration along a measurable partition makes the blocks independent conditionally on $Ω$ . For $j \geq 1$ , $Z_{j}$ is the marked Poisson configuration with intensity $2 n$ times the restriction of $P_{ω}$ to $C_{j}$ , equivalently with mean $2 n m_{n}$ and the conditional observation law on $C_{j}$ . The complement block $Z_{0}$ is the marked Poisson configuration with intensity $2 n$ times the restriction of $P_{ω}$ to $S ∖ ⋃_{k} C_{k}$ ; when this complement has positive mass this is the corresponding conditional law with that mean, and when the complement has zero mass it is the zero-intensity Poisson configuration. By the equal-cell-mass, one-bit-locality and common-outside clauses of Lemma 2, the conditional law of $Z_{j}$ depends on $Ω$ only through $Ω_{j}$ , and the conditional law of $Z_{0}$ is common for all values of $Ω$ , including the zero-intensity case.

The compressions and the decoders. The compressed blocks required by Lemma 1 are written $S_{1}, \dots, S_{M_{n}}$ here, the plain letter $S$ being reserved for the covariate support; let $S_{j} = c_{j} (Z_{j}), c_{j} : ((y, z), mark) ⟼ ((y, ∥ z - x_{j} ∥_{2}), mark),$ so that $S_{j}$ records, for each atom of the $j$ th cell block, its outcome, its distance from $x_{j}$ , and its mark.

For coordinate $j$ , superpose in mark order the compressed own block $S_{j}$ , the images under $c_{j}$ of the other cell blocks ${Z_{k}}_{k \neq = j}$ , and the image under $c_{j}$ of the complement block $Z_{0}$ ; call the result $s_{j}$ , and set $u_{j} = {T_{n, x_{j}} (first n atoms of s_{j} in mark order), 0, if s_{j} has at least n atoms, otherwise, Ω_{j} = 1 {∣ u_{j} - u_{j, 1} ∣ \leq ∣ u_{j} - u_{j, 0} ∣} .$

Each $Ω_{j}$ is a measurable ${0, 1}$ -valued function of $(S_{j}, (Z_{1}, \dots, Z_{M_{n}}), Z_{0})$ , and it is unchanged when the raw block $Z_{j}$ is altered while $(S_{j}, Z_{- j}, Z_{0})$ is held fixed, because $Z_{j}$ enters $s_{j}$ only through $S_{j}$ .

The coordinatewise divergence budget. Because $Pr {Ω_{j} = b} = 1/2$ for $b \in {0, 1}$ , the conditional law in Definition 1 is defined; write $Q_{j, b} = L (S_{j} ∣ Ω_{j} = b) .$ Then $KL (Q_{j, 0}, Q_{j, 1}) \leq 2 α lo g M_{n} for every j .$ Indeed, under $Ω_{j} = b$ , $S_{j}$ is exactly the block of radii at most $w_{n}$ of the marked Poisson experiment generated by the one-observation outcome-distance law at $x_{j}$ for the single-bit vertex with $j$ th bit $b$ . Restricting to that block is a measurable map, so it cannot increase divergence, and a marked Poisson experiment of mean $2 n$ has divergence at most twice that of the corresponding $n$ -fold product experiment, which is the compressed-distance divergence bounded by $α lo g M_{n}$ in Lemma 2.

The direct-product step. The uniform bits $Ω_{1}, \dots, Ω_{M_{n}}$ , the conditionally independent blocks $(Z_{0}, Z_{1}, \dots, Z_{M_{n}})$ , the compressions $c_{j}$ , and the decoders $Ω_{j}$ satisfy all the hypotheses of Lemma 1. Applying that lemma with $κ = 2 α < 1/4$ and using the previous display gives $Pr {at least one decoding error} \geq \frac{1}{2} [1 - exp {- M_{n}^{1 - 2 α} /2}] \geq d,$ the last inequality because $M_{n} \geq 1$ and $1 - 2 α \geq 0$ give $M_{n}^{1 - 2 α} \geq 1$ .

From decoding errors to loss, and choice of a vertex. If $Ω_{j} \neq = Ω_{j}$ , then the nearest-value decoder has picked the wrong endpoint, so the triangle inequality and the signal separation $∣ u_{j, 1} - u_{j, 0} ∣ \geq c_{1} a_{n}$ give $u_{j} - u_{j, Ω_{j}} \geq \frac{c _{1} a _{n}}{2} .$

Multiplying the previous two displays and averaging over the uniform prior on ${0, 1}^{M_{n}}$ selects a vertex $ω$ such that, writing $E_{2 n}^{ω}$ for expectation in the mean- $2 n$ marked Poisson experiment under $P_{ω}$ , $E_{2 n}^{ω} [1 \leq j \leq M_{n} max u_{j} - u_{j, ω_{j}}] \geq \frac{c _{1} a _{n}}{2} \cdot d = 2 c a_{n} .$

De-poissonization. Put $P = P_{ω}$ , so $P \in P_{NP} (L, q)$ , $X_{P} = S$ , and every $x_{j} \in bd (X_{P})$ . Define the finite packing loss of the rule at a sample point $w$ of size $n$ by $ℓ_{n, P}^{pack} (w) = 1 \leq j \leq M_{n} max μ_{n, P}^{PI} (x_{j}) - μ_{P} (x_{j}) evaluated at w,$ a measurable function of $w$ , being a finite maximum of compositions of the Borel sections $T_{n, x_{j}}$ with the measurable distance compressions of Definition 10.

On the event ${N \geq n}$ the first $n$ atoms in mark order are an exact i.i.d. sample from $P$ , and by construction $u_{j}$ is then the value of the rule at $x_{j}$ on that retained sample and $u_{j, ω_{j}} = μ_{P} (x_{j})$ ; hence the Poissonized maximum equals $ℓ_{n, P}^{pack}$ of the retained sample.

On the complementary event ${N < n}$ every $u_{j}$ equals the fallback value $0$ , so the maximum is at most $max_{j} ∣ u_{j, ω_{j}} ∣ = max_{j} ∣ μ_{P} (x_{j}) ∣ \leq L$ , the envelope coming from the zeroth-order derivative bound in $μ_{P} \in H^{q} (X_{P}, L)$ of Definition 4.

Since $N$ is Poisson with mean $2 n$ , a Chernoff bound for its lower tail gives $Pr {N < n} \leq exp {- n (1 - lo g 2)}$ , and therefore $E_{2 n}^{ω} [1 \leq j \leq M_{n} max u_{j} - u_{j, ω_{j}}] \leq E_{P} [ℓ_{n, P}^{pack}] + L exp {- n (1 - lo g 2)} .$

Because $exp {- n (1 - lo g 2)} = o (a_{n})$ , we may enlarge the threshold on $n$ so that $L exp {- n (1 - lo g 2)} \leq c a_{n}$ .

Combining the last three displays and cancelling one copy of the finite quantity $c a_{n}$ yields: for every point-indexed rule $μ_{n, P}^{PI} \in T_{n}^{PI}$ there are $P \in P_{NP} (L, q)$ , an integer $M_{n}$ , and boundary points $x_{1}, \dots, x_{M_{n}} \in bd (X_{P})$ with $E_{P^{n}} [1 \leq j \leq M_{n} max μ_{n, P}^{PI} (x_{j}) - μ_{P} (x_{j})] \geq c a_{n} .$
For such $P$ and $x_{1}, \dots, x_{M_{n}}$ , the finite maximum is bounded pointwise by the full boundary loss because each $x_{j}$ lies on $bd (X_{P})$ : $1 \leq j \leq M_{n} max μ_{n, P}^{PI} (x_{j}) - μ_{P} (x_{j}) \leq x \in bd (X_{P}) sup μ_{n, P}^{PI} (x) - μ_{P} (x) .$ The left-hand side is measurable, so its ordinary expectation is bounded above by the outer expectation of the right-hand side, that is, by the quantity appearing in Definition 14. Taking the supremum over $P \in P_{NP} (L, q)$ , and then the infimum over all rules in $T_{n}^{PI}$ , gives a sample-size threshold $n_{0}$ , distinct from the Poisson count $N$ introduced in the Poissonized experiment, such that $R_{n}^{PI} \geq c a_{n} for every n \geq n_{0} .$
For every $n \geq 2$ , $a_{n} > 0$ and $(\frac{n}{lo g n})^{1/4} R_{n}^{PI} = \frac{R _{n}^{PI}}{a _{n}},$ because $(n / lo g n)^{1/4}$ is the reciprocal of $a_{n}$ . Thus the two normalized sequences agree eventually, so monotonicity of $lim inf$ in both directions gives $n \to \infty lim inf (\frac{n}{lo g n})^{1/4} R_{n}^{PI} = n \to \infty lim inf \frac{R _{n}^{PI}}{a _{n}} .$
For every $n \geq max {n_{0}, 2}$ , the positivity of $a_{n}$ allows division in the preceding eventual lower bound: $c \leq \frac{R _{n}^{PI}}{a _{n}} .$ Taking $lim inf$ preserves this eventual lower bound. With $c_{PI} = c > 0$ , the asserted equality of normalizations and the lower bound $n \to \infty lim inf \frac{R _{n}^{PI}}{a _{n}} \geq c_{PI}$ follow.

∎

Proof of Theorem 4.

1. Apply Theorem 3 for the chosen order $p$ , and let $L_{conv} \geq 48$ be the envelope threshold it provides. Set $L_{0} = max {L_{conv}, 4} .$ Then $L_{0} \geq 48$ . Fix $ν \geq 2$ , $L \geq L_{0}$ , and the three displayed hypotheses of the theorem. The choice of $L_{0}$ gives both $L \geq L_{conv}$ and $L \geq 4$ .

2. The converse theorem at $L_{conv}$ supplies a constant $c > 0$ such that $c \leq n \to \infty lim inf \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}} .$ That theorem also records the parallel lower bound for the fixed-geometry risk $R_{n}^{12, \pm, fix} (p, ν, L)$ of Definition 27; only the full-risk statement just displayed is used below.

3. By Proposition 1, applied with $L \geq 4$ and the distance-identification, first-order-bias, and expected-maximal-bound hypotheses, there are $C_{0} > 0$ and $N$ such that, for every $n \geq N$ , $P \in P_{12} (p, ν, L) sup E_{P}^{*} [x \in B_{P} sup τ_{n, a_{n}}^{12, LP} (x) - τ_{P} (x)] \leq C_{0} a_{n} .$

Since $a_{n} > 0$ eventually, division by $a_{n}$ and passage to the limsup give $n \to \infty lim sup \frac{sup _{P \in P_{12} (p, ν, L)} E _{P}^{*} [ sup _{x \in B_{P}} τ _{n, a_{n}}^{12, LP} ( x ) - τ _{P} ( x ) ]}{a _{n}} \leq C_{0} .$

4. The explicit rule is admissible. For every $n \geq 2$ the frontier bandwidth $a_{n}$ is positive, so the winsorization level $B (a_{n}) = a_{n}^{- 1/3}$ of Definition 28 is well defined, and the estimator is generated by the law-independent family of sections $T_{n, G, x} (u) = max {- 2 L, min {2 L, e_{1}^{⊤} (β_{1}^{B} (a_{n}; u) - β_{0}^{B} (a_{n}; u))}}, u \in (R \times R)^{n},$ where $β_{t}^{B} (a_{n}; u)$ is the stabilized coefficient of Definition 28 computed from the signed-distance sample $u$ . This family does not depend on the unknown member law, and each fixed- $(G, x)$ section is Borel in $u$ , because the empirical Gram $Ψ_{t, x} (a_{n})$ , the uniform conditioning event ${(2 L)^{- 1} \sum_{j} v_{j}^{2} \leq v^{⊤} Ψ_{t, x} (a_{n}) v for all v}$ , the matrix inverse on that event, the winsorized empirical moment $m_{t, x}^{B} (a_{n})$ , and the clipping map are all Borel functions of $u$ . Moreover, for every $P \in P_{12} (p, ν, L)$ and every $x \in B_{P}$ , the signed-distance sample built from the known geometry $G_{P}$ of Definition 20 is exactly $U_{n, P}^{\pm} (x)$ of Definition 21, since the treatment indicator is the deterministic function $1 {X \in A_{1, P}}$ of the covariate recorded in $G_{P}$ . Hence $τ_{n, a_{n}}^{12, LP} \in T_{n}^{12, \pm} .$

Therefore the minimax risk of Definition 26 is bounded above by the risk of this explicit rule: $R_{n}^{12, \pm} (p, ν, L) \leq P \in P_{12} (p, ν, L) sup E_{P}^{*} [x \in B_{P} sup τ_{n, a_{n}}^{12, LP} (x) - τ_{P} (x)] .$

After normalizing by $a_{n}$ , this inequality holds eventually in $n$ , and monotonicity of limsup yields $n \to \infty lim sup \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}} \leq n \to \infty lim sup \frac{sup _{P \in P_{12} (p, ν, L)} E _{P}^{*} [ sup _{x \in B_{P}} τ _{n, a_{n}}^{12, LP} ( x ) - τ _{P} ( x ) ]}{a _{n}} \leq C_{0} .$

5. Finally set $C = max {C_{0}, c} .$ Then $0 < c \leq C$ . The lower bound from Step 2, the general inequality $lim inf \leq lim sup$ , and the two upper bounds from Steps 3 and 4 give $c \leq n \to \infty lim inf \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}} \leq n \to \infty lim sup \frac{R _{n}^{12, \pm} ( p , ν , L )}{a _{n}} \leq C,$ and $n \to \infty lim sup \frac{sup _{P \in P_{12} (p, ν, L)} E _{P}^{*} [ sup _{x \in B_{P}} τ _{n, a_{n}}^{12, LP} ( x ) - τ _{P} ( x ) ]}{a _{n}} \leq C .$ These are exactly the asserted frontier bounds.

∎

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Cattaneo, Matias D. and Idrobo, Nicolás and Titiunik, Rocío (2019). A Practical Introduction to Regression Discontinuity Designs. Cambridge University Press. doi
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Formal layer — Lean correspondence

Every object below is formalized in Lean 4. Click a verified row to view its statement; the rest link to the Lean source.

Setups (7)

Setup S-1 — Cattaneo–Titiunik–Yu nonparametric regression world on a compact bivariate random design: a law P of (Y,X) on \\mathbb R \\times \\mathbb R^2 whose DERIVED objects are tied to the law itself — \\mathcal X_P is the support of the X-marginal, f_P IS that marginal's Lebesgue density, \\mu_P IS the continuous version of x \\mapsto \\mathbb E_P[Y \\mid X = x] and \\sigma^2_P IS the continuous version of x \\mapsto \\operatorname{Var}_P(Y \\mid X = x), all realised through the regular conditional kernel of Y given X — together with the class regularity (compact support in [-L,L]^2 with rectifiable boundary, L^{-1} \\le f_P \\le L, \\mu_P in the q-Hölder ball of radius L, L^{-1} \\le \\sigma^2_P \\le L) and i.i.d. sampling (Y_i,X_i)_{i=1}^n \\sim P^{\\otimes n}. — ROUND-7 REVISION (D-F1) — REUSE-MISFIT REPAIR, and the ONLY thing that moves on this node: the decl cited as certifying the fidelity field (F3) was wrong. ProbabilityTheory.condDistrib_ae_eq_condExp (CondDistrib.lean:321), read in source this round, concerns conditional expectations of INDICATORS of a fixed measurable set — it certifies that condDistrib is a regular conditional DISTRIBUTION, not the (F3) mean equality. THE (F3) ANCHOR IS NOW ProbabilityTheory.condExp_ae_eq_integral_condDistrib' (CondDistrib.lean:389, verified in source this round): (hX : Measurable X) (hY_int : Integrable Y mu) : mu[Y | mbeta.comap X] =ᵐ[mu] fun a => ∫ y, y ∂condDistrib Y X mu (X a), all six typeclass side conditions on the codomain met at ℝ and the ambient [IsFiniteMeasure mu] met by [IsProbabilityMeasure law]. INTEGRABILITY DISCHARGE, recorded as the reviewer requires: its explicit hypothesis Integrable Prod.fst law comes from the structure's own sq_integrable : MemLp Prod.fst 2 law field by MeasureTheory.MemLp.integrable (L1Space/Integrable.lean:649, (hq1 : 1 ≤ q) [IsFiniteMeasure mu] : MemLp f q mu -> Integrable f mu) at q = 2 — so sq_integrable is load-bearing twice over: it blocks the Bochner/variance junk values (as round 6 recorded) AND it is the hypothesis that unlocks the (F3) certificate. ONE BOOKKEEPING SEAM F2 must not fumble: the anchor is stated a.e. under law for the composite x ∘ Prod.snd, whereas (F3) is stated a.e. under the pushforward law.map Prod.snd; MeasureTheory.ae_map_iff / MeasureTheory.ae_of_ae_map (Map.lean:243/:247, both verified this round, Prod.snd being measurable) convert between the two. THE MIS-CITED DECL IS RETAINED WHERE IT FITS: condDistrib_ae_eq_condExp certifies that the kernel is a regular conditional distribution, which is what licenses reading (F4) — a functional of the conditional LAW — as Var_P(Y | X = ·); for (F4)'s second-moment half F2 may additionally use ProbabilityTheory.condExp_ae_eq_integral_condDistrib (:381) at f = fun y => y^2, whose integrability hypothesis is MeasureTheory.MemLp.integrable_norm_pow (:202) from the SAME sq_integrable field. Nothing else on this node moves: the four fidelity fields, the structure layout, the define-local disposition, the PO bypass-justification, binds_symbols, and every other reuse anchor are exactly as round 6 left them. PRIOR TEXT (round 6), authoritative except for the two condDistrib citations corrected above: ROUND-6 REVISION (D-E0) — WORLD-FIDELITY REPAIR, the whole point of this round. PRIOR DEFECT (reviewer FLAG-world-infidelity): CtyLaw was planned as a bag of INDEPENDENT fields — a measure, a set, and three bare functions f_P, mu_P, sigma_P^2 — with the class then imposing envelopes and Hölder regularity on those functions but NEVER relating them to the measure. Under that encoding a CtyLaw could carry any smooth bounded triple beside an unrelated law, the estimand mu_P would not be the regression of the sampled data, and both converse theorems would be about a decorated law rather than about E_P[Y | X = x]. REPAIRED ENCODING, which F2 must follow field for field: structure CtyLaw carries (1) law : Measure (ℝ × EuclideanSpace ℝ (Fin 2)) with [IsProbabilityMeasure law] — the joint law of (Y,X), sample coordinates ordered as in def:cty-distance-data, which is UNCHANGED; (2) support : Set (EuclideanSpace ℝ (Fin 2)) and density : EuclideanSpace ℝ (Fin 2) → ℝ (the object named f_P); (3) mu, sigmaSq : EuclideanSpace ℝ (Fin 2) → ℝ; and then the FOUR FIDELITY FIELDS that make the world faithful and that are the substance of this repair — (F1) marginal_eq : law.map Prod.snd = volume.withDensity (fun x => ENNReal.ofReal (support.indicator density x)), i.e. f_P IS the Lebesgue density of X under P and it vanishes off the support; (F2) support_closed : IsClosed support together with density_pos : ∀ x ∈ support, 0 < density x, from which the non-node helper support_eq_measureSupport proves that support IS the topological support of the X-marginal, so \\mathcal X_P is the core's 'support of X under P' rather than an arbitrary labelled set; (F3) mu_condMean : ∀ᵐ x ∂(law.map Prod.snd), mu x = ∫ y, y ∂((ProbabilityTheory.condDistrib Prod.fst Prod.snd law) x); (F4) sigmaSq_condVar : ∀ᵐ x ∂(law.map Prod.snd), sigmaSq x = ProbabilityTheory.variance id ((ProbabilityTheory.condDistrib Prod.fst Prod.snd law) x); plus the integrability field sq_integrable : MemLp Prod.fst 2 law WITHOUT which (F3)/(F4) would be silently satisfied by Bochner/variance junk values. The regular-conditional-kernel route is the core's own 'regular conditional distribution of Z_j given (S_j, Omega_j)' idiom and is the one Mathlib supports: ProbabilityTheory.condDistrib (Y := Prod.fst) (X := Prod.snd) law : Kernel (EuclideanSpace ℝ (Fin 2)) ℝ type-checks here because ℝ is standard Borel and nonempty and law is finite (signature verified by loogle this round), and — CORRECTED IN ROUND 7, this clause originally named condDistrib_ae_eq_condExp — ProbabilityTheory.condExp_ae_eq_integral_condDistrib' (Mathlib/Probability/Kernel/CondDistrib.lean:389, read in source this round) certifies that (F3) really is E_P[Y | X] rather than a bespoke surrogate, its Integrable hypothesis being discharged from sq_integrable as above. AE-VERSUS-POINTWISE, the one subtlety F2 must not fumble: (F3)/(F4) hold only a.e. under the X-marginal, whereas the ESTIMAND lives on bd(\\mathcal X_P), a Lebesgue-null set. The class pins the boundary values through CONTINUITY, not through the a.e. relation: since density >= L^{-1} > 0 on the support, every nonempty relatively open subset of the support has positive marginal mass, so a continuous function on the support is determined by its a.e.-class — F2 proves this ONCE as the non-node helper continuous_version_unique_on_support in Basic.lean, and it is what makes mu_P(x) at a boundary point a well-defined functional of P (and hence makes the packing lemma's explicit boundary values u_{j,b} the genuine regression values, not a convention). SETTLED AND UNCHANGED: the disposition stays define-local; the bypass-justification vs Causalean PO stands verbatim — the core has no treatment variable, no potential outcomes and no assignment geometry, so Causalean.PO.POSystem (../Causalean/PO/Core/System.lean:34) and POSharpRDDSystem (../doc/API.md §9i'/§9i0b, a SCALAR running variable with a cutoff and a one-sided-limit target) are both at the wrong abstraction for a law-quantified minimax programme on \\mathbb R^2; Causalean.Stat.Nonparametric.HolderBallStd (Defs.lean:98) is still reused AT d = 2 for the Hölder condition, with isCompact_cube/supBall for the cube geometry; binds_symbols is untouched (D-B4). NEW REUSE from this repair (citation corrected in round 7): Mathlib ProbabilityTheory.condDistrib + condExp_ae_eq_integral_condDistrib' (the (F3) anchor) + condDistrib_ae_eq_condExp (retained as the regular-conditional-distribution certificate behind (F4)) + condDistrib_def, MeasureTheory.MemLp.integrable and MeasureTheory.ae_of_ae_map (the (F3) side conditions), ProbabilityTheory.condKernel_compProd (Mathlib/Probability/Kernel/Disintegration/Unique.lean — the a.e.-uniqueness that lets an EXPLICITLY CONSTRUCTED kernel be identified with condDistrib, consumed by t2 and t4), MeasureTheory.Measure.fst_compProd, ProbabilityTheory.variance (Mathlib/Probability/Moments/Variance.lean), MeasurableEquiv.prodComm for the (X,Y)-to-(Y,X) transport of a compProd, and Causalean's own disintegration layer Causalean.measure_eq_bind_marginal_condDistrib (Causalean/Mathlib/CondDistrib.lean:181) / Causalean.condDistrib_comp_right_measurableEquiv (:110) / Causalean.condDistrib_map_comp (:276), which are exactly the swap-and-compose lemmas the construction nodes need. SEARCH TRACE for this round's edit: grepped ../Causalean for 'condDistrib|condKernel|HasPDF|Measure\\.fst' -> 30 files, all in SCM/ID (do-calculus density factorisation), Estimation/CATE (conditional bias / pseudo-outcome means) and Causalean/Mathlib/CondDistrib.lean; grepped ../Causalean for structure .*(RegressionLaw|NonparamLaw|ObservedLaw|CondLaw|Design) -> only Panel/EventStudy ConventionalDesign/IWDesign and Experimentation FiniteDesign, none of which is a random-design regression law. So Causalean has the KERNEL PLUMBING (reused above) but NO packaged 'law with its density and conditional moments' structure. Mathlib-side: loogle MeasureTheory.HasPDF -> FOUND (Mathlib/Probability/Density.lean) and it is the right idiom for (F1) if F2 prefers HasPDF Prod.snd law volume + pdf to the explicit withDensity equation — either spelling is acceptable, the explicit equation is planned because the packing construction builds the marginal as a withDensity in the first place; loogle ProbabilityTheory.condDistrib / ProbabilityTheory.variance -> FOUND with the signatures used above. Not found: any ready-made compact-support random-design regression law world — hence still define-local.
Setup S-2 — Coordinatewise distance-compressed observation world: at a boundary point x the estimator sees only V_{n,P}(x) = ((Y_i, \\lVert X_i - x \\rVert_2) : i \\le n) \\in (\\mathbb R \\times [0,\\infty))^n, and a decision rule \\widehat\\mu_n is generated by ONE common law-independent measurable map T_n applied to that compressed sample at every boundary point. — Realises D-B2. The unsigned-distance compression is a deterministic measurable coordinate projection of the sample, encoded as distanceData : CtyLaw → (Fin n → ℝ × ℝ²) → ℝ² → (Fin n → ℝ × ℝ); the class is a Set of boundary-indexed rules carrying the single-common-map factorisation. Search trace: grepped ../doc/API.md for 'compression', 'compressed sample', 'sufficient statistic', 'decision class', 'estimator class', 'channel' -> the CausalSmith DP run (doc/API.md §3.7) compresses through a PRIVACY CHANNEL with a TV-contraction premise (an analogous compression-barrier pattern, recorded on the theorem node), and Causalean.Stat.minimaxValue takes an abstract estimator type E so it carries no class; nothing realises a deterministic coordinatewise projection class. Mathlib-side: no decision-theoretic layer. Not found.
Setup S-3 — Minimax decision-theoretic world: the same-class risk R_n^{\\mathrm{NP}} = \\inf_{\\widehat\\mu_n \\in \\mathcal T^{\\mathrm{NP}}_n} \\sup_{P \\in \\mathcal P_{\\mathrm{NP}}(L,q)} \\mathbb E_P[\\sup_{x \\in \\operatorname{bd}(\\mathcal X_P)} |\\widehat\\mu_n(x) - \\mu_P(x)|], and the frontier rate a_n against which it is normalised. — ROUND-10 REVISION (D-J0), note only — the world choice, the define-local disposition, binds_symbols and the round-7 Measure.completion encoding of R_n^NP are ALL UNCHANGED. WHAT MOVES: the certificate that this world's expectation IS the CTY ordinary completed expectation is now UNCONDITIONAL. Round 9 made it conditional on the named external assumption AnalyticSetNullMeasurable; that fact is now a PROVED Causalean theorem (MeasureTheory.AnalyticSet.nullMeasurableSet, with the function-level MeasureTheory.UpperSemianalytic.measurable_completion and MeasureTheory.UpperSemianalytic.lintegral_completion_eq_outerLIntegral, module Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability, ../doc/API.md section 10k'''', 0-sorry, source-verified this round), so S3 carries the completed expectation TOGETHER WITH an unconditional proof that it is one. The module list is unchanged and already names that Causalean module; Helpers.AnalyticMeasurability stays because the three localizing lemmas built over it live there. S4's outer-expectation world is untouched, and its outerLintegral now has an upstream twin MeasureTheory.outerLIntegral that the NEW S7 world reuses directly (see S7). PRIOR TEXT: ROUND-9 REVISION (D-H0), note only — the world choice, the define-local disposition, binds_symbols, the module list and the round-7 Measure.completion encoding of R_n^NP are ALL UNCHANGED. WHAT MOVES IS THE STATUS OF THE CERTIFICATE that this world's expectation IS the CTY ordinary completed expectation. Round 8 planned that certificate as an in-run Souslin/Choquet BUILD; the reviewer sized analyticSet_nullMeasurable as absent from both Mathlib and Causalean and as a Mathlib-grade descriptive-set-theory development, hence not bounded build-inline work. It is now the NAMED EXTERNAL SUBSTRATE ASSUMPTION AnalyticSetNullMeasurable (Helpers/AnalyticMeasurability.lean, infrastructure item I-1, LARGE), STATED and not proved, and THREADED as an explicit hypothesis through commonMapBoundarySup_completionMeasurable, ctyBoundarySup_completed_eq_outer and ctyRisk_ge_pointIndexedRisk into the exported t3. So S3 still carries the ORDINARY COMPLETED expectation as its object, and the proof that it is one is now CONDITIONAL on a named classical interface rather than claimed in-run; Helpers.AnalyticMeasurability and Mathlib.MeasureTheory.Constructions.Polish.Basic stay in the module list for exactly that chain. S4's outer-expectation world is untouched and unconditional. PRIOR TEXT: ROUND-8 REVISION (D-G0), modules and note only — the world choice, the define-local disposition, binds_symbols and the round-7 Measure.completion encoding of R_n^NP are ALL UNCHANGED, and the directive is explicit that ordinary expectation must NOT be redefined as a lower lintegral. WHAT MOVES: this world's expectation is now CERTIFIED to be the CTY ordinary completed expectation rather than merely defined against the completion. Helpers/AnalyticMeasurability.lean joins the module list because commonMapBoundarySup_completionMeasurable and ctyBoundarySup_completed_eq_outer are BLOCKING build-inline obligations of this world's risk, not optional decorations, and Mathlib.MeasureTheory.Constructions.Polish.Basic joins it as the AnalyticSet layer they are built over. S3 therefore carries the ORDINARY COMPLETED expectation together with a proof that it is one, and S4 carries the OUTER expectation, exactly as the core declares them. PRIOR TEXT: ROUND-7 REVISION (D-F0), modules and the numeric encoding only — the world choice, the define-local disposition and the bound symbols are AGAIN UNCHANGED. R_n^NP is no longer the BASE lower integral: its expectation is the core's COMPLETED expectation, encoded as ∫⁻ ... ∂((P.law.pi n).completion) against MeasureTheory.Measure.completion (Mathlib/MeasureTheory/Measure/NullMeasurable.lean:444), which is why Mathlib.MeasureTheory.Measure.NullMeasurable and the two completion-transfer helpers in Helpers/OuterExpectation.lean join the module list. The ℝ≥0∞ ambient introduced in round 5 is kept (complete lattice, no junk-value convention, no finite-risk reduction), and so is the round-5 finding that Causalean.Stat.minimaxValue cannot wrap an ℝ≥0∞ risk. This world is now the honest formal shadow of the core's own E_P / E*_P distinction: S3 carries the COMPLETED expectation, S4 the OUTER expectation, exactly as the core declares them. PRIOR TEXT (round 5): ROUND-5 REVISION (D-D0), modules and reuse only — the world choice, the define-local disposition and the bound symbols are UNCHANGED. The risk R_n^NP is now ℝ≥0∞-valued on Mathlib's LOWER integral MeasureTheory.lintegral, matching S4's R_n^PI, so that t3's planned proof route is coherent with the object it bounds; Causalean.Stat.Minimax.MinimaxValue is dropped from the module list because minimaxValue/worstCaseRisk and every subtype spelling are : ℝ and cannot wrap an ℝ≥0∞ risk, and MinimaxValue.lean:36-38 itself directs such a loss to the Mathlib complete-lattice API (iInf_le, le_iInf, le_iSup, iSup_le). PRIOR TEXT, world-level half still binding: DEFINE-LOCAL (D-A8, re-affirmed). The S3 world is this paper's coordinatewise-compression decision-theoretic world: the class \\mathcal T^{\\mathrm{NP}}_n from S2, the LAW-DEPENDENT uniform loss whose index set \\operatorname{bd}(\\mathcal X_P) moves with the law being supped over, and the rate a_n. None of that is realised by any Causalean decl. BODY-LEVEL REUSE (RETIRED THIS ROUND by D-D0 when R_n^NP moved to ℝ≥0∞; the paragraph is kept for provenance and F2 must NOT scaffold it — the replacement is the Mathlib ENNReal complete-lattice API named in MinimaxValue.lean:36-38, i.e. iInf_le/le_iInf for the rule infimum and le_iSup/iSup_le for the class supremum, with no Nonempty/BddAbove/BddBelow side conditions to discharge): Causalean.Stat.minimaxValue (../Causalean/Stat/Minimax/MinimaxValue.lean:59, namespace Causalean.Stat) is the generic (E → Θ → ℝ) → ℝ inf-sup operator with Causalean.Stat.worstCaseRisk (:50); def:cty-distance-risk instantiates it at E := the local decision-class subtype and Θ := the local law class, via the subtype spellings minimaxValue_subtype_eq_sInf_setOf (:288) / worstCaseRisk_subtype_eq_sSup_setOf (:270). SEARCH TRACE (carried from round 2, re-checked): grepped ../Causalean/Stat/ for 'decision class|DecisionClass|decisionClass|estimator class|supNorm|uniformLoss|classFrontierRisk' -> single hit Causalean.Stat.classFrontierRisk in Minimax/HonestConfidenceSet.lean, which packages expected CONFIDENCE-SET LENGTH against a strength threshold (an analogous liminf/limsup frontier PACKAGING pattern, recorded on the theorem node, not a decision-class world); grepped ../Causalean for 'structure .*(Experiment|DecisionProblem|RiskWorld|MinimaxProblem)' -> only the Experimentation-cluster design structures. Mathlib-side not applicable. Not found.
Setup S-4 — Point-indexed sectionwise-Borel decision world with outer-expectation risk: a rule is generated by a law-independent family {T_{n,x} : x in R^2} whose fixed-x section T_{n,x} : (R x [0,inf))^n -> R is Borel measurable, with NO joint measurability, continuity or common modulus in x imposed; consequently the boundary loss sup over x in bd(X_P) of |mu-hat^PI_{n,P}(x) - mu_P(x)| need not be measurable, and the risk R_n^PI is an inf-sup of OUTER expectations E*_P valued in the extended nonnegative reals. — NEW world (D-C0). It is the strict enlargement of S2 and reuses S1 verbatim for the law side (same class, same sampling model, same compression from S2 — only the decision architecture and the risk functional change), so nothing settled in S1-S3 moves. define-local for the same reason S2/S3 are (D-A8): the sectionwise-Borel family, the law-dependent uniform loss and the outer expectation are all this paper's own objects. SEARCH TRACE: Causalean Grep over the whole tree for 'outer(Integral|Expectation|Lintegral)|outer expectation' -> NO FILES; grepped ../doc/API.md for 'outer expectation', 'nonmeasurable', 'universally measurable', 'analytic set', 'sectionwise', 'Borel section', 'point-indexed' -> nothing (the Minimax layer is entirely ordinary-expectation and abstract-estimator-typed). Mathlib-side: lean_local_search 'outerLintegral' -> EMPTY; Mathlib has Measure.toOuterMeasure but no outer-integral layer, and its sectionwise-measurability lemmas all sit downstream of JOINT measurability (Fubini-Tonelli), so none applies. Not found — hence Helpers/OuterExpectation.lean is an explicit build-inline helper (D-C2), with Mathlib lintegral/lintegral_mono (the lower integral, no measurability hypothesis, verified this round) reused as its engine.
Setup S-5 — ROUND-10 NEW WORLD (D-J0), the SECOND co-main world: the Cattaneo-Titiunik-Yu Assumptions 1-2 potential-outcome geographic-boundary design. A law P carries the triple (Y(0), Y(1), X) on R x R x R^2 together with a KNOWN Borel partition of the compact rectangular support into the two assignment regions A_0,P and A_1,P; the deterministic assignment is T = 1{X in A_1,P}, the observed outcome obeys consistency Y = T Y(1) + (1 - T) Y(0), the potential-outcome regressions mu_t,P and conditional variances sigma^2_t,P are the continuous versions of the conditional mean and variance of Y(t) given X, the known interface B_P = bd(A_0,P) cap bd(A_1,P) is a compact rectifiable interior curve with H^1 mass in [L^{-1}, L], the metric is exactly Euclidean and the kernel is exactly K_square, the armwise conditional absolute (2 + nu)-moment of Y(t) given X = x is bounded by L UNIFORMLY AT EVERY POINT x OF THE SUPPORT, taken against a SELECTED conditional kernel that the law object carries as a primitive (ROUND-12/D-L0: the envelope is a genuine POINTWISE statement about that selected version, not an a.e. statement about an abstract conditional expectation, and the global (2 + nu)-integrability of each arm is a DERIVED bookkeeping consequence rather than a primitive field), and P_12(p, nu, L) is the exact displayed L-uniformized specialization of clauses (i)-(x). This is a DIFFERENT world from S1 and is NOT a subclass of it: S1 has no treatment, no potential outcomes and no assignment geometry, and its target is the observed-outcome regression on the SUPPORT boundary, whereas S5's target is the treatment-effect curve tau_P on the KNOWN INTERIOR interface. The two co-main results therefore share only the ambient rate a_n and the direct-product lemma; nothing settled in S1-S4 moves. — ROUND-12 REVISION (D-L0 finding FLAG-WORLD-F5) -- the ONLY thing that moves on this node, and it moves at the world layer again so that no consumer can bypass it. THE DEFECT, accepted in full: round 11 made condAbsMoment a REAL-VALUED FIELD tied to the conditional kernel by the a.e. fidelity field (F5) ∀ᵐ x ∂(law.map score), condAbsMoment t x = ∫ |y|^(2+nu) d(condDistrib (armCoord t) score law) x, and then let clause (v) and t7 use that identification AT EVERY POINT of the support. An a.e. equality licenses nothing pointwise: condAbsMoment is not assumed continuous anywhere in the class (unlike mu and sigmaSq, whose boundary values the S1 pattern pins THROUGH CONTINUITY), so the round-11 world still did not entitle t7 to evaluate the kernel moment at a fixed x, and the winsorization bias remained unprovable as stated. THE REPAIR -- A SELECTED CONDITIONAL KERNEL, carried as a primitive, with the envelope taken on THAT version. A1A2Law now carries, per arm, the field condKer : Bool -> ProbabilityTheory.Kernel Score R with [∀ t, IsMarkovKernel (condKer t)], together with the SINGLE fidelity field that replaces the round-11 (F5): (F5) condKer_disint : ∀ t, (law.map score).compProd (condKer t) = law.map (fun w => (score w, armCoord t w)). That is an EQUALITY OF MEASURES, carrying no a.e. quantifier, and it says exactly that condKer t IS a regular conditional distribution of Y(t) given X -- a SELECTED version, chosen once and for all by the law object. condAbsMoment is then NOT A FIELD at all but a DEFINITION on the structure, A1A2Law.condAbsMoment t x := ∫ y, |y| ^ (2 + nu) ∂(P.condKer t x), so it has a meaning at EVERY x by construction and clause (v)'s pointwise envelope is a genuine statement about the object t7 integrates. Likewise (F3)/(F4) are RESTATED against the same selected kernel -- ∀ᵐ x, muPO t x = ∫ y ∂(condKer t x) and ∀ᵐ x, sigmaSqPO t x = variance id (condKer t x) -- so the world carries ONE kernel and F2 must not mint a second. THE TWO-WAY BRIDGE TO condDistrib, verified in source this round, is what makes this a free repair rather than a new primitive: forwards, ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable (Mathlib/Probability/Kernel/CondDistrib.lean:147, (hX : Measurable X) (hY : Measurable Y) {kappa} [IsFiniteKernel kappa] (hkappa : mu.map (fun x => (X x, Y x)) = mu.map X ⊗ₘ kappa) : condDistrib Y X mu =ᵐ[mu.map X] kappa) turns (F5) into the a.e. identification with condDistrib, so every earlier round's condDistrib-anchored reasoning (including condDistrib_ae_eq_condExp as the regular-conditional-distribution certificate and condExp_ae_eq_integral_condDistrib/' as the mean anchors) survives verbatim; backwards, MeasureTheory.Measure.compProd_congr (Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean, (h : kappa =ᵐ[mu] eta) : mu.compProd kappa = mu.compProd eta) shows that ANY a.e.-modification of condDistrib satisfies (F5), so requiring a selected version restricts NOTHING about which laws inhabit the world -- it only forces the law object to name the version its envelope is stated against, which is precisely the missing content the reviewer identified. ProbabilityTheory.compProd_map_condDistrib remains the disintegration used to integrate a pointwise bound back to a global one. CONSEQUENCES F2 MUST FOLLOW: (1) the derived helper is now A1A2Law.memLp_of_condAbsMoment_le, unchanged in role but proved by rewriting the integral over law through (F5)'s measure equality -- an EQUALITY rewrite, not an a.e. transfer -- and then bounding the inner integral by the clause-(v) envelope pointwise on the support, the marginal being carried by the support by (F1); (2) every consumer that INTEGRATES over x (the winsorized-score envelope and variance computation, the mean-of-the-score display in the upper proposition) disintegrates directly against condKer via (F5), so the a.e.-versus-pointwise seam disappears from those proofs rather than being finessed; (3) t7 is STATED against condKer (see its node) and is therefore pointwise on the support with no version caveat. DEPENDENCY CLOSURE OF THIS S5 EDIT, traced through the core and patched this round: def:cty-a1-a2-class clause (v); lem:cty-winsorization-bias (t7); lem:cty-winsorized-score-maximal-bound (its conditional-Jensen step and its envelope/variance computation); lem:cty-a1-a2-rectangle-angular-hypercube-all-orders (t8), which must now EXHIBIT condKer -- trivial and in fact easier than before, since its law is BUILT as an explicit Measure.compProd of the marginal with the explicit Bernoulli-conditional kernel, so (F5) is that construction read backwards and clause (v) is the bound moment <= 1 <= L at every x; and prop:cty-a1-a2-winsorized-expected-outer-upper (t10), whose winsorization-bias step consumes t7 pointwise under the same disintegration. NOTHING ELSE ON THIS NODE MOVES: the disposition stays define-local, binds_symbols, binds_sampling_model, the PO bypass-justification and the round-10/11 search traces are exactly as they were; the module list gains only Mathlib.Probability.Kernel.Composition.MeasureCompProd for compProd_congr. PRIOR TEXT (round 11), authoritative except that its condAbsMoment-as-a-field-with-an-a.e.-(F5) design is superseded above: ROUND-11 REVISION (D-K0 finding 1, FLAG-WORLD-MISFIT) -- the ONLY thing that moves on this node, and it moves at the world layer so that no consumer can bypass it. THE DEFECT: the round-10 field moment_integrable : MemLp (arm-t coordinate) (2 + nu) law is an UNCONDITIONAL GLOBAL moment. Core clause (v) is the UNIFORM POINTWISE CONDITIONAL envelope E_P[|Y(t)|^{2+nu} | X = x] <= L for every x in X_P, which is strictly stronger and is the exact object lem:cty-winsorization-bias (t7) integrates against the conditional kernel and the exact object the winsorized-score envelope computation consumes; a global moment implies no pointwise conditional bound, so under the round-10 field t7 was unprovable as stated and the world was weaker than the class it carries. THE REPAIR: the primitive is now the CONDITIONAL-KERNEL MOMENT ENVELOPE and the global moment is derived. A1A2Law gains the armwise field condAbsMoment : Bool -> Score -> R together with the fidelity field (F5) condAbsMoment_eq : forall t, ∀ᵐ x ∂(law.map score), condAbsMoment t x = ∫ y, |y| ^ (2 + nu) ∂((ProbabilityTheory.condDistrib (armCoord t) score law) x) -- the same regular-conditional-kernel idiom as (F3)/(F4), and ProbabilityTheory.condDistrib_ae_eq_condExp is again the certificate that a functional of the conditional LAW (which the absolute (2+nu)-moment is) really is the conditional moment, with ProbabilityTheory.condExp_ae_eq_integral_condDistrib (:381) at f = fun y => |y| ^ (2 + nu) as the mean-form anchor. condAbsMoment is a REAL-valued field with (F5) stated a.e.; as with mu and sigmaSq, its values at the H^1-null interface B_P are pinned by the class's continuity/envelope clauses through the same continuous_version_unique_on_support pattern the S1 world documents, and the envelope itself is imposed on the SUPPORT (clause (v)), which is where the winsorization bias is taken. THE GLOBAL MOMENT IS NOW A LEMMA, not a field: the non-node helper A1A2Law.memLp_of_condAbsMoment_le derives MemLp (armCoord t) (2 + nu) law from clause (v) plus (F5) by integrating the pointwise envelope against the probability marginal (ProbabilityTheory.compProd_map_condDistrib for the disintegration, MeasureTheory.lintegral_mono -- verified in an earlier round to carry no measurability side condition -- for the bound), and THAT lemma is what discharges the (F3)/(F4) integrability side conditions via MeasureTheory.MemLp.integrable at exponent 2 + nu >= 4 >= 1. So the field moment_integrable is DELETED; F2 must not keep it as a second primitive, because a structure carrying both invites an inhabitant that satisfies the weak one and quietly fails the strong one. DEPENDENCY CLOSURE OF THIS S5 EDIT, traced through the core: def:cty-a1-a2-class clause (v) (now the pointwise envelope on condAbsMoment), lem:cty-winsorization-bias (t7, which consumes exactly that clause under the kernel), lem:cty-winsorized-score-maximal-bound (whose conditional-Jensen step E[Y(t)^2 | X] <= L^{2/(2+nu)} <= L and whose envelope/variance computation both come from clause (v)), and lem:cty-a1-a2-rectangle-angular-hypercube-all-orders (t8), which must now DISCHARGE (F5) for its explicit family -- trivial there, since its conditional outcome laws are Bernoulli on {0,1}, so the conditional absolute (2+nu)-moment is at most one <= L. NOTHING ELSE ON THIS NODE MOVES: the disposition stays define-local, binds_symbols, binds_sampling_model, the module list, the PO bypass-justification and the round-10 search trace are exactly as they were. PRIOR TEXT (round 10), authoritative except that its moment_integrable sentence is superseded above: SHAPE F2 MUST FOLLOW, deliberately parallel to the S1 CtyLaw repair (D-E0/D-F1) so that the two worlds share their fidelity discipline: structure A1A2Law carries (1) law : Measure (R x R x EuclideanSpace R (Fin 2)) with [IsProbabilityMeasure law], the joint law of (Y(0), Y(1), X) -- the POTENTIAL outcomes are the sampled coordinates and the observed pair is DERIVED, which is what makes consistency a definition rather than an assumption; (2) the geometry fields support, A0, A1, boundary : Set (EuclideanSpace R (Fin 2)) and density; (3) muPO, sigmaSqPO : Bool -> EuclideanSpace R (Fin 2) -> R; and (4) the SAME four fidelity fields as S1, now armwise: (F1) the X-marginal of law is volume.withDensity of the support-indicated density; (F2) support is closed with a positive density there, so it IS the topological support of the X-marginal; (F3) for each t, muPO t agrees a.e. with the mean of ProbabilityTheory.condDistrib of the arm-t coordinate given the score; (F4) likewise sigmaSqPO t and ProbabilityTheory.variance; plus moment_integrable : MemLp (arm-t coordinate) (2 + nu) law which is BOTH the (F3)/(F4) integrability discharge (via MeasureTheory.MemLp.integrable, at exponent 2 + nu >= 4 >= 1) and the literal content of clause (v). The observed outcome Y := if X in A1 then Y(1) else Y(0) and the treatment T := 1{X in A1} are DEFINITIONS on this structure, so clause (i)'s consistency and assignment equations are rfl-level and cannot be violated by an inhabitant. tau_P is then muPO true x - muPO false x on boundary, matching the core's tau_P = mu_1,P - mu_0,P exactly. BYPASS-JUSTIFIED vs Causalean PO, and this time the justification is NOT the round-6 one (which said 'the core has no treatment'), because this world DOES have treatment and potential outcomes -- so it was re-surveyed from scratch this round. Causalean.PO.POSystem (../Causalean/PO/Core/System.lean:34) is a Fintype-VARIABLE-INDEXED regime/evaluation world: a law here is a single explicit product measure on R x R x R^2 quantified over a minimax CLASS, and encoding it as a POSystem would force an artificial variable index, a Regime layer and a counterfactual-distribution layer that no statement in this core uses. Causalean.PO.POSharpRDDSystem (../doc/API.md section 9i', PO/ID/Exact/RDD/SharpRDD.lean) is the closest ANALOGOUS module and is recorded as such: it is a SCALAR running variable with a single cutoff c, its assumptions bundle is a fixed-law identification statement, and its target is the one-sided limit contrast at that cutoff -- structurally the d = 1 shadow of this world, but it cannot express a bivariate score, a Borel partition of a rectangle, a rectifiable interface curve, or a law-quantified minimax risk. Its ENGINE Causalean.PO.RDDLimits.oneSidedLimit_eq_right / oneSidedLimit_eq_left (PO/ID/Exact/RDD/RDDLimits.lean, both Proved) is the reusable analytic core of the one-sided-limit identification and is exactly the shape of lem:cty-distance-identification -- but that node is a CITED leaf in this core (gate, see its node), so the engine is recorded as the adaptation reference for a future upstream build rather than consumed now. Also noted: ../doc/API.md section 8h states the PO layer is a SKELETON with sorry proofs where marked, so it is not a substrate to build a headline on. SEARCH TRACE (ROUND-10, run for this new world): grepped ../doc/API.md for 'potential outcome', 'assignment region', 'geographic', 'boundary discontinuity', 'RDD', 'running variable', 'known geometry' -> sections 8h (PO/, skeleton), 9i0b (RDDLimits, scalar one-sided limits), 9i' (SharpRDD, scalar cutoff), 9i'' (FuzzyRDD); grepped ../Causalean for structure .*(Design|Geometry|Partition|Boundary) -> Panel/EventStudyContamination ConventionalDesign/IWDesign and Experimentation/DesignBased FiniteDesign only, none of which is a spatial assignment geometry. Mathlib-side: MeasureTheory.Measure.hausdorffMeasure (Mathlib/MeasureTheory/Measure/Hausdorff.lean, verified this round together with hausdorffMeasure_real, Measure.hausdorffMeasure_mono, Isometry.hausdorffMeasure_image) is the H^1 layer clause (vi)/(x) are encoded over; leansearch for a bivariate geographic-RDD or known-assignment-partition model -> nothing. NOT FOUND as a reusable world; define-local, with the S1 discipline copied rather than reinvented.
Setup S-6 — ROUND-10 NEW WORLD (D-J0): the SIGNED-distance compressed observation and local-polynomial world attached to S5. At an interface point x the estimator sees U^pm_{n,P}(x) = ((Y_i, D^pm_{P,x,i}) : i <= n), where D^pm_{P,x} = (1{X in A_1,P} - 1{X in A_0,P}) * dist(X, x) is the SIGNED Euclidean distance -- so, unlike the unsigned S2 compression, the arm is recoverable from the compressed observation, which is exactly why this world admits a matching upper bound while S2/S4 does not. The world also carries the degree-p polynomial basis r_p, the population Gram functional Psi_{t,P,x}(h), the winsorization map psi_B, the winsorization level B_n = a_n^{-1/3}, and the explicit winsorized, clipped, Gram-stabilized estimator at bandwidth h. — ENCODING F2 MUST FOLLOW. The Gram object is a Matrix (Fin (p+1)) (Fin (p+1)) R, and the core's lambda_min(Psi) >= L^{-1} clause is spelled as the QUADRATIC FORM forall v, L^{-1} * (norm v)^2 <= v (dot) (Psi *v v) rather than through a spectral decomposition: that spelling is what both consumers actually use (the hypercube lemma proves it by a polar-coordinate integral lower bound, and the upper proposition consumes it through Weyl plus an operator-norm bound), and it avoids the Matrix.IsHermitian.eigenvalues machinery entirely. The stabilization branch of the estimator is then if (forall v, ...) then Psihat^{-1} mhat else 0, decidable by Classical.dec, and the final clip is max (-2L) (min (2L) .), one-Lipschitz by abs_max_sub_max_le_abs-style algebra. SEARCH TRACE (ROUND-10): grepped ../doc/API.md for 'local polynomial', 'localPoly', 'bandwidth', 'kernel weight', 'Nadaraya', 'winsor', 'trimming' -> NO local-polynomial or smoothing-kernel layer anywhere in Causalean; the only Gram decls (sections 9i0a-MultipleInstrumentIV firstStageGram, 9i0a-FlexibleDIDMundlak gram, 10f FWL.gramResid, 13W WeightedSupport.Q_XX, 15 empiricalGram/populationGram in ML/Ridge) are FIXED-DESIGN or finite-index second-moment matrices with no bandwidth, no kernel and no local weighting, so they are the wrong object; ML/Ridge/Rate.lean (empiricalGram_posSemidef, sampleRidgeCoef_sub_eq) is recorded as the ANALOGOUS module for the 'invert a stabilized empirical Gram and factor the coefficient error' pattern, which is literally the shape of the upper proposition's betahat - beta^* = Psihat^{-1} Ohat step (gap to bridge: ridge stabilizes by adding lambda I and works at a fixed feature map, whereas here stabilization is a Gram eigenvalue GUARD with a zero fallback and the feature map moves with (x, h)). Mathlib-side: leansearch 'local polynomial regression', 'kernel smoothing estimator' -> nothing; Matrix.PosDef, Matrix.inv and Ring.inverse are the only pieces needed. NOT FOUND; define-local.
Setup S-7 — ROUND-10 NEW WORLD (D-J0): the A1/A2 known-geometry point-indexed decision world with outer-expectation risk. A rule is generated by one law-independent collection {T_{n,G,x}} indexed by an admissible known geometry G and a point x in R^2, with every fixed (G, x) section Borel and NO joint measurability, continuity or common modulus in x; the risk R_n^{12,pm} is the inf over that class of the sup over P_12(p, nu, L) of the OUTER expectation of the interface-supremum loss. It is the exact S4 architecture transplanted onto S5/S6 -- same sectionwise-only measurability, same outer expectation, same complete-lattice ENNReal ambient -- with the one addition that the section may also read the known geometry G_P, which is what makes the class large enough to contain the explicit winsorized estimator and hence to support a MATCHED frontier. — DISPOSITION IS define-local FOR THE WORLD AND BODY-LEVEL reuse FOR THE ONE PIECE THAT NOW EXISTS UPSTREAM (the same convention S3 uses), and this is the ROUND-10 substrate-built consequence on the risk side: the outer expectation E*_P is MeasureTheory.outerLIntegral (Causalean, Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean:33, (mu : Measure Omega) (f : Omega -> ENNReal) : ENNReal := iInf over measurable majorants of the lower integral), which is DEFINITIONALLY the core's E*_P and the same object the local D-C2 helper outerLintegral spells for the S4 world. F2 must NOT mint a third copy: the A1/A2 risk is built directly on MeasureTheory.outerLIntegral, and F2 should additionally align the already-built local Helpers/OuterExpectation.outerLintegral with it by a one-line rfl/simp bridge lemma rather than duplicating the API a second time. The decision class, the law-dependent interface-supremum loss and the geometry-indexed section family remain define-local (nothing upstream types them), so only the integral layer is a reuse. SEARCH TRACE (ROUND-10, re-run because this is a substrate-built node): lean_local_search 'UpperSemianalytic' -> MeasureTheory.UpperSemianalytic FOUND; ../doc/API.md section 10k'''' documents the whole roll-up with MeasureTheory.outerLIntegral, MeasureTheory.UpperSemianalytic, UpperSemianalytic.nullMeasurable, UpperSemianalytic.measurable_completion, UpperSemianalytic.lintegral_completion_eq_outerLIntegral, MeasureTheory.AnalyticSet.nullMeasurableSet, MeasureTheory.AnalyticSet.isCapacitable, MeasureTheory.Measure.toChoquetCapacity; source read at ../Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean, grep for sorry over the whole AnalyticSetUniversalMeasurability/ directory returns ZERO matches. For the decision-class half: grepped ../doc/API.md for 'decision class', 'point-indexed', 'known geometry', 'sectionwise' -> nothing (as in the S4 trace). Class and risk: NOT FOUND, define-local; outer integral: FOUND and reused.

Definitions (14)

Definition P-1 verified CausalSmith.Stat.BddUniformLogPenalty.frontierRate $a_n=(\log n/n)^{1/4}.$
Definition P-2 verified CausalSmith.Stat.BddUniformLogPenalty.CtyNonparametricClass $\mathcal P_{\mathrm{NP}}(L,q)=\bigl\{P:\text{conditions (i)--(iv) below hold}\bigr\}.$ Condition $\mathrm{(i)}$ : for every $n$ , $(Y_i,X_i)_{i=1}^n$ are independently and identically distributed under $P$ in $\mathbb R\times\mathbb R^2$ . Condition $\mathrm{(ii)}$ : $X$ has a Lebesgue density $f_P$ , continuous on its compact support $\mathcal X_P\subseteq[-L,L]^2$ , with $L^{-1}\leq\inf_{x\in\mathcal X_P}f_P(x)\leq\sup_{x\in\mathcal X_P}f_P(x)\leq L$ , and $\operatorname{bd}(\mathcal X_P)$ is a rectifiable curve. Condition $\mathrm{(iii)}$ : $\mu_P(x)=\mathbb E_P[Y\mid X=x]$ belongs to $\mathcal H^q(\mathcal X_P,L)$ . Condition $\mathrm{(iv)}$ : $\sigma_P^2(x)=\operatorname{Var}_P(Y\mid X=x)$ is continuous on $\mathcal X_P$ and satisfies $L^{-1}\leq\sigma_P^2(x)\leq L$ for every $x\in\mathcal X_P$ .
Definition P-3 verified CausalSmith.Stat.BddUniformLogPenalty.distanceData $V_{n,P}(x)=((Y_i,\lVert X_i-x\rVert_2):1\leq i\leq n),\qquad x\in\operatorname{bd}(\mathcal X_P).$
Definition P-4 verified CausalSmith.Stat.BddUniformLogPenalty.CtyDistanceDecisionClass For a represented rule, write $T_n$ for its selected law-independent measurable map. $\mathcal T_n^{\mathrm{NP}} =\left\{\widehat\mu_n:\begin{array}{l} \text{there exists a measurable map, denoted }T_n:(\mathbb R\times[0,\infty))^n\to\mathbb R \text{ selected independently of }P,\\ \widehat\mu_n(x)=T_n(V_{n,P}(x)) \text{ for every }x\in\operatorname{bd}(\mathcal X_P) \end{array}\right\}.$
Definition P-5 verified CausalSmith.Stat.BddUniformLogPenalty.ctyDistanceRisk $R_n^{\mathrm{NP}}=\inf_{\widehat\mu_n\in\mathcal T_n^{\mathrm{NP}}}\sup_{P\in\mathcal P_{\mathrm{NP}}(L,q)}\mathbb E_P\!\left[\sup_{x\in\operatorname{bd}(\mathcal X_P)}|\widehat\mu_n(x)-\mu_P(x)|\right].$
Definition P-6 verified CausalSmith.Stat.BddUniformLogPenalty.PointIndexedDecisionClass $\mathcal T_n^{\mathrm{PI}} =\left\{\widehat\mu_n^{\mathrm{PI}}:\begin{array}{l} \text{there is a family }\{T_{n,x}:x\in\mathbb R^2\}\text{ selected independently of }P,\\ T_{n,x}:(\mathbb R\times[0,\infty))^n\to\mathbb R \text{ is Borel measurable for every fixed }x,\\ \widehat\mu_{n,P}^{\mathrm{PI}}(x)=T_{n,x}(V_{n,P}(x)) \text{ for every }P\in\mathcal P_{\mathrm{NP}}(L,q)\\ \text{and every }x\in\operatorname{bd}(\mathcal X_P) \end{array}\right\}.$ No joint measurability, continuity, common modulus, or other regularity of $x\mapsto T_{n,x}$ is required.
Definition P-7 verified CausalSmith.Stat.BddUniformLogPenalty.pointIndexedDistanceRisk For an extended nonnegative function $Z$ on the sample space, define its outer expectation by $\mathbb E_P^*[Z] =\inf\left\{\mathbb E_P[W]: W\text{ is an extended nonnegative measurable function and }W\geq Z \text{ pointwise}\right\}.$ Then $R_n^{\mathrm{PI}} =\inf_{\widehat\mu_n^{\mathrm{PI}}\in\mathcal T_n^{\mathrm{PI}}} \sup_{P\in\mathcal P_{\mathrm{NP}}(L,q)} \mathbb E_P^*\!\left[ \sup_{x\in\operatorname{bd}(\mathcal X_P)} \left|\widehat\mu_{n,P}^{\mathrm{PI}}(x)-\mu_P(x)\right| \right].$
Definition P-8 verified CausalSmith.Stat.BddUniformLogPenalty.uniformKernel Define the fixed uniform kernel by $K_\square(u)=\mathbf 1\{|u|\leq1\},\qquad u\in\mathbb R.$
Definition P-9 verified CausalSmith.Stat.BddUniformLogPenalty.knownGeometry For a member law $P$ , the known design object is $G_P=(\mathcal X_P,\mathcal A_{0,P},\mathcal A_{1,P},\mathcal B_P,d_P,K_\square),$ where $\mathcal A_{0,P}$ and $\mathcal A_{1,P}$ are a Borel partition of $\mathcal X_P$ , $\mathcal B_P=\operatorname{bd}(\mathcal A_{0,P})\cap\operatorname{bd}(\mathcal A_{1,P})\subset\operatorname{int}(\mathcal X_P)$ , and the fixed metric is $d_P(x,z)=\lVert x-z\rVert_2$ .
Definition P-10 verified CausalSmith.Stat.BddUniformLogPenalty.signedDistanceData For $x\in\mathcal B_P$ , define $D^{\pm}_{P,x}=\{\mathbf 1(X\in\mathcal A_{1,P})-\mathbf 1(X\in\mathcal A_{0,P})\}\lVert X-x\rVert_2, \qquad U^{\pm}_{n,P}(x)=((Y_i,D^{\pm}_{P,x,i}):1\leq i\leq n).$
Definition P-11 verified CausalSmith.Stat.BddUniformLogPenalty.A1A2Class For integers $p\geq0$ , real numbers $\nu\geq2$ , and $L\geq4$ , let $\mathcal P_{12}(p,\nu,L)$ be the set of laws $P$ satisfying all the following member properties. Write $Y_i(0),Y_i(1)$ for the sampled potential outcomes of observation $i$ . $\mathrm{(i)}$ For every $n$ , the vectors $(Y_i(0),Y_i(1),X_i)$ , $1\leq i\leq n$ , are independently and identically distributed; $T=\mathbf 1\{X\in\mathcal A_{1,P}\}$ ; and $Y=TY(1)+(1-T)Y(0)$ . $\mathrm{(ii)}$ The support has the rectangular form $\mathcal X_P=[\underline x_P,\overline x_P]^2\subseteq[-L,L]^2$ . The variable $X$ has a Lebesgue density $f_P$ , continuous on $\mathcal X_P$ , such that $L^{-1}\leq f_P(x)\leq L$ for all $x\in\mathcal X_P$ . $\mathrm{(iii)}$ For each $t\in\{0,1\}$ , $\mu_{t,P}(x)=\mathbb E_P[Y(t)\mid X=x]$ has a $C^{p+1}$ extension to an open neighborhood of $\mathcal X_P$ , and $\max_{0\leq|\alpha|\leq p}\sup_{x\in\mathcal X_P}|\partial^\alpha\mu_{t,P}(x)| +\max_{0\leq|\alpha|\leq p}\sup_{x\neq z\in\mathcal X_P} { |\partial^\alpha\mu_{t,P}(x)-\partial^\alpha\mu_{t,P}(z)|\over\lVert x-z\rVert_2} \leq L.$ $\mathrm{(iv)}$ For each $t$ , $\sigma^2_{t,P}(x)=\operatorname{Var}_P(Y(t)\mid X=x)$ is continuous and satisfies $L^{-1}\leq\sigma^2_{t,P}(x)\leq L$ on $\mathcal X_P$ . $\mathrm{(v)}$ For each $t$ and every $x\in\mathcal X_P$ , $\mathbb E_P[|Y(t)|^{2+\nu}\mid X=x]\leq L.$ Write $\mathcal H^1(\mathcal B_P)$ for the one-dimensional Hausdorff measure of the common interface. $\mathrm{(vi)}$ The sets $\mathcal A_{0,P}$ and $\mathcal A_{1,P}$ form a Borel partition of $\mathcal X_P$ , and their known common interface $\mathcal B_P\subset\operatorname{int}(\mathcal X_P)$ is a compact rectifiable curve satisfying $L^{-1}\leq\mathcal H^1(\mathcal B_P)\leq L$ . $\mathrm{(vii)}$ The known metric is exactly Euclidean, $d_P(x,z)=\lVert x-z\rVert_2$ , and the known kernel is $K_\square$ . The collection $\{\{z\in\mathcal X_P:\lVert z-x\rVert_2\leq r\}:x\in\mathcal X_P, r\geq0\}$ is a VC class with VC index at most four. $\mathrm{(viii)}$ Put $r_p(u)=(1,u,\ldots,u^p)^\top$ , $I_1=[0,\infty)$ , $I_0=(-\infty,0)$ , and $\Psi_{t,P,x}(h)=\mathbb E_P\!\left[ {1\over h^2}\mathbf 1\{D^{\pm}_{P,x}\in I_t\}K_\square(D^{\pm}_{P,x}/h) r_p(D^{\pm}_{P,x}/h)r_p(D^{\pm}_{P,x}/h)^\top \right].$ For every $0<h\leq L^{-1}$ , every $x\in\mathcal B_P$ , and both $t$ , $\lambda_{\min}(\Psi_{t,P,x}(h))\geq L^{-1}.$ $\mathrm{(ix)}$ For every $0<h\leq L^{-1}$ , every $x\in\mathcal B_P$ , and both $t$ , ${1\over h^2}\int_{\mathcal A_{t,P}} K_\square(\{(2t-1)\lVert z-x\rVert_2\}/h)\,dz\geq L^{-1}.$ $\mathrm{(x)}$ For every $x\in\mathcal B_P$ , both $t$ , and every $0<s\leq L^{-1}$ , the one-dimensional Hausdorff integral of $f_P$ over $\{z\in\mathcal A_{t,P}:\lVert z-x\rVert_2=s\}$ is finite and strictly positive. This is the exact displayed $L$ -uniformized Euclidean-distance, uniform-kernel specialization of CTY Assumptions~1--2, augmented by the displayed Theorem-2 envelope restrictions and quantitative small-bandwidth thresholds. It is not CTY's entire metric/kernel regime.
Definition P-12 verified CausalSmith.Stat.BddUniformLogPenalty.A1A2PointIndexedDecisionClass Let $\mathcal T^{12,\pm}_n$ be the class of rules for which there is one collection $\{T_{n,G,x}:G\text{ is an admissible known geometry},\ x\in\mathbb R^2\}$ selected independently of the unknown member law, such that every fixed $(G,x)$ section is a Borel map from $(\mathbb R\times\mathbb R)^n$ to $\mathbb R$ and $\widehat\tau_{n,P}(x)=T_{n,G_P,x}(U^{\pm}_{n,P}(x)), \qquad P\in\mathcal P_{12}(p,\nu,L),\quad p\geq0,\quad x\in\mathcal B_P.$ No joint measurability, continuity, common modulus, or other regularity in $x$ is imposed.
Definition P-13 verified CausalSmith.Stat.BddUniformLogPenalty.a1a2OuterRisk Using the outer expectation $\mathbb E_P^*$ defined in def:point-indexed-distance-risk, for $p\geq0$ the signed-distance minimax risk is $R^{12,\pm}_n(p,\nu,L)= \inf_{\widehat\tau_n\in\mathcal T^{12,\pm}_n} \sup_{P\in\mathcal P_{12}(p,\nu,L)} \mathbb E_P^*\!\left[ \sup_{x\in\mathcal B_P}|\widehat\tau_{n,P}(x)-\tau_P(x)| \right].$
Definition P-14 verified CausalSmith.Stat.BddUniformLogPenalty.stabilizedLocalPolynomial For $B\in\mathbb R$ , define the winsorization map $\psi_B(y)=\operatorname{sign}(y)\min\{|y|,B\}.$ For each $t\in\{0,1\}$ , query point $x$ , and bandwidth parameter $h$ , set $\widehat\Psi_{t,x}(h) = {1\over nh^2}\sum_{i=1}^n \mathbf 1\{D^{\pm}_{P,x,i}\in I_t\}K_\square(D^{\pm}_{P,x,i}/h) r_p(D^{\pm}_{P,x,i}/h)r_p(D^{\pm}_{P,x,i}/h)^\top$ and, with $B(h)=h^{-1/3}$ , $\widehat m^B_{t,x}(h) = {1\over nh^2}\sum_{i=1}^n \mathbf 1\{D^{\pm}_{P,x,i}\in I_t\}K_\square(D^{\pm}_{P,x,i}/h) r_p(D^{\pm}_{P,x,i}/h)\psi_{B(h)}(Y_i).$ Define $\widehat\beta^B_{t,x}(h) = \begin{cases} \widehat\Psi_{t,x}(h)^{-1}\widehat m^B_{t,x}(h), & \forall v\in\mathbb R^{p+1},\ (2L)^{-1}\sum_{j=0}^p v_j^2\leq v^\top\widehat\Psi_{t,x}(h)v,\\ 0, & \neg\!\left(\forall v\in\mathbb R^{p+1},\ (2L)^{-1}\sum_{j=0}^p v_j^2\leq v^\top\widehat\Psi_{t,x}(h)v\right). \end{cases}$ With $e_1=(1,0,\ldots,0)^\top$ , define $\widehat\tau^{12,\mathrm{LP}}_{n,h}(x) = \max\!\left\{-2L,\min\!\left\{2L, e_1^\top(\widehat\beta^B_{1,x}(h)-\widehat\beta^B_{0,x}(h)) \right\}\right\}.$

Lemmas (7)

Lemma L-1 verified CausalSmith.Stat.BddUniformLogPenalty.coordinatewise_overlap_direct_product Let all spaces carrying $Z_0,Z_1,\ldots,Z_M,S_1,\ldots,S_M$ be standard Borel. Let $\Omega_1,\ldots,\Omega_M$ be independent uniform binary variables. Conditionally on them, let $Z_0,Z_1,\ldots,Z_M$ be independent, with the law of $Z_j$ depending only on $\Omega_j$ for $1\leq j\leq M$ , and with the law of $Z_0$ independent of $(\Omega_1,\ldots,\Omega_M)$ . Suppose decoder $j$ is measurable with respect to $(S_j,Z_{-j},Z_0)$ , where $S_j$ is a measurable compression of $Z_j$ , and write $Q_{j,b}=\mathcal L(S_j\mid\Omega_j=b)$ and $\rho_j=1-\lVert Q_{j,0}-Q_{j,1}\rVert_{\mathrm{TV}}$ . Then every such decentralized decoder obeys $\Pr\{\widehat\Omega_j=\Omega_j\text{ for every }j\} \leq\frac12\left\{1+\prod_{j=1}^M(1-\rho_j)\right\}.$ In particular, if $\operatorname{KL}(Q_{j,0},Q_{j,1})\leq\kappa\log M$ for all $j$ and a fixed $\kappa<1$ , then $\Pr\{\text{at least one decoding error}\} \geq\frac12\left[1-\exp\{-M^{1-\kappa}/2\}\right].$ Along any sequence with $M=M_n\to\infty$ and fixed $\kappa<1$ , this lower bound is $1/2-o(1)$ .
Lemma L-2 verified CausalSmith.Stat.BddUniformLogPenalty.cty_support_boundary_angular_packing For every $q\geq1$ and $L\geq4$ , there are constants $c_0,c_1>0$ and $0<\alpha<1/8$ , depending only on $q$ and $L$ , such that, for all sufficiently large $n$ , there exist $M_n\geq c_0a_n^{-1/q}$ points $x_1,\ldots,x_{M_n}$ on the boundary of the square $S=[-1/2,1/2]^2$ , separated by at least a constant multiple of $a_n^{1/q}$ , a radius $w_n\asymp a_n^{1/q}$ , and laws $P_\omega\in\mathcal P_{\mathrm{NP}}(L,q)$ , indexed by $\omega\in\{0,1\}^{M_n}$ , with the following properties. Every law has support $S$ , a continuous density $f_{P_\omega}$ satisfying $L^{-1}\leq f_{P_\omega}\leq L$ , a regression $\mu_{P_\omega}$ in the stated $q$ -H\"older ball, and a continuous conditional variance $\sigma^2_{P_\omega}$ satisfying $L^{-1}\leq\sigma^2_{P_\omega}\leq L$ . The boundary $\operatorname{bd}(S)$ is rectifiable and the observations are independently and identically distributed. The half-discs $C_j=B_2(x_j,w_n)\cap S,\qquad 1\leq j\leq M_n,$ are pairwise disjoint. The subprobability law $P_\omega\{(Y,X)\in\cdot,\ X\in C_j\}$ depends on $\omega$ only through $\omega_j$ ; the subprobability law $P_\omega\{(Y,X)\in\cdot,\ X\in S\setminus\bigcup_{k=1}^{M_n}C_k\}$ is independent of $\omega$ ; and $P_\omega(X\in C_j)=m_n$ for one common number $m_n$ independent of $j$ and $\omega$ . There are numbers $u_{j,0},u_{j,1}$ such that $\mu_{P_\omega}(x_j)=u_{j,\omega_j}, \qquad |u_{j,1}-u_{j,0}|\geq c_1a_n.$ If $\omega^{(j)}$ differs from $\omega$ only in coordinate $j$ , then the laws of $\lVert X-x_j\rVert_2$ agree, the conditional radial outcome laws agree outside radii of order $a_n$ , and $\operatorname{KL}\!\left(\mathcal L_{P_\omega}\{V_{n,P_\omega}(x_j)\}, \mathcal L_{P_{\omega^{(j)}}}\{V_{n,P_{\omega^{(j)}}}(x_j)\}\right) \leq\alpha\log M_n.$
Lemma L-3 verified CausalSmith.Stat.BddUniformLogPenalty.common_map_strict_in_point_indexed For every $n\geq1$ , $q\geq1$ , and $L\geq4$ , $\mathcal T_n^{\mathrm{NP}}\subsetneq\mathcal T_n^{\mathrm{PI}}.$
Lemma L-4 verified CausalSmith.Stat.BddUniformLogPenalty.euclidean_balls_vc The collection of all closed Euclidean balls in $\mathbb R^2$ is a VC class with VC index at most four. Hence the fixed Euclidean metric and uniform kernel in $\mathrm{def:cty\text{-}a1\text{-}a2\text{-}class}$ satisfy the actual VC alternative in CTY Assumption~2(iii), uniformly over every member law.
Lemma L-5 verified CausalSmith.Stat.BddUniformLogPenalty.cty_winsorization_bias For $B\geq1$ , define $\psi_B(y)=\operatorname{sign}(y)\min\{|y|,B\}$ . If $P\in\mathcal P_{12}(p,\nu,L)$ with $p\geq0$ , $\nu\geq2$ , and $L\geq4$ , then, for both $t\in\{0,1\}$ and every $x\in\mathcal X_P$ , $\left|\mathbb E_P\!\left[Y(t)-\psi_B\{Y(t)\}\mid X=x\right]\right| \leq L B^{-3}.$ ↗ Lean
Lemma L-6 verified CausalSmith.Stat.BddUniformLogPenalty.cty_winsorized_score_maximal_bound Let $p\geq0$ , $\nu\geq2$ , $L\geq4$ , $P\in\mathcal P_{12}(p,\nu,L)$ , $n\geq1$ , $0<h\leq L^{-1}$ , and $B\geq1$ . Let $\beta^*_{t,P,x}(h)$ solve the original, unwinsorized population normal equations, and define $\widehat O^B_{t,x}(h)={1\over nh^2}\sum_{i=1}^n \mathbf1\{D^{\pm}_{P,x,i}\in I_t\}K_\square(D^{\pm}_{P,x,i}/h) r_p(D^{\pm}_{P,x,i}/h) \left[\psi_B(Y_i)-r_p(D^{\pm}_{P,x,i}/h)^\top\beta^*_{t,P,x}(h)\right].$ There is a constant $C=C(p,L)$ such that $\mathbb E_P^*\!\left[ \sup_{t\in\{0,1\},\,x\in\mathcal B_P} \left\|\widehat O^B_{t,x}(h)-\mathbb E_P\widehat O^B_{t,x}(h)\right\| \right] \leq C\left\{ \sqrt{{\log(B/h)\over nh^2}}+{B\log(B/h)\over nh^2} \right\}.$ This is a direct bounded-envelope adaptation of the expected maximal-inequality arguments in Supplemental Appendix SA-8.3 (Proof of Lemma SA-3), printed pp. 32--33, and SA-8.4 (Proof of Lemma SA-4), printed pp. 33--35, of Cattaneo, Titiunik, and Yu, not a verbatim theorem attributed to them.
Lemma L-7 verified CausalSmith.Stat.BddUniformLogPenalty.cty_a1_a2_rectangle_angular_hypercube For every integer $p\geq0$ and every $\nu\geq2$ , there is a finite $L_0=L_0(p)\geq48$ such that, for every $L\geq L_0$ and all sufficiently small $\Delta>0$ , the class $\mathcal P_{12}(p,\nu,L)$ contains a hypercube $\{P_\omega:\omega\in\{0,1\}^M\}$ with common known geometry $\mathcal X=[-3,3]^2, \quad \mathcal A_1=[-1,1]\times[0,2], \quad \mathcal A_0=\mathcal X\setminus\mathcal A_1, \quad \mathcal B=\partial\mathcal A_1,$ Euclidean distance, and $K_\square$ . Writing $q=p+1\geq1$ , the family has $M\geq c\Delta^{-1/q}$ disjoint disks $S_j=B_2(x_j,w)$ , with $x_j$ on the middle of the bottom edge and $w=A\Delta^{1/q}$ , and satisfies: $\mathrm{(a)}$ every member obeys every clause of $\mathrm{def:cty\text{-}a1\text{-}a2\text{-}class}$ ; $\mathrm{(b)}$ the cell probabilities are bit-independent and equal to $\rho=\pi w^2/36$ , the law restricted to $S_j$ depends only on $\omega_j$ , and the law outside $\bigcup_jS_j$ is bit-independent; $\mathrm{(c)}$ adjacent vertices satisfy $|\tau_{P_\omega}(x_j)-\tau_{P_{\omega^{(j)}}}(x_j)|=\Delta$ ; and $\mathrm{(d)}$ , conditional on $X\in S_j$ , the adjacent one-observation signed-distance laws $Q_{j,0}$ and $Q_{j,1}$ have the same signed-radius marginal, agree outside $0<D^{\pm}_{P,x_j}<2C\Delta$ , and obey in both orientations $\operatorname{KL}(Q_{j,0},Q_{j,1})\leq C_0{\Delta^4\over w^2}.$ The constants are independent of $M$ , $\omega$ , $\Delta$ , and $n$ . When $p=0$ , the constants are chosen with $A>2C$ , so $2C\Delta<w=A\Delta$ .

Theorems (5)

Theorem T-1 verified CausalSmith.Stat.BddUniformLogPenalty.cty_same_class_log_converse For every $q\geq1$ and $L\geq4$ , there is a constant $c_{\mathrm{CTY}}=c_{\mathrm{CTY}}(q,L)>0$ such that $\liminf_{n\to\infty} \left(\frac{n}{\log n}\right)^{1/4} \inf_{\widehat\mu_n\in\mathcal T_n^{\mathrm{NP}}} \sup_{P\in\mathcal P_{\mathrm{NP}}(L,q)} \mathbb E_P\!\left[ \sup_{x\in\operatorname{bd}(\mathcal X_P)} |\widehat\mu_n(x)-\mu_P(x)| \right] =\liminf_{n\to\infty}\frac{R_n^{\mathrm{NP}}}{a_n} \geq c_{\mathrm{CTY}}.$ Thus the logarithmic penalty is necessary in the coordinatewise distance decision class and on the same nonparametric law class used by Cattaneo, Titiunik, and Yu, Theorem~6; the result is not a subclass transfer from the two-sided-thickness treatment-effect model.
Theorem T-2 verified CausalSmith.Stat.BddUniformLogPenalty.point_indexed_distance_log_converse For every $q\geq1$ and $L\geq4$ , there is a constant $c_{\mathrm{PI}}=c_{\mathrm{PI}}(q,L)>0$ such that $\liminf_{n\to\infty} \left(\frac{n}{\log n}\right)^{1/4} \inf_{\widehat\mu_n^{\mathrm{PI}}\in\mathcal T_n^{\mathrm{PI}}} \sup_{P\in\mathcal P_{\mathrm{NP}}(L,q)} \mathbb E_P^*\!\left[ \sup_{x\in\operatorname{bd}(\mathcal X_P)} \left|\widehat\mu_{n,P}^{\mathrm{PI}}(x)-\mu_P(x)\right| \right] =\liminf_{n\to\infty}\frac{R_n^{\mathrm{PI}}}{a_n} \geq c_{\mathrm{PI}}.$ Here $\mathcal T_n^{\mathrm{PI}}$ permits an arbitrary law-independent family $\{T_{n,x}:x\in\mathbb R^2\}$ whose fixed- $x$ sections are Borel measurable; no joint measurability, continuity, or common modulus in $x$ is imposed, and $\mathbb E_P^*$ is outer expectation. Thus the logarithmic penalty remains necessary in a decision class strictly larger than the literal common-map class displayed by CTY, $\mathcal T=\{T_n(U_n(x)):T_n\in\mathcal T_n\}$ . The theorem is a lower bound on the same nonparametric law class $\mathcal P_{\mathrm{NP}}(L,q)$ , not an Assumption~2 upper-bound result or a matched two-sided frontier.
Theorem T-3 verified CausalSmith.Stat.BddUniformLogPenalty.cty_a1_a2_point_indexed_log_converse For every integer $p\geq0$ and every $\nu\geq2$ , there is a finite $L_0=L_0(p)$ such that, for every $L\geq L_0$ , there is $c=c(p,\nu,L)>0$ satisfying $\liminf_{n\to\infty}\left({n\over\log n}\right)^{1/4}R^{12,\pm}_n(p,\nu,L)\geq c.$ The conclusion holds for arbitrary law-independent known-geometry point-indexed families with Borel fixed sections, under outer expectation and without joint regularity in the boundary index. The hard subclass has one fixed support, one fixed known assignment rectangle, and one fixed known interior treatment boundary.
Theorem T-4 verified CausalSmith.Stat.BddUniformLogPenalty.cty_a1_a2_winsorized_expected_outer_upper For every integer $p\geq0$ , every $\nu\geq2$ , and every $L\geq4$ , the explicit winsorized, Gram-stabilized estimator in $\mathrm{def:cty\text{-}stabilized\text{-}local\text{-}polynomial\text{-}estimator}$ , with $h_n=a_n=(\log n/n)^{1/4}$ and $B_n=a_n^{-1/3}$ , satisfies, for all sufficiently large $n$ , $\sup_{P\in\mathcal P_{12}(p,\nu,L)} \mathbb E_P^*\!\left[ \sup_{x\in\mathcal B_P} |\widehat\tau^{12,\mathrm{LP}}_{n,h_n}(x)-\tau_P(x)| \right] \leq Ca_n.$ The expectation theorem is derived by comparing the winsorized empirical score directly with the original population coefficient and by a bounded-envelope adaptation of the CTY supplement's expected maximal-inequality proof; it is not attributed verbatim to CTY.
Theorem T-5 verified CausalSmith.Stat.BddUniformLogPenalty.cty_a1_a2_winsorized_matched_frontier For every integer $p\geq0$ , every $\nu\geq2$ , and every $L\geq L_0(p)$ , there are constants $0<c\leq C<\infty$ such that $0<c\leq\liminf_{n\to\infty}{R^{12,\pm}_n(p,\nu,L)\over a_n} \leq\limsup_{n\to\infty}{R^{12,\pm}_n(p,\nu,L)\over a_n}\leq C<\infty.$ Thus the outer-expected minimax rate over the same exact displayed $L$ -uniformized Euclidean-distance, uniform-kernel CTY Assumptions~1--2 plus Theorem-2-envelope class is $a_n=(\log n/n)^{1/4}$ throughout $q=p+1\geq1$ and $\nu\geq2$ . The upper bound is attained by the explicit winsorized, Gram-stabilized degree- $p$ signed-distance local-polynomial estimator with $h_n=a_n$ and $B_n=a_n^{-1/3}$ ; the lower bound holds over all law-independent known-geometry point-indexed Borel sections.

Citeds (3)

Cited result lem:cty-distance-identification verified CausalSmith.Stat.BddUniformLogPenalty.CtyDistanceIdentification Under CTY Assumptions~1(i)--(iii) and 2(i), (ii), and (v), the causal boundary treatment-effect curve is identified by the signed-distance conditional limits: $\tau_P(x)=\lim_{r\downarrow0}\mathbb E_P[Y\mid D^{\pm}_{P,x}=r] -\lim_{r\uparrow0}\mathbb E_P[Y\mid D^{\pm}_{P,x}=r], \qquad x\in\mathcal B_P.$
Cited result lem:cty-uniform-first-order-bias verified CausalSmith.Stat.BddUniformLogPenalty.CtyUniformFirstOrderBias Fix an integer $p\geq0$ , a number $\nu\geq2$ , and $L\geq4$ . For $P\in\mathcal P_{12}(p,\nu,L)$ , $t\in\{0,1\}$ , $x\in\mathcal B_P$ , and $h>0$ , let $\beta^*_{t,P,x}(h)\in\mathbb R^{p+1}$ solve the original, unwinsorized population normal equations $\Psi_{t,P,x}(h)\beta^*_{t,P,x}(h) =\mathbb E_P\!\left[ {1\over h^2}\mathbf1\{D^{\pm}_{P,x}\in I_t\}K_\square(D^{\pm}_{P,x}/h) r_p(D^{\pm}_{P,x}/h)Y \right],$ and put $e_1=(1,0,\ldots,0)^\top$ . There is a constant $C=C(p,L)<\infty$ such that, for every deterministic sequence $(h_n)_{n\geq1}$ satisfying $h_n>0$ , $h_n\downarrow0$ , and $nh_n^2\to\infty$ , for all sufficiently large $n$ , $\sup_{P\in\mathcal P_{12}(p,\nu,L)} \sup_{x\in\mathcal B_P} \left| e_1^\top\{\beta^*_{1,P,x}(h_n)-\beta^*_{0,P,x}(h_n)\}-\tau_P(x) \right| \leq Ch_n.$
Cited result lem:cty-expected-local-polynomial-maximal-bounds verified CausalSmith.Stat.BddUniformLogPenalty.CtyExpectedLocalPolynomialMaximalBounds For the degree- $p$ signed-distance local-polynomial score and Gram processes with the fixed uniform kernel, the expected uniform deviations over $P\in\mathcal P_{12}(p,\nu,L)$ , both arms, and $x\in\mathcal B_P$ satisfy $\mathbb E_P^*\!\left[\sup_{t,x}\lVert\widehat\Psi_{t,x}(h)-\Psi_{t,P,x}(h)\rVert\right] \leq C\sqrt{{\log(1/h)\over nh^2}},$ and $\mathbb E_P^*\!\left[\sup_{t,x}\lVert\widehat O_{t,x}(h)\rVert\right] \leq C\left\{ \sqrt{{\log(1/h)\over nh^2}} +{\log(1/h)\over n^{(1+\nu)/(2+\nu)}h^2} \right\},$ whenever $h\downarrow0$ and $n^{(1+\nu)/(2+\nu)}h^2/\log(1/h)\to\infty$ . The constant $C$ depends only on the uniform class constants and $p$ .

Auxiliary Lean lemmas (machine-checked) (850)

Definition CtyLaw —
Definition CtyRule —
Definition normalizedRisk —
Definition PIRule —
Definition RectifiableBoundary —
Definition scaledRisk —
Definition EuclideanHolderBallStd —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scaledRisk_eventually_eq_normalizedRisk —
Lemma CausalSmith.Stat.BddUniformLogPenalty.pointIndexedDistanceRisk_eventually_lower —
Lemma CausalSmith.Stat.BddUniformLogPenalty.overlap_ge_exp_neg_klBudget —
Lemma CausalSmith.Stat.BddUniformLogPenalty.prod_one_sub_le_exp_neg_sum —
Lemma CausalSmith.Stat.BddUniformLogPenalty.prod_one_sub_le_exp_neg_card_mul —
Lemma CausalSmith.Stat.BddUniformLogPenalty.tvDist_eq_zero_of_klBudget_nonpos —
Lemma CausalSmith.Stat.BddUniformLogPenalty.card_mul_exp_neg_log_eq_rpow —
Lemma CausalSmith.Stat.BddUniformLogPenalty.coordinateOverlap_product_le_of_nonnegative_kl —
Lemma CausalSmith.Stat.BddUniformLogPenalty.ennreal_error_lower_bound_of_success_upper_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.half_integral_abs_rnDeriv_sub_le_tvDist —
Definition CausalSmith.Stat.BddUniformLogPenalty.selectCoupledRaw —
Definition CausalSmith.Stat.BddUniformLogPenalty.coupledDecoderGood —
Definition CausalSmith.Stat.BddUniformLogPenalty.coupledGoodIndicator —
Lemma CausalSmith.Stat.BddUniformLogPenalty.coupledDecoderGood_flip_exclusive —
Lemma CausalSmith.Stat.BddUniformLogPenalty.coupledDecoderGood_count_le_half —
Definition CausalSmith.Stat.BddUniformLogPenalty.measurableEqOfStandardBorel —
Lemma CausalSmith.Stat.BddUniformLogPenalty.tvDist_eq_half_integral_abs_rnDeriv_sub —
Definition CausalSmith.Stat.BddUniformLogPenalty.rnCommonPart —
Lemma CausalSmith.Stat.BddUniformLogPenalty.rnCommonPart_le_left —
Lemma CausalSmith.Stat.BddUniformLogPenalty.rnCommonPart_le_right —
Lemma CausalSmith.Stat.BddUniformLogPenalty.rnCommonPart_mass_eq_one_sub_tvDist —
Lemma CausalSmith.Stat.BddUniformLogPenalty.measure_eq_of_tvDist_eq_zero —
Definition CausalSmith.Stat.BddUniformLogPenalty.maximalCoupling —
Lemma CausalSmith.Stat.BddUniformLogPenalty.residual_mass —
Lemma CausalSmith.Stat.BddUniformLogPenalty.residual_mass_right —
Lemma CausalSmith.Stat.BddUniformLogPenalty.maximalCoupling_map_fst —
Lemma CausalSmith.Stat.BddUniformLogPenalty.maximalCoupling_map_snd —
Definition CausalSmith.Stat.BddUniformLogPenalty.maximalCoupling.instIsProbabilityMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.maximalCoupling_eq_mass_ge —
Definition CausalSmith.Stat.BddUniformLogPenalty.compressionCoupling —
Lemma CausalSmith.Stat.BddUniformLogPenalty.condDistrib_id_comp_map_compress —
Lemma CausalSmith.Stat.BddUniformLogPenalty.comap_comp_map_measure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressionCoupling_map_fst —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressionCoupling_map_snd —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressionCoupling_map_compress_pair —
Definition CausalSmith.Stat.BddUniformLogPenalty.compressionCoupling.instIsProbabilityMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressionCoupling_equal_compression_mass_ge —
Definition CausalSmith.Stat.BddUniformLogPenalty.planarRadius —
Lemma CausalSmith.Stat.BddUniformLogPenalty.halfDisc_radial_integral —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingSquare_isCompact —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingSquare_subset_scoreCube —
Lemma CausalSmith.Stat.BddUniformLogPenalty.integral_cos_sq_zero_to_pi —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressedSampleLaw_klDiv_le —
Definition CausalSmith.Stat.BddUniformLogPenalty.gaussianNoiseKernel —
Definition CausalSmith.Stat.BddUniformLogPenalty.gaussianNoiseKernel_isMarkovKernel —
Lemma CausalSmith.Stat.BddUniformLogPenalty.gaussianNoiseKernel_apply —
Lemma CausalSmith.Stat.BddUniformLogPenalty.gaussianNoiseKernel_klDiv_bind_le —
Definition CausalSmith.Stat.BddUniformLogPenalty.scorePoint —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scorePoint_apply_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scorePoint_apply_one —
Lemma CausalSmith.Stat.BddUniformLogPenalty.dist_scorePoint_same_second —
Lemma CausalSmith.Stat.BddUniformLogPenalty.dist_scorePoint_same_first —
Lemma CausalSmith.Stat.BddUniformLogPenalty.lowerEdgePoint_mem_frontier —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_apply_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_apply_one —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_first_abs_lt —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_mem_frontier —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_dist —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_separated —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCells_disjoint —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_mem_packingSquare_frontier —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridPackingCells_disjoint —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_three_radius_separated —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridPacking_three_radius —
Definition CausalSmith.Stat.BddUniformLogPenalty.planarAngle —
Lemma CausalSmith.Stat.BddUniformLogPenalty.halfDisc_weighted_polar_integral —
Lemma CausalSmith.Stat.BddUniformLogPenalty.halfDisc_weighted_cos_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.halfDisc_weighted_cos_sq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_first_abs_lt_quarter —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularGridSize —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularGridRadius —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridRadius_eventually_small —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridSize_lower —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridSize_frontier_lower —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridSize_spacing —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridSize_geometry —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGrid_frontierRate_geometry —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPacking_geometry —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingContDiffBump —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingBump —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBump_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBump_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBump_eq_zero_of_one_le_norm —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBump_contDiff —
Definition CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump_center —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump_eq_zero_of_bandwidth_le_dist —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump_contDiff —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingAffineBaseline —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAffineBaseline_mem_Icc —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingRegression —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_center —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_center_separation —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularCutoff —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularCutoff_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularCutoff_eq_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularCutoff_eq_one —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularCutoff_continuous —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularRadialProfile —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularRadialProfile_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularRadialProfile_continuous —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularTilt —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularTilt_continuous —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularTilt_abs_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularTilt_abs_le_quarter —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularDensityFactor —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDensityFactor_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDensityFactor_pos —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularTilt_eq_zero_of_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularTilt_eq_formula —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularTilt_halfDisc_mass_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAffineBaseline_contDiff —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_contDiff —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_bump_sum_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularRadialProfile_eq_zero_of_bandwidth_le_abs —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularTilt_eq_zero_of_bandwidth_le_abs —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDensityFactor_eq_one_of_bandwidth_le_abs —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularOutcomeCancellation_identity —
Definition CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianLaw —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianLaw_eq_gaussian_mixture —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianLaw_isProbabilityMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianLaw_integrable_id —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianLaw_integral_id —
Lemma CausalSmith.Stat.BddUniformLogPenalty.gaussianReal_integral_sq_sub —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianLaw_variance_id —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianLaw_memLp_id_two —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingConditionalVariance —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingConditionalVariance_continuous —
Lemma CausalSmith.Stat.BddUniformLogPenalty.one_add_mul_one_sub_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingConditionalVariance_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingConditionalVariance_envelope —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingDirectionCos —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingDirectionCos_abs_le_one —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingAngularTerm —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularTerm_abs_le_quarter —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularTerm_eq_zero_of_bandwidth_le_dist —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingAngularDensity —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularDensity_sum_abs_le_quarter —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularDensity_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularDensity_eq_on_cell —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularDensity_eq_one_off_cells —
Lemma CausalSmith.Stat.BddUniformLogPenalty.onePointDistanceLaw_isProbabilityMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressedSampleLaw_isProbabilityMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianLaw_klDiv_le_four_sq_sub —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularTerm_continuous —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularDensity_continuous —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularDensity_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_eq_on_cell —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_eq_off_cells —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingConditionalVariance_eq_on_cell —
Definition CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianKernel —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianKernel_isMarkovKernel —
Definition CausalSmith.Stat.BddUniformLogPenalty.jointBernoulliGaussianLaw —
Lemma CausalSmith.Stat.BddUniformLogPenalty.jointBernoulliGaussianLaw_isProbabilityMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.jointBernoulliGaussianLaw_map_snd —
Lemma CausalSmith.Stat.BddUniformLogPenalty.jointBernoulliGaussianLaw_condDistrib —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianLaw_integral_sq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.jointBernoulliGaussianLaw_memLp_fst_two —
Lemma CausalSmith.Stat.BddUniformLogPenalty.jointBernoulliGaussianLaw_condMean —
Lemma CausalSmith.Stat.BddUniformLogPenalty.jointBernoulliGaussianLaw_condVar —
Lemma CausalSmith.Stat.BddUniformLogPenalty.frontierRate_le_angularGridRadius —
Lemma CausalSmith.Stat.BddUniformLogPenalty.frontierRate_eventually_le_one —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularPackingDelta —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingDelta_eventually_admissible —
Lemma CausalSmith.Stat.BddUniformLogPenalty.frontierRate_fourth_power —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingDelta_fourth_power —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scoreCube_measurableSet —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularDesignDensity —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignDensity_eq_on_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignDensity_eq_zero_off_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignDensity_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignDensity_continuousOn —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignDensity_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignDensity_pos —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_absolutelyContinuous —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingSquare_volume —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_allFalse_isProbabilityMeasure —
Definition CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianCtyLaw —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianCtyLaw_map_snd —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianCtyLaw_mu —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianCtyLaw_sigmaSq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingScoreCube_convex —
Lemma CausalSmith.Stat.BddUniformLogPenalty.zero_mem_interior_packingScoreCube —
Lemma CausalSmith.Stat.BddUniformLogPenalty.closure_interior_packingScoreCube —
Lemma CausalSmith.Stat.BddUniformLogPenalty.volume_restrict_packingScoreCube_support —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_eq_restrict_withDensity —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_support —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_allFalse_eq_restrict —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_allFalse_support —
Definition CausalSmith.Stat.BddUniformLogPenalty.squareFrontierX —
Definition CausalSmith.Stat.BddUniformLogPenalty.squareFrontierY —
Definition CausalSmith.Stat.BddUniformLogPenalty.squareFrontierParam —
Lemma CausalSmith.Stat.BddUniformLogPenalty.squareFrontier_coordinates_lipschitz —
Lemma CausalSmith.Stat.BddUniformLogPenalty.squareFrontierParam_lipschitz —
Lemma CausalSmith.Stat.BddUniformLogPenalty.mem_frontier_scoreCube_half_iff —
Lemma CausalSmith.Stat.BddUniformLogPenalty.squareFrontierParam_right_edge —
Lemma CausalSmith.Stat.BddUniformLogPenalty.squareFrontierParam_left_edge —
Lemma CausalSmith.Stat.BddUniformLogPenalty.squareFrontierParam_top_edge —
Lemma CausalSmith.Stat.BddUniformLogPenalty.squareFrontierParam_bottom_edge —
Lemma CausalSmith.Stat.BddUniformLogPenalty.squareFrontierParam_image —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingSquare_rectifiableBoundary —
Definition CausalSmith.Stat.BddUniformLogPenalty.scoreCoordinates —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scoreCoordinates_measurePreserving —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scoreCoordinates_sub —
Lemma CausalSmith.Stat.BddUniformLogPenalty.planarRadius_scoreCoordinates_sub —
Lemma CausalSmith.Stat.BddUniformLogPenalty.mem_angularGrid_closedBall_iff_planarRadius —
Lemma CausalSmith.Stat.BddUniformLogPenalty.translatedHalfDisc_weighted_cos_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianCtyLaw_mem_nonparametricClass —
Lemma CausalSmith.Stat.BddUniformLogPenalty.mem_angularGrid_packingCell_iff_closedUpperHalfDisc —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingCenterValue —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_eq_packingCenterValue —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingCenterValue_true_false —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingCenterValue_frontierRate_separation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scoreCoordinates_measurableEmbedding —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingDirectionCos_eq_planarFirst_div_radius —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularTerm_integral_gridCell —
Lemma CausalSmith.Stat.BddUniformLogPenalty.planarFirst_div_radius_eq_cos —
Lemma CausalSmith.Stat.BddUniformLogPenalty.halfDisc_weighted_first_div_radius_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.closedHalfDisc_weighted_first_div_radius_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.translatedClosedHalfDisc_weighted_first_div_radius_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingScoreCube_isCompact —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularTerm_integral_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularDensity_integral_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_isProbabilityMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBump_iteratedFDeriv_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBump_iteratedFDeriv_holder —
Definition CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression_eq_of_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression_eq_on_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPacking_eventually_geometry_admissible —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_support —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_mu_eq_on_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_mem_nonparametricClass_of_holder —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression_continuous —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_map_snd —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_mu_center —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump_iteratedFDeriv —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump_iteratedFDeriv_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump_iteratedFDeriv_holder —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignDensity_eq_on_cell —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignDensity_eq_one_off_cells —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_restrict_cell_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression_eq_on_cell —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularDensity_integral_gridCell —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_gridCell_eq_volume —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_cell_mass_eq_volume —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_center_value_certificate —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressedSampleLaw_eq_pi_onePointDistanceLaw —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressedSampleLaw_klDiv_le_of_onePoint —
Lemma CausalSmith.Stat.BddUniformLogPenalty.jointBernoulliGaussianLaw_restrict_score_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_restrict_cell_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_restrict_off_cells_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_restrict_off_cells_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.planarRadius_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.halfDisc_radialSet_weighted_cos_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.translatedHalfDisc_radialSet_weighted_cos_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_locality_certificate —
Lemma CausalSmith.Stat.BddUniformLogPenalty.halfDisc_radialSet_weighted_first_div_radius_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.closedHalfDisc_radialSet_weighted_first_div_radius_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.translatedClosedHalfDisc_radialSet_weighted_first_div_radius_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularTerm_integral_gridCell_radialSet —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_map_distance_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_flip_map_distance_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularDensity_integral_gridCell_radialSet —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_gridCell_radialSet_eq_volume —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignDensity_eq_off_gridCell —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_restrict_compl_gridCell_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_map_distance_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump_support_iteratedFDeriv_subset —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump_iteratedFDeriv_pairwise_exclusive —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump_iteratedFDeriv_unique_active —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBumpSum_iteratedFDeriv_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBumpSum_contDiff —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBumpSum_iteratedFDeriv_bound_with —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBumpSum_iteratedFDeriv_lipschitz —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_onePointDistanceLaw_eq_map_compProd —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression_eventuallyEq_packingRegression_at_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression_iteratedFDeriv_eq_on_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression_contDiffOn_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.euclideanHolderBallStd_nat_iff —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAffineBaseline_iteratedFDeriv_one_norm_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAffineBaseline_iteratedFDeriv_eq_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAffineBaseline_iteratedFDeriv_sub_eq_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBumpSum_iteratedFDeriv_lipschitz_with —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression_mem_holder_of_bump_bounds —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression_mem_holder_of_scaling_bounds —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_mem_nonparametricClass_of_scaling_bounds —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_radial_representation_certificate —
Lemma CausalSmith.Stat.BddUniformLogPenalty.halfDisc_radialSet_angular_outcome_cancellation —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingBumpDerivativeBound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBumpDerivativeBound_nonneg —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBump_iteratedFDeriv_le_derivativeBound —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingBumpDerivativeScale —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBumpDerivativeScale_pos —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBumpDerivativeBound_le_scale —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularPackingScaledDelta —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingScaledDelta_derivative_scaling —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_mem_class_scaledDelta —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingScaledDelta_eventually_admissible —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingScaledDelta_eq_scale_mul_frontierRate —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingCenterValue_scaledDelta_separation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingAt_of_certificates —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingAt_scaledDelta_of_radial_certificates —
Lemma CausalSmith.Stat.BddUniformLogPenalty.translatedHalfDisc_radialSet_angular_outcome_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump_eq_delta_mul_angularRadialProfile —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingBump_eq_of_norm_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localizedPackingBump_eq_delta_mul_radial —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingScaledDelta_cutoff_fully_active —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGrid_packingCell_volume_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.map_radiusOutcome_compProd_apply_prod —
Lemma CausalSmith.Stat.BddUniformLogPenalty.map_radiusOutcome_compProd_restrict_Ici_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_onePointDistanceLaw_apply_prod —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_radialSet_angular_outcome_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianKernel_setLIntegral_eq_of_mass_and_mean —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_restrict_Ici_eq_of_radial_slices —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingScaledDelta_fourth_power —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_mul_density_flip_cell_identity —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_mul_density_flip_cell_radial_identity —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_closedRadialSet_weighted_direction_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingScaledDelta_cutoff_active_on_tail —
Lemma CausalSmith.Stat.BddUniformLogPenalty.radialSlice_mass_eq_of_map_distance_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_openRadialSet_weighted_direction_cancellation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_density_flip_integral_eq_zero_of_active —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedPackingRegression_setIntegral_angularDesignMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPacking_flip_setIntegral_eq_of_active —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingCtyLaw_flip_restrict_Ici_eq_of_active —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressedSampleLaw_klDiv_le_of_onePoint_finite_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPackingScaledDelta_eventually_klBudget —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularRadialSuccessIncrement_identity —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularRadialSuccessIncrement_abs_le —
Definition CausalSmith.Stat.BddUniformLogPenalty.angularPackingOnePointKLConstant —
Definition CausalSmith.Stat.BddUniformLogPenalty.middleHalfClip —
Lemma CausalSmith.Stat.BddUniformLogPenalty.middleHalfClip_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.middleHalfClip_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.middleHalfClip_clippedPackingRegression_ae_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularDesignMeasure_radial_Iio_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPacking_radialOutcome_klDiv_le —
Definition CausalSmith.Stat.BddUniformLogPenalty.radialSuccessMeasure —
Definition CausalSmith.Stat.BddUniformLogPenalty.radialSuccessParameter —
Lemma CausalSmith.Stat.BddUniformLogPenalty.radialSuccessMeasure_absolutelyContinuous —
Lemma CausalSmith.Stat.BddUniformLogPenalty.radialSuccessParameter_setIntegral —
Lemma CausalSmith.Stat.BddUniformLogPenalty.radialSuccessParameter_mem_Icc_ae —
Definition CausalSmith.Stat.BddUniformLogPenalty.clippedRadialSuccessParameter —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedRadialSuccessParameter_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedRadialSuccessParameter_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedRadialSuccessParameter_ae_eq —
Definition CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianKernelOn —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianKernelOn_isMarkovKernel —
Lemma CausalSmith.Stat.BddUniformLogPenalty.bernoulliGaussianKernel_setLIntegral_eq_of_mass_and_mean_general —
Lemma CausalSmith.Stat.BddUniformLogPenalty.radialOutcomeLaw_eq_map_swap_compProd —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedRadialSuccessParameter_abs_sub_le_ae —
Lemma CausalSmith.Stat.BddUniformLogPenalty.radialOutcomeLaw_klDiv_le_of_localized_success_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularRadialSuccessIncrement_setIntegral_abs_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.halfDisc_radialSet_angular_outcome_abs_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularGridCenter_radialSet_angular_outcome_abs_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPacking_flip_setIntegral_abs_le_cellArea —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPacking_flip_setIntegral_abs_le_radialMass —
Lemma CausalSmith.Stat.BddUniformLogPenalty.angularPacking_flip_success_setIntegral_abs_le —
Definition CtyDistanceIdentification —
Definition CtyExpectedLocalPolynomialMaximalBounds —
Definition CtyUniformFirstOrderBias —
Definition empiricalGramGuard —
Definition EuclideanBallsVCProperty —
Definition EuclideanCExtEnvelope —
Definition GeometryData —
Definition PopulationGramFloor —
Definition RectangularScoreSupport —
Definition RectifiableCurve —
Definition ShatteredByClosedBalls —
Definition signedArm —
Definition stabilizedLocalPolynomialOuterRisk —
Definition CausalHardGeometryLaw —
Definition CtyIdentificationAssumptions —
Definition CtyIdentificationMetric —
Definition SignedDistanceMeanVersions —
Definition CtyMetricBallsVCProperty —
Definition CtyUniformBiasKernel —
Definition CtyUniformBiasLawClass —
Definition ShatteredByMetricBalls —
Lemma CausalSmith.Stat.BddUniformLogPenalty.abs_sub_winsorize_le_abs —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorize_tail_le_moment —
Lemma CausalSmith.Stat.BddUniformLogPenalty.a1a2OuterRisk_le_stabilizedLocalPolynomialOuterRisk —
Lemma CausalSmith.Stat.BddUniformLogPenalty.stabilizedLocalPolynomial_normalizedRisk_limsup_le —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalSelectedBernoulliKernel —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalSelectedBernoulliKernel_isMarkovKernel —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalSelectedBernoulliKernel_integral_id —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalSelectedBernoulliKernel_variance_id —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalSelectedBernoulliKernel_condAbsMoment_le_one —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalBernoulliPotentialOutcomeMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalBernoulliPotentialOutcomeMeasure_map_score —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalBernoulliPotentialOutcomeMeasure_isProbabilityMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalBernoulliPotentialOutcomeMeasure_map_score_arm —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalBernoulliA1A2Law —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardSquare_eq_scoreCube —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardSquare_measurableSet —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardSquare_isCompact —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardSquare_volume —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArmOne_measurableSet —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArmZero_measurableSet —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArmOne_subset_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardAssignment_partition —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCell_subset_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCell_volume —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreDensity —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreDensity_continuous —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreDensity_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardPackingDensity_integral_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreMeasure_isProbabilityMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreMeasure_cell_mass —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularTerm_integral_closedBall —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularDensity_integral_closedBall —
Lemma CausalSmith.Stat.BddUniformLogPenalty.closure_interior_causalHardSquare —
Lemma CausalSmith.Stat.BddUniformLogPenalty.volume_restrict_causalHardSquare_support —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreMeasure_support —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreMeasure_restrict_cell_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreMeasure_restrict_off_cells_eq —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardGridCenter —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardGridCenter_mem_bottomEdge —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardGridCenter_dist —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardGridCenter_separated —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardGrid_geometry —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardGrid_scaled_geometry —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArmOne_isClosed —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArmOne_isCompact —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArmOne_subset_interior_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardFrontier_compact_and_interior —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardAssignment_frontier —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardControlProfile —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardTreatmentProfile —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardProfiles_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardProfiles_mem_unitInterval —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardPackingRegression_mem_middleHalf —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardTreatmentProfile_mem_middleHalf —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardTreatmentProfile_center_flip —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardRectangleAffineHomeomorph —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardRectangleAffineHomeomorph_image —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardRectangleFrontierParam —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardFrontier_rectifiableCurve —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_geometry —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_muPO —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_cell_mass —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_tau_center_flip —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_condAbsMoment_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardSquare_rectangularScoreSupport —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardFrontier_hausdorff_bounds_fixed —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardFrontier_hausdorff_bounds —
Lemma CausalSmith.Stat.BddUniformLogPenalty.A1A2Law.memLp_armCoord_of_condAbsMoment_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_density_certificates —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_kernel_certificates —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_class_geometry —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_restrict_cell_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_restrict_off_cells_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalBernoulliPotentialOutcomeMeasure_restrict_score_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreMeasure_restrict_compl_cells_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardControlProfile_coordinatePartial —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardControlProfile_euclideanCExtEnvelope —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_control_euclideanCExtEnvelope —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardTreatmentProfile_eq_packingRegression_on_square —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardTreatmentProfile_has_smooth_extension —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardGrid_scaled_family_geometry —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_mem_class_of_analytic_certificates —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardTreatmentProfile_euclideanCExtEnvelope_of_scaling_bounds —
Lemma CausalSmith.Stat.BddUniformLogPenalty.coordinatePartial_abs_le_iteratedFDeriv_norm —
Lemma CausalSmith.Stat.BddUniformLogPenalty.coordinatePartial_sub_abs_le_iteratedFDeriv_sub_norm —
Lemma CausalSmith.Stat.BddUniformLogPenalty.euclideanCExtEnvelope_of_iteratedFDeriv_bounds —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardTreatmentProfile_powerBandwidth_derivative_scaling —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardTreatmentProfile_euclideanCExtEnvelope_powerBandwidth —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardConstructionBandwidthConstant —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardPowerBandwidthEventuallySmall —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardSmoothGridPackage —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardHypercubeConstructorPrefix —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardHypercubeConstructorData —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_vertex_core —
Lemma CausalSmith.Stat.BddUniformLogPenalty.uniformKernel_eq_one_iff_abs_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.uniformKernel_signedDist_eq_one_iff —
Lemma CausalSmith.Stat.BddUniformLogPenalty.armLocalMass_integrand_eq_indicator_closedBall —
Lemma CausalSmith.Stat.BddUniformLogPenalty.armLocalMass_eq_normalized_inter_closedBall —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreBox —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreBox_volume —
Lemma CausalSmith.Stat.BddUniformLogPenalty.mem_closedBall_of_coordinate_diff_le_half —
Lemma CausalSmith.Stat.BddUniformLogPenalty.mem_frontier_causalHardArmOne_iff —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArmOne_inter_closedBall_volume_lower —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArmZero_inter_closedBall_volume_lower —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArm_inter_closedBall_volume_lower —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArmLocalMass_lower —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_localMass_certificate —
Lemma CausalSmith.Stat.BddUniformLogPenalty.matrixQuadratic_rankOne —
Lemma CausalSmith.Stat.BddUniformLogPenalty.matrixQuadratic_polyBasis_rankOne —
Lemma CausalSmith.Stat.BddUniformLogPenalty.matrixQuadratic_polyBasis_rankOne_nonneg —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreComplexEquiv —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardSphere_hausdorffMeasure_lt_top —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreCoordinate_lipschitz —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hausdorffMeasure_pos_of_scoreCoordinate_interval —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardArc0 —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardArc1 —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArc0_dist —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArc1_dist —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArc_sqrt_bounds —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArmOne_slice_hausdorffMeasure_pos —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardArmZero_slice_hausdorffMeasure_pos —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardDensitySlice_finite_pos —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_slice_certificate —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hasCountableEmpiricalSupReduction_of_pointwise_dense —
Definition CausalSmith.Stat.BddUniformLogPenalty.localPolynomial —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localPolynomial_eval —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localPolynomial_eq_zero_iff —
Definition CausalSmith.Stat.BddUniformLogPenalty.hardSquareRadialEnergy —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareRadialEnergy_eq_integral —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareRadialEnergy_pos —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareRadialEnergy_continuous —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareRadialEnergy_smul —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareRadialEnergy_coercive —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareSignedRadialEnergy_coercive —
Lemma CausalSmith.Stat.BddUniformLogPenalty.populationGram_quadratic_eq_integral —
Lemma CausalSmith.Stat.BddUniformLogPenalty.firstQuadrant_radial_integral —
Lemma CausalSmith.Stat.BddUniformLogPenalty.firstQuadrant_scaledPolynomial_integral —
Definition CausalSmith.Stat.BddUniformLogPenalty.scoreOfProd —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scoreOfProd_apply —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scoreOfProd_volumePreserving —
Definition CausalSmith.Stat.BddUniformLogPenalty.signedProdEquiv —
Lemma CausalSmith.Stat.BddUniformLogPenalty.signedProdEquiv_volumePreserving —
Definition CausalSmith.Stat.BddUniformLogPenalty.hardSquareSectorEquiv —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareSectorEquiv_volumePreserving —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareSectorEquiv_dist —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareSectorEquiv_apply_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareSectorEquiv_apply_one —
Lemma CausalSmith.Stat.BddUniformLogPenalty.fst_le_planarRadius —
Lemma CausalSmith.Stat.BddUniformLogPenalty.snd_le_planarRadius —
Definition CausalSmith.Stat.BddUniformLogPenalty.hardSquareSectorDomain —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareArmOne_sector_subset —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareArmOne_contains_sector —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareArmZero_sector_subset —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareArmZero_contains_sector —
Definition CausalSmith.Stat.BddUniformLogPenalty.hardSquareGramScoreIntegrand —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareGramScoreIntegrand_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareGramScoreIntegrand_nonneg —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareGramScoreIntegrand_integrable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.populationGram_quadratic_eq_scoreIntegral —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquareGramScoreIntegrand_on_arm —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquare_sector_scaledPolynomial_integral —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquare_populationGram_quadratic_lower —
Lemma CausalSmith.Stat.BddUniformLogPenalty.hardSquare_populationGramFloor_eventually —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_populationGram_certificate —
Definition CausalSmith.Stat.BddUniformLogPenalty.signedObservationAt —
Lemma CausalSmith.Stat.BddUniformLogPenalty.signedObservationAt_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationMeasure —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationMeasure_isFiniteMeasure —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationLaw —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationLaw_isProbabilityMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationMeasure_eq_mass_smul_law —
Definition CausalSmith.Stat.BddUniformLogPenalty.commonStatisticBernoulliKernel —
Lemma CausalSmith.Stat.BddUniformLogPenalty.commonStatisticBernoulliKernel_isMarkovKernel —
Lemma CausalSmith.Stat.BddUniformLogPenalty.commonStatisticBernoulli_klDiv_le_of_localized_parameter —
Lemma CausalSmith.Stat.BddUniformLogPenalty.commonStatisticBernoulliOutcome_klDiv_le_of_localized_parameter —
Definition CausalSmith.Stat.BddUniformLogPenalty.statisticSuccessMeasure —
Definition CausalSmith.Stat.BddUniformLogPenalty.statisticSuccessParameter —
Lemma CausalSmith.Stat.BddUniformLogPenalty.statisticSuccessMeasure_absolutelyContinuous —
Lemma CausalSmith.Stat.BddUniformLogPenalty.statisticSuccessParameter_setIntegral —
Lemma CausalSmith.Stat.BddUniformLogPenalty.statisticSuccessParameter_mem_Icc_ae —
Definition CausalSmith.Stat.BddUniformLogPenalty.clippedStatisticSuccessParameter —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedStatisticSuccessParameter_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedStatisticSuccessParameter_mem_Icc —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedStatisticSuccessParameter_ae_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.commonStatisticBernoulliKernel_setLIntegral_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.statisticBernoulliOutcomeLaw_eq_map_swap_compProd —
Lemma CausalSmith.Stat.BddUniformLogPenalty.clippedStatisticSuccessParameter_abs_sub_le_ae —
Lemma CausalSmith.Stat.BddUniformLogPenalty.statisticBernoulliOutcome_klDiv_le_of_localized_success_bound —
Definition CausalSmith.Stat.BddUniformLogPenalty.estimatorMatrixMeasurableSpace —
Definition CausalSmith.Stat.BddUniformLogPenalty.estimatorMatrixBorelSpace —
Lemma CausalSmith.Stat.BddUniformLogPenalty.uniformKernel_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.polyBasis_apply_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorize_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.estimatorMeasurable_matrix_iff —
Lemma CausalSmith.Stat.BddUniformLogPenalty.estimatorMatrix_inv_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.estimatorInv_mulVec_apply_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.empiricalGram_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.empiricalScore_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.empiricalGramGuard_measurableSet —
Lemma CausalSmith.Stat.BddUniformLogPenalty.guardedCoefficient_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.stabilizedLocalPolynomial_mem —
Lemma CausalSmith.Stat.BddUniformLogPenalty.statisticBernoulliOutcome_restrict_compl_eq_of_localized_success_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.signedObservationAt_eq_armOne —
Lemma CausalSmith.Stat.BddUniformLogPenalty.signedObservationAt_eq_armZero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.signedDistance_knownGeometry_eq_dist —
Lemma CausalSmith.Stat.BddUniformLogPenalty.signedDistance_knownGeometry_eq_neg_dist —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scoreCenter_radialSet_angular_outcome_abs_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardGridCenter_radialSet_angular_outcome_abs_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.measure_eq_of_pos_ofReal_smul_eq —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardObservedSuccessProfile —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardObservedSuccessProfile_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardObservedSuccessProfile_mem_middleHalf —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardSignedStatistic —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardSignedStatistic_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.signedDistance_causalHardA1A2Law_eq_causalHardSignedStatistic —
Definition CausalSmith.Stat.BddUniformLogPenalty.selectedArmSuccessProfile —
Lemma CausalSmith.Stat.BddUniformLogPenalty.selectedArmSuccessProfile_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.selectedArmOutcome —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalBernoulliPotentialOutcomeMeasure_restrict_map_selected —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationMeasure_eq_scoreBernoulli —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scoreCenter_closedUpperHalf_radial_angularTerm_integral_eq_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scoreCenter_closedBall_radial_angularTerm_integral_eq_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scoreCenter_closedLowerHalf_radial_angularTerm_integral_eq_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationLaw_eq_of_bit_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationLaw_vertex_package —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationLaw_eq_of_restrict_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scoreCenter_signedRadialSet_angularTerm_integral_eq_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardSignedStatistic_eq_verticalSignedRadius_on_cell —
Lemma CausalSmith.Stat.BddUniformLogPenalty.scoreCenter_causalHardSignedStatistic_angularTerm_integral_eq_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreMeasure_signedSlice_eq_uniform —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreMeasure_restrict_map_signedStatistic_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardScoreMeasure_signedStatistic_Ioo_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_mul_density_enable_cell_identity —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRegression_mul_density_enable_cell_radial_identity —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardObservedSuccess_setIntegral_scoreMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingAngularDensity_enable_cell_sub —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardSignedSuccess_setIntegral_abs_le_cellArea —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardSignedSuccess_localized_setwise_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationMeasure_klDiv_enable_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardHypercubeAt_of_constructorData —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardCellBitRepresentative —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardCellBitObservationLaw —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalHardCellBitObservationLaw_isProbabilityMeasure —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationMeasure_eq_bitLaw —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationMeasure_map_snd —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCellBitObservationLaw_common_signedMarginal —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCellBitObservationLaw_restrict_compl_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCellBitObservationLaw_klDiv_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardCellSignedObservationMeasure_restrict_compl_enable_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedScore_boundary_volume_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedScore_boundary_nonempty —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedScore_boundary_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.uniformKernel_div_eq_if_abs_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.separableWinsorizedScoreFunction_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.separableWinsorizedScoreFunction_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.separableWinsorizedScoreIndex_exists_approximating_sequence —
Lemma CausalSmith.Stat.BddUniformLogPenalty.separableWinsorizedScoreFunction_tendsto —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedScore_hasUniformL2Radius_at —
Lemma CausalSmith.Stat.BddUniformLogPenalty.outerLIntegral_le_lintegral_of_ae_eq_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.separableWinsorizedScore_sq_pointwise_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.separableWinsorizedScore_abs_le_envelope —
Lemma CausalSmith.Stat.BddUniformLogPenalty.separableWinsorizedScore_hasUniformL2Radius —
Lemma CausalSmith.Stat.BddUniformLogPenalty.separableWinsorizedScore_hasUniformPolynomialL2Cover —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedScore_hasVCUniformEntropy —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedScore_hasVCUniformEntropy_at —
Lemma CausalSmith.Stat.BddUniformLogPenalty.separableWinsorizedScore_hasUniformL2Radius_at —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedScore_hasCountableEmpiricalSupReduction_at —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedScore_hasCountableEmpiricalSupReduction —
Lemma CausalSmith.Stat.BddUniformLogPenalty.populationScore_apply_uniform_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.populationGram_posDef —
Lemma CausalSmith.Stat.BddUniformLogPenalty.populationGram_mulVec_populationCoefficient —
Lemma CausalSmith.Stat.BddUniformLogPenalty.populationCoefficient_uniform_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.marginal_closedBall_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localized_arm_sq_lintegral_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedScore_sq_pointwise_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedScore_hasVCUniformEntropy_all_nu —
Lemma CausalSmith.Stat.BddUniformLogPenalty.populationCoefficient_uniform_bound_explicit —
Lemma CausalSmith.Stat.BddUniformLogPenalty.separableWinsorizedScore_hasUniformL2Radius_explicit —
Lemma CausalSmith.Stat.BddUniformLogPenalty.vcExpectedMaximalInequality_explicit —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedCenteredScore_apply_eq_separableAverage —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedCenteredScore_integral_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedScoreDeviation_le_empiricalProcessSup —
Lemma CausalSmith.Stat.BddUniformLogPenalty.outerLIntegral_const_mul_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.outerLIntegral_mono —
Lemma CausalSmith.Stat.BddUniformLogPenalty.outerLIntegral_add_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.outerLIntegral_const_probability —
Lemma CausalSmith.Stat.BddUniformLogPenalty.frontierRate_shift_antitone —
Lemma CausalSmith.Stat.BddUniformLogPenalty.t4_frontierRate_fourth_power —
Lemma CausalSmith.Stat.BddUniformLogPenalty.frontierRate_shift_tendsto_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.frontierRate_shift_nh2_tendsto_top —
Lemma CausalSmith.Stat.BddUniformLogPenalty.matrixQuadratic_sub_abs_le_entrywise —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalGramDeviation_entry_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.empiricalGramGuard_of_deviation_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.empiricalGramGuard_inv_mulVec_euclidean_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.euclideanNorm_le_card_mul_piNorm —
Lemma CausalSmith.Stat.BddUniformLogPenalty.empiricalGramGuard_isUnit —
Lemma CausalSmith.Stat.BddUniformLogPenalty.guardedCoefficient_sub_populationCoefficient_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.condKer_winsorization_tail_lintegral_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.localized_arm_winsorization_tail_lintegral_le —
Definition CausalSmith.Stat.BddUniformLogPenalty.winsorizedResidualTail —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedResidualTail_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedResidualTail_enorm_le_localized_arms —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedResidualTail_lintegral_le —
Definition CausalSmith.Stat.BddUniformLogPenalty.signedLocalPolynomialWeight —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalSignedDistance_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.signedLocalPolynomialWeight_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.signedLocalPolynomialWeight_abs_le_one —
Lemma CausalSmith.Stat.BddUniformLogPenalty.observedOutcome_integrable —
Definition CausalSmith.Stat.BddUniformLogPenalty.rawPopulationResidual —
Lemma CausalSmith.Stat.BddUniformLogPenalty.rawPopulationResidual_integral_eq_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.winsorizedCenteredScore_mean_norm_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.euclideanCExtEnvelope_abs_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.stabilizedLocalPolynomial_boundaryLoss_le —
Definition CausalSmith.Stat.BddUniformLogPenalty.t4Bandwidth —
Lemma CausalSmith.Stat.BddUniformLogPenalty.t4Bandwidth_eventually_eq_frontierRate —
Lemma CausalSmith.Stat.BddUniformLogPenalty.t4Bandwidth_pos —
Lemma CausalSmith.Stat.BddUniformLogPenalty.t4Bandwidth_antitone —
Lemma CausalSmith.Stat.BddUniformLogPenalty.t4Bandwidth_tendsto_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.frontierRate_nh2_tendsto_top —
Lemma CausalSmith.Stat.BddUniformLogPenalty.t4Bandwidth_nh2_tendsto_top —
Lemma CausalSmith.Stat.BddUniformLogPenalty.t4Bandwidth_maximal_regime_tendsto_top —
Lemma CausalSmith.Stat.BddUniformLogPenalty.t4_winsorization_power —
Lemma CausalSmith.Stat.BddUniformLogPenalty.t4_frontier_rate_terms_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressedCoordinateLaw_klDiv_le —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingPartitionSet —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingPartitionSet_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingPartitionSet_pairwiseDisjoint —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingPartitionSet_iUnion —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingFinitePartition —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingFinitePartition_cellSet —
Definition CausalSmith.Stat.BddUniformLogPenalty.finiteSampleMap —
Lemma CausalSmith.Stat.BddUniformLogPenalty.measurable_finiteSampleMap —
Lemma CausalSmith.Stat.BddUniformLogPenalty.finiteSampleMap_fixedSizeEmbed —
Lemma CausalSmith.Stat.BddUniformLogPenalty.cellObservationLaw_eq_of_restrict_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.markedPoissonKL_le_two_mul_of_piKL —
Lemma CausalSmith.Stat.BddUniformLogPenalty.finitePackingLoss_measurable_of_pointIndexed —
Lemma CausalSmith.Stat.BddUniformLogPenalty.midpointDecoder_error_ge_half_separation —
Lemma CausalSmith.Stat.BddUniformLogPenalty.midpointDecoder_wrong_bit_error —
Lemma CausalSmith.Stat.BddUniformLogPenalty.finitePackingLoss_ge_of_midpointDecoder_wrong —
Definition CausalSmith.Stat.BddUniformLogPenalty.canonicalPrefixObservations —
Lemma CausalSmith.Stat.BddUniformLogPenalty.measurable_canonicalPrefixObservations —
Lemma CausalSmith.Stat.BddUniformLogPenalty.map_canonicalPrefixObservations_restrict_count_ge —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingFinitePartition_cell_measure_eq_map_snd —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingCellObservationLaw_eq_of_bit_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingCanonicalCellLaw_eq_of_bit_eq —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingMarkLaw —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingComplement_restrict_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.canonicalMarkedPoissonSampleLaw_zero —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingCanonicalComplementLaw_eq —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingMarkedDistance —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingMarkedDistance_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.compressPackingCell —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressPackingCell_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.assemblePackingDistanceBlocks —
Lemma CausalSmith.Stat.BddUniformLogPenalty.assemblePackingDistanceBlocks_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingPoissonValue —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingPoissonValue_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingPoissonDecoder —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingPoissonDecoder_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingPoissonDecoder_local —
Lemma CausalSmith.Stat.BddUniformLogPenalty.canonicalPrefix_finiteSampleMap_packingMarkedDistance —
Lemma CausalSmith.Stat.BddUniformLogPenalty.globalPackingPoissonLoss_eq_finitePackingLoss —
Lemma CausalSmith.Stat.BddUniformLogPenalty.canonicalMarkedPoissonSampleLaw_map_count —
Lemma CausalSmith.Stat.BddUniformLogPenalty.globalPackingPoissonLoss_le_on_count_lt —
Lemma CausalSmith.Stat.BddUniformLogPenalty.globalPackingPoissonRisk_le_fixedRisk_add_tail —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingSingleBit —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingSingleBit_self —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingCellExperiment —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingCommonExperiment —
Definition CausalSmith.Stat.BddUniformLogPenalty.synthesizePackingConfiguration —
Lemma CausalSmith.Stat.BddUniformLogPenalty.synthesizePackingConfiguration_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.map_finitePoissonSampleLaw_finiteSampleMap —
Lemma CausalSmith.Stat.BddUniformLogPenalty.map_finiteMarkedPoissonSampleLaw_finiteSampleMap —
Definition CausalSmith.Stat.BddUniformLogPenalty.packingRadialPartition —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingRadialPartition_cellSet_false —
Lemma CausalSmith.Stat.BddUniformLogPenalty.map_shortRadiusBlock_finiteMarkedPoissonSampleLaw —
Lemma CausalSmith.Stat.BddUniformLogPenalty.map_packingCell_restrict_eq_onePointDistance_restrict —
Lemma CausalSmith.Stat.BddUniformLogPenalty.finiteSampleMap_orderByMarks —
Lemma CausalSmith.Stat.BddUniformLogPenalty.map_packingCellObservationLaw_eq_radialCellObservationLaw —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressedPackingCellExperiment_eq_shortRadiusBlock —
Lemma CausalSmith.Stat.BddUniformLogPenalty.finiteMeasureMarkedPoissonLaw_probability_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressedPackingCellExperiment_klDiv_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingExperiment_synthesis_law —
Definition CausalSmith.Stat.BddUniformLogPenalty.finiteSamplePaddedStream —
Lemma CausalSmith.Stat.BddUniformLogPenalty.finiteSamplePaddedStream_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.streamToFiniteSample_paddedStream —
Lemma CausalSmith.Stat.BddUniformLogPenalty.finiteSamplePaddedStream_range —
Definition CausalSmith.Stat.BddUniformLogPenalty.finiteSample_standardBorelSpace —
Definition CausalSmith.Stat.BddUniformLogPenalty.globalPackingPoissonValue —
Lemma CausalSmith.Stat.BddUniformLogPenalty.globalPackingPoissonValue_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.coordinatewiseErrorProbability —
Lemma CausalSmith.Stat.BddUniformLogPenalty.coordinatewiseErrorProbability_eq_one_sub_success —
Lemma CausalSmith.Stat.BddUniformLogPenalty.packingCoordinatewiseError_lower_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.assemblePackingDistanceBlocks_eq_globalMap —
Definition CausalSmith.Stat.BddUniformLogPenalty.globalPackingPoissonLoss —
Lemma CausalSmith.Stat.BddUniformLogPenalty.globalPackingPoissonLoss_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.blockPackingPoissonLoss —
Lemma CausalSmith.Stat.BddUniformLogPenalty.blockPackingPoissonLoss_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.blockPackingPoissonLoss_eq_global —
Lemma CausalSmith.Stat.BddUniformLogPenalty.exists_vertex_poissonLoss_ge_coordinatewiseError —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalPackingPartitionSet —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingPartitionSet_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingPartitionSet_pairwiseDisjoint —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingPartitionSet_iUnion —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalPackingFinitePartition —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingFinitePartition_cellSet —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingFinitePartition_cell_measure_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingCellObservationLaw_eq_of_bit_eq —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalPackingSingleBit —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalPackingCellExperiment —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalPackingCommonExperiment —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingCanonicalCellLaw_eq_of_bit_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingCanonicalComplementLaw_eq —
Definition CausalSmith.Stat.BddUniformLogPenalty.synthesizeCausalPackingConfiguration —
Lemma CausalSmith.Stat.BddUniformLogPenalty.synthesizeCausalPackingConfiguration_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingExperiment_synthesis_law —
Lemma CausalSmith.Stat.BddUniformLogPenalty.signedObservationAt_eq_of_hard_geometry_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.signedDistanceData_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalPackingMarkedStatistic —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingMarkedStatistic_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.compressCausalPackingCell —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressCausalPackingCell_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.assembleCausalPackingBlocks —
Lemma CausalSmith.Stat.BddUniformLogPenalty.assembleCausalPackingBlocks_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalPackingPoissonValue —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingPoissonValue_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalPackingPoissonDecoder —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingPoissonDecoder_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingPoissonDecoder_local —
Definition CausalSmith.Stat.BddUniformLogPenalty.causalFinitePackingLoss —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalFinitePackingLoss_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalHardA1A2Law_tau_center_eq_of_bit_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.knownGeometry_eq_of_components_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.compressedCausalPackingCellExperiment_eq —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalScaledFrontier_eventually_klBudget —
Definition CausalSmith.Stat.BddUniformLogPenalty.globalCausalPackingPoissonValue —
Lemma CausalSmith.Stat.BddUniformLogPenalty.globalCausalPackingPoissonValue_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.globalCausalPackingPoissonLoss —
Lemma CausalSmith.Stat.BddUniformLogPenalty.globalCausalPackingPoissonLoss_measurable —
Definition CausalSmith.Stat.BddUniformLogPenalty.blockCausalPackingPoissonLoss —
Lemma CausalSmith.Stat.BddUniformLogPenalty.blockCausalPackingPoissonLoss_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.assembleCausalPackingBlocks_eq_globalMap —
Lemma CausalSmith.Stat.BddUniformLogPenalty.blockCausalPackingPoissonLoss_eq_global —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalPackingCoordinatewiseError_lower_bound —
Lemma CausalSmith.Stat.BddUniformLogPenalty.exists_vertex_causalPoissonLoss_ge_coordinatewiseError —
Lemma CausalSmith.Stat.BddUniformLogPenalty.globalCausalPackingPoissonLoss_eq_finitePackingLoss —
Lemma CausalSmith.Stat.BddUniformLogPenalty.globalCausalPackingPoissonLoss_le_on_count_lt —
Lemma CausalSmith.Stat.BddUniformLogPenalty.globalCausalPackingPoissonRisk_le_fixedRisk_add_tail —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalEuclideanCExtEnvelope_abs_le —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalOuterLIntegral_mono —
Lemma CausalSmith.Stat.BddUniformLogPenalty.causalFinitePackingLoss_le_boundaryLoss —
Lemma CausalSmith.Stat.BddUniformLogPenalty.a1a2_finite_max_eventually_lower —
Lemma CausalSmith.Stat.BddUniformLogPenalty.lintegral_le_outerLIntegral_of_measurable —
Lemma CausalSmith.Stat.BddUniformLogPenalty.a1a2OuterRisk_eventually_lower —

Symbol realizations (Lean @realizes) (43)

X verified realized by Score, Observation, CausalObservation, causalScore
Y verified realized by Observation, CtyLaw, observedOutcome
n verified realized by variable@38, variable@157, variable@26
q verified realized by variable@39, CtyNonparametricClass
L verified realized by variable@40, CtyNonparametricClass, variable@38, variable@26, A1A2ClassWitness
P verified realized by CtyLaw, A1A2Law
$\mathcal{X}_P$ verified realized by CtyLaw, CtyNonparametricClass, A1A2Law, knownGeometry, A1A2ClassWitness
f_P verified realized by CtyLaw, CtyNonparametricClass, A1A2Law, A1A2ClassWitness
$\mu_P$ verified realized by CtyLaw, CtyNonparametricClass
$\sigma_P^2$ verified realized by CtyLaw, CtyNonparametricClass
$\mathcal{X}_P(compact)$ verified realized by CtyNonparametricClass
$\mathcal{P}_{\mathrm{NP}}$ verified realized by admissibleLaws
V_{n,P}(x) verified realized by distanceData
$\widehat{\mu}_n$ verified realized by RuleFun, CtyDistanceDecisionClass
$\widehat{\mu}^{\mathrm{PI}}_n$ verified realized by RuleFun, PointIndexedDecisionClass
$\mathcal{T}^{\mathrm{NP}}_n$ verified realized by CtyDistanceDecisionClass
$\mathcal{T}^{\mathrm{PI}}_n$ verified realized by PointIndexedDecisionClass
$R_n^{\mathrm{NP}}$ verified realized by ctyDistanceRisk
$R_n^{\mathrm{PI}}$ verified realized by pointIndexedDistanceRisk
a_n verified realized by frontierRate
Y(t) verified realized by CausalObservation, armCoord, A1A2Law, A1A2Law.condAbsMoment, A1A2ClassWitness
p verified realized by variable@37, variable@26
$\nu$ verified realized by variable@38, variable@26, A1A2ClassWitness
$\mathcal{A}_{t,P}$ verified realized by A1A2Law, knownGeometry, A1A2ClassWitness
$\mathcal{B}_P$ verified realized by A1A2Law, knownGeometry, A1A2ClassWitness
$\mu_{t,P}$ verified realized by A1A2Law, A1A2ClassWitness
$\sigma^2_{t,P}$ verified realized by A1A2Law, A1A2ClassWitness
T verified realized by treatment
$\tau_P$ verified realized by A1A2Law.tau, a1a2BoundaryLoss
$K_\square$ verified realized by uniformKernel, knownGeometry, EuclideanBallsVCProperty, A1A2ClassWitness
G_P verified realized by knownGeometry
d_P verified realized by knownGeometry, EuclideanBallsVCProperty, A1A2ClassWitness
$D^{\pm}_{P,x}$ verified realized by signedDistance
h verified realized by variable@158, stabilizedLocalPolynomial
$U^{\pm}_{n,P}(x)$ verified realized by signedDistanceData
r_p verified realized by polyBasis
$\psi_B$ verified realized by winsorize, stabilizedLocalPolynomial
$\mathcal{T}^{12,\pm}_n$ verified realized by A1A2PointIndexedDecisionClass
$R_n^{12,\pm}$ verified realized by a1a2OuterRisk
B_n verified realized by winsorLevel
$\widehat{\tau}^{12,\mathrm{LP}}_{n,h}$ verified realized by stabilizedLocalPolynomial
$\Psi_{t,P,x}(h)$ verified realized by populationGram, A1A2ClassWitness
$\mathcal{P}_{12}$ verified realized by A1A2Class

Proof map 10 statements · 10 edges

Uniform Expected Risk for Distance-Based Boundary Regression Designs

Abstract

Introduction

Related Literature

Setup: Distance-Compressed Boundary Regression

Setup: Known-Geometry Signed-Distance Designs

Main Results

Conditional signed-distance frontier.

Discussion and Extensions

Open questions.

Appendices

Direct-Product and Support-Boundary Lower Bound

Signed-Distance Geometry, Maximal Inequalities, and Upper Bound

Signed-Distance Hypercube and Matched Rate

Reproducibility Note

Proofs of the main results

References