Noetherian Quantum Control: The Admissible Intervention Criterion and Topological Rupture

Standard quantum control theory characterizes interventions—measurements, unitary gates, and dissipative couplings—by their physical mechanisms and fidelity costs. In this work, we dispense with this descriptive heuristic and reclassify quantum intervention strictly as an external completely positive trace-preserving (CPTP) constraint injection. By mapping the discrete intervention channel to a continuous Hilbert-Schmidt perturbation of the system's effective generator in Liouville space, we introduce the Admissible Intervention Criterion. We prove that a quantum control protocol is mathematically legitimate if and only if it rigorously preserves the regularized determinant (det2) of the induced Birman-Schwinger operator. Interventions that violate this topological bound do not merely induce decoherence or state collapse; they trigger a structural rupture of the admissible manifold (an eigenvalue crossing at -1). Furthermore, we demonstrate that standard Quantum Error Correction (QEC) is not a mechanism of informational repair. By translating the Knill-Laflamme conditions into this topological framework, we prove that QEC is a specialized subset of Omega-Sigma gap enforcement designed specifically to avoid the det2 zero-crossing. Fault tolerance is mathematically reclassified as the physical engineering of spectral avoidance.