Omega-Sigma Control: From Discrete H3 Geometric Admissibility to Quantum Delta Continuum Mechanics

We formalize a unified admissibility principle extending from finite-dimensional control systems to the continuum limit of the three-dimensional incompressible Navier-Stokes equations. The central claim is that stable dynamical evolution is not characterized by instantaneous gain maximization, but by the preservation of an admissible manifold under perturbation. This manuscript bridges discrete geometric control and infinite-dimensional topological fluid dynamics through a multi-stage operator-theoretic framework. We first construct a finite chiral coordinate realization derived from the noncrystallographic Coxeter system H3. By embedding this system into a hyperbolic triangle group, we generate a strict 60-vertex snub dodecahedral frame. We demonstrate mathematically and computationally (via JAX simulation) that the negative curvature of this hyperbolic embedding induces structural coercivity, strictly preventing resolvent blow-up under adversarial perturbation. To elevate this bounded architecture to unbounded continuum mechanics, we utilize the Kato-Rellich theorem and a higher-order Schatten-class Birman-Schwinger framework. We apply this synthesis directly to the 3D Navier-Stokes Millennium problem. By transforming physical fluid transport into a renormalized operator flow using "Quantum Delta coordinates," we remove the need for ultraviolet shell cutoffs. Ultimately, fluid blow-up is formally reclassified. A finite-time singularity is no longer viewed as an unstructured loss of smoothness in physical space, but as a precise topological rupture in spectral space—manifesting either as the failure of Schatten-class operator compactification, or the exact zero-crossing of a regularized Fredholm determinant. This framework absorbs classical bounds (such as Prodi-Serrin and Beale-Kato-Majda) while identifying spectral admissibility as the fundamental topological governor of three-dimensional fluid transport. Notes: Includes the full mathematical framework, proofs of continuum Schatten control, and the core algorithmic logic for the associated Hyperbolic Spectral JAX Simulator.