Exact Ollivier-Ricci curvature of the associahedron flip graph, and when it equals Forman curvature

The flip graph of triangulations of a convex n-gon is the 1-skeleton of the associahedron K_n, an (n−3)-regular graph on Cat_{n−2} vertices. We compute its Ollivier-Ricci curvature (uniform measure on closed balls) exactly. For the edge given by flipping a diagonal inside a quadrilateral Q, the curvature is κ = (2 − p)/(n − 2), where p is the number of sides of Q that are diagonals, proved by an explicit coupling and an explicit 1-Lipschitz potential. The exact mean curvature is (10 − n)/((n − 2)(2n − 5)): positive for n < 10, vanishing exactly at the decagon, and negative for n > 10. Edge by edge, κ = κ_F/(D + 1), where κ_F is the Forman curvature and D = n − 3. We explain this link by a general inequality for every simple polytope, a special case of a corollary of Jost and Münch, under which Ollivier and Forman curvature agree when every 2-face through an edge has at most six sides and provably part ways at faces with seven or more. We conjecture that the inequality is always an equality and support this with a preregistered locked run on all 140,107 edges of 513 simple polytopes in dimensions 2 to 8, including 18,388 edges where the cap is active, which found the gap to be exactly zero on every edge. Finite instances are certified in Lean 4 with kernel-checked decision procedures.