Dense Neuro-Symbolic Reasoning in a Unified Geometry State

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.

Artificial Intelligence (cs.AI)

arXiv:2610.04280 [cs.AI]

(or arXiv:2610.04280v1 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2610.04280 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.