If gauge symmetry is redundancy, why does it have physical consequences?

Run status: not attempted — no run in lab/runs/ names this case in its settles: field.

Object. A lattice U(1) gauge theory on a graph. Computed: the gauge group has one parameter per vertex, V of them, and its action on the link variables has rank exactly V−1 in every case tested — triangle, square, K₄, theta graph — because the constant transformation acts trivially. Physical degrees of freedom come out as E − (V−1), which is the first Betti number, the loops. A single link variable moves under a gauge transformation (4.0021 → 4.6272) while a Wilson loop does not (6.058559 → 6.058559).

So a gauge transformation fixes every physical state. A global symmetry maps physical states to different physical states and hands you a conserved charge; gauge does neither, and Noether's second theorem says as much — a local symmetry yields identities among the field equations rather than charges.

Presenting problem (the friction). On that reading gauge "symmetry" is an equivalence relation on descriptions, not a symmetry. But redundancy of description is not supposed to have consequences, and this one does.

The Aharonov–Bohm effect is observable and is a statement about the gauge potential's holonomy. Anomalies obstruct quantization and are computed from the gauge structure. The theta angle is a physical parameter of a gauge theory with no counterpart in any gauge-fixed description. The global form of a gauge group — which quotient by a discrete centre — changes the spectrum of allowed line operators while leaving every local correlation function identical. And the modern generalized-symmetry framework treats a gauge theory's one-form symmetries as genuine symmetries, with charges measured by linking.

Either redundancy can carry physical content, which needs explaining, or "gauge" names two things that have been run together.

Differential. 1. Two things. Local gauge transformations are redundancy and carry nothing; the global structure of the gauge group — its centre, its one-form symmetries, its bundle topology — is genuine symmetry, and every listed consequence belongs to the second. The word covers both. 2. Redundancy with a remainder. Gauge is redundancy, but quotienting by it is not free: the quotient has topology (the loops above), and every physical consequence is a fact about that topology rather than about the group action. 3. The framing is wrong. Gauge really is a symmetry, and the rank deficiency shows only that it acts with stabilizers, which is ordinary for group actions and not a disqualification. 4. Category-dependent. Whether it counts as a symmetry depends on whether one works with the space of field configurations, where it acts nontrivially, or with the space of physical states, where it does not — and both are legitimate.

Discriminating tests. For each consequence in the list, determine whether it survives in a fully gauge-fixed formulation: if the theta angle and the line-operator spectrum survive gauge fixing while the redundancy does not, branch 1 or 2. Check whether the one-form symmetry can be defined without reference to the gauge redundancy — if it can, the two are separable and branch 1 is right; if it cannot, branch 2. For branch 3, ask whether the stabilizer is the whole group on physical states, which is a stronger condition than acting with stabilizers.

Stakes. A slogan of the form "symmetry determines the geometry" is true for one of these readings and empty for the other, and a great deal of structural reasoning has been built on the slogan without specifying which.