set=A L=12 coupling=a level=low c1_over_c2=1 chi=2.18493 beta=0.693117 inv_Epp_exact=227.88 inv_Epp_fd=227.881 inv_Epp_band_ring=227.683 inv_Epp_66=254.736 inv_Epp_66_plus_ring=227.683 mu_sum=177.024 mu_over_beta=255.404 rel_dev_band_ring_vs_exact=-0.00086453 rel_dev_66ring_vs_exact=-0.000866607 rel_dev_mu_over_beta_vs_exact=0.120779 ring_term_rel=0.118819 ring_bound=0.980813 f_antipode_sq=0.00940482 D_exact=50.8559 D_lat=78.3791 D_mark_formula=-0.433161 D_lat_plus_mark=77.9459 dE_cost=-0.00353514 D_over_dE_times_c1sq=-575.432 D_over_dE_times_c2sq=-575.432 set=A L=12 coupling=c level=low c1_over_c2=1 chi=2.15978 beta=0.695604 inv_Epp_exact=224.102 inv_Epp_fd=224.102 inv_Epp_band_ring=223.92 inv_Epp_66=251.399 inv_Epp_66_plus_ring=223.92 mu_sum=174.96 mu_over_beta=251.522 rel_dev_band_ring_vs_exact=-0.000810322 rel_dev_66ring_vs_exact=-0.000812665 rel_dev_mu_over_beta_vs_exact=0.122358 ring_term_rel=0.122721 ring_bound=1.01868 f_antipode_sq=0.00964338 D_exact=49.1417 D_lat=76.5623 D_mark_formula=0.424144 D_lat_plus_mark=76.9865 dE_cost=-0.0035177 D_over_dE_times_c1sq=-567.207 D_over_dE_times_c2sq=-567.207 set=A L=16 coupling=a level=low c1_over_c2=1 chi=1.63512 beta=0.79213 inv_Epp_exact=215.788 inv_Epp_fd=215.788 inv_Epp_band_ring=215.589 inv_Epp_66=222.405 inv_Epp_66_plus_ring=215.589 mu_sum=176.675 mu_over_beta=223.038 rel_dev_band_ring_vs_exact=-0.000922874 rel_dev_66ring_vs_exact=-0.000924078 rel_dev_mu_over_beta_vs_exact=0.0335974 ring_term_rel=0.0316186 ring_bound=0.340164 f_antipode_sq=0.00151956 D_exact=39.113 D_lat=46.363 D_mark_formula=-0.440081 D_lat_plus_mark=45.9229 dE_cost=-0.00312846 D_over_dE_times_c1sq=-500.093 D_over_dE_times_c2sq=-500.093 set=A L=16 coupling=c level=low c1_over_c2=1 chi=1.62269 beta=0.793088 inv_Epp_exact=214.057 inv_Epp_fd=214.057 inv_Epp_band_ring=213.867 inv_Epp_66=220.83 inv_Epp_66_plus_ring=213.866 mu_sum=175.303 mu_over_beta=221.039 rel_dev_band_ring_vs_exact=-0.000889061 rel_dev_66ring_vs_exact=-0.000890678 rel_dev_mu_over_beta_vs_exact=0.0326181 ring_term_rel=0.0325613 ring_bound=0.352012 f_antipode_sq=0.00155458 D_exact=38.7535 D_lat=45.7357 D_mark_formula=0.434106 D_lat_plus_mark=46.1698 dE_cost=-0.00312016 D_over_dE_times_c1sq=-502.419 D_over_dE_times_c2sq=-502.419 set=B L=12 coupling=a level=high c1_over_c2=0.8 chi=1.55239 beta=0.771791 inv_Epp_exact=-182.829 inv_Epp_fd=-182.829 inv_Epp_band_ring=-182.62 inv_Epp_66=-193.47 inv_Epp_66_plus_ring=-182.62 mu_sum=-149.732 mu_over_beta=-194.007 rel_dev_band_ring_vs_exact=-0.00113989 rel_dev_66ring_vs_exact=-0.00114076 rel_dev_mu_over_beta_vs_exact=0.0611391 ring_term_rel=0.0594129 ring_bound=0.339697 f_antipode_sq=0.00428455 D_exact=-33.0961 D_lat=-44.2741 D_mark_formula=0.481902 D_lat_plus_mark=-43.7922 dE_cost=0.00364334 D_over_dE_times_c1sq=-232.55 D_over_dE_times_c2sq=-363.36 set=B L=12 coupling=c level=high c1_over_c2=0.816189 chi=1.55919 beta=0.770603 inv_Epp_exact=-182.059 inv_Epp_fd=-182.06 inv_Epp_band_ring=-181.861 inv_Epp_66=-193.079 inv_Epp_66_plus_ring=-181.861 mu_sum=-148.439 mu_over_beta=-192.627 rel_dev_band_ring_vs_exact=-0.0010876 rel_dev_66ring_vs_exact=-0.00108869 rel_dev_mu_over_beta_vs_exact=0.0580434 ring_term_rel=0.0616834 ring_bound=0.354983 f_antipode_sq=0.00422347 D_exact=-33.6207 D_lat=-44.188 D_mark_formula=-0.483418 D_lat_plus_mark=-44.6714 dE_cost=0.00365744 D_over_dE_times_c1sq=-239.264 D_over_dE_times_c2sq=-359.166 set=B L=16 coupling=a level=high c1_over_c2=0.8 chi=1.16692 beta=0.776915 inv_Epp_exact=-191.462 inv_Epp_fd=-191.462 inv_Epp_band_ring=-191.165 inv_Epp_66=-192.381 inv_Epp_66_plus_ring=-191.165 mu_sum=-149.957 mu_over_beta=-193.016 rel_dev_band_ring_vs_exact=-0.00155024 rel_dev_66ring_vs_exact=-0.00155098 rel_dev_mu_over_beta_vs_exact=0.00811379 ring_term_rel=0.0063618 ring_bound=0.036297 f_antipode_sq=0.000245782 D_exact=-41.5054 D_lat=-43.0589 D_mark_formula=0.578926 D_lat_plus_mark=-42.48 dE_cost=0.00509914 D_over_dE_times_c1sq=-208.376 D_over_dE_times_c2sq=-325.588 set=B L=16 coupling=c level=high c1_over_c2=0.812106 chi=1.16811 beta=0.776063 inv_Epp_exact=-191.061 inv_Epp_fd=-191.061 inv_Epp_band_ring=-190.773 inv_Epp_66=-192.03 inv_Epp_66_plus_ring=-190.773 mu_sum=-148.578 mu_over_beta=-191.452 rel_dev_band_ring_vs_exact=-0.00150643 rel_dev_66ring_vs_exact=-0.00150725 rel_dev_mu_over_beta_vs_exact=0.00204295 ring_term_rel=0.00658724 ring_bound=0.0378638 f_antipode_sq=0.000243974 D_exact=-42.4827 D_lat=-42.873 D_mark_formula=-0.579869 D_lat_plus_mark=-43.4529 dE_cost=0.00510877 D_over_dE_times_c1sq=-215.55 D_over_dE_times_c2sq=-326.83 set=C L=12 coupling=a level=high c1_over_c2=1 chi=1.7286 beta=-0.576737 inv_Epp_exact=-289.006 inv_Epp_fd=-289.006 inv_Epp_band_ring=-288.271 inv_Epp_66=-306.072 inv_Epp_66_plus_ring=-288.27 mu_sum=176.192 mu_over_beta=-305.498 rel_dev_band_ring_vs_exact=-0.00254237 rel_dev_66ring_vs_exact=-0.0025439 rel_dev_mu_over_beta_vs_exact=0.057065 ring_term_rel=0.061755 ring_bound=0.429752 f_antipode_sq=0.00435964 D_exact=-465.197 D_lat=-481.689 D_mark_formula=-0.437579 D_lat_plus_mark=-482.127 dE_cost=0.0167361 D_over_dE_times_c1sq=-1111.84 D_over_dE_times_c2sq=-1111.84 set=C L=12 coupling=c level=high c1_over_c2=1 chi=1.70608 beta=-0.581149 inv_Epp_exact=-281.917 inv_Epp_fd=-281.918 inv_Epp_band_ring=-281.164 inv_Epp_66=-298.893 inv_Epp_66_plus_ring=-281.164 mu_sum=173.836 mu_over_beta=-299.125 rel_dev_band_ring_vs_exact=-0.00267193 rel_dev_66ring_vs_exact=-0.00267358 rel_dev_mu_over_beta_vs_exact=0.0610369 ring_term_rel=0.0630575 ring_bound=0.439796 f_antipode_sq=0.00449956 D_exact=-455.753 D_lat=-472.961 D_mark_formula=0.426336 D_lat_plus_mark=-472.534 dE_cost=0.0168845 D_over_dE_times_c1sq=-1097.77 D_over_dE_times_c2sq=-1097.77 set=C L=16 coupling=a level=high c1_over_c2=1 chi=1.29558 beta=-0.651475 inv_Epp_exact=-269.923 inv_Epp_fd=-269.924 inv_Epp_band_ring=-269.287 inv_Epp_66=-271.2 inv_Epp_66_plus_ring=-269.287 mu_sum=176.22 mu_over_beta=-270.494 rel_dev_band_ring_vs_exact=-0.00235643 rel_dev_66ring_vs_exact=-0.00235768 rel_dev_mu_over_beta_vs_exact=0.00211469 ring_term_rel=0.0071041 ring_bound=0.0579808 f_antipode_sq=0.00031463 D_exact=-446.144 D_lat=-446.715 D_mark_formula=-0.462319 D_lat_plus_mark=-447.177 dE_cost=0.0169812 D_over_dE_times_c1sq=-1050.91 D_over_dE_times_c2sq=-1050.91 set=C L=16 coupling=c level=high c1_over_c2=1 chi=1.28453 beta=-0.654037 inv_Epp_exact=-265.673 inv_Epp_fd=-265.674 inv_Epp_band_ring=-265.02 inv_Epp_66=-266.924 inv_Epp_66_plus_ring=-265.019 mu_sum=174.662 mu_over_beta=-267.053 rel_dev_band_ring_vs_exact=-0.00246045 rel_dev_66ring_vs_exact=-0.00246123 rel_dev_mu_over_beta_vs_exact=0.00519182 ring_term_rel=0.00718622 ring_bound=0.0586878 f_antipode_sq=0.000321502 D_exact=-440.336 D_lat=-441.715 D_mark_formula=0.454412 D_lat_plus_mark=-441.26 dE_cost=0.0170946 D_over_dE_times_c1sq=-1043.27 D_over_dE_times_c2sq=-1043.27 mark term check (set A, L=12): ratio of m(rho)-mubar_1-mubar_2 to the mark term of (6.7) at contact bonds coupling=a scale=1.0 ratio=1.4006 coupling=a scale=0.3 ratio=1.3097 coupling=a scale=0.1 ratio=1.1321 coupling=a scale=0.03 ratio=1.0430 coupling=c scale=1.0 ratio=-0.1904 coupling=c scale=0.3 ratio=0.4157 coupling=c scale=0.1 ratio=0.7736 coupling=c scale=0.03 ratio=0.9284 R1 gaps from the held-sector spectra (N_f = 1): delta_1 (one motion term), delta (all other levels) set=A L=12 t_f=1.0: delta1=2.0094 delta=0.2277 field_gap=0.2679 width_of_G=0.0115 set=A L=12 t_f=40.0: delta1=2.6256 delta=2.6256 field_gap=10.7180 width_of_G=0.0005 set=A L=16 t_f=1.0: delta1=1.9996 delta=0.1233 field_gap=0.1522 width_of_G=0.0099 set=A L=16 t_f=40.0: delta1=0.0242 delta=0.0242 field_gap=6.0896 width_of_G=0.0005 set=B L=12 t_f=1.0: delta1=3.9238 delta=0.2288 field_gap=0.2679 width_of_G=0.0112 set=B L=12 t_f=40.0: delta1=3.9529 delta=3.9529 field_gap=10.7180 width_of_G=0.0003 set=B L=16 t_f=1.0: delta1=3.9295 delta=0.1234 field_gap=0.1522 width_of_G=0.0125 set=B L=16 t_f=40.0: delta1=3.9645 delta=3.9645 field_gap=6.0896 width_of_G=0.0003 set=C L=12 t_f=1.0: delta1=2.0232 delta=0.2298 field_gap=0.2679 width_of_G=0.0136 set=C L=12 t_f=40.0: delta1=2.7150 delta=2.7150 field_gap=10.7180 width_of_G=0.0004 set=C L=16 t_f=1.0: delta1=2.0116 delta=0.1216 field_gap=0.1522 width_of_G=0.0132 set=C L=16 t_f=40.0: delta1=0.0230 delta=0.0230 field_gap=6.0896 width_of_G=0.0004 leading-order sign rule (6.17) on the line: chi=0.05 R=1.0 c=0.25 sigma_i=-1 s_e=-1 Q-1=63.0 sign changes of D_nu - dE_nu/c^2 in L in [100, 3000]: coupling (a): nu=0 at [564], nu=1 at [1216]; coupling (c): nu=0 at [], nu=1 at [] estimate L*_nu = 2 4^(1/3) a_nu (Q-1) chi^(-1/3): nu=0: 553, nu=1: 1269