Quantum Geometry as Admissible Operator Structure (Geometry as a Physical Referee of Realizability)

We establish a unified operator framework in which physical realizability is strictly equivalent to the admissibility of operator families acting on a Hilbert space. Geometry is fundamentally redefined not as a descriptive arena, but as the active constraint structure enforcing admissibility through spectral gap preservation, resolvent boundedness, determinant nondegeneracy, and projection invariance. A regularized admissibility functional is constructed whose zero-set defines the physically realizable sector. Boundary geometry, curvature, and dynamical phase transitions are rigorously derived from the variational structure of this functional. Core Applications: Fluid Dynamics: We prove that ultraviolet cascades and scale-to-scale energy transfer blow-ups correspond exactly to admissibility failure. General Relativity: Curvature concentration and singularity formation are structurally mapped to the divergence of the operator resolvent. Yang-Mills: In representations admitting a spectral gap, admissible eigenstates correspond directly to gauge-invariant bound states (glueballs). Control and Stochastic Dynamics: Elevating the deterministic control of operator geometry to statistical mechanics, the framework integrates Hamilton-Jacobi reachability, Input-to-State Stability (ISS), and Freidlin-Wentzell large deviations. Physical materialization is identified as the controlled convergence into the realizable sector, while minimum-action instantons represent the most probable escape trajectories from admissibility, completely mapping the geometry of catastrophic failure transitions. Finally, we establish a universal action principle under which all physical theories arise as path-integral representations of admissible operator dynamics. Admissibility acts as a fundamental selection rule, mathematically eliminating non-realizable configurations from physical existence. The Unified Equation of State: Existence = The kernel of the admissibility functional Structure = The boundary of the kernel Dynamics = Optimization of the action principle Transitions = Topological flow across the boundary of existence Geometry is a physical referee.