{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30309",
  "author_total": 1,
  "authors": [
    {
      "name": "Jian Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6nhfsf742hmm9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30309",
  "fulltext_url": null,
  "id": "6nhfsf742hmm9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30309",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214680033",
  "pdf_url": "https://arxiv.org/pdf/2609.30309",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6nhfsf742hmm9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30309",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30309"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Collision Positivity for Symmetric Orbit-Sum Inequalities in Arbitrary Dimension: A Complete Two-Variable Criterion",
  "title_text": "Collision Positivity for Symmetric Orbit-Sum Inequalities in Arbitrary Dimension: A Complete Two-Variable Criterion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/6nhfsf742hmm9",
  "venue": "arXiv",
  "versions": []
}