Spacetime and Internal Symmetry from Split Bioctonions and the Two Extra SU(3)’s of E₈ × ωE₈

1. Introduction

Over the last few years we have proposed and developed an $E_8\times \omega E_8$ theory of unification which aims to unify the standard model with gravitation described by the general theory of relativity [2,3,4]. Here, $\omega$ is the split complex number. It is assumed that each of the two $E_8$ branches as $SU\left(3\right)\times E_6$ and each of the two resulting $E_6$ undergoes a trinification $E_6\to SU\left(3\right)\times SU\left(3\right)\times SU\left(3\right)$. The split complex number plays a crucial role, enabling the emergence of a Lorentzian signature for spacetime, and enabling the emergence of chiral fermions. Its origin in our theory can be traced to (left acting) octonionic chains made from the algebra $\mathbb{O}\oplus \omega \mathbb{O}$ of split bioctonions [5].

Remark on Clifford algebras and split-complex bookkeeping. 

Left multiplication by the seven imaginary octonion units defines $8\times 8$ real matrices satisfying Euclidean Clifford relations, hence yields a representation of $\mathrm{Cl}(0,7)$. In the standard real classification one has

$$\mathrm{Cl}(0,7)\cong {\mathrm{Mat}}_8\left(\mathbb{R}\right)\oplus {\mathrm{Mat}}_8\left(\mathbb{R}\right),\mathrm{Cl}(0,6)\cong {\mathrm{Mat}}_8\left(\mathbb{R}\right),$$

so $\mathrm{Cl}(0,7)\cong \mathrm{Cl}(0,6)\oplus \mathrm{Cl}(0,6)$ as real algebras. When split-complex scalars ${\mathbb{C}}_s$ are introduced, the idempotents $e_{\pm }=(1\pm \omega )/2$ implement ${\mathbb{C}}_s\cong \mathbb{R}\oplus \mathbb{R}$, so that ${\mathbb{C}}_s\otimes A\cong A\oplus A$ for any real algebra A. In this paper, the symbol $\omega$ is used only as a sector label associated with $e_{\pm }$; no nonstandard Clifford-algebra isomorphism is assumed.

The trinification provides the following interpretation of the branching of the two $E_8$, as discussed in detail in [4]:

$$\begin{array}{cc}\hfill E_{8L}& \to SU{\left(3\right)}_L^{geom}\times E_{6L}\stackrel{E_{6L}}{\to }SU{\left(3\right)}_c\times SU{\left(3\right)}_{F,L}\times SU{\left(3\right)}_L\stackrel{SU{\left(3\right)}_L}{\to }SU{\left(2\right)}_L\times U{\left(1\right)}_{{\gamma }_1}\to U{\left(1\right)}_Y\hfill \\ \hfill E_{8R}& \to SU{\left(3\right)}_R^{geom}\times E_{6R}\stackrel{E_{6R}}{\to }SU{\left(3\right)}_{c^{\prime }}\times SU{\left(3\right)}_{F,R}\times SU{\left(3\right)}_R\stackrel{SU{\left(3\right)}_R}{\to }SU{\left(2\right)}_R\times U{\left(1\right)}_{{\gamma }_2}\to U{\left(1\right)}_{Ydem}\hfill \end{array}$$

(1)

and

$$\mathbf{248}=(\mathbf{78},\mathbf{1})\oplus (\mathbf{1},\mathbf{8})\oplus (\mathbf{27},\mathbf{3})\oplus (\overline{\mathbf{27}},\overline{\mathbf{3}})$$

(2)

Of the three $SU\left(3\right)$ factors arising from $E_{6L}$, $SU{\left(3\right)}_c$ is identified with colour. The factor $SU{\left(3\right)}_{F,L}$ is proposed as a global flavour symmetry acting on three Peirce modules of $J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$; neither the identification of those modules with the observed generations nor their coupling to the physical Clifford ideals follows from the branching alone. The remaining $SU{\left(3\right)}_L$ contains an $SU{\left(2\right)}_L\times U\left(1\right)$ subgroup with the representation labels used for the electroweak sector.

For the second $E_6$, the treatment of $SU{\left(3\right)}_{c^{\prime }}$ is not fixed by the branching. A viable completion must either identify it dynamically with visible colour or break, confine or decouple it. Likewise, interpreting $SU{\left(3\right)}_{F,R}$ as a right-handed flavour symmetry and the $SU{\left(2\right)}_R$ subgroup as a self-dual leaf spin connection are modelling identifications. The proposed $U{\left(1\right)}_{dem}$ is also incomplete until its charged spectrum, anomaly cancellation and coupling normalisation are specified. These dynamical questions are not resolved by the present scaffold; companion work develops one proposed gravi–weak mechanism [1].

Gravity statement (clarification). 

In this framework we do not claim that breaking a compact internal gauge group produces a massless spin–2 graviton. Rather, the relevant $SU\left(2\right)$ is interpreted as part of the spin-connection/Lorentz structure on a Lorentzian leaf (self-dual/Plebanski language). The Lorentz–Higgs order parameter that selects a negative 2-plane in the $(3,3)$ ambient space reduces $SO(3,3)$ to the leaf Lorentz group and renders the mixed Lorentz components massive, leaving the tangent spin connection massless. The resulting low-energy leaf dynamics is described by a 4D BF/Plebanski-type gravity sector (developed in detail in [1]).

$U{\left(1\right)}_{\mathrm{dem}}$ effective-coupling normalisation. 

To make the proposed “couples to $\sqrt{m}$” statement dimensionless, let $m_f\left(\mu \right)=y_f\left(\mu \right)v_M$ with $v_M$ a reference mass scale (e.g., a VEV). Define the low-energy effective label

$$Q_{\mathrm{dem}}\left(f\right):=\sqrt{y_f\left(\mu \right)}=\sqrt{m_f\left(\mu \right)/v_M},$$

which is necessarily scale- and scheme-dependent through $y_f\left(\mu \right)$. It cannot be a fixed charge of a fundamental compact $U\left(1\right)$; Section 10 states the consistent effective and high-scale interpretations.

The focus of the present article is the two extra factors labelled $SU{\left(3\right)}_L^{geom}$ and $SU{\left(3\right)}_R^{geom}$ in the two $E_8\supset E_6\times SU\left(3\right)$ branchings, distinct from the six $SU\left(3\right)$ factors arising after trinification of the two $E_6$ factors. We propose to use their reduced representation bundles as geometric structure. Independently, the split-biquaternionic coordinate seed supplies a selected six-dimensional real vector space which, once equipped with the quadratic form of Section 3, has signature $(3,3)$. Identifying the two triplet bundles with its tangent bundle requires a soldering form. Selecting and localising the two overlapping Lorentzian leaves requires the additional dynamics described below. Neither a non-compact frame group nor a spacetime manifold follows by group contraction or dimension counting from the compact $SU\left(3\right)$ factors alone.

Thus the goal of the note is to formulate precisely the proposed scaffold and to separate its algebraic consequences from its dynamical assumptions. The notation $E_8\times \omega E_8$ denotes an exchange-graded pair of sectors, not a tensor product with a number or a new Lie group. The phrase “atoms of space-time-matter” refers to the conjectured trace-dynamics completion [3], not to a result derived here.

Literature context and positioning. 

Embeddings of $E_8\times E_8$ have a long history in the heterotic string, where the ten-dimensional theory with gauge group $E_8\times E_8$ is compactified to four dimensions, typically on Calabi–Yau threefolds with $SU\left(3\right)$ holonomy [6,7,8,9]. In that framework, four-dimensional spacetime is assumed a priori and its detailed physics emerges from string vacua and compactification data. By contrast, our construction starts from the purely group-theoretic chain $E_8\supset E_6\times SU\left(3\right)$ on each factor of $E_8\times E_8$ and uses the two extra $SU\left(3\right)$’s as structure groups to fix a $(3,3)$ six-dimensional base and its two embedded Lorentzian 4-planes, while the $E_6\times E_6$ sector supplies fields and matter. Thus, spacetime, internal fibre space, and field content are organised by $E_8\times E_8$ data within one algebraic–geometric scaffold (with the dynamical leaf selection supplied by the localisation of [1]), without appealing to a string worldsheet or compactification.

There is also a sizable literature proposing $E_8$-based unification in four dimensions. Lisi’s “$E_8$ Theory of Everything” used a superconnection on a fixed $3+1$ base to house Lorentz and Standard Model gauge symmetries inside $E_8$ [10]; representation-theoretic obstructions were later emphasized in [11]. More recently, Manogue–Dray–Wilson realized Standard Model fermions and $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$, together with $\mathfrak{so}(3,1)$, inside ${\mathfrak{e}}_{8(-24)}$ using octonionic and Clifford-algebraic methods [12,13]. Our approach differs in scope: the two extra $SU\left(3\right)$’s from $E_8\to SU\left(3\right)\times E_6$ are taken as geometric structure groups that carve out the $(3,3)$ base, two Lorentzian leaves, and canonical 4D internal fibres; the $E_6\times E_6$ sector then populates this geometry with gauge and matter fields.

On the $E_6$ side, trinification $SU{\left(3\right)}_C\times SU{\left(3\right)}_L\times SU{\left(3\right)}_R$ is a classic intermediate stage in $E_6$ GUT breaking chains [14,15,16]. We use it in both $E_6$ factors, with one $SU\left(3\right)$ per factor breaking to $SU\left(2\right)\times U\left(1\right)$ as the electroweak or gravi-dem sector, while the remaining $SU\left(3\right)$’s play color/flavor roles. The novelty here is that the additional $SU\left(3\right)$’s coming from the $E_8$ parents are not gauged sectors but geometric engines: they furnish three-plus-three directions via $\mathrm{Adj}SU\left(2\right)\simeq \Im \mathbb{H}$ and leave real 4D fibres from $2\oplus \overline{2}$, naturally aligning with the $C{P}^2=SU\left(3\right)/[SU\left(2\right)\times U\left(1\right)]$ geometry often used in Kaluza–Klein realizations of QCD.

Finally, the octonionic underpinning of exceptional symmetry is well documented [17,18,19,20]. Division-algebra approaches to Standard Model structure, notably Furey’s program [21], extract gauge and family data from $\mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$; her recent $H_{16}\left(\mathbb{C}\right)$ construction makes the quaternion/octonion factorization between spacetime and internal symmetries explicit [22], in close parallel to the split employed here, and our use of split bioctonions extends this logic to spacetime itself. In particular, Truini’s “magic star” projection and the role of Jordan pairs expose a natural $A_2\sim SU\left(3\right)$ outer symmetry acting on three Jordan pairs within each exceptional algebra [23,24]. This provides a mathematical rationale for why the branching $E_8\to SU\left(3\right)\times E_6$ on each side, hence four $SU\left(3\right)$’s in total, is a structurally natural starting point for obtaining spacetime directions, internal fibres, and $E_6\times E_6$ matter within one unified scheme. To our knowledge, no previous $E_8\times E_8$ construction produced simultaneously a $(3,3)$ base with two Lorentzian 4D leaves, internal 4D fibres, and an $E_6\times E_6$ field sector directly from the two extra $SU\left(3\right)$’s.

Scope and the Distler–Garibaldi no-go result. 

Distler and Garibaldi [11] prove strong representation-theoretic constraints on attempts to realize Standard Model chirality directly inside a real form of $E_8$ (e.g., as components of a single $E_8$-representation with the Standard Model gauge group embedded as a subgroup). The present paper does not claim that chiral fermions arise from $E_8$ representation theory. Here $E_8$ plays a structural role through the decomposition $E_8\supset E_6\times SU{\left(3\right)}_{\mathrm{geom}}$: the two extra $SU{\left(3\right)}_{\mathrm{geom}}$ factors are used as geometric structure groups that fix the $(3,3)$ ambient sector, its Lorentzian leaves, and the canonical ${\mathbb{CP}}^2$ fibres. The matter/gauge sector is carried by $E_6^L\times E_6^R$ (which admits complex representations such as the $\mathbf{27}$), and the question of 4D chirality is tied to localisation/Spin^c structure on the leaves (outlined below and developed further in [1,2,4]).

2. Roadmap

Our mass-ratio programme [4] places matter in $E_{6L}\times \omega E_{6R}$ and uses the exceptional Jordan algebra $J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$ to derive charged-fermion square-root mass ratios and mixings. The present paper supplies the geometric scaffolding promised [see Section XII.H of [4] in the uplift to $E_8\times \omega E_8$: the two extra $\mathrm{SU}\left(3\right)$’s are taken as geometric structure groups that yield precisely a $(3,3)$ base, two 4D spacetimes, and two real 4D internal fibres. The $E_6^{L,R}$ fields then live on this scaffold. A complementary 6D gravi–weak reduction shows how two 4D leaves arise dynamically from a 6D BF theory [1].

What is new in the present article: 

(i) A detailed octonionic identification of the 4D fibres with the realification of $\mathbf{2}\oplus \overline{\mathbf{2}}$, using $\mathbb{O}=\mathbb{H}\oplus \mathbb{H}\epsilon$ and an intrinsic complex structure on $\mathbb{H}\epsilon$. (ii) A clean role for each extra $U\left(1\right)$ as the Spin^C line on the $\mathbb{C}{P}^2$ fibre. (iii) A unified presentation tying the split-bioctonion base, the $E_8$ branching, and the 6D gravi–weak localisation. (iv) An explicit account of Coleman–Mandula compliance: algebraic (direct-product structure) on each emergent leaf, and vacuous pre-geometrically, where no S-matrix exists (Section 10). (v) The exact $\mathrm{Cl}(3,3)$ Dirac operator of the base, constructed as a split-quaternionic matrix gradient on the rank-two Jordan algebra $J_2\left({\mathbb{H}}_s\right)\cong {\mathbb{R}}^{3,3}$, with exact factorisation of the $(3,3)$ wave operator and exact restriction to the Weyl operators of the two leaves (Section 5).

Plan of the paper: 

Section 3Section 4 build $M_6$ from split bioctonions and embed the two 4D spacetimes. Section 6Section 7 relate the $\mathrm{SU}{\left(3\right)}_{L,R}^{geom}$ branching to $\Im \mathbb{H}$ and $\mathbb{H}\epsilon$ and identify the 4D fibres with $T\mathbb{C}{P}^2$. Section 5 constructs the exact $\mathrm{Cl}(3,3)$ Dirac operator on the base, via the split quaternions and the Jordan algebra $J_2\left({\mathbb{H}}_s\right)$. Section 8 clarifies the two $U\left(1\right)$’s. Section 9 explains how the $E_{6L}\times E_{6R}$ fields live on the proposed scaffold, and Section 10 gives the big picture and interpretation.

Physical content and evaluability. 

Even at the present scaffold level, the construction yields several sharp structural outputs: (i) a fixed symmetry-breaking pattern for the ambient Lorentz group, $SO(3,3)\to SO(3,1)\times SO\left(2\right)$ on a leaf, with exactly eight mixed Lorentz modes becoming massive; (ii) a canonical ${\mathbb{CP}}^2$ fibre bundle associated to $SU{\left(3\right)}_{\mathrm{geom}}$, with $F_4=T^{\mathrm{vert}}\mathcal{Y}$ and an unavoidable Spin^C twist supplied by the geometric $U\left(1\right)$; (iii) a two-leaf structure with a $(1,1)$ overlap ${\Sigma }_L\cap {\Sigma }_R$ that constrains possible portal couplings. Detailed particle spectra and coupling values depend on the $E_6^L\times E_6^R$ dynamics and the localisation sector, treated in companion work [2,4].

What is derived versus what is assumed. 

To delineate the logical status of the construction, we separate conditional consequences from modelling choices. Derived once the stated geometric choices are made:

• the signature $(3,3)$ of $M_6$ from the explicitly defined quadratic form in Equation (6);
• the Clifford completion $\mathrm{Cl}(M_6,g)=\mathrm{Cl}(3,3)$ and ${\mathrm{Spin}}_0(3,3)\cong SL(4,\mathbb{R})$;
• the canonical identification $F_4\cong T\mathbb{C}{P}^2$ of the internal fibre, from the coset isotropy of $SU\left(3\right)/S\left(U\right(2)\times U(1\left)\right)$;
• the unavoidable Spin^C twist on $F_4$, since $w_2\left(T\mathbb{C}{P}^2\right)\ne 0$;
• the symmetry-breaking pattern $SO(3,3)\to SO(3,1)\times SO\left(2\right)$ on a leaf, with exactly eight mixed Lorentz modes becoming massive;
• the representation matches $\Im \mathbb{H}\cong {\mathbf{3}}_0$ and ${\left(\mathbb{H}\epsilon \right)}_{\mathbb{R}}\cong {(\mathbf{2}\oplus \overline{\mathbf{2}})}_{\mathbb{R}}$ after a quaternionic embedding is chosen.

Assumed or chosen (motivated, but not forced by the branching chain alone):

• that the two $E_8$-level $SU\left(3\right)$’s—as opposed to the six trinification $SU\left(3\right)$’s—are the geometric ones; this is motivated by the magic-star / Jordan-pair structure of [23,24] but is adopted here as an identification;
• the selection of $M_6=\Im {\mathbb{H}}_L\oplus \omega \Im {\mathbb{H}}_R$ as tangent space and the relative minus sign in Equation (6); neither follows from ${\omega }^2=+1$ or from the $E_8$ branching alone;
• the assignment of adjoint triplets to base directions and doublets to internal fibres (Section 7 shows this assignment is fixed by the invariant structures—the $\omega$-glued real form on the triplets, the intrinsic complex structure on the doublets—once a Lorentzian base and a complex fibre are required; that requirement is the actual choice);
• the left/right physical-role assignment (which leaf is gravitational and which is the weak-curved mirror), set by the placement of $\omega$ together with the localisation dynamics of [1];
• the choice of quaternionic frame $\mathbb{H}\subset \mathbb{O}$, which is canonical only up to $G_2$ (Section 7);
• the dynamical realisation of leaf localisation and of 4D chirality, supplied by [1] and outlined in Section 8 rather than established here.

3. Split-Bioctonionic Base $(M_6,g)$

In our $E_8\times \omega E_8$ theory of unification, an atom of space-time-matter (STM atom) is described by the trace dynamics action [3,25]

$${\displaystyle \frac{S}{\hslash }}={\displaystyle \frac{1}{2}}\int {\displaystyle \frac{d\tau }{{\tau }_{Pl}}}Tr{\displaystyle \frac{L_p^2}{L^2}}{\dot{Q}}_1^{†}{\dot{Q}}_2]$$

(3)

Here, $Q_1$ and $Q_2$ are two operators/matrices which together describe an STM atom. Evolution is described with respect to Connes time $\tau$. The trace Lagrangian is assumed invariant under an $E_8\times \omega E_8$ symmetry. Importantly, this action does not make any distinction between space-time and matter. Thus it is not that these two operators live on some abstract operator-valued space-time. Rather, $Q_1$ and $Q_2$ are the primitive matrix degrees of freedom of the STM atom, and the distinction between spacetime and matter emerges only after the subsequent symmetry breaking and bosonic/fermionic splitting. Prior to the proposed symmetry breaking the model assumes a very large ensemble of STM atoms. Estimates such as $N\sim {10}^{120}$ are heuristic inputs, not direct empirical measurements, and are not used in the algebraic results below. Matrix entries are taken from a Grassmann algebra. The action and Connes-time parameter are nondimensionalised with ℏ and ${\tau }_{Pl}$ respectively.

The asserted $E_8\times E_8$ invariance of Equation (3) is presently conditional. If the STM variables take values in a specified representation $\mathcal{R}$, if the involution ^† is defined by an invariant form, and if the physical fermion module is coupled equivariantly, then conjugation invariance follows from cyclicity of the trace. Those data have not yet been supplied for the combined geometric and Clifford-module state space. Moreover, a bare trace is invariant under a much larger similarity group, so trace invariance alone does not select $E_8\times E_8$. In this paper the exceptional group is therefore established as an organising symmetry of the branching labels; its promotion to an exact symmetry of the microscopic action is an open construction.

For the adjoint-valued sector one may nevertheless write the invariant quadratic form explicitly as the Killing form. This observation establishes invariance of an adjoint-valued kinetic pairing, but it does not solve the missing action on the physical fermion module. To avoid possible confusion, the word “adjoint” here refers to the adjoint representation of the Lie group, not to the Hermitian adjoint ^†. Writing

$$\mathfrak{g}:={\mathfrak{e}}_8\oplus \omega {\mathfrak{e}}_8,$$

the statement that the microscopic variables $Q_a\left(\tau \right)$ are “in the adjoint” of $E_8\times \omega E_8$ means that they are $\mathfrak{g}$-valued:

$$Q_a\left(\tau \right)=Q_a^I\left(\tau \right)T_I,$$

with $\left\{T_I\right\}$ a basis of generators of $\mathfrak{g}$. The group acts on these variables by the adjoint action

$$Q_a⟼{\mathrm{A}d}_g{Q}_a:=g{Q}_a{g}^{-1},{\delta }_{ϵ}{Q}_a=[ϵ,Q_a],$$

or, in components,

$${\delta }_{ϵ}{Q}_a^I={f^I}_{JK}{ϵ}^J{Q}_a^K,$$

where ${f^I}_{JK}$ are the structure constants of $\mathfrak{g}$.

The natural quadratic object on a Lie algebra is an invariant symmetric bilinear form. For $\mathfrak{g}={\mathfrak{e}}_8\oplus \omega {\mathfrak{e}}_8$, the canonical choice is the direct sum of the Killing forms on the two $E_8$ factors. In components this is a constant tensor ${\kappa }_{IJ}$ satisfying

$${f_{AI}}^K{\kappa }_{KJ}+{f_{AJ}}^K{\kappa }_{IK}=0,$$

equivalently,

$$K([Z,X],Y)+K(X,[Z,Y])=0,K(X,Y):={\kappa }_{IJ}{X}^I{Y}^J.$$

Hence the quadratic STM kinetic term can be written intrinsically as

$$L_{\mathrm{kin}}\propto K({\dot{Q}}_1,{\dot{Q}}_2)={\kappa }_{IJ}{\dot{Q}}_1^I{\dot{Q}}_2^J,$$

which is manifestly invariant under $E_8\times \omega E_8$.

The trace form in Equation (3) is the matrix realisation of this invariant pairing after one chooses a representation of $\mathfrak{g}$ on the trace-dynamics matrix space. In such a representation one has, up to an overall normalisation,

$${\kappa }_{IJ}\propto \mathrm{Tr}\left(T_I{T}_J\right),$$

so that the invariant bilinear form is the representation-independent object, whereas the trace is the concrete form naturally adapted to Adler trace dynamics. Thus one could start abstractly from $L\propto K$, but for trace dynamics it is preferable to retain the trace expression, since the basic variational calculus and the Adler–Millard structure are formulated in terms of trace polynomials of noncommuting matrices. The dagger in Equation (3) refers to the involution chosen on this matrix realisation and should not be confused with the word “adjoint” in “adjoint representation.”

A possible geometric motivation for the quadratic form of the STM action is the special status of octonionic projective geometry. Because of the nonassociativity of the octonions, there is no octonionic projective space $\mathbb{O}{P}^n$ for arbitrary n; the octonionic tower terminates at the projective plane $\mathbb{O}{P}^2$. Correspondingly, the exceptional geometry associated with the Albert algebra $J_3\left(\mathbb{O}\right)$ is controlled by the basic invariant tensors already present at this level: the invariant bilinear pairing and the cubic norm (equivalently, the associated symmetric trilinear form). In this sense, binary and ternary structures are geometrically privileged in the octonionic setting.

This does not imply a strict no-go theorem forbidding higher-order invariant terms in the action. Rather, it suggests that the bilinear term is the minimal and most natural kinematical choice for the primitive STM dynamics, while higher-degree invariants, if present, should be regarded as additional interaction terms. Thus the trace expression in Equation (3) may be viewed as the lowest-order exceptional invariant naturally adapted to trace dynamics, with the later spacetime–matter split and symmetry breaking supplying the richer interaction structure.

At the electroweak transition, segregation into space-time and matter takes place, and this is mathematically defined as follows:

$$\begin{array}{c}\hfill {\dot{Q}}_1^{†}={\dot{Q}}_B^{†}+{\displaystyle \frac{L_p^2}{L^2}}{\beta }_1{\dot{Q}}_F^{†};{\dot{Q}}_2={\dot{Q}}_B+{\displaystyle \frac{L_p^2}{L^2}}{\beta }_2{\dot{Q}}_F;\\ \hfill {\dot{Q}}_B={\displaystyle \frac{1}{L}}(i\alpha q_B+L{\dot{q}}_B);{\dot{Q}}_F={\displaystyle \frac{1}{L}}(i\alpha q_F+L{\dot{q}}_F)\end{array}$$

(4)

The matrices $Q_B$ are made of even grade Grassmann numbers and are called bosonic; whereas the matrices $Q_F$ are made of odd-grade Grassmann numbers and are called fermionic, as is the case in quantum field theory. ${\beta }_1$ and ${\beta }_2$ are unequal odd-grade Grassmann numbers, introduced to make the Lagrangian bosonic. L is a length scale which characterises the STM atom, and it is made dimensionless by introducing Planck length $L_p$. Along with Planck time and Planck’s constant, these are the only three fundamental constants in the theory - all dimensionless fundamental constants must be derived from this theory. Thus, the matrices $Q_1$ and $Q_2$ have been split into their bosonic and fermionic parts, which respectively describe bosons and fermions.

Furthermore, the bosonic part $Q_B$ has been split into $q_B$ and ${\dot{q}}_B$. The $q_B$ will describe QCD ($SU{\left(3\right)}_c$) and `gravi-QCD’ ($SU{\left(3\right)}_{c^{\prime }}$) and electromagnetism $U{\left(1\right)}_{em}$ and dark electromagnetism ($U{\left(1\right)}_{dem})$. Whereas ${\dot{q}}_B$ will describe the gravi-weak interaction from which general relativity (with $SU{\left(2\right)}_R$ identified, per the clarification above, as the self-dual leaf Lorentz/spin connection in Plebanski form) and the weak interaction (spontaneously broken $SU{\left(2\right)}_L$) will emerge. The gravi-weak interaction is the geometry of the 6D spacetime - this latter is one of the two ingredients to emerge from the extra $SU{\left(3\right)}_L^{geom}\times \omega SU{\left(3\right)}_R^{geom}$. The other ingredient to emerge from the two extra $SU\left(3\right)$s is the internal symmetry space for $SU{\left(3\right)}_c,SU{\left(3\right)}_{c^{\prime }},U{\left(1\right)}_{em},U{\left(1\right)}_{dem}$, as we show in the present article. $\alpha$ is the Yang-Mills coupling constant. In the fermionic sector the ${\dot{q}}_F$ will describe leptons while the $q_F$ will describe quarks. In terms of these new variables the Lagrangian becomes

$${\displaystyle \frac{S}{\hslash }}=\int {\displaystyle \frac{d\tau }{{\tau }_{Pl}}}Tr\left\{{\displaystyle \frac{L_P^2}{L^2}}\left[\left({\dot{q}}_B^{†}+i{\displaystyle \frac{\alpha }{L}}{q}_B^{†}\right)+{\displaystyle \frac{L_P^2}{L^2}}{\beta }_1\left({\dot{q}}_F^{†}+i{\displaystyle \frac{\alpha }{L}}{q}_F^{†}\right)\right]\times \left[\left({\dot{q}}_B+i{\displaystyle \frac{\alpha }{L}}{q}_B\right)+{\displaystyle \frac{L_P^2}{L^2}}{\beta }_2\left({\dot{q}}_F+i{\displaystyle \frac{\alpha }{L}}{q}_F\right)\right]\right\}$$

(5)

Via the 16D split bioctonions $\mathbb{O}\oplus \omega \mathbb{O}$ [5], the two geometric $SU\left(3\right)$s provide the scaffolding on which the dynamical matrices ${\dot{Q}}_B$ and ${\dot{Q}}_F$ live. The ${\dot{q}}_B$ live on one 8D half of the 16D split bioctonion and the $q_B$ live on the other 8D half. The same holds for ${\dot{q}}_F$ and $q_F$. We now justify these remarks in detail.

Definition 1

(Split bioctonions $\mathbb{O}\oplus \omega \mathbb{O}$ [5] and metric). Let ${\mathbb{C}}_s$ denote split-complex numbers with generator ${\omega }^2=+1$ and let $\mathbb{O}$ be the octonions. Choose quaternionic subalgebras ${\mathbb{H}}_L\subset \mathbb{O}$ and ${\mathbb{H}}_R\subset \mathbb{O}$. Define the 6D real vector space

$$M_6:=\Im {\mathbb{H}}_L\oplus \omega \Im {\mathbb{H}}_R,g(x_L\oplus \omega x_R,y_L\oplus \omega y_R):=\langle x_L,y_L\rangle -\langle x_R,y_R\rangle ,$$

(6)

with $\langle ·,·\rangle$ the Euclidean inner product induced by the octonion norm. Here ω tags the right sector; the relative minus sign is part of the definition of g.

3.0.0.10. Clifford-algebra status of the construction.

For one common quaternionic algebra, the split-biquaternion algebra is

$${\mathbb{B}}_s:={\mathbb{C}}_s\otimes \mathbb{H}\cong \mathbb{H}\oplus \mathbb{H}\cong \mathrm{Cl}(0,3),$$

with three grade-one generators. Its second imaginary quaternionic triplet is represented by bivectors, not by three additional grade-one generators. Hence the six displayed directions in Equation (6) do not constitute the grade-one space of $\mathrm{Cl}(0,3)$ and are not multiplied as six Clifford generators. Instead, after independently choosing g, their Clifford completion is

$$\mathrm{Cl}(M_6,g)=\mathrm{Cl}(3,3)\cong {\mathrm{Mat}}_8\left(\mathbb{R}\right),{\mathrm{Spin}}_0(3,3)\cong SL(4,\mathbb{R}).$$

The seed has real dimension 8 whereas $\mathrm{Cl}(3,3)$ has dimension 64. If ${\mathbb{H}}_L$ and ${\mathbb{H}}_R$ are different quaternionic subalgebras, $\Im {\mathbb{H}}_L\oplus \omega \Im {\mathbb{H}}_R$ is a selected real vector subspace, not automatically a subalgebra. Independent sectors can instead be written $e_{+}{\mathbb{H}}_L\oplus e_{-}{\mathbb{H}}_R$ in the split-idempotent basis.

Complexification gives $\mathbb{C}\otimes \mathrm{Cl}(0,3)\cong {\mathrm{Mat}}_2\left(\mathbb{C}\right)\oplus {\mathrm{Mat}}_2\left(\mathbb{C}\right)$ and $\mathrm{Spin}(3,\mathbb{C})\cong SL(2,\mathbb{C})$; it does not create a six-dimensional spacetime Clifford algebra. For either Lorentzian leaf, ${\mathrm{Spin}}_0(3,1)\cong SL(2,\mathbb{C})$ embeds in $SL(4,\mathbb{R})$ through the relevant stabiliser. The two leaves carry separately induced $SL(2,\mathbb{C})$ structures, not a direct-product subgroup $SL(2,\mathbb{C})\times SL(2,\mathbb{C})\subset SL(4,\mathbb{R})$.

The full $SO(3,3)$ is the orthonormal frame group of the defined quadratic space. Quaternionic conjugations directly realise only $SO{\left(3\right)}_L\times SO{\left(3\right)}_R$, its maximal compact identity component; the mixed noncompact generators arise only in the full frame structure.

Proposition 1

(Signature and isometries). $(M_6,g)$ has signature $(3,3)$. The action of unit quaternions by conjugation on each $\Im \mathbb{H}$ has kernel $\{\pm 1\}$ and therefore descends to a faithful isometric action of $\mathrm{SO}{\left(3\right)}_L\times \mathrm{SO}{\left(3\right)}_R$, the identity component of the maximal compact subgroup $S\left(O\left(3\right)\times O\left(3\right)\right)$ of $\mathrm{SO}(3,3)$. Equivalently, $\mathrm{SU}{\left(2\right)}_L\times \mathrm{SU}{\left(2\right)}_R=\mathrm{Spin}\left(4\right)$ acts through its $\mathrm{SO}\left(3\right)\times \mathrm{SO}\left(3\right)$ quotient; on the spin cover $\mathrm{Spin}(3,3)\cong \mathrm{SL}(4,\mathbb{R})$ this lifts to the maximal compact subgroup $\mathrm{SO}\left(4\right)=(\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right))/{\mathbb{Z}}_2$.

With the above construction, $(M_6,g)$ indeed has signature $(3,3)$, confirming the count of time vs space dimensions. Moreover, there is an interesting symmetry of this metric: each quaternionic subalgebra $H_L$ or $H_R$ has an $SU\left(2\right)$ of unit quaternions (the set of $a\in H$ with $\left|a\right|=1$ is isomorphic to $SU\left(2\right)$). These act on $\Im H$ by conjugation (i.e., $q::x\mapsto qx{q}^{-1}$ for $x\in \Im H$), which is a rotation of the 3D space of imaginary quaternions. Thus $SU{\left(2\right)}_L$ acts isometrically on $\Im H_L$ and $SU{\left(2\right)}_R$ acts on $\Im H_R$. Together these give an isometric action of $SU{\left(2\right)}_L\times SU{\left(2\right)}_R$ on $M_6$; the action is not faithful, and the effective isometry group is the quotient $SO{\left(3\right)}_L\times SO{\left(3\right)}_R$, as follows. In fact the maximal compact subgroup of $SO(3,3)$ is $S\left(O\right(3)\times O(3\left)\right)$, with identity component $SO{\left(3\right)}_L\times SO{\left(3\right)}_R$, and the conjugation action realises precisely this $SO{\left(3\right)}_L\times SO{\left(3\right)}_R$ as the rotations of the $3+3$ split subspaces. (On the spin cover, $\mathrm{Spin}(3,3)\cong SL(4,\mathbb{R})$, whose maximal compact subgroup is $SO\left(4\right)=(SU\left(2\right)\times SU\left(2\right))/{\mathbb{Z}}_2$; the two unit-quaternion groups give $SU{\left(2\right)}_L\times SU{\left(2\right)}_R=\mathrm{Spin}\left(4\right)$, acting through its $SO\left(3\right)\times SO\left(3\right)$ quotient. Note that $\mathrm{Spin}\left(4\right)\neg \cong SO\left(4\right)$, the former being a double cover of the latter.) This matches our construction: rotations in $\Im H_L$ and $\Im H_R$ separately. These $SU\left(2\right)$ symmetries will later be identified with subgroups of the $SU{\left(3\right)}^{\mathrm{geom}}$ structure groups (since $SU\left(2\right)\subset SU\left(3\right)$ in a standard way). So at this stage, we have built a 6D pseudo-Riemannian manifold $M_6$ with appropriate symmetry – we think of it as our toy “bulk” spacetime before selecting the physical 4D slices.

Next, we note that there are two overlapping 4D spacetimes, with relatively flipped signature, embedded in this 6D spacetime. After spontaneous symmetry breaking of the gravi-weak interaction, gravitation, and the weak interaction, respectively live on these two 4D spacetimes as their geometry, one each. Because the two spacetimes have a $(1,1)$ overlap, the two directions exclusively linked with the weak interaction appear as internal symmetry directions from the vantage point of our 4D spacetime - this latter being curved by gravitation.

4. Two Embedded 4D Spacetimes

Pick unit vectors $t_L\in \Im {\mathbb{H}}_L$ and $t_R\in \Im {\mathbb{H}}_R$. Define

$$\begin{array}{cc}\hfill {\mathcal{M}}_4^{\left(R\right)}& :=\omega \Im {\mathbb{H}}_R\oplus \mathrm{span}\left\{t_L\right\}(\mathrm{one}\mathrm{plus},\mathrm{three}\mathrm{minus}),\hfill \end{array}$$

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$$\begin{array}{cc}\hfill {\mathcal{M}}_4^{\left(L\right)}& :=\Im {\mathbb{H}}_L\oplus \mathrm{span}\left\{\omega t_R\right\}(\mathrm{three}\mathrm{plus},\mathrm{one}\mathrm{minus}).\hfill \end{array}$$

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These two Lorentzian 4-planes intersect in a neutral $(1,1)$ 2-plane $\mathrm{span}\{t_L,\omega t_R\}$. This is the kinematic version of the two-leaf picture; a dynamical realisation from a 6D BF theory is given in [1].

Detailed explanation:

Now that we have $M_6$ with (3,3) signature, the next task is to identify two Lorentzian 4-dimensional subspaces inside it. Geometrically, we seek two different 4D “planes” in $M_6$ that have signature (3,1) and (1, 3) respectively (one time + three space for one, and 1 space + 3 time for the other). The construction is as follows:

Pick a unit imaginary quaternion $t_L\in \Im H_L$. This is a unit 3-vector in the left 3-space, which we designate as a time-like direction for one of the 4D subspaces. Similarly pick a unit imaginary quaternion $t_R\in \Im H_R$. This will serve a similar role for the other subspace. Using these, define two 4D subspaces of $M_6$:

$M_4^{\left(R\right)}:=\omega \Im H_R\oplus \mathrm{span}\left\{t_L\right\}$. Its vectors are $\omega x_R+\alpha t_L$; the first summand contributes three negative directions and the second one positive direction. We interpret $M_4^{\left(R\right)}$ as one embedded 4D Lorentzian spacetime (with $t_L$ serving as a time direction for it, since it contributes the lone “+” in the metric on that subspace).

$M_4^{\left(L\right)}:=\Im H_L\oplus \mathrm{span}\left\{\omega t_R\right\}$. Its vectors are $x_L+\beta \omega t_R$; the first summand contributes three positive directions and the second one negative direction.

These two subspaces $M_4^{\left(R\right)}$ and $M_4^{\left(L\right)}$ each are isomorphic to ordinary 4D Minkowski space (at least locally), but note how they are oriented differently in the 6D space: one’s time axis lies in $H_L$ direction, the other’s in $H_R$ (with an $\omega$ factor). Importantly, these two 4D planes are not completely separate – they intersect along a 2-dimensional plane given by $\mathrm{s}pan\{t_L,\omega t_R\}$. This intersection has one basis vector from $\Im H_L$ and one from $\omega \Im H_R$, yielding signature (1,1) (one +, one −), which is a neutral plane. The presence of this overlap means the two 4D worlds share a common 1+1 dimensional subspace. In physical terms, one might imagine that there is a 2D “bridge” or intersection between our universe and the parallel flipped-signature universe. We call this a “kinematic version of the two-leaf picture” – i.e., we have simply chosen two leaves in the 6D bulk. The fully dynamical story (how fields and gravity localize on these leaves) is deferred to a gravi-weak theory in 6D [5] but at least kinematically we see how two 4D spacetimes can coexist and overlap in a 6D (3,3) spacetime.

4.1. Lorentz Covariance of the Embedded 4D Leaves

We now justify rigorously as to how 4D Lorentz invariance is preserved after selection of the 4D leaves. Let $(M_6,g)$ have signature $(3,3)$. Fix an oriented negative 2–plane $N\subset T_x{M}_6$ and set its orthogonal complement

$$W:=N^{\perp }\subset T_x{M}_6,\mathrm{so}{\left.g\right|}_W\mathrm{has}\mathrm{signature}(3,1).$$

Write $\eta =\mathrm{diag}(+,+,+,-,-,-)$ for a local orthonormal frame on $T_x{M}_6$ and collect an orthonormal basis of N in the $6\times 2$ matrix $U=\left[n_1{n}_2\right]$ with $U^T\eta U=-{\mathbf{1}}_2$.

Stabilizer and Lie algebra split.

The stabilizer of N in $SO(3,3)$ is

$$H:=\{g\in SO(3,3)\mid gN=N\}\cong SO(3,1)\times SO\left(2\right),$$

which acts as $SO(3,1)$ on W and as $SO\left(2\right)$ on N. At the Lie algebra level,

$$\mathfrak{so}(3,3)=\underset{6}{\underbrace{\mathfrak{so}\left(W\right)}}\oplus \underset{1}{\underbrace{\mathfrak{so}\left(N\right)}}\oplus \underset{8}{\underbrace{(W\wedge N)}}.$$

(9)

The 8 generators in $W\wedge N$ are precisely the transformations that mix the 4D axes in W with the two discarded axes in N.

Kinematic (projector) construction.

Define the metric projector onto W by

$$\Pi :={\mathbf{1}}_6+U{U}^T\eta .$$

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Then ${\Pi }^2=\Pi$, $\mathrm{Im}\Pi =W$, and $\mathrm{Ker}\Pi =N$. For any $g\in H$, one has $g\Pi g^{-1}=\Pi$, so the residual local symmetry on W is $SO(3,1)$. Impose the leaf constraint by projecting all tensors/frames:

$$V_A\mapsto {(\Pi V)}_A,T_{AB}\mapsto {(\Pi T\Pi )}_{AB},\mathrm{etc}.$$

Within this background, the mixed generators in $W\wedge N$ (which would rotate W into N) are broken: they do not preserve $\Pi$ and so are not symmetries of the leaf. Hence fields restricted by (10) transform Lorentz–covariantly under $SO(3,1)$ on the leaf.

Dynamical (Lorentz–Higgs) construction.

The order parameter must select a two-plane, not an ordered two-frame. Let $U\left(x\right)$ obey $U^T\eta U=-{\mathbf{1}}_2$ but identify $U\sim UR\left(x\right)$ for $R\left(x\right)\in SO\left(2\right)$, equivalently use the plane projector $P_N=-U{U}^T\eta$. A schematic invariant kinetic term is

$${\mathcal{L}}_P={\displaystyle \frac{\alpha }{2}}Tr\left(D_{\mu }{P}_N{D}^{\mu }{P}_N\right)-V\left(P_N\right),D_{\mu }{P}_N={\partial }_{\mu }{P}_N+[{\omega }_{\mu },P_N].$$

(11)

A vacuum $\langle P_N\rangle$ has stabiliser $SO(3,1)\times SO\left(2\right)$. The eight mixed connections in $W\wedge N$ may then acquire mass, while the tangent $SO(3,1)$ and normal $SO\left(2\right)$ connections remain in the stabiliser. By contrast, the earlier kinetic term built directly from an ordered U without a right $SO\left(2\right)$ redundancy would break the normal rotation as well and would have nine, not eight, Goldstone modes. Equation (11) is a symmetry-consistent effective ansatz; deriving it from the proposed $BF$/trace dynamics remains open [1].

Goldstone count and mixed Lorentz components.

Fixing a negative 2-plane $N\subset {\mathbb{R}}^{3,3}$ breaks

$$SO(3,3)⟶SO(3,1)\times SO\left(2\right),$$

so the number of broken generators is

$$dimSO(3,3)-\left(dimSO(3,1)+dimSO\left(2\right)\right)=15-(6+1)=8.$$

Equivalently, writing $W:=N^{\perp }$ (signature $(3,1)$), the mixing sector has dimension $dim(W\wedge N)=4\times 2=8$. These eight Goldstones are eaten by the eight mixed spin connections ${\omega }_{\mu }^{a\widehat{b}}\in W\wedge N$ (unitary gauge), while the normal $SO\left(2\right)$ acts as the structure group of the normal bundle.

Localization in the action.

To confine dynamics to the leaf, either insert projectors $\Pi$ on all tensor indices, or use a closed Thom form (or distributional Poincaré dual current) $\rho$ for an embedded leaf so that ${\int }_{M_6}{\alpha }_4\wedge \rho ={\int }_{{\Sigma }_4}{\iota }^{*}{\alpha }_4$. This localisation is an additional dynamical/geometric input, not a consequence of the projector alone. Off-plane components then drop out; the surviving symmetry is $SO(3,1)$.

Two leaves.

For the two leaves of Section 4 the normal 2-planes have opposite definiteness: the normal of ${\Sigma }_L=\Im {\mathbb{H}}_L\oplus \mathrm{span}\left\{\omega t_R\right\}$ is the negative 2-plane $N_L={\left\{\omega t_R\right\}}^{\perp }\cap \omega \Im {\mathbb{H}}_R$, while the normal of ${\Sigma }_R=\omega \Im {\mathbb{H}}_R\oplus \mathrm{span}\left\{t_L\right\}$ is the positive 2-plane $N_R={\left\{t_L\right\}}^{\perp }\cap \Im {\mathbb{H}}_L$. (The leaf-Lorentz derivation above, phrased for a negative normal plane, therefore applies verbatim to ${\Sigma }_L$; for ${\Sigma }_R$ the signs flip as follows.) For a normal frame with $U_X^T\eta U_X={\sigma }_X{\mathbf{1}}_2$, ${\sigma }_X=\pm 1$, the projector generalising Equation (10) is ${\Pi }_X={\mathbf{1}}_6-{\sigma }_X{U}_X{U}_X^T\eta$, and each leaf ${\Sigma }_X:=\mathrm{Im}{\Pi }_X$ carries its own residual Lorentz group, $SO(3,1)\times SO\left(2\right)$ for ${\sigma }_L=-1$ and the isomorphic $SO(1,3)\times SO\left(2\right)$ for ${\sigma }_R=+1$; the Goldstone count of the previous paragraph, $15-(6+1)=8$, is unchanged in both cases. Their intersection ${\Sigma }_L\cap {\Sigma }_R=\mathrm{span}\{t_L,\omega t_R\}$ is 2–dimensional with signature $(1,1)$. Localizing the left/right sector actions with ${\rho }_L,{\rho }_R$ yields an induced Lorentz structure on each leaf. Treating the two leaf connections as dynamically independent requires separate pullback fields or a specified matching condition on their intersection; it is not automatic from their common embedding in $M_6$.

Why there is no mixing. The only transformations that would mix the chosen time/spatial axes in W with the two discarded directions in N are the broken coset generators $W\wedge N$ in (9). Once N is fixed (kinematically by $\Pi$ or dynamically by $\langle U\rangle$), these do not act on physical fields on the leaf. Within the unbroken subgroup, $t_1\in W$ mixes only with the three spatial directions of W (ordinary Lorentz boosts and rotations). The unbroken local symmetry on each leaf is exactly $SO(3,1)$, which justifies 4D Lorentz covariance on the embedded leaves.

To summarize this part: one 4D spacetime (${\Sigma }_R$) uses the entire right-imaginary quaternion space plus one left-imaginary direction for time; the other 4D spacetime (${\Sigma }_L$) uses the left-imaginary quaternion space plus one right-imaginary (with $\omega$) direction for time. One can think of ${\Sigma }_R$ as primarily “right-handed” (since it uses $\Im H_R$ spatially) and ${\Sigma }_L$ as “left-handed” (uses $\Im H_L$ spatially), hinting at a connection to handedness of weak interactions. Indeed ${\Sigma }_L$ might correspond to our universe where $SU{\left(2\right)}_L$ acts (left-handed weak force on left-chiral particles), while ${\Sigma }_R$ is a hidden sector where $SU{\left(2\right)}_R$ acts (and its breaking gave gravity). We will see later that ${\Sigma }_R$ is where unbroken QCD lives (with possibly gravity), and ${\Sigma }_L$ might be a hidden mirror or vice versa. The (1,1) intersection could allow some communication or matching conditions between the two sectors. This elegant geometric picture emerges purely from the octonionic and $\omega$-complex structure we imposed.

Remark on signature $(3,3)$ and causality.

The $(3,3)$ metric on $M_6$ is used here as an ambient algebraic scaffold that naturally accommodates two Lorentzian 4D leaves. The physical sectors are assumed to be localised on these Lorentzian leaves; propagation and causal structure are therefore governed by the induced leaf metrics. We do not assert a fundamental $(3,3)$ bulk field theory with unrestricted ultrahyperbolic propagation. A full stability/causality analysis is required only once a specific bulk dynamics is postulated; in the present paper the bulk is primarily a kinematic organiser for the two-leaf structure and the internal fibres.

5. The Exact $\mathrm{Cl}(3,3)$ Dirac Operator: Split Quaternions and $J_2\left({\mathbb{H}}_s\right)$

The Clifford-algebra status paragraph of Section 3 established what the split-biquaternion seed does not provide: six grade-one generators. This section provides the missing Dirac layer exactly, on a carrier adapted to the base $(M_6,g)$: the split quaternions.

Definition 2

(Split quaternions and $J_2\left({\mathbb{H}}_s\right)$). Let ${\mathbb{H}}_s$ be the real algebra spanned by $1,\mathbf{i},\mathbf{j},\mathbf{k}$ with ${\mathbf{i}}^2=-1$, ${\mathbf{j}}^2={\mathbf{k}}^2=+1$, $\mathbf{i}\mathbf{j}=\mathbf{k}=-\mathbf{j}\mathbf{i}$, all imaginary units mutually anticommuting; conjugation negates the imaginary part, and $N\left(q\right)=q\overline{q}$ has signature $(2,2)$. As a real algebra ${\mathbb{H}}_s\cong M_2\left(\mathbb{R}\right)\cong \mathrm{Cl}(1,1)$; it is a split composition algebra with zero divisors. Let $J_2\left({\mathbb{H}}_s\right)$ denote the rank-two Jordan algebra of $2\times 2$ Hermitian matrices over ${\mathbb{H}}_s$, whose norm form (the determinant) is quadratic (real diagonal, conjugate off-diagonal entries, symmetrised product).

Proposition 2

($J_2\left({\mathbb{H}}_s\right)\cong {\mathbb{R}}^{3,3}$). Writing, in an orthonormal frame of $(M_6,g)$ with t-triple from $\Im {\mathbb{H}}_L$ and x-triple from $\omega \Im {\mathbb{H}}_R$, adapted to the leaves of Section 4 by taking $t_1=t_L$ and $x_1=\omega t_R$,

$$X=\left(\begin{array}{cc}{t}_1+x_1& q\\ \overline{q}& t_1-x_1\end{array}\right),q=x_2+\mathbf{i}{x}_3+\mathbf{j}{t}_2+\mathbf{k}{t}_3,$$

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one has $detX=t_1^2+t_2^2+t_3^2-x_1^2-x_2^2-x_3^2$. Hence $J_2\left({\mathbb{H}}_s\right)\cong {\mathbb{R}}^{3,3}$ as quadratic spaces, and, fibrewise, $J_2\left({\mathbb{H}}_s\right)$ is a Jordan-algebraic realisation of $(M_6,g)$: the Jordan completion of the six-dimensional base, supplying its determinant norm, its full $SO(3,3)$ norm symmetry, and (below) its Clifford principal symbol.

This is the split-signature member of the classical family $J_2\left(\mathbb{C}\right)\cong {\mathbb{R}}^{1,3}$, $J_2\left(\mathbb{H}\right)\cong {\mathbb{R}}^{1,5}$, $J_2\left(\mathbb{O}\right)\cong {\mathbb{R}}^{1,9}$ underlying the division-algebra description of Minkowski spacetimes [17,26,27], and an instance of the split-composition sector of the $2\times 2$ magic square [28,29]; the classical isomorphisms $\mathrm{Spin}(3,3)\cong SL(4,\mathbb{R})$ and $\mathrm{Cl}(3,3)\cong M_8\left(\mathbb{R}\right)$ are as in [30].

Proposition 3

(Exact factorisation and leaf restrictions). Choose an algebra isomorphism $\rho :{\mathbb{H}}_s\to M_2\left(\mathbb{R}\right)$ and apply it entrywise, writing ${\rho }_2\left(X\right)$ for the resulting real $4\times 4$ map between Weyl modules. Define the split-quaternionic matrix gradient and its adjugate,

$${\nabla }_{3,3}=\left(\begin{array}{cc}{\partial }_{t_1}+{\partial }_{x_1}& D_q\\ {\overline{D}}_q& {\partial }_{t_1}-{\partial }_{x_1}\end{array}\right),{\tilde{\nabla }}_{3,3}=\left(\begin{array}{cc}{\partial }_{t_1}-{\partial }_{x_1}& -D_q\\ -{\overline{D}}_q& {\partial }_{t_1}+{\partial }_{x_1}\end{array}\right),$$

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with $D_q={\partial }_{x_2}+\mathbf{i}{\partial }_{x_3}+\mathbf{j}{\partial }_{t_2}+\mathbf{k}{\partial }_{t_3}$. Then: (i) $D_q{\overline{D}}_q={\overline{D}}_q{D}_q={\partial }_{x_2}^2+{\partial }_{x_3}^2-{\partial }_{t_2}^2-{\partial }_{t_3}^2$ with no cross terms; (ii) ${\nabla }_{3,3}{\tilde{\nabla }}_{3,3}={\tilde{\nabla }}_{3,3}{\nabla }_{3,3}={\square }_{3,3}\mathbb{I}$ exactly, in both orderings, with no operator remnant; (iii) restricting to fields constant along $t_2,t_3$ gives the Weyl operator on the leaf with one plus and three minus directions, with ordered product ${\partial }_{t_1}^2-{\partial }_{x_1}^2-{\partial }_{x_2}^2-{\partial }_{x_3}^2$; restricting along $x_2,x_3$ yields the flipped leaf with three plus and one minus direction; and the joint restriction yields the $(1,1)$ bridge operator.

All three statements follow by direct computation (we have verified them symbolically); (i) holds because the imaginary units of ${\mathbb{H}}_s$ mutually anticommute, and (ii) is the $2\times 2$ adjugate identity. The Dirac system is the chiral pair

$$i\hslash {\rho }_2\left({\nabla }_{3,3}\right){\psi }_{-}=Qc{\psi }_{+},i\hslash {\rho }_2\left({\tilde{\nabla }}_{3,3}\right){\psi }_{+}=Qc{\psi }_{-},$$

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and eliminating either component places each chiral field on the $(3,3)$ Klein–Gordon shell. Here ${\psi }_{\pm }\in {\mathbb{R}}^4$; unrestricted columns in ${\mathbb{H}}_s^2$ would be eight-real-dimensional and would give a reducible doubled module. The real matrix $\Gamma \left(X\right)=\left(\begin{array}{cc}0& {\rho }_2\left(X\right)\\ {\rho }_2\left(\tilde{X}\right)& 0\end{array}\right)$ obeys $\Gamma {\left(X\right)}^2=det\left(X\right)\mathbb{I}$ and generates $\mathrm{Cl}(3,3)\cong M_8\left(\mathbb{R}\right)$. The Weyl modules $S_{\pm }\cong {\mathbb{R}}^4$ are not the two leaves: algebraic chirality and leaf support are distinct, and their correlation is dynamical. The construction is the split-signature analogue of the quaternionic $(5,1)$ formalism of [31]; earlier $(3,3)$ treatments include [32,33,34].

Soldering, made explicit. 

One structural step, implicit in Section 6, should be stated as a requirement. The two ${\mathbf{3}}_0$’s are representations of the $\mathrm{SU}\left(2\right)\times U\left(1\right)$ reductions of the geometric $\mathrm{SU}\left(3\right)$’s (Section 6); the corresponding rank-three real bundles $E_L,E_R$ are, as constructed, associated bundles of these reduced structure groups; identifying $E_L\oplus E_R$ with the tangent bundle $T{M}_6$ requires a soldering form $e:T{M}_6\stackrel{\sim }{\to }{E}_L\oplus E_R$, with the metric pulled back from the invariant forms of the two factors and the split grading fixing the relative sign, $g=e^{*}({\kappa }_L\oplus (-{\kappa }_R))$. Without e, the geometric $\mathrm{SU}\left(3\right)$’s supply two internal rank-three bundles; with it, they supply spacetime. The $BF$/Plebański variables of the companion work [1] are the natural dynamical supplier of this soldering, and its explicit construction is an open task. Proposition 2 is then to be read fibrewise over this soldered base, and the curved-space extension of ${\nabla }_{3,3}$ (frame fields plus spin connection valued in $\mathfrak{so}(3,3)$, acting on $S_{\pm }$) is the corresponding open extension of Proposition 3.

Jordan division of labour, and the Albert algebra. 

The appearance of a Jordan algebra for the base dovetails with the exceptional Jordan algebra’s role in the wider programme, and the fit is by rank and invariant: $J_2\left({\mathbb{H}}_s\right)$ is rank two, its determinant is quadratic, and that quadratic form is the spacetime interval; $J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$ is rank three, its norm is cubic, and its characteristic roots are the spectral data of the three-generation mass structure [35,36]. The two are related inside the Albert algebra: with respect to a primitive idempotent, $J_3\left(\mathbb{O}\right)$ has Peirce decomposition $27=1+16+10$, whose ten-dimensional block is $J_2\left(\mathbb{O}\right)\cong {\mathbb{R}}^{1,9}$—the rank-two corner of the rank-three algebra. At the $E_8$ level the connection is the magic-star decomposition ${\mathfrak{e}}_8={\mathfrak{sl}}_3\oplus {\mathfrak{e}}_6\oplus (\mathbf{3}\otimes \mathbf{27})\oplus (\overline{\mathbf{3}}\otimes \overline{\mathbf{27}})$ [23,24], in which the $\mathbf{27}$ is $J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$ and the mixed $(\mathbf{3},\mathbf{27})$ sectors tie the geometric $\mathrm{SU}\left(3\right)$ index to the Albert-algebraic internal index: the scaffold’s own soldering channels between geometry and matter. One real-form subtlety must be recorded: $J_2\left({\mathbb{H}}_s\right)$, being indefinite, does not embed in the formally real $J_3\left(\mathbb{O}\right)$; it embeds in the split Albert algebra $J_3\left({\mathbb{O}}_s\right)$ (since ${\mathbb{H}}_s\subset {\mathbb{O}}_s$), and both real forms live inside the complexification $J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$. Which real structure selects the split $(3,3)$ spacetime sector and which selects the Hermitian mass sector is a choice the full framework must make explicitly; we record it here as an open point. A natural composite operator, schematically $D=**\partial *{*}_{3,3}\otimes \mathbb{I}+{\Gamma }_7\otimes D_{\mathrm{int}}$ with $D_{\mathrm{int}}$ built from the $J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$ data, would act identically on the three generations through its spacetime part and distinguish them through its internal part; exhibiting the commuting actions of $\mathrm{Cl}(3,3)$ and the internal algebra on one module—which must not simply be identified—is part of the same open task.

Two qualifications. 

The mixed $(\mathbf{3},\mathbf{27})$ summands carry both geometric and Albert indices, but representation content alone does not make them a soldering form; an explicit equivariant map to $T{M}_6$ and to the physical Clifford-module bundle is still required. Real forms must also be tracked: ${\mathrm{Str}}_0{J}_3\left(\mathbb{O}\right)=E_{6(-26)}$ and ${\mathrm{Str}}_0{J}_3\left({\mathbb{O}}_s\right)=E_{6\left(6\right)}$, while the compact $E_6$ in compact $E_8\supset SU\left(3\right)\times E_6$ acts on the complex $\mathbf{27}$ with an additional invariant Hermitian form. The common statement used here is therefore the complex magic-star decomposition; a compatible physical real-structure selection remains open.

6. The Two Extra $\mathrm{SU}\left(3\right)$’s and Their Branching

On each side of $E_8\times \omega E_8$ one has the maximal chain

$$E_8\supset E_6\times \mathrm{SU}\left(3\right),\mathbf{248}=(\mathbf{78},\mathbf{1})\oplus (\mathbf{1},\mathbf{8})\oplus (\mathbf{27},\mathbf{3})\oplus (\overline{\mathbf{27}},\overline{\mathbf{3}}).$$

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We propose to use the two extra $\mathrm{SU}\left(3\right)$ factors as geometric structure groups. Pick the standard embedding $\mathrm{SU}\left(2\right)\subset \mathrm{SU}\left(3\right)$ and the complementary $U\left(1\right)$ generator proportional to $\mathrm{diag}(1,1,-2)$. With the convenient normalisation where the doublet has unit charge, the adjoint branches as

$$\mathbf{8}⟶{\mathbf{3}}_0\oplus {\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1}\oplus {\mathbf{1}}_0.$$

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After choosing a soldering form, the two real ${\mathbf{3}}_0$ bundles can be identified with the two rank-three summands of $T{M}_6$. The realification of ${\mathbf{2}}_{+1}$ gives a rank-four associated internal vector bundle on each side.

Detailed explanation:

Having built the base manifold, we now connect it to the group theory origin: the extra $SU\left(3\right)$ factors that we set aside as “geometric”. Recall that each $E_8$ breaks as $E_6\times SU\left(3\right)$, so there is an $SU{\left(3\right)}_L^{\mathrm{geom}}$ from $E_{8L}$ and an $SU{\left(3\right)}_R^{\mathrm{geom}}$ from $E_{8R}$. We do not gauge these $SU{\left(3\right)}^{\mathrm{geom}}$’s as physical forces; instead, we use them to understand how the 6D base and internal fibres emerge. This involves examining how an $SU\left(3\right)$ can act on the structures we built and how its representations decompose. First, note the well-known branching of the $E_8$ adjoint (248-dim) under $E_6\times SU\left(3\right)$: $\mathbf{248}=(\mathbf{78},\mathbf{1})\oplus (\mathbf{1},\mathbf{8})\oplus \left(\mathbf{27},\mathbf{3}\right)\oplus (\overline{\mathbf{27}},\overline{\mathbf{3}})$

Now, we “use the two extra $SU\left(3\right)$’s as geometry”. Concretely, consider one of these $SU\left(3\right)$ groups (the discussion applies to either left or right). We pick the standard maximal subgroup $SU\left(2\right)\times U\left(1\right)$ inside $SU\left(3\right)$. This is done by choosing an $SU\left(2\right)$ that sits in $SU\left(3\right)$ (for example, the upper-left $2\times 2$ submatrix inside $3\times 3$ matrices), and a $U\left(1\right)$ generator that complements it – usually taken proportional to diag$(1,1,-2)$ in $3\times 3$ space. We take the $U\left(1\right)$ generator to be $t=\frac{1}{3}\mathrm{diag}(1,1,-2)$, so that the fundamental $\mathbf{3}$ of $SU\left(3\right)$ carries charges $(\frac{1}{3},\frac{1}{3},-\frac{2}{3})$; the adjoint doublet charges, being differences of fundamental charges, are then $\frac{1}{3}-(-\frac{2}{3})=\pm 1$—the unit-charge normalisation of (16). (The same difference reappears below as the charge $+1$ of the $\mathbb{C}{P}^2$ tangent.) With this normalization, the adjoint of $SU\left(3\right)$ (dimension 8) decomposes under $SU\left(2\right)\times U\left(1\right)$ as: $\mathbf{8}\to {\mathbf{3}}_0\oplus {\mathbf{2}}_{+1}\oplus {\mathbf{2}}_{-1}\oplus {\mathbf{1}}_0.$ This is a crucial decomposition, explicitly given above. Here:

${\mathbf{3}}_0$ is the adjoint of $SU\left(2\right)$ (a triplet with zero $U\left(1\right)$ charge).

${\mathbf{2}}_{+1}$ is a doublet of $SU\left(2\right)$ with charge $+1$.

${\mathbf{2}}_{-1}$ is the $SU\left(2\right)$ doublet with opposite charge $-1$ (effectively the complex conjugate representation of the above).

${\mathbf{1}}_0$ is a singlet with zero charge (this is the $U\left(1\right)$ generator itself).

Now, the insight is to map these group representation pieces to geometric components. On the left geometric $SU{\left(3\right)}_L^{\mathrm{geom}}$, the ${\mathbf{3}}_0$ piece will be identified with the 3 real dimensions of $\Im H_L$ (the spatial directions of the left part of $M_6$), and on the right $SU{\left(3\right)}_R^{\mathrm{geom}}$, its ${\mathbf{3}}_0$ corresponds to $\Im H_R$. In other words, the $SU\left(2\right)$ adjoint triplet inside $SU{\left(3\right)}^{\mathrm{geom}}$ is essentially the three rotational degrees of freedom of one of the quaternionic subspaces. This makes sense because we earlier saw $\Im H\cong \mathbf{3}$ of $SU\left(2\right)$, and indeed Proposition 4 below will formalize $\Im H\cong {\mathbf{3}}_0$ as representations.

Next, the ${\mathbf{2}}_{+1}\oplus {\mathbf{2}}_{-1}$ (together forming a complex 2-dimensional rep or 4-dimensional real rep) is interpreted as the 4 internal dimensions at each point – essentially the tangent space of $C{P}^2$ fibre. Specifically, one $\mathbf{2}\oplus \mathbf{2}$ from $SU{\left(3\right)}_L^{\mathrm{geom}}$ will correspond to a 4D internal fibre on the left side, and similarly for the right side. Thus, each $SU{\left(3\right)}^{\mathrm{geom}}$ yields a 3-dimensional “external” piece ($\Im H$) and a 4-dimensional “internal” piece ($\mathbf{2}\oplus \mathbf{2}$ real), summing to 7 dimensions; plus a singlet. The 7 corresponds to $\Im O$, but we’ll focus on the 3+4 split.

What about the remaining ${\mathbf{1}}_0$ in (13)? That is the $U\left(1\right)$ generator itself. In group terms it’s a central generator of the $SU\left(2\right)\times U\left(1\right)$ subgroup. Geometrically, one might wonder if this corresponds to the real line in the octonions (since O is 8-dimensional: 1 real + 7 imaginary parts). However, we emphasize that we do not identify the octonion’s real unit 1 with this $U\left(1\right)$ Instead, the $U\left(1\right)$ here is seen as acting on the complex structure of the 4-dimensional fibre (essentially rotations in the $C{P}^2$ tangent – more on this in the next sections). So ${\mathbf{1}}_0$ is kept distinct as a geometric $U\left(1\right)$ gauge field needed for the $Spi{n}^c$ structure, and not as a physical scalar.

7. Octonionic Realisation: $\mathbb{O}=\mathbb{H}\oplus \mathbb{H}\epsilon$ and the 4D Fibre

Fix a quaternionic subalgebra $\mathbb{H}=\langle 1,u,v,uv\rangle \subset \mathbb{O}$ and choose $\epsilon \in \Im \mathbb{O}$ orthogonal to $\mathbb{H}$ with ${\epsilon }^2=-1$. Then

$$\mathbb{O}=\mathbb{H}\oplus \mathbb{H}\epsilon ,\Im \mathbb{O}=\Im \mathbb{H}\oplus \mathbb{H}\epsilon .$$

(17)

Define on the real 4-space $\mathbb{H}\epsilon$ the complex structure J by left multiplication with a fixed unit $u\in \Im \mathbb{H}$:

$$J\left(a\epsilon \right):=\left(ua\right)\epsilon ,J^2=-{\mathrm{Id}}_{\mathbb{H}\epsilon }.$$

(18)

The stabilizer of $\mathbb{H}$ in $G_2=\mathrm{Aut}\left(\mathbb{O}\right)$ is $SO\left(4\right)\cong (\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right))/{\mathbb{Z}}_2$, acting by

$$a+b\epsilon ⟼qa{q}^{-1}+\left(pb{q}^{-1}\right)\epsilon ,(p,q)\in \mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right),$$

(19)

for unit quaternions $p,q$ [17]; that (19) is an automorphism of $\mathbb{O}$ is verified directly from the Cayley–Dickson product. Let $\mathrm{SU}\left(2\right)$ be the q-factor—acting on $\mathbb{H}$ by conjugation and on the fibre coefficient by right multiplication, $a\epsilon \mapsto \left(a{q}^{-1}\right)\epsilon$—and let $U\left(1\right)=\left\{e^{\theta u}\right\}$ inside the p-factor, which acts trivially on $\mathbb{H}$ and as the phases $e^{\theta J}$ on $(\mathbb{H}\epsilon ,J)$. Right multiplication commutes with the left-multiplication complex structure J, so the $\mathrm{SU}\left(2\right)$ action is complex-linear on $(\mathbb{H}\epsilon ,J)$; a conjugation action on the coefficient alone, $a\epsilon \mapsto \left(qa{q}^{-1}\right)\epsilon$, would instead decompose $\mathbb{H}\epsilon$ as $\mathbf{1}\oplus \mathbf{3}$ and would not commute with J. Then:

Proposition 4

(Identifications).

(a) 

$\Im \mathbb{H}\cong \mathrm{Adj}\mathrm{SU}\left(2\right)\cong {\mathbf{3}}_0$ as real representations.

(b) 

$(\mathbb{H}\epsilon ,J)\cong {\mathbf{2}}_{+1}$ as a complex $\mathrm{SU}\left(2\right)\times U\left(1\right)$ representation. Forgetting the complex structure, the underlying real representation is

$$\mathbb{H}\epsilon \cong {\left({\mathbf{2}}_{+1}\right)}_{\mathbb{R}},$$

(20)

a real 4-vector space. Its complexification is ${\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1}$, the off-diagonal part of (16). This is the internal fibre $F_4$.

Thus the octonionic split realises the branching (16) concretely: $\Im \mathbb{H}$ gives the 3’s for spacetime directions; $\mathbb{H}\epsilon$ gives the 4 internal directions.

Why the base/fibre assignment is natural once chosen. 

Within the present construction the proposed split of the adjoint into “spacetime” and “internal” pieces is compatible with the different invariant structures carried by the two summands. The triplet ${\mathbf{3}}_0\cong \Im \mathbb{H}$ is a real representation equipped with the $SO\left(3\right)$-invariant Euclidean form inherited from the octonion norm; it is precisely such real 3-blocks that the split tag $\omega$ glues with a relative sign to produce the indefinite $(3,3)$ base metric of Definition 1. The doublet sector $\mathbb{H}\epsilon \cong {\left({\mathbf{2}}_{+1}\right)}_{\mathbb{R}}$, by contrast, carries the chosen complex structure J defined above, which identifies it with the isotropy representation ${\mathbf{2}}_{+1}\cong T_{\left[n\right]}\mathbb{C}{P}^2$ of $S\left(U\right(2)\times U(1\left)\right)$ (Section 8). The two roles cannot be interchanged without discarding structure: using $\mathbb{H}\epsilon$ as base directions would require forgetting J—the very object that identifies it as a $\mathbb{C}{P}^2$ tangent—and there is no $\omega$ sign-flip acting on the doublet sector to render it Lorentzian. Hence, once $\omega$ acts on the $\Im \mathbb{H}$ blocks and J is intrinsic to $\mathbb{H}\epsilon$, the assignment “${\mathbf{3}}_0=$ base, ${\left({\mathbf{2}}_{+1}\right)}_{\mathbb{R}}=$ fibre” is determined within the chosen ansatz—given that a Lorentzian base is to be built by $\omega$-gluing real triplets and that the fibre is to retain its complex structure. That these two structures are to be so used is itself the modelling identification recorded in the “derived versus assumed” list of Section 2; what the present argument establishes is that, once made, the identification admits no alternative distribution of roles.

Detailed explanation: The octonions O come back into play now to realize the above abstract decomposition concretely. We use the fact that octonions contain many quaternionic subalgebras and can be split by a suitable choice of a new imaginary unit. We choose a specific decomposition: $O=H\oplus Hϵ$ with H a fixed quaternionic subalgebra of O and $ϵ$ an octonionic element orthogonal to H that behaves like a new imaginary unit (satisfying ${ϵ}^2=-1$). Here $Hϵ:=\{qϵ\mid q\in H\}$ is isomorphic (as a real vector space) to H itself, but consists of “quaternions times $ϵ$”. This Cayley–Dickson decomposition of the ordinary octonions yields: $\Im O=\Im H\oplus Hϵ$ Since $\Im H$ is 3-dimensional and $Hϵ$ is 4-dimensional (because H is 4-dim real, and we are not including the H’s real unit in $\Im O$), we recover $dim\Im O=3+4=7$. This matches the decomposition $8\to 3+4+1$ we saw (with the remaining 1 corresponding to the real line spanned by the identity in O). Essentially, the octonion’s imaginary part splits into a 3D part and a 4D part, exactly what we want for spacetime vs internal fibre.

Introducing a complex structure on $Hϵ$: To make that 4-dimensional real space the realification of a charged complex doublet, we identify it with a complex 2-dimensional vector space carrying an $SU\left(2\right)\times U\left(1\right)$ action. We accomplish this by defining a complex structure J on $Hϵ$ as follows: pick a specific unit imaginary quaternion $u\in \Im H$ (one of the three quaternionic basis elements, say u akin to i). For any element $aϵ\in Hϵ$ (with $a\in H$), define $J\left(aϵ\right):=(u·a)ϵ$ Since u is an imaginary quaternion with $u^2=-1$, one can check that $J^2\left(aϵ\right)=\left(u\left(ua\right)\right)ϵ=\left(u^{2}a\right)ϵ=-aϵ$, so indeed $J^2=-\mathrm{Id}$ on $Hϵ$. This means $Hϵ$ is now a complex vector space of complex dimension 2 (real dimension 4), with J playing the role of multiplication by i. We denote this complex vector space as $(Hϵ,J)$.

Next, let’s understand the group action. The action is the restriction of the $G_2$-stabilizer of H, Equation (19): the unit quaternion q acts on H by conjugation, rotating $\Im H$, and on the fibre by right multiplication of the coefficient, $q:aϵ\mapsto \left(a{q}^{-1}\right)ϵ$. One may check directly from the Cayley–Dickson multiplication rule that the combined map $a+bϵ\mapsto qa{q}^{-1}+\left(b{q}^{-1}\right)ϵ$ is an automorphism of O. Right multiplication commutes with the left-multiplication complex structure J defined above, so the action is complex-linear on $(Hϵ,J)$. Had we instead conjugated the coefficient, $aϵ\mapsto \left(qa{q}^{-1}\right)ϵ$, the fibre would decompose as $\mathbf{1}\oplus \mathbf{3}$ rather than as a doublet pair, and J would not be preserved; the right-multiplication action is what realises the doublet structure.

A $U\left(1\right)$ group is introduced to correspond to phase rotations on the complex structure J. Specifically, let $e^{\theta J}$ denote the linear map on $Hϵ$ that rotates by angle $\theta$ in the J-complex sense (i.e., it sends $v\mapsto cos\theta ,v+sin\theta ,Jv$). Because J acts like i, $e^{\theta J}$ is essentially multiplying by a phase $e^{i\theta }$ on the complex vector $(Hϵ,J)$. We identify this action with the $U\left(1\right)$ generator we had from $SU\left(3\right)\to SU\left(2\right)\times U\left(1\right)$. In other words, this $U\left(1\right)$ acts as $e^{i\theta }$ on the 2-dimensional complex space $Hϵ$.

Now we arrive at Proposition 4 (Identifications): Given the above setup:

(a) $\Im H\cong \mathbf{Adj},SU\left(2\right)\cong {\mathbf{3}}_0$ as real representations. This simply restates that the imaginary quaternions form the adjoint (3-dim) representation of $SU\left(2\right)$, with zero $U\left(1\right)$ charge.

(b) $(Hϵ,J)\cong {\mathbf{2}}_{+1}$ as a complex representation of $SU\left(2\right)\times U\left(1\right)$. This means that the 4-dimensional real space $Hϵ$ is, when viewed as a complex vector space, the fundamental doublet of $SU\left(2\right)$ carrying charge +1 under the $U\left(1\right)$. Concretely, one can choose a basis of $Hϵ$ such that $SU\left(2\right)$ acts by the 2-dimensional spin-${\displaystyle \frac{1}{2}}$ representation and $U\left(1\right)$ multiplies vectors by a phase $e^{i\theta }$ (charge +1).

Forgetting the complex structure, $Hϵ$ is the realification ${\left({\mathbf{2}}_{+1}\right)}_{\mathbb{R}}$ and has real dimension four. It is important not to count ${\mathbf{2}}_{+1}$ and its conjugate as two independent real fibres: rather, the complexification of this one real representation is ${\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1}$, exactly the off-diagonal part of the $SU\left(3\right)$ adjoint decomposition.

Thus, the octonionic split $O=H\oplus Hϵ$ realizes the abstract decomposition (13) in a very explicit way:

$\Im H$ gives the ${\mathbf{3}}_0$ directions (spacetime 3-axes for each $SU{\left(3\right)}^{\mathrm{geom}}$),

$Hϵ$ gives the realification ${\left({\mathbf{2}}_{+1}\right)}_{\mathbb{R}}$ (the 4D fibre).

In particular, for each side $X=L,R$, we identify: $F_4^X:=H_X{ϵ}_X\cong {\left({\mathbf{2}}_{+1}\right)}_{\mathbb{R}}\left(\mathrm{real}4\mathrm{d}\right)$ which is the 4D internal fibre at each point. After an isotropy reduction it matches the tangent representation at a point of $C{P}^2$.

It’s worth giving a more tangible example of the above: Take $H=\mathbb{H}$ with basis $1,i,j,k$ (quaternions), and let’s choose $ϵ$ to be one of the octonion units outside this H (octonions have units $e_1,\dots ,e_7$; suppose $H=\mathrm{Span}1,e_1,e_2,e_3$, then pick $ϵ=e_4$, which anticommutes with $e_1,e_2,e_3$ and squares to $-1$). Then $Hϵ=\mathrm{Span}{e}_4,e_5,e_6,e_7$ which is 4D. Define J as left-multiplication by $u=e_1$ on $Hϵ$. Then, with $q=e^{\alpha e_1+\beta e_2+\gamma e_3}$ acting as $a+bϵ\mapsto qa{q}^{-1}+\left(b{q}^{-1}\right)ϵ$ and $U\left(1\right)=\left\{e^{\theta J}\right\}$ acting as $bϵ\mapsto \left(e^{\theta e_1}b\right)ϵ$, one verifies that $Hϵ$ transforms as a charged doublet under this chosen $SU\left(2\right)\times U\left(1\right)$ subgroup of the $SO\left(4\right)$ stabiliser of H.

In summary, after choosing H, u and the subgroup action, we obtain the representation match

$${\mathbf{3}}_0\oplus {\left({\mathbf{2}}_{+1}\right)}_{\mathbb{R}}\leftrightarrow \Im H\oplus Hϵ=\Im O$$

(21)

matching the real triplet and off-diagonal real four-space of the $SU{\left(3\right)}_{\mathrm{geom}}$ adjoint to subspaces of O. This supports the proposed assignment of $\Im H$ to the base and $Hϵ$ to an internal vector fibre; it does not by itself produce the soldering form or the global bundles.

7.0.0.22. Choice of quaternionic subalgebra and $G_2$-equivalence.

The decomposition $\mathbb{O}=\mathbb{H}\oplus \mathbb{H}\epsilon$ depends on a choice of quaternionic subalgebra $\mathbb{H}\subset \mathbb{O}$. Different choices are related by automorphisms of $\mathbb{O}$, i.e., by $G_2$. Thus the split is “canonical” only up to $G_2$, or equivalently natural once a $G_2$-frame (hence a choice of $\mathbb{H}$) is fixed.

7.1. Relation to $\mathbb{C}{P}^2$ and Kaluza–Klein Intuition

At a point $\left[n\right]\in \mathbb{C}{P}^2=SU\left(3\right)/S(U\left(2\right)\times U\left(1\right))$,

$$T_{\left[n\right]}\mathbb{C}{P}^2\cong Hom(\mathbb{C}n,{\mathbb{C}}^3/\mathbb{C}n)\cong {\mathbb{C}}^2\cong {\mathbf{2}}_{+1}.$$

(22)

Thus there is an isotropy-representation match $F_4\cong {\left(T_{\left[n\right]}\mathbb{C}{P}^2\right)}_{\mathbb{R}}$ after a point $\left[n\right]$—equivalently, a reduction to $H=S\left(U\right(2)\times U(1\left)\right)$—has been chosen. This does not identify the vector space $F_4$ with the manifold $\mathbb{C}{P}^2$, nor does representation matching by itself construct a $\mathbb{C}{P}^2$ bundle.

There are two distinct geometric models. The minimal, linear model uses a reduced principal H-bundle $P_H\to M_6$ and

$$F_4=P_H{\times }_H{\left({\mathbb{C}}^2\right)}_{\mathbb{R}}⟶M_6.$$

(23)

The nonlinear Kaluza–Klein model instead starts from a principal $SU\left(3\right)$ bundle $P\to M_6$ and forms

$$\pi :\mathcal{Y}=P{\times }_{SU\left(3\right)}\mathbb{C}{P}^2⟶M_6.$$

(24)

Its vertical tangent is rank four over $\mathcal{Y}$, not over $M_6$. A section $s:M_6\to \mathcal{Y}$ (a symmetry reduction) pulls it back to $s^{*}{T}^{\mathrm{vert}}\mathcal{Y}\cong F_4$. The manuscript uses the linear model by default; the full $\mathbb{C}{P}^2$ fibration is an optional additional ansatz.

8. The Two $U\left(1\right)$’s

The $U\left(1\right)$ in (16) is the isotropy phase in $H=S\left(U\right(2)\times U(1\left)\right)$ and acts on $(\mathbb{H}\epsilon ,J)$ by $e^{\theta J}$. It is a structure-group connection; it need not be a new low-energy gauge force and is not the real octonion line $\mathbb{R}1$. Abelian mixing with factors inside $E_6$ is a further model choice.

In the linear bundle model, no non-spin $\mathbb{C}{P}^2$ manifold is present, so the obstruction of $T\mathbb{C}{P}^2$ does not arise merely from having a rank-four associated vector bundle. If the nonlinear bundle $\mathcal{Y}$ is introduced and fermions propagate in its vertical directions, then a Spin^c structure must be specified on each $\mathbb{C}{P}^2$ fibre. Its determinant line L must satisfy

$$c_1\left(L\right)\equiv w_2\left(T\mathbb{C}{P}^2\right)(mod2).$$

(25)

The canonical almost-complex Spin^c structure has determinant $K_{\mathbb{C}{P}^2}^{-1}\cong \mathcal{O}\left(3\right)$. The isotropy $U\left(1\right)$ is a natural candidate from which to construct this line and connection, but the associated character and charge normalisation must be specified and shown to yield the required first Chern class. Thus Spin^c compatibility is available, not automatic.

9. How the $E_6^L\times E_6^R$ Fields Sit on the Scaffold

The two geometric $SU\left(3\right)$ factors are used here as structure groups. Separately, one may introduce principal $E_6^L$ and $E_6^R$ bundles over $M_6$ and their adjoint connection bundles. Calling these connections physical gauge fields is an additional dynamical choice: the action, real form, symmetry-breaking pattern, and localization of their four-dimensional zero modes must all be specified.

The complex $\mathbf{27}$ is naturally the carrier of the Albert algebra $J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$ and is therefore useful as an internal ledger for charges and Jordan spectral data. In the companion $E_8\times E_8$ construction, however, physical chiral fermions are assigned to minimal ideals of ${\mathrm{Cl}}_6\left(\mathbb{C}\right)$, not simply declared to be sections of an $E_6$ $\mathbf{27}$ bundle. Schematically their kinematic bundle is

$${\mathcal{S}}_{3,3}\otimes {\mathcal{I}}_{{\mathrm{Cl}}_6}⟶M_6,$$

(26)

with leaf support selected dynamically. Relating these Clifford ideals to the $\mathbf{27}$ labels requires an explicit equivariant map or interaction. The mixed $(\mathbf{3},\mathbf{27})$ sectors in the magic-star decomposition are candidate carriers of such a coupling, but their representation content alone does not supply it.

Likewise, $F_4^{L,R}\simeq {({\mathbb{H}}_{L,R}{\epsilon }_{L,R},J)}_{\mathbb{R}}$ are associated vector bundles in the default model, not automatically compact extra dimensions. Gauge-field components along a $\mathbb{C}{P}^2$ fibre and a Kaluza–Klein tower exist only in the optional nonlinear model. These distinctions keep the scaffold compatible with the division-algebraic fermion construction while making clear which links still require dynamics.

Fermion localisation and chirality (outline).

The vector projector ${{\Pi }^A}_B$ of Section 4.1 does not act on spinors. For an oriented orthonormal normal frame $(n_{\widehat{1}},n_{\widehat{2}})$, define

$${\mathcal{P}}_{\pm }={\displaystyle \frac{1}{2}}(1\pm i{\Gamma }_{\widehat{1}}{\Gamma }_{\widehat{2}}),{\Gamma }_{\widehat{a}}=\Gamma \left(n_{\widehat{a}}\right).$$

For complex or Spin^c spinors these are genuine projectors onto the normal $SO\left(2\right)$ weights. Since a leaf has codimension two, there is no canonical single signed distance and a codimension-one Jackiw–Rebbi profile is insufficient. A vortex-type normal defect is required; schematically,

$$[**D*{*}_6+{\Phi }_1\left(y\right){\mathcal{M}}_1+{\Phi }_2\left(y\right){\mathcal{M}}_2]\psi =0,\psi ={\mathcal{P}}_{\pm }\psi ,$$

where ${\Phi }_1+i{\Phi }_2$ has nonzero winding, as in a Jackiw–Rossi vortex [37]. Normalisable zero modes and net chirality require the corresponding codimension-two index analysis. The Spin^c twist supplies internal bundle data but does not by itself determine this index; eliminating mirror partners remains an open requirement.

Coleman–Mandula compliance: emergent and pre-geometric.

A natural question for any scheme that houses spacetime and internal symmetry inside one algebraic structure is its consistency with the Coleman–Mandula theorem: under standard assumptions (a Poincaré-invariant S-matrix in four dimensions, a mass gap, analyticity, finitely many particle species), any connected symmetry group of the S-matrix is locally a direct product of the Poincaré group and an internal symmetry group, so that no S-matrix symmetry may mix spacetime and internal charges; supersymmetric extensions evade this only through graded algebras, per Haag–opuszański–Sohnius. Conformity with Coleman–Mandula has rightly been emphasised as a first checkpoint for algebraic models of particle physics [38].

The proposed low-energy construction is intended to comply in two distinct ways. First, algebraically on each emergent leaf, the division-algebra split is chosen so that the represented spacetime and internal actions commute. The quaternionic directions $\Im {\mathbb{H}}_L\oplus \Im {\mathbb{H}}_R$ carry the $(3,3)$ base and, after leaf selection, the leaf Lorentz group $SO(3,1)$ arises from the spin-connection sector (Section 4.1); the internal symmetries act on separate associated internal modules, while the geometric $SU\left(3\right)$ factors are structure groups. A completed leaf theory must demonstrate that its S-matrix symmetry is the permitted direct product $SO(3,1)\times G_{\mathrm{int}}$ and that the candidate mixed sectors do not generate forbidden conserved charges. This quaternion/octonion division of labour — $\mathbb{H}$ for spacetime, $\mathbb{O}$ for internal—is the same factorization recently emphasised by Furey, whose $H_{16}\left(\mathbb{C}\right)$ construction uses commuting quaternionic and octonionic multiplication algebras precisely to obtain easy compliance with Coleman–Mandula [22].

Second, and more distinctively, pre-geometrically the theorem does not apply. Coleman–Mandula is a statement about symmetries of an S-matrix defined on a fixed Minkowski background; Poincaré invariance and asymptotic particle states are among its hypotheses. In the trace-dynamics phase of the present program there is no spacetime, no asymptotic states, and no S-matrix: the $E_8\times \omega E_8$ symmetry acts by conjugation on the pre-geometric matrix degrees of freedom, with evolution in Connes time (Section 3). The unified symmetry—which does contain would-be spacetime ($SU{\left(3\right)}^{geom}$) and internal ($E_6$) generators within one simple algebra per branch — is therefore not an S-matrix symmetry group, and is not the kind of object the theorem constrains. Coleman–Mandula becomes operative only after leaf localisation creates Lorentzian 4D leaves carrying an emergent quantum field theory, and at that point the construction is required to satisfy it in the algebraic manner just described. We stress that this is a statement about where the theorem’s hypotheses hold, not a claimed loophole in the theorem: any emergent leaf QFT must satisfy Coleman–Mandula. The unification of spacetime and internal symmetry in $E_8\times \omega E_8$ is thus pre-geometric, with Coleman–Mandula compliance recovered, rather than violated, in the emergent phase.

10. Big Picture and Interpretation

Three layers. 

1.

Geometric $\mathrm{SU}{\left(\mathbf{3}\right)}_{\mathbf{L},\mathbf{R}}^{\mathrm{geom}}$ from $E_8\supset E_6\times \mathrm{SU}\left(3\right)$ on each side: not gauged. Purpose: carve the base and fibre geometry. After $\mathrm{SU}\left(3\right)\to \mathrm{SU}\left(2\right)\times U\left(1\right)$,

$$T{M}_6\cong (\Im {\mathbb{H}}_L)\oplus (\omega \Im {\mathbb{H}}_R),F_4^X\cong {\left({\mathbf{2}}_{+1}\right)}_{\mathbb{R}},X=L,R.$$

Here $\Im \mathbb{H}$ realizes ${\mathbf{3}}_0$, while the complexification of $\mathbb{H}\epsilon$ is ${\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1}$. This is the tangent isotropy representation of $\mathbb{C}{P}^2$; a global $\mathbb{C}{P}^2$ fibre and its Spin^c determinant line require the additional bundle and character data described in Section 8.

2.

Gauge $\mathrm{SU}\left(\mathbf{3}\right)$’s inside each $E_6$ (trinification): these are dynamical. On the left: $E_6^L\to \mathrm{SU}{\left(3\right)}_c\times \mathrm{SU}{\left(3\right)}_L\times \mathrm{SU}{\left(3\right)}_{F,L}$ with $\mathrm{SU}{\left(3\right)}_L\to \mathrm{SU}{\left(2\right)}_L\times U{\left(1\right)}_Y$. On the right: $E_6^R\to \mathrm{SU}{\left(3\right)}_{c^{\prime }}\times \mathrm{SU}{\left(3\right)}_R\times \mathrm{SU}{\left(3\right)}_{F,R}$ with $\mathrm{SU}{\left(3\right)}_R\to \mathrm{SU}{\left(2\right)}_R\times U{\left(1\right)}_{\mathrm{Ydem}}$.

3.

Localization and Lorentz breaking in 6D: the model proposes two Higgs order parameters and covariant Thom-form-like densities ${\rho }_X$ to localize 4D leaves ${\Sigma }_R,{\Sigma }_L\subset M_6$. Normal 2-frames $U_X$ implement $SO(3,3)\to SO(3,1)\times SO\left(2\right)$ on ${\Sigma }_R$ and $SO(3,3)\to SO(1,3)\times SO\left(2\right)$ on ${\Sigma }_L$, eating 8 Lorentz coset modes per leaf and leaving the tangent spin connections massless [1].

Where do the unbroken gauge groups live? 

If every sector action is coupled to a normalized representative of the Poincaré dual of a leaf and the corresponding transverse zero modes are normalisable, its low-energy dynamics can be localized there. Demonstrating this for the proposed defect profiles and gauge spectrum remains a dynamical task.

What are the fibres relative to spacetime? 

$F_4^{L,R}$ are rank-4 internal vector bundles over $M_6$ with $F_4^X\simeq {\mathbb{H}}_X{\epsilon }_X\cong {\left({\mathbf{2}}_{+1}\right)}_{\mathbb{R}}$. Their complexifications contain the doublet and its conjugate, and they match a $\mathbb{C}{P}^2$ tangent space only as isotropy representations after reduction. They are not extra spacetime directions. Restriction to a leaf gives internal fibres $F_4^X{|}_{{\Sigma }_X}$ on which internal interactions act. The two 4D spacetimes come from the selected six tangent directions $(\Im {\mathbb{H}}_L)\oplus (\omega \Im {\mathbb{H}}_R)$; each leaf selects a $3+1$ subspace and the remaining two directions form its normal plane.

Two consistent options for $\mathrm{SU}{\left(\mathbf{3}\right)}_{{\mathbf{c}}^{\prime }}$. 

• Decoupled/hidden: break or confine $\mathrm{SU}{\left(3\right)}_{c^{\prime }}$ above the localization scale; only visible $\mathrm{SU}{\left(3\right)}_c$ remains on ${\Sigma }_R$.
• Gauged on a leaf: keep $\mathrm{SU}{\left(3\right)}_{c^{\prime }}$ dynamical on one leaf (typically ${\Sigma }_R$ or ${\Sigma }_L$). Portal terms can live on $X_3={\Sigma }_R\cap {\Sigma }_L$ under the BF matching conditions.

Dictionary (one line). 

$$\begin{array}{|c|}\hline \mathrm{structural}\mathrm{SU}{\left(3\right)}_{L,R}^{\mathrm{geom}}\Rightarrow (M_6,F_4^{L,R})\mathrm{vs}\mathrm{gauge}\mathrm{SU}\left(3\right)\subset E_6^{L,R}\Rightarrow 4\mathrm{D}\mathrm{forces}\mathrm{on}{\Sigma }_{L,R}.\\ \hline\end{array}$$

Anomalies. 

On each leaf, gauge and mixed anomalies match those of the $E_6$ embeddings and their standard symmetry–breaking chains; we assume usual $E_6$ anomaly freedom sector by sector. If a portal is introduced on the 2D overlap ${\Sigma }_L\cap {\Sigma }_R$, one must ensure either explicit cancellation in the 4D content or anomaly inflow from appropriate 6D counterterms; both options are available in this scaffold.

The one gauged abelian beyond the Standard Model, the dark electromagnetism $U{\left(1\right)}_{\mathrm{dem}}$, deserves a separate statement. Its charge $Q_{\mathrm{dem}}\left(f\right)=\sqrt{y_f\left(\mu \right)}$ is a function of flavour and mass alone, hence chirality-blind: the left- and right-handed components of each Dirac species carry equal dark charge, so $U{\left(1\right)}_{\mathrm{dem}}$ acts vectorially. In an all-left-handed Weyl basis every species then contributes a conjugate pair with dark charges $\pm Q_{\mathrm{dem}}\left(f\right)$, and every triangle anomaly involving $U{\left(1\right)}_{\mathrm{dem}}$ — the cubic ${\left[U{\left(1\right)}_{\mathrm{dem}}\right]}^3$, the mixed ${\left[SU{\left(3\right)}_c\right]}^{2}U{\left(1\right)}_{\mathrm{dem}}$, ${\left[U{\left(1\right)}_{\mathrm{em}}\right]}^{2}U{\left(1\right)}_{\mathrm{dem}}$ and $U{\left(1\right)}_{\mathrm{em}}{\left[U{\left(1\right)}_{\mathrm{dem}}\right]}^2$, and the mixed gravitational anomaly—cancels within each species, independently of the numerical charge values. Anomaly freedom of the dark sector is therefore automatic, for the structural reason that a vector-like symmetry sources no triangle anomaly.

A caveat must, however, be stated plainly. A charge proportional to (the square root of) mass cannot be a Cartan charge of the unbroken electroweak gauge group: the two members of an $SU{\left(2\right)}_L$ doublet have unequal masses, so $Q_{\mathrm{dem}}$ does not commute with $SU{\left(2\right)}_L$; and irrational square-root-mass ratios are incompatible with the charge quantization required of a compact $U\left(1\right)$. Hence $U{\left(1\right)}_{\mathrm{dem}}$ with mass-dependent charge is a symmetry of the broken phase (the mass-eigenstate basis, residual gauge group $SU{\left(3\right)}_c\times U{\left(1\right)}_{\mathrm{em}}$), while its high-scale parent $U{\left(1\right)}_{Y\mathrm{dem}}\subset SU{\left(3\right)}_R$ carries quantized, group-theoretic Cartan charges. Two consistent readings are available: (i) the fundamental gauge charge is the quantized Cartan charge of $U{\left(1\right)}_{Y\mathrm{dem}}$, with $\sqrt{m_f}$ entering as an effective coupling strength generated at gravi-weak breaking; or (ii) $U{\left(1\right)}_{\mathrm{dem}}$ with charge $\sqrt{y_f}$ is an emergent low-energy vector-like symmetry, matched onto $U{\left(1\right)}_{Y\mathrm{dem}}$ at the localisation scale. Neither reading introduces a new anomaly constraint: in reading (i) the dark $U\left(1\right)$ inherits the anomaly freedom of the $E_6$ chain (complete multiplets), as already assumed above for the leaf content, while in reading (ii) it is anomaly-free by vector-likeness, as just shown. Selecting between the two readings is a matching task for the localisation dynamics of [1] and is left open here.

Relation to $J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$ mass geometry. 

The fibre $F_4$ carries the realification of the same complex doublet that appears as the tangent isotropy representation of $\mathbb{C}{P}^2$. This is a representation-theoretic parallel with the $SU\left(3\right)$ actions used in the $J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$ mass construction, not an identification of the geometric $SU\left(3\right)$ with flavour or colour. An explicit homomorphism or coupling is needed to relate those distinct group actions. In this scaffold, primitive idempotents remain intact; a Majorana condition, when used, is imposed at the spinor level rather than by replacing internal projectors with non–idempotent directions. This keeps the Jordan–algebraic state geometry consistent while retaining the small symmetry–breaking effects needed for realistic spectra.

10.1. UV Completion and Trace Dynamics

We take the microscopic degrees of freedom to be matrices $Q_a\left(\tau \right)$ in the adjoint of $E_8\times \omega E_8$ evolving in Connes time $\tau$. A minimal single–atom Lagrangian is

$${\mathcal{L}}_{\mathrm{atom}}=\mathrm{Tr}\left({\dot{Q}}_1{\dot{Q}}_2\right)-{\displaystyle \frac{{\Omega }^2}{2}}\mathrm{Tr}\left(Q_1^2+Q_2^2\right)-\lambda \mathrm{Tr}\left(Q_1{Q}_2\right),\dot{}:=\frac{d}{d\tau },$$

(27)

where the quadratic terms are the lowest–degree $E_8\times \omega E_8$–invariant potentials; ${\Omega }^2\ge 0$ and $\lambda$ are real couplings. The many–atom system is intended as an interacting trace dynamics of the Adler type. A successful coarse-grained, large-N limit would have to derive: (i) emergent quantum kinematics from an appropriate conserved Adler–Millard charge, (ii) the 6D BF plus Lorentz–Higgs sector as infrared hydrodynamics, and (iii) localisation on two 4D leaves. These are programme objectives; they do not follow from the illustrative quadratic Lagrangian (27) alone.

Power counting and predictivity.

Polynomial matrix dynamics avoids assuming a fundamental continuum field at the microscopic level, but polynomiality alone is not a proof of ultraviolet finiteness. A controlled continuum limit would require a measure, scaling limit, renormalisation analysis and a demonstration that higher-dimension operators are suppressed by a scale ${\Lambda }_{\mathrm{TD}}$.

Phenomenological normalisations.

Fermion masses enter via $m_f=y_f{v}_M$; if the dark $U{\left(1\right)}_{\mathrm{dem}}$ couples to $\sqrt{m_f}$, we define the dimensionless charge $Q_{\mathrm{dem}}\left(f\right)=\sqrt{y_f}$ (with scheme dependence through $y_f\left(\mu \right)$). Because $y_f\left(\mu \right)$ runs, this quantity is scale- and scheme-dependent. It may be used as an effective coupling after symmetry breaking, but not as a fixed charge of a fundamental compact $U\left(1\right)$ without an additional matching construction.

Open UV checks.

(i) cluster/locality in the hydrodynamic limit; (ii) anomaly matching on each leaf (with possible inflow at ${\Sigma }_L\cap {\Sigma }_R$); (iii) absence of ghosts/tachyons in the mixed connection sector; (iv) independence of physical outputs from the choice of quaternionic frame inside $\mathbb{O}$ up to $G_2$ automorphisms.

11. Summary

• The split-biquaternion seed ${\mathbb{B}}_s\cong \mathrm{Cl}(0,3)$ is an 8-dimensional real algebra, distinct from $\mathrm{Cl}(M_6,g)=\mathrm{Cl}(3,3)\cong {\mathrm{Mat}}_8\left(\mathbb{R}\right)$ with ${\mathrm{Spin}}_0(3,3)\cong SL(4,\mathbb{R})$.
• The selection $M_6=\Im {\mathbb{H}}_L\oplus \omega \Im {\mathbb{H}}_R$ and the relative sign in g are model inputs; given them, $M_6$ has signature $(3,3)$ and contains the two displayed Lorentzian 4-planes.
• The extra $\mathrm{SU}\left(3\right)$’s branch as $\mathbf{8}\to {\mathbf{3}}_0\oplus {\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1}\oplus {\mathbf{1}}_0$.
• $\Im \mathbb{H}$ realises ${\mathbf{3}}_0$; $\mathbb{H}\epsilon$ realises ${\left({\mathbf{2}}_{+1}\right)}_{\mathbb{R}}$ and is a rank-four associated vector fibre. Its complexification is ${\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1}$.
• This fibre matches $T_{\left[n\right]}\mathbb{C}{P}^2$ as an isotropy representation. A global $\mathbb{C}{P}^2$ bundle and a compatible Spin^c determinant line are optional extra structures, not consequences of the dimension count alone.

We recap and slightly rephrase these summary points here for clarity:

Spacetime Emergence: We select $M_6=\Im H_L\oplus \omega \Im H_R$ and define its metric of signature $(3,3)$. This is not the grade-one space of $\mathrm{Cl}(0,3)$; its Clifford completion is $\mathrm{Cl}(3,3)$. The two 4-planes are ${\Sigma }_R=\omega \Im H_R\oplus \mathrm{span}\left\{t_L\right\}$ and ${\Sigma }_L=\Im H_L\oplus \mathrm{span}\left\{\omega t_R\right\}$, sharing $\mathrm{span}\{t_L,\omega t_R\}$. Physical localisation requires the companion dynamics.

$SU\left(3\right)$ Branching and Octonions: Each extra $SU{\left(3\right)}^{\mathrm{geom}}$ (left or right) branches as $8\to 3_0+2_{+1}+{\overline{2}}_{-1}+1_0$. Correspondingly, the octonion split $O=H\oplus Hϵ$ supplies $\Im H$ as a real triplet and $Hϵ$ as one complex doublet, hence four real directions. Its complexification contains the doublet and its conjugate. Identifying the two triplets with $T{M}_6$ still requires the soldering form stated above.

Linear fibre and optional $C{P}^2$ geometry: The 4D internal fibre $F_4$ constructed from $Hϵ$ is a rank-four vector bundle carrying the realified $C{P}^2$ tangent isotropy representation. It becomes the pullback of a vertical tangent bundle only after a $C{P}^2$ bundle and a section are supplied. This observation does not by itself geometrize Standard Model colour or flavour; those are distinct group factors until an explicit coupling is constructed.

Geometric $U\left(1\right)$ and Spin^c: In the optional $C{P}^2$ model the isotropy $U\left(1\right)$ is a natural source for a Spin^c connection, but the character defining its determinant line must satisfy $c_1\left(L\right)\equiv w_2(mod2)$; for the canonical almost-complex structure, $L\cong \mathcal{O}\left(3\right)$. No such topological requirement follows from the linear vector-fibre model alone.

Putting it together, the paper proposes a scaffold rather than a completed unified theory. Its exact algebraic core is the $(3,3)$ quadratic form and Dirac factorisation on $J_2\left({\mathbb{H}}_s\right)$; the $E_8$, $E_6$, Albert-algebra, leaf-localisation, flavour and dark-sector links are compatible modelling layers whose connecting maps and dynamics remain to be derived. In particular, localisation spectra, anomaly matching after symmetry breaking, the physical real form, and quantitative low-energy predictions are open tests of the proposal.