PASS [G] conventions: the Pauli routine equals the Kronecker product of 2x2 matrices (qubit q = bit q; Y = [[0,-i],[i,0]]) for Y0Y1Y2Y3, X0Z1Y2Y3, Y0Y3Y4Y5; CX(c,t) flips t iff c = 1; H' = [[1,1],[1,-1]]
PASS [C1] instrument: 13 leaves (9 accepted); on every path the measured Paulis are Hermitian and commute pairwise
PASS [C1] completeness: the 13 leaf projectors sum to the identity on (C^2)^(x)8 (all 256 columns, exact)
PASS [C2] rank one: each of the 9 accepted leaf operators maps u_0..u_8 (Dicke basis of Sym^8) into one fixed nonzero line
PASS [C2] sector Z: Pi_L u_w = 0 for w not in {2,6}, Pi_L u_2 = -Pi_L u_6 = chi_L/2 with chi_L = sum over 8 strings e_i+e_j of |e_i+e_j> - |complement>: A1 {2,3}x{4,5,6,7}, A2 {0,1}x{4,5,6,7}, B {0,3}x{1,2} + {4,5}x{6,7}
PASS [C3] capture: <E_P|Pi_L|E_P> = 2/7 for all 9 accepted leaves, so per sector A1+A2 keep 4/7 and A1+A2+B keep 6/7 of E_P (E_Z = (D2-D6)/sqrt2, E_X = H^(x)8 E_Z, E_Y = R^(x)8 E_Z)
PASS [C4] decoders, sector Z: the explicit circuits D_A1, D_A2, D_B (17, 17, 21 gates of CX, X, H) map chi_A1, chi_A2, chi_B exactly to multiples of |CCZ> on qubits (4,5,6), (4,5,6), (0,1,4) times |0> on the other five; control: D_B chi_A1 is not
PASS [C4] sector X: chi^X_L = H^(x)8 chi^Z_L (up to a factor; X-type leaf projectors) and D_L H^(x)8 maps chi^X_L exactly to |CCZ>(x)|0>^5
PASS [C4] sector Y: chi^Y_L = R^(x)8 chi^Z_L (Y-type projectors, Gaussian integers) and D_L R^dag(x)8 maps chi^Y_L exactly to |CCZ>(x)|0>^5; R^(x)8 in place of R^dag(x)8 also works (R^(x)8 R^(x)8 = X^(x)8 = -1 there); control: D_L H^(x)8 fails
PASS [C4] independent invariant: the Pauli spectrum of all 9 leaf states equals that of |CCZ>(x)|0>^5 (32 x |<P>| = 1, 896 x 1/2, rest 0)
PASS [C5] per-leaf forms <u_w|Pi_L|u_w'>: sector Z 4 v2v2^T, sector X (v1-v3)(v1-v3)^T, sector Y (v1+v3)(v1+v3)^T (v_k = e_k - e_{k+4})
PASS [C5] total: N = sum over the 9 accepted leaves = 6 (v1v1^T + 2 v2v2^T + v3v3^T) exactly (9x9 over Q)
PASS [C5] M_8: E_X, E_Y, E_Z lie in Sym^8, are pairwise orthogonal and orthogonal to the six tau^(x)8 (rank 6, so dim M_8 = 3); Pi_M8 has form 7(v1v1^T + 2v2v2^T + v3v3^T) and N = (6/7) Pi_M8 on Sym^8
PASS [C6] Bloch conversion (polynomial identities over Q(i)): x + iy = 2 conj(a) b, x^2+y^2+z^2 = |psi|^4 and (|psi|^12 - x^6-y^6-z^6)/2 = 6|f6|^2 = (3/2)(|psi|^4-x^2)(|psi|^4-y^2)(|psi|^4-z^2), f6 = ab(a^4-b^4); so m_3 = 6|f6|^2 for normalised psi
PASS [C6] success probability: <psi^8| sum_L Pi_L |psi^8> = sum_ww' conj(a^(8-w)b^w) N_ww' a^(8-w')b^w' = 6|f6|^2 |psi|^4 identically, so p(psi) = m_3(psi) for every pure psi
PASS [C7] direct exact evaluation of sum_L ||Pi_L psi^8||^2 at psi = (1, 2+i), (2-i, 3), (1+2i, 1-i): p = m_3 from the Bloch vector, and the A-leaves alone give (2/3) m_3
PASS [C8] positive control: the six A-leaves alone give N_RL = 4(v1v1^T + 2v2v2^T + v3v3^T), i.e. 4|f6|^2|psi|^4 = (2/3) m_3, the value of Rizzo-Leone's eight-copy protocol
PASS [C8] negative control: the leaf (+Z^E1, +Z^E2, -Z^{0,1,4,6}) is rank one with capture 1/7, but its state chi_C (pairs 03,12,45,67) is a stabilizer state (256 Paulis with |<P>| = 1), so no Clifford decoder yields CCZ
PASS [C8] negative control: B with the opposite P^E2 outcome (the rejected leaf) has rank 2 on Sym^8 in every sector, not rank one; in sector Z its image is spanned by GHZ- = |0^8>-|1^8> and chi_C, both stabilizer states
PASS [C9] values: T-type (x^2=y^2=z^2=1/3) p = m_3 = 4/9 vs (2/3) m_3 = 8/27 and (7/6) m_3 = 14/27; H-type (x^2=z^2=1/2, y=0) 3/8 vs 1/4 and 7/16; m_3 = 0 at the six stabilizer states
VERDICT PASS: an explicit U_8 protocol outputs exactly |CCZ> with probability m_3(psi) > (2/3) m_3(psi) at every non-stabilizer psi, so Eq. (2) fails
