The Parity Dual-Sphere Invariance Theorem: Constructive Multi-Sphere Packings Beyond Behrend's Bound for Arithmetic Progression-Free Sets
Year: 2026
Type: preprint
Abstract: We introduce the Parity Dual-Sphere Invariance Theorem, establishing that the union of two concentric hyperspheres S(R^2) and S(R^2 + 1) in the integer lattice [0, b-1]^m contains no non-trivial three-term arithmetic progressions. The proof rests on a fundamental parity contradiction: any progression spanning both shells forces the midpoint difference norm squared to satisfy 2||x - y||^2 = 1, which has no solution in the integer lattice. By extending this invariance to an independent radius hypergraph, we formulate maximal multi-sphere families executed on an NVIDIA GeForce RTX 3090 GPU, achieving constructive densities +250% to +402.9% beyond Behrend's classic single-sphere lower bound with 100% certified absence of 3-APs.
Author: Gad-Allah, Hussam
Institution: Cairo University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225100326