OpenAI · 113-02 · 2026-09-23 · Theoretical computer science
Entropy and Face Dimension of the Perfect-Matching Polytope
preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026
Family 113. Approximate counting and the perfect-matching entropy conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points.
We prove the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant. For every feasible vector x of perfect-matching edge marginals in a loopless labelled multigraph on vertices, the maximum entropy of a matching law with marginals x satisfies where and . This bound holds throughout the polytope, including its boundary. We also prove the sharp bound , where Fx is the minimal face of the perfect-matching polytope containing x.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 113: Approximate counting and entropy of perfect matchings. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.