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 \emph{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.

翻译：我们引入Urysohn机器，一种面向分类计算的有效模型，其中度量分离、前沿结构和压缩是计算状态的显式组成部分。其基本对象是"Urysohn三元组"：一个支撑区域、一个目标划分，以及一个存储在可重用度量库中的分离分类器。其拓扑基础是有限单纯形设定下的构造性Urysohn实现定理。该定理通过嵌套多面体区域的二元梯级构建分离器，并为其前沿配备链级演算：前沿构成循环，而层级间的壳体的边界由前沿的差给出。这一构造产生两种相关的复杂性度量：决策边界宽度（单一分类器边界的几何测度）以及Urysohn宽度（由库或实现表示的总前沿质量）。我们证明了一个摊销分离定理，表明在显式边界足迹假设下，用宽度为 \(w\) 的边界近似精度 \(\epsilon\)，需要数量正比于边界宽度、反比于分辨率的简单基三元组。我们还引入了一个对比分离算子，其图割函数能从采样的度量数据中一致地估计决策边界宽度，而其拉普拉斯谱则能认证类组分结构与电导率。最后，我们分析了动态Urysohn梯级，并证明了四个保证：商塌缩下的可分离性、已承诺前沿的稳定性、压缩下的有界容量，以及随商距离的可扩展性。这些结果共同提供了对分类复杂性、摊销推理和组合重用的度量-拓扑解释，在保留经典可计算性的同时，揭示了纯符号描述所隐藏的几何结构。