The Recursive Manifold Hypothesis for Hierarchical Data Laws

Many real-world distributions are organized at more than one resolution. A coarse conditional law captures structure shared by its descendants; conditioning more finely reveals additional variation, while the resulting laws remain related through their common parent. We call this the recursive manifold hypothesis. We formalize the hypothesis as multiscale conditional refinement of probability laws across successive coarse-to-fine relations. The energy distance metric between probability distributions compares the laws model-independently and allows us to decompose their variation exactly. Each parent–child relation has three coordinates: absolute law variation \(V\), the fraction \(F\) revealed by the children, and parent organization \(A\) relative to the expected organization of a random parent map with the same group sizes. The product \(R=FA\) yields a sufficient population criterion for one specified parent–child relation. We estimate this profile from pooled energy-distance U-statistics and quantify uncertainty in \(R\) by an observation jackknife. Experiments center on ImageNet-1k and Amazon Reviews 2023 and include pushforward laws from DINOv2-L and Qwen3-Embedding-4B. Across image, text, and spiking-speech datasets, 13 of 14 relation–representation pairs reveal additional variation that the supplied parent organizes better than a random map with the same group sizes. ImageNet-1k, Amazon Reviews, FGVC-Aircraft, and ImageNet-R satisfy the criterion on two successive relations. Spiking Heidelberg Digits distinguishes structural strength across attributes: speakers are strongly organized within words, whereas the language attribute provides little organization of word laws. Excluding singleton branches preserves these empirical findings. These results suggest objectives and models that respect the hierarchical organization in data laws.