{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-11-06-axiom-weil-conjectures-lean-formalization/","md":"https://postcutoff.com/e/2026-11-06-axiom-weil-conjectures-lean-formalization/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":"2026-11-06-axiom-weil-conjectures-lean-formalization","date":"2026-11-06","date_precision":"day","short_title":"Axiom: the Weil conjectures proved in Lean","deck":null,"takeaway":"On 6 Nov 2026 Axiom Math published AxiomMath/Weil, a kernel-checked Lean 4 proof of rationality, the functional equation, the Riemann hypothesis (Deligne) and the Betti-number statement, following Freitag–Kiehl. About 980 modules were written by AI agents; the code has not been reviewed by humans.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"AxiomMath/Weil repository and README","url":"https://github.com/AxiomMath/Weil","type":"code","group":"primary","domain":"github.com"},{"n":2,"title":"Axiom on X: release of the Weil conjectures formalization (6 Nov)","url":"https://x.com/axiommathai/status/2118689554085646813","type":"official","group":"primary","domain":"x.com"},{"n":3,"title":"lean-eval problem: Weil conjectures in terms of point counts","url":"https://lean-lang.org/eval/problems/weil_conjectures/","type":"docs","group":"primary","domain":"lean-lang.org"}],"official":3,"filed":"2026-11-07","updated":"2026-11-08","orgs":["Axiom Math"],"title":"Axiom releases a Lean 4 formalization of the proof of all four Weil conjectures, including Deligne's Riemann hypothesis over finite fields, written by AI agents on its Stacks Project library","summary":"Axiom Math's repository AxiomMath/Weil (created 6 Nov 2026) is \"A Lean 4 (v4.34.0-rc2) formalization of the proof of the Weil conjectures, following E. Freitag and R. Kiehl, Étale Cohomology and the Weil Conjecture\", built on Mathlib and on Axiom's Lean formalization of the Stacks Project (released 4 Nov). The README states that all four conjectures are proved, with no sorry, no native_decide and no added axiom, and that the statement of the lean-eval problem weil_conjectures is proved verbatim. It also states that the library's roughly 980 modules were \"written by AI agents\" and \"have not yet been reviewed by humans\".","key_facts":["Reactions by 8 Nov: web search finds no mathematicians' assessment or press coverage of the Weil release yet; Axiom's earlier claims drew caution on the Lean Zulip","Claim (README, verbatim): 'All four Weil conjectures are proved: rationality, the functional equation, the Riemann hypothesis (Deligne's theorem) and the Betti-number statement. Lean's kernel checks every proof. There is no `sorry`, no `native_decide` and no added axiom.'","Source followed: Freitag and Kiehl, 'Étale Cohomology and the Weil Conjecture' (Ergebnisse 13, Springer, 1988); Deligne's theorem is obtained from the Stacks library's LAdicSheaf.weightBound 'proved in StacksFrontier along Laumon's Fourier-transform route'; the Betti-number statement goes through Artin's comparison theorem, also formalized","Benchmark: the Mathlib-only statement of the lean-eval problem 'weil_conjectures' is proved verbatim (Weil/LeanEval/WeilConjectures.lean)","AI role and review (README, verbatim): '[Weil] about 980 modules written by AI agents. Every proof is kernel-checked; the code has not yet been reviewed by humans.' Which models were used is not stated","Announcement: Axiom on X, 6 Nov 12:58 UTC, 'Now, we release the Lean formalization of the proof of the renowned Weil conjectures (I-IV), one of which gives Riemann Hypothesis over finite fields' (about 42k views, 265 likes on 7 Nov, fxtwitter); far less reach than the 4 Nov Stacks post (6.9M)","Repository: github.com/AxiomMath/Weil, Apache 2.0, 13 stars on 6 Nov; no independent build or expert review reported as of 7 Nov"],"key_numbers":[],"tags":["math","algebraic-geometry","weil-conjectures","etale-cohomology","lean","formalization","autoformalization","agents","axiom"],"science":{"field":"mathematics","subfield":"algebraic geometry (étale cohomology), formal verification","problem":"Weil conjectures (Weil 1949; Dwork 1960, Grothendieck 1960s, Deligne 1974): machine-checked proof","result":"Claimed sorry-free Lean 4 proof of rationality, functional equation, Riemann hypothesis and Betti-number statement for smooth projective varieties over finite fields","open_since":"1949","ai_system":["Axiom agents"],"human_role":"AI agents wrote the library on top of Axiom's Stacks formalization; no human code review yet","verification":{"key":"lean","label":"Lean-verified","text":"Lean kernel-checked per the repository; not independently rebuilt or reviewed"},"status":"pending","shock":""},"body_md":"## What happened\n\nTwo days after releasing a Lean formalization of the Stacks Project, Axiom Math published a Lean 4 library that proves the Weil conjectures, following Freitag and Kiehl's textbook. Axiom presented it as the first major result built on the Stacks library (\"The true test of a foundational library is what you can build on it\"). The README lists the main theorems with their Lean statements and says the proofs use only the standard axioms.\n\nThe proof was not new mathematics: the conjectures were proved by Dwork, Grothendieck and Deligne (1960–1974). The claim is that the full proof, including Deligne's theorem, is now machine-checked. According to the README, AI agents wrote the code and humans have not reviewed it. As of 7 Nov no one outside Axiom had reported rebuilding it or checking that the Lean statements match the classical ones.\n\n## Why it matters\n\nDeligne's proof rests on a large body of étale cohomology that formalization projects had treated as out of reach for years. If the build and statements hold up, this follows Axiom's Classification of Finite Simple Groups formalization (27 Oct) as a second large body of classical mathematics checked by machine.","disputed":[{"label":"Verification","text":"Lean kernel-checked per the repository; not independently rebuilt or reviewed","sources":[]}],"related":[{"id":"2026-11-04-axiom-stacks-project-lean-formalization","url":"https://postcutoff.com/e/2026-11-04-axiom-stacks-project-lean-formalization/","date":"2026-11-04","date_precision":"day","short_title":"Axiom: the Stacks Project formalized in Lean","deck":null,"takeaway":"On 4 Nov 2026 Axiom Math released AxiomMath/Stacks, a Lean 4 formalization of 'every tag of the Stacks Project with formalizable content' (about 63,000 modules), with an agent-written StacksFrontier tier and a human-reviewed StacksCore tier. Axiom's X post had 6.9M views.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-11-07","updated":"2026-11-09","orgs":["Axiom Math"]},{"id":"2026-10-27-axiom-cfsg-classification-finite-simple-groups-lean","url":"https://postcutoff.com/e/2026-10-27-axiom-cfsg-classification-finite-simple-groups-lean/","date":"2026-10-27","date_precision":"day","short_title":"Axiom: Classification of Finite Simple Groups formalized in Lean","deck":null,"takeaway":"On 27 Oct 2026 Axiom Math published a Lean 4 formalization of the Classification of Finite Simple Groups (CFSG). The proof was originally spread over hundreds of papers and an estimated 10,000–15,000 pages. Axiom says it is sorry-free, follows the first-generation proof, and was built by AI agents working with about 30 mathematicians and engineers. Press reports put the build time at 24 days. Along the way the team found a false step in one published route to the Unbalanced Group Theorem and wrote a replacement argument. The project is self-reviewed: outside experts have not yet re-run the build or checked that the Lean statement matches CFSG.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":8,"filed":"2026-10-28","updated":"2026-11-03","orgs":["Axiom Math"]}],"people":[],"posts":[{"id":"x-axiommathai-2118689554085646813","title":"“We shared our formalization of Stacks. Is it reusable? The true test of a foundational…”","url":"https://postcutoff.com/p/x-axiommathai-2118689554085646813/"}],"videos":[],"models":[],"changes":[{"date":"2026-11-08","type":"edit","text":"Re-checked reactions (web search): none found"},{"date":"2026-11-07","type":"filed","text":"Created (science agent; README, Axiom X post via fxtwitter)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-07","run":null,"sources_read":null,"updated":"2026-11-08","human_review":null,"version":{"date":"2026-11-08"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":190,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":129,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":220,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":159,"in_training_data":false}],"short_url":"https://postcutoff.com/s/axiom-weil-conjectures-proved"}