====================================================================================================
GRAPH S_4,adjacent transpositions   n=24 d=3 diam=6 bipartite=True
 lambda_2=0.804737854124 (mult m2=3)  lambda_n=-1.0   tau(0)=5.12132034356   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.008262239192 at p=9999/10000 ; min tau/boundB = 1.18221114283024 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 7.58123196395 at p=7/40 ; max/tau(0) = 1.4803276 ; max/d = 2.5270773
 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=1:1.0241 k=2:0.26581 k=3:0.079412 k=4:0.024686 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=3  n*f0(e)^2=3.0  Gamma(f0)=0.104604721005  c0=(m2-1)/2+(n/4)Gamma=1.62762832602883   s*=max over V=1.62894023634183  (s*-c0=0.0013119)
    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 : 0.40122049 | -0.210333 | -0.225888 | tau_abs(0)=inf, tau_abs(p)=23.154828
     3 : 0.042473048 | -0.022877 | -0.0255312 | tau_abs(0)=inf, tau_abs(p)=174.32758
     4 : 0.0042698802 | -0.00230615 | -0.00258683 | tau_abs(0)=inf, tau_abs(p)=1674.4726
     5 : 0.00042721228 | -0.000230799 | -0.000259025 | tau_abs(0)=inf, tau_abs(p)=16674.487
     6 : 4.2723469e-5 | -2.30817e-5 | -2.59059e-5 | tau_abs(0)=inf, tau_abs(p)=166674.49
     8 : 4.2723715e-7 | -2.30819e-7 | -2.59063e-7 | tau_abs(0)=inf, tau_abs(p)=16666674.0
    10 : 4.2723718e-9 | -2.30819e-9 | -2.59063e-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   (2.8s)
====================================================================================================
GRAPH D_4 {r,r^-1,s,sr}   n=8 d=4 diam=2 bipartite=False
 lambda_2=0.353553390593 (mult m2=2)  lambda_n=-0.5   tau(0)=1.54691816068   tau_abs(0)=2.0
 T3 on 45 values of p: min tau/boundA = 1.00003749875012 at p=9999/10000 ; min tau/boundB = 1.76794897809192 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 3.9998999875 at p=9999/10000 ; max/tau(0) = 2.5857218 ; max/d = 0.999975
 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:-0.11226 k=2:-0.010125 k=3:-0.0010012 k=4:-0.00010001 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=2  n*f0(e)^2=2.0  Gamma(f0)=0.482668522349  c0=(m2-1)/2+(n/4)Gamma=1.46533704469724   s*=max over V=1.46533704469724  (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 : 0.035563637 | -0.0125562 | -0.0439409 | -0.024975336
     3 : 0.003511666 | -0.00115946 | -0.00451134 | -0.0025638372
     4 : 0.00035070066 | -0.000114959 | -0.000452341 | -0.00025706674
     5 : 3.5065388e-5 | -1.1486e-5 | -4.52462e-5 | -2.5713524e-5
     6 : 3.506492e-6 | -1.1485e-6 | -4.52475e-6 | -2.571421e-6
     8 : 3.5064868e-8 | -1.14849e-8 | -4.52476e-8 | -2.5714285e-8
    10 : 3.5064868e-10 | -1.14849e-10 | -4.52476e-10 | -2.5714286e-10
    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 D_5 {r,r^-1,s,sr}   n=10 d=4 diam=3 bipartite=False
 lambda_2=0.559016994375 (mult m2=2)  lambda_n=-0.559016994375   tau(0)=2.26766108273   tau_abs(0)=2.26766108273
 T3 on 45 values of p: min tau/boundA = 1.00006875249981 at p=9999/10000 ; min tau/boundB = 1.74452803431979 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 4.02435953072 at p=33/40 ; max/tau(0) = 1.7746742 ; max/d = 1.0060899
 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=1:0.019559 k=2:0.0024579 k=3:0.00024959 k=4:2.4996e-5 k=6:2.5e-7 k=8:2.5e-9 k=10:2.5e-11 k=12:2.5e-13 k=16:2.5e-17 k=20:2.5e-21 k=24:2.5e-25
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = 2.5e-25
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.322547877943  c0=(m2-1)/2+(n/4)Gamma=1.30636969485808   s*=max over V=1.30636969485808  (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 : 0.066438958 | -0.0511342 | -0.0634754 | 0.066438958
     3 : 0.0067106052 | -0.00523548 | -0.00661679 | 0.0067106052
     4 : 0.00067170182 | -0.000524769 | -0.000664498 | 0.00067170182
     5 : 6.7176567e-5 | -5.24891e-5 | -6.64782e-5 | 6.7176567e-5
     6 : 6.7177205e-6 | -5.24904e-6 | -6.6481e-6 | 6.7177205e-6
     8 : 6.7177275e-8 | -5.24905e-8 | -6.64813e-8 | 6.7177275e-8
    10 : 6.7177276e-10 | -5.24905e-10 | -6.64813e-10 | 6.7177276e-10
    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_2xZ_4 {(1,0),(0,+-1),(1,2)}   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)
====================================================================================================
GRAPH Q8 {+-i,+-j}   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)
====================================================================================================
GRAPH A_4 {(012)^+-,(013)^+-}   n=12 d=4 diam=3 bipartite=False
 lambda_2=0.5 (mult m2=3)  lambda_n=-0.5   tau(0)=2.0   tau_abs(0)=2.0
 T3 on 45 values of p: min tau/boundA = 1.00012496960374 at p=9999/10000 ; min tau/boundB = 1.73505845965966 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 4.32260508374 at p=29/40 ; max/tau(0) = 2.1613025 ; max/d = 1.0806513
 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=1:0.20476 k=2:0.028639 k=3:0.0029857 k=4:0.00029986 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=3  n*f0(e)^2=3.0  Gamma(f0)=0.333333333333  c0=(m2-1)/2+(n/4)Gamma=2.0   s*=max over V=2.0  (s*-c0=0.0)
    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 : 0.080380243 | -0.0681358 | -0.102121 | 0.080380243
     3 : 0.0080051 | -0.00670326 | -0.0106937 | 0.0080051
     4 : 0.00080005235 | -0.000668911 | -0.00107443 | 0.00080005235
     5 : 8.0000525e-5 | -6.68766e-5 | -0.000107494 | 8.0000525e-5
     6 : 8.0000052e-6 | -6.68752e-6 | -1.07499e-5 | 8.0000052e-6
     8 : 8.0000001e-8 | -6.6875e-8 | -1.075e-7 | 8.0000001e-8
    10 : 8.0e-10 | -6.6875e-10 | -1.075e-9 | 8.0e-10
    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 S_4 {(012)^+-,(23)}   n=24 d=3 diam=6 bipartite=False
 lambda_2=0.853850937603 (mult m2=3)  lambda_n=-0.666666666667   tau(0)=6.84232921921   tau_abs(0)=6.84232921921
 T3 on 45 values of p: min tau/boundA = 1.01069183970006 at p=9999/10000 ; min tau/boundB = 1.58453736494446 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 9.7045520196 at p=1/8 ; max/tau(0) = 1.4183112 ; max/d = 3.2348507
 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=1:1.2638 k=2:0.3387 k=3:0.10231 k=4:0.031907 k=6:0.0031725 k=8:0.00031707 k=10:3.1706e-5 k=12:3.1705e-6 k=16:3.1705e-8 k=20:3.1705e-10 k=24:3.1705e-12
    |tau-d| ~ eps^0.5 (from k=16,24); tau_abs(1-10^-24)-d = 3.1705e-12
 T4: m2=3  n*f0(e)^2=3.0  Gamma(f0)=0.0957739582728  c0=(m2-1)/2+(n/4)Gamma=1.57464374963667   s*=max over V=1.57464374963667  (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 : 0.66672379 | -0.276996 | -0.289676 | 0.66672379
     3 : 0.07302026 | -0.0314329 | -0.0340997 | 0.07302026
     4 : 0.0073651022 | -0.00318271 | -0.00347149 | 0.0073651022
     5 : 0.00073713857 | -0.000318666 | -0.000347776 | 0.00073713857
     6 : 7.3720137e-5 | -3.18705e-5 | -3.47839e-5 | 7.3720137e-5
     8 : 7.3720828e-7 | -3.1871e-7 | -3.47846e-7 | 7.3720828e-7
    10 : 7.3720835e-9 | -3.1871e-9 | -3.47846e-9 | 7.3720835e-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   (2.6s)
====================================================================================================
GRAPH Z_3xZ_3 {(+-1,0),(0,+-1)}   n=9 d=4 diam=2 bipartite=False
 lambda_2=0.25 (mult m2=4)  lambda_n=-0.5   tau(0)=1.33333333333   tau_abs(0)=2.0
 T3 on 45 values of p: min tau/boundA = 1.00004999500062 at p=9999/10000 ; min tau/boundB = 1.50020012655495 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 3.99999997 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=2
 T2 numeric tau(1-10^-k)-d: k=1:-0.027793 k=2:-0.00029731 k=3:-2.9973e-6 k=4:-2.9997e-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)=0.666666666667  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 : 0.053912134 | -0.0852959 | -0.130767 | -9.3345229e-61
     3 : 0.0053397768 | -0.00835658 | -0.0136053 | 9.3345229e-61
     4 : 0.00053339844 | -0.00083357 | -0.00136605 | -9.3345229e-61
     5 : 5.3333985e-5 | -8.33357e-5 | -0.000136661 | -9.3345229e-61
     6 : 5.3333399e-6 | -8.33336e-6 | -1.36666e-5 | -3.1115076e-61
     8 : 5.3333334e-8 | -8.33333e-8 | -1.36667e-7 | 0.0
    10 : 5.3333333e-10 | -8.33333e-10 | -1.36667e-9 | -3.1115076e-61
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [0, 0, 0, 0, 0, 0, 0]
 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{+-1,2}   n=9 d=4 diam=2 bipartite=False
 lambda_2=0.469846310393 (mult m2=2)  lambda_n=-0.5   tau(0)=1.88624547863   tau_abs(0)=2.0
 T3 on 45 values of p: min tau/boundA = 1.00006249711003 at p=9999/10000 ; min tau/boundB = 2.12190895386917 at p=1/1000 ; violations=0
 shape on grid: increasing ; max tau = 3.99999997594 at p=9999/10000 ; max/tau(0) = 2.1206147 ; 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=1
 T2 numeric tau(1-10^-k)-d: k=1:-0.021658 k=2:-0.00023765 k=3:-2.4032e-6 k=4:-2.4059e-8 k=6:-2.4062e-12 k=8:-2.4062e-16 k=10:-2.4062e-20 k=12:-2.4062e-24 k=16:-2.4062e-32 k=20:-2.4062e-40 k=24:-2.4062e-48
    |tau-d| ~ eps^2.0 (from k=16,24); tau_abs(1-10^-24)-d = -2.4062e-48
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.390768828766  c0=(m2-1)/2+(n/4)Gamma=1.37922986472403   s*=max over V=1.37922986472403  (s*-c0=-1.5558e-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 : 0.049093645 | -0.0343918 | -0.0532986 | -0.039076281
     3 : 0.0049076551 | -0.00344945 | -0.00551138 | -0.0040607095
     4 : 0.00049072411 | -0.000345018 | -0.000553022 | -0.00040766525
     5 : 4.9071972e-5 | -3.45026e-5 | -5.53211e-5 | -4.0782534e-5
     6 : 4.9071928e-6 | -3.45026e-6 | -5.5323e-6 | -4.0784136e-6
     8 : 4.9071923e-8 | -3.45026e-8 | -5.53232e-8 | -4.0784312e-8
    10 : 4.9071923e-10 | -3.45026e-10 | -5.53232e-10 | -4.0784314e-10
    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_12{+-1,5}   n=12 d=4 diam=3 bipartite=True
 lambda_2=0.5 (mult m2=2)  lambda_n=-1.0   tau(0)=2.0   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.00005833025493 at p=9999/10000 ; min tau/boundB = 1.38462004908286 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 4.16326163374 at p=3/4 ; max/tau(0) = 2.0816308 ; max/d = 1.0408154
 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=1:0.10574 k=2:0.013053 k=3:0.0013305 k=4:0.00013331 k=6:1.3333e-6 k=8:1.3333e-8 k=10:1.3333e-10 k=12:1.3333e-12 k=16:1.3333e-16 k=20:1.3333e-20 k=24:1.3333e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = 1.3333e-24
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.263888888889  c0=(m2-1)/2+(n/4)Gamma=1.29166666666667   s*=max over V=1.29166666666667  (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 : 0.052814335 | -0.00527818 | -0.0531851 | tau_abs(0)=inf, tau_abs(p)=33.766795
     3 : 0.0051785945 | -0.000361609 | -0.00551231 | tau_abs(0)=inf, tau_abs(p)=303.99619
     4 : 0.00051678637 | -3.45014e-5 | -0.000553247 | tau_abs(0)=inf, tau_abs(p)=3004.0206
     5 : 5.1667864e-5 | -3.43356e-6 | -5.5345e-5 | tau_abs(0)=inf, tau_abs(p)=30004.023
     6 : 5.1666786e-6 | -3.4319e-7 | -5.5347e-6 | tau_abs(0)=inf, tau_abs(p)=300004.02
     8 : 5.1666668e-8 | -3.43171e-9 | -5.53472e-8 | tau_abs(0)=inf, tau_abs(p)=30000004.0
    10 : 5.1666667e-10 | -3.43171e-11 | -5.53472e-10 | tau_abs(0)=inf, tau_abs(p)=3.0e+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   (0.4s)
====================================================================================================
GRAPH Z_13{+-1,5}   n=13 d=4 diam=2 bipartite=False
 lambda_2=0.344300713493 (mult m2=4)  lambda_n=-0.662773352234   tau(0)=1.52508935205   tau_abs(0)=2.96536470835
 T3 on 45 values of p: min tau/boundA = 1.00013744168324 at p=9999/10000 ; min tau/boundB = 1.88719165535555 at p=1/100 ; violations=0
 shape on grid: up-then-down ; max tau = 4.42798044773 at p=3/4 ; max/tau(0) = 2.9034236 ; max/d = 1.1069951
 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=1:0.29233 k=2:0.042663 k=3:0.0044752 k=4:0.00044975 k=6:4.5e-6 k=8:4.5e-8 k=10:4.5e-10 k=12:4.5e-12 k=16:4.5e-16 k=20:4.5e-20 k=24:4.5e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = 4.5e-24
 T4: m2=4  n*f0(e)^2=4.0  Gamma(f0)=0.435394024921  c0=(m2-1)/2+(n/4)Gamma=2.91503058099355   s*=max over V=2.91503058099355  (s*-c0=-1.5558e-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 : 0.072344445 | 0.0544947 | -0.141218 | -0.045113056
     3 : 0.0068293779 | 0.00811289 | -0.0147576 | -0.0047156664
     4 : 0.00067850292 | 0.000838164 | -0.00148244 | -0.00047371239
     5 : 6.7805595e-5 | 8.40848e-5 | -0.000148311 | -4.7392804e-5
     6 : 6.7801121e-6 | 8.41116e-6 | -1.48318e-5 | -4.7394961e-6
     8 : 6.7800629e-8 | 8.41146e-8 | -1.48318e-7 | -4.7395199e-8
    10 : 6.7800624e-10 | 8.41146e-10 | -1.48318e-9 | -4.7395201e-10
    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   (0.6s)
====================================================================================================
GRAPH D_7 {r,r^-1,s}   n=14 d=3 diam=4 bipartite=False
 lambda_2=0.748993201239 (mult m2=2)  lambda_n=-0.933979245268   tau(0)=3.98395583282   tau_abs(0)=15.1467520186
 T3 on 45 values of p: min tau/boundA = 1.00822373270283 at p=9999/10000 ; min tau/boundB = 1.51937931439225 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 5.54762501061 at p=11/40 ; max/tau(0) = 1.3924916 ; max/d = 1.8492083
 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=1:0.86753 k=2:0.25297 k=3:0.078225 k=4:0.02457 k=6:0.0024502 k=8:0.00024496 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)=0.189146099365  c0=(m2-1)/2+(n/4)Gamma=1.16201134777643   s*=max over V=1.16201134777643  (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 : 0.1756646 | -0.101987 | -0.107483 | -2.9091517
     3 : 0.018354176 | -0.0109204 | -0.0116129 | -0.3696022
     4 : 0.0018434404 | -0.00109972 | -0.00117071 | -0.037993676
     5 : 0.0001844244 | -0.000110049 | -0.000117166 | -0.0038100272
     6 : 1.8443243e-5 | -1.10057e-5 | -1.17176e-5 | -0.00038110965
     8 : 1.8443332e-7 | -1.10058e-7 | -1.17177e-7 | -3.8112142e-6
    10 : 1.8443333e-9 | -1.10058e-9 | -1.17177e-9 | -3.8112154e-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   (0.8s)
====================================================================================================
GRAPH Petersen (vertex-transitive, not Cayley)   n=10 d=3 diam=2 bipartite=False
 lambda_2=0.333333333333 (mult m2=5)  lambda_n=-0.666666666667   tau(0)=1.5   tau_abs(0)=3.0
 T3 on 45 values of p: min tau/boundA = 1.00818137397675 at p=9999/10000 ; min tau/boundB = 1.66685007043147 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 3.85606508734 at p=29/40 ; max/tau(0) = 2.5707101 ; max/d = 1.285355
 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=1:0.66665 k=2:0.23829 k=3:0.076909 k=4:0.024443 k=6:0.002449 k=8:0.00024494 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=5  n*f0(e)^2=5.0  Gamma(f0)=0.666666666667  c0=(m2-1)/2+(n/4)Gamma=3.66666666666667   s*=max over V=3.66666666666667  (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 : 0.081909235 | -0.214752 | -0.231408 | -2.8003569e-60
     3 : 0.0082447897 | -0.0223468 | -0.0245051 | -4.3561107e-60
     4 : 0.00082494866 | -0.00224347 | -0.00246504 | -8.4010706e-60
     5 : 8.2499487e-5 | -0.000224435 | -0.00024665 | -4.3561107e-60
     6 : 8.2499949e-6 | -2.24443e-5 | -2.46665e-5 | -2.1780553e-60
     8 : 8.2499999e-8 | -2.24444e-7 | -2.46667e-7 | -7.7787691e-60
    10 : 8.25e-10 | -2.24444e-9 | -2.46667e-9 | -2.8003569e-60
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [0, 0, 0, 0, 0, 0, 0]
 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)

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_4,adjacent transpositions        |  24 |  3 | B |  3 | 5.1213203 | up-then-down | 1.48033 | 1.6276283 | 1.6289402 | 1.0082622 | 1.1822111 | 2.45e-12 | 0.5 | OK | [None, None, None, None, None, None, None]
 D_4 {r,r^-1,s,sr}                  |   8 |  4 | - |  2 | 1.5469182 | increasing   | 2.58572 | 1.465337 | 1.465337 | 1.0000375 | 1.767949 | -1.0e-24 | 1.0 | OK | [-1, -1, -1, -1, -1, -1, -1]
 D_5 {r,r^-1,s,sr}                  |  10 |  4 | - |  2 | 2.2676611 | up-then-down | 1.77467 | 1.3063697 | 1.3063697 | 1.0000688 | 1.744528 | 2.5e-25 | 1.0 | OK | [1, 1, 1, 1, 1, 1, 1]
 Z_2xZ_4 {(1,0),(0,+-1),(1,2)}      |   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]
 Q8 {+-i,+-j}                       |   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]
 A_4 {(012)^+-,(013)^+-}            |  12 |  4 | - |  3 | 2.0 | up-then-down | 2.1613 | 2.0 | 2.0 | 1.000125 | 1.7350585 | 3.0e-24 | 1.0 | OK | [1, 1, 1, 1, 1, 1, 1]
 S_4 {(012)^+-,(23)}                |  24 |  3 | - |  3 | 6.8423292 | up-then-down | 1.41831 | 1.5746437 | 1.5746437 | 1.0106918 | 1.5845374 | 3.17e-12 | 0.5 | OK | [1, 1, 1, 1, 1, 1, 1]
 Z_3xZ_3 {(+-1,0),(0,+-1)}          |   9 |  4 | - |  4 | 1.3333333 | increasing   | 3.0 | 3.0 | 3.0 | 1.00005 | 1.5002001 | -3.0e-48 | 2.0 | OK | [0, 0, 0, 0, 0, 0, 0]
 Z_9{+-1,2}                         |   9 |  4 | - |  2 | 1.8862455 | increasing   | 2.12061 | 1.3792299 | 1.3792299 | 1.0000625 | 2.121909 | -2.41e-48 | 2.0 | OK | [-1, -1, -1, -1, -1, -1, -1]
 Z_12{+-1,5}                        |  12 |  4 | B |  2 | 2.0 | up-then-down | 2.08163 | 1.2916667 | 1.2916667 | 1.0000583 | 1.38462 | 1.33e-24 | 1.0 | OK | [None, None, None, None, None, None, None]
 Z_13{+-1,5}                        |  13 |  4 | - |  4 | 1.5250894 | up-then-down | 2.90342 | 2.9150306 | 2.9150306 | 1.0001374 | 1.8871917 | 4.5e-24 | 1.0 | OK | [-1, -1, -1, -1, -1, -1, -1]
 D_7 {r,r^-1,s}                     |  14 |  3 | - |  2 | 3.9839558 | up-then-down | 1.39249 | 1.1620113 | 1.1620113 | 1.0082237 | 1.5193793 | 2.45e-12 | 0.5 | OK | [-1, -1, -1, -1, -1, -1, -1]
 Petersen (vertex-transitive, not Cayley) |  10 |  3 | - |  5 | 1.5 | up-then-down | 2.57071 | 3.6666667 | 3.6666667 | 1.0081814 | 1.6668501 | 2.45e-12 | 0.5 | OK | [0, 0, 0, 0, 0, 0, 0]
