Dimension and local rotation groups [frap-0013]
✍️source

    The local corner groups of simplices and cubes are alternating groups. Their commutator subgroups determine the information that survives the total-orientation character [marchetti2025rubik, Sections 4--5].

    Proposition 1. Local rotations of simplices and cubes [frap-000Z]

      For \(d\geq 3\), a vertex figure of a \(d\)-simplex or a \(d\)-cube is a \((d-1)\)-simplex. Its automorphism group is \(S_d\), and its rotation subgroup is \(A_d\). These are the local corner groups used in [marchetti2025rubik, Section 4, p. 10, and Section 5, p. 14].

      Proof. 

        At a vertex of either polytope there are \(d\) incident edges. Faces containing that vertex correspond to subsets of those edges: for a simplex one chooses the other vertices, and for a cube one chooses the coordinate directions in which the face varies. This is the face poset of a simplex on \(d\) vertices, of dimension \(d-1\). An automorphism permutes its vertices, and every vertex permutation extends uniquely to the subsets, giving \(S_d\).

        A base flag orders these \(d\) vertices. Its distinguished involutions are adjacent transpositions. They generate \(S_d\), and the even words are precisely the even permutations. Thus the rotation subgroup is \(A_d\).

        Here \(d\) is abstract dimension. Varying the layer count of an \(N\times N\times N\) cube leaves the ambient dimension equal to three; it changes the piece orbits and allowed layer moves instead.

        Theorem 2. Abelianizations of the local alternating groups [frap-0010]

          Let \(A_d\) be the alternating group on \(d\) letters, with composition rightmost first. Then

          \[(A_3)_{\mathrm {ab}}\cong C_3,\qquad (A_4)_{\mathrm {ab}}\cong C_3,\qquad (A_d)_{\mathrm {ab}}=1\quad (d\geq 5).\]

          The corresponding commutator subgroups are stated after Proposition 4.1 in [marchetti2025rubik, Section 4.1, p. 11]. We give the elementary calculation, using \([g,h]=ghg^{-1}h^{-1}\).

          Proof. 

            The group \(A_3\) is cyclic of order three, so already abelian. In \(A_4\), let \(V=\{1,(12)(34),(13)(24),(14)(23)\}\). Products and inverses stay in this four-element set, and conjugation permutes its three nonidentity elements. Hence \(V\) is normal. The quotient has order three, so it is cyclic and \([A_4,A_4]\subseteq V\). Direct multiplication gives \([(123),(124)]=(12)(34)\). Conjugating this element by the three-cycle \((123)\) gives the other two double transpositions. The commutator subgroup is normal, so it contains all of \(V\); equality follows.

            For \(d\geq 5\), put \(g=(12)(34)\) and \(h=(12)(35)\) in \(A_d\). Multiplication gives \([g,h]=(345)\). Every three-cycle is conjugate to this one within \(A_d\): first choose a conjugating permutation in \(S_d\); if it is odd, compose on the right with a transposition of two symbols outside \(\{3,4,5\}\). This centralizes \((345)\) and changes the parity. Thus the normal subgroup \([A_d,A_d]\) contains every three-cycle.

            Three-cycles generate all even permutations. Write an even permutation as a product of an even number of transpositions and pair them. Equal transpositions cancel; two sharing one symbol give a three-cycle; a disjoint pair satisfies \((ab)(cd)=(acb)(acd)\). Hence every pair is a product of three-cycles, proving \([A_d,A_d]=A_d\).

            The three cases have separate formalized ingredients. alternatingGroup.isCyclic_of_card_le_three gives the cyclicity of \(A_3\). For four letters, alternatingGroup.kleinFour_eq_commutator identifies \(V\) with the commutator subgroup, and TauCeti.card_abelianization_alternatingGroup gives the quotient's cardinality three; the displayed cyclic isomorphism then uses the elementary classification of groups of prime order. For \(d\geq 5\), commutator_alternatingGroup_eq_top is the perfection statement. These links concern alternating groups, not the entire puzzle group.

            Remark 3. What the corner character can detect [frap-0011]

              For a \(d\)-simplex or \(d\)-cube with \(d\geq 3\), the local corner groups have rotation subgroup \(A_d\). Their contribution to the total-orientation character passes through \((A_d)_{\mathrm {ab}}\). The abelianization calculation therefore measures exactly what this detector can see, as in [marchetti2025rubik, Section 4.1, p. 11, and equation (19), p. 14].

              In dimension three the target is \(C_3\), the familiar corner twist sum. In dimension four the local group has grown to the nonabelian \(A_4\), but the character still sees only a \(C_3\) quotient: every local change in the Klein four group disappears from it. In dimensions five and above the local group is perfect, so this corner character is identically trivial on the ambient framed transports.

              A lost detector does not classify the legal moves. Other piece types, permutation signs and the subgroup actually generated by facet turns may still impose constraints. Even when the target is nontrivial, conservation requires a generator check; the local abelianization alone does not supply one.

              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}
}

                Context

                  Transfer and the effect of dimension [frap-000B]

                    The total-orientation character detects one conservation law, but it does not explain how to compare frames chosen at different positions. The motivating question is how to transport that comparison through a transitive action. Transfer uses the same cancellation as the total-orientation character for coset representatives [mayer2015artin, Sections 3.1--3.3]. The first section identifies the connection through an actual transitive action and its stabilizer; the second computes how the local rotation groups of simplices and cubes change with dimension, following [marchetti2025rubik, Sections 4--5].

                    Transfer and local orientation [frap-0012]

                      Left multiplication permutes the cosets of a finite-index subgroup. Representatives record the motion between cosets, leaving one residual subgroup element at each position. Abelianizing their product gives transfer [mayer2015artin, Sections 3.1--3.3]. Transported frames then identify its local image with the total-orientation character of an action.

                      Theorem. Finite-index transfer [frap-000W]
                      MonoidHom.transferMonoidHom.transfer_def
                      L∃∀N
                      transfer_eq_prod_of_bijective
                      L∃∀N

                        Let \(H\leq G\) have finite index. Choose representatives \(r_i\) for the left cosets \(r_iH\), indexed bijectively by a finite set \(I\). For \(g\in G\), left multiplication permutes the cosets, so uniquely write

                        \[g r_i=r_{\pi _g(i)}h_i(g),\qquad h_i(g)\in H.\]

                        Write \(H_{\mathrm {ab}}=H/[H,H]\) and \([h]\) for the class of \(h\in H\). The abelianization is the target of the transfer

                        \[T_{G,H}:G\longrightarrow H_{\mathrm {ab}},\qquad T_{G,H}(g)=\prod _{i\in I}[h_i(g)]\]

                        is a homomorphism independent of the chosen representatives. This is the left-coset construction of [mayer2015artin, Definition 3.1, Proposition 3.1 and Theorem 3.1, pp. 4--6]. Neither finiteness of \(G\) nor normality of \(H\) is required.

                        Proof. 

                          We compose left actions rightmost first. Multiplication on cosets gives \(\pi _{gk}=\pi _g\pi _k\). Expanding \(gkr_i\) in two steps yields

                          \[h_i(gk)=h_{\pi _k(i)}(g)h_i(k).\]

                          Taking classes and multiplying over \(i\) gives \(T_{G,H}(gk)=T_{G,H}(g)T_{G,H}(k)\), because \(\pi _k\) reorders the first factors. Replace \(r_i\) by \(r_i t_i\), with \(t_i\in H\). The new residual factor is \(t_{\pi _g(i)}^{-1}h_i(g)t_i\). Its total class contains each \([t_i]\) and its inverse exactly once, so they cancel. Reindexing the cosets also only reorders an abelian product. This proves independence of all choices.

                          The linked transfer definitions specialize the homomorphism \(\phi :H\to A\) to the quotient \(H\to H_{\mathrm {ab}}\). The indexed formula takes \(f(i)=r_i\) and \(\pi =\pi _g\); its factor \(r_{\pi _g(i)}^{-1}g r_i\) is precisely \(h_i(g)\). Its finite-index instance follows from the finite bijective family of cosets.

                          Proposition. Transfer computes total local orientation [frap-000X]

                            Let \(G\) act transitively on a finite nonempty position set \(I\), and on a tagged union of fibers \(X_i\), carrying \(X_i\) bijectively to \(X_{g i}\). Fix \(i_0\in I\) and put \(H=G_{i_0}\). Let \(K\) act faithfully on a nonempty set \(\Omega \); choose a bijective base frame \(c_0:\Omega \to X_{i_0}\) and a homomorphism \(\rho :H\to K\) satisfying \(h c_0=c_0\rho (h)\) for every \(h\in H\). Choose \(r_i\in G\) with \(r_i i_0=i\), and set \(c_i=r_i c_0\). Then the total-orientation character of this framed transport is

                            \[\chi =\rho _{\mathrm {ab}}\circ T_{G,H}:G\longrightarrow K_{\mathrm {ab}},\]

                            where \(T_{G,H}\) is transfer and \(\rho _{\mathrm {ab}}([h])=[\rho (h)]\). This identifies the coset construction in [mayer2015artin, Section 3.3, p. 6] with the local-frame coordinates of [marchetti2025rubik, Section 3.1, pp. 6--7].

                            Proof. 

                              The map \(G/H\to I\), \(rH\mapsto r i_0\), is a bijection: \(r i_0=s i_0\) exactly when \(s^{-1}r\in H\). Thus \(H\) has finite index \(|I|\), and the \(r_i\) are left-coset representatives. Write \(\pi _g(i)=g i\) and \(g r_i=r_{\pi _g(i)}h_i(g)\). Then

                              \[g c_i=g r_i c_0=r_{\pi _g(i)}h_i(g)c_0 =c_{\pi _g(i)}\rho (h_i(g)).\]

                              Consequently every local factor belongs to \(K\), and its source-indexed coordinate is \(a_i(g)=\rho (h_i(g))\). A homomorphism sends commutators to commutators, so \(\rho _{\mathrm {ab}}\) is well defined. Multiplying the local classes gives

                              \[\chi (g)=\prod _i[\rho (h_i(g))] =\rho _{\mathrm {ab}}\left (\prod _i[h_i(g)]\right ).\]

                              The vertical arrows are the transported frames. The equality \(g r_i=r_{\pi _g(i)}h_i(g)\) makes the square commute: after the destination representative is removed, the remaining motion fixes \(i_0\) and acts through \(\rho \). Changing representatives changes these frames within \(K\), so the character is unchanged.

                              The induced map is Abelianization.map. Naturality of transfer is MonoidHom.transfer_comp, with \(\phi :H\to H_{\mathrm {ab}}\) the quotient map and the target map \(\rho _{\mathrm {ab}}\). That algebraic identity supports the last equality; the identification with the actual fiber action is the transported-frame argument above.

                              Example. Corner and edge stabilizers of the cube [frap-000Y]

                                Let \(G\) be the actual permutation group generated by the six outer-face quarter-turns of the ordinary cube, with its frame of reference fixed. Use the corner and edge fibers: cyclic corner labels with local group \(C_3\), and edge labels with local group \(C_2\). This specializes the local-orientation construction of [marchetti2025rubik, Section 3.1]; the orbit argument below identifies the stabilizers needed for transfer.

                                The action on eight corner positions is transitive. Adjacent corners share a face whose quarter-turn takes one to the other, and the cube's edge graph is connected. Hence a corner stabilizer \(H_c\) has index eight. Its action on the three cyclically ordered labels gives a homomorphism \(\rho _c:H_c\to C_3\). The action on twelve edge positions is also transitive: the four edges of one face form a single orbit under its quarter-turn; neighboring faces share an edge; and the face-adjacency graph is connected. Thus an edge stabilizer \(H_e\) has index twelve, with local action \(\rho _e:H_e\to C_2\).

                                Choose a frame at the fixed corner or edge and transport it using coset representatives. The transfer comparison gives

                                \[\chi _c=(\rho _c)_{\mathrm {ab}}\circ T_{G,H_c},\qquad \chi _e=(\rho _e)_{\mathrm {ab}}\circ T_{G,H_e}.\]

                                We identify the abelianizations of \(C_3\) and \(C_2\) with those groups. The face-turn calculation proves that the twist and flip characters vanish on every legal generator, hence on \(G\). It follows that these two local images of transfer vanish. It does not follow that \(T_{G,H_c}\) or \(T_{G,H_e}\) is trivial in the larger stabilizer abelianizations. No classification of \(G\) was needed.

                                Dimension and local rotation groups [frap-0013]
                                ✍️source

                                  The local corner groups of simplices and cubes are alternating groups. Their commutator subgroups determine the information that survives the total-orientation character [marchetti2025rubik, Sections 4--5].

                                  Proposition. Local rotations of simplices and cubes [frap-000Z]

                                    For \(d\geq 3\), a vertex figure of a \(d\)-simplex or a \(d\)-cube is a \((d-1)\)-simplex. Its automorphism group is \(S_d\), and its rotation subgroup is \(A_d\). These are the local corner groups used in [marchetti2025rubik, Section 4, p. 10, and Section 5, p. 14].

                                    Proof. 

                                      At a vertex of either polytope there are \(d\) incident edges. Faces containing that vertex correspond to subsets of those edges: for a simplex one chooses the other vertices, and for a cube one chooses the coordinate directions in which the face varies. This is the face poset of a simplex on \(d\) vertices, of dimension \(d-1\). An automorphism permutes its vertices, and every vertex permutation extends uniquely to the subsets, giving \(S_d\).

                                      A base flag orders these \(d\) vertices. Its distinguished involutions are adjacent transpositions. They generate \(S_d\), and the even words are precisely the even permutations. Thus the rotation subgroup is \(A_d\).

                                      Here \(d\) is abstract dimension. Varying the layer count of an \(N\times N\times N\) cube leaves the ambient dimension equal to three; it changes the piece orbits and allowed layer moves instead.

                                      Theorem. Abelianizations of the local alternating groups [frap-0010]

                                        Let \(A_d\) be the alternating group on \(d\) letters, with composition rightmost first. Then

                                        \[(A_3)_{\mathrm {ab}}\cong C_3,\qquad (A_4)_{\mathrm {ab}}\cong C_3,\qquad (A_d)_{\mathrm {ab}}=1\quad (d\geq 5).\]

                                        The corresponding commutator subgroups are stated after Proposition 4.1 in [marchetti2025rubik, Section 4.1, p. 11]. We give the elementary calculation, using \([g,h]=ghg^{-1}h^{-1}\).

                                        Proof. 

                                          The group \(A_3\) is cyclic of order three, so already abelian. In \(A_4\), let \(V=\{1,(12)(34),(13)(24),(14)(23)\}\). Products and inverses stay in this four-element set, and conjugation permutes its three nonidentity elements. Hence \(V\) is normal. The quotient has order three, so it is cyclic and \([A_4,A_4]\subseteq V\). Direct multiplication gives \([(123),(124)]=(12)(34)\). Conjugating this element by the three-cycle \((123)\) gives the other two double transpositions. The commutator subgroup is normal, so it contains all of \(V\); equality follows.

                                          For \(d\geq 5\), put \(g=(12)(34)\) and \(h=(12)(35)\) in \(A_d\). Multiplication gives \([g,h]=(345)\). Every three-cycle is conjugate to this one within \(A_d\): first choose a conjugating permutation in \(S_d\); if it is odd, compose on the right with a transposition of two symbols outside \(\{3,4,5\}\). This centralizes \((345)\) and changes the parity. Thus the normal subgroup \([A_d,A_d]\) contains every three-cycle.

                                          Three-cycles generate all even permutations. Write an even permutation as a product of an even number of transpositions and pair them. Equal transpositions cancel; two sharing one symbol give a three-cycle; a disjoint pair satisfies \((ab)(cd)=(acb)(acd)\). Hence every pair is a product of three-cycles, proving \([A_d,A_d]=A_d\).

                                          The three cases have separate formalized ingredients. alternatingGroup.isCyclic_of_card_le_three gives the cyclicity of \(A_3\). For four letters, alternatingGroup.kleinFour_eq_commutator identifies \(V\) with the commutator subgroup, and TauCeti.card_abelianization_alternatingGroup gives the quotient's cardinality three; the displayed cyclic isomorphism then uses the elementary classification of groups of prime order. For \(d\geq 5\), commutator_alternatingGroup_eq_top is the perfection statement. These links concern alternating groups, not the entire puzzle group.

                                          Remark. What the corner character can detect [frap-0011]

                                            For a \(d\)-simplex or \(d\)-cube with \(d\geq 3\), the local corner groups have rotation subgroup \(A_d\). Their contribution to the total-orientation character passes through \((A_d)_{\mathrm {ab}}\). The abelianization calculation therefore measures exactly what this detector can see, as in [marchetti2025rubik, Section 4.1, p. 11, and equation (19), p. 14].

                                            In dimension three the target is \(C_3\), the familiar corner twist sum. In dimension four the local group has grown to the nonabelian \(A_4\), but the character still sees only a \(C_3\) quotient: every local change in the Klein four group disappears from it. In dimensions five and above the local group is perfect, so this corner character is identically trivial on the ambient framed transports.

                                            A lost detector does not classify the legal moves. Other piece types, permutation signs and the subgroup actually generated by facet turns may still impose constraints. Even when the target is nontrivial, conservation requires a generator check; the local abelianization alone does not supply one.

                                            Backlinks

                                              Further connections [frap-0007]

                                                The total-orientation character can expose a constraint, but what survives when we change the local frame or the dimension? Framed orientations and transfer and dimension develop the first connections: wreath product records positions and local transport, while abelianized characters detect constraints when the allowed generators preserve them [marchetti2025rubik, Section 3.1]. Finite-index transfer organizes related calculations through coset representatives [mayer2015artin, Sections 3.1--3.3]. The dimension calculation explains how a local abelianization can lose this detecting power without deciding reachability.

                                                The following outlooks ask different questions about transport and functions on a move group. Each needs its own construction; neither is a prerequisite for continuing to the layered cubes.

                                                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.

                                                          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.

                                                                  Which moves survive a quotient? [frap-0019]

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

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

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

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

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

                                                                    The antipodal sticker quotient

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

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

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

                                                                      Why one face turn fails

                                                                        The positive-\(z\) face turn uses

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

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

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

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

                                                                        A synchronized opposite-layer move

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

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

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

                                                                          What the cover does not provide

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

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

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

                                                                            Related

                                                                              Theorem. Abelianizations of the local alternating groups [frap-0010]

                                                                                Let \(A_d\) be the alternating group on \(d\) letters, with composition rightmost first. Then

                                                                                \[(A_3)_{\mathrm {ab}}\cong C_3,\qquad (A_4)_{\mathrm {ab}}\cong C_3,\qquad (A_d)_{\mathrm {ab}}=1\quad (d\geq 5).\]

                                                                                The corresponding commutator subgroups are stated after Proposition 4.1 in [marchetti2025rubik, Section 4.1, p. 11]. We give the elementary calculation, using \([g,h]=ghg^{-1}h^{-1}\).

                                                                                Proof. 

                                                                                  The group \(A_3\) is cyclic of order three, so already abelian. In \(A_4\), let \(V=\{1,(12)(34),(13)(24),(14)(23)\}\). Products and inverses stay in this four-element set, and conjugation permutes its three nonidentity elements. Hence \(V\) is normal. The quotient has order three, so it is cyclic and \([A_4,A_4]\subseteq V\). Direct multiplication gives \([(123),(124)]=(12)(34)\). Conjugating this element by the three-cycle \((123)\) gives the other two double transpositions. The commutator subgroup is normal, so it contains all of \(V\); equality follows.

                                                                                  For \(d\geq 5\), put \(g=(12)(34)\) and \(h=(12)(35)\) in \(A_d\). Multiplication gives \([g,h]=(345)\). Every three-cycle is conjugate to this one within \(A_d\): first choose a conjugating permutation in \(S_d\); if it is odd, compose on the right with a transposition of two symbols outside \(\{3,4,5\}\). This centralizes \((345)\) and changes the parity. Thus the normal subgroup \([A_d,A_d]\) contains every three-cycle.

                                                                                  Three-cycles generate all even permutations. Write an even permutation as a product of an even number of transpositions and pair them. Equal transpositions cancel; two sharing one symbol give a three-cycle; a disjoint pair satisfies \((ab)(cd)=(acb)(acd)\). Hence every pair is a product of three-cycles, proving \([A_d,A_d]=A_d\).

                                                                                  The three cases have separate formalized ingredients. alternatingGroup.isCyclic_of_card_le_three gives the cyclicity of \(A_3\). For four letters, alternatingGroup.kleinFour_eq_commutator identifies \(V\) with the commutator subgroup, and TauCeti.card_abelianization_alternatingGroup gives the quotient's cardinality three; the displayed cyclic isomorphism then uses the elementary classification of groups of prime order. For \(d\geq 5\), commutator_alternatingGroup_eq_top is the perfection statement. These links concern alternating groups, not the entire puzzle group.

                                                                                  Proposition. Local rotations of simplices and cubes [frap-000Z]

                                                                                    For \(d\geq 3\), a vertex figure of a \(d\)-simplex or a \(d\)-cube is a \((d-1)\)-simplex. Its automorphism group is \(S_d\), and its rotation subgroup is \(A_d\). These are the local corner groups used in [marchetti2025rubik, Section 4, p. 10, and Section 5, p. 14].

                                                                                    Proof. 

                                                                                      At a vertex of either polytope there are \(d\) incident edges. Faces containing that vertex correspond to subsets of those edges: for a simplex one chooses the other vertices, and for a cube one chooses the coordinate directions in which the face varies. This is the face poset of a simplex on \(d\) vertices, of dimension \(d-1\). An automorphism permutes its vertices, and every vertex permutation extends uniquely to the subsets, giving \(S_d\).

                                                                                      A base flag orders these \(d\) vertices. Its distinguished involutions are adjacent transpositions. They generate \(S_d\), and the even words are precisely the even permutations. Thus the rotation subgroup is \(A_d\).

                                                                                      Here \(d\) is abstract dimension. Varying the layer count of an \(N\times N\times N\) cube leaves the ambient dimension equal to three; it changes the piece orbits and allowed layer moves instead.

                                                                                      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.