A proposed E 8 × E 8 kinematic scaffolding for the standard model with pre-gravitation

Abstract

Abstract We present a proposed E 8 × E 8 kinematic scaffolding for the standard model with pre-gravitation. The notation E 8 × ωE 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) × E 6 and thence through trinification, E 6 → 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>C</m:mi> <m:msub> <m:mrow> <m:mi>l</m:mi> </m:mrow> <m:mrow> <m:mn>6</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $C{l}_{6}\left(\mathbb{C}\right)$ , 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: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${J}_{2}\left({\mathbb{H}}_{s}\right)$ realises the vector space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3,3</m:mn> </m:mrow> </m:msup> </m:math> ${\mathbb{R}}^{3,3}$ through its determinant, and after choosing <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msub> <m:mo>≅</m:mo> <m:msub> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${\mathbb{H}}_{s}\cong {M}_{2}\left(\mathbb{R}\right)$ 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">O</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${J}_{3}\left({\mathbb{O}}_{\mathbb{C}}\right)$ supplies the proposed internal, generation and cubic spectral layer through the magic-star decomposition of