{
  "schema_version": 1,
  "problem_number": "AMR-022-7069",
  "title": "A Cauchy Kernel without the Fixed-Coefficient Clifford Relations",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give an explicit counterexample to the fixed-coefficient necessity in the existence formulation of Hayman-Lingham Problem 7.69(a). In the commutative complex algebra C direct-product C, the coefficients e1=(1,1) and e2=(i,2i) define two differently scaled Cauchy-Riemann operators. The ordinary Cauchy formula in two orientation-preserving real-linear coordinates yields a real-analytic reproducing kernel for every admissible real-slice domain. The kernel is right regular, homogeneous of degree -1 and normalized by unit flux, and is unique within that normalized class. Nevertheless no elements p1,p2 satisfy p_j e_l+p_l e_j=2 delta_jl. A common-real-quadratic obstruction excludes repairing the relations by an invertible real coordinate change. Thus the printed existence criterion fails even for real-analytic kernels, and remains false under the proposed C1 weakening. The original Ryan theorem includes a uniqueness assertion omitted in the problem-book formulation; this distinction is explicit. The classical Cauchy formula and commutative-algebra literature are credited. The nonassociative part (b) is not addressed, so the whole UnsolvedMath record AMR-022-7069 is not claimed solved. The package contains the four-page proof, exact symbolic checks and a source/scope audit. AI-assisted author-self-audited preprint; no independent peer review, formal verification or exhaustive priority certification is claimed.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CV",
    "math.RA"
  ],
  "keywords": [
    "AMR-022-7069",
    "Cauchy integral formula",
    "Clifford relations",
    "commutative algebra",
    "counterexample"
  ],
  "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-022-7069/",
  "pdf_url": "https://eulersolve.org/papers/amr-022-7069/paper.pdf?v=185751539972",
  "doi": "10.5281/zenodo.23652392",
  "zenodo_record_url": "https://zenodo.org/records/23652392",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "In A=C direct-product C, the coefficients e1=(1,1), e2=(i,2i) have a real-analytic right-regular degree-minus-one Cauchy kernel with unit flux for every admissible real-slice domain, but admit no Clifford-dual family. No real invertible coordinate change repairs the common scalar quadratic factorization. The homogeneous right-regular unit-flux kernel is unique. The printed fixed-coefficient existence criterion in Problem7.69(a) fails even for real-analytic kernels. The original uniqueness wording is distinguished; nonassociative part(b) remains unaddressed. The whole source record is not claimed solved. Classical Cauchy theory is 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": "185751539972d161ed0060b4b108295d90ac262bfc8968d883da26c01936d2e6"
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    "source.zip": {
      "sha256": "59418d3990fa466efc08eb8b0b33a867cbc81c7dbba7417789229ab018682d93"
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    "verification_report.md": {
      "sha256": "8921ccd2953646323298c57eb04d10021f4e5fcb4c9544ebb70ccd8089852385"
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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."
}