Action Bounds for a Four-Level Quantum Gate with a Forbidden Coupling

This preprint bounds the action required to realize a specified four-level unitary when one entry of the control Hamiltonian is forbidden. The model allows arbitrary finite ancillary levels and measurable Hermitian controls. The action is the integrated spectral semidiameter. The endpoint error is measured in operator norm on all four code columns, including leakage, up to one common phase. An affine positive semidefinite condition characterizes the compressed paths and gives a realization on at most eight levels. Reflection of the symmetric target yields half-path certificates. Rational witnesses and interval arithmetic exclude four action–error pairs and establish a strict minimum-action interval 1.22442 < Λ* < 1.235092. Exact five- and six-level constructions are certified by pulse bounds and an analytic endpoint correction. A separate two-copy readout distinguishes hidden rank for a specified intermediate frame. The archive contains source, frozen witness data, pulse words and scripts for all checks. The results concern the declared mathematical model. The proposed two-copy protocol is prospective.