{
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  "source": "vibemath",
  "problem_id": "vibemathed:brezis-open-problem-5-6-fourier-summation",
  "title": "Brezis's Open Problem 5.6 on Universal Fourier Summation",
  "canonical_statement": "Does a universal summation process recover the degree of a circle map from its Fourier moduli, that is, does $\\sum_n \\sigma_{n,\\varepsilon} n |\\hat f(n)|^2 \\to \\deg f$ hold for Holder maps below the threshold? No. For every $0 < \\alpha < 1/3$ there is an $f \\in C^{0,\\alpha}(S^1;S^1)$ for which the sum fails to converge to $\\deg f$, answering Open Problem 5.6 from Brezis's list of favourite open problems negatively for all $p > 3$. The endpoint $C^{0,1/3}$ is left unresolved.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:brezis-open-problem-5-6-fourier-summation:event",
      "problem_id": "vibemathed:brezis-open-problem-5-6-fourier-summation",
      "type": "disproved",
      "status": "partial",
      "occurred_at": "2026-07-26T00:00:00.000Z",
      "title": "arXiv:2607.23598 - Nonexistence of universal Fourier summation formulas for the degree below the Holder threshold",
      "summary": "negative below the 1/3 threshold; the endpoint case is still open",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:brezis-open-problem-5-6-fourier-summation:attempt"
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      "id": "vibemathed:brezis-open-problem-5-6-fourier-summation:attempt",
      "problem_id": "vibemathed:brezis-open-problem-5-6-fourier-summation",
      "model": "ChatGPT",
      "provider": "OpenAI",
      "attempted_at": "2026-07-26T00:00:00.000Z",
      "human_collaborators": [
        "Michal Cieszynski"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The disclosure states that ChatGPT generated preliminary drafts of the proofs in the manuscript. The author then verified each argument in detail, revised the proofs where necessary, checked the cited sources, determined the final formulation of all results, and takes sole responsibility for the content.",
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      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2607.23598 - Nonexistence of universal Fourier summation formulas for the degree below the Holder threshold",
          "url": "https://arxiv.org/abs/2607.23598",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Brezis's Open Problem 5.6 on Universal Fourier Summation",
          "url": "https://vibemathed.com/problem/brezis-open-problem-5-6-fourier-summation",
          "kind": "source-assertion",
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        }
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      "id": "vibemathed:brezis-open-problem-5-6-fourier-summation:verification",
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      "solution_event_id": "vibemathed:brezis-open-problem-5-6-fourier-summation:event",
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      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Single-author arXiv preprint; the construction combines degree-zero quotients of Blaschke factors with a Baire category argument. Not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Brezis's Open Problem 5.6 on Universal Fourier Summation",
          "url": "https://vibemathed.com/problem/brezis-open-problem-5-6-fourier-summation",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.23598 - Nonexistence of universal Fourier summation formulas for the degree below the Holder threshold",
          "url": "https://arxiv.org/abs/2607.23598",
          "kind": "primary-mathematical-source",
          "license": null
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  "sources": [
    {
      "label": "arXiv:2607.23598 - Nonexistence of universal Fourier summation formulas for the degree below the Holder threshold",
      "url": "https://arxiv.org/abs/2607.23598",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Brezis's Open Problem 5.6 on Universal Fourier Summation",
      "url": "https://vibemathed.com/problem/brezis-open-problem-5-6-fourier-summation",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
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  "recommended_exposure": "result_only",
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    {
      "id": "vibemathed:brezis-open-problem-5-6-fourier-summation:signal:recent_partial_result",
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      "kind": "recent_partial_result",
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      "generated_at": "2026-09-13T16:28:40.178Z"
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  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}