# When better results hide missing care

Research supplement · 23 October 2026  
VANTHEON · Alejandro S. Méndez Valentín · x/acc labs

This package accompanies a public research note on provider incentives, incomplete review and the timing of information. It contains the original finite-model proof, a risk-adjustment extension, exact Python evaluators and their outputs. The purpose is to make a limited result inspectable and reproducible.

**These are exact expectations of an illustrative mathematical model, not patient measurements.** No patient data, clinical trial, RL training or trained-model evaluation is included. The result applies established economic principles to a specified contract; it does not establish new general economic theory, an operational PTI formula or clinical benefit. The note has not undergone external peer review. Research, computation and editorial preparation used AI assistance.

## Why this belongs to VANTHEON

VANTHEON is building the foundations of a new healthcare system connecting care, evidence, learning and funding across people, institutions, AI agents and robots. Its proposed primitives give those responsibilities a shared structure: care events and obligations, service-specific provider trust, Codex governance, Trust-Time capacity, Pool/Treasury and THEON healthcare intelligence.

This study tests a requirement for that architecture. A record of completed care cannot alone establish which eligible opportunities were omitted, and an average-correct risk benchmark can conceal subgroup errors. The findings motivate evaluating opportunity visibility and case-appropriate comparisons before evidence is used for incentives or learning. They do not demonstrate a VANTHEON solution, clinical improvement, complete ascertainment or trained-agent performance. The proposed response and the proof's scope remain separate.

## Start here

- [Model and proof overview](model-and-proof.md): question, findings, assumptions and reading order.
- [Finite model and proof](finite-model-and-proof.md): payoffs, action choices, complete constraints, frontier, ties and exit limitation.
- [Capture-timing proof](capture-timing-limitation.md): what changes when review capture is known before acceptance.
- [Risk adjustment](risk-adjustment/README.md): correct-adjustment control, specified error path and aggregate-calibration counterexample.
- [References and comparison](REFERENCES.md): closest inspected economic and computational research.

## Reproduce

Use Python 3.9 or later. Only the Python standard library is required; keep the two scripts beside each other. From this package directory, run:

```sh
python3 -B exact_solver.py --output reproduced
python3 -B visible_capture.py --output reproduced
```

The first command reports **173 passed assertions**. The second reruns those baseline checks and adds **140 visibility checks**, for **313 distinct assertions in total**, not 486. It writes a fresh `reproduced/` directory, preserving the published `results/` files.

The arithmetic uses `fractions.Fraction`. The base analysis intersects incentive and feasibility constraints, independently enumerates half-plane vertices, and checks reduced utility formulas against a finite event tree. The capture-visible analysis checks choices at each positive-probability information state. No Monte Carlo sampling or random seed is needed. The written proof and executable checks are not Lean kernel verification.

The public package was tested with the Python version in [the reproduction receipt](reproduction-receipt.json). Both scripts were reproduced from this directory, and all five generated output files matched the published copies byte for byte. [SHA256SUMS](SHA256SUMS) records bundle-relative file hashes.

## Files and units

| File | Contents |
|---|---|
| `exact_solver.py` | Exact finite-model evaluator and baseline assertions |
| `visible_capture.py` | Capture-visible extension; imports the baseline evaluator |
| `results/exact-results.json` | Parameters, utility lines, feasible regions, boundary ties, policies and exit checks |
| `results/visible-capture.json` | Statewise visible-capture choices, frontier, outcomes and conditional false-flag burdens |
| `results/breakpoints.csv` | Exact best-response regions and boundary utilities |
| `results/canonical-policies.csv` | All enumerated pure best-response policies at the listed parameter points, including ties |
| `results/visible-capture-states.csv` | Four information states in the article’s visible-capture comparison |

Fractions are stored as strings such as `"31/40"`. They are exact model values. Any decimal percentages in the article are presentation conversions. Utility, cost and payment quantities use arbitrary model units, not dollars or clinically estimated weights. A field named `reject` in the executable representation denotes the model's avoidable-withholding action; it does not describe lawful refusal, patient choice or justified clinical escalation.

## Reproduce the article’s numbers

Both article comparison regimes use `λ = 1`, `K = 1/5`, `q = 1/2`, the same case distribution, payments, error rates and available current budget. Realized spending differs because the optimal policies complete different amounts of care.

The unknown-capture row is `policies_at_key_points["lambda=1,K=1/5"][0]` in `results/exact-results.json`. The known-capture row is `lambda1_Konefifth[0]` in `results/visible-capture.json`.

| Quantity | Capture unknown at choice | Capture known at choice |
|---|---:|---:|
| Completion among eligible opportunities | 1 | 3/4 |
| Success among completed cases | 31/40 | 5/6 |
| Successful completion per eligible opportunity | 31/40 | 5/8 |
| Current care and review spending | 29/50 | 9/20 |

The two success measures have different denominators. Completion alone is not a patient-welfare calculation. The original frontier (recovered at θ=1 in the new phase diagram) uses the admissible class of bounded consequences, rather than a single fixed `K`: its exact endpoints are `19/15` with unknown capture and `4/5` with known capture. An accepting best response exists at each endpoint, but withholding can tie there. Strict acceptance requires staying below the boundary.

The timing proof additionally gives a constructive example at `K = 1/10`. Its payoffs and false-flag costs differ from the article's `K = 1/5` example; its acceptance pattern is the same. Both calibrations are stated explicitly in the proof.

## Scope and rights

The provider is risk neutral, all modeled opportunities are appropriate, willing and feasible, capacity is sufficient, and future opportunity value is an affine assumption. In the capture-visible regime, an uncaptured opportunity has no later discovery channel. There is no free post-case escape from the modeled consequence. Changing these assumptions requires re-solving the model. No authority to fine providers, withhold earned pay or deny care follows from these equations.

PTI concerns accountable providers, not patients. Care payment, training reward, assignment and Codex voting retain separate rules. The calculation supplies no systemwide Pool/Treasury, Trust-Time, THEON foundation-model or recursive self-improvement validation.

Copyright remains with the original authors. Public release of this supplement does not change existing rights or licenses. No new software license is granted by this package; third-party works remain subject to their own rights.

## Risk-adjustment extension

Correct reference-risk adjustment removes the original pattern at λ=1 and K=1/5. A specified error path restores it only beyond its threshold. A separate λ=2 example matches the population's expected success (77.5%) but assigns wrong subgroup benchmarks (80% and 75% instead of 95% and 60%). Completion then falls from 100% to 75% while the adjusted residual among completed cases rises from 0 to +5 percentage points. See [the extension](risk-adjustment/README.md) for assumptions and exact results.

From the package root:

```sh
python3 -B risk-adjustment/risk_adjustment.py --output reproduced-risk
python3 -B risk-adjustment/independent_solver_audit.py \
  --original exact_solver.py \
  --risk-adjustment risk-adjustment/risk_adjustment.py \
  --output reproduced-risk/independent-audit.json
```

The risk evaluator performs 1,026 parameterized checks; the separate audit performs 1,726 consistency and provenance checks. These counts are software verification, not independent scientific findings or clinical evidence. The written proof covers the stated parameter families. All three output files reproduce byte for byte. No CMS dataset was used. The ratings discussion cites external historical studies and current CMS documentation, not model validation.

## Figure reproduction

The current three figures use an expected-count grid, an exact feasibility phase diagram and paired calibration bars. Their complete generator, exact inputs, fonts and environment are in [figure-source-v2/README.md](figure-source-v2/README.md). The original two-figure source is retained in [figure-source/README.md](figure-source/README.md) for provenance. Website SVG, PDF and PNG exports are in `figures/`; `cohort`, `risk-frontier` and `calibration` identify the current figures. These are exact model expectations, not sampled patient data. Scroll reveals change visibility only, then settle into the complete chart. Static and reduced-motion views show all data immediately.

[Citation](citation.bib) · [Published research note](https://vantheon.io/writing/when-better-results-hide-missing-care/)