Euclean: Automated Geometry Problem Formalization with Unified Verification in Lean

Linbin Tang, Jingyan You, Zilin Kang, Hanzhang Liu, Sophia Zhang, Zenan Li, Chenrui Cao, Liangcheng Song, Jiaao Wu, Xian Zhang, Fan Yang
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:119351-119371, 2026.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-tang26ac, title = {Euclean: Automated Geometry Problem Formalization with Unified Verification in Lean}, author = {Tang, Linbin and You, Jingyan and Kang, Zilin and Liu, Hanzhang and Zhang, Sophia and Li, Zenan and Cao, Chenrui and Song, Liangcheng and Wu, Jiaao and Zhang, Xian and Yang, Fan}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {119351--119371}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/tang26ac/tang26ac.pdf}, url = {https://proceedings.mlr.press/v306/tang26ac.html}, 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.} }
Endnote
%0 Conference Paper %T Euclean: Automated Geometry Problem Formalization with Unified Verification in Lean %A Linbin Tang %A Jingyan You %A Zilin Kang %A Hanzhang Liu %A Sophia Zhang %A Zenan Li %A Chenrui Cao %A Liangcheng Song %A Jiaao Wu %A Xian Zhang %A Fan Yang %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-tang26ac %I PMLR %P 119351--119371 %U https://proceedings.mlr.press/v306/tang26ac.html %V 306 %X 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.
APA
Tang, L., You, J., Kang, Z., Liu, H., Zhang, S., Li, Z., Cao, C., Song, L., Wu, J., Zhang, X. & Yang, F.. (2026). Euclean: Automated Geometry Problem Formalization with Unified Verification in Lean. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:119351-119371 Available from https://proceedings.mlr.press/v306/tang26ac.html.

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