Organizations: 1SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Beijing 100190, China · University of Chinese Academy of Sciences · Institute of Logic and Cognition, Department of Philosophy, Sun Yat-sen University, Guangdong 510275, China · School of Mathematics, Shandong University, Shandong 250100, China
We present MechGeo, a Mathlib native agentic framework that jointly addresses faithful autoformalization and certified proof construction for Euclidean geometry. In this framework, GeoFormalizer represents informal problems in GeoIR, deterministically translates them into Lean 4, and iteratively repairs candidate statements using structural diagnostics and semantic evaluation. GeoProver constructs geometric proof plans, derives intermediate lemmas, and selectively algebraizes suitable subgoals through a library verified in Lean. Singular or SymPy may generate algebraic certificates, but all resulting proofs and counterexamples are checked by Lean's kernel. Experiments across seven LLM backbones show substantial improvements in autoformalization, particularly for models with weaker direct translation performance. On 43 historical IMO geometry problems, GeoFormalizer generates formal statements that GeoProver proves in 29 cases; for the remaining 14, it constructs counterexamples verified in Lean and proves all repaired statements after expert correction. Together with IMO 2026 Problem 2, this yields, to the best of our knowledge, the largest reported collection of automated, kernel-checked Lean proofs for IMO geometry problems. On the 14 geometry statements in LEAP's Lean-IMO-Bench, MechGeo proves 12 for the first time, formally refutes the remaining two, and proves both repaired statements. These results establish counterexample guided diagnosis, geometric reasoning, and certified symbolic computation as a practical foundation for trustworthy formal geometry.
Geometry problem generation is useful for AI-assisted education and multimodal mathematical reasoning, but reliable synthesis remains difficult because the problem statement, diagram, constraints, and solution should be mutually consistent. Existing methods often trade off controllability and reliability: seed-based rewriting is flexible but weakly verifiable, whereas diagram-first construction improves validity but is less suited to arbitrary user-specified constraints. We introduce VeriGeo, a controllable geometry generation framework grounded in executable reasoning traces. Given user constraints such as target concepts and difficulty, an Author agent generates a problem and diagram, and a Solver agent produces a proof-aligned solution. Both agents use a shared action sequence that connects natural language, diagrams, geometric constraints, and proof steps into a verifiable representation. A three-stage pipeline checks numerical consistency, analytical realizability, and global consistency, using verification-guided reflection to repair recoverable failures and reject unrecoverable ones. Across five LLM backbones, raw generations frequently fail these checks, while VeriGeo repairs a substantial fraction of the invalid attempts. Supervised fine-tuning on 8.7k examples generated by VeriGeo achieves the best reported GeoQA performance among end-to-end multimodal LLM-based solvers, and obtains strong results on PGPS9K and MathVista-GPS, demonstrating the effectiveness of verified synthetic data for improving multimodal geometry reasoning.
Proof autoformalization aims to translate a mathematical informal proof written in natural language into a formal proof in a formal language such as Lean~4. Several works have developed LLM-based models for proof autoformalization. However, existing evaluations have typically focused on translating well-formed informal proofs from curated datasets. We argue that a robust proof autoformalizer must remain faithful even for informal proofs that diverge from these idealized ones, and we present the first study on the robustness of proof autoformalization models. We formulate two categories of perturbations and evaluate robustness under each: a global perturbation paraphrases the informal proof in a different style, under which the formalization should remain consistent; a local perturbation alters a value, symbol, or proof step, possibly in a counterfactual way, and a robust formalization should faithfully reflect the perturbation rather than reverting to the original one or inferring a different one on its own. We build a benchmark with both perturbations on miniF2F and MATH-500, and automatically measure how stable a proof autoformalization's correctness is under global perturbations and how faithfully its output reflects local perturbations. We evaluate seven recent models, all of which are sensitive to global perturbations and mostly fail to remain faithful under local perturbations. Code and data are available via https://github.com/ucr-rai/robust-proof-autoformalization.
We introduce Goedel-Architect, an agentic framework for formal theorem proving in Lean 4 centered on blueprint generation and refinement. A blueprint is a dependency graph of definitions and lemmas that builds up to the main theorem. First, Goedel-Architect generates a blueprint of formally stated definitions and lemmas, along with declared dependencies. This blueprint is optionally guided by a natural language proof. Then, a tool-equipped Lean prover component closes each open lemma node in parallel using relevant dependencies. Failed lemmas in turn drive refinement of the global blueprint. This strategy contrasts with other mainstream approaches which use recursive lemma decomposition, and can inefficiently loop on dead-end strategies. Using the open-weight DeepSeek-V4-Flash (284B-A13B) as the backbone, Goedel-Architect attains 99.2% pass@1 on MiniF2F-test and 75.6% pass@1 on PutnamBench. With an optional natural-language proof seeding the initial blueprint on the harder problems, we additionally close the remaining two MiniF2F-test problems (reaching 100%), lift PutnamBench to 88.8% (597/672), and solve 4/6 on IMO 2025, 11/12 on Putnam 2025, and 3/6 on USAMO 2026. This represents state-of-the-art performance for an open-source pipeline at a price point up to 500x less than comparable open-source pipelines.