# Optimal transport sensitivity benchmark and audit

Edition 9 October 2026. Current decision: **Prototype diagnostic**. Prices and product uses remain hypotheses. No independent source-proof verification or buyer validation has been completed.

## Research finding

For a uniform compact convex source in dimension at least two, the source claims target-uniform one-third Holder stability of quadratic optimal maps. Its cube example uses a central atom of mass a, tilted by b=a/2: W2^2=ab^2 while the source L2 map distance squared is b/2+ab^2. The reassigned source mass is b/2. This rules out every better uniform exponent.

## Problem and buyer

Transport researchers and teams evaluating distribution-to-map pipelines need controls that separate target proximity from map sensitivity. An assumed square-root rule can be overconfident even with only three target atoms.

## What the finding could enable

An exact counterexample benchmark can test whether a numerical method resolves a thin cell and reports the correct error quantity. A domain-assumption audit can then distinguish theorem-covered continuum maps from empirical, discrete or regularized output.

## Technical and commercial limits

The manuscript does provide an explicit domain constant: C_*^2=12dR^2(1+sqrt(162))^2+28L^2 P_K/volume(K), with P_K the sum of coordinate projection volumes. On K=Y=[-1,1]^2, the conservative rational upper bound C_*^2<=9464 gives C=98. This is source-based mathematics, independently unverified here. Thin domains can make the bound loose. A sample dataset is not the required uniform continuum source; pointwise, entropic and arbitrary learned-map guarantees need separate results.

## Minimal architecture

Source/domain contract -> rational affine-cell geometry -> target and map error references -> solver adapter -> perturbation/mesh sweep -> evidence report. The implemented reference uses exact halfplane clipping and polygon areas for the two-dimensional source example. Extra cube coordinates factor out as stated by the paper. It fits no arbitrary transport map and computes no general target-distance optimization.

## Existing alternatives and differentiation

Python Optimal Transport supplies substantial transport implementations. Differentiate by an independently reproducible sharpness control, named assumptions and a measured solver/mesh failure mode. A generic solver wrapper has little novelty.

## Monetization hypothesis

Hypothesis: AUD 2,000-8,000 for a research-method assessment, followed by a supported adapter only if the workflow recurs. At AUD 6,000, 25 assumed hours at AUD 180 leave AUD 1,500 before other costs. No willingness to pay, predictive benefit or profitability is established.

## Validation experiment

131 finite checks reproduce masses and map integrals across rational parameters, the exact target distances, half-exponent divergence controls and budget rejection. For a=4^-j the required square-root-exponent constant squared is 2^(j-1)(1+a^2), while the sixth power of the one-third ratio is (1+a^2)^3/16. Next compare one existing solver across parameter/mesh scales and preserve solver error separately from theorem claims.

## Conditions to reject or defer

Defer a certified numerical product when solver or source correspondence is missing, the domain constant is useless for the intended question, or the buyer cannot act on the report. Successful reproduction alone is not evidence of a business.

## Next concrete action

Build a solver benchmark manifest using the exact cells as references. Record discretization, regularization, stopping criteria and the actual decision the report would inform.

## Source evidence

Repository sources are pinned to revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. Selected scope notes and manuscript statements have been reviewed to the extent described above; these links do not represent successful kernel checks.

### Family 374

Subject: Sharp one-third stability of Brenier maps.

- [Sharp One-Third Stability of Brenier Maps](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf)
- [Formal scope notes](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/374.md)

## Current alternative sources

Primary documentation reviewed on 8 October 2026. Product availability demonstrates alternatives, not demand or willingness to pay for this proposal.

- [Python Optimal Transport](https://pythonot.github.io/)

## New local evidence

[Computational feasibility](../../FEASIBILITY-2026-10-09-v5.md), [semantic comparison](../../plans/definition-semantics-review-2026-10-09-v2.md), [eighteen utilities](../../PROTOTYPES-2026-10-09-v6.md). All sources remain pinned to the cited revision.