The Faithfulness Gap: Certifying Semantic Equivalence Between Natural-Language and Formal Mathematical Statements

Autoformalization, translating natural-language mathematics into formal proof assistants, is bottlenecked not by translation fluency but by faithfulness: a formal statement can typecheck and be provable, yet still encode a different theorem than the source intended. We introduce Bidirectional Provability Fingerprinting (bpf), a framework that certifies faithfulness by characterizing each candidate through its forward and backward consequence neighborhoods in the ambient theory and matching these against probes derived from the natural-language statement. We further introduce four novel components: (i) Counterfactual Probe Generation (cpg), a contrastive procedure that synthesizes probes targeting specific drift directions; (ii) the Equivalence Spectrum, a continuous faithfulness score that replaces brittle binary verdicts; (iii) Adaptive Probe Budget Allocation (apba), an information-theoretic budget router; and (iv) Faithfulness-Guided Decoding (fgd), which uses bpf signals as a reward during autoformalization. We prove a drift detection theorem and a PAC-faithfulness result establishing that the equivalence class of a natural language statement is learnable from O(log(1/δ)/varepsilon) probes under mild assumptions. We release driftbench, a benchmark of 2,183 NL/Lean~4 pairs with controlled drift labels across six subfields of mathlib4. bpf,+,cpg detects 89.6% of drifted formalizations at a 3.0% false-positive rate-against 41.2% for typecheck and 63.3% for LLM-judge baselines, and fgd reduces the rate at which a state-of-the-art autoformalizer emits drifted statements by 47%. https://pmlrbd.github.io/BPF/