M22: CFOP solver — optimal cross, F2L, full OLL/PLL with named cases

[cypherair]

What

Adds CFOP as a fifth solving method: the speedcubing method, with staged playback showing Cross, the four F2L pairs, and the recognized last-layer cases by name ("OLL 27 (Sune)", "PLL · Gc"). Measured average solution: 60.6 moves over 40 seeded random states.

Design

Cross — table-optimal. An exact BFS distance table over the four cross edges (P(12,4) · 2⁴ = 190,080 entries, built once per process in milliseconds) drives a greedy descent. The cross is never longer than 8 moves, and the test suite asserts it.

F2L — macro search instead of a transcribed case table. The plan called for the standard 41-case algorithm table; the implementation instead runs a per-pair Dijkstra over the tracked corner+edge state (24 × 24 codes) with an alphabet of the standard triggers — U turns, the slot's six inserts (R U R' and F' U F families, y-conjugated per slot), and pop-out triggers for the other unsolved slots. Every macro provably preserves the cross and all solved pairs, so any path is clean by construction: no transcription risk, automatic extraction of buried pieces, complete coverage without enumerating cases — and the output still reads like textbook F2L because the macros are the textbook triggers. Pairs are inserted in greedy cheapest-first order.

OLL/PLL — the full 57 + 21 algorithm sets, with nothing resting on transcription.

• Algorithms are written in extended notation exactly as the standard sheets print them. A new ExtendedNotation parser expands wide turns, M/E/S slices, and x/y/z rotations to outer moves using the same fixed-center equivalences as the scene (M ≙ L′ R + x′ …), tracking the cumulative frame so letters after a rotation land on the right faces. The net frame must end as identity or a pure y rotation (anything else would tilt the U axis and change the algorithm's meaning). Anchor test: M2 U M2 U2 M2 U M2 must equal the H-perm.
• Recognition keys are derived from the algorithms themselves: the case an algorithm solves is the state its inverse produces from solved. validationIssues() re-checks every entry — parseability, first-two-layer preservation, case uniqueness modulo U rotation — and proves exhaustive coverage of all 216 legal last-layer orientation patterns and all 288 permutation states. The test suite requires the issue list to be empty, so a single mistyped algorithm fails CI with a message naming the entry.
• OLL recognizes by orientation-pattern lookup under the four U rotations; PLL recognizes by trial (pre-AUF × algorithm × post-AUF), which also absorbs any AUF convention differences between sheets.

Shared refactor: [Move].rotatedY / merged() and the per-move piece-action code tables moved out of BeginnerSolver into shared internal helpers (MoveTransforms.swift); BeginnerSolver delegates, its tests unchanged. The CubeState.rotatedY conjugation the plan sketched turned out to be unnecessary — slot conjugation happens at the move level only.

Verification

• swift test -c release: 85 tests pass (9 new). The new suite includes: parser anchors; the full table-validation pass (all 119 algorithms checked, both coverage proofs); by-construction tests for every OLL and PLL entry (build the case from the algorithm's inverse plus a random AUF, solve, assert the recognized case name matches); a 200-state sweep with cumulative per-stage invariants (cross → pairs → oriented → solved) and cross-length ≤ 8; average length ≤ 70 (measured 60.6).
• macOS + iOS Simulator builds green.
• Interactive (macOS app): Settings › Solving shows 5 methods; scramble → CFOP solve produced a 57-move, 7-stage solution; chips advanced through "Cross", four "F2L pair n" stages, OLL, and "PLL · Gc" to "Solved!".

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