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One-sample matroid prophet inequalities against an almighty adversary

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

For every finite matroid known in advance, gives a distribution-independent online rule using one independent sample per element and earning a universal constant fraction of the expected offline optimum. Values are independent and nonnegative, with finite expected optimum. The guarantee holds even when the arrival-order adversary sees all samples, values, and the rule’s entire random seed; no polynomial-time implementation is asserted. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments

The paper states

Theorem 1.1

In One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary, 23 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

There is a measurable online selection rule, independent of the value distributions, with the following property. Let M=(E,\mathcal I) be any finite known labeled matroid, and let (S_e,V_e:e\in E) be mutually independent nonnegative real random variables, with S_e and V_e having the same arbitrary law D_e. Assume \Ex[\OPT_M(V)]<\infty. The rule receives exactly the sample vector S before arrivals and uses a seed R independent of S,V.

For every measurable permutation rule \pi depending on M, the laws (D_e), S, V, and R, the rule accepts an independent set A, irrevocably deciding at each arrival using only the available history, and

\Ex\left[\sum_{e\in A}V_e\right] \ge 2^{-310}\Ex[\OPT_M(V)].
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
Papers and explanations
One current paper, readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary

23 September 2026, current edition.

Read it herePDF

We prove that one independent sample per element suffices for a constant-competitive prophet inequality on every finite matroid. The guarantee holds even when the arrival-order adversary observes all samples, all online values, and the algorithm’s entire random seed. The rule needs no description of the value distributions and achieves the absolute competitive ratio \(2^{-310}\).

Two Lean statements in this family

Choose one to see it beside the paper theorem it is paired with.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

MatroidProphet

The paper states

Theorem 1.1

In One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary, 23 September 2026

There is a measurable online selection rule, independent of the value distributions, with the following property. Let M=(E,\mathcal I) be any finite known labeled matroid, and let (S_e,V_e:e\in E) be mutually independent nonnegative real random variables, with S_e and V_e having the same arbitrary law D_e. Assume \Ex[\OPT_M(V)]<\infty. The rule receives exactly the sample vector S before arrivals and uses a seed R independent of S,V.

For every measurable permutation rule \pi depending on M, the laws (D_e), S, V, and R, the rule accepts an independent set A, irrevocably deciding at each arrival using only the available history, and

\Ex\left[\sum_{e\in A}V_e\right] \ge 2^{-310}\Ex[\OPT_M(V)].

The words they share: arrival, finite, independent, labeled, matroid, nonnegative, online, rule, sample, seed, selection, value, vector.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

One-sample matroid prophet inequality against an almighty adversary

MatroidProphet.lean, the release lists it as a supporting statement.

theorem one_sample.{u} : OneSampleChallenge.{u} := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

MatroidSecretary

The paper states

Theorem 1.1

In One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary, 23 September 2026

There is a measurable online selection rule, independent of the value distributions, with the following property. Let M=(E,\mathcal I) be any finite known labeled matroid, and let (S_e,V_e:e\in E) be mutually independent nonnegative real random variables, with S_e and V_e having the same arbitrary law D_e. Assume \Ex[\OPT_M(V)]<\infty. The rule receives exactly the sample vector S before arrivals and uses a seed R independent of S,V.

For every measurable permutation rule \pi depending on M, the laws (D_e), S, V, and R, the rule accepts an independent set A, irrevocably deciding at each arrival using only the available history, and

\Ex\left[\sum_{e\in A}V_e\right] \ge 2^{-310}\Ex[\OPT_M(V)].

The words they share: arrival, finite, independent, labeled, matroid, nonnegative, online, rule, sample, seed, selection, value, vector.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Hidden-vector matroid selection with worst-order guarantee

MatroidSecretary.lean, the release lists it as a supporting statement.

theorem one_sample.{u} : MatroidProphet.OneSampleChallenge.{u} := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

The file states two more theorems: hidden_vector, secretary.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

Their note: None; the hidden-vector selection result (constant 2^{-293}) is an extra supporting statement.

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).

“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalization proves a one-sample matroid prophet inequality with expected reward at least $2^{-310}$ times the expected offline optimum. Each element has one independent sample paired with an identically distributed nonnegative online value, and all coordinates are independent. The rule and finite seed law depend only on the labeled matroid; choices are irrevocable and feasible after every prefix. The arrival order may depend measurably on all samples, values, and the seed, and only the offline optimum must be integrable.

A supporting hidden-vector selection result for fixed nonnegative matroid weights gives expected worst-order reward at least $2^{-293}$ times the optimum, again with a finite seed law and prefix feasibility.

AI-drafted comparisons, with no recorded human review

  • One-sample matroid prophet inequality against an almighty adversary MatroidProphet

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem with the paper’s constant: a one-sample matroid prophet inequality with reward at least 2^{-310} times the expected offline optimum against an adversary choosing the arrival order.

  • Hidden-vector matroid selection with worst-order guarantee MatroidSecretary

    Unreviewed AI draft: states a supporting result. States the supporting hidden-vector selection result for fixed matroid weights (worst-order reward at least 2^{-293} times the optimum), not the prophet inequality itself.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

One-sample matroid prophet inequality against an almighty adversary MatroidProphet

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Hidden-vector matroid selection with worst-order guarantee MatroidSecretary

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

Follow-up work by others

  • Ryan Shea, shea256/matroid-prophet-inequality: a sharper one-sample matroid prophet guarantee (family 111), 10 October 2026. Source

    One-sample matroid prophet guarantee improved above 51/50 * 2^-31, from an initial 2^-36.

  • bottlebrushes, bottlebrushes/math: Lean sharpening of the one-sample matroid prophet constant (family 111), 8 October 2026. Source

    Lean-checked one-sample matroid prophet constant sharpened from 2^-310 to 2^-38 over five commits.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

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Offer to write an explanationCompare one Lean statement with its paper: MatroidProphet, MatroidSecretarySuggest a source attributionSuggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a algorithm and rates its significance, if it holds up, as “A strong professor’s paper of the year”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form
  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 111 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Compare the proposed paper and Lean matches

  • Family 111 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form