Resolution's AI-assisted formalization of Hironaka's resolution of singularities reported, costing ~$135,000 in tokens over one week
FollowingDay 732 sourcesupdated 26 Oct 2026
AI Systems Produce Formal Math Proofs Beyond Competition Problems
The gist
Formal verification removes human referee fallibility from mathematical checking by having a trusted kernel program verify each line, though as one item notes, the statement itself must be translated correctly and the checker must be sound. Resolution stated the formalized library is already being used to extend singular learning theory and understand model generalization.
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The full picture
Axiom Math used its multi-agent system AxiomProver to produce a machine-checkable formal verification of the 246 Prime-Gaps Theorem in Lean 4, built on the Mathlib community library. The system used a three-stage pipeline: a blueprint phase mapping every definition, lemma, and theorem with dependencies; proof generation by AxiomProver; and verification against Mathlib. Separately, Resolution (formerly Sequent) used AI agents to formalize Hironaka's resolution of singularities in Lean, spending roughly $135,000 in token costs over about one week to produce 600,000 lines of formal proof. Both projects address the same gap: most AI math benchmarks had measured performance on competition problems with short, self-contained proofs, leaving longer real mathematics unaddressed.
How it developed
Axiom Math published results of AxiomProver formally verifying the 246 Prime-Gaps Theorem in Lean 4
Sources
Related
- Related storyAxiom's AI formalization of finite simple groups
- Claude's formal verification of open-source libraries
- OpenAI's internal Astra model produced Lean 4-verified proofs for ten long-standing open mathematics problems
- Google Cogentic and Meta Muse Spark proofs
- AI proofs of Navier-Stokes and Yang-Mills
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