Part A: gap-ratio constants r_s and conductance constants K_s
  s = 1.05: r_s = 2.58367936803175479492  (residual 1.0e-59),  K_s = s(1+r_s)^(s+1) = 14.373550
  s = 1.25: r_s = 2.16209479573722235404  (residual 1.0e-59),  K_s = s(1+r_s)^(s+1) = 16.666849
  s = 1.5: r_s = 1.85354768338729305401  (residual 1.0e-59),  K_s = s(1+r_s)^(s+1) = 20.632606
  s = 1.75: r_s = 1.66439896487700603420  (residual 1.0e-59),  K_s = s(1+r_s)^(s+1) = 25.908126
  s = 2: r_s = 1.53861576354917625747  (residual 1.0e-59),  K_s = s(1+r_s)^(s+1) = 32.720574,  (1+r_2)(2+r_2) = 8.98318575848954
Part B: shift identity (Lemma 2.3), finite-range form, 60-digit arithmetic
  s = 1.25: 202 identities, max relative error 1.3e-59; symmetry/positivity failures = 0
  s = 1.5: 179 identities, max relative error 1.4e-59; symmetry/positivity failures = 0
  s = 1.75: 200 identities, max relative error 9.5e-60; symmetry/positivity failures = 0
  s = 2: 196 identities, max relative error 7.7e-60; symmetry/positivity failures = 0
Part C: inequality behind Lemma 2.2 on arbitrary finite configurations
  s = 1.25: 247 checks, failures = 0
  s = 1.5: 255 checks, failures = 0
  s = 1.75: 277 checks, failures = 0
  s = 2: 227 checks, failures = 0
Part D: Lemma 2.4 on adversarial ratio patterns (ratio bound r = r_s rounded down)
  s = 1.25, r = 2.162094: 2002 pairs; min min(D,D')(1+r)/D = 1.462515 (>= 1 needed); max c_s D^(s+1) = 3.0173 <= K_s = 16.6668: True
  s = 1.5, r = 1.853547: 2002 pairs; min min(D,D')(1+r)/D = 1.539506 (>= 1 needed); max c_s D^(s+1) = 3.3084 <= K_s = 20.6326: True
  s = 1.75, r = 1.664398: 2002 pairs; min min(D,D')(1+r)/D = 1.600818 (>= 1 needed); max c_s D^(s+1) = 3.6047 <= K_s = 25.9081: True
  s = 2, r = 1.538615: 2002 pairs; min min(D,D')(1+r)/D = 1.649935 (>= 1 needed); max c_s D^(s+1) = 3.9060 <= K_s = 32.7205: True
Part E: inequalities of Proposition 4.1 for s < 2 (floating point)
  s = 1.25, r = 2.162094, K_s = 16.6668
    profile       R     E*(f_R)    E*R^(s-1)  S1        S1-bound   S2        S2-bound   omitted<=  ok
    AP              10  3.119e+00    5.5462  1.176e+00  1.660e+01  1.319e+00  2.399e+02  6.9e+01   True
    AP              30  2.438e+00    5.7058  9.341e-01  1.261e+01  1.016e+00  1.823e+02  5.4e+01   True
    AP             100  1.831e+00    5.7912  7.089e-01  9.333e+00  7.562e-01  1.349e+02  4.0e+01   True
    alternating     10  3.133e+00    5.5707  1.101e+00  1.660e+01  1.331e+00  2.399e+02  7.5e+01   True
    alternating     30  2.440e+00    5.7096  8.958e-01  1.261e+01  1.025e+00  1.823e+02  5.3e+01   True
    alternating    100  1.832e+00    5.7928  6.967e-01  9.333e+00  7.599e-01  1.349e+02  4.0e+01   True
    dense           10  3.203e+00    5.6956  1.211e+00  1.660e+01  1.352e+00  2.399e+02  7.1e+01   True
    dense           30  2.480e+00    5.8048  9.588e-01  1.261e+01  1.026e+00  1.823e+02  5.4e+01   True
    dense          100  1.847e+00    5.8401  7.191e-01  9.333e+00  7.583e-01  1.349e+02  4.0e+01   True
    geometric       10  1.904e+00    3.3864  0.000e+00  1.660e+01  1.592e+00  2.399e+02  1.0e+02   True
    geometric       30  9.205e-01    2.1544  1.294e-01  1.261e+01  3.456e-01  1.823e+02  5.7e+01   True
    geometric      100  1.277e+00    4.0386  0.000e+00  9.333e+00  1.104e+00  1.349e+02  6.0e+01   True
    random_log      10  3.091e+00    5.4968  5.273e-02  1.660e+01  1.716e+00  2.399e+02  5.9e+01   True
    random_log      30  2.236e+00    5.2328  7.332e-01  1.261e+01  9.712e-01  1.823e+02  5.4e+01   True
    random_log     100  1.679e+00    5.3103  4.374e-01  9.333e+00  7.572e-01  1.349e+02  4.0e+01   True
  s = 1.5, r = 1.853547, K_s = 20.6326
    profile       R     E*(f_R)    E*R^(s-1)  S1        S1-bound   S2        S2-bound   omitted<=  ok
    AP              10  2.185e+00    6.9089  8.677e-01  1.263e+01  5.888e-01  9.445e+01  2.1e+01   True
    AP              30  1.356e+00    7.4295  5.577e-01  7.294e+00  3.466e-01  5.453e+01  1.2e+01   True
    AP             100  7.777e-01    7.7766  3.272e-01  3.995e+00  1.913e-01  2.987e+01  6.8e+00   True
    alternating     10  2.162e+00    6.8356  7.695e-01  1.263e+01  6.089e-01  9.445e+01  2.2e+01   True
    alternating     30  1.350e+00    7.3920  5.309e-01  7.294e+00  3.508e-01  5.453e+01  1.3e+01   True
    alternating    100  7.761e-01    7.7607  3.213e-01  3.995e+00  1.910e-01  2.987e+01  6.8e+00   True
    dense           10  2.336e+00    7.3862  9.498e-01  1.263e+01  6.078e-01  9.445e+01  2.2e+01   True
    dense           30  1.430e+00    7.8314  6.026e-01  7.294e+00  3.506e-01  5.453e+01  1.2e+01   True
    dense          100  8.050e-01    8.0499  3.447e-01  3.995e+00  1.919e-01  2.987e+01  6.8e+00   True
    geometric       10  1.172e+00    3.7075  7.925e-02  1.263e+01  7.363e-01  9.445e+01  2.0e+01   True
    geometric       30  6.697e-01    3.6682  1.518e-01  7.294e+00  3.217e-01  5.453e+01  1.4e+01   True
    geometric      100  3.897e-01    3.8970  1.063e-01  3.995e+00  1.712e-01  2.987e+01  7.9e+00   True
    random_log      10  1.881e+00    5.9482  6.066e-01  1.263e+01  5.378e-01  9.445e+01  2.4e+01   True
    random_log      30  1.299e+00    7.1161  5.034e-01  7.294e+00  3.563e-01  5.453e+01  1.3e+01   True
    random_log     100  6.529e-01    6.5294  2.190e-01  3.995e+00  1.924e-01  2.987e+01  6.5e+00   True
all checks passed
