Part 0. Rational bounds for pi and 2^(3/2)
PASS pi < 355/113 (Machin's formula with alternating-series bounds)   [pi <= 3.141592653589793]
PASS (99/70)^2 > 2, hence 2^(3/2) < 99/35
Part 1. Hypotheses of Theorems C and D
PASS alpha > 1
PASS k = 2 alpha/beta = 4/3 > 1, i.e. beta < 2 alpha
PASS k^2 = 16/9 < 3, i.e. beta > 2 alpha/sqrt(3)
Part 2. Constants of Lemma 4.1 for a = 1/4 (Table 2)
PASS q0 = k(a + 1/(12a)) = 7/9 < 1
PASS h_a = a/(k(4a^2 + 1)) = 3/20 < 1/4
PASS (a^2 + 1/4)(2k)^2/(2a^2) = 160/9, so q_a(h) = (7/9)/(1 - (160/9) h^2)
PASS denominator of q_a at h = h_a equals 1 - 1/(2(4a^2+1)) = 3/5 > 1/2
Part 3. Constants of Corollary 4.4: delta = 1/20, r = 1/100, rho = 1/100000 (Table 2)
PASS 0 < delta < h_a/2 = 3/40, and delta < 1/8
PASS q_a(2 delta) = 35/37 < 1
PASS 0 < rho < min(r, 1/2 - a)
PASS omega = (1 + rho/r)/(1 - rho/r) = 1001/999
PASS eps = (log omega)/(2k) = (3/4) artanh(1/1000) lies in [0.00075000025, 0.00075000026]   [[0.0007500002500000, 0.0007500002500002]]
PASS eps <= delta and eps < 0.00076
PASS delta + eps <= 2 delta < h_a (Lemma 4.1(ii) applies at h' <= delta + eps)
PASS q_a(delta + eps) < 0.81510   [q_a(delta+eps) <= 0.815099366821]
PASS eta = (delta/2)(1 - q_a(delta+eps)) - eps - 2 pi r^2 > 0.0032441   [eta >= 0.003244196995]
Part 4. Conditions (T1)-(T3) of Theorem C with kappa = 2^(beta/2)/c0 = 2^(3/2)/c0, T = max(10^5, ceil(100/c0))
PASS c0 = 1, T = 100000: (T1) kappa (T/2)^-2 <= eta   [lhs <= 0.000000001131]
PASS c0 = 1, T = 100000: (T2) 64/(T delta^2) + 4 kappa sum_{s>=T} s^-2 <= 1/2   [lhs <= 0.256113]
PASS c0 = 1, T = 100000: (T3) 1/T <= delta/4
PASS c0 = 1/100, T = 100000: (T1) kappa (T/2)^-2 <= eta   [lhs <= 0.000000113142]
PASS c0 = 1/100, T = 100000: (T2) 64/(T delta^2) + 4 kappa sum_{s>=T} s^-2 <= 1/2   [lhs <= 0.267314]
PASS c0 = 1/100, T = 100000: (T3) 1/T <= delta/4
PASS c0 = 1/1000, T = 100000: (T1) kappa (T/2)^-2 <= eta   [lhs <= 0.000001131428]
PASS c0 = 1/1000, T = 100000: (T2) 64/(T delta^2) + 4 kappa sum_{s>=T} s^-2 <= 1/2   [lhs <= 0.369143]
PASS c0 = 1/1000, T = 100000: (T3) 1/T <= delta/4
PASS c0 = 1/1000000, T = 100000000: (T1) kappa (T/2)^-2 <= eta   [lhs <= 0.000000001131]
PASS c0 = 1/1000000, T = 100000000: (T2) 64/(T delta^2) + 4 kappa sum_{s>=T} s^-2 <= 1/2   [lhs <= 0.113398]
PASS c0 = 1/1000000, T = 100000000: (T3) 1/T <= delta/4
PASS c0 = 1000, T = 100000: (T1) kappa (T/2)^-2 <= eta   [lhs <= 0.000000000001]
PASS c0 = 1000, T = 100000: (T2) 64/(T delta^2) + 4 kappa sum_{s>=T} s^-2 <= 1/2   [lhs <= 0.256000]
PASS c0 = 1000, T = 100000: (T3) 1/T <= delta/4
PASS every c0 >= 1, T = 10^5: kappa (T/2)^-2 < 1.2e-9 <= eta  (T1)   [lhs <= 0.0000000011314]
PASS every c0 >= 1, T = 10^5: 64/(T delta^2) + 4 kappa/(T-1) < 0.2562 <= 1/2  (T2)   [lhs <= 0.2561131]
PASS every 0 < c0 < 1: kappa (T/2)^-2 < 0.0012 <= eta  (T1)   [bound 0.00113142]
PASS every 0 < c0 < 1: 64/(T delta^2) + 4 kappa/(T-1) < 0.371 <= 1/2  (T2)   [bound 0.370285]
PASS every c0 > 0: (T3), since T >= 10^5 >= 4/delta = 80
Part 5. Consequences stated in Section 6
PASS 1/2 + delta = 11/20 < 3/4 (N_(1)(t)/t <= 11/20 on the event A)
PASS (1/2 - delta) t = (9/20) t (each city has at least 9t/20 inhabitants on A)
PASS |z_T - 1/2| <= 1/(2T) <= delta/8 for T >= 10^5
TOTAL failures: 0 (of 41 checks)
