A Power-Diagram Upper Bound for Twenty-Six Variable-Radius Disks in a Square
Abstract
Let S26 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… ≤ S26≤ √(2577/2500+11/√3)=2.716919756099889566…. 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 S26 exactly, but gives a nontrivial global upper bound for a benchmark whose best-known packing is asymmetric.
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BibTeX
@misc{agarwal2026powerdiagram,
title = {{A Power-Diagram Upper Bound for Twenty-Six Variable-Radius Disks in a Square}},
author = {Ayush Agarwal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13962},
eprint = {2610.13962},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13962},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Power-Diagram Upper Bound for Twenty-Six Variable-Radius Disks in a Square
AU - Ayush Agarwal
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.13962
UR - https://arxiv.org/abs/2610.13962
AB - 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… ≤ S_26≤ √(2577/2500+11/√3)=2.716919756099889566…. 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.
ER -