====================================================================================================
GRAPH Z_8   n=8 d=2 diam=4 bipartite=True
 lambda_2=0.707106781187 (mult m2=2)  lambda_n=-1.0   tau(0)=3.41421356237   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.01011329069844 at p=9999/10000 ; min tau/boundB = 1.13832239389461 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 4.45768468233 at p=1/4 ; max/tau(0) = 1.3056256 ; max/d = 2.2288423
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=6
 T2 numeric tau(1-10^-k)-d: k=1:0.77174 k=2:0.21304 k=3:0.064514 k=4:0.020126 k=6:0.0020013 k=8:0.00020001 k=10:2.0e-5 k=12:2.0e-6 k=16:2.0e-8 k=20:2.0e-10 k=24:2.0e-12
    |tau-d| ~ eps^0.5 (from k=16,24); tau_abs(1-10^-24)-d = 2.0e-12
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.292893218813  c0=(m2-1)/2+(n/4)Gamma=1.0857864376269   s*=max over V=1.0857864376269  (s*-c0=3.1115e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.12038156 | -0.0882481 | -0.0883512 | tau_abs(0)=inf, tau_abs(p)=28.214445
     3 : 0.012593286 | -0.00942348 | -0.009429 | tau_abs(0)=inf, tau_abs(p)=253.30311
     4 : 0.001265048 | -0.000948762 | -0.000949285 | tau_abs(0)=inf, tau_abs(p)=2503.3116
     5 : 0.00012656217 | -9.49408e-5 | -9.49928e-5 | tau_abs(0)=inf, tau_abs(p)=25003.312
     6 : 1.265679e-5 | -9.49472e-6 | -9.49993e-6 | tau_abs(0)=inf, tau_abs(p)=250003.31
     8 : 1.2656854e-7 | -9.4948e-8 | -9.5e-8 | tau_abs(0)=inf, tau_abs(p)=25000003.0
    10 : 1.2656854e-9 | -9.4948e-10 | -9.5e-10 | tau_abs(0)=inf, tau_abs(p)=2.5e+9
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [None, None, None, None, None, None, None]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.2s)
====================================================================================================
GRAPH Z_9   n=9 d=2 diam=4 bipartite=False
 lambda_2=0.766044443119 (mult m2=2)  lambda_n=-0.939692620786   tau(0)=4.27431608521   tau_abs(0)=16.5817187388
 T3 on 45 values of p: min tau/boundA = 1.01156609812845 at p=9999/10000 ; min tau/boundB = 1.23677188227977 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 5.39264993808 at p=7/40 ; max/tau(0) = 1.2616404 ; max/d = 2.696325
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=7
 T2 numeric tau(1-10^-k)-d: k=1:0.89151 k=2:0.24423 k=3:0.07386 k=4:0.023031 k=6:0.0022897 k=8:0.00022884 k=10:2.2883e-5 k=12:2.2882e-6 k=16:2.2882e-8 k=20:2.2882e-10 k=24:2.2882e-12
    |tau-d| ~ eps^0.5 (from k=16,24); tau_abs(1-10^-24)-d = 2.2882e-12
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.234338187819  c0=(m2-1)/2+(n/4)Gamma=1.02726092259293   s*=max over V=1.02726092259293  (s*-c0=6.223e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.17495729 | -0.107285 | -0.107741 | -3.9825677
     3 : 0.018635424 | -0.011675 | -0.0117026 | -0.53505467
     4 : 0.0018754535 | -0.00117786 | -0.00118044 | -0.055417706
     5 : 0.00018766499 | -0.000117891 | -0.000118147 | -0.005561657
     6 : 1.8767696e-5 | -1.17901e-5 | -1.18157e-5 | -0.00055636536
     8 : 1.8767828e-7 | -1.17902e-7 | -1.18158e-7 | -5.5638733e-6
    10 : 1.8767829e-9 | -1.17902e-9 | -1.18158e-9 | -5.5638755e-8
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [-1, -1, -1, -1, -1, -1, -1]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.3s)
====================================================================================================
GRAPH Z_10   n=10 d=2 diam=5 bipartite=True
 lambda_2=0.809016994375 (mult m2=2)  lambda_n=-1.0   tau(0)=5.2360679775   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.01158015681701 at p=9999/10000 ; min tau/boundB = 1.16391797681149 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 6.43047732323 at p=1/8 ; max/tau(0) = 1.2281119 ; max/d = 3.2152387
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=8
 T2 numeric tau(1-10^-k)-d: k=1:0.93672 k=2:0.24749 k=3:0.074151 k=4:0.023059 k=6:0.00229 k=8:0.00022884 k=10:2.2883e-5 k=12:2.2882e-6 k=16:2.2882e-8 k=20:2.2882e-10 k=24:2.2882e-12
    |tau-d| ~ eps^0.5 (from k=16,24); tau_abs(1-10^-24)-d = 2.2882e-12
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.190983005625  c0=(m2-1)/2+(n/4)Gamma=0.977457514062631   s*=max over V=0.977457514062631  (s*-c0=-9.3345e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.24406661 | -0.126884 | -0.127961 | tau_abs(0)=inf, tau_abs(p)=24.269054
     3 : 0.026545871 | -0.014094 | -0.0141693 | tau_abs(0)=inf, tau_abs(p)=204.44705
     4 : 0.0026772984 | -0.00142514 | -0.00143232 | tau_abs(0)=inf, tau_abs(p)=2004.4623
     5 : 0.00026795834 | -0.000142673 | -0.000143387 | tau_abs(0)=inf, tau_abs(p)=20004.464
     6 : 2.679812e-5 | -1.42689e-5 | -1.43403e-5 | tau_abs(0)=inf, tau_abs(p)=200004.46
     8 : 2.6798371e-7 | -1.42691e-7 | -1.43404e-7 | tau_abs(0)=inf, tau_abs(p)=20000004.0
    10 : 2.6798374e-9 | -1.42691e-9 | -1.43405e-9 | tau_abs(0)=inf, tau_abs(p)=2.0e+9
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [None, None, None, None, None, None, None]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.3s)
====================================================================================================
GRAPH Z_11   n=11 d=2 diam=5 bipartite=False
 lambda_2=0.841253532831 (mult m2=2)  lambda_n=-0.959492973614   tau(0)=6.29935278457   tau_abs(0)=24.6870750394
 T3 on 45 values of p: min tau/boundA = 1.01239448237066 at p=9999/10000 ; min tau/boundB = 1.22976326231872 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 7.5874990888 at p=1/10 ; max/tau(0) = 1.2044887 ; max/d = 3.7937495
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=9
 T2 numeric tau(1-10^-k)-d: k=1:1.01 k=2:0.26533 k=3:0.079414 k=4:0.024688 k=6:0.0024514 k=8:0.00024497 k=10:2.4495e-5 k=12:2.4495e-6 k=16:2.4495e-8 k=20:2.4495e-10 k=24:2.4495e-12
    |tau-d| ~ eps^0.5 (from k=16,24); tau_abs(1-10^-24)-d = 2.4495e-12
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.158859579297  c0=(m2-1)/2+(n/4)Gamma=0.936863843067473   s*=max over V=0.936863843067473  (s*-c0=-3.8894e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.32980204 | -0.147096 | -0.149031 | -9.199162
     3 : 0.036726502 | -0.0167045 | -0.0168495 | -1.4986766
     4 : 0.0037131165 | -0.00169334 | -0.00170726 | -0.16000271
     5 : 0.00037171951 | -0.000169566 | -0.000170952 | -0.016109294
     6 : 3.7176033e-5 | -1.6959e-5 | -1.70975e-5 | -0.0016120279
     8 : 3.7176482e-7 | -1.69592e-7 | -1.70977e-7 | -1.6121488e-5
    10 : 3.7176486e-9 | -1.69592e-9 | -1.70977e-9 | -1.61215e-7
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [-1, -1, -1, -1, -1, -1, -1]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.5s)
====================================================================================================
GRAPH Z_12   n=12 d=2 diam=6 bipartite=True
 lambda_2=0.866025403784 (mult m2=2)  lambda_n=-1.0   tau(0)=7.46410161514   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.01240297028299 at p=9999/10000 ; min tau/boundB = 1.178995666995 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 8.84463766924 at p=3/40 ; max/tau(0) = 1.1849568 ; max/d = 4.4223188
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=10
 T2 numeric tau(1-10^-k)-d: k=1:1.039 k=2:0.26733 k=3:0.07959 k=4:0.024705 k=6:0.0024516 k=8:0.00024497 k=10:2.4495e-5 k=12:2.4495e-6 k=16:2.4495e-8 k=20:2.4495e-10 k=24:2.4495e-12
    |tau-d| ~ eps^0.5 (from k=16,24); tau_abs(1-10^-24)-d = 2.4495e-12
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.133974596216  c0=(m2-1)/2+(n/4)Gamma=0.901923788646684   s*=max over V=0.901923788646684  (s*-c0=-2.3336e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.43281589 | -0.167633 | -0.170622 | tau_abs(0)=inf, tau_abs(p)=22.04584
     3 : 0.04948886 | -0.0194895 | -0.0197237 | tau_abs(0)=inf, tau_abs(p)=172.3729
     4 : 0.0050172043 | -0.00198106 | -0.00200364 | tau_abs(0)=inf, tau_abs(p)=1672.3958
     5 : 0.00050241038 | -0.000198433 | -0.000200683 | tau_abs(0)=inf, tau_abs(p)=16672.398
     6 : 5.0247944e-5 | -1.98465e-5 | -2.00714e-5 | tau_abs(0)=inf, tau_abs(p)=166672.4
     8 : 5.0248704e-7 | -1.98469e-7 | -2.00718e-7 | tau_abs(0)=inf, tau_abs(p)=16666672.0
    10 : 5.0248711e-9 | -1.98469e-9 | -2.00718e-9 | tau_abs(0)=inf, tau_abs(p)=1.6666667e+9
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [None, None, None, None, None, None, None]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.5s)
====================================================================================================
GRAPH Z_13   n=13 d=2 diam=6 bipartite=False
 lambda_2=0.885456025653 (mult m2=2)  lambda_n=-0.970941817426   tau(0)=8.73027154595   tau_abs(0)=34.4137145348
 T3 on 45 values of p: min tau/boundA = 1.01290270438527 at p=9999/10000 ; min tau/boundB = 1.22591162013729 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 10.2102699865 at p=3/40 ; max/tau(0) = 1.1695249 ; max/d = 5.105135
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=11
 T2 numeric tau(1-10^-k)-d: k=1:1.0865 k=2:0.27841 k=3:0.082828 k=4:0.025704 k=6:0.0025505 k=8:0.00025485 k=10:2.5483e-5 k=12:2.5483e-6 k=16:2.5483e-8 k=20:2.5483e-10 k=24:2.5483e-12
    |tau-d| ~ eps^0.5 (from k=16,24); tau_abs(1-10^-24)-d = 2.5483e-12
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.114585150084  c0=(m2-1)/2+(n/4)Gamma=0.872401737772892   s*=max over V=0.872401737772892  (s*-c0=1.1668e-60)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.55441259 | -0.18843 | -0.19267 | -17.013923
     3 : 0.065265008 | -0.0224577 | -0.0228013 | -3.4218639
     4 : 0.0066368314 | -0.00228955 | -0.00232277 | -0.38107522
     5 : 0.0006647998 | -0.000229402 | -0.000232712 | -0.03854605
     6 : 6.649116e-5 | -2.29446e-5 | -2.32755e-5 | -0.0038590463
     8 : 6.649239e-7 | -2.29451e-7 | -2.3276e-7 | -3.8595355e-5
    10 : 6.6492403e-9 | -2.29451e-9 | -2.3276e-9 | -3.8595404e-7
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [-1, -1, -1, -1, -1, -1, -1]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.7s)
====================================================================================================
GRAPH Z_14   n=14 d=2 diam=7 bipartite=True
 lambda_2=0.900968867902 (mult m2=2)  lambda_n=-1.0   tau(0)=10.097834679   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.01290818702419 at p=9999/10000 ; min tau/boundB = 1.188548949113 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 11.6942144763 at p=1/20 ; max/tau(0) = 1.1580913 ; max/d = 5.8471072
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=12
 T2 numeric tau(1-10^-k)-d: k=1:1.106 k=2:0.27971 k=3:0.082942 k=4:0.025715 k=6:0.0025506 k=8:0.00025486 k=10:2.5483e-5 k=12:2.5483e-6 k=16:2.5483e-8 k=20:2.5483e-10 k=24:2.5483e-12
    |tau-d| ~ eps^0.5 (from k=16,24); tau_abs(1-10^-24)-d = 2.5483e-12
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.0990311320976  c0=(m2-1)/2+(n/4)Gamma=0.846608962341533   s*=max over V=0.846608962341533  (s*-c0=-7.7788e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.69455673 | -0.209283 | -0.214929 | tau_abs(0)=inf, tau_abs(p)=20.798832
     3 : 0.084415203 | -0.0255986 | -0.0260692 | tau_abs(0)=inf, tau_abs(p)=149.94773
     4 : 0.0086132005 | -0.00261807 | -0.00266366 | tau_abs(0)=inf, tau_abs(p)=1435.6912
     5 : 0.00086306174 | -0.000262404 | -0.000266947 | tau_abs(0)=inf, tau_abs(p)=14292.836
     6 : 8.6323616e-5 | -2.62464e-5 | -2.67006e-5 | tau_abs(0)=inf, tau_abs(p)=142864.27
     8 : 8.6325535e-7 | -2.6247e-7 | -2.67012e-7 | tau_abs(0)=inf, tau_abs(p)=14285721.0
    10 : 8.6325554e-9 | -2.6247e-9 | -2.67012e-9 | tau_abs(0)=inf, tau_abs(p)=1.4285714e+9
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [None, None, None, None, None, None, None]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.7s)
====================================================================================================
GRAPH Q_1   n=2 d=1 diam=1 bipartite=True
 lambda_2=-1.0 (mult m2=1)  lambda_n=-1.0   tau(0)=0.5   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.0 at p=11/40 ; min tau/boundB = 1.0000500050005 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 0.999900009999 at p=9999/10000 ; max/tau(0) = 1.9998 ; max/d = 0.99990001
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=1
 T2 numeric tau(1-10^-k)-d: k=1:-0.090909 k=2:-0.009901 k=3:-0.000999 k=4:-9.999e-5 k=6:-1.0e-6 k=8:-1.0e-8 k=10:-1.0e-10 k=12:-1.0e-12 k=16:-1.0e-16 k=20:-1.0e-20 k=24:-1.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = 1.0e-24
 T4: m2=1  n*f0(e)^2=1.0  Gamma(f0)=2.0  c0=(m2-1)/2+(n/4)Gamma=1.0   s*=max over V=1.0  (s*-c0=-7.7788e-62)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.0025125628 | -3.1504e-59 | -4.66726e-61 | tau_abs(0)=inf, tau_abs(p)=100.0
     3 : 0.00025012506 | -2.33752e-58 | 7.7321e-59 | tau_abs(0)=inf, tau_abs(p)=1000.0
     4 : 2.500125e-5 | -8.56054e-58 | 6.99622e-58 | tau_abs(0)=inf, tau_abs(p)=10000.0
     5 : 2.5000125e-6 | -3.19711e-56 | 1.47014e-56 | tau_abs(0)=inf, tau_abs(p)=100000.0
     6 : 2.5000013e-7 | -1.87547e-55 | -3.19713e-56 | tau_abs(0)=inf, tau_abs(p)=1000000.0
     8 : 2.5e-9 | -5.30832e-53 | 9.14698e-54 | tau_abs(0)=inf, tau_abs(p)=1.0e+8
    10 : 2.5e-11 | -2.08659e-52 | -2.08659e-52 | tau_abs(0)=inf, tau_abs(p)=1.0e+10
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [None, None, None, None, None, None, None]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.0s)
====================================================================================================
GRAPH Q_2   n=4 d=2 diam=2 bipartite=True
 lambda_2=9.07016631075e-62 (mult m2=2)  lambda_n=-1.0   tau(0)=1.0   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.00004999250137 at p=9999/10000 ; min tau/boundB = 1.00010001499875 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 1.99999998 at p=9999/10000 ; max/tau(0) = 2.0 ; max/d = 0.99999999
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=2
 T2 numeric tau(1-10^-k)-d: k=1:-0.017556 k=2:-0.00019707 k=3:-1.997e-6 k=4:-1.9997e-8 k=6:-2.0e-12 k=8:-2.0e-16 k=10:-2.0e-20 k=12:-2.0e-24 k=16:-2.0e-32 k=20:-2.0e-40 k=24:-2.0e-48
    |tau-d| ~ eps^2.0 (from k=16,24); tau_abs(1-10^-24)-d = -2.0e-48
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=1.0  c0=(m2-1)/2+(n/4)Gamma=1.5   s*=max over V=1.5  (s*-c0=3.1115e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.015022913 | -0.0199435 | -0.0246064 | tau_abs(0)=inf, tau_abs(p)=51.239918
     3 : 0.0015002479 | -0.00199949 | -0.00249601 | tau_abs(0)=inf, tau_abs(p)=501.249
     4 : 0.0001500025 | -0.000199995 | -0.00024996 | tau_abs(0)=inf, tau_abs(p)=5001.2499
     5 : 1.5000025e-5 | -1.99999e-5 | -2.49996e-5 | tau_abs(0)=inf, tau_abs(p)=50001.25
     6 : 1.5000002e-6 | -2.0e-6 | -2.5e-6 | tau_abs(0)=inf, tau_abs(p)=500001.25
     8 : 1.5e-8 | -2.0e-8 | -2.5e-8 | tau_abs(0)=inf, tau_abs(p)=50000001.0
    10 : 1.5e-10 | -2.0e-10 | -2.5e-10 | tau_abs(0)=inf, tau_abs(p)=5.0e+9
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [None, None, None, None, None, None, None]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.0s)
====================================================================================================
GRAPH Q_3   n=8 d=3 diam=3 bipartite=True
 lambda_2=0.333333333333 (mult m2=3)  lambda_n=-1.0   tau(0)=1.5   tau_abs(0)=inf
 T3 on 46 values of p: min tau/boundA = 1.00006666222259 at p=9999/10000 ; min tau/boundB = 1.0001500587118 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 3.11437049533 at p=31/40 ; max/tau(0) = 2.076247 ; max/d = 1.0381235
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=3
 T2 numeric tau(1-10^-k)-d: k=1:0.077503 k=2:0.0097676 k=3:0.00099767 k=4:9.9977e-5 k=6:1.0e-6 k=8:1.0e-8 k=10:1.0e-10 k=12:1.0e-12 k=16:1.0e-16 k=20:1.0e-20 k=24:1.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = 1.0e-24
 T4: m2=3  n*f0(e)^2=3.0  Gamma(f0)=0.5  c0=(m2-1)/2+(n/4)Gamma=2.0   s*=max over V=2.0  (s*-c0=-4.6673e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.045215745 | -0.0492154 | -0.0746876 | tau_abs(0)=inf, tau_abs(p)=35.908877
     3 : 0.004502577 | -0.00484358 | -0.00772092 | tau_abs(0)=inf, tau_abs(p)=335.99069
     4 : 0.0004500262 | -0.000483437 | -0.000774708 | tau_abs(0)=inf, tau_abs(p)=3335.9991
     5 : 4.5000262e-5 | -4.83344e-5 | -7.74971e-5 | tau_abs(0)=inf, tau_abs(p)=33336.0
     6 : 4.5000026e-6 | -4.83334e-6 | -7.74997e-6 | tau_abs(0)=inf, tau_abs(p)=333336.0
     8 : 4.5e-8 | -4.83333e-8 | -7.75e-8 | tau_abs(0)=inf, tau_abs(p)=33333336.0
    10 : 4.5e-10 | -4.83333e-10 | -7.75e-10 | tau_abs(0)=inf, tau_abs(p)=3.3333333e+9
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [None, None, None, None, None, None, None]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.1s)
====================================================================================================
GRAPH Q_4   n=16 d=4 diam=4 bipartite=True
 lambda_2=0.5 (mult m2=4)  lambda_n=-1.0   tau(0)=2.0   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.00007499812505 at p=9999/10000 ; min tau/boundB = 1.00020022098278 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 4.44214996276 at p=3/5 ; max/tau(0) = 2.221075 ; max/d = 1.1105375
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=4
 T2 numeric tau(1-10^-k)-d: k=1:0.17969 k=2:0.0198 k=3:0.001998 k=4:0.00019998 k=6:2.0e-6 k=8:2.0e-8 k=10:2.0e-10 k=12:2.0e-12 k=16:2.0e-16 k=20:2.0e-20 k=24:2.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = 2.0e-24
 T4: m2=4  n*f0(e)^2=4.0  Gamma(f0)=0.25  c0=(m2-1)/2+(n/4)Gamma=2.5   s*=max over V=2.5  (s*-c0=1.8669e-60)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.10202685 | -0.0731315 | -0.154997 | tau_abs(0)=inf, tau_abs(p)=29.244364
     3 : 0.010023848 | -0.00653502 | -0.0163942 | tau_abs(0)=inf, tau_abs(p)=254.65661
     4 : 0.0010002421 | -0.000644735 | -0.00164894 | tau_abs(0)=inf, tau_abs(p)=2504.7031
     5 : 0.00010000242 | -6.43849e-5 | -0.000164989 | tau_abs(0)=inf, tau_abs(p)=25004.708
     6 : 1.0000024e-5 | -6.4376e-6 | -1.64999e-5 | tau_abs(0)=inf, tau_abs(p)=250004.71
     8 : 1.0e-7 | -6.4375e-8 | -1.65e-7 | tau_abs(0)=inf, tau_abs(p)=25000005.0
    10 : 1.0e-9 | -6.4375e-10 | -1.65e-9 | tau_abs(0)=inf, tau_abs(p)=2.5e+9
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [None, None, None, None, None, None, None]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.8s)
====================================================================================================
GRAPH Q_5   n=32 d=5 diam=5 bipartite=True
 lambda_2=0.6 (mult m2=5)  lambda_n=-1.0   tau(0)=2.5   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.00007999999997 at p=9999/10000 ; min tau/boundB = 1.00025069766378 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 5.98340174482 at p=19/40 ; max/tau(0) = 2.3933607 ; max/d = 1.1966803
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=5
 T2 numeric tau(1-10^-k)-d: k=1:0.28511 k=2:0.029859 k=3:0.0029986 k=4:0.00029999 k=6:3.0e-6 k=8:3.0e-8 k=10:3.0e-10 k=12:3.0e-12 k=16:3.0e-16 k=20:3.0e-20 k=24:3.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = 3.0e-24
 T4: m2=5  n*f0(e)^2=5.0  Gamma(f0)=0.125  c0=(m2-1)/2+(n/4)Gamma=3.0   s*=max over V=3.0  (s*-c0=-3.1115e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.19798582 | -0.064687 | -0.269023 | tau_abs(0)=inf, tau_abs(p)=26.092467
     3 : 0.018872855 | -0.00296828 | -0.0292179 | tau_abs(0)=inf, tau_abs(p)=207.52974
     4 : 0.0018762451 | -0.000259199 | -0.00294715 | tau_abs(0)=inf, tau_abs(p)=2007.7272
     5 : 0.00018751247 | -2.5542e-5 | -0.000294972 | tau_abs(0)=inf, tau_abs(p)=20007.748
     6 : 1.8750125e-5 | -2.55042e-6 | -2.94997e-5 | tau_abs(0)=inf, tau_abs(p)=200007.75
     8 : 1.8750001e-7 | -2.55e-8 | -2.95e-7 | tau_abs(0)=inf, tau_abs(p)=20000008.0
    10 : 1.875e-9 | -2.55e-10 | -2.95e-9 | tau_abs(0)=inf, tau_abs(p)=2.0e+9
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [None, None, None, None, None, None, None]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (4.7s)

SUMMARY (name | n | d | bip | m2 | tau(0) | shape | max/tau0 | c0 | s* | minA | minB | tau(1-1e-24)-d | exponent | status | abs-sign)
 Z_8                                |   8 |  2 | B |  2 | 3.4142136 | up-then-down | 1.30563 | 1.0857864 | 1.0857864 | 1.0101133 | 1.1383224 | 2.0e-12 | 0.5 | OK | [None, None, None, None, None, None, None]
 Z_9                                |   9 |  2 | - |  2 | 4.2743161 | up-then-down | 1.26164 | 1.0272609 | 1.0272609 | 1.0115661 | 1.2367719 | 2.29e-12 | 0.5 | OK | [-1, -1, -1, -1, -1, -1, -1]
 Z_10                               |  10 |  2 | B |  2 | 5.236068 | up-then-down | 1.22811 | 0.97745751 | 0.97745751 | 1.0115802 | 1.163918 | 2.29e-12 | 0.5 | OK | [None, None, None, None, None, None, None]
 Z_11                               |  11 |  2 | - |  2 | 6.2993528 | up-then-down | 1.20449 | 0.93686384 | 0.93686384 | 1.0123945 | 1.2297633 | 2.45e-12 | 0.5 | OK | [-1, -1, -1, -1, -1, -1, -1]
 Z_12                               |  12 |  2 | B |  2 | 7.4641016 | up-then-down | 1.18496 | 0.90192379 | 0.90192379 | 1.012403 | 1.1789957 | 2.45e-12 | 0.5 | OK | [None, None, None, None, None, None, None]
 Z_13                               |  13 |  2 | - |  2 | 8.7302715 | up-then-down | 1.16952 | 0.87240174 | 0.87240174 | 1.0129027 | 1.2259116 | 2.55e-12 | 0.5 | OK | [-1, -1, -1, -1, -1, -1, -1]
 Z_14                               |  14 |  2 | B |  2 | 10.097835 | up-then-down | 1.15809 | 0.84660896 | 0.84660896 | 1.0129082 | 1.1885489 | 2.55e-12 | 0.5 | OK | [None, None, None, None, None, None, None]
 Q_1                                |   2 |  1 | B |  1 | 0.5 | increasing   | 1.9998 | 1.0 | 1.0 | 1.0 | 1.00005 | -1.0e-24 | 1.0 | OK | [None, None, None, None, None, None, None]
 Q_2                                |   4 |  2 | B |  2 | 1.0 | increasing   | 2.0 | 1.5 | 1.5 | 1.00005 | 1.0001 | -2.0e-48 | 2.0 | OK | [None, None, None, None, None, None, None]
 Q_3                                |   8 |  3 | B |  3 | 1.5 | up-then-down | 2.07625 | 2.0 | 2.0 | 1.0000667 | 1.0001501 | 1.0e-24 | 1.0 | OK | [None, None, None, None, None, None, None]
 Q_4                                |  16 |  4 | B |  4 | 2.0 | up-then-down | 2.22107 | 2.5 | 2.5 | 1.000075 | 1.0002002 | 2.0e-24 | 1.0 | OK | [None, None, None, None, None, None, None]
 Q_5                                |  32 |  5 | B |  5 | 2.5 | up-then-down | 2.39336 | 3.0 | 3.0 | 1.00008 | 1.0002507 | 3.0e-24 | 1.0 | OK | [None, None, None, None, None, None, None]
