OAI.IsingPerceptron.variationalValue_real_of_continuous

The theorem states that, for every real α ≥ 0 and every continuous function f : ℝ → ℝ that is bounded (there is K with |f(x)| ≤ K for all x), the variational value of the Ising perceptron model is a finite real number, i.e. variationalValue(α,f) equals the extended real (p : EReal) for some real p. Here variationalValue(α,f) is the infimum, over all admissible overlap paths q (almost-everywhere-defined functions on (0,1) that agree a.e. with a monotone function taking values in [0,1]), of α times the pattern functional of f at q plus the Ising entropy of q. The pattern functional is the limit as n→∞ of uniformPattern, a nested Gaussian transform over a uniform grid with n+1 cells built from cell averages of q and applied to f at 0, where each transform is either a Gaussian average or a log-exponential-moment average divided by d, depending on whether d is zero. The Ising entropy is the supremum, over field steps (a partition of [0,1] with nonnegative monotone step values), of the field recursion value plus half the integral of the step function against q. The statement asserts that this infimum is neither +∞ nor −∞.