Analysis of Conjectural Improvements to Minkowski's Lower Bound on the Sphere Packing Density
Torquato and Stillinger conjectured an exponential improvement of Minkowski's classical lower bound on the maximal density of sphere packings in $\mathbb{R}^d$, with exponential rate $2^{-(0.7786524795\ldots+o(1))d}$, 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 $ϕ_{\max}\gtrsim d^β2^{-d}$ for every fixed $β>1$ in the high-$d$ limit. As the polynomial exponent increases with dimension, this family approaches the conjectured exponential improvement. We first give an explicit hyperuniform construction based on Gauss--Radau quadrature. It contains $\lfloor(d-1)/4\rfloor$ positive delta-function shells and has exponential rate $2^{-(0.622556248918\ldots+o(1))d}$, showing how growing radial complexity surpasses the Torquato--Stillinger rate. We then prove strong duality between the unrestricted pair-correlation program and its Cohn--Elkies dual: their optimal values coincide, with no duality gap. Combining this result with approximation by ordinary finite-band functions, we show that the unrestricted pair-correlation program attains the optimal Cohn--Elkies exponential rate $2^{-(0.6044005\ldots+o(1))d}$ in the high-$d$ limit, although this general result does not supply a comparable closed-form family. We further derive the Torquato--Stillinger exponential rate independently from the Cohn--Elkies dual linear-programming upper-bound formulation, showing that its radial objective test functions cannot asymptotically exclude packings with the Torquato--Stillinger density scaling. The agreement between these approaches provides evidence that exceptionally dense disordered sphere packings may exist in high dimensions and supports the Torquato--Stillinger conjectural lower bound.