{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:a-complex-structure-on-s-2-times-s-4",
  "title": "A complex structure on $S^2\\times S^4$",
  "canonical_statement": "On page 408 of Calabi's 1958 paper \"Construction and Properties of Some 6-Dimensional Almost Complex Manifolds\" we find: \n\n\"...it is still unknown whether the sphere $S^6$ or any product manifold $V^2 \\times S^4$ ($V^2 = $ any closed, orientable surface) admit any complex analytic structure.\"",
  "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",
  "solution_events": [
    {
      "id": "vibemathed:a-complex-structure-on-s-2-times-s-4:event",
      "problem_id": "vibemathed:a-complex-structure-on-s-2-times-s-4",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-30T00:00:00.000Z",
      "title": "arXiv",
      "summary": "The authors give an explicit construction of a complex manifold (called $X_+$) which is diffeomorphic to $S^2 \\times S^4$.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:a-complex-structure-on-s-2-times-s-4:attempt"
      ],
      "method_family_ids": [
        "vibemathed:a-complex-structure-on-s-2-times-s-4:method"
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      "source_assertion_ids": [
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    {
      "id": "vibemathed:a-complex-structure-on-s-2-times-s-4:attempt",
      "problem_id": "vibemathed:a-complex-structure-on-s-2-times-s-4",
      "model": "ChatGPT 5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-08-30T00:00:00.000Z",
      "human_collaborators": [
        "Ruiming Liang",
        "Chenhan Liu",
        "Yang Zhang"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The human authors write the following: \n\n\"Statement on the use of AI. At first, GPT 5.6 Sol used the fibration method similar to [7] to construct a complex manifold $X$ fibreed over $P^1$ and claimed that it is homeomorphic to $S^2 \\times S^4$. However, the proof provided by GPT 5.6 Sol is cumbersome, incomplete, and misses an essential lemma. Then by comparing the singular fibre of $X$ with the singular fibres of $X_{sph}$, we discovered that that $X$ is likely to be obtained by the blowup-quotient-flop process introduced in this paper, which led to the main theorem 1.2 of this article. Wall’s classification result is also introduced to the authors by GPT 5.6 Sol. \n\nThe proofs of lemma 2.3, lemma 3.1, proposition 4.6 are first generated by GPT 5.6 Sol, but it is checked and rewrote by the authors. \n\nThis article is also polished by GPT 5.6 Sol on grammars, language, notations, conventions, and some graphs.\"",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.29651",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: A complex structure on $S^2\\times S^4$",
          "url": "https://vibemathed.com/problem/a-complex-structure-on-s-2-times-s-4",
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      "independence": "unknown"
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      "solution_event_id": "vibemathed:a-complex-structure-on-s-2-times-s-4:event",
      "level": "unreviewed",
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      "note": null,
      "sources": [
        {
          "label": "VibeMathed: A complex structure on $S^2\\times S^4$",
          "url": "https://vibemathed.com/problem/a-complex-structure-on-s-2-times-s-4",
          "kind": "source-assertion",
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          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.29651",
          "kind": "primary-mathematical-source",
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      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.29651",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: A complex structure on $S^2\\times S^4$",
      "url": "https://vibemathed.com/problem/a-complex-structure-on-s-2-times-s-4",
      "kind": "source-assertion",
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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"
}