Proves the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension, and the scalar-dilation B-conjecture for all even log-concave Radon measures. For Lebesgue volume it also proves the additive Lp Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout \(0\lt p\lt 1\). (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 independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments
In The logarithmic Brunn–Minkowski conjecture
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
Let n\ge1 and let K,L\subset\R^n be convex bodies satisfying K=-K and L=-L. For every 0\le\lambda\le1,
In The logarithmic Brunn–Minkowski conjecture
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
Let n\ge1, let K,L\subset\R^n be full-dimensional origin-symmetric convex bodies, and let 0<p<1. For every 0\le\lambda\le1,
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.
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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.
LogBrunnMinkowskiThe paper states
Let n\ge1 and let K,L\subset\R^n be convex bodies satisfying K=-K and L=-L. For every 0\le\lambda\le1,
The words they share: bodies, brunn, convex, inequality, logarithmic, minkowski.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
LogBrunnMinkowski.lean, the release lists it as a main result.
theorem main (n : ℕ) (hn : 1 ≤ n) (K L : Set (Space n)) (hK : IsConvexBody K) (hL : IsConvexBody L) (hsK : OriginSymmetric K) (hsL : OriginSymmetric L) (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : (volume K) ^ (1 - t) * (volume L) ^ t ≤ volume (logCombination K L t) := by sorryThis 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
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
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 part of the advertised result (their term: “partial”).
Their note: Only the Lebesgue-volume log-Brunn-Minkowski inequality; the scalar-dilation B-conjecture for even log-concave measures and the L_p Brunn-Minkowski inequality for 0<p<1 are not stated.
Definitions the group asks readers to compare with the paper: wulff f := {x | ∀ u, ‖u‖ = 1 → ⟪x, u⟫ ≤ f u} and support K u := sSup (image of ⟪·,u⟫ on K) define the log combination by hand (standard Wulff-shape definition); ENNReal ^ with real exponent 1 - t on volumes
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).
The gap they name: The B-conjecture for every even log-concave Radon measure (the abstract derives it via Saroglou’s transfer theorem and a support-subspace reduction) and the L_p Brunn–Minkowski inequality for 0 < p < 1 (derivable from log-BM by a classical argument) are not stated.
“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
The logarithmic Brunn–Minkowski conjecture asserts that logarithmic interpolation of origin-symmetric convex bodies preserves the geometric-mean lower bound for volume. The formalized result proves, for every dimension $n\ge1$, such bodies $K,L\subset\mathbb R^n$, and $0\le t\le1$, that their logarithmic Wulff combination has volume at least $|K|^{1-t}|L|^t$. It assumes neither boundary smoothness nor coordinatewise unconditionality.
Logarithmic Brunn–Minkowski inequality LogBrunnMinkowski
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension. The L_p corollary is not part of the statement.
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.
LogBrunnMinkowskiThree 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. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 630 s). | 10 October 2026 | Original report |
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.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
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 origin-symmetric logarithmic Brunn-Minkowski inequality to Formal Conjectures (issue 6973, PR 6978).
Yude Liu, Qiang Sun, Ge Xiong, The centered logarithmic Brunn--Minkowski inequality, 7 October 2026. Source
Liu, Sun and Xiong extend the logarithmic Brunn-Minkowski inequality from origin-symmetric bodies, as proved by OpenAI, to bodies with centroid at the origin, with equality cases.
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
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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 “Career-defining publication”. The page says that nothing on it is verified.
Author not stated. Source
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Why this passage is shown
Theorem 1.1 (Even logarithmic Brunn–Minkowski inequality)
In The logarithmic Brunn–Minkowski conjecture
Why this passage is shown
Corollary 1.2 (Symmetric L_p Brunn–Minkowski inequality)
In The logarithmic Brunn–Minkowski conjecture
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, LogBrunnMinkowski
Proof-verification report
MathVet, LogBrunnMinkowski
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, LogBrunnMinkowski
Lean statement
LogBrunnMinkowski