Hironaka’s resolution of singularities formalized in Lean by agents
Result confirmed
Importance: major (4 of 5)The takeaway
Resolution (resolution.org) reports that an orchestrated team of agents, mostly Claude Fable 5.1 with occasional checks by GPT-6 Astra, autoformalized Hironaka’s 1964 theorem in characteristic zero between 4 and 10 Sept 2026, together with refinements by Kollár, Włodarczyk and Bierstone–Milman. That took about 100B input tokens, 300M output tokens and roughly $135k. A human mathematician intervened about 30 times. The repository says neither library contains a sorry.
Status
- Claim
Result confirmed
- Our reporting
- High confidence
- Verification
- Lean-verified
- Importance
- Major (4 of 5)
- Last verified
- 29 October 2026
Your AI and this story
- GPT-6 Astra159 days after its cutoff
- Claude Opus 5.598 days after its cutoff
- Gemini 4 Argon189 days after its cutoff
- Grok 4.7128 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 98 days before it.
Key facts
- Timeline (post): autoformalization Stage 1 ran 4–10 Sept 2026, final audit 11 Sept; repo created 30 Sept; post published 6 Oct; tag v1.0.0
- Models: ‘predominantly’ Claude Fable 5.1, with occasional checks by Astra (GPT-6 Astra). An orchestrator agent spawned agents for pen-and-paper maths, Lean writing and progress checks
- Scale: ~100B input tokens (90% cache reads), ~300M output tokens, estimated $135,000; ~600k lines of Lean in Stage 1 (refactoring figures in the post: ~506k → ~488k lines)
- Human role: Chris Elliott intervened about 30 times, mostly where agents got stuck at interfaces between different human-written sources. On refactoring: ‘Doing this well requires taste in ways the models don’t always exhibit spontaneously.’ Also: ‘We think we could have gotten by with even less human input, at the cost of a higher token spend.’
- README (verbatim): ‘Hironaka’s main theorems and their refinements by Kollár, Włodarczyk and Bierstone–Milman, for algebraic schemes and for real- and complex-analytic manifolds and spaces … Every theorem is proved: neither library contains a
sorry.’ Main theorems are restated with sorry in Challenge files and checked by a Comparator CI workflow - Caveat (post): the proof follows modern sources (Kollár, Włodarczyk) more than Hironaka’s original, and some hypotheses and conclusions differ from the sources. The Lean Main Theorem I assumes CharZero k and covers schemes locally of finite type, quasi-compact, separated and integral over a field
- Background: Hironaka’s 1964 theorem won him the 1970 Fields Medal; its proof was famously long (over 200 pages), and it had not been formalized before
- Positive characteristic (29 Oct): Taylor Dupuy posted ‘Counterexamples to Hironaka’s Characteristic-p Resolution: A Report On Queries to Frontier Agents’ (Hexagon 2610.00621), assembled by Claude Code agents (23–24 Oct) and revised by Codex agents (26–27 Oct) from a demonstration in his Rochester seminar talk of 23 Oct; it argues that for x^p + y^n (n > 2p, p ∤ n) the transformation rule of Hironaka’s 2017 characteristic-p proposal fails. Provenance: ‘The proofs have received AI review only; no human mathematical verification is recorded’. Unverified; separate from the characteristic-zero Lean formalization above
What happened
Resolution of singularities says that any algebraic variety over a field of characteristic zero can be made smooth by a sequence of blow-ups. Heisuke Hironaka proved it in 1964. The Resolution team gave agents a 112-page plan broken into atomic steps. An orchestrator then ran separate agents for paper proofs, Lean code and progress checks. Most of the work was done by Claude Fable 5.1, with GPT-6 Astra used for occasional checks. The first stage took a week. A mathematician (Chris Elliott) stepped in about 30 times, and a refactoring pass followed.
The library proves Hironaka’s Main Theorem I and II, Kollár’s functorial versions, Włodarczyk’s embedded desingularization, and analytic versions. Each main theorem is checked against a separate statement file.
Why it matters
It is one of the largest formalizations of a deep 20th-century theorem so far, and the post gives a concrete price for one: about $135k in tokens and a week of agent time. Its caveat matters too. The Lean statements follow modern sources and differ in places from the originals, so what was proved must be read from the challenge files.
Sources
5 sources from 5 sites. Numbers match the chips in the text.
5 sources: 4 primary, 1 press
Primary
- Resolution: Formalizing resolution of singularities (6 Oct 2026)resolution.org, official
- GitHub: resolutionorg/hironakagithub.com, code
- Blueprintresolutionorg.github.io, docs
- Hexagon 2610.00621: Counterexamples to Hironaka’s characteristic-p resolution, a report on queries to frontier agents (Dupuy)hexagonmath.org, paper
Press
- Import AI 478 (Jack Clark, 26 Oct 2026)importai.substack.com, press
Changes
- Added Dupuy’s AI-only report on Hironaka’s characteristic-p proposal
- Filed. Found late through Import AI 478; post and README read