Exact mutual cost on a ring (one field object, t_f = 1), gamma = g L / t_f, w = d(L-d)/L^2 rel2 = (M_exact - M2)/M2, relx = (M_exact - M2)/M_exact (error relative to the exact value), pred3 = M3/M2 from (4.8), rel3 = (M_exact - M2 - M3)/M2, pred3_L = -(gamma/5)(1-10w+20w^2)/(1-6w) (leading in 1/L); dev10 / dev12 = max deviation of the smallest zero of the pole-cleared (4.10) / of (4.12) from eigh (one- and two-source energies) -- L = 64 gamma d w M_exact M2 rel2 relx pred3 pred3_L rel3 dev10 dev12 0.03 1 0.0154 -3.30464e-08 -3.32326e-08 -5.602e-03 -5.634e-03 -5.619e-03 -5.625e-03 1.63e-05 2.2e-16 2.2e-16 0.03 8 0.1094 -1.25477e-08 -1.25796e-08 -2.535e-03 -2.541e-03 -2.530e-03 -2.540e-03 -5.34e-06 2.2e-16 2.2e-16 0.03 16 0.1875 4.54887e-09 4.58658e-09 -8.221e-03 -8.289e-03 -8.249e-03 -8.250e-03 2.78e-05 2.2e-16 2.2e-16 0.03 32 0.2500 1.82646e-08 1.83195e-08 -2.994e-03 -3.002e-03 -2.999e-03 -3.000e-03 5.02e-06 4.4e-16 4.4e-16 0.10 1 0.0154 -3.62402e-07 -3.69251e-07 -1.855e-02 -1.890e-02 -1.873e-02 -1.875e-02 1.81e-04 4.4e-16 4.4e-16 0.10 8 0.1094 -1.38586e-07 -1.39773e-07 -8.489e-03 -8.562e-03 -8.432e-03 -8.466e-03 -5.74e-05 4.4e-16 4.4e-16 0.10 16 0.1875 4.95764e-08 5.09620e-08 -2.719e-02 -2.795e-02 -2.750e-02 -2.750e-02 3.06e-04 4.4e-16 4.4e-16 0.10 32 0.2500 2.01527e-07 2.03550e-07 -9.940e-03 -1.004e-02 -9.995e-03 -1.000e-02 5.53e-05 4.4e-16 4.4e-16 0.30 1 0.0154 -3.14194e-06 -3.32326e-06 -5.456e-02 -5.771e-02 -5.619e-02 -5.625e-02 1.63e-03 2.2e-16 2.2e-16 0.30 8 0.1094 -1.22553e-06 -1.25796e-06 -2.578e-02 -2.646e-02 -2.530e-02 -2.540e-02 -4.80e-04 2.2e-16 2.2e-16 0.30 16 0.1875 4.22078e-07 4.58658e-07 -7.975e-02 -8.667e-02 -8.249e-02 -8.250e-02 2.73e-03 2.2e-16 2.2e-16 0.30 32 0.2500 1.77793e-06 1.83195e-06 -2.949e-02 -3.039e-02 -2.999e-02 -3.000e-02 4.96e-04 4.4e-16 4.4e-16 1.00 1 0.0154 -3.06651e-05 -3.69251e-05 -1.695e-01 -2.041e-01 -1.873e-01 -1.875e-01 1.78e-02 4.4e-16 4.4e-16 1.00 8 0.1094 -1.27422e-05 -1.39773e-05 -8.836e-02 -9.693e-02 -8.432e-02 -8.466e-02 -4.04e-03 4.4e-16 4.4e-16 1.00 16 0.1875 3.84439e-06 5.09620e-06 -2.456e-01 -3.256e-01 -2.750e-01 -2.750e-01 2.93e-02 4.4e-16 4.4e-16 1.00 32 0.2500 1.84310e-05 2.03550e-05 -9.452e-02 -1.044e-01 -9.995e-02 -1.000e-01 5.43e-03 4.4e-16 4.4e-16 3.00 1 0.0154 -1.94134e-04 -3.32326e-04 -4.158e-01 -7.118e-01 -5.619e-01 -5.625e-01 1.46e-01 0.0e+00 0.0e+00 3.00 8 0.1094 -9.24888e-05 -1.25796e-04 -2.648e-01 -3.601e-01 -2.530e-01 -2.540e-01 -1.18e-02 2.2e-16 2.2e-16 3.00 16 0.1875 1.88729e-05 4.58658e-05 -5.885e-01 -1.430e+00 -8.249e-01 -8.250e-01 2.36e-01 2.2e-16 2.2e-16 3.00 32 0.2500 1.36755e-04 1.83195e-04 -2.535e-01 -3.396e-01 -2.999e-01 -3.000e-01 4.64e-02 2.2e-16 2.2e-16 -0.30 1 0.0154 -3.51540e-06 -3.32326e-06 5.782e-02 5.466e-02 5.619e-02 5.625e-02 1.63e-03 4.4e-16 4.4e-16 -0.30 8 0.1094 -1.28903e-06 -1.25796e-06 2.470e-02 2.411e-02 2.530e-02 2.540e-02 -5.93e-04 0.0e+00 0.0e+00 -0.30 16 0.1875 4.97778e-07 4.58658e-07 8.529e-02 7.859e-02 8.249e-02 8.250e-02 2.81e-03 0.0e+00 0.0e+00 -0.30 32 0.2500 1.88780e-06 1.83195e-06 3.049e-02 2.958e-02 2.999e-02 3.000e-02 5.01e-04 4.4e-16 4.4e-16 -1.00 1 0.0154 -4.45014e-05 -3.69251e-05 2.052e-01 1.702e-01 1.873e-01 1.875e-01 1.79e-02 4.4e-16 4.4e-16 -1.00 8 0.1094 -1.50412e-05 -1.39773e-05 7.612e-02 7.073e-02 8.432e-02 8.466e-02 -8.20e-03 4.4e-16 4.4e-16 -1.00 16 0.1875 6.66095e-06 5.09620e-06 3.070e-01 2.349e-01 2.750e-01 2.750e-01 3.21e-02 4.4e-16 4.4e-16 -1.00 32 0.2500 2.25039e-05 2.03550e-05 1.056e-01 9.549e-02 9.995e-02 1.000e-01 5.62e-03 4.4e-16 4.4e-16 -- L = 256 gamma d w M_exact M2 rel2 relx pred3 pred3_L rel3 dev10 dev12 0.03 1 0.0039 -2.22219e-09 -2.23535e-09 -5.886e-03 -5.921e-03 -5.906e-03 -5.906e-03 1.94e-05 4.4e-16 4.4e-16 0.03 32 0.1094 -7.84745e-10 -7.86746e-10 -2.543e-03 -2.550e-03 -2.539e-03 -2.540e-03 -4.26e-06 4.4e-16 4.4e-16 0.03 64 0.1875 2.83784e-10 2.86137e-10 -8.225e-03 -8.293e-03 -8.250e-03 -8.250e-03 2.46e-05 4.4e-16 4.4e-16 0.03 128 0.2500 1.14102e-09 1.14444e-09 -2.996e-03 -3.005e-03 -3.000e-03 -3.000e-03 4.24e-06 4.4e-16 4.4e-16 0.10 1 0.0039 -2.43535e-08 -2.48372e-08 -1.948e-02 -1.986e-02 -1.969e-02 -1.969e-02 2.11e-04 2.2e-16 2.2e-16 0.10 32 0.1094 -8.66715e-09 -8.74163e-09 -8.520e-03 -8.594e-03 -8.464e-03 -8.466e-03 -5.65e-05 2.2e-16 2.2e-16 0.10 64 0.1875 3.09285e-09 3.17930e-09 -2.719e-02 -2.795e-02 -2.750e-02 -2.750e-02 3.07e-04 2.2e-16 2.2e-16 0.10 128 0.2500 1.25896e-08 1.27160e-08 -9.944e-03 -1.004e-02 -1.000e-02 -1.000e-02 5.54e-05 2.2e-16 2.2e-16 0.30 1 0.0039 -2.10755e-07 -2.23535e-07 -5.717e-02 -6.064e-02 -5.906e-02 -5.906e-02 1.89e-03 2.2e-16 2.2e-16 0.30 32 0.1094 -7.66398e-08 -7.86746e-08 -2.586e-02 -2.655e-02 -2.539e-02 -2.540e-02 -4.73e-04 2.2e-16 2.2e-16 0.30 64 0.1875 2.63314e-08 2.86137e-08 -7.976e-02 -8.668e-02 -8.250e-02 -8.250e-02 2.73e-03 2.2e-16 2.2e-16 0.30 128 0.2500 1.11068e-07 1.14444e-07 -2.950e-02 -3.040e-02 -3.000e-02 -3.000e-02 4.97e-04 2.2e-16 2.2e-16 1.00 1 0.0039 -2.04534e-06 -2.48372e-06 -1.765e-01 -2.143e-01 -1.969e-01 -1.969e-01 2.04e-02 2.2e-16 2.2e-16 1.00 32 0.1094 -7.96704e-07 -8.74163e-07 -8.861e-02 -9.722e-02 -8.464e-02 -8.466e-02 -3.97e-03 4.4e-16 4.4e-16 1.00 64 0.1875 2.39830e-07 3.17930e-07 -2.457e-01 -3.256e-01 -2.750e-01 -2.750e-01 2.93e-02 2.2e-16 2.2e-16 1.00 128 0.2500 1.15137e-06 1.27160e-06 -9.456e-02 -1.044e-01 -1.000e-01 -1.000e-01 5.44e-03 2.2e-16 2.2e-16 3.00 1 0.0039 -1.28013e-05 -2.23535e-05 -4.273e-01 -7.462e-01 -5.906e-01 -5.906e-01 1.63e-01 4.4e-16 4.4e-16 3.00 32 0.1094 -5.78102e-06 -7.86746e-06 -2.652e-01 -3.609e-01 -2.539e-01 -2.540e-01 -1.13e-02 2.2e-16 2.2e-16 3.00 64 0.1875 1.17749e-06 2.86137e-06 -5.885e-01 -1.430e+00 -8.250e-01 -8.250e-01 2.37e-01 0.0e+00 0.0e+00 3.00 128 0.2500 8.54277e-06 1.14444e-05 -2.535e-01 -3.397e-01 -3.000e-01 -3.000e-01 4.64e-02 2.2e-16 2.2e-16 -0.30 1 0.0039 -2.37166e-07 -2.23535e-07 6.098e-02 5.747e-02 5.906e-02 5.906e-02 1.92e-03 0.0e+00 0.0e+00 -0.30 32 0.1094 -8.06263e-08 -7.86746e-08 2.481e-02 2.421e-02 2.539e-02 2.540e-02 -5.85e-04 4.4e-16 4.4e-16 -0.30 64 0.1875 3.10548e-08 2.86137e-08 8.531e-02 7.860e-02 8.250e-02 8.250e-02 2.81e-03 0.0e+00 0.0e+00 -0.30 128 0.2500 1.17935e-07 1.14444e-07 3.050e-02 2.960e-02 3.000e-02 3.000e-02 5.02e-04 4.4e-16 4.4e-16 -1.00 1 0.0039 -3.02602e-06 -2.48372e-06 2.183e-01 1.792e-01 1.969e-01 1.969e-01 2.15e-02 4.4e-16 4.4e-16 -1.00 32 0.1094 -9.41059e-07 -8.74163e-07 7.653e-02 7.109e-02 8.464e-02 8.466e-02 -8.11e-03 4.4e-16 4.4e-16 -1.00 64 0.1875 4.15579e-07 3.17930e-07 3.071e-01 2.350e-01 2.750e-01 2.750e-01 3.21e-02 4.4e-16 4.4e-16 -1.00 128 0.2500 1.40593e-06 1.27160e-06 1.056e-01 9.554e-02 1.000e-01 1.000e-01 5.64e-03 4.4e-16 4.4e-16 Unequal and mixed-sign sources (pole-cleared (4.10) against eigh): L= 64 d= 8 g1= 0.00469 g2= 0.00937: E_eigh = -1.999791523936965, E_(4.10 cleared) = -1.999791523936965, F(q=0) = -2.25e-04, (M_exact - M2)/M2 = -3.896e-02 L= 64 d= 8 g1= 0.01562 g2= 0.03125: E_eigh = -1.999382359521463, E_(4.10 cleared) = -1.999382359521462, F(q=0) = -7.86e-04, (M_exact - M2)/M2 = -1.335e-01 L= 64 d= 8 g1=-0.01562 g2=-0.03125: E_eigh = -2.000878297773575, E_(4.10 cleared) = -2.000878297773575, F(q=0) = 6.79e-04, (M_exact - M2)/M2 = 1.042e-01 L= 64 d= 8 g1= 0.01562 g2=-0.00781: E_eigh = -1.999895398537231, E_(4.10 cleared) = -1.999895398537231, F(q=0) = -1.09e-04, (M_exact - M2)/M2 = -2.154e-02 L= 64 d= 20 g1= 0.04688 g2=-0.01562: E_eigh = -1.999715219474732, E_(4.10 cleared) = -1.999715219474732, F(q=0) = -3.31e-04, (M_exact - M2)/M2 = -1.396e-01 L= 4 d= 2 g1= 1.00000 g2=-0.50000: E_eigh = -2.000000000000000, E_(4.10 cleared) = -2.000000000000000, F(q=0) = 0.00e+00, (M_exact - M2)/M2 = -1.484e-01 L= 4 d= 1 g1= 1.00000 g2=-0.50000: E_eigh = -1.972818212999035, E_(4.10 cleared) = -1.972818212999035, F(q=0) = -3.12e-02, (M_exact - M2)/M2 = -1.849e-01 L= 64 d= 8 g1= 0.01562 g2=-0.01408: E_eigh = -2.000000000000000, E_(4.10 cleared) = -2.000000000000000, F(q=0) = 6.78e-21, (M_exact - M2)/M2 = -4.655e-03 Exact finite-L bracket of (4.8) on the ring: B_matrix = 2 l0 ld + ld^2 - (l0^(2) + 2 ld^(2))/L against B_closed = (L^2/60)(1 - 10w + 20w^2) - 1/12 + w/3 + 1/(15 L^2); max_d |B_matrix - B_closed| / max_d |B_closed| L= 3: 7.49e-16 L= 4: 1.48e-16 L= 5: 9.91e-16 L= 16: 8.49e-16 L= 64: 3.75e-15 L= 256: 2.26e-13 Mean over placements (4.17), g1 = g2 = g, all y = 0..L-1 (y = 0: both sources on one place): Mbar = (1/L) sum_y M(0,y); inv = E(0,d) - mean_y E(0,y) = M(d) - Mbar (computed from total contracts only); res = max_d |inv - M2 - M3| / max_d |M2| (M2, M3 from (4.5), (4.8)); (4.17): Mbar has no term of order < g^4 L gamma Mbar*L^2/t_f Mbar*L^2/(t_f gamma^4) |Mbar|/max_d|M2| res 64 0.10 1.2774e-08 1.2774e-04 7.666e-06 2.28e-04 64 0.30 9.6820e-07 1.1953e-04 6.456e-05 2.03e-03 64 1.00 9.4761e-05 9.4761e-05 5.687e-04 2.17e-02 64 -0.30 1.1827e-06 1.4601e-04 7.886e-05 2.09e-03 64 -1.00 1.8410e-04 1.8410e-04 1.105e-03 2.37e-02 256 0.10 1.2767e-08 1.2767e-04 7.660e-06 2.29e-04 256 0.30 9.6940e-07 1.1968e-04 6.463e-05 2.04e-03 256 1.00 9.4897e-05 9.4897e-05 5.694e-04 2.18e-02 256 -0.30 1.1840e-06 1.4617e-04 7.893e-05 2.10e-03 256 -1.00 1.8427e-04 1.8427e-04 1.106e-03 2.38e-02 Additive continuum scaling (4.21): F(gamma,u) = Y^2 - 2y^2 from the limiting equations; diff = M_exact L^2/t_f - F; (4.21) states diff = O(gamma^2 L^-2), i.e. diff * L^2 / gamma^2 bounded (its gamma -> 0 value is 1/6, from (4.5)) gamma u F(gamma,u) L=64: diff diff*L^2/g^2 L=256: diff diff*L^2/g^2 |diff|/|F| (L=256) 0.03 0.1250 -5.143127e-05 3.601e-08 0.1639 2.201e-09 0.1603 4.3e-05 0.03 0.2500 1.859583e-05 3.635e-08 0.1654 2.214e-09 0.1612 1.2e-04 0.03 0.5000 7.477537e-05 3.662e-08 0.1667 2.232e-09 0.1625 3.0e-05 0.10 0.1250 -5.680340e-04 3.845e-07 0.1575 2.400e-08 0.1573 4.2e-05 0.10 0.2500 2.026681e-04 3.968e-07 0.1625 2.477e-08 0.1623 1.2e-04 0.10 0.5000 8.250462e-04 4.068e-07 0.1666 2.538e-08 0.1663 3.1e-05 0.30 0.1250 -5.022856e-03 3.082e-06 0.1403 1.926e-07 0.1402 3.8e-05 0.30 0.2500 1.725440e-03 3.391e-06 0.1543 2.119e-07 0.1543 1.2e-04 0.30 0.5000 7.278729e-03 3.654e-06 0.1663 2.283e-07 0.1663 3.1e-05 1.00 0.1250 -5.221420e-02 2.213e-05 0.0907 1.383e-06 0.0907 2.6e-05 1.00 0.2500 1.571558e-02 3.105e-05 0.1272 1.940e-06 0.1272 1.2e-04 1.00 0.5000 7.545355e-02 3.976e-05 0.1628 2.484e-06 0.1628 3.3e-05 3.00 0.1250 -3.788667e-01 3.252e-05 0.0148 2.032e-06 0.0148 5.4e-06 3.00 0.2500 7.715868e-02 1.446e-04 0.0658 9.035e-06 0.0658 1.2e-04 3.00 0.5000 5.598396e-01 3.107e-04 0.1414 1.941e-05 0.1413 3.5e-05 -0.30 0.1250 -5.284192e-03 4.319e-06 0.1966 2.700e-07 0.1966 5.1e-05 -0.30 0.2500 2.034958e-03 3.939e-06 0.1793 2.463e-07 0.1793 1.2e-04 -0.30 0.5000 7.728769e-03 3.653e-06 0.1662 2.283e-07 0.1662 3.0e-05 -1.00 0.1250 -6.167754e-02 6.868e-05 0.2813 4.294e-06 0.2814 7.0e-05 -1.00 0.2500 2.723217e-02 5.109e-05 0.2092 3.194e-06 0.2093 1.2e-04 -1.00 0.5000 9.213644e-02 3.939e-05 0.1613 2.462e-06 0.1614 2.7e-05