Rotational Symmetry and Multiplication in the 96-Dimensional Khanhnion Framework: A Unit Obstruction, a Classification of Equivariant Products, and a Cross-Product Construction

This paper examines the compatibility of multiplication, spatial rotational symmetry, and a two-sided identity in the real 96-dimensional Khanhnion framework. The underlying representation is K = R³ ⊗ W, where W = R² ⊗ S is a 32-dimensional internal space and S denotes the real sedenion algebra. Proper spatial rotations act on the R³ factor, while the internal factor carries the trivial action. The paper establishes that no real-bilinear multiplication on this representation can be both SO(3)-equivariant and equipped with a two-sided identity. The argument uses the uniqueness of an identity and the absence of nonzero vectors fixed by every spatial rotation; associativity is not required. Using the standard classification of invariant rank-three Cartesian tensors, the paper characterizes all equivariant bilinear products as combinations of the three-dimensional cross product and a bilinear operation on the internal space. A concrete product based on symmetrized sedenion multiplication is constructed, with explicit structure constants. It is anticommutative and rotation-equivariant, but an explicit counterexample shows that it fails the Jacobi identity and therefore does not define a Lie algebra. The discussion also distinguishes proper rotations from reflections and presents a 97-dimensional unitization that restores a two-sided identity by adding a rotation-invariant scalar direction. This work is a mathematical exposition and framework-development note. It does not claim priority for the underlying symmetry arguments, invariant-tensor classification, or unitization. No new sphere-packing density bound, proof of the Kepler theorem, physical quantum model, or computational advantage is established. Author: Nguyen Kim KhanhProject: Khanh MathematicsDate: 29 October 2026Series: Paper 6 AI assistance was used in mathematical formulation, organization, and wording. Independent verification of the arguments, references, and relationship to prior work is required before publication.