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Same Predictions, Different Harms: Causal Auditing of Patient World Models

Yicheng Qi, Xiyi Xiong

arXiv:2610.05198v1stat.MEcs.AI

TL;DR

Patient world models can agree on transition kernels and arm-specific risks yet disagree on patient-level counterfactual harm. The paper derives bounds under shared-response models and relaxes their causal assumptions with declared budgets; its symmetric three-state frontier is sharp, while general sensitivity bounds are outer.

  • Problem

    Matching transition kernels and treatment-specific risks does not determine the fraction of patients harmed under a treatment switch, because that quantity depends on how their potential outcomes coexist.

    Source: §1 · p. 1

  • Method

    The paper models an intermediate categorical state followed by common terminal care, derives sharp bounds under independent stages, and adds budgets for dependence and response mismatch.

    Source: §2 · p. 2, §1 · p. 2, §3.2 · p. 3

  • Results

    In the symmetric three-state model, the sharp upper harm rises from 1/3 at slope 1/2 and reaches the endpoint-only upper bound 1/2 at total budget 1/3.

    Source: §4 · p. 4

  • Takeaways & Limitations

    The bounds provide a conditional audit of counterfactual-harm reliability, rather than a validated digital twin or complete treatment-selection rule.

    Source: §1 · p. 2, §8 · p. 9

    The guarantees require valid interventional primitives, sufficient states, and declared budgets; general sensitivity bounds can be loose and the sharp joint frontier is symmetric.

    Source: §8 · p. 9

Abstract

from arXiv · show

Patient world models used for clinical trial simulation can agree on transition kernels and arm-specific risks, yet disagree on the fraction of patients harmed by switching treatment---the counterfactual quantity that matters for intervention-aware reasoning. We audit this reliability gap in a two-stage shared-response SCM: a categorical intermediate health state is followed by common terminal care. Under independent stages, the sharp harm interval has closed-form endpoints for at most three intermediate states, with an exactness boundary at four states. Declared dependence and response-mismatch budgets yield calibrated outer bounds when stage independence or complete mediation is relaxed; in a symmetric three-state model the entire sensitivity frontier is sharp, $[0,\min\{1/2,1/3+(\rho+\delta)/2\}]$, and shows exactly how budgets erase the gain over endpoint-only bounds. Two eight-variable response LPs propagate interventional uncertainty for finite-sample audits. Exact witnesses verify attainability. On public clinical simulators (EpiCare; sepsis), native configurations show little resolved stage dependence and no additional joint-compatibility gain over pairwise transport---honest negative results for reliability claims. All experiments are locally reproducible; guarantees remain conditional on the stated causal model. The results provide a concrete protocol for deciding when a patient world model is safe to trust for counterfactual harm.

1 Introduction

Matching transition frequencies and treatment-specific risks does not ensure agreement on patient-level counterfactual harm. The paper audits this reliability gap with bounds that make dependence and response mismatch explicit, while keeping its guarantees conditional on the causal model.

1 Introduction

  • Agreement on transition frequencies and treatment-specific risks is insufficient to establish agreement on patient-level counterfactual harm.The harm query is the fraction of patients with an adverse outcome under regimen A but not B, and depends on how potential outcomes coexist for each patient.Source: §1 · p. 1
  • A shared random seed imposes causal assumptions beyond a simulator’s transition kernels.The paper situates its audit in a two-stage setting with an intermediate health state followed by common care.Source: §1 · p. 1
  • Identical arm-specific state/outcome tables can coexist with harm values of 0 and 1/3 in the three-state example.The example has three equiprobable intermediate states and adverse-outcome risk 1/2 at each state.Source: §1 · p. 1
  • Persistent patient heterogeneity and incomplete mediation are addressed with declared budgets for stage dependence and treatment-specific response mismatch.These budgets yield calibrated outer bounds and an attained frontier, but are generally unidentified from observational data.Source: §1 · p. 2
  • The paper studies exact three-state bounds, general outer sensitivity bounds, a restricted multistage extension, and two public-simulator audits.The guarantees concern reliability of intervention-aware simulators, not clinical deployment of a treatment policy.Source: §1 · p. 2

2 Model, information and the sharp target

The paper defines harm through a coupled pair of treatment-specific intermediate states and a shared terminal response mechanism. Identified marginal risks do not generally identify the patient-level harm probability, which is bounded over compatible mechanisms.

2 Model, information and the sharp target

  • The information primitives are treatment-specific state distributions and adverse-outcome risks under intervention on the intermediate state.Using observed state-conditionals additionally requires mediator–outcome exchangeability after included baseline covariates; initial-treatment randomization alone is insufficient.Source: §2 · p. 2
  • The model couples the two intermediate states and independently draws one shared terminal response vector whose state-specific means equal the known risks.Reusing the same response vector makes terminal outcomes coincide whenever both counterfactual regimes reach the same state.Source: §2 · p. 2
  • Although the marginal adverse-outcome risks are identified, the harm probability is generally not; its feasible values form an interval with sharp endpoints.The upper endpoint is a worst compatible probability, not evidence that this amount of harm occurs.Source: §2 · p. 2
  • Choosing a separate response coupling for each state pair is a valid outer relaxation, but does not ensure one jointly compatible response mechanism.The paper distinguishes this relaxation from methods that enforce joint response compatibility or use a different factual query.Source: §2 · p. 2

3 Sharp heterogeneous-risk bounds with three states

Under independent stages, the lower harm endpoint is transport for any number of states, while the cut-envelope upper endpoint is sharp through three states. A separate concentration condition also makes pairwise bounds sharp, and the three-state exactness does not extend unchanged to four states.

3.1 The lower endpoint is transport for every state count

  • The lower endpoint is attainable for every state count using one shared threshold response independent of the first-stage coupling.The passage notes that joint compatibility affects the upper and lower endpoints asymmetrically.Source: §3.1 · p. 2

3.2 A cut envelope and its exactness for at most three states

  • For n ≤ 3, the cut-envelope upper endpoint is sharp and has a closed form for three states.A response vector defines a cut, and the maximum first-stage probability crossing that cut is 2b(S).Source: §3.2 · p. 3
  • At four states, the cut reduction need not be sharp: one instance has exact upper endpoint 0.30 versus cut and pairwise bounds of 0.35.For three states, a single transport plan attains the diagonal lower bounds for every cut; a triangle correction accounts for incompatible pairwise disagreements.Source: §3.2 · p. 3

3.3 A second exact regime: one dominant pooled state

  • Pairwise bounds are also sharp for any state count and terminal risks when a pooled state has mass at least 1/2.The condition is sufficient but not necessary.Source: §3.3 · p. 3

4 Relaxing the causal assumptions jointly

The paper relaxes stage independence and complete mediation with budgets for dependence and treatment-specific response mismatch, yielding calibrated bounds that are generally outer. In the symmetric three-state model, the joint frontier is sharp and reaches the endpoint-only upper bound as the total budget grows.

4 Relaxing the causal assumptions jointly

  • The general sensitivity bounds calibrate deviations from stage independence and equality of treatment-specific response tables using separate budgets ρ and δ.A total-variation budget limits dependence between the state pair and response tables; the mismatch budget bounds direct response changes at a fixed state.Source: §4 · p. 4
  • The general bounds are outer rather than necessarily sharp, while the direct-effect lower endpoint remains sharp under independence despite mismatch budgets.The lower endpoint is attained by common forward thresholds with mismatch |ai − bi|.Source: §4 · p. 4
  • The exact symmetric frontier is attained by a uniform cyclic state-pair coupling and response-table changes that increase harm up to saturation.The passage specifies that the sum-only dependence on ρ + δ applies to this symmetric example, not universally.Source: §4 · p. 4
  • In the symmetric three-state model, the sharp upper harm rises from 1/3 at slope 1/2 and reaches the endpoint-only upper bound 1/2 when ρ + δ = 1/3.The frontier remains sharp after fixing both complete arm-specific state/outcome tables to probabilities 1/6 per state-outcome cell.Source: §4 · p. 4, Figure 1 · p. 5

5 Extensions

The extensions carry the three-state entrance bound into a restricted multistage chain and propagate finite-sample uncertainty through two small response LPs. The guarantees depend on the stated structural assumptions, and the confidence projection is an outer enclosure when primitive values are uncertain.

5.1 Multistage continuation

  • For a ternary entrance followed by binary transitions, both harm-interval endpoints remain jointly attainable under independent suffix noises and the specified common-care structure.The extension does not cover arbitrary multivalued trajectories or relax suffix independence; retaining known intermediate kernels can give [0, .32] rather than the collapsed-risk interval [0, .5].Source: §5.1 · p. 5

5.2 Finite-sample uncertainty

  • A shared terminal-response transport minimizes every cut crossing in the three-state model, giving an exact response-LP representation for the centered lower endpoint.Both sharp endpoints therefore have small response-LP representations, avoiding general nonlinear causal optimization or a sum of separate Lipschitz errors.Source: §5.2 · p. 5
  • Two eight-variable response LPs propagate simultaneous primitive confidence boxes, and every true primitive in those boxes satisfies the resulting enclosure.The construction assumes iid observations within relevant batches and at least 1 − ζ simultaneous coverage; compatible point boxes recover both sharp endpoints, but nonzero boxes need not yield sharp nuisance-robust extrema.Source: §5.2 · pp. 5–6

6 Local verification and public health audits

The verification combines numerical checks, finite-sample comparisons, and audits of two public simulators under a declared protocol. Results include conservatism and limited applicability: native EpiCare shows little resolved dependence, while sepsis configurations show no extra joint-compatibility gain over pairwise transport.

Auditing protocol

  • The audit protocol declares the information set, computes sharp or outer intervals, reports sensitivity budgets, retains witnesses when available, and compares baselines at equal information or observation counts.The paper also reports null results and keeps a local reproduction record.Source: §6 · p. 6

Algebra, sharpness and generality

  • The verification includes exhaustive transport/response checks, a rational four-state counterexample, and tests of joint surfaces and proposed outer intervals.The four-state counterexample has an exact rational certificate; 150 joint mechanisms were tested and every realized harm lay in the proposed outer interval.Source: §6 · p. 6
  • The general analytic upper enclosure exceeded numerical references for 126 arbitrary-risk direct-effect configurations by mean .03271 and maximum .19749, so it is not generally sharp.The maximum LP primal residual was 6.50 × 10^-15, a numerical diagnostic rather than a formal outward-rounded upper certificate.Source: §6 · p. 6

Finite-sample uncertainty

  • All 24,000 CP-projection intervals contained the full true identified range, demonstrating conservatism rather than exact 95% calibration.Mean widths improved over the retained Lipschitz/Hoeffding wrapper in all 24 strata by .00457–.11633.Source: §6 · p. 6

Finite-sample comparison

  • At equal 5N observation counts, the CP projection was narrower than the stronger three-event comparator in 12 of 24 strata and wider in 12.At N = 100, endpoint-only inference won in all six settings; at N = 10000, the projection won in five, reducing mean width by .10287–.17841, while setting 6 remained .00571 wider.Source: §6 · p. 6, Figure 2 · p. 7

Native EpiCare

  • The native EpiCare audit found mean empirical TV of 2.37 × 10^-5 for three diseases and 4.02 × 10^-5 for four, with zero full-table pointwise 99% lower dependence bounds in every case.The maximum empirical TV was 1.87 × 10^-4; these results do not establish a practically large violation.Source: §6 · p. 7

Independent public benchmark

  • The sepsis audit found no additional joint-compatibility gain beyond pairwise transport across 60 native configurations, while endpoint-only bounds may still be looser.The compared initial actions have disjoint nonabsorbing state supports; the audit used 240,000 episodes and the complete sufficient state.Source: §6 · p. 7

Remission stress experiment

  • A separate 720-case EpiCare-derived remission stress experiment had zero pairwise gap in all 240 upstream-transition cases; declared stress cases produced gaps up to .11347.Native and modified results are reported separately.Source: §6 · p. 7

7 Related work

The paper positions its results alongside work on probabilities of causation, mediator-informed bounds, sensitivity analysis, compatible response mechanisms, and confidence-region propagation. It emphasizes a specialized exact shared-response result while identifying where its scope and sharpness end.

Probability of causation

  • The paper complements Kallus [2022] by restricting mediator co-responses in a two-stage simulator rather than replacing estimation, and provides explicit three-state endpoints and finite-sample set containment.The construction does not claim optimal statistical efficiency.Source: §7 · pp. 7–8

Mediator-informed bounds

  • The closed-form multivalued shared-response result solves a health-simulation subclass exactly and marks where its cut reduction fails, complementing general optimization approaches.The four-state example is a limit of this reduction, not a general computational dichotomy.Source: §7 · p. 8

Sensitivity analysis

  • The sensitivity analysis bounds deviations from stage independence or equality of treatment-specific response tables, with an attainable joint surface specific to the symmetric half-risk three-state family.The dependence budget is a total-variation measure, not an observable correlation coefficient.Source: §7 · p. 8

Mechanism compatibility and inference

  • For compatible mechanisms and discrete causal inference, the paper connects its response-compatibility query to Haugh and Singal [2023] and propagates confidence-region uncertainty following Duarte et al. [2024].Its prospective two-regime query differs from the factual-trajectory-conditioned query of Lally et al. [2026].Source: §7 · p. 8

8 Limitations and responsible use

This section extends harm bounds to settings that relax stage independence or complete mediation, distinguishing sharp results from outer certificates and keeping guarantees conditional on the stated causal model and sensitivity budgets. It also sets limits for responsible use: explicit assumptions and information sets, sensitivity reporting, reproducible evidence, and no use for dosing, deployment, or patient-benefit claims.

8 Limitations and responsible use

  • The guarantees depend on valid interventional information, sufficient states, and declared budgets; general sensitivity bounds may be loose, and native audits show limited evidence of material gains.Confidence intervals can also be wide, so these results are not clinical thresholds or evidence of a validated digital twin.Source: §8 · pp. 8–9

8.1 Responsible use

  • Generated probabilities and public simulator code—not identifiable patient records—support the experiments, and the outputs must not guide dosing, clinical deployment, or patient-benefit claims.Misstated mediation, independence, or sensitivity assumptions can invalidate safety conclusions; explicit information sets, attaining witnesses, outer-bound labels, and reproduction records help expose this risk. Human accountability and independent scientific review remain necessary.Source: §8.1 · p. 9

A Model, realizability, and the identified interval

  • The stipulated independent-stage, shared-assignment model represents exactly its admissible mechanisms, and its attainable harm values form a compact interval; conditioning on a factual trajectory changes the feasible problem.The exact lower endpoint is evaluated by sorting risk values and transporting mass in increasing order, in O(n log n) time.Source: A Model, realizability, and the… · p. 12, B The exact lower endpoint for… · p. 12

D Equal terminal risks: partition structure, sharp gaps, and computation

  • With equal terminal risks, the pairwise endpoint exceeds the sharp endpoint by at most 1/6, with equality for uniform weights on three states.For integer-weight inputs, the sharp endpoint reduces to subset sum, is weakly NP-complete to decide at 1/2, and is computable exactly in O(nW) Boolean operations.Source: D Equal terminal risks: partition… · pp. 14–15

E Uniform intermediate states and homogeneous terminal risk

  • For uniform intermediate states and homogeneous terminal risk, a directed-cycle construction attains the upper endpoint by mixing rotated binary response patterns and adjacent grid-point constructions.The construction covers arbitrary risk p by mixing patterns at m = floor(np) and ceiling(np).Source: E Uniform intermediate states and… · p. 15

F Stability in terminal risks

  • Terminal-risk stability yields a valid enclosure of the sharp upper endpoint, while state-distribution stability bounds changes in the equal-risk partition endpoint by total variation.The state-sampling confidence interval is conditional on known terminal risks p_i = 1/2; combining uncertain state laws with uncertain terminal risks is not covered by that guarantee.Source: F Stability in terminal risks · pp. 15–16, G State-distribution stability and… · p. 16

C Cut transport and the three-state formula

  • For n ≤ 3 intermediate states, the cut envelope equals the sharp upper harm endpoint for every heterogeneous risk vector; at four states, the envelope can be loose.In the four-state example, exhaustive vertex optimization gives U = 3/10, while both the cut envelope and rectangular bound equal 7/20.Source: H A four-state cut-envelope… · pp. 16–17

I A dominant pooled state makes the pairwise bound sharp

  • A pooled state with weight at least 1/2 is sufficient to make the pairwise bound sharp for every heterogeneous risk vector, without restricting the number of states or requiring balanced risks.This is a sufficient condition, not a necessary one.Source: I A dominant pooled state makes the… · p. 17
  • A dominant pooled state makes the pairwise upper endpoint sharp under the main model’s assumptions, without implying that a zero observed gap requires that mass condition.The assumptions add no ordering, monotonicity, positivity, or equality of terminal risks.Source: I A dominant pooled state makes the… · p. 17

J Extension models and information sets

  • Dependence and response-mismatch budgets are sensitivity parameters rather than quantities identified by transition marginals, and each bound is conditional on a nonempty stated model class.The dependence extension constrains the joint law’s total-variation distance from a product of its marginals.Source: J Extension models and information sets · p. 18

K Stage dependence and mean calibration

  • Fixing regime means yields a calibrated dependence interval shifted by τp and widened by ρ/2 on each side of the independent-stage endpoints.The calibrated interval is not generally sharp; with a single shared state, consistency forces harm to zero. Unrestricted dependence instead has sharp LP bounds for its stated information set.Source: K Stage dependence and mean calibration · p. 18

L Bounded direct effects with independent stages

  • For feasible direct-effect budgets, the sharp lower endpoint survives; when a = b = p, it is unchanged from the shared-response model.Upper certificates can be intersected with the endpoint Fréchet bound, and enumerating transportation vertices with response LPs gives a globally sharp finite procedure for unknown C.Source: L Bounded direct effects with… · p. 19

M Joint relaxation of independence and full mediation

  • Jointly relaxing stage independence and full mediation yields outer certificates, but no general sharpness claim; arbitrary stage dependence can change the exact independent-stage lower endpoint.The joint certificates use declared dependence and direct-effect budgets and fixed observed regime risks.Source: M Joint relaxation of independence and… · p. 20

N A sharp two-parameter sensitivity surface

The section gives an attained two-budget sensitivity frontier for a symmetric three-state model, then extends common-care bounds to restricted binary suffixes while showing unrestricted folding can be loose.

N A sharp two-parameter sensitivity surface

  • Mixing response laws raises independent-stage harm from 1/3 to 1/3 + d/2, while dependence can add r/2 without changing the observed tables or budgets.The dependence step redistributes equal-response-cell mass to disagreeing cells while preserving the response law and both full state–outcome tables.Source: N A sharp two-parameter sensitivity… · p. 20
  • The symmetric three-state harm interval is [0, 1/3 + (d + r)/2], where d = min(δ, 1/3) and r = min(ρ, 1/3 − d), and every value is attained.The construction preserves the state–outcome tables while using the declared response-mismatch and dependence budgets.Source: N A sharp two-parameter sensitivity… · p. 20
  • The frontier is a specialized calibration result, not a formula for arbitrary primitive risks or state distributions.Setting either budget to zero recovers the separate sharp dependence and direct-effect frontiers.Source: N A sharp two-parameter sensitivity… · p. 20

O Multistage common care: a sharp restricted extension and a folding failure

  • Folding internal stages into terminal risks generally gives only an outer interval; sharpness requires a folded extremizer realizable by the retained kernels.Known internal transition kernels can restrict the feasible joint response law.Source: O Multistage common care: a sharp… · pp. 20–21
  • The fixed-kernel counterexample has true upper harm 8/25, below the folded upper endpoint 1/2.All transition entries are strictly positive, and a common last-stage response attains lower harm zero.Source: O Multistage common care: a sharp… · p. 21
  • With independent time-specific noises and common binary care that does not depend on unrecorded history, suffix disagreement factors yield jointly attainable endpoints after a three-state entrance.The endpoint construction combines prefix and suffix extremizers under the stated assumptions.Source: O Multistage common care: a sharp… · p. 21
  • The multistage result connects to prior binary-chain product mechanisms; its claimed scope is the solved three-state entrance, not a general multivalued multistage solution.Source: O Multistage common care: a sharp… · p. 21

P Simultaneous finite-sample outer intervals beyond balanced risks

The section extends finite-sample outer intervals beyond balanced risks and develops a small-state projection for simultaneous primitive uncertainty, while stating the assumptions and scope of its guarantees.

P Simultaneous finite-sample outer intervals beyond balanced risks

  • The centered endpoint stability bound is generally unimprovable: deterministic state-risk changes and distinct-state risk shifts attain its state and risk terms.The confidence construction propagates certified endpoint bounds through primitive uncertainty, with simultaneous coverage based on a union bound.Source: P Simultaneous finite-sample outer… · p. 22

Q Simultaneous confidence projection for the simulator audit

  • For simulator audits, simultaneous primitive confidence intervals cover all K prespecified settings by allocating η=γ/K per setting; the reported audit uses K=120 and γ=0.01.The construction uses complete simulator response vectors, which are instrumentation rather than ordinary patient observations or counterfactual-response identification from clinical records.Source: Q Simultaneous confidence projection… · pp. 23–24

R Scope and provenance of the extension arguments

  • The extensions do not solve general unknown-product total-variation optimization, arbitrary multivalued multistage sharpness, or data identification of the sensitivity budgets.The section identifies calibrated joint bounds, the sharp two-budget surface, and its connection to the exact three-state result as the dedicated contributions.Source: R Scope and provenance of the… · p. 24

S Small-state projection of simultaneous primitive confidence boxes

  • The small-state projection uses two eight-cell response LPs and is generally conservative for nonzero primitive boxes, but equals the exact interval when compatible boxes collapse to points.The guarantee covers the entire true identified range with probability at least 1−ζ under the stated sampling conditions; incompatible boxes trigger the vacuous [0,1] fallback.Source: S Small-state projection of… · pp. 24–25
  • For three states, one coupling simultaneously minimizes crossing probabilities for all response cuts, attaining the lower representation; fewer states can be padded with zero-mass states.The construction preserves maximal diagonal mass for each state pair.Source: S Small-state projection of… · pp. 24–25

T Confidence projection and equal-observation-budget evaluation

The section evaluates a confidence projection against equal-observation-budget endpoint procedures and retained baselines, alongside audits of general sensitivity bounds. Width comparisons vary by sample size and setting, while the reported containment results are conservative rather than exact calibration.

T Confidence projection and equal-observation-budget evaluation

  • The CP response projection has smaller mean width than the retained Lipschitz/Hoeffding procedure in all 24 strata, by .00457–.11633, and all 24,000 intervals contain their true structured ranges.The containment demonstrates conservatism, not exact nominal calibration, and the comparison does not establish uniform dominance.Source: T Confidence projection and… · p. 26

T.1 Comparable observation counts and stronger endpoint baselines

  • The stronger three-event endpoint comparator has empirical coverage .997–1 for the structured range and .964–.991 for its own endpoint-only range; these are not claims of exact nominal calibration.Equal observation counts are obtained by using N arm observations plus N state-intervention outcomes per state for structured methods, and 5N total for the endpoint comparator.Source: §T.1 · p. 26

T.2 Complete width comparisons

  • At N=100, all six configurations favor endpoint-only inference; at N=10000, CP projection beats the 3-event procedure in five of six settings, reducing mean width by .10287–.17841, while setting 6 remains .00571 wider.All procedures use 5N observations.Source: §T.2 · p. 26, Table 2 · p. 27

U Extended experiment protocols

  • The extended audits retain fixed-seed cases and distinguish rational certificates from floating-point optimization diagnostics.The joint-sensitivity audit covers 30 cases at each of n=1, 2, 3, 4, and 6; its containment checks are not claims of global sharpness.Source: U Extended experiment protocols · p. 27, §U.1 · p. 27

U.1 General mechanism and multistage tests

  • Across 126 arbitrary-risk direct-effect configurations spanning two, three, and four states, analytic outer upper bounds are compared with global numerical upper endpoints; positive gaps show the general enclosure can be loose.Source: Figure 3 · p. 28

U.2 Retained Lipschitz/Hoeffding baseline and information comparison

  • The retained Lipschitz/Hoeffding interval is narrower than endpoint-only inference in 12 of 24 strata and wider in 12, while both contain the true range in all 24,000 repetitions.Its largest mean reduction is .15959 and largest increase is .03423; the containment documents conservatism, not exact nominal calibration.Source: §U.2 · p. 28, Table 4 · p. 29

U.3 Native simulator audits and applicability limits

Native EpiCare and sepsis audits show limited resolved dependence or no additional joint-compatibility gain over pairwise transport. The section also characterizes the balanced-risk gap, exact algorithms, and sampling bounds, with guarantees conditional on the stated model.

U.3 Native simulator audits and applicability limits

  • Across 120 native EpiCare cases, estimated dependence with persistent modifiers was 2.37×10−5 and 4.02×10−5 for 3/4 diseases, with maximum 1.87 × 10−4; no large, well-resolved independence violation appeared.With modifiers disabled, population dependence is exactly zero; the sepsis protocol admits no coherence gain over pairwise transport because active supports are disjoint.Source: §U.3 · p. 29

V.1 Balanced risks identify a partition problem

  • For balanced risks, the pairwise upper bound exceeds the sharp upper endpoint by at most 1/6, attained for uniform three-state weights.A dominant pooled atom makes the bounds exact; otherwise a subset with mass in [1/3, 2/3] gives a sharp upper endpoint of at least 1/3.Source: §V.1 · p. 30

V.2 Exact and approximate algorithms

  • With integer pooled weights, subset-sum computes the balanced-risk upper endpoint exactly, but deciding whether it equals 1/2 is weakly NP-complete in this restricted model.The approximate algorithm has O(n2/η) dynamic-program updates and O(n/η) sum states; for maxi wi < 1/2, U ≥1/3 permits a (1 −ε) approximation with η = ε/3.Source: §V.2 · pp. 30–31

W Heterogeneous risk and sampling uncertainty

  • The balanced-risk sampling interval covers the sharp upper endpoint conditional on known pi = 1/2 and the specified causal model, not the realized individual harm.For heterogeneous risks, the continuity enclosure is generally not sharp; the three-state closed form is preferable, and unknown terminal risks require separate treatment.Source: W Heterogeneous risk and sampling… · p. 31

X Additional baseline validation and declared mechanism stress

The validation finds zero pairwise-to-sharp gap in all 240 upstream-transition EpiCare cases; positive gaps occur only in declared mixing and intermediate-efficacy stress settings. The section also reports exhaustive numerical checks, EpiCare extraction details, and a local reproduction protocol.

Y Complete experiment protocol and reproduction

Y.1 Exhaustive optimization reference

  • The exhaustive reference checked every feasible transport-vertex and response-polytope-vertex pair for n ≤4, while the three-state closed form was checked on 3,000 configurations.The generic reference uses floating-point tolerances, so it is a numerical check rather than a formal rational certificate; the supplied four-state counterexample has a separate exact certificate.Source: X Additional baseline validation and… · p. 31, §Y.1 · p. 33

Y.2 Public EpiCare extraction

  • The EpiCare audit extracts two-visit, three- or four-state subproblems and disables persistent modifiers and adverse-event termination for its oracle mechanism audit.The adaptive-care rule uses the full latent state and is not an implementable recommendation from noisy symptoms; the configurations are not a validated clinical trial.Source: X Additional baseline validation and… · p. 32, §Y.2 · p. 34

X Additional baseline validation and declared mechanism stress

  • All 240 EpiCare upstream-transition cases had zero pairwise-to-sharp gap; positive gaps appeared only in declared mixing and intermediate-efficacy stress settings with state-adaptive common care.The largest pooled-state mass was at least 0.64599 in every upstream-transition case, explaining exactness under Proposition 3.Source: X Additional baseline validation and… · p. 32, Table 5 · p. 32

Y.3 Finite-sample audit and what its coverage means

The finite-sample audit evaluates endpoint intervals over 1,000 repetitions in four balanced-risk settings. Its coverage statements concern the sharp upper endpoint under known terminal risks, not realized harm.

Y.3 Finite-sample audit and what its coverage means

  • Table 6 reports empirical coverage of the nominal 95% endpoint interval over 1,000 repetitions and notes that the union bound is conservative in these settings.The intervals target the sharp upper endpoint, not unknown realized harm, and do not establish coverage for unknown heterogeneous terminal risks.Source: Table 6 · p. 35, §Y.3 · p. 35
  • The audit uses four settings with terminal risks known and exactly 1/2, drawing two independent multinomial count vectors per repetition.Each setting is evaluated at N ∈{100, 500, 2000, 10000} samples per arm.Source: §Y.3 · p. 35
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