From your perspective, that may be. From my perspective, I would include the associative law also, since I often look at other structures that have the signature < 2 > (one binary operation) . No examples come to mind that look as nice as bands. However, your question is about representing equational theories succinctly when they lie between the trivial theory and some (not necessarily finitely equationally based) nontrivial equational theory. I don't know the literature, but you may find Austin identities of (remotely) related interest.

Do you mean exactly one identity? Or exactly one more than is needed to distinguish it as a subvariety? In the latter case, I'm confident there are more examples in other signatures.

You might try Y = 2. Or even 3.

could this approach generalize to three or four shapes? I'm thinking for the case of three shapes, draw one bisecting line, and then find a bisecting perpendicular ray of appropriate width coming off that line?

Indeed, with such a coloring, one can only have two vertices of the same color joining an edge. This gives that there is no such decomposition for d=3.

Stand inside the cube center, and look at the corner. Color a vertex black if the three paths go clockwise throuvh the corner, white otherwise. For a Hamiltonian path, each cube face has two black and two white vertices on the face. This should cut down the possibilities to within hand computation.

Letting Q be $(1+x^2)^{1/2}$, I rewrite the identity using operator notation as Q^{k+1}D^{k+1}Q^k = C for a certain nonzero constant C. I then wonder if this is part of a larger total differential or perhaps the result of some inversion formula. Just some random musings, in case a random start toward an answer proves useful.

Miodrag Zivkovic has enumerated some small binary matrices by Smith Normal Form ( arxiv.org/abs/math/0511636 ). Not quite what you ask, but it may give asymptotic hints.

I hope it works out. Given the poster's original offer, I would like Lev to update this post to hear if things concluded satisfactorily. (The detail of whether he was compensated in precious metal is less important to me than knowing that the proof withstood vetting and that the poster made good on the offer. I think others on MO would like to know the same.)

There is either a sign disagreement or some notational ambiguity. Where do the minus signs come from? what is P_{k+1-r}? What is r?

If Aaron is right, that would make the face angle of a regular tetrahedron exactly 72 degrees. Which it isn't.

Thus the comment above on balloon and pinch. If genus 1 were allowed, there would be almost nothing to work on.

Take a deflated balloon, put it in the middle, and inflate it. If you do it carefully, the balloon will be pinched by the curve for some of the curve. Elongate the balloon to maintain the pinch.

Oh, and remember to divide by 2-1 first.

Fermat-Lagrange, which says that p is 134k+1 for some integer k.

Actually, f(x) + f(-x) is in $F_D$, and one can subtract a smaller function g which leaves something not in $F_D$. This reminds me of stacking rectangles, then looking just at the height of the stack as x varies, and using that to reconstruct the rectangles. I suspect you have something similar and NP hard to untangle.

It looks like you can drop the alpha's in your definition of $F_D$. Also, all such combinations have domain contained in (-D, D). If a function $f$ majorises a D basic function $g$, hopefully $f-g$ is in $F_D$, which would then lend hope to find quickly a decomposition of $f$.