◄ WORLD V · SONNY 5DART 390 · a helldive at the net

THE EQUIVARIANCE transform then compute = compute then transform — f(g·x) = g·f(x)

A map is equivariant to a transformation when it does not matter whether you transform first or compute first: f(g·x) = g·f(x). Reverse a list then double it, or double it then reverse it — same answer; doubling is equivariant to reversal. But prefix-sum is not: reverse then accumulate gives a different sequence than accumulate then reverse. The difference is the point — equivariance is a property a correct-but-different mechanism can lack, and it is structural: a statement about how the output relates to the input under the whole group, not about any one value. It is the shape of every symmetry-respecting computation, from convolution to physics.

THE TECHNIQUE f(g·x) = g·f(x) — the output transforms as the input does

The demo tests two maps against reversal: double is equivariant (transform-order-free), prefix-sum is not: live demo







  
HISTORY & CREDIT equivariance · group-equivariant maps

  
“Order of operations is bookkeeping.” — not for symmetry: some maps commute with a transformation (equivariant) and some do not, and which is which is structural, not incidental. cited

  
the relation · f(g·x) = g·f(x) for every g in a group — the output carries the same symmetry as the input.
invariance is the special case · g·f(x)=f(x) (the output does not move; dart 391).
2016+ · Cohen & Welling — group-equivariant convolutional networks; the modern design principle.

  
Transform-then-compute equals compute-then-transform — the map wears the input's symmetry. A property a mechanism can have or lack. equivariance






  
RECOMMEND FOR I-13 equivariant vs not, on the compiler

  
On the canonical compiler, double commutes with reversal (equivariant, =1); prefix-sum does not (=0):

  
$ i13 run sy_equivariance.i13   # f(reverse x) == reverse(f x) ?
RUN OK · 823 step(s) · peak stack 9 · call depth 4
equi_double = 1     -- double commutes with reverse: equivariant
equi_prefix = 0     -- prefix-sum does NOT: a correct map that LACKS the property

  
Recommend — the batch's keeper shot, and the most genuine one since the math pivot. Equivariance is a structural invariant of the output-relation (f(g·x)=g·f(x)), it is a real supplement (i13 shows a correct map — prefix-sum — that lacks it), and it is enacted: it holds by the mechanism's form (dart 392, convolution's weight-sharing), and a differently-built correct mechanism breaks it. The honest questions for the panel: (1) is equivariance distinct from invariance (dart 391, batch 38's THE INVARIANT, which the panel ruled witnessed — a theorem the computation obeys)? Invariance was “f(g·x)=f(x)” observed after the fact; equivariance-by-construction is the mechanism causing the commutation — the exact enacted-vs-witnessed line B38 left open. (2) or does “the output has the input's symmetry” reduce to a theorem coextensive with a correctly-symmetric task? The strongest test of the near-complete read — put to the full panel.