A Finite Certificate for the Positive $n=11$ Vasc Inequality

Abstract

We establish the Vasc cyclic inequality for all strictly positive real eleven-tuples by a finite exact certificate. Cyclic rotation places a minimum at the first position, and rank words together with cumulative gaps reduce the problem to $10!=3{,}628{,}800$ homogeneous integer polynomials of degree eleven. Independent coefficient reconstruction proves $3{,}358{,}617$ roots directly. The remaining $270{,}183$ roots are covered by $267{,}952$ ordinary midpoint certificates, $2{,}195$ complete binary subdivision certificates, and $36$ complete certificates containing partial sorting nodes. Each nontrivial leaf is checked by subtracting weighted AM-GM midpoint circuits from a positive multiple of the transformed polynomial and verifying every residual coefficient. Explicit word-set comparisons certify disjointness and exact coverage. We prove the soundness of the leaves and both subdivision rules, then recover the rational inequality by positive denominator clearing and continuity. The resulting proof combines explicit mathematical reductions with independently checked exact integer certificates.