Four subgroups. No model.
Each stage establishes exactly one named invariant and hands a smaller generator set to the next. That is what makes the progress monotone: stage 2 cannot un-orient the edges stage 1 oriented, because the moves that could are no longer admissible. Not a heuristic — a restriction.
What each stage locks
Stage 1 · G0 → G1 — every edge oriented. 2,048 coordinates, max depth 7. From here F and B quarter turns are dropped: they would re-flip edges.
Stage 2 · G1 → G2 — every corner oriented and all four E-slice edges home. 1,082,565 coordinates, max depth 10. Now L and R quarter turns go too.
Stage 3 · G2 → G3 — every piece in its own orbit. 705,600 coordinates (29,400 cosets × 24 E-slice arrangements — and 29,400 is exactly the known index |G2 : G3|), max depth 13. U and D quarter turns are dropped last.
Stage 4 · G3 → {e} — solved, with only half turns left. 7,962,624 coordinates, of which exactly 663,552 are reachable, max depth 15.
Every table is a complete breadth-first search from that stage’s goal set, so each stage’s solution is the exact optimum for that stage and termination is guaranteed by construction. 7 + 10 + 13 + 15 = 45 moves is the worst case this solver can ever produce — proved, not measured. The mean over 150 random scrambles is 31.2.
Thistlethwaite’s algorithm is from 1981 — the subgroup chain, the coordinates and the depth bounds are his, and these tables reproduce them rather than discovering them. God’s number = 20 was proved by Rokicki, Kociemba, Davidson and Dethridge in 2010, so this is not an optimal solver and does not claim to be. What is ours is the wiring: reachability as an admission floor with named refusals, the shrinking generator set as an affordance card, and the solve executed through a shield. See it against the layer method →