发表机构
Beijing University of Posts and Telecommunications（北京邮电大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过有限精确证书证明了正十一元组的 Vasc 循环不等式，利用秩词与累积间隙化简为整数多项式，结合中点与细分证书及系数验证完成证明。
AI 中文摘要
我们通过一个有限的精确证书，对所有严格正实数十一元组建立了 Vasc 循环不等式。循环旋转将最小值置于首位，秩词与累积间隙将问题简化为 $10!=3{,}628{,}800$ 个次数为十一的齐次整数多项式。独立的系数重构直接证明了 $3{,}358{,}617$ 个根。剩余的 $270{,}183$ 个根由 $267{,}952$ 个普通中点证书、$2{,}195$ 个完全二叉细分证书和 $36$ 个包含部分排序节点的完全证书覆盖。每个非平凡叶节点通过从变换后多项式的正倍数中减去加权 AM-GM 中点电路，并验证每个残差系数来检查。显式的词集比较证明了不相交性和精确覆盖。我们证明了叶节点及两种细分规则的可靠性，然后通过正分母清除和连续性恢复有理不等式。所得证明结合了显式的数学化简与独立检查的精确整数证书。
英文摘要
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.
