{
  "schema_version": 1,
  "problem_number": "OWR-12975-010",
  "title": "A Rank Threshold for Exact Sums of Squares in Virtually Abelian Group Algebras",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let G have a finite-index subgroup isomorphic to Z^d. We give an explicit finite-index compression formula and use it to show that positivity in all finite-dimensional unitary representations is equivalent to a sum of Hermitian squares at every matrix level exactly when d<=2. In the positive cases, the representations induced by characters of the abelian subgroup suffice. The essential input is the corrected bivariate matrix factorization theorem of Dritschel, Klep, McCullough and Volcic. For d>=3, a trigonometric version of the classical Motzkin form gives an explicit obstruction, and coefficient projection transfers it to every group containing Z^3. We spell out the resulting two-dimensional representation test and factor-transfer formula for the Klein bottle group. The result is a deduction from credited factorization and positivity methods; it does not settle the exact-squares conjecture for general closed surface groups. No factor-degree bound or absolute priority claim is made. This is an unrefereed, AI-assisted preprint.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.FA",
    "math.GR",
    "math.AG"
  ],
  "keywords": [
    "OWR-12975-010",
    "Hermitian sums of squares",
    "virtually abelian groups",
    "Klein bottle group",
    "matrix trigonometric factorization"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-12975-010/",
  "pdf_url": "https://eulersolve.org/papers/owr-12975-010/paper.pdf?v=04c08b4678f3",
  "doi": "10.5281/zenodo.23630822",
  "zenodo_record_url": "https://zenodo.org/records/23630822",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For any group G with a finite-index subgroup H isomorphic to Z^d, positivity in all finite-dimensional unitary representations, positivity in all unitary representations, and finite sums of Hermitian squares coincide at every matrix level if and only if d<=2. Induced characters of H suffice in the positive cases. A direct finite-index compression gives the factor transfer, an explicit trigonometric Motzkin element gives the d>=3 obstruction, and the Klein bottle case has a concrete two-dimensional representation test. The general closed-surface conjecture remains unresolved. The corrected 2026 factorization theorem of Dritschel, Klep, McCullough and Volcic is the essential input; prior crossed-product transfer and the classical rank-three obstruction are credited. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "04c08b4678f3d39911c48a668ac85ff62da0befc86040f4fe7118d2de8de6cda"
    },
    "source.zip": {
      "sha256": "a0aaf52aef9f18afe40de530bb9d0686f8bdc85598f03be5adc47fdb31ec1849"
    },
    "verification_report.md": {
      "sha256": "a5e4a7f1246df8f75dda72bfc3cd9ccfc4121d9d301ff272e037490a9c4a702c"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}