The Mahler conjectures, functional inequalities and polar-product symplectic width
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product \(K\times K^\circ\) in dimension \(2n\) has Gromov width 4. (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.
The two independent assessments disagree on how much of the result described above is expressed by the Lean statements.Read the assessments
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.
Five other possible matches are listed below the card, among them Corollary 1.2(Functional Mahler inequality) in The Mahler Conjecture for General Convex Bodies and Theorem 1.1(General Mahler conjecture) in The Mahler Conjecture for General Convex Bodies.
For every integer n\ge1 and every origin-symmetric convex body K\subset\R^n,
|K|\,|K^\circ|\ge\frac{4^n}{n!}.
Equality holds if and only if K is a linear Hanner body.
Five more possible matches
Corollary 1.2(Functional Mahler inequality)in The Mahler Conjecture for General Convex Bodies, 22 September 2026.
Theorem 1.1(General Mahler conjecture)in The Mahler Conjecture for General Convex Bodies, 22 September 2026.
Corollary 1.2(Even functional Mahler inequality)in The symmetric Mahler conjecture and its equality cases, 22 September 2026.
Theorem 1.1in Symplectic Balls in Symmetric Polar Products, 22 September 2026.
Corollary 1.3(Symmetric entropy–transport inequality)in The symmetric Mahler conjecture and its equality cases, 22 September 2026.
Reported proof verification
Four Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in ℝn has volume product at least \(4^n/n!\), with equality exactly for invertible linear images of Hanner polytopes.
We resolve the Mahler conjecture for general convex bodies positively. For every convex body \(K\subset\mathbb R^n\), n ≥ 1, with Santaló point \(s(K)\), \(|K|\,|(K-s(K))^\circ|\ge (n+1)^{n+1}/(n!)^2\), with equality exactly for simplices.
For every integer n ≥ 2 and every origin-symmetric convex body \(K\subset\mathbb R^n\), we prove that the Gromov width of \(\mathop{\mathrm{int}}\nolimits K\times\mathop{\mathrm{int}}\nolimits K^\circ\) is 4. We construct smooth symplectic embeddings of standard balls of every capacity \(0\lt c\lt 4\) into this polar product. Volume preservation then resolves the symmetric Mahler conjecture positively in every dimension.
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.
MahlerConjecture
The paper states
No paper theorem is paired with this Lean statement.
no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements it shares words with:
Theorem 1.1in The symmetric Mahler conjecture and its equality cases, 22 September 2026.
Corollary 6.2(The symmetric volume-product inequality)in The symmetric Mahler conjecture and its equality cases, 22 September 2026.
Theorem 1.1(General Mahler conjecture)in The Mahler Conjecture for General Convex Bodies, 22 September 2026.
Proposition 8.4(Classification of equality)in The symmetric Mahler conjecture and its equality cases, 22 September 2026.
Theorem 1.1in Symplectic Balls in Symmetric Polar Products, 22 September 2026.
Corollary 5.3(Symmetric Mahler inequality)in Symplectic Balls in Symmetric Polar Products, 22 September 2026.
Corollary 1.2(Even functional Mahler inequality)in The symmetric Mahler conjecture and its equality cases, 22 September 2026.
Proposition 8.1(Medians of an equality body)in The symmetric Mahler conjecture and its equality cases, 22 September 2026.
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.
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.
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.
SymmetricPolar.lean, the release lists it as a main result.
theoremsymmetric_polar_main {n : ℕ} (hn : 2 ≤ n) (K : Set (Position n)) (hK : IsSymmetricConvexBody K) : gromovWidth (polarProduct K) = 4 ∧ ∀ c : ℝ, 0 < c → c < 4 → HasSymplecticEmbedding (capacityBall n c) (polarProduct K) := bysorry
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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
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.
The source scan also found 2 theorem or lemma declarations whose names contain ‘manuscript’.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
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:Symmetric and general Mahler with all equality cases and the Gromov width 4 are stated; the sharp functional Mahler inequalities of the summary are not.
Definitions the group asks readers to compare with the paper:Custom but standard encodings: coordinatePolar (polar w.r.t. the dot product), IsHanner/hannerJoin/hannerProduct, volumeProduct := sInf (... '' interior K), HasSymplecticEmbedding (C^∞ on U, embedding, omega0 pullback condition) and gromovWidth := sSup {c | HasSymplecticEmbedding (capacityBall n c) U} with capacity π r^2
“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.
The symmetric Mahler conjecture predicts $|K||K^\circ|\ge4^n/n!$ for every origin-symmetric convex body $K\subset\mathbb R^n$. The formalization establishes this for every $n\ge1$ and characterizes equality exactly by invertible linear images of Hanner bodies, built from intervals using Cartesian products and convex-hull joins.
The nonsymmetric Mahler conjecture and the functional inequalities are not included.
The general Mahler conjecture gives a sharp lower bound for the volume product of a convex body and its polar. For every $n\ge1$ and compact convex body $K\subset\mathbb R^n$ with nonempty interior, the formalization proves
$\inf_{z\in\mathrm{int}\,K}|K|\,|(K-z)^\circ|\ge (n+1)^{n+1}/(n!)^2$.
Equality holds exactly when $K$ is a simplex. The paper’s functional inequality is outside this selected statement.
For an origin-symmetric convex body $K\subset\mathbb R^n$, $n\ge2$, the formalized result determines the symplectic ball capacity of $\mathrm{int}\,K\times\mathrm{int}\,K^\circ$: its Gromov width is $4$, and every ball of capacity $0<c<4$ embeds symplectically into it. The normalization assigns capacity $\pi r^2$ to a ball of radius $r$.
No boundary smoothness or strict convexity is assumed. An embedding at capacity exactly $4$ is not asserted.
AI-drafted comparisons, with no recorded human review
Symmetric Mahler inequalityMahlerConjecture
Unreviewed AI draft: states part of a paper’s theorem. States the inequality |K||K°| ≥ 4^n/n! of the main theorem for every origin-symmetric convex body. The equality characterisation by Hanner bodies, the other half of the theorem, is the separate statement SymmetricMahlerEquality.
General Mahler inequality and simplex equality casesGeneralMahler
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the general Mahler paper: the volume product, minimised over interior centres, is at least (n + 1)^{n+1}/(n!)², with equality exactly for simplices. The functional inequality is outside the statement.
Unreviewed AI draft: states part of a paper’s theorem. States the equality case of the main theorem: the volume product equals 4^n/n! exactly for linear images of Hanner bodies. The inequality itself is the separate statement MahlerConjecture.
Gromov width of symmetric polar productsSymmetricPolar
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the symplectic paper: for origin-symmetric K in dimension n ≥ 2, int K × int K° has Gromov width 4, every ball of capacity below 4 embedding symplectically.
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.
Symmetric Mahler inequalityMahlerConjecture
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original 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.
Other people’s accounts of running proof-verification software, with links to their reports.
SUNNY99, Independent Comparator/Lean audit of two OpenAI mathematical results, 7 October 2026. Source
Symmetric Mahler inequality: Comparator exit 0, Lean kernel accepts, standard three axioms, 11:21.19 wall time.
SUNNY99, Independent Comparator/Lean audit of two OpenAI mathematical results, 7 October 2026. Source
SUNNY99’s Comparator replay accepted OAI.SymmetricMahler.symmetric_mahler with exit 0, the default Lean kernel only and the three standard axioms, in 11 min 21 s. It ran at openai/math adc7f124; the challenge file, its configuration, lean-toolchain and lake-manifest are byte-identical at fd4aeeb2.
SUNNY99, Independent Comparator/Lean audit of two OpenAI mathematical results, 7 October 2026. Source
SUNNY99’s Comparator replay accepted OAI.SymmetricPolar.symmetric_polar_main with exit 0, the default Lean kernel only and the three standard axioms, in 47 min 52 s, at adc7f124, where the challenge file is byte-identical to the pinned release.
Other proofs and exact calculations
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the finite certificates of Appendices 07-08 (the 152-node profile enclosures and every printed node bound and finite sum; the stable-formula identities; the layer rectangles; the chord-error, tail-vector, interpolation and strip arithmetic; the segment threshold sums and sample tables; the two Bernstein certificates; the cubic and larger-width tables; the tail and short-width budgets) that support Proposition prop:scalar-input. Not the analytic interpolation/tail/segment-reduction arguments, and not sections 01-06 and 09, where the convex-geometric proof of the Mahler inequality lives”
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
Gil Kalai, Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans, 7 October 2026. Source
Maps the symmetric and general Mahler conjecture papers (equality cases Hanner polytopes and simplices) to their posts on Mahler conjectures and symplectic geometry.
seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source
Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.
SUNNY99, Independent Comparator/Lean audit of two OpenAI mathematical results, 7 October 2026. Source
A definition-by-definition statement check against Theorem 1.1 of the symmetric Mahler paper finds no added hypothesis or weakened conclusion; the challenge states the inequality for every n at least 1, and the equality classification is a separate challenge (SymmetricMahlerEquality).
SUNNY99, Independent Comparator/Lean audit of two OpenAI mathematical results, 7 October 2026. Source
A statement check against Theorem 1.1 of the polar-products paper finds a full match: the Lean spells out the symplectic form, capacity balls and Gromov width by hand because Mathlib has no symplectic API, and it claims embeddings for every capacity below 4, nothing at exactly 4.
SUNNY99, Independent Comparator/Lean audit of two OpenAI mathematical results, 7 October 2026. Source
A five-step reading map of the polar-product construction behind the Gromov width 4 result, in English and Chinese.
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 cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
Follow-up work by others
anxkhn, Series of 15 issues and 15 draft PRs adding release results to Formal Conjectures (issues 6972 to 6997, PRs 6977 to 7002), 8 October 2026. Source
Proposes adding the release’s symmetric Mahler inequality |K||K°| >= 4^n/n! to Formal Conjectures (issue 6972, PR 6977).
Martin Winter, A relation between Mahler volume and flag number for convex polytopes, 7 October 2026. Source
Winter shows the Mahler conjecture implies Kalai’s flag conjecture, so OpenAI’s claimed Mahler proof, if correct, would settle the flag conjecture.
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.
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 “Would win a Fields Medal”. The page says that nothing on it is verified.
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
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
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 087The 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
Address a question raised in an independent assessment
In The symmetric Mahler conjecture and its equality cases, 22 September 2026
How it was found
Its words were compared with the words of the Lean statement MahlerConjecture.
Words they share
symmetric, integer, convex, mahler, hanner, equality, linear, body
Review
Atlas records no review confirming that the two statements mean the same thing.
Why this passage is shown
Corollary 6.2 (The symmetric volume-product inequality)
In The symmetric Mahler conjecture and its equality cases, 22 September 2026
How it was found
Its words were compared with the words of the Lean statement MahlerConjecture.
Words they share
volume, symmetric, convex, product, inequality, body
Review
Atlas records no review confirming that the two statements mean the same thing.
Why this passage is shown
Theorem 1.1 (General Mahler conjecture)
In The Mahler Conjecture for General Convex Bodies, 22 September 2026
How it was found
Its words were compared with the words of the Lean statement MahlerConjecture.
Words they share
real, mahler, integer, convex, equality, body
Review
Atlas records no review confirming that the two statements mean the same thing.
Why this passage is shown
Proposition 8.4 (Classification of equality)
In The symmetric Mahler conjecture and its equality cases, 22 September 2026
How it was found
Its words were compared with the words of the Lean statement MahlerConjecture.
Words they share
symmetric, convex, hanner, equality, linear, body
Review
Atlas records no review confirming that the two statements mean the same thing.
Why this passage is shown
Theorem 1.1
In Symplectic Balls in Symmetric Polar Products, 22 September 2026
How it was found
Its words were compared with the words of the Lean statement MahlerConjecture.
Words they share
symmetric, integer, convex, polar, body
Review
Atlas records no review confirming that the two statements mean the same thing.
Why this passage is shown
Corollary 5.3 (Symmetric Mahler inequality)
In Symplectic Balls in Symmetric Polar Products, 22 September 2026
How it was found
Its words were compared with the words of the Lean statement MahlerConjecture.
Words they share
mahler, symmetric, integer, convex, inequality, body
Review
Atlas records no review confirming that the two statements mean the same thing.
Why this passage is shown
Corollary 1.2 (Even functional Mahler inequality)
In The symmetric Mahler conjecture and its equality cases, 22 September 2026
How it was found
Its words were compared with the words of the Lean statement MahlerConjecture.
Words they share
mahler, integer, convex, inequality
Review
Atlas records no review confirming that the two statements mean the same thing.
Why this passage is shown
Proposition 8.1 (Medians of an equality body)
In The symmetric Mahler conjecture and its equality cases, 22 September 2026
How it was found
Its words were compared with the words of the Lean statement MahlerConjecture.
Words they share
symmetric, convex, equality, body
Review
Atlas records no review confirming that the two statements mean the same thing.
Independent assessment
MathVet
Their term
“full”
What their term means
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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 202 s; peak memory 6.5 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the finite certificates of Appendices 07-08 (the 152-node profile enclosures and every printed node bound and finite sum; the stable-formula identities; the layer rectangles; the chord-error, tail-vector, interpolation and strip arithmetic; the segment threshold sums and sample tables; the two Bernstein certificates; the cubic and larger-width tables; the tail and short-width budgets) that support Proposition prop:scalar-input. Not the analytic interpolation/tail/segment-reduction arguments, and not sections 01-06 and 09, where the convex-geometric proof of the Mahler inequality lives”
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.