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GRAPH Q_6   n=64 d=6 diam=6 bipartite=True
 lambda_2=0.666666666667 (mult m2=6)  lambda_n=-1.0   tau(0)=3.0   tau_abs(0)=inf
 T3 on 46 values of p: min tau/boundA = 1.0000833347222 at p=9999/10000 ; min tau/boundB = 1.00030189211073 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 7.7474947443 at p=3/8 ; max/tau(0) = 2.5824982 ; max/d = 1.2912491
 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=1:0.39227 k=2:0.039932 k=3:0.0039993 k=4:0.00039999 k=6:4.0e-6 k=8:4.0e-8 k=10:4.0e-10 k=12:4.0e-12 k=16:4.0e-16 k=20:4.0e-20 k=24:4.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = 4.0e-24
 T4: m2=6  n*f0(e)^2=6.0  Gamma(f0)=0.0625  c0=(m2-1)/2+(n/4)Gamma=3.5   s*=max over V=3.5  (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 : 0.3544971 | 0.0226055 | -0.418991 | tau_abs(0)=inf, tau_abs(p)=24.459535
     3 : 0.031958204 | 0.0134833 | -0.0468734 | tau_abs(0)=inf, tau_abs(p)=178.19874
     4 : 0.0031546232 | 0.00145499 | -0.00474366 | tau_abs(0)=inf, tau_abs(p)=1678.9059
     5 : 0.00031504627 | 0.00014655 | -0.000474937 | tau_abs(0)=inf, tau_abs(p)=16678.983
     6 : 3.1500463e-5 | 1.46655e-5 | -4.74994e-5 | tau_abs(0)=inf, tau_abs(p)=166678.99
     8 : 3.1500005e-7 | 1.46667e-7 | -4.75e-7 | tau_abs(0)=inf, tau_abs(p)=16666679.0
    10 : 3.15e-9 | 1.46667e-9 | -4.75e-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   (34.8s)

SUMMARY (name | n | d | bip | m2 | tau(0) | shape | max/tau0 | c0 | s* | minA | minB | tau(1-1e-24)-d | exponent | status | abs-sign)
 Q_6                                |  64 |  6 | B |  6 | 3.0 | up-then-down | 2.5825 | 3.5 | 3.5 | 1.0000833 | 1.0003019 | 4.0e-24 | 1.0 | OK | [None, None, None, None, None, None, None]
