PASS  A  Lemma 2.1 = direct grid-cell integration on 30 random rational point sets (d<=3, n<=4, p<=6)
PASS  B  gap formula = brute-force double integral (3000 x 3000 midpoint rule) for 12 (S, alpha), incl. translated/scaled sets: max deviation 1.54e-04 (< 2e-3)
PASS  C  phi(cS) = phi(S) exactly for c in {2, 3, -1}, alpha in {1, 2}, 40 random sets
PASS  C  phi_alpha(S) = alpha/(alpha+|S|) for |S| <= 2 (a <= 11, alpha <= 7)
PASS  D  Prop. 4.1(a) closed form and sign of the difference, alpha = 1..12
PASS  D  phi values 1/4, 19/48, 1/4, 517/1296, 11/54, 85/432: ['1/4', '19/48', '1/4', '517/1296', '11/54', '85/432']
PASS  D  gap integrals 1/2, 131/432 ({0,1,3}); 11/27 ({0,1,2,3}); 85/216 ({0,1,2,4})
PASS  E  p=4: direct integration = expansion, E_iid binomial = cumulant route, sign as claimed; E_lat(n,1,4): 1:1/15, 2:1/48, 3:19/1620, 4:139/17280, 5:31/5000, 6:647/129600, 7:274313/64827000, 8:29399/8064000
PASS  E  p=2: direct integration = expansion, E_iid binomial = cumulant route, sign as claimed; E_lat(n,1,2): 1:1/6, 2:1/12, 3:1/18, 4:1/24, 5:1/30
PASS  E  p=6: direct integration = expansion, E_iid binomial = cumulant route, sign as claimed; E_lat(n,1,6): 2:11/1344, 3:353/81648, 4:1273/435456, 5:72751/32400000
PASS  E  p=8: direct integration = expansion, E_iid binomial = cumulant route, sign as claimed; E_lat(n,1,8): 2:47/11520, 3:4399/2099520, 4:101273/71663616
PASS  E  p=10: direct integration = expansion, E_iid binomial = cumulant route, sign as claimed; E_lat(n,1,10): 3:196897/166281984
PASS  F  Theorem 1.1(i): E_lat(n,d,2) = E_iid(n,d,2) = (2^-d - 3^-d)/n for n <= 7, d <= 5
PASS  F  Theorem 1.1(ii): equality for n <= 2, d <= 3, p in {4, 6, 8}
PASS  F  Theorem 1.2: E_iid(n,d,4) formula (n <= 12) and both closed differences (n = 3, 4), d = 1..12
PASS  G  Monte Carlo d=2, n=3, p=4 (10^6 samples, exact disc per sample, no lemma): 0.0087936 +- 0.0000156 against 6829/777600 = 0.0087822 (|z| = 0.73 < 3)
PASS  H  Remark 7.3: (1/3) int (Delta^5 + (1-Delta)^5)/5 dDelta = 1/45 > 16/1620
TOTAL failures: 0
