Spectral Gap Control via Triadic Operator Closure, Regularized Determinants, and the Universal Ω-Ξ-Σ-Π Framework

We establish an exact, non-heuristic operator-theoretic framework for spectral gap amplification and invariant subspace stabilization. We introduce a complete triadic closure system consisting of a spacing operator (Ξ) preserving resolvability, a receptive operator (Σ) bounded in the clarity metric, and an alignment projector (Π) defining the admissible subspace. Coupled with an incoherent-sector variance penalty (Ω), we prove that the resulting effective operator completely suppresses off-anchor spectral intrusion. The structural admissibility conditions are shown to be sufficient and sharp within the monotone resolvent bound. We secure analytic exactness via Birman-Schwinger reductions, regularized Fredholm determinants (det₂), and heat-kernel zeta regularization. By tracing the exact μ-dependence of the reference resolvent, we prove asymptotic operator decoupling governed by second-order determinant convergence. Furthermore, we provide an exact 64-bit numerical implementation validating the absolute spectral decoupling. Log-log scaling of the empirical output confirms that the Hilbert-Schmidt leakage channel decays exactly as O(μ⁻¹) and the analytic signature converges as O(μ⁻²). This zero-slack, second-order determinant convergence numerically verifies the analytic trace-log expansion and proves exact operator-level decoupling. The framework universally governs cascade arrest in hydrodynamic manifolds and catastrophic forgetting suppression in continuous learning dynamics, uniting discrete matrices and infinite-dimensional Schrödinger operators under a single topological invariant.