Analysis of Conjectural Improvements to Minkowski's Lower Bound on the Sphere Packing Density

Abstract

Torquato and Stillinger used a pair-correlation-function optimization framework to conjecture an exponential improvement of Minkowski's lower bound on the maximal density of sphere packings in ℝ^d, with rate 2^{−(0.7786524795…+o(1))d}. Conditional on their realizability conjecture, we show that a family of hyperuniform pair correlation functions yields polynomial improvements of the form ϕ_max≳ d^β2^−d for every fixed β>1 as d→∞. As the polynomial exponent increases with dimension, this family approaches the conjectured exponential improvement. From the Cohn--Elkies dual linear-programming upper bound, we independently derive the Torquato--Stillinger rate and show that its radial test functions cannot asymptotically exclude packings with this density scaling. For the near-contact families considered, any fixed number of shells, gaps, or radial bands can improve subexponential factors but not the leading exponential rate. To surpass this rate, we introduce an explicit hyperuniform construction based on Gauss--Radau quadrature. It contains ⌊(d−1)/4⌋ positive delta-function shells and has rate 2^{−(0.622556248918…+o(1))d}, showing how growing radial complexity improves upon the Torquato--Stillinger rate. Finally, we prove strong duality between the unrestricted pair-correlation program and its Cohn--Elkies dual: their optimal values coincide, with no duality gap. By approximation with ordinary finite-band functions, we show that the unrestricted pair-correlation program attains the optimal Cohn--Elkies rate 2^{−(0.6044005…+o(1))d}. This result is optimal but supplies no comparable closed-form family. Together, these results support the possibility of exceptionally dense disordered sphere packings in high dimensions and strengthen the case for the Torquato--Stillinger realizability conjecture.

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BibTeX RIS

BibTeX

@misc{vanoni2026analysis,
  title = {{Analysis of Conjectural Improvements to Minkowski's Lower Bound on the Sphere Packing Density}},
  author = {Carlo Vanoni and Alan Guo and Salvatore Torquato},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.00920},
  eprint = {2608.00920},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.00920},
}

RIS

TY  - UNPB
TI  - Analysis of Conjectural Improvements to Minkowski's Lower Bound on the Sphere Packing Density
AU  - Carlo Vanoni
AU  - Alan Guo
AU  - Salvatore Torquato
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2608.00920
UR  - https://arxiv.org/abs/2608.00920
AB  - Torquato and Stillinger used a pair-correlation-function optimization framework to conjecture an exponential improvement of Minkowski's lower bound on the maximal density of sphere packings in ℝ^d, with rate 2^−(0.7786524795…+o(1))d. Conditional on their realizability conjecture, we show that a family of hyperuniform pair correlation functions yields polynomial improvements of the form ϕ_max≳ d^β2^−d for every fixed β>1 as d→∞. As the polynomial exponent increases with dimension, this family approaches the conjectured exponential improvement. From the Cohn--Elkies dual linear-programming upper bound, we independently derive the Torquato--Stillinger rate and show that its radial test functions cannot asymptotically exclude packings with this density scaling. For the near-contact families considered, any fixed number of shells, gaps, or radial bands can improve subexponential factors but not the leading exponential rate. To surpass this rate, we introduce an explicit hyperuniform construction based on Gauss--Radau quadrature. It contains ⌊(d−1)/4⌋ positive delta-function shells and has rate 2^−(0.622556248918…+o(1))d, showing how growing radial complexity improves upon the Torquato--Stillinger rate. Finally, we prove strong duality between the unrestricted pair-correlation program and its Cohn--Elkies dual: their optimal values coincide, with no duality gap. By approximation with ordinary finite-band functions, we show that the unrestricted pair-correlation program attains the optimal Cohn--Elkies rate 2^−(0.6044005…+o(1))d. This result is optimal but supplies no comparable closed-form family. Together, these results support the possibility of exceptionally dense disordered sphere packings in high dimensions and strengthen the case for the Torquato--Stillinger realizability conjecture.
ER  -

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