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  "problem_id": "vibemathed:banach-s-isometric-conjecture",
  "title": "Banach's isometric conjecture",
  "canonical_statement": "Banach asked in 1932 whether a real Banach space $X$ whose $n$-dimensional subspaces, for some fixed $1 < n < \\operatorname{dim}X$, are all isometric must be a Hilbert space. Gromov proved the conjecture for even n, and subsequent work settled several odd-dimensional cases. We prove the conjecture for every odd n, including all previously unresolved cases. Together with Gromov’s even-dimensional result, this completes Banach’s isometric conjecture in the real case. The proof combines bundle topology with Brouwer degree theory.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
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      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-13T00:00:00.000Z",
      "title": "arXiv",
      "summary": "The paper proves the previously unresolved odd-dimensional real cases of Banach's isometric conjecture. Combined with Gromov's earlier theorem for even dimensions and previous results, this completes the conjecture for real Banach spaces.",
      "ai_contribution": "ai_co_developed",
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      "provider": "OpenAI",
      "attempted_at": "2026-08-13T00:00:00.000Z",
      "human_collaborators": [
        "Xinbao Lu",
        "Kaiwen Yang"
      ],
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      "prompt_public": null,
      "outcome": "The authors state that they had already reduced the main problem to proving Theorem 3.10 before using generative AI. An approach to that theorem then emerged through extensive interactions with ChatGPT 5.5 Pro and ChatGPT 5.6 Pro. GPT-5.6 Sol generated the initial draft of Section 3 and corresponding material in Section 2 following this approach; the authors subsequently checked and rewrote it. GPT-5.6 Sol was also used to improve the exposition.",
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      "sources": [
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          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.13536",
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      "level": "unreviewed",
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      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.13536, Lu-Yang, 21 pages). The AI declaration is verbatim as this entry quotes it: the authors reduced the problem to one theorem before using AI, the approach to that theorem emerged through interactions with ChatGPT 5.5 Pro and 5.6 Pro, and GPT-5.6 Sol drafted Section 3 and parts of Section 2, subsequently checked and rewritten by the authors. The mathematics - bundle topology plus Brouwer degree - was not checked here and needs a geometer. One-day-old preprint, no independent review.",
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      "url": "https://arxiv.org/abs/2608.13536",
      "kind": "primary-mathematical-source",
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  "generated_at": "2026-09-13T16:28:40.178Z"
}