lambda = 0.5549534868173658903729023436864483775714
mu     = 0.8900930263652682192541953126271032448573
c_H    = 0.0744657545579490787341402419257298672613374455277810684338133
1/c_H  = 13.4289917014377703320425863458
PASS phi(6) - 2 lambda = c_H
PASS tangency: phi(6) = 2 - mu + c_H
PASS c_H = 2 |H \ B| by polar quadrature (60 digits) (diff 4.41e-62)
PASS c_H = 2 min_z |H \ B(z)| (float, exact polygon-disc area, 25 starts) (2*min = 0.074465754557949, c_H = 0.074465754557949)
PASS grid over the disc centre: no value below the concentric one

 k   tan(pi/k)   phi(k)          Psi(k)          Psi(k)+0.03(k-6)
    3  1.7320508  1.46222822176  0.277855493563  0.1878554936
    4  1.0  1.29014387611  0.105771147919  0.04577114792
    5  0.72654253  1.21995331016  0.0355805819633  0.005580581963
    6  0.57735027  1.18437272819  0.0  0.0
    7  0.48157462  1.16376533892  -0.0206073892751  0.009392610725
    8  0.41421356  1.15072443166  -0.0336482965285  0.02635170347
    9  0.36397023  1.14193337825  -0.0424393499433  0.04756065006
   10  0.3249197  1.1357191321  -0.0486535960922  0.07134640391
   11  0.29362649  1.13116067682  -0.0532120513706  0.09678794863
   12  0.26794919  1.12771589206  -0.0566568361313  0.1233431639
   20  0.15838444  1.11625786892  -0.0681148592702  0.3518851407
   40  0.078701707  1.11148838371  -0.072884344483  0.9471156555
   60  0.052407779  1.1106093043  -0.0737634238914  1.546236576
  100  0.031426266  1.11015971721  -0.0742130109819  2.745786989
 1000  0.003141603  1.10990950054  -0.0744632276536  29.74553677
10000  0.00031415928  1.1099069989  -0.074465729289  299.7455343
PASS Psi(k)+nu(k-6) >= s_* = 0.0055 for k = 3,4,5,7,8 (min = 0.005580582 at k = 5)
PASS exact Psi(k)+nu(k-6) for 9 <= k <= 60, 100, 1000, 10000: min = 0.04756065 (> s_*)
PASS k = 6: Psi(6) = 0
exact admissible range of nu (all k <= 60 and 100,1000,10^4): [0.0206073892751, 0.0355805819633]
PASS nu = 0.03 lies in the exact range
paper's k >= 9 argument needs 3 nu >= c_H (c_H/3 = 0.024821918) and 3 nu - c_H = 0.015534245 > s_*
PASS c_H < 3 nu and 3 nu - c_H > s_*
PASS mu - c_H > s_* and mu in (0,2)
PASS c_H < 0.075 < 1 and lambda in (0,1)
so with the paper's proof (k >= 9 via the trivial bound) the admissible nu are [c_H/3, hi] = [0.024821918, 0.035580582]
   k=  3  min_t G_k(t) = 0.277855493563   Psi(k) = 0.277855493563   min_t G_k + 0.03(k-6) = 0.1878554936   argmin t = 0.859696
   k=  4  min_t G_k(t) = 0.105771147919   Psi(k) = 0.105771147919   min_t G_k + 0.03(k-6) = 0.0457711479   argmin t = 0.973445
   k=  5  min_t G_k(t) = 0.035580581963   Psi(k) = 0.035580581963   min_t G_k + 0.03(k-6) = 0.0055805820   argmin t = 0.994633
   k=  6  min_t G_k(t) = -0.000000000000   Psi(k) = 0.000000000000   min_t G_k + 0.03(k-6) = -0.0000000000   argmin t = 1.000000
   k=  7  min_t G_k(t) = -0.020607389275   Psi(k) = -0.020607389275   min_t G_k + 0.03(k-6) = 0.0093926107   argmin t = 1.001499
   k=  8  min_t G_k(t) = -0.033648296528   Psi(k) = -0.033648296528   min_t G_k + 0.03(k-6) = 0.0263517035   argmin t = 1.001849
   k=  9  min_t G_k(t) = -0.042439349943   Psi(k) = -0.042439349943   min_t G_k + 0.03(k-6) = 0.0475606501   argmin t = 1.001825
   k= 10  min_t G_k(t) = -0.048653596092   Psi(k) = -0.048653596092   min_t G_k + 0.03(k-6) = 0.0713464039   argmin t = 1.001683
   k= 20  min_t G_k(t) = -0.068114859270   Psi(k) = -0.068114859270   min_t G_k + 0.03(k-6) = 0.3518851407   argmin t = 1.000577
   k= 30  min_t G_k(t) = -0.071651453703   Psi(k) = -0.071651453703   min_t G_k + 0.03(k-6) = 0.6483485463   argmin t = 1.000268
   k=100  min_t G_k(t) = -0.074213010982   Psi(k) = -0.074213010982   min_t G_k + 0.03(k-6) = 2.7457869890   argmin t = 1.000025
   k=200  min_t G_k(t) = -0.074402578729   Psi(k) = -0.074402578729   min_t G_k + 0.03(k-6) = 5.7455974213   argmin t = 1.000006
PASS min_t G_k(t) (direct geometry, k = 3..30, 40, 50, 64, 100, 200) equals Psi(k): max |diff| = 4.77e-15
PASS G_k(t) + nu(k-6) >= s_* for all grid t and all those k != 6: min = 0.00558058
PASS G_k(t) >= -c_H on all grid points (checked inside the loop)
PASS h_k'(t) of Lemma 3.4(a) vs central difference of the geometric h_k (k = 3,4,5,6,7,8,12): max |diff| = 1.12e-07
PASS h_k convex: minimal second difference on a 3000-point grid = -2.66e-15 (>= -1e-12)
ALL TANGENT/CONSTANT CHECKS PASSED
