Causal Inference in Metric Spaces via Distance Profiles

• Existing causal methods often rely on domain-specific structure, while metric statistics describes random-object laws without addressing treatment assignment or confounding.Scalar summaries can miss changes in tails or dispersion even when mean-level summaries remain nearly unchanged.Source: §1 · pp. 1, 3
• The profile causal effect compares potential-outcome probabilities inside the same anchor-centered metric ball, providing a common contrast across random-object spaces.At fixed anchors, distance-profile coordinates describe where and at what scale probability mass shifts.Source: §1 · pp. 2–3
• The inferential grid is fixed through a deterministic or auxiliary design, while the main sample is reserved for cross-fitted score evaluation.Causal adjustment uses orthogonal doubly robust scores under the paper’s stated assumptions.Source: §1 · p. 2
• Cross-fitted doubly robust scores support simultaneous bands and global tests on design-fixed inferential grids, with bands extending to all radii at fixed anchors.The method addresses joint calibration across anchors and radii, including coordinates with zero variance.Source: §1 · p. 3
• The paper interprets bands as probability statements about balls and shells and provides a protocol for choosing anchors and radii.The protocol distinguishes primary designs from sensitivity designs.Source: §1 · p. 3
• Across simulations and semi-synthetic benchmarks, the max test detects concentration effects missed by mean-quantile and Fréchet-mean proxies, with empirical size near or below nominal, including under confounding.Its power depends on anchor design.Source: §1 · p. 3

• Metric statistics uses distance profiles to explore and test random-object laws, but does not address treatment assignment, confounding, or potential outcomes.This paper adapts those metric-neighborhood coordinates for potential outcomes and calibrates them jointly on a fixed grid.Source: §2 · p. 3
• Wasserstein distributional causal methods exploit one-dimensional quantile geometry, whereas profile effects compare potential-outcome probabilities in anchor-centered metric balls.The two targets describe different aspects of change in W2(R).Source: §2 · p. 3
• Geodesic causal inference targets displacement between Fréchet means when usable geodesics and stable means are available; profiles can detect dispersion, tail, and multimodality changes.Pointwise profile identification requires no outcome-space geometry beyond the metric.Source: §2 · p. 3
• Unlike scalar distributional effects or single kernel statistics, the paper jointly calibrates many distance-indexed effects with simultaneous intervals for retained coordinates.At a fixed anchor, a profile curve is the distribution function of the scalar distance to the outcome.Source: §2 · p. 4

• A distance profile Fω(t) records the probability that a random object lies within radius t of anchor ω.If the space is totally bounded and has strong negative type, agreement of complete profiles at every anchor and radius implies equality of laws.Source: §3 · p. 4
• The continuous anchor-radius domain motivates the profile process, while formal inference uses finite grids and, at fixed anchors, all radii.The visualization grid, candidate grid, and retained inferential grid serve distinct roles.Source: §3 · p. 4
• Grid construction must be deterministic or auxiliary-design-measurable so main-sample observations do not select anchors, radii, or inferential cells.The setup treats the inferential grid as fixed.Source: §3 · p. 4
• Pilot trimming removes cells with near-zero or near-one marginal ball probabilities, but does not itself guarantee a lower bound on score variance.The stated theorem assumes a variance bound separately.Source: §3 · p. 4

• Identification assumes consistency, no interference, conditional exchangeability given covariates, and positivity.Under these assumptions, each fixed profile coordinate is identified as a binary treatment effect.Source: §4 · pp. 4–5
• The profile effect compares the probabilities that treated and control potential outcomes fall inside the same anchor-centered metric ball.The potential-outcome indicators encode whether each outcome lies inside that ball.Source: §4 · pp. 4–5
• The profile contrast can be nonzero when mean-level summaries are unchanged: the examples separate it from both mean-quantile and Fréchet-mean displacement.In the examples, the profile effect equals 1 for radii between zero and the specified distance.Source: §4 · p. 5
• The efficient influence-function score is Neyman orthogonal and doubly robust, with bias governed by a product of propensity and outcome-regression errors.Its variance determines the scale used to standardize and compare coordinates.Source: §4 · p. 5

5.1 Finite-Grid Inference

• Cross-fitted AIPW scores estimate profile effects on a finite inferential grid, with simultaneous bands and a global max test calibrated by Gaussian approximation and multiplier bootstrap.The fixed-coordinate scores are standard AIPW scores for binary profile indicators; inference jointly calibrates many such scores.Source: §5 · p. 5, §5.1 · p. 6
• Under the stated regularity conditions, finite-grid inference gives simultaneous bands and a global max test with conditional size tending to α.The procedure conditions on a nonempty, nondegenerate inferential grid; cross-fitting and Gaussian approximation support the calibration.Source: §5.1 · pp. 6–7
• If the finite inferential grid is empty or an estimated standard deviation vanishes, the procedure returns trivial intervals and does not reject; this event becomes asymptotically negligible under stated conditions.Source: §5.1 · p. 7

5.2 Design and Interpretation

• Anchors, radii, and trimming rules are fixed before main-sample score evaluation, and the design determines which ball probabilities are compared.The authors recommend declaring one primary design and treating independently fixed alternatives as sensitivity analyses; joint claims across designs require a common critical value.Source: §5.2 · p. 7
• Simultaneous bands support joint conclusions about coordinate signs, shell-probability effects, and equivalence within a specified tolerance on the retained finite grid.A positive profile effect means treatment raises the potential-outcome probability of the ball, not that individual outcomes move toward its anchor; a coordinate pattern need not identify a unique mechanism.Source: §5.2 · p. 7

5.3 All Radii at Fixed Anchors

• With a small fixed anchor set, the method extends simultaneous inference to all radii, including when some coordinates have zero efficient-score variance.The theorem uses two fixed floors: one stabilizes studentization and the other prevents the cutoff from collapsing under a degenerate limiting process.Source: §5.3 · pp. 7–8

6 Experiments

• The four experiment questions cover calibration across object spaces, differences from mean-quantile and Fréchet-mean summaries, confounding, and real-geometry performance.Source: §6 · p. 8

6.1 Finite-Grid Calibration and Convergence

• Across five object spaces, null rejection stays below 0.05; power differs between anchor designs only for trees, at 0.900 versus 0.944.The designs are Medoid MaxMin and Guarded Shell-Signal; the latter also uses pilot estimates of standardized signal within distance shells.Source: §6.1 · p. 9, Table 1 · p. 9
• Under the Euclidean null, rejection rates are 0.046, 0.052, and 0.056 at n = 3200, 6400, and 12800, while simultaneous-band coverage remains near nominal 95%.Source: §6.1 · p. 9
• Under the controlled alternative, rejection is 1.000 throughout; as n increases, weighted L2 error falls from 0.0109 to 0.0051 and mean band width from 0.0806 to 0.0378.The trend is decreasing estimation error and band width, not increasing power.Source: §6.1 · p. 9

6.2 Estimand Separation and Confounding Adjustment

• In Wasserstein collapse, the profile test rejects at 1.000 for both sample sizes, while the mean-quantile proxy rejects at only 0.040–0.045; under confounded nulls, the doubly robust test stays near 0.05 while naive rejection reaches 0.674.In graph and SPD mean-zero examples, profile rejection reaches 0.980 and 0.990 at n = 400 while the Fréchet-mean proxy remains near nominal.Source: §6.2 · pp. 9–10, Figure 2 · p. 10

6.3 Real-Geometry Semi-Synthetic Benchmarks

• On ABIDE connectomes, rejection rises from 0.430 at effect 0.05 to 0.955 at effect 0.10 under collapse, and from 0.485 at effect 0.10 to 0.970 at effect 0.20 under prototype shift.At effect zero, rejection is 0.025; the benchmark retains real connectomes and covariates while injecting controlled alternatives.Source: §6.3 · p. 10, Figure 3 · p. 11
• For the insurance-network structural alternative, power is 0.416 for Medoid MaxMin, 0.562 for Guarded Shell-Signal, and 0.586 for Balanced LHS, with null sizes of 0.004, 0.018, and 0.014, respectively.The alternative changes local motif and bridge organization rather than only edge mass.Source: §6.3 · pp. 10–11

7 Discussion and Limitations

• Each anchor–radius coordinate is a binary potential-outcome contrast, while the framework’s object-specific challenge is simultaneous calibration, grid design, and interpreting bands as ball and shell probability statements.The approach extends potential-outcome reasoning to heterogeneous random objects without requiring a common algebraic structure on the outcome space.Source: §7 · pp. 11–12
• Guarantees cover finite grids fixed deterministically or by auxiliary design and all radii at fixed anchors, but continuum-anchor inference and valid same-sample adaptive anchor search remain open.Uniformity over laws approaching degeneracy is also open; radius-wise control uses nested ball indicators, whereas continuum-anchor bands would require ball-class entropy, uniformly accurate conditional distance-CDF nuisances, and a domain-wide variance floor.Source: §7 · p. 11
• Finite profile coordinates need not determine the outcome law: effects can fall between grid radii, and distinct laws can share profiles at finitely many anchors.A non-rejection therefore does not establish equality of potential-outcome laws; full anchor families separate laws under strong negative type, while weaker local separation conditions remain a possible direction.Source: §7 · p. 11
• At n = 200, the mean-quantile proxy rejects at 1.000 versus 0.470 for the profile test under quantile-location shifts, while profile tests detect concentration effects missed by mean-level proxies.Under confounded causal nulls, doubly robust adjustment keeps rejection near nominal, whereas a naive metric two-sample test rejects frequently; the test is conservative under the null on semi-synthetic connectome and insurance-network outcomes, and insurance-benchmark power depends on anchor design.Source: §7 · pp. 11–12

A.4 Proofs of Identification and Double Robustness

• For each profile coordinate, identification follows under SUTVA, conditional exchangeability, and positivity; strong negative type is not required for pointwise identification.The appendix also states that the AIPW contrast score is unbiased for the profile effect if either the propensity model or both arm-specific outcome regressions are correct.Source: §A.1 · p. 15, §A.4.3 · p. 19

Then the simultaneous confidence band

• The appendix states that Gaussian multiplier bootstrap bands attain asymptotic coverage 1 − α for the inferential grid under its assumptions.The bootstrap is applied to centered cross-fitted scores, with unconditional coverage obtained by averaging over auxiliary information.Source: §A.2 · p. 17, Then the simultaneous confidence band · p. 17

A.3 Global Omnibus Testing via a Max-Statistic

• The max-statistic omnibus test is calibrated under the grid-indexed null, and the corresponding coordinatewise band rule asymptotically controls familywise error strongly.The stated control applies for every law satisfying the assumptions and every configuration of true coordinate nulls.Source: §A.3 · p. 17

A.5.2 Uniform Remainder under Assumption A.6

• The uniform remainder bound vanishes after √n and √log L_n scaling, while its conditional tail control also implies the asserted order in joint probability.Source: §A.5.2 · pp. 20–21

A.5.3 Studentization and Covariance Control

• Explicit sufficient rate conditions imply Assumptions A.6 and A.8, so Theorem A.9, Corollary A.10, and Theorem A.11 apply.The feasible studentized maximum stays close to its oracle counterpart when estimated standard-deviation errors are sufficiently small relative to the logarithmic dimension.Source: §A.5.3 · pp. 21–22

A.5.4 Proof of Theorem A.9 (High-Dimensional Gaussian Approximation)

• Conditioning on the auxiliary information makes the inferential grid fixed and finite, with conditionally i.i.d. main-sample score vectors.The proof uses linearization and a high-dimensional Gaussian approximation for the maximum.Source: §A.5.4 · pp. 22–23
• The multiplier statistic is conditionally Gaussian, with covariance equal to the empirical correlation matrix of the centered estimated standardized scores.Gaussian comparison and quantile transfer establish conditional multiplier-bootstrap validity.Source: §A.5.4 · pp. 23–24
• The simultaneous coverage event for the inferential grid yields a global test whose rejection probability approaches the nominal level under the null.The coverage statement also controls false coordinate rejections.Source: §A.5.4 · p. 24

B Inference over All Radii at Fixed Anchors

• For a fixed law, inference covers all radii at a fixed set of anchors; an independent auxiliary sample may choose those anchors once.The results condition on that choice when the stated conditions hold; uniformity over design laws changing with sample size is not considered.Source: B Inference over All Radii at Fixed… · p. 24, §B.1 · p. 25

B.1 Scores and Training Regimes

• The all-radii scores use cross-fitting and allow known propensity, correctly specified parametric propensity, or two consistent nuisance estimates.The limiting influence function depends on the regime: it is u_h, v_h, or the efficient influence function ψ_h, respectively.Source: §B.1 · p. 25

B.2 Atoms and the Complexity of the Radius Path

• Monotone radius paths have controlled covering complexity even with jumps, so atoms in distance distributions do not disrupt the entropy argument.The argument applies to each frozen fit separately and does not claim that the union of all possible fits is a Donsker class.Source: §B.2 · pp. 25–26

B.3 Linearization and Process Limit

• Under Assumption B.1, the estimator has a uniform process limit to a tight centered Gaussian process, without requiring any coordinate variance to be positive.The covariance is given by P(f_h f_h′).Source: §B.3 · p. 26
• The projected covariance result applies to balanced cross-fitting; unequal fold proportions or an independently fitted propensity model change the influence function or covariance.With two folds, the coefficient is −1 at equal proportions and 1.25 when one fold has proportion 0.2.Source: §B.3 · p. 27

B.4 Multiplier Calibration and Protected Bands

• The multiplier calibration covers all radii at fixed anchors, with the entropy bound independent of the radius interval’s length.For finite-support weighted empirical CDF fits, the supremum can be computed exactly over their jump points and endpoint states; smooth fits require a different algorithm.Source: §B.4 · p. 27, §B.6 · p. 30
• Simultaneous bands have asymptotic coverage at least 1 − α even for degenerate laws; when the limit process is zero, coverage converges to one.For a nonzero limit process, the band without the lower cutoff has coverage converging to 1 − α.Source: §B.4 · p. 28

B.5 Learners That Satisfy the Conditions

• A fixed-kernel weighted conditional empirical CDF satisfies the stated stability condition without requiring the true conditional CDF, a density for X, or distance moments.A fixed Gaussian kernel is one example, but its limit is a kernel-smoothed conditional CDF; data-tuned kernels and forest-based weights need separate stability arguments.Source: §B.5 · p. 29

B.6 Measurability and Computation

• For a fixed number of multiplier draws, an order-statistic rule can conservatively account for Monte Carlo quantile error.The rule requires increasing the draw count if its specified quantile index exceeds the available draws.Source: §B.6 · p. 30

C.1 Scalable Nuisance Estimation via Conditional Distribution Learning

• Direct conditional-CDF learning preserves monotonicity in radius, unlike separate threshold-wise binary learners, and supports the proper fitted CDFs needed for all-radii inference.Separate binary learners remain compatible with finite-grid inference under its rate and correlation conditions.Source: §C.1 · pp. 30–31
• DRF weighted ECDFs and anchor-specific kernel-weighted ECDFs both produce valid profile-regression CDFs; experiments use DRF except in the five-space table.DRF errors decrease more clearly with sample size, while the kernel ECDF is simpler and faster.Source: §C.1 · p. 31

C.2 Diagnostics for Nuisance Estimation

• Structural-validity diagnostics check that each fitted nuisance path is a valid CDF in radius, a property built into the two ECDF learners but not guaranteed by threshold-wise binary learners..Source: §C.2 · p. 31

C.3 Strategy Naming and Design Rules

• Choose anchors using an independent design or pilot sample; a medoid-start MaxMin rule spreads anchors by geometry, while scientific reference objects can serve directly as anchors.The design is fixed before using the main sample for score estimation.Source: §C.3 · pp. 31, 33
• Shared radii can be chosen as pooled auxiliary-distance quantiles, while signal-aware anchor selection uses pilot estimates with separation, shell-quota, and trimming guardrails.The standardized power proxy is diagnostic, not a necessary condition for gaining power: a larger grid can raise power even when the strongest standardized signal is unchanged.Source: §C.3 · p. 33

D.1 Signs, Shells, and Largest Effects

• A simultaneous band yields bounds for shell-probability effects by differencing its endpoint bounds, while failure to reject zero does not establish equivalence.The same band supports an equivalence statement when the maximum absolute effect is bounded by a chosen probability tolerance.Source: §D.1 · p. 34

D.2 What Finitely Many Profile Coordinates Cannot Recover

• A finite radius grid can miss a mean-matched effect with sup_t |Δ(t)| = 1/2, and complete profiles at two anchors can still miss a probability difference of one.The latter occurs because the two anchors miss the joint law of the distance vector; changing the center can also reveal an effect invisible at anchor zero.Source: §D.2 · p. 34

D.3 Ball Features and Mean Distances

• Ball indicators define a positive-semidefinite kernel whose squared mean-embedding distance is the integrated squared profile contrast; for finite features, its witness is a linear transform of the profile vector.At one anchor, integrating the profile contrast to a finite radius gives a truncated mean-distance effect; the untruncated mean requires finite first distance moments and may vanish while the profile curve does not.Source: §D.3 · pp. 34–35

E.1 Anchor Selection: Theoretical Standards and Empirical Rules

• Anchor design should balance geometric coverage, nondegenerate scores, standardized signal relevance, and the max test’s multiplicity cost, with anchors and radii fixed before main-sample score evaluation.For a declared family of law changes, task-relative grids can target separation between profile signatures; coverage and task-specific grids may be combined as one frozen grid.Source: §E.1 · pp. 36, 38
• Auxiliary-sample-selected grids retain the fixed-grid inference guarantees conditionally when independent of the main sample and the stated dimension and nondegeneracy conditions hold.The resulting estimator has uniform linearization, Gaussian approximation, multiplier-bootstrap validity, simultaneous bands, and max-test level control.Source: §E.1 · p. 36
• Under uniform profile anti-concentration, finite anchor and radius nets approximate the population profile, but discrete spaces with atoms can violate the condition and the bound does not control between-grid empirical processes.A farthest-point medoid rule targets covering radius; the stated factor-two guarantee applies to exact farthest-first selection on a finite auxiliary cloud started from a cloud point.Source: §E.1 · p. 37
• With uniform pilot error, a top-K pilot-selected set contains a grid point whose standardized signal is within 2ε_n of the best guarded-grid signal.Source: §E.1 · p. 38

E.2 Efficiency–Power Pareto Frontier under a Fixed Metric

• The fixed-metric design framework evaluates simultaneous-band width and Gaussian power across admissible grids, exposing an efficiency–power tradeoff among design choices.The band-width criterion determines first-order average width, while Gaussian power approximates max-test rejection probability along local alternatives under the stated Gaussian approximations.Source: §E.2 · pp. 38–39

E.2.1 Pointwise Efficiency Implies Pointwise Local Power

• For a fixed profile location, the semiparametrically efficient estimator maximizes first-order pointwise local power among regular estimators using the same Wald calibration.The result follows because power is nonincreasing in asymptotic variance.Source: §E.2.1 · p. 39
• At a fixed anchor–radius pair, the doubly robust efficient score aligns efficiency with power, but coordinate choice can create a conflict even between two single-coordinate designs.Proposition E.6 establishes the fixed-coordinate alignment; Proposition E.10 shows the design-choice conflict.Source: §E.2.1 · p. 39

E.2.2 Global Power and Multiplicity

• A sufficient power condition compares the maximal standardized signal with the design’s simultaneous critical value, while adding coordinates can raise both.The bound is not necessary; power can also depend on the full signal vector and its dependence.Source: §E.2.2 · pp. 39–40
• A larger nested grid cannot lower the max-statistic critical value under the same Gaussian score process, and may not improve power.An example shows a power gain even when the strongest standardized shift is unchanged; comparisons across design classes are not covered by the lemma.Source: §E.2.2 · p. 40

E.2.3 Pareto Efficiency of Designs

• In nonconvex design classes, some Pareto-efficient designs need not arise from weighted scalarization.The empirical protocol evaluates candidate designs on power, band width, representation error, trimming, grid size, and cost before selecting a design.Source: §E.2.3 · pp. 40–41
• Designs trade off band width and power: the lower-variance design has narrower bands, while the other has strictly greater local power.Neither design dominates the other; design evaluation can also include representation error and computational cost.Source: §E.2.3 · pp. 40–41

F.1 Five-Space Configuration

• The five-space study freezes anchors, radii, and the inferential grid using auxiliary information, reserving the main sample for nuisance fitting and score evaluation.The implementation also screens coordinates with estimated standard deviation below 0.02; the high-sample figure reports all 300 repetitions reject at each sample size.Source: §F.1 · pp. 41–42, Figure 4 · p. 43

F.2 Exploratory Anchor-Design Screening

• Pilot-conditioned exploratory rules increased average power over Medoid MaxMin while null rejection and band coverage remained in a similar range.Medoid MaxMin remains the default design; the pilot-conditioned variants are treated as exploratory because their selection rule is more adaptive.Source: §F.2 · p. 43, Table 4 · p. 43

F.3 Additional Experimental Figures

• The null test’s simultaneous-band coverage is 0.954, 0.948, and 0.944 at the three sample sizes.The figure also reports null rejection rates and retained grid fractions over 500 repetitions.Source: Figure 5 · p. 44
• At n = 800, the DR estimator’s mean L2 error is 0.022–0.031 when either nuisance model is correct; with both misspecified, DR and IPW coincide at 0.071.When the outcome model is correct, OR alone is most accurate at 0.004.Source: Figure 6 · p. 44
• For graph pilots, trimmed fractions differ sharply by strategy: Medoid MaxMin is 0.27 under the null and 0.87 under the alternative, while Guarded Shell-Signal is 0.80 and 0.18.The figure averages trimmed fractions from null and alternative pilots by outcome space and anchor strategy.Source: Figure 7 · p. 44
• Exploratory screening plots mean power against absolute size deviation from 0.05; gray points mark other screened rules that passed the validity gate.Guarded Shell-Signal and Calibration LHS share identical coordinates and a split marker.Source: Figure 8 · p. 45

G.1 Real-Geometry Semi-Synthetic Benchmarks

• In the insurance graph benchmark, every design’s null rejection stays below nominal, while pilot-conditioned designs increase structural-alternative rejection over geometry-only Medoid MaxMin.The benchmark retains real networks, covariates, and treatment structure for known-truth graph-valued potential outcomes.Source: §G.1 · p. 46, Figure 9 · p. 46, Figure 10 · p. 46

G.2 Auxiliary-Sample Allocation

• At effect 0.10, shrinking the auxiliary sample without reallocating freed units lowers power versus state L by 0.060, while returning those units to the main sample raises it by 0.212.These are complete-procedure contrasts; state D differs in both auxiliary design and main-sample size.Source: §G.2 · p. 47, Table 5 · p. 47

G.3 Distribution-Valued Observational Analysis

• The NHANES marriage-exposure analysis is descriptive and asks whether activity distributions move toward or away from prototype neighborhoods, a different question from horizontal quantile effects.The analysis is unweighted and notes weak overlap among participants aged 14–17.Source: §G.3 · p. 48

G.4 Related-Work Stress Tests

• Related-work simulations show profile signals when mean-quantile or Fréchet-mean contrasts are weak or near zero, while a metric two-sample statistic that ignores covariates rejects frequently under confounded causal nulls.In matched randomized tasks, both procedures reject almost always under Wasserstein shifts, but the Fréchet-mean proxy is more powerful for the SPD shift at θ = 0.5; the tests target different functionals and multiplicity.Source: §G.4 · pp. 48–49, Figure 13 · p. 49, Table 6 · p. 49

H Conditional-CDF Learner Comparison: DRF and Kernel ECDF

• DRF and the anchor-specific kernel-weighted ECDF learner estimate the same conditional-CDF nuisance target and preserve monotonicity in t, but differ in finite-sample behavior and runtime.The DRF learner is used in the high-sample, separation, confounding, and real-geometry experiments; the five-space table uses the kernel-weighted learner.Source: H Conditional-CDF Learner Comparison:… · p. 50

H.1 High-n Finite-Sample Diagnostics

• Kernel ECDF null rejection is 0.020, 0.022, and 0.022, while SCB coverage is 0.980, 0.978, and 0.978 at n = 3200, 6400, and 12800.Its conditional-CDF error remains near 0.355, and product-rate diagnostics do not clearly decline.Source: §H.1 · p. 50
• Across n = 3200 to 12800, DRF's conditional-CDF error falls from 0.193 to 0.120, and its squared-rate and product-rate diagnostics also decrease.The squared-rate diagnostic falls from 0.796 to 0.305, the product-rate diagnostic from 0.447 to 0.272, and the scaled remainder from 1.554 to 1.058; these empirical errors do not verify Assumption A.6's rate conditions.Source: §H.1 · p. 50

H.2 Learner Scorecard

• Table 7's learner comparison supports using DRF beyond the five-space table, including on the ABIDE, confounding, separation, and insurance graph analyses.Kernel ECDF remains simple, fast, and structurally coherent, but has weaker nuisance-rate diagnostics in the high-sample analysis.Source: §H.2 · p. 50, Table 7 · p. 50

H.3 Relation to Reported Claims

• Under confounded causal nulls, the doubly robust profile test stays below the nominal level, whereas the naive metric two-sample test rejects frequently.The naive test compares observed groups.Source: §H.3 · p. 51

I Experimental Reproducibility Details

• Code, configuration files, and figure-rendering scripts for the reported analyses are available from the authors upon request.Reusable modules cover data generation, anchor and radius construction, nuisance estimation, inference, and visualization.Source: I Experimental Reproducibility Details · p. 51
• Configurations cover calibration, nuisance diagnostics, separating examples, confounding experiments, and the insurance graph and ABIDE semi-synthetic runs.Large raw datasets, restricted insurance-network data, and large per-repetition matrices are not redistributed; provenance and access notes are provided for public or externally governed data sources.Source: I Experimental Reproducibility Details · p. 51
• Reported experiments are CPU-based and require no GPU; repeated nuisance fitting, inferential-grid construction, and bootstrap calibration dominate cost.The distributional random forest learner uses an R backend, with no fallback reported.Source: I Experimental Reproducibility Details · p. 51
• Regenerated observational ABIDE and NHANES figures are descriptive; known-truth validation relies on simulation and semi-synthetic analyses.The rendering script regenerates main and appendix figures from lightweight derived outputs.Source: I Experimental Reproducibility Details · p. 51