{
  "schema_version": 1,
  "problem_number": "OWR-12732-005",
  "title": "A Three-Vertex Counterexample to Product-Order Lifting of TF Classes",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Does every comparison in the facial semistable order admit coordinatewise ordered representatives in the corresponding TF-equivalence classes? We give a complete counterexample for the path algebra of 3 -> 2 -> 1. The classes C = {(-a,-b,a+b): a,b > 0} and D = {(-c,0,c): c > 0} satisfy a cover relation C < D, but no point of C is coordinatewise comparable with any point of D. Both classes lie on the hyperplane perpendicular to (1,1,1). We compute the strict and weak semistable torsion classes directly from the six interval modules and prove that even the transitive closure of the representative relation misses this cover. This answers the auxiliary lifting question in Hanson, A facial order for torsion classes, Remark 4.19. It does not resolve OWR-12732-005 or the general conjecture that g-vector cones determine the support tau-tilting order. The English manuscript includes an explicit proof, scope statement, source attribution and reproducible exact-arithmetic diagnostics. It is an unrefereed preprint with no claim of formal verification or absolute priority.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "keywords": [
    "OWR-12732-005",
    "TF equivalence",
    "facial semistable order",
    "product order",
    "A3 quiver",
    "tau-tilting",
    "torsion class"
  ],
  "manuscript_version_date": "2026-10-26",
  "publication_date": "2026-10-26",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-26",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-12732-005/",
  "pdf_url": "https://eulersolve.org/papers/owr-12732-005/paper.pdf?v=c53749a968cf",
  "doi": "10.5281/zenodo.23672459",
  "zenodo_record_url": "https://zenodo.org/records/23672459",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete negative answer to the representative-lifting question for TF-equivalence classes in Hanson, A facial order for torsion classes, Remark 4.19. For k(3 -> 2 -> 1), C={(-a,-b,a+b):a,b>0} is covered by D={(-c,0,c):c>0} in the facial semistable order, but no representatives are coordinatewise comparable in either direction. Even the transitive closure of the representative relation does not recover this cover. This four-page manuscript gives a complete counterexample to the representative-lifting question for TF classes in Hanson, Remark 4.19. A cover for k(3 -> 2 -> 1) admits no coordinatewise comparable representatives in either direction, even after transitive closure. One auxiliary explicit literature question is closed; the original OWR g-vector order conjecture remains unresolved, with zero exact source-question and zero frozen-record closures. Lower-dimensional faces are essential; no chamber counterexample or absolute-priority claim is made. 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": "c53749a968cfc6173f3fe6695f0d8e3eb4b67204873454d4a0ad6958629515b7"
    },
    "source.zip": {
      "sha256": "ca47008859b819ef878da9438161e183f1b922fe90d745a556242f9024e14706"
    },
    "verification_report.md": {
      "sha256": "303c2e2dc0908cd73e7af5780d41270824c5ade4b14d36cc85457fbdc86c2f31"
    }
  },
  "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."
}