{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:boussinesq-blowup-smooth-forcing", "title": "Finite-time blowup for the inviscid Boussinesq system with smooth forcing", "canonical_statement": "Does the inviscid Boussinesq system on $\\mathbb R^2$ admit finite-time blowup from smooth data with forces that are smooth in both space and time? Córdoba, Laín-Sanclemente and Martínez-Zoroa obtained finite-time singularity for the two-dimensional Boussinesq equation with a force only of class $C^{1,\\sqrt{4/3}-1-\\epsilon}\\cap L^2$, leaving the smooth-force case open.", "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:boussinesq-blowup-smooth-forcing:event", "problem_id": "vibemathed:boussinesq-blowup-smooth-forcing", "type": "proved", "status": "resolved", "occurred_at": "2026-09-08T00:00:00.000Z", "title": "Blowup for the Boussinesq equations with smooth forcing (manuscript)", "summary": "Yes. Blowup for the inviscid Boussinesq system on $\\mathbb R^2$ with forces in $C^\\infty(\\mathbb R^2\\times[0,T_*])$ in both equations, supported in one fixed spatial ball, from smooth compactly supported initial temperature and zero initial velocity. The temperature stays bounded while $\\|\\nabla\\theta(t)\\|_\\infty\\to\\infty$ and the vorticity norm has infinite limsup as $t\\uparrow T_*$. The solution is smooth on every closed interval before blowup and unique in a finite-energy Lipschitz class. This lifts the force from barely $C^1$ to fully smooth, and is the construction the Euler paper then builds on. 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:boussinesq-blowup-smooth-forcing:attempt" ], "method_family_ids": [ "vibemathed:boussinesq-blowup-smooth-forcing:method" ], "source_assertion_ids": [ "vibemathed:boussinesq-blowup-smooth-forcing:assertion" ] } ], "attempts": [ { "id": "vibemathed:boussinesq-blowup-smooth-forcing:attempt", "problem_id": "vibemathed:boussinesq-blowup-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" ], "exposure": "unknown", "prompt_public": null, "outcome": "The Boussinesq paper devotes its section 2 to an \"AI statement\": \"We happily used both Claude and Codex to iterate on our proof. We had, using Claude, our first blowup solution, but not this one, on 8/15/26, and we Lean verified it on 8/22/26. This involved inputting ideas from previous joint work of ours on blowup for the IPM equation following Córdoba-Martínez-Zoroa and a number of other works of ours and others, as well as iteration with the model on various ansätze, eventually leading to blowup in Boussinesq. The first writeup that we produced iterating with Claude was, in our opinion, the worst writeup we had ever seen in the history of mathematics (topped soon after by the writeups for 3d Euler and then for hypodissipative Navier-Stokes).\" The statement continues that after iterating with Claude and Codex on alternative proof architectures, \"leading to several different proofs, we arrived at the current simplified argument, which we then iterated on using Claude and Codex, first 5.6 Sol and then Astra once we had access, to arrive at the current writeup modulo our hand editing\", and that intermediate writeups were generally fed to Codex \"for simplification, ideation, and iteration\". Co-developed rather than discovered: the first blowup solution came from Claude, but the multiscale mechanism is Córdoba and Martínez-Zoroa's and the authors fed in their own earlier IPM work.", "failed_routes": [], "artifacts": [ { "label": "Lean formalisation, comparator-checked against Challenge.lean", "url": "https://github.com/tristanbuckmaster/fluid_lean/tree/main/boussinesq-blowup", "kind": "lean-proof", "license": null }, { "label": "Challenge.lean: the trusted statement, in plain Mathlib", "url": "https://github.com/tristanbuckmaster/fluid_lean/blob/main/boussinesq-blowup/Challenge.lean", "kind": "lean-statement", "license": null }, { "label": "Buckmaster's public statement on the work and its release", "url": "https://cims.nyu.edu/~tristanb/statement.pdf", "kind": "announcement", "license": null }, { "label": "affinecore: the second Lean formalisation, normalised data", "url": "https://github.com/tristanbuckmaster/fluid_lean/tree/main/affinecore", "kind": "lean-proof", "license": null } ], "sources": [ { "label": "Blowup for the Boussinesq equations with smooth forcing (manuscript)", "url": "https://cims.nyu.edu/~tristanb/boussinesq.pdf", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Finite-time blowup for the inviscid Boussinesq system with smooth forcing", "url": "https://vibemathed.com/problem/boussinesq-blowup-smooth-forcing", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:boussinesq-blowup-smooth-forcing:verification", "problem_id": "vibemathed:boussinesq-blowup-smooth-forcing", "solution_event_id": "vibemathed:boussinesq-blowup-smooth-forcing:event", "level": "lean_verified_statement_audited", "mathematical_correctness": "supported", "statement_fidelity": "audited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Formalised in Lean 4 in tristanbuckmaster/fluid_lean, twice over: boussinesq-blowup carries the general theorem and affinecore the normalised construction with explicit odd initial data. Checked here on 8 September 2026 by reading both READMEs: in each, Challenge.lean is the only file a reader must trust, no other file contains a sorry, the proof rests on no axiom beyond propext, Classical.choice and Quot.sound, and leanprover/comparator is configured to type-check the statement independently and confirm the solution inhabits exactly it. Not peer reviewed. Tao's public commentary describes the Boussinesq case as the model case whose amplitude-frequency dynamics reduce to \"a remarkably simple ODE\", but records that he is still working through the arguments.", "sources": [ { "label": "VibeMathed: Finite-time blowup for the inviscid Boussinesq system with smooth forcing", "url": "https://vibemathed.com/problem/boussinesq-blowup-smooth-forcing", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Blowup for the Boussinesq equations with smooth forcing (manuscript)", "url": "https://cims.nyu.edu/~tristanb/boussinesq.pdf", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Blowup for the Boussinesq equations with smooth forcing (manuscript)", "url": "https://cims.nyu.edu/~tristanb/boussinesq.pdf", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Finite-time blowup for the inviscid Boussinesq system with smooth forcing", "url": "https://vibemathed.com/problem/boussinesq-blowup-smooth-forcing", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "replay", "recommended_exposure": "method_aware", "opportunity_signals": [ { "id": "vibemathed:boussinesq-blowup-smooth-forcing:signal:replay_ready", "problem_id": "vibemathed:boussinesq-blowup-smooth-forcing", "kind": "replay_ready", "reason": "A public source and a non-unreviewed verification label support replay triage.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "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" }