p, r_1(p) rounded down to 5 decimals, exact check  P_1(c) >= p > P_1(c + 1e-5)
p =     2.01:  r_1(p) >= 0.49877   check: True
p =      2.1:  r_1(p) >= 0.48941   check: True
p =      2.5:  r_1(p) >= 0.46608   check: True
p =        3:  r_1(p) >= 0.45226   check: True
p =        4:  r_1(p) >= 0.43932   check: True
p =        5:  r_1(p) >= 0.43302   check: True
p =        6:  r_1(p) >= 0.42927   check: True
p =        8:  r_1(p) >= 0.42498   check: True
p =       10:  r_1(p) >= 0.42260   check: True
p =       12:  r_1(p) >= 0.42108   check: True
p =  12.0068:  r_1(p) >= 0.42108   check: True
p =     12.1:  r_1(p) >= 0.42102   check: True
p =       13:  r_1(p) >= 0.42051   check: True
p =       14:  r_1(p) >= 0.42003   check: True
p =       16:  r_1(p) >= 0.41926   check: True
p =       18:  r_1(p) >= 0.41867   check: True
p =       20:  r_1(p) >= 0.41820   check: True
p =       24:  r_1(p) >= 0.41751   check: True
p =       30:  r_1(p) >= 0.41683   check: True
p =       40:  r_1(p) >= 0.41616   check: True
p =       50:  r_1(p) >= 0.41576   check: True
p =       76:  r_1(p) >= 0.41522   check: True
p =      100:  r_1(p) >= 0.41498   check: True
p =      150:  r_1(p) >= 0.41472   check: True
p =      200:  r_1(p) >= 0.41459   check: True
p =      300:  r_1(p) >= 0.41446   check: True
p =      500:  r_1(p) >= 0.41436   check: True
p =     1000:  r_1(p) >= 0.41428   check: True
p =    10000:  r_1(p) >= 0.41422   check: True
consistency of the certificates of Theorem D with Theorem C:  p_0 > P_1(r_0)
p_0 =    14, r_0 = 0.4932:  P_1(r_0) =   2.0604 < p_0: True
p_0 =    16, r_0 = 0.4872:  P_1(r_0) =   2.1255 < p_0: True
p_0 =    18, r_0 = 0.4821:  P_1(r_0) =   2.1919 < p_0: True
p_0 =    20, r_0 = 0.4779:  P_1(r_0) =   2.2559 < p_0: True
p_0 =    24, r_0 = 0.4710:  P_1(r_0) =   2.3848 < p_0: True
p_0 =    30, r_0 = 0.4635:  P_1(r_0) =   2.5710 < p_0: True
p_0 =    40, r_0 = 0.4553:  P_1(r_0) =   2.8597 < p_0: True
p_0 =    50, r_0 = 0.4498:  P_1(r_0) =   3.1329 < p_0: True
p_0 =    76, r_0 = 0.4415:  P_1(r_0) =   3.7638 < p_0: True
p_0 =   100, r_0 = 0.4372:  P_1(r_0) =   4.2754 < p_0: True
p_0 =   150, r_0 = 0.4322:  P_1(r_0) =   5.1840 < p_0: True
p_0 =   200, r_0 = 0.4294:  P_1(r_0) =   5.9576 < p_0: True
p_0 =   300, r_0 = 0.4263:  P_1(r_0) =   7.2358 < p_0: True
p_0 =   500, r_0 = 0.4234:  P_1(r_0) =   9.2170 < p_0: True
p_0 =  1000, r_0 = 0.4207:  P_1(r_0) =  12.6596 < p_0: True
ALL CHECKS PASSED
