A = [[1, 0, 0, 0, 1], [0, 1, 0, 0, 1], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1], [1, 1, 1, 1, 1]]  primitive: True
model non-commutative: max ||[q_ab,q_cd]|| = 1.104
(I) P', Q' magic: True
    PF vector [0.1667 0.1667 0.1667 0.1667 0.3333] preserved: True
(II) A P' = Q' A: True
(III) level 1: max partial-isometry defect over admissible pairs = 9.242e-16
(III) level 2: max partial-isometry defect over admissible pairs = 1.834e-15
(III) level 3: max partial-isometry defect over admissible pairs = 6.223e-01
      length-2 products (alpha admissible, beta not) max |entry| = 3.9892373755459124e-17
=> (a) w defines a surjection C(G^2_A) -> C(S_4^+): S_4^+ IS a quantum subgroup of G^2_A.
(b) Step 4 with the loop path a->a (a=0): max ||[Q'_ab, P'_ad]|| over b,d in T = 1.42e-16
(c) Step 5 premise: max ||[Q'_(x,b), P'_(y,d)]|| over ALL x,b,y,d in T = 1.104  (should be 0 if Step 5 were valid)
    => Steps 1-5 run at level m+1 = 2 would 'prove' C(S_4^+) commutative: the proof is invalid as written.
(e) fixed points of w: [4]
    every x in [0] -> hub and hub -> every y in [0] (so alpha=(x,4,y) admissible for ALL x,y): True
    level-3 requirement w_(x,b) w_(y,d) partial isometry, max defect = 0.622 (>0: fails, as Thm C/fixed Thm A predict)
DONE v1
