Numina Fuse: Interactive Autoformalization in Lean 4

Long-horizon autoformalization of research mathematics fails not only at hard lemmas, but at scale: statements drift, dependencies tangle, context decays, and local repairs corrupt distant work. We present LeanMarathon, a multi-agent harness for reliable research-level Lean autoformalization. Its core abstraction is an evolving blueprint: a Lean file that serves simultaneously as formal proof skeleton, natural-language proof graph, and shared system of record. Four contract-scoped agents construct, audit, prove, and repair this blueprint. These agents are coordinated by a two-stage orchestrator that first stabilizes target fidelity through adversarial review and then discharges the proof directed acyclic graph (DAG) from its dynamic leaves upward in parallel CI-gated rounds. LeanMarathon turns one brittle multi-hour run into many local, recoverable, parallel transactions. We evaluate LeanMarathon on two recent research papers spanning four Erdős problems (#1051, #1196, #164, #1217). Across three autonomous runs, it formalizes all seven target theorems with no sorry, proving 258 lemmas and theorems. These results show that reliable AI co-mathematics requires not only stronger provers, but durable harnesses that preserve target fidelity across long mathematical developments. The code can be found at https://github.com/YuanheZ/LeanMarathon.

Autoformalization - automatically translating natural language mathematical texts into formal proof language such as Lean4 - can help accelerate AI-assisted mathematical research, be it via proof verification or proof search. I fine-tune Qwen3.5-2B with LoRA for natural language to Lean4 formalization on FineLeanCorpus and consider three training regimes: supervised fine-tuning (SFT) with curriculum learning (difficulty 1 to 10), SFT without curriculum ordering, and reinforcement learning using group relative policy optimization (GRPO) with a cycle consistency reward. Cycle consistency measures how well the meaning of a statement is preserved through a NL to Lean4 to NL' loop, computed as cosine similarity of off-the-shelf sentence embeddings. On an unseen subset of FineLeanCorpus (FLC) and on PutnamBench, RL substantially outperforms both SFT variants (mean cycle consistency 0.669 vs. 0.513 on FLC; 0.561 vs. 0.422 on PutnamBench), while increasing cross-entropy loss by only 0.011 nats, with minimal impact on formalization quality. Curriculum ordering provides no measurable benefit over shuffled training.

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.