State Space Analysis as First-Object Mathematics

This paper introduces State Space Analysis (SSA) as a first-object mathematical framework in which the primary object of analysis is the existence, geometry, and persistence of the admissible state space itself, rather than the evolution equations defined on it. By making admissibility explicit and time-dependent, the paper formalizes a class of failure modes—regime collapse, loss of objecthood, and non-jurisdiction—that occur without instability, singularity, or blow-up of classical norms. Central to the framework is the distance-to-admissibility function and its extension to a unified admissibility diagnostic vector, which together provide field-independent criteria for regime validity across fluids, control, optimization, artificial intelligence, and finance. Vortex breakdown is reinterpreted as the canonical physical manifestation of admissibility collapse, demonstrating how equations may remain valid while meaningful objects cease to exist. The work establishes admissibility geometry as a foundational analytical layer, upstream of dynamics, PDEs, and algorithms, and derives practical implications for computation, control, and AI safety under hazard-driven regimes.