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 $SU\left(3\right)$s arising from the branching of $E_{6L}$, the $SU{\left(3\right)}_c$ implements the color gauge symmetry of QCD. Furthermore, $SU{\left(3\right)}_{F,L}$ is the non-gauged global flavor symmetry which is responsible for three left-handed fermion generations described by the exceptional Jordan algebra $J_3\left({\mathbb{O}}_c\right)$. $SU{\left(3\right)}_L$ branches as $SU{\left(2\right)}_L\times U{\left(1\right)}_Y$ giving rise to the electroweak sector, and the $SU{\left(2\right)}_L$ acts only on left-handed particles.

As regards the branching of the second $E_6$, which we label $E_{6R}$, the $SU{\left(3\right)}_{c^{\prime }}$ symmetry is preferably not gauged (so as to be consistent with phenomenology)—it is global and explicitly broken at the electroweak scale. Its role is to rotate the so-called Jordan frame which arises in the Peirce decomposition of the exceptional Jordan algebra matrices. The $\omega SU{\left(3\right)}_{F,R}$ is the global flavor symmetry which describes three generations of right-handed fermions. The $SU{\left(3\right)}_R$ branches as $SU{\left(2\right)}_R\times U{\left(1\right)}_{Ydem}$. Here, the $SU{\left(2\right)}_R$—interpreted (see the clarification below) as the self-dual part of the leaf Lorentz/spin connection rather than as a compact internal gauge group whose breaking would produce a graviton—yields general relativity in 4D Plebanski form after Lorentz–Higgs leaf selection, while the unbroken $U{\left(1\right)}_{dem}$ arising from $U{\left(1\right)}_{Ydem}\to U{\left(1\right)}_{dem}$ is a new force, dubbed dark electromagnetism. It is sourced by the square-root of mass and can be made cosmologically and phenomenologically safe. The emergence of general relativity in this manner, and the preceding gravi-weak unification, was briefly discussed in [2] and is analysed in detail in a companion work [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 the companion work).

$U{\left(1\right)}_{\mathrm{dem}}$ charge normalisation.

To make the “couples to $\sqrt{m}$” statement precise and 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 dark charge as

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

so that $Q_{\mathrm{dem}}$ is RG–scheme aware via $y_f\left(\mu \right)$ and the $U{\left(1\right)}_{\mathrm{dem}}$ coupling remains dimensionless. Phenomenology then fixes $g_{\mathrm{dem}}$ and the relevant scale $\mu$.

The focus of the present article is the two extra $SU\left(3\right)$s which we label $SU{\left(3\right)}_L^{geom}$ and $SU{\left(3\right)}_R^{geom}$, which arise in the decomposition of the two $E_8$, and which are apart from the six $SU\left(3\right)s$ arising in the trinification of the two $E_6$. As was already proposed by us in the foundational $E_8\times \omega E_8$ papers, the role of $SU{\left(3\right)}_L^{geom}$ and $\omega SU{\left(3\right)}_R^{geom}$ is to describe spacetime and internal symmetry space. The split complex number is crucial in enabling a Lorentzian signature to emerge from combination of two compact groups, as we rigorously demonstrate below. The split bioctonions also play an important role here: their split quaternionic subalgebra here gives rise to a 6D spacetime with signature (3,3), which upon electroweak symmetry breaking gives rise to two overlapping 4D spacetimes with signature (3,1) and (1,3) respectively. One of these is our familiar 4D spacetime, curved by gravitation. The other 4D spacetime has flipped signature and a (1,1) intersection with our spacetime—its directions are interpreted as being curved by the weak force (companion work). The complementary remaining quaternionic subalgebras describe internal symmetry space for holding the unbroken symmetries $SU{\left(3\right)}_c$ and $SU{\left(3\right)}_{c^{\prime }}$. A decomposition of the adjoint $\mathbf{8}$ of $SU{\left(3\right)}_{L,R}^{geom}$ shows an elegant mapping to the octonions, as described below.

Thus the goal of the present short note is to rigorously demonstrate how $SU{\left(3\right)}_L^{geom}$ and $\omega SU{\left(3\right)}_R^{geom}$ provide the scaffolding on which the fields contained in $E_{6L}\times E_{6R}$ live. This construction realises the all-encompassing role of $E_8\times \omega E_8$, making the unified symmetry a source of space-time as well as of matter. It is in this spirit that we call the primitive entities possessing $E_8\times \omega E_8$ symmetry `atoms of space-time-matter’ [3].

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 the companion work), 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 [17].

Finally, the octonionic underpinning of exceptional symmetry is well documented [18,19,20,21]. Division-algebra approaches to Standard Model structure, notably Furey’s program [22], 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 [23], 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 [24,25]. 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 the companion and mass-ratio works).

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 8).

Plan of the paper:

Section 3Section 4 build $M_6$ from split bioctonions and embed the two 4D spacetimes. Section 5Section 6 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 7 clarifies the two $U\left(1\right)$’s. Section 8 explains how the $E_{6L}\times E_{6R}$ fields live on the proposed scaffold, and Section 9 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 the works cited above.

What is derived versus what is assumed.

To delineate the logical status of the construction, we separate its forced consequences from its modelling choices. Forced (given the $E_8\times \omega E_8$ starting point and the split-complex structure):

• the signature $(3,3)$ of $M_6$, once $\omega$ (with ${\omega }^2=+1$) tags the right $\Im \mathbb{H}$ block;
• 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 assignment ${\mathbf{3}}_0=$ base, ${(\mathbf{2}\oplus \overline{\mathbf{2}})}_{\mathbb{R}}=$ fibre, which is fixed by the invariant structures the summands carry (Section 6).

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 [24,25] but is adopted here as an identification;
• 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 the companion work;
• the choice of quaternionic frame $\mathbb{H}\subset \mathbb{O}$, which is canonical only up to $G_2$ (Section 6);
• 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 [3,26]

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

(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 electroweak symmetry breaking our observable universe is made of a very large collection of STM atoms $(N\sim {10}^{120}$- this number is an empirical input). The entries in these matrices are Grassmann numbers. The action is made dimensionless by introducing Planck’s constant ℏ, and Connes time $\tau$ is made dimensionless by introducing Planck time ${\tau }_{Pl}$.

More precisely, the statement that the trace Lagrangian in Eq. (3) is invariant under $E_8\times \omega E_8$ is to be understood representation-theoretically. The symbol $\omega$ is here a split-complex sector label for the second copy, and the matrix indices of ${\dot{Q}}_1$ and ${\dot{Q}}_2$ are taken to furnish a chosen representation $\mathcal{R}$ of $E_8\times \omega E_8$. The action of the symmetry on the primitive STM variables is by conjugation,

$${\dot{Q}}_A⟼\mathcal{R}\left(g\right){\dot{Q}}_A\mathcal{R}{\left(g\right)}^{-1},A=1,2,$$

for $g\in E_8\times \omega E_8$. The adjoint ^† in Eq. (3) is understood with respect to the invariant form preserved by $\mathcal{R}\left(g\right)$, so that

$${\left(\mathcal{R}\left(g\right){\dot{Q}}_A\mathcal{R}{\left(g\right)}^{-1}\right)}^{†}=\mathcal{R}\left(g\right){\dot{Q}}_A^{†}\mathcal{R}{\left(g\right)}^{-1}.$$

It then follows that

$$\mathrm{Tr}\left({\dot{Q}}_1^{†}{\dot{Q}}_2\right)⟼\mathrm{Tr}\left[\mathcal{R}\left(g\right){\dot{Q}}_1^{†}\mathcal{R}{\left(g\right)}^{-1}\mathcal{R}\left(g\right){\dot{Q}}_2\mathcal{R}{\left(g\right)}^{-1}\right]=\mathrm{Tr}\left[\mathcal{R}\left(g\right){\dot{Q}}_1^{†}{\dot{Q}}_2\mathcal{R}{\left(g\right)}^{-1}\right]=\mathrm{Tr}\left({\dot{Q}}_1^{†}{\dot{Q}}_2\right),$$

by cyclicity of the trace. Thus, the exceptional symmetry acts not because $E_8$ is itself a classical unitary group, but because it admits a representation on the STM matrix space which preserves the form defining ^†.

Of course, the bare trace is invariant under a larger class of similarity transformations. The point of the present construction is that the split-bioctonionic/octonionic scaffold singles out $E_8\times \omega E_8$ as the distinguished physical symmetry of the STM atom. At this pre-geometric stage, $Q_1$ and $Q_2$ are therefore not fields living on a prior operator-valued spacetime; rather, they are the primitive matrix degrees of freedom from which the later separation into spacetime, internal symmetry space, and matter variables emerges.

Thus the exceptional symmetry acts by preserving the invariant bilinear form on the representation space. In the adjoint representation this invariant quadratic form is naturally the Killing form. 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 Eq. (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 Eq. (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 Eq. (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. The split tag ω implements the sign flip for the right copy, matching the $\omega SU{\left(3\right)}_R^{geom}$ language.

What we are doing here is that we restrict to the two quaternionic subalgebras $H_L$ and $H_R$ in the two parts of the split bioctonion. The imaginary parts $x_L$ and $x_R$ of these two quaternions define, via the inner product, a 6D spacetime of signature $(3,3)$. The gravi-weak interaction lives on this 6D spacetime.

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 one gets an $SU{\left(2\right)}_L\times SU{\left(2\right)}_R$ isometry group of $M_6$. 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

$$\begin{array}{cc}\hfill {\mathcal{M}}_4^{\left(R\right)}& :=\Im {\mathbb{H}}_R\oplus \mathrm{span}\left\{t_L\right\}(\mathrm{signature}(3,1)),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\mathcal{M}}_4^{\left(L\right)}& :=\Im {\mathbb{H}}_L\oplus \mathrm{span}\left\{\omega t_R\right\}(\mathrm{signature}(1,3)).\hfill \end{array}$$

(8)

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 the companion work.

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 $M_6$:

$M_4^{\left(R\right)}:=\Im H_R\oplus \mathrm{span}\left\{t_L\right\}$. This consists of all vectors of the form $x_R+\alpha ,t_L$ with $x_R\in \Im H_R$ and $\alpha \in \mathbb{R}$. By construction, $\Im H_R$ is a 3D space contributing with a negative sign in the metric (from the $\omega$ factor), and $t_L$ lies in $\Im H_L$ contributing with a positive sign. Therefore, the metric on $M_4^{\left(R\right)}$ has signature $(3,1)$: three negative (from $\Im H_R$) and one positive (from $t_L$). 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).

$M_4^{\left(L\right)}:=\Im H_L\oplus \mathrm{span}\{\omega ,t_R\}$. This is all vectors of the form $x_L+\beta ,\left(\omega t_R\right)$ with $x_L\in \Im H_L$ and $\beta \in \mathbb{R}$. Here $\Im H_L$ part has positive metric, and $\omega t_R$ lies in $\omega ,\Im H_R$ hence contributes a negative metric component. So $M_4^{\left(L\right)}$ has signature $(1,3)$ – three positive (from $\Im H_L$) and one negative (from $\omega t_R$). This is the second embedded 4D Lorentzian spacetime, and in this one $t_R$ (via $\omega t_R$) effectively acts as the time-like 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

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

(10)

Then ${\Pi }^2=\Pi$, $\mathrm{Im}\Pi =W$, and $\mathrm{Ker}\Pi =\eta N$. 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:

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

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.

Promote U to a field $U\left(x\right)$ with the constraint $U^T\eta U=-{\mathbf{1}}_2$ and couple it to the 6D spin connection ${\omega }_{\mu }$:

$${\mathcal{L}}_U={\displaystyle \frac{\alpha }{2}}\mathrm{Tr}\left[{\left(D_{\mu }U\right)}^T\eta \left(D^{\mu }U\right)\right]-V\left(U\right),D_{\mu }U:={\partial }_{\mu }U+{\omega }_{\mu }U.$$

(11)

A vacuum $\langle U\rangle$ picks N and spontaneously breaks $SO(3,3)\to SO(3,1)\times SO\left(2\right)$. The eight mixed connections ${\omega }_{\mu }^{a\widehat{b}}\in W\wedge N$ (here $a=0,\dots ,3$ along W, $\widehat{b}=1,2$ along N) eat the eight Goldstones and become massive. The normal rotation ${\omega }_{\mu }^{\widehat{1}\widehat{2}}\in \mathfrak{so}\left(N\right)$ is either left as a spectator $SO\left(2\right)$ or can be fixed/decoupled. At low energy, the residual local symmetry on the leaf is exactly $SO(3,1)$. This is the effective-field-theory implementation of the leaf localisation mechanism (see also [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 (e.g., $g_4=\Pi g\Pi$, $R_4=\Pi R\Pi$), or wedge sector actions with a 2–form density $\rho$ supported on the leaf so that ${\int }_{M_6}(·)\wedge \rho ={\int }_{{\Sigma }_4}(·)$. Off–plane components drop out; the surviving symmetry is $SO(3,1)$.

Two leaves.

Choose two negative 2–planes $N_L,N_R$ with projectors ${\Pi }_L,{\Pi }_R$. Each leaf ${\Sigma }_X:=\mathrm{Im}{\Pi }_X$ ($X=L,R$) carries its own residual $SO(3,1)$. 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 two independent 4D Lorentz symmetries, one on each leaf.

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 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

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

(12)

We use the two extra $\mathrm{SU}\left(3\right)$’s as geometry. Pick the standard embedding $\mathrm{SU}\left(2\right)\subset \mathrm{SU}\left(3\right)$ and take 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

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

(13)

The ${\mathbf{3}}_0$ from $SU{\left(3\right)}_L^{geom}$ will supply the three directions of $\Im {\mathbb{H}}_L$ and the ${\mathbf{3}}_0$ from $SU{\left(3\right)}_R^{geom}$ will supply the three directions of $\Im {\mathbb{H}}_R$; together forming $M_6$. The $\mathbf{2}\oplus \overline{\mathbf{2}}$ will furnish a real 4D internal fibre, one each from the two $SU{\left(3\right)}^{geom}$.

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 (13). (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 2 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.

6. 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

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

(14)

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

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

(15)

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

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

(16)

for unit quaternions $p,q$ [18]; that (16) 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 2 

(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

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

(17)

a real 4-vector space. This is the internal fibre $F_4$.

Thus the octonionic split realises the branching (13) 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 forced.

Within the present construction the split of the adjoint into “spacetime” and “internal” pieces is not a free labelling but is fixed by the invariant structures the two summands carry. 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 $\mathbb{H}\epsilon \cong {({\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1})}_{\mathbb{R}}$, by contrast, carries the intrinsic complex structure J defined above, which makes it canonically 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 7). 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, ${(\mathbf{2}\oplus \overline{\mathbf{2}})}_{\mathbb{R}}=$ fibre” is determined rather than chosen.

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 consisting of “quaternions times $ϵ$”. This particular split, sometimes called the “split-octonion” decomposition, 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 space $Hϵ$ look like a ${\mathbf{2}}_{+1}\oplus {\mathbf{2}}_{-1}$ representation, we need to 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 (16): 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 2 (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 (i.e., as a real space), $Hϵ$ then corresponds to ${\mathbf{2}}_{+1}\oplus {\mathbf{2}}_{-1}$. Why both? Because a complex 2D rep is real 4D and contains the vector and its complex conjugate. In other words, if $(Hϵ,J)$ is ${\mathbf{2}}_{+1}$, the same real space can also be seen as ${\mathbf{2}}_{-1}$ by using $-J$ as the complex structure; effectively J provides an orientation for $U\left(1\right)$ charge. So $Hϵ$ as a real rep carries a doublet of charge +1 and an equivalent doublet of charge -1 – exactly the pair ${\mathbf{2}}_{+1}\oplus {\mathbf{2}}_{-1}$ we found in 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 $\mathbf{2}\oplus \mathbf{2}$ internal directions (the 4D fibre).

In particular, for each side $X=L,R$, we identify: $F_4^X:=H_X{ϵ}_X\cong {\mathbf{2}}_{+1}\oplus {\mathbf{2}}_{-1}\left(\mathrm{real}4\mathrm{d}\right)$ which is the 4D internal fibre at each point. We will shortly see this is isomorphic to the tangent space 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 can verify that $Hϵ$ transforms as a charged doublet. This matches the algebraic relationships inside octonions (a known fact: $G_2$, the automorphism group of octonions, has an $SU\left(3\right)$ subgroup that precisely preserves a chosen split like this, acting as rotations on a $S^6$ etc.).

In summary, through the lens of the exceptional algebra O, we have found a canonical isomorphism between:

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

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matching the representation content of $SU{\left(3\right)}_{\mathrm{geom}}$ to subspaces of O. This cements the idea that the extra $SU\left(3\right)$ naturally splits into a 3D space (for $M_6$ base) and a 4D internal fibre. It also justifies why, earlier, we could assign the “3 directions of $\Im H_L$” to ${\mathbf{3}}_0$ and “4 directions of $Hϵ$” to the doublets. The octonions essentially provide the coordinate system for this split.

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.

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

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

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

(19)

so the real tangent is 4-dimensional. The octonionic identification above gives a canonical isomorphism

$$F_4\simeq T\mathbb{C}{P}^2(\mathrm{real}\mathrm{rank}4\mathrm{at}\mathrm{each}\mathrm{point}).$$

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This matches the minimal Kaluza–Klein choice for an $\mathrm{SU}\left(3\right)$ internal, explaining why four internal real directions are “right” for a QCD-like sector at the level of geometry.

Detailed explanation: Above, we clarify the significance of the 4D internal fibre by relating it to a well-known geometric space: $C{P}^2$, the complex projective plane. $C{P}^2$ is a compact 4-real-dimensional manifold (complex dimension 2) often studied in grand unification and Kaluza–Klein models as a candidate internal space (notably, $C{P}^2$ appears in certain $SU\left(3\right)$ Kaluza–Klein coset models for unified interactions). By definition,

$$C{P}^2=SU\left(3\right)/S(U\left(2\right)\times U\left(1\right)),$$

(21)

which means at each point $\left[n\right]\in C{P}^2$ (think of n as a 1-dimensional complex subspace in ${\mathbb{C}}^3$), the tangent space can be identified with the space of homomorphisms from that line $Cn$ to the orthogonal complement $C^3/Cn$. In formula:

$$T_{\left[n\right]}C{P}^2\simeq {\mathrm{Hom}}_{\mathbb{C}}(Cn,C^3/Cn)$$

(22)

Since $Cn$ is 1-dimensional and $C^3/Cn$ is 2-dimensional over $\mathbb{C}$, $\mathrm{Hom}(Cn,C^3/Cn)$ is isomorphic to $C^2$. Therefore complex-dimension($TC{P}^2$) = 2, or real-dimension $=4$. In fact one can say:

$$T_{\left[n\right]}C{P}^2\simeq {\mathbb{C}}^2\simeq {\mathbf{2}}_{+1}$$

(23)

as a representation of the stabilizer $S\left(U\right(2)\times U(1\left)\right)$ (where the $U\left(1\right)$ acts with charge +1 on that ${\mathbb{C}}^2$). This is exactly the same structure we have for our internal fibre! The ${\mathbf{2}}_{+1}$ in our model corresponds to the complex tangent at a point of $C{P}^2$ (and ${\mathbf{2}}_{-1}$ would be the opposite charge because the $U\left(1\right)$ in $S\left(U\right(2)\times U(1\left)\right)$ would also have a -1 action on the conjugate). Thus the 4D real internal space we obtained can be viewed as the tangent space of $C{P}^2$.

In fact, we assert a canonical isomorphism $F_4\cong TC{P}^2$ (as real 4-spaces). The octonionic model gives a specific identification, not just an abstract isomorphism, thereby providing a concrete model of $C{P}^2$’s tangent bundle inside $E_8$’s structure. The phrase “realification of $2\oplus \overline{2}$” earlier also alluded to this $C{P}^2$ tangent (since $2\oplus \overline{2}$ is the realified form of the complex 2). The upshot is: each point in the 6D base $M_6$ can be thought of as carrying an internal fibre isomorphic to $C{P}^2$’s tangent space. If one were to imagine a Kaluza–Klein scenario, $C{P}^2$ might be the internal manifold – but here, the internal space is not global $C{P}^2$ per se, it’s an oriented plane (tangent) at each base point. In other words, we have a fibre bundle with fibre $TC{P}^2$. The minimal dimensionality (4 real dims) of this fibre is pleasing: it’s exactly what we need for embedding an $SU\left(3\right)$ (color-like) gauge sector if we follow Kaluza–Klein arguments. We note that having four internal dimensions is the “right” number for a QCD-like sector at the level of geometry – referencing that an $SU\left(3\right)$ gauge theory in 4D could emerge from a $C{P}^2$ compactification (since $C{P}^2$ has isometry $SU\left(3\right)$ and requires a $Spi{n}^c$ twist to admit fermions, which is exactly what we are setting up with the $U\left(1\right)$ lines).

Summarizing: The internal 4D space provided by the split-bioctonion construction is identified with the tangent space of the coset $SU\left(3\right)/S\left(U\right(2)\times U(1\left)\right)$, i.e., $C{P}^2$. This not only validates the choice of 4 internal dimensions but also situates our model in the context of known geometry (where $C{P}^2$ often appears in grand unification). It’s a nice consistency check and provides intuition: just as Kaluza–Klein theory might use $M_4\times C{P}^2$ as spacetime, here we have $M_6$ base and an internal fibre that behaves like $C{P}^2$ at each point. The difference is that $C{P}^2$ itself is 4D and non-trivial (non-spin), but as a fibre attached to each point of $M_6$, it’s more like an internal degrees-of-freedom space rather than additional global dimensions.

7. The two $U\left(1\right)$’s

Global formulation: $F_4$ as a vertical tangent bundle over a ${\mathbb{CP}}^2$-bundle.

Let $P\to M_6$ be the principal $SU{\left(3\right)}_{\mathrm{geom}}$ bundle. Define the associated ${\mathbb{CP}}^2$-bundle

$$\pi :\mathcal{Y}:=P{\times }_{SU\left(3\right)}{\mathbb{CP}}^2⟶M_6,{\mathbb{CP}}^2\simeq SU\left(3\right)/S(U\left(2\right)\times U\left(1\right)).$$

Its vertical tangent bundle $T^{\mathrm{vert}}\mathcal{Y}:=ker\left(d\pi \right)$ is the associated bundle

$$T^{\mathrm{vert}}\mathcal{Y}\cong P{\times }_{S\left(U\right(2)\times U(1\left)\right)}{\mathbb{C}}^2,$$

where ${\mathbb{C}}^2$ carries the standard isotropy representation of $S\left(U\right(2)\times U(1\left)\right)$. The underlying real bundle has rank 4; we denote it by $F_4$. In particular, the identification “$F_4\simeq T{\mathbb{CP}}^2$” should be read as the canonical identification of each fibre ${\left(F_4\right)}_x$ with the tangent space of the ${\mathbb{CP}}^2$ fibre over x, i.e., as a vertical tangent.

The $U\left(1\right)$ in (13) is the isotropy phase in $S\left(U\right(2)\times U(1\left)\right)$. On $(\mathbb{H}\epsilon ,J)$ it acts by $e^{\theta J}$. For each extra $\mathrm{SU}\left(3\right)$ we therefore obtain a natural line bundle whose connection is the Spin^c twist needed on $\mathbb{C}{P}^2$ (which is non-spin). We do not identify the octonion real line $\mathbb{R}·1\subset \mathbb{O}$ with this $U\left(1\right)$; rather, the $U\left(1\right)$ from $SU{\left(3\right)}_{L,R}^{geom}\to SU\left(2\right)\times U\left(1\right)$ is a Lie-algebra direction acting on the ${\mathbb{C}}^2$ tangent. It is the Spin^c line on $T\mathbb{C}{P}^2$. In model-building one may consider mixing these geometric $U\left(1\right)$’s with abelian factors inside $E_6$, but the Spin^c role is canonical and model-independent.

Detailed explanation: Each $SU{\left(3\right)}_{\mathrm{geom}}$ gave us not only an $SU\left(2\right)$ (which we used for rotations in $\Im H$) but also a $U\left(1\right)$ (the extra $U\left(1\right)$ generator in the decomposition). These $U\left(1\right)$’s provide the necessary $Spi{n}^c$ structure on the $C{P}^2$ fibres. Why $Spi{n}^c$? It is a known topological fact that $C{P}^2$ is not a spin manifold (its second Stiefel–Whitney class $w_2$ is nonzero, preventing a spin structure). However, $C{P}^2$ is a $Spi{n}^c$ manifold, meaning it can admit spinors if supplemented with a $U\left(1\right)$ gauge field (a complex line bundle) whose field strength compensates for the obstruction.

In practical terms, to have fermions live on $C{P}^2$ or its tangent, one needs a $U\left(1\right)$ connection – often called a $Spi{n}^c$ line bundle – twisting the spin structure. Here, the $U\left(1\right)$ we got from $SU\left(3\right)\to SU\left(2\right)\times U\left(1\right)$ does exactly that: it rotates the tangent $C^2$ fibre by a phase and thus corresponds to the natural $U\left(1\right)$ in the stabilizer $S\left(U\right(2)\times U(1\left)\right)$ of a point in $C{P}^2$. In other words, this $U\left(1\right)$ is exactly the isotropy $U\left(1\right)$ that appears in the coset $SU\left(3\right)/S\left(U\right(2)\times U(1\left)\right)$, whose connection can be seen as the $Spi{n}^c$ connection on $C{P}^2$.

For each extra $SU\left(3\right)$, we obtain a natural complex line bundle over $M_6$ whose connection is the $Spi{n}^c$ twist needed on $C{P}^2$. We should not confuse this $U\left(1\right)$ with the real scalar in octonions (the $R·1$). It’s not some extra dimension; it is literally the $U\left(1\right)$ subgroup of $SU{\left(3\right)}^{\mathrm{geom}}$ that acts as phase rotations on the $C^2$ fibre. So, each $SU{\left(3\right)}^{\mathrm{geom}}$ yields a principal $U\left(1\right)$-bundle over $M_6$ (the fibration of $C{P}^2$ tangents) whose curvature is what’s required to define spinor fields on those fibres. In effect, we have built a consistent $SU{\left(3\right)}^{\mathrm{geom}}$-structured fibre bundle $F_4\to M_6$ which is a $Spi{n}^c$ bundle (not spin, but $Spi{n}^c$).

From a model-building perspective, these geometric $U\left(1\right)$’s could potentially mix with other $U\left(1\right)$ factors from the physical gauge groups (for instance, hypercharge or a $B-L$ symmetry in $E_6$). But such mixing would be an additional consideration; intrinsically, their role is fixed: they ensure the fermions on the internal fibre have the right twist to exist.We emphasize that this $Spi{n}^c$ role is canonical and model-independent – any theory with $C{P}^2$ fibre would need such a $U\left(1\right)$, so it’s not an arbitrary choice but a topological necessity satisfied neatly by the $SU{\left(3\right)}^{\mathrm{geom}}$ splitting.

In summary, the presence of those $U\left(1\right)$ factors is not an extra complication but in fact a crucial feature allowing spinors (matter fields) to propagate on the internal space. Each geometric $U\left(1\right)$ becomes a kind of background field (a part of the geometry) rather than a new gauge force to be identified with, say, the Standard Model $U{\left(1\right)}_Y$. We deliberately say “we do not identify it with the octonion real line” to avoid the misconception that it’s a trivial scalar; it’s part of the Lie algebra direction acting on the $C^2$ tangent. This $U\left(1\right)$ connection can be thought of as the field that, if we were to compactify on $C{P}^2$, would be needed to satisfy the Dirac equation on that space. Thus, the unified framework naturally includes the gravitationally necessary $U\left(1\right)$ fibres.

Spin^c on ${\mathbb{CP}}^{\mathbf{2}}$.

One has $w_2\left(T{\mathbb{CP}}^2\right)\ne 0$, so ${\mathbb{CP}}^2$ is not spin, but it is Spin^c: there exists a determinant line $L\to {\mathbb{CP}}^2$ with $c_1\left(L\right)\equiv w_2(mod2)$ [27]. The geometric $U\left(1\right)$ factor in $SU{\left(3\right)}_{\mathrm{geom}}\to SU\left(2\right)\times U\left(1\right)$ furnishes precisely the Spin^c connection on the ${\mathbb{CP}}^2$ fibres. Hence the two geometric $U\left(1\right)$’s are not optional decorations; they implement the canonical Spin^c twists needed for fermions on $F_4\simeq T{\mathbb{CP}}^2$.

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

We treat the two extra $\mathrm{SU}\left(3\right)$’s as structure only. Over $M_6$ take principal bundles for $E_6^L$ and $E_6^R$; matter sits in associated $\mathbf{27}$ vector bundles and gauge fields in adjoint bundles. Tangent/internal decomposition uses $T{M}_6=(\Im {\mathbb{H}}_L)\oplus (\Im {\mathbb{H}}_R)$ and the two fibres $F_4^{L,R}\simeq {\mathbb{H}}_{L,R}{\epsilon }_{L,R}$ from above. The visible interactions come from $E_6^{L,R}$; the extra $\mathrm{SU}\left(3\right)$’s supply geometry and Spin^c lines on the internal fibres.

Detailed explanation: With the geometric “scaffold” – consisting of the 6D base $M_6$, two 4D $C{P}^2$-like fibres, and $U\left(1\right)$$Spi{n}^c$ connections – now established, we turn to how the physical fields (fermions, gauge bosons) reside in this setup. Essentially, the two $E_6$ groups (one on the left, one on the right) will provide the usual matter content and forces, but now they propagate in a restricted manner on $M_6$.

We propose to treat the $SU{\left(3\right)}_{\mathrm{geom}}$ factors as pure structure, not gauge fields. That means we do not include dynamical gauge bosons for them in the action; instead, we only keep the metric and connection associated with the $M_6$ and the $Spi{n}^c$ line. Meanwhile, we take principal bundles for $E_6^L$ and $E_6^R$ over $M_6$. In other words, imagine on the 6D base we have two sets of gauge fields: one with group $E_6^L$ and one with $E_6^R$. These are genuine gauge fields (with their field strengths, etc.) but their presence is tied to the geometry. Matter fields (like fermions) are then sections of associated vector bundles – specifically, in the 27-dimensional representation of $E_6$ (since each generation of fermions can fit in an $E_6$27, in many $E_6$ GUT models). Gauge bosons are connection fields in the adjoint (78) bundles.

Now, how do these 6D fields give us effective 4D physics? The concept introduced is that of a tangent/internal decomposition of everything, leveraging the splitting:

$$T{M}_6=\Im H_L\oplus \Im H_R$$

(24)

and

$$F_4^L\simeq H_L{ϵ}_L,F_4^R\simeq H_R{ϵ}_R$$

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This splitting means that at each point of $M_6$, one can distinguish the directions along $M_6$ (six of them, the “spacetime directions”) and the directions along the internal fibres (four on the left fibre and four on the right fibre). So any field can be classified by how it transforms under rotations of these subspaces. Concretely:

• The gauge fields of $E_6^L$ and $E_6^R$ will have components that could a priori point along $M_6$ or along the fibre. But if $M_6$ is eventually effectively 4D (due to localization on leaves), the physically observed gauge fields will be those components tangential to the 4D leaves. The components along the internal fibre might manifest as Higgs fields or heavy modes (similar to Kaluza–Klein modes).

• The fermionic fields living in 27s can be decomposed likewise. For instance, an $E_6$ 27 contains Standard Model fermions; those would be functions on $M_6$ that also take values in an internal spinor representation on the fibre. The presence of the $Spi{n}^c$ structure ensures we can define these spinor fields properly on $M_6$ with internal $C{P}^2$ fibres.

We note that the visible (low-energy) interactions come from $E_6^{L,R}$ only. That is, $E_6$ gauge fields include the Standard Model and possibly additional $Z^{\prime }$ or other exotics, and those are what mediate forces in 4D. The $SU{\left(3\right)}^{\mathrm{geom}}$ do not add new forces; instead, they “supply geometry and $Spi{n}^c$ lines” as we discussed. This separation of roles is crucial to ensure no unwanted gauge fields from $SU{\left(3\right)}^{\mathrm{geom}}$ clutter the low-energy spectrum.

So at this point, the picture is:

• We have a fibre bundle structure: base $M_6$ (with two embedded 4D sub-manifolds ${\Sigma }_L,{\Sigma }_R$) and internal fibres that are 4D (tangent $C{P}^2$ spaces).

• Over this entire structure, we have $E_6^L\times E_6^R$ gauge fields. They are free to propagate in the six base directions, but ultimately they will be confined to the 4D subspaces due to a mechanism in the next section (localization).

• Matter fields (quarks, leptons, etc.) live in these bundles – presumably, a chiral projection will leave them effectively on one of the 4D leaves (e.g., left-chiral matter on ${\Sigma }_R$ and maybe some mirror on ${\Sigma }_L$).

It’s worth noting: since $E_6$ contains the Standard Model gauge group, once we break $E_6$ at some high scale, we would get $SU{\left(3\right)}_c,SU{\left(2\right)}_L,U{\left(1\right)}_Y$ (and perhaps additional stuff like $U{\left(1\right)}_{B-L}$, etc.). The trinification breakdown mentioned in the introduction will occur inside these $E_6$ groups. That means on ${\Sigma }_L$ leaf, we’ll see something like $SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$ as active gauge symmetries (the Standard Model), and on ${\Sigma }_R$ we might see $SU{\left(3\right)}_{c^{\prime }}$ and $SU{\left(2\right)}_R$ etc., depending on how the symmetry breaking is arranged (we propose $SU{\left(2\right)}_R$ breaks giving gravity, so perhaps $SU{\left(3\right)}_R$ breaks at high scale on the right side). We’ll clarify this in the Big Picture section.

In essence, this section doesn’t introduce new formulas but sets the stage: we embed the “theory of everything” ($E_6\times E_6$ gauge fields + matter) into the hybrid space constructed by $SU{\left(3\right)}^{\mathrm{geom}}$. The geometrical $SU\left(3\right)$’s determine how spacetime and internal spaces are glued together, while the $E_6$’s bring in the particle content. This is somewhat analogous to how in string theory one has an internal manifold and gauge fields on it; here the role of “internal manifold” is played by the $C{P}^2$ fibre and “spacetime manifold” is the 6D $M_6$. The difference is we have two overlapping 4D spacetimes rather than one global 6D-to-4D compactification.

8.1. Fermion Localisation and Chirality (Outline)

Let ${\Gamma }^A$ be 6D gamma matrices for $\mathrm{Cl}(3,3)$ and let $\Pi$ denote the projector onto a chosen leaf W (Section 4). A minimal localisation ansatz uses a domain–wall mass profile $m\left(\sigma \right)$ depending on signed distance $\sigma$ to the leaf and the Spin^c connection on $F_4\simeq T{\mathbb{CP}}^2$:

$$**D*{*}_6\psi :=i{\Gamma }^A{e}_A^{\mu }{D}_{\mu }\psi +i{\Gamma }^{\widehat{a}}{n}_{\widehat{a}}^M{\partial }_M\psi +m\left(\sigma \right)\psi =0,\psi =\Pi \psi .$$

Standard domain–wall arguments (Jackiw–Rebbi type) produce normalisable 4D zero–modes of definite leaf chirality; the opposite chirality localises on the other leaf or is lifted by boundary conditions. The Spin^c twist on $F_4$ provides the internal index needed to obtain the desired family multiplicities. A full zero–mode and index analysis will appear elsewhere. We stress that 4D chirality is here outlined, not established: the zero-mode ansatz above indicates how chiral families could localise, but a complete index computation—and in particular a demonstration that the two-leaf spectrum is not vector-like, i.e., that mirror partners across ${\Sigma }_L,{\Sigma }_R$ are projected out or lifted—remains an open requirement of the programme rather than a result of the present paper.

8.2. 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 [28]: 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 [29]. Conformity with Coleman–Mandula has rightly been emphasised as a first checkpoint for algebraic models of particle physics [30].

The present construction complies in two distinct ways. First, algebraically on each emergent leaf: the division-algebra split is engineered so that spacetime and internal generators 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 (the $E_6^{L,R}$ gauge content and its unbroken descendants) act on the fibres $F_4\simeq \mathbb{H}\epsilon$; and the geometric $SU\left(3\right)$’s are structure groups, not gauged. On a leaf the residual symmetry is therefore a direct product $SO(3,1)\times G_{\mathrm{int}}$, exactly the form Coleman–Mandula permits. 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 [23].

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 satisfies 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—and, by the commutant structure above, does—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.

9. 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 $\mathrm{SU}\left(3\right)\to \mathrm{SU}\left(2\right)\times U\left(1\right)$,

$$T{M}_6\cong (\Im {\mathbb{H}}_L)\oplus (\Im {\mathbb{H}}_R),F_4^X\cong {({\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1})}_{\mathbb{R}}\simeq T\mathbb{C}{P}^2,X=L,R,$$

with $\Im \mathbb{H}$ realizing ${\mathbf{3}}_0$ and $\mathbb{H}\epsilon$ realizing ${(\mathbf{2}\oplus \overline{\mathbf{2}})}_{\mathbb{R}}$. The two $U\left(1\right)$’s act as Spin^c line connections on the $\mathbb{C}{P}^2$ fibres.

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: two Higgs order parameters define localized 4D leaves ${\Sigma }_R,{\Sigma }_L\subset M_6$ via a covariant two-form density ${\rho }_X$. 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 (companion work).

Where do the unbroken gauge groups live?

Because every sector action is wedged with ${\rho }_X$, dynamics is localized on the leaves. Hence unbroken $\mathrm{SU}{\left(3\right)}_c$ (and $\mathrm{SU}{\left(3\right)}_{c^{\prime }}$ if retained) are 4D gauge symmetries on the relevant $\Sigma$’s. The 6D base $M_6$ persists as the ambient bundle base, but low-energy fields do not propagate in the bulk.

What are the fibres relative to spacetime?

$F_4^{L,R}$ are rank-4 internal vector bundles over $M_6$, canonically $F_4^X\simeq {\mathbb{H}}_X{\epsilon }_X\cong {({\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1})}_{\mathbb{R}}\simeq T\mathbb{C}{P}^2$. 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 six tangent directions $(\Im {\mathbb{H}}_L)\oplus (\Im {\mathbb{H}}_R)$ plus one opposite-side normal on each leaf.

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).

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

Detailed explanation: This section consolidates the whole framework, breaking it into three conceptual layers and addressing important questions about how the effective 4D physics emerges.

Layer 1: Geometric $SU{\left(3\right)}^{\mathrm{geom}}$’s carving out spacetime and internal space. These are structural, not dynamical. Each $E_8$ branch provided one such $SU\left(3\right)$, and using them we constructed $M_6$ and the internal fibres. After breaking $SU\left(3\right)\to SU\left(2\right)\times U\left(1\right)$, we had:

$T{M}_6\cong (\Im H_L)\oplus (\Im H_R)$, the 6 real dimensions of the base.

$F_4^X\cong {(2_{+1}\oplus 2_{-1})}_R\cong TC{P}^2$ for $X=L,R$, the real 4D internal fibre on each side.

$\Im H$ realizes the $3_0$ and $Hϵ$ realizes ${(2\oplus \overline{2})}_R$, as we already detailed.

The two $U\left(1\right)$’s from each $SU{\left(3\right)}^{\mathrm{geom}}$ act as the $Spi{n}^c$ line bundle connections on those $C{P}^2$ fibres.

In simpler terms, Layer 1 is: “the extra $SU\left(3\right)$s create a 6D world (with two embedded 4D slices) and give each 4D slice a 4D internal space (tangent to $C{P}^2$) with the necessary $U\left(1\right)$ field for fermions.” This layer is purely about geometry and internal degrees of freedom—no Standard Model forces here yet, no direct dynamics.

Layer 2: Gauge $SU\left(3\right)$’s inside each $E_6$ (the trinification). These are the usual gauge groups we think of in particle physics, and they are dynamical (they have field strengths, particles, etc.). On the left side, $E_6^L$ splits as $(SU{\left(3\right)}_c\times SU{\left(3\right)}_L\times SU{\left(3\right)}_{F,L})$, and on the right $E_6^R$ splits as $(SU{\left(3\right)}_{c^{\prime }}\times SU{\left(3\right)}_R\times SU{\left(3\right)}_{F,R})$. Then further breakings yield:

Left: $SU{\left(3\right)}_L\to SU{\left(2\right)}_L\times U{\left(1\right)}_Y$ (electroweak interactions for left-handed fermions), while $SU{\left(3\right)}_c$ is QCD and $SU{\left(3\right)}_{F,L}$ is a global flavor symmetry for generations.

Right: $SU{\left(3\right)}_R\to SU{\left(2\right)}_R\times U{\left(1\right)}_{Y\mathrm{dem}}$ (the $SU{\left(2\right)}_R$, interpreted as the self-dual leaf Lorentz/spin connection, yields gravity in Plebanski form rather than as a broken compact gauge group, and $U{\left(1\right)}_{\mathrm{dem}}$ is the dark electromagnetism), and $SU\left(3\right)c^{\prime }$ is either hidden or very high-scale, and $SU{\left(3\right)}_{F,R}$ is a global flavor symmetry for right-handed fermions.

So Layer 2 comprises all the familiar gauge forces (and some new ones like the right-handed sector’s), but crucially these gauge fields will ultimately be confined to the 4D slices ${\Sigma }_L$ or ${\Sigma }_R$ (our “two worlds”). They do not propagate in the full 6D bulk at low energies; how that happens is explained by Layer 3.

Layer 3: Localization and Lorentz symmetry breaking in 6D. This is perhaps the most novel layer, describing how two separate 4D spacetimes emerge dynamically from the 6D and how standard 4D physics is confined to them. We envision using two Higgs-like order parameters (presumably scalar fields or two-form fields) that develop expectation values to define two 4D “leaves” ${\Sigma }_R$ and ${\Sigma }_L$ inside $M_6$. We suggest a covariant two-form density ${\rho }_X$ for each leaf $X=L,R$. One can imagine ${\rho }_R$ is like a localized 2-form that is peaked on ${\Sigma }_R$ and similarly for ${\rho }_L$ on ${\Sigma }_L$. By wedging all sector actions with these ${\rho }_X$, the dynamics (kinetic terms, etc.) are essentially restricted to the leaves. This is analogous to fields living on domain walls or branes in higher-dimensional theories.

Lorentz breaking: The normal 2-frames $U_X$ mentioned are likely fields that pick out a preferred 2D plane (normal to each 4D leaf) in the 6D tangent space. The effect is to break $SO(3,3)$ (the 6D Lorentz group) down to $SO(3,1)\times SO\left(2\right)$ on ${\Sigma }_R$ and to $SO(1,3)\times SO\left(2\right)$ on ${\Sigma }_L$ (the latter is the same Lorentz factor with the sign flip in the metric for time). This breaking eats exactly 8 Lorentz coset modes per leaf: $dimSO(3,3)-[dimSO(3,1)+dimSO(2\left)\right]=15-(6+1)=8$, matching the count $dim(W\wedge N)=4\times 2=8$ of the mixing sector. These 8 would-be Goldstone components are absorbed by the eight mixed spin connections (a gravitational-Higgs-type mechanism), leaving the tangent $SO(3,1)$ spin connection massless and the normal $SO\left(2\right)$ as the structure group of the normal bundle. In other words, the full 6D local Lorentz symmetry is broken such that each 4D leaf has its own local Lorentz invariance (gravity on each leaf), while the extra degrees of freedom that would mix the leaf with the two normal directions are made heavy by these Higgs fields (this relates to the graviweak unification in 6D of the companion work).

A key question addressed: Where do the unbroken gauge groups live? The answer: because the action for each sector is weighted by ${\rho }_X$, the gauge fields and matter fields effectively only “see” their respective leaf ${\Sigma }_X$. Thus, the unbroken gauge symmetries like $SU{\left(3\right)}_c$ (QCD) – which we want to exist in our 4D world – end up confined to (say) ${\Sigma }_R$ (we suggest ${\Sigma }_R$ for visible sector). If $SU{\left(3\right)}_{c^{\prime }}$ is retained (not broken entirely), it could live on either ${\Sigma }_R$ or ${\Sigma }_L$ as a hidden QCD sector. But either way, in low-energy 4D physics, fields do not propagate in all 6 dimensions, only on their localized 4D slice. The 6D base still exists as an “ambient space” but is mostly empty of propagating degrees of freedom at low energy (think of it like two branes in a higher-dimensional bulk, with bulk gravity perhaps but gauge fields on the branes).

Another question: What are the fibres relative to spacetime? We clarify that $F_4^L$ and $F_4^R$ (the internal 4D fibres) are internal degrees of freedom, not extra spacetime dimensions. If we stand on one 4D leaf and look around, we see 3 space + 1 time; we do not directly perceive those 4 internal dimensions as large spatial directions – they are like an internal symmetry space at each point. When we “restrict to a leaf”, each point of the 4D spacetime still has an attached $C{P}^2$-tangent-like internal space where internal symmetries (like color, etc.) act. This is akin to saying: on the 4D leaf, physics has gauge symmetries that can be thought of as arising from motion in those internal fibre directions, but those directions aren’t freely accessible dimensions for propagation. The Standard Model forces thus act on internal fibre indices rather than as extra spacetime dimensions.

We also phrase: “The two 4D spacetimes come from the six tangent directions $(\Im H_L)\oplus (\Im H_R)$ plus one opposite-side normal on each leaf.”. This is exactly how we constructed $M_4^{\left(R\right)}$ and $M_4^{\left(L\right)}$ earlier: each took the 3 from one side’s $\Im H$ and added the normal from the other side. The “opposite-side normal” means ${\Sigma }_R$ uses a direction in $H_L$ as its 4th dimension (time), and ${\Sigma }_L$ uses a direction in $H_R$ as its time. Thus each leaf’s 4D tangent is not just the naive splitting $H_L$ or $H_R$, but rather a mix: it’s 3 from its own side + 1 from the other.

The discussion also lists two consistent options for the second color group $SU{\left(3\right)}_{c^{\prime }}$ (the one from $E_6^R$ that we said is hidden or global):

Decoupled/hidden: Break or confine $SU{\left(3\right)}_{c^{\prime }}$ at high scale so that only the ordinary QCD $SU{\left(3\right)}_c$ remains in low energy. In this scenario, $SU{\left(3\right)}_{c^{\prime }}$ might not appear in 4D at all, or it might be a confined hidden sector (perhaps giving dark bound states, etc.) that doesn’t interfere with known physics. This is likely preferred to avoid mirror quarks etc. Only $SU{\left(3\right)}_c$ on ${\Sigma }_R$ is then the QCD we see.

Gauged on a leaf: Alternatively, we could allow $SU{\left(3\right)}_{c^{\prime }}$ to remain and assign it to one of the two leaves (maybe the other 4D world ${\Sigma }_L$ or also ${\Sigma }_R$). It would then be like a shadow QCD in a parallel sector. We mention “portal terms can live on $X^3={\Sigma }_R\cap {\Sigma }_L$ under BF matching conditions”, indicating that if both leaves have color forces, their intersection (which is 2D) might host interactions connecting them (perhaps something like a common 2D defect where fields from both sectors meet, reminiscent of a brane intersection scenario). This is a more speculative option and would mean our world might interact weakly with a hidden world via this intersection.

Finally, we present a one-line Dictionary:

Structural $SU{\left(3\right)}_{L,R}^{\mathrm{geom}}$ correspond to $(M_6,F_4^L,F_4^R)$, defining geometry (base + fibres).

Gauge $SU\left(3\right)$ inside $E_6^{L,R}$ correspond to 4D forces on ${\Sigma }_{L,R}$. This neatly separates “geometry group” vs “force group” roles of the various $SU\left(3\right)$ factors.

The Big Picture is that we have a cohesive theory where gravity and weak interactions cause a splitting of spacetime into two sheets, while the gauge interactions of the Standard Model are confined to those sheets, and the internal symmetries (like color) are interpreted as rotations in an abstract 4D internal space attached to each spacetime point. It’s like a blend of Kaluza–Klein (internal space for gauge forces) and brane-world (fields localized on sub-manifolds) scenarios, all orchestrated by the exceptional algebra structure of $E_8\times E_8$.

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 the companion work and is left open here.

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

The geometric fibre $F_4\simeq T{\mathbb{CP}}^2$ realises the same $SU\left(3\right)$–flavour geometry that underlies the $J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$ mass–ratio construction: the adjoint $8\to 3_0\oplus 2_{+1}\oplus 2_{-1}\oplus 1_0$ maps to $\Im \mathbb{H}\oplus \mathbb{H}\epsilon$, with ${(2\oplus \overline{2})}_R$ providing the internal complex 2. In this scaffold, internal 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.

9.1. 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

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

(26)

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 an interacting trace dynamics of the Adler type. In the coarse–grained, large–N limit one recovers: (i) emergent quantum kinematics from the conserved Adler–Millard charge (canonical commutators and unitary evolution), (ii) the 6D BF+Lorentz–Higgs sector as the IR geometric hydrodynamics, and (iii) localisation on two 4D leaves via the normal 2–frame condensate, which breaks $SO(3,3)\to SO(3,1)\times SO\left(2\right)$ and gives mass to the eight mixed connections.

Power counting and predictivity.

The UV theory is polynomial in the matrices and free of short–distance field singularities; the continuum fields arise as collective variables. The 6D effective action inherits a finite set of relevant/marginal operators at low dimension, while higher–dimension operators are suppressed by the trace–dynamics 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 $y_f\left(\mu \right)$). This keeps $U{\left(1\right)}_{\mathrm{dem}}$ RG–safe.

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.

10. Summary

• The $(3,3)$ base $M_6$ is $\Im {\mathbb{H}}_L\oplus \omega \Im {\mathbb{H}}_R$; the two 4D spacetimes are 4-planes obtained by adding a single normal from the opposite side.
• 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 ${(\mathbf{2}\oplus \overline{\mathbf{2}})}_{\mathbb{R}}$ and is the 4D fibre $F_4\simeq T\mathbb{C}{P}^2$.
• Each geometric $U\left(1\right)$ is the Spin^c line on $\mathbb{C}{P}^2$; we do not confuse it with the octonion real line.

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

Spacetime Emergence: The 6D base $(M_6,g)$ has signature (3,3) and is explicitly given by $\Im H_L\oplus \omega \Im H_R$. Two overlapping Lorentzian 4D spacetimes (leaves) are embedded as 4-planes in $M_6$ by including one normal direction from the opposite $\Im H$ (as in equations (4) and (5) earlier). One leaf ${\Sigma }_R$ ends up with signature (3,1) using $(\Im H_R+t_L)$ and the other ${\Sigma }_L$ uses $(\Im H_L+\omega t_R)$. These share a 2D (1,1) intersection. Thus, our familiar 4D spacetime is one “leaf” of a 6D space, and there’s a second, hidden 4D leaf intertwined via a 2D bridge.

$SU\left(3\right)$ Branching and Octonions: Each extra $SU\left(3\right)\mathrm{geom}$ (left or right) when broken to $SU\left(2\right)\times U\left(1\right)$ gives $8\to 3_0+2_{+1}+2_{-1}+1_0$. Correspondingly, the octonion split $O=H\oplus Hϵ$ realizes this: $\Im H$ provides the $3_0$ (tied to spatial directions in $M_6$), and $Hϵ$ provides the $2_{+1}\oplus 2_{-1}$ (the 4 internal directions). The real unit $1\in O$ is separate from the $U\left(1\right)$ generator and is not used directly as it would correspond to the singlet $1_0$. Thus octonionic algebra explains why 6+4 dimensions naturally appear from $SU\left(3\right)$.

Internal $C{P}^2$ fibre: The 4D internal fibre $F_4$ constructed from $Hϵ$ is canonically isomorphic to the tangent space of $C{P}^2$. This means each point in spacetime has an internal structure equivalent to a small $C{P}^2$ direction. The ${(2\oplus \overline{2})}_R$ representation of $SU\left(2\right)\times U\left(1\right)$ is exactly what acts on $TC{P}^2$. So the model finds the minimal internal space for an $SU\left(3\right)$ symmetry. In effect, the Standard Model’s “internal” symmetries (like color, flavor) are geometrized as symmetries of a tiny $C{P}^2$ fibre attached to spacetime.

Geometric $U\left(1\right)$’s as $Spi{n}^c$ Connections: Each of the two $U\left(1\right)$ factors from $SU{\left(3\right)}^{\mathrm{geom}}\to SU\left(2\right)\times U\left(1\right)$ serves as the $Spi{n}^c$ line bundle on the $C{P}^2$ fibre. In other words, these are background $U\left(1\right)$ gauge fields ensuring that the internal space can host spinor fields (quarks, leptons). They are not to be confused with any scalar or physical $U\left(1\right)$ in the octonions. They could mix with model $U\left(1\right)$’s in principle, but fundamentally their role is fixed by geometry. Thus, the existence of a “dark” $U\left(1\right)$ (or two of them) in the unified group is not arbitrary: it’s required to allow spin structure on the internal fibre.

Putting it all together: The paper presents a unified theory scaffold in which space, time, and internal quantum numbers all stem from a common $E_8\times \omega E_8$ symmetry structure. Spacetime (including possibly an extra hidden timelike dimension sector) emerges from using the extra $SU\left(3\right)$ factors as a frame-Higgs that breaks 6D down to $4D+4D$ in a controlled way, while the internal symmetry space (needed for gauge forces like color) is identified with $C{P}^2$ directions coming from the same $SU\left(3\right)$ factors. Meanwhile, the $E_6\times E_6$ part of the symmetry contains the known Standard Model forces and matter, which are placed on these 4D slices and benefit from the geometric structuring (e.g., the existence of three generations from $SU{\left(3\right)}_F$ flavor, mass ratios from Jordan algebra, etc., as referenced). Gravity emerges from the Lorentz–Higgs leaf-localization mechanism, with the relevant $SU{\left(2\right)}_R$ playing the role of the self-dual leaf spin connection (Plebanski form) rather than a compact gauge group whose breaking yields a graviton, and a new pseudo-force (dark electromagnetism) appears related to mass. All fields of the $E_6$ sectors are now living in a higher-dimensional space but effectively constrained to our 4D due to the Higgs localization.