Pseudo-Formalization for Automatic Proof Verification

A team of researchers has introduced Pseudo-Formalization, a new proof format designed to make automatic verification of mathematical reasoning more reliable by combining the precision of formal logic with the flexibility of natural language [1]. The work, posted to the arXiv preprint server, addresses a persistent bottleneck in artificial intelligence: verifying the correctness of proofs generated by AI systems. Fully formal proofs written in languages like Lean are unambiguous and modular, making them straightforward to check, but most proofs — especially those produced by AI — lack these properties. Translating them into formal languages remains difficult in many frontier mathematics settings [1]. Proof theory, a major branch of mathematical logic, treats proofs as formal mathematical objects, often represented as inductively defined data structures such as lists or trees constructed according to axioms and inference rules [3]. The new approach seeks a middle ground. Pseudo-Formalization, or PF, decomposes a proof into self-contained modules. Each module states its premises, conclusion, and proof in natural language. An algorithm called Block Verification then checks each module independently. To verify a regular natural-language proof, a large language model first translates it into the Pseudo-Formal format, after which Block Verification assesses correctness [1]. An algorithm is a finite sequence of mathematically rigorous instructions used to perform a computation or solve a class of problems [4]. The researchers evaluated the combined PF and Block Verification method on two benchmarks covering olympiad and research-level mathematics. The system pareto-dominated LLM-as-judge baselines on both precision and recall for finding errors [1]. The scientific method relies on empirical testing and experimental validation, with hypotheses adjusted or discarded based on results [5]. Alongside the method, the team released ArxivMathGradingBench, a new benchmark for research-level proof verification, to support further work in the field [1]. The paper was submitted by Luke Bailey on May 19, 2026, with a revised version following on June 17, 2026. The initial submission was 676 KB; the updated version is 698 KB [1]. arXiv, where the paper appears, is an open-access repository of electronic preprints that has grown to host over two million articles as of late 2021 and receives about 24,000 submissions per month [10].