The Signal-Coverage Matrix: Stratifying Type and Semantic Errors in Statement Autoformalization

Lean 4 autoformalization has become increasingly popular in recent years, with frontier language models and open-weight autoformalizers now producing valid formalizations of mathematical theorems. However, these evaluations often rely on single canonical phrasings of theorems and rarely probe whether outputs are robust to natural variation in inputs, while prior work has shown that semantically equivalent paraphrases often induce divergent formal outputs. We study the structure of these divergences in Lean 4 by applying deterministic paraphrase rules to datasets of undergraduate and Olympiad-level math problems. Across four frontier models and three open-weight autoformalizers, we find that paraphrase sensitivity is dominated by failures at the code-generation layer, and that these failures are typed differently by dataset. Furthermore, these patterns generalize to open-weight models, showing that state-of-the-art autoformalizers still struggle to generate valid Lean code. Our results provide a failure-mode taxonomy for autoformalization and motivate training-time interventions targeted at specific compilation failures.

We present Trellis: an autoformalization system that leverages LLM agents in a deterministically constrained workflow to enforce incremental progress in Lean autoformalization tasks through iterative refinement of natural language proofs. Our approach is motivated by the common mathematician's notion of what it means to have a rigorous proof in the first place: namely, that it would be routine to elaborate any part of the proof in further detail. The result is a system which aims to achieve reliable autoformalization on a modest budget and with generalist agents, with specialization to autoformalization coming not from any task-specific agent training but instead from a meaning-of-rigor inspired workflow enforced by process semantics. We link to an end-to-end Lean formalization of a recent Ramsey theory breakthrough produced by the process.

Lean verifies that a generated declaration is well typed, but not that it expresses the statement a user intended. We study two questions for autoformalization without canonical Lean targets: whether LLM judges can provide a usable proxy for human semantic review, and how much compilation overstates faithfulness across systems. Our criterion combines Lean compilation with strict semantic consensus between GPT-5.2 and Gemini-2.5-Pro. On an independently audited random sample, it agrees with human majority on 89.7% of cases (Wilson 95% CI: 82.1--94.3%). Across eight systems evaluated on 400 graduate-level statements, every system has a nonzero compile--faithfulness gap, whose observed magnitude ranges from 3.0 to 29.0 percentage points. The full GPT-5.2 tool-augmented agent shows the largest gap, compiling 89.5% while satisfying the semantic criterion on 60.5%. Human review, an independent third-family judge, and a BEq formal cross-check provide complementary evidence that the accepted core is reliable and that most audited outputs in the gap are genuine semantic mismatches. A secondary $2^3$ factorial analysis shows that elaboration feedback is the largest validity intervention, yet does not eliminate semantic drift. LLM judging is therefore useful as a human-calibrated, conservative aggregate measure, not as an equivalence oracle.