A Connectivity-Order Law and Conditional Minimal-Mechanism Identification in Many-Body Geometric Phases
The value of a multiqubit geometric phase at a single operating point does not reveal whether it was generated directly or through a connected sequence of lower-body interactions. For analytic, gapped, nondegenerate Abelian holonomies, we jointly resolve logical support and independently calibrated coupling support. Every nonzero connected response must involve an active set that connects and covers the target and must contain one factor of each active coupling, yielding the sharp onset bound $\nu_S \geq \tau_E(S)$. The same selection law applies locally to projected Berry curvature. Conversely, an unrestricted finite-dimensional construction realizes all responses allowed by the connected-cover condition simultaneously, making the condition necessary and sufficient within this theorem class. Operationally, simultaneous confidence bands certify detected mixed responses while controlling false positives; a factor-of-two separation condition additionally recovers exact library-relative response minima. In a complete dictionary, nontriviality of every minimal-cover response is generically necessary and sufficient for these minima to coincide with the minimal calibrated mechanisms. Three-qubit Wilson calculations distinguish direct and pair-mediated routes with the same endpoint phase, and a synthetic Ramsey audit locates the finite-resolution boundary.