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This research aims to establish a differential algebraic framework for dense sphere packings across multiple geometrical settings.

February 16, 2026Open Access

Unified Differential Algebraic Theory of Dense Sphere Packing: From First Principles to High-Dimensional Constructions

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This framework demonstrates optimal packing density in non-euclidean and hyperbolic spaces, indicating new computational complexities.

This research aims to establish a differential algebraic framework for dense sphere packings across multiple geometrical settings.
Developed a packing differential closure for dense sphere packings.
Utilized model theory to establish geometric differential closures with constant fields.
Applied linear programming methods to derive an upper bound for the kissing number in various settings.
Proved the existence of locally optimal packing configurations in fixed containers.
Established bounds for the minimum fidelity of maximally entangled states in quantum state space.
Demonstrated that solving high-density packing problems is equivalent to the existential theory of the reals.

shifa liu (2025) studied this question.

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