Monotonicity of moment ratios for the six-variable Dumont-Foata polynomials: a computer-assisted argument with exact certificates

We study the six-variable Dumont-Foata moments m_k = (J^k)_00, where J is the Jacobi matrix with b_n = 2n^2 + (X+Y+Z-1)n + delta and lambda_n = n(n+X-1)(n+Y-1)(n+Z-1) (Josuat-Verges, Sem. Lothar. Combin. 64, eq. 1.5). We give a computer-assisted argument that m_k / m_(k-1)^2 strictly decreases in each of the six variables on [1/2, 1]^6 for every k >= 3. The claim reduces to two inequalities. L1, Q_k = 2(k-1) m_k m_(k-2) - k m_(k-1)^2 > 0, is proved for all k (Stieltjes moment theory, a Hankel determinant formula and Cauchy-Schwarz, with exact finite checks) and independently replayed. L2, P_k < 0, is certified by exact rational arithmetic over the whole parameter box for k = 3 to 20. For k >= 21 it is covered by an asymptotic ledger whose total, about 139.20, stays below the available budget of 680, with every row bounded for all k >= 20. Two natural routes were refuted exactly, and the record includes their failure laws. A step condition (S), equivalent to positivity of certain Hankel minors, fails at (Y, Z, delta) = (1, 1, 5/4) from n = 70; its failure law predicted, before computation, the exact failure at n = 45 at a new point. A real-rootedness condition (RR_A) fails there at every n checked from 163 to 300, shown by exact violations of Newton's inequalities. Both failures trace to a 2x2 model chain whose eigenvalues coalesce on the line Y = Z, an exceptional point in Kato's sense; the completeness of the failure laws and this mechanism are conjectures with evidence, labelled as such. Status: every step is written, but the argument for large k is conditional on five conditions stated in the route note (paragraph 35), which are reserved for expert review. No professional mathematician has yet reviewed any part of it, and an independent clean-room replication is still outstanding. Method and credit: designed and directed by the author. Most of the computation was carried out with AI systems (Claude and Gemini, with Grok supporting) and checked by exact arithmetic: standard-library Python, hashed scripts, and negative controls that must fail. Claude wrote both sides of most automated checks, which limits their independence. Contents: README, the L1 proof, the route note, the ledger notes (items 128 to 142a), the refutation note, a theory note and a list of open questions.