{
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  "problem_id": "vibemathed:ipm-blowup-spacetime-smooth-forcing",
  "title": "Finite-time blowup for the IPM equation with a uniformly space-time smooth force",
  "canonical_statement": "Córdoba and Martínez-Zoroa proved finite-time singularity formation for the two-dimensional incompressible porous media equation from smooth initial data with a force smooth in space but merely bounded in time, that is in $L^\\infty_t C^\\infty_x$. Their Remark 1 anticipates joint smoothness in space and time but does not prove it. Can the force be taken uniformly smooth in space and time?",
  "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": "2026-09-08T00:00:00.000Z",
      "title": "Extending the Córdoba-Martínez-Zoroa IPM blow-up to uniformly space-time smooth forcing (manuscript)",
      "summary": "Yes. Theorem 2.1: there are a smooth odd initial density $\\rho_{in}\\in C^\\infty(\\mathbb T^2)$ of zero spatial mean, an odd force $F\\in C^\\infty([0,1]\\times\\mathbb T^2)$, and a solution smooth on $[0,T]$ for every $T<1$ with $\\rho(t)\\to\\rho_*$ in $C^\\eta$ for every $0\\le\\eta<1$, yet $\\|\\nabla\\rho(t)\\|_\\infty$ and $\\|D_xu_{\\mathbb T}(\\rho(t))\\|_\\infty$ both diverging as $t\\uparrow1$. The advance over Córdoba and Martínez-Zoroa is precisely the force class, from $L^\\infty_t C^\\infty_x$ to uniformly space-time smooth, on the torus rather than the plane. Not the Clay Millennium problem: that problem is Navier-Stokes with viscosity, and Fefferman's official description states that the Euler equation \"is not on the Clay Institute's list of prize problems\".",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:ipm-blowup-spacetime-smooth-forcing:attempt"
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      "source_assertion_ids": [
        "vibemathed:ipm-blowup-spacetime-smooth-forcing:assertion"
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  "attempts": [
    {
      "id": "vibemathed:ipm-blowup-spacetime-smooth-forcing:attempt",
      "problem_id": "vibemathed:ipm-blowup-spacetime-smooth-forcing",
      "model": "Claude, Codex with GPT-5.6 Sol",
      "provider": "Anthropic / OpenAI",
      "attempted_at": "2026-09-08T00:00:00.000Z",
      "human_collaborators": [
        "Levent Alpöge",
        "Tristan Buckmaster",
        "Matei P. Coiculescu"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "This is the earliest of the three results and the one the authors describe as feeding the others: the Boussinesq AI statement says the Boussinesq work \"involved inputting ideas from previous joint work of ours on blowup for the IPM equation following Córdoba-Martínez-Zoroa\". Buckmaster's public statement covers the whole project: \"For most of the past year progress was slow. We worked through the literature and upgraded various preliminary results, up to obtaining finite time blow up for the Incompressible Porous Media equation (with smooth forcing)\", using \"Anthropic's Claude, OpenAI's Codex, especially with GPT-5.6 Sol\". The IPM paper itself does not break the contribution down per step, and defers a fuller account: \"The complete human-readable proofs will be released shortly by the first and second authors, together with an account of the role of artificial intelligence in this work.\"",
      "failed_routes": [],
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          "kind": "announcement",
          "license": null
        },
        {
          "label": "Córdoba and Martínez-Zoroa: the theorem extended, and Remark 1's open case",
          "url": "https://arxiv.org/abs/2410.22920",
          "kind": "problem-record",
          "license": null
        }
      ],
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          "label": "Extending the Córdoba-Martínez-Zoroa IPM blow-up to uniformly space-time smooth forcing (manuscript)",
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        {
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      "level": "unreviewed",
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      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "No independent check. Unlike the Boussinesq and Euler results, this one has no Lean formalisation: tristanbuckmaster/fluid_lean contains projects for Boussinesq (twice) and Euler and none for IPM, confirmed by listing the repository tree on 8 September 2026. The paper is a 57-page manuscript on the second author's university page, not on arXiv and not peer reviewed, and no independent expert reading is on record. It is the most conventional of the three write-ups, being the one the authors had time to prepare.",
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          "label": "Extending the Córdoba-Martínez-Zoroa IPM blow-up to uniformly space-time smooth forcing (manuscript)",
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      "label": "Extending the Córdoba-Martínez-Zoroa IPM blow-up to uniformly space-time smooth forcing (manuscript)",
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    {
      "label": "VibeMathed: Finite-time blowup for the IPM equation with a uniformly space-time smooth force",
      "url": "https://vibemathed.com/problem/ipm-blowup-spacetime-smooth-forcing",
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      "generated_at": "2026-09-13T16:28:40.178Z"
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    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
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  "generated_at": "2026-09-13T16:28:40.178Z"
}