The Ford-Circle Packing Has Maximum Area: An Answer to a Question of Propp and Kenyon

Two disks of radius 1 centred at (±1, 1) and the x-axis enclose a curvilinear triangle. Propp and Kenyon asked in the 2015 Oberwolfach workshop on discrete differential geometry whether the greedy packing has maximum total area among all packings of this triangle by disks touching the x-axis. We prove that the answer is yes. The greedy packing is a rescaled copy of the Ford circles and has area π(ζ(3)/ζ(4) − 1), approximately 0.34754. More generally, for tangent disks A and B resting on a line, the greedy packing G of their gap maximizes the sum of radius powers r^α for every α > 1 among all finite or countable packings of the gap by disks resting on that line. The main inequality is first proved for the weight 1/(e^(1/√r) − 1). Moving two consecutive disks in a tangent chain reduces it to log-convexity, whose proof uses an explicit double power series with nonnegative coefficients. Scope: the theorem concerns disks touching the boundary line in the gap between tangent boundary disks. It does not claim uniqueness of the maximizing packing, optimality for arbitrary disks that do not touch the line, or a theorem for gaps between non-tangent boundary disks. No absolute priority claim is made. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Author: Alper Ferudun, Mercury Software GmbH. Corpus identifier: OWR-13498-011 (Oberwolfach Report 13/2015, Problem 7 by J. Propp and R. Kenyon). Paper page: https://eulersolve.org/papers/owr-13498-011/