| import OAI.Combinatorics.MatchingEntropy.ExpansionSequences | |
| /-! | |
| # Sharp analytic bounds for perfect-matching entropy | |
| For a finite loopless labelled multigraph on `2m` vertices, with `m ≥ 1` and a perfect | |
| matching, every point of the matching polytope satisfies | |
| `F - (2 - 2 / m) * B ≤ H ≤ F`. Here `F` and `B` are the coordinate entropy and | |
| complement entropy with natural logarithms, and `H` is the attained supremum of entropy | |
| over all perfect-matching laws with the prescribed mean. | |
| The development includes the odd-cut description of the matching polytope, sharp face | |
| rank bounds, minimal faces, fractional and connected-component reductions, attained | |
| entropy and weighted partition objectives, matching counts, degenerate cases, colored | |
| triangle-expansion families, and concrete coefficient obstructions. | |
| -/ | |