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.