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Shareshian–Wachs elementary positivity

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

Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in \(\mathbb N[q]\). The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity. (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

Two problems have been reported by others; the atlas has not judged them.

What the papers state

Theorem 1.1 (Elementary positivity and a permutation witness)

In Elementary positivity of chromatic quasisymmetric functions, 24 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 deterministic terminating procedure which, given a natural unit interval graph G on [n] and \sigma\in D_G^0, returns a partition \theta_G(\sigma)\vdash n such that

\chi_G(X;q)=\sum_{\sigma\in D_G^0} q^{\ginv_G(\sigma)}e_{\theta_G(\sigma)}(X).

In particular every elementary coefficient of \chi_G(X;q) belongs to \N[q]. The assertion holds as a formal symmetric-function identity, uniformly in n and G.

Corollary 1.2 (Young permutation modules in Hessenberg cohomology)

In Elementary positivity of chromatic quasisymmetric functions, 24 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

Let n\ge1, let h be as above with graph G=G(h), and let S\in M_n(\C) be regular semisimple. Write

\operatorname{Hess}(S,h)= \{F_\bullet\in\operatorname{Flag}(\C^n): SF_i\subseteq F_{h(i)}\text{ for all }i\}.

For \lambda=(\lambda_1,\ldots,\lambda_\ell)\vdash n, let M^\lambda=\operatorname{Ind}_{S_\lambda}^{S_n}\mathbf{1}_{\C} be the Young permutation module, where S_\lambda=S_{\lambda_1}\times\cdots\times S_{\lambda_\ell} is the standard Young subgroup, and put

m_{\lambda,d}(h)= \#\{\sigma\in D_G^0:\theta_G(\sigma)=\lambda,\ \ginv_G(\sigma)=d\}.

Then for every integer d\ge0, under Tymoczko’s dot action on ordinary cohomology there is an abstract \C[S_n]-module isomorphism

H^{2d}(\operatorname{Hess}(S,h);\C) \cong\bigoplus_{\lambda\vdash n}(M^\lambda)^{\oplus m_{\lambda,d}(h)}.

No Lean statement in the release is paired with this theorem.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.

Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file 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 Lean statement in this family

Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.

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.

ElementaryPositivity

The paper states

Theorem 1.1 (Elementary positivity and a permutation witness)

In Elementary positivity of chromatic quasisymmetric functions, 24 September 2026

There is a deterministic terminating procedure which, given a natural unit interval graph G on [n] and \sigma\in D_G^0, returns a partition \theta_G(\sigma)\vdash n such that

\chi_G(X;q)=\sum_{\sigma\in D_G^0} q^{\ginv_G(\sigma)}e_{\theta_G(\sigma)}(X).

In particular every elementary coefficient of \chi_G(X;q) belongs to \N[q]. The assertion holds as a formal symmetric-function identity, uniformly in n and G.

The words they share: positivity, witness, permutation, natural, partition, interval, elementary, graph, unit, theta, chromatic.

Possible Lean counterpart

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

Elementary positivity for natural unit interval graphs

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

def elementaryPositivityWitness (n : ℕ)    (G : ElementaryPositivity.NaturalUnitIntervalGraph n) :    G.PermutationWitness := by  sorry

Comparator compares one definition by its name and type only: elementaryPositivityWitness.

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: elementaryPositivityWitness n G : G.PermutationWitness (a def closed by sorry, not a theorem) must supply theta : {σ // G.Nondescent σ} → Nat.Partition n with ∀ r, G.chromatic r = Σ_σ C(X^(graphInversions σ)) * esymmPart (theta σ) in MvPolynomial (Fin r) ℕ[X]: an explicit ℕ[q]-positive e-expansion for every natural unit interval graph (stronger than bare positivity). Chromatic quasisymmetric function and Hessenberg-function graphs are the standard definitions.

“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 elementary-positivity part of the Shareshian–Wachs conjecture asks whether the chromatic quasisymmetric function of every natural unit interval graph has nonnegative coefficients in the elementary basis. The formalization proves this over $\mathbb N[q]$. It constructs an explicit elementary-basis expansion indexed by the graph’s permitted nondescent permutations, with each term weighted by a nonnegative power of $q$ determined by its graph inversions. The expansion holds for every finite number of color variables.

AI-drafted comparisons, with no recorded human review

  • Elementary positivity for natural unit interval graphs ElementaryPositivity

    Unreviewed AI draft: states a paper’s theorem as the paper does. States elementary positivity over N[q] with the permutation-indexed expansion. The file has no theorem: Comparator checks the type of one definition hole, whose type is the witness structure carrying the expansion. That the partition is produced by a deterministic procedure, as the main theorem states, is not part of the checked type.

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.

Elementary positivity for natural unit interval graphs ElementaryPositivity

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

Other reports of proof verification

Other people’s accounts of running proof-verification software, with links to their reports.

  • From github.com/BenKnill/openai-math-hol-cards, 8 October 2026. Source

    Passed its release configuration, whose challenge leaves the witness body as sorry by design; whether the witness type expresses the intended theorem remains a separate reading task.

  • Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source

    The review reports that Comparator accepted OpenAI’s solution for this challenge, run end to end on Lean 4.34.1 with the default kernel and only the three standard axioms.

Problems reported by others

Reports of errors and corrections from outside readers, as they state them. The atlas has not judged them.

  • Jack McCarthy, OpenAI math: Comparator fixes (mirror of openai/math at fd4aeeb), 11 October 2026. Source

    The configuration had no theorem target, only a definition hole; the fork states the claim as a theorem that a witness exists (elementaryPositivityWitness_nonempty) and the release’s construction proves it.

  • Jack McCarthy, OpenAI math: Comparator fixes (mirror of openai/math at fd4aeeb), 9 October 2026. Source

    ElementaryPositivity challenge had no theorem target, so Comparator checked nothing.

Missing or incomplete source records

The program that collects Atlas’s source records, Statement Map, could not obtain or interpret some source material. These notes describe those limits; they do not identify mathematical errors.

  • ElementaryPositivity: Tracked source lean/OAI/Combinatorics/Chromatic/Main.lean unavailable: bytes were not captured at fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb

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.

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.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanationCompare the Lean statement with the paperSuggest 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 proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

    Author not stated. Source

  • tparker, Philip Weiss, Will Sawin, Nik Weaver, Pace Nielsen, mathworker21, Joshua P. Swanson, Timothy Chow, Daniel Soltesz, MathOverflow: how many of the released results went against the conventional wisdom?, 10 October 2026. Source

    The same expert has read this family in detail and finds the key step clever and quick, relating an existing programme to a highly structured setting with powerful tools, but not a new technique.

  • Alex Meiburg, Several of the comparator challenges in the OpenAI release are broken and hackable, 8 October 2026. Source

    The post notes a definitional hole here but judges it acceptable, because the main objective is a data-carrying term rather than a proposition.

  • Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source

    The review notes that the Lean statement leaves the rule as a definition hole that accepts any function meeting the expansion identity, so it states positivity but not the paper’s explicit rule.

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

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

Read the definitions the solution supplies

  • OpenAI’s proof file supplies some definitions. Atlas records no comparison confirming that they mean the same objects as the paper’s definitions. elementaryPositivityWitness Open the form

Identify the theorem the catalogue describes

  • Family 169 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 169 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form