=== (1) primitivity, L_A, out-degrees, |Aut(A)|
    1_4      primitive=True  L_A=2  out-degrees=[4, 4, 4, 4]  |Aut|=24
OK   1_4: primitive, L_A = 2, |Aut(A)| = 24
    1_5      primitive=True  L_A=2  out-degrees=[5, 5, 5, 5, 5]  |Aut|=120
OK   1_5: primitive, L_A = 2, |Aut(A)| = 120
    J_4-I_4  primitive=True  L_A=2  out-degrees=[3, 3, 3, 3]  |Aut|=24
OK   J_4-I_4: primitive, L_A = 2, |Aut(A)| = 24
    J_5-I_5  primitive=True  L_A=2  out-degrees=[4, 4, 4, 4, 4]  |Aut|=120
OK   J_5-I_5: primitive, L_A = 2, |Aut(A)| = 120
    A_D      primitive=True  L_A=3  out-degrees=[3, 3, 2, 2]  |Aut|=4
OK   A_D: primitive, L_A = 3, |Aut(A)| = 4
    A_E      primitive=True  L_A=3  out-degrees=[3, 3, 3, 3]  |Aut|=8
OK   A_E: primitive, L_A = 3, |Aut(A)| = 8
    A_hub    primitive=True  L_A=3  out-degrees=[2, 2, 2, 2, 5]  |Aut|=24
OK   A_hub: primitive, L_A = 3, |Aut(A)| = 24
    A_7      primitive=True  L_A=5  out-degrees=[1, 1, 1, 1, 2, 1, 4]  |Aut|=24
OK   A_7: primitive, L_A = 5, |Aut(A)| = 24
OK   A_D: out-degrees (3,3,2,2); A_hub: hub is the only vertex of out-degree 5
OK   C_4: B^2 = 2I + 2R (R = antipodal permutation)
=== (2) exact block model of a non-commutative 4x4 magic unitary
OK   P, Q projections, PQ not a partial isometry, [P,Q] != 0 (Lemma 2.2(b))
OK   q magic, non-commutative
OK   1_4: q commutes with A; canonical relations first fail at level 2 = L_A (Theorem 1.1)
OK   J_4-I_4: q commutes with A; canonical relations first fail at level 2 = L_A (Theorem 1.1)
OK   A_hub: right PF vector (1,1,1,1,2)/6 with eigenvalue 3
OK   A_hub: q (+) 1 magic, commutes with A_hub, preserves the PF vector
OK   A_hub: levels 1, 2 hold and level 3 = L_A fails for u -> q (+) 1 (Example 4.2)
OK   A_7: q (+) I_3 magic, commutes with A_7
OK   A_7: levels 1-4 hold, level 5 = L_A fails (sharpness, Remark 3.2)
=== (3) Proposition 5.1 (A_D)
OK   p = diag(E,F) magic, commutes with A_D, non-commutative
OK   w = diag(s,t,1,1): range/source matrices = identity, A_D P' = Q' A_D
OK   w = diag(s,t,1,1): all words of length <= 6 are partial isometries
OK   Delta(s) = s (x) s and Delta(t) = t (x) t in the model; [s,t] != 0
=== (4) Lemma 5.2 and Proposition 5.3 (A_E)
OK   model: a,b,c,d self-adjoint; a^2 = d^2 = e, b^2 = c^2 = 1 - e; e projection
OK   model: ab = cd = ac = bd = 0 (cubic relations)
OK   model: p magic, commutes with B = adj(C_4) and with A_E = B + I
OK   model: p_{i+2,j+2} = p_ij and p_{i+2,j} = p_{i,j+2}
OK   model: [p_11, p_22] != 0, so C(QAut(C_4)) is non-commutative
OK   model: e commutes with every p_ij
OK   model: a = p_11 - p_13, b = p_12 - p_14, c = p_21 - p_23, d = p_22 - p_24
OK   coproducts: Delta(a) = a(x)a + b(x)c, Delta(b) = a(x)b + b(x)d, Delta(c) = c(x)a + d(x)c, Delta(d) = c(x)b + d(x)d
OK   g = 2e - 1 is a group-like unitary, g != 1 (so H_2^+ lacks property (G))
OK   embedding w: entries partial isometries; P' = Q' magic; A_E P' = Q' A_E; constant PF vector preserved
OK   embedding w: all words of length <= 6 are partial isometries
OK   embedding w: Delta(w_ij) = sum_k w_ik (x) w_kj for all i, j (checked in the model)
OK   canonical u -> p for A_E fails at level 2 <= L_A = 3 (Theorems 3.1, 1.1)
=== summary: 36 of 36 checks passed
ALL CHECKS PASSED
