Khanhnions: A 96-Dimensional Tensor Framework with a Componentwise Sedenion Product

This paper introduces “Khanhnions” as proposed terminology for a real 96-dimensional tensor framework combining a two-channel index, a three-dimensional spatial index, and a 16-dimensional sedenion index.

The underlying space is defined as K = R² ⊗ R³ ⊗ S, where S is the real sedenion algebra. A Euclidean inner product and a componentwise multiplication are specified explicitly. Under this multiplication, the resulting unital real algebra is isomorphic to the direct product of six sedenion algebras.

The paper develops:

• A basis and coordinate representation for the 96-dimensional space.
• Explicit structure constants using a recursive Cayley–Dickson sign rule and binary exclusive OR.
• The block decomposition, conjugation, associator coefficients, and zero-product degeneracies.
• A counterexample showing that componentwise multiplication is not equivariant under general spatial rotations.
• A rotation-invariant sedenion-valued pairing.
• A Gram-matrix formulation of finite unit-sphere packing constraints and reconstruction up to an orthogonal transformation.
• A distinction between the real two-channel framework and its complex extension, which has 96 complex dimensions and 192 real dimensions.

The construction is presented as a mathematical framework and methodological research note. It does not claim a new proof of the Kepler theorem, a new packing-density bound, or a demonstrated advantage over conventional coordinate, tensor, or Gram-matrix methods.

“Khanhnions” is proposed terminology; the underlying tensor construction, direct-product algebra, and elementary propositions are not asserted to be original mathematical results. No comprehensive priority assessment has been completed.

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

AI assistance was used in mathematical formulation, organization, translation, and wording. Independent verification of the arguments, references, and relationship to prior work is recommended before publication.