A Proposed $E_8\times E_8$ Kinematic Scaffolding for the Standard Model with Pre-Gravitation

Abstract

We present a proposed $E_8\times E_8$ kinematic scaffolding for the standard model with pre-gravitation. The notation $E_8\times \omega E_8$ used below records a split-complex exchange grading of two factors; it is not a tensor product of groups. Each factor branches through $SU(3)\times E_6$ and thence through trinification, $E_6\to SU(3{)}^3$. The first factor supplies representation labels associated with the standard-model lineage, and the second supplies a mirror, pre-gravitational lineage. The identification of a single vector-like colour group across the two factors, and the proposed gravitational reading of the right-sector $SU(2)$, remain dynamical hypotheses. Physical chiral fermions are not assigned to the $E_8$ adjoints: they are modelled by minimal ideals of the complex Clifford algebra $C{l}_6(\mathbb{C})$, while the $496$ adjoint dimensions are used only as a representation-label ledger. The numerical split $208+288$ is therefore a declared roster-matching convention, not an invariant decomposition and not a particle count. Spacetime enters through a selected six-dimensional split-biquaternionic vector space with a separately specified quadratic form of signature $(3,3)$. Its full frame group is $SO(3,3)$; deriving a soldering form and the proposed $BF$/Plebański dynamics remains open. The conventional Clifford--Dirac operator is constructed independently and exactly: $J_2({\mathbb{H}}_s)$ realises the vector space ${\mathbb{R}}^{3,3}$ through its determinant, and after choosing ${\mathbb{H}}_s\cong M_2(\mathbb{R})$ its matrix gradient and adjugate act between real four-dimensional Weyl modules and factorise the wave operator in both orderings. The rank-two algebra supplies the quadratic spacetime layer, whereas $J_3({\mathbb{O}}_{\mathbb{C}})$ supplies the proposed internal, generation and cubic spectral layer through the magic-star decomposition of ${\mathfrak{e}}_8^{\mathbb{C}}$.

Comments: v4: 23 pages, one figure, significant revisions, supersedes earlier versions. Accepted for publication in Zeitschrift fur Naturforschung A