Spectral Gap Control via Exact Feshbach Reduction, Birman-Schwinger Structure, and the Universal Ω-Σ Theorem

We establish an exact, non-heuristic operator-theoretic framework for spectral gap amplification, uniting finite-dimensional discrete Laplacian networks with infinite-dimensional continuous Schrödinger operators. By utilizing an exact Feshbach-Schur reduction and a reduced Birman-Schwinger formulation, we isolate the spectral shift equation governing structural coherence. We prove that the structural admissibility condition (γΔ > β²) is sufficient and sharp within the monotone resolvent bound for strict spectral gap amplification, yielding a closed-form optimal lower bound. We establish necessity under commuting constraints and derive the exact geometric admissibility condition for simple sectors. Advancing beyond finite matrices, we extend these exact bounds to self-adjoint operators with compact resolvents. We employ regularized Fredholm determinants (det2) to track exact analytic zero-crossings in Hilbert space, removing the trace-class requirement and securing an exact topological boundary for the invariant subspace independently of ambient dimension. Finally, we formulate the Grand Unified Ω-Σ Operator Theorem. We prove that this invariant admissibility law universally governs phase closure across both discrete networks and continuous quantum manifolds. The geometric boundary γΔ > β² dictates that an injected constraint field (the Ω-layer) will only stabilize a system if its spectral alignment with the target invariant subspace (γ) combined with the intrinsic spectral isolation of that space (Δ) strictly dominates the squared variance of orthogonal leakage (β²). This establishes that dynamic coherence is not merely persistence, but an exactly tunable operator geometry spanning from discrete nxn matrices to infinite-dimensional Hilbert space.