$\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)$ can be written as follows:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{E}}^{6}}$ $\displaystyle=$
$\displaystyle\int
d^{11}x\,\\{\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varSigma_{-}^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)-\lambda_{1}\phi_{1}(x)\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)\tilde{\varSigma}_{-}\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)\\},$
(304)

where $\varSigma_{-}^{\mathsf{A}}$ and $\tilde{\varSigma}_{-}$ matrices are
defined with $\varGamma$-matrices
$\varGamma^{\hat{\mathsf{A}}}=(\varGamma^{a},\varGamma^{A},\varGamma^{12})$
($\hat{\mathsf{A}}=0,1,2,3,5,\cdots,12$, $a=0,1,2,3,A=5,\cdots,11$). Their
explicit forms are given by,

$\displaystyle\varSigma^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},\quad\tilde{\varSigma}_{-}=\frac{1}{2}\tilde{\varGamma}\varGamma_{-},$
$\displaystyle\varGamma^{\mathsf{A}}=\;\sigma_{0}\otimes\Gamma^{\mathsf{A}},\quad\mathsf{A}=0,1,2,3,5,\cdots,10,$
$\displaystyle\varGamma^{11}=\,i\sigma_{1}\otimes\hat{\gamma}_{11},$
$\displaystyle\varGamma^{12}=-i\sigma_{2}\otimes\hat{\gamma}_{11}\equiv\tilde{\varGamma},$
$\displaystyle\varGamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{13}),\quad\hat{\gamma}_{13}=\sigma_{3}\otimes\hat{\gamma}_{11},$
(305)

with $\Gamma^{\mathsf{A}}$ ($\mathsf{A}=0,1,2,3,5,\cdots,10$) defined in
Eq.(VIII.3). The entangled hyperqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)$ satisfies the following self-
conjugated chiral conditions:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}^{c}(x)=C_{\mathsf{Q}_{\mathsf{E}}^{6}}\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{6}}^{T}(x)=\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x),$
$\displaystyle\hat{\gamma}_{13}\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)=\hat{\gamma}_{13}\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}^{c}(x)=-\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x),$
$\displaystyle C_{\mathsf{Q}_{\mathsf{E}}^{6}}=\sigma_{0}\otimes
C_{\mathsf{Q}_{\mathsf{E}}^{5}}=\sigma_{0}\otimes\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D}.$ (306)

We may refer to $\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)$ as entangled
hexaqubit-spinor field.

The action in Eq.(304) possesses an associated symmetry in which the
inhomogeneous hyperspin symmetry WS(1,10) is in association with inhomogeneous
Lorentz-type/Poincaré-type group symmetry PO(1,10). The whole associated
symmetry of the action can be expressed as follows:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,10)\,\widetilde{\Join}\,WS(1,10)\rtimes SG(1)\times SP(1)$ (307)
$\displaystyle=$ $\displaystyle SC(1)\ltimes P^{1,10}\ltimes
SO(1,10)\,\widetilde{\Join}\,SP(1,10)\rtimes W^{1,10}\rtimes SG(1)\times
O(1),$

where the group transformations of hyperspin symmetry SP(1,10) and Lorentz-
type group symmetry SO(1,10) must be coincidental as indicated by the symbol
“$\,\widetilde{\Join}\,$”. The group generators of WS(1,10) are simply
presented by the commutators of anti-commuting $\varGamma$-matrices,

$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}=\frac{i}{4}[\varGamma^{\mathsf{A}},\varGamma^{\mathsf{B}}],\quad\varSigma_{-}^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},$

with $\mathsf{A},\mathsf{B}=0,1,2,3,5,\cdots,11$. The ${\mathcal{Q}}_{c}$-spin
symmetry SP(1)$\cong$O(1) becomes manifest with transformation
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)\to-\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)$,
which is coincidental to the chirality property of entangled hyperqubit-spinor
field
$\hat{\gamma}_{13}\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)=-\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)$.

The hyperspin symmetry SP(1,10) can be decomposed into the following
subgroups:

$\displaystyle SP(1,10)\supset SP(1,3)\times SP(7)\cong SO(1,3)\times SO(7)$
$\displaystyle\supset SO(1,3)\times SU_{C}(4)\supset SO(1,3)\times SU_{C}(3),$
(308)

with the subgroup generators,

$\displaystyle\Sigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=0,1,2,3)\in
sp(1,3)\cong so(1,3)$
$\displaystyle\Sigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=5,\cdots,10)\in
sp(6)\cong so(6)\cong su_{C}(4).$ (309)

The unitary symmetry SUC(4) is regarded as color-spin charge symmetry of
lepton-quark state, which is correlated to extra six dimensions in Minkowski
hyper-spacetime of coordinates. The subgroup symmetry SUC(3) of SUC(4)
produces the strong interaction symmetry of three vector-like chromo-quarks in
SM.

To show explicitly the vector-like property of color-spin symmetry SUC(3), it
is useful to express the action in Eq.(304) into the following equivalent one
by using the self-conjugated pentaqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{5}}(x)$:

$\displaystyle{\mathcal{S}}_{\mathbf{Q}_{\mathsf{S}}^{5}}\equiv{\mathcal{S}}_{\mathsf{Q}_{\mathsf{E}}^{6}}$
$\displaystyle=$ $\displaystyle\int
d^{11}x\,\\{\frac{1}{2}\bar{\Psi}_{\mathbf{Q}_{\mathsf{S}}^{5}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\Gamma^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathbf{Q}_{\mathsf{S}}^{5}}(x)-\frac{1}{2}\lambda_{1}\phi(x)\bar{\Psi}_{\mathbf{Q}_{\mathsf{S}}^{5}}(x)\tilde{\Gamma}\Psi_{\mathbf{Q}_{\mathsf{S}}^{5}}(x)\\},$
(310)

where the $\Gamma$-matrices are given by,

$\displaystyle\Gamma^{\mathsf{A}}=\Gamma^{\mathsf{A}},\;\;\mathsf{A}=0,1,2,3,5,\cdots,10,$
$\displaystyle\Gamma^{11}=i\hat{\gamma}_{11},\quad\tilde{\Gamma}=I_{32},$
(311)

with $\Gamma^{\mathsf{A}}$ ($\mathsf{A}=0,1,2,3,5,\cdots,10$) defined in
Eq.(VIII.3) and $I_{32}$ the $32\times 32$ unit matrix. In obtaining the above
action, we have used the following identity resulting from the definitions
given in Eq.(VIII.4):

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}}^{5}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}-}^{5}}(x)+\Psi_{\mathsf{Q}_{\mathsf{E}+}^{5}}(x),$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}}^{5}}^{c}(x)=C_{\mathsf{Q}_{\mathsf{E}}^{5}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{S}}^{5}}^{T}(x)=\Psi_{\mathbf{Q}_{\mathsf{S}}^{5}}(x),$
(312)

where $\Psi_{\mathbf{Q}_{\mathsf{S}}^{5}}(x)$ is defined in Eqs.(VIII.4) and
(VIII.4) with lepton-quark state formed by Dirac fermions.

###  VIII.5  Entangled Heptaqubit-spinor field in Category-2 as entangled
hyperqubit-spinor field and inhomogeneous hyperspin symmetry as unified spin-
color-flavor symmetry of lepton-quark states in 12D hyper-spacetime

The lepton-quark state appearing in category-1 entangled hexaqubit-spinor
field does characterize the vector-like strong interaction in SM, but it
cannot reflect the flavor property of leptons and quarks in the weak
interaction of SM, which brings us to consider the next category-2 entangled
hyperqubit-spinor field with ${\mathcal{Q}}_{c}$-spin charge
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=2$. Such an entangled hyperqubit-spinor
field is referred to as entangled heptaqubit-spinor field with the following
qubit-spinor structure:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{E}}^{7}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{W}}^{6}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{W}}^{6}}^{\,c}(x)}\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{S}}^{6}}^{L}(x)}{\Psi_{\mathbf{Q}_{\mathsf{S}}^{6}}^{L\,c}(x)},$
(313)

where $\Psi_{\mathbf{Q}_{\mathsf{W}}^{6}}(x)$ is regarded as a chiral
hexaqubit-spinor field with $\Psi_{\mathbf{Q}_{\mathsf{W}}^{6}}^{\,c}(x)$ as
its complex charge-conjugated one. $\Psi_{\mathbf{Q}_{\mathsf{W}}^{6}}(x)$ and
$\Psi_{\mathbf{Q}_{\mathsf{W}}^{6}}^{\,c}(x)$ can be provided from a self-
conjugated hexaqubit-spinor field $\Psi_{\mathbf{Q}_{\mathsf{S}}^{6}}(x)$ via
the following definitions:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}}^{6}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{S}}^{6}}^{L}(x)\equiv\varGamma_{-}^{5}\Psi_{\mathbf{Q}_{\mathsf{S}}^{6}}(x),$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}}^{6}}^{\,c}(x)=C_{\mathbf{Q}_{\mathsf{W}}^{6}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{6}}^{T}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{S}}^{6}}^{L\,c}(x)=\varGamma_{+}^{5}\Psi_{\mathbf{Q}_{\mathsf{S}}^{6}}(x),$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}}^{6}}^{\,c}(x)=C_{\mathbf{Q}_{\mathsf{S}}^{6}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{S}}^{6}}^{T}(x)=\Psi_{\mathbf{Q}_{\mathsf{S}}^{6}}(x),$
$\displaystyle\varGamma_{\pm}^{5}=\frac{1}{2}(1\pm\tilde{\gamma}_{5}),\quad\tilde{\gamma}_{5}=\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
$\displaystyle
C_{\mathbf{Q}_{\mathsf{W}}^{6}}=C_{\mathbf{Q}_{\mathsf{S}}^{6}}=\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D}.$ (314)

which have the following explicit forms:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}}^{6}}^{L}(x)\equiv\begin{pmatrix}\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}^{{}^{\prime}\;\bar{c}}(x)\\\
\Psi^{\prime}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}^{\;\bar{c}}(x)\end{pmatrix}_{L},\quad\Psi_{\mathbf{Q}_{\mathsf{S}}^{6}}^{L\;c}(x)\equiv\begin{pmatrix}\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}^{{}^{\prime}\;\bar{c}}(x)\\\
\Psi^{\prime}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}^{\;\bar{c}}(x)\end{pmatrix}_{R},$ (315)

where $\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)$ and
$\Psi^{\prime}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)$ are two complex tetraqubit-
spinor fields with definitions presented in Eq.(VIII.4). They are formed from
two flavors of lepton-quark states $Q^{\alpha}(x)$ and
$Q^{{}^{\prime}\alpha}(x)$ with superscript $\alpha=(r,b,g,w)$ labeling four
color-spin charges. The projection operators $\varGamma_{\mp}^{5}$ give rise
to definitions for the left-handed and right-handed lepton-quark states
$Q^{\alpha}_{L,R}(x)$.

The entangled heptaqubit-spinor field $\Psi_{\mathsf{Q}_{\mathsf{E}}^{7}}(x)$
satisfies the following self-conjugated chiral conditions:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{E}}^{7}}^{c}(x)=C_{\mathsf{Q}_{\mathsf{E}}^{7}}\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{7}}^{T}(x)=\Psi_{\mathsf{Q}_{\mathsf{E}}^{7}}(x),$
$\displaystyle\hat{\gamma}_{15}\Psi_{\mathsf{Q}_{\mathsf{E}}^{7}}(x)=\hat{\gamma}_{15}\Psi_{\mathsf{Q}_{\mathsf{E}}^{7}}^{c}(x)=-\Psi_{\mathsf{Q}_{\mathsf{E}}^{7}}(x),$
$\displaystyle C_{\mathsf{Q}_{\mathsf{E}}^{7}}=\sigma_{1}\otimes
C_{\mathbf{Q}_{\mathsf{W}}^{6}}=\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D},$
$\displaystyle\hat{\gamma}_{15}=\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5}.$
(316)

The action for such an entangled heptaqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{7}}(x)$ is constructed to be,

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{E}}^{7}}=\int
d^{12}x\,\\{\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{7}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varSigma_{-}^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{E}}^{7}}(x)-\lambda_{2}\phi_{p}(x)\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{7}}(x)\tilde{\varSigma}_{-}^{p}\Psi_{\mathsf{Q}_{\mathsf{E}}^{7}}(x)\\},$
(317)

with $\mathsf{A},\mathsf{M}=0,1,2,3,5,\cdots,12$. The
$\varSigma_{-}^{\mathsf{A}}$ and $\tilde{\varSigma}_{-}^{p}$ matrices are
defined with $\varGamma$-matrices
$\varGamma^{\hat{\mathsf{A}}}=(\varGamma^{a},\varGamma^{A},\varGamma^{13},\varGamma^{14})$
($\hat{\mathsf{A}}=0,1,2,3,5,\cdots,14$, $a=0,1,2,3,A=5,\cdots,12$) as
follows:

$\displaystyle\varSigma_{-}^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},\quad\tilde{\varSigma}_{-}^{p}=\frac{1}{2}\tilde{\varGamma}^{p}\varGamma_{-},$
$\displaystyle\varGamma^{a}=\;\sigma_{0}\otimes\sigma_{0}\otimes\Gamma^{a},\quad
a=0,1,2,3,$
$\displaystyle\varGamma^{A}=\;\sigma_{1}\otimes\sigma_{0}\otimes\Gamma^{A},\quad
A=5,\cdots,10,$
$\displaystyle\varGamma^{11}=i\sigma_{1}\otimes\sigma_{1}\otimes\hat{\gamma}_{11},$
$\displaystyle\varGamma^{12}=i\sigma_{1}\otimes\sigma_{2}\otimes\hat{\gamma}_{11},$
$\displaystyle\varGamma^{13}=i\sigma_{1}\otimes\sigma_{3}\otimes\hat{\gamma}_{11}\equiv\tilde{\varGamma}^{1},$
$\displaystyle\varGamma^{14}=i\sigma_{2}\otimes\sigma_{0}\otimes\tilde{\gamma}_{5}\;\equiv\tilde{\varGamma}^{2},$
$\displaystyle\hat{\gamma}_{15}=i\sigma_{3}\otimes\sigma_{0}\otimes\tilde{\gamma}_{5},\quad\varGamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{15}),$
$\displaystyle\hat{\gamma}_{11}=\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
$\displaystyle\tilde{\gamma}_{5}=\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
(318)

with $\Gamma^{\mathsf{A}}$ ($A=0,1,2,3,5,\cdots,10$) defined in Eq.(VIII.3).
It is easy to check that all $\varGamma$-matrices
$\varGamma^{\hat{\mathsf{A}}}$ become anti-commuting.

The action in Eq.(317) possesses an associated symmetry in which the
inhomogeneous hyperspin symmetry WS(1,11) is in association with inhomogeneous
Lorentz-type/Poincaré-type group symmetry PO(1,11) together with
${\mathcal{Q}}_{c}$-spin symmetry SP(2) as well as global scaling symmetry,
i.e.:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,11)\,\widetilde{\Join}\,WS(1,11)\rtimes SG(1)\times SP(2)$ (319)
$\displaystyle=$ $\displaystyle SC(1)\ltimes P^{1,11}\ltimes
SO(1,11)\,\widetilde{\Join}\,SP(1,11)\rtimes W^{1,11}\rtimes SG(1)\times
SP(2),$

where the symbol “$\,\widetilde{\Join}\,$” is used to indicate the associated
symmetry with coincidental transformations between symmetry groups SP(1,11)
and SO(1,11), which is ensured by the isomorphic property SP(1,11)$\cong$
SO(1,11).

The group generators of inhomogeneous hyperspin symmetry WS(1,11) and
${\mathcal{Q}}_{c}$-spin symmetry SP(2) are directly given by the commutators
of $\varGamma$-matrices,

$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}=\frac{i}{4}[\varGamma^{\mathsf{A}},\varGamma^{\mathsf{B}}],\quad\varSigma_{-}^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},$
$\displaystyle\varSigma^{D_{h}+p\,D_{h}+q}=\frac{i}{4}[\varGamma^{12+p},\varGamma^{12+q}]=\frac{i}{4}[\tilde{\varGamma}^{p},\tilde{\varGamma}^{q}]\equiv\tilde{\varSigma}^{pq},$
(320)

with $\mathsf{A},\mathsf{B}=0,1,2,3,5,\cdots,12$ and $p,q=1,2$. The
${\mathcal{Q}}_{c}$-spin symmetry characterizes the chiral flavor-spin
symmetry SP(2)$\cong$U(1) between two flavors of lepton-quark states
$\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)$ and
$\Psi^{\prime}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)$.

The hyperspin symmetry SP(1,11) can generally be decomposed into the following
subgroup structure:

$\displaystyle SP(1,11)\supset SP(1,3)\times SP(8)\supset SP(1,3)\times
SP(6)\times SP(2)$ $\displaystyle\cong SO(1,3)\times SO(6)\times SO(2)\cong
SO(1,3)\times SU_{C}(4)\times U(1),$ (321)

where the symmetry SUC(4)$\times$U(1) characterizes the vector-like four
color-spin symmetry SUC(4) and flavor-spin symmetry U(1) for two type of
flavors of lepton-quark states. The corresponding subgroup algebras are given
by,

$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=0,1,2,3)\in
sp(1,3)\cong so(1,3)$
$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=5,6,7,8,9,10)\in
sp(6)\cong so(6)\cong su_{C}(4),$
$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=11,12)\in
sp(2)\cong so(2)\cong u(1).$

To show the vector-like color-spin charge symmetry in SM, the action in
Eq.(317) for the entangled heptaqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{7}}(x)$ can equivalently be expressed into the
following form by using the self-conjugated hexaqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)$:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{S}}^{6}}$ $\displaystyle=$
$\displaystyle\int
d^{12}x\,\\{\frac{1}{2}\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varGamma^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)-\frac{1}{2}\lambda_{2}\phi_{p}(x)\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)\tilde{\varGamma}^{p}\Psi_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)\\},$
(322)

with $\mathsf{A},\mathsf{M}=0,1,2,3,5,\cdots,12$. The $\Gamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{a},\varGamma^{A})$
($a=0,1,2,3,A=5,\cdots,12$) and $\tilde{\varGamma}$-matrices
$\tilde{\varGamma}^{p}$ (p=1,2) are explicitly given as follows:

$\displaystyle\varGamma^{\mathsf{A}}=\sigma_{0}\otimes\Gamma^{\mathsf{A}},\quad\mathsf{A}=0,1,2,3,5,\cdots,10,$
$\displaystyle\varGamma^{11}=i\sigma_{1}\otimes\hat{\gamma}_{11},$
$\displaystyle\varGamma^{12}=i\sigma_{2}\otimes\hat{\gamma}_{11},$
$\displaystyle\varGamma^{13}=i\sigma_{3}\otimes\hat{\gamma}_{11}=i\hat{\gamma}_{13}\equiv\tilde{\varGamma}^{1},$
$\displaystyle\varGamma^{14}=\;\sigma_{0}\otimes
I_{64}=\,\,I_{128}\equiv\tilde{\varGamma}^{2}.$ (323)

with $\Gamma^{\mathsf{A}}$ ($\mathsf{A}=0,1,2,3,5,\cdots,10$) defined in
Eq.(VIII.3). The ${\mathcal{Q}}_{c}$-spin symmetry reflects the chiral flavor-
spin symmetry SP(2) $\cong$ U(1) with the group generator,

$\displaystyle\tilde{\Sigma}^{12}_{0}=-\frac{1}{2}\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5}.$

It is shown that the entangled heptaqubit-spinor field in 128-dimensional
Hilbert space spanned by 7-product qubit basis states unifies two type of
flavors of lepton-quark states into a single entangled hyperqubit-spinor
field. Nevertheless, the maximal subgroup symmetry
SUC(4)$\times$UV(1)$\times$UA(1) deduced from hyperspin symmetry SP(1,11) and
${\mathcal{Q}}_{c}$-spin symmetry SP(2) cannot produce the basic symmetries
SUC(3)$\times$SUL(2)$\times$UY(1) of leptons and quarks in SM. So that the
entangled heptaqubit-spinor field as entangled hyperqubit-spinor field of
leptons and quarks remains incomplete to realize the SM.

##  IX  Entangled octoqubit-spinor field as ultra-grand unified qubit-spinor
field in 14D hyper-spacetime with inhomogeneous hyperspin symmetry as ultra-
grand unified symmetry and the comprehension on lepton-quark state with one
more family and the observed universe with 4D spacetime

Three entangled hyperqubit-spinor fields with respective to
${\mathcal{Q}}_{c}$-spin charges $\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=0,1,2$
and hyper-spacetime dimensions $D_{h}=10,11,12$ have been analyzed to show how
they characterize the basic properties of lepton-quark states, such as color-
spin and flavor-spin charges. Nevertheless, their maximal inhomogeneous
hyperspin symmetry and ${\mathcal{Q}}_{c}$-spin symmetry cannot yet produce
the basic symmetries of chiral type leptons and quarks in SM though the
entangled heptaqubit-spinor field contains ${\mathcal{D}}_{H}=64$ independent
degrees of freedom which is equal to the degrees of freedom involved in each
family of leptons and quarks in SM. To realize the basic symmetries of leptons
and quarks in SM as subgroup symmetries of hyperspin symmetry, it is natural
to take into account, from the qubit-spinor categorization theorem, the next
entangled hyperqubit-spinor field in category-3 based on the locally
entangled-qubits motion postulate. Such an entangled hyperqubit-spinor field
is referred to as entangled octoqubit-spinor field which spans 256-dimensional
Hilbert space and has a free motion in 14-dimensional Minkowski hyper-
spacetime. Its action will be shown to possess the inhomogeneous hyperspin
symmetry WS(1,13) which contains the subgroup symmetry SO(10) as grand unified
group symmetry GUT1 ; GUT2 , so that the basic symmetries of leptons and
quarks in SM can be reproduced as subgroup symmetries of hyperspin symmetry
SP(1,13). The ${\mathcal{Q}}_{c}$-spin symmetry SP(3)$\cong$SU(2) is found to
characterize the family-spin symmetry for two families of lepton-quark states
in SM, which enables us to comprehend the basic issue on why there exist more
than one family of leptons and quarks in nature. Meanwhile, we are able to
provide a natural explanation on the longstanding open question why our
observed universe is only four-dimensional spacetime.

###  IX.1  Entangled octoqubit-spinor field in 14D hyper-spacetime with
inhomogeneous hyperspin and ${\mathcal{Q}}_{c}$-spin symmetries in producing
basic symmetries of leptons and quarks in SM and comprehension on lepton-quark
state beyond one family in nature

In the weak interaction of SM, either leptons or quarks as left-handed Weyl
fermions always form weak isospin doublet. In the strong interaction of SM,
quarks appear as vector-like Dirac fermions. In the electromagnetic
interaction of SM, both charged leptons and quarks behave as vector-like Dirac
fermions. In the entangled qubit-spinor representation, the hyperspin symmetry
of entangled pentaqubit-spinor field in category-0 reflects a unified color-
spin symmetry of chiral lepton-quark state, which is not suitable to
characterize either the electromagnetic interaction of vector-like charged
leptons and quarks in SM or the strong interaction of vector-like quarks in
SM. Though the hyperspin symmetries for both entangled hexaqubit-spinor field
in category-1 and entangled heptaqubit-spinor field in category-2 can describe
the vector-like properties of charged leptons and quarks in the
electromagnetic and/or strong interactions, but they cannot realize the weak
isospin symmetry of chiral leptons and quarks in SM. Following along the
qubit-spinor categorization theorem, we are going to present a detailed
analysis and discussion on the entangled octoqubit-spinor field as hyperqubit-
spinor field in category-3.

To characterize the chiral weak isospin property, let us begin with a
category-3 chiral complex heptaqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)$ defined as follows:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{W}-}^{6}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{W}+}^{6}}(x)}\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{L}(x)}{\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{R}(x)},$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}(x)\equiv\varGamma_{\pm}^{5}\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{R,L}(x),$
$\displaystyle\varGamma_{\pm}^{5}=\frac{1}{2}(1\pm\tilde{\gamma}_{5}),\quad\tilde{\gamma}_{5}=\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
(324)

where $\Psi_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}(x)$ are chiral complex hexaqubit-
spinor fields defined from the complex hexaqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}(x)$. They have the following explicit
forms:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}(x)\equiv\begin{pmatrix}\Psi^{u}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\Psi^{d^{\prime}\,\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\Psi^{d}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
-\Psi^{u^{\prime}\,\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\end{pmatrix},\quad\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{R,L}(x)\equiv\begin{pmatrix}\Psi^{u}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\Psi^{d^{\prime}\,\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\Psi^{d}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
-\Psi^{u^{\prime}\,\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\end{pmatrix}_{R,L}\equiv\Psi_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}(x),$
(325)

with $\Psi^{q}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)$
($q=u,u^{\prime},d,d^{\prime}$) regarded as complex tetraqubit-spinor fields
and $\Psi^{q\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)$ as complex charge-
conjugated ones. Their explicit forms are given as follows:

$\displaystyle\Psi^{q}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\equiv\begin{pmatrix}Q^{r}(x)\\\
Q^{b}(x)\\\ Q^{g}(x)\\\
Q^{w}(x)\end{pmatrix}^{q},\quad\Psi^{q\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\equiv\begin{pmatrix}Q^{r\,\bar{c}}(x)\\\
Q^{b\,\bar{c}}(x)\\\ Q^{g,\bar{c}}(x)\\\ Q^{w\,\bar{c}}(x)\end{pmatrix}^{q},$
(326)

where the superscripts $q=u,u^{\prime},d,d^{\prime}$ represent four kinds of
lepton-quark states with respective to the notations
$\left(Q^{\alpha}\right)^{q}=(U^{\alpha},U^{{}^{\prime}\alpha},D^{\alpha},D^{{}^{\prime}\alpha})$.
The projection operators $\varGamma^{5}_{\mp}$ define the left-handed and
right-handed lepton-quark states, respectively. There are two kinds of up-type
and down-type lepton-quark states with unified four color-spin charges
$\alpha=(r,b,g,w)$ corresponding to ‘red’, ‘blue’, ‘green’ and ‘white’.

The action for such a chiral heptaqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)$ in category-3 is constructed as
follows:

$\displaystyle{\mathcal{S}}_{\mathbf{Q}_{\mathsf{W}}^{7}}$ $\displaystyle=$
$\displaystyle\int
d^{14}x\,\frac{1}{2}[\bar{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varGamma^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)+H.c.],$
(327)

which gets a free motion in 14-dimensional hyper-spacetime with
$\mathsf{A},\mathsf{M}=0,1,2,3,5,\cdots,14$ by following along the maximum
coherence motion principle. The $\varGamma$-matrices $\varGamma^{\mathsf{A}}$
($\mathsf{A}=0,1,2,3,5,\cdots,14$) have the following explicit forms:

$\displaystyle\varGamma^{0}=\;\,\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{1}\otimes\sigma_{0},$
$\displaystyle\varGamma^{1}=\;i\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{1},$
$\displaystyle\varGamma^{2}=\;i\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{2},$
$\displaystyle\varGamma^{3}=\;i\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{3},$
$\displaystyle\varGamma^{5}=\;i\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\varGamma^{6}=\;i\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\varGamma^{7}=\;i\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\varGamma^{8}=\;i\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\varGamma^{9}=\;i\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{1}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\varGamma^{10}=i\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\varGamma^{11}=i\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\varGamma^{12}=i\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\varGamma^{13}=i\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\varGamma^{14}=i\sigma_{1}\otimes\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\sigma_{0}.$
(328)

It can be verified that the hermitian action in Eq.(IX.1) can be expressed as
follows:

$\displaystyle{\mathcal{S}}_{\mathbf{Q}_{\mathsf{W}}^{7}}$ $\displaystyle=$
$\displaystyle\int
d^{14}x\,\frac{1}{2}\\{\bar{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varGamma^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)+\bar{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)\delta_{a}^{\;\mu}\varGamma^{a}i\partial_{\mu}\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)$
$\displaystyle-$
$\displaystyle\bar{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)\delta_{A}^{\;M}\varGamma^{A}i\partial_{M}\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)\\},$

where we have used the notations $\mathsf{A}=(a,A),\mathsf{M}=(\mu,M)$ and
$\varGamma^{\mathsf{A}}=(\varGamma^{a},\varGamma^{A})$ with
$a,\mu=0,1,2,3;\;A,M=5,\cdots,14$. $\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)$
is defined as the complex charge-conjugation of
$\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)$ and has the following properties:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{W}}^{7}}\bar{\Psi}^{T}_{\mathbf{Q}_{\mathsf{W}}^{7}}(x),$
$\displaystyle\hat{\gamma}_{15}\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)=-\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x),\;\;\hat{\gamma}_{15}\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)=-\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x),$
$\displaystyle
C_{\mathbf{Q}_{\mathsf{W}}^{7}}=\varGamma_{2}\varGamma_{0}\varGamma_{6}\varGamma_{8}\varGamma_{10}\varGamma_{12}\varGamma_{14}\gamma_{5}$
$\displaystyle\qquad\;=\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D},$
$\displaystyle\hat{\gamma}_{15}=\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5}.$
(329)

The above formalism of the action indicates that it is appropriate to
introduce an entangled hyperqubit-spinor field with the following qubit-spinor
structure:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)=\binom{\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)},$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{W}-}^{6}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{W}+}^{6}}(x)}\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{L}(x)}{\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{R}(x)},$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{W}+}^{6}}^{c}(x)}{\Psi_{\mathbf{Q}_{\mathsf{W}-}^{6}}^{c}(x)}\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{R\,c}(x)}{\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{L\,c}(x)},$
(330)

where $\Psi_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}^{c}(x)$ are the complex charge-
conjugation of chiral complex hexaqubit-spinor fields
$\Psi_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}(x)$ with the following definition:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}\bar{\Psi}^{T}_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}(x)\equiv\varGamma_{\mp}^{5}\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{c}(x),$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{C}}^{6}}\bar{\Psi}^{T}_{\mathbf{Q}_{\mathsf{C}}^{6}}(x),$
$\displaystyle
C_{\mathbf{Q}_{\mathsf{C}}^{6}}=C_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}=\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D}.$ (331)

The projection operators $\varGamma^{5}_{\mp}$ given in Eq.(IX.1) define the
left-handed and right-handed lepton-quark states, respectively.
$\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{c}(x)$ is the complex charge-conjugation
of $\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}(x)$ with the following explicit form:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{c}(x)\equiv\begin{pmatrix}\Psi^{u^{\prime}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\Psi^{d\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\Psi^{d^{\prime}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
-\Psi^{u\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\end{pmatrix},\quad\Psi_{\mathbf{Q}_{\mathsf{C}}^{6}}^{L,R\;c}(x)\equiv\begin{pmatrix}\Psi^{u^{\prime}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\Psi^{d\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\Psi^{d^{\prime}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
-\Psi^{u\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\end{pmatrix}_{R,L}.$ (332)

In terms of the entangled hyperqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)$, the action in Eq.(IX.1) can be
rewritten into the following simple form:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{E}}^{8}}$
$\displaystyle\equiv$ $\displaystyle\int
d^{14}x\,\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varSigma_{-}^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x).$
(333)

with $\mathsf{A},\mathsf{M}=0,1,2,3,5,\cdots,14$. $\varSigma_{-}^{\mathsf{A}}$
matrices are defined by $\varGamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{a},\varGamma^{A})$
($a=0,1,2,3,A=5,\cdots,14$). All $\varGamma$-matrices
$\varGamma^{\hat{\mathsf{A}}}=(\varGamma^{\mathsf{A}},\varGamma^{D_{h}+p})$
($\mathsf{A}=0,1,2,3,5,\cdots,14$, $p=1,2,3$) and ${\mathcal{Q}}_{c}$-matrices
$\tilde{\varGamma}^{p}\equiv\varGamma^{D_{h}+p}$ ($p=1,2,3$) are found to have
the following structures:

$\displaystyle\varSigma_{-}^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},\quad\varGamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{17}),$
$\displaystyle\varGamma^{a}=\,\sigma_{0}\otimes\varGamma^{a},\quad a=0,1,2,3,$
$\displaystyle\varGamma^{\mathsf{A}}=\,\sigma_{3}\otimes\varGamma^{A},\quad
A=5,\cdots,14,$
$\displaystyle\varGamma^{15}=i\sigma_{1}\otimes\tilde{\gamma}_{5}\equiv\tilde{\varGamma}^{1},$
$\displaystyle\varGamma^{16}=i\sigma_{2}\otimes\tilde{\gamma}_{5}\equiv\tilde{\varGamma}^{2},$
$\displaystyle\varGamma^{17}=i\sigma_{3}\otimes\hat{\gamma}_{15}\equiv\tilde{\varGamma}^{3},$
$\displaystyle\tilde{\gamma}_{5}=\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
$\displaystyle\hat{\gamma}_{17}=\sigma_{0}\otimes\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5}\
\equiv\sigma_{0}\otimes\hat{\gamma}_{15},$ (334)

where $\varGamma^{a}$ and $\varGamma^{A}$ ($a=0,1,2,3,A=5,\cdots,14$) on the
right-hand side of equality are defined in Eq.(IX.1).

The entangled hyperqubit-spinor field $\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)$
is composed of four kinds of lepton-quark states and satisfies the following
self-conjugated chiral conditions:

$\displaystyle\Psi^{c}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)=C_{\mathsf{Q}_{\mathsf{E}}^{8}}\bar{\Psi}^{T}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)=\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x),$
$\displaystyle
C_{\mathsf{Q}_{\mathsf{E}}^{8}}=i\varGamma_{2}\varGamma_{0}\varGamma_{6}\varGamma_{8}\varGamma_{10}\varGamma_{12}\varGamma_{14}\varGamma_{16}=\sigma_{1}\otimes
C_{\mathbf{Q}_{\mathsf{W}}^{7}}$
$\displaystyle\quad\quad=\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D},$
$\displaystyle\hat{\gamma}_{17}\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)=-\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x),\quad\hat{\gamma}_{17}=\sigma_{0}\otimes\hat{\gamma}_{15},$
(335)

where $\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)$ is referred to as entangled
octoqubit-spinor field in category-3.

All $\varGamma$-matrices are anti-commuting and satisfy special commutation
relations with the matrix $\hat{\gamma}_{17}$,

$\displaystyle\\{\varGamma^{\mathsf{A}},\varGamma^{\mathsf{B}}\\}=2\eta^{\mathsf{A}\mathsf{B}},\qquad\qquad\mathsf{A},\mathsf{B}=0,1,2,3,5,\cdots,17,$
$\displaystyle\\{\varGamma^{\mathsf{A}},\hat{\gamma}_{17}\\}=0,\qquad\qquad\quad\;\mathsf{A}=0,1,2,3,5,\cdots,14,$
$\displaystyle[\varGamma^{\mathsf{A}},\hat{\gamma}_{17}]\equiv[\varGamma^{D_{h}+p},\hat{\gamma}_{17}]\equiv[\tilde{\varGamma}^{p},\hat{\gamma}_{17}]=0,\quad\mathsf{A}=15,16,17,\;p=1,2,3,$
(336)

where the commutator property in the last equality results in the absence of
scalar coupling term in association with the ${\mathcal{Q}}_{c}$-matrices
$\tilde{\varGamma}^{p}$ ($p=1,2,3$), which becomes a manifest feature for
entangled octoqubit-spinor field $\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)$ in
category-3.

The action in Eq.(333) possesses an associated symmetry in which the
inhomogeneous hyperspin symmetry WS(1,13) and ${\mathcal{Q}}_{c}$-spin
symmetry SP(3) in association with inhomogeneous Lorentz-type/Poincaré group
symmetry PO(1,13) together with the scaling symmetries SG(1) and SC(1), i.e.:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,13)\,\widetilde{\Join}\,WS(1,13)\rtimes SG(1)\times SP(3),$ (337)
$\displaystyle=$ $\displaystyle SC(1)\ltimes P^{1,13}\ltimes
SO(1,13)\,\widetilde{\Join}\,SP(1,13)\rtimes W^{1,13}\rtimes SG(1)\times
SP(3),$

which indicates from the symbol ‘$\,\widetilde{\Join}\,$’ that the group
transformation of hyperspin symmetry SP(1,13) in Hilbert space should be
coincidental to that of Lorentz-type group symmetry SO(1,13) in Minkowski
hyper-spacetime via the isomorphic property of two group symmetries, i.e.,
SP(1,13)$\cong$ SO(1,13).

The group generators of inhomogeneous hyperspin symmetry WS(1,13) and
${\mathcal{Q}}_{c}$-spin symmetry SP(3) can directly be expressed by the
commutators of $\varGamma$-matrices,

$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}=\frac{i}{4}[\varGamma^{\mathsf{A}},\varGamma^{\mathsf{B}}],\quad\varSigma_{-}^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},$
$\displaystyle\varSigma^{D_{h}+p\,D_{h}+q}=\frac{i}{4}[\varGamma^{14+p},\varGamma^{14+q}]=\frac{i}{4}[\tilde{\varGamma}^{p},\tilde{\varGamma}^{q}]\equiv\tilde{\varSigma}^{p,q},$
(338)

with $\mathsf{A},\mathsf{B}=0,1,2,3,5,\cdots,14$ and $p,q=1,2,3$.

By decomposing the hyperspin symmetry SP(1,13) into possible subgroup
symmetries, we arrive at the following appropriate subgroup symmetry:

$\displaystyle SP(1,13)$ $\displaystyle\supset$ $\displaystyle SP(1,3)\times
SP(10)\supset SP(1,3)\times SP(6)\times SP(4)$ (339) $\displaystyle\cong$
$\displaystyle SO(1,3)\times SO(10)\supset SO(1,3)\times SO(6)\times SO(4)$
$\displaystyle\cong$ $\displaystyle SO(1,3)\times SU_{C}(4)\times
SU_{L}(2)\times SU_{R}(2)$ $\displaystyle\supset$ $\displaystyle SO(1,3)\times
SU_{C}(3)\times SU_{L}(2)\times U_{Y}(1),$

where the direct product subgroup symmetry $SO(1,3)\times SU_{C}(3)\times
SU_{L}(2)\times U_{Y}(1)$ in the last equality produces the basic symmetries
in SM, which reflects the Lorentz-spin charge, color-spin charge, isospin
charge and hyper-electric charge in SM. The direct product subgroup algebra is
in correspondence to the following group generators:

$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=0,1,2,3)\in
sp(1,3)\cong so(1,3)$
$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=5,6,7,8,9,10)\in
sp(6)\cong so(6)\cong su_{C}(4),$
$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=11,12,13,14)\in
sp(4)\cong so(4)\cong su_{L}(2)\times su_{R}(2).$

It is clear that the hyperspin symmetry SP(1,13) of entangled octoqubit-spinor
field $\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)$ as entangled hyperqubit-spinor
field in category-3 does bring on the basic symmetries of leptons and quarks
in SM as subgroup symmetries. From the entangled qubit-spinor representation
of $\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)$ and the definitions of
$\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)$ and
$\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)$, it can be verified that the
${\mathcal{Q}}_{c}$-spin symmetry SP(3) describes the family-spin symmetry
SUF(2)$\cong$SP(3) for two kinds of lepton-quark states, i.e.,
$Q^{\alpha}=(U^{\alpha},D^{\alpha})$ and
$Q^{{}^{\prime}\alpha}=(U^{{}^{\prime}\alpha},D^{{}^{\prime}\alpha})$.

So far we come to the observation that when a locally entangled state of
qubits as entangled hyperqubit-spinor field is considered as fundamental
building block of nature with following along the locally entangled-qubits
motion postulate, the resulting inhomogeneous hyperspin symmetry of entangled
hyperqubit-spinor field naturally reflects the fundamental symmetry of nature.
To realize the basic symmetries of leptons and quarks in SM as subgroup
symmetries of hyperspin symmetry, the entangled octoqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)$ in category-3 is found to be the
minimal entangled hyperqubit-spinor field, which leads the hyperspin symmetry
SP(1,13) $\cong$ SO(1,13) as minimal unified symmetry to produce the basic
symmetries of SM as subgroup symmetries presented via a direct product group
symmetry, i.e., SO(1,3)$\times$SUC(3)$\times$SUL(2)$\times$ UY(1). Meanwhile,
the intrinsic ${\mathcal{Q}}_{c}$-spin symmetry SP(3) as family-spin symmetry
SUF(2) indicates the simultaneous existence of two families of leptons and
quarks in nature.

It becomes manifest that the category-3 entangled octoqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)$ in 14-dimensional hyper-spacetime
brings about the prediction on the simultaneous presence of two families of
chiral type leptons and quarks in nature, which provides a natural answer to
the best-known question “who ordered that?” exclaimed when I.I. Rabi was
informed in 1936 the discovery of the muon as a new particle with an
unexpected surprising property111In the middle of 1930s, all the subatomic
particles of nature were thought to be composed of the electron, proton and
neutron, it was surprised to find that the muon was not the hadron predicted
by theoretical particle physicist, but a new and entirely unexpected type of
charged lepton, a heavier relative to the electron, which made I.I. Rabi posed
his famous question in pondering the existence of the muon: “who ordered
that?” .

Therefore, it is the entangled octoqubit-spinor field that provides a minimal
entangled hyperqubit-spinor field to unify simultaneously two families of
chiral type leptons and quarks in SM into a single entangled hyperqubit-spinor
field and meanwhile produce exactly the basic symmetries of SM
SO(1,3)$\times$SUC(3)$\times$SUL(2)$\times$UY(1) as a direct product subgroup
symmetry of hyperspin symmetry SP(1,13)$\cong$SO(1,13). It becomes a necessity
to merge automatically two families of chiral type leptons and quarks in SM
into such a minimal entangled hyperqubit-spinor field in 14-dimensional hyper-
spacetime. The two families of chiral type lepton-quark states are
characterized by the intrinsic ${\mathcal{Q}}_{c}$-spin symmetry SP(3) as
family-spin symmetry SUF(2), which brings on a natural explanation on why
there exist leptons and quarks beyond one family in SM. It is clear that the
hyperspin symmetry SP(1,13)$\cong$SO(1,13) for the unification model UM
contains a maximal subgroup symmetry SP(10)$\cong$SO(10) as grand unified
symmetryGUT1 ; GUT2  which unifies all internal symmetries
SUC(3)$\times$SUL(2)$\times$UY(1) of leptons and quarks in SM.

###  IX.2 Ultra-grand unified qubit-spinor field in 14D hyper-spacetime with
inhomogeneous hyperspin symmetry as ultra-grand unified symmetry and
comprehension on the observed universe with only 4-dimensional spacetime

The entangled octoqubit-spinor field is shown to be the minimal entangled
hyperqubit-spinor field, which realizes all basic symmetries in SM as a direct
product subgroup symmetry of hyperspin symmetry and obtains the action with a
free motion in 14-dimensional hyper-spacetime. Nevertheless, our currently
observed universe is only four-dimensional spacetime. How to understand such a
universe with four-dimensional spacetime from 14-dimensional hyper-spacetime
becomes an inevitable question to be answered in exploring the foundation of
the hyperunified field theory. In fact, the question why our observed universe
is only four-dimensional spacetime has been a longstanding and essential
issue. Following along the locally entangled-qubits motion postulate and
noticing the fact that our observed universe is composed of the atomic matter
as basic constituent of observable matter world, we are able to provide a
natural explanation on such a basic question.

Let us first rewrite the chiral complex heptaqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)$ in Eq.(IX.1) into the sum of two
typical self-conjugated chiral hyperqubit-spinor fields,

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)\equiv\frac{1}{\sqrt{2}}[\Psi_{\mathbf{Q}_{\mathsf{E}_{1}^{-}}^{7}}(x)+i\Psi_{\mathbf{Q}_{\mathsf{E}_{2}^{-}}^{7}}(x)],$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)\equiv\frac{1}{\sqrt{2}}[\Psi_{\mathbf{Q}_{\mathsf{E}_{1}^{-}}^{7}}(x)-i\Psi_{\mathbf{Q}_{\mathsf{E}_{2}^{-}}^{7}}(x)],$
(340)

where $\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$ ($i=1,2$) have the
following qubit-spinor structure:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{W}_{i}^{-}}^{6}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{W}_{i}^{-}}^{6}}^{c}(x)}\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}^{L}(x)}{\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}^{L\,c}(x)}.$
(341)

$\Psi_{\mathbf{Q}_{\mathsf{W}_{i}^{-}}^{6}}(x)$ ($i=1,2$) are two chiral
complex hexaqubit-spinor fields defined from two typical self-conjugated
hexaqubit-spinor fields $\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}(x)$ as
follows:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}_{i}^{-}}^{6}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}^{L}(x)=\varGamma^{5}_{-}\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}(x),$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}_{i}^{-}}^{6}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{W}_{i}}^{6}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{W}_{i}^{-}}^{6}}^{T}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}^{L\,c}(x)=\varGamma^{5}_{+}\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}(x),$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}^{T}(x)=\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}(x),$
$\displaystyle
C_{\mathbf{Q}_{\mathsf{W}_{i}}^{6}}=C_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}=\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D},$
$\displaystyle\varGamma^{5}_{\pm}=\frac{1}{2}(1\pm\tilde{\gamma}_{5}),\quad\tilde{\gamma}_{5}=\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
(342)

where $\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}(x)$ ($i=1,2$) have the following
explicit structures:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}^{L}(x)\equiv\begin{pmatrix}\Psi_{\mathbf{Q}_{\mathsf{C}_{i}}^{4}}^{u}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{C}_{i}}^{4}}^{d\,\bar{c}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{C}_{i}}^{4}}^{d}(x)\\\
-\Psi_{\mathbf{Q}_{\mathsf{C}_{i}}^{4}}^{u\,\bar{c}}(x)\end{pmatrix}_{L},\quad\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}^{L,c}(x)\equiv\begin{pmatrix}\Psi_{\mathbf{Q}_{\mathsf{C}_{i}}^{4}}^{u}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{C}_{i}}^{4}}^{d\,\bar{c}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{C}_{i}}^{4}}^{d}(x)\\\
-\Psi_{\mathbf{Q}_{\mathsf{C}_{i}}^{4}}^{u\,\bar{c}}(x)\end{pmatrix}_{R}.$
(343)

It is noted that $\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}(x)$ ($i=1,2$) should
be distinguished from the self-conjugated hexaqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{S}}^{6}}(x)$ defined in Eq.(315) as they exhibit a
new kind of twisted hyperqubit-spinor structure. Where
$\Psi_{\mathbf{Q}_{\mathsf{C}_{i}}^{4}}^{q}(x)$ ($q=u,d$) are defined as
complex tetraqubit-spinor fields with
$\Psi_{\mathbf{Q}_{\mathsf{C}_{i}}^{4}}^{q\,\bar{c}}(x)$ as complex charge-
conjugated ones, they are composed of lepton-quark states with the following
explicit forms:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{C}_{i}}^{4}}^{q}(x)\equiv\begin{pmatrix}Q_{i}^{r}(x)\\\
Q_{i}^{b}(x)\\\ Q_{i}^{g}(x)\\\
Q_{i}^{w}(x)\end{pmatrix}^{q},\quad\Psi_{\mathbf{Q}_{\mathsf{C}_{i}}^{4}}^{q\,\bar{c}}(x)\equiv\begin{pmatrix}Q^{r\,\bar{c}}_{i}(x)\\\
Q^{b\,\bar{c}}_{i}(x)\\\ Q^{g\,\bar{c}}_{i}(x)\\\
Q^{w\,\bar{c}}_{i}(x)\end{pmatrix}^{q}.$ (344)

where the superscripts $q=u$ and $q=d$ label up-type and down-type lepton-
quark states, i.e., $Q^{\alpha\,u}_{i}(x)\equiv U_{i}^{\alpha}(x)$ and
$Q^{\alpha\,d}_{i}(x)\equiv D_{i}^{\alpha}(x)$, with four color-spin charges
$\alpha=r,b,g,w$ in correspondence to ‘red’, ‘blue’, ‘green’ and ‘white’. The
subscripts $i=1,2$ reflect two classes of lepton-quark states. The projection
operators $\varGamma^{5}_{\mp}$ define the left-handed and right-handed
lepton-quark states, i.e., $Q_{i\,L,R}^{\alpha\,u}(x)\equiv
U_{i\,L,R}^{\alpha}(x)$ and $Q_{i\,L,R}^{\alpha\,d}(x)\equiv
D_{i\,L,R}^{\alpha}(x)$.

From the definitions presented in Eqs.(IX.2) and (341), it can be checked that
$\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$ satisfy the following self-
conjugated chiral conditions:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}^{c}(x)\equiv
C_{\mathbf{Q}_{\mathsf{E}_{i}}^{7}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}^{T}(x)=\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x),$
$\displaystyle\hat{\gamma}_{15}\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)=\hat{\gamma}_{15}\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}^{c}(x)=-\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x),$
$\displaystyle
C_{\mathbf{Q}_{\mathsf{E}_{i}}^{7}}=C_{\mathbf{Q}_{\mathsf{W}}^{7}}=\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D}.$ (345)

So that the action in Eq.(IX.1) can be rewritten into the following form:

$\displaystyle{\mathcal{S}}_{\mathbf{Q}_{\mathsf{W}}^{7}}$
$\displaystyle\equiv$ $\displaystyle\int
d^{14}x\,\frac{1}{2}\\{\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{1}^{-}}^{7}}(x)\delta_{a}^{\;\mu}\varGamma^{a}i\partial_{\mu}\Psi_{\mathbf{Q}_{\mathsf{E}_{1}^{-}}^{7}}(x)+\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{2}^{-}}^{7}}(x)\delta_{a}^{\;\mu}\varGamma^{a}i\partial_{\mu}\Psi_{\mathbf{Q}_{\mathsf{E}_{2}^{-}}^{7}}(x)$
(346) $\displaystyle+$ $\displaystyle
i\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{1}^{-}}^{7}}(x)\delta_{A}^{\;M}\varGamma^{A}i\partial_{M}\Psi_{\mathbf{Q}_{\mathsf{E}_{2}^{-}}^{7}}(x)-i\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{2}^{-}}^{7}}(x)\delta_{A}^{\;M}\varGamma^{A}i\partial_{M}\Psi_{\mathbf{Q}_{\mathsf{E}_{1}^{-}}^{7}}(x)\\}$
$\displaystyle+$
$\displaystyle\frac{1}{2}i\partial_{M}[\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{1}^{-}}^{7}}(x)\delta_{A}^{\;M}\varGamma^{A}\Psi_{\mathbf{Q}_{\mathsf{E}_{1}^{-}}^{7}}(x)+\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{2}^{-}}^{7}}(x)\delta_{A}^{\;M}\varGamma^{A}\Psi_{\mathbf{Q}_{\mathsf{E}_{2}^{-}}^{7}}(x)],$

with $a,\mu=0,1,2,3$ and $A,M=5,\cdots,14$. Where we have used the following
identity:

$\displaystyle\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)\delta_{A}^{\;M}\varGamma^{A}i\partial_{M}\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)\equiv
i\partial_{M}[\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)]\delta_{A}^{\;M}\varGamma^{A}\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$
$\displaystyle\;=\frac{1}{2}i\partial_{M}[\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)\delta_{A}^{\;M}\varGamma^{A}\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)],\quad
A,M=5,\cdots,14,$

due to the self-conjugated feature of
$\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$, and also the following
property of $\varGamma$-matrices:

$\displaystyle
C_{\mathbf{Q}_{\mathsf{E}_{i}}^{7}}\varGamma^{a}C_{\mathbf{Q}_{\mathsf{E}_{i}}^{7}}^{-1}=-(\varGamma^{a})^{T},\quad
C_{\mathbf{Q}_{\mathsf{E}_{i}}^{7}}\varGamma^{A}C_{\mathbf{Q}_{\mathsf{E}_{i}}^{7}}^{-1}=(\varGamma^{A})^{T},$

with $a=0,1,2,3$ and $A=5,\cdots,14$.

The total derivative appearing in the last term of the action in Eq.(346)
indicates that the free motion of
$\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$($i=1,2$) in extra ten spatial
dimensions can only occur through the cross term between
$\Psi_{\mathbf{Q}_{\mathsf{E}_{1}^{-}}^{7}}(x)$ and
$\Psi_{\mathbf{Q}_{\mathsf{E}_{2}^{-}}^{7}}(x)$. Such a feature is attributed
to the specific twisted qubit-spinor structure of
$\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$($i=1,2$), which distinguishes
from the structure of entangled heptaqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{7}}(x)$ given in Eq.(313). It is noted that
the letter in bold type ‘$\mathbf{Q}_{\mathsf{E}_{i}}$’ has been used to
distinguish them. For convenience of mention in terminology, we may refer to
such typical self-conjugated chiral hyperqubit-spinor fields
$\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$ as twisting entangled
hyperqubit-spinor fields.

In general, the action in Eq.(346) can be expressed into the following simple
form:

$\displaystyle{\mathcal{S}}_{\mathbf{Q}_{\mathsf{E}}^{8}}$
$\displaystyle\equiv$ $\displaystyle\int
d^{14}x\,\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varSigma_{-}^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x),$
(347)

with $\varSigma_{-}^{\mathsf{A}}$ ($\mathsf{A}=0,1,2,3,5,\cdots,14$) matrices
given by $\varGamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{a},\varGamma^{A})$
($a=0,1,2,3,A=5,\cdots,14$). The $\varGamma$-matrices $\varGamma^{\mathsf{A}}$
($\mathsf{A}=0,1,2,3,5,\cdots,14$) and ${\mathcal{Q}}_{c}$-matrices
$\tilde{\varGamma}^{p}$ ($p=1,2,3$) can be combined to form the full
$\varGamma$-matrices
$\varGamma^{\hat{\mathsf{A}}}\equiv(\varGamma^{\mathsf{A}},\varGamma^{D_{h}+p})\equiv(\varGamma^{\mathsf{A}},\tilde{\varGamma}^{p})$
($\hat{\mathsf{A}}=0,1,2,3,5,\cdots,17$) with the following forms:

$\displaystyle\varSigma_{-}^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},\quad\varGamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{17}),$
$\displaystyle\varGamma^{a}=\;\;\;\,\sigma_{0}\otimes\varGamma^{a},\qquad
a=0,1,2,3,$ $\displaystyle\varGamma^{A}=-\sigma_{2}\otimes\varGamma^{A},\qquad
A=5,\cdots,14,$
$\displaystyle\varGamma^{15}=\;i\sigma_{1}\otimes\tilde{\gamma}_{5}\equiv\tilde{\varGamma}^{1},$
$\displaystyle\varGamma^{16}=\;i\sigma_{2}\otimes\hat{\gamma}_{15}\equiv\tilde{\varGamma}^{2},$
$\displaystyle\varGamma^{17}=\;i\sigma_{3}\otimes\tilde{\gamma}_{5}\equiv\tilde{\varGamma}^{3},$
$\displaystyle\hat{\gamma}_{17}=\sigma_{0}\otimes\hat{\gamma}_{15},$ (348)

where the matrices $\varGamma^{a}$ and $\varGamma^{A}$ on the right-hand side
of equality are presented in Eq.(IX.1).

The entangled hyperqubit-spinor field $\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)$
in Eq.(347) is defined as follows:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{E}_{1}^{-}}^{7}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{E}_{2}^{-}}^{7}}(x)}\equiv\begin{pmatrix}\Psi_{\mathbf{Q}_{\mathsf{S}1}^{6}}^{L}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{S}_{1}}^{6}}^{L\,c}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{S}_{2}}^{6}}^{L}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{S}_{2}}^{6}}^{L\,c}(x)\end{pmatrix},$ (349)

which satisfies the following self-conjugated chiral conditions:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{E}}^{8}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}^{T}(x)=\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x),\quad\hat{\gamma}_{17}\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)=-\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x),$
$\displaystyle C_{\mathbf{Q}_{\mathsf{E}}^{8}}=\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{E}_{i}}^{7}}=\sigma_{0}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D}.$ (350)

Note that $\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)$ defines an alternative
entangled octoqubit-spinor field, which has a different qubit-spinor structure
from that of entangled octoqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)$ given in Eq.(IX.1).
$\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)$ is distinguished symbolically by the
letter in bold type ‘$\mathbf{Q}_{\mathsf{E}}$’.

As all $\varGamma$-matrices in Eq.(IX.2) become anti-commuting, the group
generators of inhomogeneous hyperspin symmetry WS(1,13) and
${\mathcal{Q}}_{c}$-spin symmetry SP(3) for the action in Eq.(347) can
directly be given by the commutators of $\varGamma$-matrices,

$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}=\frac{i}{4}[\varGamma^{\mathsf{A}},\varGamma^{\mathsf{B}}],\quad\varSigma_{-}^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},$
$\displaystyle\varSigma^{D_{h}+p\,D_{h}+q}\equiv\frac{i}{4}[\varGamma^{14+p},\varGamma^{14+q}]=\frac{i}{4}[\tilde{\varGamma}^{p},\tilde{\varGamma}^{q}]\equiv\tilde{\varSigma}^{p,q},$
(351)

with $\mathsf{A},\mathsf{B}=0,1,2,3,5,\cdots,14$ and $p,q=1,2,3$.

It is seen from the action in Eq.(346) that each twisting entangled
hyperqubit-spinor field $\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$ has a
free motion only in four-dimensional spacetime, which motivates us to build
the action for each $\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$ in four
dimensional spacetime. The action for each twisting entangled hyperqubit-
spinor field is found to have the following general form:

$\displaystyle{\mathcal{S}}_{\mathbf{Q}_{\mathsf{E}_{i}}^{7}}$
$\displaystyle\equiv$ $\displaystyle\int
d^{4}x\,\\{\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)\delta_{a}^{\;\mu}\varSigma_{-}^{a}i\partial_{\mu}\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$
(352) $\displaystyle+$
$\displaystyle\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)[H^{A}(x)+i\gamma_{5}H_{5}^{A}(x)]\varSigma_{-}^{A}\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)\\},$

with $a,\mu=0,1,2,3$, and the $\varGamma$-matrices,

$\displaystyle\varSigma_{-}^{a}=\frac{1}{2}\varGamma^{a}\varGamma_{-},\quad\varSigma_{-}^{A}=\frac{1}{2}\varGamma^{A}\varGamma_{-},$
$\displaystyle\varGamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{15}),\quad
a=0,1,2,3;\;A=5,\cdots,14.$ (353)

The vector fields $H^{A}(x)$ and $H_{5}^{A}(x)$ in Hilbert space are regarded
as Higgs-like scalar fields.

It can be verified that the action in Eq.(352) possesses the following
associated symmetry:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,3)\,\widetilde{\Join}\,WS(1,3)\rtimes SG(1)\times SP(10),$ (354)
$\displaystyle=$ $\displaystyle SC(1)\ltimes P^{1,3}\ltimes
SO(1,3)\,\widetilde{\Join}\,SP(1,3)\rtimes W^{1,3}\rtimes SG(1)\times SO(10),$

which is considered as a subgroup symmetry of the whole associated symmetry in
which the inhomogeneous hyperspin symmetry and Poincaré-type group symmetry
together with global scaling symmetry, i.e., SC(1) $\ltimes$ PO(1,13)
$\,\widetilde{\Join}\,$ WS(1,13) $\rtimes$ SG(1).

It is interesting to notice that each twisting entangled hyperqubit-spinor
field $\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$ contains 64 independent
degrees of freedom, which characterizes exactly all intrinsic features for
each family of chiral type leptons and quarks in SM. So that the hyperspin
symmetry SP(10) brings on the grand unified symmetry with the following
appropriate subgroup symmetries:

$\displaystyle G$ $\displaystyle=$ $\displaystyle SP(10)\cong SO(10)\supset
SU_{C}(4)\times SU_{L}(2)\times SU_{R}(2)\supset SU_{C}(3)\times
SU_{L}(2)\times U_{Y}(1),$ $\displaystyle G$ $\displaystyle=$ $\displaystyle
SP(10)\cong SO(10)\supset SU(5)\times U(1)\supset SU_{C}(3)\times
SU_{L}(2)\times U_{Y}(1),$ (355)

which brings about the grand unified theoryGUT1 ; GUT2  for leptons and quarks
in SM with including right-handed neutrinos. Where SU(5) is regarded as a
minimal grand unified theorySU5 , and SUC(4)$\times$SUL(2)$\times$SUR(2) as a
lepton-quark unified theoryPS  which gives rise to the unified four color-spin
symmetry and left-right isospin symmetry.

As each twisting entangled hyperqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$ unifies each family of chiral
type leptons and quarks in SM into a single qubit-spinor field with grand
unified symmetry SP(10)$\cong$SO(10), it is appropriate to call
$\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$ as grand unified qubit-spinor
field denoted as follows:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{G}_{i}}^{-}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x).$
(356)

To be more explicit, the sixty four independent degrees of freedom in each
family of chiral type leptons and quarks in SM can be expressed into a single
westward entangled hyperqubit-spinor fieldHUFT :

$\displaystyle\Psi_{W\mspace{1.0mu}i}^{T}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{G}_{i}}^{-}}^{T}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}^{T}(x)$
$\displaystyle=[(U_{i}^{r},U_{i}^{b},U_{i}^{g},U_{i}^{w},D^{r}_{ic},D^{b}_{ic},D^{g}_{ic},D^{w}_{ic},D_{i}^{r},D_{i}^{b},D_{i}^{g},D_{i}^{w},-U^{r}_{ic},-U^{b}_{ic},-U^{g}_{ic},-U^{w}_{ic})_{L}\,,$
$\displaystyle\;\;(U_{i}^{r},U_{i}^{b},U_{i}^{g},U_{i}^{w},D^{r}_{ic},D^{b}_{ic},D^{g}_{ic},D^{w}_{ic},D_{i}^{r},D_{i}^{b},D_{i}^{g},D_{i}^{w},-U^{r}_{ic},-U^{b}_{ic},-U^{g}_{ic},-U^{w}_{ic})_{R}]^{T},$
(357)

with $i=1,2$ for two families. The superscript $T$ denotes the transposition
of a column matrix. $U_{i}^{\alpha}$ and $D_{i}^{\alpha}$ are up-type and
down-type lepton-quark states in SM with $\alpha=(r,\,b\,,g\,,w)$ representing
the trichromatic (red, blue, green) and white colors, respectively.
$Q_{ic}^{\alpha}=(U_{ic}^{\alpha},D_{ic}^{\alpha})$ are defined as complex
charge-conjugated Dirac spinors, i.e.,
$Q_{i\mspace{1.0mu}c}^{\alpha}=C_{D}\bar{Q}_{i}^{T}=C_{D}\gamma_{0}Q_{i}^{\ast}$
with $C_{D}$ the charge-conjugation matrix
$C_{D}=-i\sigma_{3}\otimes\sigma_{2}$. The subscripts “L” and “R” in
$Q_{iL,R}^{\alpha}=(U_{i}^{\alpha},D_{i}^{\alpha})_{L,R}$ denote the left-
handed and right-handed Dirac spinors, respectively, i.e.,
$Q_{iL,R}^{\alpha}=\frac{1}{2}(1\mp\gamma_{5})Q_{i}^{\alpha}$ with
$\gamma_{5}Q_{iL,R}^{\alpha}=\mp Q_{iL,R}^{\alpha}$.

As the entangled octoqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)$ is composed of two grand unified
qubit-spinor fields, it can be verified that the intrinsic
${\mathcal{Q}}_{c}$-spin symmetry SP(3) characterizes the family-spin symmetry
SUF(2)$\cong$SP(3) for two families of chiral type leptons and quarks in SM,
i.e., $Q_{1}^{\alpha}=(U_{1}^{\alpha},D_{1}^{\alpha})$ and
$Q_{2}^{\alpha}=(U_{2}^{\alpha},D_{2}^{\alpha})$, with respective to two grand
unified qubit-spinor fields $\Psi_{\mathbf{Q}_{\mathsf{G}_{1}}^{-}}(x)$ and
$\Psi_{\mathbf{Q}_{\mathsf{G}_{2}}^{-}}(x)$.

From the above analyses, we come to the observation that if there exists only
one family of chiral type leptons and quarks in nature, their maximally
correlated motion brings on a free motion just in four-dimensional spacetime.
Whereas the locally entangled-qubits motion postulate and maximum coherence
motion principle inevitably lead to the prediction on the simultaneous
existence of two families of chiral type leptons and quarks in nature with
free motion in 14-dimensional hyper-spacetime. Let us suppose that all chiral
type leptons and quarks become massive with different masses, and meanwhile
the heavy leptons and quarks decay into light leptons and quarks. Such a
situation does occur in the real case of SM, all heavy massive charged leptons
and quarks decay eventually into the lightest massive electron and neutrinos
as well as up quark and down quark. All the lightest leptons and quarks belong
to a single family as the first family in SM. In fact, the observed matter in
our living universe consists of the atomic matter which is just composed of
the lightest up and down quarks together with electron. When combining such a
fact with the observation that each twisting entangled hyperqubit-spinor field
as grand unified qubit-spinor field
($\Psi_{\mathbf{Q}_{\mathsf{G}_{i}}^{-}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$)
contains exactly a single family of chiral type lepton-quark state and has a
free motion only in four-dimensional spacetime, we arrive at the conclusion
that the lightest up and down quarks together with electron can only keep a
free motion in four-dimensional spacetime since they belong to a single family
of lepton-quark state as the first family in SM and constitute the observed
atomic matter in our universe. In other word, all existing matter in our
living universe can only move in four-dimensional Minkowski spacetime, which
brings about a natural explanation on the longstanding issue why our observed
universe is only four-dimensional spacetime.

Therefore, the entangled octoqubit-spinor field spanned in 256-dimensional
Hilbert space is regarded as an ultra-grand unified qubit-spinor field which
is composed of two grand unified qubit-spinor fields. Such an ultra-grand
unified qubit-spinor field keeps a free motion in 14-dimensional hyper-
spacetime determined by ${\mathcal{M}}_{c}$-spin charge, which may generally
be denoted as follows:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{U}}^{-}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{G}_{1}}^{-}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{G}_{2}}^{-}}(x)}\equiv\binom{\Psi_{W\mspace{1.0mu}1}(x)}{\Psi_{W\mspace{1.0mu}2}(x)},$
$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{U}}^{-}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)=\binom{\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)},$
(358)

where the superscript ‘-’ of the symbols $\mathbf{Q}_{\mathsf{U}}$ and
$\mathsf{Q}_{\mathsf{U}}$ characterizes the chirality property, which is
referred to as U-parity of grand unified qubit-spinor field for each family of
lepton-quark state in SM. The inhomogeneous hyperspin symmetry and
${\mathcal{Q}}_{c}$-spin symmetry of ultra-grand unified qubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{U}}^{-}}(x)$
($\Psi_{\mathsf{Q}_{\mathsf{U}}^{-}}(x)$) with negative U-parity is regarded
as ultra-grand unified symmetry denoted as follows:

$\displaystyle G_{U}=WS(1,13)\times SP(3)\cong SO(1,13)\rtimes W^{1,13}\times
SU_{F}(2),$

which is in association with inhomogeneous Lorentz-type/Poincaré-type group
symmetry PO(1,13) in 14-dimensional Minkowski hyper-spacetime to provide an
associated symmetry. The subgroup symmetry SP(10) of ultra-grand unified
symmetry SP(1,13)$\cong$SO(1,13) brings about grand unified symmetry
SO(10)$\cong$SP(10) for each family of chiral type leptons and quarks in SM.
Each grand unified qubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{G}_{i}}^{-}}(x)$ defined by the twisting entangled
hyperqubit-spinor field $\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$
characterizes exactly a single family of chiral type leptons and quarks in SM,
which keeps a free motion only in four-dimensional spacetime.

##  X  Hyperunified qubit-spinor field as fundamental building block of nature
and inhomogeneous hyperspin symmetry as hyperunified symmetry from maximum
locally entangled-qubits motion principle

Based on the locally entangled-qubits motion postulate, four categories of
entangled hyperqubit-spinor fields with respective to categoric
${\mathcal{Q}}_{c}$-spin charges
$\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c},k)}=q_{c}=0,1,2,3$ in the first
period ($k=1$) have been investigated in detail, which are shown to have free
motions in hyper-spacetime with respective to dimensions $D_{h}=10,11,12,14$.
The ${\mathcal{Q}}_{c}$-spin charges not only provide the categorization for
entangled hyperqubit-spinor fields as unified qubit-spinor fields of lepton-
quark states, but also reflect the flavor-spin and family-spin charges of
lepton-quark states via the intrinsic ${\mathcal{Q}}_{c}$-spin symmetries.
Unlike the hyperspin symmetry that reflects a motion-correlation spin
symmetry, the ${\mathcal{Q}}_{c}$-spin symmetry characterizes a motion-
irrelevance intrinsic symmetry. For the entangled octoqubit-spinor field in
category-3, the hyperspin symmetry SP(1,13)$\cong$SO(1,13) as ultra-grand
unified symmetry in 14-dimensional hyper-spacetime is verified to describe
consistently a minimal unified symmetry of helicity-spin, boost-spin, color-
spin and isospin charges of lepton-quark states in SM, while the intrinsic
${\mathcal{Q}}_{c}$-spin symmetry SP(3) presents as a direct product group
symmetry to hyperspin symmetry SP(1,13) and describes the family-spin symmetry
SUF(2) for two families of chiral type leptons and quarks in SM. From the
unification point of view, it appears a necessity to attribute such an
intrinsic family-spin symmetry into a motion-correlation hyperspin symmetry so
as to obtain a single hyperunified symmetry in hyper-spacetime, which comes to
the main purpose in this section. For that, we are led to make the least
${\mathcal{Q}}_{c}$-spin postulate and propose the maximum locally entangled-
qubits motion principle as guiding principle, which enables us to construct a
hyperunified qubit-spinor field of lepton-quark states as fundamental building
block of nature.

###  X.1 Least ${\mathcal{Q}}_{c}$-spin postulate and maximum locally
entangled-qubits motion principle

The entangled hyperqubit-spinor fields in category-2 and category-3 possess
the ${\mathcal{Q}}_{c}$-spin symmetries SP(2) and SP(3), which indicates that
the maximal symmetry of entangled hyperqubit-spinor field with
${\mathcal{Q}}_{c}$-spin charges $\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=2,3$
is not yet characterized by a single inhomogeneous hyperspin symmetry as the
intrinsic ${\mathcal{Q}}_{c}$-spin symmetry SP($q_{c}$) appears as a direct
product group symmetry to inhomogeneous hyperspin symmetry WS(1,$D_{h}$-1). To
get inhomogeneous hyperspin symmetry as a single hyperunified symmetry, we are
motivated to make a least ${\mathcal{Q}}_{c}$-spin postulate and propose a
maximum locally entangled-qubits motion principle as guiding principle, so
that it brings on an entangled hyperqubit-spinor field as a hyperunified
qubit-spinor field.

Least ${\mathcal{Q}}_{c}$-spin postulate: To make inhomogeneous hyperspin
symmetry as a single hyperunified symmetry of entangled hyperqubit-spinor
field, a hyperunified qubit-spinor field is postulated to have the least
${\mathcal{Q}}_{c}$-spin charge in building hyperunified field theory. This
postulate indicates that such a hyperunified qubit-spinor field should belong
to either category-0 or category-1 entangled hyperqubit-spinor field with
${\mathcal{Q}}_{c}$-spin charge
$\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c},k)}=q_{c}=0$ or $1$. With such a
least ${\mathcal{Q}}_{c}$-spin postulate, we are led to propose the following
guiding principle.

Maximum locally entangled-qubits motion principle: when combing the least
${\mathcal{Q}}_{c}$-spin postulate and locally entangled-qubits motion
postulate together with local coherent-qubits motion postulate, we arrive at a
maximum locally entangled-qubits motion principle as guiding principle. We
should demonstrate that it is such a maximum locally entangled-qubits motion
principle together with the maximum coherence motion principle that brings
about an entangled hyperqubit-spinor field with least ${\mathcal{Q}}_{c}$-spin
charge as hyperunified qubit-spinor field. Such a hyperunified qubit-spinor
field is regarded as fundamental building block of nature.

Following along the maximum locally entangled-qubits motion principle and
noticing the categorization theorems on both qubit-spinor field with categoric
qubit number and Minkowski spacetime with categoric dimension, we should come
to investigate locally entangled states of nine qubits ($Q_{N}=9$) and ten
qubits ($Q_{N}=10$) with respective to ${\mathcal{Q}}_{c}$-spin charges
$\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c},k)}=q_{c}=0$ and
$\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c},k)}=q_{c}=1$ in the second period
$k=2$, they may be referred to as entangled enneaqubit-spinor field in
category-0 and entangled decaqubit-spinor field in cetegory-1.

###  X.2  Entangled enneaqubit-spinor field with zero ${\mathcal{Q}}_{c}$-spin
charge as minimal hyperunified qubit-spinor field and inhomogeneous hyperspin
symmetry as minimal hyperunified symmetry in 18D hyper-spacetime

The entangled octoqubit-spinor field as ultra-grand unified qubit-spinor field
has a maximal ${\mathcal{Q}}_{c}$-spin charge
$\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c},k)}=q_{c}=3$ in the first period
$k=1$, which leads the maximum ${\mathcal{Q}}_{c}$-spin symmetry SP(3) to
characterize the family-spin symmetry SUF(2)$\cong$SP(3) of two families of
chiral type leptons and quarks in SM. To embed such a ${\mathcal{Q}}_{c}$-spin
symmetry into a motion-correlation hyperspin symmetry in hyper-spacetime, it
is a necessity to explore the next entangled hyperqubit-spinor field with
least ${\mathcal{Q}}_{c}$-spin charge. Following along the periodic feature of
${\mathcal{Q}}_{c}$-spin charge, we come to study the locally entangled state
of nine qubits with zero ${\mathcal{Q}}_{c}$-spin charge in the second period
$k=2$, which is referred to as entangled enneaqubit-spinor field in
category-0.

Let us begin with the construction of entangled enneaqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)$ with zero ${\mathcal{Q}}_{c}$-spin
charge $\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c},k)}=q_{c}=0$ in the second
period $k=2$. It is straightforward to build
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)$ via the following qubit-spinor
structure:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)=\binom{\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)}{\tilde{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)}\equiv\binom{\Psi_{\mathsf{Q}_{\mathsf{U}}^{-}}(x)}{\Psi_{\mathsf{Q}_{\mathsf{U}}^{+}}(x)}\equiv\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x),$
(359)

where
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{U}}^{-}}(x)$
is an entangled octoqubit-spinor field which is regarded as ultra-grand
unified qubit-spinor field for two families of chiral type lepton-quark states
with negative U-parity as shown in Eq.(IX.1).
$\tilde{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{U}}^{+}}(x)$
is an additional entangled octoqubit-spinor field with positive U-parity,
which is defined as follows:

$\displaystyle\tilde{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)\equiv\binom{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)}{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)}\equiv\Psi_{\mathsf{Q}_{\mathsf{U}}^{+}}(x),$
$\displaystyle\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)\equiv\binom{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}+}^{6}}(x)}{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}-}^{6}}(x)}\equiv\binom{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}}^{6}}^{R}(x)}{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}}^{6}}^{L}(x)},$
$\displaystyle\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)\equiv\binom{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}-}^{6}}^{c}(x)}{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}+}^{6}}^{c}(x)}\equiv\binom{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}}^{6}}^{L\,c}(x)}{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}}^{6}}^{R\,c}(x)},$
(360)

where $\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)$ is a chiral complex
heptaqubit-spinor field with
$\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)$ as its complex charge-
conjugated one. They all have positive U-parity,

$\displaystyle\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{W}}^{7}}\bar{\tilde{\Psi}}^{T}_{\mathbf{Q}_{\mathsf{W}}^{7}}(x),$
$\displaystyle
C_{\mathbf{Q}_{\mathsf{W}}^{7}}=\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D},$
$\displaystyle\hat{\gamma}_{15}\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)=+\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}(x),\quad\hat{\gamma}_{15}\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)=+\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x).$
(361)

$\tilde{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{U}}^{+}}(x)$
with positive U-parity satisfies the following self-conjugated chiral
conditions:

$\displaystyle\tilde{\Psi}^{c}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)=C_{\mathsf{Q}_{\mathsf{E}}^{8}}\bar{\tilde{\Psi}}^{T}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)=\tilde{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{U}}^{+}}^{c}(x)=C_{\mathsf{Q}_{\mathsf{U}}}\bar{\Psi}^{T}_{\mathsf{Q}_{\mathsf{U}}^{+}}(x),$
$\displaystyle\hat{\gamma}_{17}\Psi_{\mathsf{Q}_{\mathsf{U}}^{+}}(x)\equiv\hat{\gamma}_{17}\tilde{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)=+\tilde{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)\equiv+\Psi_{\mathsf{Q}_{\mathsf{U}}^{+}}(x),$
$\displaystyle C_{\mathsf{Q}_{\mathsf{U}}}\equiv
C_{\mathsf{Q}_{\mathsf{E}}^{8}}\equiv\sigma_{1}\otimes
C_{\mathbf{Q}_{\mathsf{W}}^{7}},\quad\hat{\gamma}_{17}=\sigma_{0}\otimes\hat{\gamma}_{15}.$
(362)

It is noticed that
$\Psi_{\mathsf{Q}_{\mathsf{U}}^{+}}(x)\equiv\tilde{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)$
and
$\Psi_{\mathsf{Q}_{\mathsf{U}}^{-}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)$
have a similar qubit-spinor structure but with opposite U-parity. The chiral
complex heptaqubit-spinor field
$\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}(x)$ are composed of chiral complex
hexaqubit-spinor fields $\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}(x)$,
and similarly for their complex charge conjugated ones
$\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}}^{7}}^{c}(x)$ and
$\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}^{c}(x)$, i.e.:

$\displaystyle\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}(x)\equiv\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}}^{6}}^{R,L}(x)\equiv\varGamma_{\pm}^{5}\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}}^{6}}(x),$
$\displaystyle\tilde{\Psi}_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}\bar{\tilde{\Psi}}^{T}_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}(x)\equiv\varGamma_{\mp}^{5}\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}}^{6}}^{c}(x),$
$\displaystyle\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}}^{6}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{C}}^{6}}\bar{\tilde{\Psi}}^{T}_{\mathbf{Q}_{\mathsf{C}}^{6}}(x),$
$\displaystyle
C_{\mathbf{Q}_{\mathsf{C}}^{6}}=C_{\mathbf{Q}_{\mathsf{W}\pm}^{6}}=\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D},$
$\displaystyle\varGamma_{\pm}^{5}=\frac{1}{2}(1\pm\tilde{\gamma}_{5}),\quad\tilde{\gamma}_{5}=\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
(363)

where $\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}}^{6}}^{R,L}(x)$ are chiral complex
hexaqubit-spinor fields with
$\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}}^{6}}^{R,L\,c}(x)$ their complex charge-
conjugated ones. They have the following explicit structures:

$\displaystyle\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}}^{6}}^{R,L}(x)\equiv\begin{pmatrix}\tilde{\Psi}^{u}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\tilde{\Psi}^{d^{\prime}\,\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\tilde{\Psi}^{d}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
-\tilde{\Psi}^{u^{\prime}\,\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\end{pmatrix}_{R,L},\quad\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}}^{6}}^{R,L\,c}(x)\equiv\begin{pmatrix}\tilde{\Psi}^{u^{\prime}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\tilde{\Psi}^{d\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
\tilde{\Psi}^{d^{\prime}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\\\
-\tilde{\Psi}^{u\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\end{pmatrix}_{L,R},$
(364)

where $\tilde{\Psi}^{q\,R,L}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)$
($q=u,u^{\prime},d,d^{\prime}$) are chiral complex tetraqubit-spinor fields
with $\tilde{\Psi}^{q\,\bar{c}\;L,R}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)$ the
complex charge-conjugated ones. Their explicit forms can be expressed as
follows:

$\displaystyle\tilde{\Psi}^{q\,R,L}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\equiv\begin{pmatrix}\tilde{Q}^{r}(x)\\\
\tilde{Q}^{b}(x)\\\ \tilde{Q}^{g}(x)\\\
\tilde{Q}^{w}(x)\end{pmatrix}^{q}_{R,L},\quad\tilde{\Psi}^{q\,\bar{c}\,L,R}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\equiv\begin{pmatrix}\tilde{Q}^{r\,\bar{c}}(x)\\\
\tilde{Q}^{b\,\bar{c}}(x)\\\ \tilde{Q}^{g,\bar{c}}(x)\\\
\tilde{Q}^{w\,\bar{c}}(x)\end{pmatrix}^{q}_{L,R},$ (365)

with $q=u,u^{\prime},d,d^{\prime}$ labeling four kinds of lepton-quark states
presented by the following notations:

$\displaystyle\left(\tilde{Q}^{\alpha}\right)^{u,u^{\prime},d,d^{\prime}}_{R,L}\equiv\tilde{U}^{\alpha}_{R,L},\tilde{U}^{{}^{\prime}\alpha}_{R,L},\tilde{D}^{\alpha}_{R,L},\tilde{D}^{{}^{\prime}\alpha}_{R,L}.$
(366)

Note that the lepton-quark states $\left(\tilde{Q}^{\alpha}\right)^{q}_{R,L}$
always appear to have opposite chirality to
$\left(Q^{\alpha}\right)^{q}_{L,R}$. We may refer to
$\left(\tilde{Q}^{\alpha}\right)^{q}_{R,L}$ as mirror lepton-quark states.
They involve two kinds of up-type ($q=u,u^{\prime}$) and down-type
($q=d,d^{\prime}$) mirror lepton-quark states with four color-spin charges
$\alpha=r,b,g,w$ in correspondence to ‘red’, ‘blue’, ‘green’ and ‘white’. The
projection operators $\varGamma^{5}_{\pm}$ define the right-handed and left-
handed mirror lepton-quark states notated by the subscripts ‘R’ and ‘L’,
respectively.

Therefore,
$\Psi_{\mathsf{Q}_{\mathsf{U}}^{+}}(x)\equiv\tilde{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)$
with positive U-parity is regarded as mirror entangled octoqubit-spinor field
for two families of mirror lepton-quark states. It will be demonstrated below
that the ${\mathcal{Q}}_{c}$-spin symmetry SP(3) as family-spin symmetry
SUF(2) of entangled octoqubit-spinor field does turn into inhomogeneous
hyperspin symmetry of entangled enneaqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)$, so that we will refer to
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)$ as minimal hyperunified qubit-spinor
field denoted as
$\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)$.

From the above analyses, we are able to obtain the following action for the
minimal hyperunified qubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)$
in 18-dimensional hyper-spacetime:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{H}}^{0}}\equiv{\mathcal{S}}_{\mathsf{Q}_{\mathsf{E}}^{9}}$
$\displaystyle\equiv$ $\displaystyle\int
d^{18}x\,\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varSigma_{-}^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)$
(367) $\displaystyle\equiv$ $\displaystyle\int
d^{18}x\,\bar{\Psi}_{\mathsf{Q}_{\mathsf{H}}^{0}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varSigma_{-}^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x).$

with $\mathsf{A},\mathsf{M}=0,1,2,3,5,\cdots,18$. The matrices
$\varSigma_{-}^{\mathsf{A}}$ are defined from the $\varGamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{a},\varGamma^{A})$
($a=0,1,2,3,A=5,\cdots,18$) with the following explicit structures:

$\displaystyle\varSigma_{-}^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},\quad\varGamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{19}),$
$\displaystyle\varGamma^{a}=\,\sigma_{0}\otimes\sigma_{0}\otimes\varGamma^{a},\quad
a=0,1,2,3,$
$\displaystyle\varGamma^{\mathsf{A}}=\,\sigma_{0}\otimes\sigma_{3}\otimes\varGamma^{A},\quad
A=5,\cdots,14,$
$\displaystyle\varGamma^{15}=i\sigma_{2}\otimes\sigma_{1}\otimes\tilde{\gamma}_{5},$
$\displaystyle\varGamma^{16}=i\sigma_{2}\otimes\sigma_{2}\otimes\tilde{\gamma}_{5},$
$\displaystyle\varGamma^{17}=i\sigma_{2}\otimes\sigma_{3}\otimes\hat{\gamma}_{15},$
$\displaystyle\varGamma^{18}=i\sigma_{1}\otimes\sigma_{0}\otimes\hat{\gamma}_{15},$
$\displaystyle\hat{\gamma}_{19}=\sigma_{3}\otimes\sigma_{0}\otimes\hat{\gamma}_{15}=\sigma_{3}\otimes\hat{\gamma}_{17},$
$\displaystyle\tilde{\gamma}_{5}=\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
(368)

with $\varGamma^{a}$ and $\varGamma^{A}$ ($a=0,1,2,3,A=5,\cdots,14$) on the
right-hand side of equality defined in Eq.(IX.1).

It can be checked that
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x)$
satisfies the following self-conjugated chiral conditions:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}^{c}(x)=C_{\mathsf{Q}_{\mathsf{E}}^{9}}\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{9}}^{T}(x)=\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}^{c}(x)=C_{\mathsf{Q}_{\mathsf{H}}^{0}}\bar{\Psi}_{\mathsf{Q}_{\mathsf{H}}^{0}}^{T}(x),$
$\displaystyle\hat{\gamma}_{19}\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x)\equiv\hat{\gamma}_{19}\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)=-\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)\equiv-\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x),$
$\displaystyle C_{\mathsf{Q}_{\mathsf{H}}^{0}}\equiv
C_{\mathsf{Q}_{\mathsf{E}}^{9}}=\sigma_{0}\otimes
C_{\mathsf{Q}_{\mathsf{E}}^{8}}=\sigma_{0}\otimes\sigma_{1}\otimes
C_{\mathbf{Q}_{\mathsf{W}}^{7}},$ (369)

where
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x)$
has a negative chirality in 18-dimensional hyper-spacetime, which may be
referred to as negative H-parity.

To characterize explicitly the family-spin charge of lepton-quark states in
SM, let us consider an alternative qubit-spinor structure of entangled
enneaqubit-spinor field based on the twisting entangled hyperqubit-spinor
field as grand unified qubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{G}_{i}}^{-}}(x)$.
Such an entangled enneaqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)$ as minimal hyperunified qubit-spinor
field $\Psi_{\mathbf{Q}_{\mathsf{H}}^{0}}(x)$ is simply defined as follows:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)=\binom{\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)}{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)}\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{U}}^{-}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{U}}^{+}}(x)}\equiv\Psi_{\mathbf{Q}_{\mathsf{H}}^{0}}(x),$
(370)

where
$\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{U}}^{-}}(x)$
is an entangled octoqubit-spinor field, which is viewed as an ultra-grand
unified qubit-spinor field with negative U-parity as shown in Eq.(349).
$\tilde{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{U}}^{+}}(x)$
with positive U-parity is defined as follows with a similar qubit-spinor
structure:

$\displaystyle\tilde{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{E}_{3}^{+}}^{7}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{E}_{4}^{+}}^{7}}(x)}\equiv\begin{pmatrix}\tilde{\Psi}_{\mathbf{Q}_{\mathsf{S}3}^{6}}^{R}(x)\\\
\tilde{\Psi}_{\mathbf{Q}_{\mathsf{S}_{3}}^{6}}^{R\,c}(x)\\\
\tilde{\Psi}_{\mathbf{Q}_{\mathsf{S}_{4}}^{6}}^{R}(x)\\\
\tilde{\Psi}_{\mathbf{Q}_{\mathsf{S}_{4}}^{6}}^{R\,c}(x)\end{pmatrix}\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{G}_{3}}^{+}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{G}_{4}}^{+}}(x)}\equiv\Psi_{\mathbf{Q}_{\mathsf{U}}^{+}}(x),$
(371)

which satisfies the following self-conjugated chiral conditions:

$\displaystyle\tilde{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{E}}^{8}}\bar{\tilde{\Psi}}_{\mathbf{Q}_{\mathsf{E}}^{8}}^{T}(x)=\tilde{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{U}}^{+}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{U}}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{U}}^{+}}^{T}(x),$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{E}_{j}^{+}}^{7}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{E}_{j}}^{7}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}_{j}^{+}}^{7}}^{T}(x)=\Psi_{\mathbf{Q}_{\mathsf{E}_{j}^{+}}^{7}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{G}_{j}}^{+}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{G}}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{G}_{j}}^{+}}^{T}(x),$
$\displaystyle\hat{\gamma}_{17}\Psi_{\mathbf{Q}_{\mathsf{U}}^{+}}(x)\equiv\hat{\gamma}_{17}\tilde{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)=+\tilde{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)\equiv+\Psi_{\mathbf{Q}_{\mathsf{U}}^{+}}(x),$
$\displaystyle C_{\mathbf{Q}_{\mathsf{U}}}\equiv
C_{\mathbf{Q}_{\mathsf{E}}^{8}}=\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{E}_{i}}^{7}}=\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{W}}^{7}}\equiv\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{G}}},$ $\displaystyle
C_{\mathbf{Q}_{\mathsf{G}}}=C_{\mathbf{Q}_{\mathsf{E}_{j}}^{7}}\equiv
C_{\mathbf{Q}_{\mathsf{W}}^{7}}=C_{\mathbf{Q}_{\mathsf{E}_{i}}^{7}}=\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D},$ (372)

with $j=3,4$.

It is noticed that
$\Psi_{\mathbf{Q}_{\mathsf{U}}^{+}}(x)\equiv\tilde{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)$
has an opposite chirality to
$\Psi_{\mathbf{Q}_{\mathsf{U}}^{-}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)$
and characterizes mirror lepton-quark states with positive U-parity. So that
the entangled octoqubit-spinor field with positive U-parity
$\tilde{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)$ is regarded as mirror ultra-
grand unified qubit-spinor field $\Psi_{\mathbf{Q}_{\mathsf{U}}^{+}}(x)$.
Correspondingly, the twisting entangled hyperqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{E}_{j}^{+}}^{7}}(x)$ with positive U-parity is
regarded as mirror grand unified qubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{G}_{j}}^{+}}(x)$ ($j=3,4$), which is defined as
follows:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{G}_{j}}^{+}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}_{j}^{+}}^{7}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{W}_{j}^{+}}^{6}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{W}_{j}^{+}}^{6}}^{c}(x)}\equiv\binom{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{S}_{j}}^{6}}^{R}(x)}{\tilde{\Psi}_{\mathbf{Q}_{\mathsf{S}_{j}}^{6}}^{R\,c}(x)}.$
(373)

where $\Psi_{\mathbf{Q}_{\mathsf{W}_{j}^{+}}^{6}}(x)$ ($j=3,4$) are mirror
chiral complex hexaqubit-spinor fields and
$\tilde{\Psi}_{\mathbf{Q}_{\mathsf{S}_{j}}^{6}}^{R}(x)$ ($j=3,4$) are mirror
self-conjugated hexaqubit-spinor field with the following explicit forms:

$\displaystyle\tilde{\Psi}_{\mathbf{Q}_{\mathsf{S}_{j}}^{6}}^{R}(x)\equiv\begin{pmatrix}\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{j}}^{4}}^{u}(x)\\\
\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{j}}^{4}}^{d\,\bar{c}}(x)\\\
\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{j}}^{4}}^{d}(x)\\\
-\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{j}}^{4}}^{u\,\bar{c}}(x)\end{pmatrix}_{R},\quad\tilde{\Psi}_{\mathbf{Q}_{\mathsf{S}_{j}}^{6}}^{R\;c}(x)\equiv\begin{pmatrix}\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{j}}^{4}}^{u}(x)\\\
\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{j}}^{4}}^{d\,\bar{c}}(x)\\\
\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{j}}^{4}}^{d}(x)\\\
-\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{j}}^{4}}^{u\,\bar{c}}(x)\end{pmatrix}_{L}.$
(374)

$\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{j}}^{4}}^{q}(x)$ ($j=3,4$) are regarded
as mirror complex tetraqubit-spinor fields with the complex charge-conjugated
ones $\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{j}}^{4}}^{q\,\bar{c}}(x)$
($j=3,4$). Their explicit forms are given as follows:

$\displaystyle\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{j}}^{4}}^{q}(x)\equiv\begin{pmatrix}\tilde{Q}_{j}^{r}(x)\\\
\tilde{Q}_{j}^{b}(x)\\\ \tilde{Q}_{j}^{g}(x)\\\
\tilde{Q}_{j}^{w}(x)\end{pmatrix}^{q},\quad\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{j}}^{4}}^{q\,\bar{c}}(x)\equiv\begin{pmatrix}\tilde{Q}^{r\,\bar{c}}_{j}(x)\\\
\tilde{Q}^{b\,\bar{c}}_{j}(x)\\\ \tilde{Q}^{g\,\bar{c}}_{j}(x)\\\
\tilde{Q}^{w\,\bar{c}}_{j}(x)\end{pmatrix}^{q}$ (375)

where $q=u,d$ represent the up-type and down-type mirror lepton-quark states
$\tilde{Q}^{\alpha\,u}_{j}(x)=\tilde{U}_{j}^{\alpha}(x)$ and
$\tilde{Q}^{\alpha\,u}_{j}(x)=\tilde{D}_{j}^{\alpha}(x)$ with four color-spin
charges $\alpha=r,b,g,w$ corresponding to ‘red’, ‘blue’, ‘green’ and ‘white’.
The subscripts $j=3,4$ denote two families of mirror lepton-quark states. The
subscripts ‘R’ and ‘L’ label the right-handed and left-handed mirror lepton-
quark states
$\tilde{Q}_{j\,R,L}^{\alpha\,u,d}(x)=(\tilde{U}_{j\,R,L}^{\alpha}(x),\tilde{D}_{j\,R,L}^{\alpha}(x))$.

In light of entangled octoqubit-spinor fields as ultra-grand unified qubit-
spinor fields with the mirror pair of negative and positive U-parities, i.e.,
$\Psi_{\mathbf{Q}_{\mathsf{U}}^{-}}\equiv\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)$
and
$\Psi_{\mathbf{Q}_{\mathsf{U}}^{+}}(x)\equiv\tilde{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)$,
it is easy to verify that
$\Psi_{\mathbf{Q}_{\mathsf{H}}^{0}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)$
with negative H-parity satisfies the following self-conjugated chiral
conditions:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{E}}^{9}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{9}}^{T}(x)=\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{H}}^{0}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{H}}^{0}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{H}}^{0}}^{T}(x),$
$\displaystyle\hat{\gamma}_{19}\Psi_{\mathbf{Q}_{\mathsf{H}}^{0}}(x)\equiv\hat{\gamma}_{19}\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)=-\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)\equiv-\Psi_{\mathbf{Q}_{\mathsf{H}}^{0}}(x),$
$\displaystyle C_{\mathbf{Q}_{\mathsf{H}}^{0}}\equiv
C_{\mathbf{Q}_{\mathsf{E}}^{9}}=\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{E}}^{8}}=\sigma_{0}\otimes\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{E}_{i}}^{7}}\equiv\sigma_{0}\otimes\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{G}}}.$ (376)

So that the action in Eq.(367) can be rewritten into the following form:

$\displaystyle{\mathcal{S}}_{\mathbf{Q}_{\mathsf{H}}^{0}}\equiv{\mathcal{S}}_{\mathbf{Q}_{\mathsf{E}}^{9}}$
$\displaystyle\equiv$ $\displaystyle\int
d^{18}x\,\bar{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varSigma_{-}^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x),$
(377) $\displaystyle\equiv$ $\displaystyle\int
d^{18}x\,\bar{\Psi}_{\mathbf{Q}_{\mathsf{H}}^{0}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varSigma_{-}^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathbf{Q}_{\mathsf{H}}^{0}}(x)$

with $\mathsf{A},\mathsf{M}=0,1,2,3,5,\cdots,18$. The matrices
$\varSigma_{-}^{\mathsf{A}}$ are defined from $\varGamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{a},\varGamma^{A})$
($a=0,1,2,3,A=5,\cdots,18$) with the following structures:

$\displaystyle\varSigma_{-}^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},\quad\varGamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{19}),$
$\displaystyle\varGamma^{a}=\;\;\;\,\sigma_{0}\otimes\sigma_{0}\otimes\varGamma^{a},\quad
a=0,1,2,3,$
$\displaystyle\varGamma^{\mathsf{A}}=\,-\sigma_{0}\otimes\sigma_{2}\otimes\varGamma^{A},\quad
A=5,\cdots,14,$
$\displaystyle\varGamma^{15}=\;i\sigma_{2}\otimes\sigma_{1}\otimes\tilde{\gamma}_{5},$
$\displaystyle\varGamma^{16}=\;i\sigma_{2}\otimes\sigma_{2}\otimes\hat{\gamma}_{15},$
$\displaystyle\varGamma^{17}=\;i\sigma_{2}\otimes\sigma_{3}\otimes\tilde{\gamma}_{5},$
$\displaystyle\varGamma^{18}=\;i\sigma_{1}\otimes\sigma_{0}\otimes\hat{\gamma}_{15},$
$\displaystyle\hat{\gamma}_{19}=\sigma_{3}\otimes\sigma_{0}\otimes\hat{\gamma}_{15}=\sigma_{3}\otimes\hat{\gamma}_{17},$
$\displaystyle\tilde{\gamma}_{5}=\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
(378)

where $\varGamma^{a}$ and $\varGamma^{A}$ ($a=0,1,2,3,A=5,\cdots,14$) on the
right-hand side of equality are given in Eq.(IX.1).

As there exists no other $\varGamma$-matrix that can be anti-commuting with
all $\varGamma$ matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{a},\varGamma^{A})$
($a=0,1,2,3,A=5,\cdots,18$) and couple with scalar fields, the entangled
enneaqubit-spinor field as minimal hyperunified qubit-spinor field,
$\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{H}}^{0}}(x)$
or
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x)$,
is a category-0 entangled hyperqubit-spinor field with
${\mathcal{Q}}_{c}$-spin charge $\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=0$.

It can be verified that the actions in Eqs.(367) and (377) possess an
associated symmetry in which the inhomogeneous hyperspin symmetry WS(1,17) is
in association with inhomogeneous Lorentz-type/Poincaré-type group symmetry
PO(1,17) as well as global scaling symmetries SG(1) and SC(1), i.e.:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,17)\,\widetilde{\Join}\,WS(1,17)\rtimes SG(1)$ (379) $\displaystyle=$
$\displaystyle SC(1)\ltimes P^{1,17}\ltimes
SO(1,17)\,\widetilde{\Join}\,SP(1,17)\rtimes W^{1,17}\rtimes SG(1),$

where the symbol “$\,\widetilde{\Join}\,$” is adopted to indicate the
associated symmetry so that the transformation of hyperspin symmetry SP(1,17)
should be coincidental to that of Lorentz-type group symmetry SO(1,17).

The group generators of inhomogeneous hyperspin symmetry WS(1,17) are directly
given by the commutators of $\varGamma$-matrices as follows:

$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}=\frac{i}{4}[\varGamma^{\mathsf{A}},\varGamma^{\mathsf{B}}],\quad\varSigma_{-}^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},$
(380)

with $\mathsf{A},\mathsf{B}=0,1,2,3,5,\cdots,18$. The hyperspin symmetry
SP(1,17) can be decomposed into the following subgroup symmetries:

$\displaystyle SP(1,17)$ $\displaystyle\supset$ $\displaystyle SP(1,3)\times
SP(14)\supset SP(1,3)\times SP(10)\times SP(4)$ (381) $\displaystyle\cong$
$\displaystyle SO(1,3)\times SO(14)\supset SO(1,3)\times SO(10)\times SO(4)$
$\displaystyle\supset$ $\displaystyle SO(1,3)\times SO(6)\times SO(4)\times
SO(4)$ $\displaystyle\cong$ $\displaystyle SO(1,3)\times SU_{C}(4)\times
SU_{L}(2)\times SU_{R}(2)\times SU_{F}(2)\times SU_{M}(2)$
$\displaystyle\supset$ $\displaystyle SO(1,3)\times SU_{C}(3)\times
SU_{L}(2)\times U_{Y}(1),$

with the group generators of subgroup symmetries given as follows:

$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=0,1,2,3)\in
sp(1,3)\cong so(1,3)$
$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=5,6,7,8,9,10,11,12,13,14)\in
sp(10)\cong so(10),$
$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=15,16,17,18)\in
sp(4)\cong so(4)\cong su_{F}(2)\times su_{M}(2),$
$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=5,6,7,8,9,10)\in
sp(6)\cong so(6)\cong su_{C}(4),$
$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=11,12,13,14)\in
sp(4)\cong so(4)\cong su_{L}(2)\times su_{R}(2).$

The subgroup symmetry SP(10)$\cong$SO(10) leads to the grand unified symmetry
and the subgroup symmetry SP(4)$\cong$SO(4)$\cong$SUF(2)$\times$SUM(2)
characterizes the family-spin symmetry SUF(2) and mirror family-spin symmetry
SUM(2) for two families of chiral type lepton-quark states and mirror lepton-
quark states, $Q_{i}^{\alpha}=(U_{i}^{\alpha},D_{i}^{\alpha})$ ($i=1,2$) and
$\tilde{Q}_{j}^{\alpha}=(\tilde{U}_{j}^{\alpha},\tilde{D}_{j}^{\alpha})$
($j=3,4$).

It is clear that the motion-irrelevant intrinsic ${\mathcal{Q}}_{c}$-spin
symmetry SP(3) as family-spin symmetry SUF(2) of entangled octoqubit-spinor
field $\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)$
($\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)(x)$) in 14-dimensional hyper-spacetime
does transmute into a motion-correlation hyperspin symmetry SP(1,17) of
entangled enneaqubit-spinor field $\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)$
($\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)$) in 18-dimensional hyper-spacetime.

Therefore, the least ${\mathcal{Q}}_{c}$-spin postulate brings us to construct
entangled enneaqubit-spinor field $\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)$ (or
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)$) which provides a minimal hyperunified
qubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{H}}^{0}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)$
(or
$\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)$)
with negative H-parity. Its maximal inhomogeneous hyperspin symmetry WS(1,17)
leads to a minimal hyperunified symmetry. Such a minimal hyperunified qubit-
spinor field brings about the prediction for the existence of mirror lepton-
quark states. Specifically, two twisting entangled hyperqubit-spinor fields as
grand unified qubit-spinor fields
$\Psi_{\mathbf{Q}_{\mathsf{G}_{i}}^{-}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{-}}^{7}}(x)$
($i=1,2$) with negative U-parity describe two families of chiral type lepton-
quark states in SM, whereas two mirror twisting entangled hyperqubit-spinor
fields as mirror grand unified qubit-spinor fields
$\Psi_{\mathbf{Q}_{\mathsf{G}_{j}}^{+}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}_{j}^{+}}^{7}}(x)$
($j=3,4$) with positive U-parity characterize two families of mirror lepton-
quark states beyond the SM.

###  X.3  Entangled decaqubit-spinor field in 19D hyper-spacetime as
hyperunified qubit-spinor field for fundamental building block of nature and
inhomogeneous hyperspin symmetry WS(1,18) as hyperunified fundamental symmetry
of nature

The entangled enneaqubit-spinor field as minimal hyperunified qubit-spinor
field
$\Psi_{\mathbf{Q}_{\mathsf{H}}^{0}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)$
(or
$\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)$
) provides a unified description on two families of chiral type leptons and
quarks in SM together with two families of mirror leptons and quarks beyond
SM. Nevertheless, the experimental observation in high energy physics
indicates that there exist at least three families of chiral type leptons and
quarks as basic constituents of nature. To unify all discovered leptons and
quarks into a single hyperunified qubit-spinor field, we are led to consider
further the next entangled hyperqubit-spinor field. Such an entangled
hyperqubit-spinor field should be a locally entangled state of ten qubits with
${\mathcal{Q}}_{c}$-spin charge $\mathcal{C}_{{\mathcal{Q}}_{c}}=1$ based on
the periodic feature of ${\mathcal{Q}}_{c}$-spin charge and categorization of
qubit-spinor field, which is referred to as entangled decaqubit-spinor field
in category-1.

The entangled decaqubit-spinor field can be constructed in a straightforward
way by taking entangled enneaqubit-spinor field as a minimal hyperunified
qubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x)$
(or
$\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{H}}^{0}}(x)$).
Let us first construct entangled decaqubit-spinor field with the following
qubit-spinor structure:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{E}}^{10}}(x)$ $\displaystyle\equiv$
$\displaystyle\binom{\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)}{\hat{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)}=\begin{pmatrix}\Psi_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)\\\
\tilde{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)\\\
\tilde{\Psi}^{\prime}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)\\\
\Psi^{\prime}_{\mathsf{Q}_{\mathsf{E}}^{8}}(x)\end{pmatrix}\equiv\Psi_{\mathsf{Q}_{\mathsf{H}}}(x),$
(382)

with the identical definition notated as follows:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{H}}}(x)$ $\displaystyle\equiv$
$\displaystyle\binom{\Psi_{\mathsf{Q}_{\mathsf{H}}^{-}}(x)}{\Psi_{\mathsf{Q}_{\mathsf{H}}^{+}}(x)}=\begin{pmatrix}\Psi_{\mathsf{Q}_{\mathsf{U}}^{-}}(x)\\\
\Psi_{\mathsf{Q}_{\mathsf{U}}^{+}}(x)\\\
\hat{\Psi}_{\mathsf{Q}_{\mathsf{U}}^{+}}(x)\\\
\hat{\Psi}_{\mathsf{Q}_{\mathsf{U}}^{-}}(x),\end{pmatrix},$ (383)

which is regarded as hyperunified qubit-spinor field denoted by
$\Psi_{\mathsf{Q}_{\mathsf{H}}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}}^{10}}(x)$.
We have introduced two entangled enneaqubit-spinor fields as minimal
hyperunified qubit-spinor fields,
$\Psi_{\mathsf{Q}_{\mathsf{H}}^{-}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{H}}^{0}}(x)$
and
$\Psi_{\mathsf{Q}_{\mathsf{H}}^{+}}(x)\equiv\hat{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{9}}(x)$,
with negative and positive H-parities, respectively, i.e.:

$\displaystyle\hat{\gamma}_{19}\Psi_{\mathsf{Q}_{\mathsf{H}}^{-}}(x)=-\Psi_{\mathsf{Q}_{\mathsf{H}}^{-}}(x),\quad\hat{\gamma}_{19}\Psi_{\mathsf{Q}_{\mathsf{H}}^{+}}(x)=+\hat{\Psi}_{\mathsf{Q}_{\mathsf{H}}^{+}}(x)$
$\displaystyle\hat{\gamma}_{19}=\sigma_{3}\otimes\sigma_{0}\otimes\hat{\gamma}_{15}=\sigma_{3}\otimes\hat{\gamma}_{17}.$

The action for such an entangled decaqubit-spinor field as hyperunified qubit-
spinor field is obtained, based on the maximum coherence motion Principe, as
follows:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{H}}}$ $\displaystyle\equiv$
$\displaystyle\int
d^{19}x\,\\{\bar{\Psi}_{\mathsf{Q}_{\mathsf{H}}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varSigma_{-}^{\mathbf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{H}}}(x)-\lambda_{1}\phi_{1}(x)\bar{\Psi}_{\mathsf{Q}_{\mathsf{H}}}(x)\tilde{\varSigma}_{-}\Psi_{\mathsf{Q}_{\mathsf{H}}}(x)$
(384) $\displaystyle\equiv$ $\displaystyle\int
d^{19}x\,\\{\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{10}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varSigma_{-}^{\mathbf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{E}}^{10}}(x)-\lambda_{1}\phi_{1}(x)\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{10}}(x)\tilde{\varSigma}_{-}\Psi_{\mathsf{Q}_{\mathsf{E}}^{10}}(x)\\},$

which has free motion in 19-dimensional hyper-spacetime with
$\mathsf{A},\mathsf{M}=0,1,2,3,5,\cdots,19$. The matrices
$\varSigma_{-}^{\mathbf{A}}$ and $\tilde{\varSigma}_{-}$ are defined from
$\varGamma$-matrices
$\varGamma^{\hat{\mathsf{A}}}\equiv(\varGamma^{\mathsf{A}},\varGamma^{20})\equiv(\varGamma^{a},\varGamma^{A},\tilde{\varGamma})$
($\hat{\mathsf{A}}=0,1,2,3,5,\cdots,20$) with the following explicit forms:

$\displaystyle\varSigma_{-}^{\mathbf{A}}=\frac{1}{2}\varGamma^{\mathbf{A}}\varGamma_{-},\quad\mathsf{A}=0,1,2,3,5,\cdots
19,$
$\displaystyle\tilde{\varSigma}_{-}\equiv\varSigma_{-}^{20}=\frac{1}{2}\varGamma^{20}\varGamma_{-}=\frac{1}{2}\tilde{\varGamma}\varGamma_{-},\quad\varGamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{21}),$
$\displaystyle\varGamma^{a}=\;\;\,\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\varGamma^{a},\quad
a=0,1,2,3,$
$\displaystyle\varGamma^{\mathsf{A}}=\;\;\,\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\varGamma^{A},\quad
A=5,\cdots,14,$
$\displaystyle\varGamma^{15}=\;i\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{1}\otimes\tilde{\gamma}_{5},$
$\displaystyle\varGamma^{16}=\;i\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\tilde{\gamma}_{5},$
$\displaystyle\varGamma^{17}=\;i\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\hat{\gamma}_{15},$
$\displaystyle\varGamma^{18}=\;i\sigma_{0}\otimes\sigma_{1}\otimes\sigma_{0}\otimes\hat{\gamma}_{15},$
$\displaystyle\varGamma^{19}=\;i\sigma_{1}\otimes\sigma_{3}\otimes\sigma_{0}\otimes\hat{\gamma}_{15},$
$\displaystyle\varGamma^{20}=\;i\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{0}\otimes\hat{\gamma}_{15}\equiv\tilde{\varGamma},$
$\displaystyle\hat{\gamma}_{21}=\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{0}\otimes\hat{\gamma}_{15}=\sigma_{3}\otimes\hat{\gamma}_{19},$
$\displaystyle\tilde{\gamma}_{5}=\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
(385)

where $\varGamma^{a}$ and $\varGamma^{A}$ ($a=0,1,2,3,A=5,\cdots,14$) on the
right-hand side of equality are defined in Eq.(IX.1).

The entangled decaqubit-spinor field $\Psi_{\mathsf{Q}_{\mathsf{E}}^{10}}(x)$
as hyperunified qubit-spinor field,
$\Psi_{\mathsf{Q}_{\mathsf{H}}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}}^{10}}(x)$,
satisfies the following self-conjugated chiral conditions:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{H}}}^{c}(x)=C_{\mathsf{Q}_{\mathsf{H}}}\bar{\Psi}_{\mathsf{Q}_{\mathsf{H}}}^{T}(x)=\Psi_{\mathsf{Q}_{\mathsf{H}}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}}^{10}}^{c}(x)=\Psi_{\mathsf{Q}_{\mathsf{E}}^{10}}(x),$
$\displaystyle C_{\mathsf{Q}_{\mathsf{H}}}=\sigma_{0}\otimes
C_{\mathsf{Q}_{\mathsf{H}}^{0}}\equiv\sigma_{0}\otimes
C_{\mathsf{Q}_{\mathsf{E}}^{9}}=\sigma_{0}\otimes\sigma_{0}\otimes
C_{\mathsf{Q}_{\mathsf{E}}^{8}},$
$\displaystyle\hat{\gamma}_{21}\Psi_{\mathsf{Q}_{\mathsf{H}}}(x)\equiv\hat{\gamma}_{21}\Psi_{\mathsf{Q}_{\mathsf{E}}^{10}}(x)=-\Psi_{\mathsf{Q}_{\mathsf{E}}^{10}}(x)\equiv-\Psi_{\mathsf{Q}_{\mathsf{H}}}(x),$
(386)

where $\Psi_{\mathsf{Q}_{\mathsf{H}}}(x)$ is defined by convention to have
negative chirality in the spinor representation of twenty dimensions.

The scalar coupling with a single $\tilde{\varGamma}$ matrix indicates that
the entangled decaqubit-spinor field as hyperunified qubit-spinor field
belongs to category-1 hyperqubit-spinor field with ${\mathcal{Q}}_{c}$-spin
charge $\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=1$. As the hyperunified qubit-
spinor field $\Psi_{\mathsf{Q}_{\mathsf{H}}}(x)$ is composed of two minimal
hyperunified qubit-spinor fields $\Psi_{\mathsf{Q}_{\mathsf{H}}^{-}}(x)$ and
$\Psi_{\mathsf{Q}_{\mathsf{H}}^{+}}(x)$, it concerns four families of chiral
type lepton-quark states in two ultra-grand unified hyperqubit-spinor fields
$\Psi_{\mathsf{Q}_{\mathsf{U}}^{-}}(x)$ and
$\hat{\Psi}_{\mathsf{Q}_{\mathsf{U}}^{-}}(x)$ with negative U-parity and also
four families of mirror lepton-quark states in two ultra-grand unified
hyperqubit-spinor fields $\Psi_{\mathsf{Q}_{\mathsf{U}}^{+}}(x)$ and
$\hat{\Psi}_{\mathsf{Q}_{\mathsf{U}}^{+}}(x)$ with positive U-parity.

Let us now turn to analyze alternative entangled decaqubit-spinor field as
hyperunified qubit-spinor field, which provides a distinguished qubit-spinor
structure so as to characterize explicitly four families of chiral type
lepton-quark states and mirror lepton-quark states. Such an entangled
decaqubit-spinor field as hyperunified qubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}}^{10}}(x)$
is constructed straightforwardly from grand unified qubit-spinor fields which
are composed of twisting entangled hyperqubit-spinor fields with both negative
and positive U-parities.

In general, such an entangled decaqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{E}}^{10}}(x)$ contains two entangled enneaqubit-
spinor fields $\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)$ and
$\hat{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)$ which are regarded as minimal
hyperunified qubit-spinor fields with negative and positive H-parities, i.e.,
$\Psi_{\mathbf{Q}_{\mathsf{H}}^{-}}(x)$ and
$\Psi_{\mathbf{Q}_{\mathsf{H}}^{+}}(x)$. Explicitly,
$\Psi_{\mathbf{Q}_{\mathsf{E}}^{10}}(x)$ has the following structure:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{E}}^{10}}(x)$ $\displaystyle\equiv$
$\displaystyle\binom{\Psi_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)}{\hat{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{9}}(x)}=\begin{pmatrix}\Psi_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)\\\
\tilde{\Psi}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)\\\
\tilde{\Psi}^{\prime}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)\\\
\Psi^{\prime}_{\mathbf{Q}_{\mathsf{E}}^{8}}(x)\end{pmatrix}=\begin{pmatrix}\Psi_{\mathbf{Q}_{\mathsf{E}_{1}^{-}}^{7}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{E}_{2}^{-}}^{7}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{E}_{3}^{+}}^{7}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{E}_{4}^{+}}^{7}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{E}_{1}^{+}}^{7}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{E}_{2}^{+}}^{7}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{E}_{3}^{-}}^{7}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{E}_{4}^{-}}^{7}}(x)\end{pmatrix}\equiv\Psi_{\mathbf{Q}_{\mathsf{H}}}(x),$
(387)

with the identical expression notated as follows:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)$ $\displaystyle\equiv$
$\displaystyle\binom{\Psi_{\mathbf{Q}_{\mathsf{H}}^{-}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{H}}^{+}}(x)}=\begin{pmatrix}\Psi_{\mathbf{Q}_{\mathsf{U}}^{-}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{U}}^{+}}(x)\\\
\hat{\Psi}_{\mathbf{Q}_{\mathsf{U}}^{+}}(x)\\\
\hat{\Psi}_{\mathbf{Q}_{\mathsf{U}}^{-}}(x),\end{pmatrix}=\begin{pmatrix}\Psi_{\mathbf{Q}_{\mathsf{G}_{1}}^{-}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{G}_{2}}^{-}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{G}_{3}}^{+}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{G}_{4}}^{+}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{G}_{1}}^{+}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{G}_{2}}^{+}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{G}_{3}}^{-}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{G}_{4}}^{-}}(x)\end{pmatrix},$ (388)

where the subscripts with letters in bold style ‘$\mathbf{Q}_{\mathsf{E}}$’
and ‘$\mathbf{Q}_{\mathsf{H}}$’ are adopted to distinguish from the
hyperunified qubit-spinor field discussed in previous analysis. The two mirror
pairs of twisting entangled hyperqubit-spinor fields as grand unified qubit-
spinor fields with negative and positive U-parity,
$\Psi_{\mathbf{Q}_{\mathsf{G}_{i}}^{\pm}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{\pm}}^{7}}(x)$
($i=1,2$) and
$\Psi_{\mathbf{Q}_{\mathsf{G}_{j}}^{\pm}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}_{j}^{\pm}}^{7}}(x)$
($j=3,4$), are defined as follows:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{G}_{i}}^{\pm}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}_{i}^{\pm}}^{7}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{W}_{i}^{\pm}}^{6}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{W}_{i}^{\pm}}^{6}}^{c}(x)}\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}^{\tilde{R},L}(x)}{\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}^{\tilde{R},L\;c}(x)},$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{G}_{j}}^{\pm}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}_{j}^{\pm}}^{7}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{W}_{j}^{\pm}}^{6}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{W}_{j}^{\pm}}^{6}}^{c}(x)}\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{S}_{j}}^{6}}^{\tilde{R},L}(x)}{\Psi_{\mathbf{Q}_{\mathsf{S}_{j}}^{6}}^{\tilde{R},L\;c}(x)},$
(389)

where $\Psi_{\mathbf{Q}_{\mathsf{W}_{i}^{\pm}}^{6}}(x)$ ($i=1,2$) and
$\Psi_{\mathbf{Q}_{\mathsf{W}_{j}^{\pm}}^{6}}(x)$ ($j=3,4$) are two mirror
pairs of chiral complex hexaqubit-spinor fields. They are given by two mirror
pairs of self-conjugated hexaqubit-spinor fields
$\Psi_{\mathbf{Q}_{\mathsf{S}_{i}}^{6}}^{\tilde{R},L}(x)$ ($i=1,2$) and
$\Psi_{\mathbf{Q}_{\mathsf{S}_{j}}^{6}}^{\tilde{R},L}(x)$ ($j=3,4$) with the
following explicit forms:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}_{f}}^{6}}^{L}(x)\equiv\begin{pmatrix}\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{u}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{d\,\bar{c}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{d}(x)\\\
-\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{u\,\bar{c}}(x)\end{pmatrix}_{L},\quad\Psi_{\mathbf{Q}_{\mathsf{S}_{f}}^{6}}^{L\;c}(x)\equiv\begin{pmatrix}\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{u}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{d\,\bar{c}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{d}(x)\\\
-\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{u\,\bar{c}}(x)\end{pmatrix}_{R},$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}_{f}}^{6}}^{\tilde{R}}(x)\equiv\begin{pmatrix}\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{u}(x)\\\
\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{d\,\bar{c}}(x)\\\
\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{d}(x)\\\
-\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{u\,\bar{c}}(x)\end{pmatrix}_{R},\quad\Psi_{\mathbf{Q}_{\mathsf{S}_{f}}^{6}}^{\tilde{R}\;c}(x)\equiv\begin{pmatrix}\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{u}(x)\\\
\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{d\,\bar{c}}(x)\\\
\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{d}(x)\\\
-\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{u\,\bar{c}}(x)\end{pmatrix}_{L}$
(390)

with $f=i,j=1,2,3,4$ ($i=1,2$, $j=3,4$). Where
$\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{q}(x)$ and
$\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{q}(x)$ ($q=u,d$) are the
mirror pairs of complex tetraqubit-spinor fields,

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{q\,L,R}(x)\equiv\begin{pmatrix}Q_{f}^{r}(x)\\\
Q_{f}^{b}(x)\\\ Q_{f}^{g}(x)\\\
Q_{f}^{w}(x)\end{pmatrix}^{q}_{L,R},\quad\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{q\,L,R}(x)\equiv\begin{pmatrix}\tilde{Q}_{f}^{r}(x)\\\
\tilde{Q}_{f}^{b}(x)\\\ \tilde{Q}_{f}^{g}(x)\\\
\tilde{Q}_{f}^{w}(x)\end{pmatrix}^{q}_{L,R},$ (391)

with their complex charge-conjugated ones
$\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{q\,\bar{c}}(x)$ and
$\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{q\,\bar{c}}(x)$,

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{q\,\bar{c}\,L,R}(x)\equiv\begin{pmatrix}Q^{r\,\bar{c}}_{f}(x)\\\
Q^{b\,\bar{c}}_{f}(x)\\\ Q^{g\,\bar{c}}_{f}(x)\\\
Q^{w\,\bar{c}}_{f}(x)\end{pmatrix}^{q}_{L,R},\quad\tilde{\Psi}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}^{q\,\bar{c}\,L,R}(x)\equiv\begin{pmatrix}\tilde{Q}^{r\,\bar{c}}_{f}(x)\\\
\tilde{Q}^{b\,\bar{c}}_{f}(x)\\\ \tilde{Q}^{g\,\bar{c}}_{f}(x)\\\
\tilde{Q}^{w\,\bar{c}}_{f}(x)\end{pmatrix}^{q}_{L,R},$ $\displaystyle
Q^{\alpha\,\bar{c}}_{f}(x)=C_{D}\bar{Q}^{\alpha\,T}_{f}(x),\quad\tilde{Q}^{\alpha\,\bar{c}}_{f}(x)=C_{D}\bar{\tilde{Q}}^{\alpha\,T}_{f}(x),\quad
C_{D}=-i\sigma_{3}\otimes\sigma_{2},$ (392)

where $q=u,d$ represent the up-type and down-type lepton-quark states in SM,
i.e., $Q^{\alpha\,u}_{f\,L,R}(x)=U_{f\,L,R}^{\alpha}(x)$ and
$Q^{\alpha\,d}_{f\,L,R}(x)=D_{f\,L,R}^{\alpha}(x)$, and also the up-type and
down-type mirror lepton-quark states, i.e.,
$\tilde{Q}^{\alpha\,u}_{f\,L,R}(x)=\tilde{U}_{f\,L,R}^{\alpha}(x)$ and
$\tilde{Q}^{\alpha\,d}_{f\,L,R}(x)=\tilde{D}_{f\,L,R}^{\alpha}(x)$. The four
color-spin charges $\alpha=r,b,g,w$ correspond to ‘red’, ‘blue’, ‘green’ and
‘white’, and the subscripts $f=1,2,3,4$ denote four families of chiral type
lepton-quark states and mirror lepton-quark states. The letters ‘L’ and ‘R’
label the left-handed and right-handed lepton-quark states and the
corresponding mirror lepton-quark states with opposite chirality.

In such a representation of qubit-spinor structure, the action of hyperunified
qubit-spinor field $\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)$ can be expressed as
follows:

$\displaystyle{\mathcal{S}}_{\mathbf{Q}_{\mathsf{H}}}$ $\displaystyle\equiv$
$\displaystyle\int
d^{19}x\,\\{\bar{\Psi}_{\mathbf{Q}_{\mathsf{H}}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varSigma_{-}^{\mathbf{A}}i\partial_{\mathsf{M}}\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)-\lambda_{1}\phi_{1}(x)\bar{\Psi}_{\mathbf{Q}_{\mathsf{H}}}(x)\tilde{\varSigma}_{-}\Psi_{\mathbf{Q}_{\mathsf{H}}}(x),$
(393)

with $\mathsf{A},\mathsf{M}=0,1,2,3,5,\cdots,19$. The matrices
$\varSigma_{-}^{\mathsf{A}}$ and $\tilde{\varSigma}_{-}$ are given by
$\varGamma$-matrices
$\varGamma^{\hat{\mathsf{A}}}\equiv(\varGamma^{\mathsf{A}},\varGamma^{20})\equiv(\varGamma^{a},\varGamma^{A},\tilde{\varGamma})$
($\hat{\mathsf{A}}=0,1,2,3,5,\cdots,20$). They have the following explicit
forms:

$\displaystyle\varSigma_{-}^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},\quad\mathsf{A}=0,1,2,3,5,\cdots
19,$
$\displaystyle\varSigma_{-}^{20}=\frac{1}{2}\varGamma^{20}\varGamma_{-}=\frac{1}{2}\tilde{\varGamma}\varGamma_{-}\equiv\tilde{\varSigma}_{-},\quad\varGamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{21}),$
$\displaystyle\varGamma^{a}=\;\;\,\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\varGamma^{a},\quad
a=0,1,2,3,$
$\displaystyle\varGamma^{\mathsf{A}}=-\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{2}\otimes\varGamma^{A},\quad
A=5,\cdots,14,$
$\displaystyle\varGamma^{15}=\;i\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{1}\otimes\tilde{\gamma}_{5},$
$\displaystyle\varGamma^{16}=\;i\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\hat{\gamma}_{15},$
$\displaystyle\varGamma^{17}=\;i\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\tilde{\gamma}_{5},$
$\displaystyle\varGamma^{18}=\;i\sigma_{0}\otimes\sigma_{1}\otimes\sigma_{0}\otimes\hat{\gamma}_{15},$
$\displaystyle\varGamma^{19}=\;i\sigma_{1}\otimes\sigma_{3}\otimes\sigma_{0}\otimes\hat{\gamma}_{15},$
$\displaystyle\varGamma^{20}=\;i\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{0}\otimes\hat{\gamma}_{15}\equiv\tilde{\varGamma},$
$\displaystyle\hat{\gamma}_{21}=\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{0}\otimes\hat{\gamma}_{15}=\sigma_{3}\otimes\hat{\gamma}_{19},$
$\displaystyle\tilde{\gamma}_{5}=\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
(394)

which have a different structure in comparison to those in Eq.(X.3). Where
$\varGamma^{a}$ and $\varGamma^{A}$ ($a=0,1,2,3,A=5,\cdots,14$) on the right-
hand side of equality are defined in Eq.(IX.1).

The entangled decaqubit-spinor field as hyperunified qubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{E}}^{10}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)$
satisfies the following self-conjugated chiral conditions:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{H}}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{H}}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{H}}}^{T}(x)=\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}}^{10}}^{c}(x)=\Psi_{\mathbf{Q}_{\mathsf{E}}^{10}}(x),$
$\displaystyle C_{\mathbf{Q}_{\mathsf{H}}}=\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{H}}^{0}}\equiv\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{E}}^{9}}=\sigma_{0}\otimes\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{E}}^{8}},$ $\displaystyle
C_{\mathbf{Q}_{\mathsf{E}}^{8}}=\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{E}_{i}}^{7}}=\sigma_{0}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D},$
$\displaystyle\hat{\gamma}_{21}\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)\equiv\hat{\gamma}_{21}\Psi_{\mathbf{Q}_{\mathsf{E}}^{10}}(x)=-\Psi_{\mathbf{Q}_{\mathsf{E}}^{10}}(x)\equiv-\Psi_{\mathbf{Q}_{\mathsf{H}}}(x),$
(395)

which indicates that the entangled decaqubit-spinor field as hyperunified
qubit-spinor field has only a trivial ${\mathcal{Q}}_{c}$-spin symmetry
SP(1)$\cong$O(1) with transformation
$\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)\to-\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)$ which
coincides with the chiral condition
$\hat{\gamma}_{21}\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)=-\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)$.

It is easy to check that the actions in Eqs.(393) and (384) possess an
associated symmetry characterized by the inhomogeneous hyperspin symmetry
WS(1,18) associated with inhomogeneous Lorentz-type/Poincaré-type group
symmetry PO(1,18) together with global scaling symmetry SG(1) and SC(1), which
is given as follows:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,18)\,\widetilde{\Join}\,WS(1,18)\times SG(1)$ (396) $\displaystyle=$
$\displaystyle SC(1)\ltimes P^{1,18}\ltimes
SO(1,18)\,\widetilde{\Join}\,SP(1,18)\rtimes W^{1,18}\times SG(1),$

where the transformation of hyperspin symmetry SP(1,18) should be coincidental
to that of Lorentz-type group symmetry SO(1,18) as expressed by the symbol
“$\,\widetilde{\Join}\,$” which is adopted to indicate the associated
symmetry. The group generators of inhomogeneous hyperspin symmetry WS(1,18)
are provided directly by the following commutators of $\varGamma$-matrices:

$\displaystyle\varSigma^{\mathsf{A}\mathsf{B}}=\frac{i}{4}[\varGamma^{\mathsf{A}},\varGamma^{\mathsf{B}}],\quad\varSigma_{-}^{\mathsf{A}}=\frac{1}{2}\varGamma^{\mathsf{A}}\varGamma_{-},$
(397)

with $\mathsf{A},\mathsf{B}=0,1,2,3,5,\cdots,19$.

In general, the hyperspin symmetry SP(1,18) gets a natural decomposition into
the following subgroup symmetries:

$\displaystyle SP(1,18)$ $\displaystyle\supset$ $\displaystyle SP(1,3)\times
SP(15)\supset SP(1,3)\times SP(10)\times SP(5)$ (398) $\displaystyle\cong$
$\displaystyle SO(1,3)\times SO(15)\supset SO(1,3)\times SO(10)\times
SO_{F}(5)$ $\displaystyle\supset$ $\displaystyle SO(1,3)\times SO(6)\times
SO(4)\times SO_{F}(4)$ $\displaystyle\cong$ $\displaystyle SO(1,3)\times
SU_{C}(4)\times SU_{L}(2)\times SU_{R}(2)\times SU_{F}(2)\times SU_{M}(2),$

with the subgroup algebras given by the following group generators:

$\displaystyle\varSigma^{\mathbf{A}\mathbf{B}}(\mathbf{A},\mathbf{B}=0,1,2,3)\in
sp(1,3)\cong so(1,3)$
$\displaystyle\varSigma^{\mathbf{A}\mathbf{B}}(\mathbf{A},\mathbf{B}=5,\cdots,19)\in
sp(15)\cong so(15),$
$\displaystyle\varSigma^{\mathbf{A}\mathbf{B}}(\mathbf{A},\mathbf{B}=5,\cdots,14)\in
sp(10)\cong so(10),$
$\displaystyle\varSigma^{\mathbf{A}\mathbf{B}}(\mathbf{A},\mathbf{B}=15,\cdots,19)\in
sp(5)\cong so_{F}(5),$
$\displaystyle\varSigma^{\mathbf{A}\mathbf{B}}(\mathbf{A},\mathbf{B}=5,\cdots,10)\in
sp(6)\cong so(6)\cong su_{C}(4),$
$\displaystyle\varSigma^{\mathbf{A}\mathbf{B}}(\mathbf{A},\mathbf{B}=11,\cdots
14)\in sp(4)\cong so(4)\cong su_{L}(2)\times su_{R}(2),$ (399)

where the subgroup symmetry SPF(5)$\cong$SOF(5) characterizes the family-spin
symmetry for four families of mirror pairs of lepton-quark states.

Therefore, the entangled decaqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{E}}^{10}}(x)$ (or
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{10}}(x)$) enables us to unify all discovered
leptons and quarks into a single hyperunified qubit-spinor field, i.e.,
$\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{E}}^{10}}(x)$
(or
$\Psi_{\mathsf{Q}_{\mathsf{H}}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}}^{10}}(x)$),
which is regarded as the fundamental building block of nature. The maximal
inhomogeneous hyperspin symmetry WS(1,18) in association with inhomogeneous
Lorentz-type/Poincaré-type group symmetry PO(1,18) together with global
scaling symmetry SG(1) and SC(1) brings on a hyperunified symmetry as
fundamental symmetry of nature.

###  X.4  Conservation of H-parity and U-parity in hyperunified field theory
with vector-like lepton-quark state and comprehension on P-parity violation in
SM

The category-1 entangled decaqubit-spinor field as hyperunified qubit-spinor
field $\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)$ brings about four families of chiral
type lepton-quark states with negative U-parity and four families of mirror
lepton-quark states with positive U-parity, i.e.,
$\Psi_{\mathbf{Q}_{\mathsf{G}_{f}}^{-}}(x)$ and
$\Psi_{\mathbf{Q}_{\mathsf{G}_{f}}^{+}}(x)$ ($f=1,2,3,4$), which motivates us
to express such a hyperunified qubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)$ into four families of vector-like lepton-
quark states in 19-dimensional hyper-spacetime.

To be explicit and specific, let us reconstruct the entangled decaqubit-spinor
field
$\Psi_{\mathbf{Q}_{\mathsf{E}}^{10}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{H}}}(x)$
into a self-conjugated enneaqubit-spinor field, which can be expressed as a
superposition of two entangled enneaqubit-spinor fields given by minimal
hyperunified qubit-spinor fields with the following form:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}}^{9}}(x)$ $\displaystyle\equiv$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{H}}^{-}}(x)+\Psi_{\mathbf{Q}_{\mathsf{H}}^{+}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{U}}}(x)}{\hat{\Psi}_{\mathbf{Q}_{\mathsf{U}}}(x)}\equiv\begin{pmatrix}\Psi_{\mathbf{Q}_{\mathsf{G}_{1}}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{G}_{2}}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{G}_{3}}}(x)\\\
\Psi_{\mathbf{Q}_{\mathsf{G}_{4}}}(x)\end{pmatrix},$ (400)

with the definitions,

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{U}}}(x)$ $\displaystyle\equiv$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{U}}^{-}}(x)+\hat{\Psi}_{\mathbf{Q}_{\mathsf{U}}^{+}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{S}}^{8}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{S}_{1}}^{7}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{S}_{2}}^{7}}(x)},$
$\displaystyle\hat{\Psi}_{\mathbf{Q}_{\mathsf{U}}}(x)$ $\displaystyle\equiv$
$\displaystyle\hat{\Psi}_{\mathbf{Q}_{\mathsf{U}}^{-}}(x)+\Psi_{\mathbf{Q}_{\mathsf{U}}^{+}}(x)\equiv\hat{\Psi}_{\mathbf{Q}_{\mathsf{S}}^{8}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{S}_{3}}^{7}}(x)}{\Psi_{\mathbf{Q}_{\mathsf{S}_{4}}^{7}}(x)},$
(401)

and

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}_{f}}^{7}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{G}_{f}}}(x)\equiv\binom{\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{6}}(x)}{\Psi^{c}_{\mathbf{Q}_{\mathsf{C}_{f}}^{6}}(x)},\;\;f=1,2,3,4.$
(402)

All above defined hyperqubit-spinor fields
$\Psi_{\mathbf{Q}_{\mathsf{S}}^{9}}(x)$,
$\Psi_{\mathbf{Q}_{\mathsf{S}}^{8}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{U}}}(x)$,
$\hat{\Psi}_{\mathbf{Q}_{\mathsf{S}}^{8}}(x)\equiv\hat{\Psi}_{\mathbf{Q}_{\mathsf{U}}}(x)$
and
$\Psi_{\mathbf{Q}_{\mathsf{S}_{f}}^{7}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{G}_{f}}}(x)$
($f=1,2,3,4$) are self-conjugated hyperqubit-spinor fields and satisfy the
following self-conjugation conditions:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}}^{9}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{S}}^{9}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{S}}^{9}}^{T}(x)=\Psi_{\mathbf{Q}_{\mathsf{S}}^{9}}(x),$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}}^{8}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{S}}^{8}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{S}}^{8}}^{T}(x)=\Psi_{\mathbf{Q}_{\mathsf{S}}^{8}}(x),$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}_{f}}^{7}}^{c}(x)=C_{\mathbf{Q}_{\mathsf{S}}^{7}}\bar{\Psi}_{\mathbf{Q}_{\mathsf{S}_{f}}^{7}}^{T}(x)=\Psi_{\mathbf{Q}_{\mathsf{S}_{f}}^{7}}(x),$
$\displaystyle C_{\mathbf{Q}_{\mathsf{S}}^{9}}\equiv\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{S}}^{8}},\quad
C_{\mathbf{Q}_{\mathsf{S}}^{8}}\equiv\sigma_{0}\otimes
C_{\mathbf{Q}_{\mathsf{S}}^{7}},$ $\displaystyle
C_{\mathbf{Q}_{\mathsf{S}}^{7}}=\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D},$ (403)

where the self-conjugated enneaqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{S}}^{9}}(x)$ provides a hyperunified qubit-spinor
field with different spinor structure. The four self-conjugated hyperqubit-
spinor fields $\Psi_{\mathbf{Q}_{\mathsf{S}_{f}}^{7}}(x)$ ($f=1,2,3,4$)
characterize four families of grand unified qubit-spinor fields
$\Psi_{\mathbf{Q}_{\mathsf{G}_{f}}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{S}_{f}}^{7}}(x)$
($f=1,2,3,4$), which are described by four complex hexaqubit-spinor fields
$\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{6}}(x)$ together with their complex
charge-conjugated ones $\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{6}}^{c}(x)$.
Explicitly, $\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{6}}(x)$
($\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{6}}^{c}(x)$) are composed of four
families of vector-like lepton-quark states which are given by the following
forms:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{6}}(x)\equiv\begin{pmatrix}\Psi^{\hat{u}}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}(x)\\\
\Psi^{\check{d}\,\bar{c}}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}(x)\\\
\Psi^{\hat{d}}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}(x)\\\
-\Psi^{\check{u}\,\bar{c}}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}(x)\end{pmatrix},\quad\Psi_{\mathbf{Q}_{\mathsf{C}_{f}}^{6}}^{c}(x)\equiv\begin{pmatrix}\Psi^{\check{u}}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}(x)\\\
\Psi^{\hat{d}\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}(x)\\\
\Psi^{\check{d}}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}(x)\\\
-\Psi^{\hat{u}\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}(x)\end{pmatrix},$
(404)

where $\Psi^{\hat{q}}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}(x)$
($\hat{q}=\hat{u},\hat{d},\check{u},\check{d}$ and $f=1,2,3,4$) are complex
tetraqubit-spinor fields with
$\Psi^{\hat{q}\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}(x)$ their complex
charge-conjugated ones. Their explicit forms are given as follows:

$\displaystyle\Psi^{\hat{q}}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}(x)\equiv\begin{pmatrix}Q_{f}^{r}(x)\\\
Q_{f}^{b}(x)\\\ Q_{f}^{g}(x)\\\
Q_{f}^{w}(x)\end{pmatrix}^{\hat{q}},\quad\Psi^{\hat{q}\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}_{f}}^{4}}(x)\equiv\begin{pmatrix}Q_{f}^{r\,\bar{c}}(x)\\\
Q_{f}^{b\,\bar{c}}(x)\\\ Q_{f}^{g,\bar{c}}(x)\\\
Q_{f}^{w\,\bar{c}}(x)\end{pmatrix}^{\hat{q}},$ (405)