{
  "schema_version": 1,
  "problem_number": "OWR-4531-016",
  "title": "The Cubic Fermat Equation in Model Spaces: A Negative Answer to a Question of Dyakonov",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For an inner function θ on the unit disc and 1 ≤ p ≤ ∞, let K^p_θ = H^p ∩ θ·conj(H^p_0) be the associated model (star-invariant) subspace of the Hardy space H^p. K. M. Dyakonov asked (Oberwolfach Reports 7 (2010); Integral Equations Operator Theory 78 (2014)) whether \"Fermat's last theorem\" holds in these spaces: does fⁿ + gⁿ = hⁿ with f, g, h ∈ K^p_θ and n ≥ 3 force f, g, h to lie in a one-dimensional subspace? Dyakonov noted that the answer is yes when θ is a finite Blaschke product, and S. R. Garcia (2026) proved that it is yes for n ≥ 4 when the boundary spectrum of θ has logarithmic capacity zero, and for n = 3 when the boundary spectrum has at most one point. We show that the answer is no for n = 3. We construct an explicit inner function θ₀ = zBB̃, where B and B̃ are Blaschke products whose zeros accumulate on the unit circle exactly at −1 and 1, and functions F, G, H ∈ K^∞_θ₀, hence in K^p_θ₀ for every p, such that F³ + G³ = H³ and F, G, H are linearly independent; moreover F + iH = B, and the inner factors of F, G, H are pairwise coprime. The construction composes Dixon's elliptic functions sm and cm with a conformal map of the disc onto a period strip on whose two boundary lines both functions are real. Consequently the one-point hypothesis in Garcia's cubic theorem cannot be relaxed to two points, and every closed subset of the unit circle with at least two points is the boundary spectrum of an inner function θ for which K^p_θ contains a nontrivial solution of the cubic Fermat equation. The case n ≥ 4 and Dyakonov's companion question on extreme points are not addressed. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.CV",
    "math.FA"
  ],
  "keywords": [
    "Fermat equation",
    "Fermat's last theorem for model spaces",
    "model spaces",
    "star-invariant subspaces",
    "backward shift",
    "inner functions",
    "Blaschke products",
    "boundary spectrum",
    "Dixon elliptic functions",
    "Fermat cubic",
    "counterexample",
    "Dyakonov problems",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-4531-016",
    "math.CV",
    "math.FA",
    "open mathematics"
  ],
  "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/owr-4531-016/",
  "pdf_url": "https://eulersolve.org/papers/owr-4531-016/paper.pdf?v=86759dbf585c",
  "doi": "10.5281/zenodo.23431590",
  "zenodo_record_url": "https://zenodo.org/records/23431590",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Dyakonov's Problem 2 negatively for n = 3 and every p by an explicit triple; Problem 1 on extreme points and the case n at least 4 are not addressed.",
  "files": {
    "paper.pdf": {
      "sha256": "86759dbf585cd71bf091a091a39ead1944a777435e616b86161df8d18183873a"
    },
    "source.zip": {
      "sha256": "e4fde82e545f190d4dc043b8a234ad18b11b7fabf220892ec75c18a00bd08e51"
    },
    "verification_report.md": {
      "sha256": "02c2290d64e44f705506c4b12122c4ff4ae6c8b90fac5fcea0499a84c3876839"
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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."
}