The Geometry of Abstraction: Continual Learning via Recursive Quotienting
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.
Summary
This paper addresses the fundamental challenge of continual learning: how to learn new information without catastrophically forgetting old knowledge in a fixed-dimensional space. The authors argue that the problem stems from the "flat manifold problem," where the geodesic distance between temporal events grows linearly with time, leading to trajectory overlap and interference. Their proposed solution, "Recursive Metric Contraction," involves topologically deforming the data manifold by collapsing validated temporal neighborhoods into single points (tokens) via quotient maps. This process creates a hierarchy of quotient manifolds, trading linear metric growth for logarithmic topological depth. The authors provide four theoretical results to support their framework. First, the Bounded Capacity Theorem demonstrates that recursive quotient maps enable embedding arbitrarily long trajectories into bounded representational volumes. Second, the Topological Collapse Separability Theorem proves that recursive quotienting renders non-linearly separable temporal sequences linearly separable in the limit, obviating the need for infinite-dimensional kernel projections. Third, the Parity-Partitioned Stability Theorem solves catastrophic forgetting by partitioning the state space into orthogonal flow and scaffold manifolds, ensuring that metric deformations during active learning do not disrupt stored memories. Finally, the Correctness under Abstraction theorem ensures that semantic discriminability is preserved throughout the condensation hierarchy. The paper argues that tokens in neural architectures can be physically realized as wormholes. The significance of this work lies in its geometric perspective on continual learning and abstraction. It reframes memory and learning as operations on the metric tensor, suggesting that biological and artificial intelligence can achieve scalability not by expanding representational dimensions but by transforming the topology of the representational manifold. The work introduces a novel approach to continual learning, offering a compelling alternative to existing methods based on geometric expansion.
Key Insights
- •The paper introduces the concept of "Recursive Metric Contraction" as a geometric solution to the flat manifold problem in continual learning.
- •The Bounded Capacity Theorem demonstrates that recursive quotient maps allow embedding arbitrarily long trajectories into bounded representational volumes, trading linear metric growth for logarithmic topological depth (D = O(logL)).
- •The Topological Collapse Separability Theorem proves that recursive quotienting renders non-linearly separable temporal sequences linearly separable in the limit, bypassing the need for infinite-dimensional kernel projections.
- •The Parity-Partitioned Stability Theorem solves the catastrophic forgetting problem by partitioning the state space into orthogonal flow and scaffold manifolds, ensuring that metric deformations during active learning do not disturb stored memories. The guarantee of zero topological interference provides the rigorous justification for the separate Search and Structure phases observed in the cortical column.
- •The 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.
- •The work provides a geometric interpretation of Savitch's theorem and dynamic programming, linking them to inference on flat and recursively folded manifolds, respectively.
- •The paper connects its theoretical framework to biological plausibility, arguing that the severe capacity limit of working memory and the cytoarchitectonic uniformity of the neocortex are dual manifestations of the same topological principle. The analysis suggests that two fundamental biological phenomena arise from the necessity of maintaining bounded local covering numbers (N (ε,M k ) ≤ C bio ≈ 7).
Practical Implications
- •This research has implications for designing more efficient and scalable continual learning systems, particularly for resource-constrained devices. The framework suggests that future AI architectures should focus on "Folding Laws" that maximize topological density rather than "Scaling Laws" that demand massive parameter counts.
- •The proposed method could be applied to various real-world applications, such as robotics, autonomous vehicles, and personalized learning systems, where agents need to adapt to new environments and tasks without forgetting previously learned skills.
- •Practitioners and engineers can use the theoretical results to guide the development of novel continual learning algorithms based on recursive metric contraction and quotient maps. This could lead to the design of neural network architectures that dynamically transform their topology during learning.
- •Future research directions include exploring different types of quotient maps and metric contraction operators, investigating the trade-offs between hierarchical depth and computational complexity, and developing practical algorithms for partitioning the state space into orthogonal flow and scaffold manifolds.