OAI.SphericalPerceptron.spherical_linear_field_formula

The theorem states that the defined proposition SphericalLinearFieldFormula holds, a two-part formula for the field value of a spherical model, in the SphericalPerceptron namespace, built from a cascade of Poisson point processes with Gaussian marks. Here a trial is a monotone measurable function m on [0,1] with values in [0,1], its entropy is half the integral over t of 1/∫_{[t,1]} m minus 1/(1−t), and finiteSphericalFieldValue(n,k,w,h) is, for weights w on k+1 levels and a field vector h, the expectation of the normalized logarithm (divided by n+1) of an angular average of exponentiated Gaussian-cascade fields over the sphere of radius √(n+1) in dimension n+1, minus h at the last level. Part one concerns every k and every strictly positive weight vector w summing to 1. First, for every monotone h with h(0) ≥ 0, there exists a monotone q from the k+1 levels into [0,1] with q(last) < 1 that minimizes the dual objective, namely the entropy of the step trial with jumps of size w_i at q_i, minus Σ w_i h_i q_i, over all such monotone q with q(last) < 1. Second, on every compact set K of monotone vectors h with h(0) ≥ 0, the finite field values converge uniformly, as the dimension parameter n tends to infinity, to the infimum of these dual values. Part two concerns every B with 0 ≤ B < 1 and every ε > 0: for all sufficiently large n, simultaneously for every monotone q from [0,1] to [0,1] with q ≤ B almost everywhere, the entropy of the quantile trial t ↦ Leb{q ≤ t} is finite, the stationary field, defined as half of the integral of 1/(∫(1−max(q,t))du)² over t from 0 to the clipped quantile, satisfies 2·field(u) = quantileA(q)(q(u)) almost everywhere, and the limiting field value of that stationary field, defined as a limit along finite-level rounding approximations, lies within ε of the entropy of the quantile trial minus ∫ field(u)q(u)du.