GAUGE INVARIANCE a redundancy in the description that the physics ignores — only differences matter
Gauge invariance is a symmetry of the description, not the thing: you can change the bookkeeping and the physics does not notice. The electric potential has no absolute zero — add any constant and every voltage difference (which is all you can measure) is unchanged; the force, being the derivative of the potential, is untouched. That freedom to relabel is the gauge, and demanding the physics be invariant to it — a local version, different at every point — is what forces the existence of the electromagnetic field (and, generalised, the whole Standard Model, via Yang–Mills). A symmetry that is a redundancy, and yet dictates the forces of nature.
THE TECHNIQUE potential + constant → same force (the derivative kills the constant)
The demo adds a constant to a potential and shows the force (its discrete derivative) is unchanged — gauge-invariant: live demo
HISTORY & CREDIT Weyl 1929 · Yang–Mills 1954
“Every symmetry is of the world.” — gauge symmetry is of the description: a redundancy in how we write the potential. Yet demanding it locally forces the existence of the forces. cited
the redundancy · potential +C changes no measurable difference; the force (derivative) is unchanged. 1929 · Hermann Weyl — gauge invariance of electromagnetism. 1954 · Yang & Mills — non-abelian gauge theory; demanding local gauge symmetry generates the forces (the Standard Model).
Relabel the potential and nothing observable moves — a redundancy in the description that nonetheless dictates the forces. Weyl / Yang–Mills
RECOMMEND FOR I-13 potential + C, same force, on the compiler
On the canonical compiler, adding a constant to the potential leaves its discrete derivative (the force) unchanged (gauge_invariant=1):
$ i13 run sy_gaugeinvariance.i13 # force = d(potential); +C leaves it fixed
RUN OK · 47 step(s) · peak stack 4 · call depth 1
f0 = 1 f0b = 1
gauge_invariant = 1 -- potential + constant -> identical force
Recommend: gauge invariance is a symmetry of the description — add a constant, the force (the derivative) does not move — and i13 confirms it. Not a keeper (invariance-of-the-observable-under-a-relabeling is witnessed, and the “add a constant” freedom is a representation choice, near the B44 encoding-ban). But the dart with the deepest reach: demanding this redundancy hold locally is what generates the forces of nature — a symmetry of our bookkeeping that writes the laws.