Gauge Completeness in Multiply-Connected Electromagnetism: Differential Cohomology and Operational No-Go for Field-Only Models

This paper articulates a structural principle for gauge theories in multiply-connected spacetime domains. We formulate a field-mediated hypothesis class based on causality, local reconstructibility from Maxwell dynamics, and bounded response, and derive a universal envelope that bounds endpoint-defined phase excursions using only metrology-controlled parameters. By casting the operational memory observable in the language of degree-2 differential cohomology, we identify a gauge-complete description in which the relevant phase is encoded by a differential character . The resulting topological prediction, expressed through linking numbers and net flux increments, is invariant under intermediate-time gauge choices and separated from the field-mediated class by the metrology bound. Together, these results establish that any verified violation of the envelope inequality under a window-cycle protocol demonstrates the operational incompleteness of field-only accounts and isolates the differential character as an essential structural datum.