= Solution
A <gauge redundancy> relates different field representatives of the same physical configuration. For example, the transformation just exhibited changes $A_\mu$ and the scalar phase but leaves $F_{\mu\nu}$, $|\varphi|^2$ and all other <gauge-invariant operators> unchanged. Fixing a gauge removes this freedom; it does not remove a physical symmetry-related state. In <canonical quantization>, local transformations trivial at the boundary are generated by the <Gauss law constraint in gauge theory>, and physical states satisfy that constraint.
A <global symmetry in field theory> instead has a spacetime-independent parameter and can act nontrivially on physical states. For a continuous <global symmetry in field theory>, <Noether theorem> gives a conserved charge that generates the transformation and labels the states; discrete global symmetries do not acquire a <Noether current> from the same infinitesimal theorem. These are statements about physical transformations, rather than merely whether a parameter is written as a constant or a function.
Boundary conditions matter in <gauge theory>. <gauge transformations> that vanish at spatial infinity are redundancies, while transformations nontrivial at infinity may act through a physical electric charge. In particular a constant phase transformation can survive as a global charge symmetry rather than being quotiented out with the local <gauge redundancy>. Thus \b[local <gauge redundancy> is a redundancy of description; genuine <global symmetry in field theory> can distinguish physical states and charges], with the boundary qualification made explicit.
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