Learned Interventions in Lean 4 grind

Lean~4's \grind{} tactic combines congruence closure, \ematch{}ing, and case-splitting into a single automated solver, and like any such solver, it relies on hand-tuned heuristics to decide what to instantiate and where to case-split. These heuristics are tempting targets for learning, but there is a catch: because \grind{}'s search is non-monotone, a learned heuristic that helps one proof can break another, and an always-on replacement usually nets out near zero. We avoid this by invoking a learned intervention only after stock \grind{} has already failed: a failure-triggered cascade that, by construction, cannot lose a proof \grind{} already had. We apply it to two of \grind{}'s internal decisions. A cost-aware \ematch{} filter solves slightly more problems and runs about 5\% faster. A lookahead step, proves five theorems it otherwise times out on. We also report the negative result that motivated the design: across four feature-based models, statically predicting the correct case split is no better than random, because whether a split explodes is a runtime property that the features do not capture. Our results suggest that learning within theorem-proving tactics is most effective as a mechanism for deciding when and how to spend bounded search, backed by a reliable symbolic fallback.

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.