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Benefit Ledger · provisional · Mathematics

Buckmaster and Alpöge, using Sol and Claude, prove smooth-forced blowup for Euler, IPM, and Boussinesq

On 8–11 September 2026 Tristan Buckmaster and Levent Alpöge released Lean-backed finite-time blowup results with smooth forcing for incompressible porous media, 2D Boussinesq, and 3D Euler, crediting Anthropic’s Claude and especially OpenAI’s GPT-5.6 Sol. Astra was used only for writeups and auditing.

8 Sep 2026Tier 3 MajorMethodology 0.1

Current score

+1.13

10 base · Major (tier 3 of 5, 10 pts)
× 0.5500 attribution · Critical contribution
× 0.5000 evidence · External expert evaluation
× 0.6000 realization · Experimentally validated
× 0.7500 durability
Event-level product before credit split: 1.24

Lean-checked smooth-forced Euler/IPM/Boussinesq blowup is a major PDE result (tier 3) and not Clay Navier–Stokes. Humans led a pre-existing program (AI attribution 0.55/0.50/0.25). Evidence is an NYU release plus the authors’ statement and Lean claims (0.50). Realization is public theorems with machine-checked arguments (0.60).

What happened

Buckmaster (NYU Courant) and Alpöge (an Anthropic mathematician working in a personal capacity) pushed the Córdoba–Martínez-Zoroa forced-blowup program from rough forces to smooth forces and onto 3D incompressible Euler. They say the first LLM-generated proofs of the Boussinesq and Euler results arrived around 15 August 2026 after slower IPM progress. Public release came in the same week as OpenAI’s Navier–Stokes announcement. Their statement is explicit about tools: several LLMs, especially GPT-5.6 Sol in Codex, Anthropic’s Claude, and more recently Astra only for writeups and auditing. They believe they have hypodissipative Navier–Stokes blowup but withheld that paper because Lean verification was unfinished. This is not a solution of the Clay Navier–Stokes problem. AI credit is split among the named releases they documented, with Sol as the primary model and Claude as a version-unspecified family member because they did not name an Opus/Fable/Sonnet release. Human mathematicians remain the scientific authors; the attribution multipliers reflect material acceleration, not sole authorship.

Model attribution

+0.68

GPT-5.6

Flagship GPT-5.6 release the authors say they used especially, including via Codex, for the core LLM-assisted proofs.

Buckmaster’s statement singles out GPT-5.6 Sol as the primary LLM used in the successful August proofs.

Attribution 0.5500 · Credit share 55.00000000000001% · OpenAI

Claude
Version unspecified
+0.39

Claude

Anthropic Claude, version unspecified by the authors, used throughout the collaboration.

Authors name Claude as a tool they used throughout; they did not identify Opus, Sonnet, or Fable, so the record stays at family granularity.

Attribution 0.5000 · Credit share 35% · Anthropic

+0.06

GPT-6

Used only for writeups and auditing arguments after the Euler/Boussinesq proofs existed.

Buckmaster states Astra was used more recently, and only for writeups and auditing, not for the August breakthroughs.

Attribution 0.2500 · Credit share 10% · OpenAI

Claims

  • The authors released finite-time blowup results with smooth forcing for IPM, 2D Boussinesq, and 3D Euler, with Lean verification described for the released papers.

    outcome · supported

  • This work solves the Clay Navier–Stokes Millennium Prize problem.

    significance · disputed

  • GPT-6 Astra was the primary model that found the Euler proofs.

    attribution · disputed

Sources

primary sources

independent sources

other sources

Secondary domains: Physics, Computer Science

Revision history

  • 13 Sep 2026 · 0.00 → 1.13

    Imported September 2026 mathematics events under methodology 0.1 credit-unit rules.

Buckmaster and Alpöge, using Sol and Claude, prove smooth-forced blowup for Euler, IPM, and Boussinesq · NetGoodIndex