Euclean: Automated Geometry Problem Formalization with Unified Verification in Lean

arXiv:2607.19374v1 Announce Type: new Abstract: Recent formal reasoning systems have reached IMO-level performance, yet they leave a fragmented landscape: algebra and number theory are handled in Lean, while geometry still relies on domain-specific languages with limited formal guarantees. This split increases the trusted computing base and hinders unified model development. Existing geometry-in-Lean efforts (LeanEuclid, LeanGeo) introduce custom axiom systems incompatible with standard Mathlib, and their small scale ($<$ 1,100 problems) limits large-scale training. Native Mathlib autoformalization of geometry, however, poses distinct challenges: implicit diagrammatic assumptions (e.g., topological configuration and non-degeneracy) must be made explicit rather than deferred to external solvers, and models must adapt to Mathlib's small, rapidly evolving geometry infrastructure. We present Euclean, a four-stage framework - constraint explication, configuration anchoring, formalization mapping, and iterative repair - for automatically formalizing geometry in native Mathlib. We construct OMNI-Geometry (768 competition problems) and Numina-Geometry (177,597 problems), the largest geometry formalization dataset in Lean. Human evaluation shows 48.89% TOP1 and 73.33% TOP5 accuracy. Training Goedel v2 on our formalizations improves proof success from 13.6% to 15.1%, validating dataset quality for unified neural theorem proving. Code and datasets: https://github.com/tlb-22/Euclean.

Abstract:Recent formal reasoning systems have reached IMO-level performance, yet they leave a fragmented landscape: algebra and number theory are handled in Lean, while geometry still relies on domain-specific languages with limited formal guarantees. This split increases the trusted computing base and hinders unified model development. Existing geometry-in-Lean efforts (LeanEuclid, LeanGeo) introduce custom axiom systems incompatible with standard Mathlib, and their small scale ($<$ 1,100 problems) limits large-scale training. Native Mathlib autoformalization of geometry, however, poses distinct challenges: implicit diagrammatic assumptions (e.g., topological configuration and non-degeneracy) must be made explicit rather than deferred to external solvers, and models must adapt to Mathlib's small, rapidly evolving geometry infrastructure. We present Euclean, a four-stage framework - constraint explication, configuration anchoring, formalization mapping, and iterative repair - for automatically formalizing geometry in native Mathlib. We construct OMNI-Geometry (768 competition problems) and Numina-Geometry (177,597 problems), the largest geometry formalization dataset in Lean. Human evaluation shows 48.89% TOP1 and 73.33% TOP5 accuracy. Training Goedel v2 on our formalizations improves proof success from 13.6% to 15.1%, validating dataset quality for unified neural theorem proving. Code and datasets: this https URL.

Artificial Intelligence (cs.AI)

arXiv:2607.19374 [cs.AI]

(or arXiv:2607.19374v1 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2607.19374 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Euclean: Automated Geometry Problem Formalization with Unified Verification in Lean

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