Gauge Phases Beyond the Line Integral: Nonlocal Corrections and Flux Memory from Field Traversals -- Part I: Foundational obstruction

This work introduces nonlocal corrections to gauge potentials arising when quantum paths traverse regions of nonzero electromagnetic field strength. These corrections restore single-valuedness of the gauge function and resolve inconsistencies that appear in standard treatments of phase accumulation when paths cross field regions. Explicit analytic expressions are derived for static and time-dependent configurations, including solvable stripe models. The results suggest a geometric path dependence beyond conventional formulations of gauge phases and may have implications for the Aharonov–Bohm effect, Berry phases, and related gauge-theoretic constructions. This manuscript represents the first part of a broader investigation of nonlocal gauge corrections and their physical applications.