arXiv:2610.33123v1 Announce Type: new Abstract: This paper develops a foundational measure-geometric theory of robustness in machine learning. Robustness is formulated as escape from a retained task state relative to a reference measure on a data-bearing manifold, a perturbation gauge, and an admissible realization mechanism. The theory distinguishes the structural possibility of escape on the reference manifold from its ambient realization under mechanism-dependent observable readouts, while placing both within a unified geometric account. It introduces a mesoscopic organization of escape geometry, through which vulnerable mass, finite-scale morphology, delivered progress, and branch coherence are characterized by explicit laws as linked but non-identical robustness phenomena. By recasting robustness as the geometric formation, organization, realization, and persistence of escape, the theory provides a structural basis for analysis beyond scalar criteria.