== Certificate A ==
d/ds [sqrt(g+2s)-sqrt(g)] - q(s) = 0
Q(1) - sqrt(g) = 0   (equality at s = 1; Q increasing, so Q(s) <= sqrt(g) on [0,1])
kappa_A = 2*sqrt(1+g) = 2*sqrt(15)/3 = 2.58198889747161
== Certificate B ==
check of the substitution: (weight) * dr/dy = 1
E(Y) = -2*Y**5/25 + 3*Y**4/10 + 2*Y**3/5 - 8*Y**2/5 - 8*Y/5 + 209/50
E - claimed quintic = 0
E'(Y) = -2*(Y - 2)*(Y**3 - Y**2 - 5*Y - 2)/5
E'(Y) + (2/5)(Y-2)(Y^3-Y^2-5Y-2) = 0
cubic = Y(Y^2-5) - (Y^2+2) : 0  -> negative for 0 < Y <= sqrt(5)
real roots of the cubic: [-1.47283390900, -0.462598422975, 2.93543233197]
E(1) = 8/5   E(2) = 1/50   E(sqrt5) = 92/25 - 8*sqrt(5)/5 = 0.102291236000
=> E decreasing on [1,2], increasing on [2,sqrt5], min E = E(2) = 1/50 > 0 (attained at s = 3/4)
number of real roots of E - 1/100 in [1, 9/4] (sympy count_roots): 0
number of real roots of E in [1, sqrt5]: 0
kappa_B = 2*sqrt(1+g) = sqrt(6) = 2.44948974278318
== direct numerical cross-check of (C) from the integrals (mpmath, 30 digits) ==
certificate A: min over s=i/200 of sqrt(gamma) phi(s) - lower integral = 0.0,  - upper integral = 0.0
certificate B: min over s=i/200 of sqrt(gamma) phi(s) - lower integral = 0.0141421356237,  - upper integral = 0.0141421356237
(for B the exact minimum is sqrt(1/2)*E(2) = sqrt(1/2)/50 = 0.0141421356237 at s = 3/4 resp. 1/4)
