Check of 04-weight-of-bond v1 (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