{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-hironaka-resolution-of-singularities-lean-fable-agents/","md":"https://postcutoff.com/e/2026-10-06-hironaka-resolution-of-singularities-lean-fable-agents/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-06-hironaka-resolution-of-singularities-lean-fable-agents","date":"2026-10-06","date_precision":"day","short_title":"Hironaka's resolution of singularities formalized in Lean by agents","deck":null,"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"Resolution: Formalizing resolution of singularities (6 Oct 2026)","url":"https://resolution.org/post/formalizing-resolution-of-singularities","type":"official","group":"primary","domain":"resolution.org"},{"n":2,"title":"GitHub: resolutionorg/hironaka","url":"https://github.com/resolutionorg/hironaka","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"Blueprint","url":"https://resolutionorg.github.io/hironaka/blueprint/","type":"docs","group":"primary","domain":"resolutionorg.github.io"},{"n":4,"title":"Hexagon 2610.00621: Counterexamples to Hironaka's characteristic-p resolution, a report on queries to frontier agents (Dupuy)","url":"https://hexagonmath.org/abs/2610.00621v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":5,"title":"Import AI 478 (Jack Clark, 26 Oct 2026)","url":"https://importai.substack.com/p/import-ai-478-math-for-alignment","type":"press","group":"press","domain":"importai.substack.com"}],"official":4,"filed":"2026-10-26","updated":"2026-10-29","orgs":["Resolution","Anthropic"],"title":"Hironaka's resolution of singularities formalized in Lean 4 by AI agents (mainly Claude Fable 5.1) in about a week: ~600k lines, ~$135k in tokens, no sorry","summary":"A 6 Oct 2026 post by Billy Snikkers, Adam Newgas, Jesse Hoogland and Chris Elliott describes a Lean 4 + Mathlib formalization of resolution of singularities in characteristic zero (github.com/resolutionorg/hironaka). It covers Hironaka's Main Theorems I and II and corollaries, Kollár's functorial resolution and principalization, Włodarczyk's embedded desingularization, and analytic versions (Atiyah, Bierstone–Milman). Agents wrote it from a 112-page plan of atomic steps in about a week. The cost was about 100B input tokens (90% cache reads), 300M output tokens and about $135k, and the result is about 600k lines of Lean. The main theorems are checked against separate challenge files with Comparator. It was found late, through Import AI 478 (26 Oct).","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"],"key_numbers":[],"tags":["math","algebraic-geometry","lean","autoformalization","agents","claude","claude-fable-5-1","gpt-6-astra","mathlib"],"science":{"field":"mathematics","subfield":"algebraic geometry (resolution of singularities)","problem":"Formalization of Hironaka's resolution of singularities in characteristic zero (1964)","result":"Sorry-free Lean 4 + Mathlib library (~500–600k lines) of Hironaka's main theorems and modern refinements, for schemes and analytic spaces","open_since":"1964","ai_system":["Claude Fable 5.1","GPT-6 Astra"],"human_role":"Humans wrote the 112-page plan, chose sources and intervened ~30 times; agents wrote the proofs and Lean code","verification":{"key":"lean","label":"Lean-verified","text":"Formal proof in Lean (no sorry per README; Comparator checks against challenge statements); not independently audited by us"},"status":"confirmed","shock":"A Fields-Medal-level theorem whose proof filled hundreds of pages was formalized by agents in about a week for about $135k."},"body_md":"## What happened\n\nResolution of singularities says that any algebraic variety over a field of characteristic zero can be made smooth by a sequence\nof blow-ups. Heisuke Hironaka proved it in 1964. The Resolution team gave agents a 112-page plan broken into atomic steps. An\norchestrator then ran separate agents for paper proofs, Lean code and progress checks. Most of the work was done by Claude Fable\n5.1, with GPT-6 Astra used for occasional checks. The first stage took a week. A mathematician (Chris Elliott) stepped in about 30\ntimes, and a refactoring pass followed.\n\nThe library proves Hironaka's Main Theorem I and II, Kollár's functorial versions, Włodarczyk's embedded desingularization, and\nanalytic versions. Each main theorem is checked against a separate statement file.\n\n## Why it matters\n\nIt is one of the largest formalizations of a deep 20th-century theorem so far, and the post gives a concrete price for one:\nabout $135k in tokens and a week of agent time. Its caveat matters too. The Lean statements follow modern sources and differ in\nplaces from the originals, so what was proved must be read from the challenge files.","disputed":[],"related":[{"id":"2026-10-26-ai-assisted-results-october-26-listing","url":"https://postcutoff.com/e/2026-10-26-ai-assisted-results-october-26-listing/","date":"2026-10-26","date_precision":"day","short_title":"AI-assisted results in the 26 Oct 2026 sweep","deck":"Claude-written Lean proof of a frame-indifferent rank-one convex function that is not quasiconvex, a sculptor and Claude Code prove Takemura's conjecture, Claude's contribution to deterministic root-finding, and Zenodo records","takeaway":"Monday's arXiv announcement (papers submitted 22–23 Oct) brought four results with their own entries: OpenAI's propeller conjecture reproved by Heilman, Tokareva's conjecture disproved by MiniMax agents, the triangle lower tail with GPT-6 Astra and Claude, and graph-connectivity communication complexity with ChatGPT. This page lists the next tier, from Lean proofs written entirely by Claude to papers where the AI only checked the text.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-26","updated":"2026-10-26","orgs":["Anthropic","OpenAI","various researchers"]},{"id":"2026-10-09-geometrization-conjecture-lean-formalization","url":"https://postcutoff.com/e/2026-10-09-geometrization-conjecture-lean-formalization/","date":"2026-10-09","date_precision":"day","short_title":"Thurston's geometrization conjecture formalised in Lean","deck":"The Hamilton–Perelman proof machine-checked with GPT-6, Claude and DeepSeek agents (Chow's group)","takeaway":"Twelve days after the Poincaré conjecture, the same group machine-checked the full geometrization theorem, the result Perelman actually proved. The paper names the models: GPT-6 Astra reviews, and GPT-6 Luna, Sol, Astra, Claude Opus 5.5 and DeepSeek V4.1 Flash build.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-12","updated":"2026-10-12","orgs":["UC San Diego","Cornell University","Princeton University","Tsinghua University","Amazon AGI","NVIDIA Research","University of Oregon"]},{"id":"2026-09-01-claude-fable-5-1-mythos-5-1","url":"https://postcutoff.com/e/2026-09-01-claude-fable-5-1-mythos-5-1/","date":"2026-09-01","date_precision":"day","short_title":"Anthropic releases Claude Fable 5.1 and Claude Mythos 5.1","deck":null,"takeaway":"Fable/Mythos 5.1 was Anthropic's capability frontier until Opus 5.5 matched it three weeks later at less than half the price.","category":"model-release","category_label":"Model releases","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":9,"official":5,"filed":"2026-09-29","updated":"2026-10-19","orgs":["Anthropic"]}],"people":[{"id":"jack-clark","name":"Jack Clark","url":"https://postcutoff.com/person/jack-clark/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"edit","text":"Added Dupuy's AI-only report on Hironaka's characteristic-p proposal"},{"date":"2026-10-26","type":"filed","text":"Created. Found late through Import AI 478; post and README read"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-26","run":null,"sources_read":"Found late through Import AI 478; post and README read","updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/hironaka-resolution"}