# What changes when capture is known before the decision

23 October 2026 · Timing corollary and proof

This is an exact limitation of the stated finite model and an application of a known monitoring principle. It uses no patient data and supplies no claim of new economic theory or clinical validation.

[Supplement overview](model-and-proof.md) · [Reproduction instructions](README.md) · [Related research](REFERENCES.md)

## Statement

Use the finite model and calibration in [the finite model](finite-model-and-proof.md): `q=1/2`, `d=4/5`, `f=1/10`, `Kmax=1/5`, `ε=1/100`, `a=1/20`, `B1=3/4`. There is no enrollment transfer. The provider enrolls before observing the independently drawn group and capture status. All toy patients are willing, clinically appropriate and feasible; no valid refusal, escalation or capacity exception is modeled as withholding.

Let `C~Bernoulli(q)` be an exogenously assigned review status, independent of case group and the randomness governing potential outcomes. The provider cannot alter `C`. This does not assert independence between `C` and the realized action or observed outcome: in the revealed-capture regime, the provider can condition the action on `C`. Conditional on `C=1`, an adverse flag has probability `d` after avoidable withholding and `f` after appropriate acceptance. It is common knowledge that `C=0` entails **no subsequent detection or consequence** through any other channel. Only the revealed-capture regime tells the provider whether `C=0` actually occurred before the choice. The consequence is bounded by `K`. Enrollment does not allow cost-free post-type exit that evades the contract.

Compare two information structures while preserving all distributions, payments, action sets, error rates, resource limits and the score-to-opportunity continuation rule:

1. **Capture hidden when acceptance is chosen.** The provider must choose whether to accept without knowing `C`; no other signal reveals it. Capture can be drawn before the decision and concealed, or drawn after it, provided the distribution and available information are identical at choice.
2. **Capture revealed before acceptance is chosen.** The provider observes both group and `C` and can condition acceptance on them.

**Corollary.** With hidden capture, the weak full-access frontier is `λ≤19/15`, with strict acceptance achievable for `λ<19/15`, as derived in the [finite-model proof](finite-model-and-proof.md). With revealed capture, weak full access is feasible **if and only if `λ≤4/5`**, and strict acceptance for every positive-probability information state is achievable **if and only if `λ<4/5`**.

“Weak full access” means some best-response policy accepts every positive-probability group/capture state; an indifferent provider may choose that compliant policy. “Strict” means acceptance is strictly preferred to withholding in every such state. Neither statement treats the weak endpoint as guaranteed behavior under arbitrary tie-breaking.

## Proof

For revealed capture, utilities conditional on group `g` and capture `c∈{0,1}` are

```
U_g(W | C=c)  = −c d K
U_g(Ae | C=c) = A_ge(λ) − c f K.
```

The optimal accepted effort still maximizes `A_ge(λ)`: the false-flag term is independent of effort. Hence revealing `C` does not change either effort threshold (`3/5` for H, `14/15` for E) conditional on acceptance.

In the positive-probability state `(H,C=0)`, all consequence terms vanish. Therefore the best accepted action has utility

```
A_H(λ) = max(9/50−λ/4, 3/25−3λ/20),
```

whereas withholding has utility zero. For `λ>4/5`, `A_H(λ)<0`; consequently every best response withholds in that state, regardless of `K`, error protection, captured-case behavior or positive ex ante enrollment surplus. Full access is impossible.

At `λ=4/5`, `A_H=0`, so in the uncaptured H state acceptance and withholding tie. A weak full-access policy exists, but no `K` can make acceptance strictly preferable there.

For `0≤λ<4/5`, select `K=0`. Then both groups strictly prefer acceptance in both capture states. At `λ=4/5`, the same contract yields weak acceptance in both groups. It satisfies consequence and false-flag protection constraints trivially. Mean enrolled utility is `m(λ)`, which is at least `11/100>0` throughout `0≤λ≤4/5`. The maximum current gross care/review cost on this interval is `27/50+1/40=113/200<3/4`. Thus ex ante participation and the current-period budget do not obstruct sufficiency. ∎

## Constructive comparison at λ=1

With hidden capture and `K=1/10`, H acceptance exceeds withholding by `1/200>0`; both groups accept at high effort. Mean enrollment utility is `7/80`, current gross cost is `29/50`, and expected false-flag loss on appropriate acceptance is `1/200≤ε`.

Reveal the same capture draw before acceptance while retaining the same `K`. Then:

| Information state | Best accepted utility | Withholding utility | Strict decision |
|---|---:|---:|---|
| E, C=0 | 43/200 | 0 | Accept, high effort |
| E, C=1 | 41/200 | −2/25 | Accept, high effort |
| H, C=0 | −3/100 | 0 | Withhold |
| H, C=1 | −1/25 | −2/25 | Accept, high effort |

All four states have probability `1/4`. The optimal revealed-capture policy therefore completes `3/4` of eligible opportunities, compared with full completion under hidden capture. Its mean enrollment utility is `19/200>0`, so the access loss is not caused by exit at enrollment. This example only demonstrates the information-timing mechanism; selected outcomes and realized spending differ because its optimal policy withholds one state. It is not a matched-realized-spending or population-welfare comparison.

## Limits and implications

- Positive ex ante enrollment surplus cannot repair a strictly dominated action in the uncaptured H state. Expected utility is separable across information states here: no cross-state commitment restriction, capacity coupling or reward for a complete portfolio links them. The provider is free to condition its action on known capture status.
- For any fixed `0≤q<1`, provided uncaptured H has positive probability, the same necessity argument applies. The calibrated sufficiency and budget statements above use `q=1/2`. At `q=1` there is no uncaptured state, so this limitation does not apply; hidden and revealed capture provide identical acceptance-relevant information.
- The result does not apply unchanged if an uncaptured opportunity can later be independently rediscovered with positive probability, if a population-level instrument carries a consequence despite missing capture, or if another binding action-independent obligation changes payoffs. Those are different detection or contract models and need re-solving.
- Revealing capture only after acceptance is irrevocably selected does not create this acceptance loophole. If acceptance can be reversed after that revelation, the reversal must be modeled as an additional decision.
- A later independent outcome record, a patient complaint or a valid linked referral might defeat the zero-detection premise in reality. Conversely, a signed record of captured cases alone does not establish that uncaptured cases can be detected.
- The protection condition bounds the ex-ante expected consequence for a predetermined compliant action before capture is known. It does not bound expected loss conditional on acceptance after selection by visible capture. For example, at λ=1,K=1/5 the visible policy has loss 1/100 per eligible opportunity but 1/75 per accepted case and 1/50 per accepted H case. These are different denominators. It does not prove that clinical refusals, referrals, patient choice or resource constraints are classified safely.
- An equally informed conventional contract has the same theorem. This is a monitoring/information limitation, not evidence of a blockchain or VANTHEON-specific advantage. Its research value is to expose a necessary assumption and prevent an overstated guarantee; the related monitoring, inspection and provider-selection research is summarized in [References](REFERENCES.md).
- The provider proxy remains separate from patient rights, actual PTI estimation, payment, routing and Codex voting. The result supplies a possible evidence requirement for a proposed rule without narrowing VANTHEON's Pool/Treasury, Trust-Time, agent/robot or THEON foundation-model and RSI program.