I'm currently attempting to explain the concept of Gauge Symmetry to a friend. Copied and pasted pretty much directly from MathIM,

(And the same applies for any other potential field, such as gravitational potential.)

Would this be correct? I've tried explaining Gauge Symmetry multiple times to no avail (don't worry, it's not a technicality barrier, I think they're familiar with elementary electrodynamics,) so does anyone have a suggestion of an easier way to explain this?

Electrostatic and gravitational potential are the easiest systems, I think.
"Change the potential by the same amount everywhere, and physics stays the same".

These potentials that we use in physics are just devices of our mathematical models used to describe the universe, and they aren't "real". Sometimes our mathematical models have more degrees of freedom than exist in nature, and there is some redundancy in the choice of numbers. If you picture a physics model as a relation whose domain is the values in a model and whose range is possible realities, then we would have multiple values mapping to the same reality.

From a more mathematical viewpoint, you could view sets of values pertaining to the same reality as an equivalence class. Then, the "correct" theory (from an Ockham's razor stance) would take parameters from the quotient set of all parameters modulo the gauge symmetry. The gauge symmetry is just meaningless excess.

Nevertheless, sometimes it's easier to work with mathematical structures with extra redundancy because the rules for those math structures have already been worked out. For example, we might use a 2x2 matrix to represent something with 3 degrees of freedom when a 2x2 matrix has 4 degrees of freedom because physicists don't (always) want to invent a whole new math structure for the 3 degree of freedom object when 2x2 matrix works.