Stronger Lean provers make autoformalization checks weaker - a fix that judges the proof term, not the tactic script
AspireAI community · arXiv (Xinrui Li, Yongxi Lin, Stella Biderman et al.)
As statement autoformalization moves mathematics from prose into Lean 4, the correctness burden shifts from the proof to the statement: does the formal statement faithfully translate its source? The natural test, logical equivalence P<->Q, proves too much - a prover can prove each statement separately and pass through True, so the stronger the prover the more unrelated pairs pass. SemantEq instead requires each statement to turn into the other only through a chain of allowed steps such as logical rearrangement and rewriting with the library's own equations, judging a proof by the proof term Lean checks rather than by its tactic script. Its model-free search reached 100% precision and 65.6% recall on the ProofNetVerif test split under adjudicated labels, against 95.8% and 61.8% for BEq+, at roughly 8.9x the CPU time for at most three extra pairs. A language model finds more proofs but under two of four prompts proves both directions of about two in five non-equivalent pairs; the audit mode accepted none of the 104 non-equivalent pairs and raised recall on 407 equivalent pairs from 65.4% to 81.1%. The one cost is that stricter judging rejects some proofs a weaker prover would have let through.
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Two sparse-autoencoder analyses have resisted scaling: matching semantically similar features across layers, and compressing large feature circuits into interpretable supernodes. This paper argues both are the same underlying problem, estimating semantic distances between SAE features that sit on different activation manifolds. Its representation treats each feature not as a single decoder vector but as an activation-weighted distribution over the hidden states expressing it, projects those distributions into a shared reference space, and compares them with Wasserstein distance. The representation is proved invariant to activation rescaling and stable under perturbation, and recovers true matches under finite-sample margin conditions. Empirically it outperformed decoder-vector and LLM-based baselines and captured subtle functional distinctions between related features, and it compressed large feature circuits into interpretable supernodes automatically. The limitation is that this is interpretability tooling: the paper demonstrates better feature matching and circuit compression, not that downstream interpretability claims become more accurate.