{
  "schema_version": 1,
  "problem_number": "AMR-022-7058",
  "title": "A Zero-or-Infinity Modulus Dichotomy on the Real Line",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let E be a compact subset of [0,1], regarded as a subset of the real axis L in the plane. We prove that the 2-modulus of all rectifiable curves joining E to L minus E is infinite whenever E has positive conformal capacity. This gives an affirmative answer to A. A. Gonchar's question recorded as Hayman-Lingham Problem 7.58 (UnsolvedMath AMR-022-7058). When E has zero capacity, the modulus is zero, giving a zero-or-infinity dichotomy. The proof applies the classical condenser capacity-modulus identity to separated compact approximations, uses the H^1 to H^(1/2) trace theorem and critical rigidity of indicator functions, and retains the capacitary constraint on null-length sets by finite Mazur combinations and approximation away from the line. The four-page English manuscript supplies the complete analytic argument and supporting exact algebra checks. Classical tools are credited. AI-assisted, self-audited preprint; no independent peer review, formal verification or exhaustive priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CV",
    "math.CA"
  ],
  "keywords": [
    "AMR-022-7058",
    "Gonchar problem",
    "conformal capacity",
    "curve modulus",
    "Sobolev trace",
    "potential theory"
  ],
  "manuscript_version_date": "2026-10-26",
  "publication_date": "2026-10-26",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-26",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-022-7058/",
  "pdf_url": "https://eulersolve.org/papers/amr-022-7058/paper.pdf?v=359b8ccdb72f",
  "doi": "10.5281/zenodo.23654353",
  "zenodo_record_url": "https://zenodo.org/records/23654353",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every compact E in [0,1], viewed in the real line L in the plane, the 2-modulus of all rectifiable curves joining E to L minus E is infinity when cap(B_2,E)>0, and zero when cap(B_2,E)=0. No positive-length or regularity assumption is added. Separated condenser tests, critical Sobolev trace rigidity, finite Mazur combinations and zero-trace approximation prove the positive-capacity case, including null-length positive-capacity sets. Complete affirmative answer to the Gonchar real-line modulus question, Hayman-Lingham Problem 7.58. The zero-or-infinity dichotomy covers every compact subset of [0,1], including null-length sets of positive capacity. Classical capacity and trace tools are credited. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "359b8ccdb72ff1288e423a041d17982ef87d1283b8bc50dcb462812fbd831656"
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    "source.zip": {
      "sha256": "6851ae7314e416c0e481e4984de9988b921e0e0e29fd0786a3e6bad166c7adfe"
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    "verification_report.md": {
      "sha256": "468a55c9ff431e90a1dbb945be52037169cff0b4b971cf161dbced3a4ba25c92"
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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."
}