Category-Native Solomonoff Approximation: From Algorithmic Geometry to Kernels, Operators, and Induction.

Solomonoff induction mixes all computable explanations with description-length weights, but it is incomputable. This theory-and-position paper argues that practical approximation must be category-native: one should first declare the mathematical category in which a computable shadow will live, then use that category's native comparison functional, complexity code, and inductive object. The proposal is not an omnibus theorem asserting that all categories are equivalent. It is a research architecture that separates comparison, representation, and prediction and makes the information lost by each projection explicit. The metric-measure branch supplies the developed realization. Compression data define an empirical Solomonoff space; Gromov-Wasserstein (GW) distance supplies relational distortion; minimum description length (MDL) controls candidate complexity; and distance-to-kernel embedding produces a positive-semidefinite predictor. For finite or countable coded classes, we prove existence and stability results, a held-out validation oracle inequality, a Kolmogorov-Solomonoff kernel unification, conditional empirical-GW consistency, and a coding-redundancy bound. Stronger learning-oracle statements remain conditional on marked/predictive selection, candidate-family adequacy, and kernel stability. Topological, Banach/Barron, graph, tree, and operator branches are presented as a constructional and testable research programme, with their maturity stated explicitly.