SL_2 standard  q=53 n=148824 d=4 diam=22  lambda_2=0.956388 lambda_min=-0.958701 rho=0.958701 kappa=0.0304 tau(0)=22.9295
    Step 1 (K^j(e,x) <= 2/n + rho^(j-1), j=1..13200): True  (max ratio lhs/rhs = 0.5000)
    2 max_p Var_p(dist) = 46.9391 at p = 0.02330 (mean time 42.9);  ratio to tau(0): 2.0471 ; ratio to d: 11.7348 ; (log_d n)^2 = 73.82
    hypothesis n >= 32 d^6 holds: r=6, t=330, p=1/t
    Step 2 (P(|X(j)|<=r) <= 2 d^r (2/n+rho^(j-1))): True ; Step 3 (P(|X(j)|<=r) <= 1/4 for t<=j<=13200): True (value at j=t: 0.00374)
    P(A)=0.01207 >= kappa/5=0.00608: True ; P(B)>=0.36595 >= 1/4: True ; Var(Y)=10.7538 >= kappa r^2/80=0.01369: True ; hence tau(1/t) >= 21.5076 >= kappa r^2/40 = 0.02738
    (13.2s)
SL_2 random pair (7, 20, 32, 31) (32, 41, 6, 16)  q=53 n=148824 d=4 diam=14  lambda_2=0.872749 lambda_min=-0.889002 rho=0.889002 kappa=0.0849 tau(0)=7.8585
    Step 1 (K^j(e,x) <= 2/n + rho^(j-1), j=1..4760): True  (max ratio lhs/rhs = 0.5000)
    2 max_p Var_p(dist) = 29.1742 at p = 0.05430 (mean time 18.4);  ratio to tau(0): 3.7125 ; ratio to d: 7.2935 ; (log_d n)^2 = 73.82
    hypothesis n >= 32 d^6 holds: r=6, t=119, p=1/t
    Step 2 (P(|X(j)|<=r) <= 2 d^r (2/n+rho^(j-1))): True ; Step 3 (P(|X(j)|<=r) <= 1/4 for t<=j<=4760): True (value at j=t: 0.00979)
    P(A)=0.03319 >= kappa/5=0.01697: True ; P(B)>=0.36274 >= 1/4: True ; Var(Y)=6.6021 >= kappa r^2/80=0.03819: True ; hence tau(1/t) >= 13.2042 >= kappa r^2/40 = 0.07638
    (5.6s)
done
