Sedenions and Sphere-Packing Geometry: Does an Algebraic Encoding Preserve Geometric Information?

This methodological note examines the potential role and limitations of sedenion algebra in representing three-dimensional sphere-packing configurations.

The paper constructs an isometric encoding of four ordered neighbor vectors into the underlying 16-dimensional Euclidean space of the sedenions. It proves that the encoding preserves the product Euclidean distance between ordered configurations, while emphasizing that this property uses only the vector-space structure, not sedenion multiplication.

An explicit example shows that the scalar norm of an encoded configuration does not determine whether its constituent spheres satisfy pairwise non-overlap constraints. The paper then uses the existence of nonzero zero divisors to establish that no universal positive lower bound of the form ||ab|| ≥ c||a||||b|| can hold for all sedenions.

This obstruction is carefully distinguished from a domain-specific claim: the note does not establish that a zero-divisor pair occurs within the particular packing-encoding domain.

The discussion identifies requirements for a meaningful algebraic application, including geometric interpretation, rotational and permutation invariance where appropriate, control of degeneracies, and a demonstrable contribution beyond coordinate repackaging.

The work does not claim a new proof of the Kepler theorem, a new sphere-packing density bound, or an established advantage over existing methods. It is a methodological investigation with elementary propositions, explicit limitations, and directions for further research. References to sedenion algebra and the established Kepler theorem are included.

Author: Nguyen Kim Khanh
Project: Khanh Mathematics
Date: 28 October 2026

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