FIRMA: Bidirectional Formal-Informal Mathematical Language Alignment with Proof-Theoretic Grounding

AbstractWhile large language models excel at gener- ating plausible mathematical text, they often produce subtly incorrect formal translations that violate proof-theoretic constraints. We present FIRMA (Formal-Informal Reasoning in Mathematical Alignment), a bidirectional translation system between formal and informal mathematical language that leverages proof-theoretic interpretability hierarchies and specialized architectural components for proof preservation. Unlike existing approaches that treat this as pure sequence-to-sequence translation, FIRMA introduces a hierarchical architecture with complexity-aware routing, proof-preserving attention mechanisms, and multi-objective training that balances formal correctness with natural readability. Through progressive complexity training on curated datasets from Lean 4 and formal mathematics repositories, we evaluate FIRMA on 200 translation samples across complexity levels and compare against two baseline systems. Our analysis shows statistically significant improvements of 277.8% over BFS-Prover- V1-7B and 6307.5% over REAL-Prover on overall translation quality metrics. Ablation studies on 50 samples demonstrate that each architectural component contributes substan- tially to performance, with removal of any component resulting in 83-85% performance degradation. We release our code at https://github.com/smfatima3/FIRMA