The coordinate signatures of the orbits
The useful invariant is the distance of a coordinate from the nearest
boundary. A layer turn permutes two coordinates, possibly changes their
signs relative to the cube center, and leaves the multiset of these
distances unchanged. It therefore preserves corners, edges and centers.
On an edge, the one nonboundary coordinate gives a ring index. On a center,
the two interior coordinates give an unordered pair of distances. Equal
distances give a center-corner position in a concentric ring; unequal
distances give a center-edge position. The central pair in an odd cube is
the fixed center.
Write \(K_e=N/2-1\) for even \(N\) and
\(K_o=(N-3)/2\) for odd \(N\). The position orbits under the selected layer
turns have the following sizes. For even \(N\), the coupled-edge orbits
\(E_1,...,E_{K_e}\) each have \(24\) positions. For odd \(N\), there is a
single-edge orbit \(E_s\) of size \(12\), together with coupled-edge orbits
\(E_1,...,E_{K_o}\) of size \(24\) each. The term coupled records
that the edge is away from the odd cube's middle edge position; it does not
by itself choose an orientation convention.
For even \(N\), each center-corner ring \(C_k\),
\(1\le k\le K_e\), has \(24\) positions. The center-edge ring \(D_k\),
\(2\le k\le K_e\), has \(48(k-1)\) positions, split into
\(2(k-1)\) distinct orbits of \(24\). For odd \(N\), there are \(6\) fixed
centers, center-corner rings \(C_k\) of size \(24\), and center-edge rings
\(D_k\) of size \(24(2k-1)\), each for \(1\le k\le K_o\). Each odd ring is
likewise a union of \(2k-1\) orbits of \(24\). Summing these rows gives
\(6(N-2)^2\) centers in both parities. The formulas expose the distinction
between a position orbit and an orientation coordinate: a one-sticker
center has no orientation unless it is specially marked, while a coupled
edge can have a convention-dependent orientation record.
These sets are not merely a count. Face turns carry a representative
edge around the four faces of its ring; an inner layer at depth \(k\),
conjugated by outer turns, carries it to the other axis directions. The
same construction carries a center-corner representative around its ring
and a center-edge representative around every position with the same
ordered distance-pair or chirality label. Hence each displayed
\(24\)-position set is a move orbit, while the distance signature proves
that different rows cannot mix. This is the coordinate part of the model;
legality constraints on the permutations are a separate question.