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

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

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 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
\[I_R=U R U' R' U' F' U F,\]

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

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
\[P=R U' R U R U R U' R' U' R^2\]

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
\[C=R' F R' B^2 R F' R' B^2 R^2\]

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.

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.

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

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.

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

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.

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.

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

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

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

The corresponding upper estimate for a radius-15 ball is

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.

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

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

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.

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

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.

A different compression searches through a goal subgroup. Let

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

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.

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.

Let \(S\) be a declared state space and let every legal move \(m\) be a reversible map \(S\to S\). A notation parser is correct only if it maps a token to one such map, with an inverse token satisfying

For a word \(w=m_1m_2\cdots m_k\), we read moves from left to right as in the complete layer-first method:

A replay test therefore records the initial state, the token list, the convention for viewing each face, and the final state. Replaying \(w\) followed by \(w^{-1}\) must return to \(s\); two displays are equivalent only after their resulting states, rather than their typography, agree.

The tetrahedron turn gives a short noncommuting trace. Starting at the recorded incidence position \((1,124)\), the \(H\)-move followed by the \(K\)-move reaches \((4,134)\). Reversing the order reaches \((3,234)\). The two words are therefore different maps, even though each move separately has order three. A player can check this by stepping through the two token lists and comparing the final records.

The cube face turn fixes the frame and turns only the chosen layer. The global shape map \(q(x,y,z)=(y,-x,z)\) sends the four \(z=1\) corners around a four-cycle; the local puzzle move restricts this map to that layer and fixes the opposite layer pointwise. This is a useful display test: a global rotation of the drawing is not the same operation as the local puzzle move, and the unturned center must not move merely because the picture was rotated.

A reproducible solve has five recorded stages: identify the state in the chosen coordinates; state the target condition; choose a move word; replay it from the recorded state; and check the target and preserved invariants. The layer-first method supplies a human version: each stage fixes information and uses a stabilizer for what remains. The computer account supplies a search version: a returned word is an upper bound, while a projected-distance table gives a lower bound.

These certificates should be reported separately. A valid replay shows that the word reaches the target. A short word improves an upper bound in the named metric. A shortest-path claim additionally needs an exhaustive search or a matching lower bound. The distinction also applies to a scramble: scrambling and spectra requires the move law and length before a distributional claim can be evaluated.

The worked replay and projected lower bound makes this separation concrete: one fixed-frame scramble is replayed by its inverse, a projected search supplies a lower bound, and the ten-token word supplies an upper bound. The example closes the replay loop while leaving the all-state method and any optimality claim at their stated scopes.

A small interface can make the contract visible without pretending to replace it. It can reject an unknown token, show the affected pieces, step through a word, undo by inverse tokens, and compare the final state with a target. It can also display parity, orientation, or subgroup coordinates as diagnostics, provided each coordinate is tied to the declared state model.

An animation illustrates a claim; the move map and replay log are the evidence for it. The interface should therefore expose the convention, the exact token sequence, and the before-and-after state so another person or program can reproduce the check. The existing Rubik's Cube resources collect notation, methods, and demonstrations; the companion FRAP repository can later host executable versions of these checks.

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].

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.

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

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

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.

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

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

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 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.

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:

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

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,

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.

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.

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.

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

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

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].

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.

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.

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.

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

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

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

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.

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

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

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

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.

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.

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

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

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].

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

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

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

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.

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

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.

Choose the six coordinate directions as a fixed 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 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.

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. A necessary invariant will not be promoted to a sufficiency theorem until that reconciliation is written out.

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

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

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

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

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

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

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

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

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

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

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

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

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

The positive-\(z\) face turn uses

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

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.

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:

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.

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.