The complete three-layer cube [frap-0016]
✍️source

    The all-size law has two reasons to be followed by a separate three-layer account. A \(3\times 3\times 3\) cube has no moving centers or coupled edges, and its familiar three tests can be checked directly on the six outer-face generators. We use a fixed frame and unmarked centers. A labelled state is therefore

    \[(\sigma ,x;\tau ,z)\in S_8\times (\mathbb Z_3)^8 \times S_{12}\times (\mathbb Z_2)^{12}.\]

    Here \(\sigma \) and \(\tau \) record corner and edge positions. The corner frame uses the U/D sticker to record a twist in \(\mathbb Z_3\); an edge frame uses the U/D or F/B sticker convention to record a flip in \(\mathbb Z_2\). The six centers fix the frame but have no recorded orientation. This is the ordinary cube model, rather than a supercube model with marked centers.

    1. Generators and the three obstructions

      Write \(R,L,U,D,F,B\) for clockwise quarter-turns as seen from the corresponding face; inverses and squares are allowed. Each turn is a four-cycle on corners and a four-cycle on edges. Thus it changes both permutation signs, leaving

      \[\operatorname {sgn}(\sigma )=\operatorname {sgn}(\tau )\]

      unchanged. A turn of U or D changes no corner twist in this convention; a turn of another face adds one and subtracts one at opposite corners. The four affected edge frames change in pairs. Consequently every reachable state satisfies

      \[\sum _{i=1}^{8}x_i=0\pmod 3,\qquad \sum _{j=1}^{12}z_j=0\pmod 2.\]

      These checks are generator checks, so they also cover inverses, powers and arbitrary words. They give immediate impossible states: one twisted corner has corner sum \(1\); one flipped edge has edge sum \(1\); and a swap of two corners alone has odd corner sign but even edge sign. A state can pass all three tests and still require a construction; necessity is not sufficiency. The coordinate convention and invariant calculation agree with the 3-layer case of [salkinder2021nxnxn, Theorems 8--18].

      2. Localized effects and sufficiency

        We now describe a constructive cleanup. Let \([a,b]=aba^{-1}b^{-1}\) and \([a:b]=aba^{-1}\). The word

        \[c=[[R:U],D]\]

        is a three-cycle on corners and fixes the other corner positions. A direct coordinate replay gives this cycle; conjugating by face turns moves it through the connected corner orbit. Hence these conjugates generate \(A_8\). For edges, the outer-turn word

        \[e=R\,U^{-1}\,R\,U\,R\,U\,R\,U^{-1}\,R^{-1}\,U^{-1}\,R^2\]

        is a three-cycle on three edges and fixes the corners (the familiar U-permutation). Conjugates generate \(A_{12}\). The displayed words are useful witnesses, not a claim that they are short in every metric.

        Given a state satisfying the sign equation, apply one face turn when the common sign is odd. The remaining corner and edge permutations are even. Use the conjugated corner cycles to clear \(\sigma \), then the conjugated edge cycles to clear \(\tau \); each cleanup is chosen in the subgroup that fixes the already-cleared piece type. The sign equation is exactly what makes both cleanups possible.

        After positions are restored, the orientation kernels are generated by local commutators. The standard corner trigger \((R^{-1}D^{-1}RD)^2\) restores positions while twisting two corners in opposite directions; setup, trigger, and inverse setup transport this effect to any pair. For edges, the local trigger \(F R U R^{-1}U^{-1}F^{-1}\) is combined with its conjugate and inverse setup to give a pure two-edge flip. A direct sticker replay checks that support and the two flipped signs. Their transported effects generate the kernels

        \[\ker ((\mathbb Z_3)^8\to \mathbb Z_3)\quad \text {and}\quad \ker ((\mathbb Z_2)^{12}\to \mathbb Z_2).\]

        The two displayed sums are therefore the only orientation obstructions. Reversing the cleanup word reaches every tuple satisfying the three tests. This is a constructive subgroup argument; it does not identify a shortest solution.

        3. Count and a worked reachability check

          There are \(8!\) corner permutations, \(12!\) edge permutations, \(3^8\) raw corner orientations and \(2^{12}\) raw edge orientations. The sign equation removes a factor of two, and the two orientation sums remove factors of three and two. Thus the number of labelled reachable states is

          \[8!\,12!\,3^7\,2^{10} =43\,252\,003\,274\,489\,856\,000.\]

          For a concrete reachability check, start from solved and apply the corner cycle \(c\). Its corner permutation is a 3-cycle, its edge permutation is the identity, and its orientation sums remain zero, so it passes every test. Applying \(c^{-1}\) returns to solved. More generally, the cleanup above factors any passing state into a sign correction, even corner and edge permutations, and orientation-pair operations; this gives an explicit solve word once the required cycles and setup positions have been selected.

          Unmarked centers explain why a whole-cube rotation is not counted as a generator and why the center colors fix a frame rather than add twelve more coordinates. The all-\(N\) chapter keeps these choices visible when moving centers and coupled edges return. The next chapters can now discuss human methods and computer search using this completely specified 3-layer action.

          References

            Reference. Nxnxn rubik’s cubes and god’s number [salkinder2021nxnxn]

            @misc{salkinder2021nxnxn,
 title = {nxnxn Rubik's Cubes and God's Number},
 author = {Daniel Salkinder},
 year = {2021},
 url = {https://arxiv.org/abs/2112.08602},
 primaryclass = {math.CO},
 archiveprefix = {arXiv},
 eprint = {2112.08602}
}

            Backlinks

              A complete layer-first method [frap-0017]

                The three-layer model in the complete cube account turns a human solve into a sequence of stabilizer problems. We keep the fixed frame and write \(U,D,F,B,R,L\) for clockwise quarter-turns as seen from the named face. A prime is an inverse and a superscript \(2\) is a half-turn; a word is read from left to right. The bottom face \(D\) is the first layer, so its center fixes the target colour and the four side centres fix the cross alignment. The notation and face-view convention agree with the cube notation notes.

                Effects, setup, and preservation

                  A slot is a destination for one corner or edge. A setup word \(s\) brings a chosen case to the working slot; a trigger \(t\) performs the local change there; and \(s t s^{-1}\) transports that change to another location. This is conjugation in the move group, not a promise that the transported word has the same visible support before the replay is checked. A commutator

                  \[ [a,b]=a b a^{-1}b^{-1} \]

                  is useful when \(a\) and \(b\) overlap in a controlled way. Its support must still be computed in the current convention: cancellation in the group word does not by itself prove that every other piece is fixed.

                  Let \(H_i\) be the subgroup fixing everything completed before stage \(i\). A stage is valid when each of its words lies in \(H_i\) after the setup data have been restored. Thus a solve follows a descending chain

                  \[G\supseteq H_{\rm cross}\supseteq H_{\rm first}\supseteq H_{\rm middle}\supseteq H_{\rm last\,orientation}.\]

                  The inclusions describe preservation, while the case table below gives the progress that makes the chain terminate. This separates the algebraic question “what does the word change?” from the practical questions of recognition, memory, and execution.

                  The complete seven-stage method

                    The following method is complete under the stated case conventions. At every stage, rotate only the working view or use a setup word; never rotate the whole cube while a positional convention is being recorded. A displayed algorithm is a witness for its local effect. Its inverse and a conjugate supply the other orientations and slots.

                    1. Cross. Solve the four \(D\)-edges one at a time, matching each side colour to its centre. If a target edge is in the top layer, bring it above its slot and insert it with \(F^2\) or a quarter-turn setup. If it is in the first layer with the wrong orientation, eject it to the top and insert it again. Earlier cross edges are kept out of the working slot.
                    2. First-layer corners. Put a target corner above its \(D\) slot at \(UFR\). Repeat \(T=R' D' R D\) until the target sticker faces down and the two side stickers match. The word \(T\) fixes the already solved cross after the setup is restored; at most four repetitions are needed. A corner incorrectly occupying the slot is first ejected with the same trigger, then treated as a top-layer case.
                    3. Middle edges. With the solved layer on the bottom, align a top-layer edge with its side centre. To insert it on the right use
                      4. \[I_R=U R U' R' U' F' U F,\]

                      and to insert it on the left use

                      \[I_L=U' L' U L U F U' F'.\]

                      Both words restore the first layer. If a middle edge is misplaced, use either insertion to eject it, then solve the resulting top-layer case.

                    4. Last-layer cross. Ignore the side colours and orient the four \(U\)-edges with
                      5. \[X=F R U R' U' F'.\]

                      With the \(U\)-face viewed from above, a line is held horizontal and an L is held at the back-left. The same word changes the zero-, two-, or four-oriented-edge cases to a case with strictly more oriented edges after the indicated setup; repeat until the cross is present.

                    5. Last-layer edges. First restore the side colours of the four cross edges. The checked three-edge cycle
                      6. \[P=R U' R U R U R U' R' U' R^2\]

                      fixes the corners and cycles three \(U\)-edges. Together with a \(U\)-turn, its conjugates generate all permutations of the four cross edges: a 4-cycle and a 3-cycle generate \(S_4\). Apply a setup, one or more copies of \(P\) or \(P^{-1}\), and undo the setup.

                    6. Last-layer corners. Put a correctly located corner at \(UFR\); if none is located, apply the word once and inspect again. The corner cycle
                      7. \[C=R' F R' B^2 R F' R' B^2 R^2\]

                      fixes the cross edges and cycles the three other \(U\)-corners. Its conjugates and \(U\)-turns generate \(S_4\), so repeated applications place all four corners without changing their orientations.

                    7. Last-layer corner orientation. Keep the cube fixed. At \(UFR\), repeat \(T=R'D'RD\) until that corner is oriented; turn only \(U\) to bring the next unsolved corner to \(UFR\), and repeat. During this stage the visible lower layers look disturbed, but after the last corner the accumulated orientation sum is zero and every earlier piece returns to its solved position.

                    The method uses only a small vocabulary of triggers. The setup words are the memory burden; each trigger has a recognizable local effect. The two places where an algorithm appears to disrupt completed material (the corner orientation stage and an ejection of a misplaced piece) are deliberate: the word is applied until the current local case is solved, after which the preservation invariant is restored.

                    Why the cases terminate

                      We now justify completeness rather than infer it from successful demonstrations. A cross edge can always be moved to the top layer or ejected from its slot, and a quarter-turn or half-turn inserts it into a matching slot. Removing one solved edge leaves a connected set of unsolved slots, so four repetitions finish the cross. The corner trigger has a nonzero twist effect at \(UFR\) and fixes the cross after each full insertion; the eject-and-reinsert rule therefore solves four corners. The two middle insertions are inverses after reflection of the working view, and the same ejection argument solves four middle edges.

                      For the last layer, the edge-orientation word \(X\) changes the orientation pattern in the standard line/L/point cases while fixing both completed layers. The finite list of patterns has representatives point, line, and L; the prescribed setup makes the number of oriented edges increase, so the process reaches the cross. After orientation, \(P\) is a 3-cycle on edges and \(U\) is a 4-cycle. These generate \(S_4\), proving that the edge-position stage reaches every permutation. The analogous checked effect of \(C\), together with \(U\), gives \(S_4\) on corners.

                      Finally, \(T\) changes the orientation of the working corner while the temporary permutation cancels after the corner is finished and the next corner is brought into place. The corner orientation sum from the cube constraints is zero, so the final corner is forced when the other three have been oriented. Thus every state satisfying the three reachability tests has a finite word supplied by these stages. This is a completeness proof for the method, not an optimality claim.

                      One annotated local solve

                        Suppose the first two layers are solved, the \(U\)-cross is oriented, and the three-edge cycle needed to place the \(U\)-edge at the back is visible. Hold that edge at the slot to be fixed and apply \(P\); if the cycle runs the other way, apply \(P^{-1}\). The corners and all lower pieces return to their previous states. If the desired edge is not in the working support, choose a setup \(s\) that brings it there and replay

                        \[s P s^{-1}.\]

                        After the edge positions are correct, use the same inspect--setup--cycle pattern for \(C\). Only then use \(T\) at \(UFR\), counting quarter-turns until the corner's \(U\)-sticker points up. The visible disorder during this last replay is expected; the final inverse setup and the orientation sum certify that the solved state has returned. This is the smallest complete trace of the method: recognize a case, transport a checked effect, and finish inside the next stabilizer.

                        A block-building or Roux-style organization can reduce the number of stages for an experienced solver, but it changes the recognition and preservation cases. It should be compared through the same questions—what is fixed, what subgroup remains, which effects are checked, and what is remembered—rather than by treating a lower move count as a complete measure of usability. The cube resource notes collect contrasting organizations for that later comparison.

                        A worked replay and projected lower bound [frap-001B]

                          The fixed-frame model has a concrete replay on a three-layer puzzle. Work in the fixed-frame HTM convention of the complete cube account: a face quarter-turn, its inverse, or a half-turn costs one move, and words are read from left to right. Start at the solved state and apply the declared scramble

                          \[w=R\ U\ R'\ U'\ F\ R\ U\ R'\ U'\ F'.\]

                          The inverse word is

                          \[w^{-1}=F\ U\ R\ U'\ R'\ F'\ U\ R\ U'\ R'.\]

                          A sticker-level replay of the six fixed-frame face maps sends the solved state to the recorded scramble state and then returns to solved. The check represents each of the 54 pairs in how the cube models connect as a slot, applies the six quarter-turn permutations and their inverses, and enumerates the projected corner states breadth first through depth two. The token-by-token states and hashes are preserved in the replay receipt. Running the tracked replay checker with `node scripts/check-frap-b9b.js trees/frap-001B-replay.json` reproduces and compares the stored `replay_log`. The complete cube account supplies the move convention; the replay, hashes and 262-state count are the deterministic calculation reported here. The second replay is therefore a certificate for the move convention and the inverse, not merely an animation of a familiar sequence.

                          What the small search certifies

                            Project the scrambled state to its corner positions and orientations. Exhaustive projected search finds 262 distinct projected states at depths zero, one, and two, while the target is absent; a projected distance of three is therefore a lower bound for this state. The declared word has ten HTM moves, so the same replay gives the bounded report

                            \[3\le d_{\rm HTM}(w,\mathrm {id})\le 10.\]

                            The lower bound is admissible because projection forgets edge data and every full move projects to one projected move. The upper bound is a valid word because the inverse replay reaches the target. Neither bound claims that ten is shortest; a full optimality claim would need exhaustive search through depths four through nine or a matching certificate.

                            Relation to layer-first and computer methods

                              A player can recognize the displayed ten-move word as a repeated trigger pattern, but recognition is not a replacement for the state record. The complete layer-first method solves an arbitrary legal state by preserving completed layers and applying checked setup, trigger, and inverse-setup words; the computer account gives the corresponding upper-bound word and projected lower bound. This worked scramble shows how the two accounts meet: name the frame, replay the exact tokens, check the target, then state separately what has and has not been proved.

                              The example is deliberately bounded. It demonstrates a complete scramble-and-replay loop and a concrete comparison of certificates; it does not claim to be a full optimal solve, a universal human algorithm, or a replacement for the all-state completeness argument.

                              Computer solving and efficiency [frap-0018]

                                A computer solver answers a sharper question than a human method: from a recorded state, what is the shortest path to solved, or what short path can be found quickly? The answer depends on the move metric, the state coordinates, and the information retained by the search. We use the fixed-frame ordinary 3-layer action from the complete cube account, with the six outer face turns as generators. The methods chapter supplies the group-action graph viewpoints; here they become finite search procedures.

                                Move graphs and lower bounds

                                  For a move metric \(M\), let \(\Gamma _M\) be the undirected graph whose vertices are legal cube states and whose edges apply one allowed move, with each edge weighted by that move's cost in \(M\). A graph distance is the function

                                  \[d_M(x,y)=\text {the minimum total edge weight on a path from }x\text { to }y.\]

                                  The metric diameter is the maximum of \(d_M(x,y)\) over legal states; in a transitive move graph it is also the maximum distance from the solved state. For unit-cost moves, breadth-first search from solved labels every reachable vertex by exact distance, because the first time a vertex is reached there is no shorter path left in the earlier layers. With non-unit weights, Dijkstra's algorithm supplies the analogous exact labels. A depth-first method can use the same graph without storing all of it; optimal branch-and-bound and IDA* need a certified lower bound to prune safely.

                                  The usual half-turn metric (HTM) charges each face quarter-turn, inverse quarter-turn, or half-turn one move. The quarter-turn metric (QTM) charges a half-turn two moves. A word can therefore be shorter in HTM while representing the same group element; an efficiency claim is incomplete until its metric and move set are named.

                                  A simple counting bound already separates search from proof. In HTM there are at most eighteen first moves and at most seventeen choices after the first if an immediate inverse is suppressed. Hence a ball of radius \(r\) has at most

                                  \[1+18\sum _{i=0}^{r-1}17^i\]

                                  vertices. The reachable 3-layer state count from the complete cube account is

                                  \[N_3=8!\,12!\,3^7\,2^{10} =43\,252\,003\,274\,489\,856\,000.\]

                                  The corresponding upper estimate for a radius-15 ball is

                                  \[1+18\sum _{i=0}^{14}17^i =3\,220\,225\,932\,948\,542\,767<N_3.\]

                                  Thus not every state can lie within fifteen moves, so the HTM diameter is at least \(16\). This is only a counting lower bound: it does not exhibit a state at distance sixteen, and it says nothing about a particular scramble.

                                  One exact small search

                                    The same procedure can be made exhaustive on the corner-only cube. In a fixed-frame labelling (there are no centers on a \(2\times 2\times 2\)), a legal state is a corner permutation together with seven independent twists, so there are

                                    \[8!\,3^7=88\,179\,840\]

                                    labelled states. Quotienting by the twenty-four orientations of the whole cube leaves \(3\,674\,160\) rotation classes. Enumerate the six face quarter-turns, their inverses, and the six face half-turns; canonicalize each state by the whole-cube rotation action, and keep the first depth at which each rotation class is seen. The resulting layer sizes in HTM are

                                    \[1,9,54,321,1847,9992,50136,227536, 870072,1887748,623800,2644.\]

                                    They sum to \(3\,674\,160\); the last nonempty layer is \(11\). Thus this finite breadth-first computation proves that the rotation-quotient corner-only cube has diameter eleven in HTM. It is an exact search: every rotation class is covered and every recorded depth is a shortest distance. The calculation agrees with the published exhaustive layer table for the pocket cube, p. 3. The calculation also illustrates why the full 3-layer graph is not attacked by naïve BFS: its state count is larger by many orders of magnitude.

                                    Admissible heuristics and pattern databases

                                      A projection heuristic forgets part of a state. If \(\pi \) maps the full move graph to a smaller graph and every full move projects to a projected path of no greater total cost, then

                                      \[h_\pi (x)=d_{\pi }(\pi (x),\pi (\mathrm {id}))\]

                                      never overestimates the full distance: projecting a full solution gives a projected solution of no greater cost. For example, a table entry \(h_\pi (x)=2\) certifies that every full solution from \(x\) costs at least two. Such a function is an admissible heuristic. A maximum of admissible heuristics is admissible; a sum needs an additional cost-partition argument and is not automatically safe. The graph maps here are specified by these equations and move lists, so a separate vector diagram would add no state or cost data.

                                      A pattern database stores these projected distances in a table indexed by a selected set of cubies or coordinates. Reverse breadth-first search fills the table once; an iterative-deepening A* (IDA*) search then explores depth thresholds while pruning a node \(x\) whenever \(g+h_\pi (x)\) exceeds the threshold. Korf's Rubik's-cube computation used this combination of pattern databases and IDA* for optimal solutions to random instances the AAAI paper, abstract and sections “Problem Space” and “Pattern Databases”. The table certifies a lower bound, while the exhaustive threshold search is what certifies optimality.

                                      For the 3-layer coordinates, a corner-orientation table has \(3^7\) entries, an edge-orientation table has \(2^{11}\), and a chosen edge-position table can be indexed separately. Taking the maximum of their exact projected distances is safe and often much stronger than either table alone. Memory, lookup cost, and the number of expanded nodes are measurable resources; none of them changes the definition of the shortest distance.

                                      Subgroup reduction and two phases

                                        A different compression searches through a goal subgroup. Let

                                        \[G_1=\langle U,D,R^2,L^2,F^2,B^2\rangle .\]

                                        In this subgroup the corner and edge orientations are fixed, and the four middle-slice edges stay in their slice. A first phase searches the full generator set for a word taking the scramble into \(G_1\); a second phase solves inside \(G_1\) using only its generators. The phase-1 coordinates are corner orientation, edge orientation, and the choice of the four slice edges, with raw size

                                        \[3^7\,2^{11}\binom {12}{4}=2\,217\,093\,120.\]

                                        Pruning tables and symmetry reductions store only useful projections of this space.

                                        Kociemba's two-phase account describes IDA* over these coordinates and explicitly distinguishes a fast near-optimal solver from an optimal solver the Phase 1, Phase 2 and optimality paragraphs. A solution found by one phase-1 word followed by a phase-2 word is a valid upper bound. It becomes a proof of optimality only when every shorter possibility under the declared metric has been excluded. Restricting phase two to \(G_1\) makes the search practical, but can omit an overall shortest route that leaves and later re-enters \(G_1\).

                                        This reduction is the computer analogue of the preservation chain in the complete layer-first method: both deliberately fix information before solving what remains. The difference is the certificate. A human stage records a finite case argument; a search stage records a lower bound, an upper-bound word, and the unexplored threshold.

                                        Efficiency and growing cubes

                                          A useful report separates four measurements: solution length in a named metric, number of expanded nodes, peak table memory, and wall-clock time on named hardware. Recognition and move execution belong to the human method; node expansions and table construction belong to the computer method. A shorter returned word is evidence of a better upper bound, not by itself a proof that the word is shortest.

                                          For the growing \(N\times N\times N\) family, the asymptotic question is how the diameter changes with \(N\) when a move rotates one legal slice. Demaine, Demaine, Eisenstat, Lubiw and Winslow prove a scoped result for this family: the diameter is \(\Theta (N^2/\log N)\), with a counting lower bound and a parallelized upper-bound construction their abstract and Section 2, “Common Definitions”. Their model allows a legal slice rotation as defined there; that move convention must be checked before transferring the statement to a different layered-cube convention.

                                          The fixed-frame 3-layer count and the exact corner-only search give two finite anchors for that asymptotic discussion. The asymptotic theorem does not make a particular 3-layer word optimal, and the exact small search does not predict the table size or diameter of larger cubes. Across all sizes, correctness means reaching the target; optimality means proving a shortest path; efficiency means reporting the resources used to obtain either one.

                                          Computer solving therefore extends the earlier construction in three controlled directions: exact graph distances on a small quotient, admissible lower bounds for the full state space, and subgroup reductions that trade completeness of the search for speed. These distinctions let a later return to play report a solver's output without confusing a valid solve, a short solve, and a certified optimum.

                                          This chapter is a mathematical specification, not a Lean-certified solver. For comparison, rubik-lean4 at commit b8a5a09 formalizes the fixed-frame move and sticker correspondence, while rubiks-cube-group at commit 3baebd73 formalizes the legal-state subgroup; neither repository is used here as a formal proof of the search, diameter, or optimality claims above.

                                          Five encounters with a cube [frap-001C]

                                            Put a solved three-layer cube on a table and turn its right face once. The move is familiar, but it raises concrete puzzles: which labelled pieces moved, why can a visually plausible arrangement be impossible, and how can another person undo the turn without watching the first hand? Begin with these experiences, then introduce only the distinctions needed to explain them.

                                            One turn is larger than the visible face

                                              The shaded strip is only the visible part of the layer: the same stickers cross the front, right, back, top, and bottom faces; viewed from above, the top-face part is the corresponding rightmost strip. A whole-cube rotation would change the viewing frame, whereas this face move keeps the six centres fixed and changes one layer. The layered-cube family makes this distinction for every \(N\); the complete three-layer cube spells it out for the six outer generators.

                                              A sticker needs a home and a frame

                                                A player sees colours; a proof must also track each sticker's cubie and transported local frame. How the cube models connect follows one sticker through incidence position, cubie fibre, and fixed-frame coordinate. A colour quotient forgets labels, while a larger cube adds positions, so the state space changes with the question.

                                                A plausible state can still be impossible

                                                  Take a solved cube and twist one corner in place. Every sticker still has a cubie-sized home, and the three colours at that corner look locally consistent. But the corner orientations now have sum \(1\) modulo \(3\), whereas every face turn preserves sum \(0\). No sequence of legal turns can produce this state.

                                                  The complete three-layer cube derives the matching edge and permutation restrictions from the six generators. The calculation is useful at the table: it tells a solver when a scrambled-looking object was assembled illegally, and it tells a proof which state space is worth searching.

                                                  A human algorithm has to preserve these legal states while it borrows pieces from completed areas.

                                                  A formula often disturbs pieces that were already placed. The disturbance is acceptable only when the setup is undone and the completed pieces return. A working slot, a trigger, and an inverse setup make that promise visible. The complete layer-first method repeats this pattern through the cross, the first layer, the middle layer, and the last layer. Its completeness depends on the preservation and termination checks, not on the formula's familiarity.

                                                  Undoing a scramble checks the notation

                                                    The worked replay and projected lower bound fixes the viewing convention, records the ten tokens, and applies their inverse. Sticker replay returns to solved. A corner-only projected search finds no solution of depth at most two, so the evidence is the bounded statement

                                                    \[3\le d_{\rm HTM}\le 10.\]

                                                    The inverse word is an upper-bound witness; the projected search is a lower-bound witness. The computer account explains why these checks do not prove that ten is shortest. A state, a word, and a replay log are enough to test the notation before asking for a general solver.

                                                    Changing one ingredient changes the problem

                                                      Once the move, its frame, its local repair, and its replay are explicit, three variations separate the effects of enlarging the cube, changing the shape, and forgetting labels.

                                                      The layered-cube family develops the first variation; the incidence and facet construction in incidence, symmetry and moves develops the second; and which moves survive a quotient? gives the descent condition for the third. Framed orientations, label transport and topology, and scrambling, spectra and search then study the extra information each variation introduces or forgets. Their hypotheses stay attached to the concrete move that prompted the question.

                                                      A face turn can therefore be drawn, named, replayed, and checked before it is generalized. The resulting state maps and invariants support later formal certification or measured comparisons, while the examples here make clear which claim each later programme would have to establish.

                                                      How the cube models connect [frap-001A]

                                                        An abstract cube turn records incidence positions and restricts a shape symmetry to one facet. A layered-cube turn records cubies, stickers, frames and allowed layers. For the ordinary three-layer cube these are two descriptions of the same action; the map below identifies their coordinates.

                                                        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.

                                                          What changes when layers are added

                                                            For larger \(N\), the same surface-cell description retains the corner and edge fibers but adds coupled-edge and center orbits. Inner layers move those new fibers, and even and odd sizes have different orbit signatures. The all-\(N\) coordinates in the layered-cube law therefore extend the three-layer state description rather than replace it. Their parity and orientation equations are stated for a labelled fixed-frame model; they are not obtained by treating every center sticker as one of the twelve ordinary edges.

                                                            The transition is consequently a specialization in one direction: the \(N=3\) outer-layer case of the family of layered cubes gives the ordinary fixed-frame action, while the abstract facet action explains why that outer turn moves precisely the selected surface fibers. Whole-cube rotations, marked centers, central slices and colour-only identifications are additional choices, not silent changes of coordinates.

                                                            What the identification supports

                                                              The explicit identification supports this chain of state descriptions:

                                                              1. incidence, flags and a local facet action explain what a move is;
                                                              2. the layered model and its first law explain which fibres and constraints appear as \(N\) changes;
                                                              3. the fixed-frame three-layer coordinates make the three familiar constraints and their constructive sufficiency explicit;
                                                              4. human methods and computer searches use the same state action while assigning different meanings to preservation, cost and optimality;
                                                              5. a declared move word can be replayed and checked against these state descriptions.

                                                              Three linked comparisons use the same distinctions. Label transport and branched unfolding study return maps on simplex labels; compatible quotients test whether an identification of sticker observations inherits a move; and scrambling and spectra study which functions retain memory under a declared move law. These are comparisons of transport, forgetting and randomization, not identifications with the fixed-frame cube action.

                                                              How the notes are organized [frap-000C]

                                                                The developed mathematics follows one practical question: what data must a move carry so that we can later compose moves and test reachability? Incidence, symmetry and moves answers it in order. First, polytopes, flags and facet symmetry describe the shape and show their role in triangle and tetrahedron examples. Next, facet moves turn a local facet rotation into an action on incidence positions, with tetrahedron and cube calculations that make the construction concrete. Only then do positions, orientations and global constraints separate a piece's destination from transport in its local frame, while transfer and the effect of dimension connect that transport with a group homomorphism and compute the relevant local abelianizations.

                                                                The cube examples above connect the general mathematics with solving and play. The later accounts use those concrete states and moves as their starting point:

                                                                The transition between the abstract facet action and the fixed-frame cube coordinates is recorded in how the cube models connect. Read it before the all-\(N\) law if the distinction between incidence positions, cubie fibres and observations is new.

                                                                1. General Rubik puzzles: the incidence actions, framed orientations and transfer developed below.
                                                                2. The layered-cube family: piece types, layer turns and conventions as \(N\) varies.
                                                                3. The layered-cube law: all-\(N\) state coordinates and a constructive characterization of reachability.
                                                                4. The complete three-layer cube: explicit outer generators, impossible states, the standard count and a worked cleanup.
                                                                5. Human methods and computer search: recognizable cases, algebraic effects, usability, move costs and optimality. The complete layer-first method supplies one full human construction, and computer solving and efficiency supplies exact small search, admissible bounds and subgroup reduction.
                                                                6. From move sequences to play: notation, scrambling, recognition, replay and complete solves, including the worked replay and projected lower bound.

                                                                Further connections relate the developed orientation and transfer calculations to label transport and topology and scrambling and search functions. The quotient-compatible move account gives the criterion for an identification of observations to inherit a puzzle action. Together these results compare transport, randomization and information loss around the same concrete move.

                                                                Five encounters with a cube [frap-001C]

                                                                  Put a solved three-layer cube on a table and turn its right face once. The move is familiar, but it raises concrete puzzles: which labelled pieces moved, why can a visually plausible arrangement be impossible, and how can another person undo the turn without watching the first hand? Begin with these experiences, then introduce only the distinctions needed to explain them.

                                                                  One turn is larger than the visible face

                                                                    The shaded strip is only the visible part of the layer: the same stickers cross the front, right, back, top, and bottom faces; viewed from above, the top-face part is the corresponding rightmost strip. A whole-cube rotation would change the viewing frame, whereas this face move keeps the six centres fixed and changes one layer. The layered-cube family makes this distinction for every \(N\); the complete three-layer cube spells it out for the six outer generators.

                                                                    A sticker needs a home and a frame

                                                                      A player sees colours; a proof must also track each sticker's cubie and transported local frame. How the cube models connect follows one sticker through incidence position, cubie fibre, and fixed-frame coordinate. A colour quotient forgets labels, while a larger cube adds positions, so the state space changes with the question.

                                                                      A plausible state can still be impossible

                                                                        Take a solved cube and twist one corner in place. Every sticker still has a cubie-sized home, and the three colours at that corner look locally consistent. But the corner orientations now have sum \(1\) modulo \(3\), whereas every face turn preserves sum \(0\). No sequence of legal turns can produce this state.

                                                                        The complete three-layer cube derives the matching edge and permutation restrictions from the six generators. The calculation is useful at the table: it tells a solver when a scrambled-looking object was assembled illegally, and it tells a proof which state space is worth searching.

                                                                        A human algorithm has to preserve these legal states while it borrows pieces from completed areas.

                                                                        A formula often disturbs pieces that were already placed. The disturbance is acceptable only when the setup is undone and the completed pieces return. A working slot, a trigger, and an inverse setup make that promise visible. The complete layer-first method repeats this pattern through the cross, the first layer, the middle layer, and the last layer. Its completeness depends on the preservation and termination checks, not on the formula's familiarity.

                                                                        Undoing a scramble checks the notation

                                                                          The worked replay and projected lower bound fixes the viewing convention, records the ten tokens, and applies their inverse. Sticker replay returns to solved. A corner-only projected search finds no solution of depth at most two, so the evidence is the bounded statement

                                                                          \[3\le d_{\rm HTM}\le 10.\]

                                                                          The inverse word is an upper-bound witness; the projected search is a lower-bound witness. The computer account explains why these checks do not prove that ten is shortest. A state, a word, and a replay log are enough to test the notation before asking for a general solver.

                                                                          Changing one ingredient changes the problem

                                                                            Once the move, its frame, its local repair, and its replay are explicit, three variations separate the effects of enlarging the cube, changing the shape, and forgetting labels.

                                                                            The layered-cube family develops the first variation; the incidence and facet construction in incidence, symmetry and moves develops the second; and which moves survive a quotient? gives the descent condition for the third. Framed orientations, label transport and topology, and scrambling, spectra and search then study the extra information each variation introduces or forgets. Their hypotheses stay attached to the concrete move that prompted the question.

                                                                            A face turn can therefore be drawn, named, replayed, and checked before it is generalized. The resulting state maps and invariants support later formal certification or measured comparisons, while the examples here make clear which claim each later programme would have to establish.

                                                                            Scrambling, spectra and search coordinates [frap-0015]

                                                                              Random moves raise a different question from solving: when, and how quickly, does a scramble lose information about its starting state? A random walk depends on its allowed moves and their probabilities. We use the fixed-frame three-layer move group from the complete cube account, and write \(G\) for its reachable state group. A walk with step law \(\mu \) has distribution

                                                                              \[\Pr [X_k=g]=\mu ^{*k}(g)\]

                                                                              from the solved state, where convolution records the successive moves.

                                                                              A parity calculation before mixing claims

                                                                                Let \(\varepsilon :G\to \{+1,-1\}\) be the sign of the corner permutation. Each face quarter-turn or inverse is a four-cycle on corners, so \(\varepsilon =-1\); each face half-turn is a product of two transpositions, so \(\varepsilon =+1\). If the walk chooses uniformly from the twelve quarter-turns and inverse quarter-turns, then

                                                                                \[\varepsilon (X_k)=(-1)^k\]

                                                                                with probability one. Its support is therefore contained in one parity coset at every fixed time, while the uniform distribution on \(G\) gives equal mass to the two cosets. The total-variation distance from uniform is at least \(1/2\) for every \(k\). This walk cannot converge to uniform: the obstruction is periodicity, not a slow estimate of a mixing time.

                                                                                The usual eighteen-move HTM law gives a different exact observable. Six half-turns have sign \(+1\) and twelve quarter-turns or inverses have sign \(-1\), hence one step has

                                                                                \[\mathbb E[\varepsilon (X_1)]=\frac 6{18}-\frac {12}{18}=-\frac 13.\]

                                                                                Independence gives \(\mathbb E[\varepsilon (X_k)]=(-1/3)^k\). Since uniform parity has expectation zero, the parity marginal alone gives the lower bound

                                                                                \[\lVert \mathcal L(X_k)-\mathcal U_G\rVert _{\mathrm {TV}}\geq \frac 12\left (\frac 13\right )^k.\]

                                                                                This bound decays and does not prove convergence by itself. A lazy walk, which includes the identity with positive probability, removes this specific periodicity; a convergence statement still needs a generated reachable component and an aperiodicity argument.

                                                                                Averaging and finite-group spectra

                                                                                  For a function \(f:G\to \mathbb C\), the one-step averaging operator is

                                                                                  \[(T_\mu f)(g)=\sum _{s\in G}\mu (s)f(gs).\]

                                                                                  Constants are fixed by \(T_\mu \). For a finite-dimensional unitary representation \(\rho \), its matrix coefficients

                                                                                  \[f_{u,v}(g)=\langle u,\rho (g)v\rangle \]

                                                                                  form invariant blocks for this operator: averaging a coefficient replaces \(\rho (g)\) by multiplication by the Fourier block

                                                                                  \[\widehat \mu (\rho )=\sum _{s\in G}\mu (s)\rho (s).\]

                                                                                  Thus the nonconstant spectrum describes which observables retain memory. If the step support generates the reachable group and every nontrivial block contracts on the relevant quotient, then these blocks decay and the walk converges to uniform on that component. The hypotheses matter: the quarter-turn walk above fails the aperiodicity test, so no spectral slogan can erase its parity obstruction.

                                                                                  The same calculation explains why a prescribed scramble length is not a universal guarantee of randomness. A distance bound answers how far a state can be from solved; a distributional claim asks how the whole law projects onto every persistent observable. These are different certificates.

                                                                                  Fourier coordinates for learned search

                                                                                    Pan and Kondor use irreducible-representation matrix coefficients as a Fourier basis for learned value functions on permutation puzzles [pan2021fourier, Sections 4--4.1]. Their experiments compare learned scores that choose moves with conventional heuristic and value-based search; the paper explicitly discusses the distinction from an admissible lower bound [pan2021fourier, Section 6]. A learned score can rank promising moves while overestimating or underestimating the remaining distance. It is therefore evidence about search guidance, not a shortest-path certificate.

                                                                                    For the cube, a safe lower bound can still be obtained from the exact projected distances in a pattern database as in computer solving and efficiency. A Fourier score may be combined with that bound as a tie-breaker or move-ordering device, but replacing the bound with the score changes the guarantee. The representation coordinates explain how a search function is organized; they do not, without an error theorem, make it admissible.

                                                                                    The displayed parity bounds and finite-operator framework are the results established here. They do not claim a full-cube mixing time, a universal scramble length, or a reproduced machine-learning benchmark. Those would require a declared move law, reachable component, statistical or spectral analysis, and independently checked computational evidence.

                                                                                    The first law for the layered cube [frap-0004]

                                                                                      The layered model needs a reachability test. A tuple of permutations and orientations can describe a dismantled cube without being obtainable by legal slice turns. We now choose the labelled, fixed-frame state space from the layered-cube model and state the first law for that space. The theorem is a statement about labelled cubies; an ordinary colour pattern is its observation after equal-looking labels have been forgotten.

                                                                                      Coordinates for one state space

                                                                                        Let \(K_o=(N-3)/2\) for odd \(N\) and \(K_e=N/2-1\) for even \(N\). For every cube there are a corner permutation \(\sigma \in S_8\) and corner orientation vector \(x\in (\mathbb Z_3)^8\). For odd \(N\), there is also a single-edge permutation \(\tau _s\in S_{12}\) and flip vector \(z\in (\mathbb Z_2)^{12}\). Each coupled-edge orbit indexed by \(i\in I\) has a permutation \(\tau _i\in S_{24}\) and a labelled orientation vector \(y_i\in (\mathbb Z_2)^{24}\). Each center-corner orbit has a permutation \(\rho _i\in S_{24}\). Center-edge orbits are indexed by a finite set \(J\); write \(\rho _{ij}\) for the orbit whose two distance signatures are \(i,j\). In the fixed-center convention, \(I=\{1,\ldots ,K_o\}\) and \(J=I^2\) when \(N\) is odd, while \(I=\{1,\ldots ,K_e\}\) and \(J=\{(i,j)\in I^2:i\ne j\}\) when \(N\) is even. This indexing records which coupled-edge signatures border a center-edge orbit; it is the coordinate version of the rings in the preceding chapter.

                                                                                        The full labelled configuration space is the product of these factors, with the solved state represented by identity permutations and zero orientation vectors. It is deliberately larger than the move group. For example, a vector \(y_i\) records all sticker orientations allowed by a labelled reassembly, even though the legal slice action fixes the relevant coupled-edge orientation in its orbit. Whole-cube rotations are still outside the generator set, and the six face directions remain part of the frame.

                                                                                        The invariant law

                                                                                          For odd \(N\), a configuration is legal exactly when

                                                                                          \[\begin {aligned} \operatorname {sgn}(\sigma )&=\operatorname {sgn}(\tau _s)= \operatorname {sgn}(\rho _i) &&(i\in I),\\ \operatorname {sgn}(\rho _{ij})&=\operatorname {sgn}(\sigma ) \operatorname {sgn}(\tau _i)\operatorname {sgn}(\tau _j) &&((i,j)\in J),\\ \sum _r x_r&=0\pmod 3,\qquad \sum _r z_r=0\pmod 2,\\ y_i&=0 &&(i\in I). \end {aligned}\]

                                                                                          For even \(N\), the corresponding law is

                                                                                          \[\begin {aligned} \operatorname {sgn}(\sigma )&=\operatorname {sgn}(\rho _i) &&(i\in I),\\ \operatorname {sgn}(\rho _{ij})&=\operatorname {sgn}(\sigma ) \operatorname {sgn}(\tau _i)\operatorname {sgn}(\tau _j) &&((i,j)\in J),\\ \sum _r x_r&=0\pmod 3,\\ y_i&=0 &&(i\in I). \end {aligned}\]

                                                                                          The empty-index cases are part of the statement: \(N=2\) has only corners, \(N=3\) has the usual twelve single edges and fixed centers, and \(N=4\) has one coupled-edge and one center-corner orbit but no center-edge orbit. The \(N=5\) instance has one orbit of each coupled-edge, center-corner and center-edge type, giving the 98 visible cubies counted earlier.

                                                                                          Necessity follows by checking generators. An outer face turn is a 4-cycle on corners, single edges, center corners and each relevant center-edge orbit, and two 4-cycles on each affected coupled-edge orbit. An inner slice turn is the identity on corners and single edges, and is a 4-cycle on its coupled edges and on the matching center-edge orbits, with two 4-cycles on center corners. These cycle signs give the two parity rows. The corner character changes by two opposite units on a non-\(U\)/\(D\) face and not at all on a \(U\)/\(D\) face, so its sum is fixed modulo 3. An outer turn flips four single edges, so their sum is fixed modulo 2. The chosen coupled-edge sticker convention makes every legal \(y_i\) zero. The solved state satisfies the rows, hence every word in the generators does.

                                                                                          Bonzio--Loi--Peruzzi give the labelled Professor's-Cube version of the same tuple and its corrected coupled-edge condition [bonzio2018nxnxn, Theorem 5]. Salkinder states the odd and even families and checks these generator effects in [salkinder2021nxnxn, Theorems 8--9]. Their notation indexes center-edge orbits by the two adjacent coupled-edge signatures; our \(J\) makes that indexing explicit. The sources also use a shifted notation for a middle slice when constructing a single-edge 3-cycle. That auxiliary word is a composition of parallel slice turns in the fixed-center model; it does not add a central generator to the law above.

                                                                                          Why the conditions are sufficient

                                                                                            The converse uses localized effects rather than a counting guess. First use a face turn, or an internal slice turn, to correct the common parity signs. The sign equations ensure that after this step every permutation factor is even. Commutators and conjugates then supply the required local subgroups:

                                                                                            • a corner 3-cycle, for example \([[R:U],D]\), generates \(A_8\) on corners;
                                                                                            • the single-edge 3-cycle \([F,[R:S]]\) generates \(A_{12}\) in the odd case, with the middle-slice symbol interpreted by the parallel-slice composition just described;
                                                                                            • \([[R_j^{-1},D_i],F^{-1}]\) gives 3-cycles on a center-corner orbit when \(i=j\) and on a center-edge orbit when \(i\ne j\);
                                                                                            • the analogous coupled-edge commutator gives \(A_{24}\) on each coupled-edge orbit. In even size the source's explicit odd permutation on that orbit upgrades it to \(S_{24}\); in odd size the parity row keeps it at \(A_{24}\).

                                                                                            Conjugating these words moves each 3-cycle through its whole orbit. Thus the local subgroups clear the corner, single-edge, center-corner, center-edge and coupled-edge permutations in turn. Finally, commutators that twist two corners in opposite directions generate the kernel of the corner-sum map \((\mathbb Z_3)^8\to \mathbb Z_3\); the corresponding edge-flip commutators generate the kernel of the single-edge sum. The coupled-edge orientation vector is already zero by its law. The parity and orientation equations are exactly what is needed for each cleanup step, so reversing the resulting word reaches every tuple satisfying the displayed conditions. This is the constructive sufficiency argument of [salkinder2021nxnxn, Theorems 10--18]; it is a subgroup generation argument, not an assertion that one universal short algorithm solves every size.

                                                                                            There are two boundaries to keep visible. The theorem concerns labelled positions and a fixed frame; a colour-only observation quotients by equal labels and may identify several legal labelled states. Also, Bonzio's and Salkinder's orbit labels are source conventions, so a numerical state count must be transported through the index sets before comparison. The next three-layer chapter specializes the same action to \(N=3\), where all center and coupled-edge factors disappear and the three familiar constraints can be worked through with explicit face generators.

                                                                                            Which moves survive a quotient? [frap-0019]

                                                                                              A quotient of observations is useful only when the moves still have a well-defined action on the observations. Let \(X\) be a declared sticker-state set, let \(G\) act on \(X\), and let the surjection \(q:X\twoheadrightarrow Y\) identify the observations we want to retain (equivalently, take \(Y=q(X)\)). The move \(g\in G\) preserves the fibers of \(q\) when

                                                                                              \[q(x)=q(x')\Longrightarrow q(gx)=q(gx')\qquad (x,x'\in X).\]

                                                                                              Exactly in this case there is a unique induced map \(\bar g:Y\to Y\) with

                                                                                              \[q\circ g=\bar g\circ q.\]

                                                                                              If \(g\) is a bijection and \(X\) is finite, then \(\bar g\) is surjective and therefore bijective. In an infinite setting, require the inverse move to preserve fibers as well. Thus, for the finite cube action below, the fiber-preserving bijections form a subgroup; a quotient does not automatically inherit the whole puzzle action.

                                                                                              The antipodal sticker quotient

                                                                                                For the ordinary cube, use the surface-cell and frame convention in how the cube models connect and the complete three-layer cube. Represent a sticker by a pair \((p,n)\), where \(p\in \{-1,0,1\}^3\) satisfies \(\lVert p\rVert _\infty =1\) and \(n=\operatorname {sgn}(p_i)e_i\) for each nonzero coordinate \(p_i\). Thus corners, edges and face centers contribute \(8\cdot 3\), \(12\cdot 2\) and \(6\cdot 1\) pairs, respectively, for 54 stickers. The antipodal identification is the involution

                                                                                                \[a(p,n)=(-p,-n),\qquad q(p,n)=\{(p,n),a(p,n)\}.\]

                                                                                                Its sticker fibres pair opposite stickers. This is a quotient of the labelled sticker action; it is not the same as forgetting equal colors or quotienting by whole-cube rotations.

                                                                                                Why one face turn fails

                                                                                                  The positive-\(z\) face turn uses

                                                                                                  \[u(x,y,z)=(y,-x,z)\]

                                                                                                  on every sticker whose cubie has \(z=1\), rotating both its position and its normal, and fixes the other cubies. Take a top sticker and its antipode, for example

                                                                                                  \[x=((1,1,1),e_z),\qquad a(x)=((-1,-1,-1),-e_z).\]

                                                                                                  The turn sends \(x\) to the sticker at \(((1,-1,1),e_z)\), while it fixes \(a(x)\). These outputs are not an antipodal pair, so \(q(ux)\ne q(u\,a(x))\). The single \(U\) move therefore does not descend to the antipodal quotient.

                                                                                                  A synchronized opposite-layer move

                                                                                                    The clockwise turn of the negative-\(z\) face, viewed from that face, is the inverse global rotation on \(z=-1\). Write it as \(d\); its inverse \(d^{-1}\) uses the same global rotation as \(u\). The composite \(u d^{-1}\) rotates both opposite layers by the same linear map and fixes the middle cubies. It commutes with the antipodal involution:

                                                                                                    \[a\,(u d^{-1})=(u d^{-1})\,a.\]

                                                                                                    Consequently it preserves every antipodal fiber and induces a bijection on \(Y\). A direct check of all 54 stickers gives the same result: \(U\) fails on the displayed pair, while the synchronized \(U D'\) move maps every antipodal pair to an antipodal pair. The surviving action is therefore an action of the fiber-preserving subgroup, not of the six ordinary face turns individually.

                                                                                                    What the cover does not provide

                                                                                                      A covering or quotient of the underlying shape supplies a map of shape data. It does not by itself supply a map between sticker-state spaces, a chosen fiber relation, or an equivariance proof for each legal move. Those data are separate: the explicit cube action records which stickers a face turn moves, while the quotient records which states are identified. A move descends only after the displayed fiber test has been checked.

                                                                                                      This is the precise limit of the comparison with the complete unfolding in label transport and topology. Projectivity and branched unfolding describe transport of simplex labels; the antipodal example describes descent of a particular sticker action. They pose analogous questions about information lost by identification, but neither construction is being identified with the other. A general lifting or descent classification is outside this note.

                                                                                                      The practical rule is short: state the state space, the quotient map and the move action before saying that a move survives identification. The antipodal calculation supplies both a negative and a positive finite example, while preserving the distinction between a shape cover and a compatible puzzle action.