@techreport{causalsmith_stat_finitep_homogeneity_densegamma_v1,
  title       = {Testing treatment effect homogeneity under finite conditional moments},
  author      = {{CausalSmith}},
  year        = {2026},
  month       = oct,
  institution = {CausalSmith Working Papers},
  number      = {CSWP-2026-041},
  url         = {https://causalsmith.org/papers/stat_finitep_homogeneity_densegamma_v1/},
  note        = {Version 3, 4 Oct 2026. Machine-generated; formal statements verified in Lean 4 at commit 1efc342}
}

Introduction

This paper establishes matching minimax separation bounds for conditional mean treatment-effect homogeneity under finite conditional outcome moments and independently specified nuisance smoothness. The statistical target is constancy of the difference between the treated and untreated conditional means, with an unknown common effect. Under a potential-outcome representation satisfying consistency and conditional independence of treatment assignment, this contrast is the conditional average treatment effect. We measure heterogeneity by its population $L_2$ distance from its average, which equals the standard deviation of the conditional effect under the stipulated covariate distribution. The resulting frontier quantifies how outcome tails and assignment information jointly determine the smallest uniformly detectable departure from homogeneity.

The model has a known uniform covariate distribution on $[0,1]$, a propensity between $1/4$ and $3/4$, and separate Hölder restrictions on the propensity, baseline conditional mean, and mean effect. Write $p$, the conditional moment order, for an exponent satisfying $1<p\le 2$; write $\alpha$ and $\beta$, the propensity and baseline smoothness exponents, for numbers in $(0,1]$; and write $\gamma$, the effect smoothness exponent, for a number in $[1/4,1]$. These exponents are public. The Hölder radii are fixed at $20$, the baseline mean and effect have supremum norms at most $1/2$, and each treatment arm has conditional raw $p$-moment at most $10$, uniformly in the covariate up to null sets. These restrictions define the class in Definition 1. Testing errors are assessed uniformly over that class: size covers every admissible constant effect and nuisance function, while power covers every law separated by the stipulated population distance, as specified in Definition 5.

Our main result gives the critical separation radius up to positive constants depending on the public tuple. Define $q=(p-1)/p$, the moment-derived exponent, and $S=\alpha +\beta$, the sum of nuisance smoothness exponents. The two separation exponents are $$E_0(p)=\frac{2\gamma q}{2\gamma +q},E_4(p)=\frac{2S}{1+2\alpha +\beta /q+S/(2\gamma )}.$$ For sample size $n\ge 2$, the capped critical radius has order $$n^{-\min \{E_0(p),E_4(p)\}},$$ as established in Theorem 4. The first exponent balances effect approximation with outcome-moment control. The second incorporates the interaction of propensity and baseline regularity. Their exact equality boundary is determined by $$F(p)=\frac{2\alpha }{p-1}+\frac{\beta }{q}+\frac{S}{2\gamma },$$ the phase-comparison quantity: effect smoothness determines the rate when $F(p)\ge 1$, and rough confounding determines it when $F(p)<1$. The bounds are pure powers of sample size, including at equality, with constants locally uniform on compact subsets of the admissible exponent domain.

The attaining procedure operates directly on the original records. It clips outcomes, forms histogram scores with ordered-pair projection corrections, and takes the inner product of scores from two independent evaluation blocks. Profiling this quadratic statistic over the admissible common-effect interval accommodates the composite null. The public construction in Algorithm 1 specifies all ranks, clipping levels, bias and covariance bounds, and rejection cutoffs. Its rejection probability is at most $0.0075$ throughout the null for every $n\ge 2$. Its type-II error is at most $0.0075$ at the stated upper separation when that separation lies below the model’s maximal heterogeneity distance; fully evaluable sample-size thresholds ensure a nonempty power domain. These guarantees appear in Theorems 1 and 2. Matching converses apply to all admissible randomized tests observing the complete records.

The matched outcome comparison retains the same design, overlap, smoothness, and mean envelopes while imposing bounded outcomes. Its separation exponent is the original exponent evaluated at $p=2$. Every admissible $p<2$ strictly reduces that exponent, so the finite-moment critical radius exceeds its bounded counterpart by a polynomial factor. At $p=2$, the ratio remains bounded across sample sizes. Moreover, mean-preserving sign randomization gives exact equality of bounded and signed-binary minimax risks and capped radii, as shown in Proposition 1. The full-record witnesses in Theorem 3 connect the slower finite-moment frontier to growing outcome marks: for each fixed tuple with $p<2$, close finite-moment null and alternative mixtures coexist with eventual total-variation separation of every normalized finite prior pair supported on the bounded null and bounded alternatives at the same lower separation.

Supplying the entire true continuous propensity gives a complementary frontier. Over the continuous-carrier class in Definition 22, the propensity and conditional arm means may have unrestricted moduli of continuity, while uniform design, overlap, effect Hölder smoothness, mean caps, and the conditional moment envelope are retained. A clipped inverse-propensity test attains the matching order $n^{-E_0(p)}$ on this broader class, with whole-null calibration and a public nonempty-power threshold, by Theorem 5. Comparing supplied and unknown propensity on the original shared class then yields an exact preservation boundary: their critical-radius ratio remains bounded over $n\ge 2$ precisely when $F(p)\ge 1$. For $F(p)<1$, supplying the propensity improves separation by a factor of order $n^{E_0(p)-E_4(p)}$. This comparison, including its equality case, follows from the matched frontiers in Theorems 6 and 7.

These results connect econometric homogeneity testing with quadratic nonparametric testing and higher-order causal methods. Crump et al. (2008) develop sieve Wald tests of conditional mean-effect homogeneity. Recent orthogonal-score procedures provide asymptotic calibration and power under their respective outcome and nuisance-learning conditions (Lu et al., 2026); (Lapenta et al., 2026). Directional testing also explicitly pursues uniform asymptotic calibration, with power tied to alignment of a learned direction and the population signal (Dhawan et al., 2026). Our contribution concerns finite-sample whole-null calibration and uniform population separation over the primitive class specified above. The approximation–variability balance has antecedents in quadratic goodness-of-fit theory (Ingster et al., 2009), while projection corrections connect to higher-order conditional-effect estimation (Kennedy et al., 2024).

The paper proceeds through related work, setup and assumptions, the matched separation rates and outcome-tail comparison, the calibrated original-record construction, and the supplied-propensity frontiers and assignment-information comparison. A discussion considers interpretation and extensions. The appendices develop causal identification and benchmark equivalence, calibration and attainment, full-record lower bounds, supplied-propensity arguments and comparison algebra, followed by proofs of the main results.

Related work

The closest literature tests whether conditional average treatment effects are constant across covariate-defined subpopulations. Crump et al. (2008) develop a sieve Wald approach that compares series estimates of the two conditional outcome means and removes the constant component when testing homogeneity. Their analysis combines smoothness and sieve-growth restrictions with conditions on conditional variances and third moments of outcome disturbances to obtain asymptotic calibration and consistency against specified smooth alternatives. This establishes an important precedent for treating homogeneity as a global restriction on conditional means. The present paper studies that restriction through finite-sample control over the entire composite null and a population separation criterion that is uniform over the stipulated model class; the corresponding guarantees are stated in Theorems 1 and 2.

Orthogonal conditional moments provide a second route to the same substantive question. Lu et al. (2026) construct a marked empirical process from a doubly robust score and centered covariate indicators. Their theory specifies series estimators for the propensity score and outcome regressions, with primitive smoothness and series-growth conditions ensuring a feasible-to-oracle approximation uniform over the process index. The outcome conditions include finite second moments and uniformly bounded conditional variances. The resulting tests have asymptotic Gaussian-process calibration, multiplier-bootstrap validity, consistency against fixed alternatives, and nontrivial power along specified local alternatives approaching the null at the inverse square root of sample size. Uniformity over the process index supports these distributional approximations. Our separation criterion instead takes a supremum over all admissible population laws whose effect function is sufficiently far from its average, as specified in Definition 5.

Lapenta et al. (2026) integrate orthogonal unconditional moments to obtain an omnibus test that accommodates flexible nuisance estimation. Their formulation separates the controls used for identification from the covariates used to assess heterogeneity. Under unconfoundedness, the stated theory assumes bounded outcomes and nuisance estimators converging faster than the inverse fourth root of sample size in population $L_2$, together with boundedness and overlap conditions on the fitted functions. It establishes an asymptotic null distribution, consistency against fixed alternatives, and a bootstrap implementation that retains the fitted nuisance functions across replications. These conditions make explicit the nuisance-learning and outcome envelopes supporting their calibration. Here, the primitive restrictions in Definition 1 determine the public tuning and finite-sample bounds directly.

A closely related scalar target is the variance of the conditional average treatment effect. Yu (2026) address its zero boundary using intra-fold sample splitting: distinct evaluation subsamples share the same out-of-fold nuisance estimates, producing cancellation of nuisance-bias terms. The stated asymptotic normality result uses finite fourth outcome moments, a nondegeneracy condition, and convergence faster than the inverse fourth root of sample size for the propensity, outcome-regression, and conditional-effect estimators. Under the uniform covariate marginal in Assumption 1, the squared population $L_2$ distance of the effect from its average also equals its variance. Our testing risk therefore measures separation using a closely related population quantity, with calibration and power assessed uniformly over the model’s conditional-moment envelope.

Earlier variance-target work studies estimation and confidence intervals rather than minimax separation. Levy et al. (2021) define the variance of the full conditional treatment-effect function and develop TMLE and cross-validated TMLE estimators; their efficiency conditions include control of an outcome-regression remainder that is not removed merely by knowing the treatment mechanism. Sanchez-Becerra (2023) instead work in experiments with known assignment probabilities and a predictive first stage. They establish efficiency for the full variance under additional learning conditions and uniform asymptotic coverage for a fold-specific proxy variance under their stated fourth-moment and regularity conditions. Our target is the same full population variance under the uniform design, but the criterion is finite-sample testing risk over Hölder and conditional-moment classes; it does not replace the variance by a fold-specific proxy or assume a learned first stage.

Directional testing offers a complementary way to concentrate power. Dhawan et al. (2026) learn a promising score direction on one sample and evaluate a debiased score on an independent sample. Their general theory incorporates conditional score-moment bounds, variance conditions, uniform nuisance-estimation requirements, and the additional moment conditions stated in their asymptotic theorem. It delivers uniform asymptotic null calibration and a power guarantee requiring sufficient correlation between the learned direction and the population signal. Thus, uniform asymptotic inference is already an explicit objective in this literature. The present analysis adds finite-sample whole-null calibration and uniform separation guarantees indexed by primitive moment and smoothness restrictions. The distinction concerns both the error criterion and the alternatives over which power is guaranteed.

The quadratic structure connects these causal tests to nonparametric goodness-of-fit theory. Ingster et al. (2009) analyze minimax separation in multivariate regression using increasing-dimensional U-statistics and ellipsoidal smoothness classes. Comminges et al. (2013) study composite regression nulls defined by quadratic functionals, obtaining sharp asymptotic testing results for nonnegative diagonal functionals under their stated conditions. These analyses explain how approximation error and the stochastic cost of a quadratic statistic jointly determine detectable signal size. In the present setting, that balance also incorporates unknown treatment assignment and conditional outcome tails. The comparisons in Theorems 4 and 7 characterize their consequences for critical separation radii within the stipulated classes.

The projection corrections also have a close methodological antecedent in higher-order causal estimation. Kennedy et al. (2024) derive minimax bounds for pointwise conditional-effect estimation in Hölder models and construct a higher-order local polynomial R-learner, with attainment under the stated additional conditions. Its second-order corrections use projections to control nuisance-product bias, linking the accuracy of causal estimation to separate smoothness restrictions on treatment assignment and outcome means. Our corrected scores in Definition 21 and Algorithm 1 use related projection ideas for a global homogeneity test. The population targets and loss criteria distinguish the two analyses: pointwise estimation measures recovery of the effect function at a covariate value, whereas Definitions 5 and 6 assess rejection errors for constancy against population-separated alternatives.

Outcome-tail handling connects the construction to robust U-statistics. Joly et al. (2016) use median-of-means aggregation to estimate U-statistic expectations; their canonical-statistic result permits moment orders strictly above one and at most two. This supplies a precise comparator for finite-moment control of nonlinear statistics. Our construction combines clipping, projection corrections, and independent evaluation blocks, with explicit bias and covariance bounds in Algorithm 1. Together with the full-record converse in Theorem 3, the analysis relates the attainable separation scale to the conditional outcome-moment restriction.

Other heterogeneity procedures address distinct scientific targets. Dukes et al. (2026) study quantitative and qualitative effect heterogeneity through contrasts tied to the population value of treatment rules, with asymptotic testing guarantees under their regularity conditions. Jain et al. (2026) target equality of conditional potential-outcome distributions and establish asymptotic bootstrap calibration and consistency against fixed alternatives under their stated conditions. Ashouri et al. (2026) assess the predictive advantage of fitted models that allow heterogeneous effects over models imposing a constant effect. Policy-value contrasts, conditional distributional equality, and predictive performance each give heterogeneity a specific operational meaning. The contribution developed here concerns constancy of the conditional mean contrast under fixed overlap, a known uniform covariate marginal, stipulated smoothness restrictions, and a conditional raw-moment envelope. Within that scope, it combines whole-null finite-sample calibration with separation-rate comparisons that quantify the roles of outcome tails and supplied propensity information.

Setup and assumptions

Throughout, generic positive constants may depend on the fixed public exponents and numerical envelopes, while remaining independent of the sample size $n$. We write $\Vert f{\Vert }_{\infty }={\sup }_{x\in [0,1]}|f(x)|$ and $\Vert f{\Vert }_2=(\int f^{2}d\lambda {)}^{1/2}$, where $\lambda$ is Lebesgue probability measure on $[0,1]$. The relations $\lesssim$ and $\asymp$ denote inequalities up to such constants and their two-sided counterpart, respectively. Sample indices range over $i=1,\dots ,n$, treatment-arm indices over $a\in \{0,1\}$, and all sample sizes satisfy $n\in \mathbb{N}$ with $n\ge 2$.

Records, public exponents, and conditional means

An original record is $O=(X,A,Y)$, with covariate $X\in [0,1]$, treatment indicator $A\in \{0,1\}$, and observed outcome $Y\in \mathbb{R}$. An original-record law $P$ is a Borel probability measure on this product space; the restrictions defining the statistical model are collected in Definition 1. Its covariate marginal is denoted by $P_X$.

The public exponent tuple $v=(p,\alpha ,\beta ,\gamma )$ consists of the conditional moment order $p$, the propensity smoothness exponent $\alpha$, the baseline smoothness exponent $\beta$, and the effect smoothness exponent $\gamma$. The admissible public domain $\mathcal{V}$ is $$\mathcal{V}=\left\{(p,\alpha ,\beta ,\gamma )\in {\mathbb{R}}^4:1<p\le 2,0<\alpha \le 1,0<\beta \le 1,1/4\le \gamma \le 1\right\}.$$ For benchmark comparisons, the matched smoothness tuple $w=(\alpha ,\beta ,\gamma )$ belongs to the domain $\mathcal{W}$, where $$\mathcal{W}=(0,1]\times (0,1]\times [1/4,1].$$ These exponents are public parameters indexing the classes over which testing errors are controlled.

For $s\in (0,1]$, the Hölder ball ${\mathcal{H}}^s(20)$ uses the convention $$[f{]}_s=\underset{\begin{array}{c}x,z\in [0,1]\\ x\ne z\end{array}}{\sup }\frac{|f(x)-f(z)|}{|x-z{|}^s},{\mathcal{H}}^s(20)=\left\{f\in C([0,1]):\Vert f{\Vert }_{\infty }\le 20,[f{]}_s\le 20\right\},$$ where $C([0,1])$ denotes continuous real functions with the uniform topology. Thus the convention at $s=1$ is Lipschitz continuity. We write $[f{]}_{\lambda }$ for the equivalence class of a function under Lebesgue almost-everywhere equality; the same convention applies to sums of functions and to each conditional mean.

Let $Q_{a,P}(dy\mid x)$ be a conditional outcome probability kernel given $X=x$ and $A=a$. The propensity equivalence class ${\mathcal{E}}_P$ is the class of $x\mapsto P(A=1\mid X=x)$, and the conditional arm-mean classes ${\mathcal{G}}_{0,P}$ and ${\mathcal{G}}_{1,P}$ are the classes of $$x⟼\int y{Q}_{0,P}(dy\mid x)\text{and}x⟼\int y{Q}_{1,P}(dy\mid x),$$ respectively. Under the restrictions below, these classes have continuous representatives: the propensity $e_P$, the baseline mean $m_{0,P}$, and the treated mean $m_{1,P}$, as specified in Definition 1. Their canonical mean contrast is ${\tau }_P=m_{1,P}-m_{0,P}$. Continuity and the full support of Lebesgue measure make each representative unique, so pointwise restrictions on these functions have an unambiguous meaning.

Primitive restrictions

The design conditions fix the distribution over which heterogeneity is measured and ensure that both treatment arms are represented throughout the covariate interval.

The known uniform random design condition supplies a fixed integration measure for the population target and for approximation errors, following the random-design goodness-of-fit formulation (Ingster et al., 2009). Treatment assignment is further restricted by a fixed overlap envelope.

Uniform overlap keeps both conditional arm distributions observable across the entire interval. The overlap form follows the conditional-effect model in Kennedy et al. (2024); the numerical bounds $1/4$ and $3/4$ are fixed normalizations stipulated for the present class.

Smoothness is indexed separately for treatment assignment, the baseline mean, and the mean contrast. We begin with regularity of the propensity.

In our model, independent propensity Hölder smoothness controls variation in assignment probabilities through its own exponent, with radius $20$ fixing the scale of that variation. Baseline outcome variation receives a separate regularity restriction.

In our model, the independent baseline Hölder condition controls the conditional mean in the untreated arm, allowing its regularity to differ from propensity regularity. The third exponent governs the conditional mean contrast directly.

Independent effect Hölder smoothness describes the treated mean as the baseline mean plus a regular contrast. This separates the target regularity from the two nuisance regularities. By comparison, the pointwise conditional-effect model of Kennedy et al. (2024) summarizes propensity and baseline smoothness through an average exponent. The Hölder radius $20$ used here is a fixed normalization of the present model.

We also fix the scales of the baseline mean and the mean effect. The first envelope bounds the baseline level.

The bounded conditional baseline mean condition uses the fixed normalization $1/2$, placing baseline levels on a common scale across the indexed models. This numerical envelope is stipulated here. A corresponding envelope fixes the scale of the contrast.

The bounded conditional effect condition uses a second stipulated normalization $1/2$, keeping the target on the same scale across smoothness and moment classes. Together, the two mean envelopes imply $\Vert m_{1,P}{\Vert }_{\infty }\le 1$.

Outcome tails are controlled separately from conditional mean levels. The following restriction bounds a raw conditional moment in each arm.

This conditional raw-moment envelope is specific to the present analysis. It controls outcome magnitudes uniformly across covariate values, up to null sets, and across treatment arms. The range $1<p\le 2$ ensures integrability of the arm means while allowing the moment order to index the strength of tail control.

The original-record model ${\mathcal{M}}_v$ in Definition 1 collects these design, smoothness, mean-envelope, and moment restrictions.

Causal interpretation and the homogeneity target

A potential-outcome representation connects the canonical mean contrast to a conditional average treatment effect. Let $Y(0)$ and $Y(1)$ denote the baseline and treated potential-outcome coordinates. The completion ${\mathbb{Q}}_P$ in Definition 3 combines the observed arm distributions with assignment that is conditionally independent of these coordinates given the covariate.

Under this completion, consistency and conditional independence of assignment give $${\mathbb{E}}_{{\mathbb{Q}}_P}[Y(1)-Y(0)\mid X=x]={\tau }_P(x)\text{for }\lambda \text{-almost every }x.$$ The observed-record marginal is $P$, so the same conditional mean contrast admits the potential-outcome interpretation associated with consistency and unconfounded assignment (Rubin, 1974); (Rosenbaum et al., 1983). The auxiliary justification for this completion is given in Lemma 1.

To measure heterogeneity, define the average contrast ${\bar{\tau }}_P$ and its population distance from constancy $d(P)$ by $${\bar{\tau }}_P={\int }_{[0,1]}{\tau }_P(x)d\lambda (x),d(P)={\left({\int }_{[0,1]}\{{\tau }_P(x)-{\bar{\tau }}_P{\}}^{2}d\lambda (x)\right)}^{1/2}.$$ Equivalently, $d(P)={\inf }_{b\in \mathbb{R}}\Vert {\tau }_P-b{\Vert }_2$. The uniform design fixes the population weighting, and continuity makes zero distance equivalent to constancy on the entire interval. Homogeneity therefore permits an unknown common effect $c$ in the public mean envelope.

The unknown-constant conditional-average-effect homogeneity condition treats the common effect level as a nuisance parameter, matching the mean-homogeneity target of (Crump et al., 2008). The constant-effect null $H_0(v)$ in Definition 2 imposes this condition within the original-record model.

Thus the null consists precisely of model laws with $d(P)=0$. Under the causal completion, its target is constancy of the conditional average treatment effect.

The original-record testing experiment

Testing uses an independent sample ${\mathcal{D}}_n=((X_i,A_i,Y_i){)}_{i=1}^n$ from $P$, together with an independent public uniform variable $U$ on $[0,1]$. An original-record randomized test is a jointly Borel rejection-probability map $\varphi$ with domain and range $$\varphi :([0,1]\times \{0,1\}\times \mathbb{R}{)}^n\times [0,1]⟶[0,1].$$ Its expected rejection probability is denoted by ${\mathsf{R}}_n(P,\varphi )$, the expectation over both the sample and the public randomization. The experiment and this expectation are specified together in Definition 4.

Tests may depend on the public tuple and sample size. Their size is controlled uniformly over all laws in the constant-effect null, including every admissible common effect level and nuisance function.

Let $D_v$ denote the original-model distance supremum, $$D_v=\underset{P\in {\mathcal{M}}_v}{\sup }d(P),$$ and let $r$ be a separation satisfying $0<r<D_v$. The minimax type-II error $B_n(v,r)$ in Definition 5 compares tests of size at most $1/10$ against every model law whose population heterogeneity distance is at least $r$.

The size constraint and the alternative supremum give the risk a uniform interpretation over the stipulated model. To summarize the separation at which this risk reaches $1/10$, we use the capped critical radius $r_n^{\ast }(v)$ in Definition 6.

Including the model supremum makes the defining set nonempty for every admissible sample size and public tuple. When the risk threshold is achieved at an interior separation, the radius records the infimum of such separations; otherwise, the cap supplies the value $D_v$.

Matched bounded and signed-binary benchmarks

The benchmark family retains the same design, overlap, smoothness exponents, and mean envelopes while imposing public outcome support restrictions. Its bounded member uses the following unit envelope.

The bounded-outcome condition places every observed outcome within a public unit envelope, a fixed-envelope version of bounded-outcome regularity used in homogeneity testing (Lapenta et al., 2026). The bounded model ${\mathcal{M}}_w^{\mathrm{b}}$ in Definition 7 combines this support restriction with the primitive restrictions at moment order two.

The signed-binary member specializes the support further.

Signed-binary support is the affine recoding $Y=2B-1$ of a conventional binary outcome $B\in \{0,1\}$, the outcome convention used in the binary conditional-effect model of Kennedy et al. (2024). Either support restriction implies a conditional second raw moment bounded by one almost everywhere, and hence satisfies the moment envelope at $p=2$. The signed-binary model ${\mathcal{M}}_w^{\mathrm{bin}}$ in Definition 8 is consequently indexed by the matched smoothness tuple.

These definitions give $${\mathcal{M}}_w^{\mathrm{bin}}\subseteq {\mathcal{M}}_w^{\mathrm{b}}\subseteq {\mathcal{M}}_{(2,\alpha ,\beta ,\gamma )}.$$ Both benchmarks preserve the unknown common effect as a nuisance parameter. Their nulls, denoted by $H_0^{\mathrm{b}}(w)$ and $H_0^{\mathrm{bin}}(w)$, are defined in Definitions 9 and 10.

For their separation domains, define the bounded and signed-binary distance suprema $D_w^{\mathrm{b}}$ and $D_w^{\mathrm{bin}}$ by $$D_w^{\mathrm{b}}=\underset{P\in {\mathcal{M}}_w^{\mathrm{b}}}{\sup }d(P),D_w^{\mathrm{bin}}=\underset{P\in {\mathcal{M}}_w^{\mathrm{bin}}}{\sup }d(P).$$ The corresponding minimax type-II errors $B_n^{\mathrm{b}}(w,r)$ and $B_n^{\mathrm{bin}}(w,r)$, defined in Definitions 11 and 12, use the same original-record experiment, population distance, and size threshold.

Finally, the benchmark critical radii $r_n^{\ast ,\mathrm{b}}(w)$ and $r_n^{\ast ,\mathrm{bin}}(w)$ in Definitions 13 and 14 apply the same capping convention to the respective risks.

Throughout the benchmark comparisons, these quantities are evaluated at $n\ge 2$ and $w\in \mathcal{W}$. Holding the primitive smoothness tuple fixed aligns the population target and error thresholds across the original, bounded, and signed-binary families, providing a common basis for assessing the role of outcome tails.

Separation rates and the cost of outcome tails

Holding the primitive smoothness tuple fixed gives a direct comparison of the precision attainable under the finite conditional moment restriction and the bounded-outcome benchmark. The comparison concerns the population heterogeneity distance and the capped critical radii defined in Definitions 6 and 13. Two exponents determine the original-record separation rate: one depends on effect smoothness and moment order, while the other also reflects the smoothness of the propensity and baseline mean.

We collect the scales and their finite-sample multipliers before stating the comparison. Write $C_v^{\mathrm{att}}$ and $N_v^{\mathrm{att}}$ for the fully evaluable attainment multiplier and nonempty-power threshold specified in Theorem 2; evaluation at $(2,w)$ gives their matched bounded counterparts. The following definition combines these attainment quantities with the lower-bound multipliers.

The original-model scale ${\rho }_n(v)$ and the matched bounded scale ${\rho }_n^{\mathrm{b}}(w)$ summarize the sample-size exponents. Their lower multipliers, $c_v$ and $c_w^{\mathrm{b}}$, and upper multipliers, $C_v$ and $C_w^{\mathrm{b}}$, retain the finite-sample content of the comparison. The associated public thresholds $N_v$ and $N_w^{\mathrm{b}}$ place the upper separations below a fixed attainable heterogeneity distance. Thus the power statements below are attached to an explicit nonempty separation domain.

To measure the cost of outcome tails at the same smoothness tuple, we use the ratio of the capped radii.

The ratio ${\Delta }_n(v)$ in Definition 16 compares experiments with the same mean-effect target and error thresholds. We distinguish persistent enlargement of the original-model radius from a comparison that remains bounded over all sample sizes.

The tail region $\mathcal{T}$ and equality region $\mathcal{E}$ describe the behavior of this matched comparison. The equality region concerns bounded radius ratios; its definition permits different finite-sample constants.

Before comparing rates, the bounded benchmark can be identified exactly with its signed-binary submodel. Let $\lambda$ denote Lebesgue probability measure on $[0,1]$, and let $d_0=1/(8\sqrt{2})$ be the fixed positive witness distance supplied by Lemma 1. The next result shows that both benchmarks have the same attainable heterogeneity distances and the same minimax testing risks.

The sign randomization in Proposition 1 preserves the conditional outcome means, and hence the effect function, its heterogeneity distance, and null membership. Integrating the randomization into a test transfers its rejection probabilities to the bounded experiment. The resulting equality of minimax risks holds throughout the common nonempty separation domain $0<r<D_v$, and equality of capped radii holds for every $n\ge 2$. This gives an exact decision-theoretic benchmark for the mean-effect target.

The lower-bound comparison uses finite mixtures of original-record laws. Their public tuning selects a copula-frame construction in the rough-confounding regime once its sample-size threshold is reached, and an opposite paired-tent construction otherwise. The following selection rule fixes the priors used in the main result; their support laws and complete-record comparison bounds are developed in Lemma 5, Algorithm 5, Lemma 6, Lemma 7, and Lemma 8.

The selected null and alternative priors ${\pi }_{0,n,v}$ and ${\pi }_{1,n,v}$ in Algorithm 6 induce the complete independent-record mixtures ${\mathbb{P}}_{0,n,v}$ and ${\mathbb{P}}_{1,n,v}$. Their rare-mark probability ${\epsilon }_{n,v}$ and outcome magnitude $L_{n,v}$ describe how the lower witnesses use the finite moment allowance.

For the next statement, ${\mathsf{R}}_n(P,\varphi )$ denotes expected rejection in the original-record experiment, $Q_{a,P}(dy\mid x)$ denotes the conditional outcome probability kernel in arm $a$, and $\mathrm{TV}$ denotes the supremum of the absolute difference in event probabilities. The threshold $N(v)$ identifies the sample sizes at which every bounded-model finite prior pair at the original-model lower separation has mixture total variation at least one half. The theorem combines the matched separation bounds, the power guarantees, and this tail-essential comparison.

The exponent comparison in Theorem 4 explains the two statistical regimes. When $F(p)\ge 1$, the effect-smoothness exponent $E_0(p)$ determines the separation scale: moment order and effect smoothness govern its sample-size dependence. When $F(p)<1$, the rough-confounding exponent $E_4(p)$ determines the scale, bringing the propensity and baseline smoothness exponents into the rate. The identity for $E_4(p)-E_0(p)$ makes $F(p)=1$ their exact equality boundary. Across both regimes, the strict inequality $E(p)<E(2)$ for $p<2$ translates the finite moment allowance into a polynomial enlargement of the matched critical radius.

The two denominators reflect different balances. At histogram rank $J$, approximating the effect contributes the usual Hölder term of order $J^{-\gamma }$; clipping at level $T$ contributes a moment-tail term of order $T^{1-p}$, while the clipped quadratic variability grows with $T^{2-p}$. Balancing these terms gives $E_0(p)$. When the nuisance functions are rough, the ordered-pair correction removes the first-order propensity error, leaving a product of the fine-scale propensity and baseline approximation errors. Its rank cost depends on $2\alpha +\beta /q$: propensity regularity enters twice through treatment weighting and projection correction, whereas the baseline appears once and is amplified by the tail factor $1/q$. The additional term $S/(2\gamma )$ records the coarse resolution needed to retain the effect signal. Correction-specific clipping prevents the fine increments from paying the main cutoff at every resolution, which is what makes the multiresolution covariance sum compatible with $E_4(p)$. These statements summarize the verified mean and covariance bounds in Lemmas 2 and 3; their full calculations are in the attainment appendix.

For example, $v=(3/2,1,1,1)$ has $F(p)=8$, so the effect channel governs with $E=E_0=2/7$. In contrast, $v=(3/2,1/20,1/20,1)$ has $F(p)=2/5$, so rough confounding governs with $E=E_4=2/13$. The first tuple uses smooth nuisance functions and the second deliberately makes them much rougher than the effect. These statistical regimes are distinct from the procedure’s representation choice in Algorithm 1, which uses the conditions $S\ge \gamma$ or $\alpha \ge q$ to choose between a single correction and a multiresolution expansion.

The regimes can be summarized by substituting the formulas in Definition 15. In the following table, $q=(p-1)/p$ and $S=\alpha +\beta$, with the primitive smoothness tuple held fixed.

Here $F(2)=2S+S/(2\gamma )$. Each row describes the power of $n$ in the corresponding scale; the finite-sample radius bounds retain the multipliers and capping convention of Theorem 4. The original and bounded exponents are selected separately at their respective moment orders.

For every fixed admissible tuple, the two-sided bounds on ${\Delta }_n(v)$ quantify the cost of tails through $n^{E(2)-E(p)}$. The moment-order-two face has equal original and bounded exponents and a uniformly bounded ratio over $n\ge 2$, together with the ordering ${\Delta }_n(v)\ge 1$. Every admissible tuple with $p<2$ instead belongs to the tail region, where the ratio diverges. This distinction between moment orders is separate from the equality boundary $F(p)=1$, which identifies the meeting of the two rate mechanisms within a given moment class.

The finite-sample cap remains part of the interpretation. The radius inequalities hold for every $n\ge 2$, while the power guarantees for the attaining tests ${\varphi }_{n,v}^{\ast }$ and ${\varphi }_{n,w}^{\ast ,\mathrm{b}}$ from Definition 15 apply when their upper separations lie below the corresponding model distance suprema. The public thresholds ensure this condition through the fixed witness distance $d_0$. Size is controlled over the whole null at every $n\ge 2$. On compact subsets of the admissible exponent domains, common lower and upper multipliers and a common bound on the public thresholds make these comparisons locally uniform.

Finally, the tail-essential witnesses identify the statistical content of the enlargement. For $p<2$, their outcome magnitudes grow with sample size, while the selected null and separated alternative mixtures remain within total variation one half in the complete-record experiment. At the same original-model lower separation, every bounded-model finite prior pair has mixture total variation at least one half once the tuple-dependent threshold is reached. Thus the witnesses connect the slower finite-moment rate to outcome tails while preserving the stipulated primitive restrictions and the mean-effect target. The full support-law constructions and the quantified finite-prior comparison appear in the proof appendix through Definition 29, Definition 30, Algorithm 5, and Theorem 3.

A calibrated test from the original records

The upper separation guarantee in Theorem 4 is attained by a procedure whose inputs are the original records and the public moment and smoothness tuple. The construction combines histogram projection, clipped outcomes, and ordered-pair corrections within two independent evaluation blocks. Minimization over the candidate constant then accommodates the composite null of a homogeneous mean effect. We first specify the projection geometry and score ingredients, and then give the complete calculation.

At a positive integer rank $J$, the histogram projection averages functions over equal-width cells. Its endpoint convention is part of the procedure.

The normalized feature vector $F_J(x)$ represents cellwise constant functions in Euclidean coordinates, and the projection kernel ${\Pi }_J(x,z)$ selects pairs in the same cell. An internal bin endpoint belongs to the cell on its right, while the final cell contains $1$. These conventions assign a value to every record, including records at cell boundaries.

To obtain independent score vectors, partition the independent sample into two deterministic blocks of equal size.

The evaluation blocks ${\mathcal{I}}_1,{\mathcal{I}}_2$ each contain $s_n$ records. When the sample size is odd, the last record is left outside these blocks. Independence between the blocks permits their score inner product to estimate a squared population mean.

Within each block, a single-record score is paired with an ordered-pair correction. Both depend affinely on the candidate constant $c\in [-1/2,1/2]$. The following ingredients use a coarse histogram rank $J_{\mathrm{c}}$, whose public selection is specified in Algorithm 1.

The clipping function ${\ell }_T(y)$ bounds each outcome contribution before averaging. The single-record score $H_{b,T}(c)$ uses treated records, while the ordered-pair score $U_{b,G,T}(c)$ combines the treatment indicator of one record with the clipped outcome and treatment indicator of a distinct record. The correction kernel $G$ supplies the projection used in this pairing. The explicit dependence on $c$ makes profiling a finite quadratic calculation.

The construction uses two resolutions for different tasks. The coarse rank $M$ retains the effect signal tested by the cross-block inner product, while the fine rank $K$ controls the product of propensity and baseline approximation errors left after the ordered-pair correction. A single correction suffices when $S\ge \gamma$ or $\alpha \ge q$. Otherwise the dyadic increments between $M$ and $K$ receive separate cutoffs $T_j$: coarse increments can tolerate more clipping variability, while fine increments use smaller cutoffs to keep their covariance contribution summable. This representation condition determines how the same statistic is assembled; it is separate from the phase condition $F(p)=1$, which determines which separation exponent is smaller.

The complete procedure now selects these ranks and clipping thresholds, subtracts the pair corrections, and computes the bias and covariance bounds used for calibration. The fine histogram rank $J_{\mathrm{f}}$ governs the correction resolution. Each selection depends on the sample size and public tuple.

The corrected block score $Z_b(c)$ uses either a single fine-rank correction or a sum of dyadic projection increments. In the latter construction, each increment receives its own clipping threshold; the main clipping threshold is $T_0$. This allocation enters both the public bias ledger ${\mathfrak{b}}_{n,v}$ and the public covariance ledger ${\Lambda }_{n,v}$. The rejection cutoff $h_{n,v}$ combines their contributions and is rounded upward.

The statistical comparison and the computational selection have distinct roles. The comparison of $E_0$ and $E_4$, as defined in Definition 15 and Algorithm 1, determines the exponent used for tuning. The condition $S\ge \gamma$ or $\alpha \ge q$ selects the single-correction representation; when $S<\gamma$ and $\alpha <q$, the procedure uses the multiresolution representation. Both representations use the same minimum exponent $E$.

All calculations in Algorithm 1 are finite. The public dyadic grids specify the rank and outcome selections, and the upper-dyadic rule specifies the rejection cutoff. Scores are normalized by the block size and number of ordered pairs. Consequently, empty histogram cells and observations at bin endpoints have stipulated values throughout the calculation. For $n=2,3$, the zero-score convention and zero rejection rule complete the procedure.

The cross-block statistic $W(c)$ is quadratic because each block score is affine in $c$. Its minimum is therefore obtained by comparing the endpoints and, when applicable, the interior vertex. Rejection requires the statistic to exceed its cutoff for every admissible constant, which gives the profiling step its role in testing the whole null.

For the calibration statement, write ${\psi }_{n,v}:={\varphi }_{n,v}^{\ast }$ for the rule in Algorithm 1, and define the population score mean ${\theta }_P(c)$ by $${\theta }_P(c):={\mathbb{E}}_{P^{\otimes n}}[Z_b(c)].$$ Let $\lambda$ be Lebesgue probability measure on $[0,1]$, and use $$d(P):={\left\{{\int }_0^1{\left({\tau }_P(x)-{\int }_0^1{\tau }_P(t)d\lambda (t)\right)}^{2}d\lambda (x)\right\}}^{1/2},D_v:=\underset{P\in {\mathcal{M}}_v}{\sup }d(P),$$ for the heterogeneity distance and its model supremum. Set $C:=128$ for the simultaneous profiling multiplier, and define the calibrated separation by $$R_{\mathrm{att}}(n,v):=\frac{16}{3}\left\{16J_{\mathrm{c}}^{-\gamma }+5{\mathfrak{b}}_{n,v}+1024J_{\mathrm{c}}^{1/4}\sqrt{{\Lambda }_{n,v}}\right\}.$$ The following result connects these public quantities to uniform size control and power.

Under a homogeneous mean effect, an admissible constant makes the population score norm at most the bias ledger. Profiling retains this comparison while the covariance bounds control fluctuations simultaneously over the candidate constants. Independence between the evaluation blocks gives the population identity $\mathbb{E}[W(c)]=\Vert {\theta }_P(c){\Vert }_2^2$. The calibrated separation combines coarse-projection approximation, original-mean bias, and stochastic fluctuation. For $n\ge 4$ with $R_{\mathrm{att}}(n,v)<D_v$, effects at or above that separation are detected with type-II error at most $0.0075$; the size bound applies throughout the null for every $n\ge 2$.

The appendix develops the three parts of this calibration. The original-mean analysis in Lemma 2 controls the bias introduced by projection and clipping. The coefficient covariance bounds are supplied by Lemma 3, and Lemma 4 controls the simultaneous profiling operation. Their combination yields the attainable separation result in Theorem 2, connecting the explicit procedure to the power scale ${\rho }_n(v)$ in Definition 15 and the upper separation bound in Theorem 4.

Supplied propensity scores and assignment information

The calibrated procedure in Algorithm 1 obtains assignment information from the original records. We now consider an experiment in which the entire true continuous propensity is supplied together with those records. This experiment supports two complementary conclusions: an attaining test over a broader class with continuous propensity and conditional mean functions, and an exact comparison of testing sensitivity on the original model.

A supplied-function test ${\varphi }^{\mathrm{e}}$ is a jointly Borel map $${\varphi }^{\mathrm{e}}:C([0,1])\times ([0,1]\times \{0,1\}\times \mathbb{R}{)}^n\times [0,1]⟶[0,1].$$ The function space carries the uniform topology. The remaining arguments are the independent original-record sample and an independent public uniform randomization variable. Under a law $P$, the supplied function is $e_P$; the formal risk definition below specifies the corresponding rejection expectation.

The continuous-carrier experiment

Let $\mathcal{V}$ denote the admissible domain of public moment and smoothness tuples in Definition 1, and let $\lambda$ be Lebesgue probability measure on $[0,1]$. The broader model retains uniform design, overlap, effect smoothness, the mean caps, and the conditional moment envelope. Its propensity and both conditional arm means have continuous representatives. The following definition specifies these restrictions and the maximum heterogeneity distance.

The continuous-carrier class ${\mathcal{M}}_v^{\mathrm{e},\mathrm{br}}$ in Definition 22 permits continuous propensity and baseline mean functions with unrestricted moduli of continuity. Effect smoothness controls the approximation of heterogeneity, while overlap bounds the inverse-propensity weights. The conditional moment envelope controls outcome clipping uniformly across covariate values and treatment arms. The caps bound the scale of the conditional means and effect. Within this class, the null retains the unknown constant effect used in the original experiment.

Size is controlled uniformly over the constant-effect null $H_0^{\mathrm{e},\mathrm{br}}(v)$ in Definition 23. For a supplied-function test, write ${\mathsf{R}}_n^{\mathrm{e}}(P,{\varphi }^{\mathrm{e}})$ for its rejection expectation when it receives $e_P$. The following risk and radius measure the separation required to attain type-II error at most $1/10$, subject to the same bound on size.

The broader-class risk $B_n^{\mathrm{e},\mathrm{br}}(v,r)$ in Definition 24 uses the original outcome records throughout. Its capped radius includes the model supremum as a saturation value, making the sensitivity measure well defined at every stipulated sample size.

The capped radius $r_n^{\ast ,\mathrm{e},\mathrm{br}}(v)$ and its saturation value $D_v^{\mathrm{e},\mathrm{br}}$ are defined in Definitions 25 and 22. To attain the separation rate, the test uses clipped inverse-propensity outcomes and the histogram features of Definition 19. Removing the constant histogram direction targets deviations from an unknown constant effect. The procedure then takes the inner product of two independent block averages.

The supplied-propensity rule ${\varphi }_{n,v}^{\ast ,\mathrm{e}}$ in Algorithm 2 uses public clipping-bias and covariance bounds to set its rejection threshold. The tuning depends on the moment order and effect smoothness. The next result gives matching bounds for the broader-class critical radius, uniform calibration over its entire null, and power whenever the stated separation lies below the saturation value.

The conditional-mean identity in Theorem 5 explains the role of supplied assignment information: inverse weighting recovers the mean effect directly, and constant centering isolates its heterogeneity. Clipping controls the contribution of outcome tails, while effect smoothness controls histogram approximation. Their balance yields the supplied-propensity separation scale ${\rho }_n^{\mathrm{e}}(v)$ defined in that result. The matching lower bound uses null and alternative laws with the same constant propensity and close complete-record distributions. Thus the stated separation order remains necessary even when assignment information is identical across the comparison laws.

The radius bounds apply to every $n\ge 2$, with the cap accounting for finite-sample saturation. The procedure’s power guarantee applies when its upper separation is strictly below the broader-model supremum. The public sample-size threshold in Theorem 5 ensures this condition by placing that separation at most halfway to the fixed witness distance. These statements distinguish an all-sample radius bound from a power guarantee over a nonempty range of alternatives.

Comparison on the original model

To measure the value of supplied assignment information on a fixed statistical class, we now use the original model of Definition 1. Its quantitative propensity and baseline smoothness restrictions are retained in both experiments being compared. The supplied-function test carrier remains the same, while the null and alternative suprema are taken over the original model. The following definitions make this shared-class comparison precise.

The shared-class risk $B_n^{\mathrm{e}}(v,r)$ in Definition 26 uses the original-model null for its size constraint. Its radius therefore uses the original-model supremum as the saturation value.

The shared-class capped radius $r_n^{\ast ,\mathrm{e}}(v)$ of Definition 27 is compatible with the broader-class attainment result. The original model is contained in the continuous-carrier class, and the finite lower-bound witnesses in Theorem 5 belong to both classes. The next result records the matching separation bounds and attainment on the original model, providing the supplied-propensity benchmark for the radius comparison.

The compatibility in Theorem 6 rests on both attainment and necessity. The supplied-propensity procedure controls size and power on the original class, and common-propensity witnesses establish the corresponding lower separation bound within that class. Consequently, the original-model comparison uses matched statistical classes in its numerator and denominator. Its finite-sample power condition is expressed relative to the original-model supremum; the broader-class result uses its own supremum.

We measure preservation of separation order by a uniformly bounded ratio of the unknown- and supplied-propensity capped radii. The following region includes every admissible tuple for which that ratio remains bounded across sample sizes.

The preservation region $\mathcal{R}$ in Definition 28 follows from comparing the matched rates in Theorems 4 and 6. The scalar comparison quantity $F(p)$ defined in Definition 15 determines whether the separation exponents coincide. The next result gives the exact boundary, including its equality case and bounds for every capped finite-sample radius ratio.

The boundary in Theorem 7 expresses the value of assignment information through the difference between the two separation exponents. When $F(p)\ge 1$, the unknown-propensity experiment attains the supplied-propensity separation order on the same original class. At $F(p)=1$, the ratio remains between positive, sample-size-independent constants. When $F(p)<1$, supplying the true propensity improves sensitivity by a polynomial factor whose exponent is stated explicitly in the result.

These comparisons concern orders of critical separation, with finite-sample constants and saturation incorporated into the capped radii. They hold for every $n\ge 2$, and their constants admit uniform bounds on compact subsets of the admissible parameter domain. Together with the broader-class attainment result, they distinguish testing with supplied assignment information from the exact region where the original unknown-propensity experiment attains the same separation order. Requirements for a particular estimated-propensity procedure are outside this experiment comparison.

Discussion and extensions

The separation comparisons provide two benchmarks for the sensitivity of homogeneity testing: one varies the outcome envelope at matched primitive smoothness, and the other varies the assignment information available to the test on a shared model class. The first comparison, in Theorem 4, measures the cost of allowing outcomes subject to a finite conditional moment bound relative to bounded outcomes. The second, in Theorem 7, measures the value of supplying the true propensity function. Together, they characterize how outcome tails and assignment information affect the smallest detectable departure from a constant conditional mean effect.

Matching primitive smoothness makes the outcome comparison interpretable. The propensity and baseline smoothness exponents describe nuisance complexity, whereas the effect smoothness exponent describes the complexity of the departure being tested. The bounded benchmark in Definition 7 retains these three restrictions. Consequently, the radius ratio in Definition 16 compares outcome envelopes while keeping the stipulated effect and nuisance complexity fixed. The regions in Definitions 17 and 18 distinguish a diverging separation penalty from a uniformly bounded radius ratio. These are benchmarks for worst-case sensitivity over the specified classes: they identify when the conditional moment envelope changes the order of the testing problem.

The assignment comparison separates the complexity of the effect from the information needed to handle the nuisance functions. On the shared original class, Theorem 7 identifies the exact region where the unknown-propensity experiment attains the supplied-propensity separation order. The broader-class result in Theorem 5 gives a complementary interpretation. Its model, defined in Definition 22, permits continuous propensity and baseline mean functions subject to the retained overlap and mean envelopes, together with the conditional moment and effect smoothness restrictions. Attainment over this class supports a broader nuisance scope when the true propensity is supplied. The resulting separation order is governed by the moment order and effect smoothness, clarifying the role of assignment information in accommodating nuisance variation.

The population target also has a direct variance interpretation. Writing the distance from a constant effect as $$d(P)={\left[{\int }_0^1{\left\{{\tau }_P(x)-{\int }_0^1{\tau }_P(z)dz\right\}}^{2}dx\right]}^{1/2},$$ the uniform covariate design in Assumption 1 gives $$d(P{)}^2={\mathrm{Var}}_P\{{\tau }_P(X)\}.$$ Thus separation is measured in the standard deviation of the conditional mean treatment effect across covariate values. Under the causal completion in Definition 3, this contrast equals the conditional expectation of the potential-outcome difference. The target therefore quantifies heterogeneity in conditional mean effects, with the covariate distribution determining the population weighting.

Public finite-sample quantities connect these comparisons to calibrated rejection rules. The construction in Algorithm 1 specifies the projection ranks, clipping levels, bias and covariance bounds, and rejection cutoff. Whole-null calibration in Theorem 1 accounts for the unknown constant effect through profiling. Combined with the attainment guarantees in Theorem 4, the public sample-size thresholds identify when the stated power separation lies inside the model’s nonempty alternative domain. The analogous supplied-propensity guarantees appear in Theorems 5 and 6.

The explicit sufficient constants are deliberately conservative. The following values come directly from the displayed formulas; logarithms make their scale readable. Here $N_v^{\mathrm{att}}$ is the original-record nonempty-power threshold and $N_v^{\mathrm{e}}$ is its supplied-propensity counterpart.

For the first row, for example, the supplied-propensity formula gives $N_v^{\mathrm{e}}=7,183,754,725,838$. These are analytic sufficient thresholds for the stated finite-sample power guarantee, rather than estimates of the sample size needed in practice. The theorem’s main quantitative content is the minimax power of $n$ and the matching lower bound; practical calibration of the constants would require implementation and simulation beyond the present analysis.

Limitations and future work

The analysis treats a known uniform covariate design on the unit interval and public moment and smoothness exponents. Extending the construction to an unknown covariate distribution would require handling the population weighting and the estimation of projection geometry together. Higher-dimensional covariates raise a related question about how approximation complexity and nuisance smoothness interact in the separation comparison.

Adaptive selection of the moment and smoothness exponents is another research direction. A useful extension would quantify the cost of choosing clipping levels and projection ranks from the data while maintaining calibration over the entire constant-effect null and establishing nonempty power domains.

Finally, the supplied-function experiment provides a benchmark for studying estimated assignment functions. Such an extension would quantify how propensity approximation error and the dependence introduced by its estimation enter the bias bounds, rejection threshold, and attainable separation. Comparing those guarantees with the exact supplied-function benchmark would clarify the assignment accuracy needed to retain its sensitivity to effect heterogeneity.

Appendices

Causal interpretation and benchmark equivalence

The population target in Definitions 1 and 2 uses continuous representatives of conditional means. Its interpretation therefore requires both invariance to conditional-distribution versions and a causal representation that reproduces the observed law. The following result establishes these properties and supplies legal constant-effect and heterogeneous-effect witnesses for every admissible parameter tuple.

The uniqueness assertion makes every pointwise restriction on the propensity and mean functions a property of the observed law. The uniform design and overlap conditions in Assumptions 1 and 2 give the conditional arm distributions a common Lebesgue almost-everywhere interpretation; continuity makes their canonical mean representatives unambiguous across the whole interval. In particular, the treated mean inherits the regularity stated in Lemma 1 from the separate baseline and contrast restrictions.

Version invariance also applies to the potential-outcome completion. As specified in Definition 3, changing either conditional outcome distribution on a Lebesgue-null set leaves ${\mathbb{Q}}_P$ unchanged. The completion combines the observed arm distributions with conditionally independent assignment and reproduces the original-record marginal. Its integrable potential-outcome difference has conditional mean ${\tau }_P(X)$, so $d(P)$ measures variation in conditional average treatment effects under this completion. The equivalence between zero distance and membership in Definition 2 connects the population criterion to the pointwise constant-effect restriction.

The two witnesses in Lemma 1 give the testing problem a nonempty null and a positive separation domain for every admissible tuple. The heterogeneous witness attains the same fixed distance $d_0$ throughout the parameter domain, while the constant-contrast witness permits a nonzero common effect. Together with the upper bound on $D_v$, these properties give the cap in Definition 6 a finite, positive population scale.

Mean preservation and the benchmark experiment

The bounded and signed-binary models in Definitions 7 and 8 retain the primitive restrictions on assignment and conditional means. Their comparison in Proposition 1 uses two related operations: a model-level conversion specified by conditional arm means, and a record-level randomization available for bounded observations. At the model level, the converted outcome has probabilities $$\mathrm{Pr}(+1\mid X=x,A=a)=\frac{1+m_{a,P}(x)}{2},\mathrm{Pr}(-1\mid X=x,A=a)=\frac{1-m_{a,P}(x)}{2},a\in \{0,1\}.$$ These probabilities describe the signed-binary conversion in Proposition 1. It retains the propensity, both conditional arm means, and hence the contrast and its distance from constancy. The same result establishes that the bounded, signed-binary, and original models have equal distance suprema.

For a bounded original record, the conversion can instead use its observed outcome directly. To express this randomization on the full real-outcome test carrier, define the measurable sign map $$s(y,t)=21\left\{t\le \frac{1+\max \{-1,\min \{y,1\}\}}{2}\right\}-1,(y,t)\in \mathbb{R}\times [0,1].$$ For $|y|\le 1$, a uniform $t$ gives the probabilities $(1+y)/2$ and $(1-y)/2$ for the two signs, with conditional mean $y$. The truncation in this definition specifies the map on the entire original-record carrier. Its restriction to the bounded model is precisely the record-level conversion in Proposition 1.

Write $\mathbf{V}=(V_1,\dots ,V_n)$ for independent uniform variables on $[0,1]$, independent of the sample and of the public seed $U$ in Definition 4. The randomized signed-binary sample is $${\mathcal{D}}_n^s(\mathbf{V})=((X_i,A_i,s(Y_i,V_i)){)}_{i=1}^n.$$ For a test $\varphi$ on the carrier of Definition 5, its integrated rejection-probability map is $$\tilde{\varphi }({\mathcal{D}}_n,u)={\int }_{[0,1{]}^n}\varphi \left(((X_i,A_i,s(Y_i,t_i)){)}_{i=1}^n,u\right){\lambda }^{\otimes n}(d\mathbf{t}),\mathbf{t}=(t_1,\dots ,t_n).$$ Thus $\tilde{\varphi }$ is a jointly Borel map from $$([0,1]\times \{0,1\}\times \mathbb{R}{)}^n\times [0,1]\text{to}[0,1].$$ It uses the original-record experiment and its public seed; the additional sign randomization is averaged into the rejection probability.

For $P\in {\mathcal{M}}_w^{\mathrm{b}}$, let $P^s$ denote its signed-binary conversion from Proposition 1. The rejection-probability preservation assertion there takes the explicit form $${\mathsf{R}}_n(P,\tilde{\varphi })={\mathsf{R}}_n(P^s,\varphi ).$$ This identity expresses the statistical role of mean-preserving randomization: rejection probabilities under each bounded law transfer exactly to those under its signed-binary conversion, including laws in the constant-effect null and laws at a specified heterogeneity distance.

Proposition 1 consequently gives the exact benchmark identity $$B_n^{\mathrm{bin}}(w,r)=B_n^{\mathrm{b}}(w,r),n\ge 2,0<r<D_v,$$ for the risks in Definitions 12 and 11. With the common distance supremum and the capping conventions in Definitions 14 and 13, the same comparison gives equal capped critical radii for every $n\ge 2$. The signed-binary benchmark therefore represents the bounded mean-effect testing problem at the level of attainable distances, minimax risks, and critical radii.

Bias, covariance, profiling, and attainment

The calibration in Theorem 1 rests on three population quantities: the mean of the original-record score, the covariance of its affine coefficients, and the fluctuation of the cross-block inner product uniformly over candidate constants. This appendix organizes those quantities around the public construction in Algorithm 1. The resulting bounds connect the finite score calculation to the explicit separation guarantee.

Calibration and population decomposition

The deterministic calibration handle collects the score construction, its affine coefficients, the public bounds, and the profiling operation. Keeping these components together makes their dependence on the sample size and public exponent tuple explicit.

The affine coefficient $Z_{b,A}$ records the score’s response to the candidate constant, while $Z_{b,0}$ records its outcome contribution. Their joint population behavior is therefore sufficient to describe the entire profiled statistic. The mean handle below retains both coefficients, their singleton and canonical pair projections, and all four combinations of single-record and pair contributions.

The conventions in Algorithms 4 and 3 use the same one-based blocks ${\mathcal{I}}_1,{\mathcal{I}}_2$. In the mean calculation, $T_{\mathrm{main}}$ denotes the procedure’s main cutoff $T_0$, while $T_j^{\mathrm{inc}}$ denotes its increment cutoff $T_j$ for $1\le j\le L$. Thus the two cutoff families and the two block scores have distinct names without a positional shift. At the public selections, the coarse rank is $J=M$ and the multiresolution fine rank is $2^{L}J=K$.

For clarity about the symmetric pair convention, write $o=(x,a,y)$ and $z=(x^{\prime },a^{\prime },y^{\prime })$. For a correction function $G$ and cutoff $T$, define its signed, symmetrized contribution by $$k_{G,T,c}(o,z):=-\frac{1}{2}\left[F_M(x)aG(x,x^{\prime })\{{\ell }_T(y^{\prime })-c{a}^{\prime }\}+F_M(x^{\prime })a^{\prime }G(x^{\prime },x)\{{\ell }_T(y)-ca\}\right].$$ The single-record contribution is $$h_c(o):=F_M(x)a\{{\ell }_{T_0}(y)-c\}.$$ In the single-correction branch, $g_c:=k_{{\Pi }_K,T_0,c}$. In the multiresolution branch, using the increment indices of Algorithm 1, $$g_c:=k_{{\Pi }_M,T_0,c}+\sum _{j=1}^L{k}_{D_j,T_j,c}.$$ The affine coefficients in Algorithm 4 are thus $h_0=h_c{|}_{c=0}$, $h_1=h_c{|}_{c=0}-h_c{|}_{c=1}$, and the corresponding two coefficients of $g_c$. Symmetrization preserves the ordered-pair average because that average contains both orientations of every distinct pair.

The four contractions in Algorithm 4 retain the single–single, single–pair, pair–single, and pair–pair contributions separately. Their independent-record arguments match the four finite sums defining $W(c)$. Both orientations of the mixed contribution are retained, including their dependence on the two affine coefficients.

Original-mean bias

The population calculation separates a weighted effect contrast, a baseline approximation term, and clipping tails. To specify the latter explicitly, let $\lambda$ denote Lebesgue probability measure on $[0,1]$, and define the conditional clipping discrepancy and its unconditional conditional mean by $${\delta }_{a,T}(x):={\mathbb{E}}_P[{\ell }_T(Y)-Y\mid X=x,A=a],r_T(x):=e_P(x){\delta }_{1,T}(x)+\{1-e_P(x)\}{\delta }_{0,T}(x).$$ These quantities are defined up to $\lambda$-almost-everywhere equality; their integrated contributions use that convention. Under the mean handle’s conditions, the clipping-tail remainders for the two branches are $$\begin{array}{rl}{\mathcal{T}}_{\mathrm{single}}& :={\int }_0^1{F}_M(x)\left[e_P(x){\delta }_{1,T_0}(x)-e_K(x)r_{T_0}(x)\right]d\lambda (x),\\ {\mathcal{T}}_{\mathrm{multi}}& :={\int }_0^1{F}_M(x)\left[e_P(x){\delta }_{1,T_0}(x)-({\Pi }_M{e}_P)(x)r_{T_0}(x)-\sum _{j=1}^L(D_j{e}_P)(x)r_{T_j}(x)\right]d\lambda (x).\end{array}$$ Here $(D_j{e}_P)(x):={\int }_0^1{D}_j(x,z)e_P(z)d\lambda (z)$, and the increment indices follow Algorithm 1. Each remainder is independent of the candidate constant.

The retained mean in Algorithm 4 therefore identifies the baseline approximation contribution through $(e_P-e_K)m_{0,P}$, while the clipping remainder records the outcome-tail contribution at each selected resolution. The public bound $B$ in Algorithm 1 aggregates these terms. The next statement controls that bound uniformly over the candidate constants and relates the score mean to heterogeneity.

The mean bounds supply the population distinction used by the test. Under a constant effect, evaluating at that constant gives a score mean of norm at most $B$. Under heterogeneity, every admissible candidate constant has a score mean bounded below by a multiple of $d(P)$, after accounting for coarse approximation and original-mean bias. Uniformity in the candidate constant is what makes these bounds compatible with profiling.

Singleton, pair, and cross-level covariance bounds

The singleton projection in Algorithm 4 combines the centered single-record contribution with twice the pair-kernel singleton projection. Its canonical pair projection retains the component centered with respect to either record. These two components have separate public bounds, $L_1$ and $L_2$, in Algorithm 1.

For the multiresolution construction, both bounds aggregate the main correction and every retained increment. In particular, $L_1$ contains the sum of the increment-specific smoothness and clipping contributions, while $L_2$ contains the sum of their rank-weighted second-moment bounds. The squared sums in $${\Lambda }_{n,v}=\frac{L_1^2}{s}+\frac{2L_2^2}{s(s-1)}$$ retain the cross-level contributions within each component. The covariance statement below controls both affine coefficients and their covariance in arbitrary, potentially different, directions.

The directional form gives a common covariance bound for the outcome coefficient and the constant-effect coefficient. The mixed bound retains their dependence within a block, including the dependence contributed by the different correction levels. Independence between the two evaluation blocks then provides the setting for the cross-block profiling statement.

Simultaneous affine profiling and calibration

The profiling multiplier $C=128$ and the population mean ${\theta }_P(c)$ were introduced with Theorem 1. The following bound applies to the entire interval of candidate constants on one measurable event.

Simultaneous control permits minimization over the candidate constant within the same event. Its fluctuation bound separates a term proportional to the population score norm from a term involving the coarse dimension and covariance ledger. When the covariance ledger vanishes, the simultaneous identity gives the population target exactly.

The public computation retains this affine structure explicitly. Using the one-based block convention of Algorithm 3, set $Z_{b,1}:=Z_{b,A}$ and $$w_{r{r}^{\prime }}:=⟨Z_{1,r},Z_{2,r^{\prime }}⟩,r,r^{\prime }\in \{0,1\}.$$ Then the computed quadratic is $$W(c)=w_{00}-c(w_{01}+w_{10})+c^2{w}_{11}.$$ Its minimum over $[-1/2,1/2]$ is obtained by comparing the two endpoints and, when $w_{11}>0$ and the vertex lies in the interval, the value at $$c_{\mathrm{vert}}:=\frac{w_{01}+w_{10}}{2w_{11}}.$$ This formula preserves both mixed affine-coefficient contributions through their sum. The same calculation applies under the zero-based convention after translating the block indices.

Together, Lemmas 2, 3, and 4 supply the population and fluctuation bounds used in Theorem 1. The cutoff in Algorithm 1 rounds $$2B^2+2^{16}\sqrt{M}{\Lambda }_{n,v}$$ upward to a dyadic value. The calibration theorem gives size at most $0.0075$ throughout the constant-effect null and type-II error at most $0.0075$ at its calibrated separation, for $n\ge 4$ when that separation is below the model distance supremum. The zero rejection rule supplies the stated size guarantee at $n=2,3$.

The calibrated separation $R_{\mathrm{att}}(n,v)$ in Theorem 1 groups coarse approximation, the bias ledger, and the covariance contribution. The explicit attainment statement below expresses their combined requirement as a power of the sample size with a fully specified multiplier.

Explicit attainment constants and separation

Fix an admissible tuple $v=(p,\alpha ,\beta ,\gamma )$ from Definition 1, and use the exponents $q,S,t,E_0,E_4,E$ of Algorithm 1. Before stating attainment, define the minimum smoothness $s_0$, the clipping allocation exponent ${\lambda }_0$, the increment summability exponent ${\eta }_0$, and the geometric-series factor $A(x)$ by $$s_0:=\min \{\alpha ,\beta ,\gamma \},{\lambda }_0:=\frac{(p-1{)}^2}{4p},{\eta }_0:=\frac{(p-1)(p+6)}{4p},A(x):=(1-2^{-x}{)}^{-1}(x>0).$$ Here $A(x)$ denotes a scalar series factor; its argument distinguishes it from the treatment coordinate. Define the grouped bias, summation, singleton, and canonical-pair constants by $$\begin{array}{rl}{B}_{\ast }& :=420+800\{A(\alpha )+A({\lambda }_0)\},\\ D_{\ast }& :=1+\{\exp (1)S\log 2{\}}^{-1},\\ L_{\ast }& :=16+880A(s_0)+160D_{\ast }+960A(\alpha ),\\ G_{\ast }& :=\sqrt{64}\{2^{t/2}A(1/2)+A({\eta }_0/2)\}.\end{array}$$ The aggregate covariance constant $A_{\ast }$ and attainment multiplier $C_v^{\mathrm{att}}$ are $$A_{\ast }:=\sqrt{2}\{24576+64L_{\ast }^2+16384+256G_{\ast }^2\},C_v^{\mathrm{att}}:=16\{16+5B_{\ast }+1024\sqrt{A_{\ast }}\}.$$ These definitions resolve every constant entering the public attainment multiplier.

Using the heterogeneity distance and model supremum defined in Theorem 1, define the fixed witness distance $d_0$ and public threshold $N_v^{\mathrm{att}}$ by $$d_0:=\frac{1}{8\sqrt{2}},N_v^{\mathrm{att}}:=\lceil \max \left\{4,{\left(\frac{2C_v^{\mathrm{att}}}{d_0}\right)}^{1/E}\right\}\rceil .$$ For the rejection expectation below, ${\mathcal{D}}_n$ denotes $n$ independent original records with law $P$, and $U$ denotes an independent uniform variable on $[0,1]$, as in Definition 4. Set $${\mathsf{R}}_n(P,\varphi ):=\mathbb{E}[\varphi ({\mathcal{D}}_n,U)].$$

For an admissible bounded-model tuple $w=(\alpha ,\beta ,\gamma )$ from Definition 7, define the matched exponent, multiplier, and distance supremum by $$E_{\mathrm{b}}:=\min \left\{\frac{2\gamma }{4\gamma +1},\frac{2(\alpha +\beta )}{1+2(\alpha +\beta )+(\alpha +\beta )/(2\gamma )}\right\},C_{(2,w)}^{\mathrm{att}}:=C_{(2,\alpha ,\beta ,\gamma )}^{\mathrm{att}},D_w^{\mathrm{b}}:=\underset{P\in {\mathcal{M}}_w^{\mathrm{b}}}{\sup }d(P).$$ The following result gives attainment for the original model and its matched bounded submodel, together with public tuning bounds and uniform control of the constants on compact sets of admissible tuples.

The attainment guarantee combines the mean separation in Lemma 2 with the simultaneous fluctuation bound in Lemma 4. Its exponent is the minimum of the two public separation exponents, while its multiplier accounts explicitly for projection, clipping, and covariance contributions. Evaluating the same construction at moment order two gives the bounded-model exponent and multiplier. The power clauses apply when the stated separation is below the corresponding model distance supremum; the capped-radius bounds apply with the capping conventions in Definitions 6 and 13.

The tuning bounds place every selected rank and outcome cutoff within the finite public grids of Algorithm 1. Compact uniformity supplies common bounds for the attainment constants and thresholds over compact sets of admissible tuples. Thus the explicit procedure links a finite original-record calculation to a separation guarantee whose dependence on the public moment and smoothness parameters is fully specified.

Full-record lower bounds and tail essentiality

The lower bounds complement the attainment guarantee in Theorem 2 by constructing null and alternative mixtures that remain close when the entire original record is observed. Two constructions supply the required comparisons: paired tents with constant propensity, and coupled continuous coefficient fields in the rough phase. Their separate bounded versions provide witnesses at moment order two.

We use total variation in the convention $$\mathrm{TV}(P,Q)=\underset{E\mathrm{measurable}}{\sup }|P(E)-Q(E)|.$$ For probability laws $P\ll Q$, the directed chi-square divergence is $${\chi }^2(P\Vert Q)=\int {\left(\frac{dP}{dQ}-1\right)}^{2}dQ=\int {\left(\frac{dP}{dQ}\right)}^{2}dQ-1.$$ These divergences compare probability laws on the same measurable space.

A finite prior draws one original-record law and then generates the independent sample from that law. Specifically, for $$\pi =\sum _{j=1}^k{\omega }_j{\delta }_{P_j},k\ge 1,{\omega }_j\ge 0,\sum _{j=1}^k{\omega }_j=1,$$ write $${\mathrm{supp}}_{+}(\pi )=\{P_j:{\omega }_j>0\},{\mathbb{P}}_{\pi ,n}=\sum _{j=1}^k{\omega }_j{P}_j^{\otimes n}.$$ Thus ${\mathbb{P}}_{\pi ,n}$ mixes product laws after the common latent law has been drawn. Repeated support laws have their weights combined. Normalization ensures that the positive-weight support is nonempty.

For the paired-tent comparison, let $\lambda$ be Lebesgue probability measure on $[0,1]$, and let $Q_{a,P}(dy\mid x)$ be the conditional outcome probability kernel in treatment arm $a$. The following result supplies both rare-mark witnesses under a finite conditional moment and signed-binary witnesses in the matched bounded model.

The fixed propensity in Lemma 5 makes the comparison applicable to both observation regimes. Every alternative support law meets the separation requirement, while the mixture remains close to the null in the complete original-record experiment. The growing mark magnitudes coexist with a conditional $p$-moment bound of one. The signed-binary package supplies a separate bounded comparison at the moment-order-two exponent.

Marked records and continuous coefficient fields

The rough-phase construction uses a normalized marked table $\mathsf{T}$ to specify treatment, outcome sign, and mark occurrence jointly. Keeping its six categories explicit preserves the assignment information and the zero outcomes in the likelihood comparison.

In Definition 29, the treatment sign $\ell$ and marked-outcome sign $h$ index the four nonzero-outcome categories. The coordinate $\xi$ governs treatment probabilities, $\upsilon$ governs marked-outcome signs, and $\zeta$ governs their interaction. Summing over the categories gives total probability one and nonzero-outcome probability $\epsilon$; the positive mark magnitude is $L$. The remaining two categories retain the treatment coordinate when the outcome is zero.

Continuous coefficient fields are formed from the finite-overlap frame $(f_i{)}_{i=0}^K$ below, where $K$ is the fine rank. Its adjacent-cell support gives a continuous construction on the fine partition.

On each fine cell, the two active frame coordinates have squared values summing to one. The prior construction uses these squares to interpolate coarse paired tents, while the frame coordinates themselves carry the random propensity and outcome coefficients. The following algorithm specifies both branches through a finite collection of sign draws.

The copula-frame prior ${\pi }_{\nu ;v,a,u,\epsilon ,L}^{K,M}$ in Algorithm 5 changes the dependence between the fine coefficient signs across branches. Its propensity-coordinate amplitude is $a$, and its marked-outcome-coordinate amplitude is $u$. Each coefficient draw determines one original-record law, so the pushforward is a finite probability prior even when several draws determine the same law.

For the model tuning, write $b$ for the baseline amplitude, and denote the resulting null and alternative priors by ${\pi }_0$ and ${\pi }_1$. The next result gives their support placement under the phase and rank conditions, together with a separate bounded tuning.

The support conclusion in Lemma 6 applies to every coefficient realization. The rank-compatibility condition links the coarse effect resolution to the product of the two fine-scale amplitudes. The displayed effect is centered and has distance of order $K^{-S}$. In the finite-moment tuning, mark rarity and magnitude yield conditional $p$-moment exactly one; in the bounded tuning, unit mark magnitude and mark probability one half yield conditional second moment one half. Both tunings retain the same effect identity and distance bounds.

Augmentation and complete-record likelihood budgets

The component comparison retains the complete record through a common augmentation. Write $B_i=1\{Y_i\ne 0\}$ for the mark indicator, and let ${\mathcal{D}}_{\partial }$ disclose coefficient pairs at the coarse-cell boundary nodes. The augmentation consists of the covariates, these indicators, and this boundary disclosure. Its common probability law is denoted by $\mu$.

Conditional on the augmentation, use $P_0$ and $P_1$ for the two branch label laws. These are local conditional laws in the component comparison. Each label contains treatment and an outcome sign, including a retained sign when its mark indicator is zero. The intermediate conditional label law is denoted by $Q$, so that the intermediate augmented probability law is $\mu \otimes Q$. Recovering the observed outcome from the sign and mark indicator gives the original-record mixtures.

Conditioning on the augmentation fixes the allocation of covariates to fine and coarse cells and identifies which records carry nonzero marks. Boundary disclosure is part of this conditioning. The first and second conditional likelihood budgets, denoted by $\mathcal{A}$ and $\mathcal{B}$, bound the comparisons from $P_0$ to $Q$ and from $Q$ to $P_1$, respectively. The following result averages these budgets over the same augmentation law and gives the resulting complete-record comparison.

The common augmentation in Lemma 7 includes the disclosed boundary coefficients, and the coarse signs remain independent of it under both branches. All three conditional label laws are normalized probability laws. The first expected budget has sample-size and rank factor $n^2/K$; the second has factor $n^4/(M{K}^2)$. Both include the amplitude and mark factor $a^4{u}^4{\epsilon }^2$. Their stated sum condition gives total variation below one eighth for the recovered complete-record mixtures, including the treatment labels and zero outcomes.

Selected priors and the rough lower bound

The selected converse construction in Algorithm 6 now has its ingredients. For $F(p)<1$ and sample size at least the public lower-construction threshold, it selects the copula-frame priors at its specified ranks and finite-moment tuning. The other branch uses the paired-tent package. This selection combines the rough-phase construction with witnesses available for every integer sample size at least two.

At moment order two, the bounded companion uses the bounded copula tuning in the large-sample rough branch and the signed-binary paired-tent package otherwise. Let $\mathcal{V}$ denote the admissible tuple domain stipulated in Definition 1. The following definition fixes the bounded prior pair and its complete-record mixtures.

The bounded prior pair ${\pi }_{\nu ,n,v}^{\mathrm{b}}$ in Definition 31 is defined for every admissible moment-order-two tuple and every $n\ge 2$. Its complete-record mixture ${\mathbb{P}}_{\nu ,n,v}^{\mathrm{b}}$ uses the same draw-once convention as ${\mathbb{P}}_{\pi ,n}$. The next result assembles support placement, likelihood control, and the testing-risk lower bound in the rough phase.

Under $F(p)<1$, Lemma 8 gives the rough separation exponent for every $n\ge 2$. Above the public lower-construction threshold, the selected ranks also meet the component-comparison conditions and give the stronger total-variation bound of one eighth. The deterministic budget bounds control the upper bounds on the expectations of $\mathcal{A}$ and $\mathcal{B}$ in Lemma 7. The bounded conclusion uses its own outcome tuning at $p=2$, preserving the rough exponent and the finite-sample risk guarantee.

Tail essentiality and the moment-order-two face

The final result combines the rough and paired-tent branches with attainment. Its bounded-prior comparison quantifies over normalized finite priors in the convention fixed above: all support restrictions apply to positive-weight laws, and each mixture draws its latent law once for the complete sample. For each fixed admissible tuple with moment order below two, the comparison has a tuple-dependent eventual threshold. On the moment-order-two face, the selected bounded constructions give witnesses for every $n\ge 2$.

The distinction in Theorem 3 concerns the outcome distributions that sustain close full-record comparisons at the finite-moment separation scale. The selected finite-moment witnesses have growing mark magnitudes and total variation below one half for every $n\ge 2$. For each fixed tuple with $p<2$, the bounded scale is asymptotically smaller, and every normalized finite bounded-prior pair satisfying the stated support restrictions has total variation at least one half once $n\ge N(v)$. This threshold depends on the fixed tuple.

At $p=2$, the bounded copula construction and the signed-binary paired-tent construction together supply normalized, nonempty bounded witnesses for every sample size. The common scales and constants then connect these lower comparisons to the attained upper bounds. The power conclusions apply when the common upper separation is below the model distance supremum, a condition guaranteed by the public threshold $N_v$; the capped-radius conclusions retain the conventions of Definitions 6, 13, and 14.

Supplied-propensity proofs and comparison algebra

The supplied-propensity results combine inverse weighting, removal of the constant effect, and a quadratic comparison of independent blocks. The construction in Algorithm 2 applies over the continuous-carrier class of Definition 22. Its upper bounds and the common-propensity lower witnesses give the matching separation guarantees in Theorems 5 and 6. The comparison algebra then distinguishes the value of supplied assignment information on the original model from the cost of outcome tails at matched primitive smoothness.

Inverse weighting and the public error bounds

Fix an admissible tuple $v=(p,\alpha ,\beta ,\gamma )$ and a law $P\in {\mathcal{M}}_v^{\mathrm{e},\mathrm{br}}$, as defined in Definition 22. Throughout this section, $\lambda$ denotes Lebesgue probability measure on $[0,1]$. The conditional-mean identity in Theorem 5 identifies the mean effect through the untruncated inverse-propensity score. The treatment probability cancels its inverse weight separately in each arm, leaving the difference between the treated and baseline conditional means. This applies the inverse-probability weighting principle of Horvitz et al. (1952) to the conditional mean contrast.

Outcome clipping makes this identity usable under the conditional moment envelope. The score $R_i$, cutoff $T$, and public bounds $b,\Lambda$ are those of Algorithm 2: $$b=20T^{1-p},\Lambda =\frac{80T^{2-p}}{s},s=\lfloor n/2\rfloor .$$ The bias bound accounts for clipping in both treatment arms. The covariance bound combines the clipped second-moment envelope with the bounded inverse weights and averaging over an evaluation block. These calculations use uniform design, overlap, and the conditional moment restriction in Assumptions 1, 2, and 8.

The statistical roles of the two bounds are distinct. Increasing the cutoff reduces clipping bias through $T^{1-p}$, while the covariance allowance grows through $T^{2-p}$. At $p=2$, the latter factor equals one. Effect smoothness in Assumption 5 controls histogram approximation of the mean contrast. Thus the broader-class attainment result depends on the moment order and effect smoothness, with continuous propensity and arm-mean representatives supplied by Definition 22.

Removing the unknown constant

The histogram representation in Definition 19 makes the constant-effect direction explicit. Let $$u=J^{-1/2}(1,\dots ,1{)}^T,C_J=I_J-u{u}^T,$$ where $I_J$ is the $J\times J$ identity matrix and $C_J$ is the orthogonal projection onto the complement of the constant direction. In the notation of Algorithm 2, $u$ is a unit vector and $$C_J{F}_J(x)=F_J(x)-u.$$ Consequently, the centered block vectors and their inner product have the equivalent representations $$V_b=\frac{1}{s}\sum _{i\in {\mathcal{I}}_b}{C}_J{F}_J(X_i)R_i,b\in \{1,2\},$$ and $$⟨V_1,V_2⟩=\frac{1}{s^2}\sum _{i\in {\mathcal{I}}_1}\sum _{k\in {\mathcal{I}}_2}\{{\Pi }_J(X_i,X_k)-1\}{R}_i{R}_k.$$ These are representations of the statistic already specified in Algorithm 2.

Constant centering targets the distance of the effect from its average. Every unknown constant in the null of Definition 23 occupies the same removed direction. Clipping bias remains covered by the public bound $b$, while effect smoothness governs the approximation of the remaining heterogeneity. The two disjoint evaluation blocks provide independent score vectors, so their quadratic comparison measures the squared population signal subject to the stated bias and covariance allowances.

The construction also specifies the test on its full supplied-function domain. For an arbitrary supplied continuous function $f$, define its overlap-clipped version by $$\tilde{f}(x)=\max \{1/4,\min \{f(x),3/4\}\}.$$ The score $R_i(f)$ and block vector $V_b(f)$ denote the expressions in Algorithm 2 evaluated with $\tilde{f}$ in place of $\tilde{e}$. This convention bounds the denominators throughout the test domain. For a legal true supplied propensity, it agrees with the propensity itself, as required by the rejection expectation in Definition 24.

Calibration, attainment, and common-propensity witnesses

The threshold in Algorithm 2 is $$2b^2+1024\sqrt{J}\Lambda .$$ Its first term accommodates the population discrepancy caused by clipping under a constant effect. The second supplies the quadratic fluctuation allowance. The resulting calibration in Theorem 5 controls rejection probability by $0.01$ uniformly over the entire broader-class null for every $n\ge 2$. At the smallest sample sizes, the public tuning gives $J=1$, for which constant centering makes the statistic identically zero.

The tuning coordinates clipping, projection, and quadratic variability. With the target scale $a=s^{-E_0}$ from Algorithm 2, the cutoff and rank are upper dyadic roundings of $$a^{-1/(p-1)}\text{and}{a}^{-1/\gamma },$$ respectively. Their common separation exponent is $$E_0(p)=\frac{2\gamma q}{2\gamma +q},q=\frac{p-1}{p},$$ as defined in Definition 15. The attained scale is ${\rho }_n^{\mathrm{e}}(v)=n^{-E_0(p)}$. At moment order two, $$E_0(2)=\frac{2\gamma }{4\gamma +1}.$$ This agrees numerically with the familiar one-dimensional quadratic-testing exponent in the Gaussian Sobolev setting of Ingster et al. (2009), where the stated smoothness condition is strictly above $1/4$. The supplied-propensity guarantees here have the conditional-moment and continuous-carrier scope specified in Theorem 5, including $\gamma =1/4$.

The lower and upper multipliers $c_v^{\mathrm{e}},C_v^{\mathrm{e}}$ and threshold $N_v^{\mathrm{e}}$ are defined in Theorem 5. Their radius bounds hold for every $n\ge 2$. The power guarantee applies when $$C_v^{\mathrm{e}}{\rho }_n^{\mathrm{e}}(v)<D_v^{\mathrm{e},\mathrm{br}}.$$ The public threshold places this upper separation at most $d_0/2$, where $d_0=1/(8\sqrt{2})$ is the positive witness distance in that result. The lower bound $D_v^{\mathrm{e},\mathrm{br}}\ge d_0$ then supplies a nonempty separation range.

The common-propensity witnesses connect this attainment result to both supplied experiments. In Theorem 5, the null law and every positive-weight alternative law belong to the original and broader classes and share the propensity $x\mapsto 1/2$. Their complete-record distributions have total variation below one half. Supplying that same function preserves the comparison underlying the lower error bound.

Class inclusion transfers the broader-class calibration and power guarantees to the original model: $${\mathcal{M}}_v\subseteq {\mathcal{M}}_v^{\mathrm{e},\mathrm{br}},H_0(v)\subseteq H_0^{\mathrm{e},\mathrm{br}}(v),$$ using Definitions 1, 2, 22, and 23. The witnesses also lie within the original class, where Theorem 6 supplies the corresponding necessity statement. Its multipliers $c_v^{\mathrm{e}},C_v^{\mathrm{e}}$ have the same displayed values as their broader-class counterparts; its power condition uses the original supremum $D_v$.

The preservation boundary on the original class

Hold $w=(\alpha ,\beta ,\gamma )$ fixed when evaluating the exponents in Definition 15. The comparison identity in Theorem 4 is $$E_4(p)-E_0(p)=\frac{2\gamma q}{(2\gamma +q)D_p}\{F(p)-1\}.$$ Its positive prefactor makes $F(p)=1$ the equality boundary between the two separation exponents. The original-model exponent is their minimum.

The nonnegative comparison exponent $G(v)$, defined in Theorem 7, therefore has the representations $$G(v)=E_0(p)-E(p)=\frac{2\gamma q}{(2\gamma +q)D_p}\max \{0,1-F(p)\}.$$ The same-class radius ratio is bounded on both sides by positive multiples of $n^{G(v)}$, with the exact multipliers given in that result. For $F(p)\ge 1$, this exponent equals zero and the ratio remains bounded over all $n\ge 2$. Equality points $F(p)=1$ are included. For $F(p)<1$, the positive exponent quantifies the polynomial gain from supplied assignment information. This yields precisely the preservation region in Definition 28 and Theorem 7.

The finite-moment and bounded comparison

The bounded comparison evaluates the original-record exponent at moment order two while preserving $w$. For presentation, let ${\delta }_0(p)$ and ${\delta }_4(p)$ denote the within-mechanism exponent differences. The formulas in Definition 15 give $${\delta }_0(p):=E_0(2)-E_0(p)=\frac{4{\gamma }^2(2-p)}{p(4\gamma +1)(2\gamma +q)},$$ and $${\delta }_4(p):=E_4(2)-E_4(p)=\frac{2S\beta (1/q-2)}{D_2{D}_p},D_2=1+2S+\frac{S}{2\gamma }.$$ Both differences are strictly positive for $p<2$ and vanish at $p=2$.

At fixed smoothness, $F(p)\ge F(2)$. The selected exponent difference consequently has three compatible representations: $$E(2)-E(p)=\{\begin{array}{ll}{\delta }_0(p),& F(2)\ge 1,\\ {\delta }_4(p)+\frac{2\gamma q}{(2\gamma +q)D_p}\{F(p)-1\},& F(p)\ge 1>F(2),\\ {\delta }_4(p),& F(p)<1.\end{array}$$ At $F(p)=1$, the additional term in the middle expression vanishes. At $F(2)=1$, the bounded exponents coincide, so the adjoining expressions agree. These equality points preserve the exponent comparison across the change in the rate-determining mechanism.

The matched capped-radius inequalities in Theorem 4 attach the factor $n^{E(2)-E(p)}$ to ${\Delta }_n(v)$, the ratio defined in Definition 16. Strict positivity for $p<2$ gives the diverging ratio characterized by Definition 17. At $p=2$, the exponent difference is zero and the ratio remains uniformly bounded over sample sizes, as characterized by Definition 18. The phase equality $F(p)=1$ and the moment-order equality $p=2$ thus describe distinct comparisons.

All these statements retain the finite-sample multipliers and saturation conventions of the cited radius results. Their local uniformity is expressed through common positive lower multipliers, finite upper multipliers, and finite bounds on the public power thresholds over compact subsets of the admissible parameter domains, as stipulated in Theorems 4, 5, 6, and 7. These uniform constants accompany the parameter-dependent powers, including at the stated equality boundaries.

Verification note

The theorem statements and their proofs are machine-checked in Lean 4. The anchors attached to mathematical environments connect the displayed definitions and results to their corresponding formal declarations. Checking establishes the stated conclusions from the hypotheses and dependencies of each result, preserving its parameter domain, quantifiers, constants, and sample-size conditions.

The restrictions defining the statistical models, including those in Definitions 1, 7, 8, and 22, are stipulated hypotheses. The verified conclusions concern deductions under these restrictions, including calibration, attainment, lower bounds, and separation-radius comparisons. Where a result uses a published conclusion as a dependency, its theorem-local verification-scope footnote identifies that conclusion and records whether the source proof was formalized. These local disclosures specify the dependency boundary alongside the checked statements and proofs.

Proofs of the main results

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