Introduction

Transporting an instrumental-variable effect requires learning both the population weights and the encouragement contrasts that identify the effect among target-population compliers. When transported compliance is small, uncertainty in those contrasts is amplified by the compliance denominator. This paper characterizes that interaction through the optimal expected length of confidence intervals with finite-sample coverage uniformly over a specified continuous-covariate model.

Our experiment combines two independent samples of equal size: source records containing covariates, encouragement, receipt and outcomes, and target records containing covariates. The covariate is scalar and lies in the unit interval. Source and target densities, the source encouragement propensity, and outcome and receipt arm means belong to fixed Hölder classes with exponent $1/8$. Density envelopes are fixed and positive, encouragement probabilities lie between $1/4$ and $3/4$, and potential outcomes lie in $[0,1]$. Conditional instrument randomization, consistency, exclusion and monotonicity connect observed source contrasts to complier quantities. Two transport equalities, holding for target-almost every covariate value, carry the conditional complier share and the conditional complier-weighted outcome effect across populations. Under these restrictions, Lemma 1 identifies the target complier average treatment effect as the ratio of target-averaged source outcome and receipt contrasts.

The main result concerns intervals whose coverage is at least $1-\alpha$ at every law in this model, where $\alpha \in (0,1)$ is the fixed noncoverage probability. Precision is evaluated on laws whose transported first stage lies between $a$, the evaluation floor, and $1/4$. Thus global coverage and strength-specific expected length are distinct requirements. For fixed density envelopes and Hölder radius, Theorem 1 establishes $$L_n(a)\asymp \min \{1,n^{-1/3}/a\},n\ge 256,0<a\le 1/4,$$ where $L_n(a)$ is the minimax worst-case expected length among globally honest connected intervals. The comparison constants depend only on the fixed regularity parameters and noncoverage probability. The bounds are uniform across the stated strength range and contain no leading logarithmic factor.

A single observable interval sequence attains this frontier. The construction expresses the transported contrasts as smooth rational functionals of seven observable marked densities. A clipped histogram pilot, followed by linear, quadratic and cubic corrections using separate evaluation blocks, yields the uniform mean-square bound in Lemma 3. Inverting an affine score with a deterministic radius calibrated from that bound gives the interval in Definition 9. Its coverage holds for every positive sample size, and its expected length is bounded by a constant times the rough estimation scale divided by the actual transported first stage, capped at constant order. The evaluation floor enters the performance criterion; the same procedure serves every strength slice.

The converse preserves the causal structure of the problem. The construction in Algorithm 2 and Lemma 4 supplies admissible observed probability laws together with compatible potential receipts and outcomes satisfying monotonicity and the transport restrictions. Its actual compliance share is the larger of the evaluation floor and the rough estimation scale. A continuum of common effect values across sign mixtures, combined with observational proximity to a center experiment, produces the matching expected-length lower bound. Theorem 3 applies to arbitrary measurable confidence sets contained in the bounded effect domain. The constant-order branch therefore reflects the information available under global coverage when the strength floor is at or below $n^{-1/3}$.

This characterization joins three established lines of work. Transported complier-effect identification and regular inference are developed by Chen et al. (2025), while Clark et al. (2024) establish regular inference for transported principal effects under their identification and nuisance-rate conditions. Rough transport-functional estimation has a direct antecedent in Zeng et al. (2025); their lower-bound exponent specializes to one third under the stated one-dimensional smoothness and design conditions. Higher-order correction follows the functional-estimation approach of Robins et al. (2008). Identification-robust score inversion is developed for LATE by Ma (2026), and its confidence-set geometry is studied by Smucler et al. (2026). The immediate expected-length benchmark is CausalSmith (2026), which treats supplied geometry and specified cell-based weight-learning settings. The present result connects rough functional estimation and score inversion to a sharp expected-length criterion with fully learned continuous geometry, equal sample sizes and uniform coverage under the declared causal model.

The analysis covers scalar covariates, equal source and target sample sizes, fixed Hölder exponent (1/8), and the causal and transport restrictions stated in Section 3. Section 2 gives the closest methodological comparisons; Section 3 specifies the observed experiment and inference criterion; and Section 4 states the frontier and its mechanism. Section 5 interprets the calibration and formulates the unequal-sample problem. The technical construction and supporting bounds appear in Section A, the legal lower family in Section B, and the auxiliary comparators and verification scope in Section C. Full proofs of the anchored results are collected in Section D.

Related work

The closest comparisons concern transported complier effects, estimation of rough transport functionals, and inference under weak identification. Our immediate benchmark is CausalSmith (2026), which characterizes asymptotically honest expected length through first-stage strength and transport-weight dispersion when the covariate and assignment geometry is supplied. Its feasible cell-based procedures learn target weights under uniform source cells and balanced assignment, or under regular nonuniform cells with supplied source probabilities and assignment propensities. Those results assume proportional source and target sample sizes and cell counts growing more slowly than the square root of the source sample size. The present comparison concerns the continuous-covariate model in Definition 2: the densities, encouragement propensity and arm means have Hölder exponent $\beta$, fixed at $1/8$, and the two samples each contain $n$ records.

For identification and regular inference, Chen et al. (2025) provide a particularly close comparison. Their observation scheme combines study records containing covariates, encouragement, receipt and outcomes with target covariates. Theorem 3.1 identifies the target complier average causal effect as a ratio of target-averaged source encouragement contrasts under their sampling overlap, target-population IV validity, and mean exchangeability restrictions for the outcome contrast and first stage. Their regular inference results distinguish supplied and estimated selection probabilities. Theorem 4.2 assumes known selection and assignment models, a fixed source-to-target sample-size ratio, and finite weighted second moments. Theorem 4.3 treats a correctly specified logistic selection model estimated by maximum likelihood, with known assignment probabilities and the stated moment and full-rank conditions. These theorems support delta-method inference at a fixed positive first stage. Our identification uses the transport restrictions on the complier-weighted outcome contrast and complier share in Assumptions 16 and 17, while the inference criterion in Definition 3 requires coverage uniformly over the model class.

A distinct observation scheme underlies Rudolph et al. (2017). They transport outcomes to a site where covariates, encouragement and receipt are observed, so that the target first stage can be estimated directly. Their identification combines a common conditional outcome model across sites with target encouragement exchangeability, exclusion, monotonicity and positivity. Section 5 derives the transported-CACE influence curve and constructs a ratio of targeted estimators, with inference based on the influence curve or the multivariate delta method. Section 7 applies this construction to the Moving to Opportunity experiment. In the covariate-only target scheme studied here, both the outcome and receipt contrasts are obtained by averaging source contrasts over the target covariate distribution, as stated in Lemma 1. This difference determines which transport restrictions identify the denominator as well as the numerator.

Transported principal-effect inference supplies another close comparison. Within their principal-stratification identification framework, Clark et al. (2024) establish asymptotic linearity, normality and efficiency for an influence-function estimator using trial assignment, receipt and outcome data together with target covariates. Their theorem assumes a positive limiting source-to-target sample-size ratio, nuisance estimation on a separate independent sample, uniform positivity of the relevant true and estimated probabilities, bounded outcome regressions, and consistency of the component influence functions. Efficient regular inference further requires each of the three specified products of nuisance estimation errors to be negligible relative to the inverse square root of sample size. These conditions describe a regular estimation regime for transported principal effects. The rough-class analysis here instead calibrates the observable estimator through the uniform mean-square bound in Lemma 3, which supplies the error control used by the score interval.

The reduced-form estimation scale has a direct methodological antecedent in Zeng et al. (2025). Their Theorem 6 gives a minimax lower bound for generalization when source and target covariates coincide, the joint covariate density for source membership and a specified treatment arm is constant, and the compound selection–treatment probability and outcome regression satisfy the stated positivity bounds. Its exponent specializes to one third for one-dimensional covariates and average nuisance smoothness one eighth. Their Theorem 8 gives conditional bias and variance bounds for a quadratic estimator of a transportation functional. That result assumes independent training data, uniform positivity, and uniformly well-conditioned true and estimated projection Gram matrices. Its bias bound includes both a product of projection approximation errors and a remainder multiplying two nuisance errors by the inverse-Gram estimation error. Thus, the exponent and the use of higher-order corrections are established antecedents, with the design, approximation and remainder conditions forming part of their results.

The cubic construction in Algorithm 1 belongs to the higher-order functional-estimation approach developed by Robins et al. (2008). The marked densities in Definition 6 express each transported contrast as the integral of a smooth rational function, as specified in Definition 7. This representation permits a clipped pilot followed by observable linear, quadratic and cubic corrections computed from separate sample blocks. The construction specializes higher-order bias correction to the joint source–target observation scheme and provides the uniform mean-square control needed for finite-sample calibration.

Score inversion connects that estimation problem to identification-robust inference. Ma (2026) establish uniformly correct asymptotic size for an orthogonalized test of a LATE moment under their high-dimensional nuisance, sparsity, overlap, moment and empirical regularity conditions. The result accommodates weak identification within that class. Our score interval in Definition 9 uses a finite-sample calibration derived from a uniform mean-square bound over the specified rough class. The shared principle is to test an affine moment in the candidate effect; the calibration and the model over which coverage is required determine the resulting guarantee.

The geometry of this inversion is clarified by Smucler et al. (2026). Their studentized score inversion reduces to a quadratic inequality and can produce bounded or unbounded intervals, disconnected sets, the whole line, an empty set or a singleton. Under their independent nuisance estimation, boundedness, overlap, consistency and product-rate conditions, Theorem 1 shows agreement with the doubly robust Wald interval up to stochastic endpoint errors of order $O_P(n^{-1})$, with probability tending to one, at a fixed law with a nonzero first stage. Theorem 2 gives a nonnormal ratio limit under their local-to-zero moment and nuisance-negligibility conditions. In our construction, a deterministic calibration radius and the bounded causal domain yield the connected acceptance interval and fallback rules specified in Definition 9. These comparisons locate the inversion principle within established weak-identification methodology.

Two older principles organize the criterion used here. Anderson–Rubin inversion constructs confidence sets by testing an affine reduced-form moment without dividing by a possibly weak first stage (Anderson et al., 1949); Fieller’s ratio analysis likewise allows denominator uncertainty to determine confidence-set geometry (Fieller, 1954). The present interval specializes that logic to the bounded causal domain and replaces an asymptotic critical value with the finite-sample rough-functional radius in Definition 9. On the decision-theoretic side, weak identification can force unbounded confidence sets on an unrestricted parameter space (Dufour, 1997), while honest expected length over bounded functional classes is governed by between-class modulus arguments (Cai et al., 2004). Here bounded outcomes cap length at constant order, and the legal transported-IV family supplies the testing comparison that yields the second branch of the frontier.

The contribution is the joint equal-size, rough-geometry, uniform-coverage and expected-length characterization in Theorem 1. Its upper bound uses an observable cubic estimator and a calibrated score interval; its lower bound is realized by the primitive IV family in Algorithm 2 and Lemma 4. That family retains the causal and transport restrictions of the model while producing the observed experiments needed for the length comparison. Together, these results connect the established rough-functional exponent and score-inversion principle to a sharp expected-length criterion for transported complier effects with unknown continuous covariate and assignment geometry.

Model, transport restrictions and inference criterion

Throughout, sample sizes are positive integers, and all random variables and conditional expectations are measurable. We use $\Vert \cdot {\Vert }_{\infty }$ for the supremum norm and $|\cdot |$ for absolute value. Generic positive constants may depend on the fixed density envelopes, smoothness radius and coverage level; their values may change between occurrences. The notation $u_n\lesssim v_n$ denotes an inequality up to such a constant, uniformly over the stated model class and strength range, and $u_n\asymp v_n$ denotes inequalities in both directions.

Primitive variables and the observed experiment

The scalar covariate domain is $\mathcal{X}$=[0,1]. Population membership is coded by $S=1$ for the source and $S=0$ for the target. A primitive full-data record under the full-data law $P_n^F$ is $$O^F=(S,X,D(0),D(1),Y(0),Y(1)),$$ where $D(0),D(1)\in \{0,1\}$ are potential receipts under encouragement, $Y(0),Y(1)\in [0,1]$ are potential outcomes under receipt, and $C=1\{D(1)>D(0)\}$ is the complier indicator. The target conditional effect among compliers is represented by $$\theta =\frac{{\mathbb{E}}_{P_n^F}[(Y(1)-Y(0))C\mid S=0]}{{\mathbb{E}}_{P_n^F}[C\mid S=0]},$$ whenever the denominator is positive.

The source assignment law augments a source full-data draw with encouragement $Z$, using propensity $e$, and consistency induces the observed receipt $D=D(Z)$ and outcome $Y=Y(D)$. Its observed record is $O^{\mathrm{S}}=(X,Z,D,Y)$ with one-record law $P_{\mathrm{S}}$. The target observation is only $X^{\mathrm{T}}=X$, with one-record law $P_{\mathrm{T}}$. Thus $P_n^F$ determines the causal variables, the source assignment mechanism determines the observed source law, and the target-population covariate marginal determines $P_{\mathrm{T}}$. The samples ${\mathcal{S}}_n$ and ${\mathcal{T}}_n$ contain $n$ records from these two observed laws. Their densities and conditional arm means receive the regularity restrictions below.

We first impose independent sampling within each population and independence between the two samples.

Independent source sampling makes the source records repeated observations from a common population law, as in standard two-sample transport designs (Zeng et al., 2025). The corresponding target condition fixes the population represented by the covariate sample.

Independent target sampling allows the target covariate distribution to be learned from repeated draws under the same two-sample framework (Zeng et al., 2025). We complete the sampling design by requiring the two collections of records to be independent.

Source–target sample independence describes separately drawn samples and gives the joint observed law a product structure (Zeng et al., 2025).

Density, overlap and smoothness restrictions

The regularity conditions use fixed envelopes and a common Hölder convention. We define these quantities together so that density, propensity and regression restrictions share the same scale.

The lower density envelope $c_f$ and upper density envelope $C_f$ control covariate support across populations. The common radius $L$ bounds both components of the Hölder ball ${\mathcal{H}}^{\beta }([0,1],L)$. At exponent $\beta =1/8$, the class permits rough continuous nuisance functions. We first impose density bounds for the source and target designs.

A uniformly positive bounded source design density supplies source observations throughout the covariate domain, a standard support condition for transport analysis (Zeng et al., 2025). The full-data support clause also fixes the scale of the potential outcomes. The target density obeys the same envelopes.

Uniformly positive bounded target density gives both populations the common support used in transport comparisons (Zeng et al., 2025). Together, the two density envelopes bound the target-to-source density ratio. We next control variation within that support.

The one-dimensional isotropic Hölder ball controls the source density’s local variation while allowing low regularity, the smoothness framework used in higher-order functional estimation (RobinsLiMukherjeeTchetgenVanDerVaart2017HigherOrder, 2017). The target density receives the same exponent and radius.

Target density Hölder regularity places the target design in the same one-dimensional smoothness class as the source density, giving common control over the covariate distributions entering our transported functional. The related transport rate comparison specifies smoothness of outcome regressions and inverse selection–treatment weights (Zeng et al., 2025). For source encouragement, we impose positivity before its smoothness restriction.

Source instrument positivity ensures that both encouragement arms are represented at every covariate value (Chen et al., 2025). Its fixed bounds concern assignment probabilities, whereas compliance strength will be governed by a separate condition. The propensity also satisfies the common Hölder restriction.

A Hölder-smooth propensity controls how assignment probabilities vary with covariates in our model. The related transport rate comparison imposes smoothness on an inverse selection–treatment weight and an outcome regression (Zeng et al., 2025). The remaining observed nuisance functions are the receipt and outcome regressions within each encouragement arm.

Hölder-smooth observed arm regressions control the covariate variation of the conditional receipt and outcome means (Zeng et al., 2025). Their range reflects the bounded responses, and the arm-moment identities tie the chosen regression versions to the observed source law.

Instrument validity and complier transport

The causal restrictions connect observed encouragement-arm contrasts to effects among compliers. We begin with consistency of treatment receipt and outcomes in the source population.

Receipt consistency connects realized treatment receipt to its potential value under the assigned encouragement (Chen et al., 2025). The outcome restriction completes this link between observed records and potential variables.

Outcome consistency identifies the observed response with the potential outcome under realized receipt (Chen et al., 2025); boundedness fixes its scale for the inference problem. We then require conditional randomization of encouragement.

Conditional instrument exchangeability makes source encouragement comparable across potential receipt and outcome types within covariate strata (Chen et al., 2025). This permits a covariate-dependent randomized encouragement design. The next restriction specifies the causal channel through which encouragement affects outcomes.

Instrument exclusion assigns encouragement’s outcome effect to the receipt it induces (Chen et al., 2025). Combined with conditional randomization and consistency, it supplies the outcome interpretation of the source arm contrast. Monotonicity supplies its principal-stratum interpretation.

The no-defiers monotonicity condition makes individual receipt weakly increasing in encouragement (Chen et al., 2025). Under these instrument restrictions, the source receipt contrast represents a conditional complier share, and the source outcome contrast represents a conditional complier-weighted effect; their transported interpretation is given in Lemma 1.

Transport is imposed through two equalities at target-almost-every covariate value. The first carries the complier-weighted outcome contrast across populations.

After aligning encouragement potential outcomes with receipt-mediated outcomes under exclusion and monotonicity, this complier-weighted outcome restriction corresponds to Assumption 2 of Chen et al. (2025). It transports the conditional contribution to the effect numerator, including local complier prevalence. The second equality transports that prevalence itself.

The pointwise complier-share equality is the transport condition chosen for the present model and implies the averaged equality in Assumption 4 of Chen et al. (2025). It is substantively plausible when source participation changes covariate composition but, within the measured covariate strata, does not change how encouragement shifts receipt. Together, the two equalities carry the numerator and denominator of the target complier effect through the target covariate distribution. The contribution developed below concerns estimation over the rough nuisance class and the honest expected-length frontier under these identification conditions.

The source contrast ${\Delta }_A$ and target-averaged contrast $T_A$ separate the source conditional response from target covariate weighting. The target complier share $\mu$ determines the normalization of the target complier effect $\theta$. Because both potential outcomes lie in $[0,1]$, the effect domain $\Theta$ is $[-1,1]$. Positive compliance gives the conditional target effect a well-defined denominator.

Target instrument relevance requires a positive transported complier share (Chen et al., 2025). The model permits this share to approach zero as the sample-size index varies. Under the preceding causal and transport restrictions, Lemma 1 gives $$\mu =T_D,\theta =\frac{T_Y}{T_D}.$$ Transported compliance therefore determines both the population represented by the causal effect and the first stage that scales the observed outcome contrast. The bounded domain gives confidence intervals a common length scale even when that first stage is small.

Global honesty and expected length

We collect the sampling, regularity and causal restrictions into a single law-level class before defining coverage.

A law triple $P$ in the model class ${\mathcal{M}}_n$ specifies both the causal target and the observed two-sample experiment. For probabilities and expectations involving a data-dependent interval, the observed law is $P_{\mathrm{S}}^{\otimes n}\otimes P_{\mathrm{T}}^{\otimes n}$. The following conventions fix the coverage level, the estimator’s sample-size threshold and the measure used to evaluate interval length.

The noncoverage probability $\alpha$ is fixed across the model class. The sample-size threshold $n_0$ governs use of the cubic estimator, and restricted Lebesgue length ${\lambda }_{\Theta }$ measures uncertainty on the causal domain. A generic random interval $C_n$ is a measurable function of the two observed samples: formally, its graph $\{(\omega ,t):t\in C_n(\omega )\}$ is measurable in the product of the observed sample space and the Borel $\sigma$-field on $\Theta$. We require its coverage to hold uniformly over the full model class.

The honest interval class ${\mathfrak{H}}_n$ imposes finite-sample coverage at every law in ${\mathcal{M}}_n$, including laws with arbitrarily small positive transported compliance. Connectedness makes the reported set an interval, and containment in $\Theta$ bounds its length by two. To evaluate how its expected length changes with first-stage strength, we next select a strength slice.

The evaluation floor $a$ indexes the strength slice ${\mathcal{M}}_n(a)$, whose transported first stages lie between $a$ and $1/4$. It enters the laws over which expected length is evaluated. Coverage continues to be required over all of ${\mathcal{M}}_n$, and the attaining interval in Definition 9 uses a single construction across these slices. This ordering separates global validity from the strength-specific evaluation of precision.

We express the resulting optimal expected-length criterion as follows.

The minimax expected length $L_n(a)$ compares globally honest intervals by their worst expected length on a specified strength slice. Its infimum ranges over procedures satisfying the full coverage requirement in Definition 3, while its supremum evaluates precision on Definition 4. This criterion links the roughness of the observed nuisance functions to the uncertainty remaining about a transported complier effect at a given first-stage scale.

Main results and an attaining interval

The minimax expected length has order $$L_n(a)\asymp \min \{1,n^{-1/3}/a\},n\ge n_0,quad0<a\le 1/4.$$ A single observable interval attains the upper bound while maintaining coverage over the full model class, and the lower bound applies to every measurable confidence set, including disconnected ones. This section states the exact interval and frontier. The technical construction and supporting bounds appear in Sections A and B; full proofs are collected in Section D.

Observable representation and cubic correction

The construction collects the observable density components in a seven-coordinate vector. Its source arm densities and response-marked arm densities encode the conditional contrasts without estimating conditional means separately.

Weighting the arm contrast by the target density gives the transported functional.

Under the causal and transport assumptions, $T_D=\mu (P)$ is the transported first stage and $T_Y=\mu (P)\theta (P)$; see Lemma 1. Thus inference can invert the affine moment $T_Y-v{T}_D$ without dividing by an estimated first stage.

For $n\ge n_0=256$, each source and target sample is split into one pilot block and three evaluation blocks. Blockwise marked histograms estimate all seven coordinates. The pilot resolution satisfies $K_0\asymp n^{4/5}$, while the quadratic and cubic correction resolutions satisfy $K_2\asymp n^{4/3}$ and $K_3\asymp n$, up to dyadic rounding. The transported-contrast estimator ${\hat{T}}_A$ expands ${\Phi }_A$ around the clipped pilot and adds linear, quadratic, and cubic terms formed from separate evaluation blocks.

The three correction orders cancel the corresponding Taylor terms while retaining independence among repeated residual factors. At smoothness $\beta =1/8$, the fourth-order remainder and correction variances are controlled by the uniform mean-square envelope constant $C_{\mathrm{mse}}=C_{\mathrm{mse}}(c_f,C_f,L)$. Specifically, $$\underset{P\in {\mathcal{M}}_n}{\sup }{\mathbb{E}}_P\left[({\hat{T}}_A-T_A{)}^2\right]\le C_{\mathrm{mse}}{n}^{-2/3},A\in \{Y,D\}.$$ The exact histogram conventions, estimator, and computable constant are given in Algorithm 1 and Lemma 3.

Score interval and sharp frontier

The mean-square scale yields the following calibrated affine-score acceptance set and interval.

The full-domain fallback supplies coverage below the estimator threshold. For larger samples, the procedure retains a nonempty affine-score acceptance interval and uses the stated singleton convention when the acceptance set is empty. The exact result combines all-positive-sample-size coverage with the two-sided frontier for $n\ge 256$.

The two regimes follow by comparing the score radius with the transported first stage. Above $n^{-1/3}$, affine inversion turns contrast error into effect uncertainty of order $n^{-1/3}/\mu (P)$. At or below that scale, the bounded effect domain caps length at constant order. The legal instrumental-variable mixture in Lemma 4 produces matching separations in both regimes. The supporting upper and lower statements are Theorems 2 and 3.

Discussion and open questions

The expected-length bounds in Theorem 1 provide precision benchmarks for the declared source–target experiment: independent samples with $n$ source records and $n$ target covariate records, under the restrictions in Definition 2. For $n\ge n_0$, fixed calibration and regularity parameters, and an evaluation floor $0<a\le 1/4$, the two regimes describe the interaction between transported compliance and the rough estimation scale. When $a>n^{-1/3}$, the optimal worst-case expected length has order $n^{-1/3}/a$. When $a\le n^{-1/3}$, it has constant order. Thus increasing the common sample size improves the precision benchmark once the strength floor exceeds the estimation scale, while global honesty requires positive worst-case expected length across the weaker-strength slices.

The floor $a$ indexes the performance criterion in Definitions 4 and 5; the interval in Definition 9 uses the observed transported contrasts. Its law-specific expected-length guarantee in Theorem 2 relates precision to the actual transported first stage $\mu (P)$. A single interval therefore supplies global coverage and responds to the compliance strength of the underlying law, with inverse-strength control of expected length above the rough estimation scale and bounded length throughout the effect domain. This guarantee concerns expected length under repeated sampling; it is compatible with variation in the reported interval across realizations.

The fixed value $\beta =1/8$ specifies the rough-nuisance regularity class analyzed here. At this regularity, the cubic correction is used to control the nonlinear remainder at the required scale. The split-sample estimator includes linear, quadratic, and cubic Taylor corrections across all seven learned marked-density coordinates and has mean-square error bounded by $C_{\mathrm{mse}}{n}^{-2/3}$ under the model assumptions for $n\ge 256$. The resulting contrast-estimation radius is therefore of order $n^{-1/3}$, which becomes the elbow in the length frontier.

The displayed finite-sample radius, $2\sqrt{2C_{\mathrm{mse}}/\alpha }{n}^{-1/3}$, makes the hidden rate constant explicit: it depends on the density envelopes, the common Hölder radius, and the desired coverage level through $C_{\mathrm{mse}}$ and $\alpha$. The threshold $n_0=256$ ensures that the deterministic four-block construction and its resolution choices are defined. Numerical precision is governed by the calibration constant and the transported first stage. For example, take the admissible inputs $c_f=0.9$, $C_f=1.1$, $L=1.01$, and $\alpha =0.05$. Substitution into the displayed formula in Lemma 3 gives $C_{\mathrm{mse}}\approx 1.16\times 10^{27}$, so the certified score tolerance is approximately $4.31\times 10^{14}{n}^{-1/3}$: about $6.79\times 10^{13}$ at $n=256$ and $4.31\times 10^{12}$ at $n=10^6$. These values exceed the effect-domain diameter and show that the explicit worst-case envelope is highly conservative; a realized affine-score interval can be shorter, but that realized length is not the uniform certificate. An analyst supplies fixed, scientifically defensible bounds on density overlap and Hölder size before applying this finite-sample calibration. The expected-length bound then uses the corresponding $C_0$ and the law-specific first stage.

An application of this benchmark requires the source encouragement experiment to satisfy conditional randomization, exclusion, and monotonicity, together with the sampling and regularity conditions of the model. Transport to the target population rests on the conditional equalities for the complier-weighted outcome contrast and the complier share in Assumptions 16 and 17. Under these conditions, the precision benchmark concerns the target-population average treatment effect among compliers, using source receipt and outcome observations together with target covariate observations.

Separate recruitment costs motivate the unequal-sample extension of the same decision problem in Definition 10. Let $N_{\mathrm{S}}$ denote the source sample size and $N_{\mathrm{T}}$ the target sample size. Retaining the law-level restrictions while allowing these counts to differ gives the following criterion.

The unequal-sample minimax expected-length criterion $R_{N_{\mathrm{S}},N_{\mathrm{T}}}(a)$ preserves the distinction between global coverage and performance on a strength slice. Coverage is required over the entire unequal-sample model, while the supremum defining expected length evaluates laws whose transported first stage lies between the specified floor and $1/4$, inclusive. The two sample counts consequently enter through the observation experiment, with the population restrictions and the estimand held fixed.

Open questions

The unequal-sample criterion makes allocation between source outcomes and target covariates a statistical design question. Characterizing its order would provide a basis for comparing the precision gains from additional records in either sample, subject to their respective recruitment costs. The central question is the joint behavior of the criterion across sample-size and strength regimes.

Resolving Remark 1 would connect an attaining procedure and an information bound for each allocation regime. Requiring the lower family to satisfy the no-defiers restriction keeps that comparison within the same causal model as the proposed intervals.

Limitations and future work

The established benchmark uses a scalar covariate, bounded outcomes, fixed Hölder smoothness with exponent $\beta =1/8$, and independent records within and across the two samples. Its matching expected-length bounds concern equal source and target sample sizes. The unequal-allocation problem in Definition 10 defines an extension of the criterion; its general rate characterization and an interval attaining that characterization remain the questions in Remark 1.

Further work could examine higher-dimensional covariates, inference under unknown smoothness, and sampling structures with dependent records. These extensions require corresponding coverage and expected-length analyses. Applications with alternative population transport restrictions likewise require an identification analysis suited to those restrictions before the precision benchmark can be extended.

Appendices

Identification and upper-bound proofs

Identification from primitive variables

The causal interpretation of the transported contrasts rests on consistency, conditional instrument randomization, exclusion, and monotonicity. The two transport restrictions then carry the source conditional contrasts to target complier quantities.

The common-version qualification makes the conditional identities compatible with target integration. Positive density envelopes provide common covariate support, monotonicity identifies the receipt contrast with complier prevalence, and exclusion identifies the outcome contrast with the complier-weighted receipt effect. Positive transported strength makes their ratio the target complier average effect.

Coordinate and sampling conventions

The estimator uses one target-density coordinate and six marked source coordinates. The next two definitions fix the coordinate names, observable marks, channels, and index conversion once for the remaining construction.