Memory-Amortized Inference: A Topological Unification of Search, Closure, and Structure

Contemporary ML separates the static structure of parameters from the dynamic flow of inference, yielding systems that lack the sample efficiency and thermodynamic frugality of biological cognition. In this theoretical work, we propose \textbf{Memory-Amortized Inference (MAI)}, a formal framework rooted in algebraic topology that unifies learning and memory as phase transitions of a single geometric substrate. Central to our theory is the \textbf{Homological Parity Principle}, which posits a fundamental dichotomy: even-dimensional homology ($H_{even}$) physically instantiates stable \textbf{Content} (stable scaffolds or ``what''), while odd-dimensional homology ($H_{odd}$) instantiates dynamic \textbf{Context} (dynamic flows or ``where''). We derive the logical flow of MAI as a topological trinity transformation: \textbf{Search $\to$ Closure $\to$ Structure}. Specifically, we demonstrate that cognition operates by converting high-complexity recursive search (modeled by \textit{Savitch's Theorem} in NPSPACE) into low-complexity lookup (modeled by \textit{Dynamic Programming} in P) via the mechanism of \textbf{Topological Cycle Closure}. We further show that this consolidation process is governed by a topological generalization of the Wake-Sleep algorithm, functioning as a coordinate descent that alternates between optimizing the $H_{odd}$ flow (inference/wake) and condensing persistent cycles into the $H_{even}$ scaffold (learning/sleep). This framework offers a rigorous explanation for the emergence of fast-thinking (intuition) from slow-thinking (reasoning) and provides a blueprint for post-Turing architectures that compute via topological resonance.