Example. Corner twists and edge flips on the cube [frap-000V]
Example. Corner twists and edge flips on the cube [frap-000V]
Use the unmarked-center, fixed-frame sticker model of one cube face turn. We apply the canonical character of [marchetti2025rubik, Section 3.1, p. 7] to this explicit action. At a corner the three incident sticker directions have a cyclic order inherited from the ambient orientation. Choose bijective cyclic frames at all eight positions. Rotations preserve that order, giving local group \(C_3\); at each of the twelve edges an ordering of its two stickers gives local group \(C_2\). Both actions on the local stickers are faithful. Writing these groups additively, the two characters are
\[\chi _c=\sum _{i=1}^{8}a_i\pmod 3,\qquad \chi _e=\sum _{j=1}^{12}b_j\pmod 2.\]A quarter-turn has one four-cycle of affected corners and one of affected edges. Four applications restore every sticker, so the return transport on each of these cycles is the identity. Every other corner and edge is fixed with identity local transport. The cycle criterion gives \(\chi _c=0\) and \(\chi _e=0\) for every face quarter-turn. The kernel argument extends this to every word in those turns, including inverses and half-turns.
For a concrete cyclic frame, choose any frame at corner \(i_0\) of the upper face and define the next three by transporting it with the shape rotation \(q\). Since \(q^4=1\), all four local factors of this turn become zero. This frame checks that one generator; the frame-independence theorem transfers the conclusion to any fixed choice of cyclic frames used for the other generators.
A configuration with a corner twist of just one corner by \(1\) has \(\chi _c=1\) and is impossible under these moves. Twisting two corners by \(1\) and \(-1\) passes this test, but the calculation alone does not construct a legal sequence. Likewise, flipping exactly one edge fails the second test. An edge flip of exactly one edge therefore fails the second test. Reachability needs additional arguments.