Palomar – a registry of Lean verified mathematics
In recent months there has been a proliferation of AI-generated proofs of various old and new results, some of which have been formalized in the proof assistant language Lean. However, checking tha…
Anatomy of a Lean Proof for Software Engineers
Intro I recently worked through a problem from a theory of computation textbook that asked me to prove a property of a language using finite automata. The informal proof is a simple constructive proof where you build an automaton and show that it recognizes the language. This is kind of similar to program verification, so I thought it’d be interesting to see what it takes to formalize the proof. Lean is a good choice for this, because its Mathlib has all the theorems for the problem.
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Lean certificates accompanying proofs in mathematics and theoretical computer science - . · openai/ten-proofs@d0e1ae7
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Lean certificates accompanying proofs in mathematics and theoretical computer science - . · openai/ten-proofs@5a102c1
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Lean certificates accompanying proofs in mathematics and theoretical computer science - . · openai/ten-proofs@e9dce9a
Palomar — Lean-verified mathematics
A public registry of Lean-verified mathematical results.
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Lean certificates accompanying proofs in mathematics and theoretical computer science - . · openai/ten-proofs@94bc0fe
Mathematics with large language models as provers and verifiers
During 2024 and 2025 the discussion about the theorem-proving capabilities of large language models started reporting interesting success stories, mostly to do with difficult exercises (such as problems from the International Mathematical Olympiad), but also with conjectures [Feldman & Karbasi, arXiv:2509.18383v1] formulated for the purpose of verifying whether the artificial intelligence could prove it. In this paper we report a theorem proving feat achieved by ChatGPT by using a protocol involving different prover and verifier instances of the gpt-5 model working collaboratively. To make sure that the produced proofs do not suffer from hallucinations, the final proof is formally verified by the lean proof assistant, and the conformance of premises and conclusion of the lean code is verified by a human. Our methodology is by no means complete or exact. It was nonetheless able to solve five out of six 2025 IMO problems, and close about a third of the sixty-six number theory conjectures in [Cohen, Journal of Integer Sequences, 2025].
Mathematics with large language models as provers and verifiers
During 2024 and 2025 the discussion about the theorem-proving capabilities of large language models started reporting interesting success stories, mostly to do with difficult exercises (such as problems from the International Mathematical Olympiad), but also with conjectures [Feldman & Karbasi, arXiv:2509.18383v1] formulated for the purpose of verifying whether the artificial intelligence could prove it. In this paper we report a theorem proving feat achieved by ChatGPT by using a protocol involving different prover and verifier instances of the gpt-5 model working collaboratively. To make sure that the produced proofs do not suffer from hallucinations, the final proof is formally verified by the lean proof assistant, and the conformance of premises and conclusion of the lean code is verified by a human. Our methodology is by no means complete or exact. It was nonetheless able to solve five out of six 2025 IMO problems, and close about a third of the sixty-six number theory conjectures in [Cohen, Journal of Integer Sequences, 2025].
Tutorial: Introduction to Formal Verification with Lean (Part 1) - HashCloak
A tutorial Formal Verification using Lean for cryptographic engineers. Implement and prove correctness of the One-Time pad following a cryptography book.
Fermat's Last Theorem in Lean 4 (Lean 4.33.1, Mathlib v4.33.0) · anthropics/fermats-last-theorem@aa2d8b3
A complete machine-checked proof of Fermat's Last Theorem: the import closure of FinalCheck.lean - 1,450 definition modules (Definitions/), 29,511 statement modules (Theorems/), 29,511 proo...
CausalSmith · AI Causal Scientist
Working papers from CausalSmith, an AI causal scientist: econometric theory in which every theorem, assumption, and lemma is machine-verified in Lean 4.
Advancing Mathematics Research with AI-Driven Formal Proof Search
Large language models (LLMs) increasingly excel at mathematical reasoning, but their unreliability limits their utility in mathematics research. A mitigation is using LLMs to generate formal proofs in languages like Lean. We perform the first large-scale evaluation of this method's ability to solve open problems. Our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars, proved 44/492 OEIS conjectures, and is being deployed in combinatorics, optimization, graph theory, algebraic geometry, and quantum optics research. A basic agent alternating LLM-based generation with Lean-based verification replicated the Erdős successes but proved costlier on the hardest problems. These findings demonstrate the power of AI-aided formal proof search and shed light on the agent designs that enable it.
Verification in the Age of AI Scientists
AI Scientists now operate at a scale that outpaces human capacity for manual review. Systems such as Sakana’s AI Scientist write entire workshop papers end-to-end. Lila Sciences runs autonomous ``AI Science Factories’’ that hypothesize, experiment, and iterate without human guidance. FutureHouse’s Kosmos and Robin generate thousands of candidate hypotheses in a single run, and Google’s Co-Scientist proposes testable experiments at a rate no laboratory can fully evaluate. Each of these systems emphasizes verified results, yet that standard ranges from near-perfect formal proof in mathematics to decade-long clinical trials in medicine, with no shared framework for judging sufficiency across domains. As outputs scale beyond what humans can manually inspect, the question of which results to trust becomes as hard as generating them. The bottleneck for AI for Science is no longer hypothesis generation, it is verification.
Navier-Stokes and Lean
I was working on this week's post on Lean after reading Kevin Hartnett's book The Proof in the Code: How a Truth Machine Is Transforming Ma...