A Dimensional Map of the Khanhnion Framework: From the 96-Dimensional Obstruction to the 97-Dimensional Unitization and the 128-Dimensional Scalar–Vector Extension

This paper gives a unified dimensional map of the Khanhnion framework developed in Papers 6–13. The starting representation is K = R³ ⊗ W, where W is a 32-dimensional internal space. Its real dimension is 96. Proper spatial rotations act only on the R³ factor, while W carries the trivial action. Because this representation has no nonzero rotation-invariant vector, no bilinear multiplication on K can be both SO(3)-equivariant and unital. This is the basic obstruction of the framework. The first response is to add one rotation-invariant scalar line: R ⊕ K. This gives a 97-dimensional unital, rotation-equivariant extension. For any rotation-equivariant product on K, the extended product preserves the original product exactly on the 96-dimensional vector sector. Within the class of extensions obtained by adjoining a rotation-invariant scalar line, 97 dimensions are minimal. This does not prove that 97 is minimal among all conceivable unital equivariant extensions. The second response is to add a full copy of the internal space: (R ⊕ R³) ⊗ W. This gives a 128-dimensional scalar–vector extension. It supports a quaternion-inspired product and, for a commutative internal product, induces a bracket on the 96-dimensional vector sector. The paper compares the two extensions: - 96 dimensions: original rotation-equivariant vector sector, no unit.- 97 dimensions: minimal scalar-line unitization, unit present, original product preserved.- 128 dimensions: scalar–vector extension, unit present, quaternion-inspired product, internal scalar sector added. The behavior of the 128-dimensional extension depends on the internal product. If the internal product is commutative and associative, the vector-sector commutator satisfies the Jacobi identity. For W = R³² with componentwise multiplication, the algebra is isomorphic to H³². If the internal product is symmetrized sedenion multiplication, the algebra is nonassociative and its vector-sector bracket generally fails Jacobi. The failure is governed by nonzero associators. The paper does not claim that 97 is the unique minimal dimension among all possible extensions, that 128 dimensions are necessary, that either construction gives a division algebra, that either construction solves sphere packing, or that either construction gives a physical quantum model. No new sphere-packing density bound, proof of the Kepler theorem, physical quantum model, gauge symmetry, conservation law, or computational advantage is established. Author: Nguyen Kim KhanhProject: Khanh MathematicsDate: 31 October 2026Series: Paper 14 AI assistance was used in mathematical formulation, organization, and wording. Independent verification of the statements, definitions, and relationship to prior work is required before publication.