The Geometry of Abstraction: Continual Learning via Recursive Quotienting

TL;DR

This paper introduces a geometric framework using recursive quotienting to enable continual learning in fixed-dimensional spaces, overcoming the flat manifold problem and catastrophic interference.

Contribution

It formalizes abstraction as a topological deformation via quotient maps, providing rigorous theorems that address trajectory embedding, separability, and stability in continual learning.

Findings

Recursive quotient maps embed long trajectories into bounded volumes.

They make non-linearly separable sequences linearly separable.

Partitioning state space prevents catastrophic forgetting.

Abstract

Continual learning systems operating in fixed-dimensional spaces face a fundamental geometric barrier: the flat manifold problem. When experience is represented as a linear trajectory in Euclidean space, the geodesic distance between temporal events grows linearly with time, forcing the required covering number to diverge. In fixed-dimensional hardware, this volume expansion inevitably forces trajectory overlap, manifesting as catastrophic interference. In this work, we propose a geometric resolution to this paradox based on Recursive Metric Contraction. We formalize abstraction not as symbolic grouping, but as a topological deformation: a quotient map that collapses the metric tensor within validated temporal neighborhoods, effectively driving the diameter of local sub-manifolds to zero. We substantiate our framework with four rigorous results. First, the Bounded Capacity Theorem establishes that recursive quotient maps allow the embedding of arbitrarily long trajectories into bounded representational volumes, trading linear metric growth for logarithmic topological depth. Second, the Topological Collapse Separability Theorem, derived via Urysohn's Lemma, proves that recursive quotienting renders non-linearly separable temporal sequences linearly separable in the limit, bypassing the need for infinite-dimensional kernel projections. Third, the Parity-Partitioned Stability Theorem solves the catastrophic forgetting problem by proving that if the state space is partitioned into orthogonal flow and scaffold manifolds, the metric deformations of active learning do not disturb the stability of stored memories. Our analysis reveals that tokens in neural architectures are physically realizable as singularities or wormholes, regions of extreme positive curvature that bridge distant points in the temporal manifold.