OK    area of the wedge triangle = (1/2) d^2 (ta+tb)  [with x = theta - delta]
OK    (d^2)' = - d^2 w
OK    ta + tb = sin 2g / (cos(g+dl) cos(g-dl)) > 0
OK    d/d(delta) R^2 = - d^2 sec^2(x) (w + 2 tan x)
OK    d/dx [ w tan x + tan^2 x ] = sec^2 x (w + 2 tan x)
OK    p(tb) = ta tb
OK    p(-ta) = ta tb
OK    both components:  -(1/2) d^2 (left+right) = d^2 w T0
OK    left component only:  -(1/2) d^2 left = (1/2) d^2 (ta - T0)(tb + T0)
OK    whole interval one component:  p(tb) - p(-ta) = 0
OK    K'(delta): formula vs central difference of the quadrature, 400 random cases {'both': 114, 'left': 131, 'inside': 94, 'whole': 61}, max rel. err 3.12e-22
OK    symbolic derivative of closed-form K (left only regime) equals the K' formula (12 exact-rational points, max err 8.68e-165)
OK    symbolic derivative of closed-form K (both regime) equals the K' formula (12 exact-rational points, max err 1.34e-138)
OK    F'(s) = -B  (exact-rational points)
OK    B'(s) = -A  (so F'' = A)  (exact-rational points)
OK    eps_tautau = A cot^2 gamma  (12 points in the middle regime, max err 1.41e-38)
OK    eps_taugamma = csc^2 gamma (B - A tau cot gamma)  (12 points in the middle regime, max err 3.57e-39)
OK    eps_gammagamma = A tau^2 csc^4 gamma - 2 B tau csc^2 gamma cot gamma  (12 points in the middle regime, max err 1.63e-39)
OK    det Hess = csc^2 gamma B^2 ( rho^2/(rho^2-s) - csc^2 gamma )  (12 points in the middle regime, max err 6.64e-37)
OK    rho^2/(rho^2-s) - csc^2 g = (s - rho^2 cos^2 g)/((rho^2-s) sin^2 g)
OK    regime II/III boundary: eps = 0, eps_tau = 0, eps_gamma = 0 (value and gradient glue to the zero function)
      I/II boundary: F(rho^2) = rho^2 acos(1) - 0 = 0 and F'(s) = -sqrt((rho^2-s)/s) -> 0 as s -> rho^2: gradient of eps_II = (1, -rho^2) = gradient of eps_I.
OK    F(rho^2) limit = 0
OK    F'(s) -> 0 as s -> rho^2
OK    eps in the middle regime = direct integral (60 random points, max err 9.18e-41)
OK    tan'' > 0 on (0,pi/2)
OK    (sin g cos g)'' = -2 sin 2g < 0 on (0,pi/2)
OK    d/da [ a w + rho^2 (pi/k - arccos(a/rho)) ] = 2 w,  w = sqrt(rho^2 - a^2)
OK    dh/dt = 2 (1 - w/(a tan(pi/k)))
OK    w/a = lam q at a = rho/sqrt(1+lam^2 q^2)
OK    min_t [h_k(t) - mu t] = 2 k rho^2 arctan(lam tan(pi/k)) - 2  (t*-terms cancel)
OK    w/(a tan(pi/6)) at (k,t)=(6,1) equals sqrt(3 (2 sqrt3/pi - 1))

ALL SYMBOLIC CHECKS PASSED
