PART A: ring L=32, c0=1.0, m_h=0.25 (ell=4.0), F0=1, G0=1.985916, w0=0.550116, Omega_max=2.0156 second-order coefficients: formula (6.2), (6.3), (6.5) vs ED restricted to <= 2 quanta grip Mk sig r | Cost2 form Cost2 ED | E2pp form E2pp ED | Cost1 ED-form Epp1 ED-form square (F1=0.3, F2=F1^2/4=0.0225) 3 +1 0.5 | -0.79709146 -0.79709146 | 0.00272483 0.00272483 | -5.88e-11 -7.56e-13 square (F1=0.3, F2=F1^2/4=0.0225) 3 -1 0.5 | -0.81355379 -0.81355379 | 0.06469109 0.06469109 | 6.38e-11 -1.30e-11 square (F1=0.3, F2=F1^2/4=0.0225) 3 +1 1.0 | -0.77188596 -0.77188596 | -0.01790101 -0.01790101 | -5.00e-11 -1.63e-11 square (F1=0.3, F2=F1^2/4=0.0225) 3 -1 1.0 | -0.83695453 -0.83695453 | 0.24473099 0.24473099 | 6.85e-11 -5.10e-11 square (F1=0.3, F2=F1^2/4=0.0225) 3 +1 1.5 | -0.73621933 -0.73621933 | -0.12253651 -0.12253651 | -3.93e-11 -5.93e-11 square (F1=0.3, F2=F1^2/4=0.0225) 3 -1 1.5 | -0.88257932 -0.88257932 | 0.47513821 0.47513820 | 7.74e-11 -1.10e-10 square (F1=0.3, F2=F1^2/4=0.0225) 4 +1 0.5 | -1.42156754 -1.42156754 | -0.00839540 -0.00839540 | -4.41e-10 -1.88e-11 square (F1=0.3, F2=F1^2/4=0.0225) 4 -1 0.5 | -1.44333641 -1.44333642 | 0.07578325 0.07578325 | 4.69e-10 -6.91e-11 square (F1=0.3, F2=F1^2/4=0.0225) 4 +1 1.0 | -1.38830326 -1.38830326 | -0.06039710 -0.06039710 | -3.94e-10 -1.21e-10 square (F1=0.3, F2=F1^2/4=0.0225) 4 -1 1.0 | -1.47480499 -1.47480499 | 0.28739467 0.28739468 | 4.99e-10 -2.70e-10 square (F1=0.3, F2=F1^2/4=0.0225) 4 +1 1.5 | -1.33921483 -1.33921484 | -0.21748355 -0.21748356 | -3.33e-10 -3.65e-10 square (F1=0.3, F2=F1^2/4=0.0225) 4 -1 1.5 | -1.53381757 -1.53381758 | 0.57028552 0.57028551 | 5.55e-10 -5.68e-10 other (F1=0.3, F2=-0.15) 3 +1 0.5 | -0.88255089 -0.88255088 | 0.00244496 0.00244496 | 1.97e-09 1.22e-10 other (F1=0.3, F2=-0.15) 3 -1 0.5 | -0.90108819 -0.90108819 | 0.07033359 0.07033359 | -2.17e-09 6.41e-10 other (F1=0.3, F2=-0.15) 3 +1 1.0 | -0.85386846 -0.85386846 | -0.01963079 -0.01963079 | 1.67e-09 8.37e-10 other (F1=0.3, F2=-0.15) 3 -1 1.0 | -0.92705583 -0.92705583 | 0.26807672 0.26807672 | -2.47e-09 2.53e-09 other (F1=0.3, F2=-0.15) 3 +1 1.5 | -0.81331009 -0.81331009 | -0.12802719 -0.12802719 | 1.32e-09 2.48e-09 other (F1=0.3, F2=-0.15) 3 -1 1.5 | -0.97758359 -0.97758359 | 0.53083127 0.53083125 | -3.16e-09 4.96e-09 other (F1=0.3, F2=-0.15) 4 +1 0.5 | -1.57419026 -1.57419024 | -0.01025575 -0.01025575 | 1.49e-08 1.09e-09 other (F1=0.3, F2=-0.15) 4 -1 0.5 | -1.59864379 -1.59864377 | 0.08292655 0.08292654 | -1.60e-08 3.27e-09 other (F1=0.3, F2=-0.15) 4 +1 1.0 | -1.53652382 -1.53652380 | -0.06819105 -0.06819104 | 1.32e-08 5.66e-09 other (F1=0.3, F2=-0.15) 4 -1 1.0 | -1.63363731 -1.63363729 | 0.31662572 0.31662569 | -1.77e-08 1.28e-08 other (F1=0.3, F2=-0.15) 4 +1 1.5 | -1.48093983 -1.48093982 | -0.23595428 -0.23595426 | 1.11e-08 1.53e-08 other (F1=0.3, F2=-0.15) 4 -1 1.5 | -1.69916986 -1.69916983 | 0.63864993 0.63864987 | -2.13e-08 2.58e-08 max relative deviation of second-order coefficients (formula vs ED): 2.26e-07 zeroth order: ED level = sigma*F0*M_k and E''(0) = sigma*tau^2/(F0*M_k) asserted to 1e-10 / 1e-8 PART B: formula (6.3) on rings with L = 8 ell, c0 = 1, F0 = 1; rho_1, rho_2 = I^mark tau^2/C^mark of the F1^2 and F2^2 parts; prediction (6.9): 2 - r^2 held mark: g1 = G0*2c0^2/ell, g2 = J2*16c0^3/ell (prediction (6.7): -> 1); Cratio = C^mark(tau)/C^mark(0) for the F1^2 part t_i = rho_i + 2 sigma eps_k R_i: rho with the term (iv) of Step 6 removed, R_i = <>/<<1/E>> of the part i ell g1 g2 Mk sig r 2-r^2 rho_1 rho_2 t_1 t_2 Cratio 16 1.00018 0.96912 3 +1 0.5 1.75 1.02947 0.27353 1.71048 1.55478 0.99488 16 1.00018 0.96912 3 +1 1.0 1.00 0.42042 -0.09180 1.10143 1.18945 0.98053 16 1.00018 0.96912 3 +1 1.5 -0.25 -0.49901 -0.61591 0.18200 0.66535 0.95935 16 1.00018 0.96912 3 -1 0.5 1.75 2.34232 2.68110 1.66131 1.39984 1.00443 16 1.00018 0.96912 3 -1 1.0 1.00 1.77840 2.40227 1.09739 1.12101 1.01848 16 1.00018 0.96912 3 -1 1.5 -0.25 0.72788 1.84891 0.04687 0.56766 1.04497 16 1.00018 0.96912 4 +1 0.5 1.75 0.79609 -0.17873 1.70411 1.52961 0.99624 16 1.00018 0.96912 4 +1 1.0 1.00 0.19283 -0.53165 1.10085 1.17669 0.98551 16 1.00018 0.96912 4 +1 1.5 -0.25 -0.73980 -1.05627 0.16821 0.65207 0.96919 16 1.00018 0.96912 4 -1 0.5 1.75 2.57525 3.12404 1.66724 1.41570 1.00337 16 1.00018 0.96912 4 -1 1.0 1.00 2.00585 2.83561 1.09784 1.12727 1.01394 16 1.00018 0.96912 4 -1 1.5 -0.25 0.97571 2.28919 0.06769 0.58085 1.03324 32 1.00055 0.98473 3 +1 0.5 1.75 1.30991 0.66603 1.73271 1.63703 0.99686 32 1.00055 0.98473 3 +1 1.0 1.00 0.63602 0.17247 1.05882 1.14347 0.98799 32 1.00055 0.98473 3 +1 1.5 -0.25 -0.41564 -0.56204 0.00716 0.40895 0.97466 32 1.00055 0.98473 3 -1 0.5 1.75 2.12805 2.50760 1.70526 1.53660 1.00278 32 1.00055 0.98473 3 -1 1.0 1.00 1.48043 2.07982 1.05764 1.10882 1.01154 32 1.00055 0.98473 3 -1 1.5 -0.25 0.31874 1.26875 -0.10406 0.29775 1.02776 32 1.00055 0.98473 4 +1 0.5 1.75 1.16546 0.32687 1.72918 1.62153 0.99769 32 1.00055 0.98473 4 +1 1.0 1.00 0.49491 -0.15757 1.05863 1.13709 0.99105 32 1.00055 0.98473 4 +1 1.5 -0.25 -0.56826 -0.89831 -0.00454 0.39635 0.98081 32 1.00055 0.98473 4 -1 0.5 1.75 2.27233 2.84200 1.70860 1.54734 1.00211 32 1.00055 0.98473 4 -1 1.0 1.00 1.62149 2.40670 1.05776 1.11204 1.00869 32 1.00055 0.98473 4 -1 1.5 -0.25 0.47630 1.60897 -0.08742 0.31431 1.02055 64 1.00064 0.99254 3 +1 0.5 1.75 1.49021 0.99884 1.74289 1.68674 0.99814 64 1.00064 0.99254 3 +1 1.0 1.00 0.78115 0.41067 1.03383 1.09857 0.99285 64 1.00064 0.99254 3 +1 1.5 -0.25 -0.35207 -0.49218 -0.09939 0.19572 0.98478 64 1.00064 0.99254 3 -1 0.5 1.75 1.98048 2.31156 1.72780 1.62366 1.00168 64 1.00064 0.99254 3 -1 1.0 1.00 1.28617 1.76929 1.03349 1.08139 1.00692 64 1.00064 0.99254 3 -1 1.5 -0.25 0.07381 0.77769 -0.17887 0.08979 1.01650 64 1.00064 0.99254 4 +1 0.5 1.75 1.40406 0.76016 1.74097 1.67736 0.99863 64 1.00064 0.99254 4 +1 1.0 1.00 0.69686 0.17823 1.03377 1.09543 0.99467 64 1.00064 0.99254 4 +1 1.5 -0.25 -0.44489 -0.73290 -0.10798 0.18430 0.98849 64 1.00064 0.99254 4 -1 0.5 1.75 2.06655 2.54781 1.72965 1.63061 1.00127 64 1.00064 0.99254 4 -1 1.0 1.00 1.37043 2.00020 1.03352 1.08300 1.00521 64 1.00064 0.99254 4 -1 1.5 -0.25 0.16961 1.02296 -0.16730 0.10576 1.01223 128 1.00066 0.99645 3 +1 0.5 1.75 1.60052 1.25165 1.74753 1.71538 0.99893 128 1.00066 0.99645 3 +1 1.0 1.00 0.87314 0.59979 1.02014 1.06352 0.99585 128 1.00066 0.99645 3 +1 1.5 -0.25 -0.30818 -0.42551 -0.16118 0.03822 0.99111 128 1.00066 0.99645 3 -1 0.5 1.75 1.88631 2.14051 1.73931 1.67678 1.00098 128 1.00066 0.99645 3 -1 1.0 1.00 1.16705 1.51888 1.02005 1.05514 1.00403 128 1.00066 0.99645 3 -1 1.5 -0.25 -0.06643 0.41326 -0.21343 -0.05048 1.00955 128 1.00066 0.99645 4 +1 0.5 1.75 1.55048 1.09149 1.74648 1.70980 0.99921 128 1.00066 0.99645 4 +1 1.0 1.00 0.82412 0.44367 1.02012 1.06198 0.99690 128 1.00066 0.99645 4 +1 1.5 -0.25 -0.36296 -0.58958 -0.16696 0.02873 0.99328 128 1.00066 0.99645 4 -1 0.5 1.75 1.93632 2.29943 1.74032 1.68112 1.00074 128 1.00066 0.99645 4 -1 1.0 1.00 1.21606 1.67424 1.02005 1.05592 1.00303 128 1.00066 0.99645 4 -1 1.5 -0.25 -0.00998 0.58111 -0.20599 -0.03720 1.00709 256 1.00067 0.99840 3 +1 0.5 1.75 1.66579 1.43035 1.74963 1.73137 0.99939 256 1.00067 0.99840 3 +1 1.0 1.00 0.92909 0.73812 1.01293 1.03914 0.99764 256 1.00067 0.99840 3 +1 1.5 -0.25 -0.27981 -0.37076 -0.19597 -0.06974 0.99491 256 1.00067 0.99840 3 -1 0.5 1.75 1.82902 2.00926 1.74519 1.70824 1.00056 256 1.00067 0.99840 3 -1 1.0 1.00 1.09673 1.33612 1.01290 1.03510 1.00230 256 1.00067 0.99840 3 -1 1.5 -0.25 -0.14469 0.16335 -0.22853 -0.13767 1.00542 256 1.00067 0.99840 4 +1 0.5 1.75 1.63728 1.32673 1.74906 1.72810 0.99955 256 1.00067 0.99840 4 +1 1.0 1.00 0.90114 0.63703 1.01292 1.03839 0.99824 256 1.00067 0.99840 4 +1 1.5 -0.25 -0.31142 -0.47842 -0.19964 -0.07706 0.99616 256 1.00067 0.99840 4 -1 0.5 1.75 1.85751 2.11225 1.74573 1.71089 1.00043 256 1.00067 0.99840 4 -1 1.0 1.00 1.12468 1.43684 1.01290 1.03548 1.00173 256 1.00067 0.99840 4 -1 1.5 -0.25 -0.11219 0.27373 -0.22397 -0.12763 1.00403 512 1.00067 0.99938 3 +1 0.5 1.75 1.70349 1.55041 1.75057 1.74011 0.99966 512 1.00067 0.99938 3 +1 1.0 1.00 0.96211 0.83367 1.00920 1.02337 0.99868 512 1.00067 0.99938 3 +1 1.5 -0.25 -0.26229 -0.32977 -0.21521 -0.14007 0.99713 512 1.00067 0.99938 3 -1 0.5 1.75 1.79528 1.91619 1.74819 1.72649 1.00032 512 1.00067 0.99938 3 -1 1.0 1.00 1.05627 1.21113 1.00919 1.02143 1.00130 512 1.00067 0.99938 3 -1 1.5 -0.25 -0.18766 0.00087 -0.23475 -0.18883 1.00303 512 1.00067 0.99938 4 +1 0.5 1.75 1.68749 1.48528 1.75027 1.73821 0.99975 512 1.00067 0.99938 4 +1 1.0 1.00 0.94642 0.77007 1.00920 1.02300 0.99901 512 1.00067 0.99938 4 +1 1.5 -0.25 -0.28022 -0.39831 -0.21744 -0.14537 0.99784 512 1.00067 0.99938 4 -1 0.5 1.75 1.81127 1.98101 1.74849 1.72807 1.00024 512 1.00067 0.99938 4 -1 1.0 1.00 1.07197 1.27454 1.00919 1.02160 1.00097 512 1.00067 0.99938 4 -1 1.5 -0.25 -0.16927 0.07122 -0.23205 -0.18171 1.00226