# Matroid online allocation research simulator

Edition: 8 October 2026. Initial decision: **Defer direct product**. The product interpretation and pricing are hypotheses; independent proof and buyer validation remain open.

## Research finding

The one-sample matroid prophet result guarantees only an absolute reward fraction 2^-310 and does not assert a polynomial-time implementation. Independence and matched sample/value distributions are part of the statement.

## Problem and buyer

A marketplace or resource-allocation researcher may want irrevocable choices under quotas with sparse historical information. Real demand can be correlated and can drift away from the independent-sample model.

## What the finding could enable

A research simulator could isolate structural insights and test better behavior in a restricted matroid family. The theorem currently supports a theoretical feasibility result, not a commercially meaningful minimum revenue fraction.

## Technical and commercial limits

The extremely small guaranteed fraction is around 4.8 times 10^-94. It does not imply useful empirical reward, learning under distribution shift or arbitrary quota systems. A feasible set family must actually satisfy matroid axioms.

## Minimal architecture

Matroid model validator -> independent distribution fixtures -> online-policy simulator -> offline optimum reference -> expected reward report. Keep correlations and adversarial arrival policies as separate tests.

## Existing alternatives and differentiation

Existing allocation policies and optimization modeling are substitutes. A restriction with a much stronger effective ratio is necessary to justify a business application.

## Monetization hypothesis

Hypothesis: a research partnership rather than a revenue-optimization SaaS. Commercial development is deferred until a practical rule and meaningful empirical improvement are established.

## Validation experiment

Compare partition-matroid cases with simple thresholds and greedy policies under independent and correlated values. Measure runtime and reward over many independent trials, not a single favorable run.

## Conditions to reject or defer

Reject the direct product if the ratio remains the only performance evidence or if the independence assumptions fail for all reachable buyers.

## Next concrete action

Inspect the hidden-vector selection construction for a implementable restricted case and record where the constant losses occur.

## 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 111

Subject: One-sample matroid prophet inequalities against an almighty adversary.

- [One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026/final.pdf)
- [Formal scope notes](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/111.md)