Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 113-02. Entropy and Face Dimension of the Perfect-Matching Polytope
Manuscript map · lines 4665–4674
OpenAI@2026-10-07#rem:oai-p-113-02
Authors: OpenAI. Family 113.
Title: Entropy and Face Dimension of the Perfect-Matching Polytope.
Abstract: 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 $2m{}ge2$ vertices, the maximum entropy $H(x)$ of a matching law with marginals x satisfies ${}displaystyle F(x)-(2-2/m)B(x){}le H(x){}le F(x),$ where $F(x)=-{}sum_e x_e{}log x_e$ and $B(x)=-{}sum_e(1-x_e){}log(1-x_e)$. This bound holds throughout the polytope, including its boundary. We also prove the sharp bound $|{}mathop{{}mathrm{supp}}{}nolimits x|-{}dim F_x{}le3m-2$, where Fx is the minimal face of the perfect-matching polytope containing x.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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