Minimal Unital, Rotation-Equivariant Extensions of the Khanhnion Framework: A 97-Dimensional Unitization, a 128-Dimensional Scalar–Vector Algebra, and the Role of the Invariant Sector

This paper studies unital, rotation-equivariant extensions of the 96-dimensional Khanhnion framework. The starting representation is K = R³ ⊗ W, where W is a 32-dimensional internal space. Proper spatial rotations act only on the R³ factor, while W carries the trivial action. The representation therefore has no nonzero vector fixed by every rotation. As shown in Paper 6, this implies that no real-bilinear multiplication on K can be both SO(3)-equivariant and equipped with a two-sided identity. This paper examines two ways to restore a unit. The first is the ordinary unitization R ⊕ K, which has real dimension 97. Its unit is a rotation-invariant scalar direction. For any rotation-equivariant bilinear product on K, the extended product (α, x) ⋆ (β, y)=(αβ, αy + βx + x ⋄ y) is unital and rotation-equivariant. On the original 96-dimensional vector sector, this product preserves the original product exactly. The paper proves that this 97-dimensional unitization is minimal among extensions obtained by adjoining a rotation-invariant scalar line. It does not prove that 97 is minimal among all conceivable unital equivariant extensions under arbitrary changes of the representation. The second construction is the 128-dimensional scalar–vector extension from Paper 7: A = (R ⊕ R³) ⊗ W. Here the added invariant sector is a copy of W, not merely one real line. The product combines internal scalar multiplication, a contracted dot-type term, and a W-valued cross-type term. Its vector projection recovers a cross-type product on the original 96-dimensional sector. The paper compares the two constructions. The 97-dimensional unitization preserves the original product exactly and uses the smallest possible rotation-invariant scalar addition. The 128-dimensional extension gives scalar and vector components the same internal coefficient space and supports a quaternion-inspired multiplication. For a concrete internal choice using componentwise symmetrized sedenion multiplication, both extensions can be nonassociative and may have zero divisors. Neither construction is a division algebra. 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 8 AI assistance was used in mathematical formulation, organization, and wording. Independent verification of the arguments, definitions, and relationship to prior work is required before publication.