A Uniform Upper Bound on the Sum of Radii in Variable-Radius Disk Packings of the Unit Square

This paper investigates the problem of obtaining a uniform upper bound on the sum of the radii of pairwise interior-disjoint disks of variable radii contained in the unit square. Let S_n denote the supremum of the sum of the radii of n such disks. The main result establishes a uniform upper bound for every integer n ≥ 15, with an explicit positive algebraic constant. The proof uses clipped power diagrams to convert geometric containment into inequalities involving polygon areas, exterior angles and side counts. Its central refinement is a coupled optimization of the four corner cells in the four-boundary-transition case. The maximizing parameter is characterized by a degree-seven polynomial over Q(sqrt(2)) and an isolating rational interval. The manuscript also provides exact-rational computational certificates for the finite arithmetic inequalities arising in the proof. These computations support the geometric arguments but do not replace their mathematical justification. The algebraic parameter arising from the four-corner optimization also motivates its decimal and continued-fraction expansions, which are being submitted as related integer sequences to the On-Line Encyclopedia of Integer Sequences (OEIS). This is an author-version preprint of a manuscript submitted to the Journal of Geometry. It has not yet been accepted or published.