PART A: lattice sums by mode classes and pair histograms vs brute-force mode and ordered-pair sums, torus L=12, ell=3.0 dE=0.02: max relative deviation S_1 4.11e-14, S_2 3.98e-10 (cases (sigma,r,eps) = ((-1, 1.0, 1.0), (-1, 1.5, 3.0), (1, 1.0, 5.0), (1, 2.0, 6.0)), P = 0, 0.02) dE=0.002: max relative deviation S_1 4.11e-14, S_2 1.43e-13 (cases (sigma,r,eps) = ((-1, 1.0, 1.0), (-1, 1.5, 3.0), (1, 1.0, 5.0), (1, 2.0, 6.0)), P = 0, 0.02) R_1: classes 0.229832921964, brute force 0.229832921964; J2: histogram 0.011638555558, brute force 0.011638555558 PART B: convergence in ell and L (c0 = 1); rho_1, rho_2 of the F1^2 and F2^2 parts L ell G0 w0 J2 R1 R2 | s=-1 r=1.0 e=1.0: rho_1, rho_2 s=-1 r=1.0 e=4.0: rho_1, rho_2 s=-1 r=0.5 e=2.0: rho_1, rho_2 s=+1 r=1.0 e=4.0: rho_1, rho_2 s=+1 r=1.0 e=2.5: rho_1, rho_2 32 4.0 0.232239 0.22480 0.011949 0.23251 0.12474 | 1.03105 0.45186 2.60889 1.38144 1.57378 0.78110 -0.83750 0.18378 0.01669 nan 48 6.0 0.239180 0.22625 0.012219 0.23251 0.12634 | 1.04864 0.45934 2.62184 1.39875 1.59684 0.79160 -0.83211 0.16405 0.01721 nan 48 8.0 0.242641 0.22682 0.012329 0.23197 0.12701 | 1.05586 0.46260 2.62348 1.40612 1.60690 0.79612 -0.82529 0.15616 0.02013 nan 64 8.0 0.242616 0.22682 0.012329 0.23200 0.12701 | 1.05570 0.46260 2.62347 1.40610 1.60686 0.79611 -0.82583 0.15617 0.01967 nan 128 16.0 0.247714 0.22742 - 0.23035 - | 1.06325 nan 2.61780 nan 1.61911 nan -0.81032 nan 0.02707 nan 256 32.0 0.250233 0.22760 - 0.22901 - | 1.06527 nan 2.61022 nan 1.62351 nan -0.79887 nan 0.03300 nan (rho_2 is nan where the two-quantum part is not isolated by (6.4) or not computed) PART C: main table, torus L=64, ell=8.0 (m_h=0.125), c0=1, F0=1, eps_k=M_k G0=0.242616, w0=0.226815, J2=0.012329, R1=0.231996, R2=0.127005, Omega_max=3.4664 edge of (U): F2=0 at eps=Omega_max/2: eps*R1 = 0.4021; F2!=0 at eps=Omega_max: eps*R1 = 0.8042, eps*R2 = 0.4402 C_i = S_i(0)/S_i(0)|_{tau=0} (moving mark over held mark), rho_i of the parts, t = T/G0, xi* = F2^2/(c0 F1^2) at which rho = 1 (- if none >= 0) r eps sig gap | C_1 C_2 | rho_1 rho_2 | t xi* 0.25 0.5 -1 0.1250 | 1.00550 1.00171 | 0.6521 0.2474 | 0.3278 - 0.25 1.0 -1 0.1250 | 1.00392 1.00139 | 1.0202 0.4490 | 0.5121 0.1811 0.25 2.0 -1 0.1250 | 1.00252 1.00102 | 1.6032 0.7931 | 0.8036 14.3639 0.25 2.0 +1 0.5336 | 0.99251 - | 0.5102 - | 0.2532 - 0.25 3.0 -1 0.1250 | 1.00187 1.00081 | 2.1224 1.1002 | 1.0632 - 0.25 3.0 +1 2.5336 | 0.99643 - | -0.2254 - | -0.1123 - 0.25 4.0 -1 0.1250 | 1.00148 1.00067 | 2.6185 1.3894 | 1.3112 - 0.25 4.0 +1 4.5336 | 0.99762 0.99792 | -0.7811 0.1632 | -0.3896 - 0.25 6.0 -1 0.1250 | 1.00105 1.00050 | 3.5823 1.9414 | 1.7930 - 0.25 6.0 +1 8.5336 | 0.99857 0.99900 | -1.7836 -0.6903 | -0.8905 - 0.25 8.0 -1 0.1250 | 1.00081 1.00040 | 4.5299 2.4758 | 2.2668 - 0.25 8.0 +1 12.5336 | 0.99897 0.99934 | -2.7441 -1.2970 | -1.3706 - 0.25 16.0 -1 0.1250 | 1.00043 1.00022 | 8.2736 4.5550 | 4.1386 - 0.25 16.0 +1 28.5336 | 0.99952 0.99972 | -6.4999 -3.4363 | -3.2484 - 0.25 32.0 -1 0.1250 | 1.00022 1.00012 | 15.7146 8.6470 | 7.8590 - 0.25 32.0 +1 60.5336 | 0.99977 0.99987 | -13.9439 -7.5421 | -6.9703 - 0.50 0.5 -1 0.1250 | 1.02269 1.00691 | 0.6616 0.2494 | 0.3383 - 0.50 1.0 -1 0.1250 | 1.01603 1.00562 | 1.0270 0.4516 | 0.5217 0.2446 0.50 2.0 -1 0.1250 | 1.01023 1.00412 | 1.6069 0.7961 | 0.8116 14.7325 0.50 2.0 +1 0.5336 | 0.97111 - | 0.4846 - | 0.2353 - 0.50 3.0 -1 0.1250 | 1.00754 1.00326 | 2.1245 1.1034 | 1.0703 - 0.50 3.0 +1 2.5336 | 0.98595 - | -0.2385 - | -0.1176 - 0.50 4.0 -1 0.1250 | 1.00598 1.00270 | 2.6196 1.3927 | 1.3176 - 0.50 4.0 +1 4.5336 | 0.99059 0.99177 | -0.7905 0.1617 | -0.3915 - 0.50 6.0 -1 0.1250 | 1.00423 1.00201 | 3.5823 1.9447 | 1.7987 - 0.50 6.0 +1 8.5336 | 0.99431 0.99603 | -1.7904 -0.6882 | -0.8901 - 0.50 8.0 -1 0.1250 | 1.00327 1.00160 | 4.5293 2.4791 | 2.2721 - 0.50 8.0 +1 12.5336 | 0.99592 0.99736 | -2.7497 -1.2943 | -1.3692 - 0.50 16.0 -1 0.1250 | 1.00172 1.00089 | 8.2720 4.5583 | 4.1431 - 0.50 16.0 +1 28.5336 | 0.99808 0.99887 | -6.5040 -3.4332 | -3.2458 - 0.50 32.0 -1 0.1250 | 1.00088 1.00047 | 15.7124 8.6503 | 7.8631 - 0.50 32.0 +1 60.5336 | 0.99907 0.99947 | -13.9473 -7.5389 | -6.9672 - 1.00 0.5 -1 0.1250 | 1.10472 1.02891 | 0.7065 0.2579 | 0.3903 - 1.00 1.0 -1 0.1250 | 1.07031 1.02324 | 1.0557 0.4626 | 0.5650 0.5334 1.00 2.0 -1 0.1250 | 1.04324 1.01685 | 1.6212 0.8088 | 0.8457 16.3994 1.00 2.0 +1 0.5336 | 0.89837 - | 0.3965 - | 0.1781 - 1.00 3.0 -1 0.1250 | 1.03137 1.01326 | 2.1322 1.1166 | 1.0996 - 1.00 3.0 +1 2.5336 | 0.94730 - | -0.2862 - | -0.1356 - 1.00 4.0 -1 0.1250 | 1.02465 1.01095 | 2.6235 1.4061 | 1.3441 - 1.00 4.0 +1 4.5336 | 0.96399 0.96826 | -0.8258 0.1562 | -0.3980 - 1.00 6.0 -1 0.1250 | 1.01728 1.00813 | 3.5819 1.9582 | 1.8219 - 1.00 6.0 +1 8.5336 | 0.97785 0.98441 | -1.8159 -0.6801 | -0.8879 - 1.00 8.0 -1 0.1250 | 1.01330 1.00646 | 4.5264 2.4927 | 2.2933 - 1.00 8.0 +1 12.5336 | 0.98399 0.98958 | -2.7711 -1.2839 | -1.3634 - 1.00 16.0 -1 0.1250 | 1.00693 1.00356 | 8.2652 4.5717 | 4.1613 - 1.00 16.0 +1 28.5336 | 0.99240 0.99550 | -6.5199 -3.4210 | -3.2352 - 1.00 32.0 -1 0.1250 | 1.00354 1.00188 | 15.7036 8.6636 | 7.8796 - 1.00 32.0 +1 60.5336 | 0.99629 0.99789 | -13.9607 -7.5262 | -6.9545 - 1.50 0.5 -1 0.1012 | 1.34842 1.07115 | 0.6742 0.2776 | 0.4546 - 1.50 1.0 -1 0.1250 | 1.19218 1.05566 | 1.0962 0.4847 | 0.6534 1.0371 1.50 2.0 -1 0.1250 | 1.10791 1.03948 | 1.6407 0.8323 | 0.9089 20.0338 1.50 2.0 +1 0.5336 | 0.80839 - | 0.2846 - | 0.1150 - 1.50 3.0 -1 0.1250 | 1.07583 1.03076 | 2.1414 1.1404 | 1.1519 - 1.50 3.0 +1 2.5336 | 0.89239 - | -0.3537 - | -0.1578 - 1.50 4.0 -1 0.1250 | 1.05860 1.02524 | 2.6268 1.4299 | 1.3904 - 1.50 4.0 +1 4.5336 | 0.92432 0.93255 | -0.8774 0.1481 | -0.4055 - 1.50 6.0 -1 0.1250 | 1.04036 1.01861 | 3.5787 1.9818 | 1.8616 - 1.50 6.0 +1 8.5336 | 0.95228 0.96593 | -1.8545 -0.6674 | -0.8830 - 1.50 8.0 -1 0.1250 | 1.03080 1.01475 | 4.5198 2.5160 | 2.3295 - 1.50 8.0 +1 12.5336 | 0.96510 0.97703 | -2.8040 -1.2673 | -1.3531 - 1.50 16.0 -1 0.1250 | 1.01583 1.00807 | 8.2530 4.5945 | 4.1918 - 1.50 16.0 +1 28.5336 | 0.98315 0.98997 | -6.5452 -3.4010 | -3.2175 - 1.50 32.0 -1 0.1250 | 1.00803 1.00424 | 15.6883 8.6859 | 7.9072 - 1.50 32.0 +1 60.5336 | 0.99171 0.99528 | -13.9825 -7.5052 | -6.9333 - 2.00 2.0 -1 0.1250 | 1.23048 1.07474 | 1.6093 0.8717 | 0.9901 26.7521 2.00 2.0 +1 0.5336 | 0.72022 - | 0.1702 - | 0.0613 - 2.00 3.0 -1 0.1250 | 1.15137 1.05724 | 2.1326 1.1781 | 1.2277 - 2.00 3.0 +1 2.5336 | 0.83011 - | -0.4303 - | -0.1786 - 2.00 4.0 -1 0.1250 | 1.11345 1.04653 | 2.6184 1.4665 | 1.4577 - 2.00 4.0 +1 4.5336 | 0.87662 0.88865 | -0.9382 0.1388 | -0.4112 - 2.00 6.0 -1 0.1250 | 1.07589 1.03394 | 3.5670 2.0171 | 1.9188 - 2.00 6.0 +1 8.5336 | 0.91983 0.94176 | -1.9018 -0.6512 | -0.8747 - 2.00 8.0 -1 0.1250 | 1.05710 1.02675 | 4.5053 2.5504 | 2.3813 - 2.00 8.0 +1 12.5336 | 0.94052 0.96025 | -2.8453 -1.2455 | -1.3380 - 2.00 16.0 -1 0.1250 | 1.02874 1.01450 | 8.2334 4.6272 | 4.2350 - 2.00 16.0 +1 28.5336 | 0.97067 0.98239 | -6.5782 -3.3740 | -3.1926 - 2.00 32.0 -1 0.1250 | 1.01443 1.00758 | 15.6657 8.7176 | 7.9459 - 2.00 32.0 +1 60.5336 | 0.98542 0.99166 | -14.0118 -7.4763 | -6.9038 - PART B2: convergence over the configurations of Part C: change of rho_1 and t from (L,ell)=(64,8) to (256,32) [one quantum], of rho_2 from (48,6) to (64,8) [two quanta]; absolute for eps <= 4, relative (|change|/|value at ell=8|) for eps > 4 eps <= 4, max |change|: rho_1 0.0928, t 0.0581, rho_2 0.0081 eps > 4, max relative change: rho_1 0.0341, t 0.0341, rho_2 0.0120 R1: ell=8 0.23200, ell=32 0.22901; G0: ell=8 0.24262, ell=32 0.25023 PART D: fine grid, 71 values of eps in [0.1, 64] (geometric), torus L=64, ell=8.0 lower branch where lower gap > m_h/2; upper branch where (U) holds with margin 0.1 item 2: rho_{i,-} increasing in eps? window [e1,e2]: rho_{1,-}(e1)=1, rho_{2,-}(e2)=1; xi*_- monotone? max rho_{i,+} < 1? item 3: phi = eps*t_- increasing and strictly convex? psi = eps*(t_- - t_+) increasing? y_b = (t_+ + t_-)/4 r=0.10: lower on eps in [0.100,64.0] (71 pts); rho_1,- 0.1999..30.572 incr True; rho_2,- 0.0553..16.790 incr True xi*_- >= 0 on eps in [0.986,2.658] (81-point refinement, 73 pts), xi* 0.0821..1552.27, increasing True, decreasing False phi increasing True; phi'' min 0.464 max 1.321; t_- 0.1001..15.286 upper F1^2 on eps >= 1.918: max rho_1,+ = 0.6050; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.4345 both branches on eps >= 1.918: psi increasing True; y_b in [0.2216,0.2703]; min(rho_1,- - rho_1,+) = 0.9526; (rho_1,- - rho_1,+)/(4 eps R1) = 0.5353 at eps=1.918, 0.9997 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.8570 r=0.25: lower on eps in [0.100,64.0] (71 pts); rho_1,- 0.2022..30.571 incr True; rho_2,- 0.0555..16.791 incr True xi*_- >= 0 on eps in [0.986,2.658] (81-point refinement, 73 pts), xi* 0.0990..2192.64, increasing True, decreasing False phi increasing True; phi'' min 0.464 max 1.328; t_- 0.1019..15.287 upper F1^2 on eps >= 1.918: max rho_1,+ = 0.5967; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.4330 both branches on eps >= 1.918: psi increasing True; y_b in [0.2221,0.2693]; min(rho_1,- - rho_1,+) = 0.9621; (rho_1,- - rho_1,+)/(4 eps R1) = 0.5405 at eps=1.918, 0.9997 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.8593 r=0.50: lower on eps in [0.100,64.0] (71 pts); rho_1,- 0.2112..30.569 incr True; rho_2,- 0.0561..16.794 incr True xi*_- >= 0 on eps in [0.963,2.635] (81-point refinement, 73 pts), xi* 0.0245..770.92, increasing True, decreasing False phi increasing True; phi'' min 0.464 max 1.350; t_- 0.1094..15.291 upper F1^2 on eps >= 1.918: max rho_1,+ = 0.5684; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.4281 both branches on eps >= 1.918: psi increasing True; y_b in [0.2239,0.2662]; min(rho_1,- - rho_1,+) = 0.9942; (rho_1,- - rho_1,+)/(4 eps R1) = 0.5586 at eps=1.918, 0.9997 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.8675 r=0.75: lower on eps in [0.100,64.0] (71 pts); rho_1,- 0.2310..30.565 incr True; rho_2,- 0.0573..16.799 incr True xi*_- >= 0 on eps in [0.963,2.635] (81-point refinement, 73 pts), xi* 0.1334..7000.48, increasing True, decreasing False phi increasing True; phi'' min 0.464 max 1.367; t_- 0.1258..15.298 upper F1^2 on eps >= 1.918: max rho_1,+ = 0.5256; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.4204 both branches on eps >= 1.918: psi increasing True; y_b in [0.2270,0.2621]; min(rho_1,- - rho_1,+) = 1.0434; (rho_1,- - rho_1,+)/(4 eps R1) = 0.5862 at eps=1.918, 0.9997 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.8808 r=1.00: lower on eps in [0.100,64.0] (71 pts); rho_1,- 0.2783..30.559 incr True; rho_2,- 0.0593..16.807 incr True xi*_- >= 0 on eps in [0.933,2.605] (81-point refinement, 70 pts), xi* 0.1195..2169.05, increasing True, decreasing False phi increasing True; phi'' min 0.464 max 1.282; t_- 0.1658..15.307 upper F1^2 on eps >= 1.918: max rho_1,+ = 0.4730; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.4103 both branches on eps >= 1.918: psi increasing True; y_b in [0.2312,0.2586]; min(rho_1,- - rho_1,+) = 1.1047; (rho_1,- - rho_1,+)/(4 eps R1) = 0.6207 at eps=1.918, 0.9997 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.8987 r=1.25: lower on eps in [0.209,64.0] (63 pts); rho_1,- 0.4854..30.551 incr True; rho_2,- 0.1241..16.817 incr True xi*_- >= 0 on eps in [0.884,2.556] (81-point refinement, 70 pts), xi* 0.0591..705.35, increasing True, decreasing False phi increasing True; phi'' min 0.016 max 0.654; t_- 0.3196..15.318 upper F1^2 on eps >= 1.918: max rho_1,+ = 0.4149; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.3986 both branches on eps >= 1.918: psi increasing True; y_b in [0.2367,0.2567]; min(rho_1,- - rho_1,+) = 1.1734; (rho_1,- - rho_1,+)/(4 eps R1) = 0.6593 at eps=1.918, 0.9998 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.9208 r=1.50: lower on eps in [0.438,64.0] (55 pts); rho_1,- 0.4346..30.542 incr True; rho_2,- 0.2490..16.829 incr True xi*_- >= 0 on eps in [0.860,2.532] (81-point refinement, 70 pts), xi* 0.0967..7303.24, increasing True, decreasing False phi increasing True; phi'' min -13.908 max 0.483; phi'' <= 0 at eps in [0.480,0.634]; t_- 0.3042..15.333 upper F1^2 on eps >= 1.918: max rho_1,+ = 0.3546; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.3859 both branches on eps >= 1.918: psi increasing True; y_b in [0.2434,0.2573]; min(rho_1,- - rho_1,+) = 1.2438; (rho_1,- - rho_1,+)/(4 eps R1) = 0.6988 at eps=1.918, 0.9998 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.9469 r=2.00: lower on eps in [1.209,64.0] (44 pts); rho_1,- 0.5484..30.518 incr True; rho_2,- 0.6043..16.860 incr True xi*_- >= 0 on eps in [1.341,2.394] (81-point refinement, 71 pts), xi* 0.0853..1011.35, increasing True, decreasing False phi increasing True; phi'' min -12.032 max 0.464; phi'' <= 0 at eps in [1.326,1.749]; t_- 0.3913..15.369 upper F1^2 on eps >= 1.918: max rho_1,+ = 0.2356; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.3594 both branches on eps >= 1.918: psi increasing True; y_b in [0.2605,0.2629]; min(rho_1,- - rho_1,+) = 1.3247; (rho_1,- - rho_1,+)/(4 eps R1) = 0.7443 at eps=1.918, 0.9998 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 1.0104 r=3.00: lower on eps in [3.337,64.0] (33 pts); rho_1,- -0.3053..30.445 incr True; rho_2,- 1.4165..16.950 incr True xi*_- >= 0 on eps in [3.337,3.563] (81-point refinement, 57 pts), xi* 0.0934..20.27, increasing False, decreasing True phi increasing True; phi'' min -21.133 max 0.464; phi'' <= 0 at eps in [3.660,4.827]; t_- -0.2284..15.474 upper F1^2 on eps >= 1.918: max rho_1,+ = 0.0297; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.3109 both branches on eps >= 3.337: psi increasing True; y_b in [-0.1260,0.3094]; min(rho_1,- - rho_1,+) = 0.4452; (rho_1,- - rho_1,+)/(4 eps R1) = 0.1438 at eps=3.337, 0.9999 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.9867 min over all computed points of S_i(0)/S_i(0)|_(tau=0) (both parts, both branches): 0.5574 r-dependence, eps >= 2: max |rho(r=2) - rho(r=0.25)|: rho_1,- 0.0534, rho_2,- 0.0786, rho_1,+ 0.3162, rho_2,+ 0.0737 grip y = F0 F2/F1^2 that equalises the bracket of (6.12) for two lower levels (eps_a, eps_b): r=0.10: (0.25,0.5) 0.2241 (0.5,1) 0.3470 (1,2) 0.5466 (2,4) 0.9087 (4,8) 1.6106 (1,8) 1.2580 r=0.25: (0.25,0.5) 0.2256 (0.5,1) 0.3482 (1,2) 0.5475 (2,4) 0.9094 (4,8) 1.6111 (1,8) 1.2587 r=0.50: (0.25,0.5) 0.2311 (0.5,1) 0.3526 (1,2) 0.5507 (2,4) 0.9118 (4,8) 1.6132 (1,8) 1.2610 r=0.75: (0.25,0.5) 0.2406 (0.5,1) 0.3599 (1,2) 0.5560 (2,4) 0.9158 (4,8) 1.6166 (1,8) 1.2648 r=1.00: (0.25,0.5) 0.2543 (0.5,1) 0.3699 (1,2) 0.5631 (2,4) 0.9213 (4,8) 1.6212 (1,8) 1.2701 r=1.25: (0.25,0.5) 0.2714 (0.5,1) 0.3814 (1,2) 0.5717 (2,4) 0.9280 (4,8) 1.6272 (1,8) 1.2766 r=1.50: (0.5,1) 0.4286 (1,2) 0.5821 (2,4) 0.9359 (4,8) 1.6343 (1,8) 1.2845 r=2.00: (2,4) 0.9629 (4,8) 1.6524 r=3.00: (4,8) 1.8618 PART D2: F1^2 part only (all that items 3 and the upper-branch bound of item 2 need from it), torus L=256, ell=32.0; 36 values of eps in [0.1, 64]; same criteria as Part D r=0.10: lower eps in [0.100,64.0]; rho_1,- incr True, rho_1,-=1 at eps=0.928; phi incr True, phi''>0 everywhere; max rho_1,+ = 0.6286; psi incr True; y_b in [0.2288,0.2760] r=0.50: lower eps in [0.100,64.0]; rho_1,- incr True, rho_1,-=1 at eps=0.920; phi incr True, phi''>0 everywhere; max rho_1,+ = 0.5872; psi incr True; y_b in [0.2298,0.2707] r=1.00: lower eps in [0.100,64.0]; rho_1,- incr True, rho_1,-=1 at eps=0.896; phi incr True, phi''>0 everywhere; max rho_1,+ = 0.4767; psi incr True; y_b in [0.2328,0.2593] r=1, two-level grips y_ab (interpolated): (0.25,0.5) 0.2567 (0.5,1) 0.3701 (1,2) 0.5608 (2,4) 0.9142 (4,8) 1.6049 r=1.50: lower eps in [0.438,64.0]; rho_1,- incr True, rho_1,-=1 at eps=0.892; phi incr True, phi''>0 except eps in [0.527,0.634]; max rho_1,+ = 0.3330; psi incr True; y_b in [0.2379,0.2511] r=2.00: lower eps in [1.102,64.0]; rho_1,- incr True, rho_1,-=1 at eps=1.453; phi incr True, phi''>0 except eps in [1.326,1.595]; max rho_1,+ = 0.1797; psi incr True; y_b in [0.2430,0.2454] r=3.00: lower eps in [3.337,64.0]; rho_1,- incr True, rho_1,-=1 at eps=3.851; phi incr True, phi''>0 except eps in [4.014,4.827]; max rho_1,+ = -0.1146; psi incr True; y_b in [-0.2699,0.2651] PART E: ED of the contract restricted to <= 2 field quanta, cubic torus L=4, ell=2.0, dimension 4290; M_k=5.0, F1=0.3, F2=0.1, r=1.0, Omega_max=3.5000 sigma=+1 Cost second order: ED -0.2793475807 formula -0.2793475807 sigma=+1 E''(0) second order: ED -0.0038368899 formula -0.0038368899 sigma=-1 Cost second order: ED -0.2892384565 formula -0.2892384576 sigma=-1 E''(0) second order: ED 0.0307210119 formula 0.0307210118 max relative deviation: 8.72e-09