{
  "schema_version": 1,
  "problem_number": "OWR-3392-009",
  "title": "Ambient Dependence and Extreme Extension for Exchangeable Free-Product States",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A proposed independence relation declares two subalgebras independent when their joint state is realized in two different copies of an extreme exchangeable state on the infinite unital free product of the ambient algebra. We show that this relation depends on the ambient algebra. The diagonal inclusion C^2 into M_2(C) gives an explicit example; a four-point version has distinct proper subalgebras with scalar intersection. Downstairs, the joint law forces a nontrivial mixture of two invariant characters. Upstairs, a pure vector state provides a pure folded witness. In contrast, every extreme invariant state has an extreme invariant extension along a unital C*-algebra inclusion. The latter standard compact-face argument rules out the particular forward-extension obstruction suggested in the original problem, without implying ambient invariance. The examples distinguish extremality among all invariant states from extremality at a fixed marginal.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.OA",
    "math.PR"
  ],
  "keywords": [
    "exchangeable state",
    "free product",
    "extreme invariant state",
    "ambient dependence",
    "OWR-3392-009"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-3392-009/",
  "pdf_url": "https://eulersolve.org/papers/owr-3392-009/paper.pdf?v=e44f55889474",
  "doi": "10.5281/zenodo.23486381",
  "zenodo_record_url": "https://zenodo.org/records/23486381",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "A complete counterexample to ambient invariance of the literal relation associated with OWR-3392-009. It answers the dataset's ambient-dependence question, not all neighboring questions in the 2009 report. Extremality is among all invariant states, not at a fixed marginal. The upstairs pure state need not be faithful. The standard extreme-extension property holds; the obstruction concerns restriction losing extremality. The folding mechanism, purification and extension tools are standard and attributed. AI-assisted, self-audited and unrefereed; no independent review, formal certification, novelty certificate, dataset-maintainer acceptance or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "e44f55889474e6232af91ebc4593cd6fd186dbdb9c0ff3fa4d4d0047a4c46b1b"
    },
    "source.zip": {
      "sha256": "795d1018734b9db3d27be545170daf2c71a843109281f78d9be90e3ac862b97a"
    },
    "verification_report.md": {
      "sha256": "747e6eba687f0255ff7fe1df35872f04ff03cbca6fd7e1cc497a6d225f7066f8"
    }
  },
  "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."
}