Referenced in the paper panic suggested below ( https://news.ycombinator.com/item?id=11595273 ), and suggested to me directly by @mattcmd on twitter ( https://twitter.com/mattmcd/status/726342652845809664 ), there is something called the "fusion property of fold," which would explain transformations I'm discussing here applied to folds (sided reduce). From the paper ( http://www.cs.nott.ac.uk/~pszgmh/fold.pdf ), (and some minimal editorial substitution from me) we see that:

h(fold(g, w, l)) = fold(f, h(w), l)

if

h(g(x, y)) = f(x, h(y))

This is exciting because (1) it suggests the equivalent transformation for sided reduces (folds), and (2) fold is a more general operator than the others. My somewhat silly use of "hypothetical inverses" effectively just creates a goalpost to (1) constructively find the the function f, given g (or vice-versa) and (2) prove the necessary condition.

Very exciting.