phi_5 for k=9: [252, 504, 126]  phi_9 for k=9: [630, 1680, 756, 72, 1]  phi_4 for k=7: [21, 105, 35]  phi_7 for k=7: [140, 210, 42, 1]
   {'k': 7, 'deg_phi_m+1': 2, 'deg_phi_k': 3, 'C1': True, 'C2': True, 'C3': True}
   {'k': 9, 'deg_phi_m+1': 2, 'deg_phi_k': 4, 'C1': True, 'C2': True, 'C3': True}
   {'k': 11, 'deg_phi_m+1': 3, 'deg_phi_k': 5, 'C1': True, 'C2': True, 'C3': True}
   {'k': 13, 'deg_phi_m+1': 3, 'deg_phi_k': 6, 'C1': True, 'C2': True, 'C3': True}
   {'k': 15, 'deg_phi_m+1': 4, 'deg_phi_k': 7, 'C1': True, 'C2': True, 'C3': True}
   {'k': 51, 'deg_phi_m+1': 13, 'deg_phi_k': 25, 'C1': True, 'C2': True, 'C3': True}
   {'k': 101, 'deg_phi_m+1': 25, 'deg_phi_k': 50, 'C1': True, 'C2': True, 'C3': True}
   {'k': 151, 'deg_phi_m+1': 38, 'deg_phi_k': 75, 'C1': True, 'C2': True, 'C3': True}
   {'k': 201, 'deg_phi_m+1': 50, 'deg_phi_k': 100, 'C1': True, 'C2': True, 'C3': True}
part 1: odd k in [7,201]: (C1),(C2),(C3) all hold: True; failures: []; time 0.8s
part 2: exact sympy cross-check over Q, odd k in [7,61] (phi formula checked against direct expansion for k<=15): all hold: True; time 0.6s
   k=7: m=3, exponents n=1 mod m: [1, 4, 7, 10, 13]; |sigma|=1.71462; max relative error of P_n vs (sigma,0,0,0,0) = 3.59e-89; explicit 6-term sum: max |coef - [t^n]t| = 1.26e-87 -> OK
   k=9: m=4, exponents n=1 mod m: [1, 5, 9, 13, 17]; |sigma|=6.80248; max relative error of P_n vs (sigma,0,0,0,0) = 1.43e-97; explicit 8-term sum: max |coef - [t^n]t| = 2.8e-95 -> OK
   k=11: m=5, exponents n=1 mod m: [1, 6, 11, 16, 21]; |sigma|=10.0117; max relative error of P_n vs (sigma,0,0,0,0) = 1.1e-105; explicit 10-term sum: max |coef - [t^n]t| = 2.26e-102 -> OK
   k=13: m=6, exponents n=1 mod m: [1, 7, 13, 19, 25]; |sigma|=6.91975; max relative error of P_n vs (sigma,0,0,0,0) = 1.88e-112; explicit 12-term sum: max |coef - [t^n]t| = 8.18e-110 -> OK
   k=15: m=7, exponents n=1 mod m: [1, 8, 15, 22, 29]; |sigma|=10.6549; max relative error of P_n vs (sigma,0,0,0,0) = 2.94e-120; explicit 14-term sum: max |coef - [t^n]t| = 1.96e-116 -> OK
   k=17: m=8, exponents n=1 mod m: [1, 9, 17, 25, 33]; |sigma|=6.96471; max relative error of P_n vs (sigma,0,0,0,0) = 8.05e-129; explicit 16-term sum: max |coef - [t^n]t| = 1.29e-124 -> OK
   k=19: m=9, exponents n=1 mod m: [1, 10, 19, 28, 37]; |sigma|=10.6468; max relative error of P_n vs (sigma,0,0,0,0) = 2.07e-136; explicit 18-term sum: max |coef - [t^n]t| = 9.95e-132 -> OK
   k=21: m=10, exponents n=1 mod m: [1, 11, 21, 31, 41]; |sigma|=7.00137; max relative error of P_n vs (sigma,0,0,0,0) = 2.41e-145; explicit 20-term sum: max |coef - [t^n]t| = 2.73e-139 -> OK
   k=31: m=15, exponents n=1 mod m: [1, 16, 31, 46, 61]; |sigma|=26.229; max relative error of P_n vs (sigma,0,0,0,0) = 1.17e-183 -> OK
   k=51: m=25, exponents n=1 mod m: [1, 26, 51, 76, 101]; |sigma|=25.4981; max relative error of P_n vs (sigma,0,0,0,0) = 6.22e-264 -> OK
   k=101: m=50, exponents n=1 mod m: [1, 51, 101, 151, 201]; |sigma|=43.8846; max relative error of P_n vs (sigma,0,0,0,0) = 3.99e-460 -> OK
   k=201: m=100, exponents n=1 mod m: [1, 101, 201, 301, 401]; |sigma|=43.5122; max relative error of P_n vs (sigma,0,0,0,0) = 5.65e-855 -> OK
part 3: numeric construction: all OK
   corruption: phi_k replaced by phi_{m+1}*(I+1): rejected = True
   corruption: phi_{m+1} replaced by (I+2)^2: rejected = True
   corruption: phi_k replaced by phi_{m+1}+5: rejected = True
   corruption: k=5 (construction degenerates): rejected = True
part 4: all corruptions rejected: True
OVERALL: PASS (total 56.1s)
