Check of 04-weight-of-bond v3 (N_f = 1, t_f = 1) Couplings g_i = sigma_i gamma_i M_{k_i}, gamma_1 = eta/(2.5 ell_0), gamma_2 = 0.6 gamma_1; rel.dev = |exact/formula - 1|. Expected: rel.dev of the third-order formulas proportional to eta (ratio ~2 when eta halves). --- graph D=1: L=30, x1=[6], r=[3], r'=[2], e=[1]; midpoint version: xbar=[8], r=[4], r'=[2]; ell_0=2.4972, g_s=0.004004 choice | eta | rel.dev bond profile (4.2) | rel.dev W_1 (4.5) | rel.dev W_b (4.6) | rho_ex(full grip) | rho0 (4.10) | rho_ex(a)/[(1+n g1)/(n g1) rho0] (4.13) | rho_mid_ex vs rho0_mid (4.12) A | 0.08 | 8.10e-02 | 2.52e-05 | 3.68e-01 | 1.23433 (body1), 1.23925 (body2) | 1.18588 | 1.12672, 1.11709 | 1.03307 vs 1.00000 A | 0.04 | 4.14e-02 | 1.18e-05 | 1.90e-01 | 1.21058 (body1), 1.21317 (body2) | 1.18588 | 1.06230, 1.05784 | 1.01699 vs 1.00000 A | 0.02 | 2.09e-02 | 6.29e-06 | 9.65e-02 | 1.19831 (body1), 1.19964 (body2) | 1.18588 | 1.03085, 1.02871 | 1.00861 vs 1.00000 A | dev(eta)/dev(eta/2) for eta=0.08; 0.04, columns [profile, W_1, W_b, rho_fg body1, rho_fg body2, rho_a body1, rho_a body2, rho_mid]: 1.96 2.14 1.94 1.96 1.96 2.03 2.02 1.95; 1.98 1.87 1.97 1.99 1.98 2.02 2.01 1.97 B | 0.08 | 2.56e-02 | 3.50e-04 | 1.54e-01 | 1.18085 (body1), 1.15057 (body2) | 1.18588 | 0.99252, 1.03700 | 0.99375 vs 1.00000 B | 0.04 | 1.19e-02 | 6.87e-05 | 6.98e-02 | 1.18356 (body1), 1.16883 (body2) | 1.18588 | 0.99723, 1.01914 | 0.99692 vs 1.00000 B | 0.02 | 5.76e-03 | 1.78e-05 | 3.29e-02 | 1.18472 (body1), 1.17746 (body2) | 1.18588 | 0.99882, 1.00968 | 0.99846 vs 1.00000 B | dev(eta)/dev(eta/2) for eta=0.08; 0.04, columns [profile, W_1, W_b, rho_fg body1, rho_fg body2, rho_a body1, rho_a body2, rho_mid]: 2.14 5.10 2.21 2.17 2.07 2.70 1.93 2.03; 2.07 3.87 2.12 2.00 2.02 2.34 1.98 2.00 --- graph D=1: L=60, x1=[12], r=[6], r'=[5], e=[1]; midpoint version: xbar=[16], r=[8], r'=[4]; ell_0=4.9986, g_s=0.002001 choice | eta | rel.dev bond profile (4.2) | rel.dev W_1 (4.5) | rel.dev W_b (4.6) | rho_ex(full grip) | rho0 (4.10) | rho_ex(a)/[(1+n g1)/(n g1) rho0] (4.13) | rho_mid_ex vs rho0_mid (4.12) A | 0.08 | 7.81e-02 | 4.15e-05 | 5.08e-01 | 1.22521 (body1), 1.22951 (body2) | 1.17551 | 1.12828, 1.11810 | 1.03196 vs 1.00000 A | 0.04 | 3.98e-02 | 9.43e-06 | 2.63e-01 | 1.20079 (body1), 1.20306 (body2) | 1.17551 | 1.06300, 1.05828 | 1.01643 vs 1.00000 A | 0.02 | 2.01e-02 | 3.48e-06 | 1.34e-01 | 1.18819 (body1), 1.18936 (body2) | 1.17551 | 1.03116, 1.02890 | 1.00833 vs 1.00000 A | dev(eta)/dev(eta/2) for eta=0.08; 0.04, columns [profile, W_1, W_b, rho_fg body1, rho_fg body2, rho_a body1, rho_a body2, rho_mid]: 1.96 4.40 1.93 1.97 1.96 2.04 2.03 1.95; 1.98 2.71 1.96 1.99 1.99 2.02 2.02 1.97 B | 0.08 | 2.47e-02 | 1.02e-04 | 2.10e-01 | 1.17234 (body1), 1.14548 (body2) | 1.17551 | 0.99405, 1.04153 | 0.99547 vs 1.00000 B | 0.04 | 1.15e-02 | 2.58e-06 | 9.43e-02 | 1.17425 (body1), 1.16122 (body2) | 1.17551 | 0.99811, 1.02144 | 0.99776 vs 1.00000 B | 0.02 | 5.54e-03 | 3.09e-06 | 4.44e-02 | 1.17491 (body1), 1.16849 (body2) | 1.17551 | 0.99928, 1.01083 | 0.99887 vs 1.00000 B | dev(eta)/dev(eta/2) for eta=0.08; 0.04, columns [profile, W_1, W_b, rho_fg body1, rho_fg body2, rho_a body1, rho_a body2, rho_mid]: 2.15 39.56 2.23 2.52 2.10 3.15 1.94 2.02; 2.08 0.84 2.13 2.08 2.04 2.63 1.98 1.99 --- graph D=2: L=12, x1=[4, 0], r=[1, 1], r'=[2, 2], e=[1, 0]; midpoint version: xbar=[4, 0], r=[0, 2], r'=[2, 2]; ell_0=0.4441, g_s=0.02252 choice | eta | rel.dev bond profile (4.2) | rel.dev W_1 (4.5) | rel.dev W_b (4.6) | rho_ex(full grip) | rho0 (4.10) | rho_ex(a)/[(1+n g1)/(n g1) rho0] (4.13) | rho_mid_ex vs rho0_mid (4.12) A | 0.08 | 6.62e-02 | 8.57e-04 | 3.61e-02 | 0.63843 (body1), 0.63918 (body2) | 0.63014 | 1.09582, 1.08369 | 0.96268 vs 0.95615 A | 0.04 | 3.39e-02 | 2.71e-04 | 1.81e-02 | 0.63435 (body1), 0.63474 (body2) | 0.63014 | 1.04734, 1.04144 | 0.95946 vs 0.95615 A | 0.02 | 1.72e-02 | 1.25e-04 | 9.02e-03 | 0.63226 (body1), 0.63246 (body2) | 0.63014 | 1.02353, 1.02061 | 0.95782 vs 0.95615 A | dev(eta)/dev(eta/2) for eta=0.08; 0.04, columns [profile, W_1, W_b, rho_fg body1, rho_fg body2, rho_a body1, rho_a body2, rho_mid]: 1.95 3.16 2.00 1.97 1.97 2.02 2.02 1.97; 1.98 2.17 2.01 1.99 1.98 2.01 2.01 1.99 B | 0.08 | 2.23e-02 | 3.17e-03 | 1.17e-02 | 0.62917 (body1), 0.62451 (body2) | 0.63014 | 0.99446, 1.05863 | 0.95656 vs 0.95615 B | 0.04 | 1.01e-02 | 8.50e-04 | 5.78e-03 | 0.62985 (body1), 0.62760 (body2) | 0.63014 | 0.99853, 1.02967 | 0.95647 vs 0.95615 B | 0.02 | 4.82e-03 | 2.70e-04 | 2.89e-03 | 0.63004 (body1), 0.62893 (body2) | 0.63014 | 0.99958, 1.01491 | 0.95634 vs 0.95615 B | dev(eta)/dev(eta/2) for eta=0.08; 0.04, columns [profile, W_1, W_b, rho_fg body1, rho_fg body2, rho_a body1, rho_a body2, rho_mid]: 2.20 3.73 2.03 3.30 2.22 3.77 1.98 1.29; 2.10 3.15 2.00 2.79 2.10 3.51 1.99 1.69 --- graph D=2: L=20, x1=[6, 0], r=[1, 1], r'=[2, 2], e=[1, 0]; midpoint version: xbar=[6, 0], r=[0, 2], r'=[2, 2]; ell_0=0.5255, g_s=0.01903 choice | eta | rel.dev bond profile (4.2) | rel.dev W_1 (4.5) | rel.dev W_b (4.6) | rho_ex(full grip) | rho0 (4.10) | rho_ex(a)/[(1+n g1)/(n g1) rho0] (4.13) | rho_mid_ex vs rho0_mid (4.12) A | 0.08 | 7.38e-02 | 9.64e-04 | 5.57e-02 | 0.99896 (body1), 0.99963 (body2) | 0.99147 | 1.08976, 1.07714 | 0.95031 vs 0.94577 A | 0.04 | 3.79e-02 | 3.05e-04 | 2.84e-02 | 0.99527 (body1), 0.99562 (body2) | 0.99147 | 1.04439, 1.03822 | 0.94808 vs 0.94577 A | 0.02 | 1.92e-02 | 1.40e-04 | 1.43e-02 | 0.99339 (body1), 0.99357 (body2) | 0.99147 | 1.02208, 1.01903 | 0.94688 vs 0.94577 A | dev(eta)/dev(eta/2) for eta=0.08; 0.04, columns [profile, W_1, W_b, rho_fg body1, rho_fg body2, rho_a body1, rho_a body2, rho_mid]: 1.95 3.16 1.96 1.97 1.96 2.02 2.02 1.96; 1.97 2.17 1.99 1.98 1.97 2.01 2.01 2.08 B | 0.08 | 2.50e-02 | 3.58e-03 | 2.29e-02 | 0.99338 (body1), 0.98914 (body2) | 0.99147 | 0.99788, 1.06566 | 0.93773 vs 0.94577 B | 0.04 | 1.14e-02 | 9.60e-04 | 1.08e-02 | 0.99252 (body1), 0.99049 (body2) | 0.99147 | 1.00005, 1.03282 | 0.94182 vs 0.94577 B | 0.02 | 5.41e-03 | 3.04e-04 | 5.24e-03 | 0.99201 (body1), 0.99102 (body2) | 0.99147 | 1.00029, 1.01639 | 0.94382 vs 0.94577 B | dev(eta)/dev(eta/2) for eta=0.08; 0.04, columns [profile, W_1, W_b, rho_fg body1, rho_fg body2, rho_a body1, rho_a body2, rho_mid]: 2.20 3.73 2.13 1.82 2.37 44.43 2.00 2.04; 2.10 3.16 2.05 1.94 2.16 0.16 2.00 2.03 --- graph D=3: L=8, x1=[3, 0, 0], r=[0, 1, 0], r'=[1, 1, 1], e=[1, 0, 0]; midpoint version: xbar=[3, 0, 0], r=[0, 2, 0], r'=[2, 2, 0]; ell_0=0.2246, g_s=0.04452 choice | eta | rel.dev bond profile (4.2) | rel.dev W_1 (4.5) | rel.dev W_b (4.6) | rho_ex(full grip) | rho0 (4.10) | rho_ex(a)/[(1+n g1)/(n g1) rho0] (4.13) | rho_mid_ex vs rho0_mid (4.12) A | 0.08 | 6.70e-02 | 1.10e-03 | 6.23e-02 | 1.10348 (body1), 1.10360 (body2) | 1.09927 | 1.08568, 1.07251 | 0.84421 vs 0.84329 A | 0.04 | 3.44e-02 | 3.47e-04 | 3.18e-02 | 1.10140 (body1), 1.10146 (body2) | 1.09927 | 1.04240, 1.03594 | 0.84374 vs 0.84329 A | 0.02 | 1.74e-02 | 1.60e-04 | 1.61e-02 | 1.10032 (body1), 1.10035 (body2) | 1.09927 | 1.02107, 1.01787 | 0.84352 vs 0.84329 A | dev(eta)/dev(eta/2) for eta=0.08; 0.04, columns [profile, W_1, W_b, rho_fg body1, rho_fg body2, rho_a body1, rho_a body2, rho_mid]: 1.95 3.16 1.96 1.98 1.97 2.02 2.02 2.04; 1.97 2.17 1.98 2.02 2.02 2.01 2.01 2.01 B | 0.08 | 2.28e-02 | 4.10e-03 | 2.35e-02 | 1.10014 (body1), 1.09940 (body2) | 1.09927 | 0.99671, 1.06825 | 0.84441 vs 0.84329 B | 0.04 | 1.03e-02 | 1.10e-03 | 1.08e-02 | 1.09979 (body1), 1.09943 (body2) | 1.09927 | 0.99945, 1.03399 | 0.84388 vs 0.84329 B | 0.02 | 4.90e-03 | 3.47e-04 | 5.22e-03 | 1.09955 (body1), 1.09938 (body2) | 1.09927 | 1.00000, 1.01696 | 0.84359 vs 0.84329 B | dev(eta)/dev(eta/2) for eta=0.08; 0.04, columns [profile, W_1, W_b, rho_fg body1, rho_fg body2, rho_a body1, rho_a body2, rho_mid]: 2.21 3.74 2.17 1.68 0.80 5.99 2.01 1.90; 2.11 3.16 2.07 1.84 1.52 724.93 2.00 1.95 --- graph D=3: L=10, x1=[4, 0, 0], r=[0, 1, 0], r'=[1, 1, 1], e=[1, 0, 0]; midpoint version: xbar=[4, 0, 0], r=[0, 2, 0], r'=[2, 2, 0]; ell_0=0.2302, g_s=0.04344 choice | eta | rel.dev bond profile (4.2) | rel.dev W_1 (4.5) | rel.dev W_b (4.6) | rho_ex(full grip) | rho0 (4.10) | rho_ex(a)/[(1+n g1)/(n g1) rho0] (4.13) | rho_mid_ex vs rho0_mid (4.12) A | 0.08 | 6.82e-02 | 1.10e-03 | 6.36e-02 | 1.28260 (body1), 1.28276 (body2) | 1.27776 | 1.08563, 1.07249 | 0.85054 vs 0.84964 A | 0.04 | 3.50e-02 | 3.47e-04 | 3.25e-02 | 1.28019 (body1), 1.28028 (body2) | 1.27776 | 1.04236, 1.03591 | 0.85004 vs 0.84964 A | 0.02 | 1.77e-02 | 1.60e-04 | 1.65e-02 | 1.27884 (body1), 1.27889 (body2) | 1.27776 | 1.02096, 1.01777 | 0.84932 vs 0.84964 A | dev(eta)/dev(eta/2) for eta=0.08; 0.04, columns [profile, W_1, W_b, rho_fg body1, rho_fg body2, rho_a body1, rho_a body2, rho_mid]: 1.95 3.16 1.95 1.99 1.99 2.02 2.02 2.24; 1.97 2.17 1.97 2.24 2.23 2.02 2.02 1.25 B | 0.08 | 2.33e-02 | 4.10e-03 | 2.30e-02 | 1.27649 (body1), 1.27546 (body2) | 1.27776 | 0.99492, 1.06621 | 0.84989 vs 0.84964 B | 0.04 | 1.05e-02 | 1.10e-03 | 1.05e-02 | 1.27721 (body1), 1.27672 (body2) | 1.27776 | 0.99855, 1.03300 | 0.84980 vs 0.84964 B | 0.02 | 4.99e-03 | 3.47e-04 | 5.08e-03 | 1.27754 (body1), 1.27730 (body2) | 1.27776 | 0.99958, 1.01649 | 0.84979 vs 0.84964 B | dev(eta)/dev(eta/2) for eta=0.08; 0.04, columns [profile, W_1, W_b, rho_fg body1, rho_fg body2, rho_a body1, rho_a body2, rho_mid]: 2.21 3.74 2.18 2.34 2.22 3.51 2.01 1.63; 2.11 3.16 2.08 2.51 2.27 3.41 2.00 1.04 --- mirror identity max spec h(g) = -min spec h(-g) (even L): D=1, L=30: max + min(-g) = 0.00e+00 D=2, L=12: max + min(-g) = 0.00e+00 D=3, L=8: max + min(-g) = 0.00e+00 --- sum over x1 of the full g_s g1 g2 coefficient of (4.6) (incl. -ell2(r)/X): D=1, L=30: sum = -8.88e-16 D=2, L=12: sum = 1.39e-17 D=3, L=8: sum = 3.47e-18 --- (4.3) ring: max |ell(r) - [(L^2-1)/(12L) - r(L-r)/(2L)]|: L=30: 6.66e-16 L=61: 4.44e-15 L=200: 7.11e-15 --- (4.4) 2D: ell(0)-ell(r e1) - [ln r/(2 pi) + (2 gamma_E + 3 ln 2)/(4 pi)]: L=256: r=2: -4.30e-03, r=4: -1.05e-03, r=8: -4.59e-04, r=16: -1.03e-03, r=32: -3.95e-03 L=1024: r=2: -4.28e-03, r=4: -9.89e-04, r=8: -2.30e-04, r=16: -1.13e-04, r=32: -2.57e-04 --- (4.4) 3D: ell(0)-ell(r e1) - [G_0 - 1/(4 pi r)], G_0 = 0.2527310098: L=64: r=2: -3.10e-03, r=4: -4.05e-04, r=8: -8.25e-05, r=16: -1.83e-04 L=128: r=2: -3.10e-03, r=4: -3.96e-04, r=8: -4.60e-05, r=16: -2.57e-05 --- (4.12) ring: lattice rho0 (e=+1, r'=r-1) vs 2(1-w)(1-2w)/((1-2u)(1-2v)), u=(d+1/2)/L, v=(r-1/2)/L, w=u+v: u=0.2, v=0.1: L=40: 1.18050 vs 1.18050 (diff +2.0e-15); L=80: 1.17316 vs 1.17316 (diff +2.9e-15); L=160: 1.16981 vs 1.16981 (diff +3.5e-14); L=320: 1.16821 vs 1.16821 (diff +1.2e-13) u=0.1, v=0.05: L=40: 1.65998 vs 1.65998 (diff +4.9e-15); L=80: 1.65601 vs 1.65601 (diff -2.7e-14); L=160: 1.65430 vs 1.65430 (diff -1.9e-13); L=320: 1.65352 vs 1.65352 (diff -5.0e-13) u=0.6, v=0.2: L=40: 1.70667 vs 1.70667 (diff +2.4e-15); L=80: 1.84394 vs 1.84394 (diff +7.3e-15); L=160: 1.91940 vs 1.91940 (diff -1.4e-13); L=320: 1.95903 vs 1.95903 (diff -3.6e-13) small-v limit 2(1-u): u=0.2 -> 1.6; u=0.1 -> 1.8 --- (4.11) 3D, L=128: rho0 - 1 and, in brackets, d*(rho0 - 1) vs d (x1=(d,0,0), e=e1): r=(0, 1, 0), r'=(1, 1, 1): d=8: +1.220e-01 (+0.976), d=16: +6.192e-02 (+0.991), d=32: +3.458e-02 (+1.106) r=(0, 2, 0), r'=(2, 3, 0): d=8: +2.321e-01 (+1.857), d=16: +1.238e-01 (+1.981), d=32: +6.826e-02 (+2.184) r=(0, 1, 0), r'=(0, 2, 0): d=8: +2.569e-02 (+0.205), d=16: +5.985e-03 (+0.096), d=32: +1.486e-03 (+0.048) r=(1, 0, 0), r'=(2, 0, 0): d=8: -8.986e-03 (-0.072), d=16: -4.938e-03 (-0.079), d=32: -2.950e-03 (-0.094) --- (4.11) 2D: rho0 - 1 and, in brackets, d*(rho0 - 1) vs d and L (x1=(d,0), e=e1, r=(2,1), r'=(4,3)): L=128: d=8: +3.663e-01 (+2.931), d=16: +1.958e-01 (+3.133), d=32: +9.628e-02 (+3.081), d=64: -1.832e+00 (-117.227) L=512: d=8: +6.365e-01 (+5.092), d=16: +3.619e-01 (+5.790), d=32: +1.807e-01 (+5.783), d=64: +8.655e-02 (+5.539) L=2048: d=8: +9.125e-01 (+7.300), d=16: +5.357e-01 (+8.571), d=32: +2.758e-01 (+8.825), d=64: +1.327e-01 (+8.493) --- (4.12) midpoint place: rho0_mid - 1 (ring; expected 0 up to rounding): L=40: (dbar,r,r')=(10,4,2): +2.2e-15, (dbar,r,r')=(8,10,4): +0.0e+00, (dbar,r,r')=(24,12,6): +7.1e-15 L=160: (dbar,r,r')=(40,4,2): -5.3e-14, (dbar,r,r')=(32,10,4): +1.6e-14, (dbar,r,r')=(96,12,6): +1.1e-15 --- (4.12) midpoint place, 2D: rho0_mid - 1 and, in brackets, d^2*(rho0_mid - 1) (xbar=(d,0), e=e1, r=(2,2), r'=(4,2)): L=128: d=8: -2.734e-01 (-17.5), d=16: -6.036e-02 (-15.5), d=32: -1.385e-02 (-14.2), d=64: -5.869e-03 (-24.0) L=512: d=8: -5.208e-01 (-33.3), d=16: -1.243e-01 (-31.8), d=32: -2.808e-02 (-28.8), d=64: -6.302e-03 (-25.8) L=2048: d=8: -7.729e-01 (-49.5), d=16: -1.918e-01 (-49.1), d=32: -4.520e-02 (-46.3), d=64: -1.040e-02 (-42.6) --- (4.12) midpoint place, 3D, L=128: rho0_mid - 1 and, in brackets, d^2*(rho0_mid - 1) (xbar=(d,0,0), e=e1): r=(0, 2, 0), r'=(2, 2, 2): d=8: -3.577e-02 (-2.29), d=16: -1.059e-02 (-2.71), d=32: -2.971e-03 (-3.04) r=(2, 0, 0), r'=(4, 2, 0): d=8: -3.812e-02 (-2.44), d=16: -7.741e-03 (-1.98), d=32: -1.755e-03 (-1.80) --- 3D large separations (consistency check 2): W_b / (2 n g1 g2) = S - ell2(r)/X at |r|, d ~ L/4 shrinks with L: L=16: 3.502e-05 L=32: 7.703e-06 L=64: 1.877e-06 --- (4.15): exact lowest eigenvalue (Lanczos) minus the root of the finite-rank equation (bodies and source, eta = 0.08): D=1, L=256: +8.9e-16 D=2, L=64: -5.8e-15 D=3, L=16: -1.7e-14 --- Hellmann-Feynman g_s-coefficient |psi_0(x_s)|^2 against the symmetric difference in g_s (D=2, L=32, eta = 0.08): g_s=0.0167: symmetric difference / |psi_0(x_s)|^2 - 1 = +9.90e-05 g_s=0.0083: symmetric difference / |psi_0(x_s)|^2 - 1 = +2.48e-05 === Distance sweeps (I5): exact eigenpairs, full grip, body 1; rel.dev = exact/formula - 1; [ratio] = deviation at eta over deviation at eta/2 (2 for a deviation linear in eta) === --- sweep D=1, L=64: r=[3], r'=[2]; midpoint r=[4], r'=[2]; ell_0=5.3320 choice A (k1,s1,k2,s2)=(0,+,1,+): actual eta = max|g_i| ell_0/t_f = 0.0320 and 0.0160 d | rel.dev W_1 (4.5) at eta, eta/2 | rho0 (4.10) | rho_ex-rho0 at eta, eta/2 [ratio] | |rho_ex-rho0|/eta_act, bound scale K of (4.10) | rho0_mid (4.12) | rho_mid_ex-rho0_mid at eta, eta/2 [ratio] 8 | +6.69e-04, +1.64e-04 | 1.60548 | +2.760e-02, +1.432e-02 [1.93] | 0.862, 4.4e+02 | 1.00000 | +5.000e-03, +2.889e-03 [1.73] 16 | -2.52e-04, -6.55e-05 | 1.27939 | +4.258e-02, +2.206e-02 [1.93] | 1.331, 5.8e+02 | 1.00000 | +2.700e-02, +1.410e-02 [1.91] 24 | -6.78e-04, -1.73e-04 | 0.83616 | +7.562e-02, +3.900e-02 [1.94] | 2.363, 9.7e+02 | 1.00000 | +3.939e-02, +2.053e-02 [1.92] profile (4.2), r = 2s e1 against r' = s e1; s: rel.dev at eta, eta/2 [ratio]: s=1: -1.04e-01, -5.35e-02 [1.94]; s=2: -9.47e-02, -4.86e-02 [1.95]; s=4: -7.77e-02, -3.96e-02 [1.96]; s=8: -4.88e-02, -2.45e-02 [1.99]; s=16: -1.55e-02, -7.67e-03 [2.02] choice B (k1,s1,k2,s2)=(2,-,3,+): actual eta = max|g_i| ell_0/t_f = 0.0640 and 0.0320 d | rel.dev W_1 (4.5) at eta, eta/2 | rho0 (4.10) | rho_ex-rho0 at eta, eta/2 [ratio] | |rho_ex-rho0|/eta_act, bound scale K of (4.10) | rho0_mid (4.12) | rho_mid_ex-rho0_mid at eta, eta/2 [ratio] 8 | +2.34e-03, +6.13e-04 | 1.60548 | -1.357e-02, -7.027e-03 [1.93] | 0.212, 4.4e+02 | 1.00000 | -1.503e-02, -7.747e-03 [1.94] 16 | -1.27e-03, -2.95e-04 | 1.27939 | -1.551e-02, -8.354e-03 [1.86] | 0.242, 5.8e+02 | 1.00000 | -1.171e-02, -6.482e-03 [1.81] 24 | -3.04e-03, -7.32e-04 | 0.83616 | -1.023e-02, -5.164e-03 [1.98] | 0.160, 9.7e+02 | 1.00000 | -5.475e-03, -3.373e-03 [1.62] profile (4.2), r = 2s e1 against r' = s e1; s: rel.dev at eta, eta/2 [ratio]: s=1: +3.11e-02, +1.50e-02 [2.07]; s=2: +2.92e-02, +1.39e-02 [2.10]; s=4: +2.45e-02, +1.14e-02 [2.15]; s=8: +1.39e-02, +6.74e-03 [2.07]; s=16: -1.50e-03, +6.40e-04 [-2.34] --- sweep D=1, L=256: r=[3], r'=[2]; midpoint r=[4], r'=[2]; ell_0=21.3330 choice A (k1,s1,k2,s2)=(0,+,1,+): actual eta = max|g_i| ell_0/t_f = 0.0320 and 0.0160 d | rel.dev W_1 (4.5) at eta, eta/2 | rho0 (4.10) | rho_ex-rho0 at eta, eta/2 [ratio] | |rho_ex-rho0|/eta_act, bound scale K of (4.10) | rho0_mid (4.12) | rho_mid_ex-rho0_mid at eta, eta/2 [ratio] 32 | +7.30e-04, +1.79e-04 | 1.71486 | +9.297e-03, +5.203e-03 [1.79] | 0.291, 6.8e+03 | 1.00000 | -2.173e-03, -7.220e-04 [3.01] 64 | -2.20e-04, -5.76e-05 | 1.44669 | +3.091e-02, +1.628e-02 [1.90] | 0.966, 9.2e+03 | 1.00000 | +2.110e-02, +1.121e-02 [1.88] 96 | -6.64e-04, -1.69e-04 | 1.15171 | +7.050e-02, +3.654e-02 [1.93] | 2.203, 1.6e+04 | 1.00000 | +3.420e-02, +1.804e-02 [1.90] profile (4.2), r = 2s e1 against r' = s e1; s: rel.dev at eta, eta/2 [ratio]: s=1: -1.11e-01, -5.73e-02 [1.93]; s=2: -1.08e-01, -5.61e-02 [1.93]; s=4: -1.04e-01, -5.35e-02 [1.94]; s=8: -9.47e-02, -4.87e-02 [1.95]; s=16: -7.77e-02, -3.96e-02 [1.96] choice B (k1,s1,k2,s2)=(2,-,3,+): actual eta = max|g_i| ell_0/t_f = 0.0640 and 0.0320 d | rel.dev W_1 (4.5) at eta, eta/2 | rho0 (4.10) | rho_ex-rho0 at eta, eta/2 [ratio] | |rho_ex-rho0|/eta_act, bound scale K of (4.10) | rho0_mid (4.12) | rho_mid_ex-rho0_mid at eta, eta/2 [ratio] 32 | +2.58e-03, +6.73e-04 | 1.71486 | -3.837e-02, -1.982e-02 [1.94] | 0.600, 6.8e+03 | 1.00000 | -4.121e-02, -2.080e-02 [1.98] 64 | -1.14e-03, -2.63e-04 | 1.44669 | -2.993e-02, -1.629e-02 [1.84] | 0.468, 9.2e+03 | 1.00000 | -2.539e-02, -1.371e-02 [1.85] 96 | -2.98e-03, -7.18e-04 | 1.15171 | -1.625e-02, -9.221e-03 [1.76] | 0.254, 1.6e+04 | 1.00000 | -4.079e-03, -3.392e-03 [1.20] profile (4.2), r = 2s e1 against r' = s e1; s: rel.dev at eta, eta/2 [ratio]: s=1: +3.23e-02, +1.59e-02 [2.03]; s=2: +3.19e-02, +1.56e-02 [2.05]; s=4: +3.11e-02, +1.50e-02 [2.07]; s=8: +2.92e-02, +1.39e-02 [2.10]; s=16: +2.46e-02, +1.14e-02 [2.15] --- sweep D=2, L=64: r=[2, 1], r'=[4, 3]; midpoint r=[2, 2], r'=[4, 2]; ell_0=0.7107 choice A (k1,s1,k2,s2)=(0,+,1,+): actual eta = max|g_i| ell_0/t_f = 0.0320 and 0.0160 d | rel.dev W_1 (4.5) at eta, eta/2 | rho0 (4.10) | rho_ex-rho0 at eta, eta/2 [ratio] | |rho_ex-rho0|/eta_act, bound scale K of (4.10) | rho0_mid (4.12) | rho_mid_ex-rho0_mid at eta, eta/2 [ratio] 4 | +1.62e-03, +4.04e-04 | 1.37592 | +8.995e-03, +4.553e-03 [1.98] | 0.281, 2.7e+02 | 0.47681 | -1.335e-02, -6.762e-03 [1.97] 8 | +1.31e-03, +3.27e-04 | 1.22839 | +3.346e-03, +1.711e-03 [1.96] | 0.105, 5.0e+02 | 0.84304 | -5.490e-03, -2.776e-03 [1.98] 16 | +1.01e-03, +2.52e-04 | 1.09635 | +3.904e-03, +1.994e-03 [1.96] | 0.122, 1.1e+03 | 0.96673 | +5.793e-04, +3.046e-04 [1.90] profile (4.2), r = 2s e1 against r' = s e1; s: rel.dev at eta, eta/2 [ratio]: s=1: -8.90e-02, -4.60e-02 [1.93]; s=2: -8.04e-02, -4.14e-02 [1.94]; s=4: -7.16e-02, -3.68e-02 [1.95]; s=8: -6.29e-02, -3.22e-02 [1.95] choice B (k1,s1,k2,s2)=(2,-,3,+): actual eta = max|g_i| ell_0/t_f = 0.0640 and 0.0320 d | rel.dev W_1 (4.5) at eta, eta/2 | rho0 (4.10) | rho_ex-rho0 at eta, eta/2 [ratio] | |rho_ex-rho0|/eta_act, bound scale K of (4.10) | rho0_mid (4.12) | rho_mid_ex-rho0_mid at eta, eta/2 [ratio] 4 | +6.42e-03, +1.61e-03 | 1.37592 | +1.239e-02, +6.464e-03 [1.92] | 0.194, 2.7e+02 | 0.47681 | -2.960e-02, -1.469e-02 [2.02] 8 | +5.23e-03, +1.31e-03 | 1.22839 | +5.690e-03, +2.935e-03 [1.94] | 0.089, 5.0e+02 | 0.84304 | -8.871e-03, -4.385e-03 [2.02] 16 | +4.03e-03, +1.01e-03 | 1.09635 | +1.150e-03, +6.080e-04 [1.89] | 0.018, 1.1e+03 | 0.96673 | -3.399e-03, -1.677e-03 [2.03] profile (4.2), r = 2s e1 against r' = s e1; s: rel.dev at eta, eta/2 [ratio]: s=1: +2.94e-02, +1.36e-02 [2.16]; s=2: +2.72e-02, +1.24e-02 [2.19]; s=4: +2.44e-02, +1.11e-02 [2.21]; s=8: +2.13e-02, +9.64e-03 [2.21] --- sweep D=2, L=256: r=[2, 1], r'=[4, 3]; midpoint r=[2, 2], r'=[4, 2]; ell_0=0.9313 choice A (k1,s1,k2,s2)=(0,+,1,+): actual eta = max|g_i| ell_0/t_f = 0.0320 and 0.0160 d | rel.dev W_1 (4.5) at eta, eta/2 | rho0 (4.10) | rho_ex-rho0 at eta, eta/2 [ratio] | |rho_ex-rho0|/eta_act, bound scale K of (4.10) | rho0_mid (4.12) | rho_mid_ex-rho0_mid at eta, eta/2 [ratio] 4 | +1.99e-03, +4.94e-04 | 1.72475 | +1.902e-02, +9.558e-03 [1.99] | 0.594, 5.2e+02 | -0.15929 | -3.863e-02, -1.952e-02 [1.98] 8 | +1.75e-03, +4.37e-04 | 1.50024 | +5.507e-03, +2.791e-03 [1.97] | 0.172, 9.1e+02 | 0.60418 | -1.963e-02, -9.946e-03 [1.97] 16 | +1.51e-03, +3.77e-04 | 1.27770 | -1.266e-03, -6.238e-04 [2.03] | 0.040, 1.6e+03 | 0.90859 | -1.013e-02, -5.151e-03 [1.97] 32 | +1.26e-03, +3.15e-04 | 1.13720 | -1.657e-03, -8.216e-04 [2.02] | 0.052, 3.1e+03 | 0.97980 | -4.742e-03, -2.422e-03 [1.96] profile (4.2), r = 2s e1 against r' = s e1; s: rel.dev at eta, eta/2 [ratio]: s=1: -9.44e-02, -4.89e-02 [1.93]; s=2: -8.79e-02, -4.55e-02 [1.93]; s=4: -8.15e-02, -4.20e-02 [1.94]; s=8: -7.50e-02, -3.85e-02 [1.95]; s=16: -6.84e-02, -3.52e-02 [1.95] choice B (k1,s1,k2,s2)=(2,-,3,+): actual eta = max|g_i| ell_0/t_f = 0.0640 and 0.0320 d | rel.dev W_1 (4.5) at eta, eta/2 | rho0 (4.10) | rho_ex-rho0 at eta, eta/2 [ratio] | |rho_ex-rho0|/eta_act, bound scale K of (4.10) | rho0_mid (4.12) | rho_mid_ex-rho0_mid at eta, eta/2 [ratio] 4 | +7.81e-03, +1.96e-03 | 1.72475 | +2.263e-03, +1.839e-03 [1.23] | 0.035, 5.2e+02 | -0.15929 | -1.059e-01, -5.254e-02 [2.02] 8 | +6.94e-03, +1.74e-03 | 1.50024 | +1.925e-04, +4.825e-04 [0.40] | 0.003, 9.1e+02 | 0.60418 | -4.242e-02, -2.093e-02 [2.03] 16 | +6.01e-03, +1.50e-03 | 1.27770 | +1.030e-03, +6.905e-04 [1.49] | 0.016, 1.6e+03 | 0.90859 | -1.415e-02, -6.923e-03 [2.04] 32 | +5.05e-03, +1.26e-03 | 1.13720 | +6.941e-04, +4.143e-04 [1.68] | 0.011, 3.1e+03 | 0.97980 | -5.110e-03, -2.498e-03 [2.05] profile (4.2), r = 2s e1 against r' = s e1; s: rel.dev at eta, eta/2 [ratio]: s=1: +3.05e-02, +1.43e-02 [2.14]; s=2: +2.91e-02, +1.34e-02 [2.17]; s=4: +2.75e-02, +1.25e-02 [2.19]; s=8: +2.56e-02, +1.16e-02 [2.21]; s=16: +2.33e-02, +1.05e-02 [2.22] --- sweep D=3, L=16: r=[0, 1, 0], r'=[1, 1, 1]; midpoint r=[0, 2, 0], r'=[2, 2, 2]; ell_0=0.2386 choice A (k1,s1,k2,s2)=(0,+,1,+): actual eta = max|g_i| ell_0/t_f = 0.0320 and 0.0160 d | rel.dev W_1 (4.5) at eta, eta/2 | rho0 (4.10) | rho_ex-rho0 at eta, eta/2 [ratio] | |rho_ex-rho0|/eta_act, bound scale K of (4.10) | rho0_mid (4.12) | rho_mid_ex-rho0_mid at eta, eta/2 [ratio] 2 | +1.20e-03, +3.00e-04 | 1.15431 | +1.559e-03, +7.997e-04 [1.95] | 0.049, 1.9e+02 | 0.95983 | +2.024e-03, +1.039e-03 [1.95] 3 | +1.11e-03, +2.76e-04 | 1.14713 | +1.023e-03, +5.236e-04 [1.95] | 0.032, 4.2e+02 | 0.93780 | +3.179e-04, +1.677e-04 [1.90] 4 | +1.06e-03, +2.64e-04 | 1.14101 | +1.243e-03, +6.330e-04 [1.96] | 0.039, 7.7e+02 | 0.94019 | +1.913e-04, +9.999e-05 [1.91] profile (4.2), r = 2s e1 against r' = s e1; s: rel.dev at eta, eta/2 [ratio]: s=1: -6.97e-02, -3.58e-02 [1.95]; s=2: -6.22e-02, -3.18e-02 [1.95]; s=4: -5.86e-02, -2.99e-02 [1.96] choice B (k1,s1,k2,s2)=(2,-,3,+): actual eta = max|g_i| ell_0/t_f = 0.0640 and 0.0320 d | rel.dev W_1 (4.5) at eta, eta/2 | rho0 (4.10) | rho_ex-rho0 at eta, eta/2 [ratio] | |rho_ex-rho0|/eta_act, bound scale K of (4.10) | rho0_mid (4.12) | rho_mid_ex-rho0_mid at eta, eta/2 [ratio] 2 | +4.82e-03, +1.20e-03 | 1.15431 | +5.541e-03, +2.798e-03 [1.98] | 0.087, 1.9e+02 | 0.95983 | -6.895e-04, -2.840e-04 [2.43] 3 | +4.43e-03, +1.10e-03 | 1.14713 | +1.697e-03, +8.658e-04 [1.96] | 0.027, 4.2e+02 | 0.93780 | -8.827e-04, -4.261e-04 [2.07] 4 | +4.24e-03, +1.06e-03 | 1.14101 | +8.373e-05, +6.336e-05 [1.32] | 0.001, 7.7e+02 | 0.94019 | -8.300e-04, -4.054e-04 [2.05] profile (4.2), r = 2s e1 against r' = s e1; s: rel.dev at eta, eta/2 [ratio]: s=1: +2.38e-02, +1.08e-02 [2.21]; s=2: +2.11e-02, +9.53e-03 [2.21]; s=4: +1.96e-02, +8.92e-03 [2.20] --- sweep D=3, L=32: r=[0, 1, 0], r'=[1, 1, 1]; midpoint r=[0, 2, 0], r'=[2, 2, 2]; ell_0=0.2457 choice A (k1,s1,k2,s2)=(0,+,1,+): actual eta = max|g_i| ell_0/t_f = 0.0320 and 0.0160 d | rel.dev W_1 (4.5) at eta, eta/2 | rho0 (4.10) | rho_ex-rho0 at eta, eta/2 [ratio] | |rho_ex-rho0|/eta_act, bound scale K of (4.10) | rho0_mid (4.12) | rho_mid_ex-rho0_mid at eta, eta/2 [ratio] 2 | +1.26e-03, +3.15e-04 | 1.20904 | +1.604e-03, +8.254e-04 [1.94] | 0.050, 2.0e+02 | 1.04934 | +2.472e-03, +1.265e-03 [1.95] 4 | +1.12e-03, +2.79e-04 | 1.17605 | +4.431e-04, +2.304e-04 [1.92] | 0.014, 7.5e+02 | 0.94747 | -7.771e-04, -3.922e-04 [1.98] 8 | +1.04e-03, +2.61e-04 | 1.11165 | +7.610e-04, +3.886e-04 [1.96] | 0.024, 3.0e+03 | 0.96916 | -2.509e-04, -1.303e-04 [1.93] profile (4.2), r = 2s e1 against r' = s e1; s: rel.dev at eta, eta/2 [ratio]: s=1: -7.10e-02, -3.65e-02 [1.95]; s=2: -6.36e-02, -3.26e-02 [1.95]; s=4: -5.99e-02, -3.07e-02 [1.96]; s=8: -5.83e-02, -2.99e-02 [1.95] choice B (k1,s1,k2,s2)=(2,-,3,+): actual eta = max|g_i| ell_0/t_f = 0.0640 and 0.0320 d | rel.dev W_1 (4.5) at eta, eta/2 | rho0 (4.10) | rho_ex-rho0 at eta, eta/2 [ratio] | |rho_ex-rho0|/eta_act, bound scale K of (4.10) | rho0_mid (4.12) | rho_mid_ex-rho0_mid at eta, eta/2 [ratio] 2 | +5.05e-03, +1.26e-03 | 1.20904 | +5.876e-03, +2.976e-03 [1.97] | 0.092, 2.0e+02 | 1.04934 | -2.253e-03, -1.043e-03 [2.16] 4 | +4.48e-03, +1.12e-03 | 1.17605 | +3.624e-04, +1.938e-04 [1.87] | 0.006, 7.5e+02 | 0.94747 | -2.125e-03, -1.065e-03 [2.00] 8 | +4.18e-03, +1.04e-03 | 1.11165 | -8.162e-04, -3.942e-04 [2.07] | 0.013, 3.0e+03 | 0.96916 | -9.861e-04, -4.938e-04 [2.00] profile (4.2), r = 2s e1 against r' = s e1; s: rel.dev at eta, eta/2 [ratio]: s=1: +2.43e-02, +1.10e-02 [2.21]; s=2: +2.16e-02, +9.79e-03 [2.21]; s=4: +2.02e-02, +9.16e-03 [2.20]; s=8: +1.95e-02, +8.88e-03 [2.20] --- (4.14) exact ratios, highest state (sigma) vs lowest state (-sigma) and (sigma), nominal eta=0.08: graph | choice | rho_fg: max(s), min(-s), min(s), rho0 | rho_a: max(s), min(-s) | max|diff| of max(s) and min(-s) | (rho_fg max(s) - rho0)/(rho_fg min(s) - rho0) D=1, L=30 | A | 1.132834, 1.132834, 1.234330, 1.185882 | 2447.748830, 2447.748830 | 0.0e+00 | -1.095 D=1, L=30 | B | 1.189073, 1.189073, 1.180850, 1.185882 | 2775.917380, 2775.917380 | 0.0e+00 | -0.634 D=2, L=12 | A | 0.621265, 0.621265, 0.638428, 0.630142 | 1144.781210, 1144.781210 | 0.0e+00 | -1.071 D=2, L=12 | B | 0.629656, 0.629656, 0.629173, 0.630142 | 1253.793494, 1253.793494 | 0.0e+00 | +0.502 D=3, L=8 | A | 1.094787, 1.094787, 1.103481, 1.099269 | 3626.786245, 3626.786245 | 0.0e+00 | -1.064 D=3, L=8 | B | 1.097713, 1.097713, 1.100140, 1.099269 | 3929.840964, 3929.840964 | 0.0e+00 | -1.785 --- (4.17) square torus, fixed places, x_s=0, e=e1: L | ell_0 | rho0 | beta_L | rho_L = rho0 - beta_L ell_0 | K/ell_0^3 | |Delta_e ell2(R)| / [(pi^2/8)(2d+1) ell_0] L=128 | place x1=(8,0), r=(2,1), r'=(4,3) | 0.8210 | 1.36635 | 1.30347 | 0.29622 | 1224.5 | 0.075 L=128 | place x1=(16,0), r=(2,1), r'=(4,3) | 0.8210 | 1.19583 | 0.89910 | 0.45768 | 2303.1 | 0.049 L=128 | midpoint xbar=(8,0), r=(2,2), r'=(4,2) | 0.8210 | 0.72658 | -1.16409 | 1.68228 | 1590.2 | 0.075 L=512 | place x1=(8,0), r=(2,1), r'=(4,3) | 1.0416 | 1.63650 | 1.25930 | 0.32478 | 1058.9 | 0.102 L=512 | place x1=(16,0), r=(2,1), r'=(4,3) | 1.0416 | 1.36189 | 0.80287 | 0.52559 | 1850.9 | 0.081 L=512 | midpoint xbar=(8,0), r=(2,2), r'=(4,2) | 1.0416 | 0.47921 | -1.14639 | 1.67332 | 1057.3 | 0.102 L=2048 | place x1=(8,0), r=(2,1), r'=(4,3) | 1.2623 | 1.91253 | 1.25661 | 0.32636 | 964.4 | 0.119 L=2048 | place x1=(16,0), r=(2,1), r'=(4,3) | 1.2623 | 1.53569 | 0.79738 | 0.52918 | 1634.5 | 0.102 L=2048 | midpoint xbar=(8,0), r=(2,2), r'=(4,2) | 1.2623 | 0.22711 | -1.14529 | 1.67277 | 723.1 | 0.119