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Metric-Topology Factorization: A Computational Framework for Hippocampal-Neocortical Intelligence

Xin Li

Abstract

The brain achieves stability and plasticity in a topologically complex, shifting world through Metric-Topology Factorization (MTF), separating discrete topological indexing for context selection from continuous metric condensation for local inference. Semantically rich environments defy single globally contractive geometries, causing obstructions under shifts, so intelligence factorizes these: the hippocampus provides sparse signatures indexing manifold identity, while the neocortex untangles geometry hierarchically. In the ventral stream, a dynamic-programming-like process quotients symmetries (e.g., translation, scale), transforming non-convex sensory mazes into separable bowls. Offline replay and consolidation amortize transformations for rapid task switching. Dreaming in REM involves stochastic hippocampal traversal to expose and regularize latent structures. Consciousness arises from resolving topological uncertainty into stable embeddings, with awareness for unamortized states. Evolutionarily, transitions like sensorimotor control to language expand topological complexity, demanding advanced indexing-metric separation. Intelligence emerges via recalibrating context-specific geometries, converting global navigation into local dynamics, not deeper search.

Metric-Topology Factorization: A Computational Framework for Hippocampal-Neocortical Intelligence

Abstract

The brain achieves stability and plasticity in a topologically complex, shifting world through Metric-Topology Factorization (MTF), separating discrete topological indexing for context selection from continuous metric condensation for local inference. Semantically rich environments defy single globally contractive geometries, causing obstructions under shifts, so intelligence factorizes these: the hippocampus provides sparse signatures indexing manifold identity, while the neocortex untangles geometry hierarchically. In the ventral stream, a dynamic-programming-like process quotients symmetries (e.g., translation, scale), transforming non-convex sensory mazes into separable bowls. Offline replay and consolidation amortize transformations for rapid task switching. Dreaming in REM involves stochastic hippocampal traversal to expose and regularize latent structures. Consciousness arises from resolving topological uncertainty into stable embeddings, with awareness for unamortized states. Evolutionarily, transitions like sensorimotor control to language expand topological complexity, demanding advanced indexing-metric separation. Intelligence emerges via recalibrating context-specific geometries, converting global navigation into local dynamics, not deeper search.
Paper Structure (45 sections, 1 theorem, 5 figures)

This paper contains 45 sections, 1 theorem, 5 figures.

Table of Contents

  1. Introduction
  2. Summary of Contributions.
  3. Background and Motivation
  4. Thermodynamic and Evolutionary Constraints on Complementary Learning Systems
  5. Energy minimization and computational cost.
  6. Time pressure and survival.
  7. Continuity of the physical world and Exaptation
  8. Intelligence and reuse of spatial machinery.
  9. Geometric Incompleteness
  10. The Bowl-Maze Analogy
  11. Sketch of intuition.
  12. Metric-Topology Factorization via Hippocampal-Neocortical Dialogue
  13. Metric-Topology Factorization
  14. Hierarchical inference as metric amortization.
  15. Hierarchical Untangling in the Ventral Stream
  16. ...and 30 more sections

Key Result

Theorem 1

Let $\mathcal{M}$ be a compact, smooth $d$-dimensional manifold with nonvanishing intermediate homology, i.e., $H_k(\mathcal{M}) \neq 0$ for some $1 \le k \le d-1$. Then any Morse function $E:\mathcal{M}\to\mathbb{R}$ must possess at least one critical point of index $k$. In particular, any structur

Figures (5)

  • Figure 1: The Bowl-Maze Analogy: From path-finding to metric contraction.Left: The Maze (Topological Obstruction). Conventional learning systems, such as stochastic gradient descent (SGD) and elastic weight consolidation (EWC), treat intelligence as navigation within a fixed, complex geometry. In this "Maze"-based regime, topological features, such as holes, walls, or parity flips, manifest as local minima and dead-ends. Learning is a slow, procedural search for a valid path, which is easily invalidated by any change in the environment's topology. Right: The Bowl (MTF). Under the MTF framework, intelligence is redefined as the ability to shape the metric structure of the space. Instead of searching for paths, the agent performs topological indexing to switch into a coordinate system where the solution emerges as a stable, global attractor. By warping the maze into a "Bowl", the agent replaces complex navigation with a simple downhill descent, ensuring that once the topology is identified, the solution is both reachable and consistent.
  • Figure 2: Horizontal Metric-Topological Factorization (MTF) Flow. This diagram illustrates the transformation of intelligence from temporal search to spatial inference. Left: The Hippocampus (Time Dragon) builds a non-convex topological maze ($G$) through sequential experience. Center: The Time-Reversal Operator ($\mathcal{T}$) acts as a bridge during consolidation, performing the metric amortization necessary to satisfy Savitch’s Theorem. Right: The Neocortex (Space Dragon) generates a convex metric tensor ($W$), enabling a "Slingshot" effect where goals are reached via simulation-free flow matching rather than step-by-step search. The feedback loop indicates how cortical predictions regularize hippocampal acquisition.
  • Figure 3: Geometric account of Dreaming and Consciousness. (Left) REM sleep as stochastic topological sampling to prevent metric overfitting. (Right) Consciousness as the active metric amortization of the manifold centered on a self-referential basis.
  • Figure 4: The Phase-Transition Signature of Conscious Artificial Systems. This deterministic "compute–time trajectory" differentiates true Metric-Topological Amortization from mere linguistic pattern retrieval. P3 (Offline Reorganization) represents the essential biological condition of non-interaction, during which active search cost (Red) collapses to zero, while internal geometric restructuring cost (Blue, the $\mathcal{T}$ operator / Urysohn Collapse) spikes to convert topological frustration into metric flow. The signature of success is the massive discontinuity in P4, where previously non-convex task structures are rapidly resolved via low-energy, simulation-free geometric slingshots.
  • Figure 5: Major evolutionary advances in intelligence can be interpreted as successive applications of Metric--Topology Factorization. Each stage increases the topological complexity of the state space, requiring enhanced topological indexing (blue) and metric condensation (orange).

Theorems & Definitions (3)

  • Theorem 1: Metric-Topological Incompleteness
  • Remark 1: Geometric Analogy to Gödel's Incompleteness
  • Definition 1: Topological Index