Sage formalizes math into Lean 4 and cuts answer leakage to 2.7%

Neural theorem provers have hit real milestones in formal mathematics, but they mostly assume that faithful Lean 4 formal statements already exist. A new arXiv paper argues that producing those statements from informal natural language is a critical data bottleneck, and that it suffers from an "illusion of rigor": standard type-checkers accept statements that compile but drop hypotheses, introduce vacuous truths, or subtly alter mathematical bounds.

The paper's answer is Sage (Semantic Agent-Guided Formalization Engine), an agentic framework. Instead of translating a problem in one monolithic step, Sage uses a four-stage decomposed generation pipeline, coupled with a dual-signal semantic correction loop. The loop pairs Lean 4 compiler diagnostics with multi-dimensional semantic feedback, so the output has to be mathematically faithful as well as syntactically valid.

The authors also target a second problem: the gap between open-ended queries (questions that ask for an answer) and declarative formal targets (statements that assert one). Without care for that gap, models can reach high formalization rates by guessing unverified answers; the abstract gives a 70.9% answer leakage rate for this behaviour. The authors say Sage prevents it and suppresses leakage to 2.7%.

On Omni-MATH without proofs, Sage reaches 73.3% pass@4 joint compilation and semantic fidelity, compared with 42.0% for a fine-tuned Goedel-Formalizer-V2 baseline.

The paper also introduces IMO-Unformalized, a new set of 175 unformalized International Mathematical Olympiad problems. There Sage shows what the authors call effective zero-shot generalization: 87.4% pass@4 verified fidelity against just 19.4% for the baseline. Sage also won over 79% of blind pairwise evaluations against the baseline.

Key facts

Sage (Semantic Agent-Guided Formalization Engine) turns informal math into Lean 4 statements using a four-stage decomposed pipeline and a dual-signal semantic correction loop.
The correction loop combines Lean 4 compiler diagnostics with multi-dimensional semantic feedback, aiming at fidelity as well as compilation.
Answer leakage, where models guess unverified answers, is cited at 70.9% in the abstract; Sage suppresses it to 2.7%.
On Omni-MATH without proofs, Sage scores 73.3% pass@4 joint compilation and semantic fidelity versus 42.0% for a fine-tuned Goedel-Formalizer-V2 baseline.
On the new 175-problem IMO-Unformalized set, Sage reaches 87.4% pass@4 verified fidelity versus 19.4% for the baseline, and wins over 79% of blind pairwise evaluations.

Why it matters

Theorem provers can only prove what they are given, and the paper says they largely assume faithful Lean 4 statements are already provided. Getting those statements from natural language is framed as a critical data bottleneck. The danger is statements that compile yet say something different: hypotheses dropped, vacuous truths, bounds subtly changed. Sage tries to catch those cases by checking meaning alongside compilation, and it targets answer leakage, where a model looks successful by guessing answers nobody verified.

Who it affects

The work is aimed at people building autoformalization and neural theorem proving systems, who need faithful Lean 4 statements to work from. It also bears on anyone judging formalization quality by compile rate alone, since the paper argues that compiling is not enough.

How to use it

The abstract describes the method and results only. No code, dataset release, or licence is mentioned, so there is nothing in it to download or run yet. The practical takeaway is the design: split formalization into stages, and feed both compiler errors and semantic feedback back into a correction loop.

How solid is it

The numbers come from the paper's abstract and are the authors' own results. The comparison is against a single baseline, a fine-tuned Goedel-Formalizer-V2; no comparison against any other system is reported. The gaps are large: 73.3% versus 42.0% on Omni-MATH without proofs, and 87.4% versus 19.4% on IMO-Unformalized. The abstract does not say who performed the blind pairwise evaluations (humans or models) or how many comparisons were made.

Risks and caveats

The abstract does not say which system the 70.9% leakage rate was measured on, so the 70.9% to 2.7% drop should not be read as a head-to-head result against a named baseline. It does not name the underlying language models Sage uses, and it does not list the four stages or the dimensions of the semantic feedback. It also does not say whether IMO-Unformalized is publicly released. IMO-Unformalized is a new set introduced by the same paper, so results on it are not yet independently reproduced.

“standard type-checkers accept statements that compile but drop hypotheses, introduce vacuous truths, or subtly alter mathematical bounds”

— Sage paper abstract, arXiv 2609.35790