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

Xin Li

TL;DR

The paper introduces Memory-Amortized Inference (MAI), a topological framework unifying memory and inference as phase transitions on a single geometric substrate. Central is the Homological Parity Principle, separating Content ($H_{even}$) and Context ($H_{odd}$), and a Scaffold-Flow memory model that grounds semantic/episodic memory in topology. MAI is formalized as a memory-based Wake-Sleep-like process with retrieval and bootstrapping operators, underpinned by a sheaf-theoretic view of coherence and cycle closure, enabling rapid inference via topological condensation. The approach offers a principled path to energy-efficient, sample-efficient computation and lays groundwork for post-Turing architectures that exploit topological resonance to achieve general intelligence.

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 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.

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

TL;DR

The paper introduces Memory-Amortized Inference (MAI), a topological framework unifying memory and inference as phase transitions on a single geometric substrate. Central is the Homological Parity Principle, separating Content () and Context (), and a Scaffold-Flow memory model that grounds semantic/episodic memory in topology. MAI is formalized as a memory-based Wake-Sleep-like process with retrieval and bootstrapping operators, underpinned by a sheaf-theoretic view of coherence and cycle closure, enabling rapid inference via topological condensation. The approach offers a principled path to energy-efficient, sample-efficient computation and lays groundwork for post-Turing architectures that exploit topological resonance to achieve general intelligence.

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 () physically instantiates stable \textbf{Content} (stable scaffolds or ``what''), while odd-dimensional homology () instantiates dynamic \textbf{Context} (dynamic flows or ``where''). We derive the logical flow of MAI as a topological trinity transformation: \textbf{Search Closure 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 flow (inference/wake) and condensing persistent cycles into the 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.

Paper Structure

This paper contains 8 sections, 3 theorems, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Homological Parity Principle
  3. The Scaffold-Flow Memory Model
  4. Memory-Amortized Inference (MAI)
  5. Dual-Mode and Logical Trinity of MAI
  6. Sheaf Theoretical Perspective of MAI
  7. Phase Dynamics of What and Where
  8. Toward Topological Equilibration

Key Result

Proposition 2.2

An open, persistent $H_{\mathrm{odd}}$ cycle represents a non-convergent prediction error or surprisal (high free energy). The topological closure of this cycle, either by canceling it with a boundary or condensing it into an $H_{\mathrm{even}}$ component, is isomorphic to the compression of informa

Figures (3)

  • Figure 1: Dot-Cycle Dichotomy.Left: A cycle that is the boundary of a filled region ($\gamma=\partial S$) is trivial in $H_1$: it can be canceled as a boundary when becoming a dot. Right: A cycle encircling a hole is not the boundary of any 2-chain in the space, so it represents a nontrivial class in $H_1$. In our framework, nontrivial cycles correspond to high-entropy, short-lived flows ($\Psi$) that collapse under boundary cancellation, whereas trivial cycles correspond to low-entropy content scaffolds ($\Phi$) that persist as memory.
  • Figure 2: Cycle of memory-amortized inference. Instead of recomputing $\Phi^* = \arg\min \mathcal{L}(\Psi, \Phi)$, the system reuses prior trajectories: $\Phi_{t+1}$ and $\Psi_t$ guide memory-based retrieval via $\mathcal{R}$, and bootstrapping $\mathcal{F}$ updates the latent state $\Phi_t$. The process forms a self-consistent loop grounded in structured memory.
  • Figure 3: Dynamic alignment via CCUP-based dual-mode MAI filtering. The outer synchronization layer (context filter $\Psi$) gates which evidences can be glued (sheaf coherence), forming the admissible inner region. Within that region, recurrence (content filter $\Phi$) tests for cycle closure and persistence ($H_1$). Memory consolidation/retrieval (topological condensation) occurs only when context gluing succeeds and a nontrivial recurrent cycle is validated.

Theorems & Definitions (9)

  • Definition 2.1: Homological Capacity
  • Proposition 2.2: Closure-Compression Equivalence
  • Definition 3.1: Semantic Scaffold and Episodic Flow
  • Proposition 3.2: Recursive Condensation
  • Definition 4.1: MAI
  • Definition 4.2: Sheaf of Memory Traces
  • Proposition 4.3: Topological Closure $\Longleftrightarrow$ Information Consistency
  • Example 1: Multistable Perception
  • Example 2: The Magical Number Seven