The Geometry of Abstraction: Continual Learning via Recursive Quotienting

TL;DR

The paper tackles the problem that continual learning in fixed-width systems suffers from catastrophic interference due to linear growth in the geodesic distance of experiences. It introduces Recursive Metric Contraction, a geometric operation that collapses validated temporal submanifolds via quotient maps, creating a hierarchy of compact quotient spaces and wormhole-like shortcuts. Three main theorems establish bounded capacity, topological separability without kernel methods, and interference-free stability through parity partitioning of the state space. The framework reinterprets abstraction as metric singularity and links it to biological principles like working memory and cortical uniformity, arguing for folding-based scalability over dimensional growth. Overall, the work provides a capacity-theoretic foundation for scalable continual learning on fixed-width hardware through topological deformation rather than expansion.

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.

Figures (3)

• Figure 1: The Topological Trinity Transformation. The diagram demonstrates the core mechanism of MAI. Left: The raw input stream ($\mathcal{H}_{\text{odd}}$) contains complex, intertwined temporal trajectories that violate Cover's Theorem for linear separability. Right: After applying the Condensation Operator $\Psi$ (Metric Collapse), the trajectories are topologically quotiented into single points in the Scaffold space ($\mathcal{H}_{\text{even}}$). In this collapsed metric, the classes become trivially linearly separable, effectively resetting the system's memory capacity.
• Figure 2: Geometric Expansion vs. Topological Contraction/Folding. (A) The Flat Manifold Problem: In standard continual learning, the temporal manifold $\mathcal{M}$ grows linearly in geodesic diameter as the input stream length $L \to \infty$. A system with fixed representational capacity (dashed box) eventually fails to maintain a valid $\epsilon$-cover of the expanding volume, resulting in catastrophic interference (red starburst) where new experience overwrites the old. (B) The Quotient Solution: The proposed Token-Limited Capacity framework applies a condensation operator $\Psi$ that functions as a topological quotient map. It introduces "wormhole" connections (orange arcs) that contract the metric between causally related states. This collapses continuous sub-manifolds into discrete singularities or tokens ($\tau_1, \tau_2$) within the quotient space $\mathcal{M}_1$. By recursively folding the manifold, the system ensures the effective covering number remains bounded ($O(1)$), allowing unbounded temporal history to be represented within fixed-dimensional hardware.
• Figure 3: Logarithmic Scaling via Recursive Condensation. The diagram illustrates how a linear input stream of length $L$ on the base manifold $\mathcal{M}_0$ is progressively compressed through a hierarchy of quotient maps $q_k$. Each contraction reduces the metric volume by a factor $\rho$. To achieve a bounded effective diameter at the top manifold $\mathcal{M}_D$ (necessary for constant-width inference), the depth of the hierarchy $D$ must scale logarithmically with the input length $L$. The shaded cones represent the collapse of submanifolds into points at the next level.

Theorems & Definitions (19)

• Definition 1: Metric Contraction Operator
• Definition 2: Parity Partitioning
• Definition 3: Effective Capacity Demand
• Lemma 1: Linear Capacity Growth on Flat Manifolds
• Definition 4: Recursive $\rho$-Compressibility
• Theorem 1: Bounded Capacity under Recursive Metric Contraction
• Lemma 2: Urysohn's Lemma
• Theorem 2: Quotient Collapse Preserves Separability
• Definition 5: Orthogonal Parity Decomposition
• Theorem 3: Parity-Partitioned Stability