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

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