A. Rates of atomic Xi: formula (1) vs brute force; consistency; identities (5), (6)
  a=1/3, (1/4,1/8)                     brute force = formula (b<=7): True; consistency violations: 0; (5): True; (6): True
  a=0, 2*(1/3,1/5,1/7) + (1/2)/2       brute force = formula (b<=7): True; consistency violations: 0; (5): True; (6): True
  a=1/5, (1/2,1/2) + 3*(1/6,1/6,1/6)   brute force = formula (b<=7): True; consistency violations: 0; (5): True; (6): True

B. Paintbox forms (4), (10) of T, Bin, X, W against the type sums (m <= 8)
  a=1/3, (1/4,1/8)                     agree for m = 2..8: True
  a=0, 2*(1/3,1/5,1/7) + (1/2)/2       agree for m = 2..8: True
  a=1/5, (1/2,1/2) + 3*(1/6,1/6,1/6)   agree for m = 2..8: True

C. Lemma 2.1: Kingman gives the Ewens EPPF; Kingman and star are regenerative (n <= 7)
  Kingman a=1 rho=1: EPPF = Ewens(theta=2rho/a) and (R_n) holds, n <= 7: True
  Kingman a=3/2 rho=5/2: EPPF = Ewens(theta=2rho/a) and (R_n) holds, n <= 7: True
  star c=1 rho=1: (R_n) holds for n <= 7: True
  star c=2 rho=1/3: (R_n) holds for n <= 7: True

D/E. Lemmas 2.2, 2.3, the q(n;2) formula, (9) and Lemma 3.1 for n = 4..8
  arbitrary consistent array 0         rho=2     hook q>=0: True; Lemma 2.3: True; kappa: True; q(n;2): True; (9): True; Lemma 3.1: True
  arbitrary consistent array 1         rho=2/3   hook q>=0: True; Lemma 2.3: True; kappa: True; q(n;2): True; (9): True; Lemma 3.1: True
  arbitrary consistent array 2         rho=9/2   hook q>=0: True; Lemma 2.3: True; kappa: True; q(n;2): True; (9): True; Lemma 3.1: True
  a=1/3, (1/4,1/8)                     rho=2     hook q>=0: True; Lemma 2.3: True; kappa: True; q(n;2): True; (9): True; Lemma 3.1: True
  a=0, 2*(1/3,1/5,1/7) + (1/2)/2       rho=8     hook q>=0: True; Lemma 2.3: True; kappa: True; q(n;2): True; (9): True; Lemma 3.1: True
  a=1/5, (1/2,1/2) + 3*(1/6,1/6,1/6)   rho=8     hook q>=0: True; Lemma 2.3: True; kappa: True; q(n;2): True; (9): True; Lemma 3.1: True
  perturbed (inconsistent) array: identity broken at n = 7, as it must be: True

F. Level 4 (Theorem A, Corollary 3.2)
  Lambda = delta_{1/2}                 j_4 = 0, levels 2-3 regenerative, level 4 fails exactly at (2,2): True
  Kingman + star                       j_4 = 0, levels 2-3 regenerative, level 4 fails exactly at (2,2): True
  a=1/2 + delta_{1/3}                  j_4 = 0, levels 2-3 regenerative, level 4 fails exactly at (2,2): True
  certificate Xi                       j_4 > 0; rho* = 13833/5120; level 4 regenerative iff rho = rho* (tested): True
  a=1/3, (1/4,1/8)                     j_4 > 0; rho* = 2057/480; level 4 regenerative iff rho = rho* (tested): True
  a=0, 2*(1/3,1/5,1/7) + (1/2)/2       j_4 > 0; rho* = 38639752051273/34885001592000; level 4 regenerative iff rho = rho* (tested): True
  a=1/5, (1/2,1/2) + 3*(1/6,1/6,1/6)   j_4 > 0; rho* = -11/75; level 4 regenerative iff rho = rho* (tested): True

G. Proposition 7.1: Xi = (17/1024) delta_0 + delta_(1/4,1/8), rho = 13833/5120
  rho equals rho* of Theorem A: True; consistency violations: 0
  level 2: q = (9222/10957, 1735/10957); all (R_2) hold, q >= 0, sum q = 1, as printed: True
  level 3: q = (53/70, 10229/54785, 43063/766990); all (R_3) hold, q >= 0, sum q = 1, as printed: True
  level 4: q = (106/149, 53/298, 421721/4897779, 241301/9795558); all (R_4) hold, q >= 0, sum q = 1, as printed: True
  level 5: q = (23055/33517, 5300/33517, 9805/100551, 16183550/367245769, 1245097/100157937); all (R_5) hold, q >= 0, sum q = 1, as printed: True
  level 6: hook q >= 0, (R_6) fails exactly at [(4, 2), (3, 3), (3, 2, 1), (2, 2, 2), (2, 2, 1, 1)], residuals as printed: True
     residual at (4, 2): -1072617989416/62545017657659229
     residual at (3, 3): 3435702810172/187635052972977687
     residual at (3, 2, 1): 39121971094/62545017657659229
     residual at (2, 2, 2): -92094918940/62545017657659229
     residual at (2, 2, 1, 1): 367050883928/62545017657659229
  level 7: (R_7) fails at 8 partitions: [(5, 2), (4, 3), (4, 2, 1), (3, 3, 1), (3, 2, 2), (3, 2, 1, 1), (2, 2, 2, 1), (2, 2, 1, 1, 1)]
  Delta_n from the rate array = Delta_n from the closed forms, n = 4..7: True

H. Signs of Delta_n for the example of Proposition 7.1, n = 4..60
  signs: 00++++++++++---------------------------------------------
  Delta_4 = Delta_5 = 0, Delta_n > 0 for 6 <= n <= 15, Delta_n < 0 for 16 <= n <= 60: True
  n = 60: Bin(n)W(n-1)/C(n,2) = 0.2124999993 (a*M = 17/80 = 0.2125); Phi(n-1)X(n)/C(n,2) = 1.537e-08

I. Theorem C example: a = 0, Xi_0 = delta_(1/3,1/3,1/3)
  rho = 1: Bin(5) = 0, X(5) = 10/9, Delta_5 = 70/9 > 0; (R_5) fails at [(3, 2), (2, 2, 1)]
  rho = 5/2: Bin(5) = 0, X(5) = 10/9, Delta_5 = 130/9 > 0; (R_5) fails at [(3, 2), (2, 2, 1)]

J. Eventual sign of Delta_n (n <= 60): Theorem D (a = 0) and Theorem B (a > 0)
  D: (1/2,1/4)                         rho=1: predicted sign +1 holds for 4 <= n <= 60: True
  D: (1/2,1/4)                         rho=3: predicted sign +1 holds for 4 <= n <= 60: True
  D: (1/2) + (1/3,1/3)                 rho=1: predicted sign -1 holds for 4 <= n <= 60: True
  D: (1/2) + (1/3,1/3)                 rho=3: predicted sign -1 holds for 16 <= n <= 60: True
  D: (1/3,1/3) + 2*(1/5,1/5,1/5)       rho=1: predicted sign +1 holds for 4 <= n <= 60: True
  D: (1/3,1/3) + 2*(1/5,1/5,1/5)       rho=3: predicted sign +1 holds for 4 <= n <= 60: True
  D: (1/2,1/2) + (1/4)                 rho=1: predicted sign -1 holds for 5 <= n <= 60: True
  D: (1/2,1/2) + (1/4)                 rho=3: predicted sign -1 holds for 5 <= n <= 60: True
  D: star atom + (1/4,1/4)             rho=1: predicted sign +1 holds for 7 <= n <= 60: True
  D: star atom + (1/4,1/4)             rho=3: predicted sign +1 holds for 5 <= n <= 60: True
  D: 3*(1/2) + (1/4,1/4)               rho=1: predicted sign +1 holds for 12 <= n <= 60: True
  D: 3*(1/2) + (1/4,1/4)               rho=3: predicted sign +1 holds for 10 <= n <= 60: True
  B: a=1/10, (1/4,1/8)                 rho=1: predicted sign -1 holds for 4 <= n <= 60: True
  B: a=1/10, (1/4,1/8)                 rho=3: predicted sign -1 holds for 4 <= n <= 60: True
  B: a=2, (1/2,1/3)                    rho=1: predicted sign -1 holds for 4 <= n <= 60: True
  B: a=2, (1/2,1/3)                    rho=3: predicted sign -1 holds for 5 <= n <= 60: True
  B: a=1/3, (1/3,1/3,1/3) + (1/2)      rho=1: predicted sign -1 holds for 5 <= n <= 60: True
  B: a=1/3, (1/3,1/3,1/3) + (1/2)      rho=3: predicted sign -1 holds for 6 <= n <= 60: True

K. Lemmas 4.1, 4.2, (Q1) on random paintboxes; constants of Lemma 4.3 and Proposition 5.1
  60 random paintboxes, m = 3..30: violations of Lemma 4.1: 0, Lemma 4.2: 0, (Q1): 0
  smallest P(Z_m >= 3) seen with lambda_m >= 1/4: 0.01399 (Lemma 4.3 asserts a positive lower bound)
  exact constants for 4 <= m <= 3000 and the scales m(J), 3 <= J <= 60, in the proof of Proposition 5.1: True
  Poisson bounds, Chernoff bound P(N >= m) <= (e/4)^(m/2) (4 <= m <= 400), Bernoulli (floating point): True

checks run: 55; failed: 0 
ALL CHECKS PASSED
