An Analytical Solution to the Generalized Belgian Chocolate Problem

We study the generalized Belgian chocolate problem in its polynomial formulation of strong stabilization. We prove that the set of parameters admitting a finite strictly stable controller is an open interval (0, δ∗), and hence the optimal bound cannot be attained by any such controller. We give an exact characterization of δ∗ in terms of harmonic measure on a family of slit domains. More precisely, if r∗ is the unique solution of the balance equation 2p_L(r) + p_O(r) = 1, then the critical parameter is δ∗ = cos(π p_O(r∗) / 2). We also determine the corresponding sharp threshold for the associated holomorphic interpolation problem. The proof combines conformal mapping, the elliptic modular function, a monodromy estimate, and a Cauchy integral argument. Numerical evaluation gives δ∗ ≈ 0.987237027. Source code:https://github.com/Panda-China666/belgian-chocolate-lean