Five encounters with a cube [frap-001C]
✍️source

    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.

    1. 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.

      2. A sticker needs a home and a frame

        Definition. Cubies, sticker states, and fixed local frames [frap-001D]

          A description with a fixed global viewing frame needs three pieces of vocabulary. A cubie is one physical unit of the layered cube, with a home position and the visible stickers incident to that position. For a three-layer cube these are the eight corners, twelve edges and six centres described in how the cube models connect. Bonzio, Loi and Peruzzi use this cubie decomposition in their general model [bonzio2018nxnxn, Sections 2-3]; the visible-sticker formulation in this fixed global viewing frame is our adaptation to the present cube convention.

          For \(N=3\), let \(\mathcal S\) be the 54 labelled visible stickers and let \(\mathcal P=\{(p,n):p\in \{-1,0,1\}^3,\ \lVert p\rVert _\infty =1,\ n \text { is an incident outward normal at }p\}\). Thus \(\lvert \mathcal P\rvert =54\). A sticker state is a bijection \(s:\mathcal S\xrightarrow {\sim }\mathcal P\): every valid surface slot has exactly one occupant. The solved assignment \(s_0\) is the reference state, and \(s_0^{-1}\mathbin {\circ }s:\mathcal S\to \mathcal S\) records the state as a permutation of sticker labels, while \(s\mathbin {\circ }s_0^{-1}:\mathcal P\to \mathcal P\) records the same state as a permutation of the 54 slots. A move \(m:\mathcal P\to \mathcal P\) acts on the state by \(m\mathbin {\circ }s:\mathcal S\to \mathcal P\), so it left-composes the slot permutation by \(m\) and left-composes the sticker-label permutation by the transported move \(s_0^{-1}\mathbin {\circ }m\mathbin {\circ }s_0\). Marchetti's position-and-label transport gives the structural pattern [marchetti2025rubik, Section 3.1, Definition 7 and equation (5)]; the present 54-slot bijection is our adaptation of that pattern to the three-layer cube. The correspondence between these slots, cubie fibres, and the coordinates \((\sigma ,x;\tau ,z)\) is made explicit in how the cube models connect.

          A fixed local frame specializes the local frame: it is a chosen reference ordering of the incident sticker directions at each cubie. A move records both the cubie's destination and how its labels are transported relative to that ordering. The six centre directions keep the global viewing frame fixed; a whole-cube rotation would change that frame and is a separate symmetry. These definitions distinguish the sticker state from a colour observation, which forgets labels that look alike.

          A complete cube description tracks each sticker's cubie and transported fixed local frame. How the cube models connect follows one sticker through incidence position, cubie fibre, and fixed-local-frame coordinate. A colour quotient forgets labels, while a larger cube adds positions, so the state space changes with the question.

          3. A plausible state can still be impossible

            Take a solved cube and twist one corner in place. The resulting sticker state still puts every sticker in 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.

            4. 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.

              5. 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.

                References

                  Reference. Rubik's Abstract Polytopes [marchetti2025rubik]

                  @misc{marchetti2025rubik,
 title = {Rubik's Abstract Polytopes},
 author = {Marchetti, Giovanni Luca},
 year = {2025},
 archiveprefix = {arXiv},
 eprint = {2502.13518},
 primaryclass = {math.CO},
 url = {https://arxiv.org/abs/2502.13518v1}
}

                  Reference. On the nxnxn rubik’s cube [bonzio2018nxnxn]

                  @article{bonzio2018nxnxn,
 title = {On the nxnxn Rubik's Cube},
 author = {Bonzio, Stefano and Loi, Andrea and Peruzzi, Luisa},
 year = {2018},
 journal = {Mathematica Slovaca},
 volume = {68},
 number = {5},
 pages = {957--974},
 publisher = {De Gruyter}
}

                  Context

                    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 longer motivation and formalization scope live in why these notes connect polytopes and puzzles; 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 source-backed sufficiency sketch for 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]
                      ✍️source

                        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

                            Definition. Cubies, sticker states, and fixed local frames [frap-001D]

                              A description with a fixed global viewing frame needs three pieces of vocabulary. A cubie is one physical unit of the layered cube, with a home position and the visible stickers incident to that position. For a three-layer cube these are the eight corners, twelve edges and six centres described in how the cube models connect. Bonzio, Loi and Peruzzi use this cubie decomposition in their general model [bonzio2018nxnxn, Sections 2-3]; the visible-sticker formulation in this fixed global viewing frame is our adaptation to the present cube convention.

                              For \(N=3\), let \(\mathcal S\) be the 54 labelled visible stickers and let \(\mathcal P=\{(p,n):p\in \{-1,0,1\}^3,\ \lVert p\rVert _\infty =1,\ n \text { is an incident outward normal at }p\}\). Thus \(\lvert \mathcal P\rvert =54\). A sticker state is a bijection \(s:\mathcal S\xrightarrow {\sim }\mathcal P\): every valid surface slot has exactly one occupant. The solved assignment \(s_0\) is the reference state, and \(s_0^{-1}\mathbin {\circ }s:\mathcal S\to \mathcal S\) records the state as a permutation of sticker labels, while \(s\mathbin {\circ }s_0^{-1}:\mathcal P\to \mathcal P\) records the same state as a permutation of the 54 slots. A move \(m:\mathcal P\to \mathcal P\) acts on the state by \(m\mathbin {\circ }s:\mathcal S\to \mathcal P\), so it left-composes the slot permutation by \(m\) and left-composes the sticker-label permutation by the transported move \(s_0^{-1}\mathbin {\circ }m\mathbin {\circ }s_0\). Marchetti's position-and-label transport gives the structural pattern [marchetti2025rubik, Section 3.1, Definition 7 and equation (5)]; the present 54-slot bijection is our adaptation of that pattern to the three-layer cube. The correspondence between these slots, cubie fibres, and the coordinates \((\sigma ,x;\tau ,z)\) is made explicit in how the cube models connect.

                              A fixed local frame specializes the local frame: it is a chosen reference ordering of the incident sticker directions at each cubie. A move records both the cubie's destination and how its labels are transported relative to that ordering. The six centre directions keep the global viewing frame fixed; a whole-cube rotation would change that frame and is a separate symmetry. These definitions distinguish the sticker state from a colour observation, which forgets labels that look alike.

                              A complete cube description tracks each sticker's cubie and transported fixed local frame. How the cube models connect follows one sticker through incidence position, cubie fibre, and fixed-local-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. The resulting sticker state still puts every sticker in 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.

                                    Related

                                      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 global viewing frame fixed and use its fixed local frame conventions. We 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.

                                                                  Definition. Cubies, sticker states, and fixed local frames [frap-001D]

                                                                    A description with a fixed global viewing frame needs three pieces of vocabulary. A cubie is one physical unit of the layered cube, with a home position and the visible stickers incident to that position. For a three-layer cube these are the eight corners, twelve edges and six centres described in how the cube models connect. Bonzio, Loi and Peruzzi use this cubie decomposition in their general model [bonzio2018nxnxn, Sections 2-3]; the visible-sticker formulation in this fixed global viewing frame is our adaptation to the present cube convention.

                                                                    For \(N=3\), let \(\mathcal S\) be the 54 labelled visible stickers and let \(\mathcal P=\{(p,n):p\in \{-1,0,1\}^3,\ \lVert p\rVert _\infty =1,\ n \text { is an incident outward normal at }p\}\). Thus \(\lvert \mathcal P\rvert =54\). A sticker state is a bijection \(s:\mathcal S\xrightarrow {\sim }\mathcal P\): every valid surface slot has exactly one occupant. The solved assignment \(s_0\) is the reference state, and \(s_0^{-1}\mathbin {\circ }s:\mathcal S\to \mathcal S\) records the state as a permutation of sticker labels, while \(s\mathbin {\circ }s_0^{-1}:\mathcal P\to \mathcal P\) records the same state as a permutation of the 54 slots. A move \(m:\mathcal P\to \mathcal P\) acts on the state by \(m\mathbin {\circ }s:\mathcal S\to \mathcal P\), so it left-composes the slot permutation by \(m\) and left-composes the sticker-label permutation by the transported move \(s_0^{-1}\mathbin {\circ }m\mathbin {\circ }s_0\). Marchetti's position-and-label transport gives the structural pattern [marchetti2025rubik, Section 3.1, Definition 7 and equation (5)]; the present 54-slot bijection is our adaptation of that pattern to the three-layer cube. The correspondence between these slots, cubie fibres, and the coordinates \((\sigma ,x;\tau ,z)\) is made explicit in how the cube models connect.

                                                                    A fixed local frame specializes the local frame: it is a chosen reference ordering of the incident sticker directions at each cubie. A move records both the cubie's destination and how its labels are transported relative to that ordering. The six centre directions keep the global viewing frame fixed; a whole-cube rotation would change that frame and is a separate symmetry. These definitions distinguish the sticker state from a colour observation, which forgets labels that look alike.

                                                                    From incidence to moves [frap-0002]

                                                                      To describe a puzzle move without committing to coordinates, we need an incidence record that says which pieces and labels belong together. Faces and incidence define an abstract polytope without choosing lengths or an embedding [schulte2017regular, Section 2]. Regularity supplies shape symmetries; the Rubik construction applies facet rotations locally to incidence positions [marchetti2025rubik, Definitions 5--6]. The sections below first make the shape visible in triangle and tetrahedron examples, then build this action and calculate its tetrahedron and cube consequences.

                                                                      Faces, flags, and the symmetry of a shape [frap-0008]

                                                                        A local turn needs a shape-level answer to two questions: which faces are incident, and which symmetries preserve those incidences? An abstract polytope retains the incidence of vertices, edges and faces without prescribing distances or angles. Schulte's flag formulation [schulte2017regular, Section 2] leads from these incidences to symmetries and to the extension of a facet symmetry used in local puzzle moves.

                                                                        Definition. Abstract polytopes and flags [schulte2017regular, Section 2] [frap-000D]

                                                                          An abstract \(n\)-polytope, for \(n\geq 0\), is a partially ordered set \(P\) of faces with a least face \(F_{-1}\) and greatest face \(F_n\). Every chain extends to a maximal chain, called a flag, with exactly \(n+2\) faces. Faces in each flag have ranks \(-1,0,\ldots ,n\), consistently across flags. Faces other than the least and greatest are proper; vertices, edges and facets have ranks \(0,1,n-1\).

                                                                          The diamond condition requires exactly two faces of rank \(i\) between faces \(F<G\) of ranks \(i-1\) and \(i+1\), for \(0\leq i<n\). Two flags are adjacent if they differ in exactly one proper face. Strong flag connectivity requires any two flags to be joined by adjacent flags, each containing their intersection.

                                                                          Thus, for each \(0\leq i<n\), each flag has a unique adjacent flag differing in rank \(i\). Strong flag connectivity restricts paths to these adjacent flags, each containing the endpoint flags' intersection; arbitrary paths through the least face would connect every pair without imposing an axiom. These conditions prescribe incidence without choosing lengths or angles.

                                                                          A rank-zero polytope is a two-element chain. A rank-one polytope has two vertices between its least and greatest faces. These low-rank cases occur as sections of higher-rank polytopes.

                                                                          Definition. Sections and vertex figures [schulte2017regular, Section 2] [frap-000E]

                                                                            Let \(P\) be a abstract polytope. For faces \(F\leq G\) of ranks \(i,j\), its section \(G/F\) is the interval \(\{H:F\leq H\leq G\}\), with ranks shifted so that \(F\) has rank \(-1\). For \(F<G\), it is an abstract polytope of rank \(j-i-1\geq 0\). For \(F=G\), the singleton section is called the rank-\(-1\) polytope. These low ranks allow all intervals, including vertex figures of polygons and intervals between consecutive faces.

                                                                            The section \(F_n/v\) at a vertex \(v\) is its vertex figure. Its vertices are the edges through \(v\); its faces record the incidences above \(v\). A facet \(H\) is viewed as the section \(H/F_{-1}\).

                                                                            Example. Flags of a triangle and a tetrahedron [frap-000F]

                                                                              The face-poset construction of [schulte2017regular, Section 2] gives the following elementary examples of flags. A triangle has the empty face, three vertices, three edges and the whole triangle, of ranks \(-1,0,1,2\). The flag \(\varnothing <1<12<123\) has rank-zero neighbor \(\varnothing <2<12<123\) and rank-one neighbor \(\varnothing <1<13<123\). There are \(3\cdot 2=6\) flags.

                                                                              Each vertex in this flag graph records the vertex and edge of one flag; the common empty and greatest faces are suppressed. Edge labels give the rank changed by adjacency.

                                                                              A tetrahedron has all subsets of \(\{1,2,3,4\}\) as faces, ordered by inclusion; a nonempty face with \(k\) vertices has rank \(k-1\). Its flag \(\varnothing <1<12<123<1234\) exhibits all five ranks. The vertex figure at \(1\) is a triangle: its vertices correspond to the three incident edges. A cube also has a triangular vertex figure, although its facets are squares. Local and global shapes carry different information.

                                                                              The triangle and tetrahedron calculation now supplies a concrete check: the flag graph records which flags are adjacent; a vertex figure records the local neighborhood seen from one vertex. With those objects in hand, the next steps ask which incidence-preserving maps act on all flags and how a symmetry of one facet can be extended.

                                                                              Definition. Automorphisms and regularity [schulte2017regular, Section 2] [frap-000G]

                                                                                An automorphism of a abstract polytope \(P\) is a bijection \(a:P\to P\) satisfying \(F\leq G\iff a(F)\leq a(G)\). Automorphisms form a group \(\Gamma (P)\) under composition; in \(ab\), the map \(b\) acts first. They preserve ranks and act on flags face by face.

                                                                                The polytope is regular if this action is transitive on flags. The tetrahedron is regular: a permutation of its four vertices sends any flag to any other.

                                                                                The underlying order automorphism is represented by OrderIso with equal source and target. This declaration does not impose the polytope axioms or assert regularity.

                                                                                Lemma. A flag determines an automorphism [schulte2017regular, Section 2, p. 4] [frap-000H]

                                                                                  An automorphism of an abstract polytope fixing every face of one flag is the identity. Consequently the automorphism group of a regular polytope acts freely and transitively on its flags.

                                                                                  The cited paragraph preceding Lemma 2.1 states simple transitivity for regular polytopes. The elementary argument below also proves freeness without assuming regularity.

                                                                                  Proof. 

                                                                                    For a flag adjacent to a pointwise fixed flag, only one rank has changed. The unchanged neighboring ranks bound a diamond condition with two possible faces. One is fixed, so the other is fixed too. The adjacent flag is therefore fixed pointwise. Propagating along the connected flag graph fixes every flag. Every face belongs to a flag, so every face is fixed. Transitivity is the definition of being regular.

                                                                                    Remark. Distinguished generators and the intersection property [frap-000I]

                                                                                      Choose a base flag \(\Phi \) in a regular abstract polytope \(P\) of rank \(n\). By flag freeness, there is a unique automorphism \(\rho _i\) taking \(\Phi \) to its rank-\(i\) neighbor, for \(0\leq i<n\). The diamond condition and uniqueness give \(\rho _i^2=1\). For \(|i-j|>1\), the two rank changes are independent; \(\rho _i\rho _j\) and \(\rho _j\rho _i\) agree on \(\Phi \), hence are equal. Connectivity shows that the \(\rho _i\) reach every flag, so they generate \(\Gamma (P)\). These are the distinguished generators of [schulte2017regular, Section 5, pp. 13--14].

                                                                                      For \(I\subseteq \{0,\ldots ,n-1\}\), put \(G_I=\langle \rho _i:i\in I\rangle \). They also satisfy

                                                                                      \[G_I\cap G_J=G_{I\cap J}.\]

                                                                                      This intersection property is part of the string C-group description [schulte2017regular, Section 5, equation (18)]. Involution and nonadjacent commutation relations alone do not reconstruct a polytope; the group-theoretic characterization includes this additional condition [schulte2017regular, Section 5, p. 14].

                                                                                      Lemma. Extending a symmetry of one facet [frap-000J]

                                                                                        Let \(H\) be a facet of a regular abstract polytope \(P\) of rank \(n\geq 2\). Restriction gives an isomorphism

                                                                                        \[\operatorname {Stab}_{\Gamma (P)}(H)\cong \Gamma (H/F_{-1}).\]

                                                                                        In particular the facet is regular, and each of its automorphisms has a unique extension fixing \(H\). This is a consequence of the flag action in [schulte2017regular, Section 2 and Lemma 2.1]; the argument follows.

                                                                                        Proof. 

                                                                                          Extend a flag of the facet section by the greatest face of \(P\). A stabilizer element restricting to the identity fixes this whole flag and is the identity by flag freeness. Restriction is therefore injective.

                                                                                          Given a facet automorphism \(\alpha \), the regular action supplies an automorphism of \(P\) taking the extended flag to the extension of its image under \(\alpha \). It fixes \(H\). Its restriction agrees with \(\alpha \) on the chosen facet flag, so flag freeness in the facet makes these restrictions equal. This proves surjectivity. The same use of same regular action with any two facet flags shows that the facet is regular.

                                                                                          Turning a facet into a puzzle move [frap-0009]

                                                                                            Once the shape symmetries are known, the next question is local: how can a turn change only the positions incident to one chosen facet while carrying their labels along? A facet rotation extends to a symmetry of the whole shape, but a puzzle move applies it only at positions incident to the chosen facet. Marchetti's construction retains both entries of each incidence pair [marchetti2025rubik, Definitions 5--6]. We use finite regular abstract polytopes of rank at least three; the extension lemma itself does not require finiteness.

                                                                                            Definition. The rotation subgroup [marchetti2025rubik, Definition 4] [frap-000K]

                                                                                              Let \(Q\) be a regular abstract polytope of rank \(m\geq 2\). Choose a base flag and its distinguished generators (involutions) \(\rho _0,\ldots ,\rho _{m-1}\). Its rotation subgroup is

                                                                                              \[\Gamma ^+(Q)=\langle \rho _{j-1}\rho _j:1\leq j<m\rangle .\]

                                                                                              The index starts at \(1\): for a polygon, the single adjacent pair \(\rho _0\rho _1\) generates its rotations. For triangles and squares these are the usual cyclic rotation groups.

                                                                                              Equivalently, \(\Gamma ^+(Q)\) consists of the elements represented by even-length words in the distinguished involutions. Indeed every pair \(\rho _i\rho _j\) is a product of adjacent pairs or their inverses, by cancellation along the index interval. Conjugating an even word by a distinguished involution again gives an even word, so this subgroup is normal. A change of base flag conjugates the distinguished generators and therefore leaves the subgroup unchanged. We do not require it to have index two for every abstract polytope.

                                                                                              Definition. Incidence positions and facet moves [marchetti2025rubik, Definitions 5--6] [frap-000L]

                                                                                                Let \(P\) be a finite regular abstract polytope of rank \(n\geq 3\), with least and greatest faces \(F_{-1},F_n\). Write \(P^\circ =P\setminus \{F_{-1},F_n\}\). The incidence positions are

                                                                                                \[\mathcal R(P)=\{(F,G)\in P^\circ \times P^\circ :F\leq G\}.\]

                                                                                                Choose a facet \(H\) and an element \(\alpha \) of the rotation subgroup \(\Gamma ^+(H/F_{-1})\). The facet extension lemma gives its unique extension \(\widehat \alpha \in \Gamma (P)\) fixing \(H\). The corresponding facet move is

                                                                                                \[\mu _{H,\alpha }(F,G)= \begin {cases} (\widehat \alpha (F),\widehat \alpha (G)),&F\leq H,\\ (F,G),&F\not \leq H. \end {cases}\]

                                                                                                The first entry decides where the shape automorphism is applied; the second entry is transported with it. The facet is part of the move's name: one shape automorphism may stabilize several facets and give different local moves. Each move is a permutation of \(\mathcal R(P)\).

                                                                                                Lemma. The inverse of a facet move [frap-000M]

                                                                                                  Let \(P\) be a finite regular abstract polytope of rank \(n\geq 3\), \(H\) a facet and \(\alpha \in \Gamma ^+(H/F_{-1})\) a facet rotation. The facet move \(\mu _{H,\alpha }\) is a permutation of \(\mathcal R(P)\), with

                                                                                                  \[\mu _{H,\alpha }^{-1}=\mu _{H,\alpha ^{-1}}.\]

                                                                                                  This follows directly from the construction in [marchetti2025rubik, Definitions 5--6].

                                                                                                  Proof. 

                                                                                                    An automorphism preserves proper faces and the order relation, so the moving branch remains in \(\mathcal R(P)\). Since \(\widehat \alpha (H)=H\), the condition \(F\leq H\) holds exactly when \(\widehat \alpha (F)\leq H\). Thus the moving and fixed subsets are preserved separately. On the moving subset the extension of \(\alpha ^{-1}\) is \(\widehat \alpha ^{-1}\) by uniqueness, and cancels the first map. On the fixed subset both maps are the identity.

                                                                                                    The general construction combining permutations of complementary subsets is Equiv.Perm.subtypeCongr; its inverse formula is Equiv.Perm.subtypeCongr.symm. Here the predicate is \(F\leq H\), the moving permutation is the restricted shape automorphism and the other permutation is the identity. These declarations supply that general combination; the polytope-specific preservation argument is the proof above.

                                                                                                    The inverse lemma makes the local action reversible. The tetrahedron calculation that follows is a small witness for why the facet is part of a move's identity: two locally valid turns can act differently on the same incidence record and need not commute.

                                                                                                    Example. Two turns of a tetrahedron [frap-000O]

                                                                                                      Represent the faces of a tetrahedron by subsets of \(\{1,2,3,4\}\). We calculate facet moves from [marchetti2025rubik, Definitions 5--6]. For the facet \(H=123\), take an element of the rotation subgroup with global extension \(\widehat \alpha =(123)\), fixing \(4\). The vertex-edge records follow the cycle

                                                                                                      \[(1,12)\longmapsto (2,23)\longmapsto (3,13)\longmapsto (1,12).\]

                                                                                                      Every arrow is the local move \(\mu _{H,\alpha }\). The record \((4,14)\) stays fixed because \(4\not \leq H\), although the global shape symmetry sends it to \((4,24)\). The diagram records this permutation of incidence positions, rather than the tetrahedron's face graph.

                                                                                                      For records whose second entry is a facet, the same move gives

                                                                                                      \[(1,124)\longmapsto (2,234),\qquad (12,124)\longmapsto (23,234),\qquad (4,124)\longmapsto (4,124).\]

                                                                                                      The last record is fixed even though the global symmetry changes its second entry. Now let \(K=124\) and take an element of the rotation subgroup with global extension \(\widehat \beta =(124)\). Applying the \(H\)-move and then the \(K\)-move sends

                                                                                                      \[(1,124)\xrightarrow {\mu _{H,\alpha }}(2,234) \xrightarrow {\mu _{K,\beta }}(4,134).\]

                                                                                                      Reversing their order gives \((1,124)\mapsto (2,124)\mapsto (3,234)\), so the two moves do not commute. Each has order three: its cube is the identity on both branches, and its displayed action is nontrivial.

                                                                                                      The tetrahedron witness motivates the generated move group: reachable labelled configurations are products of these reversible local turns. The ordinary cube calculation then shows how the same definition compares with visible stickers and which data it leaves out.

                                                                                                      Definition. The Rubik group and labelled configurations [frap-000N]

                                                                                                        For a finite regular abstract polytope \(P\) of rank \(n\geq 3\), the Rubik group is the subgroup generated by all facet moves:

                                                                                                        \[\operatorname {Rub}(P)= \langle \mu _{H,\alpha }:H\text { a facet},\quad \alpha \in \Gamma ^+(H/F_{-1})\rangle \leq \operatorname {Sym}(\mathcal R(P)).\]

                                                                                                        This is Marchetti's generated group of permutations [marchetti2025rubik, Section 3, p. 6, following Definition 6]. The inverse lemma ensures that the generators are permutations.

                                                                                                        Give each incidence position its own initial label. A labelled configuration can be represented by the permutation carrying each initial label to its current position. The identity is the solved configuration; the reachable labelled configurations are the elements of \(\operatorname {Rub}(P)\). A new move acts on the left: applying \(\mu \) to configuration \(g\) gives \(\mu g\), with \(g\) acting first. A visible coloring may identify several labelled configurations, so a solved visible coloring need not determine the labelled permutation.

                                                                                                        Example. One ordinary cube face turn [frap-000P]

                                                                                                          Realize the cube as \([-1,1]^3\) and let \(H\) be the facet \(z=1\). The facet move construction of [marchetti2025rubik, Definitions 5--6] gives the following explicit outer-face action. The shape rotation

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

                                                                                                          is a clockwise quarter-turn when viewed from positive \(z\), with \(x\) rightward and \(y\) upward. Its restriction to \(H\) supplies our move. The corner \((1,1,1)\) goes to \((1,-1,1)\), and its color on \(y=1\) goes to the color position on \(x=1\). The corresponding corner at \(z=-1\) is fixed by the puzzle move, although the global map \(q\) moves it.

                                                                                                          The signs give the \(x,y\) coordinates of the four corners on \(z=1\). The arrows show their clockwise motion; the central incidence record \((H,H)\) stays fixed. No orientation of a marked center is recorded.

                                                                                                          Restricting the second entry to facets gives \(8\cdot 3+12\cdot 2+6=54\) incidence positions: three per corner, two per edge and one per face center. The turn moves twelve corner stickers and eight edge stickers, fixing the other thirty-four positions, including the unmarked center of the turned face. This is the usual outer-face action on visible stickers of a \(3\times 3\times 3\) cube in a fixed frame.

                                                                                                          The full incidence set has \(8\cdot 7+12\cdot 3+6=98\) elements: above each vertex are that vertex, three edges and three facets; above each edge are itself and two facets. The facet-colored subset is invariant under every move. Its restricted action supplies the comparison with ordinary stickers; a faithful restriction for all abstract polytopes is not needed. Marked center orientations and inner-layer turns are additional data, absent from this comparison.

                                                                                                          The local action determines which positions participate and how their incident labels travel. Frames and orientation separate this transport into a destination and a local orientation.

                                                                                                          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.

                                                                                                                  Label transport and topology [frap-0014]

                                                                                                                    A sequence of puzzle moves may return pieces to their positions while changing their orientations. The first question is what local return map records that change; the cycle criterion computes it. A related question asks how labels move between neighboring simplices. The projectivity construction of Izmestiev and Joswig makes that question precise, and the complete unfolding records the topology of one such transport system [izmestiev2002branched, Sections 2.1--2.3.3].

                                                                                                                    Perspectivities along facet paths

                                                                                                                      Let \(K\) be a pure \(d\)-dimensional simplicial complex. If neighboring facets \(\sigma \) and \(\tau \) share a ridge, \(\sigma \) has one vertex \(v(\sigma ,\tau )\) not in \(\tau \). The perspectivity

                                                                                                                      \[h_{\sigma ,\tau }:V(\sigma )\longrightarrow V(\tau )\]

                                                                                                                      swaps that exceptional vertex with \(v(\tau ,\sigma )\) and fixes the common ridge vertices. For a facet path \(\gamma =(\sigma _0,\ldots ,\sigma _n)\), its projectivity is the composite

                                                                                                                      \[\langle \gamma \rangle =h_{\sigma _0,\sigma _1}\cdots h_{\sigma _{n-1},\sigma _n}.\]

                                                                                                                      A closed facet path based at \(\sigma _0\) gives a permutation of \(V(\sigma _0)\). The set of these permutations is the projectivity group \(\Pi (K,\sigma _0)\leq \operatorname {Sym}(V(\sigma _0))\); changing the base facet conjugates the group when the dual graph is connected. The inverse path supplies the inverse permutation, so this is transport with a genuine composition law [izmestiev2002branched, Section 2.1].

                                                                                                                      Coloring as a trivial-transport test

                                                                                                                        A proper coloring of the vertices of \(K\) with \(d+1\) colors gives every facet a consistently labeled vertex set. Each perspectivity preserves those labels, so every projectivity is the identity. Thus proper coloring is a sufficient condition for \(\Pi (K)=1\); a nontrivial projectivity group is an obstruction to such a global labeling [izmestiev2002branched, Section 2.1].

                                                                                                                        This is a useful comparison with a framed puzzle. A local frame assigns labels in one fiber, and a move transports them to another fiber. The framed transport and its total-orientation character measure the return action in the puzzle model. Projectivities do the analogous bookkeeping for simplex vertices, but the two constructions have different state spaces and move laws. No identification between their groups is being assumed.

                                                                                                                        Complete unfolding as a transport cover

                                                                                                                          Fix a base facet and a labeling of its vertices by \(0,\ldots ,d\). A facet reached by a path inherits a labeling by transporting along that path. Different paths can give different admissible labelings, precisely when their projectivity differs. The complete unfolding keeps these admissibly labeled facets as separate sheets and glues neighboring sheets only when their labels agree on the common ridge. It carries a canonical map back to \(K\); the base projectivity group acts by permuting these sheets. The complete unfolding itself has trivial projectivity group [izmestiev2002branched, Section 2.2, Proposition 2.2.2].

                                                                                                                          The construction is therefore a topological record of transport failure: a loop in the base may close geometrically while changing the sheet label. The complete unfolding removes that ambiguity by remembering the transported labeling. This is analogous to a covering-space monodromy calculation; the branch locus is additional local-combinatorial data of the unfolding and is not inferred here from the puzzle model.

                                                                                                                          The tetrahedral boundary example

                                                                                                                            For the boundary complex \(\partial \Delta ^3\), the projectivity group is \(S_3\). Its complete unfolding is glued from 24 triangles. Identifying opposite sides of the triangulated hexagon gives a torus \(T^2\); the canonical map to the tetrahedral boundary, topologically \(S^2\), is a six-fold branched covering. There are four branch points on \(S^2\); each has three preimages, each of branching index two [izmestiev2002branched, Section 2.3.3, Figure 3].

                                                                                                                            The example separates three pieces of data that are easy to conflate: the base triangulation, the permutation group generated by facet-path transport, and the topology of the unfolded surface. The torus is not a claim about a cube solver. It is a concrete model showing how local label transport can assemble a global branched cover.

                                                                                                                            A comparison between these constructions must name the fiber, the local transport map, and the allowed paths before relating topology to puzzle orientation. The projectivity example supplies one transport vocabulary; the framed cube account supplies a different finite-state calculation. A bridge between them would require an explicit map between those data and a proof that it preserves composition, return maps, and the claimed topology.

                                                                                                                            Positions, orientations, and global constraints [frap-000A]

                                                                                                                              After a facet move, a piece can reach the right position with the wrong local orientation. The first question is therefore what data records both the destination and the frame carried by the piece. Local frames turn this transport into group coordinates, as in [marchetti2025rubik, Section 3.1]. We then derive the composition law, a frame-independent total-orientation character, and the additional test that makes this character a conserved quantity for cube face turns.

                                                                                                                              Definition. Framed transports [frap-000Q]

                                                                                                                                Let \(I\) be a finite set of positions, \(\Omega \) a nonempty set, and \(K\) a group acting faithfully on \(\Omega \). At each position \(i\), choose a set \(X_i\) of local labels and a bijective local frame \(c_i:\Omega \to X_i\). Keep the fibers distinct by using the tagged union

                                                                                                                                \[E=\coprod _{i\in I}X_i=\{(i,y):i\in I,\ y\in X_i\}.\]

                                                                                                                                A framed transport is a permutation of \(E\) given by a position permutation \(\pi \in \operatorname {Sym}(I)\) and elements \(a_i\in K\), with

                                                                                                                                \[g(i,c_i(x))=(\pi (i),c_{\pi (i)}(a_i x)).\]

                                                                                                                                Write \(g=(\pi ,a)\). Its local factors are indexed by the source position. Nonempty fibers determine \(\pi \) uniquely, and faithfulness determines each \(a_i\) uniquely. Conversely every such pair gives a permutation: in framed coordinates its inverse sends \((j,x)\) to \((\pi ^{-1}(j),a_{\pi ^{-1}(j)}^{-1}x)\).

                                                                                                                                This finite-position model isolates the coordinate mechanism of [marchetti2025rubik, Section 3.1, Definition 7 and equation (5)]. Which local label sets and local group describe a particular puzzle must still be justified for that puzzle. The requirement that a transport moves whole fibers is part of the model.

                                                                                                                                Under composition these transports form a wreath product \(K\wr \operatorname {Sym}(I)\). The normal subgroup \(K^I\) acts within fibers, and position permutations reorder its factors. In TauCeti.WreathProduct, a pair is stored using the destination-indexed function \(A_j=a_{\pi ^{-1}(j)}\). Its permutation action is TauCeti.WreathProduct.imprimitiveToPerm_apply; the nonempty-fiber and faithful-action hypotheses give its injectivity in TauCeti.WreathProduct.imprimitiveToPerm_injective. The bijections \(c_i\) identify that action on \(I\times \Omega \) with the action on \(E\) above.

                                                                                                                                Lemma. Composition of framed transports [frap-000R]
                                                                                                                                mul_left
                                                                                                                                L∃∀N

                                                                                                                                  Let \(I\) be finite and let \(K\) act faithfully on a nonempty \(\Omega \), with bijective frames \(c_i:\Omega \to X_i\). Write two framed transports as \(g=(\pi ,a)\) and \(h=(\sigma ,b)\), acting in coordinates by \(g(i,x)=(\pi (i),a_i x)\) and \(h(i,x)=(\sigma (i),b_i x)\). With left actions, applying \(g\) and then \(h\) gives the composition law

                                                                                                                                  \[hg=(\sigma \pi ,(b_{\pi (i)}a_i)_{i\in I}).\]

                                                                                                                                  Proof. 

                                                                                                                                    The intermediate coordinates are \((\pi (i),a_i x)\). Applying \(h\) evaluates its local map at that intermediate position, giving \((\sigma \pi (i),b_{\pi (i)}a_i x)\). Thus a local factor must be multiplied at its transported position index.

                                                                                                                                    The vertical arrows are the chosen frames; the bottom arrows are the restrictions of the actual transports. Each square commutes. Along the top row, the second factor is evaluated at \(\pi (i)\), explaining the shifted index in the composite. This expands the wreath-product coordinates of [marchetti2025rubik, Section 3.1, pp. 6--7]. The wreath product uses destination-indexed factors in the linked formalization.

                                                                                                                                    The linked multiplication theorem uses destination-indexed factors. Set \(A_j=a_{\pi ^{-1}(j)}\) and \(B_j=b_{\sigma ^{-1}(j)}\). Its factor for \(hg\) at destination \(j\) is

                                                                                                                                    \[B_j A_{\sigma ^{-1}(j)} =b_{\sigma ^{-1}(j)}a_{\pi ^{-1}\sigma ^{-1}(j)}.\]

                                                                                                                                    Putting \(i=(\sigma \pi )^{-1}(j)\) makes this exactly \(b_{\pi (i)}a_i\). Thus the theorem and the displayed source-indexed law use the same multiplication order.

                                                                                                                                    Theorem. The abelianized total orientation [frap-000S]

                                                                                                                                      Let \(I\) be finite, let \(K\) act faithfully on a nonempty \(\Omega \), and choose bijective frames \(c_i:\Omega \to X_i\). For a framed transport \(g=(\pi ,a)\), let \(K_{\mathrm {ab}}=K/[K,K]\) be its abelianization, where \([K,K]\) is generated by \(uvu^{-1}v^{-1}\), and write \([a]\) for the class of \(a\). The total-orientation character

                                                                                                                                      \[\chi (g)=\prod _{i\in I}[a_i]\quad \hbox {in }K_{\mathrm {ab}}\]

                                                                                                                                      is a homomorphism, independent of replacing the frames by \(c_i'=c_i t_i\) for \(t_i\in K\). This is the canonical character in [marchetti2025rubik, Section 3.1, p. 7]; the proof below gives both assertions directly in source-indexed coordinates.

                                                                                                                                      Proof. 

                                                                                                                                        For \(h=(\sigma ,b)\), the composition law and commutativity of the target give

                                                                                                                                        \[\chi (hg)=\prod _i[b_{\pi (i)}][a_i] =\left (\prod _j[b_j]\right )\left (\prod _i[a_i]\right ) =\chi (h)\chi (g).\]

                                                                                                                                        The middle equality reindexes using the bijection \(\pi \). In the new frames the local map is \(a_i'=t_{\pi (i)}^{-1}a_i t_i\). Its abelianized product contains each \([t_i]\) and its inverse exactly once, so these cancel and leave \(\chi (g)\).

                                                                                                                                        Frame independence here concerns changes within the chosen group of frames. Reversing a common orientation convention can also change the identification of the target group. The target and its quotient map are Abelianization and Abelianization.of; the total product and its frame independence are the argument above.

                                                                                                                                        Remark. When a character is conserved [frap-000T]

                                                                                                                                          For finite \(I\), the ambient coordinate group \(W=K\wr \operatorname {Sym}(I)\) has the total-orientation character \(\chi (\pi ,a)=\prod _i[a_i]\in K_{\mathrm {ab}}\). If a puzzle's allowed moves form a set \(S\subseteq W\), its legal group is \(G=\langle S\rangle \), usually a proper subgroup of \(W\). Then

                                                                                                                                          \[G\subseteq \ker \chi \quad \Longleftrightarrow \quad \chi (s)=1\quad \hbox {for every }s\in S.\]

                                                                                                                                          The forward implication includes each generator. Conversely the kernel of a homomorphism is a subgroup, so it contains all products and inverses of generators lying in it. This is the generator-check step discussed after the canonical character in [marchetti2025rubik, Section 3.1, p. 7]. Its premise must be verified for the chosen moves.

                                                                                                                                          The nonabelian product of local factors need not be independent of order or frames. Abelianization makes reindexing and frame cancellation valid, but neither of those operations says that \(\chi (s)=1\). The cycle criterion supplies a sufficient test, applied explicitly to cube face turns.

                                                                                                                                          Lemma. Orientation checked cycle by cycle [frap-000U]

                                                                                                                                            Let \(g=(\pi ,a)\) be a framed transport on finitely many positions \(I\), with a faithful local group \(K\) on nonempty \(\Omega \) and bijective local frames. On a cycle

                                                                                                                                            \[i_0\longmapsto i_1\longmapsto \cdots \longmapsto i_{r-1} \longmapsto i_0\]

                                                                                                                                            of \(\pi \), the return transport in the first fiber has framed map \(a_{i_{r-1}}\cdots a_{i_1}a_{i_0}\). Its class in \(K_{\mathrm {ab}}\) is that cycle's contribution to \(\chi (g)=\prod _{i\in I}[a_i]\). Consequently, identity return transport on every position cycle implies \(\chi (g)=1\). This is the cycle criterion.

                                                                                                                                            Proof. 

                                                                                                                                              Apply the coordinate action \(r\) times. Each local map is applied after the preceding one, giving the stated product with the first map on the right. In the abelianization its factors may be reordered; multiplying over disjoint cycles uses each factor exactly once. A fixed position is a cycle of length one and must also be checked.

                                                                                                                                              This is a sufficient test derived from the canonical character of [marchetti2025rubik, Section 3.1, p. 7]. A generator's finite order alone does not make its return map on each shorter position cycle the identity; the local transport on that cycle is the quantity to check.

                                                                                                                                              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.

                                                                                                                                                Transfer gives a group-theoretic form of the same frame cancellation and shows how the local group limits what total orientation can detect.

                                                                                                                                                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 complete three-layer cube [frap-0016]

                                                                                                                                                          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 the fixed local frame convention and keep the global viewing frame fixed. Unmarked centers give the following coordinate record.

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

                                                                                                                                                          Here \(\sigma \) and \(\tau \) record corner and edge positions. The corner local frame uses the U/D sticker to record a twist in \(\mathbb Z_3\); an edge local 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.

                                                                                                                                                          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].

                                                                                                                                                            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.

                                                                                                                                                              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.

                                                                                                                                                                The family of layered cubes [frap-0003]

                                                                                                                                                                  To pass from one cube size to another, we first fix a physical model. Let \(C_N\) be an \(N\times N\times N\) array of physical units, and call each unit a cubie. Give the units coordinates \((i,j,k)\in {1,...,N}^3\). A visible cubie is a surface cell, so at least one coordinate is \(1\) or \(N\). A cell with three boundary coordinates is a corner; with exactly two it is an edge; with exactly one it is a center. This is a description of positions, before we decide which labels a player can see.

                                                                                                                                                                  The resulting counts are immediate: there are \(8\) corners, \(12(N-2)\) edge cubies, and \(6(N-2)^2\) center cubies. Their sum is the number of surface cells. The count already explains why changing \(N\) is more than making a larger drawing. New edge and center positions appear, and those positions can fall into different move orbits. Bonzio, Loi and Peruzzi use this cubie decomposition in their general model [bonzio2018nxnxn, Sections 2--3]; Salkinder gives a parallel orbit and configuration analysis [salkinder2021nxnxn, Section 2].

                                                                                                                                                                  The abstract-to-cube handoff is made explicit in how the cube models connect. In the \(N=3\) case, the 54 facet-colored incidence positions group into corner and edge sticker fibers, with six fixed centers supplying the frame; the resulting action is the coordinate model in the global viewing frame used later. For larger \(N\), the new center and coupled-edge orbits prevent this from being a single ordinary-cube coordinate space.

                                                                                                                                                                  Layer turns and the global viewing frame

                                                                                                                                                                    Choose the six coordinate directions as a global viewing frame. For an axis \(a\in {x,y,z}\), a depth \(r\) with \(1\le r\le \lfloor N/2\rfloor \), and either boundary of that axis, turn the layer \(a=r\) or \(a=N+1-r\) by a quarter-turn. The turn rotates the other two coordinates and carries every sticker on a moved cubie with it. For odd \(N\), the central layer is deliberately omitted; this is the usual fixed-center convention. Thus \(N=3\) has the six outer-face turns, while \(N=4\) has the outer and adjacent inner layers.

                                                                                                                                                                    This choice distinguishes a layer turn from a whole-cube rotation. Whole-cube rotations change the chosen frame and are not generators of the state group here. They can be added as a separate symmetry action, or used to identify observations later. Keeping them out makes the six face directions, and hence a fixed center color scheme, part of the state description.

                                                                                                                                                                    A labelled state assigns a distinct name to every physical cubie and, when needed, to every sticker. A move acts by permuting these labels and their local frames. An ordinary colored observation forgets distinctions between equal-colored stickers; a marked or 'supercube' observation retains them. These are different quotients of the same labelled action. In particular, a statement that two center cubies 'look the same' is an observation claim, not a claim that the labelled pieces are interchangeable by a move.

                                                                                                                                                                    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.

                                                                                                                                                                      Small cubes make the parity split visible

                                                                                                                                                                        For \(N=2\), \(K_e=0\): there are only the eight corners. For \(N=3\), \(K_o=0\): the pieces are eight corners, twelve single edges and six fixed centers. This is the familiar cube, but the description explains why its centers do not move in the chosen layer family.

                                                                                                                                                                        For \(N=4\), \(K_e=1\): there are eight corners, one coupled-edge orbit of \(24\), and one center-corner orbit of \(24\); there are no fixed centers and no center-edge ring. For \(N=5\), \(K_o=1\): there are eight corners, twelve single edges, one coupled-edge orbit of \(24\), six fixed centers, one center-corner orbit of \(24\), and one center-edge orbit of \(24\). The latter has \(8+36+54=98\) visible cubies, the familiar Professor's Cube count. These examples show what changes with parity without claiming that the same labels or orientation coordinates are used by every source.

                                                                                                                                                                        The ordinary cube action is the \(N=3\) outer-layer case of this model. The next chapter can therefore state constraints for a fixed three-layer convention, while the all-\(N\) first law must still reconcile the labelled, colored and coupled-edge conventions in Bonzio--Loi--Peruzzi and Salkinder. Salkinder writes all slice rotations, including a central slice in odd size, and then treats its use through a shifted middle-slice convention; this is a genuine convention choice, not a harmless notation change. We use the fixed-center generators above and leave the translation between these move families explicit. The first law records the invariant equations with a source-backed sufficiency sketch; the full constructive convention transfer remains a separate proof obligation.

                                                                                                                                                                        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 local frame convention in how the cube models connect and the complete three-layer cube. Represent a sticker in the sticker state 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 sticker state action on labelled stickers; 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.