Does the Proof Prove It That Way? Faithful Formalization of Elements Proofs

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 $\mathcal{O}(\log(1/δ)/\varepsilon)$ 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/

Lean verifies that a generated declaration is well typed, but cannot check whether it states what the user intended. We measure how far compilation overstates faithfulness in statement autoformalization without canonical Lean targets. Our criterion requires Lean compilation and agreement of two LLM judges from different model families. On an independent random audit, it agrees with the human majority on 91.5% of cases; it rarely accepts an unfaithful output but rejects some faithful ones. Across eight systems on 227 university-level statements, the gap between compiling and accepted outputs ranges from 1.3 to 29.5 percentage points: our tool-augmented agent compiles 87.2% of the statements, but only 57.7% are accepted. Human review of all the agent's rejections confirms that 23.8 of its 29.5 points are real translation errors. For the gaps of the seven one-shot systems, a two-stage review by Claude Opus 5.5 with Lean and GPT-6.1 Sol, validated on these human labels, finds that 78-100% are real translation errors for six of them. When GPT-5.2, one of the two judges, grades the translations of two models released after it, GPT-6 Astra and Claude Opus 5.5, it accepts 83% of the statements for each, against 21% for one-shot GPT-5.2. For these two models, the same review classifies only 4 of their 54 rejections (7%) as translation errors, so most of their measured gap is judge errors rather than translation errors. This is consistent with these systems erring less while the judge does not: the fewer errors a system makes, the smaller the share of its rejections that are translation errors. Finally, a 2^3 tool ablation of the agent shows that Lean feedback drives most of its compilation gain, and that nearly half of this gain fails the semantic criterion. Code and data are available at https://github.com/grenadeComing/beyond-compilation.

Reliable verification of proofs remains a bottleneck for training and evaluating AI systems on hard mathematical reasoning. Fully formal proofs, in languages like Lean, are easy to verify because they are unambiguous and modular. Most proofs, particularly those written by AI systems, have neither property, and translating them into formal languages remains challenging in many frontier math settings. We propose Pseudo-Formalization (PF), a proof format that captures the modularity and precision of formal proofs while retaining the flexibility of natural language. A Pseudo-Formal proof is decomposed into self-contained modules, each stating its premises, conclusion, and proof in natural language. To verify the correctness of a regular natural language proof, an LLM translates it to Pseudo-Formal and then verifies each module independently, an algorithm we call Block Verification (BV). We evaluate PF+BV on two benchmarks spanning olympiad and research-level mathematics, where it pareto-dominates LLM-as-judge baselines on error-finding precision and recall. To support future work, we release our research-level proof verification benchmark ArxivMathGradingBench.