Beyond Static Crosstalk Cancellation: Hidden-Mediator Gram Curvature in Tunable Qubit Processors

Static cancellation of mediated crosstalk controls the reduced computational Hamiltonian at a chosen operating point, but it does not by itself determine the geometry of the dressed computational subspace during multiparameter control. This raises a practical question: can two tunable qubit processors appear identical under static spectroscopy and one-parameter calibration while implementing different coherent processes around a closed control loop? This work develops an operator-geometric framework for identifying the missing information. The normal motion of the dressed computational subspace is organized by a full Hermitian Gram structure. Its diagonal blocks describe the costs of tuning along individual control directions, whereas its mixed blocks retain the relative orientation that generates geometric curvature. An exact qubit–mediator model shows that two control orientations can have the same reduced Hamiltonian, spectrum, spectral gap, and full quantum metric while carrying opposite curvature and producing oppositely oriented conditional loop phases. Static cancellation and metric agreement therefore do not determine the coherent response of a closed control cycle. The missing orientation becomes observable through small-loop process tomography. The resulting channel determines the traceless, physically observable part of the curvature without direct measurement of the mediator. A quantitative stability law balances tomographic error against finite-loop corrections and yields an optimal loop scale. The framework therefore separates static decoupling from coherent cycle errors and provides an operational test of hidden mediator orientation.