Verification run 2 -- D: the two lower bounds and the hypercube
PASS  Theorem 1.3, proof: E_p(1_e) = (m-p)/q, m > p, and E_p(|x|) <= 1/2   (exact)
PASS  Theorem 1.3(a): tau(p) >= (1-p) m (1-m)/(m-p)   (40 graphs x 25 values of p; smallest ratio 1.0)
PASS  Theorem 1.3(b): tau(p) >= 2 Var|X(T_p - 1)|   (smallest ratio 1.000050005)
PASS  Proposition 5.1(a): mu_p depends on |x| only; w strictly decreasing and w(k+1)^2 <= w(k) w(k+2); Fourier coefficients a_j = pd/(pd+2jq)   (exact, d = 2..7)
PASS  Proposition 5.1(b): mean and variance of |x| under mu_p   (exact)
PASS  Proposition 5.1(c): E_p(|x|,|x|) = E_p|x| / d   (exact)
PASS  Corollary 1.4: Var/E = H_d(p) exactly, and tau(p) > H_d(p) (d <= 6, 40 digits)
PASS  Proposition 5.1(a): the integral representation of w(k)  (numerical quadrature; largest deviation 3.59e-43)
PASS  Corollary 1.4(c): identity (11) for H_d(p) - d   (symbolic)
PASS  Remark 6.3(b): 1/H_d(p) = 2/d - ((d+1)/2) p + O(p^2)   (symbolic)
PASS  H_d(0+) = d/2 and H_d(1-) = d   (symbolic)
PASS  Corollary 1.4(a): H_d(1/d) >= d(d-1)/15   (exact, d = 2..199 and 500, 1e3, 1e4, 1e6)
PASS  Corollary 1.4(b): H_d(p) >= (1-p)d/(3p) whenever pd >= 4   (exact, same d)
PASS  Corollary 1.4(c): H_d(p) > d iff p > 4/(d^2-2d+4), and this threshold is < 1 iff d >= 3   (exact, same d)
PASS  d(d-1)/15 > d iff d >= 17
PASS  Proposition 5.2: relaxation time of the level chain = relaxation time of the full chain   (d = 2..6 with 40 digits, d = 7, 8 in double precision)
   d, tau(1/d), d(d-1)/15:
   2  1.6809466  0.1333
   3  2.6748799  0.4000
   4  3.8740875  0.8000
   5  5.2959135  1.3333
   6  6.9627794  2.0000
   7  8.8947303  2.8000
   8  11.107562  3.7333
   Q_8: tau(0) = 4, max over the grid k/400 is 11.992773 at p = 0.2625
PASS  Figure 1(b): on Q_8 the maximum of tau is 11.99 near p = 0.26
   d, tau(1/d)/d^2, max over p*d in {0.05,0.1,...} of tau/d^2 (level chain, double precision):
   24  0.1538  0.1538   (H_d(1/d)/d^2 = 0.0931, 1/15 = 0.0667)
   32  0.1540  0.1558   (H_d(1/d)/d^2 = 0.0865, 1/15 = 0.0667)
   48  0.1549  0.1610   (H_d(1/d)/d^2 = 0.0799, 1/15 = 0.0667)
   64  0.1556  0.1652   (H_d(1/d)/d^2 = 0.0766, 1/15 = 0.0667)
   96  0.1566  0.1710   (H_d(1/d)/d^2 = 0.0733, 1/15 = 0.0667)
   128  0.1572  0.1748   (H_d(1/d)/d^2 = 0.0716, 1/15 = 0.0667)
   192  0.1578  0.1791   (H_d(1/d)/d^2 = 0.0700, 1/15 = 0.0667)
   256  0.1582  0.1819   (H_d(1/d)/d^2 = 0.0691, 1/15 = 0.0667)
   384  0.1586  0.1847   (H_d(1/d)/d^2 = 0.0683, 1/15 = 0.0667)
   512  0.1588  0.1861   (H_d(1/d)/d^2 = 0.0679, 1/15 = 0.0667)
PASS  Remark 7.3: for d = 24..512, max tau/d^2 lies in [0.15, 0.19] and tau(1/d) is about 0.16 d^2

checks passed: 18   TOTAL failures: 0
