On the Topological Foundation of Learning and Memory

We propose a formal foundation for cognition rooted in algebraic topology, built on a Homological Parity Principle. This posits that even-dimensional homology represents stable Structure/Context (e.g., generative models), while odd-dimensional homology represents dynamic Flow/Content (e.g., sensory/memory data). Cognition is governed by the Context-Content Uncertainty Principle (CCUP), a dynamical cycle aligning these parities. This framework distinguishes two modes: Inference (waking), where the scaffold predicts the flow (a Context-before-Content process); and Learning (sleep), an inverted Structure-before-Specificity process where memory traces sculpt the scaffold. This parity interpretation unifies cognitive functions like semantic and episodic memory and provides a structural generalization of existing theories, recasting Friston's Free Energy Principle and Tonini's Integrated Information in topological terms.