The issue is that the ‘orthogonal’ step in the Hex lattice (between two hexes sharing a side) is identified with a Bishop step (between cubes sharing an edge but no face) on the cubic one (which I suppose is what you mean by “the adjaceny relation”?). And since all spaces on a Hex board can be reached by orthogonal steps, it follows that the 3D extrapolation is also the set of cubes reachable by such steps: a single Bishop binding (only half the cubes).
The observation that a single cubic Bishop binding corresponds to the Xyrixa board is made by Gilman a few times, f.ex. in Oddly Oblique (¶5 of the main text, following the Senhelm diagrams) and in the introduction (¶2) to OctHex 146; probably also in comments and such over the years.
Hence also the fact that a cubic Unicorn step (which is colourswitching) doesn't resolve to a Hex move with this identification (any more than a Rook step does).
The issue is that the ‘orthogonal’ step in the Hex lattice (between two hexes sharing a side) is identified with a Bishop step (between cubes sharing an edge but no face) on the cubic one (which I suppose is what you mean by “the adjaceny relation”?). And since all spaces on a Hex board can be reached by orthogonal steps, it follows that the 3D extrapolation is also the set of cubes reachable by such steps: a single Bishop binding (only half the cubes).
The observation that a single cubic Bishop binding corresponds to the Xyrixa board is made by Gilman a few times, f.ex. in Oddly Oblique (¶5 of the main text, following the Senhelm diagrams) and in the introduction (¶2) to OctHex 146; probably also in comments and such over the years.
Hence also the fact that a cubic Unicorn step (which is colourswitching) doesn't resolve to a Hex move with this identification (any more than a Rook step does).