From facet incidence to the 54-sticker state
The facet calculation in one ordinary cube face turn
selects the facet-colored part of the incidence set. Put the cube's surface
cells at
\(p\in \{-1,0,1\}^3\ \text {with}\ \lVert p\rVert _\infty =1\).
A visible sticker is a pair \((p,n)\), where \(n=\operatorname {sgn}(p_i)e_i\)
for each nonzero coordinate \(p_i\). Thus a corner has three visible
normals, an edge has two, and a face center has one:
\[8\cdot 3+12\cdot 2+6=54.\]The vertex--facet incidences in the facet-colored set map to the pairs
above by sending a cube vertex, edge, or facet to its surface-cell coordinate
and its incident facet normal. Conversely, the zero-coordinate pattern of
\(p\) determines the cubie (three nonzero entries for a
corner, two for an edge, one for a
centre), and \(n\) determines the incident facet, so this is
a bijection between the 54 incidences and the 54 pairs. The cubie counts and
this incidence decomposition follow the general model in
[bonzio2018nxnxn, Sections 2--3]; the displayed pairing and 54-count
are the calculation used here. For \(N=3\), group these pairs into eight
corner and twelve edge sticker fibres, leaving the six center
positions as fixed frame marks. A legal outer-face move preserves this
partition and carries the labels in a corner fibre or edge fibre to the
corresponding destination fibre. Write
\(C_p=\{(p,\operatorname {sgn}(p_i)e_i):p_i\ne 0\}\) for a corner fibre and
\(E_p\) for an edge fibre. For a corner, order its normals as
\(L_p=(u_p,f_p,r_p)\), where \(u_p=\operatorname {sgn}(p_3)e_3\),
\(f_p=\operatorname {sgn}(p_2)e_2\), and \(r_p=\operatorname {sgn}(p_1)e_1\).
For an edge, put first the U/D normal when \(p_3\ne 0\), otherwise the F/B
normal, and put the other normal second. A move sends \(C_p\) or \(E_p\) to a
destination fibre. Let \(k_p\in \{0,1,2\}\) be the index of the transported
U/D normal in \(L_{p'}\). If the source cubie has \(p_3=+1\), set
\(x_p=k_p\); if it has \(p_3=-1\), set \(x_p=-k_p\) in
\(\mathbb Z_3\). Thus each side turn gives two \(+1\) and two \(-1\)
corner twists, while U/D turns give zero, matching the U/D-sticker
convention in the complete three-layer cube. For an edge,
its distinguished normal agrees or disagrees with the first destination
label, giving \(z\in \mathbb Z_2\). This explicit extraction fixes each
per-fibre bijection and the destination permutation without asserting an
unstated cyclic convention.
Therefore it induces exactly
the fixed-frame coordinates
\[(\sigma ,x;\tau ,z)\in S_8\times (\mathbb Z_3)^8
\times S_{12}\times (\mathbb Z_2)^{12}\]used in the complete three-layer cube. More explicitly, if
a move sends a corner fibre \(C_p\) to \(C_{p'}\), set \(\sigma (p)=p'\);
the destination edge fibre similarly gives \(\tau \). For the positive-\(z\) turn
\(q(x,y,z)=(y,-x,z)\) from one ordinary cube face turn,
\[((1,1,1),e_z)\longmapsto ((1,-1,1),e_z),\qquad
((1,1,1),e_y)\longmapsto ((1,-1,1),e_x).\]This exhibits both the destination corner and the transported local
frame. The induced map is a change of state description for the legal
action: it sends a sticker permutation to its corner and edge destination
permutations together with the local frame changes. It does not assert that
every permutation of the 54 positions is a legal cube move.
This also separates two kinds of forgetting. Passing from the incidence
action to the displayed coordinates forgets the individual sticker
addresses inside each recorded local frame. Passing from labelled cubies to
an ordinary colour observation forgets equal-looking labels. The latter is
the observation quotient discussed in which moves survive a quotient,
and it is a separate identification of states rather than a change in these
coordinates.
