Noetherian Precision and the Spectral-Commutator Criterion for Structural Legitimacy Across Physical Scales

We formalize a universal algebraic criterion to distinguish structurally legitimate operator evolution from spectrally inert extensions. We establish a strict topological distinction between admissibility, precision, and exactness in constraint-preserving dynamical systems. By lifting perturbation theory into the Hilbert-Schmidt class, we prove that spectral gap preservation and a nonvanishing regularized determinant (det2) provide the necessary conditions for the convergence of precise dynamics to exact states. Mapping commutator annihilation to Noetherian closure via an Omega-Sigma refinement flow, we prove that legitimate mechanisms must exhibit effective transversality followed by exact commutator annihilation. We apply this criterion to macroscopic and open systems, demonstrating a rigorous, scale-invariant duality: ultraviolet instability (Navier-Stokes finite-time blowup), infrared instability (galactic phase-space deconfinement), and dissipative instability (Lindblad non-convergence) are all isomorphic manifestations of a det2 zero-crossing. The resulting framework provides a closed, invariant description of operator evolution. It definitively reclassifies physical singularities—whether in fluid dynamics, cosmology, or quantum decoherence—not as localized heuristic failures, but as mathematically detectable algebraic topology ruptures.