Collision Positivity for Symmetric Orbit-Sum Inequalities in Arbitrary Dimension: A Complete Two-Variable Criterion

Abstract

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.