Therefore, we come to the statement that the laws of nature should be independent of the choice of coordinate systems, which is indicated from emergent general linear group symmetry GL(1,$D_h$-1,R) which brings on the emergence of Riemann geometry in curved Riemannian hyper-spacetim.  So that we achieve a generalization of gauge invariance principle that the gauge invariance principle is not only essential to the fundamental symmetry of basic fields in Hilbert space but also applicable to the fundamental symmetry of coordinates in Minkowski spacetime. In this sense, we may refer to such an extension of gauge invariance principle as generalized gauge invariance principle. It is intriguing to notice that when the fundamental symmetry of basic fields in Hilbert space is gauged to be local symmetry by following along the gauge invariance principle together with scaling invariance hypothesis, the fundamental symmetry of coordinates in Minkowski spacetime automatically obeys generalized gauge invariance principle with the emergence of general linear group symmetry GL(1,$D_h$-1,R) and genesis of Riemann geometry in curved Riemannian hyper-spacetime. Such a feature implies a gravity-geometry correspondence which will be discussed below.

§.§ Flowing unitary gauge with symmetric Goldstone-like hyper-gravigauge field and the gravity-geometry correspondence in hyper-spacetime

In the action formulated in hidden gauge formalism shown in Eq.(<ref>), the hyperspin gauge symmetry SP(1,$D_h$-1) for the hyperunified qubit-spinor field $\fPsi_{\fQH}$ is preserved by the scaling gauge invariant totally antisymmetric hyper-spacetime gravigauge field $\fOm_{[\fM\fP\fQ]}$ which is characterized by the Goldstone-like hyper-gravigauge field $\fA_{\fM}^{\;\; \fA}$. The hyper-spacetime gravimetric-gauge field $ \fGa_{\fM\fN}^{\fP}$ as Christoffel symbols decouples from the hyperunified qubit-spinor field due to the self-conjugated chiral spinor representation of hyperunified qubit-spinor field and the symmetric property of Christoffel symbols $\fGa_{\fM\fN}^{\fP} =\fGa_{\fN\fM}^{\fP}$.  To see that explicitly, let us rewrite the action in Eq.(<ref>) into the following form:

_ ≡∫[d^D_hx]   (x)  = ∫[d^D_hx]   (x)

·{ _ _-^i_ _ -  β_Q sinhχ_s _ _-_

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

- ^'  ^' g_H^-21/4 _^  _'' +  1/2 g_H^-2β_G^2 β_Q^2 sinh^2χ_s ^' _^ _'

+ g_H^-2 - ^'  ^' 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)    }  + 2__ (  ^) ,
where we have used the decomposition formulation for the hyper-spacetime gauge field strengths given in Eqs.(<ref>) and (<ref>). The covariant derivative is given as follows:
i_ ≡i_ + (_[] +  _[])1/2^ .
where $\fA_{[\fM\fP\fQ]}$ is a totally antisymmetric hyper-spacetime covariant-gauge field instead of totally antisymmetric hyper-spacetime gauge field $\cA_{[\fM\fP\fQ]} $.

It is noticed that in the gravitational interaction of hyperunified qubit-spinor field, the hyper-spacetime gravimetric-gauge field $\fGa_{\fM\fP\fQ}$($\fGa_{\fM\fQ}^{\fP}$) decouples due to its symmetric property and the self-conjugated chiral feature of hyperunified qubit-spinor field. Only the totally antisymmetric hyper-spacetime covariant-gauge field $\fA_{[\fM\fP\fQ]}$ and totally antisymmetric hyper-spacetime gravigauge field $\fOm_{[\fM\fP\fQ]}$ interact with the hyperunified qubit-spinor field. Note that the hyper-spacetime gravimetric-gauge field $\fGa_{\fM\fP\fQ}$ characterized by the hyper-gravimetric field reflects the geometric property of gravitational interaction, while $\fOm_{[\fM\fP\fQ]}$ is actually characterized by the hyper-gravigauge field rather than hyper-gravimetric field. Therefore, the gravitational interaction of hyperunified qubit-spinor field is essentially governed by the Goldstone-like hyper-gravigauge field $\fA_{\fM}^{\;\;\fA}$ ($\hfA_{\fA}^{\;\; \fM}$) through the hyper-spacetime gravigauge field $\fOm_{[\fM\fP\fQ]}$ and hyper-spacetime group generators $\hfSi_{-}^{\fM}$ and $\hfSi^{\fP\fQ}$ of inhomogeneous hyperspin gauge symmetry WS(1,$D_h$-1).

On the other hand, the bosonic hyper-spacetime gauge interaction is described by antisymmetric hyper-spacetime covariant gauge field $\fA_{\fM}^{ \fP \fQ}$ and symmetric hyper-spacetime gravimetric-gauge field $\fGa_{\fM\fN}^{\fP}$ as Christoffel symbols, which is governed by the emergent general linear group symmetry GL(1,$D_h$-1,R). Namely, the hyperspin gauge symmetry is completely transmuted into a hidden gauge symmetry in the bosonic interactions. As $\fGa_{\fM\fN}^{\fP}$ is characterized by the symmetric Goldstone-like hyper-gravimetric field $\mH_{\fM\fN}$, it concerns $N_{\mH} =D_h(D_h+1)/2$ degrees of freedom.

As the fundamental gravitational field is essentially characterized by the Goldstone-like hyper-gravigauge field $\fA_{\fM}^{\;\;\fA}$ which has $N_{\fA} = D_h\times D_h$ degrees of freedom. In comparison to the symmetric hyper-gravimetric field $\mH_{\fM\fN}$, the hyper-gravigauge field $\fA_{\fM}^{\;\;\fA}$ contains extra degrees of freedom given by $N_E = D_h^2 - D_h(D_h+1)/2=D_h(D_h -1)/2$, which exactly reflects the equivalence classes of hyperspin gauge symmetry SP(1, $D_h$-1) and has the same numbers as those of group parameters of SP(1, $D_h$-1), i.e., $N_G=D_h(D_h -1)/2=N_E$.

In general, the presence of gauge symmetry in the action involves redundant degrees of freedom. In order to eliminate the redundant degrees of freedom arising from the gauge symmetry, it is necessary to make a gauge prescription with imposing gauge fixing condition. For the hyperspin gauge symmetry which is unlike usual internal gauge symmetry as it has a gravitational origin characterized by the $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\;\;\fA}$ as hyper-gravigauge field, we are able to realize a simple gauge prescription. That is just to take an appropriate hyperspin gauge transformation $\bar{\Lambda}_{\;\, \fA}^{\fB}(x)$ to transmute the Goldstone-like hyper-gravigauge field into symmetric one, i.e.:
_(x) →_(x) = _(x) Λ̅^_    (x)
=  _(x) ,

^(x) →^(x) = ^(x) Λ̅_^   (x)
=  ^(x) ,
which sets an explicit gauge prescription as a gauge fixing condition. We may refer to such a gauge prescription as flowing unitary gauge, which holds locally with a chosen coordinate system at point to point in hyper-spacetime. A detailed discussion on such a flowing unitary gauge will be presented later on.

When fixing the hyperspin gauge symmetry with flowing unitary gauge, both Goldstone-like hyper-gravigauge field $\bfA_{\fM\fA}(x)$ and hyper-gravimetric field $\mH_{\fM\fN}(x)$ are symmetric fields, they involve exactly the same degrees of freedom. Explicitly, they are correlated determinately each other with the following relation:
_ = _ _ η^ = _η^ _  ≡()^2_    .
By adopting a nonlinearly realized exponential representation, we can express them as matrices with the following forms:
_ = ϕ(e^G/)_  ,   G_ = G_ ,

= ^2 = ϕ^2 e^G/ e^G/ = ϕ^2 e^2G/ ,
where the matrix $G_{\fM\fA}$ is regarded as nonlinearly realized symmetric Goldstone-like hyper-gravigauge field and $\Mka$ as basic mass scale.

In such a flowing unitary gauge, both hyper-spacetime gravigauge field $\fOm_{[\fM\fP\fQ]}$ and hyper-gravigauge $\Gamma$-matrices $\fGa^{\fM}$ are described by the symmetric Goldstone-like hyper-gravigauge field $\bfA_{\fM\fA}$ and its dual field $\hbfA^{\fA\fM}$,
_[] →_ ≡1/3 ( _ _^    _ + _ _^    _  + _ _^    _),

_-^ →_-^ ≡^ _-,   ^ ≡^ _ = ^ _

^ →^ ≡i/4 [^, ^ ] = ^ ^ _ = ^ ^ _ .

In the hyperunified field theory formulated in a hidden gauge formalism, when setting the gauge fixing condition to be flowing unitary gauge, both hyper-spacetime gravigauge field $\bar{\fOm}_{\fM\fP\fQ}$ and hyper-spacetime gravimetric field $\fGa_{\fM\fQ}^{\fP}$ are basically characterized by the symmetric Goldstone-like hyper-gravigauge field $\bfA_{\fM\fA}(x)= \bfA_{\fA\fM}(x)$ (or nonlinearly realized symmetric Goldstone-like hyper-gravigauge field $G_{\fM\fA}(x)=G_{\fA\fM}(x)$). Therefore, the independent degrees of freedom concerning the hyper-spacetime gauge interaction are represented by the symmetric Goldstone-like hyper-gravigauge field $\bfA_{\fM\fA}(x)= \bfA_{\fA\fM}(x)$ (or equivalently the symmetric Goldstone-like hyper-gravimetric field $\mH_{\fM\fN}(x)= (\bfA)^2_{\; \fM\fN}(x)$) and antisymmetric hyper-spacetime covariant-gauge field $\fA_{\fM}^{\fP\fQ}(x)$. Nevertheless, the total independent degrees of freedom in the hyperunified field theory should remain unchanged as the extra degrees of freedom in a general Goldstone-like hyper-gravigauge field $\fA_{\fM}^{\;\;\fA}(x)$ are considered to be absorbed into the antisymmetric hyper-spacetime covariant-gauge field $\fA_{\fM}^{\fP\fQ}(x)$, which behaves as a massive-like gauge field with $\cQ_c$-spin scaling field regarded as mass-like term shown from the scalar interaction term $\beta_{G}$ in the action given in Eq.(<ref>).

As the symmetry groups GL(1,$D_h$-1, R) and WS(1,$D_h$-1) in Eq.(<ref>) cannot act as a direct product group, the gauge fixing condition is actually a local one and holds only for a given reparametrization of coordinate systems in hyper-spacetime. To make such a gauge fixing condition valid at any reparametrization of coordinate systems in hyper-spacetime, the gauge fixing operation must always associate with any given reparametrization of coordinate systems and proceed with an appropriate hyperspin gauge transformation at any given reparametrization so as to preserve symmetric hyper-gravigauge field. Such a gauge fixing condition is regarded as flowing unitary gauge in the gravigauge hyper-fiber bundle structure when carrying out general transformations under symmetry groups GL($D_h$, R) and WS(1,$D_h$-1), which makes the gauge fixing procedure to run with a flowing gauge fixing condition, so that it ensures the symmetric property of hyper-gravigauge field at arbitrary point of coordinate systems in hyper-spacetime. Namely, for every general coordinate transformation defined as an arbitrary reparametrization of coordinate systems, $x' \equiv x'(x)$, one can always perform an appropriate hyperspin gauge transformation associated with a distinct local reparametrization at every point in hyper-spacetime.

Such a gauge fixing procedure can always be realized with two-step transformations. Namely, carrying out a general coordinate transformation as follows:
dx^ →dx^' = T^_        dx^  ,   _ →'_ =  T_^    _ ,

T_^    ,   T^_       ∈GL(1, D_h-1, R)
as the first step, and then performing an appropriate hyperspin gauge transformation at point $x' \equiv x'(x)$,
^ →S'(Λ̅') ^ S^'-1(Λ̅') =  Λ̅^'_    (x') ^  ,

Λ̅^'_    (x') ∈SP(1,D_h1) ,
as the second step, so that we arrive at a symmetric hyper-gravigauge field at every point in hyper-spacetime, i.e.:
_(x) = _(x) →'_(x') = T_^      _(x) ≠'_(x') ,

^”_(x') = Λ̅_^'  (x')  '_(x') = ^”_(x') .

For the gravitational interaction, the flowing unitary gauge always enables us to take symmetric Goldstone-like hyper-gravigauge field as fundamental gravitational field at every point in hyper-spacetime. The introduction of flowing unitary gauge allows us to regard all points in hyper-spacetime to be equivalent when taking gauge fixing condition to be a unitary gauge.

To keep Goldstone-like hyper-gravigauge field $\bfA_{\fM\fA}(x)$ be always symmetric under flowing unitary gauge for a chosen coordinate system, the action is turned out to possess only an associated symmetry in which global hyperspin symmetry SP(1,$D_h$-1) of hyperunified qubit-spinor field must correlate with global Lorentz-type group symmetry SO(1,$D_h$-1) of coordinates, so that their group transformations have to coincide with each other as indicated from their isomorphic property SP(1,$D_h$-1)$\cong$SO(1,$D_h$-1).

Therefore, when the hyperunified field theory is formulated in light of hidden gauge formalism and presented under flowing unitary gauge, the gauge interactions for both hyperunified qubit-spinor field and bosonic fields are described by the symmetric Goldstone-like hyper-gravigauge field $\bfA_{\fM\fA}(x)$ and antisymmetric hyper-spacetime covariant-gauge field $\fA_{\fM}^{\fP\fQ}(x)$. The hyper-spacetime gravimetric-gauge field $\fGa_{\fM\fN}^{\fP}(x)$ as Christoffel symbols characterized by hyper-gravimetric field is completely determined by $\bfA_{\fM\fA}(x)$ via $\mH_{\fM\fN}(x) = \bfA^2_{\;\fM\fN}(x)$. Therefore, the geometry and dynamics of hyper-spacetime are essentially governed by symmetric Goldstone-like hyper-gravigauge field $\bfA_{\fM\fA}(x)$, which leads to the gravity-geometry correspondence in hyper-spacetime.

Therefore, for a chosen gauge fixing condition under flowing unitary gauge with the symmetric hyper-gravigauge field $\bfA_{\fM\fA}(x)=\bfA_{\fA\fM}(x)$, the action of hyperunified field theory in hidden gauge formalism given in Eq.(<ref>) possesses the following associated symmetry with global inhomogeneous hyperspin symmetry WS(1,$D_h$-1) in association with global Poincaré-type group symmetry PO(1,$D_h$-1), i.e.:
G_S = SC(1)⋉PO(1,D_h-1) WS(1,D_h-1) , which reproduces the fundamental symmetry in free-motion Minkowski hyper-spacetime, where the transformations of global Lorentz-type group symmetry SO(1,$D_h$-1) and hyperspin symmetry SP(1,$D_h$-1) have to be coincidental as indicated by the symbol `$\wtjoin$'.

As the flowing unitary gauge is valid locally at point by point, to fix the flowing property of general linear group symmetry GL(1,$D_h$-1, R), it remains to impose an additional gauge fixing condition to achieve a whole unitary gauge, which will be discussed later on.

§  HYPERUNIFIED FIELD THEORY IN HIDDEN COORDINATE FORMALISM WITH GEOMETRY-GAUGE CORRESPONDENCE AND THE DYNAMICS OF FUNDAMENTAL FIELDS WITH HYPERSPIN GRAVIGAUGE FIELD AS AUXILIARY FIELD AND STRUCTURE FACTOR OF NON-COMMUTATIVE GEOMETRY IN LOCALLY FLAT GRAVIGAUGE HYPER-SPACETIME

The hyperunified field theory is shown to be independent of the choice of coordinate systems and hold in any coordinate systems so as to bring about the generalized gauge invariance principle, which inspires us to reformulate the hyperunified field theory in light of hidden coordinate formalism. We will apply for again the scaling gauge invariant hyper-gravigauge field as Goldstone-like boson to represent the hyperunified field theory in locally flat gravigauge hyper-spacetime, where the hyper-spacetime of coordinates becomes a hidden hyper-spacetime. It is intriguing to explore the nature of gravitational interaction in locally flat gravigauge hyper-spacetime and corroborate the geometry-gauge correspondence of gravitational interaction in hidden coordinate formalism. Relative to the gravity-geometry correspondence shown in hidden gauge formalism in which the gravitational interaction can equivalently be characterized by the dynamics of Riemann geometry in curved Riemannian hyper-spacetime, we will demonstrate that the geometry of locally flat gravigauge hyper-spacetime emerges as non-commutative geometry which is characterized by non-Abelian Lie algebra structure via non-commutative relation of hyper-gravicoordinate derivative operator. It is unlike globally flat free-motion hyper-spacetime described by Minkowski hyper-spacetime with commutative geometry, the gravitational interaction in locally flat gravigauge hyper-spacetime appears as an emergent interaction which is characterized through non-commutative geometry. In fact, we will verify that the hyperspin gravigauge field in locally flat gravigauge hyper-spacetime behaves as an auxiliary field and determines completely the structure factor of non-Abelian Lie algebra for characterizing non-commutative geometry. So that the dynamics of all fundamental fields is associated with gravitational interaction via emergent non-commutative geometry in locally flat gravigauge hyper-spacetime.

§.§ Dimensionless hyperspin gauge field and field strength in hidden coordinate formalism with zero global and local scaling charges

Taking the scaling gauge invariant hyper-gravigauge field as Goldstone-like boson, it is straightforward to project the hyperspin gauge field $\cA_{\fM}^{\fA\fB}$ as vector field in free-motion hyper-spacetime $\fM_h$ into the vector field in locally flat gravigauge hyper-spacetime $\fG_h$,

_^ ≡_^    _^ .
The gravitational origin of hyperspin gauge symmetry shown in Eq.(<ref>) enables us to express the above hyperspin gauge field into the following two parts:

_^ =  _^ + _^ ,
where $\fOm_{\fC}^{\fA\fB}$ and $\fA_{\fC}^{\fA\fB}$ define the dimensionless hyperspin gravigauge field and hyperspin covariant-gauge field, respectively, in locally flat gravigauge hyper-spacetime. $\fOm_{\fC}^{\fA\fB}$ reveals the gravitational origin of gauge symmetry, which is explicitly given by dual hyper-gravigauge fields $\fA_{\fM}^{\;\, \fA}$ and $\hfA_{\fA}^{\;\,\fM}$ as follows:

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

+     ( ^^ -  ^^ ) _],
where $\heth_{\fC} \equiv \hfA_{\fC}^{\;\, \fM} \p_{\fM}$ is the dimensionless hyper-gravicoordinate derivative operator defined in locally flat gravigauge hyper-spacetime.

Correspondingly, the dimensionless field strength of hyperspin gauge field in hidden coordinate formalism can be expressed into the following two parts:
_^ = _^ + _^  ,
with the explicit forms given as follows:

_^ = _ _^ - _ _^ + ( _'^ _^'  -  _'^ _^'  )  ,

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

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

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

_^ = _ _^ - _ _^  +  _'^ _^'  -  _'^ _^'

+ _'^ _^'  + _'^ _^'  -  _'^ _^'  - _'^ _^',

_  _^ = _  _^  -  _^' _'^ ,
where $\fR_{\fC\fD}^{\fA\fB}$ and $\fF_{\fC\fD}^{\fA\fB}$ are the dimensionless hyperspin gravigauge field strength and hyperspin covariant-gauge field strength, respectively. $\ckcD_{\fC}$ defines the dimensionless covariant derivative of hyper-gravicoordinate derivative $\heth_{\fC}$ under the presence of hyperspin gravigauge field in hidden coordinate formalism. For convenience, $\ckcD_{\fC}$ is mentioned as dimensionless hyper-gravigauge covariant derivative.

Similarly, the dimensionless scaling gauge field strength is defined as follows:

_ = __ - __  ,   _ = _^    _ ,

__ = __ - _^' _' .

The hyperspin gravigauge field strength $\fR_{\fC\fD}^{\fA\fB}$ defines Riemann-like tensor in locally flat gravigauge hyper-spacetime, which can be verified from the relation between the field strength of hyperspin gravigauge field defined in hidden coordinate formalism and that defined in hidden gauge formalism, i.e.:
_ = ^ _ ^ _  _ =  ^ _ ^ _ _ ^ _ ^ _ ,
which enables us to define the symmetric Ricci-like curvature tensor by contracting the Riemann-like tensor in locally flat gravigauge hyper-spacetime,

_ = _^ η_^   = - _^ η_^   = - _^ = _ ,
and also the Ricci-like scalar curvature given by the trace of Ricci-like curvature tensor,

≡η^ _ = η^   _^ η_^    = - η^   _η^  .

From the symmetry properties and identities of Riemann tensors shown in Eqs.(<ref>) and (<ref>), we arrive at the following analogous relations:

_ = -_  ,   _ = -_ ,

_ = _ ,   _ = _ ,
for the symmetry properties, and
_ + _ + _ = 0  ,

__^ + __^ + __^ = 0 ,
for the Bianchi identities.

In general, we have the following relations between the field strength defined in locally flat gravigauge hyper-spacetime and that in biframe hyper-spacetime with gravigauge hyper-fiber bundle structure:
_^ = ^ _ ^ _ _^  ;   _ =  ^ _ ^ _ _  ,

_^ =  ^ _ ^ _ _^ ;   _ = ^ _ ^ _ _   ,

_^  = ^ _ ^ _ _^ ;   _ = ^ _ ^ _ _ .
By lowering and raising the indices with the constant metric matrices $\eta^{\fA\fB}$ and $\eta_{\fA\fB}$, we can define the following gauge field and field strength in the hidden coordinate formalism:

^ ≡η^' _'^ ≡η^' η^' η^'_''' ,

^ ≡η^' η^'_''^ ≡η^' η^' η^' η^' _'''' .

It is interesting to note that all fields defined in the hidden coordinate formalism become dimensionless with zero global and local scaling charges $\cC_s=0$ and $\check{\cC}_s=0$.

§.§  Hyperunified field theory in hidden coordinate formalism with geometry-gauge correspondence and the hyperspin gravigauge field as auxiliary field in locally flat gravigauge hyper-spacetime

From the above analysis, we are able to represent the action of hyperunified field theory in hidden coordinate formalism as follows:

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

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

-     1/4 g_H^-2 _  ^ + g_H^-2 η^ _

+     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)    }
with the definitions:
i_≡i_ + _1/2^ ≡_^  i_,

[δ^D_h ] ≡[d^D_hx]   (x) ,   δ^(x) = _^   dx^,
where $[\delta^{D_h} \vka]$ is regarded as the integral measure in locally flat gravigauge hyper-spacetime, and $\delta \vka^{\fA}(x)$ is considered as the hyper-gravicoordinate displacement of hyper-gravicoordinate vector field $\vka^{\fA}(x)$ in locally flat gravigauge hyper-spacetime. $\vka^{\fA}(x)$ is also called as dimensionless hyper-gravivector field for short.

The action in hidden coordinate formalism shown in Eq.(<ref>) gets a similar form as the action in hidden gauge formalism presented in Eq.(<ref>). Unlike the action in Eq.(<ref>), the action in Eq.(<ref>) does not explicitly involve the hyper-gravigauge field $\hfA_{\fA}^{\;\fM}$ and hyper-gravimetric field $\hmH^{\fM\fN}$, all Latin alphabet indices for vectors and tensors are raised and lowered by the constant metric matrices $\eta^{\fA\fB}$ or $\eta_{\fA\fB} =\diag.(1,-1,\ldots,-1)$ in locally flat gravigauge hyper-spacetime. It is interesting to note that all basic fields defined in hidden coordinate formalism become dimensionless and have zero global and local scaling charges $\cC_s=0$ and $\check{\cC}_s=0$.

The fundamental gauge interactions in hidden coordinate formalism are governed by the hyperspin gauge field $\cA_{\fC}^{\fA\fB}$ and hyperspin gravigauge field $\fOm_{\fC}^{\fA\fB}$. In comparison to the action in hidden gauge formalism given in Eq.(<ref>), the dimensionless hyperspin gravigauge field $\fOm_{\fC}^{\fA\fB}$ in hidden coordinate formalism replaces both hyper-gravigauge field $\fA_{\fM}^{\;\fA}$ and hyper-gravimetric field $\mH_{\fM\fN}$ to characterize the gravitational interaction in locally flat gravigauge hyper-spacetime. The dynamics of $\fOm_{\fC}^{\fA\fB}$ is characterized by the Ricci-like scalar tensor $\fR \equiv \eta^{\fC\fB} \fR_{\fC\fB} = \eta^{\fC\fB}  \fR_{\fC\fD\fA\fB}\eta^{\fA\fD}$, which is analogous to the scaling gauge invariant Einstein-Hilbert type action in hidden gauge formalism. Nevertheless, they have distinguishable features that the Ricci-like scalar tensor $\fR \equiv \eta^{\fC\fB}  \fR_{\fC\fD\fA\fB}\eta^{\fD\fA}$ in hidden coordinate formalism is governed by hyperspin gauge symmetry SP(1,$D_h$-1) via hyperspin gravigauge field $\fOm_{\fC}^{\fA\fB}$, whereas the Ricci-like scalar tensor $\fR \equiv \hmH^{\fM\fQ}  \fR_{\fM\fN\fP\fQ}\hmH^{\fN\fP}$ in hidden gauge formalism is governed by general linear group symmetry GL(1,$D_h$-1) via hyper-spacetime gravimetric-gauge field $\fGa_{\fM\fQ}^{\fP}$ as Christoffel symbols in curved Riemannian hyper-spacetime.

In such a hidden coordinate formalism, the fundamental gravitational interaction is characterized by the gauge interaction through the hyperspin gravigauge field $\fOm_{\fC}^{\fA\fB}$. Therefore, when comparing the action in hidden coordinate formalism in Eq.(<ref>) with that in hidden gauge formalism in Eq.(<ref>), we come to the geometry-gauge correspondence.

To show explicitly such a geometry-gauge correspondence, it is useful to corroborate the following relation:
≡      ^  _ =   ^  _^

≡       η^ _ =   η^  _η^ ,

≡    -  _ ^ -   _^ _^ + 2_ (   _^  _^ ),
which enables us to rewrite the action in Eq.(<ref>) into the following form:

_     ≡     ∫[δ^D_h ]  {  _ _-^ i_ _ - β_Q sinhχ_s _ _-_

-     1/4 g_H^-2_  ^ - g_H^-2 ( _ ^ + _^ _^ )

+     1/2g_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 $2\p_{\fM} ( \fkA\, \hfA_{\fA}^{\;\fM} \fOm_{\fC}^{\fC\fA} )$. Such an action represents the hyperunified field theory in locally flat gravigauge hyper-spacetime, which explicitly displays the geometry-gauge correspondence for gravitational interaction. It is noticed that when taking the hyperspin gauge field $\cA_{\fC}^{\fA\fB}$ as the fundamental field, there appears no kinetic term for the hyperspin gravigauge field $\fOm_{\fC}^{\fA\fB}$ in locally flat gravigauge hyper-spacetime.

Therefore, we come to the observation that the hyperunified field theory formulated in locally flat gravigauge hyper-spacetime as shown in Eq.(<ref>) brings on an intriguing feature that the gravitational interaction appears to decouple from the hyperunified qubit-spinor field and is solely characterized by the hyperspin gravigauge field $\fOm_{\fC}^{\fA\fB}$ which behaves as an auxiliary field .

§.§  Emergence of non-commutative geometry with non-Abelian Lie algebra structure and the constraint equation of hyper-gravigauge field as auxiliary field in locally flat gravigauge hyper-spacetime

Let us begin with the nontrivial commutation relation of hyper-gravicoordinate derivative operator which forms gravigauge hyper-basis $\{\heth_{\fC}\}$ for locally flat gravigauge hyper-spacetime $\fG_h$, i.e.:

[_  , _ ] = f̂_^ _ ,

f̂_^ ≡_[]^ ≡_ ^ - _ ^ ≡-η^ _[] ,
which leads to non-Abelian Lie algebra of hyper-gravicoordinate derivative operator $\heth_{\fC}$. Unlike the usual non-Abelian Lie algebra, the structure factor $\hat{f}_{\fC\fD}^{\fA}$ for such a non-Abelian Lie algebra is no longer a constant, which is determined by the hyperspin gravigauge field $\fOm_{[\fC\fD]}^{\fA}$ and brings on the emergence of non-commutative geometry in locally flat gravigauge hyper-spacetime $\fG_h$. To explore the non-commutative geometric property of locally flat gravigauge hyper-spacetime $\fG_h$, it needs to resolve the dynamics of hyperspin gravigauge field $\fOm_{[\fC\fD]}^{\fA}$ from the geometry-gauge correspondence.

In the action of hyperunified field theory formulated in hidden coordinate formalism with geometry-gauge correspondence shown in Eq.(<ref>), the gravitational interaction is completely characterized by the hyperspin gravigauge field $\fOm_{\fC\fD}^{\fA}$. Nevertheless, when taking the hyperspin gauge field $\cA_{\fC}^{\fA\fB}$ as fundamental gauge field in locally flat gravigauge hyper-spacetime, the hyperspin gravigauge field $\fOm_{\fC\fD}^{\fA}$ should not act as an independent field in order to preserve the total independent degrees of freedom in the hyperunified field theory. In fact, $\fOm_{\fC\fD}^{\fA}$ couples only to the hyperspin gauge field $\cA_{\fC}^{\fA\fB}$ and scaling gauge field via dimensionless hyper-gravigauge covariant derivative appearing in their gauge field strengths shown in Eqs.(<ref>) and (<ref>) and meanwhile possesses the $\cQ_c$-spin scalar-gauge interaction term indicated by $\beta_G^2$. To show explicitly the hyperspin gravigauge field interaction, it is useful to rewrite Eqs.(<ref>) and (<ref>) into the following forms:

_^ ≡_^ + _[]^ _^  ,

_^ ≡_ _^  -  _ _^ + ( _'^ _^'  -  _'^ _^'  ) ,

_ ≡_ + _[]^ _ ,

_ ≡_ _  -  _ _ .

To study the dynamics of non-commutative geometry through the structure factor $\hat{f}_{\fC\fD}^{\fA}\equiv \fOm_{[\fC\fD]}^{\fA}$ of non-Abelian Lie algebra for hyper-gravicoordinate derivative operator $\heth_{\fC}$, it is useful to express the hyperspin gravigauge field $\fOm_{\fC\fD\fA}$ appearing in the action (Eq.(<ref>) ) as antisymmetric structure factor $\fOm_{[\fC\fD]\fA}$. For that, we should use the following relations:

_ ≡η_^  ''' _[' ']' ,

_ ^ = 1/2 _[] ^[] - _ ^,

_ ^  ≡_[] ^['']' η̅_'''^    ,
with the constant tensors defined as follows:

η_^  ''' = 1/2 ( η_^ ' η_^ ' η_^ ' - η_^ ' η_^ ' η_^ ' - η_^ ' η_^ ' η_^ ' ) ,

η̅_'''^    = 1/4 ( 3 η_'^  η_'^  η_'^  + 3η_'^  η_'^  η_'^  +η_'^  η_'^  η_'^  ) ,
which enables us to rewrite the action in Eq.(<ref>) into the following form:

_     ≡     ∫[δ^D_h ] {  _ _-^ i_ _ - β_Q sinhχ_s _ _-_

-     1/4 g_H^-2(_^  - 2 ^[]  _''_^'' )

+     g_H^-2 _[] ^['']' η̃_'''^
- 1/4 g_H^-2_[] ^[]  ^'' _''

+     1/2 g_H^-2β_G^2β_Q^2 sinh^2χ_s   ( _[] ^['']' + (_[] - 2_[] ) ^['']' ) η̅_'''^

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

-     1/4 g_W^-2_[] ^[]  ^ _ + 1/2 λ_S^2 _ χ_s ^χ_s  - λ_D^2 (χ_s)    } ,
with the definitions,

_[]  ≡_  - _ ,

η̃_'''^    = η̅_'''^
- 1/2 η_'^  η_'^  η_'^  - η_''η_'^  η^ ,

= 1/4 (  η_'^  η_'^  η_'^  + 3η_'^  η_'^  η_'^  +η_'^  η_'^  η_'^  ) - η_''η_'^  η^ ,
where the tensor $\bar{\eta}_{\fC'\fD'\fA'}^{\;\; \fC\fD\fA}$ is presented in Eq.(<ref>).

In the above action, the hyperspin gravigauge field $\fOm_{[\fC\fD]\fA}$ gets no kinetic term and appears as an auxiliary field in locally flat gravigauge hyper-spacetime. From the least action principle, we obtain the following relation:

η̃_'''^   []  ^['']' + f̃_'''^   []   ^['']' = F̃^[] ,
which presents a constraint equation of hyperspin gravigauge field $\fOm_{[\fC\fD]\fA}$. Where we have introduced the following definitions:

f̃_'''^   [] = 1/2  η̅_'''^   []  β_G^2 β_Q^2 sinh^2χ_s  - 1/4η_''^   [] (_''^' + g_H^2/g_W^2  _'^ ) ,

F̃^[] = 1/2 η̅_'''^   []  β_G^2 β_Q^2 sinh^2χ_s   ^['']'  - 1/2 (^''η^' _''' + g_H^2/g_W^2  ^^ ) ,

η̃_'''^   [] ≡η̃_'''^    - η̃_'''^    ,   η_''^   [] ≡η_'^  η_'^  - η_'^  η_'^  ,

η̅_'''^   [] ≡η̅_'''^    - η̅_'''^     .

It is clear that the hyperspin gravigauge field $\fOm_{[\fC\fD]\fA}$ is no longer an independent gauge field in locally flat gravigauge hyper-spacetime when the hyperspin gauge field $\cA_{\fC}^{\fA\fB}$ is chosen as fundamental gauge field in hidden coordinate formalism. It is shown from Eqs.(<ref>) and (<ref>) that $\fOm_{[\fC\fD]\fA}$ is in generally given by the hyperspin gauge field and $\cQ_c$-spin scaling field $\chi_s$ as well as scaling gauge field $\mW_{\fC}$. $\fOm_{[\fC\fD]\fA}$ in the constraint equation presented in Eq.(<ref>) appears in formal to be irrelevant to the hyperunified qubit-spinor field $\fPsi_{\fQH}$, while it is indirectly correlated to $\fPsi_{\fQH}$ through the dynamic equations of hyperspin gauge field and $\cQ_c$-spin scaling field shown as follows.

Based on the hyperunified field theory in locally flat gravigauge hyper-spacetime with geometry-gauge correspondence as shown in Eq.(<ref>), we come to the observation that the gravitational interaction behaves as an emergent interaction described by the hyperspin gravigauge field $\fOm_{[\fC\fD]\fA}$. $\fOm_{[\fC\fD]\fA}$ as auxiliary field in locally flat gravigauge hyper-spacetime is given by the fundamental bosonic fields through the constraint equation presented in Eqs.(<ref>) and (<ref>). There are three fundamental bosonic fields in the hyperunified field theory including the hyperspin gauge field $\cA_{\fC}^{\fA\fB}$ and $\cQ_c$-spin scaling field $\chi_s$ as well as scaling gauge field $\mW_{\fC}$. The constraint equation for $\fOm_{[\fC\fD]\fA}$ enables us to determine in principle the structure factor of non-Abelian Lie algebra for hyper-gravicoordinate derivative operator $\heth_{\fC}$, which characterizes an emergent non-commutative geometry as genesis of gravitational interaction in locally flat gravigauge hyper-spacetime.

§.§ Dynamics of fundamental fields in hyperunified field theory with emergent gravitational interaction in locally flat gravigauge hyper-spacetime

The hyperspin gravigauge field $\fOm_{[\fC\fD]}^{\fA}$ characterizing the geometry-gauge correspondence and emergence of non-commutative geometry is verified to be determined by the fundamental bosonic fields through the constraint equation presented in Eqs.(<ref>) and (<ref>). The gravitational interaction in locally flat gravigauge hyper-spacetime appears to decouple in formal from the hyperunified qubit-spinor field $\fPsi_{\fQH}$ when taking the hyperspin gauge field $\cA_{\fC}^{\fA\fB}$ as fundamental gauge fields in hidden coordinate formalism of the action shown in Eq.(<ref>). We will demonstrate how the dynamics of all fundamental fields gets gravitational effects through emergent gravitational interaction caused from the appearance of non-commutative geometry in locally flat gravigauge hyper-spacetime.

To derive the dynamic equations of motion for the hyperunified qubit-spinor field $\fPsi_{\fQH}$ and hyperspin gauge field $\cA_{\fC\fA\fB}$, it is useful to decompose the hyperspin gauge field $\cA_{\fC}^{\fA\fB}$ into the following two orthogonal parts:

_ ≡_[] + _(] ,

_[] ^(] = 0,
with the explicit definitions,
_[] ≡1/3 ( _ + _ + _  )

_(]  ≡1/3 ( 2_ - _ - _ ),
where $\cA_{[\fC\fA\fB]}$ denotes totally antisymmetric hyperspin gauge field and $\cA_{(\fC\fA\fB]}$ represents symmetric-antisymmetric mixing hyperspin gauge field. Correspondingly, the hyperspin gauge field strength $\cF_{\fC\fD\fA\fB}$ can also be decomposed into two orthogonal parts:

_ ≡_[] + _(] ,

_[] ≡1/3 ( _  +  _  + _ )

_(] ≡1/3 ( 2_  -  _  -_ ) ,

_[] ^(] = 0 .

With the above decompositions, the action in Eq.(<ref>) can be expressed into the following form,

_     ≡     ∫[δ^D_h ]  {  _ _-^ i__ - β_Q sinhχ_s   _ _-_

-     1/4 g_H^-2 ( _[]  ^[] +  _(]  ^(]  )

-     g_H^-2 ( _ ^ + _^ _^ )

+     1/2g_H^-2β_G^2β_Q^2 sinh^2χ_s   ( _[] - _[] )(^[] - ^[])

+     1/2g_H^-2β_G^2 β_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)    } ,

i_ ≡i_ + _[] 1/2^ .

It is noticed that in hidden coordinate formalism of the action, only the totally antisymmetric hyperspin gauge field $\cA_{[\fC\fA\fB]}$ couples to the hyperunified qubit-spinor field $\fPsi_{\fQH}$ due to its self-conjugated chiral property.

Based on the least action principle, we are able to derive the dynamic equations of motion for hyperspin gauge field in locally flat gravigauge hyper-spacetime, which can be written into the following two types:

_ ^[] = - g_H^2^[] ,

_ ^(] = - g_H^2 ^(] ,

_^[] ≡1/3 ( _^  +  _^  + _^ )

_^(] ≡1/3 ( 2_^  - _ ^  - _^ ) ,
where $\cJ^{[\fD\fA\fB]}$ represents the totally antisymmetric covariant tensor current and $\cJ^{(\fD\fA\fB]}$ the symmetric-antisymmetric mixing covariant tensor current. Their explicit forms are given as follows:

^[] =  1/4 _ { _-^, ^ } _ -  g_H^-2β_G^2β_Q^2 sinh^2χ_s   (^[] - ^[]) ,

^(] =   -  g_H^-2β_G^2 β_Q^2 sinh^2χ_s   (^(] - ^(]) .
The covariant derivative in Eq.(<ref>) is defined as follows:
_ ^ ≡_ ^ +  _'^^' + _'^^',

_ ^ ≡_ ^ + _'^^' + _'^^' ,

^ ≡^ - ^[] _^ ,
where the hyper-gravigauge covariant derivative $\ckcD_{\fC}$ and additional field strength term $\fOm^{[\fC\fD]\fE} \cA_{\fE}^{\fA \fB}$ reflect the geometry-gauge correspondence and the emergence of non-commutative geometry characterized by the hyperspin gravigauge field $\fOm_{[\fC\fD]}^{\fA}$ in locally flat gravigauge hyper-spacetime.

In general, the above equations of motion in locally flat gravigauge hyper-spacetime can also be formulated from Eq.(<ref>) obtained in biframe hyper-spacetime.

For the hyperunified qubit-spinor field $\fPsi_{\fQH}$, the equation of motion is simply given by,

_-^(i_ - i_ ) _
-β_Q sinhχ_s   _- _ =   0 ,
with the covariant derivative $\cD _{\fC}$ given in Eq.(<ref>), which appears to be irrelevant to the gravitational interaction. While its quadratic form is found to have the following form:

(_ -_ ) ( ^ -^ ) Ψ_

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

+  ^ ( _1/2^ - (_[]^- _[]^) i ( _ - _ ) - i _  ) Ψ_ ,
which indicates that the quadratic form of equation of motion for the hyperunified qubit-spinor field does generate gravitational effects characterized by the hyperspin gravigauge field $\fOm_{[\fC\fD]}^{\fA}$ due to the emergence of non-commutative geometry in locally flat gravigauge hyper-spacetime. Where $\fV_{\fC}$ and $\fV_{\fC\fD}$ are graviscaling induced gauge invariant field and field strength defined as follows:
_ ≡1/2 (_^ - _^ ),

_  = __-  __=  __-  __ - _[]^ _.

The equation of motion for the scaling gauge field $\mW_{\fC}$ can simply be expressed as follows:

_  ^  =  g_W^2  χ_s^2 ^ ,

_ ^ ≡_ ^ + _'^^' + _'^^' .

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

λ_S^2 _^χ_s =  _s ,

_^ ≡_^ + _'^ ^' ,
where the scalar current is given by,
_s     =     - β_Qcoshχ_s   _ _-_   +  β_Q^2 sinhχ_scoshχ_s   _ ^  -  λ_D^2 '(χ_s)

+     g_H^-2β_G^2 β_Q^2 sinhχ_scoshχ_s   ( _ - _ )(^ - ^)   .

To associate with the relation given in Eq.(<ref>) for the hyperspin gravigauge field $\fOm_{[\fC\fD]\fA}$, we can adopt the following identities:
_[] ≡η_[]^  ''' _[''],   _(] ≡η_(]^  ''' _[''] ,

η_[]^  ''' ≡1/3 ( η_^ ' η_^ ' η_^ ' + η_^ ' η_^ ' η_^ ' + η_^ ' η_^ ' η_^ ' ) ,

η_(]^  '''≡1/3 ( 2η_^ ' η_^ ' η_^ ' - η_^ ' η_^ ' η_^ ' - η_^ ' η_^ ' η_^ ' ) .

Therefore, the hyperunified field theory formulated in light of hidden coordinate formalism exhibits the gravitational geometry-gauge correspondence. When taking the hyperspin gauge field $\cA_{\fC}^{\fA\fB}$ as fundamental gauge field, the gravitational interaction in locally flat gravigauge hyper-spacetime appears in formal to decouple from the hyperunified qubit-spinor field. Meanwhile,  the hyperspin gravigauge field $\fOm_{\fC}^{\fA\fB}$ or $\fOm_{[\fC\fD]}^{\fA}$ behaves as an auxiliary field, which is shown to be presented by fundamental bosonic fields through the constraint equation given in Eq.(<ref>). Nevertheless, the dynamics of all fundamental fields turns out to get gravitational effects due to the emergence of non-commutative geometry characterized by the hyperspin gravigauge field via the non-Abelian Lie algebra of hyper-gravicoordinate derivative operator in locally flat gravigauge hyper-spacetime.

§  HYPERUNIFIED FIELD THEORY IN FRAMEWORK OF GQFT WITH FUNDAMENTAL SYMMETRY UNDER ENTIRETY UNITARY GAUGE AND THE DYNAMICS OF ELECTROMAGNETIC-LIKE GRAVITATIONAL FIELD WITH GAUGE-TYPE AND EINSTEIN-TYPE GRAVITATIONAL EQUATIONS AND FIRST ORDER GRAVITATIONAL DIFFERENTIAL EQUATION BASED THE GAUGE-GEOMETRY DUALITY

In this section, we are going to further investigate the foundation of the hyperunified field theory within the framework of gravitational quantum field theory (GQFT)[5] by making an appropriate gauge prescription under entirety unitary gauge. With the demonstration of gauge-geometry duality in biframe hyper-spacetime, we will present a detailed discussion on the dynamics of hyper-gravigauge field and hyper-gravimetric gauge field with electric-like and magnetic-like gravitational interactions in both gauge-type and geometric Einstein-type gravitational equations.

§.§  Hyperunified field theory within framework of GQFT and gauge-geometry duality with respect to hyper-gravigauge field and hyper-gravimetric gauge field

It has been shown that the $\cW_e$-spin gauge group symmetry W$^{1,D_h-1}$ that causes a sign flip in chirality for the transformed hyperunified qubit-spinor field brings on the $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\;\;\fA}$. Such an entanglement-correlated translation-like $\cW_e$-spin gauge group W$^{1,D_h-1}$ represents the Abelian subgroup of inhomogeneous hyperspin gauge symmetry WS(1,$D_h$-1), so that the $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\;\;\fA}$ as hyper-gravigauge field reveals the genesis of gravitational interaction and elaborates the gauge-gravity correspondence. In the hyperunified field theory with the action formulated in hidden gauge formalism, the gravitational interaction is shown to be described by the symmetric hyper-gravimetric field $\mH_{\fM\fN}$ which is composed of hyper-gravigauge field, i.e., $\mH_{\fM\fN} = \fA_{\fM}^{\;\;\fA} \fA_{\fN}^{\;\;\fB}\eta_{\fA\fB}$. When making the gauge prescription to flowing unitary gauge, the symmetric hyper-gravimetric field is determined directly by the symmetric hyper-gravigauge field $\bfA_{\fM\fA}=\bfA_{\fA\fM}$, i.e., $\mH_{\fM\fN} = (\bfA)^2_{\fM\fN}$, with $\bfA_{\fM\fA}$ representing physical degrees of freedom under flowing unitary gauge, which exposits the gravity-geometry correspondence in curved Riemannian hyper-spacetime. Furthermore, in the hyperunified field theory with the action formulated in hidden coordinate formalism, the gravitational interaction appears to be an emergent one characterized via the emergent non-commutative geometry in locally flat gravigauge hyper-spacetime. Such a non-commutative geometry is described by the hyperspin gravigauge field $\fOm_{\fC\fD}^{\fA}$ which behaves as an auxiliary field determined through the constraint equation concerning only basic bosonic fields, which elucidates the geometry-gauge correspondence.

Let us now turn to consider the hyperunified field theory within the framework of GQFT. By noticing the following relations,
_'^''  _^_''^' =
^'  ^' _ _''^

+ ^'( 2 _'^   _^  '  _^ _''^' - 4 _^   _'^  '  _^ _''^' )

^'_^   _'^  '  _^ _''^'  ≡^2 _(  _^   ) _(  ^ ) ,
we are able to rewrite the basic action for the hyperunified field theory as follows:

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

-     g_H^-2 1/4 ^' ^'   ( _  _''^  -  __''^ )

+     g_H^-2 1/2 ^' _'^   _^  '  _^ _''^' - g_H^-2^2 _(  _^   ) _(  ^ )

+     1/2 g_H^-2β_G^2 β_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)    } ,
with the definitions,
i_ = i_ + _^ 1/2_,

_ ≡_^     _^ ,  _^ = __^   - __^  ,

_^     = 1/2 _^     ( _^    _^   η_ - _^    _^   η_ - _^    _^   η_  ) ,
where the hyperspin gauge field $\cA_{\fM}^{\fA\fB}$ and hyper-gravigauge field $\fA_{\fM}^{\; \; \fA }$ are taken to be the fundamental gauge fields, which is governed by the inhomogeneous hyperspin gauge symmetry and also exposits the gauge-gravity correspondence.

It is interesting to notice that the gravitational interaction of all fundamental bosonic fields involves solely the dual Goldstone-like hyper-gravimetric field $\hmH^{\fM\fN}$. Inspired from the gauge-gravity and gravity-geometry correspondences, it is useful to take the hyper-gravimetric field $\mH_{\fM\fN}$ and hyperspin gauge field $\cA_{\fM}^{\fA\fB}$ as physical degrees of freedom in the gravigauge hyper-fiber bundle structure of biframe hyper-spacetime. In general, we can prove the following relations:

1/4   ^''_' _^_''^'   ≡  ^ _ -  2 _ (^ _^  _^ ) ,

1/4   ^''_' _^_''^'   ≡1/8  ^' ^' _' _^ _''^'

-1/4 ^ _(ln) _(ln) + 1/4 _ ^2 ^( _^   ) ^(  _^  )

+ 1/2 ^2 _^   _(ln) ^ ( _^   ) - 1/4 ^2 ^ ^ _ ( _ )

-1/4 _ (  _ _ (^ ^) - _ ( _ ) ^ ^ )

+ _ ( ^ _ ln)  - 2 _ (_^   _^ )  + 3/2 __ ( ^ ) ,
with $\hat{\fkA}=1/\fkA$ and $\hp^{\fM}\equiv \hmH^{\fM\fM'}\p_{\fM'}$. Note that the last four terms appear as surface terms in the action. Where we have introduced the following Abelian-type field strength:
_^ = __^   - __^   ,
which indicates that the hyper-gravimetric field $\mH_{\fM}^{\; \fP}$ behaves as an Abelian-type gauge field. To be physically more meaningful, the hyper-gravimetric field $\mH_{\fM}^{\; \fP}$ and field strength $\mF_{\fM\fN}^{\fP}$ may be referred to as hyper-gravimetric gauge field and hyper-gravimetric gauge field strength, which brings on useful duality relations with respective to the hyper-gravigauge field $\fA_{\fM}^{\; \fA}$ and field strength $\mF_{\fM\fN}^{\fA}$.

The relations in Eqs.(<ref>) and (<ref>) not only explicate the gauge-gravity and gravity-geometry as well as geometry-gauge correspondences,  but also corroborate that the geometric hyper-gravimetric field $\mH_{\fM}^{\; \fP}$ in itself can be regarded as an Abelian-type gauge field, which reveals in general gauge-geometry duality.

With such a gravitational gauge-geometry duality, we can take the hyperspin gauge field $\cA_{\fM}^{\fA\fB}$ and hyper-gravimetric gauge field $\mH_{\fM}^{\; \fP}$ as fundamental gauge fields to express the action of the hyperunified field theory in light of gravigauge hyper-fiber bundle structure of biframe hyper-spacetime. Explicitly, such an action can be represented within the framework of GQFT as follows:

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

-      β_Q sinhχ_s    _ _-_  -  ^' ^'  g_H^-2 1/4 _  _''^

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

+     1/8  ^' ^' _' g_H^-2 _^ _''^'

-     1/4 g_H^-2 ^ ^ _ ( _ )  - 1/4 g_H^-2 ^(ln) _(ln)

+     1/4 g_H^-2 _ ^2 ^( _^   ) ^(  _^  ) + 1/2 g_H^-2 _(ln) ^ ( _^   )

-     ^' ^' 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 terms. The hyperspin gravigauge field can be expressed as follows:
_ = _[] _^  _^   - 1/2 _ [  _(_^  ) _^   -  _(_^  ) _^     ] ,
which indicates that $\fOm_{\fM\fA\fB}$ concerns both hyper-gravimetric gauge field $\mH_{\fP}^{\; \fM}$ and hyper-gravigauge field $\fA_{\fM}^{\; \fA}$. It is also manifest that the gravitational interaction of hyperqubit-spinor field always involves the hyper-gravigauge field $\fA_{\fM}^{\; \fA}$. It is natural to take flowing unitary gauge so as to relate directly the symmetric hyper-gravigauge field to the symmetric hyper-gravimetric field, $\mH_{\fM\fN} = (\bfA)^2_{\fM\fN}$ with $\bfA_{\fM\fA}=\bfA_{\fA\fM}$.

Therefore, within the framework of GQFT based on the basic structure of biframe hyper-spacetime, the dynamics of gravitational interaction can be described either by the symmetric hyper-gravigauge field or equivalently by the symmetric hyper-gravimetric gauge field under flowing unitary gauge with maintaining physical degrees of freedom. Such an equivalent description exposits the gauge-geometry duality for the gravitational interaction.

§.§  Hyperunified field theory with maximal joint symmetry and the entirety unitary gauge in gravitational quantum field theory with fundamental symmetry of nature

The hyperunified field theory in the framework of GQFT is presented to show the gravitational gauge-geometry duality. Before discussing such an action, let us briefly outline the main properties of hyperunified field theory formulated in various formalisms.

As demonstrated in the hidden gauge formalism of the action given in Eq.(<ref>), such an action of hyperunified field theory holds in any coordinate system due to the emergence of general linear group symmetry GL(1,$D_h$-1,R), which does indicate that the laws of nature should be independent of the choice of coordinate systems. Geometrically, the gravitational interactions concerning fundamental bosonic fields are described by the dynamics of emergent Riemann geometry in curved Riemannian hyper-spacetime, which is characterized by Christoffel symbols $\fGa_{\fM\fN}^{\fP}$ defined from the hyper-gravimetric field $\mH_{\fM\fN}$. Such a gravitational interaction is governed by the emergent general linear group symmetry GL(1,$D_h$-1,R) and meanwhile the hyperspin gauge symmetry becomes a hidden gauge symmetry, which shows the gravity-geometry correspondence. As a consequence, the basic action of hyperunified field theory possesses emergent general linear group symmetry GL(1,$D_h$-1,R) under general coordinate transformation,
G_S=  1,D_h-1, , which represents the real Lie group of dimension $N_D= D_h^2 $ and consists of matrices that have non-zero determinant. It is known that such a general linear group symmetry lays the foundation of Einstein's general theory of relativity in four dimensional spacetime.

It is proved in the hidden coordinate formalism of the action that, locally, at every point of hyper-spacetime, the basic action of hyperunified field theory possesses inhomogeneous hyperspin gauge symmetry WS(1,$D_h$-1). Whereas the general linear group symmetry GL($D_h$,R) goes to be a hidden symmetry. The gravitational interaction is characterized by the hyperspin gravigauge field $\fOm_{[\fC\fD]}^{\fA}$ which behaves as an auxiliary field determined through the constraint equation concerning only basic bosonic fields. Therefore, the gravitational interaction acts as an emergent interaction in correspondence to the emergence of non-commutative geometry in locally flat gravigauge hyper-spacetime. The hyperspin gauge field $\cA_{\fC\fA\fB}$ as fundamental gauge field is taken to be dynamic field, so that the basic action exhibits the inhomogeneous hyperspin gauge symmetry as a maximal gauge symmetry,
G_S =  ,D_h , which has been applied to construct the basic action by following along the gauge invariance principle.

It is straightforward to justify that when the action is formulated in light of gravigauge hyper-fiber bundle structure of biframe hyper-spacetime as shown in Eqs.(<ref>) and (<ref>), it gets a bimaximal gauge symmetry,
G_S= 1,D_h-1,  1, D_h-1),
which is regarded as a joint Lie group gauge symmetry since they cannot be operated as a direct product group. In general, such a joint gauge symmetry can be adopted as generalized gauge invariance principle to construct the basic action of hyperunified field theory. One is general linear group symmetry GL(1,$D_h$-1, R) operating on the coordinates in curved Riemannian hyper-spacetime, and the other is inhomogeneous hyperspin gauge symmetry WS(1,$D_h$-1) operating on the entangled hyperqubit-spinor field and gauge field in locally flat gravigauge hyper-spacetime.

In practice, the basic action of hyperunified field theory is built within the framework of quantum field theory based on the gauge invariance principle and scaling invariance hypothesis in light of gravigauge hyper-fiber bundle structure of biframe hyper-spacetime. Where the free-motion hyper-spacetime described by globally flat Minkowski hyper-spacetime is taken as a base spacetime in gravigauge hyper-fiber bundle structure, so that the basic action of hyperunified field theory is built to possess global Poincaré-type group symmetry PO(1,$D_h$-1) and inhomogeneous hyperspin gauge symmetry WS(1,$D_h$-1) as well as global and local scaling symmetries,
G_S = ⋉,D_h,D_h⋊ ,
which has a total group dimension $N_D = 2\times D_h(D_h-1)/2 + 2D_h + 2 = D_h(D_h+1) +2 $. Where the Lorentz-type group SO(1,$D_h$-1) is a subgroup of GL($D_h$, R). In such a consideration, the global and local scaling symmetries play an essential role in the construction of the basic action for the hyperunified field theory.

From the gauge invariance principle, the gauge field is introduced to reveal the dynamics of fundamental building block of nature. Locally, the gauge symmetry means that the laws of nature should be independent of the choice of field configurations. Namely, the gauge symmetry leads each physically distinct configuration of the system to be as an equivalence class of detailed local field configurations, which indicates that for any two detailed configurations that are in the same equivalence class, they are related by a gauge transformation. Therefore, the gauge symmetry concerns redundant unphysical degrees of freedom.

Physically, all observables should be gauge independent.  In order to remove unphysical degrees of freedom arising from local gauge symmetry, it is necessary to fix gauge symmetry by making appropriate gauge prescription. Therefore, a gauge fixing is required to provide a mathematical procedure for dealing with redundant degrees of freedom in field variables caused by gauge symmetry.

In the hidden gauge formalism of the action, it has been shown that it is appropriate to choose a gauge prescription under flowing unitary gauge. This is because there exists always a hyperspin gauge transformation $\bar{\Lambda}_{\;\, \fA}^{\fB}(x)$ to transform the hyper-gravigauge field into a symmetric bicovariant vector field at any chosen point of hyper-spacetime,  $\bfA_{\fM\fA}(x) = \fA_{\fM\fB}(x) \bar{\Lambda}^{\fB}_{\; \, \fA}(x)=\bfA_{\fA\fM}(x)$, which is referred to as flowing unitary gauge.  In such a flowing unitary gauge, both symmetric hyper-gravigauge field $\bfA_{\fM\fA}(x)$ and hyper-gravimetric field $\mH_{\fM\fN}(x)$ involve exactly the same degrees of freedom with a simple relation $\mH_{\fM\fN} \equiv (\bfA)^2_{\;\, \fM\fN}$.

Let us turn to the basic action of the hyperunified field theory with the gauge-geometry duality shown in Eq.(<ref>) or (<ref>). Such an action built in biframe hyper-spacetime with a gravigauge hyper-fiber bundle structure gets a maximal joint symmetry of two groups GL(1,$D_h$-1, R)$\Join$WS(1,$D_h$-1) which cannot form a direct product group. It is noticed that only when setting up flowing unitary gauge to keep the symmetric property of hyper-gravigauge field at arbitrary point of coordinate systems in hyper-spacetime, we are able to obtain the same physical degrees of freedom to bring on both gauge-gravity and gravity-geometry correspondences. Namely, once the inhomogeneous hyperspin gauge symmetry WS(1,$D_h$-1) which governs the dynamics of hyperunified quibit-spinor field and gauge field in locally flat gravigauge hyper-spacetime is fixed appropriately to unitary gauge, the general linear symmetry group GL(1,$D_h$-1, R) which characterizes the dynamics of Riemann geometry in curved Riemannian hyper-spacetime should also be fixed simultaneously to ensure the symmetric property of hyper-gravigauge field. Namely, under the flowing unitary gauge, the basic action (<ref>) or (<ref>) formulated in light of gravigauge hyper-fiber bundle structure of biframe hyper-spacetime possesses only the following associated symmetry:
G_S = ⋉,D_h,D_h, which brings about the fundamental symmetry achieved from the so-called $\cM_c$-spin charge of entangled hyperqubit-spinor field in free-motion hyper-spacetime.

To freeze over the flowing feature of unitary gauge for the hyperspin gauge symmetry SP(1,$D_h$-1) and provide a whole gauge prescription, we should further make a gauge prescription to fix the general linear group symmetry GL(1,$D_h$-1, R). From the basic action presented in Eq.(<ref>) with gauge-geometry duality, it is reasonable to take the following gauge prescription:

(D_h-2)^-1^ (  _  ) - D_h^-1 _  =0,
,     ^_ - 2/D_h _ln= 0 ,
which leads the flowing unitary gauge to be frozen into a usual unitary gauge.

With the above gauge prescription, the action presented in Eq.(<ref>) within the framework of GQFT is modified to be as follows:

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

-      β_Q sinhχ_s    _ _-_  -  ^' ^'  g_H^-2 1/4 _  _''^

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

+      1/8  ^' ^' _'  g_H^-2 _^ _''^'

-     1/4 g_H^-2 ^ ^ _ ( _ )  - 1/4 g_H^-2 ^(ln) _(ln)

+     1/4 g_H^-2 ^2_ ^( _^   ) ^(  _^  ) + 1/2 g_H^-2 _(ln) ^ ( _^   )

+     λ_G/8   g_H^-2  ^ [ ^(_) + D_h-2/D_h _ ln]  [^(_) + D_h-2/D_h _ ln]

-     ^' ^' 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 introduced an additional term with arbitrary constant $\lambda_G$ to provide the gauge fixing condition.

The above action of hyperunified field theory with imposing a gauge prescription on the general linear group symmetry GL(1,$D_h$-1, R) shown in Eq.(<ref>) exhibits a manifest gauge-geometry duality within the framework of GQFT. For short, we may call such a gauge prescription as duality gauge prescription in hyper-spacetime.

It is clear that the combination of duality gauge for the group symmetry GL(1,$D_h$-1, R) in curved Riemannian hyper-spacetime with flowing unitary gauge for the gauge symmetry SP(1,$D_h$-1) in Hilbert space leads to a full gauge prescription, which may be referred to as entirety unitary gauge prescription for convenience. By choosing the gauge fixing conditions under entirety unitary gauge, we should be able to remove in principle all unphysical degrees of freedom.

Therefore, to remove all unphysical degrees of freedom, we should take the entirety unitary gauge which combines duality gauge and flowing unitary gauge together. Consequently, the fundamental symmetry of hyperunified field theory turns out to be an associated symmetry in which the global inhomogeneous hyperspin symmetry WS(1,$D_h$-1) is always in association with global Poincaré-type group symmetry PO(1,$D_h$-1) together with global scaling symmetry SC(1), i.e.:
G_S = ⋉,D_h,D_h, where the isomorphic group SP(1,$D_h$-1)$\cong$SO(1,$D_h$-1) must keep coincidental symmetry transformations.

§.§  Gauge-type and geometric Einstein-type gravitational equations and beyond for the dynamics of gravitational field

Let us first revisit the equation of motion for the hyper-gravigauge field $\fA_{\fM}^{\;\; \fA}$. From the hyperunified field theory presented in Eq.(<ref>) within the framework of GQFT, we obtain the following equation of motion:

_ _^  =  _^   ,
with the definition,

_^ ≡^[]''_' _'' ^'  = -  _^,

_'' ^' = _'_'^   ' - _'_'^   ' ,
where the tensor $\thmH^{[\fM\fN]\fM'\fN'}_{\fA\fA'}$ is defined in Eq.(<ref>).

We may refer to $\hmF_{\fA}^{\fM\fN}$ as hyper-gravigauge field strength tensor. $\hmJ_{\fA}^{\;\;\fM}$ is called hyper-gravigauge field current with the following form:

_^    ≡_^    - _^ _ln≡_^    + g_H^2 _^   ,
with $\mJ_{\fA}^{\;\;\fM}$ the bicovariant vector current defined in Eq.(<ref>). $\mG_{\fA}^{\;\; \fM}$ and $\mT_{\fA}^{\;\; \fM}$ are explicitly given as follows:
_^        ≡     ( _^     ^[]''_”'  - 1/4 _^   ^''_”')  _^”  _''^' - ^[]''_' _'' ^' _ln,

_^        ≡     -   _^    +    _^      {   _^     _ _-^ i _ _

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

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

+     λ_S^2 ^ _χ_s  _ χ_s  - _ (   ^[]''_' g_H^-2β_G^2β_Q^2 sinh^2χ_s   _'' ^' )     },
where $\mG_{\fA}^{\;\; \fM}$ is referred to as pure hyper-gravigauge field current caused solely by the hyper-gravigauge field itself. $\mT_{\fA}^{\;\; \fM}$ is referred to as hyper-gravigauge field bicovariant vector current resulting from all other basic field in the hyperunified field theory.

We may refer to Eq.(<ref>) as gauge-type gravitational equation for the hyper-gravigauge field. Such a gauge-type gravitational equation leads to the following conserved current:
__^    = 0 ,
due to the antisymmetry property of the hyper-gravigauge field strength tensor. Therefore, in the hyperunified field theory with the action presented in the framework of GQFT, the hyper-gravigauge field does behave as an Abelain gauge field.

To obtain Einstein-type equation in hyper-spacetime, we should utilize Eqs.(<ref>) and (<ref>) and take into account the gauge-gravity and gravity-geometry correspondences. It can be verified that the gauge-type gravitational equation presented in Eq.(<ref>) can be reformulated into the following formalism in light of Ricci curvature tensor $\fR_{\fM\fN}$:
-2 g_H^-2_''_^  '^'  = _^     - g_H^-2 _^  ^[]''_”'  _^” _''^' .
When projecting the above bicovariant vector tensors into hyper-spacetime tensors by adopting the hyper-gravigauge field as Goldstone-like boson, we are able to obtain two types of dynamic equations of motion.

The first dynamic equation of motion is resulted from the symmetric tensor, which has the following explicit form:

_ - 1/2 _ = -1/2 g_H^2 _ ,
and the second dynamic equation of motion is caused from the antisymmetric tensor, which gets the following explicit form:

g_H^-2β_G^2 β_Q^2 sinh^2χ_s ∇̂_ ^_        =  _[] .

It can be checked that the symmetric tensor $\mT_{\fM\fN}$ presents a generalized Einstein-type symmetric hyper-stress energy-momengtum tensor in hyper-spacetime, which is determined by the hyper-gravigauge field bicovariant vector current $\mT_{\fA}^{\;\; \fM}$ as follows:

_     ≡      _^    _^    _

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

-      g_H^-2 ( _^ _  - _ 1/4 ^''_'^ _'^  )  ^

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

+     1/2 g_H^-2β_G^2β_Q^2 sinh^2χ_s  ( _          ^() _()  - _          ^[] _[])

+     1/2 g_H^-2β_G^2β_Q^2 sinh^2χ_s    ( _^ _[]  + _^ _[] + 2 ∇̂_ _()^ )

-      g_W^-2( _ _  -  _ 1/4 ^''_' _' ) ^

+     (W_W_ - _  1/2 ^W_W_ )

+     λ_S^2 (_χ_s_χ_s - _  1/2 ^ _χ_s_χ_s  )

+     (β_Q sinhχ_s    _ _-_ + λ_D^2 (χ_s) ) _,
where $\fA_{\fP\fM\fN}$ is the hyper-spacetime covariant-gauge field defined in Eq.(<ref>). We have used the following notations and definitions,
_ ≡_ _^ = - _,   _^ ≡^ _^ = - _^,

_  ≡_ ( __^    + _^ _^  - _^_^   ) ≡_ - _

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

_[] ≡_ - _ ,   _() ≡_ + _ ,

_        ^[] ≡_        ^ -  _        ^,   _        ^() ≡_        ^ +  _        ^,

_        ^ ≡^'  ^_      '  ≡^' ^' _'' ,

^_       ≡^ _ ≡^_' _^' ,

∇̂_ _^ ≡_  _^ + _^ _^ -   _^ _^ -  _^ _^ .

The antisymmetric tensor $\mT_{[\fM\fN]}$ defines antisymmetric hyper-stress tensor in hyper-spacetime with the following explicit form:
_[] ≡_ _-^ _ i_  _  - _ _-^ _ i_  _ ,

Therefore, two equations of motion presented in Eqs.(<ref>) and (<ref>) are referred to as geometric gravitational equations. Eq.(<ref>) with symmetric tensor is a generalized Einstein-type equation, which is regarded as the extension of general theory of relativity in four-dimensional spacetime to hyperunified field theory in hyper-spacetime. Eq.(<ref>) with antisymmetric tensor brings on an additional dynamic equation beyond Einstein-type equation, which arises from hyper-gravigauge field, instead of hyper-gravimetric field, as fundamental gravitational field in the hyperunified field theory with inhomogeneous hyperspin gauge symmetry that governs the fundamental interaction of nature.

§.§  The first oder gravitational differential equation of hyper-gravigauge field from the constraint equation of hyperspin gravigauge field as auxiliary field in locally flat gravigauge hyper-spacetime

Let us now turn to the geometry-gauge correspondence in the hidden coordinate formalism of the action for the hyperunified field theory, which leads to the following relations:
^'  ( ^' _ _''^ + 2 _'^   _^  '  _^ _''^' ) = 4 _ ^ ,

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

1/4   ^''_' _^_''^'  ≡  η^ _ - 2_ (   _^  _^ ),

≡  (_ ^ + _^ _^ ) .
In such a formalism, the gravitational interaction is solely characterized by the hyperspin gravigauge field $\fOm_{[\fC\fD]}^{\fA}$ which behaves as an auxiliary field in locally flat gravigauge hyper-spacetime, so that $\fOm_{[\fC\fD]}^{\fA}$ is determined through the constraint equation which concerns only basic bosonic fields as shown in Eqs.(<ref>) and (<ref>). Meanwhile, the hyperspin gravigauge field $\fOm_{[\fC\fD]}^{\fA}$ is shown in Eq.(<ref>) to present as the structure factor of non-Abelian Lie group for the hyper-gravicoordinator derivative operator, which brings on the emergence of non-commutative geometry in locally flat gravigauge hyper-spacetime. Therefore, the gravitational interaction acts as an emergent interaction in locally flat gravigauge hyper-spacetime characterized by non-commutative geometry.

The hyperspin gravigauge field $\fOm_{[\fC\fD]}^{\fA}$ is actually presented by Abelian-type gauge field strength $\mF_{\fM\fN}^{\fA}$ as follows:
_[]^ ≡-_^    _^    _^ ,   _^ = __^    - __^    .
When projecting Eq.(<ref>) into hyper-spacetime of coordinates by using the hyper-gravigauge field as Goldstone-like bicovariant vector field in biframe hyper-spacetime, we obtain the following constraint equation:

_ - χ_[]^''' _''' = - _[]  ,
where we have introduced the following definitions:

_[]      ≡      β_G^2 β_Q^2 sinh^2χ_s [  3/2 (_ - 1/3 _[]^''' _''' )

+      _ _^   - _ _^   + _^  ' _^  '   _^  ' _''' ]

-     _ _^' ^' (_'^'' _''' + g_H^2g_W^-2 _'_' )

+      ( _^'' _'''   + g_H^2g_W^-2 _ _'  ) _^  ' ,
χ_[]^''' ≡η_^  ' ( _^  ' _^' - 2 _ ^'' ) - η_^  ' ( _^  ' _^' - 2 _ ^'' ) ,

_[]^''' ≡η_^  ' _^  ' _^' - η_^  ' _^  ' _^' .
In four-dimensional spacetime, such a constraint equation in Eq.(<ref>) brings about twenty four first order partial differential equations.

Therefore, we come to the observation that the constraint equation given in Eq.(<ref>) for hyperspin gravigauge field as auxiliary field in locally flat gravigauge hyper-spacetime brings on the first order partial differential equation of hyper-gravigauge field in biframe hyper-spacetime, which is regarded as the first order gravitational differential equation.

In the special case with keeping only hyperspin gravigauge field by turning away all other fields, the first order gravitational differential equation presented in Eq.(<ref>) is simplified to be as follows:

_ - χ_[]^''' _''' = 0 ,
which provides the first order gravitational differential equation purely for the hyper-gravigauge field.

§.§  Dynamic equation of hyper-gravimetric gauge field as geometric gauge-type gravitational equation  and the electric-like and magnetic-like gravitational interactions based on gauge-geometry duality

From the gauge-geometry duality, the gravitational interaction is described by Abelian-type hyper-gravigauge field or equivalently by hyper-gravimetric field which can also be regarded as hyper-gravimetric gauge field. We are inspired to investigate the electric-like and magnetic-like gravitational interactions in the hyperunified field theory within the framework of GQFT.

Once regarding $\mH_{\fN}^{\; \fP}$ as hyper-gravimetric gauge field, we are able to derive from the action of hyperunified field theory presented in Eq.(<ref>) the following geometric gauge-type gravitational equation:

_ _^ = _^   ,
where $\hmF_{\fP}^{\fM\fN} $ is defined as follows:
_^ ≡^'^'_'_''^' .
which is referred to as hyper-gravimetric gauge field strength tensor. $\hat{\mJ}_{\fP}^{\; \fM}$ defines the corresponding tensor current,
_^   ≡2 _^   + 2 g_H^2 _^   ,
which is referred to as hyper-gravimetric gauge field current. Where $\hmT_{\fP}^{\; \fM}$ is given by the generalized Einstein-type hyper-stress energy-momentum tensor $\mT_{\fM\fN}$ (Eq.(<ref>)) as follows:

_^   ≡_^   _ ^ = _  ^ _^   ,
and $\hat{\mG}_{\fP}^{\; \fM}$ has the following explicit form:

_^       ≡    - ^'^'_'_''^' _ln+ 1/4 ^ _^' ^' _''  _^' _''^'

+     1/8 _'^ _'  ^' ^' _^' _''^' +  1/4  ^  _^   _(ln) _(ln)

+     1/4 ^2 ^  _^'  ^( _ )  ^'(  _'') +  1/2  ^  _^  _ _(ln)

+     1/2 (  ^  _ln+ _^  ^ln) ^( _ )

+    1/4 (  ^  _ + _^  ^ ) ^( _ )

-     1/8 (  ^  _^'  +  ^' _^ ) ^_' ( _ )

-     1/8 ^' (  ^ _  +  _^ ^) _' ( _ )

-     1/8 _^ [   ^' ^' _''  _^' _''^' -2  ^  _(ln) _(ln)

+      2 ^2  ^'' ^( _' ) ^(  _')  + 4 ^(ln) ^( _) )

-     4 ^  _ _(ln) -  ^' ^ _' ( _ )

+     ^2 _ _' _' ( ^' _' )   ] ,
which defines the pure hyper-gravimetric gauge field current resulting just from the hyper-gravimetric gauge field itself.

Such an introduced hyper-gravimetric gauge field current $\hat{\mJ}_{\fP}^{\; \fM}$ is a conserved current,
_ _^   = 0,
which is attributed to the antisymmetric property of hyper-gravimetric gauge field strength tensor $\hmF_{\fP}^{\fM\fN} = - \hmF_{\fP}^{\fN\fM}$.

As either geometric gauge-type gravitational equation (<ref>) or gauge-type gravitational equation (<ref>) is Abelian-type gauge equation in correspondence to hyper-gravimetric gauge field $\mH_{\fM}^{\;\; \fP}$ or hyper-gravigauge field $\fA_{\fM}^{\;\; \fA}$, we are allowed to introduce electric-like and magnetic-like gravitational field strengths in analogous to Maxwell electromagnetic field strength. For the hyper-gravimetric gauge field strength tensor, let us make the following definitions:
_^0 =  ^0'^'_'_''^' = - _^0 ≡-_^,

_^ =  ^'^'_'_''^' ≡f^_      _^,
where $f^{\fI\fJ}_{\;\;\fK\fL}$ is a constant tensor with the following property:
f^_   = -  f^_   = -  f^_   ,

f^_    f^_  '' = 1/2 (η^_  ' η^_  ' - η^_  ' η^_  ' ) .
Similar definitions hold for the hyper-gravigauge field strength tensor,
_^0 = ^[0]''_' _'' ^'  = - _^0  ≡- _^,

_^ = ^[]''_' _'' ^'  ≡f^_   _^ .
The gravitational field strengths $\mE_{\fA}^{\fI}$ and $\mE_{\fP}^{\fI}$ are regarded as electric-like hyper-gravigauge field strength and hyper-gravimetric gauge field strength, the antisymmetric gravitational field strengths $\mB_{\fA}^{\fK\fL}= -\mB_{\fA}^{\fL\fK} $ and $\mB_{\fP}^{\fK\fL}= -\mB_{\fP}^{\fL\fK}$ are viewed as generalized magnetic-like hyper-gravigauge field strength and hyper-gravimetric gauge field strength, respectively.

In terms of the electric-like and magnetic-like gravitational field strengths, the geometric gauge-type gravitational equations can be expressed as follows:
__^ = - _^0,

f^[] __ + f^(] __ + _0_^ = _^ ,
and similarly, the gauge-type gravitational equation can be written as follows:
__^ = - _^0,

f^[] __ + f^(] __ + _0_^ = _^.
We have introduced the following definitions:
f^[] ≡1/3 ( f^ + f^ + f^ ),

f^(] ≡1/3 ( 2 f^ - f^ - f^ ),
where $f^{\fI[\fJ\fK\fL]}$ and $f^{\fI(\fJ\fK\fL]}$ denote the total antisymmetric tensor and symmetric-antisymmetric mixing tensor, respectively, for the superscripts `$\fJ\fK\fL$' in the brackets.

As a simple consideration, let us suppose that the hyperunified field theory in hyper-spacetime evolves dynamically into an effective theory in four dimensional spacetime, where the gauge-geometry duality leads the dynamics of gravigauge field to be equivalent to Einstein's general theory of relativity characterized by gravimetric gauge field. So that, the geometric gauge-type gravitational equation in four dimensional spacetime is simplified to be:
_i_ρ^i = -_ρ^0 ,

ε^ij_    k _j_ρ^k +_0_ρ^i = _ρ^i ,
with $i, j, k=1,2,3$ and $\varepsilon^{ijk}$ the totally antisymmetric constant tensor $\varepsilon^{123}=1$. The electric-like gravimetric field $\mE_{\rho}^{i}$ and magnetic-like gravimetric field $\mB_{\rho}^{i}$ are given as follows:
_ρ^0i =  ^0μ'^iν'_ρρ'_μ'ν'^ρ' ≡- _ρ^i,

_ρ^ij =  ^iμ'^jν'_ρρ'_μ'ν'^ρ' ≡ε^ij_     k _ρ^k .

By fixing the scaling gauge symmetry to Einstein basis at low energy, the dual gravimetric fields can be expressed as follows:

_μν ≡^2 ( η_μν + 1/M_ph_μν ),   M_p ≡√(2) /g_h  ,

^μν =( _μν )^-1 = ^-2 ( η^μν - 1/M_p η^μμ' η^νν' h_μ' ν' + O(1/M_p^2) ),
where $\Mka$ is regarded as fundamental mass scale, $M_{p}$ and $g_h$ are considered as the basic mass scale and coupling constant in four-dimensional spacetime.

In an approximation that the inverse of gravimetric gauge field is taken at the leading-order, we obtain the following relations:
_ρ^0i ≃^-6 η^0μ'η^iν'η_ρρ'_μ'ν'^ρ' = ^-6 ^0i_ρ ≡- ^-6 E_ρ^i,

_ρ^ij ≃^-6 η^iμ'η^jν'η_ρρ' _μ'ν'^ρ' = ^-6 ^ij_ρ ≡^-6 ε^ij_     k B_ρ^k ,
for the electric-like gravimetric field $E_{\rho}^{i}$ and magnetic-like gravimetric field $B_{\rho}^{i}$, and
_ρ^  μ     ≡    2 _ρ^  μ + g_h^2 2 _ρ^  μ ≃^-6 _ρ^  μ,   _ρ^  μ = _ρ^  μ + g_h^2 2 _ρ^  μ ,
for the hyper-gravimetric gauge field current $\hJ_{\rho}^{\; \mu}$. Where $\hT_{\rho}^{\; \mu}$ represents Einstein-type energy-momentum tensor of matter field in four-dimensional spacetime, which can be deduced from $\hmT_{\fP}^{\; \fM}$ defined in Eq.(<ref>) via the hyper-stress energy-momentum tensor $\mT_{\fM\fN}$ given in Eq.(<ref>). While $\hG_{\rho}^{\; \mu}$ denotes pure gravimetric field tensor, which can be resulted from the tensor $\hat{\mG}_{\fP}^{\; \fM}$ given in Eq.(<ref>) by fixing the scaling gauge symmetry to Einstein basis shown in Eq.(<ref>).

In light of the vector notations for both electromagnetic-like gravitational field and current, i.e., $\fE_{\rho} \equiv (E_{\rho}^1, E_{\rho}^2, E_{\rho}^3)$, $\fB_{\rho} \equiv  (B_{\rho}^1, B_{\rho}^2, B_{\rho}^3)$ and $\hfJ_{\rho} \equiv (\hJ_{\rho}^1, \hJ_{\rho}^2, \hJ_{\rho}^3)$, the geometric gauge-type gravitational equations can be rewritten as follows:

∇·_ρ ≃-_ρ^0 ,

∇×_ρ - _0_ρ ≃-_ρ .
From the Bianchi identity,
_μ _ρ^νσ  + _ν _ρ^σμ + _σ _ρ^μν = 0, we obtain the following gauge-type gravitational equations:

∇·_ρ = 0 ,

∇×_ρ + _0_ρ = 0 .

Therefore, the Abelian-like geometric gauge-type gravitational equations presented in Eqs.(<ref>) and (<ref>) in four dimensional spacetime are truly analogous to Maxwell equations of electromagnetic field, which provide electric-like and magnetic-like gravimetric field equations in an appropriate approximation. Unlike the electromagnetic field with source current arising solely from electric charged field, the gravimetric gauge field current $\hat{\mJ}_{\rho}^{\; \mu}$ as source current contains not only the current tensor $\hat{\mT}_{\rho}^{\; \mu}$ arising from all basic matter fields but also from the pure gravimetric gauge field current tensor $\hat{\mG}_{\rho}^{\; \mu}$.

§  CONFORMALLY FLAT GRAVIGAUGE HYPER-SPACETIME WITH EVOLVING GRAVISCALAR FIELD AS NONSINGULARITY BACKGROUND FOR EVOLUTION OF EARLY UNIVERSE AND SCALING GAUGE FIELD AND $\CQ_C$-SPIN SCALING FIELD AS BASIC CONSTITUENTS OF DARK UNIVERSE AND QUANTUM COSMIC MATTER IN HYPERUNIFIED FIELD THEORY

In the previous sections, we have demonstrated that the entirety unitary gauge as an appropriate gauge prescription allows us to remove all unphysical degrees of freedom which are caused from inhomogeneous hyperspin gauge symmetry WS(1,$D_h$-1) and hidden general linear group symmetry GL(1,$D_h$-1, R) in hyperunified field theory built within the framework of gravitational quantum field theory (GQFT)[5]. In this section, we shall discuss various gauge prescriptions for the scaling gauge symmetry in the hyperunified field theory under entirety unitary gauge. In Einstein-type basis, we will show that the conformally flat gravigauge hyper-spacetime with evolving graviscalar field provides nonsingularity background spacetime to describe the evolution of the early universe[31, 32, 33, 34]. Furthermore, we will demonstrate how the early universe undergoes an inflationary expansion in light of the co-moving cosmic time. In particular, the scaling gauge field is found to be as a dark matter candidate when inflationary universe comes to an end. It is interesting to corroborate that once the potential of $\cQ_c$-spin scaling field falls into the minimum, it not only stops inflationary expansion of universe but also provides cosmic energy density as dark energy candidate.

§.§  Hyperunified field theory under entirety unitary gauge with scaling gauge fixing and conformally flat gravigauge hyper-spacetime under background basis with fundamental mass scale

It has been shown that the entanglement-correlated and translation-like $\cW_e$-spin group symmetry W$^{1,D_h-1}$ is an Abelian-type subgroup of inhomogeneous hyperspin symmetry WS(1,$D_h$-1) of entangled qubit-spinor field. The $\cW_e$-spin invariant-gauge field $\fA_{\fM}^{\;\;\fA}$ as genesis of hyper-gravigauge field reveals the gauge-gravity correspondence. Meanwhile, the gravity-geometry correspondence in curved Riemannian hyper-spacetime indicates that the gravitational interaction can equivalently be described by the hyper-gravimetric gauge field when fixing the hyperspin gauge symmetry under flowing unitary gauge, so that the hyper-gravimetric gauge field is determined from the symmetric hyper-gravigauge field via the simple relation $\mH_{\fM\fN} \equiv (\bfA)^2_{\fM\fN}$ ($\bfA_{\fM\fA}=\bfA_{\fA\fM}$) in which the symmetric hyper-gravigauge field has the same physical degrees of freedom as hyper-gravimetric gauge field. To remove all unphysical degrees of freedom, the duality gauge prescription for general linear group symmetry GL(1,$D_h$-1, R) is proposed to freeze over the flowing unitary gauge for hyperspin gauge symmetry SP(1,$D_h$-1), which leads to the entirety unitary gauge prescription and corroborates the gauge-geometry duality. When taking the gauge fixing conditions under entirety unitary gauge as basic gauge prescription, we are allowed to remove in principle all unphysical degrees of freedom. Eventually, the fundamental symmetry of hyperunified field theory turns out to be an associated symmetry characterized by the global inhomogeneous hyperspin symmetry WS(1,$D_h$-1) in association with global Poincaré-type group symmetry PO(1,$D_h$-1) together with scaling symmetry SC(1),  where the symmetry transformations of groups SP(1,$D_h$-1) and SO(1,$D_h$-1) must be coincidental so as to keep the hyperunified field theory under entirety unitary gauge prescription.

The scaling gauge symmetry appears to be a hidden gauge symmetry in the hyperunified field theory expressed in terms of hidden scaling gauge formalism, which indicates that we can in principle choose any convenient scaling gauge fixing condition via appropriate scaling gauge transformation. Namely, under the entirety unitary gauge, we are allowed to further make a typical scaling gauge transformation $\xi_u(x)$ for both graviscalar field $\phi(x)$ and dimensionless symmetric hyper-gravigauge field $\bchi_{\fM\fA}(x)$, i.e.:
ϕ(x) →ϕ_u(x) ≡ξ_u(x) ϕ(x) ,   _(x) →_^u(x) ≡ξ_u^-1(x)_(x) ,

_=_ = ϕ(x)_(x) ≡ϕ_u(x)_^u(x) ,

_= ()^2_ = ϕ^2(x)χ_(x) ≡ϕ_u^2(x)χ_^u(x) ,
so that the determinant of $\bchi_{\fM\fA}^u(x)$ or $\chi_{\fM\fN}^u(x)$ is fixed to be the unit,
χ^u ≡( _^u(x) η^ ) = √( ( (-1)^D_h-1 χ_^u(x)) )  = 1  ,
which is referred to as unitary basis for the scaling gauge symmetry.

When fixing the scaling gauge symmetry to be in unitary basis under the entirety unitary gauge, we are able to express the action of the hyperunified field theory given in Eq.(<ref>) or Eq.(<ref>) and (<ref>) into the following form:

_     ≡    ∫[d^D_hx]   {  ^u  _ _^u  _-^ i_  _ - β_Q ϕ_u sinh(χ_s)    _ _-_

+     ϕ_u^D_h-4 [ -  ^u ' ^u ' 1/4g_H^2 _  _''^

+     _^u _^u  _'^u '_'^u ' η̃^''_'  1/4g_H^2  _^_''^'

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

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

+      1/2 ^u  ϕ_u^2λ_S^2 _ χ_s _χ_s - λ_D^2 ϕ_u^4 (χ_s)  ]   }

≡    ∫[d^D_hx]   {  ^u  _ _^u  _-^ i_  _ -  β_Q ϕ_u sinh(χ_s)    _ _-_

+     ϕ_u^D_h-4 [ -  ^u ' ^u ' 1/4g_H^2 _  _''^

+     g_H^-2 ^u  ( ϕ_u^2  _^u - (D_h-1)(D_h-2) _ϕ_u _ϕ_u )

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

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

+      1/2 ^u  ϕ_u^2λ_S^2 _ χ_s _χ_s - λ_D^2 ϕ_u^4 (χ_s)  ]   } ,
where we have used the following redefinitions:
_ →ϕ_u^-D_h-1/2 _ ,

_ ≡χ_^[]  ^ lnϕ^u + _ ,

χ_^[] ≡(_^u   ^u  -  _^u   ^u  )η_ ,
χ_s(x)     ≡     sinh^-1λ_Qϕ_1(x)/λ_S ϕ_u(x),

ϕ_1(x)/ϕ_u(x)      ≡    β_Q sinhχ_s(x) ,   β_Q ≡λ_S/λ_Q .
Where $\mOm_{\fM\fA\fB}$ is the hyperspin gravigauge field governed by the symmetric hyper-gravigauge field $\bchi_{\fM\fA}(x)$ in unitary basis. The rank-two tensor $\mR^u_{\fM\fN}$ is Ricci-type curvature tensor defined from the hyper-gravimetric field $\chi_{\fM\fN}^u(x)$ in unitary basis. In unitary basis, the duality gauge fixing condition given in Eq.(<ref>) is simplified to be,
^ χ_^u = 0 .
Such a unitary basis is considered to be a physically meaningful basis in calculating quantum effects within the framework of GQFT.

Alternatively, the scaling gauge invariance also allows us to make a special scaling gauge transformation $\xi_s(x)$ so as to relate the graviscalar field $\phi(x)$ with fundamental mass scale $\Mka$ as follows:

ϕ(x) →≡ξ_s(x) ϕ(x) .
When making the same scaling gauge transformation for $\bchi_{\fM\fA}(x)$,
_(x) →_^e(x) ≡ξ_s^-1(x)_(x),
we can rewrite the scaling gauge invariant symmetric hyper-gravigauge field and hyper-gravimetric gauge field as follows:
_(x)     =     _(x) ≡_^e(x) ,

_(x)     =     ()^2_(x) ≡^2 χ_^e(x) ,
which is referred to as Einstein-type basis for the scaling gauge symmetry. Where $\bchi_{\fM\fA}^{e}(x)$ and $\chi_{\fM\fN}^e(x)$ denote the symmetric hyper-gravigauge field and hyper-gravimetric field in Einstein-type basis.

When the scaling gauge fixing condition is chosen to be in Einstein-type basis, the action of the hyperunified field theory presented in Eq.(<ref>) can be rewritten into the following form:

_     ≡    ∫[d^D_hx] χ^e  ^D_h-4{ ^e  _ ^e_  _-^ i_  _ -  M_S sinh(χ_s)    _ _-_

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

+     1/2g_H^2 β_G^2  M_S^2 sinh^2χ_s    ^e ' ( _ - _ )(_'^ - _'^ )

-     ^e ' ^e ' 1/4g_W^2 _ _'' + 1/2 λ_Q^2 M_S^2 ( 1+ sinh^2χ_s ) ^e   _ _

+      1/2 ^e λ_Q^2 M_S^2 _ χ_s _χ_s - λ_D^2 ^4 (χ_s)    } ,
where $\mR^e_{\fM\fN}$ is Ricci-type curvature tensor defined from the hyper-gravimetric field $\chi_{\fM\fN}^e(x)$ in Einstein-type basis. We have also used the following redefinitions:
χ_s(x)  ≡sinh^-1ϕ_1(x)/M_S,

ϕ_1(x)/M_S   ≡sinhχ_s(x) ,

M_S ≡λ_S/λ_Q = β_Q ,   β_Q ≡λ_S/λ_Q ,
with $M_S$ regarded as a fundamental graviscaling mass scale. Where $\chi_s(x)$ is characterized by the $\cQ_c$-spin scalar field $\phi_1(x)$ in Einstein-type basis.

§.§  Geometry of gravigauge hyper-spacetime and the dynamics of graviscalar and $\cQ_c$-spin scalar fields with conformally flat gravigauge hyper-spacetime as background spacetime

To understand the evolution of universe, let us turn to study the geometry of locally flat gravigauge hyper-spacetime $\fG_h$. In such a locally flat gravigauge hyper-spacetime, the gravitational interaction is considered as an emergent interaction via non-commutative geometry characterized by the hyperspin gravigauge field $\fOm_{[\fC\fD]}^{\fA}$ shown in Eq.(<ref>), i.e.:
[_  , _ ] = _[]^ _ , with $\heth_{\fC}$ the hyper-gravicoordinate derivative operator in locally flat gravigauge hyper-spacetime $\fG_h$. The hyperspin gravigauge field $\fOm_{[\fC\fD]}^{\fA}$ as structure factor of non-Abelian Lie algebra is shown to behave as an auxiliary field in the hyperunified field theory formulated in terms of hidden coordinate formalism, which is determined through the constraint equation of hyperspin gravigauge field $\fOm_{[\fC\fD]}^{\fA}$ as shown in Eq.(<ref>).

Let us now define the gauge and scaling invariant line element in locally flat gravigauge hyper-spacetime with the gravigauge hyper-basis $\{\delta\vka^{\fA}\}$ or $\{\delta\zeta^{\fA}\}$ as follows:
δ^2 ≡^-2 δ^ η_ δ^ = ϕ^2/^2 δζ^ η_ δζ^,
where the fundamental mass scale $\Mka$ as multiplying factor is introduced to make the gauge and scaling invariant line element to have the length dimension. Based on the biframe hyper-spacetime structure and biframe displacement correspondence from which the hyper-gravicoordinate displacement for hyper-gravivector field $\vka^{\fA}(x)$ in locally flat gravigauge hyper-spacetime is associated to the ordinary coordinate displacement in globally flat Minkowski hyper-spacetime through $\cW_e$-spin invariant-gauge field as shown in Eq.(<ref>), we can express the above invariant line element in terms of the coordinate hyper-basis as follows:
δ^2    =     ^-2dx^ _^   η_ _^    dx^= ^-2 dx^ _ dx^

=     ϕ^2/^2 dx^  χ_^   η_ χ_^    dx^= ϕ^2/^2 dx^ χ_  dx^ ≡d^2l_S ,
with $\mH_{\fM\fN}$ (or $\chi_{\fM\fN}$) the hyper-gravimetric tensor field,
_ =  _^   η_ _^   = ϕ^2 χ_^  η_  χ_^   = ϕ^2 χ_ .

By taking the special scaling gauge transformation $\xi_s(x)$ as scaling gauge fixing condition, we can relate the graviscalar field $\phi(x)$ to the fundamental mass scale $\Mka$ as follows:
ϕ(x) →ξ_s(x)ϕ(x) = , and meanwhile, the dimensionless symmetric hyper-gravigauge field under the entirety unitary gauge is fixed to be in Einstein-type basis,
_^  (x) →ξ^-1_s(x) _^  (x) = _^e  (x). So that the line element can be rewritten into the following form:
δ^2≡d^2l_S =  χ_^e dx^ dx^,

χ_^e (x) =_^e  (x) η_ _^e  (x)  ,
which defines the line element in Einstein-type basis.

To study the evolution of early universe with the hypothesis of conformally flat hyper-spacetime as background gravigauge hyper-spacetime, we shall analyze the basic properties of background fields and find out their solutions by solving their field equations of motion. It is familiar and useful to work in Einstein-type basis with the following background structure:
⟨_(x) ⟩= 0 ,   ⟨_μ ⟩= 0 ,   ⟨χ_s(x) ⟩= ν_S(x) ,

⟨_(x) ⟩≡⟨^e_(x) ⟩= a_s(x) η_ ,

⟨_(x) ⟩≡^2 ⟨χ_^e(x) ⟩= ^2 a_s^2(x) η_ ,

⟨_^(x) ⟩= η_^[]  ^ ,   η_^[] ≡η_^   η_^   - η_^   η_^   ,
where $a_s(x)$ is regarded to be a background graviscaling field and $\nu_S(x)$ characterizes the scaling invariant background $\cQ_c$-spin scaling field. From the gravitational origin of hyperspin gauge symmetry through the hyperspin gravigauge field $\fOm_{\fM}^{\fA\fB}(x)$, we can obtain the following relations:
⟨_^(x) ⟩=  η_^[] ^,   ^ ≡^lna_s(x),

^ ≡^ + ^ = ^lna_s(x) + ^,
where $\mA_{\fP}$ reflects the background feature of hyperspin gauge field and $\tmA^{\fP}$ denotes a possible difference from the background feature of hyperspin gravigauge field $\mOm^{\fP}$.

Such a background structure forms background gravigauge hyper-spacetime with the following background line element:
⟨d^2l_S ⟩=  η_  ⟨_^e  (x) _^e  (x) ⟩dx^ dx^  = a_s^2(x)  η_   dx^ dx^  ,
which brings on conformally flat Minkowski hyper-spacetime governed by the background graviscaling field $a_s(x)$. Such a background gravigauge hyper-spacetime distinguishes from globally flat Minkowski hyper-spacetime introduced as inertial reference frame of coordinates.  Both Ricci-type tensor and Ricci-type scalar tensor are found to have the following forms:
⟨_^e ⟩    =     (D_h-2) d__ + ( d_^ + (D_h-1) _^ ) η_

=     (D_h-2) ( __lna_s - _lna_s _lna_s )

+     (^2lna_s +  (D_h-2) _lna_s ^lna_s ) η_ ,

⟨^e ⟩    =     2(D_h-1) d_^ + D_h(D_h-1) _^

=     2 (D_h-2) ^2lna_s +  (D_h-1)(D_h-2) _lna_s ^lna_s ,

d__     ≡    __ - __ = __ lna_s -
_lna_s _lna_s ,
and the background field strength of hyperspin gauge field is given by,
⟨_^ ⟩=  d_^ η_^[] - d_^ η_^[] - _^ η_^[] ,

⟨_^ ^_ ⟩= 4(D_h-2) d__ d^^ +
4 (d_^)^2

+ 8(D_h-1) d_^ _^ + 2D_h(D_h-1)  (_^)^2,

d__ ≡__ - __ .

From the equation of motion presented in Eq.(<ref>) for the hyperspin gauge field $\cA_{\fM}^{\fA\fB}(x)$, we arrive at the following equation of motion for the background field $\langle \cA_{\fM}^{\fA\fB}(x) \rangle = \eta_{\fM\fP}^{[\fA\fB]} \mA^{\fP}$:

[d_^ ^ - d_d^^ - 2_d^^ - (D_h-4) _d^^ + (D_h-4) _^ ^  ]  η_^[]

+  [ d^d_^ - _d^^ - (D_h-4) d_^ ^  ] η_^[] = β_G^2M_S^2 a_s^2  sinh^2ν_S   ^ η_^[].

Following along the equation of motion presented in Eq.(<ref>) or Eq.(<ref>) for the hyper-gravigauge field $\fA_{\fM}^{\; \fA}(x)$ and also the alternative equation of motion shown in Eq.(<ref>) or Eq.(<ref>) for the hyper-gravimetric field $\mH_{\fM\fN}(x)$, we obtain the following equation of motion for the background field:

^2 a_s^2 { (D_h-2) d__ - [ (D_h-2) d_^ + 1/2(D_h-1)(D_h-2) _^ ] η_  }

=  (D_h-3) d__ d_^ - d__ d^_  +  ( d__ + d__) ( d_^ + (D_h-2) _^ )

- [  1/2(D_h-4) d__ d^^ + (D_h-2) d_^ _^ ) + 1/4(D_h +1)(D_h - 4)  (_^)^2

+ 1/2 (d_^ )^2  ] η_  + 1/2 β_G^2 M_S^2 sinh^2ν_S   [ (  2d_^ + (D_h-1) _^ + (D_h-2) _^ ) η_

- d_ _  - d_ _ - (D_h-1) ( _ _ + _ _ - 2_ _  ) ]

- 1/2g_H^2 ^2 a_s^2 [ λ_S^2 _ν_S _ ν_S   - (1/2 λ_S^2 _ν_S ^ ν_S  - λ_D^2 ^2 a_s^2 (ν_S) ) η_ ] ,
where $\cF(\nu_S)$ represents the total dimensionless potential function.

The equation of motion for the dimensionless scaling field $\chi_s(x)$ presented in Eq.(<ref>) leads to the following equation of motion for the background $\cQ_c$-spin scaling field $\nu_S(x)$:
^2 ν_S + (D_h-2) ^_ν_S     =     -λ_D^2 λ_S^-2^2 a_s^2 /ν_S (ν_S)

+     2 (D_h-1)g_H^-2 β_G^2λ_Q^-2 sinhν_Scoshν_S   ^2 .

It is still hard to solve analytically the above equations of motion in the general case. In particular, the equations of motion for the background hyperspin gauge field and hyper-gravigauge field remain to be quite complicated. We should examine the most interesting case by firstly concentrating on the equation of motion for the background $\cQ_c$-spin scaling field $\nu_S(x)$. For that, let us begin with considering rational hypotheses for the total dimensionless potential function and background hyperspin gauge field $\mA^{\fP}$.

For the potential function $\cF(\nu_S)$ of background $\cQ_c$-spin scaling field $\nu_S(x)$, it is proposed to have the following feature:

/ν_S (ν_S) →0,   (ν_S) →1,   ν_S ≫1 ,
which indicates that $\cF(\nu_S)$ is expected to vary slowly as the function of background $\cQ_c$-spin scaling field $\nu_S(x)$ for $\nu_S \gg 1$, so that the potential function can approximately be regarded as a slowly varying cosmological constant.

For the background hyperspin gauge field $\mA^{\fP}$, it is simply supposed to be the same as the background hyperspin gravigauge field $\mOm^{\fP}$ based on the gravitational origin of hyperspin gauge field $\cA_{\fM}^{\fA\fB} \equiv \fOm_{\fM}^{\fA\fB} + \fA_{\fM}^{\fA\fB}$, which indicates the following relation:
⟨_^ -_^ ⟩= ⟨_^ ⟩→0,      ^ - ^ ≡^ →0 .

From the above hypotheses and analyses, we arrive at the following simplified equation for the background $\cQ_c$-spin scaling field $\nu_S(x)$:
^2 ν_S + (D_h-2) ^_ν_S ≃0 ,
which brings on the following solution in a good approximation,
_ν_S ≃0 .
Meanwhile, the equation of motion for the background hyperspin gauge field $\langle \cA_{\fM}^{\fA\fB}(x) \rangle$ presented in Eq.(<ref>) is simplified for $\tmA^{\fP} \to 0$ to be as follows:
d_^ ^ - d_d^^ - 2_d^^ - (D_h-4) _d^^ →0 ,
which can be rewritten into the following form:

3^2 a_s _ a_s - a_s^2 _ a_s - (D_h-3) ( a_s__ a_s - 2_a_s _a_s ) ^lna_s = 0 .

It can be checked that the equation of motion for the background hyper-gravigauge field presented in Eq.(<ref>) can be simplified in a good approximation into the following two equations:

^2 a_s^2 (D_h-2) d__  ≃(D_h-3) d__ d_^ - d__ d^_

+  ( d__ + d__) ( d_^ + (D_h-2) _^ )

^2 a_s^2 [ (D_h-2) d_^ + 1/2(D_h-1)(D_h-2) _^ ]

≃1/2(D_h-4) d__ d^^ + (D_h-2) d_^ _^

+ 1/4(D_h +1)(D_h - 4)  (_^)^2  + 1/2 (d_^ )^2  +  1/2g_H^2 λ_D^2 ^4 a_s^4  .

To obtain an analytic solution, let us first consider the simplified equation (<ref>). Suppose that Eq.(<ref>) holds in any dimension $D_h$ of hyper-spacetime, which leads the background scaling factor $a_s(x)$ to satisfy the following equations:
a_s__ a_s - 2_a_s _a_s = 0 ,

3^2 a_s _ a_s - a_s^2 _ a_s = 0 ,
which are found to be equivalent to the following two identities:
d__ = 0,   2^ _^  - ^2^ =0 .
With these two identities, two equations in Eq.(<ref>) can be simplified further into the following single equation:

^2 a_s^2 (D_h-1)(D_h-2) _^ ≃g_H^2 λ_D^2 ^4 a_s^4 + 1/2(D_h +1)(D_h - 4)  (_^)^2 .

A general solution for the background scaling factor $a_s(x)$ is found to be as follows:

a_s(x) = ± m_κ// 1 ∓κ_( x^ - x_o^ )   ,
where $\kappa_{\fM}$ is regarded as basic cosmic vector, which defines fundamental cosmic mass scale $m_{\kappa}$ via the Lorentz-type invariant scalar product as follows:
κ^2 = κ_κ^ ≡m_κ^2 ^2,
where $\cka$ may be referred to as invariant proper speed for convenience of mention (we will take it as a basic unit with the convention $\cka=1$). $x_o^{\fM}$ is viewed as a reference coordinate position chosen appropriately by an observer in conformally flat background hyper-spacetime, which leads the solution to be translational invariant,
x^ →x^'  =  x^ + ϖ^,    x_o^ →x_o^'  =  x_o^ + ϖ^,

a_s(x) →a_s (x + ϖ) = a_s(x) .
It can be verified that the general solution presented in Eq.(<ref>) becomes consistent when the constant $\apk$ meets to the constraint obtained from the equation given in Eq.(<ref>), i.e.:

4g_H^2λ_D^2 ^4  - 4 (D_h-1)(D_h-2) ^2 +  (D_h+1)(D_h-4) = 0 ,
which implies that the constant $\apk$ depends not only on the coupling constants but also on the dimension of hyper-spacetime. For convenience of mention, $\apk$ is referred to as conformal scaling constant.

It is easy to check that the equations of motion have a discrete symmetry under $Z_2$ transformation: $a_s(x) \to - a_s(x)$, and also reflection symmetry for coordinates: $x^{\fM}\to - x^{\fM}$, $a_s(x)\to a_s(-x)$. So that there are in general four solutions for the background graviscalar field,

ϕ_o(x) ≡a_s(x) →±ϕ_o^±(x) = ± m_κ / 1 ∓κ_( x^ - x_o^ )   ,
where the conformal scaling constant $\apk$ is easily obtained from Eq.(<ref>) as follows:

^2 = (D_h-1)(D_h-2)/2g_H^2λ_D^2 { 1 ±( 1-  (D_h+1)(D_h-4) g_H^2 λ_D^2/(D_h-1)^2(D_h-2)^2  )^1/2 } .

It is obvious that when hyper-spacetime is reduced to be four-dimensional one, the conformal scaling constant $\apk$ gets the usual solution in four dimensional spacetime[5],
= √(6)/(g_Hλ_D) .
It is interesting to notice that for hyper-spacetime beyond four dimensions ($D-h>4$), the general solution Eq.(<ref>) for the background scaling factor is still valid even if the potential of background $\cQ_c$-spin scaling field is absent ($\lambda_Q =0$). In this case, the conformal scaling constant $\apk$ is simply given by,
^2 =  (D_h+1)(D_h-4)/4  (D_h-1)(D_h-2) .
In fact, for $g_H\lambda_D \alt O(1)$, the two solutions in Eq.(<ref>) can be expressed in a good approximation as follows:

α_κ+^2     ≃    (D_h-1)(D_h-2)/g_H^2λ_D^2 [   1 -  (D_h+1)(D_h-4) g_H^2 λ_D^2/2(D_h-1)^2(D_h-2)^2    ] ,

α_κ-^2     ≃     (D_h+1)(D_h-4)/4  (D_h-1)(D_h-2) [   1 +  (D_h+1)(D_h-4) g_H^2 λ_D^2/2 (D_h-1)^2(D_h-2)^2     ]  .
For the hyper-spacetime dimension $D_h=19$ in the hyperunified field theory, we can write down the conformal scaling constant $\apk$ as follows:
α_κ+     ≃    3√(34)/g_Hλ_D (  1 -  150 g_H^2 λ_D^2/18^2·17^2    )^1/2 ,

α_κ-     ≃     5√(3)/3√(34) (  1 +  150 g_H^2 λ_D^2/18^2·17^2     )^1/2  .

§.§  Evolution of early universe with evolving graviscalar and $\cQ_c$-spin scalar field in conformally flat gravigauge hyper-spacetime as nonsingularity background spacetime

To study the evolution of early universe, let us demonstrate firstly the basic property of background gravigauge hyper-spacetime with the following background line element:
⟨d^2l_S ⟩=  a_s^2(x)  dx^ η_  dx^  ,

a_s^2 (x) →(a_s^±(x))^2 =  ^2 m_κ^2/^2 / ( 1 ∓κ_( x^ - x_o^ ) )^2  ,
which is invariant under the following Poincaré-type transformation:
x^ →x^' = L^_     x^,   x_o^ →x_o^' = L^_     x_o^,   κ_ →κ'_= L_^     κ_,

x^ →x^' = x^ + ϖ^ ,   x_o^ →x_o^' = x_o^ + ϖ^ .
In general, the existence of basic cosmic vector $\kappa_{\fM}$ indicates that the background gravigauge hyper-spacetime is not isometric and homogeneous.

The Poincaré-type invariance of the scalar product  $\kappa_{\fM} ( x^{\fM} - x_o^{\fM} )$ enables us to define a conformal proper time $\htt$ as follows:
κ_ ( x^ - x_o^ ) ≡^2 (- _o) ,
where $\hmk$ is regarded as conformal proper cosmic mass scale, $\htt_o$ is considered as reference conformal proper time. In general, the definitions of $\htt$ ($\htt_o$) and $\hmk$ allow to have the following intrinsic conformal scaling invariant transformation:
→' = λ^-α  ,    _o →'_o = λ^-α  _o,    →'  = λ^α ,
where the power $\alpha$ is constant parameter. For $\kappa^2 = \kappa_{\fM}\kappa^{\fM} = m_{\kappa}^2 > 0$, one can always choose a typical reference frame by making a special Lorentz-type transformation, so that

κ_ =0,   = 1,2,⋯, D_h ,

= κ_0 = ,   = η,   _o = η_o,   α=0 ,
which is referred to as cosmic co-moving reference frame. Where $\eta$ is referred to as cosmic co-moving Lorentz time and $\eta_o$ as cosmic co-moving reference Lorentz time. Only in such a cosmic co-moving reference frame, the intrinsic conformal scaling transformation is fixed with $\alpha=0$.

In light of conformal proper time $\htt$, the temporal coordinate $x^0$ can be expressed as follows:
x^0 = /κ_0  - κ_I/κ_0 x^I ≡1/u_0 ( - u_I x^I )  ,

u_0 ≡κ_0/  ,   u_I ≡κ_I / ,
so that the coordinates $x^{\fM}$ and displacement $dx^0$ are replaced by,
x^ →^ = (, x^I),

dx^0 = 1/u_0 ( d- u_I dx^I ) .
The line element can be rewritten as follows:
⟨d^2l_S ⟩= a_s^2()  d^ χ_^c d^   ,

a_s^2(x) →(a_s^±())^2 =  ^2 m_κ^2/^2 / ( 1 ∓(- _o) ^2 )^2  ,
where $a_s(\htt)$ evolves as the graviscaling field of conformal proper time $\htt$ with $\hx^{\fM}$ as the coordinates in terms of conformal proper time $\htt$.  $\chi_{\fM\fN}^c$ is the conformal proper time background metric with the following form:
χ_^c = 1/u_0^2

1      -u_J

-u_I         u_0^2  χ_IJ^c

)   ,   χ_IJ^c = η_IJ + u_I u_J/u_0^2  ,
which is determined by the basic cosmic vector $\kappa_{\fM}$.

In general, the conformal proper time background metric is non-diagonal. Only in the cosmic co-moving reference frame, the background gravigauge hyper-spacetime becomes isotropic and homogenous with the following features:
κ_ →κ_0 =,   →η,   ^ →x_η^ = (η, x^I) ,   d^ →dx_η^ ,

χ_^c →η_ ,   ⟨d^2l_S ⟩= a_s^2()  d^  χ_^c d^ = a_s^2(η) dx_η^ η_ dx_η^   ,
where the Lorentz-type symmetry is fixed to specify both Lorentz time $t$ and conformal proper time $\htt$ to be the same as cosmic co-moving Lorentz time, i.e., $\htt = t = \eta$.

Let us now investigate the feature of background graviscaling field in the case $\nu_S(x) \gg 1$. Formally, it appears to have a singularity when the conformal proper time or cosmic co-moving Lorentz time relative to its reference time approaches to the epoch given by the inverse of conformal proper cosmic mass scale or fundamental cosmic mass scale. Namely, when the conformal proper time or cosmic co-moving Lorentz time goes to the following critical point:
→_c^± ≡_o ±1/^2 ≡_o ±,   ≡1/^2,

η→η_c^± ≡η_o ±1/^2 ≡η_o ±η_κ ,   η_κ ≡1/^2 ,
the conformal size of background gravigauge hyper-spacetime becomes infinitely large,
a_s(η)→a_s^±(η_c^± ) ,    a_s^±(_c^± ) →∞.
In this situation, the distance relative to a reference point traveled by a field with invariant proper speed $\cka$ is given as follows:
= η_κ = 1/,      = = 1/  ,
where $\Lk$ is referred to as co-moving cosmic horizon or $\hLk$ as conformal proper cosmic horizon in background gravigauge hyper-spacetime.

Nevertheless, we will show that in view of conformal proper cosmic time or co-moving cosmic time, it takes an infinitely long time to reach the conformal proper cosmic horizon $\hLk$ or co-moving cosmic horizon $\Lk$. Let us now introduce conformal proper cosmic time $\htau$ and co-moving cosmic time $\tau$ as follows:
d≡a_s() d,       dτ= a_s(η) dη,

a_s() →a_s^±() = m_κ // 1 ∓^2 (- _o)   = m_κ // 1 ∓(- _o)/ ,

a_s(η) →a_s^±(η) = m_κ // 1 ∓^2 (η- η_o)   = m_κ // 1 ∓(η- η_o)/ ,
which enables us to obtain the following relation between the conformal proper time $\htt$ and conformal proper cosmic time $\htau$ or between the cosmic co-moving Lorentz time $\eta$ and co-moving cosmic time $\tau$:
1/1 - ^2 (- _o) = e^^2 (- _o) ,    -∞≤(-_o) ≤,     -∞≤-_o ≤∞,

1/1 - ^2 (η- η_o) = e^^2 (τ- τ_o) ,    -∞≤(η-η_o) ≤,     -∞≤τ-τ_o ≤∞,

1/1 + ^2 (- _o)  = e^-^2 (- _o) ,    -≤(-_o) ≤∞,     -∞≤-_o ≤∞,

1/1 + ^2 (η- η_o)  = e^-^2 (τ- τ_o) ,    -≤(η-η_o) ≤∞,     -∞≤τ-τ_o ≤∞,
where the normalization is fixed by the integration condition at initial points, i.e, $\htt = \htt_o$ and $\htau = \htau_o$ or $\eta = \eta_o$ and $\tau = \tau_o$ when carrying out the integrations over $\htt$ and $\htau$ or $\eta$ and $\tau$. $\tau_o$ is referred to as co-moving reference cosmic time and $\htau_o$ as conformal proper reference cosmic time.

The conformal scaling factor in light of conformal proper cosmic time or co-moving cosmic time is found to have the following form:
a_s() →a_s^±() = a_o e^±c(-_o) = a_o e^±c (τ- τ_o) = a_s^±(τ),

a_o = / ≡/ = / ≡/ = /  ,
where $a_o$ is regarded as reference conformal scaling factor at co-moving reference cosmic time. We have introduced the following definitions:

≡/,   ≡1/ ,

≡/ /  ,    ≡1/  ,
where $\Hka$ is referred to as primordial Hubble-type mass scale and $\hHk$ as conformal proper primordial Hubble-type mass scale, and $\lka$ is referred to as primordial Hubble-type length scale and $\hlka$ as conformal proper primordial Hubble-type length scale.

It is noted that the conformal scaling factor $a_s(x)$ is regarded as intrinsic scaling invariant quantity with $a_s(\htau) = a_s(\tau)$, while the conformal proper primordial Hubble-type mass scale $\hHk$ is not an intrinsic scaling invariant quantity as its value depends on the choice of conformal proper cosmic time $\htau$. Only in the cosmic co-moving reference frame, the conformal proper primordial Hubble-type mass scale $\hHk$ turns to be the primordial Hubble-type mass scale $\Hka = \Mka/\apk$ which is regarded as primordial Hubble-type constant.

It is clear that when the distance traveled via the invariant proper speed $\cka$ in background spacetime is closing to the co-moving cosmic horizon $\cka |\eta -\eta_o| \to \Lk$, it needs to take an infinitely large co-moving cosmic time $|\tau -\tau_o| \to \infty$.

The line element in light of the conformal proper cosmic time can be expresses as follows:
⟨d^2l_S ⟩=  χ_^c() d_τ^ d_τ^  ,   _τ^ = (c, x^I) ,
where the metric tensor $\chi_{\fM\fN}^c (\htau)$ is given by,
χ_^c () = 1/u_0^2

1      - a_s()  u_J

- a_s()  u_I          a_s^2()  u_0^2  χ_IJ^c

)  ,   χ_IJ^c = η_IJ + u_I u_J/u_0^2  ,
which indicates that the background gravigauge hyper-spacetime is in general not isotropic in terms of the conformal proper cosmic time $\htau$.

Only in the cosmic co-moving reference frame with $u_I =0$ and $u_0=1$, the co-moving cosmic time $\tau$ defined from the cosmic co-moving Lorentz time $\eta$ describes an isotropic and homogeneous background gravigauge hyper-spacetime with the following line element:
⟨d^2l_S ⟩=  χ_^c(τ) dx_τ^dx_τ^ =  ^2 dτ^2 -  a_s^2(τ)  (dx^I)^2 ,
with the definitions,
x_τ^ = (τ, x^I),   χ_^c (τ) =

1         0

0          a_s^2(τ) η_IJ

)  .
For the positive graviscaling factor,
a_s^+(τ) = a_o  e^^2 (τ-τ_o) ,
which brings on the well-known Friedmann-Lemaître-Robertson-Walker (FLRW) metric for describing inflationary expansion of early universe as the time arrow from the past or reference point to the future. Obviously, the negative solution for the graviscaling factor $a_s^{-}(\tau) = a_o\, e^{-\Hka \cka^2 (\tau-\tau_o)}$ characterizes deflationary expansion of early universe when the time arrow from the past or reference point to the future. Whereas once regarding such a negative solution as a time reversal one, it reflects inflationary expansion of universe for the time arrow from the future or reference point to the past.

In general, such an inflationary expansion of early universe is going to break down and end up eventually only when the potential $\cF(\nu_S)$ of background $\cQ_c$-spin scaling field $\nu_S(x)$ falls into a minimum value, where the background field $\nu_S(x)$ decreases and approaches to its minimum point $\nu_S(x) \to \nu_0 $, i.e.:

'(ν_S) ≡/ν_S (ν_S) = 0 ,      ν_S = ν_0 ,

(ν_S = ν_0 ) ≪1 .
So that the background graviscaling field as conformal scaling factor will not get into an infinitely large value since an inflationary expansion of early universe will stop before the conformal proper time (or cosmic co-moving Lorentz time) relative to its reference time reaches to the singularity point characterized by the conformal proper cosmic horizon (or co-moving cosmic horizon).

Let us suppose that the universe evolves from conformally flat gravigauge hyper-spacetime as nonsingularity background spacetime in which the potential of background $\cQ_c$-spin scaling field varies slowly for a large value $\langle \chi_s(x)\rangle = \nu_S(x) \gg 1$. Begin with such a nonsingularity background hyper-spacetime as inflationary early universe, the hyperunified qubit-spinor field and hyperspin gauge field as well as $\cQ_c$-spin scalar field and scaling gauge field behave as very heavy massive particles, so that all fundamental particles except hyper-gravigauge field are considered to decouple from interactions at the very beginning of universe. Nevertheless, when approaching to the end of inflationary expansion of early universe once the potential reaches to the minimum point $\nu_S = \nu_0$ with $\nu_0 \ll 1$, the hyperunified qubit-spinor field and hyperspin gauge field are expected to receive negligible masses in proportional to an extra small factor $ \sinh\nu_S \simeq \nu_S=\nu_0 \ll 1$.

It is noted that the above analytic solution for the background graviscaling field $a_s(x)$ is obtained in a good approximation for the slowly varying potential of background $\cQ_c$-spin scaling field, which describes the evolution of early universe and will be discussed further below. Similar inflationary scenarios with two or more general scalar fields based on the scaling gauge invariance have been studied extensively in refs.[35, 36, 37], which demonstrated that such kind of inflationary scenarios can provide a consistent explanation for the observed data. We will examine below an explicit potential of $\cQ_c$-spin scaling field to verify all required slow-roll conditions.

§.§   Evolution of early universe with inflationary expansion governed by the potential energy of $\cQ_c$-spin scalar field

At the very beginning of inflationary expansion of universe, all fundamental fields except the hyper-gravigauge field appears as very heavy particles with masses characterized by the fundamental graviscaling mass scale $M_S$, such an early universe is governed by the slowly varying potential energy of background scaling field $\nu_S$. The kinetic terms of all heavy massive particles are expected to be negligible smaller in comparison with their mass-like terms, so that all such basic fields are thought to decouple from interactions at the beginning of early universe. The slow-roll conditions are generally characterized by the following slow-roll parameters:
ϵ    =      1/2^2/g_H^2λ_Q^2 M_S^2 ( '(ν_S)/(ν_S) )^2 ≪1  ,

η    =     ^2/g_H^2λ_Q^2 M_S^2 ”(ν_S)/(ν_S) ≪1 .
When the background field $\nu_S$ goes through down to the magnitude which makes $\epsilon \sim 1$ and $|\eta| \sim 1$, the kinetic energy of $\cQ_c$-spin scaling field becomes compatible with the potential energy, which invalidates the slow-roll conditions. In this situation, the inflationary expansion of early universe starts to break down and the potential of $\cQ_c$-spin scaling field is quickly falling into the minimum point at $\nu_S=\nu_0$ with $\nu_0 \ll 1$. Meanwhile, the varying masses of hyperunified qubit-spinor field and hyperspin gauge field as well as $\cQ_c$-spin scalar field become negligible small as their masses are proportional to the small factor $\sinh\nu_0 \simeq \nu_0 \ll 1$, only the scaling gauge field obtains a finite mass characterized by the fundamental mass scale $\Mka$.

To see explicitly, let us rewrite the $\cQ_c$-spin scaling field around the minimum point $\nu_0$ as follows:
χ_s(x) ≡ν_0 + Φ(x)/M_S ,
where $\Phi(x)$ is regarded as quantized $\cQ_c$-spin scalar field. In this case, the action in Einstein-type basis is easily read from the action given in Eq.(<ref>) to be as follows:

_     ≡     ∫[d^D_hx]   χ^e ^D_h-4 _D_h ≡∫[d^D_hx]   χ^e ^D_h-4 {  ^e  _ ^e_  _-^ i_  _

-      m_ _ _-_
- 2M_S sinh( Φ/2M_S ) cosh(ν_0 + Φ/2M_S )    _ _-_

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

+     1/2g_H^2 μ_^2  ( _ -  _ )(_'^ -_'^ ) ^e '

+     1/2g_H^2 β_G^2  M_S^2 [sinh^2(Φ/M_S)  +  2 sinh(ν_0)sinh(Φ/M_S) cosh(ν_0 + Φ/M_S )   ]

·    ( _ -  _ )(_'^ -  _'^ ) ^e '

-     ^e ' ^e ' 1/4g_W^2 _ _'' + 1/2 μ_^2  _ _   ^e

+      1/2λ_Q^2 M_S^2 [sinh^2(Φ/M_S)  +  2 sinh(ν_0)sinh(Φ/M_S) cosh(ν_0 + Φ/M_S )   ]    _ _ ^e ,

+      1/2 ^e λ_Q^2 _ Φ_Φ-  Λ_Q^4
- 1/2 μ_Q^2 Φ^2

-      λ_D^2/β_Q^4 M_S^4 [  (ν_0 + Φ/M_S)  - (ν_0)  - 1/2 ”(ν_0) Φ^2/M_S^2   ] } ,
with the following definitions:
m_ = M_S sinh(ν_0) ,   μ_ = β_G M_S sinh(ν_0) ,

μ_ = λ_Q M_S  (1 + sinh^2(ν_0) )^1/2,   Λ_Q^4 = λ_D^2 ^4 (ν_0) ,

μ_Q^2 = λ_D^2/β_Q^4 M_S^2 ”(ν_0) ,   ”(ν_0) ≡ ^2 (χ_s)/^2 χ_s |_χ_s=ν_0 ,
where $m_{\fQH}$ is regarded as the mass of hyperunified qubit-spinor field in Einstein-type basis. $\mu_{\fA}$, $\mu_{\mW}$ and $\mu_{Q}$ characterize the masses of hyperspin gauge field, scaling gauge field and $\cQ_c$-spin scalar field in hyper-spacetime, respectively. $\cF(\nu_0 + \frac{\Phi}{M_S})$ is the potential of $\cQ_c$-spin scalar field. The $\Lambda_Q^4$ term which arises from the minimum potential reflects cosmic energy density in Einstein-type basis after inflation.

For a more concrete and realistic consideration, let us examine the following simple potential $\cF(\chi_s)$ of $\cQ_c$-spin scaling field $\chi_s(x)$:
(χ_s) = sinh^4(χ_s) + ^2/( ^-2 + sinh^2(χ_s) )^2  ,
with $\lak$ and $\lachi$ two constant parameters. The minimal condition of the potential is found to be as follows:
/χ_s (χ_s) =    2 sinh(2χ_s) ( ^-2 sinh^2(χ_s) - ^2 ) / ( ^-2 + sinh^2(χ_s) )^3 = 0,
which leads to the following solution on the minimum point $\chi_s \equiv \nu_0$:
sinh^2 χ_s = ^2 ^2 ,   χ_s ≡ν_0 = ±sinh^-1() .

For the background $\cQ_c$-spin scaling field $\langle \chi_s(x)\rangle = \nu_S(x)$, it can be checked that the above simple potential can satisfy all required properties shown in Eqs.(<ref>) and (<ref>) for inflationary expansion of early universe as long as the constant parameters $\lak$ and $\lachi$ meet to the following conditions:
≪1 ,   > 1 ,
which brings on the needed feature as follows:
/ν_S (ν_S) →0,     (ν_S) →1,    ν_S ≫1,

(ν_0) = ^4^2/1 + ^4^2  ≪1 .

In general, let us verify the slow-roll conditions by considering the so-called slow-roll parameters $\epsilon$ and $\eta$ (it should not be confused with the cosmic co-moving Lorentz time $\eta$ defined in Eq.(<ref>) for $\nu_S \gg 1$,

ϵ    =      1/2^2/g_H^2λ_Q^2 M_S^2 ( '(ν_S)/(ν_S) )^2 = 2/g_H^2λ_S^2 sinh^2(2ν_S) ( sinh^2ν_S -  ^2^2)^2 /(1 + ^2sinh^2ν_S )^2 ( sinh^4(ν_S) + ^2 )^2,

≃    8cosh^2(ν_S) /g_H^2λ_S^2 (1 + ^2sinh^2ν_S )^2 sinh^2ν_S  ≃8 ^2 ν_S/g_H^2λ_S^2^4 (1 + ^2 sinh^2ν_S)^2
∼128 g_H^-2λ_S^-2 ^-4 e^- 4ν_S ,

η    =     ^2/g_H^2λ_Q^2 M_S^2 ”(ν_S)/(ν_S)

=      4 (1 + ^2 sinh^2(ν_S) )cosh(2ν_S) ( sinh^2(ν_S) -^2^2 ) + 2sinh^2(2ν_S) ( 1 - 2^2sinh^2(ν_S) + 3^4^2 ) / g_H^2λ_S^2 ( 1+ ^2 sinh^2(ν_S) )^2 ( sinh^4(ν_S) + ^2 ),

≃     4 cosh(2ν_S) (1 + ^2 sinh^2(ν_S) ) + 8 cosh^2(ν_S) ( 1 - 2 ^2 sinh^2(ν_S) ) / g_H^2λ_S^2 (1 + ^2 sinh^2(ν_S) )^2 sinh^2(ν_S)

≃     4^2 cosh(2ν_S) - 16 ^2cosh^2ν_S /g_H^2λ_S^2 (1 + ^2 sinh^2ν_S)^2 ∼-32 g_H^-2λ_S^-2 ^-2 e^- 2ν_S ,
The slow-roll process holds for the case $\epsilon \ll 1$ and $|\eta| \ll 1$, which corresponds to the following conditions:

e^2ν_S ^2  g_Hλ_S ≫8√(2),    ϵ≪1 ,

e^ν_S g_Hλ_S  ≫4√(2) ,   |η| ≪1 ,
which show that when the slow-roll parameter $\eta$ satisfies the slow-roll condition $|\eta| \ll 1$, the slow-roll parameter $\epsilon$ automatically fits to the condition $\epsilon \ll 1$ as $e^{\nu_S}\lachi > 1$.

When the slow-roll parameters increase to close to the order of one, i.e., $\epsilon \sim 1$ and $|\eta| \sim 1$, the slow-roll conditions become invalid and the inflationary expansion of early universe starts to break down. As a consequence, the potential of $\cQ_c$-spin scaling field quickly runs into the minimum value around the minimum point $\nu_S = \nu_0 \ll 1$. In this case, the relevant cosmic energy density $\Lambda_Q^4$ after inflation is given by,

Λ_Q^4 ≡λ_D^2 ^4 (ν_0) =  ^4^2/1 + ^4^2  λ_D^2 ^4  ≪^4 ,      ^2≪1 .
The relevant masses of hyperunified qubit-spinor field and hyperspin gauge field as well as $\cQ_c$-spin scalar field and scaling gauge field in hyper-spacetime are characterized as follows:
m_ = M_S ≪M_S,

μ_ =  β_G M_S ≪M_S,

μ_Q = 2√(2) ^2 ( 1 + ^2 ^2)^1/2 / (1+^4^2 )  λ_D/β_Q^2 M_S  ≃2√(2) ^2 λ_D/β_Q^2 M_S ≪M_S ,

μ_ = λ_Q M_S  (1 + ()^2 )^1/2 ≃λ_Q M_S ∼M_S,
where we have used the following relation in obtaining the relevant mass of $\cQ_c$-spin scalar boson:
”(ν_0) ≡^2 (ν_S) /ν_S |_ν_S =  ν_0  =
8^4 ^2( 1 + ^2 ^2) / (1+^4^2 )^2  .
It is seen that the hyperunified qubit-spinor field and hyperspin gauge field as well as $\cQ_c$-spin scalar field get relevant masses much smaller than the fundamental graviscaling mass scale $M_S$, while the scaling gauge field can receive a very heavy mass characterized by the mass scale $M_S$.

Therefore, when the inflation ends up and the potential falls into the minimum, the initial high potential energy characterized by the fundamental mass scale $\Mka$ is transmuted into kinetic energies of fundamental fields which have much smaller masses in comparison with the fundamental mass scale $\Mka$. So that the gauge interactions of inhomogeneous hyperspin gauge field and hyperunified qubit-spinor field together with the interactions of $\cQ_c$-spin scalar field become active as fundamental interactions of nature, which are thought to play a significant role in radiative and matter dominated evolutions of the universe. In general, the scaling gauge field may still be out of the thermal equilibrium due to its very heavy mass.

§.§  Evolving universe with scaling gauge field as dark matter candidate and $\cQ_c$-spin scalar field as quantum cosmic matter and dark energy candidate

Let us suppose that all fundamental fields as functionals of coordinates in hyper-spacetime have localizing distribution amplitudes in extra spatial dimensions beyond four-dimensional spacetime. In general, the fundamental fields as functionals of coordinates are distributed with different localizing length scales $l_{A}$ ($A=5,\cdots, D_h$) in different extra spatial dimensions of hyper-spacetime. These localizing length scales in extra spatial dimensions are thought to form a manifold with a finite integrable volume $\cV_{D_h-4}\propto l_1\times l_2\times \cdots \times l_{D_h-4}$. As the present consideration is focusing on the evolution of universe, we shall not investigate extensively on the localizing distribution amplitudes of fundamental fields.

To simplify the discussion, let us consider the case that all fundamental fields will run from the high energy scale characterized by the fundamental mass scale $\Mka$ down to the mass scale $\mu_M$ determined by the maximal localizing length scale $\l_{M}=\mbox{Max}.\{ \l_{A} \} =1/\mu_M$. When the running energy scale undergoes to be below the minimal localizing energy scale $\mu_M$, i.e., $\mu < \mu_M$, all fundamental fields as effective fields appear to have motions and interactions in four-dimensional spacetime, which leads in principle the hyperunified field theory in hyper-spacetime to be effective hyperunified field theory at low energies in four-dimensional spacetime. In formal, we may display such an effective hyperunified field theory in Einstein-type basis as follows:

_     ≡    ∫[d^D_hx]   χ^e(x) ^D_h-4 _D_h = ∫[d^4x]   χ^e(x)  _D_h-4  ^D_h-4 _4D

≡    ∫[d^4x]   χ^e(x) _4D^eff ,
where $\cL_{4D}^{eff}$ represents the effective Lagrangian at low energies in four-dimensional spacetime.

It is useful to introduce a characteristic localizing length scale $\la$ or characteristic localizing mass scale $\Ma = 1/\la$ to characterize the whole volume $\cV_{D_h-4}$ as follows:
_D_h-4 ≡^D_h-4 = 1/^D_h-4 ∝l_1×l_2×⋯×l_D_h-4 ,

_D_h-4  ^D_h-4 ≡(/)^D_h-4 ≡ξ_α^-(D_h-4) ,   ≡/ ,
where $\xia$ is regarded as dimension-reduction scaling factor, which enables us to define, from the basic coupling constants $g_H$, $g_W$, $\lambda_S$ and $\lambda_Q$ of hyperunified field theory in hyper-spacetime, the effective coupling constants $g_h$, $g_w$, $\lambda_s$ and $\lambda_q$ in four-dimensional spacetime as follows:
g_h^2 ≡g_H^2 (/)^D_h-4 ≡g_H^2 ^D_h-4  ,

g_w^2 ≡g_W^2 (/)^D_h-4 ≡g_W^2 ^D_h-4 ,

λ_s^2 ≡λ_S^2 (/)^D_h-4 ≡λ_S^2 ^-(D_h-4),

λ_q^2 ≡λ_Q^2 (/)^D_h-4 ≡λ_S^2 ^-(D_h-4),

λ_d^2 ≡λ_D^2 (/)^D_h-4 ≡λ_D^2 ^-(D_h-4) .
It is manifest that the following combinations of coupling constants are independent of the dimension-reduction scaling factor:
g_Hλ_S = g_hλ_s,   g_Hλ_Q = g_hλ_q,   g_Hλ_D = g_hλ_d,

g_Wλ_S = g_wλ_s ,   g_Wλ_Q = g_wλ_q ,

β_Q≡λ_S/λ_Q = λ_s/λ_q,   λ_D/λ_Q = λ_d/λ_q ,
which are referred to as combined dimension-independent coupling constant.

It is noticed that the slow-roll conditions presented in Eqs.(<ref>)-(<ref>) are spacetime dimension independent as they are given by the combined dimension-independent coupling constant $g_H\lambda_S = g_h\lambda_s$.

To formulate the effective Lagrangian $\cL_{4D}^{eff}$ in four-dimensional spacetime, the kinetic terms of hyperspin gauge field and scaling gauge field as well as $\cQ_c$-spin scalar field are normalized via the following replacements:
(x) →g_h (x) ,   (x) →g_w (x) ,

χ_s(x) →λ_s^-1 χ_s(x)   or    Φ(x) →λ_q^-1Φ(x) ,
and the gravitational coupling $G_N$ in four-dimensional spacetime is defined as follows:
1/16πG_N ≡^2/g_h^2 = ^2/g_H^2 ^4-D_h ≡M_P^2/16π  = 1/2M_p^2 ,
which indicates that the gravitational coupling $G_N$ or Planck mass $M_P$ ( reduced Planck mass $M_p$) presented in four-dimensional spacetime depends in general on the dimension-reduction scaling factor $\xia$ and hyper-spacetime dimension $D_h$ when relating Planck mass $M_P$ to the fundamental mass scale $\Mka$ and basic coupling constant $g_H$ in the hyperunified field theory in hyper-spacetime.

With the above considerations and analyses, we now turn to study the scaling gauge field and $\cQ_c$-spin scalar field as basic constituents of universe and discuss their roles for the evolution of universe in an effective hyperunified field theory at low energies in four-dimensional spacetime.

Let us discuss firstly the property of scaling gauge field $\mW_{\fM}$. It is noticed that the scaling gauge field has no direct interactions with hyperunified qubit-spinor field and hyperspin gauge field. In the hidden scaling gauge formalism of hyperunified field theory as shown in Eq.(<ref>) with the redefinition $\mW_{\fM} \to \mW_{\fM} - \frac{1}{2}\p_{\fM}\ln (\lambda_S^2 \phi^2 + \lambda_Q^2\phi_1^2)$, the scaling gauge field interacts only with the scaling field $\chi_s(x)$ and hyper-gravigauge field. As the action of Eq.(<ref>) or Eq.(<ref>) possesses a discrete symmetry under $Z_2$ transformation, i.e., $\mW_{\fM} \to - \mW_{\fM}$, so that $\mW_{\fM}$ becomes stable and survives to present once it  is produced in the early universe. Therefore, the scaling gauge field is expected to be as dark matter candidate in the hyperunified field theory.

It can be checked that the primordial Hubble-type mass scale $\Hka$ and scaling gauge boson mass $M_W$ are given by,
=     / ≃(g_H λ_D) ((D_h-1)(D_h-2))^-1/2

=     (g_h λ_d) ((D_h-1)(D_h-2))^-1/2 ,

M_W     =     (g_w λ_q)( 1+ ^2^2 )^1/2 M_S  ≃(g_w λ_q) M_S

=     (g_W λ_Q) M_S =  (g_W λ_S) =  (g_w λ_s),
which indicates that both $\Hka$ and $M_W$ are independent of the dimension-reduction scaling factor $\xia$ as they are given by the combined dimension-independent coupling constants $g_h\lambda_q = g_H\lambda_Q$ and $g_w \lambda_q = g_W \lambda_Q$ or  $g_w \lambda_s = g_W \lambda_S$, respectively. It is clear that the magnitudes for both $\Hka$ and $M_W$ are characterized by the fundamental mass scale $\Mka$.

Let us examine a possible mechanism to produce the heavy scaling gauge boson $\mW(x)$ via the $\cQ_c$-spin scalar annihilation, i.e., $\Phi + \Phi\rightarrow \mW +\mW $, after inflation in the early universe. As analyzed in ref.[37], the cross section of the annihilation process dominated by the longitudinal mode of scaling gauge boson is given by,
σ∼g_w^4/E^2E^4/M_W^4 = E^2/^4λ_s^4, with $E$ as the typical energy of final scaling gauge boson $\mW(x)$.

To obtain the correct relic density, a high reheating temperature $T_R$ is expected to be reached when the potential energy is transmuted into the kinetic energies of fundamental fields which get very small masses after inflation. It was demonstrated in ref.[37] that in the instantaneous reheating limit, the reheating temperature $T_R$ is related to the primordial Hubble-type mass scale $\Hka$ through the relation, $T_R\sim \sqrt{\Hka M_p}$. The reheating temperature $T_R$ can maximally reach to be $T_R\sim 4\times 10^{15}\GeV$, which can produce abundantly the heavy scaling gauge boson $\mW(x)$ with the mass up to $M_W \sim 5\times 10^{16}\GeV$.

It is attributed to the discrete symmetry $Z_2$ with $\mW(x) \to -\mW(x)$ that the scaling gauge boson $\mW(x)$ becomes stable, which brings naturally such a heavy scaling gauge boson to be as dark matter candidate in the hyperunified field theory.

We now come to investigate basic properties arising from $\cQ_c$-spin scalar field. Let us consider a simple case that the smallness of constant $\lak$ is characterized just by two fundamental mass scales $M_S$ and $\mk$ in the present theory with the following relation:
= /M_S =   λ_Q/λ_S  =  λ_q/λ_s ,
where the fundamental graviscaling mass scale $M_S$ reflects ultraviolet energy scale in early universe and the fundamental cosmic mass scale $\mk$ describes infrared energy scale of present universe. Such a constant $\lak$ is regarded as basic cosmic constant.

The cosmic energy density $\Lambda_{\cQ_c}^4$ in four-dimensional spacetime is given as follows:

Λ__c^4     ≡   ^D_h-4 Λ_Q^4 = λ_d^2 ^4 (ν_0) =  λ_d^2 ^4^4^2/ 1+^4^2

≃    λ_q^2/λ_s^2 λ_d^2 ^4^2 ^2 =  1/2λ_q^2/λ_s^2 (λ_d g_h)^2 ^4 M_p^2 ^2 ≡λ_q^2/λ_s^2 λ_d^2 ^4^2 ^-2

=     1/2λ_q^2/λ_s^2 (λ_d g_h)^2^4 M_p^2 ^-2 =  1/2λ_Q^2/λ_S^2 (λ_D g_H)^2^4 M_p^2 ^-2 ≡3 ^2 M_p^2 ^-2,
which indicates that the cosmic energy density $\Lambda_{\cQ_c}^4$ in four dimensional spacetime depends on the dimension-reduction scaling factor $\xia$. Such a dependence is the same as that of Planck mass square in four-dimensional spacetime. Note that $\lambda_d g_h = \lambda_Dg_H$ and $\lambda_q/\lambda_s = \lambda_Q/\lambda_S$ are the combined dimension-independent coupling constants.

As the fundamental cosmic mass scale $\mk$ reflects the co-moving cosmic horizon of the universe, i.e., $\mk=\frac{1}{\Lk}= \frac{1}{\cka \eka}$, it is natural to expect that the cosmic energy density $\Lambda_{\cQ_c}^4$ brings on the basic constituent of universe as dark energy candidate, which can be achieved when the cosmic energy scale $\Lambda_{\cQ_c}$ fits to the current experimental observation with the magnitude $\Lambda_{\cQ_c} \sim 10^{-2.5}$eV.

In fact, it was shown extensively in ref.[38] that the dark energy characterized by the total co-moving cosmic horizon is viewed as holographic dark energy, which can provide a consistent explanation on the accelerated expansion of present universe and meanwhile overcome the so-called fine-tuning problem and coincidence problem[39]. When applying a similar analysis to the present consideration, the cosmic energy density $\Lambda_{\cQ_c}^4$ can present a consistent dark energy candidate as long as the combination of coupling constants satisfies the following condition:

≡1/√(6) λ_q/λ_s (λ_d g_h) ^2 10^1∼10^2,
which may be referred to as $\cQ_c$-spin cosmic dark energy for a short of mention.

As the scaling gauge field can serve as dark matter candidate in the hyperunified field theory, so that $\cQ_c$-spin cosmic dark energy not only becomes compatible with the so-called $\Lambda$CDM model but also enables us to understand both the fine-tuning problem and coincidence problem directly in connection with the inflationary expansion of early universe.

When the $\cQ_c$-spin cosmic dark energy $\Lambda_{\cQ_c}^4$ becomes dominant as it is just the case in the present universe, the cosmic energy scale $\Lambda_{\cQ_c}$ can be regarded as an ultraviolet (UV) cutoff of effective hyperunified field theory at low energies of present universe, which is proposed to meet a relationship between UV cutoff and infrared (IR) cutoff ($1/L$) with $L$ the size of volume $L^3$ such that an effective field theory should be a good description of Nature[40]. Such a relationship indicates that all physical states except those that have already collapsed to a black hole should be described by conventional quantum field theory in four-dimensional spacetime. From such a relationship, we arrive at the following condition in our present considerations:
Λ__c^4 = 3 ^2 M_p^2 ^-2 ≡M_p^2 L__c^-2 < M_p^2 L^-2 ,
which implies that the IR cutoff $1/L$ is constrained to be,
L < L__c ≡M_p/Λ__c^2 ∼10^33   ^-1 ,

L__c ≡/√(3)  ( 10^-1 -10^-2) ,
where $L_{\cQ_c}$ is regarded as the size of visible universe within the gravitational quantum field theory, which is shown to be less than the co-moving cosmic horizon $\Lk$ by one to two orders of magnitude. In general,  $L_{\cQ_c}$ may be referred to as visible co-moving cosmic horizon of present universe. It is clear that the effective hyperunified field theory as effective local gravitational quantum field theory can provide a good approximate description of present universe within the visible co-moving cosmic horizon $L_{\cQ_c}$. Meanwhile, such a cosmic energy density $\Lambda_{\cQ_c}^4$ as $\cQ_c$-spin cosmic dark energy matches to the spirit of holographic principle[41, 42] in four-dimensional spacetime.

As the cosmic energy scale $\Lambda_{\cQ_c} \sim 10^{-2.5}$eV presents an UV cutoff of present universe, we are able to determine the fundamental cosmic mass scale $\mk$ characterized by the co-moving cosmic horizon $\Lk^{-1}= (\cka \eka)^{-1}=\mk $ from both $\cQ_c$-spin cosmic dark energy and coupling constant parameter presented in Eqs.(<ref>) and (<ref>). The magnitude value of $\mk$ is given as follows:
= 1/ = 1/c = Λ__c^2/√(3)M_p  (10^-34 - 10^-35)   .

The mass of $\cQ_c$-spin scalar boson $\Phi(x)$ is found to be,
m__c     =      2√(2)λ_d/λ_s ^2 √(”(ν_0)) =
2√(2) λ_d/λ_s   ^2(1 + ^2 ^2)^1/2/(1+^4^2) ,

≃     2√(2) λ_dλ_q/λ_s^2 ^2 =  2√(2) λ_Dλ_Q/λ_S^2 ^2 ,
which is independent of the dimension-reduction scaling factor $\xia$ as it is given by the combined dimension-independent coupling constants $\lambda_d/\lambda_s = \lambda_D/\lambda_S$ and $\lambda_q/\lambda_s = \lambda_Q/\lambda_S$. The magnitude of $\cQ_c$-spin scalar boson mass is determined by the fundamental cosmic mass scale $\mk$ and combined dimension-independent coupling constants as follows:
m__c ≃2√(2) λ_d λ_q /λ_s^2 ^2 = 4Λ__c^2/λ_s g_hM_p ∼4/λ_s g_h ×10^-33  .
which indicates that the $\cQ_c$-spin scalar boson $\Phi(x)$ receives a tiny cosmic mass.

We may refer to $\cQ_c$-spin scalar boson $\Phi(x)$ as quantum cosmic matter, which is expected to play an essential role on the evolution of universe. This is because the $\cQ_c$-spin scalar boson $\Phi(x)$ as quantum cosmic matter not only has universal gravitational interaction with the hyper-gravigauge field but also interacts with the scaling gauge boson as dark matter candidate, and meanwhile it couples directly to the hyperspin gauge field and hyperunified qubit-spinor field.

It is interesting to notice that the hyperunified qubit-spinor field and hyperspin gauge field also receive tiny masses:
m_ = M_S sinhν_0 =   M_S = ≤(10^-34 - 10^-35)  ,

m_ = β_G  M_S sinhν_0  = β_G  M_S =  β_G ≤β_G (10^-34 - 10^-35)   ,
which are independent of dimension-reduction scaling factor.

We would like to address that all vector-like leptons and quarks contained in the hyperunified qubit-spinor field obtain a tiny cosmic mass $m_{\fQH}$ determined by the fundamental cosmic mass scale $\mk$ and coupling constant $\lachi$. Similarly, the hyperspin gauge field also acquires a tiny cosmic mass $m_{\fA}$ governed by the fundamental cosmic mass scale $\mk$ and combined coupling constant $\beta_G \lachi$, which means that all gauge fields including electromagnetic field and strong gauge fields receive tiny cosmic masses.

It is intriguing that the gluons and photon in such a hyperunified field theory become massive gauge bosons although their tiny cosmic masses appear unobservable small. The current experiment only provides the upper limits on the mass of photon to be around[49]:
m_γ < 10^-18  ∼10^-26  .