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1. Model: 54 stickers, six face-turn generators
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N_STICKERS = 54
Faces: U D L R F B  (each 9 facelets, ids face*9 + 0..8)
  U: perm sends 20 stickers (center fixed); e.g. 0→2
  D: perm sends 20 stickers (center fixed); e.g. 0→0
  L: perm sends 20 stickers (center fixed); e.g. 0→36
  R: perm sends 20 stickers (center fixed); e.g. 0→0
  F: perm sends 20 stickers (center fixed); e.g. 0→0
  B: perm sends 20 stickers (center fixed); e.g. 0→24

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2. Start: solved (identity)
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        U U U
        U U U
        U U U
L L L  F F F  R R R  B B B
L L L  F F F  R R R  B B B
L L L  F F F  R R R  B B B
        D D D
        D D D
        D D D
solved=True  stickers_off_home=0
Three interlocking rings (edge 4-cycles from U / R / F):
  U-ring: (F → L → B → R → …)
  R-ring: (U → B → D → F → …)
  F-ring: (U → R → D → L → …)
  (shared stickers at face meetings = interlocking)

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3. Scramble: R U R' U'
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tokens: ['R', 'U', "R'", "U'"]
invariants: corner/edge parity match, corner twist sum ≡ 0 (mod 3)
        U U L
        U U F
        U U F
B L L  F F D  R R U  B R R
L L L  F F U  B R R  B B B
L L L  F F F  U R R  B B B
        D D R
        D D D
        D D D
solved=False  stickers_off_home=12
Three interlocking rings (edge 4-cycles from U / R / F):
  U-ring: (F → L → R → R → …)
  R-ring: (F → B → D → U → …)
  F-ring: (U → B → D → L → …)
  (shared stickers at face meetings = interlocking)
--- sticker list (index:color) ---
00:U 01:U 02:L 03:U 04:U 05:F 06:U 07:U 08:F 09:D 10:D 11:R 12:D 13:D 14:D 15:D 16:D 17:D
18:B 19:L 20:L 21:L 22:L 23:L 24:L 25:L 26:L 27:R 28:R 29:U 30:B 31:R 32:R 33:U 34:R 35:R
36:F 37:F 38:D 39:F 40:F 41:U 42:F 43:F 44:F 45:B 46:R 47:R 48:B 49:B 50:B 51:B 52:B 53:B

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4. Inverse sequence → expect solved
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inverse tokens: ['U', 'R', "U'", "R'"]
        U U U
        U U U
        U U U
L L L  F F F  R R R  B B B
L L L  F F F  R R R  B B B
L L L  F F F  R R R  B B B
        D D D
        D D D
        D D D
solved=True  stickers_off_home=0
Three interlocking rings (edge 4-cycles from U / R / F):
  U-ring: (F → L → B → R → …)
  R-ring: (U → B → D → F → …)
  F-ring: (U → R → D → L → …)
  (shared stickers at face meetings = interlocking)

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5. Result
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scramble then inverse returns to solved: True
OK — face turns permute stickers; inverse restores identity.