PASS  J1 q5 < q4 < 25/169 < q3 < q2 < 1/2  0.105573 0.133975 0.147929 0.183503 0.292893 0.500000
PASS  J1 q2 = 1 - 1/sqrt2 (A.6, A.11)  
PASS  J1 q4 = 1 - sqrt3/2, q5 = 1 - 2/sqrt5, q6 = 1 - sqrt(5/6)  
PASS  J2 tie T1 = T2 at q2  |diff| < 1.1e-76
PASS  J2 T1 is the strict max over j <= 39 inside band k (q = 0.39645)  
PASS  J2 tie T2 = T3 at q3  |diff| < 2.5e-76
PASS  J2 T2 is the strict max over j <= 39 inside band k (q = 0.23820)  
PASS  J2 tie T3 = T4 at q4  |diff| < 2.1e-76
PASS  J2 T3 is the strict max over j <= 39 inside band k (q = 0.15874)  
PASS  J2 tie T4 = T5 at q5  |diff| < 3.3e-76
PASS  J2 T4 is the strict max over j <= 39 inside band k (q = 0.11977)  
PASS  J2 tie T5 = T6 at q6  |diff| < 3.7e-76
PASS  J2 T5 is the strict max over j <= 39 inside band k (q = 0.09635)  
      (analytic: T_{k+1}/T_k = sqrt((k+1)/k) (1-q) is >= 1 iff q <= q_{k+1}, and ln T_k is concave in k, so M = T_k exactly on [q_{k+1}, q_k] for q <= 1/2)
PASS  J3/J5 C_5 = g(4 s(q5)) = g(2 ln(5/4)) = (32/25) sqrt(2 ln(5/4))  C_5 = [0.855100455470819 +/- 2.46e-16]
PASS  J3 tie g(4 s(q5)) = g(5 s(q5))  
PASS  J3 a(k-1) = (k-1) s(q_k) increases and stays < 1/2 for k = 5..399  
PASS  J3 a(k-1) = (k-1) s(q_k) exactly  
PASS  J3 C(q_k) = g(a(k-1)) increases in k from C(q5) = C_5 (k = 5..399), limit sqrt(2/e)  C(q6) = [0.855984489174 +/- 1.85e-13], C(q_399) = [0.857763547369 +/- 2.45e-13]
      (analytic: d/dx [x ln(1 + 1/x)] = ln(1 + 1/x) - 1/(x+1) > 0 and ln(1 + 1/x) < 1/x, so a increases to 1/2; g'/g = 1/(2x) - 1 > 0 on (0, 1/2); on band k, k s runs over [a(k), (k/(k-1)) a(k-1)], which contains 1/2, and ln g is concave, so the minimum over the band is min(C(q_{k+1}), C(q_k)) = C(q_k) for k >= 5)
PASS  J3 on band k = 5..59, k s runs over [k s(q_{k+1}), k s(q_k)] around 1/2, with end values C(q_{k+1}), C(q_k)  
PASS  J4 A'(q5) < 77/50  A'(q5) = [1.538044392 +/- 3.50e-10]
PASS  J4 A' increasing on a 2000-point grid of (0, q5] (analytic: d/dq ln A'^2 > 0 iff s(1+q) > q, true since s > q)  A'(q5/2000) = [1.4142696 +/- 4.66e-8]
PASS  J5 171/200 < C_5  margin [0.000100455 +/- 4.71e-10]
PASS  J5 A.38 worst 0.8548832 < C_5 and A.39 worst 0.8549969806 <= 171/200  
PASS  J6 A.29 obstruction: a* > sqrt2 with g(1/a*^2) = C_5 is a* = (2 ln(5/4))^(-1/2), and theta_max > 17/20  a* = [1.4969001 +/- 5.00e-8], theta_max = [1.0062780 +/- 3.59e-8]
A40_SEAMS_PASS

real	0m0.050s
user	0m0.046s
sys	0m0.004s
