The sharp threshold for the multiplicative Merino–Welsh inequality on matroids

This preprint determines the least nonnegative x for which T_M(x,0)T_M(0,x) ≥ T_M(1,1)^2 holds for every finite matroid M without loops or coloops. The threshold is x_* = 2.2266815969…, the largest real root of x^3 = 9(x−1). A quantitative bound is proved for 2 ≤ x ≤ x_*, and parallel doublings of uniform matroids show that its exponential rate is sharp throughout this interval. The proof uses a weighted broken-circuit comparison, the Cunningham–Edmonds decomposition, and exact polynomial checks of a finite rational replacement table. The multiplicative Merino–Welsh conjecture for graphs at x=2 remains open. Data and verification code: https://github.com/michaeliu4/tutte-threshold/releases/tag/V1.1