lambda = 0.5549534868173658903729023436864483775714
mu     = 0.8900930263652682192541953126271032448573
c_H    = 0.07446575455794907873414024192572986726134   (closed form  (12/pi) atan(lambda/sqrt3) - 2 lambda)
1/c_H  = 13.428991701437770332
c_H (explicit h_6(1))     = 0.07446575455794907873414024192572986726134  diff to closed form: -7.78e-61
c_H (polar quadrature)    = 0.07446575455794907873414024192572986726134  diff to closed form: 8.75e-62
h_6'(1) = mu (tangency)    : OK

k    tan(pi/k)                          phi(k)                   Psi(k)=phi(k)-phi(6)     Psi(k)+0.03(k-6)        
3    1.7320508075688772935              1.46222822175568         0.277855493562999        0.187855493562999       
4    1.0                                1.29014387611205         0.105771147919365        0.0457711479193647      
5    0.7265425280053608859              1.21995331015596         0.0355805819632781       0.00558058196327809     
6    0.57735026918962576451             1.18437272819268         0.0                      0.0                     
7    0.48157461880752864433             1.1637653389176          -0.0206073892750787      0.00939261072492125     
8    0.4142135623730950488              1.15072443166419         -0.0336482965284881      0.0263517034715119      
9    0.36397023426620236135             1.14193337824939         -0.042439349943293       0.047560650056707       
10   0.32491969623290632616             1.13571913210044         -0.048653596092242       0.071346403907758       
11   0.29362649293836671402             1.13116067682204         -0.0532120513706365      0.0967879486293635      
12   0.26794919243112270647             1.12771589206139         -0.0566568361312932      0.123343163868707       
20   0.15838444032453629384             1.1162578689225          -0.0681148592701789      0.351885140729821       
50   0.062914667253649757225            1.11091859241615         -0.0734541357765284      1.24654586422347        
100  0.031426266043351147819            1.11015971721073         -0.0742130109819481      2.74578698901805        
1000 0.003141602989056156126            1.10990950053906         -0.0744632276536248      29.7455367723464        

admissible nu from k=3..8: [0.0206073892751, 0.0355805819633]
k >= 9:  G_k >= -c_H  and  c_H / 3 = 0.024821918186 < nu = 0.03 : True
smallest margin over k != 6, k in 3..8: 0.00558058196328 (k = 5)
Psi(infinity) = -c_H: -0.0744657545579490787341402419257 vs -0.0744657545579490787341402419257

min_t [h_k(t) - mu t] = phi(k) - 2 and minimiser t* = k rho^2 tan(pi/k)/(1+lambda^2 tan^2(pi/k)), k = 3..12: OK
k=3: min over the grid of D_k(t) = 0.18785577 at t = 0.860333
k=4: min over the grid of D_k(t) = 0.045771737 at t = 0.972747
k=5: min over the grid of D_k(t) = 0.0055817164 at t = 0.993879
k=6: min over the grid of D_k(t) = 2.2318714e-31 at t = 1.0
k=7: min over the grid of D_k(t) = 0.0094019278 at t = 1.0
k=8: min over the grid of D_k(t) = 0.026360028 at t = 1.00307
k=9: min over the grid of D_k(t) = 0.047571739 at t = 1.00307
G_k(t) >= -c_H on the whole grid for k = 3..9: OK
TANGENT TABLE CHECKS PASSED
