OAI.SoftChannel204.fullMain

The theorem states that the defined proposition FullMain holds, which is the conjunction of seven claims about functions on the Boolean cube {0,1}^n (n any natural number) with entropies in nats. Notation: avg is the uniform average over the cube; psi(r) = ((1+r)/2)log(1+r) + ((1-r)/2)log(1-r); H(r) = log 2 - psi(r); for u : cube -> R, info(u) = H(avg u) - avg(H∘u); psiInv inverts psi on [0,1]; noise(ρ,u) is u smoothed by the product kernel giving each coordinate agreement weight (1+ρ)/2 and disagreement weight (1-ρ)/2; softCoordinate(i,a)(x) = a·(±1 according to bit x_i). (1) SoftContraction: for every u with values in [-1,1] and every ρ in [-1,1], info(noise ρ u) ≤ psi(|ρ|·psiInv(info u)). (2) SoftAttainment: for every coordinate i and |a| ≤ 1, this inequality is an equality for u = softCoordinate(i,a). (3) SoftInformationIdentity: for every such u and ρ, the mutual information (natural log, with terms of zero probability dropped) of the joint law softJoint(ρ,u)(b,y) = avg_x[(1+sign(b)u(x))/2 · kernel(ρ,x,y)] on Bool × cube equals info(noise ρ u). (4) RefinedBoolean: for every f : cube -> Bool and ρ in [-1,1], with m the average of the ±1 signs of f and bound = psi(|ρ|·psiInv(H m)), the mutual information of the joint law boolJoint(ρ,f) equals info(noise ρ (sign∘f)), is at most bound, bound is at most psi|ρ|, and bound is strictly less than psi|ρ| whenever 0<|m|<1 and ρ≠0. (5) BooleanAttainment: for every coordinate i and ρ in [-1,1], the dictator function x_i and its negation both have mutual information exactly psi|ρ|. (6) BitsConversion: for every f and ε in [0,1/2], psi(1-2ε)/log 2 = 1 - h(ε), where h is binary entropy in bits, and the mutual information of boolJoint(1-2ε,f), divided by log 2, is at most 1 - h(ε). (7) SharpProduction: for every g with all values strictly in (-1,1), with production(g) the average of artanh(g(x)) times (1/2)Σ_i(g(x)-g(x with bit i flipped)), the quantity hybrid(avg g, info g) = max(reserve(avg g, info g), F(info g)) is at most production(g), and F(info g) is at most that hybrid; here F(s) = psiInv(s)·artanh(psiInv(s)), and reserve(m,I) = 2I + (1-m²)·S(log 2 - (H m - I)/(1-m²)) with S(s) = F(s)-2s for s ≥ 0 and 0 for s < 0. The theorem is stated with its proof admitted (sorry).