Memory-Amortized Inference: A Topological Unification of Search, Closure, and Structure (2512.05990v1)

Abstract: 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 orwhere''). 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.

• The paper presents a novel MAI framework that unifies search, closure, and structure using algebraic topology.
• It leverages dual-mode operations to balance static memory scaffolds and dynamic context flows, reducing inference cost.
• The approach provides insights into neural coding by associating even and odd homologies with stable content and dynamic information.

Introduction

In the work titled "Memory-Amortized Inference: A Topological Unification of Search, Closure, and Structure" (2512.05990), the author proposes a novel theoretical framework for machine learning and cognitive processes, grounded in algebraic topology. This framework, termed Memory-Amortized Inference (MAI), aims to unify learning and memory through the lens of geometric phase transitions, addressing inefficiencies that arise from the separation of static parameter structures from dynamic inference in contemporary machine learning systems.

Theoretical Foundations

Central to the concept of MAI is the Homological Parity Principle, which distinguishes between the roles of even- and odd-dimensional homology in cognitive systems. Even-dimensional homology represents stable content, functioning as the system's structural scaffolds, while odd-dimensional homology captures dynamic context, reflecting transient flows of information. The proposed framework articulates a transformation process referred to as the topological trinity: Search → Closure → Structure, drawing parallels to the Wake-Sleep algorithm by alternating between optimizing inference and learning via coordinate descent. This process is posited to provide an explanation for the emergence of intuitive, fast-thinking processes from deliberate, slow-thinking ones.

Scaffold-Flow Memory Model

In this model, the scaffold (content) and flow (context) are elevated to dual roles, where the scaffold is represented by stable, low-entropy invariants and the flow by high-entropy dynamic cycles. The model suggests that intelligence is manifested through the Context-Content Uncertainty Principle (CCUP), which seeks a balance between dynamic context flows and static content scaffolds, thereby achieving memory-amortized inference. This mechanism is likened to a topological version of the Wake-Sleep algorithm, engaging in inference through Savitch's theorem and consolidating knowledge through dynamic programming. It offers a blueprint for creating architectures that compute via topological resonance, extending the definitions of semantic and episodic memory.

Memory-Amortized Inference (MAI)

MAI is introduced as an operational framework for reducing the inference cost by leveraging memory to transform search problems into structured latency-based resolutions. This process assumes that inference corresponds to optimizing a system's content within its structural parameters, facilitating dual-mode operations that alternate between context-driven and content-driven modes to effectively manage computational resources. The study elaborates on the search-closure-structure transformation, converting complex recursive search into efficient memory lookup through cycle closure and dynamic programming, thereby proposing a topological resonance engine.

Biological Parity and Neural Codes

The parity framework provides insights into the neural coding strategies of the cortex, distinguishing between rate coding (even homology) and phase coding (odd homology). The structure is analogous to the What/Where pathways in visual processing, offering solutions to the binding problem and invariant object recognition. The work also provides explanations for cognitive phenomena such as multistable perception and working memory limits, suggesting that these arise from the topological organization of the brain's computational and memory structures.

Conclusion

This theoretical exposition on MAI and its topological underpinnings posits a fundamental shift in understanding intelligent systems and their architectures. By exploiting the physical dynamics of memory and learning as phase transitions of a geometric substrate, the framework suggests a path towards constructing post-Turing architectures that mimic the efficiency and adaptability of biological cognition. The MAI framework represents a significant step toward conceptualizing machines that achieve coherence and generalization akin to human intelligence by constructing and relying on topological structures.