[P1] {'assoc_fail': 0, 'orders': {1: 1, 3: 80, 9: 648}, 'exponent': 9, 'centre_is_Z': True, 'commutators_eq_Z': True, 'nclasses': 89, 'class_sizes': {1: 9, 9: 80}, 'camina': True, 'mho1_size': 9, 'mho1_eq_Z': True, 'omega_size': 81, 'omega_subgroup': True, 'omega_abelian': True}
[P1] induced-character check: {'A_abelian': True, 'A_normal': True, 'phi(e1),phi(e2)': ((0, 0), (0, 0)), 'mu_hom_and_Ind_mu_eq_chi_lambda': True}
[P1] linear class functions are homomorphisms (all pairs): True; orthonormality failures among 89: 0; sum of squared degrees = 729
[P1] indicator multiset (nu_1..nu_9): 80 x (0, 0, 1, 0, 0, 1, 0, 0, 1); 8 x (0, 0, 0, 0, 0, 0, 0, 0, 9); 1 x (1, 1, 1, 1, 1, 1, 1, 1, 1)
[P1] literal value set (j=1..9): [0, 1, 9]; nu_n periodic mod 9 (n<=18): True
[P2] {'assoc_fail': 0, 'orders': {1: 1, 3: 80, 9: 648}, 'exponent': 9, 'centre_is_Z': True, 'commutators_eq_Z': True, 'nclasses': 89, 'class_sizes': {1: 9, 9: 80}, 'camina': True, 'mho1_size': 9, 'mho1_eq_Z': True, 'omega_size': 81, 'omega_subgroup': True, 'omega_abelian': False}
[P2] induced-character check: {'A_abelian': True, 'A_normal': True, 'phi(e1),phi(e2)': ((0, 0), (1, 0)), 'mu_hom_and_Ind_mu_eq_chi_lambda': True}
[P2] linear class functions are homomorphisms (all pairs): True; orthonormality failures among 89: 0; sum of squared degrees = 729
[P2] indicator multiset (nu_1..nu_9): 80 x (0, 0, 1, 0, 0, 1, 0, 0, 1); 8 x (0, 0, 0, 0, 0, 0, 0, 0, 9); 1 x (1, 1, 1, 1, 1, 1, 1, 1, 1)
[P2] literal value set (j=1..9): [0, 1, 9]; nu_n periodic mod 9 (n<=18): True
bijection (tau on classes via alpha, sigma on characters): class sizes True; character table True; power maps n=0..18 True; indicators n=1..18 True
fusion rules preserved by sigma (all 89^2*89/2 inner products): True
[P1] |P:P'| = 81, |P'| = 9, |mho_1(P)| = 9, |Omega_1(P)| = 81, Omega_1 abelian: True
[P2] |P:P'| = 81, |P'| = 9, |mho_1(P)| = 9, |Omega_1(P)| = 81, Omega_1 abelian: False
ALL VERIFIER-A CHECKS PASSED in 10.1 s
