{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0263",
  "title": "One-Point Extensions of Multiplicative Subgroups as Self-Sumsets",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let p be an odd prime, G a proper multiplicative subgroup of F_p^*, and e outside G. This preprint classifies every unrestricted self-sumset A+A=G union {e}, including diagonal sums. They occur exactly when p>=5, G={-1,1}, and (e,A) is (0,{-1/2,1/2}), (3,{-1/2,3/2}), or (-3,{-3/2,1/2}). A differential polynomial built from the Hanson-Petridis auxiliary polynomial forces an almost reflection symmetry; a shifted-subgroup intersection bound reduces the argument to at most five elements, and exact moments finish the classification.\n\nThe zero-exception case and the impossibility of exact equality with a nontrivial proper subgroup are credited consequences of Hanson and Petridis. The theorem addresses the one-point enlargement regime of AIM Problem 4.7, recorded as AIM-COMBINATORICS-0263 in UnsolvedMath. It does not resolve the entire source question, arbitrary near-equality, or the small-exponent containment problem. No absolute priority is claimed. The preprint is self-audited, not independently reviewed or formally verified.\n\nThe accompanying source and exact-arithmetic verification materials include 749,359 checks and finite enumeration for odd primes through 101. These supplement the written all-primes proof. AI assistance was used for mathematical exploration, drafting and reproducibility checks; the author is responsible for the claims.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "keywords": [
    "multiplicative subgroup",
    "sumset",
    "finite field",
    "polynomial method",
    "additive combinatorics",
    "AIM-COMBINATORICS-0263"
  ],
  "manuscript_version_date": "2026-10-07",
  "publication_date": "2026-10-07",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-07",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-combinatorics-0263/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0263/paper.pdf?v=945aa4917181",
  "doi": "10.5281/zenodo.23201551",
  "zenodo_record_url": "https://zenodo.org/records/23201551",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete classification of unrestricted self-sumsets A+A=G union {e} for a proper multiplicative subgroup G of an odd prime field and e outside G. The theorem handles the one-point enlargement regime of AIM Problem 4.7; it does not resolve arbitrary near-equality, multiple exceptional values, or the small-exponent containment question. The zero-exception and exact-equality cases are credited to Hanson and Petridis. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed. Novelty remains undetermined after a bounded primary-source review.",
  "files": {
    "paper.pdf": {
      "sha256": "945aa4917181cef433cf660e6eb25545a17ba884d6e5223deebe69299d304405"
    },
    "source.zip": {
      "sha256": "ef99e6d466294f6bdee6fb9511a17c624c0a46221567e78ce5da5d7c278eee12"
    },
    "verification_report.md": {
      "sha256": "e33dce37244a190135599f98ac2473e5cfd0cc41e43120afd6c4b718cbee82fc"
    }
  },
  "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."
}