Sage: Formalization with Semantic Correction

While neural theorem provers have achieved impressive milestones in formal mathematics, they largely operate on the assumption that faithful formal statements are already provided. Translating informal natural language into a formal language is a critical data bottleneck plagued by an "illusion of rigor": standard type-checkers accept statements that compile but drop hypotheses, introduce vacuous truths, or subtly alter mathematical bounds. To resolve this, we introduce Sage (Semantic Agent-Guided Formalization Engine), an agentic framework that replaces monolithic translation with a four-stage decomposed generation pipeline coupled with a dual-signal semantic correction loop. By pairing formal compiler diagnostics with multi-dimensional semantic feedback, our correction loop enforces mathematical fidelity alongside syntactic validity. By explicitly accounting for the gap between open-ended queries and declarative formal targets, our pipeline prevents models from achieving high formalization rates by guessing unverified answers (exhibiting a 70.9% answer leakage rate in monolithic baselines). Consequently, Sage suppresses leakage to 2.7% while achieving 73.3% pass@4 joint compilation and semantic fidelity on the Omni-MATH benchmark without proofs (compared to 42.0% for a fine-tuned Goedel-Formalizer-V2-32B baseline). Finally, on IMO-Unformalized, a novel frontier of 175 unformalized International Mathematical Olympiad problems, Sage demonstrates effective zero-shot generalization with 87.4% pass@4 verified fidelity compared to just 19.4% for the baseline, winning over 79% of blind pairwise evaluations.