# The Urysohn Machine: A Metric-Topological Model of Computation

arXiv: https://arxiv.org/abs/2508.14143 · HTML: https://mlsanity.com/paper/2508.14143/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 48 references that are in the map.
- First posted on arXiv 19 Aug 2025.

## Builds on (most cited first, 10 of 10)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks](https://mlsanity.com/paper/1807.03888/) (2018)
- [Continual Lifelong Learning with Neural Networks: A Review](https://mlsanity.com/paper/1802.07569/) (2018)
- [Out-of-Distribution Detection with Deep Nearest Neighbors](https://mlsanity.com/paper/2204.06507/) (2022)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [Iterative Amortized Inference](https://mlsanity.com/paper/1807.09356/) (2018)
- [Continuum limit of total variation on point clouds](https://mlsanity.com/paper/1403.6355/) (2014)
- [Deep Amortized Inference for Probabilistic Programs](https://mlsanity.com/paper/1610.05735/) (2016)

## Built on by (most cited first, 1 of 1)

- [Cycle is All You Need: More Is Different](https://mlsanity.com/paper/2509.21340/) (2025)

## Abstract (arXiv)

We introduce the Urysohn Machine, an effective model of classification-oriented computation in which metric separation, frontier structure, and contraction are explicit parts of the computational state. Its basic object is a Urysohn Triple: a support region, a target partition, and a separating classifier stored in a reusable Metric Library. The topological foundation is a constructive Urysohn Realization theorem for finite simplicial settings. It builds separators from dyadic ladders of nested polyhedral regions and equips their frontiers with a chain-level calculus: frontiers are cycles, and shells between levels have boundaries given by differences of frontiers. This construction yields two related complexity measures: decision-boundary width, the geometric measure of a single classifier's boundary, and Urysohn width, the total frontier mass represented by a library or realization. We prove an Amortized Separation Theorem showing that approximating a boundary of width to accuracy requires a number of simple basis triples proportional to boundary width and inversely proportional to resolution, under explicit boundary-footprint assumptions. We also introduce a contrastive separation operator whose graph-cut functional consistently estimates decision-boundary width from sampled metric data, while its Laplacian spectrum certifies class-component structure and conductance. Finally, we analyze the dynamic Urysohn ladder and prove four guarantees: separability under quotient collapse, stability of committed frontiers, bounded capacity under contraction, and scalability with quotient distance. Together, these results give a metric-topological account of classification complexity, amortized inference, and compositional reuse that preserves classical computability while exposing geometric structure hidden by purely symbolic descriptions.