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.

翻译：AlphaGeometry代表了神经符号推理领域的一个里程碑，然而其架构中的符号推演引擎面临对数线性扩展瓶颈，导致问题复杂度提升时效率受限。近期技术报告表明，当前领域特定语言与自然语言作为输入表示具有同构性，两者互换相当于性能不变变换，暗示了现有神经引导依赖于表层编码而非结构理解。本文针对这一表示瓶颈，提出一种逻辑-拓扑编码方法，旨在揭示模型潜空间在输入空间变换下的结构不变性。通过运用观察逻辑，我们利用可观测理论中的可证性与拓扑之间的对偶性，提出了输入空间的逻辑-拓扑编码器。引入"数据集的拓扑对偶"概念，这是一种连接形式逻辑、拓扑与神经处理的变换。该框架作为神经符号人工智能的"罗塞塔石碑"，为模型如何导航复杂发现路径的机制可解释性提供了原则性路径。