Generative Geometry as Canonical Stratified Second-Variation Structure

This paper formulates a geometric theory of generative behavior in deep learning, rooted in the second-variation structure of a functional on a configuration manifold. It decomposes variational flows into reversible (antisymmetric) and dissipative (symmetric) components, where neutral directions—spanned by the antisymmetric kernel—dictate expressive capacity through their dimensionality. These directions form hierarchical strata, yielding a stratified foliation of the space and associated reversible transport groups, with dimensionality conserved under antisymmetric evolution but collapsible via irreversible singularities. Generative depth aligns with this hierarchy, imposing intrinsic bounds on expressiveness that resist nontrivial reduction or extension. Common architectures like VAEs, GANs, flows, and diffusion models emerge as projections or constrained flows within this unified framework. Empirical issues, such as mode collapse, are interpreted as symmetric dominance over antisymmetric structure, while perturbations restore reversibility. The theory provides architecture-agnostic insights, suggesting avenues like curvature-preserving optimization and antisymmetric regularization for enhanced generative stability. v2: Removed undefined section references.