check_note.py -- OWR-11129-006 (exact program of the note)

== A. constants of Lemma 4.1 and of the proofs of Theorems 1.1 and 1.2 ==
  GL,GU: c=1.0 theta=(0.3333,0.6667,0.3333) Sigma_H=57.6286 g(0)=2.0639 g'(0)=-0.1139 K=48.3714 mode of H=14.463
    dyadic suprema of H (j=0..9): 3.845, 7.000, 11.157, 13.189, 13.108, 8.021, 1.294, 0.014, 0.000, 0.000
PASS  GL,GU: g'(0) < 0 (so g is decreasing)  [-0.11392]
PASS  GL,GU: values as printed in the note
PASS  GL,GU: dense sampling never exceeds the dyadic suprema  [0.00e+00]
PASS  GL,GU: d/dy log H decreasing on a grid
  n=2m (Sp, SO+-, GSp, GO0): c=0.5 theta=(0.3333,0.6667,0.3333) Sigma_H=116.0403 g(0)=1.8956 g'(0)=-0.0219 K=37.4955 mode of H=29.343
    dyadic suprema of H (j=0..9): 3.723, 7.234, 13.358, 21.623, 25.862, 25.743, 15.892, 2.576, 0.029, 0.000
PASS  n=2m (Sp, SO+-, GSp, GO0): g'(0) < 0 (so g is decreasing)  [-0.02185]
PASS  n=2m (Sp, SO+-, GSp, GO0): values as printed in the note
PASS  n=2m (Sp, SO+-, GSp, GO0): dense sampling never exceeds the dyadic suprema  [0.00e+00]
PASS  n=2m (Sp, SO+-, GSp, GO0): d/dy log H decreasing on a grid
  n=2m+1 (SO, GO odd): c=0.5 theta=(0.3333,0.7500,0.2500) Sigma_H=121.4112 g(0)=2.0406 g'(0)=-0.0376 K=40.6163 mode of H=29.073
    dyadic suprema of H (j=0..9): 3.984, 7.705, 14.135, 22.699, 26.961, 26.809, 16.439, 2.651, 0.029, 0.000
PASS  n=2m+1 (SO, GO odd): g'(0) < 0 (so g is decreasing)  [-0.03759]
PASS  n=2m+1 (SO, GO odd): values as printed in the note
PASS  n=2m+1 (SO, GO odd): dense sampling never exceeds the dyadic suprema  [0.00e+00]
PASS  n=2m+1 (SO, GO odd): d/dy log H decreasing on a grid
PASS  max over n>1 of (n/2)^(-1/4) log n = 4*2^(1/4)/e <= 1.7500  [1.749939]
PASS  Theorem 1.1: 1.75 + 2K < 100  [98.493]
PASS  Theorem 1.1: 100/log 8 > 1 (the cases 2 <= n <= 8)
PASS  Theorem 1.2, Sp_2m: M(e^45) < 45, with 2^(k0+1)/c <= 45 and log rho < 45  [M = 39.253]
PASS  Theorem 1.2, GSp_2m: M(e^45) < 45, with 2^(k0+1)/c <= 45 and log rho < 45  [M = 42.207]
PASS  Theorem 1.2, SO_2m+1, GO_2m+1: M(e^45) < 45, with 2^(k0+1)/c <= 45 and log rho < 45  [M = 42.904]
PASS  Theorem 1.2, SO+-_2m: M(e^80) < 80, with 2^(k0+1)/c <= 80 and log rho < 80  [M = 76.718]
PASS  Theorem 1.2, GO+-_2m(q)^0: M(e^80) < 80, with 2^(k0+1)/c <= 80 and log rho < 80  [M = 77.968]
PASS  remark: GL, GU by the majorant: M(e^50) < 50  [49.458]
PASS  lower bound: 2^(-2(1+1/log 100)) > 1/8  [0.1850]
PASS  lower bound: R >= 100/(6 log 100) >= 3.6 and 1/4 - 1/(2*3.6) >= 0.111  [R = 3.6191]
PASS  lower bound: 0.111/32 > 1/300
PASS  lower bound: ceil(log2 log 100) = 3

== B. the function Phi of Theorem 1.4 ==
  grid of 20000 points: mean = 1.000000000000000, max-1 = 9.884448e-06, 1-min = 9.884435e-06
  |c_1| = 4.942221e-06 (formula), 4.942221e-06 (discrete Fourier sum); |c_2| = 4.5771e-12
  2 * sum_(j>=1) |c_j| = 9.884451e-06
PASS  Phi: mean over a period = 1
PASS  Phi: |c_1| formula = discrete Fourier coefficient
PASS  Phi: |Phi - 1| <= 2 sum |c_j| < 1e-5 (and the grid agrees)
PASS  Phi: not constant (|c_1| > 0; grid max - min > 1.9e-5)
PASS  Phi: values as printed (max-1, 1-min = 9.884e-6)

== C. the inequalities (9) of Section 6 on a finite range ==
PASS  (9), first: |sum_(mu<=M) (1-2^-v2(mu))/mu - (1/3) log M| <= 2 for M <= 3e5  [max 0.1278]
PASS  (9), second: Y |sum_(mu in (Y,2Y]) 2^-(v2(mu)+1)/mu - (log 2)/3| <= 5 for 4 <= Y <= 1.5e5  [max 0.2500]

== D. the torus model in exact arithmetic ==
PASS  valuations (2): a(j), aU(j), b(j) (12 values of q, j <= 24)
PASS  Lemma 2.3 (law of v2 of the order in a cyclic group), N < 400
PASS  weights of signed cycle types sum to 1, and to 1/2 over each parity (m <= 8)
PASS  (6): E[u^N_k] = prod_{i<=N}(1-s/i) <= (N+1)^-s, m <= 20
PASS  (7): A_m(v) >= exp(-2 log 2 (1 + log+(m/2^k))/2^k), k <= 5, m <= 80
PASS  P(no odd cycle) = binom(m,m/2)/2^m <= m^(-1/2)
PASS  GL_n(q), GU_n(q): sum over cycle types = witness formula, level by level (74 cases)
PASS  Sp, SO_2m+1, SO+-_2m: signed sum = witness formula, level by level (56 cases)
PASS  Lemma 2.5: conformal model = base model at the levels v >= a1+1 (GSp, GO0), v >= a1 (GO odd); GSp_2 = GL_2 (40 cases)
PASS  Lemma 2.5: P(M <= a1) <= E[(3/4)^N_2] for GSp_2m(q)

== D5. levels of GL_n(q) and Sp_2m(q) in floating point: bounds of Section 4, values of Table 1 ==
PASS  GL_100(3): (log n)P = 0.8877445; low levels <= (n/2+1)^(-1/4); C_v <= T_k for every k >= 1; P >= 1/(300 log n)
PASS  GL_1000(3): (log n)P = 1.0152381; low levels <= (n/2+1)^(-1/4); C_v <= T_k for every k >= 1; P >= 1/(300 log n)
PASS  P(GL_1000(3)) = 0.1470 (about 15% in [LNP])  [0.146971]
PASS  GL_100(5): (log n)P = 0.8988311; low levels <= (n/2+1)^(-1/4); C_v <= T_k for every k >= 1; P >= 1/(300 log n)
PASS  GL_1000(5): (log n)P = 1.0325923; low levels <= (n/2+1)^(-1/4); C_v <= T_k for every k >= 1; P >= 1/(300 log n)
PASS  Sp_200(3): (log 200)P = 0.9456048 (Table 1: 0.9456048)

== E. brute force over small groups, and the recorded brute-force values ==
PASS  model = recorded brute-force value for all 35 recorded groups
PASS  brute force GL n=2 q=3: |G| = 48 (formula 48), |Q|/|G| = 1/4, model 1/4  [0.0s]
PASS  brute force GL n=3 q=3: |G| = 11232 (formula 11232), |Q|/|G| = 15/32, model 15/32  [0.1s]
PASS  brute force GL n=2 q=5: |G| = 480 (formula 480), |Q|/|G| = 5/16, model 5/16  [0.0s]
PASS  brute force GL n=2 q=7: |G| = 2016 (formula 2016), |Q|/|G| = 1/4, model 1/4  [0.0s]
PASS  brute force Sp n=2 q=5: |G| = 120 (formula 120), |Q|/|G| = 0, model 0  [0.0s]
PASS  brute force Sp n=4 q=3: |G| = 51840 (formula 51840), |Q|/|G| = 21/64, model 21/64  [1.0s]
PASS  brute force GSp n=2 q=5: |G| = 480 (formula 480), |Q|/|G| = 5/16, model 5/16  [0.0s]
PASS  brute force SOodd n=3 q=3: |G| = 24 (formula 24), |Q|/|G| = 5/8, model 5/8  [0.0s]
PASS  brute force SOodd n=3 q=5: |G| = 120 (formula 120), |Q|/|G| = 5/8, model 5/8  [0.0s]
PASS  brute force SOodd n=3 q=7: |G| = 336 (formula 336), |Q|/|G| = 11/16, model 11/16  [0.1s]
PASS  brute force SOodd n=5 q=3: |G| = 51840 (formula 51840), |Q|/|G| = 21/64, model 21/64  [1.4s]
PASS  brute force SO+ n=2 q=5: |G| = 4 (formula 4), |Q|/|G| = 0, model 0  [0.0s]
PASS  brute force SO- n=2 q=5: |G| = 6 (formula 6), |Q|/|G| = 0, model 0  [0.0s]
PASS  brute force SO+ n=4 q=3: |G| = 576 (formula 576), |Q|/|G| = 9/32, model 9/32  [0.0s]
PASS  brute force SO- n=4 q=3: |G| = 720 (formula 720), |Q|/|G| = 3/8, model 3/8  [0.0s]
PASS  brute force GOodd n=3 q=3: |G| = 48 (formula 48), |Q|/|G| = 7/16, model 7/16  [0.0s]
PASS  brute force GOodd n=3 q=5: |G| = 480 (formula 480), |Q|/|G| = 7/32, model 7/32  [0.0s]
PASS  brute force GO+0 n=2 q=5: |G| = 16 (formula 16), |Q|/|G| = 5/8, model 5/8  [0.0s]
PASS  brute force GO-0 n=2 q=5: |G| = 24 (formula 24), |Q|/|G| = 0, model 0  [0.0s]
PASS  brute force GO+0 n=4 q=3: |G| = 1152 (formula 1152), |Q|/|G| = 21/64, model 21/64  [0.0s]
PASS  brute force GO-0 n=4 q=3: |G| = 1440 (formula 1440), |Q|/|G| = 7/16, model 7/16  [0.0s]
PASS  brute force GSp n=4 q=3: |G| = 103680 (formula 103680), |Q|/|G| = 49/128, model 49/128  [1.9s]
PASS  brute force GOodd n=5 q=3: |G| = 103680 (formula 103680), |Q|/|G| = 21/64, model 21/64  [2.8s]
PASS  brute force SO+ n=4 q=5: |G| = 14400 (formula 14400), |Q|/|G| = 9/32, model 9/32  [1.6s]
PASS  brute force SO- n=4 q=5: |G| = 15600 (formula 15600), |Q|/|G| = 3/8, model 3/8  [1.7s]
PASS  brute force GO+0 n=4 q=5: |G| = 57600 (formula 57600), |Q|/|G| = 51/128, model 51/128  [6.0s]
PASS  brute force GO-0 n=4 q=5: |G| = 62400 (formula 62400), |Q|/|G| = 13/32, model 13/32  [7.1s]
PASS  brute force GU n=2 q=3: |G| = 96 (formula 96), |Q|/|G| = 5/16, model 5/16
PASS  brute force GU n=2 q=5: |G| = 720 (formula 720), |Q|/|G| = 1/4, model 1/4

TOTAL failures: 0
running time 27 s
