====================================================================================================
GRAPH S_3,all transpositions   n=6 d=3 diam=2 bipartite=True
 lambda_2=0.0 (mult m2=4)  lambda_n=-1.0   tau(0)=1.0   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.00003332777865 at p=9999/10000 ; min tau/boundB = 1.05902700283958 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 2.99999997 at p=9999/10000 ; max/tau(0) = 3.0 ; 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=3
 T2 numeric tau(1-10^-k)-d: k=1:-0.027858 k=2:-0.00029754 k=3:-2.9975e-6 k=4:-2.9998e-8 k=6:-3.0e-12 k=8:-3.0e-16 k=10:-3.0e-20 k=12:-3.0e-24 k=16:-3.0e-32 k=20:-3.0e-40 k=24:-3.0e-48
    |tau-d| ~ eps^2.0 (from k=16,24); tau_abs(1-10^-24)-d = -3.0e-48
 T4: m2=4  n*f0(e)^2=4.0  Gamma(f0)=1.22222222222  c0=(m2-1)/2+(n/4)Gamma=3.33333333333333   s*=max over V=3.33333333333333  (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 : 0.033408709 | -0.100468 | -0.11588 | tau_abs(0)=inf, tau_abs(p)=44.980528
     3 : 0.0033342081 | -0.0102053 | -0.0119575 | tau_abs(0)=inf, tau_abs(p)=430.73071
     4 : 0.00033334221 | -0.00102205 | -0.00119957 | tau_abs(0)=inf, tau_abs(p)=4287.8772
     5 : 3.3333422e-5 | -0.000102221 | -0.000119996 | tau_abs(0)=inf, tau_abs(p)=42859.306
     6 : 3.3333342e-6 | -1.02222e-5 | -1.2e-5 | tau_abs(0)=inf, tau_abs(p)=428573.59
     8 : 3.3333333e-8 | -1.02222e-7 | -1.2e-7 | tau_abs(0)=inf, tau_abs(p)=42857145.0
    10 : 3.3333333e-10 | -1.02222e-9 | -1.2e-9 | tau_abs(0)=inf, tau_abs(p)=4.2857143e+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 S_4,all transpositions   n=24 d=6 diam=3 bipartite=True
 lambda_2=0.333333333333 (mult m2=9)  lambda_n=-1.0   tau(0)=1.5   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.00004999694466 at p=9999/10000 ; min tau/boundB = 1.1373967256612 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 6.26625360404 at p=3/4 ; max/tau(0) = 4.1775024 ; max/d = 1.0443756
 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=1:0.16353 k=2:0.019634 k=3:0.0019963 k=4:0.00019996 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=9  n*f0(e)^2=9.0  Gamma(f0)=0.4375  c0=(m2-1)/2+(n/4)Gamma=6.625   s*=max over V=6.625  (s*-c0=6.8453e-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 : 0.16196663 | -0.128013 | -0.656864 | tau_abs(0)=inf, tau_abs(p)=32.139956
     3 : 0.015059394 | 0.00153604 | -0.0720861 | tau_abs(0)=inf, tau_abs(p)=267.8671
     4 : 0.0014921795 | 0.000318249 | -0.00727953 | tau_abs(0)=inf, tau_abs(p)=2615.8071
     5 : 0.00014907807 | 3.3487e-5 | -0.00072867 | tau_abs(0)=inf, tau_abs(p)=26094.08
     6 : 1.4906406e-5 | 3.36534e-6 | -7.28742e-5 | tau_abs(0)=inf, tau_abs(p)=260876.69
     8 : 1.4906252e-7 | 3.36717e-8 | -7.2875e-7 | tau_abs(0)=inf, tau_abs(p)=26086964.0
    10 : 1.490625e-9 | 3.36719e-10 | -7.2875e-9 | tau_abs(0)=inf, tau_abs(p)=2.6086957e+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   (2.4s)
====================================================================================================
GRAPH D_4 {r,r^-1,s}   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 45 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 D_4 {s,sr}   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 D_5 {r,r^-1,s}   n=10 d=3 diam=3 bipartite=False
 lambda_2=0.539344662917 (mult m2=2)  lambda_n=-0.87267799625   tau(0)=2.17082039325   tau_abs(0)=7.85410196625
 T3 on 45 values of p: min tau/boundA = 1.00579038924869 at p=9999/10000 ; min tau/boundB = 1.34008875308781 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 3.79163189872 at p=3/5 ; max/tau(0) = 1.7466355 ; max/d = 1.2638773
 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=1:0.50021 k=2:0.16833 k=3:0.054275 k=4:0.017271 k=6:0.0017316 k=8:0.0001732 k=10:1.732e-5 k=12:1.7321e-6 k=16:1.7321e-8 k=20:1.7321e-10 k=24:1.7321e-12
    |tau-d| ~ eps^0.5 (from k=16,24); tau_abs(1-10^-24)-d = 1.7321e-12
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.3527864045  c0=(m2-1)/2+(n/4)Gamma=1.38196601125011   s*=max over V=1.38196601125011  (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 : 0.065091564 | -0.0409125 | -0.0630767 | -0.62654349
     3 : 0.0065129371 | -0.00403311 | -0.0065634 | -0.069049927
     4 : 0.00065125172 | -0.000402584 | -0.000659013 | -0.0069763821
     5 : 6.5124669e-5 | -4.0251e-5 | -6.59281e-5 | -0.00069836041
     6 : 6.5124617e-6 | -4.02503e-6 | -6.59308e-6 | -6.9843271e-5
     8 : 6.5124612e-8 | -4.02502e-8 | -6.59311e-8 | -6.9844067e-7
    10 : 6.5124612e-10 | -4.02502e-10 | -6.59311e-10 | -6.9844075e-9
    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   (0.4s)
====================================================================================================
GRAPH D_5 {s,sr}   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_2xZ_4 {(1,0),(0,+-1)}   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 45 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 Z_2xZ_4 {(1,+-1),(0,+-1)}   n=8 d=4 diam=2 bipartite=True
 lambda_2=0.0 (mult m2=6)  lambda_n=-1.0   tau(0)=1.0   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.00002499562566 at p=9999/10000 ; min tau/boundB = 1.14321223499953 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 3.99999996 at p=9999/10000 ; max/tau(0) = 4.0 ; 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=4
 T2 numeric tau(1-10^-k)-d: k=1:-0.037964 k=2:-0.0003977 k=3:-3.9977e-6 k=4:-3.9998e-8 k=6:-4.0e-12 k=8:-4.0e-16 k=10:-4.0e-20 k=12:-4.0e-24 k=16:-4.0e-32 k=20:-4.0e-40 k=24:-4.0e-48
    |tau-d| ~ eps^2.0 (from k=16,24); tau_abs(1-10^-24)-d = -4.0e-48
 T4: m2=6  n*f0(e)^2=6.0  Gamma(f0)=1.375  c0=(m2-1)/2+(n/4)Gamma=5.25   s*=max over V=5.25  (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 : 0.052631974 | -0.249964 | -0.277151 | tau_abs(0)=inf, tau_abs(p)=42.968291
     3 : 0.0052516478 | -0.0257879 | -0.0290889 | tau_abs(0)=inf, tau_abs(p)=403.05076
     4 : 0.00052501683 | -0.00258663 | -0.00292338 | tau_abs(0)=inf, tau_abs(p)=4003.0591
     5 : 5.2500169e-5 | -0.000258741 | -0.000292484 | tau_abs(0)=inf, tau_abs(p)=40003.06
     6 : 5.2500017e-6 | -2.58749e-5 | -2.92498e-5 | tau_abs(0)=inf, tau_abs(p)=400003.06
     8 : 5.25e-8 | -2.5875e-7 | -2.925e-7 | tau_abs(0)=inf, tau_abs(p)=40000003.0
    10 : 5.25e-10 | -2.5875e-9 | -2.925e-9 | tau_abs(0)=inf, tau_abs(p)=4.0e+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   (0.1s)

SUMMARY (name | n | d | bip | m2 | tau(0) | shape | max/tau0 | c0 | s* | minA | minB | tau(1-1e-24)-d | exponent | status | abs-sign)
 S_3,all transpositions             |   6 |  3 | B |  4 | 1.0 | increasing   | 3.0 | 3.3333333 | 3.3333333 | 1.0000333 | 1.059027 | -3.0e-48 | 2.0 | OK | [None, None, None, None, None, None, None]
 S_4,all transpositions             |  24 |  6 | B |  9 | 1.5 | up-then-down | 4.1775 | 6.625 | 6.625 | 1.00005 | 1.1373967 | 2.0e-24 | 1.0 | OK | [None, None, None, None, None, None, None]
 D_4 {r,r^-1,s}                     |   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]
 D_4 {s,sr}                         |   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]
 D_5 {r,r^-1,s}                     |  10 |  3 | - |  2 | 2.1708204 | up-then-down | 1.74664 | 1.381966 | 1.381966 | 1.0057904 | 1.3400888 | 1.73e-12 | 0.5 | OK | [-1, -1, -1, -1, -1, -1, -1]
 D_5 {s,sr}                         |  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_2xZ_4 {(1,0),(0,+-1)}            |   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]
 Z_2xZ_4 {(1,+-1),(0,+-1)}          |   8 |  4 | B |  6 | 1.0 | increasing   | 4.0 | 5.25 | 5.25 | 1.000025 | 1.1432122 | -4.0e-48 | 2.0 | OK | [None, None, None, None, None, None, None]
