Commutator Obstruction to Quantum Advantage: A Spectral Confinement Theorem for Quantum Heuristics

We establish a necessary condition for quantum heuristics to achieve nontrivial computational advantage relative to a classical problem generator A. We prove that any unitary procedure whose dynamics nearly commute with A is spectrally confined: its ability to transfer amplitude across separated spectral sectors is bounded strictly by the commutator defect divided by the spectral gap. This yields a quantitative operator-theoretic obstruction to global state-space exploration. Consequently, repeated application of such procedures (e.g., in deep variational circuits) produces apparent algorithmic activity without generating meaningful information gain. We classify this regime as a "degenerate quantum heuristic". We apply this framework to Variational Quantum Eigensolvers (VQE) and the Quantum Approximate Optimization Algorithm (QAOA), demonstrating that circuit depth cannot overcome transversality failure. The theoretical results are fully closed within trace-ideal (S₂) operator theory. We extend the spectral confinement bound to the entire resolvent geometry, proving that degenerate heuristics preserve the Birman-Schwinger kernel and regularized Fredholm determinants (det₂), thereby mathematically obstructing any computational phase transition or new eigenvalue emergence. Finally, we demonstrate that genuine quantum speedup requires structural noncommutativity, which can be deterministically enforced via an Ω-Σ dissipative control layer. The manuscript includes complete finite-dimensional numerical validation, exhibiting the exact quantitative rigidity between commutator defect, spectral transport, and resolvent deformation.