The foundation of the hyperunified field theory II
- fundamental interaction and evolving universe

Yue-Liang<EMAIL_ADDRESS>$^1$Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China

$^2$International Centre for Theoretical Physics Asia-Pacific (ICTP-AP)
(Beijing/Hangzhou), UCAS, Beijing 100190, China

$^3$ Taiji Laboratory for Gravitational Wave Universe (Beijing/Hangzhou), University of Chinese Academy of Sciences (UCAS), Beijing 100049, China

$^4$ School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China

In part I of the foundation of the hyperunified field theory, we have shown the presence of entangled hyperqubit-spinor field as fundamental building block with appearance of Minkowski hyper-spacetime as free-motion spacetime and emergence of inhomogeneous hyperspin symmetry as fundamental symmetry. In this paper as part II, we demonstrate by following along gauge invariance principle and scaling invariance hypothesis that the inhomogeneous hyperspin gauge symmetry and scaling gauge symmetry govern fundamental interactions and reveal the nature of gravity and spacetime. With the fiber bundle structure of biframe hyper-spacetime and emergence of non-commutative geometry, we explore methodically the gauge-geometry duality and genesis of gravitational interaction in locally flat gravigauge hyper-spacetime. A whole hyperunified field theory in 19-dimensional hyper-spacetime is established to unify not only all discovered leptons and quarks into hyperunified qubit-spinor field but also all known basic forces into hyperunified interaction governed by inhomogeneous hyperspin gauge symmetry. We present a systematic investigate on the hyperunified field theory with deriving the dynamics of fundamental fields as basic laws of nature and conservation laws relative to basic symmetries and showing Higgs-like bosons and three families of lepton-quark states. We provide a detailed analysis on inflationary early universe with evolving graviscalar field and a discussion on scaling gauge field as dark matter candidate and $\cQ_c$-spin scalar field as source of dark energy.

12.10.-g, 04.50.+h, 04.50.-h,11.10.Kk

Keywords: gauge and scaling invariance principle, inhomogenous hyperspin gauge symmetry,
unification of all basic forces, gravigauge hyper-space-time, inflationary universe, dark matter and
dark energy.

§ INTRODUCTION

In the previous paper as part I of the foundation of the hyperunified field theory[1], starting from the motional property of functional field by proposing the maximum coherence motion principle and maximum entangled-qubits motion principle as two guiding principles, we have provided a detailed analysis and systematic investigation on the existence of entangled hyperqubit-spinor field and $\cQ_c$-spin scalar field as fundamental building blocks of nature and also the appearance of Minkowski hyper-spacetime and emergence of inhomogeneous hyperspin symmetry and scaling symmetry as fundamental symmetries of nature. In light of advantages of the action principle with path integral formulation, the two guiding principles presented in [1] have been shown to lay the essential foundation of the hyperunified field theory[2, 3, 4] within the framework of gravitational quantum field theory[5, 6]. It has been demonstrated that a locally entangled state of qubits obeying maximum coherence motion principle brings about an entangled qubit-spinor field with automatic appearance of canonical anticommutative relation, and meanwhile both spacetime dimension and fundamental symmetry are determined by the motion-correlation $\cM_c$-spin charge $\cC_{\cM_c}=D_h$ of entangled qubit-spinor field. The $\cQ_c$-spin charge $\cC_{\cQ_c}=q_c$ has been shown to possess a periodic feature given by the mod 4 qubit number, which allows us to classify all entangled qubit-spinor fields and spacetime dimensions into four categories characterized by four $\cQ_c$-spin charges $\cC_{\cQ_c}=0,1,2,3$. In general, we come to a better understanding on the basic constituent of matter as locally entangled state of qubits and also the motional nature of matter, which includes the appearance of Minkowski spacetime with only one temporal dimension and the emergence of fundamental symmetry of matter and spacetime. To get a better understanding on the fundamental symmetry of nature, we have distinguished symmetry groups in Hilbert space from those in Minkowski spacetime, so that we arrive at an associated symmetry in which inhomogeneous hyperspin symmetry group and global scaling symmetry group in Hilbert space as internal space of basic fields are in association with inhomogeneous Lorentz-type symmetry group and scaling symmetry group in Minkowski spacetime as external spacetime of coordinates, where their symmetry group transformations must be coincidental so as to ensure the invariance of the action. In particular, it has been verified that the entangled decaqubit-spinor field with $\cM_c$-spin charge $\cC_{\cM_c}=19$ and $\cQ_c$-spin charge $\cC_{\cQ_c}=1$ brings on a hyperunified qubit-spinor field which unifies all known leptons and quarks into a single fundamental building block with prediction on the existence of mirror lepton-quark states, and its action possesses an associated symmetry in which the inhomogeneous hyperspin symmetry WS(1,18) is in association with inhomogeneous Lorentz-type/Poincaré-type symmetry PO(1,18). Such an associated symmetry provides a hyperunified symmetry as fundamental symmetry. So that we are able to provide reliable explanations on some longstanding open questions, such as why there exist leptons and quarks beyond one family in SM and why the currently observed universe is only four-dimensional spacetime.

In the present paper as part II of the foundation of the hyperunified field theory, we will further study the basic properties of entangled hyperqubit-spinor field and $\cQ_c$-spin scalar field as fundamental building blocks and also the essential features of inhomogeneous hyperspin symmetry and scaling symmetry as fundamental symmetries. As there remain some open questions to be issued, such as: what is acted as the fundamental interaction of nature? how does the fundamental symmetry govern basic forces? what is the nature of gravity? what is the basic structure of spacetime? how does early universe get inflationary expansion? what is a dark matter candidate? what is the nature of dark energy?  what is the nature of Higgs boson? how can we understand three families of chiral type leptons and quarks discovered by current experiments?

To make issues on such open questions come to the main purpose of present paper as the part II of the foundation of the hyperunified field theory. We will begin with considering gauge invariance principle as guiding principle, which brings global symmetry in Hilbert space into local symmetry with the introduction of gauge field[7, 8] as basic field for characterizing fundamental interactions. When taking global inhomogeneous hyperspin symmetry and scaling symmetry in Hilbert space to be local symmetries based on the gauge invariance principle, we arrive at local inhomogeneous hyperspin gauge symmetry and scaling gauge symmetry as fundamental gauge symmetries which are shown to govern the fundamental interactions of nature.

A biframe hyper-spacetime with fiber bundle structure is deduced from gauging inhomogeneous hyperspin symmetry with the introduction of hyper-gravigauge field, which leads locally flat gravigauge hyper-spacetime in Hilbert space to emerge as hyper-fiber bundle with globally flat Minkowski hyper-spacetime as base spacetime obeying special relativity[9]. We will demonstrate that the local scaling gauge symmetry of basic fields in Hilbert space together with global scaling symmetry of coordinates in Minkowski spacetime play an essential role for building a full hyperunified field theory. Therefore, the gauge invariance principle and scaling invariance hypothesis together with the maximum coherence motion principle and maximum entangled-qubits motion principle proposed in the previous paper[1] will be shown to lay a whole foundation of the hyperunified field theory. In light of a fully constructed hyperunified field theory within the framework of gravitational quantum field theory, by corroborating both the gauge-gravity correspondence and gravity-geometry correspondence, we will come to a better understanding on the basic structure of spacetime and the nature of basic forces, which includes the genesis of gravity and the relation to general relativity[10, 11]. The equations of motion of fundamental fields are derived to characterize the laws of nature and meanwhile various conservation laws are resulted from relevant symmetries. It will be shown that in locally flat gravigauge hyper-spacetime, the hyperspin gravigauge field behaves as an auxiliary field with emergence of non-commutative geometry characterized by the hyperspin gravigauge field. The gauge-geometry duality is verified and applied to obtain equivalent gauge-type and geometric Einstein-type gravitational equations of motion, which allows us to analyze electric-like and magnetic-like gravitational interactions. Furthermore, it makes us to achieve a better understanding on the evolution of universe with inflationary early universe and currently accelerated expanding universe from evolving graviscalar field and $\cQ_c$-spin scalar field, and meanwhile get a better comprehension on three families of chiral type leptons and quarks in the standard model (SM) of electroweak and strong interactions[12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] in the framework of quantum field theory[23, 24, 25]. In particular, such a hyperunified field theory is shown to result in dark matter and dark energy candidates and bring on more Higgs-like bosons in four-dimensional spacetime. Anyway, we are going to make a systematic analysis and detailed investigation on the whole hyperunified field theory in this paper.

Our paper is organized as follows: a brief introduction is presented in Sect.1. In Sect. 2, we will make global $\cM_c$-spin/hyperspin symmetry and $\cQ_c$-spin symmetry as well as scaling symmetry as local gauge symmetries in Hilbert space by proposing gauge invariance principle as guiding principle, and the corresponding gauge fields as basic fields are introduced to characterize gauge interactions. The gauge covariant equation of motion for entangled hyperqubit-spinor field with scaling gauge invariance is derived to describe gravitational relativistic quantum theory of entangled hyperqubit-spinor field with the presence of hyper-gravigauge field and $\cQ_c$-spin scalar field. In Sect. 3, we will show that bicovariant vector field $\chih_{\fA}^{\;\; \fM}(x)$ and dual bicovariant vector field $\chi_{\fM}^{\; \fA}(x)$ as hyper-gravigauge field form locally flat gravigauge hyper-spacetime with emergence of non-commutative geometry, which brings on the introduction of biframe hyper-spacetime with hyper-fiber bundle structure. The fundamental interaction is shown to be governed by inhomogeneous hyperspin gauge symmetry WS(1,$D_h$-1). In particular, Abelian-type $\cW_e$-spin invariant-gauge field brings on the genesis of hyper-gravigauge field as bicovariant vector field $\chi_{\fM}^{\; \fA}(x)$ associated with graviscalar field and meanwhile leads to gravitational origin of gauge symmetry in hyper-spacetime, which demonstrates that gravitational interaction is truly characterized by gauge interaction. In Sect. 4, we are going to build a whole hyperunified field theory for describing both fundamental building blocks of nature and fundamental interactions of nature by following along gauge invariance principle and scaling invariance hypothesis. The hyperunified qubit-spinor field $\fPsi_{\fQH}(x)$ in 19-dimensional hyper-spacetime is regarded as fundamental building block of nature and the inhomogeneous hyperspin gauge interaction is viewed as fundamental interaction mediated through inhomogeneous hyperspin gauge field $\bvcA_{\fM}(x)$.  In Sect. 5, by reformulating such a hyperunified field theory in biframe hyper-spacetime to have hidden scaling gauge invariance, we are able to demonstrate the gauge-gravity correspondence and derive equations of motion of fundamental fields by applying for the least action principle. Meanwhile, various conserved currents are obtained with respect to gauge symmetries. In Sect. 6, we will apply Noether's theorem to such a whole hyperunified field theory in the presence of gravitational interaction and derive conservation laws and dynamic evolution equations from both global Poincaré-type group symmetry and conformal scaling symmetry of Minkowski hyper-spacetime.

By defining hyper-spacetime gauge field from hyperspin gauge field in Sect. 7, we will reformulate an equivalent action for the whole hyperunified field theory in a hidden gauge formalism, where the hyper-gravigauge field plays an essential role as Goldstone-like boson. Such a hidden gauge formalism implies that dynamics of hyper-gravigauge field is equivalently characterized by that of Einstein-Hilbert type action which is governed by a hidden general linear group symmetry GL(1,$D_h$-1, R). So that we are able to demonstrate the gravity-geometry correspondence under flowing unitary gauge and also the profound correlation between gravitational interaction and emergent Riemann geometry in curved Riemannian hyper-spacetime. As the hyperunified field theory is in principle independent of the choice of coordinate systems, we will rewrite in Sect.8 the whole hyperunified field theory in a hidden coordinate formalism by utilizing the scaling gauge invariant hyper-gravigauge field as Goldstone-like boson. Such a hyperunified field theory reveals the geometry-gauge correspondence, so that the gravitational interaction is described by hyperspin gravigauge field. In locally flat gravigauge hyper-spacetime, such a hyperspin gravigauge field behaves as an auxiliary field and characterizes emergent non-commutative geometry. In Sect. 9, by taking entirety unitary gauge as gauge prescription to fix both hyperspin gauge symmetry and hidden general linear group symmetry, we will present a more radical investigation on the foundation of the hyperunified field theory within the framework of gravitational quantum field theory. By revealing the gauge-geometry duality in biframe hyper-spacetime, we are able to explore the dynamics of both hyper-gravigauge field and hyper-gravimetric gauge field with electric-like and magnetic-like gravitational interactions through gauge-type and geometric Einstein-type gravitational equations. In Sect. 10, we will analyze various gauge fixing conditions for scaling gauge symmetry in the whole hyperunified field theory under entirety unitary gauge. By choosing Einstein-type basis in the presence of fundamental mass scale $\Mka$, we will show how a conformally flat gravigauge hyper-spacetime as nonsingularity background enables us to describe the evolution of early universe with evolving graviscalar field in hyper-spacetime. In light of the so-called co-moving cosmic time,  the universe undergoes an inflationary expansion. We will show that the minimum potential of $\cQ_c$-spin scaling field not only prevents early universe from keeping inflationary expansion but also provides a tiny cosmic energy density as dark energy candidate characterized by fundamental cosmic mass scale $\mk$. In particular, a heavy scaling gauge field is found to be as dark matter candidate when inflationary universe comes to an end. In Sect. 11, we will analyze hyperunified symmetry structure and possible symmetry breaking mechanisms in the whole hyperunified field theory in 19D hyper-spacetime. To illustrate possible symmetry breaking mechanisms for hyperspin symmetry, we are going to focus on the action of entangled decaqubit-spinor field as hyperunified qubit-spinor field with showing the explicit group decomposition structure for hyperspin gauge field in locally flat gravigauge hyper-spacetime and examining a specific bulk structure of 19th spatial dimension in hyper-spacetime, which enables us to get a better comprehension on SM with Higgs-like bosons and three families of chiral type lepton-quark states in four-dimensional spacetime. Our conclusions and discussions will be presented in the final section.

§  INHOMOGENEOUS HYPERSPIN AND $\CQ_C$-SPIN GAUGE SYMMETRIES AND GRAVITATIONAL RELATIVISTIC QUANTUM THEORY WITH SCALING GAUGE SYMMETRY BASED ON GAUGE INVARIANCE PRINCIPLE

In the previous paper as the part I of the foundation of the hyperunified field theory[1], we have presented an extensive analysis on the categorization for qubit-spinor field and spacetime dimension and made a detailed discussion on the symmetry properties of entangled qubit-spinor field and Minkowski spacetime. In particular, it has been shown that the category-1 entangled decaqubit-spinor field $\Psi_{\fQE^{10}}(x)$ enables us to merge all known leptons and quarks in SM into a single hyperunified qubit-spinor field $\fPsi_{\fQH}(x)$ in 19-dimensional hyper-spacetime. Such a hyperunified qubit-spinor field possesses inhomogeneous hyperspin symmetry WS(1,18). Nevertheless, all previous discussions are mainly focused on a free motion of entangled qubit-spinor field without involving dynamics through interactions. In this section, we are going to demonstrate how the fundamental interaction can be introduced by making $\cM_C$-spin and $\cQ_c$-spin symmetries of entangled qubit-spinor field to be gauge symmetries in Hilbert space by proposing the gauge invariance principle as guiding principle. A gauge covariant equation of motion for entangled hyperqubit-spinor field with scaling gauge invariance will be derived to characterize gravitational relativistic quantum theory of entangled hyperqubit-spinor field in the presence of hyper-gravigauge field and $\cQ_c$-spin scalar field.

§.§ Inhomogeneous hyperspin and $\cQ_c$-spin gauge symmetries with introduction of bicovariant vector field and gauge field by making gauge invariance principle

It has been shown in ref.[1] that the action of entangled hyperqubit-spinor field in category-$q_c$ possesses an associated symmetry which is presented by inhomogeneous hyperspin symmetry WS(1,$D_h$-1) and global scaling symmetry SG(1) in association with inhomogeneous Lorentz-type/Poincaré-type group symmetry PO(1,$D_h$-1) and conformal scaling symmetry SC(1) together with $\cQ_c$-spin symmetry SP($q_c$),
G_S     =     SC(1)⋉P^1,D_h-1⋉SO(1,D_h-1)  SP(1,D_h-1)⋊W^1,D_h-1 ×SG(1) ×SP(q_c)

=      SC(1)⋉PO(1,D_h-1) WS(1,D_h-1) ×SG(1) ×SP(q_c),
where the symbol “$ \wtjoin$" has been adopted to denote such an associated symmetry in which the group transformations of hyperspin group symmetry SP(1, $D_h$-1) and global scaling symmetry SG(1) in Hilbert space of basic fields must be coincidental with those of Lorentz-type group symmetry SO(1, $D_h$-1) and conformal scaling symmetry SC(1) in Minkowski spacetime of coordinates .

It is noticed that the inhomogeneous hyperspin symmetry and global scaling symmetry SG(1) as well as $\cQ_c$-spin symmetry operate on the entangled hyperqubit-spinor field and $\cQ_c$-spin scalar field in Hilbert space, while the inhomogeneous Lorentz-type/Poincaré-type group symmetry together with scaling symmetry SC(1) act on the coordinates in Minkowski hyper-spacetime. In general, they are distinguishable group symmetries. While when imposing symmetry invariance on the action under global transformations of hyperspin group symmetry SP(1, $D_h$-1) and scaling symmetry SG(1) for freely moving hyperqubit-spinor field in Hilbert space, such group symmetry transformations have to coincide with those of Lorentz-type group symmetry SO(1, $D_h$-1) and scaling symmetry SC(1) in Minkowski hyper-spacetime. Nevertheless, distinguishing group symmetries in Hilbert space of basic fields with those in Minkowski spacetime of coordinates becomes more essential when applying a gauge invariance principle to the action of entangled qubit-spinor field and scalar field as fundamental fields in Hilbert space. Moreover, such a distinguishment enables us to avoid automatically the longstanding obstacle on the unified field theory arising from the so-called Coleman-Mandula no-go theorem[26] about the impossibility of combining spacetime and internal symmetries in any except a trivial way.

Gauge invariance principle: the laws of nature should be independent of the choice of local field configurations. So that all symmetries of entangled hyperqubit-spinor field and $\cQ_c$-spin scalar field as fundamental fields are postulated to be local gauge symmetries in Hilbert space with the introduction of relevant gauge fields.

Based on such a gauge invariance principle as guiding principle, inhomogeneous hyperspin symmetry WS(1, $D_h$-1) and scaling symmetry SG(1) as well as $\cQ_c$-spin symmetry SP($q_c$) should be gauged to be local gauge symmetries which are referred to as inhomogeneous hyperspin gauge symmetry and scaling gauge symmetry as well as $\cQ_c$-spin gauge symmetry, respectively, and all their group operations act on entangled hyperqubit-spinor field and $\cQ_c$-spin scalar field in Hilbert space. Whereas inhomogeneous Lorentz-type/Poincaré-type group symmetry PO(1,$D_h$-1) and scaling symmetry SC(1) remain global symmetries in globally flat Minkowski hyper-spacetime, their group operations act on coordinate system. In such a consideration, the local inhomogeneous hyperspin gauge symmetry WS(1, $D_h$-1) and scaling gauge symmetry SG(1) in Hilbert space of basic fields do distinguish from the global inhomogeneous Lorentz-type/Poincaré-type group symmetry PO(1,$D_h$-1) and scaling symmetry in Minkowski hyper-spacetime of coordinates.

Following along the gauge invariance principle, we should introduce gauge field $\fkA_{\fM}(x)$ relevant to inhomogeneous hyperspin gauge symmetry WS(1, $D_h$-1) and $\cQ_c$-spin gauge symmetry SP($q_c$) so as to preserve gauge invariance of the action. Concretely, it is realized by replacing the usual derivative operator of coordinates in globally flat Minkowski hyper-spacetime with the following covariant derivative operator:
i_ →i_ ≡i_ + _(x) ,

_(x) ≡_(x) +  _(x) + _(x),

_(x) ≡_(x) +  _(x) ≡_^  (x)  1/2_  + _^(x)1/2_-  ,

_(x) ≡_(x) + _(x) ≡_^  (x)  1/2_ + _^  pq(x)  1/2_pq ,
with $\fM, \fB, \fC = 0,1,2,3,5,\ldots, D_h$ and $p,q=1,\cdots, q_c$. Where $\bvcA_{\fM}(x)$ is inhomogeneous hyperspin gauge field with $\cA_{\fM}^{\; \fB\fC}(x)$ denoting hyperspin gauge field of hyperspin gauge symmetry SP(1, $D_h$-1) and $\ckcA_{\fM}^{\fB}(x)$ representing $\cW_e$-spin gauge field relevant to translation-like $\cW_e$-spin Abelian gauge symmetry $W^{1,D_h-1}$. $\tcA_{\fM}^{\; pq}(x)$ is $\cQ_c$-spin gauge field of $\cQ_c$-spin gauge symmetry SP($q_c$). $\varSigma_{\fB\fC}$, $\varSigma_{- \fB}$ and $\tvSi_{pq}$ are relevant gauge group generators given by the commutators of $\vGa$-matrices[1],
_ = i/4 [ _, _ ],   _-  = 1/2 _ _-,   _- = 1/2 ( 1 - γ̂_D_h + 1 ),

_D_h+p  D_h+q =  i/4 [ _D_h + p, _D_h + q ] = i/4 [ _p, _q ] ≡_pq.
Where the hyperspin gauge field $\cA_{\fM}(x)$ belongs to adjoint representation of hyperspin gauge group symmetry SP(1, $D_h$-1).

To ensure both local inhomogeneous hyperspin gauge symmetry WS(1, $D_h$-1) and global inhomogeneous Lorentz-type/Poincaré-type group symmetry PO(1,$D_h$-1), we should introduce a bicovaraint vector field $\chih_{\fA}^{\;\, \fM}(x)$ in addition to the gauge field $\fkA_{\fM}(x)$. So that the kinematic term of the action is extended to be as follows:
^ δ_^    i_ →^_^   (x) i_ ,
with $\fA, \fM = 0,1,2,3,5,\ldots, D_h$. Where the constant bicovariant vector $\delta_{\fA}^{\;\; \fM}$ is replaced by the bicovaraint vector field $\chih_{\fA}^{\;\, \fM}(x)$.

With the above considerations, the action for a freely moving entangled hyperqubit-spinor field with global inhomogeneous hyperspin symmetry and scaling symmetry in associating with global inhomogeneous Lorentz-type/Poincaré-type group symmetry and scaling symmetry should be generalized to be gauge invariant action, which possesses local inhomogeneous hyperspin gauge symmetry WS(1, $D_h$-1) and scaling gauge symmetry SG(1) jointly together with global inhomogeneous Lorentz-type/Poincaré-type group symmetry PO(1,$D_h$-1) and scaling symmetry SC(1).

Let us begin with the following gauge invariant action:

_^(q_c)     ≡    ∫[d^D_h^(q_c,k)x]    { Ψ̅_^(q_c,k)(x)  _-^_^   (x) i_  Ψ_^(q_c,k)(x)

+     ϕ_p(x)  Ψ̅_^(q_c,k)(x) _-^D_h+p Ψ_^(q_c,k)(x) },
with $\fA, \fM= 0,1,2,3,5, \cdots, D_h^{(q_c,k)}$ and $p=0,\cdots, q_c$. As discussed in the previous paper as the part I[1], $q_c$ represents $\cQ_c$-spin charge $\CQc=q_c$ with $q_c=0,1,2,3$ as four categoric charges, and $\lamq$ is a coupling constant with $\lambda_0=\lambda_3=0$ since there exist no such coupling terms for $q_c=0,3$. $D_h^{(q_c,k)}$ denotes the categoric dimension of hyper-spacetime given by the relation $D_h^{(q_c,k)} = D_{q_c} + 8k$ with $D_{q_c}= 2,3,4,6$ as four basic dimensions and $k=0,1,\cdots$ as periodic number. $\Psi_{\fQE}^{(q_c,k)}(x)$ denotes the categoric entangled hyperqubit-spinor field in category-$q_c$ and $k$-th period, which has dimensions $2^{Q_N^{(q_c,k)}+1}$ in qubit-spinor representation of Hilbert space with $Q_N^{(q_c,k)}$ labeling the categoric qubit number given by the relation $Q_N^{(q_c,k)}=q_c + 4k$.

The matrices $\vSi_{-}^{\fA}$ and $\vSi_{-}^{D_h+p} $ are defined via the anticommuting $\vGa$-matrices $\vGa^{\hfA}= (\vGa^{\fA}, \vGa^{D_h + p}) $ ($\hfA= 0,1,2,3,5,\cdots, D_h+q_c$) as follows:
_-^ = 1/2 ^_-,   _- = 1/2 ( 1 - γ̂_D_h + 1 ),

_-^D_h+p = 1/2 ^D_h + p_- =  1/2 ^p_- ≡_-^p.

The action in Eq.(<ref>) is invariant under global inhomogeneous Lorentz-type/Poincaré-type group symmetry PO(1,$D_h$-1) with the following Lorentz-type group transformation:
x^ →x^' = L^_        x^,   L^_      ∈(1, D_h1),

_^   (x) →_^'   (x') = L^_       _^    (x),

_(x) →'_(x') = L_^        _(x) ,

Ψ_^(q_c,k)(x) →Ψ_^' (q_c,k)(x') =  Ψ_^(q_c,k)(x) ,
and translation group transformation:

\begin{eqnarray}
x^{\fM} \to x^{'\fM} = x^{\fM} + a^{\fM}\, , \qquad  a^{\fM} \in P^{1, D_h-1}\, ,
\end{eqnarray}

with $a^{\fM}$ a constant vector in globally flat Minkowski hyper-spacetime.

The action in Eq.(<ref>) possesses the local inhomogeneous hyperspin gauge symmetry WS(1, $D_h$-1) which is separated to the global inhomogeneous Lorentz-type group symmetry. The local hyperspin gauge transformation is made as follows:
'_(x) =   S(Λ) i ∂_ S^-1(Λ) +  S(Λ) _(x) S^-1(Λ)

Ψ_^' (q_c,k)(x) =  S(Λ) Ψ_^(q_c,k)(x) ,    S(Λ) = e^iϖ_(x) ^/2 ∈(1, D_h1),
with the following explicit transformation property:
S^-1(Λ) _-^ S(Λ) = Λ^_    (x)  _-^,   _^   (x) →_^'   (x) = Λ_^    (x) _^    (x),

_^' (x) =  1/2 (Λ^_    _  Λ^ -  Λ^_     _  Λ^ ) + Λ^_     Λ^_     _^ (x) ,

_^' (x) =  Λ^_    (x) _^(x),   Λ^_    (x)  ∈(1, D_h1) .
The local $\cW_e$-spin gauge transformation is given as follows:
'_(x) =   (x) i ∂_ ^-1(x) +  (x)_(x) ^-1(x),

Ψ_^' (q_c,k)(x) =  (x) Ψ_^(q_c,k)(x) ,   (x) = e^i ϖ_(x) _-^/2   ∈W^1,D_h-1,
with the following explicit form:

(x) = 1 + i ϖ_(x) _-^/2,   _^' (x) ≡_^(x) ,

Ψ_^' (q_c,k)(x) = Ψ_^(q_c,k)(x) + Ψ̌_^(q_c,k)(x),   Ψ̌_^(q_c,k)(x) ≡i 1/2ϖ_(x) _-^ Ψ_^(q_c,k)(x),

_^' (x) =   _^(x)  + _  ϖ^(x),

_  ϖ^(x) ≡_  ϖ^(x) + _^  (x)  ϖ^(x) .
The intrinsic $\cQ_c$-spin gauge invariance of the action in Eq.(<ref>) is easily checked under the following gauge transformation:
'_(x) =  S̃(Λ)i _ S̃^-1(Λ) +  S̃(Λ) _(x) S̃^-1(Λ) ,

Ψ_^' (q_c,k)(x) =  S̃(Λ) Ψ_^(q_c,k)(x) ,    S̃(Λ) = e^iϖ_pq(x) ^pq/2  ∈SP(q_c),
with the explicit forms given by,
S̃^-1(Λ) _-^p S̃(Λ) = Λ^p_    q(x)  _-^q ,   Λ^p_     q(x)  ∈SP(q_c),

_^' pq(x) =  1/2 (Λ^p_  p' _  Λ^p' q -  Λ^q_  p' _  Λ^p' p ) +  Λ^p_   p'  Λ^q_  q' _^ p' q'(x),

ϕ_p(x) →ϕ'_p(x) = Λ_p^     q(x) ϕ_q(x) .
In obtaining the above transformation properties, we have used the following identities:

S^-1(Λ) ^ S(Λ)= Λ^_   (x) ^,   ^ _  _ = (D_h-4) _,

S(Λ) i_ S^-1(Λ) =  1/4 (Λ^_    _  Λ^ -  Λ^_     _  Λ^ ) _ ,

^-1(Λ) ^p (Λ)= Λ^p_   q(x) ^q,   ^p' _p q  _p' = (D_h-4) _p q,

(Λ) i_ ^-1(Λ) =  1/4 (Λ^p_    p'_  Λ^p' q -  Λ^q_   p' _  Λ^p' p ) _p q ,
where the matrices $S(\Lambda)$ and $\tilde{S}(\Lambda)$ as well as $\Lambda^{\fA}_{\;\;\; \fB}(x)$ and $\Lambda^{p}_{\;\;\; q}(x)$ correspond to local group elements in the qubit-spinor representations as well as adjoin representations of gauge groups SP(1,$D_h$-1) and SP($q_c$), which should be manifest as the categoric entangled hyperqubit-spinor field is the presented in qubit-spinor representation and both hyperspin gauge field and $\cQ_c$-spin gauge field belong to the adjoint representations of gauge symmetry groups SP(1, $D_h$-1) and SP($q_c$), respectively. The bicovariant vector field and scalar field are in the vector representations of gauge symmetry groups SP(1, $D_h$-1) and SP($q_c$), respectively.

From the general covariant derivative $\fkD_{\fM}$, the field strength of gauge field $\fkA_{\fM}(x)$ is defined as follows:

_ ≡i [_, _ ] = _ + _ + _,

_(x) ≡_^(x)1/2_ = _ _(x) - _ _(x) - i [ _(x) ,  _(x) ],

_(x) ≡_^(x)1/2_- =_ _(x) - _ _(x) - i (_(x)_(x) - _(x)_(x) ),

_(x) ≡_^p q(x)1/2_p q = _ _(x) - _ _(x) - i [ _(x) ,  _(x) ],

In general, the action in Eq.(<ref>) possesses the following joint symmetry,

G_S     ≡     PO(1, D_h-1)WS(1, D_h-1) ×SP(q_c) ,
where PO(1,$D_h$-1) remains global inhomogeneous Lorentz-type/Poincaré-type group symmetry,
PO(1, D_h-1) ≡P^1, D_h-1 ⋉SO(1, D_h-1) ,
which characterizes symmetry property of globally flat Minkowski hyper-spacetime.

We would like to address that the global inhomogeneous Lorentz-type/Poincaré-type group PO(1, $D_h$-1) and local inhomogeneous hyperspin gauge group WS(1, $D_h$-1) do not form a direct product group symmetry as the former is a global symmetry and the later is a local symmetry, they are presented as a joint symmetry denoted by the symbol “$\Join$.

§.§  Scaling gauge symmetry based on gauge invariance principle and the gauge covariant equation of motion of entangled hyperqubit-spinor field as gravitational relativistic quantum theory

It can be verified that the action in Eq.(<ref>) is invariant under the following global scaling transformation:
x^ →x^' = λ^-1  x^,

_(x) →'_(x') =  λ _(x),

Ψ_^(q_c,k)(x) →Ψ_^' (q_c,k)(x') = λ^D_h-1/2 Ψ_^(q_c,k)(x) ,
with $\lambda$ a constant scaling factor.

To distinguish scaling symmetry of coordinate system in Minkowski hyper-spacetime from that of entangled hyperqubit-spinor field in Hilbert space, we should extend the global scaling symmetry of entangled hyperqubit-spinor field to be local scaling gauge symmetry based on the gauge invariance principle, which has the following scaling gauge transformation:
Ψ_^(q_c,k)(x) →Ψ_^' (q_c,k)(x) = ξ^D_h-1/2(x) Ψ_^(q_c,k)(x),

χ̂_^   (x) →χ̂_^'  (x) = ξ(x) χ̂_^   (x),

ϕ_p(x) →ϕ_p(x) = ξ(x) ϕ_p(x) ,
where $\xi(x)$ is a functional scaling factor. Following along the gauge invariance principle, we should introduce a scaling gauge field $\mW_{\fM}(x)$ with the following gauge transformation:
_(x) →'_(x) = _(x) - _lnξ(x) ,
which was initiated by Weyl[7] to characterize an electromagnetic field though it was turned out to be incorrect.

To preserve the scaling gauge symmetry, the action in Eq.(<ref>) is extended in formal to be as follows:

_^(q_c)     ≡    ∫[d^D_h^(q_c,k)x]   χ(x)   { Ψ̅_^(q_c,k)(x)  _-^_^   (x) i_  Ψ_^(q_c,k)(x)

+     ϕ_p(x)Ψ̅_^(q_c,k)(x) _-^p Ψ_^(q_c,k)(x) },
with the generalized covariant derivative,
i_ ≡i_ + _(x) +  _(x) + _(x) + D_h-1/2_(x) .
It is noted that the multiplier factor $\chi(x)$ as the inverse of the determinant of bicovaraint vector field $\chih_{\fA}^{\;\, \fM}(x)$ has been introduced by regarding $\chih_{\fA}^{\;\, \fM}(x)$ as a matrix field, i.e.:
χ(x) ≡1/(x),   (x) = _^    (x) ≠0 ,
which is multiplied to the action so as to ensure the scaling gauge invariance with the following scaling gauge transformation:
χ(x) →χ'(x) = ξ^-D_h(x) χ(x) .

From the matrix property,  $\chi(x)$ can be expressed as the determinant of dual bicovaraint vector field $\chi_{\fM}^{\;\; \fA} (x)$ which is viewed as an inverse matrix field of the matrix field $\chih_{\fA}^{\;\, \fM}(x)$. Such a dual bicovaraint vector field is defined as follows:
χ_^    (x)  ^   _(x)   =  χ_^    (x) _  (x) η^ = η^   _ ,

χ_^   (x) ^  _(x) = χ_   (x) _^   (x)  η^ = η_^   ,

χ(x) = χ_^   (x) = ^-1(x),
where the matrix fields $\chi_{\fM}^{\;\; \fA} (x)$ and $\chih^{\;\; \fM}_{\fB}(x)$ satisfy orthonormality conditions.

From the self-conjugated chiral property of entangled hyperqubit-spinor field in category-$q_c$ and the transpose property of $\vGa$-matrices under complex charge conjugation, i.e.:
Ψ_^(q_c,k)  c(x)= C_^(q_c,k) Ψ̅_^(q_c,k)  T(x) = Ψ_^(q_c,k)(x) ,

C_^(q_c,k)^(C_^(q_c,k))^-1 = - (^)^T,

C_^(q_c,k)^D_h + p(C_^(q_c,k))^-1 = (^D_h + p)^T ,
together with the chirality property of entangled hyperqubit-spinor field, it can be verified that the gauge interactions of entangled hyperqubit-spinor field with both $\cW_e$-spin gauge field $\ckcA_{\fM}(x)$ and scaling gauge field $\mW_{\fM}(x)$ should be absent in the hermitian action presented in Eq.(<ref>).

The gauge invariant hermitian action in Eq.(<ref>) can actually be rewritten as follows:

_^(q_c)     ≡    ∫[d^D_h^(q_c,k)x]   χ(x)   {  Ψ̅_^(q_c,k)(x)  _-^_^   (x) i_  Ψ_^(q_c,k)(x)

+     ϕ_p(x)Ψ̅_^(q_c,k)(x) _-^p Ψ_^(q_c,k)(x) } .
with the definitions,
i_ ≡i_ + _(x),   _(x) ≡_(x) + _(x) .

Such an action possesses the following joint symmetry:

G_S= SC(1)⋉PO(1,D_h-1)WS(1,D_h-1)⋊SG(1)×SP(q_c),
where SC(1) and SG(1) denote global scaling symmetry and local scaling gauge symmetry, respectively.

From the gauge invariant action in Eq.(<ref>), we can derive the following equation of motion for categoric entangled hyperqubit-spinor field:

_-^ _^   (x) (i_ - i_(x) ) Ψ_^(q_c,k)(x)
-ϕ_p(x) _-^p Ψ_^(q_c,k)(x)  =   0 ,
where $\mV_{\fM}(x)$ represents an induced gauge field arising from the bi-covariant vector field and inhomogeneous hyperspin gauge field via a hidden gauge formalism, which has the following explicit form:
_(x)     ≡     1/2(χ(x) _^   (x))_((x)χ_^   (x) ),
where $\cD_{\fN}\left(\chih(x)\chi_{\fM}^{\;\; \fB}(x)\right)$ defines the covariant derivative of dual bicovaraint vector field $\chi_{\fM}^{\;\; \fB} (x)$,

_((x)χ_^   (x)) ≡_ ((x)χ_^   (x)) + _  ^(x) ((x)χ_^   (x)) .
It can be checked that $\mV_{\fM}(x)$ preserves local scaling gauge invariance in the above equation of motion with the following transformation property:
_(x)     →    '_ (x) = _(x) + 1/2(D_h-1)_ξ(x) .

It can be verified that the gauge covariant equation of motion Eq.(<ref>) is local scaling gauge invariant. Let us now derive the quadratic form for the equation of motion Eq.(<ref>), which is found to have the following form:

^ (∇̂_ - _ ) ( _ - _ ) Ψ_^(q_c,k)(x) =   ^_^    _^

·( _+ _ - G_^ _^    i ( _ - _ ) - i _  ) Ψ_^(q_c,k)(x)

- ( ^2 ϕ^2_p  +  ^ ^p _^    (i_ϕ_p)    ) Ψ_^(q_c,k)(x),
where we have introduced the following definition:

^(x) = _^   (x) _^   (x) η^ ,
for the symmetric tensor field, and

∇̂_ ( _  - _ ) ≡_ ( _ - _ ) - 𝐴_^ ( _ - _ ) ,

𝐴_^  ≡_^     _χ_^   =  _^   (_ χ_^    + _  ^  χ_^    ) ,

_ϕ_p ≡_ ϕ_p - _  p ^  q ϕ_q,
for the covariant derivatives $\hat{\nabla}_{\fM}$ and $\tcD_{\fM}$. On the right-hand side of Eq.(<ref>), $\cF_{\fM\fN}(x)$ and $\tcF_{\fM\fN}(x)$ define the field strengths of hyperspin gauge field $\cA_{\fM}(x)$ and $\cQ_c$-spin gauge field $\tcA_{\fM}(x)$, respectively, with the following explicit forms:
_(x) ≡F_^(x)  1/2_  = ∂_ A_(x) - ∂_ A_(x) - i [A_(x) ,  A_(x) ],

_(x) ≡F_^p q(x)  1/2_p q  = _ _(x) - _ _(x) - i [_(x) ,  _(x) ],
and $\mV_{\fM\fN}$ is the induced gauge field strength defined as follows:
_ ≡__ -  __ .

The tensor field ${\cal G}_{\fM\fN}^{\fA}$ is defined as follows:

_^      ≡    _ χ_^    - _ χ_^

=      _ χ_^   (x) - _ χ_^   (x) + _  ^(x)  χ_^  (x)  - _  ^(x)  χ_^  (x),
which provides the gauge covariant field strength of dual bicovaraint vector field $\chi_{\fM}^{\; \fA}(x)$ with $\cD_{\fM} \chi_{\fN}^{\;\; \fA}$ representing the covariant derivative of $\chi_{\fM}^{\; \fA}(x)$. It is noticed that the dual bicovaraint vector field $\chi_{\fM}^{\; \fA}(x)$ behaves as a gauge-type vector field and its inverse as bicovaraint vector field $\chih_{\fA}^{\;\; \fM}(x)$ couples to the motion of entangled hyperqubit-spinor field, so that $\chi_{\fM}^{\; \fA}(x)$ is expected to characterize gravitational interaction in hyper-spacetime. For convenience, $\chi_{\fM}^{\;\; \fA}$ is referred to as hyper-gravigauge field in hyper-spacetime, which will be demonstrated below for its genesis as gauge field.

The equations of motion presented in Eqs.(<ref>) and (<ref>) for categoric entangled hyperqubit-spinor field $\Psi_{\fQE}^{(q_c,k)}(x)$ possess global Poincaré-type group symmetry and scaling symmetry jointly with local inhomogeneous hyperspin symmetry and $\cQ_c$-spin gauge symmetry as well as local scaling gauge symmetry. The gauge symmetries are described correspondingly by the hyperspin gauge field $\cA_{\fM}^{\fA\fB}(x)$ and $\cQ_c$-spin gauge field $\tcA_{\fM}^{\fA\fB}(x)$ as well as hyper-gravigauge field $\chi_{\fM}^{\; \fA}(x)$ and scaling gauge field $\mW_{\fM}$. Such equations characterize the dynamics of entangled hyperqubit-spinor field, which brings on gravitational relativistic quantum theory in hyper-spacetime and is regarded as a generalization to Dirac relativistic quantum theory[29]. The dynamics of entangled hyperqubit-spinor field is generally governed by the hyperspin gauge field strength $\cF_{\fM\fN}^{\fA\fB}$ and hyper-gravigauge field strength ${\cal G}_{\fM\fN}^{\fA}$ as well as $\cQ_c$-spin gauge field strength $\tcF_{\fM\fN}^{\fA\fB}$.

It is noticed that the induced gauge field $\mV_{\fM}(x)$ and its field strength $\mV_{\fM\fN}$ distinguish from other gauge fields and field strengths due to the presence of an associated imaginary factor $``i"$, which appears naturally so as to keep the local scaling gauge invariance in the equation of motion of entangled hyperqubit-spinor field. For convenience of mention, we may refer to $\mV_{\fM}(x)$ as graviscaling induced gauge field and $\mV_{\fM\fN}$ as graviscaling induced gauge field strength. It is clear that the appearance of graviscaling induced gauge field and its field strength will lead distribution amplitudes of entangled hyperqubit-spinor field to be boosted by the induced scaling gauge factor.

Therefore, the gauge invariance principle as guiding principle brings all symmetries of basic fields in Hilbert space to be local gauge symmetries with the introduction of gauge fields, which enables us to reveal the dynamics of entangled hyperqubit-spinor field as fundamental building block.

§  HYPER-FIBER BUNDLE STRUCTURE OF BIFRAME HYPER-SPACETIME WITH GRAVITATIONAL ORIGIN OF GAUGE SYMMETRY AND $\CW_E$-SPIN INVARIANT-GAUGE FIELD AS GENESIS OF HYPER-GRAVIGAUGE FIELD WITH GRAVISCALAR AND GRAVIVECTOR FIELDS

The gauge invariance of the action of categoric entangled hyperqubit-spinor field brings on the introduction of bicovariant vector field $\chih_{\fA}^{\;\; \fM}(x)$. The dual bicovariant vector field $\chi_{\fM}^{\; \fA}(x)$ leads to a gauge-type field strength, which makes $\chi_{\fM}^{\; \fA}(x)$ to behave as a hyper-gravigauge field in hyper-spacetime. We are going to further demonstrate that such bicovariant vector fields form locally flat gravigauge hyper-spacetime and bring about the concept of biframe hyper-spacetime. Mathematically, such a biframe hyper-spacetime sets up a hyper-fiber bundle structure with Minkowski hyper-spacetime as base spacetime. Such a locally flat gravigauge hyper-spacetime is shown to be characterized by non-commutative geometry with Lie algebra structure factor determined by the field strength of hyper-gravigauge field. We will show that it is the hyper-gravigauge field $\chi_{\fM}^{\; \fA}(x)$ that brings on the gravitational origin of gauge symmetry in hyper-spacetime. Furthermore, we will verify that the hyper-gravigauge field $\chi_{\fM}^{\; \fA}(x)$ is identified to be the $\cW_e$-spin invariant-gauge field which is introduced via the translation-like $\cW_e$-spin Abelian-type gauge symmetry as subgroup gauge symmetry of inhomogeneous hyperspin gauge symmetry. Therefore, we get better understanding on the nature of gravity that the gravitational interaction is truly a gauge interaction and also better comprehension on the nature of spacetime that the observable spacetime is actually a biframe spacetime.

§.§  Locally flat gravigauge hyper-spacetime with emergence of non-commutative geometry and biframe hyper-spacetime with hyper-fiber bundle structure

The Poincaré-type group symmetry PO(1, $D_h$-1) = P$^{1, D_h\mbox{-}1}$ $\ltimes$ SO(1, $D_h$-1) characterizes globally flat Minkowski hyper-spacetime $M_h$ which is known to be an affine spacetime. The ordinary  derivative vector operator $\partial_{\fM} \equiv \partial/\partial x^{\fM}$ at point $x$ of $M_h$ defines a tangent basis  $\{\partial_{\fM}\}\equiv \{\partial/\partial x^{\fM}\} $ for tangent hyper-spacetime $T_h$ over $M_h$. A displacement vector $dx^{\fM}$ at point $x$ of $M_h$ provides a dual tangent basis $\{dx^{\fM}\}$ for dual tangent hyper-spacetime $T_h^{\ast}$ over $M_h$. The tangent basis and dual tangent basis satisfy the following dual condition:
⟨dx^,  ∂/∂x^  ⟩= ∂x^/∂x^ = η_^   .

In Hilbert space, the introduction of bicovaraint vector field $\chih_{\fA}^{\;\, \fM}(x)$ and its dual bicovariant vector field $\chi_{\fM}^{\;\; \fA}(x)$ not only preserves inhomogeneous hyperspin gauge symmetry for characterizing the dynamics of entangled hyperqubit-spinor field, but also allows us to define field vector operator $\eth_{\fA}$ and its dual field vector operator $ \delta\zeta^{\fA}$ at point $x$ of $M_h$ with respective to ordinary coordinate derivative vector operator $\partial_{\fM}$ and displacement vector operator $dx^{\fM}$ as follows:

ƪ_ ≡_^   (x) ∂_ ,

δζ^ ≡χ_^    (x) dx^ ,
which forms the field basis $\{\eth_{\fA} \}$ and dual field basis $ \{\delta\zeta^{\fA}\}$ for locally flat hyper-spacetime over globally flat Minkowski hyper-spacetime $M_h$.

The dual field bases satisfy the following dual condition:
⟨δζ^,  ƪ_⟩≡χ_^  (x)  _^  (x)  ⟨dx^ , ∂_ ⟩= χ_^   (x)  _^    (x)  η_^   = η_^      .
Such a pair of dual field bases $\{\eth_{\fA} \}$ and $ \{\delta\zeta^{\fA}\} $ form a pair of dual locally flat hyper-spacetimes over globally flat Minkowski hyper-spacetime $M_h$. It is easy to verify that the field basis $\{\eth_{\fA}\}$ does not commute and satisfies the following non-commutation relation:

[ i ƪ_ ,  iƪ_] = i f_^      (x)   iƪ_,

f_^      (x) ≡- _^   (x) _^   (x) _^  (x) ,   _^  (x) ≡_χ_^   (x) - _χ_^   (x).
which shows that such a non-commutation relation brings on a Lie algebra with non-constant group structure factor $f_{\fA\fB}^{\; \fC}(x)$ characterized by the tensor field $\mG_{\fM\fN}^{\; \fC}(x)$. Such a tensor field is presented as an Abelian-type field strength of hyper-gravigauge field $\chi_{\fM}^{\;\; \fC}(x)$. Therefore, the above non-commutation relation brings about a non-commutative geometry in locally flat hyper-spacetime, which is described by Abelian-type hyper-gravigauge field strength $\mG_{\fM\fN}^{\; \fC}(x)$.

For the convenience of mention, the dual field bases $ \{\eth_{\fA} \} $ and $ \{\delta\zeta^{\fA}\} $ formed by the hyper-gravigauge field are called as a pair of dual hyper-gravigauge field bases. Correspondingly, the dual locally flat hyper-spacetimes formed by the dual hyper-gravigauge field bases are referred to as locally flat dual gravigauge hyper-spacetimes. Specifically, the gravigauge hyper-spacetime formed from hyper-gravigauge field basis $\{\eth_{\fA}\}$ is called as tangent-like hyper-spacetime denoted by $G_h$, and the dual gravigauge hyper-spacetime formed from dual hyper-gravigauge field basis $ \{\delta\zeta^{\fA}\} $ is called as dual tangent-like hyper-spacetime denoted by $G_h^{\ast}$.

In general, the hyper-gravigauge field $\chi_{\fM}^{\;\; \fA}(x)$ is considered to be sided on the dual tangent Minkowski hyper-spacetime $T_h^{\ast}$ and valued on the dual gravigauge hyper-spacetime $G_h^{\ast}$. The dual hyper-gravigauge field $\hat{\chi}_{\fA}^{\;\;\fM}(x)$ is defined on the gravigauge hyper-spacetime $G_h$ and valued on the tangent Minkowski hyper-spacetime $T_h$. Namely, $\hat{\chi}_{\fA}^{\;\;\fM}(x)$ transforms as bicovariant vector field under transformations of both hyperspin gauge symmetry SP(1, $D_h$-1) and global Lorentz-type group symmetry SO(1, $D_h$-1). The non-commutation relation for hyper-gravigauge field basis $\{\eth_{\fA}\}$ displays that locally flat gravigauge hyper-spacetime $G_h$ is correlated to non-commutative geometry.

The gravigauge hyper-spacetime $G_h$ and tangent Minkowski hyper-spacetime $T_h$ form a biframe hyper-spacetime $T_h\times G_h$ associated with a dual biframe hyper-spacetime $T_h^{\ast}\times G_h^{\ast}$ over coordinate hyper-spacetime $M_h$.

Mathematically,  globally and locally flat vector spacetimes allow for a canonical identification of vectors in tangent Minkowski hyper-spacetime $T_h$ at points with vectors in Minkowski hyper-spacetime itself $M_h$, and also for a canonical identification of vectors at points with its dual vectors at the same points. For the meaning in physics, either tangent or dual tangent Minkowski hyper-spacetime over globally flat Minkowski hyper-spacetime is regarded as free-motion hyper-spacetime $\fM_h$, while locally flat gravigauge hyper-spacetime is viewed as emergent hyper-spacetime $\fGh$.  The canonical identification for the vector spacetimes brings about a simple structure of biframe hyper-spacetime,
_h =  _h ×G_h ,

_h ≡T_h ≅T_h^∗ ≅M_h ,   ≡G_h ≅G_h^∗ .
Such a biframe hyper-spacetime structure forms gravigauge hyper-fiber bundle $\fE_h$ with locally flat emergent gravigauge hyper-spacetime as a hyper-fiber $\fGh$ and globally flat free-motion Minkowksi hyper-spacetime as a base spacetime $\fM_h$.

In mathematics, the hyper-fiber bundle $\fE_h$ is related to the product hyper-spacetime $\fM_h\times \fG_h$ as biframe hyper-spacetime via a continuous surjective map $\Pi_{\chi}$, which projects the bundle $\fE_h$ to the base spacetime $\fM_h$. Geometrically, the gravigauge hyper-fiber bundle structure of biframe hyper-spacetime is expressed as  $(\fE_h, \fM_h, \Pi_{\chi}, \fG_h) $. In general, the hyper-fiber bundle $\fE_h$ with the surjective map $\Pi_{\chi}$ is expressed as follows with the trivial case:
Π_χ:     _h →_h ,

_h ∼_h = _h ×_h .

§.§ Gravitational origin of gauge symmetry and the hyperspin gravigauge and covariant-gauge fields with gauge covariant field strengths

In the usual gauge theory, the internal gauge symmetry is attributed in formal to a pure gauge field, one can always choose a configuration via a gauge transformation to eliminate unphysical part arising from gauge symmetry. By making such a gauge prescription in unitary gauge, the gauge field keeps only independent degrees of freedom. Nevertheless, for hyperspin gauge symmetry SP(1,$D_h$-1), the situation appears to be distinguishable as it is resulted from gauging global hyperspin symmetry which is initiated from a motion-correlation $\cM_c$-spin symmetry. In fact, when building the gauge invariant action of hyperqubit-spinor field, we are led to introduce not only the hyperspin gauge field $\cA_{\fM}(x)$ but also the bicovariant vector field $\chih_{\fA}^{\;\; \fM}(x)$. It is conceivable that such a bicovariant vector field must be in correlation to hyperspin gauge symmetry. To corroborate such an idea,  let us decompose the hyperspin gauge field $\cA_{\fM}(x)$ into two parts $\mOm_{\fM}(x)$ and $\mA_{\fM}(x)$ as follows:

_(x) = _(x) + _(x) ≡_^1/2_ ,
so that $\mOm_{\fM}(x)$ obeys an inhomogeneous transformation of hyperspin gauge symmetry and $\mA_{\fM}(x)$ transforms homogeneously under hyperspin gauge transformation, i.e.:

_(x) →'_(x) = S(Λ) i_ S^-1(Λ)  + S(Λ) _(x) S^-1(Λ) ,

_(x) →'_(x) = S(Λ) _(x) S^-1(Λ),   S(Λ) ∈SP(1,D_h-1).
For a usual internal gauge field, $\mOm_{\fM}$ is generally taken as a pure gauge field with vanishing field strength. In the present case, the gauge field $\mOm_{\fM}$ with inhomogeneous gauge transformation is presumed to correlate with the hyper-gravigauge field $\chi_{\fM}^{\; \fA}$. This is because when turning hyperspin gauge symmetry to be global symmetry with group transformation being coincidental to that of global Lorentz-type group symmetry, the hyperspin gauge field $\cA_{\fM}$ disappears and hyper-gravigauge field $\chi_{\fM}^{\; \fA}$ becomes the constant bicovariant vector $\delta_{\fM}^{\; \fA}$.

It can be verified that $\mOm_{\fM}$ does correlate to the hyper-gravigauge field $\chi_{\fM}^{\; \fA}$ and its explicit form is uniquely given as follows:

_^     =     1/2( χ̂^ _^ - χ̂^ _^ -  χ̂^  χ̂^  _^ χ_ ) ,
where the antisymmetric tensor $\mG_{\fM\fN}^{\fA}$ is defined as Abelian-type field strength of hyper-gravigauge field $\chi_{\fM}^{\; \fA}$ shown in Eq.(<ref>). The gauge transformation of hyper-gravigauge field $\chi_{\fM}^{\; \fA}$ in the vector representation of hyperspin gauge symmetry SP(1,$D_h$-1) does lead $\mOm_{\fM}^{\fA\fB}$ to get the following proper gauge transformation:

χ_^'   =  Λ^_    χ_^   ,   _^'   =  Λ_^    _^   ,  Λ^_    ∈ ,

_^' =  1/2 ( Λ^_    _  Λ^- Λ^_    _  Λ^ )+ Λ^_    Λ^_   _^ ,
where $\mOm_{\fM}^{\fA\fB}$ displays an inhomogeneous gauge transformation in the adjoint representation of hyperspin gauge symmetry SP(1,$D_h$-1).

As $\mOm_{\fM}^{\fA\fB}(x)$ is completely determined by the hyper-gravigauge field $\chi_{\fM}^{\; \fA}(x)$, it involves no additional independent degrees of freedom in the decomposition of hyperspin gauge field $\cA_{\fM}(x)$. For convenience, we may refer to $\mOm_{\fM}^{\fA\fB}(x)$ as hyperspin gravigauge field and $\mA_{\fM}^{\fA\fB}(x)$ as hyperspin covariant-gauge field.

It is not possible to make a gauge transformation to eliminate the hyperspin gravigauge field $\mOm_{\fM}^{\fA\fB}(x)$, which is completely different from the usual internal gauge field. To check that explicitly, let us decompose the hyperspin gauge field strength into the following two parts:
_^    ≡    _^ + _^   ,
with the explicit forms,

_^ = ∂_ _^ - ∂_ _^ + _^ _^ -  _^ _^,

_^ =  _ _^ - _ _^ +  _^ _^ -  _^ _^  ,

_ _^ = _  _^  +  _^ _^ + _^ _^  ,
where $\mR_{\fM\fN}^{\fA\fB}$ is purely the field strength of hyperspin gravigauge field $\mOm_{\fM}^{\fA\fB}(x)$, which is not vanishing.

The decomposition of hyperspin gauge field enables us to obtain two gauge covariant field strengths $\mR_{\fM\fN}^{\fA\fB}$ and $\mF_{\fM\fN}^{\fA\fB}$ in correspondence to the hyperspin gravigauge field $\mOm_{\fM}^{\fA\fB}(x)$ and hyperspin covariant-gauge field $\mA_{\fM}^{\fA\fB}(x)$. The hyperspin gravigauge field $\mOm_{\fM}^{\fA\fB}(x)$ and its gauge covariant field strength $\mR_{\fM\fN}^{\fA\fB}$ characterize the gravitational interaction originated from hyper-gravigauge field.

From the above analysis, we come to the statement about the gravitational origin of gauge symmetry that the hyperspin gauge symmetry SP(1,$D_h$-1) of entangled qubit-spinor field is essentially governed by the hyper-gravigauge field $\chi_{\fM}^{\; \fA}(x)$ with gauge transformation in the vector representation of SP(1,$D_h$-1).

§.§  $\cW_e$-spin invariant-gauge field as genesis of hyper-gravigauge field in the presence of graviscalar field

The hyper-gravigauge field $\chi_{\fM}^{\; \fA}(x)$ is defined as the dual field of bicovariant vector field $\chih_{\fA}^{\; \fM}(x)$ which is introduced when gauging inhomogeneous hyperspin symmetry based on the gauge invariance principle as guiding principle. Although $\chi_{\fM}^{\; \fA}(x)$ behaves as a gauge-type vector field indicated from its field strength shown either in the equation of motion (Eq.(<ref>)) or in the non-commutation relation (Eq.(<ref>)), while it remains inapparent whether $\chi_{\fM}^{\; \fA}(x)$ can be regarded as a gauge field introduced directly from a gauge symmetry by just following along the gauge invariance principle. For this purpose, it is interesting to notice that the field strength in Eq.(<ref>) for $\chi_{\fM}^{\; \fA}(x)$ has an analogous formalism with the field strength in Eq.(<ref>) for $\cW_e$-spin gauge field $\ckcA_{\fM}^{\fA}(x)$. As $\ckcA_{\fM}^{\fA}(x)$ is introduced via the $\cW_e$-spin gauge symmetry W$^{1,D_h-1}$ which is the subgroup symmetry of inhomogeneous hyperspin gauge symmetry WS(1,$D_h$-1), we are motivated to explore the correlation between the hyper-gravigauge field $\chi_{\fM}^{\; \fA}(x)$ and $\cW_e$-spin gauge field $\ckcA_{\fM}^{\fA}(x)$.

In analogous to the decomposition of hyperspin gauge field $\cA_{\fM}^{\fA\fB}(x)$, let us decompose the $\cW_e$-spin gauge field $\ckcA_{\fM}^{\fA}(x)$ into the following two parts:
_(x) ≡_^(x) 1/2_- =  (_^(x) +  _^(x) ) 1/2_- ,
where $\fOm_{\fM}^{\fA}(x)$ is supposed to have inhomogeneous gauge transformation and $\fA_{\fM}^{\fA}(x)$ becomes gauge invariant under the $\cW_e$-spin gauge transformation shown in Eq.(<ref>), while they all transform as covariant vector fields under the hyperspin gauge transformation. Their explicit transformations are presented as follows:
_^(x) →_^' (x) =  _^(x) + _ϖ^(x),

_^(x) →_^' (x) =  _^(x) ,
under the $\cW_e$-spin gauge transformation, and
_^(x) →_^' (x) = Λ^_    _^(x),

_^(x) →_^' (x) = Λ^_    _^(x),
under the hyperspin gauge transformation.

For convenience of mention, $\fOm_{\fM}^{\fA}(x)$ is referred to as $\cW_e$-spin gravigauge field and $\fA_{\fM}^{\fA}(x)$ as $\cW_e$-spin invariant-gauge field. In general, the combination of $\fA_{\fM}^{\fA\fB}(x)$ and $\fA_{\fM}^{\fA}(x)$, i.e., $\breve{\fA}_{\fM}(x)\equiv \fA_{\fM}^{\fA\fB}(x)\frac{1}{2}\vSi_{\fA\fB} + \fA_{\fM}^{\fA}(x)\frac{1}{2}\vSi_{-\fA}$, is regarded as inhomogeneous hyperspin covariant-gauge field.

In order to keep the initial degrees of freedom for the inhomogeneous hyperspin gauge field, there must exist certain relations among $\cW_e$-spin gravigauge field $\fOm_{\fM}^{\fA}(x)$, $\cW_e$-spin invarian-gauge field $\fA_{\fM}^{\fA}(x)$, hyper-gravigauge field $\chi_{\fM}^{\; \fA}(x)$ and hyperspin gauge field $\cA_{\fM}^{\fA\fB}(x)$. Indeed, it is natural to propose the following relations:

_^(x) ≡_^(x) = _ ^(x) + _^(x) ^(x),

_^(x) ≡ϕ(x) χ_^  (x),
where $\vka^{\fA}(x)$ is introduced as a vector field in locally flat gravigauge hyper-spacetime and $\phi(x)$ is taken as a real scalar field to keep a canonical mass dimension. To ensure the $\cW_e$-spin gauge transformation, the vector field $\vka^{\fA}(x)$ must transform as follows:

^(x) →^' (x) = ^(x) + ϖ^(x) .
which is regarded as a local translation of vector field $\vka^{\fA}(x)$ in locally flat gravigauge hyper-spacetime. As the above transformation reflects the symmetry property of translation-like $\cW_e$-spin Abelian-type gauge group, $\vka^{\fA}(x)$ is viewed as a $\cW_e$-spin vector field in locally flat gravigauge hyper-spacetime.

From the above considerations, it is the $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\fA}(x)$ that brings on the hyper-gravigauge field $\chi_{\fM}^{\; \fA}(x)$ in association with a scalar field $\phi(x)$. Therefore, the hyper-gravigauge field $\chi_{\fM}^{\; \fA}(x)$ arises from the $\cW_e$-spin invariant-gauge field associated with scalar field $\phi(x)$. Such a scalar field is considered to be a gravitational scalar field and referred to as graviscalar field for short.

It can be checked that the $\cW_e$-spin vector field $\vka^{\fA}(x)$ has zero scaling charges for both global and local scaling symmetries SC(1) and SG(1). Under the transformations of SC(1) and SG(1) symmetries, the graviscalar field $\phi(x)$ transforms as follows:
ϕ(x) →ϕ'(x') = λ  ϕ(x),   x' = λ^-1 x,   λ∈SC(1),

ϕ(x) →ϕ'(x') = ξ(x)ϕ(x),   ξ(x) ∈SG(1) .

The field strength of $\cW_e$-spin gauge field can also be decomposed into the following two parts:

_^(x) ≡_^(x) + _^(x),

_^(x) = __^(x) -  __^(x) =  _^(x) _(x) ,

_^(x) = __^(x) -  __^(x) ≡ϕ(x) _^(x),
where $\cG_{\fM\fN}^{\fA}(x)$ defines a hyperspin gauge and scaling gauge covariant field strength of hyper-gravigauge field $\chi_{\fM}^{\fA}(x)$,

_^(x)     ≡    _χ_^(x) -  _χ_^(x) = d_χ_^(x) -  d_χ_^(x)  +  _^χ_^(x) -  _^χ_^(x)

≡    _^(x) + _^χ_^(x) -  _^χ_^(x) ,
with $d_{\fM}$ defined as scaling gauge covariant derivative. Where we have introduced the gauge-type field strength of hyper-gravigauge field as follows:

_^(x) ≡d_χ_^(x) -  d_χ_^(x) ,

d_≡_ + _,   _ ≡1/2_lnϕ^2(x),
which becomes scaling gauge covariance but not hyperspin gauge covariance. Where $\fS_{\fM}$ is regarded as a pure graviscaling gauge field given by the graviscalar field $\phi(x)$.

Therefore, we arrive at the statement that it is the $\cW_e$-spin gauge symmetry as subgroup gauge symmetry of inhomogeneous hyperspin gauge symmetry that brings on the genesis of dual bicovariant vector field $\chi_{\fM}^{\; \fA}(x)$ as hyper-gravigauge field, which provides a natural explanation on the gauge property of hyper-gravigauge field appearing either in the equation of motion shown in Eq.(<ref>) or in the non-commutative relation presented in Eq.(<ref>).

§.§  Biframe displacement correspondence in biframe hyper-spacetime and the dimensionless hyper-gravicoordinate displacement and derivative with respect to hyper-gravivector field in locally flat gravigauge hyper-spacetime

The $\cW_e$-spin gauge transformation is simply presented by a local translation of $\cW_e$-spin vector field $\vka^{\fA}(x)$ in locally flat gravigauge hyper-spacetime as shown in Eq.(<ref>). It is manifest that under the $\cW_e$-spin gauge transformation, $\fF_{\fM\fN}^{\fA}(x)$ becomes $\cW_e$-spin invariant-gauge field strength and   the field strength $\fR_{\fM\fN}^{\fA}(x)$ transforms as follows,
_^(x) →_^' (x) = _^(x) + _^(x) ϖ_(x) .

Let us regard $\cW_e$-spin vector field $\vka^{\fA}(x)$ as dimensionless hyper-gravivector field in locally flat gravigauge hyper-spacetime, so that the local translation can be represented via a local translation operator defined in formal by the following coordinate-like field derivative:

ƪ̂_ ≡δ/δ^,
which is in analogous to the ordinary translation operator `$\p_{\fM}$' of coordinates in globally flat Minkowski hyper-spacetime. Similarly, we can introduce the coordinate-like field displacement $\delta\vka^{\fA}$ in locally flat gravigauge hyper-spacetime. For convenience of mention, the displacement $\delta\vka^{\fA}$ is referred to as dimensionless hyper-gravicoordinate displacement and the derivative $\hat{\eth}_{\fA}$ is called as dimensionless hyper-gravicoordinate derivative.

As the basic structure of biframe hyper-spacetime is described solely by the hyper-gravigauge field originated from the $\cW_e$-spin invariant-gauge field, it is natural to consider biframe displacement correspondence which states that the hyper-gravicoordinate displacement of hyper-gravivector field $\vka^{\fA}(x)$ in locally flat gravigauge hyper-spacetime is directly associated to the ordinary coordinate displacement in globally flat Minkowski hyper-spacetime through the $\cW_e$-spin invariant-gauge field. Namely, when the locally flat gravigauge hyper-spacetime becomes a globally flat one, the hyper-gravicoordinate displacement $\delta\vka^{\fA}$ should be coincidental to the ordinary coordinate displacement. Based on such a biframe displacement correspondence and biframe hyper-spacetime structure, we are able to express the dimensionless hyper-gravicoordinate displacement $\delta\vka^{\fA}$ and derivative $\hat{\eth}_{\fA}$ as follows:

δ^    ≡     _^  (x)dx^ = ϕ(x)χ_^  (x)dx^ ≡ϕ(x) δζ^ ,

ƪ̂_     ≡    _^  (x)_ = ϕ^-1(x)_^  (x) _ = ϕ^-1(x)ƪ_ ,
with $\phi(x)$ the graviscalar field. Where the dimensionless hyper-gravicoordinate displacement $\delta\vka^{\fA}$ and derivative $\hat{\eth}_{\fA}$ with respect to hyper-gravivector field are related to the corresponding dimensionful vector field displacement $\delta\zeta^{\fA}$ and derivative $\eth_{\fA}$ via the graviscalar field. They are all characterized by the hyper-gravigauge field which is regarded as bicovariant vector field defined in biframe hyper-spacetime.

Once locally flat gravigauge hyper-spacetime approaches to a globally flat one, we are led to the following special case:

_^  (x) →η_^   ,   ϕ(x) →,   δζ^ →η_^    dx^ = dx^,

δ^ →δζ^ = η_^    dx^,
where $\Mka$ is supposed to be a fundamental mass scale. Correspondingly,  $\delta\zeta^{\fA}$ and $\eth_{\fA}$ are considered as dimensionful hyper-gravicoordinate displacement and derivative, respectively, in locally flat gravigauge hyper-spacetime with the following forms:

δζ^ ≡ϕ^-1(x) _^  (x)dx^ = χ_^  (x)dx^,

ƪ_ ≡δ/δζ^ ≡ϕ(x) _^  (x)/x^ = _^  (x)_.

§ HYPERUNIFIED FIELD THEORY FOR FUNDAMENTAL BUILDING BLOCK AND FUNDAMENTAL INTERACTION BASED ON GAUGE INVARIANCE PRINCIPLE AND SCALING INVARIANCE HYPOTHESIS

Following along the maximum entangled qubit-spinor principle as guiding principle, we have shown in the previous paper as part I of the foundation of the hyperunified field theory[1] that the hyperunified qubit-spinor field is obtained as entangled hyperqubit-spinor field with least $q_c$-spin charge. The category-0 entangled enneaqubit-spinor field with $q_c$-spin charge $q_c=0$ is regarded as the minimal hyperunified qubit-spinor field, and the category-1 entangled decaqubit-spinor field with $q_c$-spin charge $q_c=1$ is demonstrated to be the hyperunified qubit-spinor field that can unify all known leptons and quarks[27, 28] in SM. Based on the gauge invariance principle as guiding principle, we present in previous sections a general analysis that the fundamental interaction is governed by inhomogeneous hyperspin gauge symmetry WS(1,$D_h$-1). In particular, the gravitational interaction is shown to be governed by the translation-like $\cW_e$-spin Abelian-type gauge symmetry which is the subgroup gauge symmetry of inhomogeneous hyperspin gauge symmetry. The hyper-gravigauge field is identified to be the $\cW_e$-spin invariant-gauge field in association with the graviscalar field. Therefore, we get a better understanding on the gravitational origin of gauge symmetry and genesis of gravitational interaction. In this section, we are going to build a whole hyperunified field theory for characterizing both fundamental building block and fundamental interaction based on the gauge invariance principle and scaling invariance hypothesis. Specifically, we will take the entangled decaqubit-spinor field as hyperunified qubit-spinor field $\fPsi_{\fQH}(x)$ to be the fundamental building block of nature and meanwhile the inhomogeneous hyperspin gauge interaction as hyperunified gauge interaction to be the fundamental interaction mediated via the inhomogeneous hyperspin gauge field $\bvcA_{\fM}(x)$ in 19-dimensional hyper-spacetime. In addition, we will consider scaling gauge interaction and $\cQ_c$-spin scalar interaction as fundamental interactions mediated through the scaling gauge field $\mW_{\fM}(x)$ and $\cQ_c$-spin scalar field $\phi_1(x)$.

§.§  Hyperunified qubit-spinor structure of entangled decaqubit-spinor field as fundamental building block and the inhomogeneous hyperspin symmetry WS(1,18) as hyperunified symmetry in gravigauge hyper-spacetime

Let us begin with returning to the entangled decaqubit-spinor field in category-1 with $\cM_c$-spin charge $\cC_{\cM_c}=19$ and $\cQ_c$-spin charge $\cC_{\cQ_c}=1$, which has been shown in the previous paper[1] to be the minimal entangled hyperqubit-spinor field that unifies all known leptons and quarks into a single hyperunified qubit-spinor field $\fPsi_{\fQH}(x)$. To explicitly reflect the basic properties of chiral type lepton-quark states and mirror lepton-quark states, it is useful to represent it in an alternative qubit-spinor structure as follows:

_(x)     ≡     _(x) _(x)  =
_(x)  ,

where $\fPsi_{\fQUn}(x)$ is composed of two ultra-grand unified qubit-spinor fields $\Psi_{\fQUn}(x)$ and $\hPsi_{\fQUn}(x)$ with negative U-parity, which corresponds to four families of chiral type lepton-quark states,  and $\fPsi_{\fQUp}(x)$ consists of two ultra-grand unified qubit-spinor fields $\Psi_{\fQUp}(x)$ and $\hPsi_{\fQUp}(x)$ with positive U-parity, which provides four families of mirror lepton-quark states, i.e.:

_(x) = Ψ_(x)_(x) =

_(x) = Ψ_(x)_(x) =

where both $\fPsi_{\fQUn}(x)$ and $\fPsi_{\fQUp}(x)$ are regarded as paired ultra-grand unified qubit-spinor field with negative U-parity and positive U-parity, respectively.

$\Psi_{\fQGfn}(x)$ and $\Psi_{\fQGfp}(x)$ ($f=1,2,3,4$) represent four families of grand unified qubit-spinor fields with respective to chiral type lepton-quark states and mirror lepton-quark states, they are explicitly given as follows:

Ψ_(x) ≡
Ψ___f^4^u  L(x)
Ψ___f^4^d  R  (x)
Ψ___f^4^d  L(x)
- Ψ___f^4^u  R  (x)

Ψ___f^4^u  R(x)
Ψ___f^4^d  L  (x)
Ψ___f^4^d  R(x)
- Ψ___f^4^u  L  (x)
,   Ψ_(x) ≡
___f^4^u  R(x)
___f^4^d  L  (x)
___f^4^d  R(x)
- ___f^4^u  L  (x)

___f^4^u  L(x)
___f^4^d  R  (x)
___f^4^d  L(x)
- ___f^4^u  R  (x)
where $\Psi_{\fQ_{\mC_f}^4}^{q\, L, R}(x)$ and $\tPsi_{\fQ_{\mC_f}^4}^{q\, L, R}(x)$ ($f=1,2,3,4$) are four kinds of mirror pairs of chiral complex tetraqubit-spinor fields with complex charge-conjugated ones $\Psi_{\fQ_{\mC_f}^4}^{q\, L, R\, \bc}(x)$ and $\tPsi_{\fQ_{\mC_f}^4}^{q\, L, R\, \bc}(x)$. They have the following explicit forms:

Ψ___f^4^q  L, R(x) ≡
^q_L, R,   ___f^4^q  L, R(x) ≡
^q_L, R,

Ψ___f^4^q    L, R(x) ≡
Q^r  _f(x)
Q^b  _f(x)
Q^g  _f(x)
Q^w  _f(x)
^q_L, R ,    ___f^4^q    L, R(x) ≡
^r  _f(x)
^b  _f(x)
^g  _f(x)
^w  _f(x)
^q_L, R ,

Q^α  _f(x) = C_D Q̅^α  T_f(x),   ^α  _f(x)= C_D ^α  T_f(x),   C_D = - iσ_3⊗σ_2 .
The superscripts $q=u$ and $q=d$ represent the up-type and down-type lepton-quark states $Q^{\alpha\, u}_{f\, L, R}(x) = U_{f\, L, R}^{\alpha}(x)$ and $Q^{\alpha\, d}_{f\, L, R}(x) = D_{f\, L, R}^{\alpha}(x)$ with the corresponding mirror lepton-quark states $\tQ^{\alpha\, u}_{f\, L, R}(x) = \tU_{f\, L, R}^{\alpha}(x)$ and $\tQ^{\alpha\, d}_{f\, L, R}(x) = \tD_{f\, L, R}^{\alpha}(x)$. The superscript $\alpha$ labels four color-spin charges $\alpha = r, b, g, w$ with respective to `red', `blue', `green' and `white', and the subscript $f=1,2, 3,4$ denotes four families of chiral type lepton-quark states and mirror lepton-quark states.  The letters `L' and `R' stand for the left-handed and right-handed lepton-quark states and mirror lepton-quark states.

Explicitly, each family of chiral type lepton-quark states in SM and mirror lepton-quark states beyond SM can be expressed by the so-called westward and eastward entangled hyperqubit-spinor fields[2] with the following explicit forms:

Ψ_W_f^T(x) ≡Ψ___f^-^T(x)

=[ (U_f^r, U_f^b, U_f^g, U_f^w, D^r_fc, D^b_fc, D^g_fc, D^w_fc, D_f^r, D_f^b, D_f^g, D_f^w, -U^r_fc, -U^b_fc, -U^g_fc, -U^w_fc)_L ,

(U_f^r, U_f^b, U_f^g, U_f^w, D^r_fc, D^b_fc, D^g_fc, D^w_fc, D_f^r,  D_f^b, D_f^g, D_f^w, -U^r_fc, -U^b_fc, -U^g_fc, -U^w_fc)_R ]^T ,

Ψ_E_f^T(x) ≡Ψ___f^+^T(x)

=[ (_f^r, _f^b, _f^g, _f^w, ^r_fc, ^b_fc, ^g_fc, ^w_fc, _f^r, _f^b, _f^g, _f^w, -^r_fc, -^b_fc, -^g_fc, -^w_fc)_R ,

(_f^r, _f^b, _f^g, _f^w, ^r_fc, ^b_fc, ^g_fc, ^w_fc, _f^r,  _f^b, _f^g, _f^w, -^r_fc, -^b_fc, -^g_fc, -^w_fc)_L ]^T ,
with $f=1,2,3,4$. The superscript $T$ denotes the transposition of column matrix.

In such a hyperqubit-spinor structure, the kinetic action of hyperunified qubit-spinor field $\Psi_{\fQH}(x)$ in globally flat Minkowski hyper-spacetime can be represented as follows:

_     ≡    ∫d^19x    { Ψ̅_(x)  δ_^   _-^ i_ Ψ_(x) - ϕ_1(x)  Ψ̅_(x)   _- Ψ_(x) },
with $\fA, \fM= 0,1,2,3,5, \cdots, 19$. The matrices $\vSi_{-}^{\fA}$ and $\tvSi_{-}$ are defined via $\vGa$-matrices $\vGa^{\hfA}\equiv (\vGa^{\fA}, \vGa^{20})\equiv (\vGa^a, \vGa^A, \tvGa) $ ($\hfA=0,1,2,3,5,\cdots, 20$) with the following explicit structure:

_-^ = 1/2 ^_-,   = 0,1,2,3,5, ⋯19,

_-^20 = 1/2 ^20_- ≡_- = 1/2 _-,   _- = 1/2 ( 1 - γ̃_21 ) ,

^0 =     σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_1 ⊗σ_0,

^1 =    i σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0⊗σ_2⊗σ_1,

^2 =     i σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0⊗σ_2⊗σ_2,

^3 =    i σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0⊗σ_2⊗σ_3,

^5 = -i σ_0 ⊗σ_0 ⊗σ_2 ⊗σ_1 ⊗σ_0 ⊗σ_1 ⊗σ_0 ⊗σ_2⊗σ_3⊗σ_0,

^6 = -i σ_0 ⊗σ_0 ⊗σ_2 ⊗σ_1 ⊗σ_0 ⊗σ_2 ⊗σ_3 ⊗σ_2⊗σ_3⊗σ_0,

^7 = -i σ_0 ⊗σ_0 ⊗σ_2 ⊗σ_1 ⊗σ_0 ⊗σ_1 ⊗σ_2 ⊗σ_3⊗σ_3⊗σ_0 ,

^8 = -i σ_0 ⊗σ_0 ⊗σ_2 ⊗σ_1 ⊗σ_0 ⊗σ_2 ⊗σ_2 ⊗σ_0⊗σ_3⊗σ_0,

^9 = -i σ_0 ⊗σ_0 ⊗σ_2 ⊗σ_1 ⊗σ_0 ⊗σ_1 ⊗σ_2 ⊗σ_1⊗σ_3⊗σ_0 ,

^10 =  -i σ_0 ⊗σ_0 ⊗σ_2 ⊗σ_1 ⊗σ_0 ⊗σ_2 ⊗σ_1 ⊗σ_2⊗σ_3⊗σ_0 ,

^11 =  -i σ_0 ⊗σ_0 ⊗σ_2 ⊗σ_2 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0⊗σ_3⊗σ_0 ,

^12 =  -i σ_0 ⊗σ_0 ⊗σ_2 ⊗σ_1 ⊗σ_1 ⊗σ_3 ⊗σ_0 ⊗σ_0⊗σ_3⊗σ_0 ,

^13 =  -i σ_0 ⊗σ_0 ⊗σ_2 ⊗σ_1 ⊗σ_2 ⊗σ_3 ⊗σ_0 ⊗σ_0⊗σ_3⊗σ_0 ,

^14 =  -i σ_0 ⊗σ_0 ⊗σ_2 ⊗σ_1 ⊗σ_3 ⊗σ_3 ⊗σ_0 ⊗σ_0⊗σ_3⊗σ_0 ,

^15 =  iσ_1 ⊗σ_2 ⊗σ_1⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0⊗σ_3⊗σ_0 ,

^16 =  iσ_1 ⊗σ_2 ⊗σ_2 ⊗σ_3 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0⊗σ_3⊗σ_0,

^17 =  iσ_1 ⊗σ_2 ⊗σ_3⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0⊗σ_3⊗σ_0 ,

^18 =   i σ_1 ⊗σ_1 ⊗σ_0 ⊗σ_3 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0⊗σ_3⊗σ_0 ,

^19 =   i σ_1 ⊗σ_3 ⊗σ_0 ⊗σ_3 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0⊗σ_3⊗σ_0 ,

^20 =   i σ_2 ⊗σ_0 ⊗σ_0 ⊗σ_3 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0⊗σ_3⊗σ_0 ≡,

γ̃_21 =   σ_3⊗σ_0⊗σ_0⊗σ_3 ⊗σ_0 ⊗σ_0 ⊗σ_0 ⊗σ_0⊗σ_3⊗σ_0

=    σ_3⊗σ_0⊗γ̂_17=  σ_3⊗σ_0⊗σ_0⊗γ̂_15 .
Such a hyperunified qubit-spinor field $\fPsi_{\fQH}(x)$ satisfies the following self-conjugated chiral conditions:
_^c(x) = __^T(x) =  _(x) ,  γ̃_21 _(x) = - _(x),

_ = σ_0⊗σ_0⊗C_ = σ_0⊗σ_0⊗σ_0⊗C_

=  σ_0⊗σ_0⊗σ_0⊗σ_1 ⊗σ_2 ⊗σ_2 ⊗σ_0 ⊗σ_0⊗C_D ,
with $C_D= - i\sigma_3\times \sigma_2$ defined as the usual charge conjugation matrix in four-dimensional spacetime.

§.§ Scaling properties of biframe hyper-spacetime and fundamental fields with scaling gauge invariant hyperspin gravigauge field and field strength

To characterize the basic properties of fundamental fields and biframe hyper-spacetime formed from globally flat Minkowski hyper-spacetime and locally flat gravigauge hyper-spacetime, we are going to assign global and local scaling charges for all fundamental fields and biframe hyper-spacetime bases.

The global scaling charge $\cC_s$ is defined as the power of scaling factor $\lambda$ under the following global conformal scaling transformation:
x^ →x^' = λ^-1 x^,   _s = -1,

_(x) →'_(x') = λ^D_h-1/2 _(x),   _s = D_h-1/2 ,

_(x) →'_(x') = λ_(x),   _(x) →'_(x') = λ_(x),    _s = 1,

[d^D_hx] →[d^D_hx'] = λ^-D_h [d^D_hx],   _s = -D_h .
Similarly, the local scaling charge $\ckcC_s$ is defined as the power of scaling factor $\xi(x)$ under the following local scaling gauge transformation:
χ_^  (x) →χ_^'  (x) =  ξ^-1(x) χ_^  (x) ,   _s = -1,

_(x) →'_(x) = ξ^D_h-1/2(x) _(x),   _s = D_h-1/2 ,

_(x) →'_(x) =  _(x),    _(x) →'_(x) =  _(x),   _s = 0 ,

_^  (x) →_^'  (x) =  ξ(x) _^  (x) ,   _s = 1,

χ(x)≡χ_^  (x) →χ'(x) =  ξ^-D_h(x) χ(x) ,   _s = -D_h .
The fields and operators with both global and local scaling charges are presented as follows:
ϕ_p(x) →ϕ'_p(x') =  λϕ_p(x),   ϕ_p(x) →ϕ'_p(x) =  ξ(x) ϕ_p(x),   _s = 1,   _s = 1,

ϕ(x) →ϕ'(x') =  λ  ϕ(x),   ϕ(x) →ϕ'(x) =  ξ(x) ϕ(x),   _s = 1,  _s = 1,

δζ^≡χ_^   (x) dx^ →δζ^'  = ξ^-1(x)λ^-1  δζ^ ,   _s= - 1,    _s = -1 ,

ƪ_ ≡_^   (x)_ →ƪ'_ = ξ(x)λ   ƪ_ ,   _s=1,    _s = 1  ,

[δ^D_hζ]  →[δ^D_hζ'] =  ξ^-D_h(x)  λ^-D_h [δ^D_hζ],   _s = -D_h,   _s = -D_h,
where $[\delta^{D_h}\zeta]\equiv [d^{D_h}x]\, \chi(x)$ is regarded as the hyper-gravicoordinate integral measure in locally flat gravigauge hyper-spacetime.

It is noticed that the inhomogeneous hyperspin gauge field $\bvcA_{\fM}(x)=\cA_{\fM}(x)+\ckcA_{\fM}(x)$ and scaling gauge field $\mW_{\fM}(x)$ are in general invariant under the local scaling transformation, which has zero local scaling charge $\ckcC_s=0$. Nevertheless, when the hyperspin gauge field is decomposed into the hyperspin gravigauge field and covariant-gauge field as shown in Eq.(<ref>), the hyperspin gravigauge field $\mOm_{\fM}^{\fA\fB}(x)$ in Eq.(<ref>) is no longer local scaling gauge invariant, so that the hyperspin covariant-gauge field $\mA_{\fM}^{\fA\fB}(x)$ is also not local scaling gauge invariant. It is appropriate to decompose the hyperspin gauge field into local scaling gauge invariant hyperspin gravigauge field $\fOm_{\fM}^{\fA\fB}(x)$ and covariant-gauge field $\fA_{\fM}^{\fA\fB}(x)$ as follows:

_^(x) = _^(x) + _^(x),

_^ = 1/2( χ̂^ _^ - χ̂^ _^ -  χ̂^  χ̂^  _^ χ_ )

≡1/2( ^ _^ - ^ _^ -  ^  ^  _^ _ ) ,
where $\fG_{\fM\fN}^{\fA}(x)$ is local scaling gauge covariant field strength defined in Eq.(<ref>) for the hyper-gravigauge field $\chi_{\fM}^{\;\;\fA}(x)$, which is initiated from the local scaling invariant and $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\;\;\fA}(x)$ with the following relations:

_^  (x) ≡ϕ(x) χ_^  (x) ,   _^  (x) ≡ϕ^-1(x) _^  (x),

_^(x)=  __^(x) -  __^(x) ≡ϕ(x) _^(x) ,
where $\hat{\fA}_{\fA}^{\;\;\fM}(x)$ is viewed as dual $\cW_e$-spin invariant-gauge field. It becomes manifest that the field strength $\mF_{\fM\fN}^{\fA}(x)$ is both local scaling invariant and $\cW_e$-spin gauge invariant but no longer hyperspin gauge covariant. We have used the different letter style to distinguish $\mF_{\fM\fN}^{\fA}(x)$ from inhomogeneous hyperspin gauge covariant and local scaling invariant field strength $\fF_{\fM\fN}^{\fA}(x)$ defined in Eq.(<ref>).

In terms of the local scaling invariant hyperspin gravigauge field $\fOm_{\fM}^{\fA\fB}(x)$ and covariant-gauge field $\fA_{\fM}^{\fA\fB}(x)$, their corresponding local scaling invariant field strengths are defined as follows:
_^(x) ≡_^(x) + _^(x) ,

_^ = _ _^ - _ _^ + _^ _^ -  _^ _^,

_^ =  _ _^ - _ _^ +  _^ _^ -  _^ _^  ,

_ _^ = _  _^  +  _^ _^ + _^ _^  ,
It is easy to check that both $\fR_{\fM\fN}^{\fA\fB}$ and $\fF_{\fM\fN}^{\fA\fB}(x)$ become local scaling gauge invariant field strengths.

§.§  Hyperunified field theory for fundamental building block and fundamental interaction based on gauge invariance principle and scaling invariance hypothesis

We now turn to build a whole hyperunified field theory for the category-1 entangled decaqubit-spinor field as hyperunified qubit-spinor field with inhomogeneous hyperspin gauge symmetry WS(1,$D_h$-1) and global inhomogeneous Lorentz-type/Poincaré-type group symmetry PO(1,$D_h$-1) in $D_h=19$ dimensional biframe hyper-spacetime.

Considering the hyperunified qubit-spinor field $\fPsi_{\fQH}(x)$ in 19-dimensional hyper-spacetime as fundamental building block which involves 512 independent degrees of freedom in the qubit-spinor representation of Hilbert space, and taking the inhomogeneous hyperspin symmetry WS(1,18) as gauge symmetry by following along the gauge invariance principle, we are led to introduce the inhomogeneous hyperspin gauge field $\bvcA_{\fM}(x)$ which mediates the fundamental interaction. Such a gauge field concerns both hyperspin gauge field $\cA_{\fM}^{\fA\fB}(x)$ and $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\fA}(x)$, which belong to the adjoint representation and vector representation of hyperspin gauge symmetry SP(1,18), respectively. The global and local scaling charges corresponding to global scaling symmetry SC(1) and scaling gauge symmetry SG(1) allow us to identify the graviscalar field $\phi(x)$ and hyper-gravigauge field $\chi_{\fM}^{\fA}(x)$ from the $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\fA}(x) \equiv \phi(x) \chi_{\fM}^{\fA}(x)$ as well as $\cQ_c$-spin scalar field $\phi_1(x)$ relevant to the $\cQ_c$-spin charge $\cQ_c=q_c=1$. In addition, the scaling gauge symmetry SG(1) brings on the scaling gauge field $\mW_{\fM}(x)$. For convenience, we should list all relevant fields as follows:
_(x),   _^(x),   χ_^(x),   ϕ(x),    ϕ_1(x),   _(x), which are all basic fields concerned in establishing a whole hyperunified field theory. The gravitational origin of gauge symmetry indicates that the inhomogeneous hyperspin gauge symmetry is governed by hyper-gravigauge field $\chi_{\fM}^{\fA}(x)$ via the hyperspin gravigauge field $\fOm_{\fM}^{\fA\fB}(x)$ defined in Eq.(<ref>) and also by $\cW_e$-spin gravigauge field  $\fOm_{\fM}^{\fA}(x)$ given in Eq.(<ref>).

The gauge invariant and scaling invariant action for hyperunified qubit-spinor field $\fPsi_{\fQH}(x)$ in category-1 can be read from Eq.(<ref>) in 19-dimensional biframe hyper-spacetime. In constructing the action in Eq.(<ref>), the scaling invariance for both global and local scaling symmetries plays a crucial role in addition to the gauge invariance principle. To construct the basic action for all fundamental fields, the scaling symmetry is still supposed to play an essential role. For that, we are motivated to make a scaling invariance hypothesis which postulates that the action is built to keep scaling invariance for both global and local scaling symmetries in addition to the inhomogeneous hyperspin gauge symmetry by following along gauge invariance principle. Furthermore, it is supposed that the basic gauge field and scalar field also obey the simplest motion postulate that they all involve just the lowest quadratic derivative in the gauge invariant and scaling invariant action.

Following along the scaling invariance hypothesis and gauge invariance principle together with the maximum coherence motion principle and maximum entangled quibit-spinor principle, we obtain definitively the following basic action for a whole hyperunified field theory:

_     ≡    ∫[d^D_hx]   χ(x)   {  _ _-^_^    i_  _ - ϕ_1 _ _-_

+     ϕ^D_h-4 [  - _^ _^  _'^ '_'^ '   (   η̂^''_'' g_H^-2  1/4  _^ _''^''

+     η̃^''_' g_H^-2 1/4  _^_''^'   - η̂^''  g_W^-2  1/4 _ _''  )

+      1/2  _^ _^  η^ g_H^-2β_G^2 ϕ_1^2  (_^ - _^)(_ - _)

+      1/2 _^ _^  η^  (λ_S^2 D_ ϕD_ϕ+  λ_Q^2 D_ ϕ_1 D_ϕ_1 ) - λ_D^2 ϕ_1^4 (ϕ_1/ϕ)   ]   } ,
with $\fA, \fM= 0,1,2,3,5, \cdots, D_h$ and $D_h=19$. Where $g_H$ and $g_W$ are two gauge coupling constants, $\beta_G$, $\lambda_S$, $\lambda_Q$ and $\lambda_D$ are scalar coupling parameters and $\mF(\frac{\phi_1}{\phi})$ represents a general scaling invariant potential relevant to two scalar fields. $\vSi_{-}^{\fA}$ and $\tvSi_{-}$ matrices are defined via the $\vGa$-matrices $\vGa^{\hfA}= (\vGa^a, \vGa^A, \vGa^{20}=\tvGa ) $ ($\hfA=0,1,2,3,5,\cdots, 20$) presented in Eq.(<ref>). All definitions in the basic action are already given in previous relevant sections, we represent them as follows just for convenience:
i_ ≡i_ + _(x),   _(x) = _^(x) 1/2_ ,

_^ = 1/2( χ̂^ _^ - χ̂^ _^ -  χ̂^  χ̂^  _^ χ_ )

D_ϕ= (_ + _)ϕ,   D_ϕ_1 = (_ + _)ϕ_1
for the covariant derivatives and gauge fields, and

_^ =  _ _^ - _ _^ +  _^ _^ -  _^ _^  ,

_^ =  __^  -  __^   = ϕ^2_^,

_^ =  d_χ_^  -  d_χ_^   ,   d_ ≡_ + _lnϕ,

_ =  __ -  __,
for the gauge covariant field strengths.

The constant tensor factors involved in the action are given by

η̂^''_'' ≡η^' η^' η_' η_'   ,   η̂^'' ≡η^' η^' ,

η̃^''_'  ≡η̂^''_'    - η̂^ ''_(')  + 3 η̂^ ''_['],
with the definitions,

η̂^ ''_(') ≡η^' 1/2 (  η_^  ' η_'^   + η_^   η_'^  ' )  + 1/2 ( η_^  ' η_'^   + η_^   η_'^  ' ) η^' ,

η̂^ ''_['] ≡η^' 1/2( η_^  ' η_'^   - η_^   η_'^  ' ) +  1/2 (η_^  ' η_'^   - η_^   η_'^  '  ) η^' ,

η̂^''_'  ≡η^' η^' η_' .

The tensor factor $\tilde{\eta}^{\fC\fD\fC'\fD'}_{\fA\fA'} $ may be called as hypespin gauge invariance tensor[2] as it brings the dynamic term of Abelian-type hyper-gravigauge field strength
η̃^''_' _^_''^',
to be gauge invariant under hyperspin gauge transformation although the Abelian-type hyper-gravigauge field strength $\mF_{\fM\fN}^{\fA}$ is not a gauge covariant one. The quadratic interaction term between gauge field and $\cQ_c$-spin scalar field $\phi_1$ is manifestly gauge invariant as it concerns the difference between hyperspin gauge field and hyperspin gravigauge field, i.e., $(\cA_{\fM}^{\fA\fB} - \fOm_{\fM}^{\fA\fB})$, so that their inhomogeneous gauge transformations cancel each other. Note that such a quadratic term appears as a mass-like term of gauge field.

The action in Eq.(<ref>) possesses the following joint symmetry:

G_S= SC(1)⋉PO(1,18)WS(1,18)⋊SG(1) ,
where SC(1) and SG(1) represent the global scaling symmetry and local scaling gauge symmetry, respectively. $\mF_{\fM\fN}$ in Eq.(<ref>) defines the field strength of scaling gauge field $\mW_{\fM}$. PO(1,18) remains global inhomogeneous Lorentz-type/Poincaré-type group symmetry and WS(1,18)  represents inhomogeneous hyperspin gauge symmetry, which brings about a joint global and local symmetry in biframe hyper-spacetime. Only when locally flat gravigauge hyper-spacetime goes to be globally flat one, then the motion-correlation inhomogeneous hyperspin symmetry WS(1,18) becomes to be associated with inhomogeneous Lorentz-type/Poincaré-type group symmetry PO(1,18), so that the group transformations of SP(1,18) and SO(1,18) have to be coincidental, which is realized from their isomorphic property SP(1,18)$\cong$SO(1,18).

§ HYPERUNIFIED FIELD THEORY IN HIDDEN SCALING GAUGE FORMALISM WITH GAUGE-GRAVITY CORRESPONDENCE AND THE DYNAMICS OF FUNDAMENTAL FIELDS WITH CONSERVED CURRENTS

The action of hyperunified field theory presented in Eq.(<ref>) is fully built based on the gauge invariance principle and scaling invariance hypothesis, which enables us to demonstrate the gauge-gravity correspondence. As such a hyperunified field theory is formulated in biframe hyper-spacetime, we are able to derive equations of motion for all fundamental fields by following along the least action principle and obtain various conserved currents with respect to gauge symmetries.

§.§ Hyperunified field theory in hidden scaling gauge formalism with gauge-gravity correspondence and the kinetic motion of fundamental fields as the source of gravitational interaction

It is noticed from the action in Eq.(<ref>) that the $\cW_e$-spin gauge field $\ckcA_{\fM}^{\;\;\fA}$ does not interact with the hyperunified qubit-spinor field $\fPsi_{\fQH}$ via the gauge covariant derivative as like the usual case just following along the gauge invariance principle, this is because the group generators of $\cW_e$-spin gauge symmetry get the nilpotent condition $(\vSi_{-}^{\fA})^2=0$. In another word, the $\cW_e$-spin gauge interaction via the gauge covariant derivative becomes vanishing, the reason is that such an interaction term brings on the change of chirality for the hyperunified qubit-spinor field $\fPsi_{\fQH}$ which is an entangled hyperqubit-spinor field with self-conjugated chiral structure. Instead, the $\cW_e$-spin gauge field gets to interact with the hyperunified qubit-spinor field $\fPsi_{\fQH}$ through coupling the dual $\cW_e$-spin invariant-gauge field $\hat{\fA}_{\fA}^{\;\;\fM}\equiv \phi^{-1}\chih_{\fA}^{\;\;\fM}$ to the covariant derivative operator $\cD_{\fM}$ as kinetic term of hyperunified qubit-spinor field $\fPsi_{\fQH}$, i.e.:
_-^ _^  _≡^ _^   _-  _ ,
where we have introduced the scaling gauge invariant dual tensors,

_^   _^    η_^   = η_^   ,   _^   _^    η_^   = η_^   ,

^ ≡_^ _^  η^ ≡ϕ^-2 ^ ,

^ _ = η^_    ,   _  ^ = η_^     ,

_ ≡_^ _^  η_ ≡ϕ^2 χ_ .
Geometrically, the dual tensors $\mH_{\fM\fN}$ ($\chi_{\fM\fN}$) and $\hmH^{\fM\fN}$ ($\chih^{\fM\fN}$) are referred to as dual hyper-gravimetric tensor fields.

In the whole hyperunified field theory presented in Eq.(<ref>), the bicovariant vector field $\chih_{\fA}^{\;\;\fM}$ corresponding to the dual $\cW_e$-spin invariant-gauge field $\chih_{\fA}^{\;\;\fM}\equiv \phi \hat{\fA}_{\fA}^{\;\;\fM}$ couples to the covariant derivative operator $\cD_{\fM}$, which arises not only from supplying both the local hyperspin gauge symmetry and global Lorentz-type group symmetry, but also from preserving the local scaling gauge symmetry. Therefore, the local scaling gauge invariance plays a crucial role in the building of a whole hyperunified field theory.

The gauge symmetry often involves redundant degrees of freedom, which demands to make a gauge prescription with setting up gauge fixing condition. In order to eliminate the redundant degree of freedom arising from local scaling gauge symmetry, let us rewrite the action given in Eq.(<ref>) into the following formalism:

_     ≡    ∫[d^D_hx]   (x)

=     ∫[d^D_hx]   (x)   {   ^ _ _  _-^ i_  _  - β_Q sinhχ_s   _ _-_

-      1/4 _^ _^  _'^ '_'^ ' [ η̂^''_'' g_H^-2 _^  _''^'' - η̃^''_' g_H^-2 _^_''^'

-     η̅^''_' g_H^-2β_G^2 β_Q^2 sinh^2χ_s  _^_''^'  +  η̂^'' g_W^-2_ _'' ]

+      1/2 λ_S^2( 1 + sinh^2χ_s )   ^ _ _  +  1/2 ^ λ_S^2 _ χ_s _χ_s  - λ_D^2 (χ_s)    } ,
with the definitions,

(x) ≡_^   = ϕ^D_h χ_^   = ϕ^D_h(x) χ(x),
(x) ≡_^   = ϕ^-D_h _^   = ϕ^-D_h(x) (x) = 1/(x) ,

_^ =  __^ -  __^ ≡ϕ(x) _^,

_^ = __^ -  __^ = _^ + _^ _^ - _^ _^ ,
where $\fF_{\fM\fN}^{\fA}$ defines gauge covariant field strength under hyperspin gauge transformations, which is also local scaling gauge invariant field strength.

We have also made the following definitions:
_ →ϕ^D_h-1/2 _,

_ →_ - 1/2_ln(λ_S^2 ϕ^2 + λ_Q^2ϕ_1^2) ,

λ_S^2/1 + λ_Q^2ϕ_1^2/λ_S^2ϕ^2  _(λ_Qϕ_1/λ_Sϕ) _(λ_Qϕ_1/λ_Sϕ)  ≡λ_S^2 _χ_s(x)_χ_s(x) ,
where the explicit relation between the field $\chi_s(x)$ and the ratio $\phi_1(x)/\phi(x)$ is obtained as follows:
χ_s(x)     ≡    sinh^-1λ_Qϕ_1(x)/λ_S ϕ(x),

ϕ_1(x)/ϕ(x)      ≡     β_Q sinhχ_s(x) ,
β_Q ≡λ_S/λ_Q.
The dimensionless field $\chi_s(x)$ may be referred to as scaling invariant $\cQ_c$-spin scaling field which has zero global and local scaling charges. $\cF(\chi_s)$ is considered to be a general potential of $\cQ_c$-spin scaling field.

The tensor factor $\bar{\eta}^{\fC\fD\1 \fC'\fD'}_{\fA\fA'}$ is defined as follows:

η̅^ ''_'   ≡η̂^''_'    + 1/2 ( η̂^''_' + η̂^ ''_(')  + η̂^ ''_['] ) ,
with the tensors $\hat{\eta}^{\fC\fD\fC'\fD'}_{\fA\fA'}$, $\hat{\eta}^{\fC\fD\1 \fC'\fD'}_{(\fA\fA')}$ and $\hat{\eta}^{\fC\fD\1 \fC'\fD'}_{[\fA\fA']}$ given in Eqs.(<ref>) and (<ref>). Such a constant tensor $\bar{\eta}^{\fC\fD\1 \fC'\fD'}_{\fA\fA'}$ couples to the gauge covariant dynamic term of hyper-gravigauge field to reproduce the gauge invariant quadratic interaction term concerning the difference between the hyperspin gauge field $\cA_{\fM}^{\fA\fB}$ and hyperspin gravigauge field $\fOm_{\fM}^{\fA\fB}$, i.e.:
1/4_^ _^  _'^ '_'^ ' η̅^''_' _^_''^' = 1/2^ (_^ - _^) (_ - _). It will be seen below that the gauge covariant dynamic formalism of hyper-gravigauge field becomes more convenient in deriving the equation of motion of hyper-gravigauge field.

In the action Eq.(<ref>), the scaling gauge symmetry becomes a hidden symmetry. Namely, all basic fields have zero local scaling charge $\ckcC_s=0$. In addition, the redefined hyperunified qubit-spinor field $\fPsi_{\fQH}$ and $\cQ_c$-spin scaling field $\chi_s(x)$ get zero global scaling charge $\cC_s=0$. We may refer to the action in Eq.(<ref>) as hyperunified field theory in hidden scaling gauge formalism.

It is noticed that the graviscalar field $\phi(x)$ does not appear explicitly in the action presented in Eq.(<ref>). We arrive at five fundamental fields which include the hyperunified qubit-spinor field $\fPsi_{\fQH}$, the hyperspin gauge field $\cA_{\fM}^{\fA\fB}$, the $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\;\;\fA}$, the scaling gauge field $\mW_{\fM}$ and the $\cQ_c$-spin scaling field $\chi_s(x)$. Nevertheless, the independent degrees of freedom should be unchanged as the graviscalar field is considered to be absorbed into the scaling gauge field $\mW_{\fM}$, which appears to be a massive-like gauge field.

In such a hidden scaling gauge formalism of hyperunified field theory, the dual $\cW_e$-spin invariant-gauge field $\hat{\fA}_{\fA}^{\;\;\fM}$ couples to either the coordinate derivative operator $\p_{\fM}$ that reflects the motion of fundamental fields or the fundamental gauge field that characterizes the interaction of fundamental fields. So that the $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\;\;\fA}$ as hyper-gravigauge field with the field strength $\mF_{\fM\fN}^{\fA}$ describes the fundamental gravitational interaction for all motional fields and vector fields in Minkowski hyper-spacetime.

Therefore, we come to the statement that the kinetic motion of fundamental fields emerges as the source of gravitational interaction which is truly characterized by the gauge interaction. As a consequence, the fundamental gravitational interaction exists in all motional fields and a whole hyperunified field theory in hidden scaling gauge formalism reveals the gauge-gravity correspondence.

§.§ Equations of motion and dynamics of fundamental fields

The action in Eq.(<ref>) for the hyperunified field theory enables us to derive equations of motion for all fundamental fields from the least action principle, which can be applied to describe the dynamics of fundamental fields and study gravitational relativistic quantum theory in hyper-spacetime.

Let us first present the equation of motion for the hyperunified qubit-spinor field $\fPsi_{\fQH}$,

_-^ _^   i (_ -_ ) _
-β_Q sinhχ_s _- _ =   0 ,
where $\fV_{\fM}$ is regarded as graviscaling induced gauge field defined as follows:
_     ≡    1/2   _^   _( _^   ) ,
which is local scaling gauge invariant. Where $\cD_{\fN}(\hat{\fkA} \fA_{\fM}^{\;\; \fB})$ concerns the covariant derivative for the $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\;\;\fB}(x)$,

__^   (x)  ≡_ _^   (x)
+ _  ^(x)  _^  (x) .

The quadratic form for the above equation of motion is found to be.

^ (∇_ - _ ) ( _ - _ ) Ψ_

= - ( β_Q^2 sinh^2χ_s +  ^ _^   β_Q coshχ_s   (i_χ_s)    ) Ψ_

+ ^_^    _^     ( _ - _^ _^    i ( _ - _ ) - i _  ) Ψ_,
with the definition for the covariant derivative,

∇_ ( _  - _ ) ≡_ ( _ -_ ) - _^ ( _ -_ ) ,

_^  ≡_^     __^   =  _^   (_ _^    + _  ^  _^    ) .
Where $\cF_{\fM\fN}(x)$ is the field strength of hyperspin gauge field $\cA_{\fM}(x)$ and $\fF_{\fM\fN}^{\fC}$ the field strength of $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\; \fC}(x)$ presented in Eq.(<ref>).  The graviscaling induced gauge field strength $\fV_{\fM\fN}$ is defined as follows:
_  =  __-  __ .

We now turn to the equation of motion for the hyperspin gauge field $\cA_{\fM}^{\fA\fB}$, which is found to be,

_ (  ^'^'g_H^-2_'' ) =  _^   ,
where $\cJ_{\fA\fB}^{\;\fM}$ is bicovariant tensor current with the following explicit form:

_^  =     1/2 _^
_ { _-^      _ } _ +    1/2 ^''_[]' g_H^-2β_G^2β_Q^2 sinh^2χ_s  _''^'  .
We have introduced the tensor $\bhmH^{\fM\fM'\fN'}_{[\fA\fB]\fA'}$ defined as follows:

^''_[]' ≡^[]''_'  _ - ^[]''_' _ ,

^[]''_'  ≡1/2 (   ^ ''_'  -  ^ ''_'    ),

^''_'  ≡_^ _^  _'^ '_'^ ' η̅^''_' .

Let us now derive the equation of motion for the $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\;\; \fA}$ as hyper-gravigauge field $\fA_{\fM}^{\;\; \fA}= \phi \chi_{\fM}^{\;\; \fA}$. The explicit form is found to be as follows:

_ (  ^[]''_' g_H^-2_'' ^' ) + _ (   ^[]''_' g_H^-2β_G^2  β_Q^2 sinh^2χ_s  _'' ^' )= _^     ,
with $\fJ_{\fA}^{\;\fM}$ the bicovariant vector current given by,

_^        =      -   _^   +    _^      {   _^     _ _-^ i _ _

-    ^[]''_”  '' g_H^-2 _^” _''^'' + ^[]''_”' g_H^-2 _^” _''^'

+      ^[]''_”'g_H^-2β_G^2 β_Q^2 sinh^2χ_s  _^” _''^'  +  ^ λ_S^2_χ_s  _ χ_s

-     g_W^-2^' ^'_ _'' +  λ_S^2(1+ sinh^2χ_s ) ^ _ _   } ,
where the tensors $\hmH^{[\fM\fN]\fM'\fN'}_{\fA''\fB\; \fA'\fB'}$ and $\thmH^{[\fM\fN]\fM'\fN'}_{\fA\fA'}$ are defined as follows:

^[]''_”  '' =  1/2 ( ^''_”  '' -^''_”  '' ),

^''_”  '' ≡_^ _^  _'^ '_'^ ' η̂^''_”'' ,

^[]''_'  ≡1/2 (  ^ ''_'  - ^ ''_'    ),

^''_'  ≡_^ _^  _'^ '_'^ ' η̃^''_' ,
and the tensor $\bhmH^{[\fM\fN]\fM'\fN'}_{\fA\fA'}$ is given in Eq.(<ref>).

It is straightforward to obtain the equation of motion for the scaling gauge field $\mW_{\fM}$

_ (  ^'^' g_W^-2 _'' )  =   ^ ,
with the vector current
^ = -   λ_S^2 (1 + sinh^2χ_s)    ^ _  .

It is easy to write down the equation of motion for the $\cQ_c$-spin scaling field,

_  (  ^λ_S^2 _χ_s ) =  _s ,
with the scalar current,
_s     =     -    β_Qcoshχ_s  _ _-_   +   β_Q^2 sinhχ_scoshχ_s    ^ _ _

+       1/2 _^ _^  _'^ '_'^ '  η̅^''_' g_H^-2β_G^2 β_Q^2sinhχ_scoshχ_s   _^_''^'

-   λ_D^2 '(χ_s) .
'(χ_s) ≡/χ_s  (χ_s) .

It is noticed that all equations of motion for five fundamental fields are global scaling invariant.

§.§  Conserved currents in correspondence to gauge symmetries in hyperunified field theory

From Noether's theorem[30] that for every differentiable symmetry generated by the local action, there exists the corresponding conserved current. We are going to show the conserved currents generated by the inhomogeneous hyperspin gauge symmetry WS(1,18) and the scaling gauge symmetry SG(1).

The bicovariant tensor current $\cJ^{\;\fM}_{ \fA\fB}$ generated from the hyperspin gauge symmetry SP(1,$D_h$-1) is a conserved current with respect to the covariant derivative,

_^ _  = 0 ,
which can be verified from the equation of motion of hyperspin gauge field,
_ _ ( ^'^'_'' ^   )

=_ _ (   ^'^') _'' ^    + _ (   ^'^' ) _  _'' ^

+  _ (   ^'^' ) _ _'' ^      +   ^'^' _ _ _'' ^

=    ^'^' (  _^     _'' ^   '  - _'' ^    _^   ' ) η_' = 0 . In obtaining the above identity, the symmetric and antisymmetric properties of the tensor and gauge field strength have been used to bring the cancellation.

From the conserved current of bicovariant tensor current $\cJ^{\fM}_{\;\; \fA\fB}$ presented in Eq.(<ref>), we are able to derive the following dynamical equation:

_ ^_    =  - _[],
for the bicovariant tensor current $\cS^{\fM}_{\;\; \fA\fB}$ and antisymmetric tensor current $\cJ_{[\fA\fB]}$ with the following definitions:

^_    =     1/2 _^
_ { _-^      _ } _,

_[] ≡_[]  -    1/4g_H^-2β_G^2 β_Q^2 sinh^2χ_s   η_[]^''  ^''_' _'_''^ ,

_[] = _^   _ - _^  _ ,   η_[]^'' = η_^  ' η_^  ' - η_^  ' η_^  ',

_^   ≡_^   - _ (  ^[]''_' g_H^-2_'' ^' ) .
where $\cS^{\fM}_{\;\; \fA\fB}$ is regarded as bicovariant hyperspin angular momentum tensor.

It is noticed that the dynamics of bicovariant hyperspin angular momentum tensor $\cS^{\fM}_{\;\; \fA\fB}$ is governed by the antisymmetric tensor current $\cJ_{[\fA\fB]}$. Such a tensor current is correlated to the hyper-gravigauge field strength and bicovariant vector current $\fJ_{\fA}^{\; \fM}$ which determines the dynamics of hyper-gravigauge field $\fA_{\fM}^{\;\; \fA}$.

The dynamical equation presented in Eq.(<ref>) may be referred to as dynamic evolution equation for the hyperspin angular momentum tensor. The bicovariant vector current $\tfJ_{\fA}^{\; \fM}$ provides the source for the equation of motion of hyper-gravigauge field $\fA_{\fM}^{\;\; \fA}$ as follows:

_ (   ^[]''_' g_H^-2β_G^2 β_Q^2 sinh^2χ_s   _'' ^' ) = _^   .
Note that neither $\fJ_{\fA}^{\;\;\fM}$ or $\tfJ_{\fA}^{\;\; \fM}$ is conserved.

It is useful to define the following alternative bicovariant vector current:

_^   ≡_^   - _ (   ^[]''_' g_H^-2β_G^2 β_Q^2 sinh^2χ_s   _'' ^' ) ,
so that the equation of motion presented in Eq.(<ref>) for the hyper-gravigauge field $\fA_{\fM}^{\;\; \fA}$ can be rewritten into the following simple form:

_ (  ^[]''_' g_H^-2_'' ^' ) = _^  .
It can be verified that the bicovariant vector current $\mJ_{\fA}^{\;\;\fM}$ is conserved due to the following identity:
_ _^   =  _ _ (  ^[]''_' g_H^-2_'' ^' ) = 0 ,
which is attributed to the antisymmetric property of the tensor factor $\thmH^{[\fM\fN]\fM'\fN'}_{\fA\fA'}$. Such a conserved bicovariant vector current reflects the translation-like $\cW_e$-spin Abelian-type gauge symmetry.

The equations of motion shown in Eqs.(<ref>) and (<ref>) enable us to obtain the following dynamical equations for both bicovariant vector currents $\tfJ_{\fA}^{\;\;\fM}$ and $\fJ_{\fA}^{\;\;\fM}$,
__^   = - 1/2    ^[]''_”' g_H^-2β_G^2 β_Q^2 sinh^2χ_s    _^”_'' ^' ,

__^   = - _ (   ^[]''_”' g_H^-2β_G^2 β_Q^2 sinh^2χ_s   _^” _'' ^' ) .

For convenience of mention, $\mJ_{\fA}^{\;\;\fM}$ is regarded as conserved bicovariant vector current of hyper-gravigauge field, $\fJ_{\fA}^{\;\;\fM}$ as total bicovariant vector current and $\tfJ_{\fA}^{\;\;\fM}$ as twisting bicovariant vector current which governs the bicovariant hyperspin angular momentum tensor.

The local scaling gauge symmetry leads to the conserved vector current as follows:

∂_ ^  = ∂_ ∂_ ( ^'^'  _'') ≡0 ,
which provides the following condition for the scaling gauge field when applying it to the explicit form of the current in Eq.(<ref>),
_ (   (1 + sinh^2χ_s) ^ _ ) =0  .

§  CONSERVATION LAW AND DYNAMIC EVOLUTION EQUATION FROM GLOBAL SYMMETRY OF MINKOWSKI HYPER-SPACETIME IN HYPERUNIFIED FIELD THEORY

The hyperunified field theory is formulated in globally flat Minkowski hyper-spacetime which is regarded as free-motion hyper-spacetime $\fM_h$, the action in Eq.(<ref>) possesses the global Poincaré-type group symmetry PO(1,18) and global scaling symmetry SC(1). In the absence of gravitational interaction, it is well known that the global Poincaré symmetry leads to the conservation laws for both energy momentum tensor and total angular momentum tensor, which follows Noether's theorem[30] that every differentiable symmetry of the action has the corresponding conservation law. We should apply Noether's theorem to hyperunified field theory in the presence of gravitational interaction and derive the conservation laws and dynamic evolution equations arising from global Poincaré-type group symmetry and scaling symmetry of Minkowski hyper-spacetime.

§.§  Conservation law of hyper-stress energy-momentum tensor from translation group symmetry of Minkowski hyper-spacetime in hyperunified field theory

Let us first derive the conservation law for global translation group symmetry. Consider the global translation group transformation of coordinates in globally flat Minkowski hyper-spacetime as follows:
x^ →x^' = x^ + a^ , with $a^{\fM}$ an arbitrary constant vector. When applying for the variational principle, the invariance of the action under such a translation leads to the following relation:
Δ_= ∫[d^D_hx]   _ (_^   ) a^ = 0 .
which results in the following conservation law:

_ _^   = 0 ,
where $\cT_{\fM}^{\;\, \fP}$ is referred to as bicovariant hyper-stress energy-momentum tensor in hyper-spacetime. From the action of hyperunified field theory given in Eq.(<ref>), we obtain the following explicit form for the bicovariant hyper-stress energy-momentum tensor:

_^        =      -   η_^   +    { _^   _ _-^ i _ _

-    ^[]''_  '' g_H^-2 _^ _''^'' + ^[]''_' g_H^-2 _^ _''^'

+     ^[]''_'g_H^-2β_G^2 β_Q^2 sinh^2χ_s   _^ _''^' +   λ_S^2 _χ_s  _ χ_s  ^

-     g_W^-2^' ^' _ _''+ λ_S^2(1 + sinh^2χ_s )  _ _  ^      } ,
which is gauge invariant. In deriving such a gauge-invariant formalism, the equations of motion for the fundamental fields $\cA_{\fM}^{\fA\fB}$, $\fA_{\fM}^{\;\fA}$, $\mW_{\fM}$ and $\chi_s$ have been used and the surface terms have been ignored.

It is interesting to notice the following relation between the bicovariant hyper-stress energy-momentum tensor and bicovariant vector current:
_^   ≡_^  _^   ,
which indicates that the conserved bicovariant hyper-stress energy-momentum tensor $\cT_{\fM}^{\;\;\fP}$ generated from the global translation group symmetry of coordinates in globally flat Minkowski hyper-spacetime correlates to the bicovariant vector current $\fJ_{\fA}^{\; \;\fP}$ which governs the dynamics of $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\;\;\fA}$ resulted from the local translation-like $\cW_e$-spin group symmetry of entangled hyperqubit-spinor field in locally flat gravigauge hyper-spacetime.

Therefore, we arrive at the statement that for the hyperunified field theory built based on the gauge invariance principle and scaling invariance hypothesis in light of biframe hyper-spacetime, the gauge-invariant bicovariant hyper-stress energy-momentum tensor $\cT_{\fM}^{\;\, \fN}$ derived from the translation group symmetry of Minkowski hyper-spacetime is a conserved tensor even in the presence of gravitational gauge interaction.

§.§  Dynamic evolution equations of hyper-rotation angular momentum tensor from Lorentz-type group symmetry and hyper-conformal scaling momentum tensor from conformal scaling symmetry in hyperunified field theory

We now consider the global Lorentz-type group symmetry in Minkowski hyper-spacetime with the following infinitesimal group transformation of coordinates:
x'^ = x^ + δL^_     x^ . Based on the principle of least action, the invariance of the action under such a transformation leads to the following dynamic evolution equation :

_ ^, =  ^[] ,
with $\cL^{\fP,\fM\fN}$ and $\cT^{[\fM\fN]}$ given explicitly as follows,

^, ≡^  _'η^' x^ -  ^   _'  η^'x^ ,

^[] ≡η^'_'^      - η^'_'^     ,
where $\cL^{\fP,\fM\fN}$ defines bicovariant hyper-rotation angular momentum tensor in association with the bicovariant hyper-stress energy momentum tensor in Minkowski hyper-spacetime. As the antisymmetric part of bicovariant hyper-stress energy-momentum tensor is not necessary to vanish in the hyperunified field theory due to the presence of antisymmetric hyper-gravigauge field, the covariant hyper-rotation angular momentum tensor obeys a dynamic evolution equation presented in Eq.(<ref>) rather than a homogeneously conserved law.

It is useful to express the antisymmetric bicovariant hyper-stress energy-momentum tensor $\cT^{[\fM\fN]}$ into the following two parts:

^[]  ≡_a^[] + _s^[],

_a^[]  ≡_''^  [] ^[''],   _s^[] ≡_''^  [] ^(''),

^[] ≡^ - ^,   ^() ≡^ + ^ ,

_''^  [] ≡1/2 ( _'^   η_'^    - _'^   η_'^    ),

^ ≡^_^   ≡_^    ^,
which presents an alternative tensor $\hfT^{\fM\fN}$ defined either from the bicovariant hyper-stress energy-momentum tensor $\cT_{\fP}^{\;\;\fN}$ raised with the tensor $\hmH^{\fM\fP}$ or from the total bicovariant vector current $\fJ_{\fA}^{\; \;\fN}$ projected by the dual $\cW_e$-spin invariant-gauge field $\hfA_{\fA}^{\;\;\fM}$.

To distinguish the tensor $\hfT^{\fM\fN}$ from the bicovariant hyper-stress energy-momentum tensor $\cT_{\fP}^{\;\;\fN}$, we may refer to $\hfT^{\fM\fN}$ as covariant hyper-stress energy-momentum tensor, which can be discriminated from their global scaling charges as $\cT_{\fP}^{\;\;\fN}$ and $\hfT^{\fM\fN}$ have global scaling charges $\cC_s=D_h$ and $\cC_s=D_h-2$, respectively.  It is useful to decompose the covariant hyper-stress energy-momentum tensor $\hfT^{\fM\fN}$ into antisymmetric part $\hfT^{[\fM'\fN']}$ and symmetric part $\hfT^{(\fM'\fN')}$, which brings the antisymmetric bicovariant hyper-stress energy-momentum tensor $\cT^{[\fM\fN]}$ into two parts $\cT_a^{[\fM\fN]}$ and $\cT_s^{[\fM\fN]}$. $\cT_a^{[\fM\fN]}$ is the antisymmetric bicovariant hyper-stress energy-momentum tensor associated with antisymmetric covariant hyper-stress energy-momentum tensor $\hfT^{[\fM\fN]}$, and $\cT_s^{[\fM\fN]}$ is the antisymmetric bicovariant hyper-stress energy-momentum tensor associated with symmetric covariant hyper-stress energy-momentum tensor $\hfT^{(\fM\fN)}$.

In general, the dynamic evolution equation for the bicovariant hyper-rotation angular momentum tensor $\cL^{\fP,\fM\fN}$ can be rewritten as follows,
_ ^, =  _a^[] + _s^[],
which is governed by two kinds of sources $\cT_a^{[\fM\fN]}$ and $\cT_s^{[\fM\fN]}$. It is straightforward to
show that when turning locally flat gravigauge hyper-spacetime into globally flat Minkowski hyper-spacetime, the source part $\cT_s^{[\fM\fN]}$ goes to be vanishing, i.e.:
_'^   ∝η_'^    ,   _''^  [] ∝η_''^[]  = - η_''^[],

_s^[] ∝η_''^[] ^('') = - _s^[] →0 .

Let us turn to the global conformal scaling symmetry. The invariance of the action leads to the following dynamical equation:

(x^ /x^ + D_h) (  ) + _ ^  - = 0 ,
with $\cT^{\fM}$ and $\cT$ defined as follows:

^ ≡^   _  x^    ,     ≡_^    η_^    ,
where $\cT^{\fM}$ is regarded as hyper-conformal scaling momentum tensor and $\cT= \cT_{\fM}^{\; \fM}$ is the trace of bicovariant hyper-stress energy-momentum tensor.

From the property that the integral $\int [d\hx] \lambda^{D_h}\,  \chi(\lambda x)\, \fkL (\lambda x) $ is actually independent of $\lambda$, and by taking the differentiation with respect to $\lambda$ at $\lambda =1$, we obtain the following identity:
∫[d] (x^ ∂/∂x^ + D_h) (  ) = 0 , which leads the dynamical equation in Eq.(<ref>) to be simplified as follows:
_ ^ = .
Such an equation brings on dynamic evolution equation for the global conformal scaling symmetry in Minkowski hyper-spacetime. Only when the hyper-stress energy-momentum tensor becomes traceless $\cT= 0$, the tensor vector becomes conserved, i.e., $\p_{\fM} \cT^{\fM} = 0$. In general, the existence of $\cQ_c$-spin scaling field $\chi_s(x)$ and its coupling to the hyperunified qubit-spinor field $\fPsi_{\fQH}(x)$ gives a nonzero tensor scalar $\cT\neq 0$ in hyperunified field theory.

Therefore, the dynamic evolution equation for the bicovariant hyper-rotation angular momentum tensor $\cL^{\fP,\fM\fN}$ is in general governed by the antisymmetric bicovariant hyper-stress energy-momentum tensor  $\cT^{[\fM\fN]}$, and the dynamic evolution equation for the hyper-conformal scaling momentum tensor $\cT^{\fM}$ is determined by the trace of bicovariant hyper-stress energy-momentum tensor $\cT$.

§.§ Dynamic evolution equation for the total bicovariant hyper-angular momentum tensor in hyperunified field theory

In globally flat four-dimensional Minkowski spacetime with the absence of gravitational interaction, the total angular momentum tensor as the sum of rotation and spin angular momentum tensors is known to be a conserved tensor. In the hyperunified field theory, the bicovariant hyper-rotation angular momentum tensor and bicovariant hyperspin angular momentum tensor are shown to obey two independent dynamic evolution equations with respective to global and local symmetries in globally flat Minkowski hyper-spacetime and locally flat gravigauge hyper-spacetime, respectively. To verify the total hyper-angular momentum conservation law when turning locally flat gravigauge hyper-spacetime to be globally flat one, it is useful to introduce the total bicovariant hyper-angular momentum tensor in a hidden locally flat gravigauge hyper-spacetime.

Let us first make a redefinition for the bicovariant hyperspin angular momentum tensor $\cS^{\fP}_{\;\; \fA\fB}$. It is appropriate to project $\cS^{\fP}_{\;\; \fA\fB}$ into the following tensor form via the dual $\cW_e$-spin invariant-gauge field $\hfA_{\fA}^{\;\;\fM}$:

^,      =     ^_   ^'^' _''^  []

=        1/2 _^    _ { _-^      _ } _^'^' _''^  [],
with $\mH_{\fM'\fN'}^{\; [\fM\fN]}$ defined in Eq.(<ref>). In terms of the above tensor form, we are able from the dynamical equation in Eq.(<ref>) to derive the following dynamic evolution equation for the bicovariant hyperspin angular momentum tensor:

_ ^,      =      - _a^[]  - _^[] + _χ_s^[]   + _c^[] ,
with $\cT_a^{[\fM\fN]}$ defined in Eq.(<ref>). Where we have introduced the following definitions:
_^[] ≡_   ''”^[]”_”^” ^,'' ,   ^,'' ≡_^^'^' ,

_    ''”^[]” ≡1/2 [  _'” (η_  '^ η^” -  η_  '^ η^”)   -  (_  '^ η_  ”^ -  _  '^ η_  ”^) η_'^  ”  ],

_”^” ≡_^   ”__”^  = _^   ”(__”^  + _^_”^  ) ,
for the antisymmetric bicovariant tensor $\cT_{\fS}^{[\fM\fN]}$ which arises from the inhomogeneous hyperspin gauge interaction of hyperunified qubit-spinor field, and
_χ_s^[] ≡η̅^[] ^””''_' g_H^-2β_G^2 β_Q^2 sinh^2χ_s   1/4_””_''^' ,

η̅^[] ≡_''^  []^'^' η_[]^,
for the antisymmetric bicovariant tensor $\cT_{\chi_s}^{[\fM\fN]}$ which comes from the $\cQ_c$-spin scaling field interacting with the hyperspin gauge field, as well as

_c^[] ≡_''^  [] ^[''],   ^[] =   _[]^^ ≡^ - ^ ,

^ ≡^ _^    = ^_ (  ^[]''_' g_H^-2_'' ^' ) ,
for the antisymmetric bicovariant tensor $\cT_{c}^{[\fM\fN]}$ which results from the antisymmetric covariant tensor $\hmT^{[\fM\fN]}$. Where $\hmT^{[\fM\fN]}$ is defined by the conserved bicovariant vector current $\mJ_{\fA}^{\; \fM}$ with $\hmT^{\fM\fN} \equiv \hfA^{\fA\fM} \mJ_{\fA}^{\; \fN}$.

In terms of the bicovariant hyper-rotation and hyperspin angular momentum tensors $\cL^{\fP,\fM\fN}$  and $\cS^{\fP,\fM\fN}$, we are able to define the total bicovariant hyper-angular momentum tensor as follows:
^,      ≡    ^,  + ^, ,
which satisfies the following dynamic evolution equation:
_ ^,      =     _s^[]  -  _^[] + _χ_s^[]   + _c^[] ,
which shows that the total hyper-angular momentum tensor is no longer homogeneously conserved due to the presence of gauge invariant gravitational interaction in hidden locally flat gravigauge hyper-spacetime. It is noticed that the source term of antisymmetric bicovariant hyper-stress energy-momentum tensor $\cT_{a}^{[\fM\fN]}$ in both $\cL^{\fP,\fM\fN}$ and $\cS^{\fP,\fM\fN}$ cancels each other due to the sign flip in their dynamic evolution equations.

To verify the conservation law of total bicovariant hyper-angular momentum tensor in globally flat Mincowski hyper-spacetime with the absence of gravitational interaction, let us turn the local inhomogeneous hyperspin gauge symmetry into global symmetry by taking gauge fields to approach the following limiting case:
_^ →0,   _^   →η_^   ,
which leads to the following relations:
_s^[],  _^[] ,   _χ_s^[] ,   _c^[] →0 ,

_ ^,  = -  _a^[],

_ ^,  =  + _a^[] .
As a consequence, we arrive at the following equation:
_ J^,      =     _ ( ^.  + ^,  )  =0 ,
which does verify the conservation law of total bicovariant hyper-angular momentum tensor in globally flat Minkowski hyper-spacetime.

Therefore, we come to the observation that for the hyperunified qubit-spinor field as fundamental building block with inhomogeneous hyperspin gauge interaction as fundamental interaction, its covariant hyper-stress energy-momentum tensor is generally asymmetric. Neither the bicovariant hyper-rotation angular momentum tensor nor the bicovariant hyperspin angular momentum tensor is homogeneously conserved due to the antisymmetric property of the covariant hyper-stress energy-momentum tensor in hyperunified field theory, they all get dynamic evolution equations which are governed by the antisymmetric bicovariant hyper-stress energy-momentum tensor. Only in the absence of inhomogeneous hyperspin gauge interaction with turning locally flat gravigauge hyper-spacetime to globally flat Minkowski hyper-spacetime, the total hyper-angular momentum tensor becomes homogeneously conserved due to the appearance of cancellation, where the bicovariant hyper-rotation angular momentum tensor and hyperspin angular momentum tensor are governed by the same antisymmetric hyper-stress energy-momentum tensor but with an opposite sign.

§  HYPERUNIFIED FIELD THEORY IN HIDDEN GAUGE FORMALISM WITH GRAVITY-GEOMETRY CORRESPONDENCE AND EINSTEIN-HILBERT TYPE ACTION WITH APPEARANCE OF RIEMANN GEOMETRY AND EMERGENT GROUP SYMMETRY GL(D$_H$,R) AS GENERALIZATION OF GAUGE INVARANCE PRINCIPLE UNDER FLOWING UNITARY GAUGE

The basic action of hyperunified field theory presented in Eq.(<ref>) is built based on the gauge invariance principle and scaling invariance hypothesis with inhomogeneous hyperspin gauge symmetry and global Poincaré-type group symmetry. The $\cW_e$-spin invariant-gauge field as hyper-gravigauge field is taken as a projection operator to define the bicovariant hyperspin angular momentum tensor given in Eq. (<ref>). In this section, we are going to define a hyper-spacetime gauge field from the hyperspin gauge field and present an equivalent action of hyperunified field theory in a hidden gauge formalism. The hyper-gravigauge field as projection operator is shown to play an essential role as Goldstone-like boson. Such a hidden gauge formalism of the action enables us to investigate more profound correlations between gravitational interaction and Riemann geometry of hyper-spacetime. We will demonstrate how the dynamics of hyper-gravigauge field can equivalently be characterized by the dynamics of emergent Riemann geometry in light of Einstein-Hilbert type action, which is governed by an emergent general linear group symmetry GL($D_h$, R) and regarded as the generalization of gauge invariance principle. So that we are able to corroborate the gravity-geometry correspondence in curved Riemannian hyper-spacetime.

§.§  Hyper-gravigauge field as Goldstone-like boson and the hyper-spacetime gauge field with gauge covariant field strength in hidden gauge formalism

The $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\;\;\fA}$ as hyper-gravigauge field is a bicovariant vector field spanned in biframe hyper-spacetime, which allows us to project the tensors in the representation of locally flat gravigauge hyper-spacetime into those in the representation of globally flat Minkowski hyper-spacetime.

Let us firstly define a hidden gauge invariant field from the hyperspin gauge field by applying for the hyper-gravigauge field as projection operator, i.e.:

_^   ≡_^    _ _^   + _^      _ ^ _^   ,
which already appears in the covariant derivative when deriving the quadratic form of equation of motion given in Eq.(<ref>) for the hyperunified qubit-spinor field, we may refer to such a gauge field $\cA_{\fM\fQ}^{\fP}$ as hyper-spacetime gauge field.

As the hyperspin gauge field $ \cA_{\fM}^{\fA\fB}$ exhibits gravitational origin of gauge symmetry by decomposing it into the sum of hyperspin gravigauge field $\fOm_{\fM}^{\fA\fB}$ and covariant-gauge field $\fA_{\fM}^{\fA\fB}$ presented in Eqs.(<ref>)-(<ref>), we should also decompose the hyper-spacetime gauge field $\cA_{\fM\fQ}^{\fP}$ into the following two parts in an analogous way:

_^   ≡_^  +  _^ ,
with the explicit forms,

_^      ≡     _^    _ _^   +  _^      _ ^ _^   ,

=     1/2^ (_ _ + _ _ - __ ) =_^,

_^     ≡     _^    _^ _^    ,
which are all local scaling gauge invariant.

The hyper-gravigauge field $\fA_{\fM}^{\;\;\fA}$ may be regarded as Goldstone-like boson field which transforms as bicovariant vector field under hyperspin gauge group and Lorentz-type group transformations. In light of the hyper-spacetime gauge field and field strength, the hyperspin gauge symmetry SP(1,$D_h$-1) is transmuted into a hidden gauge symmetry via the Goldstone-like hyper-gravigauge field $\fA_{\fM}^{\; \fA}$.

It is noticed that the symmetric tensor fields $\mH_{\fM\fN}$ and $\hmH^{\fM\fN}$ defined in Eq.(<ref>) are presented as the product of two hyper-gravigauge fields by contracting the vector indices in locally flat gravigauge hyper-spacetime. Such symmetric tensor fields are considered as composite fields with both hidden hyperspin gauge symmetry and scaling gauge symmetry, which are referred to as Goldstone-like dual hyper-gravimetric fields geometrically.

The gauge field $\fGa_{\fM\fQ}^{\fP}$ is a symmetric gauge field $\fGa_{\fM\fQ}^{\fP} = \fGa_{\fQ\fM}^{\fP}$ which is completely determined by the dual Goldstone-like hyper-gravimetric fields $\mH_{\fM\fN}$ and $\hmH^{\fM\fN}$. We may refer to such a gauge field as scaling gauge invariant hyper-spacetime gravimetric-gauge field. For convenience of mention, the gauge field $\fA_{\fM\fQ}^{\fP} $ is called as hyper-spacetime covariant-gauge field .

From the hyper-spacetime gauge field, we are able to define the corresponding field strength as follows:
_^  = ∇_ _^ - ∇_ _^ + _^ _^   - _^ _^ ,

∇_ _^ = _ _^ - _^ _^ + _^  _^ ,
which can also be decomposed into the following two parts:

_^ ≡_^ + _^ ,

_^   = _ _^ - _ _^  + _^ _^  - _^ _^,

_^   = ∇_ _^ - ∇_ _^ + _^ _^   - _^ _^  ,

∇_ _^ = _ _^ - _^ _^ + _^  _^ .
Where the tensor fields $\fR_{\fM\fN\fQ}^{\;\fP}$ and $\fF_{\fM\fN\fQ}^{\;\fP}$ defined in the hidden gauge formalism are referred to as scaling gauge invariant hyper-spacetime gravimetric-gauge field strength and hyper-spacetime covariant-gauge field strength, respectively.

In obtaining the above field strengths in a hidden gauge formalism, we have used the following identities:

∇__^   = _ _^   + _^ _^
- _^ _^   =0 ,

∇_ _^    =  _ _^
- _^ _^   + _^ _^  =0 ,

∇_ _ = _ _  - _^ _ - _^ _ = 0 ,

∇_ ^ = _ ^  + _^ ^ + _^^ = 0 ,
which enable us to verify the following relations:

_^ = _'^^' = _''^'^' =  _^ _^   _^   =  - _^,

_^ = _'^^' = _''^'^' =  _^ _^   _^   =  - _^,

_^ = _'^^' = _''^'^' =  _^ _^   _^   =  - _^,

_^ =  _^ _^   _^  = ^ ^  _ .

It is seen that the scaling gauge invariant hyper-spacetime gravimetric-gauge field strength $\fR_{\fM\fN\fQ}^{\fP}$ is described by the scaling gauge invariant Goldstone-like hyper-gravimetric field $\mH_{\fM\fN}$, which defines scaling gauge invariant Riemann-type curvature tensor. $\mH_{\fM\fN}$ is regarded as a basic field for hyper-spacetime gravimetric-gauge interaction in the hidden gauge formalism.

§.§ Hyperunified field theory in hidden gauge formalism and Einstein-Hilbert type action on gravitational interaction of hyper-spacetime gravimetric-gauge field

To formulate the basic action of hyperunified field theory in the hidden gauge formalism, it is useful to corroborate the following identity:

1/4   ^''_' _^_''^'  + 2 _ (^ _^  _^ ) =   ,
where the second term on the left-hand side is a total derivative. The scalar tensor $\fR$ on the right-hand side defines scaling gauge invariant Ricci-type scalar curvature in hyper-spacetime. It is useful to notice the following general relations for the hyper-spacetime gravimetric-gauge field:
_^ = η_  ^_^ = _ln,   _^ ≡^_^ = -  _ (  ^) .

In general, the scaling gauge invariant Ricci-type curvature tensor can uniquely be resulted from contracting the scaling gauge invariant Riemann-type curvature tensor,
_ ≡_^ η^_   = - _^ η^_   = - _^ ,
which is a symmetric tensor with the following explicit form:

_ = _ = ∇_ _ln- _ _^  + _^ _^ .
The scaling gauge invariant Ricci-type scalar curvature in hyper-spacetime is obtained from contracting the Ricci-type curvature tensor,

= ^ _ = ^  _^ = -  ^ _ ^ .

It is interesting to notice the following identity:

1/4 ^''_' _^_''^'     =     - 1/2 ^' ( _^ - _^ )(_'^ - _'^) ,
which brings on the quadratic interaction of hyper-spacetime gauge field.

Applying the identities and relations in Eqs.(<ref>)-(<ref>), we are able to rewrite the basic action in Eq.(<ref>) into the following form formulated in the hidden gauge formalism:

_     ≡    ∫[d^D_hx]   (x)  = ∫[d^D_hx]   (x)

·    {  _ _-^  i_  _ -  β_Q sinhχ_s   _ _-_

+      ^' ^' 1/4 g_H^-2 _^  _''^  +  g_H^-2

-     1/2 g_H^-2β_G^2β_Q^2 sinh^2χ_s    ^' ( _^ - _^ )(_'^ - _'^)

-     ^' ^' 1/4 g_W^-2 _ _'' + 1/2 λ_S^2 (1 +sinh^2χ_s ) ^'  _ _'

+      1/2 ^ λ_S^2 _ χ_s _χ_s  - λ_D^2 (χ_s)    } ,
where we have ignored the surface term $2 \p_{\fM} (\fkA \hmH^{\fM\fP} \hfA_{\fA}^{\;\fQ} \mF_{\fP\fQ}^{\fA} )$. The following definitions in the hidden gauge formalism have implicitly adopted:
i_ ≡i_ + (_[] +  _[])1/2^,

_[] ≡1/3 (_ + _  + _ ) ,

_[] ≡1/3 (_ + _  + _ )

≡1/3 ( _ _^    _ + _ _^    _  + _ _^    _) ,

_ ≡_ _^    _^    ,  _  ≡_'_^',

_-^ ≡^ _- ,   ^ ≡_^   ^,

^ ≡_^   _^   ^ = i/4 [^, ^ ] ,
where $\fOm_{[\fM\fP\fQ]}$ and $\cA_{[\fM\fP\fQ]}$ are total antisymmetric fields since the hyperunified qubit-spinor field couples only to totally antisymmetric gauge fields due to its self-conjugated chiral property. $\fOm_{\fM\fP\fQ}$ is referred to as hyper-spacetime gravigauge field which characterizes the gravitational origin of gauge symmetry. It is noticed that the above defined $\vGa$-matrices are no longer constant matrices, the matrices $\hfGa^{\fM}$ are regarded as hyper-gravigauge $\Gamma$-matrices and $\hfSi^{\fP\fQ}$ together with $\hfSi_{-}^{\fM}$ are viewed as hyper-gravigauge inhomogeneous hyperspin group generators.

In the action shown in Eq.(<ref>), the Ricci-type scalar curvature $\fR$ brings about the scaling gauge invariant Einstein-Hilbert type action which describes the gravitational interaction as dynamics of scaling gauge invariant hyper-gravimetric field $\mH_{\fM\fN}$.

§.§  Hyperunified field theory in hidden gauge formalism with emergent general linear group symmetry GL($D_h$,R) as generalization of gauge invariance principle and the appearance of Riemann geometry in curved hyper-spacetime

The action in hidden gauge formalism shown in Eq.(<ref>) is obtained from the action in Eq.(<ref>) via projecting locally flat gravigauge hyper-spacetime into globally flat free-motion hyper-spacetime $\fM_h$ by utilizing the hyper-gravigauge field $\fA_{\fM}^{\;\fA}(x)$ ($ \hfA_{\fA}^{\;\fM}(x)$) as Goldstone-like boson. The action in Eq.(<ref>) is built based on the gauge invariance principle and scaling invariance hypothesis within the framework of quantum field theory, so that it possesses global Poincaré-type group symmetry. In general, the laws of nature should be independent of the choice of coordinate systems, which indicates that the action should hold in any frame of coordinates. To explicitly verify that, let us turn to the geometric analysis.

Geometrically, when considering the hyperspin gauge and scaling gauge invariant tensor field $\mH_{\fM\fN}(x)$ as hyper-gravimetric field of hyper-spacetime, the hyper-spacetime gravimetric-gauge field $\fGa_{\fM\fQ}^{\fP}(x)$ defined in Eq.(<ref>) is regarded as scaling gauge invariant Levi-Civita hyper-connection or scaling gauge invariant Christoffel symbols of the second kind in hyper-spacetime. By lowering the index of the second kind Christoffel symbols $\fGa_{\fM\fQ}^{\fP}(x)$ via the hyper-gravimetric field $\mH_{\fM\fN}(x)$, we obtain scaling gauge invariant Christoffel symbols of the first kind as follows:

_     =     __^  = 1/2 (  _ _ + _ _ - __   )

=     _ ≡_[]  + 1/2 _ _ .
Similarly, by raising the index of the second kind Christoffel symbols $\fGa_{\fM\fQ}^{\fP}(x)$ via the dual hyper-gravimetric field $\hmH_{\fM\fN}(x)$, we obtain the following Christoffel symbols:

^     =     ^'_''^ ^' = 1/2 (  ^ ^ - ^ ^ - ^ ^   )

=     ^  ≡^[]  - 1/2 ^ ^ .
Where the antisymmetric part of hyper-spacetime gravimetric-gauge field can simply be expressed as follows:
_[] = - 1/2 ( _ _ - _ _ ) ,

^[]= 1/2 ( ^ ^ - ^ ^ ),    ^≡^ _ .

Christoffel symbols in hyper-spacetime introduces the principal connection of general linear group symmetry GL(1,$D_h$-1, R), which inspires us to generalize global inhomogeneous Lorentz-type transformation of Poincaré-type group symmetry PO(1,$D_h$-1) in globally flat Minkowski hyper-spacetime to general coordinate transformation of general linear group symmetry GL(1,$D_h$-1, R). The general coordinate transformation is defined as an arbitrary reparametrization of coordinate systems, i.e.:

x^ →x^' ≡x^'(x),
which presents the local transformation for describing a distinct reparametrization at every point in curved Riemannian hyper-spacetime. Such an arbitrary reparametrization leads the displacement and derivative of coordinate systems to obey the following transformation laws:

dx^     →     dx^'= T^_       dx^  ,   T^_       ≡x^'/x^ ,

_     →    '_ = T_^     _   ,  T_^    ≡x^/x^' ,
which keeps the following inner product:
⟨'_, dx^'' ⟩= T_^    T^'_     '
⟨_, dx^' ⟩= T_^    T^'_     ' η_^   ' = η_^   ' = ⟨_, dx^' ⟩,
where $\p_{\fM}$ is regarded as covariant vector and $dx^{\fM}$ as contravariant vector.

With the above transformation laws, any covariant vector field $\cA_{\fM}(x)$ with a lower index and contravariant vector field $\hcA^{\fM}(x)$ with an upper index will get the following general transformation properties:

'_(x') = T_^     _(x),   ^'(x') = T^_       ^(x),
which leads the scalar product of covariant vector field with contravariant vector field to be invariant, i.e.:
_(x) ^(x) = '_(x') ^'(x') .

The hyper-gravigauge field $\fA_{\fM}^{\;\fA}(x)$ and its dual vector $\hfA_{\fA}^{\; \fM}(x)$ correspond to the covariant and contravariant vector fields, respectively, in hyper-spacetime. The hyper-gravimetric field $\mH_{\fM\fN}(x)$ defined  by the covariant vector field $\fA_{\fM}^{\;\fA}(x)$ is a covariant tensor field with the following transformation law:
'_(x')  = x^/x^' x^/x^'    _(x) =  T_^     T_^     _(x).
Similarly, the dual hyper-gravimetric field $\hmH^{\fM\fN}(x)$ defined  by the contravariant vector field $\hfA_{\fA}^{\;\fM}(x)$ is a contravariant tensor with the corresponding transformation law. It can be checked that the Christoffel symbols transform as follows:
_^' (x')     =      ( x^'/x^'x^'/x^' ) (  x^'/x^' _''^' (x) - ^2 x^'/x^'x^'   )

≡    T_^   '  T_^   ' T^_     '  (  _''^' (x) -  T^'_     ' _' T^'_     '  ) ,
which behaves as local gauge-type transformation of general linear group symmetry GL(1,$D_h$-1, R) with appearance of inhomogeneous transformation term.

The covariant derivatives for covariant vector field $\cA_{\fM}(x)$ and contravariant vector field $\hcA^{\fM}(x)$ obey the general covariance under general coordinate transformations, i.e.:

(∇__)^' =  ( x^/x^'x^/x^' ) (∇__) ≡T_^     T_^    (∇__) ,

(∇_^)^' =   ( x^/x^')  ( x^'/x^ ) (∇_^) ≡T_^     T^_     (∇_^)  ,
with the definition of covariant derivatives as follows:
∇__ = _ _ - _^ _  ,

∇_^ = _ ^ + _^ ^  .

Therefore, the scaling gauge invariant hyper-spacetime gravimetric-gauge field strength $\fR_{\fM\fN\fQ}^{\;\fP}$ defines Riemann curvature tensor which characterizes curved Riemannian hyper-spacetime with the emergence of Riemann geometry. When lowering indices with the hyper-gravimetric field, we can express the Riemann curvature tensor as follows:
_     ≡    _^ ' _'

=     1/2 (__ _ -  __ _ + __ _ - __ _  )

+     ^'( __'-  __' ).
which has the following symmetry properties as both gauge field strength and Riemann curvature tensor:

_ = -_ ,   _ = -_,

_ = _,   _ = _ ,
and satisfies two kind of Bianchi identities:

∇__^  + ∇__^  + ∇__^  = 0 ,

_ + _ + _ = 0 .

It is straightforward to verify that the action in Eq.(<ref>) becomes invariant under the general coordinate transformation defined as an arbitrary reparametrization in coordinate system presented in Eqs.(<ref>) and (<ref>). The verification can easily be carried out from the property that the transformation laws of tensors and Christoffel symbols as well as covariant derivatives lead the scalar product between covariant tensor and contravariant tensor to be invariant under general coordinate transformation.

It can be proved that the action in Eq.(<ref>) brings on the following emergent general linear group symmetry:
G_S = 1,D_h-1,  ,
which is thought to govern gravitational interaction in curved Riemannian hyper-spacetime with emergence of Riemann geometry. Such a group symmetry appears actually as a hidden local symmetry in the action built by following along the gauge invariance principle and scaling invariance hypothesis in biframe hyper-spacetime as shown in Eq.(<ref>)

In general, the hyperunified field theory formulated in hidden gauge formalism should possess the following maximal joint symmetry:

G_S = 1,D_h-1,  1, D_h-1),
which brings on the extension of global Poincaré-type group symmetry PO(1,$D_h$) in globally flat Minkowski hyper-spacetime to a general linear group symmetry GL(1,$D_h$-1,R) in curved Riemannian hyper-spacetime. The inhomogeneous hyperspin gauge symmetry WS(1,$D_h$-1) still holds for the hyperunified qubit-spinor field, while it appears as a hidden gauge symmetry in the action concerning purely bosonic interactions of hyper-spacetime gauge field and scalar field.