A Scalar–Vector Extension of the Khanhnion Framework: A 128-Dimensional Unital, Rotation-Equivariant Algebra and Its 96-Dimensional Vector Sector

This paper develops a 128-dimensional scalar–vector extension of the real Khanhnion framework. The preceding 96-dimensional representation is K = R³ ⊗ W, where W is a 32-dimensional internal space. Proper spatial rotations act on the R³ factor, while the internal factor is rotation-invariant. Under this action, no equivariant bilinear multiplication can possess a two-sided identity. To address that obstruction, this paper considers A = (R ⊕ R³) ⊗ W. Its real dimension is 128. The added sector is a copy of W, not merely one real coordinate. A quaternion-inspired multiplication is defined for elements written as scalar–vector pairs (p, x), where p ∈ W and x ∈ W³. The product combines internal scalar multiplication, a contracted vector product, and a W-valued cross-product term: (p, x) ⊙ (q, y)=( β(p, q) − D_β(x, y), β(p, y) + β(x, q) + C_β(x, y) ). Here β is the internal bilinear product, D_β is a contracted dot-type operation, and C_β is a cross-type operation. The paper proves that this multiplication is real-bilinear, has a two-sided identity, and is equivariant under every proper spatial rotation. The identity lies in the rotation-invariant scalar sector. For a commutative internal product, the vector projection of the product of pure vector elements recovers the cross-product construction of Paper 6. The commutator of two arbitrary elements has no scalar component and lies in the 96-dimensional vector sector. An explicit example uses W = S ⊕ S, where S denotes the real sedenion algebra, with componentwise symmetrized multiplication. In this example, the resulting 128-dimensional algebra is nonassociative and has zero divisors, so it is not a division algebra. The paper also gives the structure constants of the extended product. They are the products of quaternion structure constants and the structure constants of the chosen internal algebra. The construction is a tensor-product algebra built from quaternions and a specified internal algebra. It does not establish a new independent number system. The paper does not claim that 128 dimensions are minimal. A 97-dimensional ordinary unitization of the preceding framework also restores a two-sided identity. The purpose of the present extension is to match scalar and vector internal coefficients and to produce a quaternion-inspired scalar–vector multiplication. 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 7 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.