The Dottie Number

The Leiden Declaration: Mathematics, AI, and Making Our Values Explicit

Reseña de «What it means to be a mathematician when AI does the math?»

El artículo «What it means to be a mathematician when AI does the math?» analiza cómo la IA ha transformado las matemáticas en pocos años, pasando de imitar soluciones básicas a resolver problemas de

AI in Mathematics Is Forcing Big Questions

Researchers debate motivation, purpose, and the field’s future

AXLE: A Cloud Infrastructure for Lean 4 Theorem Proving Utilities

We present AXLE (Axiom Lean Engine), a cloud service for Lean 4 proof manipulation, extraction, and verification. Recent progress in AI for mathematics -- reinforcement learning pipelines, agentic proving workflows, dataset curation -- demands Lean 4 tooling that scales to millions of requests while remaining correct and robust; existing infrastructure offers parallel compilation but not scalable proof verification, higher-level proof manipulation, multi-version support, or per-request isolation at the throughput modern AI workflows require. AXLE provides 14 Lean 4 metaprogramming tools spanning strict proof verification, declaration metadata extraction, semantic source manipulation, deterministic proof repair and simplification, and lemma extraction. The service runs as a multi-tenant cloud deployment with per-request isolation and concurrent support for multiple Lean 4 and Mathlib versions, accessible via a Python SDK, command-line interface, web UI, MCP server, and raw HTTP API. AXLE is publicly available and free to use at https://axle.axiommath.ai and via the axiom-axle PyPI package, with no local Lean 4 installation required. It has served over 500 million requests to date and is the underlying infrastructure for Axiom Math's proving efforts, including its 12/12 score on the 2025 Putnam competition.

Discovering New Theorems via LLMs with In-Context Proof Learning in Lean

Large Language Models (LLMs) have demonstrated significant promise in formal theorem proving. In this study, we investigate the ability of LLMs to discover novel theorems and produce verified proofs. We propose a pipeline called Conjecturing-Proving Loop (CPL), which iteratively generates mathematical conjectures and attempts to prove them in Lean 4. A key feature of CPL is that each iteration conditions the LLM on previously generated theorems and their formal proofs, enabling parameter-free improvement of proof strategies via in-context learning. We provide both theoretical and experimental evidence that CPL increases the discovery rate of hard-to-prove theorems compared to frameworks that generate statements and proofs simultaneously. Moreover, our experiments show that reusing the LLM's own formally verified outputs as context consistently improves subsequent proof success, demonstrating the effectiveness of self-generated in-context learning for neural theorem proving. The source code is available at https://github.com/auto-res/ConjecturingProvingLoop.

Readings shared June 21, 2026

The readings shared in Bluesky on 21 June 2026 are: A Lean 4 formalization of euclidean domain algorithms from a 1986 icon experimentation package. ~ Lars Warren Ericson. #LeanProver #ITP #Math A Lea

Formalizing Numerical Analysis: An Agent Pipeline and Quality Audit Beyond Kernel Acceptance

Recent work has demonstrated that coding agents can formalize entire advanced mathematics textbooks in Lean 4, yet existing efforts concentrate on branches of mathematics already well-represented in mathlib and measure success solely through kernel acceptance. We address both limitations by applying a coding agent to formalize Numerical Methods for Ordinary Differential Equations, a textbook in numerical analysis that is largely absent from mathlib, stressing the agent's capacity to develop new theory from scratch. We further introduce a systematic, reproducible three-dimensional framework for evaluating the quality of agent-produced formalizations beyond compilation: semantic correctness, Mathlib reuse, and cross-file reuse via LLM-as-judge methods. Applying this framework to our own formalization and to the released outputs of RepoProver and M2F, we uncover recurring unfaithful formalization patterns, including incomplete multi-part statements, added weakening hypotheses, and parameter restrictions, that kernel acceptance entirely obscures. Our results suggest that compilation-based metrics substantially overstate formalization quality, and we provide a reproducible audit methodology to support more rigorous evaluation of future autoformalization systems.