1. F = 0 iff r' = r or r' = L - r (reflection-equivalent); Xi of (5.10) against the closed form Xi = -(rho^2 + rho'^2 - 1)/(3 L t_f), rho = r - L/2; pairs 0 <= r < r' < L with F != 0 and Xi = 0; at those pairs Psi_0 - K_L of (5.15) (full grip, N_f = 1). L= 5: F = 0 exactly on the 2 reflection-equivalent pairs: True; max|Xi - closed form| = 1.4e-16; Xi = 0 on: [] L= 6: F = 0 exactly on the 2 reflection-equivalent pairs: True; max|Xi - closed form| = 2.5e-15; Xi = 0 on: [(2, 3), (3, 4)]; Psi_0 - K_L = -0.55903/t_f^2 L= 7: F = 0 exactly on the 3 reflection-equivalent pairs: True; max|Xi - closed form| = 3.3e-16; Xi = 0 on: [] L= 8: F = 0 exactly on the 3 reflection-equivalent pairs: True; max|Xi - closed form| = 8.9e-16; Xi = 0 on: [(3, 4), (4, 5)]; Psi_0 - K_L = -1.02539/t_f^2 L= 9: F = 0 exactly on the 4 reflection-equivalent pairs: True; max|Xi - closed form| = 8.9e-16; Xi = 0 on: [] L=10: F = 0 exactly on the 4 reflection-equivalent pairs: True; max|Xi - closed form| = 2.0e-15; Xi = 0 on: [(4, 5), (5, 6)]; Psi_0 - K_L = -1.62525/t_f^2 L=11: F = 0 exactly on the 5 reflection-equivalent pairs: True; max|Xi - closed form| = 5.1e-16; Xi = 0 on: [] L=12: F = 0 exactly on the 5 reflection-equivalent pairs: True; max|Xi - closed form| = 2.0e-15; Xi = 0 on: [(5, 6), (6, 7)]; Psi_0 - K_L = -2.35851/t_f^2 L=13: F = 0 exactly on the 6 reflection-equivalent pairs: True; max|Xi - closed form| = 2.3e-15; Xi = 0 on: [] L=14: F = 0 exactly on the 6 reflection-equivalent pairs: True; max|Xi - closed form| = 3.8e-15; Xi = 0 on: [(6, 7), (7, 8)]; Psi_0 - K_L = -3.22513/t_f^2 L=15: F = 0 exactly on the 7 reflection-equivalent pairs: True; max|Xi - closed form| = 7.2e-15; Xi = 0 on: [] L=16: F = 0 exactly on the 7 reflection-equivalent pairs: True; max|Xi - closed form| = 7.3e-15; Xi = 0 on: [(7, 8), (8, 9)]; Psi_0 - K_L = -4.22510/t_f^2 2. (5.15) at L = 8, (r, r') = (3, 4): F = -0.125000, Xi = 1.1e-16, Psi_0 = -0.369141, K_L = 0.656250, Psi_0 - K_L = -1.025391 coup M_k gamma sig exact ratio-1 prediction (exact-pred)/gamma^3 branch-odd part: exact, pred fg 1.0 0.01 +1 -1.023603e-04 -1.025391e-04 +0.1788 +1.7575e-07 +0.0000e+00 fg 1.0 0.01 -1 -1.027118e-04 -1.025391e-04 -0.1727 +1.7575e-07 +0.0000e+00 fg 1.0 0.02 +1 -4.087030e-04 -4.101562e-04 +0.1817 +1.4053e-06 +0.0000e+00 fg 1.0 0.02 -1 -4.115135e-04 -4.101562e-04 -0.1697 +1.4053e-06 +0.0000e+00 fg 1.4 0.01 +1 -2.004829e-04 -2.009766e-04 +0.4937 +4.8218e-07 +0.0000e+00 fg 1.4 0.01 -1 -2.014472e-04 -2.009766e-04 -0.4706 +4.8218e-07 +0.0000e+00 fg 1.4 0.02 +1 -7.998683e-04 -8.039062e-04 +0.5047 +3.8534e-06 +0.0000e+00 fg 1.4 0.02 -1 -8.075751e-04 -8.039062e-04 -0.4586 +3.8534e-06 +0.0000e+00 a 1.2 0.01 +1 -5.760587e-05 -5.814538e-05 +0.5395 +9.0476e-06 +8.4392e-06 a 1.2 0.01 -1 -7.570116e-05 -7.502376e-05 -0.6774 +9.0476e-06 +8.4392e-06 a 1.2 0.02 +1 -2.282679e-04 -2.325815e-04 +0.5392 +3.8623e-05 +3.3757e-05 a 1.2 0.02 -1 -3.055144e-04 -3.000950e-04 -0.6774 +3.8623e-05 +3.3757e-05 a 1.6 0.01 +1 -1.040105e-04 -1.053571e-04 +1.3466 +1.4319e-05 +1.2853e-05 a 1.6 0.01 -1 -1.326495e-04 -1.310623e-04 -1.5872 +1.4319e-05 +1.2853e-05 a 1.6 0.02 +1 -4.106866e-04 -4.214286e-04 +1.3427 +6.3131e-05 +5.1410e-05 a 1.6 0.02 -1 -5.369483e-04 -5.242492e-04 -1.5874 +6.3131e-05 +5.1410e-05 c 1.2 0.01 +1 -7.571484e-05 -7.640625e-05 +0.6914 -6.8473e-06 -7.5000e-06 c 1.2 0.01 -1 -6.202020e-05 -6.140625e-05 -0.6140 -6.8473e-06 -7.5000e-06 c 1.2 0.02 +1 -3.000968e-04 -3.056250e-04 +0.6910 -2.4776e-05 -3.0000e-05 c 1.2 0.02 -1 -2.505440e-04 -2.456250e-04 -0.6149 -2.4776e-05 -3.0000e-05 c 1.6 0.01 +1 -1.308947e-04 -1.325000e-04 +1.6053 -8.4627e-06 -1.0000e-05 c 1.6 0.01 -1 -1.139693e-04 -1.125000e-04 -1.4693 -8.4627e-06 -1.0000e-05 c 1.6 0.02 +1 -5.171705e-04 -5.300000e-04 +1.6037 -2.7699e-05 -4.0000e-05 c 1.6 0.02 -1 -4.617732e-04 -4.500000e-04 -1.4717 -2.7699e-05 -4.0000e-05 (c): predicted branch-odd part -sigma gamma^2 M_k n (L-2)(L-4)/(6 L t_f) at gamma = 0.02, M = 1.2: -3.0000e-05