Minimum central circles: an effective characterization of the global asymptotic constant
Abstract
Let R∗(n) be the least radius of a central circle to which nonoverlapping circles of radii 1,…,n are externally tangent. We prove that R∗(n)=C∗ n2+o(n2) and characterize C∗ by finite linear programs with an explicit error tending to zero. The reduction preserves arbitrary orders and all pairwise constraints: the limiting problem places marked points on a line at pairwise separation at least the geometric mean of their marks. Concatenation with a bounded boundary cost proves existence, and balanced finite-word programs supply matching effective upper and lower bounds. Their certified gap is (1/k+1/r)/π, before directed arithmetic error, for k mark types and words of length r. A quantitative reflected-block recovery theorem supplies genuine permutations and full ring geometry, with a countable extension and a strict four-block improvement. The explicit interval is Cterm+ηwidth≤C∗≤ U4; neither endpoint is asserted sharp. In particular, the coefficient 1/8 proposed in the preceding finite study is false. An elementary expression for C∗, efficient high-precision evaluation and global floating-circle structure remain open.
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BibTeX
@misc{falconi2026minimum,
title = {{Minimum central circles: an effective characterization of the global asymptotic constant}},
author = {Maurizio Falconi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13630},
eprint = {2609.13630},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13630},
}
RIS
TY - UNPB
TI - Minimum central circles: an effective characterization of the global asymptotic constant
AU - Maurizio Falconi
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13630
UR - https://arxiv.org/abs/2609.13630
AB - Let R^∗(n) be the least radius of a central circle to which nonoverlapping circles of radii 1,…,n are externally tangent. We prove that R^∗(n)=C_∗ n^2+o(n^2) and characterize C_∗ by finite linear programs with an explicit error tending to zero. The reduction preserves arbitrary orders and all pairwise constraints: the limiting problem places marked points on a line at pairwise separation at least the geometric mean of their marks. Concatenation with a bounded boundary cost proves existence, and balanced finite-word programs supply matching effective upper and lower bounds. Their certified gap is (1/k+1/r)/π, before directed arithmetic error, for k mark types and words of length r. A quantitative reflected-block recovery theorem supplies genuine permutations and full ring geometry, with a countable extension and a strict four-block improvement. The explicit interval is C_term+η_width≤C_∗≤ U_4; neither endpoint is asserted sharp. In particular, the coefficient 1/8 proposed in the preceding finite study is false. An elementary expression for C_∗, efficient high-precision evaluation and global floating-circle structure remain open.
ER -