Abstract
Geometry reasoning is naturally stateful: solving a problem repeatedly alternates between structural proposals and exact deductions. We formulate this process as dense neural-symbolic coupling, in which neural guidance and symbolic execution share a typed state and communicate through executable actions at every search step. Neural proposals contribute theorem instances, constructions, and algebraic bridges; the symbolic runtime applies registered rules, propagates exact constraints, and records provenance. A nested controller allocates computation first between neural and symbolic proposal sources and then among admitted actions. We instantiate the framework in OmniGeo, a single solver for plane, analytic, and solid geometry. With Claude Sonnet 4.6, OmniGeo reaches 94.2%, 88.5%, and 89.8% on FormalGeo7K, Conic10K, and SolidFGeo, respectively (90.8% macro average), and solves 21/30 IMO-AG-30 problems.
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Publication details
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- Open access
- Green open access
Cite this article
APA 7
Xu, R., Gao, W., Cheng, H., Kang, X., & Wang, Q. (2026). Dense Neuro-Symbolic Reasoning in a Unified Geometry State. https://omanscience.com/en/articles/dense-neuro-symbolic-reasoning-in-a-unified-geometry-state
MLA 9
Xu, Ruoran, et al. "Dense Neuro-Symbolic Reasoning in a Unified Geometry State." https://omanscience.com/en/articles/dense-neuro-symbolic-reasoning-in-a-unified-geometry-state.
Chicago (author–date)
Xu, Ruoran, Wending Gao, Haoyu Cheng, Xiaoqiang Kang, and Qiufeng Wang. 2026. "Dense Neuro-Symbolic Reasoning in a Unified Geometry State." https://omanscience.com/en/articles/dense-neuro-symbolic-reasoning-in-a-unified-geometry-state.
Harvard
Xu, R., Gao, W., Cheng, H., Kang, X. and Wang, Q. (2026) 'Dense Neuro-Symbolic Reasoning in a Unified Geometry State', Available at: https://omanscience.com/en/articles/dense-neuro-symbolic-reasoning-in-a-unified-geometry-state.
Vancouver
Xu R, Gao W, Cheng H, Kang X, Wang Q. Dense Neuro-Symbolic Reasoning in a Unified Geometry State. https://omanscience.com/en/articles/dense-neuro-symbolic-reasoning-in-a-unified-geometry-state
IEEE
R. Xu, W. Gao, H. Cheng, X. Kang, and Q. Wang, "Dense Neuro-Symbolic Reasoning in a Unified Geometry State," https://omanscience.com/en/articles/dense-neuro-symbolic-reasoning-in-a-unified-geometry-state.