Theorem L: eigenvalues of qI + beta Pi_0 - Pi_1 - c T_rho* are 0, 0, 0 at levels 0, 1, 2 and q(1 - (-1)^k (q/(1-q))^(k-2)) at level k >= 3  [EXACT, general q]
rho*-coupling: nonnegative kernel, q-biased marginals, correlation -q/(1-q)  [EXACT]
joint law (0, q, q, 1-2q), E[x_i y_i] = 1-4q, correlation -4q^2/(4q(1-q)) = rho*, e_0 = e_1 = e_2 = 0, e_k > 0 (q < 1/2) and e_k = 0 / 2q for even / odd k (q = 1/2)  [EXACT]
1{x_1=1}, 1{x_1=x_2=1}, 1{x_1=1,x_2=-1}: equality in both inequalities of Theorem L (q in {3/8,2/5,5/12,1/10,1/2})  [EXACT]
(float) 300 random f >= 0, n <= 6: identity holds (max error 4.6e-16), all terms >= 0
(float) equality for f = 1{x_1 = 1} h with h >= 0 of degree <= 1
g2: depends on all coordinates: True  mu = -3/8  level-1 = ['-1/8', '-1/8', '-1/8', '0', '0']
h (order +++,++-,...,---) = ['-1/2', '-1', '-1', '1/2', '0', '-1/2', '-1/2', '0']
inner sums of u_1 over A''_w, w = +++, +--, -++, -+-, --+, ---: [1, -1, 0, 1, -1, 0]
inner sums of u_2 over A''_w, w = +++, +--, -++, -+-, --+, ---: [1, -1, 0, -1, 1, 0]
2h is in hard288.txt; its orbit has 12 elements, all in hard288
g_{2,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) = -p1-p2-p3+2p2p3+2p1(1-p2)(1-p3) and the reduction adds 2p1(1-p2)p3 >= 0  [EXACT]
pi_+ = ['1/10', '0', '0', '3/10', '1/5', '1/10', '1/10', '1/5']
pi_- = ['3/22', '2/11', '2/11', '1/22', '1/11', '3/22', '3/22', '1/11']
beta_+ = 2/5  beta_- = 6/11
lambda^{12}_+-, lambda^{23}_+- = ['-1/4', '1/12'] ['7/12', '-17/60']
law of (w1,w2) on (++,+-,-+,--): pi_+: ['1/10', '3/10', '3/10', '3/10']  pi_-: ['7/22', '5/22', '5/22', '5/22']
law of (w2,w3) on (++,+-,-+,--): pi_+: ['3/10', '1/10', '1/10', '1/2']  pi_-: ['5/22', '7/22', '7/22', '3/22']
pi_+-, the four pair laws and lambda^{12}_+-, lambda^{23}_+- (displayed numerators/denominators)  [EXACT]
a_S, |S|=1:  nu=+: 5/6   nu=-: -11/30 ;  |S|=2 (++, +-, --): 5/18 -13/90 11/150
a_S <= 5/18 for every nonempty S other than a singleton {i} with nu_i = +1; (7/12)^3 + (1/4)^3 = 185/864  [EXACT]
(5/6 - 5/18)(q E P^2 + beta p^2) + (5/18)(E P^2 - p^2) = E P^2 / 2 for q = 2/5, beta = 1/2  [EXACT]
k<=2, values in {-1,0,1}: Gamma <= reduction = spectral <= (E P^2 - E P)/2 <= 0, max Gamma = 0  [EXACT]
  dictator: Gamma = reduction bound = final bound = 0; p = 2/5, P1(1-P2)P3 = 0 a.s., W_+ = 6/25 = (2/5) E P^2 + (1/2) p^2
  AND2: Gamma = reduction bound = final bound = 0; p = 4/25, P1(1-P2)P3 = 0 a.s., W_+ = 48/625 = (2/5) E P^2 + (1/2) p^2
  AND2 with W_+ <= 6/25 (Theorem K) instead of Theorem L: the final bound would be 2 Gamma <= 68/375, not <= 0
(float) 400 random decoders, k<=4: reduction = spectral <= final <= 0, max Gamma = 0.000e+00
Theorem L and D12 verified.
B12: 2h = (-2, -1, -1, 1, 0, 0, 0, -1), mu = -1/4, cubic coefficient of 2h = 2, pi_+ vanishes at [(1, 1, 1)]
  raising h at +++ increases the (positive) cubic coefficient: no quadratic h' >= h agrees with h on supp pi_+
C12: 2h = (-1, -2, -2, 1, -1, 0, 0, -1), mu = -3/8, cubic coefficient of 2h = 6, pi_+ vanishes at [(1, 1, -1), (1, -1, 1)]
  h' = h + (3/2)(B_++- + B_+-+); k=1 decoder P(1)=1, P(-1)=1/6 (p = 1/2): quadratic bound 1/72, true Gamma_+ = -29/144
Section 11 claims on the orbits B12 and C12 verified.  [EXACT]
