Holographic Observation Quotients and Fractal Boundaries: A Model-Agnostic Design Theory for Compute-Optimal Learning

This preprint develops a model-agnostic design theory for compute–performance trade-offs in large-scale learning systems, built on metric gradient flows, observation quotients, and fractal geometry. Starting from the evolution variational inequality (EVI) formulation of gradient flows on Wasserstein spaces, the paper introduces holographic observation quotients (HOQs): abstract objects that couple a continuous “bulk” latent space with a lower-dimensional “boundary” observation space through a metric quotient map with controlled fibers. On the bulk side, the theory assumes an EVI gradient flow of an energy functional on P(X) and a Lipschitz performance functional whose near-optimal sublevel sets have finite Minkowski dimension d_pre. On the boundary side, observation-induced functionals on P(Z) inherit an EVI flow and are constrained by a fractal boundary geometry with dimension d_b. Under these structural assumptions, the paper proves dimension-based scaling ceilings for both bulk and boundary learning and derives a holographic compute law: if d_b < d_pre and empirical power-law exponents saturate their dimension-based ceilings, then boundary-based learning can achieve a target error with asymptotically smaller training compute than bulk learning, by a power-law factor governed by the dimension gap d_pre − d_b. The framework is formulated at the level of metric spaces, Minkowski dimensions, and EVI flows, without committing to a particular neural architecture. It is designed to be instantiated by bulk–boundary systems such as sparse mixture-of-experts, hierarchical or multi-scale attention, and dendritic/fractal representations. The paper also discusses fractal dendritic truncations (which affect only logarithmic factors in compute) and outlines practical design principles and empirical strategies for estimating effective dimensions and scaling exponents in real models.