Verification run 2 -- F: the increase at p = 0 (Section 6)
PASS  Lemma 6.1: the bound with the constant 2a^2/gamma + C_0 holds in 7200 random cases (N = 2..8, multiplicity 1..N, 6 values of p up to gamma/(4a)); largest left side / right side = 1.0000
   worst case (N, k, multiplicity, gamma, a, C_0, p/p_max): (6, 1, 5, 2.788091002470387, 0.010248699260031727, 6.397786301257022, 0.7)
PASS  Lemma 6.1: 2x2 example A_0 = diag(0, gamma), A_1 = [[0,a],[a,0]] (nu = 0): bound holds, left/right up to 0.500
   illustration: A_0 = diag(0,0,1), A_1 = diag(1,-1,0): sigma_1(A(p)) = -0.0100 = sigma + p nu with nu = -1; sigma_2(A(p)) = +0.0100 (index 2 is not covered by the lemma: sigma_1 = sigma_2)
PASS  Lemma 6.2(d): <f, S_1 f> = -(n f(e)^2 - sum f^2)/2 - (n/4) Gamma(f) for arbitrary f   (exact, 40 graphs, 3 random rational f each)
PASS  Lemma 6.2(c): ||S(p) - (I-K) - p S_1|| / p^2 stays bounded as p -> 0 (p = 1e-2, 1e-3, 1e-4, 1e-6; 50 digits)
PASS  Theorem 1.6: n f_0(e)^2 = m_2, Gamma(f_0) > 0, Q(f_0) > (m_2-1)/2 and s* >= Q(f_0) on all graphs
PASS  Theorem 1.6: (1/tau(0) - 1/tau(p))/p agrees with s* to 1e-4 at p = 1e-8 and to 1e-8 at p = 1e-12 on all graphs
PASS  Theorem 1.6: (tau(p) - tau(0))/p > 0 at p = 1e-4, 1e-8, 1e-12 and tends to s* tau(0)^2
PASS  Lemma 6.1 applied to S(p) (k = 2) with the measured remainder constant: |sigma_2(S(p)) - 1/tau(0) + s* p| <= (2a^2/gamma + C_0) p^2
   name | n | d | m_2 | tau(0) | Q(f_0) | s* | diff. quotient at 1e-8 minus s* | ||remainder||/p^2 at 1e-6 | Lemma 6.1 left/right
   K_2 | 2 | 1 | 1 | 0.5 | 1.0 | 1.0 | 4.52e-43 | 0.125 | 9.63e-48
   K_3 | 3 | 2 | 2 | 0.66666667 | 2.0 | 2.0 | -2.0e-8 | 2.061 | 0.229
   K_4 | 4 | 3 | 3 | 0.75 | 3.0 | 3.0 | -6.0e-8 | 6.189 | 0.271
   K_5 | 5 | 4 | 4 | 0.8 | 4.0 | 4.0 | -1.2e-7 | 12.32 | 0.286
   K_6 | 6 | 5 | 5 | 0.83333333 | 5.0 | 5.0 | -2.0e-7 | 20.46 | 0.294
   K_7 | 7 | 6 | 6 | 0.85714286 | 6.0 | 6.0 | -3.0e-7 | 30.6 | 0.298
   C_3 | 3 | 2 | 2 | 0.66666667 | 2.0 | 2.0 | -2.0e-8 | 2.061 | 0.229
   C_4 | 4 | 2 | 2 | 1.0 | 1.5 | 1.5 | -2.0e-8 | 7.382 | 0.0659
   C_5 | 5 | 2 | 2 | 1.4472136 | 1.38196601 | 1.38196601 | -3.8e-8 | 17.12 | 0.0484
   C_6 | 6 | 2 | 2 | 2.0 | 1.25 | 1.25 | -5.48e-8 | 32.44 | 0.0324
   C_7 | 7 | 2 | 2 | 2.6559706 | 1.16201135 | 1.16201135 | -7.42e-8 | 54.53 | 0.0229
   C_8 | 8 | 2 | 2 | 3.4142136 | 1.08578644 | 1.08578644 | -9.49e-8 | 84.58 | 0.0176
   C_9 | 9 | 2 | 2 | 4.2743161 | 1.02726092 | 1.02726092 | -1.18e-7 | 123.8 | 0.0135
   C_10 | 10 | 2 | 2 | 5.236068 | 0.977457514 | 0.977457514 | -1.43e-7 | 173.3 | 0.0106
   C_11 | 11 | 2 | 2 | 6.2993528 | 0.936863843 | 0.936863843 | -1.7e-7 | 234.4 | 0.00852
   C_12 | 12 | 2 | 2 | 7.4641016 | 0.901923789 | 0.901923789 | -1.98e-7 | 308.1 | 0.00698
   C_13 | 13 | 2 | 2 | 8.7302715 | 0.872401738 | 0.872401738 | -2.29e-7 | 395.9 | 0.00581
   Q_2 | 4 | 2 | 2 | 1.0 | 1.5 | 1.5 | -2.0e-8 | 7.382 | 0.0659
   Q_3 | 8 | 3 | 3 | 1.5 | 2.0 | 2.0 | -4.83e-8 | 53.42 | 0.0218
   Q_4 | 16 | 4 | 4 | 2.0 | 2.5 | 2.5 | -6.44e-8 | 269.6 | 0.00504
   Q_5 | 32 | 5 | 5 | 2.5 | 3.0 | 3.0 | -2.55e-8 | 1188.0 | 0.000456
   S_3 all transpositions | 6 | 3 | 4 | 1.0 | 3.33333333 | 3.33333333 | -1.02e-7 | 22.9 | 0.124
   S_4 all transpositions | 24 | 6 | 9 | 1.5 | 6.625 | 6.625 | 3.37e-8 | 584.9 | 0.00038
   S_4 adjacent transpositions | 24 | 3 | 3 | 5.1213203 | 1.62762833 | 1.62894024 | -2.31e-7 | 904.5 | 0.00306
   A_4 {(012)^+-,(013)^+-} | 12 | 4 | 3 | 2.0 | 2.0 | 2.0 | -6.69e-8 | 142.2 | 0.01
   S_4 {(0123)^+-,(01)} | 24 | 3 | 3 | 5.1213203 | 1.62762833 | 1.62894024 | -2.31e-7 | 904.5 | 0.00306
   D_5 {r,r^-1,s} | 10 | 3 | 2 | 2.1708204 | 1.38196601 | 1.38196601 | -4.03e-8 | 99.31 | 0.00386
   D_6 {s,sr,sr^3} | 12 | 3 | 2 | 2.3660254 | 1.50552027 | 1.52793291 | -3.98e-8 | 155.9 | 0.00296
   D_7 {r,r^-1,s} | 14 | 3 | 2 | 3.9839558 | 1.16201135 | 1.16201135 | -1.1e-7 | 254.1 | 0.00581
   Z_3 x Z_3 {(+-1,0),(0,+-1)} | 9 | 4 | 4 | 1.3333333 | 3.0 | 3.0 | -8.33e-8 | 66.21 | 0.0334
   Z_2 x Z_4 {(1,0),(0,+-1)} | 8 | 3 | 3 | 1.5 | 2.0 | 2.0 | -4.83e-8 | 53.42 | 0.0218
   Z_2 x Z_6 {(1,0),(0,+-1),(1,3)} | 12 | 4 | 6 | 1.3333333 | 4.56613757 | 4.56741649 | -1.41e-7 | 131.2 | 0.0116
   Z_9 {+-1,+-2} | 9 | 4 | 2 | 1.8862455 | 1.37922986 | 1.37922986 | -3.45e-8 | 69.97 | 0.0101
   Z_10 {+-1,5} | 10 | 3 | 2 | 2.1708204 | 1.38196601 | 1.38196601 | -4.18e-8 | 99.05 | 0.00614
   Z_12 {+-1,+-5} | 12 | 4 | 2 | 2.0 | 1.29166667 | 1.29166667 | -3.43e-9 | 135.8 | 0.000783
   Z_13 {+-1,+-5} | 13 | 4 | 4 | 1.5250894 | 2.91503058 | 2.91503058 | 8.41e-8 | 160.0 | 0.006
   Z_15 {+-1,+-4} | 15 | 4 | 4 | 1.6792851 | 2.76535914 | 2.76535914 | 2.74e-7 | 226.9 | 0.00537
   Z_16 {+-1,+-3} | 16 | 4 | 2 | 2.884184 | 1.29809586 | 1.29809586 | -5.92e-8 | 280.3 | 0.00335
   Z_16 {+-3,+-5,8} | 16 | 5 | 2 | 4.267767 | 1.08578644 | 1.08578644 | -1.15e-7 | 311.9 | 0.00467
   Petersen (vertex-transitive, not Cayley) | 10 | 3 | 5 | 1.5 | 3.66666667 | 3.66666667 | -2.24e-7 | 94.48 | 0.0568
PASS  Remark 6.3(b): S_4 with adjacent transpositions: Q(f_0) = 1.62763 (rounded)
PASS  Remark 6.3(b): S_4 with adjacent transpositions: s* = 1.62894
PASS  Remark 6.3(b): s* = n - 1 on K_n (n = 2..7) from the formula (6)
PASS  Remark 6.3(b): on Q_d, d = 2..8, s* = Q(f_0) = (d+1)/2   (double precision): 1.5000000000, 2.0000000000, 2.5000000000, 3.0000000000, 3.5000000000, 4.0000000000, 4.5000000000
PASS  Remark 7.2(c): on C_5, ..., C_13 (odd), tau_abs(0) = 1/(1 - cos(pi/n)) and tau_abs(p) < tau_abs(0) at p = 1e-2, 1e-4, 1e-6, while tau(p) > tau(0)   [C_5: tau_abs(0) = 5.236068, tau_abs(0.01) = 5.047196; C_7: tau_abs(0) = 10.09783, tau_abs(0.01) = 8.891446; C_9: tau_abs(0) = 16.58172, tau_abs(0.01) = 12.59915; C_11: tau_abs(0) = 24.68708, tau_abs(0.01) = 15.48791; C_13: tau_abs(0) = 34.41371, tau_abs(0.01) = 17.39979]
PASS  Remark 7.2(c): on the graphs with |lambda_n| < lambda_2, tau_abs = tau near p = 0 and tau_abs(1e-6) > tau_abs(0)

checks passed: 14   TOTAL failures: 0
