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Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

For every positive dimension n, a compact convex body with nonempty interior, barycenter 0 and no interior lattice point other than 0 has volume at most (n+1)ⁿ/n!. A centered simplex attains equality. This is the inequality, separate from the later equality classification.

Sources for this record

What was already known

Ehrhart’s sharp volume conjecture was open in general dimension. Earlier special-dimensional and Fano-geometric bounds did not establish this convex-body inequality.

AI involvement AI-led

OpenAI attributes the proof and Lean formalization to Astra. Human staff selected the problem and prepared the manuscript; specialists reviewed portions of the analytic argument.

OpenAI’s original disclosure assigns the argument and subsequent Lean formalization to Astra; human manuscript preparation is separately credited.

Importance Major

If the formal statement matches the intended body class, the result settles a longstanding all-dimensional volume conjecture.

Limitations

The project reports a Lean certificate for the encoded inequality. A specialist reviewer requested major revision of the manuscript’s uniform asymptotics and limiting-slope steps; no counterexample was reported. The formal-to-published interpretation still needs specialist review.

Technical verification Formal (Lean 4.32.0 with Mathlib and Comparator + human)

The project reports a complete Lean certificate for the stated formal theorem. Human assessments have narrower scope; semantic matching and any manuscript gaps remain separately noted.

Independent checking

chapter-specific mathematical review of the published argument, with the limited coverage summarized in the audit

Checker
Chapter 8 human reviewers in the Sienicki et al. review corpus
Scope
partial analytic review of the inequality; major-revision reservation
Basis
Reported by the linked checker
Evidence source

Formal checking

Lean 4.32.0 formal certificate in EhrhartVolumeInequality.lean; comparator target for Ehrhart.Volume.ehrhart_volume_inequality_for_sets; manifest reports zero sorry and only standard axioms

Checker
OpenAI ten-proofs project, Lean kernel and Comparator configuration
Scope
sharp volume inequality under convexity, compactness, barycenter and interior-lattice hypotheses
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Partial

Human reviewers explain the potential, lattice Bergman-space and slope strategy, but one specialist review finds the uniform asymptotics and limiting slope passages too compressed to regard the analytic mechanism as fully explained.

Correction and publication history