GPT-6 Astra proves Barvinok's log-concavity question for contingency tables on lines, giving lattice-point bounds for all totally unimodular polytopes
Jonathan Leake and Maryam Mohammadi Yekta's Sept 30, 2026 preprint gives a new lower bound on the number of lattice points of every totally unimodular polytope, and with it a deterministic approximate-counting algorithm. The key ingredient, resolving Barvinok's log-concavity conjecture for contingency tables on lines, "was proven using ChatGPT 6 Astra". The result also implies a conjecture of Ferroni and Higashitani on Ehrhart polynomials of unimodular polytopes.
Key facts
- arXiv 2609.39917 (math.CO), 'Log-concavity and Approximate Counting for Totally Unimodular Polytopes', Sept 30, 2026
- New class of 'variable-wise log-concave' (VLC) polynomials; bounds based on Gurvits capacity, so counting up to an explicit exponential factor by convex optimisation
- Barvinok's 2007 question was resolved in its 'line' version for all (totally) unimodular polytopes; the full-strength question for TU polytopes remains open
- AI declaration: 'The resolution of Barvinok's conjecture/question on the log-concavity on lines of the contingency tables counts (Theorem 1.4) was the main AI input to this paper. We generalized this result to all TU polytopes … and in fact the AI suggested in a remark that this should be possible.'
Science result
- Field
- mathematics / combinatorics / polytopes / counting
- Problem
- Barvinok's log-concavity conjecture for contingency-table counts (line version); Ferroni–Higashitani conjecture on Ehrhart polynomial evaluations
- Result
- Log-concavity on lines for lattice-point counts of unimodular polytopes, a lower bound for all TU polytopes and a deterministic approximate-counting algorithm.
- AI system
- GPT-6 Astra
- Human role
- Key theorem proven by ChatGPT 6 Astra; authors generalised it (following an AI-suggested remark), verified everything and wrote the paper
- Verification
- Unrefereed preprint
- Status
- pending
What happened
Barvinok asked in 2007 whether counts of contingency tables are log-concave as the margins vary. Leake and Mohammadi Yekta report that ChatGPT 6 Astra proved the "on lines" version, which they call the paper's main AI input. They extended it to all totally unimodular polytopes with entrywise bounds. That gives lattice-point lower bounds generalising earlier work on contingency tables and integer flows, and a deterministic approximate-counting algorithm. They also state a generalised conjecture.
Why it matters
It is a named question in a central area of combinatorial counting, with the core proof credited to the model. It is an unrefereed preprint, and we found no reactions as of Oct 5.
Changelog
- 2026-10-05: created (06:30 run; the sweep missed it and the 06:30 manual arXiv scan found it)
Related events
Sources (1)
id: 2026-09-30-barvinok-log-concavity-totally-unimodular-astra · updated 2026-10-05 · open in the interactive timeline