PASS A1: wedge area = d^2 (t_a+t_b)/2 (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 0.0)
PASS A2: d/ddelta (t_a+t_b) = (t_a-t_b)(t_a+t_b) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 1.34e-51)
PASS A3: (d^2)' = -w d^2   (eq. d2 in the paper) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 1.07e-50)
PASS A4: d R^2/d delta = -d^2 sec^2(x) (w + 2 tan x) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 1.34e-51)
PASS A5: d/dx(tan^2 x + w tan x) = sec^2 x (w + 2 tan x) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 2.14e-50)
PASS A6a: p(t_b) = t_a t_b (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 3.34e-52)
PASS A6b: p(-t_a) = t_a t_b (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 1.34e-51)
PASS A7a: case 2 sum = -2 w T0 (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 2.67e-51)
PASS A7b: case 3 sum = (T0-t_a)(T0+t_b) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 5.35e-51)
PASS A7c: case 4 sum = p(t_b) - p(-t_a) = 0 (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 5.35e-51)
PASS A8: four-case K'(delta) vs central difference of K (quadrature), 1200 random cases (max rel. error 1.97e-17; cases hit: {1: 338, 2: 363, 3: 381, 4: 118})
PASS A8b: all four regimes occurred
PASS A9: K(delta) nondecreasing on [0, pi/2-gamma) (400 random (gamma,tau,rho), 25 tilts each, 40 digits) (violations: 0)
PASS B1: F'(s) = -B(s) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 4.01e-51)
PASS B2: F''(s) = A(s) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 4.28e-50)
PASS B3: A*B = rho^2/(2 s^2) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 0.0)
PASS B4: eps_tau = 1 - B cot(gamma) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 1.07e-50)
PASS B5: eps_gamma = -rho^2 + B tau csc^2(gamma) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 5.35e-51)
PASS B6: eps_tautau = A cot^2(gamma) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 2.14e-50)
PASS B7: eps_taugamma = csc^2(gamma) (B - A tau cot gamma) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 1.07e-50)
PASS B8: eps_gammagamma = A tau^2 csc^4 - 2 B tau csc^2 cot (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 1.47e-50)
PASS B9: det Hess(eps) = csc^4 B^2 (s - rho^2 cos^2 g)/(rho^2 - s) = csc^2 B^2 (rho^2/(rho^2-s) - csc^2 g), 200 points of the middle regime (max rel diff 4.95e-45; min det 0.0965 >= 0: True)
PASS B9b: csc^2(g) B^2 (rho^2/(rho^2-s) - csc^2 g) = csc^4 B^2 (s - rho^2 cos^2 g)/(rho^2-s) (algebra) (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 5.35e-50)
PASS B10a: on s = rho^2 (tau = rho^2 tan g): eps_mid equals tau - rho^2 g and the gradient is (1, -rho^2) (max dev 3.12e-25; the 1e-25 level is the sqrt-amplified rounding of acos at argument 1 +- 1e-50)
PASS B10b: on s = rho^2 cos^2 g (tau = rho^2 sin g cos g): eps_mid = 0 and the gradient is (0, 0) (max dev 2.27e-49)
PASS B11: eps closed form (three regimes) vs quadrature of the isosceles wedge, 600 random cases (max abs diff 3.16e-30, regimes {1: 224, 2: 158, 3: 218})
PASS B12: joint convexity of eps(tau,gamma): 600000 random chords, min (chord - value) = -3.753e-16
PASS B13: convexity of eps, 600000 short chords near the middle regime, min (chord - value) = 2.613e-16
PASS B14a: d2/dg2 (rho^2 tan g) > 0 (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 6.84e-49)
PASS B14b: d2/dg2 (sin(2g)/2) = -2 sin(2g) < 0 (sympy simplify==0: True; max |expr| at 12 random points, 50 digits: 0.0)
ALL APPENDIX A CHECKS PASSED
