{
  "schema_version": 1,
  "problem_number": "AMR-040-0001",
  "title": "Euclidean Realizability of Fundamental-Polytope Types",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a finite metric d, its fundamental polytope is the convex hull of the vectors (e_i-e_j)/d_ij. We answer the universality clause of Vershik's question about Euclidean metrics: not every abstract combinatorial type is Euclidean. The four-cycle metric gives a cube, whereas four Euclidean points give a fundamental polytope with 6, 10, or 12 vertices, never 8. Four points is minimal. In contrast, every strict metric has a Euclidean representative of the same labelled type. Adding a common constant c to its off-diagonal distances preserves its face lattice, and c >= 2n diam(d) guarantees an affinely independent Euclidean realization. The argument combines the known four-cycle cube, the Gordon-Petrov face criterion, and the classical additive-constant embedding method. It does not classify all degenerate Euclidean types or close the full source record. This is a self-audited, unrefereed preprint; no absolute priority claim, independent review or formal verification is asserted.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "keywords": [
    "finite metric spaces",
    "fundamental polytopes",
    "Euclidean realizability",
    "Kantorovich-Rubinstein polytopes",
    "strict metrics",
    "additive constant problem"
  ],
  "manuscript_version_date": "2026-10-13",
  "publication_date": "2026-10-13",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-13",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-040-0001/",
  "pdf_url": "https://eulersolve.org/papers/amr-040-0001/paper.pdf?v=442566a31e0d",
  "doi": "10.5281/zenodo.23328317",
  "zenodo_record_url": "https://zenodo.org/records/23328317",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete answer to the universality clause of Vershik Problem 1(4), negative in general and positive for all strict types. The full degenerate-type classification and other parts of the five-part source remain unresolved. The known cube, Gordon-Petrov criterion and Cailliez method are credited; novelty, independent review and formal verification are not certified.",
  "files": {
    "paper.pdf": {
      "sha256": "442566a31e0de5110ed385350f6187f8b0e6d2ef357f0d269b9ec398e4751473"
    },
    "source.zip": {
      "sha256": "3cb3fc8f169b16ac7f82189e445dd922442bb3c670d558d46fc9dc8c29271539"
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    "verification_report.md": {
      "sha256": "cf09aaf86ff2dbf646453aa400cb46ecbd04bfa3854bf483e69877d261bc6c49"
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  },
  "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."
}