== S1 (A.26) ==
PASS (a) 2 tau sqrt(D) <= sqrt(q/((1-q)s)) E|Z_Kf| and lambda+- = s k+- on 89 random/Boolean f (min slack -2.220e-16)
PASS Stein-Chen E[F(N)(N_i - s)] = s(1-q) Delta_i (n <= 3, max err 6.38e-16)
PASS (a) is an equality at f = OR_k when ks <= 1 (max |difference| 3.33e-16)
PASS (b) sqrt(q/((1-q)s)) g(ks) = T_k(q) for k <= 14 and five q (ball arithmetic)
== S2 (A.27) ==
PASS C20 exactly: E|chi_1+...+chi_k|^2/k <= M(q)^2 for 21993 (q, k), q in (0, q4], k <= 300 (violations: [])
PASS de Moivre formula E|S - kq| = 2 v C(k,v) q^v (1-q)^(k-v+1) (k <= 120): 0 mismatches
  equality cases found: 164 (all with kq < 1 and k maximising)
== S3 (A.28) ==
  q=0.14, n=3: R_n = 0.8889979694, M = 0.8889979694, max u|alpha_i| = 1.430950, Lambda_n = 1.524986
  q=0.14, n=4: R_n = 0.8889979694, M = 0.8889979694, max u|alpha_i| = 1.430950, Lambda_n = 1.524986
  q=0.14, n=5: R_n = 0.8889979694, M = 0.8889979694, max u|alpha_i| = 1.430950, Lambda_n = 1.524986
  q=0.2, n=3: R_n = 0.9050966799, M = 0.9050966799, max u|alpha_i| = 1.414214, Lambda_n = 1.581139
  q=0.2, n=4: R_n = 0.9050966799, M = 0.9050966799, max u|alpha_i| = 1.414214, Lambda_n = 1.581139
  q=0.2, n=5: R_n = 0.9050966799, M = 0.9050966799, max u|alpha_i| = 1.414214, Lambda_n = 1.581139
  q=0.3, n=3: R_n = 0.9165151390, M = 0.9165151390, max u|alpha_i| = 1.527525, Lambda_n = 1.690309
  q=0.3, n=4: R_n = 0.9165151390, M = 0.9165151390, max u|alpha_i| = 1.527525, Lambda_n = 1.690309
  q=0.3, n=5: R_n = 0.9165151390, M = 0.9165151390, max u|alpha_i| = 1.527525, Lambda_n = 1.690309
  q=0.45, n=3: R_n = 0.9949874371, M = 0.9949874371, max u|alpha_i| = 1.105542, Lambda_n = 1.906925
  q=0.45, n=4: R_n = 0.9949874371, M = 0.9949874371, max u|alpha_i| = 1.105542, Lambda_n = 1.906925
  q=0.45, n=5: R_n = 0.9949874371, M = 0.9949874371, max u|alpha_i| = 1.105542, Lambda_n = 1.906925
PASS C15(a): E[chi_i sgn L] = R_n alpha_i and P(L=0) = 0 at the maximisers (max err 4.24e-08)
PASS C15(b): R_n u alpha_i = 2(1-q) P(-alpha_i u < L' < alpha_i v) (max err 6.48e-08)
PASS C15(c): stop-loss bounds on a grid of a (min slack 2.58e-05)
PASS C15(d): p+-^2 <= R(R_n + delta/q)/(2(1-q)) and R_n^2 <= 4 p+ p- (min slack -3.33e-16)
PASS C15(b),(e): u|alpha_i| <= 2 p-+ / R_n and <= Lambda_n (min slack -3.33e-08)
== S4 (A.29) ==
PASS C16(i): E|Z_K| = 2 m e^(-lambda) for two-atom K with a >= b > m (max err 7.77e-16)
== S5 (A.30) ==
PASS D1: |y + w| = |y| + sgn(y) w + 2(-sgn(y) w - |y|)_+ on 20,000 random rationals (exact)
PASS D2: identity E|Z_K| = 2me^-L + 2e^-L E(W-m)_+ + 2E[(-W-Y)_+; N_S>=1] on 40 random (K_S, K_R) (max err 9.77e-15)
PASS D2: equality 2 m e^-Lambda = sqrt(2/e) sqrt(V_S) at K_S = delta_sqrt2
  K_S = 0: m = Lambda = V_S = 0, so 2 m e^-Lambda = 0 = sqrt(2/e) sqrt(V_S): equality without a single atom.
ALL PASS
