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The Urysohn Ladder: Recursive Metric Contraction for Scalable Continual Learning

Continual learning systems face a fundamental geometric obstacle: as experience accumulates on a fixed-capacity manifold, covering numbers grow linearly with time, eventually forcing representational overlap and catastrophic interference. Prevailing approaches attack this problem by \emph{expansion} - projecting into higher-dimensional spaces via kernels, overparameterization, or replay. We argue the solution is the opposite: \emph{contraction}. We formalize abstraction as the \textbf{Urysohn Ladder}, a hierarchy of quotient maps that recursively collapse validated metric neighborhoods into compact tokens, converting unbounded ambient-space search into bounded navigation on a low-dimensional intrinsic scaffold. Geometrically, each collapsed token acts as a shortcut - a region of extreme metric contraction that bridges distant experiences, much like a wormhole in the representational manifold. We establish four results that collectively guarantee \emph{separability} (metric contraction renders nonlinearly entangled structure linearly separable at each quotient level, and this separability propagates faithfully through the entire hierarchy), \emph{bounded capacity} (covering numbers remain $O(1)$ per quotient level, independent of stream length), \emph{stability} (parity-partitioned flow/scaffold subspaces enable unbounded plasticity without catastrophic interference), and \emph{scalability} (inference cost scales with quotient distance, not ambient distance). We validate each claim empirically with pretrained models and real-world datasets. Moreover, we demonstrate the potential of Urysohn Ladder for scalable continual learning via scaffold amortization.

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