THE INVARIANCE — a dart

THE INVARIANCE the output that does not move — f(g·x) = f(x), the special case of equivariance

Invariance is equivariance's still twin: the output is unchanged by the transformation, f(g·x) = f(x). The sum of a list is invariant to reversal — scramble the order, the total holds; the area of a shape is invariant to rotation; energy is invariant to a shift in time. Formally it is equivariance where the group acts trivially on the output (g·y = y). Batch 38 fired a whole quiver at invariance (the cross-ratio, the determinant, Noether) and the panel ruled it witnessed — a theorem every correct implementation obeys, not a mechanism it enacts. This dart marks that verdict, and sets up the sharper question its cousin equivariance (dart 390) asks.

THE TECHNIQUE f(g·x) = f(x) — equivariance with the output fixed

The demo shows sum is reflection-invariant (unchanged by reversal), while the first element is not: live demo

HISTORY & CREDIT invariance · batch 38, ruled witnessed

“Invariance is the deep symmetry.” — it is the special case (the output does not move); equivariance (the output moves with the input) is the general one. cited

the relation · f(g·x)=f(x) — the transformation leaves the output alone.
vs equivariance · invariance = trivial action on the output; equivariance = the output transforms too.
batch 38 · THE INVARIANT — ruled NULL/witnessed: a conserved quantity is a theorem the computation observes, not one it causes.

The quantity that does not budge when the world is transformed — the still special case; a theorem, witnessed. invariance

RECOMMEND FOR I-13 the unmoved sum, on the compiler

On the canonical compiler, sum(reverse x) = sum(x) (reflection-invariant, =1); the first element is not:

$ i13 run sy_invariance.i13 # f(reverse x) == f(x) ? RUN OK · 323 step(s) · peak stack 6 · call depth 4 sum_inv = 1 -- sum is reflection-invariant (output unchanged) first_inv = 0 -- the first element is not

Recommend as the marker: invariance is the still special case of equivariance — i13 shows the sum unmoved by reversal — and batch 38 already put it to the panel: NULL, witnessed (a conserved quantity is a theorem, not an enacted mechanism). The dart exists to draw the line: if invariance is witnessed, the open question is whether its general cousin, enacted equivariance (dart 390), is any different — the whole batch turns on that.