Collision Positivity for Symmetric Orbit-Sum Inequalities in Arbitrary Dimension: A Complete Two-Variable Criterion
Abstract
Let λ≻γ≻μ be equal-degree nonnegative integer exponent vectors with at most n parts, let Jη(n) be the labeled symmetric orbit sum associated with η, and set Pn=Jλ(n)+Jμ(n)−2Jγ(n). For every n≥4, we prove that global nonnegativity of Pn on the positive orthant is equivalent to nonnegativity on the fixed two-variable section (x,y,1,…,1), x,y>0. 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 Ck introduced below, the hypothesis is C2≥0, global positivity is equivalent to Cn−1≥0, and the main step is the stable order-raising implication Ck≥0Longrightarrow Ck+1≥0, 2≤ k≤ n−2. For n≥5, this rank-two hypothesis is strictly weaker than positivity on the normalized full collision wall, which corresponds to Cn−2≥0. 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 C1⇒ C2. 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.
Related
- Collision Positivity for Three-Variable Symmetric Monomial Inequalities
- Total positivity and symmetric spaces
- Irreducibility of the Bloch variety for periodic Schrödinger operators in arbitrary dimension
- A construction of Kakeya Sets in Arbitrary Dimension
- Diagonalization of the Toda flow for arbitrary isospectral symmetric matrices
BibTeX
@misc{sun2026collision,
title = {{Collision Positivity for Symmetric Orbit-Sum Inequalities in Arbitrary Dimension: A Complete Two-Variable Criterion}},
author = {Jian Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30309},
eprint = {2609.30309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30309},
}
RIS
TY - UNPB
TI - Collision Positivity for Symmetric Orbit-Sum Inequalities in Arbitrary Dimension: A Complete Two-Variable Criterion
AU - Jian Sun
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.30309
UR - https://arxiv.org/abs/2609.30309
AB - Let λ≻γ≻μ 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 P_n=J_λ^(n)+J_μ^(n)−2J_γ^(n). For every n≥4, we prove that global nonnegativity of P_n on the positive orthant is equivalent to nonnegativity on the fixed two-variable section (x,y,1,…,1), x,y>0. 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≥0, global positivity is equivalent to C_n−1≥0, and the main step is the stable order-raising implication C_k≥0Longrightarrow C_k+1≥0, 2≤ k≤ n−2. For n≥5, this rank-two hypothesis is strictly weaker than positivity on the normalized full collision wall, which corresponds to C_n−2≥0. 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⇒ 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.
ER -