A Power-Diagram Upper Bound for Twenty-Six Variable-Radius Disks in a Square

Abstract

Let $S_{26}$ denote the supremum of the sum of the radii of 26 pairwise interior-disjoint closed disks of positive radius contained in the unit square. We prove $2.635983084917607783\dots \le S_{26}\le \sqrt{\frac{2577}{2500}+\frac{11}{\sqrt{3}}}=2.716919756099889566\dots$. The lower bound is accompanied by an exact rational feasibility certificate. For the upper bound, we assign the disks their clipped power cells, convert cell area into a reciprocal exterior-angle cost, and combine this local inequality with Euler side counts. The extremal case for the resulting relaxation has four boundary transitions and a four-cell boundary cycle; its continuous angle allocation is reduced to a concave one-variable problem. All finite numerical inequalities are rigorously enclosed by a reproducible certificate. The argument does not determine $S_{26}$ exactly, but gives a nontrivial global upper bound for a benchmark whose best-known packing is asymmetric.

arXiv comment15 pages, 1 figure; exact rational feasibility and upper-bound verification scripts and data included as ancillary source files