The Aharonov-Bohm Phase is Gauge Dependent, But Its Gauge-Invariant Core is Observable

Contrary to long-standing textbook doctrine, we demonstrate that the Aharonov-Bohm (AB) phase for a closed interferometric loop — the relative phase governing interference — becomes gauge dependent when backreaction of the test charge on the electromagnetic source is included in a fully quantum treatment. The standard Stokes-theorem argument fails because different interfering paths become entangled with distinct field configurations, permitting branch-dependent gauge transformations. We prove that the total phase decomposes into (1) a gauge-invariant core tied to the fixed background flux, which fully accounts for the observed interference shift, and (2) a purely gauge-dependent backreaction term that can be eliminated by an appropriate gauge choice and carries no observable consequences. This decomposition follows from a general framework for observables in gauge theories, which we establish from a foundational axiom that purely gauge-dependent quantities have no observational consequences. From this axiom, we derive theorems showing that any gauge-dependent quantity contains an extractable gauge-invariant core that constitutes its measurable content, while the remaining gauge-dependent component is physically irrelevant. This framework resolves the apparent paradox between formal gauge dependence and robust experimental observations, and offers a conceptually sharper understanding of observables in quantum electrodynamics, non-abelian gauge theories, general relativity, and other gauge-based theories.