== Part A: constants
  s=1.01: r_s=2.70050994115730077212715962191145  h_s(1)=2.49654624>1  |h_s(r_s)-1|=0.0E-98  K_s=14.012871517  sharp sup cD^(s+1)=2.74332665
  s=1.05: r_s=2.58367936803175479492466067125267  h_s(1)=2.48296816>1  |h_s(r_s)-1|=0.0E-98  K_s=14.373550168  sharp sup cD^(s+1)=2.78855535
  s=1.1: r_s=2.45589749006191330380333963077492  h_s(1)=2.46651649>1  |h_s(r_s)-1|=0.0E-98  K_s=14.872037572  sharp sup cD^(s+1)=2.84535546
  s=1.25: r_s=2.16209479573722235404161653288201  h_s(1)=2.42044820>1  |h_s(r_s)-1|=0.0E-98  K_s=16.666849169  sharp sup cD^(s+1)=3.01731420
  s=1.5: r_s=1.85354768338729305401027412212670  h_s(1)=2.35355339>1  |h_s(r_s)-1|=1.0E-99  K_s=20.632606443  sharp sup cD^(s+1)=3.30843267
  s=1.75: r_s=1.66439896487700603420440935312378  h_s(1)=2.29730177>1  |h_s(r_s)-1|=1.0E-100  K_s=25.908126172  sharp sup cD^(s+1)=3.60470854
  s=1.9: r_s=1.58338861889199092125399113410053  h_s(1)=2.26794336>1  |h_s(r_s)-1|=0.0E-98  K_s=29.792287659  sharp sup cD^(s+1)=3.78486650
  s=1.99: r_s=1.54281222964363323154707021302305  h_s(1)=2.25173888>1  |h_s(r_s)-1|=1.0E-99  K_s=32.414759819  sharp sup cD^(s+1)=3.89381033
  s=2: r_s=1.53861576354917625747479266226455  h_s(1)=2.25>1  |h_s(r_s)-1|=7.0E-100  K_s=32.720573956  sharp sup cD^(s+1)=3.90595423
  r_2 = 1.5386157635491762574747926622645598344713
  (1+r_2)(2+r_2) = 8.983185758489543434   (paper: 8.983...)
  K_2 = 2(1+r_2)^3 = 32.72057395690330238
  r_2 rounded as in abstract: 1.5386...  -> 1.5386 ; Lemma 2.2: 1.538615... -> 1.538615
  h_s strictly decreasing on grid (0.025..9.975) for s=1.1,1.5,2: True
  (informational) r_s decreases as s increases on this list: True; r_1.01=2.700509
== Part B: point count after Definition 1.1; Lemma 2.1
  #(X cap [x0-M,x0+M]) <= 1 + M^s sum|y-x0|^-s : 1500 cases, 0 failures
  Lemma 2.1: net force sum sgn(y-x)|y-x|^-2 < 0 at the largest point, > 0 at the smallest: 200 sets, 0 failures
== Part C: finite-range identity behind Lemma 2.3
  s=2 (exact rational): 922 identities, worst relative error 0 (exact equality); c symmetric: True; c > 0: True; c = (D+D')/(D^2 D'^2): True
  s=5/4 (100-digit): 162 identities, worst relative error 8.2E-100; c symmetric: True; c > 0: True
  s=3/2 (100-digit): 140 identities, worst relative error 1.4E-99; c symmetric: True; c > 0: True
  s=7/4 (100-digit): 160 identities, worst relative error 9.3E-100; c symmetric: True; c > 0: True
  s=11/10 (100-digit): 154 identities, worst relative error 9.3E-100; c symmetric: True; c > 0: True
  s=19/10 (100-digit): 151 identities, worst relative error 1.1E-99; c symmetric: True; c > 0: True
  total identities checked: 1689
== Part D: the inequalities in the proof of Lemma 2.2 (arbitrary finite configurations)
  s=2: 1103 cases, failures 0; (sum of the two termwise inequalities) - (combined) max |diff| = 0 (exact)
  s=5/4: 274 cases, failures 0; (sum of the two termwise inequalities) - (combined) max |diff| = 6.0E-93
  s=3/2: 275 cases, failures 0; (sum of the two termwise inequalities) - (combined) max |diff| = 4.0E-90
  s=7/4: 239 cases, failures 0; (sum of the two termwise inequalities) - (combined) max |diff| = 1.0E-90
  reflection: gaps of -X are the gaps of X in reverse order: True
== Part E: Lemma 2.4 on gap sequences with consecutive ratios in [1/r, r]
  s=2, r=1.538615: 6496 pairs; min(D,D')>=D/(1+r) failures 0; D'>=D/r and D>=D'/r failures 0; c<=K_s D^-(s+1) failures 0; c<=(1+r)(2+r)D^-3 failures 0; max c D^(s+1) = 3.905951 (sharp sup 3.905951, K_s = 32.72054, (1+r)(2+r) = 8.9831)
  s=5/4, r=2.162094: 1680 pairs; min(D,D')>=D/(1+r) failures 0; D'>=D/r and D>=D'/r failures 0; c<=K_s D^-(s+1) failures 0; max c D^(s+1) = 3.017312 (sharp sup 3.017312, K_s = 16.66683)
  s=3/2, r=1.853547: 1680 pairs; min(D,D')>=D/(1+r) failures 0; D'>=D/r and D>=D'/r failures 0; c<=K_s D^-(s+1) failures 0; max c D^(s+1) = 3.308431 (sharp sup 3.308431, K_s = 20.63259)
  s=7/4, r=1.664398: 1680 pairs; min(D,D')>=D/(1+r) failures 0; D'>=D/r and D>=D'/r failures 0; c<=K_s D^-(s+1) failures 0; max c D^(s+1) = 3.604705 (sharp sup 3.604705, K_s = 25.90810)
== Part F: Green's identity and Lemma 3.1 (Dirichlet principle), complete-graph networks, exact
  40 random complete networks: Green's identity failures 0; E(h) = pi(x0) q failures 0; q from absorbing-chain solve differs 0; E(f) = E(h) + E(f-h) >= pi(x0) q failures 0 (240 test functions)
== Part G: Lemma 3.3 (Doob transform), exact
  30 networks (interior A, boundary values > 0, h harmonic on A): (a) failures 0; (b) failures 0; (c) killed-kernel entries differing 0 (t = 1..5); return-before-exit probabilities differing 0; (d) failures 0 (120 test functions); negative control (h not harmonic at supp phi) detected in 30/30
  birth-death chain c(n,n+1)=2^n, h(n)=1+2^-n: h harmonic on [-30,30]: True; P^h stochastic: True; (P^h)^t(0,0) = P^t(0,0) for t <= 16: True
== Part H: Remark 5.3 exponent arithmetic (sketch, not claimed)
  (3+sqrt5)/2 = 2.6180339887; s^2-3s+1 there = 1.0E-99; exponent -2 + s(s-2) + (3-s) - (s^2-3s+1) = 0 identically: True
  s^2-3s+1 < 0 on (2, 2.618): True
runtime 8.4 s
