Hypostructures: A Categorical Formalism for Dynamical Coherence and Structural Learning

This monograph presents Hypostructures, a unified categorical framework for analyzing dynamical stability, structural learning, and physical emergence. By treating mathematical rigor as a differentiable geometric constraint, the text establishes an isomorphism between the singularities of analysis (PDEs), the failure modes of artificial intelligence (Alignment), and the fundamental laws of physics (QFT/GR).

The work introduces:
-   A set of 8 structural axioms (Conservation, Stiffness, Scaling, etc.) determining system realizability.
-   The Structural Sieve: A diagnostic taxonomy of 15 fundamental failure modes.
-   The Universal Regularity Engine: A rigorous toolkit (including Ghosts, Surgery, and Lifts) for resolving singularities.
-   The Fractal Gas: A generative solver that replaces traditional analytic integration with algorithmic geometry.

This framework serves as a blueprint for "Trainable Axiomatic Systems," bridging the gap between abstract category theory and practical engineering in complex adaptive systems.