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

Abstract

Torquato and Stillinger conjectured an exponential improvement of Minkowski's classical lower bound on the maximal density of sphere packings in high-dimensional Euclidean space $\mathbb{R}^d$ using a pair-correlation-function optimization framework. Conditional on their realizability conjecture, we show that a simple family of hyperuniform pair correlation functions yields polynomial improvements over Minkowski's lower bound of the form $\phi_{\mathrm{max}} \gtrsim d^\beta 2^{-d}$ for every fixed $\beta>1$ in sufficiently high dimensions. As the polynomial exponent is allowed to increase with dimension, this family continuously approaches the previously conjectured exponential improvement. We further derive the same exponential asymptotic rate independently from the Cohn--Elkies dual linear programming upper bound formulation, demonstrating that its radial objective test functions cannot asymptotically exclude packings with the Torquato--Stillinger density scalings. The agreement between these alternative approaches provides new evidence that exceptionally dense disordered sphere packings may exist in high dimensions and strengthens the case for the Torquato--Stillinger conjectural lower bound.