Asymptotic Sufficiency Depth, Volume III: Geometric Universality, Statistical Experiments, and Falsification

This volume develops geometric, analytic, algebraic, and statistical extensions of asymptotic sufficiency depth beyond the finite-state weak-noise setting. It treats flat and iterated-flat localization, recursive boundary profiles, Newton and Mellin asymptotics, invariant-theoretic and selected real-algebraic certificates, finite-complexity bounds, and connections to statistical-experiment comparison, PAC-Bayes and e-process methods, smoothness inference, and operator theory. Diagnostics, exact and numerical certificate classes, counterexamples, withdrawn candidates, proof-path obstructions, and open problems are retained explicitly to define the scope and falsifiability of the theory. Reproducibility software and validation materials are archived separately in Zenodo Software, version 1.0.0, DOI 10.5281/zenodo.22289467.