Boundary-contact uniqueness and quadratic contact

Prove that for every parameter b with -1<b<0, the boundary density D_{A(b),-b}(t) has exactly one positive zero t_b, and prove that this zero has exactly order two, equivalently that its first derivative vanishes and its second derivative is strictly positive.

Background

The paper establishes an exact implicit description of the Bernstein-function boundary: for each -1<b<0, the density D_{A(b),-b} is nonnegative on [0,∞) and has a nonempty finite set of positive zeros. Every such zero is proved to have finite even order at least two, but the results do not determine whether there is only one contact point or whether higher-order contact can occur.

The conjecture asks for the stronger global geometry expected by the numerical formulation and local continuation theorem. If established, uniqueness and strict quadratic contact would allow the implicit boundary A(b) and the contact point t_b to be treated as a single real-analytic critical branch governed by the tangency equations, together with the required global nonnegativity condition.