{
  "schema_version": 1,
  "problem_number": "EP-668",
  "title": "Two Congruence Classes of Extremal Twenty-Point Sets",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Alexeev, Mixon and Parshall proved that twenty points in the plane have at most 54 unit-distance pairs and that every maximizing set has the same abstract unit-distance graph. We determine its Euclidean realization classes: there are exactly two, even when reflections are allowed. Both are explicit Minkowski sums of the unit diamond and a five-point triangular fan. Cartesian-product squares force every injective realization to have this form, and one additional edge leaves two relative rotations. A fourth distance moment separates the resulting sets. The proof uses the established abstract-graph enumeration and otherwise consists of elementary geometry and exact algebra. This is a finite congruence classification; it does not resolve the asymptotic question in Erdős Problem 668.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "keywords": [
    "unit distances",
    "congruence classes",
    "extremal geometry",
    "Minkowski sums",
    "Cartesian product",
    "EP-668"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/ep-668/",
  "pdf_url": "https://eulersolve.org/papers/ep-668/paper.pdf?v=f24b95689f4f",
  "doi": "10.5281/zenodo.23440649",
  "zenodo_record_url": "https://zenodo.org/records/23440649",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete finite classification C(20)=2, modulo all Euclidean isometries. General/asymptotic Erdős Problem 668 remains open. The global bound and unique abstract extremal graph are due to Alexeev–Mixon–Parshall. AI-assisted, self-audited and unrefereed; no independent review, formal certification or certified absolute priority.",
  "files": {
    "paper.pdf": {
      "sha256": "f24b95689f4fc3aa2a7be4f68f782093d2d2a147d226a7953032dcbdd5451ed7"
    },
    "source.zip": {
      "sha256": "6ed80669e4ffb3ff27cb0cca78d3b983cac03b2df23e8a7684b8b5dcbd962fef"
    },
    "verification_report.md": {
      "sha256": "f697e625bf7c1b6f60d10f2e924c6e51363cf1152bc369b4928b8f4cc8782d18"
    }
  },
  "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."
}