Part A: bound level of H(P) (11.1), sector Theta=+1, open chain r in [-L, L], eta = 0.50 FD: 7-point stencil for E_b''(0), step h = 0.050; control step 2h = 0.100 D1: tau^=(1.00,1.60) M=(1.00,2.50) w^=(w11 -0.40, w12 -0.50, w21 0.30, w22 0.70) u=0.7905 L=112 eps=0.200 E_bind=-5.537956235082e-03 E''(0)=1.588965060821e-02 -1=-2.168838e-03 -1=-8.828018e-04 |change L->1.5L| max=1.5e-12 |E''(h)-E''(2h)|=8.4e-10 eps=0.100 E_bind=-1.388688933774e-03 E''(0)=3.969821660361e-03 -1=-5.396400e-04 -1=-2.206895e-04 |change L->1.5L| max=5.9e-12 |E''(h)-E''(2h)|=2.0e-10 eps=0.050 E_bind=-3.474340889640e-04 E''(0)=9.922902739865e-04 -1=-1.347498e-04 -1=-5.517182e-05 |change L->1.5L| max=2.8e-12 |E''(h)-E''(2h)|=5.1e-11 deviation from leading form (11.14) | from next-order form (14.6)-(14.9); slopes in eps: E_bind lead: +2.238e-05 +1.396e-06 +8.723e-08 slopes 4.00 4.00 next: +5.791e-08 +9.031e-10 +1.410e-11 slopes 6.00 6.00 E''(0) lead: +1.389e-05 +8.819e-07 +5.533e-08 slopes 3.98 3.99 next: -2.914e-07 -4.547e-09 -7.229e-11 slopes 6.00 5.98 lead: -1.370e-05 -8.544e-07 -5.338e-08 slopes 4.00 4.00 next: -3.410e-08 -5.349e-10 -8.365e-12 slopes 5.99 6.00 lead: -5.545e-08 -2.872e-09 -1.703e-10 slopes 4.27 4.08 next: -1.263e-08 -1.953e-10 -3.043e-12 slopes 6.01 6.00 E_b(0.3) lead: +2.289e-05 +1.428e-06 +8.924e-08 slopes 4.00 4.00 next: +4.492e-08 +7.008e-10 +1.095e-11 slopes 6.00 6.00 item 3: D = m_b - (m1+m2-4 m1 m2 E_bind), numerical vs (14.11), both divided by m_b^(0): eps=0.200 D_num/m_b0=-6.629865e-06 D_pred/m_b0=-2.799318e-05 difference=+2.14e-05 eps=0.100 D_num/m_b0=-5.663204e-06 D_pred/m_b0=-6.998295e-06 difference=+1.34e-06 eps=0.050 D_num/m_b0=-1.664859e-06 D_pred/m_b0=-1.749574e-06 difference=+8.47e-08 relative correction of m_b: numerical m_b/m_b^(0)-1 vs (14.10): eps=0.200 num=-8.742482e-04 pred=-8.933671e-04 eps=0.100 num=-2.221467e-04 pred=-2.233418e-04 eps=0.050 num=-5.575948e-05 pred=-5.583544e-05 D2: tau^=(1.30,0.80) M=(1.50,0.70) w^=(w11 0.60, w12 0.40, w21 -0.20, w22 -0.50) u=1.1759 L=80 eps=0.200 E_bind=5.136989357678e-02 E''(0)=-1.275968735072e-02 -1=-1.199641e-02 -1=-2.033606e-02 |change L->1.5L| max=6.5e-13 |E''(h)-E''(2h)|=8.0e-10 eps=0.100 E_bind=1.294382462252e-02 E''(0)=-3.249295987270e-03 -1=-3.069918e-03 -1=-5.303397e-03 |change L->1.5L| max=4.2e-13 |E''(h)-E''(2h)|=2.0e-10 eps=0.050 E_bind=3.242458562345e-03 E''(0)=-8.161338742853e-04 -1=-7.720324e-04 -1=-1.340283e-03 |change L->1.5L| max=7.5e-13 |E''(h)-E''(2h)|=5.0e-11 deviation from leading form (11.14) | from next-order form (14.6)-(14.9); slopes in eps: E_bind lead: -5.444e-04 -3.474e-05 -2.183e-06 slopes 3.97 3.99 next: +1.546e-05 +2.482e-07 +3.905e-09 slopes 5.96 5.99 E''(0) lead: +3.189e-04 +2.036e-05 +1.279e-06 slopes 3.97 3.99 next: -9.094e-06 -1.453e-07 -2.283e-09 slopes 5.97 5.99 lead: +3.806e-04 +2.433e-05 +1.529e-06 slopes 3.97 3.99 next: -1.168e-05 -1.869e-07 -2.938e-09 slopes 5.97 5.99 lead: +1.187e-03 +7.725e-05 +4.879e-06 slopes 3.94 3.98 next: -6.696e-05 -1.091e-06 -1.724e-08 slopes 5.94 5.98 E_b(0.3) lead: -5.304e-04 -3.385e-05 -2.127e-06 slopes 3.97 3.99 next: +1.507e-05 +2.419e-07 +3.807e-09 slopes 5.96 5.99 item 3: D = m_b - (m1+m2-4 m1 m2 E_bind), numerical vs (14.11), both divided by m_b^(0): eps=0.200 D_num/m_b0=+4.227327e-02 D_pred/m_b0=+4.284997e-02 difference=-5.77e-04 eps=0.100 D_num/m_b0=+1.067556e-02 D_pred/m_b0=+1.071249e-02 difference=-3.69e-05 eps=0.050 D_num/m_b0=+2.675800e-03 D_pred/m_b0=+2.678123e-03 difference=-2.32e-06 relative correction of m_b: numerical m_b/m_b^(0)-1 vs (14.10): eps=0.200 num=+2.499426e-02 pred=+2.508015e-02 eps=0.100 num=+6.264615e-03 pred=+6.270038e-03 eps=0.050 num=+1.567169e-03 pred=+1.567510e-03 D3: tau^=(1.20,0.90) M=(0.80,1.30) w^=(w11 -0.12, w12 0.20, w21 0.10, w22 0.30) u=0.1981 L=408 eps=0.200 E_bind=-4.699931522243e-04 E''(0)=1.816024850353e-02 -1=-4.283860e-04 -1=-9.169974e-05 |change L->1.5L| max=1.6e-12 |E''(h)-E''(2h)|=2.0e-09 eps=0.100 E_bind=-1.176597100687e-04 E''(0)=4.540273684594e-03 -1=-1.071905e-04 -1=-2.286127e-05 |change L->1.5L| max=5.3e-13 |E''(h)-E''(2h)|=4.8e-10 eps=0.050 E_bind=-2.942500201675e-05 E''(0)=1.135081467153e-03 -1=-2.680349e-05 -1=-5.711356e-06 |change L->1.5L| max=4.9e-13 |E''(h)-E''(2h)|=1.2e-10 deviation from leading form (11.14) | from next-order form (14.6)-(14.9); slopes in eps: E_bind lead: +8.606e-07 +5.373e-08 +3.357e-09 slopes 4.00 4.00 next: +1.308e-09 +2.039e-11 +3.179e-13 slopes 6.00 6.00 E''(0) lead: -1.124e-06 -6.953e-08 -4.335e-09 slopes 4.02 4.00 next: -1.606e-08 -2.582e-10 -6.230e-12 slopes 5.96 5.37 lead: +5.012e-07 +3.132e-08 +1.957e-09 slopes 4.00 4.00 next: +9.991e-11 +1.581e-12 +2.442e-14 slopes 5.98 6.02 lead: -3.392e-07 -2.113e-08 -1.319e-09 slopes 4.00 4.00 next: -1.538e-09 -2.397e-11 -3.744e-13 slopes 6.00 6.00 E_b(0.3) lead: +6.392e-07 +3.991e-08 +2.494e-09 slopes 4.00 4.00 next: +7.306e-10 +1.139e-11 +1.786e-13 slopes 6.00 5.99 item 3: D = m_b - (m1+m2-4 m1 m2 E_bind), numerical vs (14.11), both divided by m_b^(0): eps=0.200 D_num/m_b0=+9.675148e-05 D_pred/m_b0=+9.581065e-05 difference=+9.41e-07 eps=0.100 D_num/m_b0=+2.401307e-05 D_pred/m_b0=+2.395266e-05 difference=+6.04e-08 eps=0.050 D_num/m_b0=+5.993875e-06 D_pred/m_b0=+5.988166e-06 difference=+5.71e-09 relative correction of m_b: numerical m_b/m_b^(0)-1 vs (14.10): eps=0.200 num=+6.191205e-05 pred=+6.102416e-05 eps=0.100 num=+1.531314e-05 pred=+1.525604e-05 eps=0.050 num=+3.819513e-06 pred=+3.814010e-06 Part B: spectrum of the B_++ cluster of H(P) on a ring of N_r = 40 sites vs the effective operator (14.4) built from its explicit coefficients; P = 0.37, eta = 0.3, data D1 max |eigenvalue difference|, (11.11) : 2.618e-04 1.657e-05 1.039e-06 slopes 3.98 4.00 max |eigenvalue difference|, (14.4) without w12,w21: 6.825e-06 2.367e-07 1.231e-08 slopes 4.85 4.26 max |eigenvalue difference|, (14.4) complete : 4.233e-06 6.754e-08 1.060e-09 slopes 5.97 5.99 Part C: ordered limit u -> 0 (w11 < 0) of the eps^4 coefficients of (14.7)-(14.10) C1: exact Taylor coefficients in u of the closed forms (sympy) vs the coefficients of (14.12): E4 through u^3: identical = True nuC through u^4: identical = True rel through u^4: identical = True Xi1 through u^5: identical = True Xi2 through u^5: identical = True C2: t_* (root of 2t^3-9t^2-3t+8 in (1/2,2)) = 0.8625413912, (2-t_*)/(2t_*-1) = 1.5687293044; 50-digit arithmetic leading power = first displayed coefficient with |c| > 1e-30; ratio = closed form / leading term D3 a^=(0.450000,0.155769) M=(0.800000,1.300000) (gc-2gm)/gm=-6.852e-01 (gs-3gm)/gm=-1.080e+00 max|skipped c|=0.0e+00 leading powers: E4 u^3 (c=+7.077016e-02) nuC u^2 (c=-3.705082e-02) rel u^2 (c=+8.004808e-02) Xi1 u^3 (c=+9.585799e-02) Xi2 u^3 (c=-2.769231e-01) u=1e-02 ratios: E4 0.999750 nuC 0.974152 rel 0.974201 Xi1 0.990132 Xi2 0.990624 u=1e-03 ratios: E4 0.999980 nuC 0.997423 rel 0.997424 Xi1 0.999029 Xi2 0.999063 equal a^=(0.250000,0.250000) M=(1.000000,1.000000) (gc-2gm)/gm=-1.000e+00 (gs-3gm)/gm=-2.000e+00 max|skipped c|=0.0e+00 leading powers: E4 u^3 (c=+6.250000e-02) nuC u^2 (c=-6.250000e-02) rel u^2 (c=+1.250000e-01) Xi1 u^4 (c=-3.125000e-02) Xi2 u^4 (c=-3.125000e-02) u=1e-02 ratios: E4 0.997450 nuC 0.970077 rel 0.970126 Xi1 1.039844 Xi2 1.039844 u=1e-03 ratios: E4 0.999750 nuC 0.997001 rel 0.997001 Xi1 1.003998 Xi2 1.003998 S2 a^=(0.250000,1.000000) M=(1.142857,1.000000) (gc-2gm)/gm=+0.000e+00 (gs-3gm)/gm=-7.500e-01 max|skipped c|=0.0e+00 leading powers: E4 u^3 (c=+2.343750e-01) nuC u^3 (c=+1.125000e-01) rel u^3 (c=-1.406250e-01) Xi1 u^3 (c=-8.125000e-01) Xi2 u^3 (c=+2.031250e-01) u=1e-02 ratios: E4 1.000575 nuC 1.014814 rel 1.014865 Xi1 0.991945 Xi2 0.992550 u=1e-03 ratios: E4 1.000062 nuC 1.001498 rel 1.001499 Xi1 0.999197 Xi2 0.999268 S3 a^=(0.208712,1.000000) M=(1.558258,1.000000) (gc-2gm)/gm=+7.800e-51 (gs-3gm)/gm=+1.560e-50 max|skipped c|=2.2e-51 leading powers: E4 u^3 (c=+1.713258e-01) nuC u^4 (c=+2.202760e-01) rel u^4 (c=-3.189205e-01) Xi1 u^3 (c=-7.165151e-01) Xi2 u^3 (c=+1.495454e-01) u=1e-02 ratios: E4 1.002450 nuC 0.995417 rel 0.995467 Xi1 0.990726 Xi2 0.993474 u=1e-03 ratios: E4 1.000249 nuC 0.999554 rel 0.999555 Xi1 0.999075 Xi2 0.999361 X3 a^=(0.862541,1.000000) M=(1.568729,1.000000) (gc-2gm)/gm=-1.022e+00 (gs-3gm)/gm=-1.956e+00 max|skipped c|=8.4e-53 leading powers: E4 u^3 (c=+7.061879e-01) nuC u^2 (c=-7.177271e-01) rel u^2 (c=+3.874586e-01) Xi1 u^5 (c=-3.791529e-01) Xi2 u^4 (c=-1.688121e-01) u=1e-02 ratios: E4 0.997560 nuC 0.970927 rel 0.970975 Xi1 0.998819 Xi2 1.022334 u=1e-03 ratios: E4 0.999760 nuC 0.997086 rel 0.997087 Xi1 0.999895 Xi2 1.002245