A Unified Framework for High-Dimensional Pure Root Lattices, Sphere Packing, and Cosmological Implications

TL;DR

This paper introduces a comprehensive framework combining high-dimensional lattice theory, advanced algorithms, and classical sphere packing results to explore mathematical structures and potential cosmological implications, including universe emergence theories.

Contribution

It presents a unified approach integrating lattice theory, computational algorithms, and sphere packing to analyze high-dimensional structures and their cosmological significance.

Findings

Development of a new dimension formula for pure root lattices

Implementation of polynomial-time approximation algorithms for SVP

Discussion of cosmological implications related to universe emergence from a white hole

Abstract

We propose a unified framework that synthesizes advances in high-dimensional lattice theory with novel computational algorithms for the shortest vector problem (SVP) to model pure root lattices and compute sphere packing densities. Building on our pure root lattice formulation characterized by a novel dimension formula and minimal vector length scaling. we integrate the recent polynomial-time approximation algorithm for SVP and discrete Gaussian sampling techniques. Our work also draws on classical results in sphere packing bounds via spherical codes and the rich structure of exceptional lattices such as the Leech lattice. Finally, we discuss how these results may have cosmological implications specifically, supporting the possibility that our universe emerges from a white hole.