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), and max|Psi_0 - closed form| (v4), closed form Psi_0 = -(L^2-1)(2L^2+7)/(360 L^2 t_f^2); K_L = 2 n S_2 + s_3 with S_2 > 0 and s_3 > 0. 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; max|Psi_0 - closed form| = 1.4e-15; S_2 = 0.5185, s_3 = 0.1728 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; max|Psi_0 - closed form| = 7.2e-16; S_2 = 1.3125, s_3 = 0.3281 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; max|Psi_0 - closed form| = 2.0e-15; S_2 = 2.6400, s_3 = 0.5280 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; max|Psi_0 - closed form| = 2.8e-15; S_2 = 4.6343, s_3 = 0.7724 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; max|Psi_0 - closed form| = 3.3e-15; S_2 = 7.4286, s_3 = 1.0612 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; max|Psi_0 - closed form| = 1.1e-14; S_2 = 11.1562, s_3 = 1.3945 closed form of Psi_0 at the pairs (L/2 - 1, L/2), (L/2, L/2 + 1), even 18 <= L <= 40: max relative |Psi_0 - closed form| and max L|Xi| over these pairs: 5.3e-12 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 (a): closed form gamma^2 G(sigma M_k/t_f) against the prediction column: max relative deviation 6.4e-16 3. (a) at Xi = 0 (even L): G(y) = y^2 [Psi_0 t_f^2 - n A_L + (n/2) psi(y^2)] + (n/2) y^3 phi(y^2), y = sigma M_k/t_f, A_L = (L-1)(L-2)/(6L), phi(u) = _q, psi(u) = _q, c = cos q. Common to all levels and branches iff (i) phi(x_k^2) = 0 for every level, x_k = M_k/t_f, and (ii) the branch-even part is equal at every level; (ii) for two zeros u_a, u_b of phi fixes n = N_f/L uniquely. identity of the even/odd split against G (L = 6..16 even, four y, N_f = 3): max relative deviation 1.2e-13 L= 6: 2 poles; zeros of phi: 1 (roots), 1 (sign changes), max|u phi(u)| at roots 3.5e-16; x = M_k/t_f at the zeros: [1.58114] N_f required by (ii) for the 0 pairs: []; positive: 0; min distance of a positive one to an integer >= 1: nan; equally spaced triples: 0; zeros at u = 1 or 4: 0 L= 8: 2 poles; zeros of phi: 1 (roots), 1 (sign changes), max|u phi(u)| at roots 1.5e-15; x = M_k/t_f at the zeros: [1.73205] N_f required by (ii) for the 0 pairs: []; positive: 0; min distance of a positive one to an integer >= 1: nan; equally spaced triples: 0; zeros at u = 1 or 4: 0 L=10: 4 poles; zeros of phi: 3 (roots), 3 (sign changes), max|u phi(u)| at roots 1.0e-14; x = M_k/t_f at the zeros: [0.66814 1.32142 1.81864] N_f required by (ii) for the 3 pairs: [-3.5787 -3.7575 -3.9206]; positive: 0; min distance of a positive one to an integer >= 1: nan; equally spaced triples: 0; zeros at u = 1 or 4: 0 L=12: 4 poles; zeros of phi: 3 (roots), 3 (sign changes), max|u phi(u)| at roots 1.6e-14; x = M_k/t_f at the zeros: [0.47368 1.50903 1.87041] N_f required by (ii) for the 3 pairs: [-4.7065 -4.9007 -5.2661]; positive: 0; min distance of a positive one to an integer >= 1: nan; equally spaced triples: 0; zeros at u = 1 or 4: 0 L=14: 6 poles; zeros of phi: 5 (roots), 5 (sign changes), max|u phi(u)| at roots 4.3e-14; x = M_k/t_f at the zeros: [0.35207 0.77116 1.22662 1.62926 1.90323] N_f required by (ii) for the 10 pairs: [-5.0186 -5.6406 -6.0239 -6.2276 -6.027 -6.3129 -6.47 -6.559 -6.681 -6.8319]; positive: 0; min distance of a positive one to an integer >= 1: nan; equally spaced triples: 0; zeros at u = 1 or 4: 0 L=16: 6 poles; zeros of phi: 5 (roots), 5 (sign changes), max|u phi(u)| at roots 1.0e-13; x = M_k/t_f at the zeros: [0.27149 0.60844 1.38676 1.71078 1.92515] N_f required by (ii) for the 10 pairs: [-5.6322 -7.1414 -7.5263 -7.7361 -7.5264 -7.8318 -8.0018 -8.3565 -8.4675 -8.615 ]; positive: 0; min distance of a positive one to an integer >= 1: nan; equally spaced triples: 0; zeros at u = 1 or 4: 0 summary, even 6 <= L <= 40: L, zeros (roots), zeros (sign changes), pairs, pairs with N_f > 0, max required N_f, equally spaced triples, zeros at u = 1 or 4, max|psi + | at the zeros, max over x in [1e-4, 1e3] of kappa'(u) - A_L (negative: (ii) needs N_f < 0), A_L L= 6: 1 1 pairs 0 positive 0 max N_f +nan triples 0 resonant 0 1.9e-16 -0.5557 0.5556 L= 8: 1 1 pairs 0 positive 0 max N_f +nan triples 0 resonant 0 8.9e-16 -0.8752 0.8750 L=10: 3 3 pairs 3 positive 0 max N_f -3.5787 triples 0 resonant 0 5.8e-15 -1.2004 1.2000 L=12: 3 3 pairs 3 positive 0 max N_f -4.7065 triples 0 resonant 0 8.7e-15 -1.5283 1.5278 L=14: 5 5 pairs 10 positive 0 max N_f -5.0186 triples 0 resonant 0 2.3e-14 -1.8579 1.8571 L=16: 5 5 pairs 10 positive 0 max N_f -5.6322 triples 0 resonant 0 5.3e-14 -2.1884 2.1875 L=18: 7 7 pairs 21 positive 0 max N_f -6.1680 triples 0 resonant 0 3.9e-14 -2.5196 2.5185 L=20: 7 7 pairs 21 positive 0 max N_f -6.6308 triples 0 resonant 0 1.6e-13 -2.8512 2.8500 L=22: 9 9 pairs 36 positive 0 max N_f -7.0284 triples 0 resonant 0 2.7e-13 -3.1832 3.1818 L=24: 9 9 pairs 36 positive 0 max N_f -7.3692 triples 0 resonant 0 6.5e-13 -3.5154 3.5139 L=26: 11 11 pairs 55 positive 0 max N_f -7.6616 triples 0 resonant 0 7.9e-13 -3.8478 3.8462 L=28: 11 11 pairs 55 positive 0 max N_f -7.9128 triples 0 resonant 0 1.0e-12 -4.1804 4.1786 L=30: 13 13 pairs 78 positive 0 max N_f -8.1294 triples 0 resonant 0 4.6e-13 -4.5131 4.5111 L=32: 13 13 pairs 78 positive 0 max N_f -8.3168 triples 0 resonant 0 3.2e-12 -4.8459 4.8438 L=34: 15 15 pairs 105 positive 0 max N_f -8.4795 triples 0 resonant 0 4.0e-12 -5.1788 5.1765 L=36: 15 15 pairs 105 positive 0 max N_f -8.6214 triples 0 resonant 0 5.2e-12 -5.5118 5.5093 L=38: 17 17 pairs 136 positive 0 max N_f -8.7457 triples 0 resonant 0 6.2e-12 -5.8448 5.8421 L=40: 17 17 pairs 136 positive 0 max N_f -8.8550 triples 0 resonant 0 7.0e-12 -6.1778 6.1750