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Ah, I've been slower than I wanted to be with these posts. Charles, I
liked the idea of your Redistribution 3d Chess very much, and started a
comment, but wound up with this, instead, some days later. My apologies for
the rather offhand reference to your style of game design. I was almost up
to R3d [need a better shorthand for this design, it sounds like an
unfinished robot] in this commentary, and had hoped to engage you in a
discussion of shortrange 3D pieces, since we have designed different kinds.
Let me post the little [heh, as if my comments are 'little'] I had
completed before I saw your comment, then take this up again.

The outside of a 5x5x5 is 98 cubes, and the inside is 27 - just under 80%
edge cells on this board. Going up a dimension makes it much worse. A
4x4x4x4 has 256 cells, with 16 in the center, and 240 on at least 1 edge.
Edge effects cripple diagonal pieces, reducing the number of cells they can
reach compared to starting in the center, but don't affect orthogonal
pieces at all in that way. This does counterbalance the increasing mobility
of diagonal pieces in higher dimensions, but is an order of magnitude too
small to be truly effective, unless you make the edges so close together
diagonal pieces diverge off the board within 1 or 2 cells in any direction.
This is somewhat like using the alfil in shatranj. It has a certain freedom
of motion, but it doesn't really go anywhere much. It can either kill you
or miss you completely, and is rather easy to avoid.