A guest post by mathematician Thomas Hales surveys the state of formal proof verification, noting that major theorems—including Fermat's Last Theorem, the sphere packing problem, and Navier-Stokes forced blowup—have recently been formally verified using computer proof assistants. Hales identifies Lean, created by Leo de Moura at Microsoft in 2013 and later open-sourced, as the proof assistant now most favored by mathematicians among several competing systems like Coq, Isabelle, and HOL Light.
terrytao.wordpress.com
· 2026-10-09
OpenAI revised its public GitHub repository of formalized mathematical results, adding six new Lean formalizations and modifying 19 existing ones. It also withdrew three results previously listed as formalized. The company said roughly 42% of its top-line results are now formalized and that it will keep updating the repo as it finds errors.
twitter.com
· 2026-10-08
OpenAI announced the release of a proof for the Unique Games Conjecture (UGC), a major problem in theoretical computer science and mathematics, as part of a batch of 372 breakthrough results. The proof includes a Lean certificate, but initial reactions suggest it is difficult for humans to interpret without AI assistance, with some describing it as unclear and poorly written. The proof has significant implications for the understanding of NP-hardness in optimization problems.
scottaaronson.blog
· 2026-10-07
A formal verification project has completed a machine-checked proof in the Lean theorem prover establishing the optimal side length for packing 11 unit squares into a square container, with the exact value defined as the root of a polynomial equation. The verification run, tied to a specific pinned commit, accepted all 7,920 local Lean modules with zero admissions, using native numerical certificates for computationally expensive exact checks alongside standard Lean proofs for geometry and proof assembly.
github.com
· 2026-10-07
A recent gathering of innovative startups supported by Convergent Research highlighted the positive outlook among mathematicians regarding AI's influence. While acknowledging significant changes to traditional workflows, many see potential for AI to complement and enhance mathematical research. The discussion reflects a community adapting to rapid technological advancements that challenge conventional problem-solving methods.
terrytao.wordpress.com
· 2026-10-05
A new book recounts how Lean, a program originally developed to catch bugs in Microsoft's software, was repurposed by mathematicians to formally verify proofs. The review describes its journey from an internal engineering tool to a system now used to check mathematical arguments with machine-level rigor.
wsj.com
· 2026-10-02
A software engineer walked through a theory-of-computation textbook exercise asking for a proof, via finite automata, of a language property, then reformalized that informal constructive proof in the Lean proof assistant. The write-up builds a deterministic finite automaton (DFA) and leans on existing theorems in Lean's Mathlib library to formally verify it recognizes the target language.
agostbiro.net
· 2026-10-01
Ten Claude Opus 5.5 agents were tasked with finding a faster exact shortest-path algorithm for directed graphs with non-negative real weights and proving it correct in the Lean theorem prover. Over roughly 15 hours and 733 messages, the agents produced an algorithm called C-HD, which its creators say formally improves on previously published complexity bounds, including Dijkstra's algorithm and two recent 2025 and 2026 papers.
vals.ai
· 2026-09-22
Trail of Bits reports that ahead of auditing the Miden zero-knowledge VM, its team spent six months using AI agents to build an LSP server, decompiler, static analysis engine, and a Lean formal model of the VM executor from scratch, since almost no developer tooling existed for Miden's custom assembly language. The resulting tools and formal proofs helped uncover a security flaw allowing a malicious prover to forge Falcon signatures and steal funds from Miden account holders, plus 95 machine-checked correctness proofs covering much of the Miden core library.
blog.trailofbits.com
· 2026-09-21
An amateur mathematician spent a month using a frontier AI model to attempt a formal Lean proof of John Conway's refinement conjecture about omnific integers in surreal number theory, a problem posed 50 years ago. The proof reportedly passed mechanical verification through the Palomar registry, and some Lean and field experts said the statement appears correct, though full independent mathematical verification is still pending.
overreacted.io
· 2026-09-18
Bend is a newly introduced programming language that compiles to native code, runs on multiple CPU cores or GPUs without manual threading, and includes a built-in proof checker similar to Lean and Rocq. Developers can write rules in a LAWS.bend file, and Bend will refuse to accept any AI-generated code change that violates those rules, checking each change in about a second. Its creators demonstrate this by showing an AI agent attempt to break a game's win condition, which Bend blocks until the AI produces a compliant fix.
bend-lang.com
· 2026-09-17
Alongside its human-readable proof resolving a longstanding Navier-Stokes question, OpenAI also released a machine-checkable Lean 4 formal proof, completed in about 17 hours. The author contrasts this with historical estimates that formalizing research-level math could take tens of thousands of person-hours by hand.
johndcook.com
· 2026-09-10
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