Derivative-Induced Topology (DIT) Vol. II: Computational Topology and Parameterized Kernels

Vol. I of this series established the axiomatic foundation of Derivative-Induced Topology (DIT), introducing the perception scale λ = (δ, Δt), the all-order discernibility spectrum S_λ^{(n)}(x,y), and the four universality classes of dynamic separation. That framework was definitional: it specified what it means for two points to be behaviorally indistinguishable, but did not provide a computational realization.

Vol. II bridges this gap by operationalizing DIT as a learnable computational engine. We make three contributions.

First, we replace the abstract variational operator Δ_λ^{(n)} with a parameterized bank of convolution kernels κ_θ^{(n)}_λ, each having compact support and vanishing moments up to order n-1. The DIT feature embedding Φ_λ(x) = {κ_θ^{(n)}_λ[f](x)} maps raw data into a Sobolev feature space where the Vol. I discernibility metric becomes a Euclidean distance.

Second, we prove that the topological quotient X/∼_λ is equivalent to density-based clustering in this feature space under the DIT-induced metric. Topological boundaries correspond to cluster margins, and the separation axioms map directly to cluster cohesion and separation conditions.

Third, we introduce the DIT Topology Loss, a contrastive objective where positive pairs are points indistinguishable under the current kernel bank and negative pairs are distinct points. This loss serves as a topological regularizer for self-supervised learning, replacing heuristic data augmentations with an endogenous, theoretically grounded objective.

All components are differentiable, enabling end-to-end optimization. The renormalization group flow R_θ of Vol. I is realized by dilated convolutions and pooling; fractals emerge as scale-covariant fixed points of this flow. Vol. II thus transforms DIT from an axiomatic framework into a computational and learnable topology ready for integration into machine learning pipelines.