g1^{-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), (1, -1, -1, 1, -1), (1, -1, 1, 1, 1)]
depends on all coordinates: True  mu = -3/8  level-1 = ['-1/8', '-1/8', '-1/8', '0', '0']
h (order +++,++-,...,---) = ['-1', '-1', '-1/2', '1/2', '0', '0', '-1/2', '-1/2']
inner sums of u_1 over A'_w, w = +-+, +--, -++, -+-, --+, ---: [1, -1, 0, 0, 1, -1]
inner sums of u_2 over A'_w, w = +-+, +--, -++, -+-, --+, ---: [1, -1, 0, 0, -1, 1]
2h is in hard288.txt (288 entries parsed); its orbit has 24 elements, all in hard288
g_{1,m}, m = 6, 7, 8: same h and mu, level-one coefficients -1/8 on w and 0 on every u_i, depends on all coordinates
2h(p) = -1 + 2p1 + p2 - 4p1p2 - 2p1p3 + 2p1p2p3 (values -2,-2,-1,1 and 0,0,-1,-1 at the vertices)  [EXACT]
pi_+ = ['0', '0', '1/10', '3/10', '1/5', '1/5', '1/10', '1/10']
pi_- = ['2/11', '2/11', '3/22', '1/22', '1/11', '1/11', '3/22', '3/22']
beta_+ = 2/5  beta_- = 6/11
lambda^{12}_+-, lambda^{13}_+-, lambda^{23}_+- = ['-2/3', '4/15'] ['-1/4', '1/12'] ['1/6', '-1/10']
law of (w1,w2) on (++,+-,-+,--): pi_+: ['0', '2/5', '2/5', '1/5']  pi_-: ['4/11', '2/11', '2/11', '3/11']
law of (w1,w3) on (++,+-,-+,--): pi_+: ['1/10', '3/10', '3/10', '3/10']  pi_-: ['7/22', '5/22', '5/22', '5/22']
pi_+-, the four pair laws and lambda^{12}_+-, lambda^{13}_+- (displayed numerators/denominators)  [EXACT]
a_S, |S|=1:  nu=+:  11/6   nu=-:  -7/10
a_S, |S|=2:  (++, +-, --):  -73/72 143/360 -281/1800
a* = 2((2/3)^3 + (1/4)^3) = 539/864 bounds a_S for |S| >= 3  [EXACT]
case split: -1+3p-4p^2+(11/6)(p-p^2) = -(5p-2)(7p-3)/6;  G(2/5) = 0, G'(2/5) = (a*-1)/5 < 0, G'' < 0  [EXACT]
dictator (k=1, nu=+1, F(z)=z): p = 2/5, P1P2 = 0 a.s., W_+ = Var P = 6/25, Gamma_+ = reduction = spectral = 0  [EXACT]
k<=2, values in {-1,0,1}: Gamma <= reduction bound = spectral formula, max Gamma = 0  [EXACT]
(float) 400 random decoders, k<=4: reduction bound == spectral formula, max Gamma = -1.388e-17 (<= 0)
g1 verified.
