The Topological Dual of a Dataset: A Logic-to-Topology Encoding for AlphaGeometry-Style Data

Abstract:AlphaGeometry represents a milestone in neuro-symbolic reasoning, yet its architecture faces a log-linear scaling bottleneck within its symbolic deduction engine that limits its efficiency as problem complexity increases. Recent technical reports suggest that current domain-specific languages may be isomorphic as input representations to natural language, interchanging them acts as a performance-invariant transformation, implying that current neural guidance relies on superficial encodings rather than structural understanding. This paper addresses this representation bottleneck by proposing a logic-to-topology encoding designed to reveal the structural invariants of a model's latent space under a transformation of its input space. By leveraging the Logic of Observation, we utilize the duality between provability in observable theories and topologies to propose a logic-to-topology encoder for the input space. We introduce the concept of the "topological dual of a dataset", a transformation that bridges formal logic, topology, and neural processing. This framework serves as a Rosetta Stone for neuro-symbolic AI, providing a principled pathway for the mechanistic interpretability of how models navigate complex discovery paths.

Artificial Intelligence (cs.AI); Logic in Computer Science (cs.LO)

arXiv:2604.18050 [cs.AI]

(or arXiv:2604.18050v2 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2604.18050 Focus to learn more arXiv-issued DOI via DataCite

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