对称轨道和不等式的碰撞正性在任意维数下的完整双变量判据
Collision Positivity for Symmetric Orbit-Sum Inequalities in Arbitrary Dimension: A Complete Two-Variable Criterion
AI总结：
本文证明任意维数下对称轨道和不等式 $P_n=J_\lambda^{(n)}+J_\mu^{(n)}-2J_\gamma^{(n)}$ 的全局非负性等价于固定双变量截面上的非负性，建立了秩二假设 $C_2\ge0$ 到全局正性的升阶路径，补充了先前三变量定理并给出独立证明。
AI中文摘要：
设 $\lambda\succ\gamma\succ\mu$ 为至多 $n$ 个部分、次数相等的非负整数指数向量，$J_\eta^{(n)}$ 为与 $\eta$ 相关的标记对称轨道和，并设 \begin{equation*} P_n=J_\lambda^{(n)}+J_\mu^{(n)}-2J_\gamma^{(n)}. \end{equation*} 对于每个 $n\ge4$，我们证明 $P_n$ 在正象限上的全局非负性等价于其在固定双变量截面 \begin{equation*} (x,y,1,\ldots,1),\qquad x,y>0 \end{equation*} 上的非负性。因此，该族的正性由固定的二维截面决定，与次数和背景维数均无关。在下面引入的单射层级 $C_k$ 中，假设为 $C_2\ge0$，全局正性等价于 $C_{n-1}\ge0$，主要步骤是稳定的升阶蕴含 \begin{equation*} C_k\ge0\Longrightarrow C_{k+1}\ge0, \qquad 2\le k\le n-2. \end{equation*} 对于 $n\ge5$，该秩二假设严格弱于归一化全碰撞壁上的正性，后者对应于 $C_{n-2}\ge0$。该结果补充了先前的三变量定理《三变量对称单项式不等式的碰撞正性：完整边界判据》，该定理证明了例外的一秩步骤 $C_1\Rightarrow C_2$。这里的证明在逻辑上是独立的：高秩所需的局部三标记成分被直接建立。主要工具包括 Green 第二复合储备、相邻子集秩上的固定并交换、秩三零/一/二标记收缩以及终端边界重组。该定理通过清除分母推广到非负有理指数。
英文摘要：
Let $λ\succγ\succμ$ be equal-degree nonnegative integer exponent vectors with at most $n$ parts, let $J_η^{(n)}$ be the labeled symmetric orbit sum associated with $η$, and set \begin{equation*} P_n=J_λ^{(n)}+J_μ^{(n)}-2J_γ^{(n)}. \end{equation*} For every $n\ge4$, we prove that global nonnegativity of $P_n$ on the positive orthant is equivalent to nonnegativity on the fixed two-variable section \begin{equation*} (x,y,1,\ldots,1),\qquad x,y>0. \end{equation*} Thus the positivity of this family is determined by a fixed two-dimensional section, independently of both degree and ambient dimension. In the injective hierarchy $C_k$ introduced below, the hypothesis is $C_2\ge0$, global positivity is equivalent to $C_{n-1}\ge0$, and the main step is the stable order-raising implication \begin{equation*} C_k\ge0\Longrightarrow C_{k+1}\ge0, \qquad 2\le k\le n-2. \end{equation*} For $n\ge5$, this rank-two hypothesis is strictly weaker than positivity on the normalized full collision wall, which corresponds to $C_{n-2}\ge0$. The result complements the preceding three-variable theorem \emph{Collision Positivity for Three-Variable Symmetric Monomial Inequalities: A Complete Boundary Criterion}, which proves the exceptional rank-one step $C_1\Rightarrow C_2$. The proof here is logically independent: the local three-label ingredients needed in higher rank are established directly. The main tools are a Green second-compound reserve, fixed-union exchange on adjacent subset ranks, a rank-three zero/one/two-mark contraction, and terminal boundary recombination. The theorem extends to nonnegative rational exponents by clearing denominators.
