The Base-Derivative of the Kneser Family: Selection Principle, Uniqueness, and Existence via a Riemann–Hilbert Problem

Let the regular Kneser/Paulsen–Cowgill superlogarithm family depend on its base through the parameter given by the logarithm of the logarithm of the base. Its derivative with respect to this parameter defines the response kernel of the mixed-base phase law. We formulate a selection principle that characterizes this response kernel through a cohomological equation, reality on the real axis, a prescribed fixed-point singularity with specified pole and logarithmic coefficients, and a normalization at the value one. We prove uniqueness under these conditions, reduce existence to a scalar Riemann–Hilbert conjugate-function problem for the base derivative of the Kneser theta map, and then solve the resulting boundary problem. The solution rests on three structural facts established in the paper. First, a geometric field constructed from the Koenigs data satisfies an exact cocycle identity, producing solutions of the cohomological equation after the addition of arbitrary periodic terms. Second, the associated boundary data are 1-periodic and extend continuously to the closed circle with a Dini modulus; the iterated-logarithm behavior of the Koenigs coordinate at the punctures is suppressed by the collapse of the reciprocal derivative. Third, the one-sided limits of the geometric field at the punctures are the exact response-kernel values along the tetration orbit, including a value verified numerically to ten digits for the natural base. The conjugate-function step consequently falls within the classical Privalov–Dini theory, without requiring a weighted function class. Interior recovery follows through Schwarz reflection, removability at the punctures, and the fixed-point pole expansion. The result is an existence-and-uniqueness theorem for the selection principle under the classical branch-geometry assumptions of the Kneser solution, together with identification of the selected kernel as the base derivative of the regular superlogarithm under Paulsen’s base holomorphy. The construction coincides with the implemented graded-mesh solver, which reproduces the independently measured response kernel to 17.8 digits in the first Fourier mode and 20.0 digits in the mean.