{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0257",
  "title": "Polynomial Unit-Distance Lower Bounds over Prime Fields",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For each a in {1,3}, we prove that infinitely many primes p congruent to a modulo 4 admit a set of exactly p points in F_p^2 with at least p^{1.0058} unordered pairs at quadratic distance one. We transfer Sawin's number-field unit-distance construction to prime fields using a field-uniform quantitative Chebotarev estimate of Zaman. A relative ideal-norm bound prevents collisions without requiring a bounded global denominator. Reduction at split or inert primes gives the required quadratic form, and disjoint translates supply exactly p points. The exponent is certified by rational interval arithmetic. This is a lower-bound result for AIM-COMBINATORICS-0257, not a determination of the optimal exponent or a result for every sufficiently large prime. The AI-assisted preprint is self-audited and unrefereed; novelty and absolute priority are not certified.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "keywords": [
    "unit distances",
    "prime fields",
    "Chebotarev",
    "CM fields",
    "additive combinatorics",
    "AIM-COMBINATORICS-0257"
  ],
  "manuscript_version_date": "2026-10-05",
  "publication_date": "2026-10-05",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-05",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-combinatorics-0257/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0257/paper.pdf?v=73e6dd0b4442",
  "doi": "10.5281/zenodo.23169763",
  "zenodo_record_url": "https://zenodo.org/records/23169763",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "A complete lower-bound theorem for AIM-COMBINATORICS-0257 (T. Tao's Problem 4.1 in the AIM additive-combinatorics list): for each residue class 1 and 3 modulo 4, infinitely many primes admit exactly p points with at least p^1.0058 unordered unit-distance pairs. The optimal extremal order and the statement for every sufficiently large prime remain unresolved. Sawin's construction and Zaman's field-uniform Chebotarev theorem are credited. The written general proof is supplemented by 440 exact checks, not replaced by them. AI-assisted, self-audited and unrefereed; no independent review or proof-assistant formalization is claimed. Novelty and absolute priority remain undetermined.",
  "files": {
    "paper.pdf": {
      "sha256": "73e6dd0b4442054579a298663b3455bcb5e68dfe179a8d4a60265f993510e993"
    },
    "source.zip": {
      "sha256": "0934d4950ec20ae3cee4581e1d75977ed130b10c5d4fe3302991a410d3c21589"
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    "verification_report.md": {
      "sha256": "2f13e8da0834f633035743c6d9b15cb9319a7c401a8a589e1b8c6920795b9339"
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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."
}