Introduction

Proxy causal models use measurements related to latent confounding or latent classes to recover causal objects from observational data. In the potential-outcomes tradition (Splawa-Neyman et al., 1990; Rubin, 1974; Holland, 1986; Imbens et al., 2015), latent heterogeneity is typically organized through assumptions linking treatment assignment, outcomes, and observed covariates. Negative-control and proximal approaches enrich that structure by introducing proxy measurements with conditional-independence and rank properties (Miao et al., 2018; Shi et al., 2020; Tchetgen Tchetgen et al., 2024; Cui et al., 2024; Miao et al., 2023; Qi et al., 2024; Liu et al., 2024; Li et al., 2024; Ai et al., 2025; Saha et al., 2026). The finite-class proxy model of Virk et al. (2026) supplies a compressed observable operator whose spectral structure identifies latent-class mean treatment effects under separated recovery conditions. This paper develops the law-level inference theory for the same proxy moment geometry when equal or nearly equal class-average effects are represented through the probability law of effect values.

The central estimand is the quotient latent-effect law $${\nu }_P=\sum _{u=1}^k{p}_u{\delta }_{{\tau }_u},$$ the distribution that places latent-class mass $p_u$ at the class treatment effect ${\tau }_u=E[Y(1)\mid U=u]-E[Y(0)\mid U=u]$ and aggregates the mass of classes with equal class means. The support points are latent-class average treatment effects; unit-level contrasts $Y(1)-Y(0)$ enter through these conditional means. The law records how much population mass has each class-average effect value, with collisions interpreted as shared support points. The observable input is the five-block summary $$S(P)=\{M_0(P),M_1(P),N_0(P),N_1(P),m_X(P)\},$$ formed from armwise proxy moments, outcome-weighted proxy moments, and the target-proxy mean.

The model class $\mathcal{M}$, the uniformly conditioned proxy causal class, fixes $k,d_x,d_z$, imposes the VMW proxy conditional-independence and consistency restrictions, and uses supplied quantitative bounds: the observable envelope $L$, a lower bound ${\pi }_0$ on every joint latent-arm probability, and a singular-value margin ${\sigma }_0$ for the reference- and target-proxy feature matrices. These conditions yield the observed proxy factorizations and rank margins in Proposition 1, compactness of the feasible summary closure in Proposition 2, and a common compact support interval for latent-class mean effect laws. The fixed-margin formulation is the population domain for all upper bounds and confidence statements in the paper.

The first main result is the gap-free modulus in Theorem 1. It establishes a constant $C_{\mathrm{mod}}$, depending only on the fixed dimensions and conditioning constants, such that $$W_1({\nu }_P,{\nu }_Q)\le C_{\mathrm{mod}}{d}_S(S(P),S(Q))$$ for all $P,Q\in \mathcal{M}$. It also extends the summary-to-law map continuously to the compact summary closure. The statement is formulated in the one-Wasserstein metric $W_1$, so perturbations move aggregate probability mass between effect values and treat simultaneous merges or splits of atoms through the same law-level metric.

This population result leads to two estimators. The nearest-summary repair estimator in Algorithm 1 projects an empirical summary to the compact feasible summary closure and evaluates the continuous extension. The structured-lattice estimator in Algorithm 2 searches over finite grids of signal bases, conditioning matrices, latent masses, and effect locations, using a criterion that matches the empirical compressed operator, target-proxy mean, and anchor equation. Theorem 2 establishes that both estimators are Borel sample maps and achieve uniform root-$n$ $W_1$ tail bounds over $\mathcal{M}$. The lattice construction in Proposition 4 has polynomial candidate growth in $n$ for fixed dimensions and constants.

The confidence results report law-level uncertainty in forms aligned with the quotient target. Algorithm 3 defines an exact repaired-image summary-inversion set through the repaired map and a lattice-centered Wasserstein outer set with a finite constrained exact-real representation. Theorem 3 gives simultaneous coverage and root-$n$ diameter bounds for these sets. Algorithm 4 and Theorem 4 then translate the Wasserstein confidence set into support-component intervals and aggregate mass intervals. The data-facing report uses the empirical lattice law, supplied constants, and the algorithmic confidence set; the true-law objects $K_C(P)$ and ${\Delta }_C(P)$ name the validity quantities used to state coverage and width. Empirical atoms are connected at threshold $4{\rho }_{n,\alpha }$, while support association and interval expansion use the radius ${\rho }_{n,\alpha }$; the resulting mass intervals sharpen according to the external separation of the associated true component.

The paper also characterizes effect-ordered latent masses on separated strata. The ordered target $p^{\uparrow }(P)$, the vector of latent masses ordered by increasing effect, is studied on $\mathcal{M}(g)$, the gap-local stratum with positive effect gaps at scale $g$. Theorem 5 gives the upper risk bound $$E\Vert {\hat{p}}_n^{\uparrow }-p^{\uparrow }(P){\Vert }_1\le C\min \{1,(\sqrt{n}g{)}^{-1}\}.$$ This rate expresses the conversion from law-level transport error to ordered mass recovery when effect locations are separated at scale $g$.

The lower bounds use explicit two-class proxy experiments. The witness law in Definition 55 verifies that a collision can occur inside the full-rank, strictly positive proxy model. Theorem 6 gives local two-point lower bounds for quotient laws and ordered weights, and Propositions 6 and 7 convert them into same-class minimax rates on the displayed two-class specialization. The quotient-law minimax rate is exactly root-$n$, while the ordered-weight minimax rate is the clipped inverse-gap rate. Theorem 7 records the corresponding transfer to comparator classes stated in the published VMW vocabulary when those classes contain the relevant witness pair.

The paper proceeds as follows. The related-work section positions the argument within proxy causal identification, spectral finite-mixture recovery, and Wasserstein inference. The setup section defines the proxy model, observable summaries, quotient law, and compact summary closure. The main-results section proves the law-level modulus and develops the repair and lattice estimators. The confidence section constructs Wasserstein confidence sets, cluster reports, and ordered-weight inference on gap-local strata. The lower-bound section gives the local and same-class minimax converses and the VMW transfer statement. The discussion section collects extensions and open questions, and the appendices provide auxiliary probability, selection, spectral, finite-net, proof, and verification material.

Related work

This paper connects three literatures: proxy causal identification, finite-mixture and spectral recovery, and Wasserstein inference for singular discrete laws. In the Neyman–Rubin potential-outcome tradition (Splawa-Neyman et al., 1990; Rubin, 1974; Holland, 1986; Imbens et al., 2015), latent treatment-effect heterogeneity is usually interpreted through assumptions that relate treatment assignment, outcomes, and observed covariates. Proxy causal models enrich that structure by using measurements linked to latent confounding or latent classes, as in negative-control and proximal identification arguments (Miao et al., 2018; Tchetgen Tchetgen et al., 2024; Shi et al., 2020; Cui et al., 2024; Miao et al., 2023; Qi et al., 2024; Liu et al., 2024; Li et al., 2024; Ai et al., 2025; Saha et al., 2026). The present analysis studies the finite-class proxy setting in which the observed proxy moments identify the quotient latent-effect law ${\nu }_P$ of Definition 40: the probability law that aggregates latent classes sharing the same class-average treatment effect. This target is tailored to collisions in effect values, because the inferential object is the distribution of class-average effect values and their aggregate masses.

The closest point of contact is Virk et al. (2026). Their framework supplies the proxy moment structure, the compressed observable operator formalized here in Definition 39, and separated spectral recovery guarantees that make latent-class mean effects estimable when the relevant spectral quantities are well resolved. The present paper uses that operator as the population bridge from observable summaries to the quotient law and develops a $W_1$ modulus, with $W_1$ defined in Definition 19, for the law target. This perspective keeps the metric aligned with the identifiable probability measure: when several latent classes have the same class-average effect, their masses are combined at a common atom, and the stability statement is made for the quotient law. Label-sensitive lists enter through the separated ordered-weight results.

The identification argument also draws on classical and modern work on finite mixtures and tensor or spectral decompositions. Moment-based identification begins with Pearson (1894) and continues through finite-mixture identifiability results such as Teicher (1963), Lindsay (1995), and Allman et al. (2009). Econometric proxy and measurement-error models use related rank and completeness ideas (Hu, 2008; Hu et al., 2008; Kasahara et al., 2009; Bonhomme et al., 2016), while algorithmic spectral methods exploit low-rank decompositions and perturbation theory (Hsu et al., 2009; Anandkumar et al., 2014; Sidiropoulos et al., 2000; Bauer et al., 1960; Kato, 1995; Davis et al., 1970; Stewart et al., 1990; Mazaheri et al., 2025). Within this lineage, the contribution here is a stability and inference theory for the atomic effect law induced by the proxy causal model, with fixed known dimensions and uniform conditioning constants.

A second comparison is with Wasserstein rates for finite mixtures near singularities. Collision phenomena are central in the mixture literature: when component locations approach one another, labeled parameters and weak distributional metrics exhibit different local geometries (Nguyen, 2013; Ho et al., 2016; Heinrich et al., 2018; Ho et al., 2016; Manole et al., 2022; Nguyen et al., 2026). The rates in Heinrich et al. (2018) clarify how overfitting and merging components affect Wasserstein estimation. The present setting is causal and proxy-based, but the same mathematical distinction matters: law-level estimation remains regular for the quotient distribution of latent-class mean effects, while effect-ordered weights are governed by local separation. The paper makes that distinction explicit by pairing collision-uniform quotient-law inference with inverse-gap behavior for ordered weights on the stated gap-local strata.

A quantitative comparison makes the difference in local geometry explicit. The closest results differ from the present paper in three respects at once: the experiment that is observed, the target the loss is measured on, and the conditions under which the stated rate holds.

The table locates the source of the difference. The ordinary-mixture results observe draws from a mixture distribution, so the map from the sampling law back to the mixing measure degenerates exactly where components merge, and the exponents above deteriorate with the number of colliding components. The present experiment observes a uniformly conditioned proxy-moment operator instead, and the compressed observable operator of Virk et al. (2026) carries the latent effects in its spectrum. Aggregating the mass of coinciding effects into the quotient law ${\nu }_P$ removes the labelling that the collision destroys, and the surviving map from observable summaries to ${\nu }_P$ stays Lipschitz across merges and splits by Theorem 1. Root-$n$ quotient-law recovery and slower Wasserstein rates for ordinary mixing measures are therefore compatible: they measure different targets in different experiments. The labelled quantity that does retain collision sensitivity here is the effect-ordered mass vector, whose inverse-gap behaviour on gap-local strata is the direct analogue of the mixture exponents, and whose matching converse appears in Theorem 6.

Finally, the confidence-set construction is related to projection and inversion methods for weakly identified or singular models. Generalized method-of-moments traditions emphasize confidence regions obtained by inverting sample restrictions (Hansen, 1982). Recent Wasserstein approaches for finite mixtures, including Wu et al. (2020), Doss et al. (2023), Deo et al. (2023), Bing et al. (2026), and Niles-Weed et al. (2022), develop distributional metrics, singular-rate analyses, and confidence constructions for nonregular finite-mixture settings. Here the observable proxy summary is the primitive sample object, and the confidence reports are expressed directly in Wasserstein distance for the quotient law and as cluster-adaptive support and mass summaries. This yields an econometric reporting target that matches the causal estimand while retaining the finite-mixture insight that separated labels and quotient laws have different local statistical behavior.

Setup and assumptions

Throughout the paper, $k$, $d_x$, and $d_z$ are fixed finite dimensions, and treatment-arm and latent-class indices range over $t\in \{0,1\}$ and $u\in \{1,\dots ,k\}$. Vector norms are Euclidean, matrix norms are operator norms, and singular values are ordered decreasingly. Generic constants may depend on the fixed dimensions and conditioning constants, while asymptotic statements vary only with the sample size $n$. Expectations and conditional laws are taken under the full-data probability law under discussion unless a subscript specifies otherwise.

The section is organized around one dependency chain, which the reader may find it useful to keep in view: the observed and full-data records fix the sampling object; the causal and proxy assumptions constrain it; the latent masses, means, and effects are the parameters those assumptions govern; the observable population moments and the quotient law are the two derived targets; the model class and the gap-local stratum collect the uniform conditions under which the theorems hold; and the empirical summary is the sample counterpart of the observable moments. The spectral, lattice, and confidence objects that appear alongside these are the machinery that later sections consume, and each is stated where the theorem citing it can be checked against it.

For reading, the logical dependencies are as follows. The observed record and full-data law supply the variables; the causal and proxy assumptions define the uniformly conditioned class $\mathcal{M}$; the population moments $M_t(P)$, $N_t(P)$, and $m_X(P)$ form the observable summary $S(P)$; the quotient law ${\nu }_P$ is the population-weighted law of latent-class average treatment effects; and the empirical summary, spectral thresholding, lattice, confidence radii, and cluster objects are derived from those primitives. Some auxiliary objects are displayed where they are needed by later statements; the surrounding prose identifies their role in this dependency chain.

The statistical procedures are calibrated by supplied values of $k,d_x,d_z,L,{\pi }_0,{\sigma }_0$. A full-data law receives the stated guarantees when it satisfies the restrictions of $\mathcal{M}(k,d_x,d_z,L,{\pi }_0,{\sigma }_0)$ with those supplied constants. A larger valid envelope $L$, or smaller valid lower margins ${\pi }_0$ and ${\sigma }_0$, preserves membership after the constants and radii are recomputed, with the numerical bounds adjusted accordingly. The joint latent-arm margin encodes overlap within each latent class and arm, while the proxy-rank margin encodes stable variation in the reference- and target-proxy feature matrices.

The observed data consist of the treatment, two proxy vectors, and the realized outcome. The displayed objects below include both primitives and derived objects used later; the prose after each display identifies how it fits with the record spaces, observed margin, transport metric, and inferential target.

The observed record collects the binary treatment $T$, target proxy $X$, reference proxy $Z$, and observed outcome $Y$. The fixed target- and reference-proxy dimensions are $d_x$ and $d_z$.

The distance $W_1$ compares atomic laws by the minimum absolute-distance transport cost, so it measures the law of effect values. Label-specific comparisons enter later through the ordered-weight target on separated strata.

The generic full-data law $P$ carries the latent class $U$, treatment, proxies, potential outcomes $Y(t)$, and observed outcome. The observed map $O$ induces the observed-data marginal $P_O$, which is the sampling law for the observable record.

Latent treatment-effect heterogeneity is described through class masses, class-specific potential-outcome means, and their differences.

The mass $p_u$ records how much of the population belongs to latent class $u$.

The mean ${\mu }_{tu}$ is the class-specific potential-outcome level for arm $t$.

The latent effect ${\tau }_u$ is the class-level treatment contrast.

The proxy structure is encoded by reference- and target-feature matrices. These matrices are the finite-dimensional objects through which the observable conditional moments factor.

The matrix $A_t$ summarizes the reference proxy within latent classes and treatment arms.

The matrix $B$ summarizes the target proxy within latent classes.

The radius $L_{\tau }$ is the common compact interval radius used for latent-class mean effect laws.

The gap $\delta (P)$ measures separation among distinct positive-mass effect values and is used only for label-sensitive ordered-weight statements.

The moments $M_t(P)$, $N_t(P)$, and $m_X(P)$ are the population coordinates observed through the proxy experiment.

The analysis works with represented atomic laws while interpreting coincident atom locations as a single aggregate effect value.

The ordered target $p^{\uparrow }(P)$ is used when the effects are separated enough to support label-sensitive mass recovery.

The model assumptions combine the proxy conditional-independence structure of Virk et al. (2026) with fixed quantitative margins. We first state the structural restrictions linking proxies, potential outcomes, treatment assignment, and the anchor coordinate.

Reference-proxy separation is the VMW reference-proxy conditional independence condition (Virk et al., 2026). It makes the reference proxy vary with the latent class and treatment arm while separating it from the target proxy and outcome once those variables are fixed.

Target-proxy separation is the VMW target-proxy conditional independence condition (Virk et al., 2026). It makes $X$ a latent-class measurement whose conditional mean is stable across treatment arms.

Observed-outcome consistency is the VMW causal consistency condition (Virk et al., 2026). It connects the realized outcome to the treatment-indexed potential outcome.

Latent ignorability is the VMW armwise latent ignorability condition (Virk et al., 2026). It places treatment assignment variation within latent classes in the role needed for the armwise proxy moments.

The target-proxy anchor is the VMW anchor-coordinate normalization (Virk et al., 2026). It fixes the scale needed by the spectral anchors below.

The next conditions impose bounded contributions and uniform conditioning. The boundedness assumptions are standard VMW envelope conditions, while the positivity and rank margins are the fixed-margin structure specific to this analysis.

Bounded target-proxy contributions follow the VMW bounded target-proxy condition (Virk et al., 2026). The radius $L$ supplies the common envelope used for summary concentration and compactness.

The proxy-product bound is the VMW bounded proxy cross-moment contribution condition (Virk et al., 2026). It controls the armwise $Z{X}^{\top }$ moment coordinates.

The outcome-weighted proxy-product bound is the VMW bounded outcome-weighted proxy contribution condition (Virk et al., 2026). It controls the armwise $YZ{X}^{\top }$ moment coordinates.

Latent-arm positivity is specific to this analysis. The supplied margin ${\pi }_0$ gives every latent class and treatment arm a fixed amount of population mass, which yields uniform arm probabilities and atom-mass control.

The proxy-rank margin is specific to this analysis. The supplied singular-value margin ${\sigma }_0$ gives the proxy feature matrices uniform conditioning, which is used to form stable compressed operators and fixed-radius effect supports.

For ordered weights, the paper restricts attention to local strata where all positive-mass latent effects are separated at a prescribed scale.

The gap window is specific to this analysis. The local scale $g$ records the separation regime for ordered-weight rates.

Distinct latent effects are the VMW spectral separation condition (Virk et al., 2026). In this paper the condition is used for the gap-local ordered-weight target, while the quotient law below aggregates coincident effects.

Three representation levels are used for atomic laws. A raw labelled record consists of $k$ weight coordinates and $k$ location coordinates. A valid represented probability law is a raw record whose masses are nonnegative and sum to one, whose locations lie in the support interval, and whose repeated locations are interpreted by aggregating their masses. A quotient law is the equality class of valid represented laws that induce the same probability measure. The coordinate container in Definition 29 records the raw labelled representation; the $W_1$, atom-floor, estimator, and confidence-set statements use the valid represented-law and quotient-law levels indicated in their hypotheses and conclusions.

We can now collect the full-data restrictions into the uniformly conditioned proxy model class.

Three distinct levels of representation are in play, and it is worth separating them once here because later definitions move between them. At the first level sits the labelled coordinate container just displayed: an element records $k$ real weights and $k$ real locations with no constraint tying them to a probability law. At the second level sit the valid labelled laws, those coordinate records whose weights are nonnegative, sum to one, and whose locations lie in the stated interval; the atom floor of Definition 31, the estimators, and the confidence sets are stated for laws of this kind. At the third level sits the quotient of Definition 12, formed from the valid labelled laws by identifying records that induce the same measure. The bridge between the levels is the induced measure ${\nu }_{\xi }$ displayed above, which sends coincident labelled locations to a single atom carrying their aggregate mass; $W_1$ compares laws through this measure, so it is insensitive to the labelling that a collision destroys. The estimand ${\nu }_P$ of Definition 40 is an object of the third level, which is what makes it the regular target under collisions, while the effect-ordered mass vector of Definition 30 retains second-level labelling information and is correspondingly governed by the effect gap.

The class $\mathcal{M}$ fixes the latent cardinality $k$, the boundedness radius, and the two conditioning margins. The qualitative proxy restrictions align with Virk et al. (2026); the fixed quantitative margins make the subsequent stability and concentration statements uniform over the class.

The observable summary is the finite-dimensional population object to which the statistical results are applied.

The summary $S(P)$ and metric $d_S$ put the four matrix moments and target-proxy mean in a single product space.

The arm count $N_{t,n}$, empirical moments ${\hat{M}}_{t,n}$ and ${\hat{N}}_{t,n}$, empirical mean ${\hat{m}}_{X,n}$, and empirical summary ${\hat{S}}_n$ are total functions of the observed sample.

The vector $e_1$ implements the anchor coordinate from Assumption 5.

The compressed operator is the population spectral object inherited from the VMW moment structure.

The signal basis $V(P)$, compressed operator $\Delta Q(P)$, and anchors $a(P)$ and $c(P)$ provide the population bridge from observable summaries to latent-class mean effect moments.

The quotient law is the primary inferential target: it records the distribution of latent-class mean effect values with the aggregate mass at each distinct value.

The stratum $\mathcal{M}(g)$ is the domain for ordered-weight results, where the gap scale and distinctness conditions provide an effect ordering.

The experiment ${\mathcal{E}}_n$ contains the sample laws $Q_P^{(n)}$ generated by all model laws in $\mathcal{M}$.

The next definition records the closure of the admissible summary image and the quotient-law functional on it.

The admissible image $\mathcal{S}$, closure $\mathcal{K}$, and functional $F$ define the population domain for the stability theorem in the next section. The moment identity is the cited spectral bridge from Virk et al. (2026); the homogeneous-effect clause records the corresponding single-atom specialization with common effect ${\tau }_{\star }$.

Proposition 1 establishes the observable consequences of the full-data restrictions. It derives armwise positivity, the proxy moment factorization, compressed and stacked singular-value margins, observable envelopes, and the effect support radius. These conclusions place every $P\in \mathcal{M}$ inside the VMW moment structure under the fixed margins used here, while preserving the quantitative bounds needed for the quotient-law target.

Proposition 2 supplies the compact population summary domain. Compactness supports nearest-summary selection in the next section, and the equivalence between nonempty $\mathcal{K}$ and nonempty $\mathcal{M}$ makes the domain statement exact for each fixed collection of supplied constants.

Several auxiliary definitions used by the empirical structured-lattice estimator and confidence reports are collected here because they share the same observable summaries, norms, and atomic-law coordinates.

Definition 34 records the coordinate formulas. The theorem-facing empirical-summary object below packages the same coordinates with the total empty-arm convention and Borel measurability used in sample-level statements.

Here and throughout, the latent treatment effects in ${\nu }_P$ are the class-average contrasts ${\tau }_u=E[Y(1)\mid U=u]-E[Y(0)\mid U=u]$. Coincident class-average contrasts appear as a single support value with aggregate population mass.

The radius $r_{n,\alpha }$ depends on the miscoverage level $\alpha$ and concentration constant $C_0$; it is used for the summary event $E_P$ in the confidence results.

Together, the definitions and propositions in this section identify the population object, the observable summary space, and the compact domain on which the next section states the gap-free $W_1$ stability and estimation results.

Main results: law stability and estimation

The preceding section reduced the proxy causal model to the observable summary $S(P)$ and the quotient law ${\nu }_P$ of latent-class average treatment effects. This section gives the population stability statement that makes the quotient law a regular law-level target, then turns the modulus into two estimators: a nearest-summary repair estimator defined on the compact summary closure and a finite structured-lattice estimator evaluated from the empirical summary with supplied $k,d_x,d_z,L,{\pi }_0,{\sigma }_0$. The constants below depend only on those fixed dimensions and conditioning constants.

The constant $C_0$ calibrates a uniform finite-sample concentration inequality for all five coordinates of ${\hat{S}}_n$. The radius $r_{n,\alpha }$ is the corresponding $d_S$-radius at confidence level $1-\alpha$, with the model envelope $L$ scaling the observable moment fluctuations.

The ambient space in Definition 46 lets the empirical summary be evaluated for every sample, while the feasible population summaries remain restricted to the compact closure $\mathcal{K}$ from Proposition 2. The repair construction projects an empirical summary back to that population domain before applying the quotient-law functional.

The estimator in Algorithm 1 is the repaired-summary route: it expresses the quotient-law estimator as a continuous extension evaluated at the nearest feasible summary. This directly links observable-summary uncertainty to $W_1$ uncertainty for the law.

The next construction replaces nearest-summary projection with a finite exhaustive lattice over the spectral factors that determine the quotient law.

The lattice program in Algorithm 2 searches over discretized signal bases, conditioning matrices, latent masses, and effect locations. Its criterion matches the empirical compressed operator, target-proxy mean, and anchor equation, so the returned law is tied to the same observable structure as the population quotient law.

The proof is deferred to Section D.

Lemma 1 is the sampling input for the law estimators. It gives a single high-probability radius for the empirical five-block summary and also bounds the probability of an empty treatment arm under the total empirical-summary convention.

Theorem 1 is the population result behind the statistical theory. It establishes a Lipschitz $W_1$ modulus from observable summaries to quotient effect laws over the uniformly conditioned model class, and it extends the summary-to-law map continuously from the admissible image to the compact closure. The result is stated for the law that aggregates coincident effect values, so the same bound covers homogeneous, partially colliding, and separated latent-effect configurations.

The intuition is spectral but the conclusion is metric. The proxy moments determine a compressed effect operator through the VMW moment identity (Virk et al., 2026); perturbation control for finite-dimensional operators (Bauer et al., 1960; Kato, 1995) gives stable spectral moments; and the quotient law is compared in $W_1$, a metric that transports aggregate probability mass between effect values. The continuous extension $\overline{F}$ then makes feasible-summary projection a valid route from empirical summaries to law estimates.

Proposition 3 supplies the measurable total version of the nearest-summary construction. It also records nonemptiness of the exact repaired-image summary-inversion confidence set for every sample, preparing the confidence-set construction in the next section.

The structured-lattice estimator uses the same observable ingredients and replaces projection over the compact summary closure by a finite lattice search. The work count below fixes the computational scale certified here: a fixed-dimensional unit-cost exact-real benchmark, with arithmetic, comparison, singular-value, polar-decomposition, and fixed-degree root-isolation operations counted as unit primitives.

Proposition 4 establishes the empirical structured-lattice law estimator used for the algorithmic confidence set and cluster report. Its deterministic inequality converts summary error and lattice mesh width into $W_1$ error, while its high-probability conclusion follows by combining that inequality with the summary concentration in Lemma 1. The candidate bound records polynomial growth in $n$ for fixed $k,d_x,d_z,L,{\pi }_0,{\sigma }_0$ in the exact-real operation model stated above.

The role of the lattice is to preserve the stabilized spectral structure at empirical summaries near the model image. The singular-value threshold keeps the compressed operator in the uniformly conditioned rank-$k$ regime; the grids over $V$, $R$, $p$, and $\tau$ approximate the population spectral representation; and lexicographic tie-breaking turns the exhaustive search into a total Borel rule.

Theorem 2 is the law-estimation conclusion. Under the fixed dimension, envelope, positivity, and rank-margin conditions defining the uniformly conditioned class, both the nearest-summary repair estimator and the structured-lattice estimator achieve the same root-$n$ $W_1$ tail rate uniformly over $\mathcal{M}$. The statement also carries forward the armwise empty-sample bound, ensuring that the total empirical-summary convention is accompanied by an explicit probability control.

The theorem separates the regular law target from label-sensitive quantities. At the level of ${\nu }_P$, coincident latent-class mean effects are represented by aggregate mass at a shared support point, and the modulus in Theorem 1 transfers root-$n$ summary concentration directly to $W_1$ law error. The next section uses the structured-lattice estimator as the center of exact-real finite-constraint confidence reporting and then treats ordered weights on the gap-local stratum, where effect separation determines the relevant local scale.

Confidence sets, cluster reports, and ordered weights

The root-$n$ law estimators from Theorem 2 lead directly to confidence reporting for the quotient law of latent-class average treatment effects. The first construction keeps two complementary objects in view: an exact repaired-image set of nearby feasible summaries through the repaired summary map, and a lattice-centered finitely represented exact-real $W_1$ outer set with an atom-floor restriction. The exact-real operation model is fixed-dimensional and unit-cost, with arithmetic, comparison, singular-value, polar-decomposition, and fixed-degree root-isolation primitives counted as the operations certified in Definition 48 and Proposition 4. This follows the econometric logic of confidence sets obtained by inverting sample restrictions (Robins et al., 2006), with the reporting metric adapted to discrete laws as in Wasserstein approaches to mixture uncertainty (Deo et al., 2023).

Algorithm 3 defines the exact repaired-image summary-inversion set ${\mathcal{C}}_{n,\alpha }$ and the lattice-centered outer set ${\mathcal{C}}_{n,\alpha }^{\mathrm{alg}}$. The radius $R_{n,\alpha }$ combines the concentration scale $r_{n,\alpha }$ with the lattice mesh scale, while the atom floor $m_{\star }={\pi }_0$ carries the latent-arm positivity margin into the represented-law class. The lattice-centered set is finitely represented by the empirical lattice center, the transport metric, and fixed finite-dimensional constraints in the exact-real model.

The computational guarantee is an exact-real finite-representation statement. In the fixed-dimensional unit-cost model used in Definition 48 and Proposition 4, the lattice search has polynomial candidate growth and exact evaluation over the stated primitive operations. The constrained representation of ${\mathcal{C}}_{n,\alpha }^{\mathrm{alg}}$ gives finite atom, mass, and transport-plan constraints for membership and endpoint formulations. The exact repaired-image set ${\mathcal{C}}_{n,\alpha }$ is represented through $\overline{F}$ on the compact repaired summary image, and Remark 1 records the polynomial-time exact-image problem for that original repaired set.

The cluster report has a sample-facing layer and a theorem-side validation layer. The sample-facing output is formed from the observed ${\hat{S}}_n$, the supplied constants, the lattice center ${\hat{\lambda }}_n$, and the constrained class ${\mathcal{C}}_{n,\alpha }^{\mathrm{alg}}$: it merges empirical support points at threshold $4{\rho }_{n,\alpha }$, expands each component by ${\rho }_{n,\alpha }$, and computes mass endpoints over the same constrained class. The true-law symbols $S(P)$, $E_P$, $K_C(P)$, and ${\Delta }_C(P)$ name the concentration event, associated true support block, and external separation scale used to state coverage and width.

Theorem 3 establishes simultaneous coverage for both confidence constructions over the uniformly conditioned model class. The statement also gives samplewise nonemptiness and $W_1$-diameter bounds, so the reported law-level uncertainty has root-$n$ scale under the same constants that control the estimator in Proposition 4.

The mechanism is the combination of summary concentration and the gap-free modulus. On the event that ${\hat{S}}_n$ lies within $r_{n,\alpha }$ of $S(P)$, the feasible-summary set includes a summary whose image is ${\nu }_P$, and the structured-lattice center lies within the radius used to form ${\mathcal{C}}_{n,\alpha }^{\mathrm{alg}}$. The transport representation in the theorem gives a finite-dimensional constrained problem over atom locations, masses, and couplings.

Law-level confidence sets can be difficult to read directly when nearby support points are statistically unresolved. The next construction converts the $W_1$ set into a component report: estimated support points are connected at threshold $4{\rho }_{n,\alpha }$, and the mass attached to each empirical component is summarized by an interval over the same constrained confidence class.

The report ${\mathfrak{R}}_{n,\alpha }$ replaces individual estimated atoms by empirical components formed at connection threshold $4{\rho }_{n,\alpha }$. For each component $C$, the support interval expands the component by ${\rho }_{n,\alpha }$, and the mass interval $I_C$ ranges over all laws in ${\mathcal{C}}_{n,\alpha }^{\mathrm{alg}}$ whose atoms can be transported to the lattice center within $R_{n,\alpha }$. The external gap ${\Delta }_C(P)$ is the separation scale relevant to the component’s aggregate mass.

Theorem 4 establishes the coverage and width properties of the component report on the concentration event. Every true support point is assigned to a unique empirical component, the reported support interval contains the associated true atom, and the reported mass interval contains the true aggregate component mass. When a component is separated from the remaining true support by ${\Delta }_C(P)$, the mass-interval length contracts at the displayed inverse-gap scale.

The report is adaptive because transport cost controls how much mass can cross a gap. If estimated support points connected to nearby true atoms fall within the $4{\rho }_{n,\alpha }$ component threshold, the construction pools them and reports aggregate mass. When a component is externally separated, moving mass across that separation consumes $W_1$ budget, and the interval width reflects the ratio between the available radius and the external gap.

The final object in this section returns to effect-ordered latent weights. The quotient law treats coincident class-average effects as a single atom, while the ordered-weight target attaches masses to effect ranks on a gap-local stratum. The estimator below applies the repaired law construction and then orders the labelled representative when all labelled atoms are positive and distinct.

Algorithm 5 defines ${\hat{p}}_n^{\uparrow }$ as a simplex-valued ordered-weight rule derived from the repaired quotient-law estimator. Its target is $p^{\uparrow }(P)$ from Definition 30, so the relevant population domain is the gap-local stratum $\mathcal{M}(g)$ in Definition 41.

Theorem 5 gives the ordered-weight risk bound on the separated gap-local class. The rate is clipped at one and improves as the product $\sqrt{n}g$ grows, capturing the scale at which effect ordering becomes informative for latent-stratum masses.

The contrast with the preceding law results is the statistical role of the gap. For the quotient law ${\nu }_P$, Theorem 3 and Theorem 4 report uncertainty in $W_1$ and in aggregate component masses, with collisions represented by shared atoms. For ordered weights, Theorem 5 works on $\mathcal{M}(g)$, where distinct effects and the gap window make the effect order a stable object at the scale shown in the bound.

Lower bounds and sharp rates

Rate matching is proved on the explicit uniformly conditioned two-class specialization $k=d_x=d_z=2$, $L=2$, ${\pi }_0=1/10$, and ${\sigma }_0=1/10$. The construction uses two related two-class submodels. The first moves a collision witness away from a single class-average effect value at $n^{-1/2}$ scale; the second changes ordered latent masses along a factorization-preserving path whose observed separation is proportional to the product of the effect gap and the weight displacement. The lower bounds combine the displayed local Kullback–Leibler neighborhoods with the standard two-point testing logic of Cam (1986) and Polyanskiy et al. (2019).

The constant $c_{\mathrm{loc}}$ fixes the radius of the local experiments used below. Keeping this radius bounded away from one gives a common testing scale for the two-point comparisons.

The divergence $D_{\mathrm{KL}}$ is used only for the observed one-record margins that generate the sample laws $Q_P^{(n)}$. Thus the lower bounds compare statistical experiments through the same observed records as the estimators and confidence reports.

The local neighborhoods center the two testing problems at the relevant collision or separated path endpoint.

The quotient-law lower bound applies the Le Cam local testing framework of Cam (1986) and Polyanskiy et al. (2019) to Assumption 13. It keeps the observed experiment within $n^{-1}$ divergence of the collision witness, so an $n$-sample comparison has bounded information distance.

The ordered-weight lower bound applies the Le Cam local testing framework of Cam (1986) and Polyanskiy et al. (2019) to Assumption 14. The center has effect separation indexed by $g$, matching the gap-local target used in Definition 41. The next display repeats the same divergence notation for the local-experiment statements that follow.

The witness laws are finite Bernoulli proxy models with explicit masses, treatment probabilities, proxy features, and potential-outcome means.

Proposition 5 places the collision witness inside the same uniformly conditioned class used for the upper bounds. The determinants and arm probabilities show that the testing pair is an interior proxy model under the displayed margins, while the quotient law moves from a single atom at $\epsilon =0$ to two nearby atoms for positive $\epsilon$.

For labeled weights, the path changes latent masses while preserving the proxy factorizations that determine the observable $Z{X}^{\top }$ moments.

The family $P_{g,h}$ supplies the local perturbation for the ordered-weight converse. At $h=0$ it coincides with the separated witness at displacement $g/2$, and changing $h$ changes the ordered masses while retaining the displayed armwise proxy factorizations.

The two-class witness also gives a compact picture of the reporting geometry. In this specialization the class masses are $0.4$ and $0.6$, and the class-average effects are $0.25-\epsilon$ and $0.25+\epsilon$. With association radius ${\rho }_{n,\alpha }$, the empirical graph in Algorithm 4 connects estimated atoms at threshold $4{\rho }_{n,\alpha }$; the support interval then expands each empirical component by ${\rho }_{n,\alpha }$.

$$\begin{array}{cccc}\text{witness regime}& \text{quotient law}& \text{component rule at radius }{\rho }_{n,\alpha }& \text{reported mass target}\\ \epsilon =0& {\delta }_{0.25}& \text{one component around }0.25& 1\\ 0<2\epsilon \le 4{\rho }_{n,\alpha }& 0.4{\delta }_{0.25-\epsilon }+0.6{\delta }_{0.25+\epsilon }& \text{connected empirical component}& 0.4+0.6\\ 2\epsilon >4{\rho }_{n,\alpha }& 0.4{\delta }_{0.25-\epsilon }+0.6{\delta }_{0.25+\epsilon }& \text{two empirical components when the center is separated}& 0.4,0.6\end{array}$$

The table serves as a reading guide for the existing witness. It illustrates how quotient aggregation and component reporting use the same masses while the merge threshold determines whether the report presents one aggregate interval or two separated component intervals.

The two local experiments distinguish the metrics under study. ${\mathcal{L}}_n^{\nu }$ is centered at a collision and measures the law-level $W_1$ difficulty, whereas ${\mathcal{L}}_{n,g}^p$ is centered on the gap-local stratum and measures the difficulty of recovering effect-ordered masses.

It is worth tracing the collision through the witness once, since every object in the paper is visible on it. The latent effects of $P_{\epsilon }^{\mathrm{wit}}$ are read off the displayed conditional means as ${\tau }_1=0.25-\epsilon$ and ${\tau }_2=0.25+\epsilon$, carried by the class masses $0.4$ and $0.6$, so the effect gap is $2\epsilon$ and the quotient law is $${\nu }_{P_{\epsilon }^{\mathrm{wit}}}=0.4{\delta }_{0.25-\epsilon }+0.6{\delta }_{0.25+\epsilon }(\epsilon >0),{\nu }_{P_0^{\mathrm{wit}}}={\delta }_{0.25}.$$

Transporting the mass $0.4$ and the mass $0.6$ each a distance $\epsilon$ onto the common location gives $W_1({\nu }_{P_{\epsilon }^{\mathrm{wit}}},{\nu }_{P_0^{\mathrm{wit}}})=\epsilon$, so the estimand moves continuously into the collision and the modulus of Theorem 1 applies across the whole family at fixed conditioning constants. The two regimes are distinguished by which report they support. The quotient law and its $W_1$ confidence set of Algorithm 3 remain meaningful at $\epsilon =0$, where the two classes have become one atom of mass $1$. The effect-ordered masses $(0.4,0.6)$ are separated only while $2\epsilon$ is positive, which is why Theorem 5 is stated on the gap-local strata and why its converse in Theorem 6 is driven by displacement along this same family. The cluster report of Algorithm 4 interpolates between the two pictures: while the estimated support points fall within the connection threshold $4{\rho }_{n,\alpha }$ they are returned as one component whose aggregate mass interval covers the combined mass, and once the separation exceeds that threshold they are reported as two components whose mass intervals sharpen with the external separation, in the manner quantified by Theorem 4.

Theorem 6 gives the two local converse rates on the displayed two-class specialization. For quotient laws, an $n^{-1/2}$ displacement in the collision witness creates $W_1$ separation of order $n^{-1/2}$ while keeping the observed Kullback–Leibler comparison at order $n^{-1}$. For ordered weights, the factorization-preserving path makes the observed divergence scale as $g^2{h}^2$, so the largest indistinguishable mass displacement is clipped at $\min \{1,(\sqrt{n}g{)}^{-1}\}$.

These local statements calibrate the upper bounds from the preceding sections on the same explicit two-class specialization. The quotient-law lower bound matches the root-$n$ law rate from Theorem 2; the ordered-weight lower bound matches the inverse-gap upper bound in Theorem 5.

Proposition 6 converts the local quotient-law comparison into a same-class minimax statement on the displayed uniformly conditioned two-class model. The proposition states that the optimal uniform $W_1$ risk for the quotient law is of exact root-$n$ order on this specialization.

Proposition 7 gives the corresponding same-class minimax statement for ordered weights on the same two-class specialization. The rate is governed by the product $\sqrt{n}g$: when the effect gap is large relative to sampling noise, ordered masses are estimable at the inverse-gap scale, and the bound remains clipped by the diameter of the simplex-valued loss.

The final comparison records how the same witnesses transfer to classes formulated in the terminology of Virk et al. (2026). The following definitions separate the published structural gates, finite-regularity condition, and separated recovery regime used for that transfer.

Definition 52, Definition 59, Definition 60, and Definition 61 provide the interface for comparing the present lower bounds with the published VMW recovery framework. The cited model conditions and separated recovery regime are used as external publication-level gates, while the converse below is stated in terms of exact containment of the two displayed witness pairs.

The calibrated displacement in Definition 58 is the path amplitude used in the labeled-weight two-point comparison. It is the same clipped scale that appears in the ordered-weight upper and lower bounds.

Theorem 7 states the converse comparison with Virk et al. (2026). Any comparator class satisfying the cited VMW model and finite-regularity gates inherits the displayed lower bound once it contains the two quotient witness laws or the two labeled-path endpoint laws. The transfer is therefore tied to concrete witness containment: the quotient comparison uses the collision pair, while the labeled-weight comparison uses separated endpoint laws at the calibrated displacement.

Discussion, extensions, and open questions

The results above identify the quotient law as the regular object for proxy-based latent-class mean treatment-effect heterogeneity. Theorem 1 characterizes a Lipschitz map from observable moment summaries to the aggregated latent-class mean effect law, and Theorem 2 transfers this population regularity to collision-uniform root-$n$ estimation in the observed i.i.d. experiment. The estimand in Definition 40 is therefore the natural law-level target when the empirical question concerns population mass at class-average effect values, with class labels handled through the separated ordered-weight results.

The confidence constructions in Algorithm 3 give two complementary ways to report this target. The summary-inversion set ${\mathcal{C}}_{n,\alpha }$ preserves the repaired feasible-image construction induced by $\overline{F}$ on $\mathcal{K}$, while the algorithmic set ${\mathcal{C}}_{n,\alpha }^{\mathrm{alg}}$ is centered at the structured-lattice estimator ${\hat{\lambda }}_n$ and has an explicit finite transport-plan representation in the fixed-dimensional exact-real model. Theorem 3 establishes simultaneous coverage for both sets and root-$n$ diameter bounds in $W_1$, so the two reports express the same law-level concentration through different computational representations.

The cluster report in Algorithm 4 translates this law-level uncertainty into component-level statements at the resolution delivered by the confidence radius. Estimated atoms are connected at threshold $4{\rho }_{n,\alpha }$, while support association and interval expansion use ${\rho }_{n,\alpha }$; the report then pools connected empirical components and attaches both a support interval and a compatible mass interval. Theorem 4 establishes that, on the simultaneous concentration event, these components partition the true support, contain the associated true atoms, and cover the corresponding aggregate masses. The interval-length bounds show how the report becomes sharper when a component is externally separated from the remaining support, with exact mass recovery for a component whose associated block exhausts the true support.

This reporting layer also clarifies the role of the ordered-weight results. Theorem 5, Theorem 6, and Proposition 7 characterize effect-ordered mass recovery on separated strata, where the positive gap scale converts law-level transport error into label-sensitive information. The quotient-law statements remain organized around $W_1$ on aggregated laws, while the ordered-weight statements quantify the additional inverse-gap cost of resolving labels by effect order.

Open questions

The computational distinction between the two confidence constructions leaves a sharp finite-dimensional problem. The structured-lattice estimator and the outer $W_1$ set have explicit constrained representations, and the original summary-inversion set retains the exact repaired-image target. The following remark records the corresponding exact-image question.

The question in Remark 1 concerns exact computation of the sharp repaired-image set and its support-dependent extrema. A positive resolution would align the original summary-inversion report with the same kind of explicit fixed-dimensional representation already available for the lattice-centered outer report.

A second extension is adaptation to latent cardinality and unknown conditioning margins. The present model class in Definition 32 fixes $k$, and the target space, structured lattice, confidence sets, and ordered-weight stratum are all calibrated to that fixed finite value together with supplied $L,{\pi }_0,{\sigma }_0$. An adaptive formulation would need a data-dependent dimension choice or margin choice together with a target convention for comparing quotient laws across candidate values of $k$, while preserving the aggregation principle that makes ${\nu }_P$ stable at collisions.

Margin misspecification belongs to the same adaptation problem. The stated guarantees attach to the supplied model class: conservative supplied constants that still satisfy the defining inequalities yield valid conclusions after recomputing the displayed constants, whereas supplied constants violated by the data-generating law place that law outside the certified class. A diagnostic theory for selecting or stress-testing $k,L,{\pi }_0,{\sigma }_0$ is a natural extension of the fixed-margin results.

The exact-real finite representations also leave ordinary computational questions for implementation work. The operation counts in Definition 48, Proposition 4, and Theorem 8 use fixed-dimensional unit-cost primitives, and the endpoint descriptions in Algorithm 4 are finite constrained programs over masses, support locations, and transport plans. Bit complexity for these endpoint programs, numerical conditioning of the singular-value and root-isolation primitives, and practical solvers for the reported intervals are open implementation questions beyond the exact-real certification.

A third direction concerns application-specific proxy mappings. Assumptions 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 give a finite proxy-causal environment in which the quotient-law and reporting results apply. In applied settings, the substantive task is to justify how observed proxy measurements instantiate these conditional-independence, normalization, boundedness, overlap, and rank conditions, following the model-building role that proxy variables play in the broader causal literature (Virk et al., 2026; van Amsterdam et al., 2022).

Appendices

Probability, selection, and concentration

This appendix collects the auxiliary statements used by the population, estimation, confidence, and lower-bound arguments. The first group supplies measurable selection and finite-sample probability control for the observable summary space introduced in Definition 33. The compact-selector statement is the measurable nearest-point device used by the repair map in Algorithm 1; it is the form of measurable extrema needed for a compact feasible summary set, in the spirit of Brown et al. (1973).

The selection lemma turns compactness of $\mathcal{K}$ into a Borel rule, so the nearest-summary construction remains a sample-measurable estimator after applying the empirical summary map. The remaining probability statements in this subsection record the arm-mass and concentration primitives that support the total empty-arm convention.

The lower bound converts the latent-arm margin into an observed-arm margin. It is the elementary probability input behind the empty-arm probability bound in Lemma 1 and behind the observed VMW margin statement in Proposition 1.

The next statements collect the moment factorizations and envelope calculations that connect the full-data restrictions to the observable five-block summary.

Together these factorization and envelope lemmas give the deterministic bridge from the conditional-independence restrictions to the matrices used in the compressed operator. The latent mean and effect envelopes place every quotient law in the common support interval fixed by Definition 26.

The following algebraic facts are used to pass from factored proxy moments to the compressed spectral representation.

These linear-algebra statements justify the rank thresholding and anchored diagonalization used by the lattice estimator. The condition-number bound provides an explicit finite-dimensional constant for perturbing between feature coordinates and the compressed spectral coordinates.

The Wasserstein comparisons use a one-dimensional dual potential and the coordinate relation between the Euclidean summary metric and $d_S$.

The potential lemma is the dual device behind the $W_1$ modulus for represented atomic laws. The two summary-metric lemmas place the finite-dimensional summary topology and the selection loss on a common continuous footing.

The next group records the witness and path bookkeeping used in the lower-bound section. The statements give explicit masses, effects, observed arm probabilities, determinant margins, model membership, and local-experiment membership for the two-class laws.

These witness calculations locate the explicit two-class comparisons inside the same model restrictions used for the upper bounds. The path-local statement calibrates the observed Kullback–Leibler radius to the product $g^2{h}^2$, which is the scale used by Theorem 6.

The final two auxiliary statements concern ordered masses as functions of represented atomic laws. They ensure that, when the support is separated, the ordered-weight target depends on the aggregate measure and is invariant across labelled coordinate representations.

The definition matches the ordering convention in Definition 30 and Algorithm 5. The invariance lemma records the compatibility needed to read ordered masses from any valid representative of the same aggregated atomic law.

Spectral repair and finite-net benchmark

This appendix collects two auxiliary constructions used to connect the population modulus with sample-level procedures. The first is a spectral repair handle that turns empirical spectral information into a positive real atomic law even when eigenvalues are locally coalescent. The second is a class-advised finite-net benchmark that selects a stored admissible summary and then runs the exact spectral construction at that representative. Both constructions use finite-dimensional perturbation ideas familiar from matrix spectral analysis (Bauer et al., 1960; Kato, 1995; Davis et al., 1970; Stewart et al., 1990) and the moment-recovery perspective behind Wasserstein estimation from moments (Wu et al., 2020).

The repair handle records the finite moment target and the positive-law replacement used in the modulus-to-estimation route.

Algorithm 6 isolates the spectral information needed for positive real-law repair. The moment vector $m$ contains the first $2k$ empirical spectral moments, and the minimizing positive atomic law supplies a real probability measure compatible with those moments. The contour-projector clause expresses the standard perturbation treatment of clustered spectra: inside overlapping localization discs, aggregate eigenspaces carry the stable law-level mass.

The finite-net benchmark makes this repair logic an exact-real finite search after a class-dependent representative library has been fixed.

Algorithm 7 defines the library selector at the level of stored summaries. The criterion is the same five-block $d_S$-type comparison used throughout the paper, and lexicographic tie-breaking makes the selected representative a total deterministic function of the sample and the library.

The estimator below specifies how the representative library is built from an ${ϵ}_n$-net of admissible summaries and how the selected representative is converted into an atomic law.

The estimator ${\hat{\nu }}_n^{\mathrm{net}}$ is class-advised through the choice of admissible representatives $q_{n,\ell }$. Once the nearest representative is selected, the construction returns to the compressed-operator formula from Definition 39: thresholding fixes the rank-$k$ signal space, and the spectral projectors aggregate the anchor weights attached to each distinct eigenvalue.

Theorem 8 establishes the finite-net benchmark. For fixed dimensions and fixed conditioning constants, the representative library has polynomial size in $n$, the online exact-real operation count is polynomial in the same sense, and the selected summary is the lexicographically first nearest library representative. The deterministic $W_1$ inequality converts empirical summary error plus the $n^{-1/2}$ mesh scale into quotient-law error, and the final tail bound applies the same concentration scale used for the main law estimator.

The intuition is that the admissible summary image is compact and finite-dimensional, so an $n^{-1/2}$-mesh supplies a representative close enough for the gap-free modulus to absorb the discretization error. At the chosen representative, the proxy-rank margin makes singular-value thresholding recover the correct $k$-dimensional signal space under perturbations below ${\pi }_0{\sigma }_0^2/2$. The exact spectral run then returns the quotient law for a model law with that stored summary, and the modulus transfers the distance from the empirical summary to $W_1$ distance between effect laws.

Proofs and verification note

This appendix collects the proof layer for the paper’s formal results. The arguments establish the population stability statement in Theorem 1, the measurable repair and estimation claims in Proposition 3, Proposition 4, and Theorem 2, the confidence and reporting guarantees in Theorems 3, 4, and 5, and the local converse statements in Theorem 6, Proposition 6, Proposition 7, and Theorem 7.

The proof of Theorem 1 uses the compact summary closure from Proposition 2 and the quotient moment representation in Definition 43. The finite-dimensional moment map gives a Lipschitz modulus from the observable summary coordinates to the $W_1$ law target on the compact domain. The extension clause follows by completing the admissible image under $d_S$, with Algorithm 1 and Proposition 3 supplying the measurable nearest-summary selection needed to evaluate the extended map at empirical summaries.

The estimation proofs combine this deterministic modulus with concentration for the empirical five-block summary in Definition 35. For the repaired estimator, nearest-summary projection keeps the evaluated summary within the same concentration scale as the population summary. For the structured-lattice estimator, Algorithm 2 and Proposition 4 give a duplicate-free finite search whose deterministic error is bounded by the empirical summary error and the $n^{-1/2}$ lattice scale. These two routes yield the joint root-$n$ tail statement in Theorem 2.

The confidence-set proofs use the same concentration event with the objects in Algorithm 3. The summary-inversion set inherits its diameter from the modulus on $\mathcal{K}$, while the algorithmic outer set inherits its radius from the deterministic lattice bound. The transport-plan formulation in Definition 2 gives the finite representation used in Theorem 3. The cluster-report argument then applies the atom floor and $W_1$ transport budget in Algorithm 4 to associate true atoms with empirical components, yielding the support containment, mass-coverage, and interval-length conclusions in Theorem 4.

For ordered weights, the proof of Theorem 5 works on the gap-local stratum in Definition 41. The positive effect gap converts law-level transport error into an ${\ell }_1$ error bound for the effect-ordered mass vector defined in Definition 30 and estimated by Algorithm 5. This gives the clipped inverse-gap upper rate stated in the theorem.

The lower-bound proofs use the explicit two-class witness in Definition 55, its validity certificate in Proposition 5, and the local experiments in Definitions 56 and 57. Standard two-point testing inequalities (Cam, 1986; Polyanskiy et al., 2019) convert the displayed Kullback–Leibler bounds and target separations into the local risk lower bounds in Theorem 6. The same comparisons give the same-class minimax rates in Propositions 6 and 7. The transfer statement in Theorem 7 expresses the same witness-pair logic for comparator classes stated in the published VMW vocabulary (Virk et al., 2026).

Verification note.

The Lean-checked scope consists of the matched theorem, proposition, and lemma declarations listed in the verification manifest for commit $$\text{715add8e9475f5193526bce0c26baef559a47b68}.$$ Definitions marked presentation-synthesized in that manifest are expository renderings of paper notation; definitions marked matched have the Lean declarations recorded by the manifest. The proof text in this appendix follows the matched declarations and preserves the % lean: audit markers that identify certified proof steps.

The Lean toolchain is leanprover/lean4:v4.33.0. The CausalSmith package is version 0.1.0; its manifest records the path dependency Causalean at the same repository commit, inherited path dependencies optlib at ../third_party/optlib and FoML at ../third_party/lean-rademacher, and the following external revisions: mathlib db584cd6d46c92f209a44c0f1c829460d327499d, plausible b7eb3304aeae834b12dda98993a37f6a41f6f0bb, LeanSearchClient 5f4d51b81cbd3f6b32b156bfad9056621a040404, importGraph 16f02aa7642864af59f1ff0e384a015994db9118, proofwidgets 4be2e3d5087eeb272cf5a8853b8f9dd025ef5957, aesop 3448c0bcc5ce01b2d1546e483ec3620e32df3d0e, Qq 92c15be17b7caf78c2ad767ec40f89052d908d81, batteries 4488d40d070b9700d4d5a6aa342f0d40c31b2a2d, checkdecls 3d425859e73fcfbef85b9638c2a91708ef4a22d4, and Cli 6130a47896ce867c6a4a55373441e59e565bad0f. The verification build command for the paper module is $$\text{lake -d CausalSmith build CausalSmith.Stat.STAT\_ProxyEffectlawEigencollisionFrontier\_Research}.$$

Proposition 1 uses the VMW model-scope claim from (Virk et al., 2026). Theorem 7 uses that model-scope claim together with the VMW separated-recovery scope claim from (Virk et al., 2026). The remaining matched declarations listed below are derived from their displayed hypotheses and previously matched declarations.

Reader-facing sections use one-based singular-value notation. Lean-facing auxiliary displays that expose array indices translate the $k$th signal singular value into the corresponding zero-based index inside the verification layer.

Proofs of the main results

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