g0^{-1}(+1): [(-1, -1, -1, -1, 1), (-1, -1, 1, 1, -1), (-1, 1, -1, 1, 1), (-1, 1, 1, -1, -1), (-1, 1, 1, 1, 1), (1, -1, -1, -1, -1), (1, -1, -1, -1, 1), (1, -1, -1, 1, -1)]
depends on all coordinates: True  mu = -1/2  sigma = -1  triv = 3/4  level-1 = ['-1/8', '-1/8', '-1/8', '0', '0']
h (order +++,++-,...,---) = ['-1', '-1', '-1', '1/2', '0', '-1/2', '-1/2', '-1/2']
inner sums of u_1 over A_w, w = +--, -++, -+-, --+, ---: [-1, 0, 1, 1, -1]
inner sums of u_2 over A_w, w = +--, -++, -+-, --+, ---: [-1, 0, 1, -1, 1]
2h = (-2,-2,-2,1,0,-1,-1,-1) is in hard288.txt (288 entries parsed)
g_m, m = 6, 7, 8: same h, mu and level-one coefficients (-1/8 on w, 0 on every u_i), depends on all coordinates
pi_+ = ['0', '0', '0', '3/8', '1/4', '1/8', '1/8', '1/8']
pi_- = ['1/6', '1/6', '1/6', '1/24', '1/12', '1/8', '1/8', '1/8']
pi_+- = (1 +- h)/(8 (1 +- mu)) = conditional laws of w given g0 = +-1  [EXACT]
2h(p) = -1 + 2p1 - 3p1p2 - 3p1p3 + p2p3 + 2p1p2p3 (values -2,-2,-2,1,0,-1,-1,-1 at the vertices)  [EXACT]
pi_+ and pi_- are invariant under w2 <-> w3  [EXACT]
beta_+ = 3/8  beta_- = 13/24
law of (w1,w2) on (++,+-,-+,--): pi_+: ['0', '3/8', '3/8', '1/4']  pi_-: ['1/3', '5/24', '5/24', '1/4']
law of (w2,w3) on (++,+-,-+,--): pi_+: ['1/4', '1/8', '1/8', '1/2']  pi_-: ['1/4', '7/24', '7/24', '1/6']
lambda^{12}_+- = ['-3/5', '23/143']  lambda^{23}_+- = ['7/15', '-25/143']  (lambda^{13} = lambda^{12})  [EXACT]
E[chi_S(x)chi_T(x')] = 1{S=T} lambda_S for (X,Y), (Y,Z), and orthonormality of chi_S, all nu with k <= 3  [EXACT]
a_S: {+} -> 43/15  {-} -> -117/143 = -9/11  {+,+} -> -11/9  {+,-} -> 653/2145  {-,-} -> -1491/20449 ;  a* = 3259/3375
a_S <= a* for all 510 row-sign patterns with 1 <= |S| <= 8 except {+};  largest of these a_S = 3259/3375
|S| >= 3: a_S <= (7/15)^|S| + 4 (3/5)^|S| <= (7/15)^3 + 4 (3/5)^3 = a* = 3259/3375  [EXACT]
2p-3p^2+(43/15)(p-p^2) = 1-(8p-3)(11p-5)/15;  G(3/8) = 3/4-27/64+43/64 = 1, G'(3/8) = (a*-1)/4 < 0, G'' < 0  [EXACT]
remark: max G = 1 + 841/180684000 = 1 + 4.6545e-06 at p = 10009/26768 = 0.373917
right-hand side of Section 9.5 <= 1 on the grid p = j/4000, with equality only at p = 3/8  [EXACT]
dictator (k=1, nu=+1, F(z)=z): Gamma = 0, p = 3/8, P1P2 = 0 a.s., W_+ = Var P = 15/64; every inequality of the chain is an equality  [EXACT]
342 decoders, k<=2, values in {-1,0,1}: 2Gamma <= reduction bound = spectral formula - 1 <= Lemma 9.3 bound - 1, W_+ = W^1(Pbar) <= min(Var P, 15/64), final bound <= 1; max Gamma = 0  [EXACT]
(float) 400 random [-1,1]-valued decoders, k<=4: the same chain holds, max Gamma = 0.000e+00 (<= 0)
g0 verified: every number of Section 9 used for Theorem E checked.
