The assumption that (effective) generating sources $\
{}^{\shortmid}\Im_{\alpha_{s}\beta_{s}}^{\star}(\tau)$ for a $\tau$-family of
$\kappa$-parametric nonassociative geometric flow equations (38) can be
parameterized in N-adapted form as $\ {}_{\star}^{\shortmid}\Im_{\
\beta_{2}}^{\alpha_{2}}=[\ \ _{\star}^{\shortmid}\Im_{1}(\tau)\delta_{\ \
j_{1}}^{i_{1}},\ \ _{\star}^{\shortmid}\Im_{2}(\tau)\delta_{\ \
b_{2}}^{a_{2}}]$ (39) is important because it allows to integrate the system
of nonlinear PDEs (41) in explcit form. The shell $s=1,2$ data $\
{}_{\star}^{\shortmid}\Im_{2}(\tau)$ are related via certain algebraic
relations (choosing respectively the coefficients of frame transforms) to
certain $\tau$-families of energy-momentum tensors for matter fields, star
product R-flux $\kappa$-parametric deformations, and $\partial_{\tau}\
^{\shortmid}\mathbf{g}_{\alpha_{s}\beta_{s}}(\tau).$ Such algebraic relations
involve partial derivatives $\partial_{\tau}$, on geometric flow parameter and
can be prescribed for explicit symmetries of solutions, additional deformation
parameters (like eccentricity, small interactions with matter fields). They
are stated as some nonholonomic constraints for some classes of nonholonomic
transforms which can be solved explicitly for additional assumptions on
nonassociative/ nonholonomic data, symmetries of interactions, prescriptions
of some effective constants, or trivial ansatz for extra dimension components.
This refers to certain nonholonomic frame and special coordinate conditions.
Nevertheless, the effective/ $\tau$-parametric geometric flow equations (41)
can be decoupled and integrated in certain general forms applying the standard
AFCDM re-defined by $\ {}_{\star}^{\shortmid}\Im_{\
\beta_{2}}^{\alpha_{2}}(\tau)$ and certain effective $\
{}_{2}\Lambda^{\star}(\tau).$ We use a star label even for such effective
cosmological (running) cosmological constants because they encode in direct
holonomic, or in nonholonomic forms, certain nonassociative data from extra
dimensional velocity (momentum) like (co) fiber coordinates. This allows us to
prove general decoupling and integration properties of nonassociative
geometric flow and/or gravitational equations as in [36] by considering
additional dependencies of s-metrics and effective sources on $\tau.$ To model
$\tau$-evolution of 4-d quasi-stationary configurations, we can consider two
effective generating sources, for instance, when that $\
{}_{\star}^{\shortmid}\Im_{1}(\tau)=\ _{\star}^{\shortmid}\Im_{1}(\tau,x^{i})$
and $\ {}_{\star}^{\shortmid}\Im_{2}(\tau)=\
_{\star}^{\shortmid}\Im_{2}(\tau,x^{i},y^{3}).$

We generate generic off-diagonal solutions describing quasi-stationary
geometric evolution of nonholonomic Einstein systems when the effective
sources $\ {}_{\star}^{\shortmid}\Im_{\ \beta_{2}}^{\alpha_{2}}(\tau)$ are
prescribed from certain physical considerations. They can be related to
effective $\tau$-running cosmological constants $\
{}_{2}\Lambda^{\star}(\tau),$ or considered as certain symbols encoding
nonassociative data and/or encoding nonholonomic deformations and distortions.
To elaborate on explicit computations such $\ {}_{\star}^{\shortmid}\Im_{\
\beta_{2}}^{\alpha_{2}}(\tau)$ may be considered as generating sources
determined by contributions, for instance, from some effective classical
and/or quantum interactions, extra dimension etc. For small parametric
decompositions and using recurrent formulas, we can generate new classes of
parametric solutions when their physical interpretation is similar to "not-
deformed" prime models. In such cases, nonassociative geometric flow data are
encoded into certain locally anisotropic polarized physical constants, star
product deformed horizons, additional off-diagonal terms, induced nonsymmetric
components of metrics etc. In general, for certain parameterizations of
generating sources, it is not clear if a corresponding class of solutions may
have, or not, importance for physical theories. Nevertheless, using the AFCDM
we are able to investigate to construct explicit classes of exact/parametric
solutions and analyze properties of off-diagonal nonlinear gravitational and
(effective) matter field interactions subjected to nonassociative star product
R-flux deformations and their $\tau$-parametric geometric flow equations. This
is more general than in the case when the (modified) Einstein equations are
transformed into systems of nonlinear ODEs for diagonal metrics and LC-
configurations. Examples of physically important quasi-stationary solutions
are studied in section 4 for modified wormhole solutions encoding
nonassociative data.

####  A.2.3 Nonassociative quasi-stationary geometric flow solutions with
fixed energy parameter

Such quasi-stationary $\tau$-running solutions are determined by nonholonomic
flows of momentum type phase configurations modeled on cotangent Lorentz
bundles with $p_{8}=E_{0}=const.$ The Table 3 summarize the data on
nonassociative star product R-flux deformations when the effective sources $\
{}_{s}^{\shortmid}\widehat{\Upsilon}$ are changed into nonassociative and
$\tau$-running ones $\ {}_{s}^{\shortmid}\Im^{\star}(\tau).$ It also extends
the results of Table 2 above for $s=3,4$ on $T^{\ast}\mathbf{V}$ in a form
with $p_{8}=E_{0}$ keeping the quasi-stationary character of solutions.

In functional form, such nonassociative phase space s-metrics define a class
of nonlinear quadratic elements:

$\displaystyle d\widehat{s}_{[8d]}^{2}$ $\displaystyle=$
$\displaystyle\widehat{g}_{\alpha_{s}\beta_{s}}(\tau,x^{k},y^{3},p_{5},p_{7};h_{4}(\tau),\mathbf{\
{}^{\shortmid}}h^{6}(\tau),\mathbf{\ {}^{\shortmid}}h^{8}(\tau);\ \
_{2}^{\shortmid}\Im^{\star}(\tau);\ \ _{s}\Lambda^{\star}(\tau))d\mathbf{\
{}^{\shortmid}}u^{\alpha_{s}}d\mathbf{\ {}^{\shortmid}}u^{\beta_{s}}$
$\displaystyle=$ $\displaystyle e^{\psi(\tau,x^{k},\ \
_{1}^{\shortmid}\Im^{\star})}[(dx^{1})^{2}+(dx^{2})^{2}]-\frac{(h_{4}^{\ast})^{2}}{|\int
dy^{3}[\ _{2}^{\shortmid}\Im^{\star}h_{4}]^{\ast}|\
h_{4}}\\{dy^{3}+\frac{\partial_{i_{1}}[\int dy^{3}(\
_{2}^{\shortmid}\Im^{\star})\ h_{4}^{\ast}]}{\ \
{}_{2}^{\shortmid}\Im^{\star}\ h_{4}^{\ast}}dx^{i_{1}}\\}^{2}+$ $\displaystyle
h_{4}\\{dt+[\ _{1}n_{k_{1}}+\ _{2}n_{k_{1}}\int
dy^{3}\frac{(h_{4}^{\ast})^{2}}{|\int dy^{3}[\
_{2}^{\shortmid}\Im^{\star}h_{4}]^{\ast}|\ (h_{4})^{5/2}}]dx_{1}^{k}\\}+$
$\displaystyle\frac{(\mathbf{\ {}^{\shortmid}}\partial^{5}\mathbf{\
{}^{\shortmid}}h^{6})^{2}}{|\int dp_{5}\mathbf{\ {}^{\shortmid}}\partial^{5}[\
_{3}^{\shortmid}\Im^{\star}\mathbf{\ {}^{\shortmid}}h^{6}]|\ \mathbf{\
{}^{\shortmid}}h^{6}}\\{dp_{5}+\frac{\partial_{i_{2}}[\int dp_{5}(\
_{3}^{\shortmid}\Im^{\star})\mathbf{\ {}^{\shortmid}}\partial^{5}\mathbf{\
{}^{\shortmid}}h^{6}]}{\ \ {}_{3}^{\shortmid}\Im^{\star}\mathbf{\
{}^{\shortmid}}\partial^{5}\ \mathbf{{}^{\shortmid}}h^{6}}dx^{i_{2}}\\}^{2}+$
$\displaystyle\mathbf{\ {}^{\shortmid}}h^{6}\\{dp_{5}+[\ _{1}n_{k_{2}}+\
_{2}n_{k_{2}}\int dp_{5}\frac{(\mathbf{\ {}^{\shortmid}}\partial^{5}\mathbf{\
{}^{\shortmid}}h^{6})^{2}}{|\int dp_{5}\mathbf{\ {}^{\shortmid}}\partial^{5}[\
_{3}^{\shortmid}\Im^{\star}\mathbf{\ {}^{\shortmid}}h^{6}]|\ (\mathbf{\
{}^{\shortmid}}h^{6})^{5/2}}]dx^{k_{2}}\\}+$ $\displaystyle\frac{(\mathbf{\
{}^{\shortmid}}\partial^{7}\mathbf{\ {}^{\shortmid}}h^{8})^{2}}{|\int
dp_{7}\mathbf{\ {}^{\shortmid}}\partial^{7}[\
_{4}^{\shortmid}\Im^{\star}\mathbf{\ {}^{\shortmid}}h^{8}]|\ \mathbf{\
{}^{\shortmid}}h^{8}}\\{dp_{7}+\frac{\partial_{i_{3}}[\int dp_{7}(\
_{4}^{\shortmid}\Im^{\star})\ \mathbf{\ {}^{\shortmid}}\partial^{7}\mathbf{\
{}^{\shortmid}}h^{8}]}{\ {}_{4}^{\shortmid}\Im^{\star}\ \mathbf{\
{}^{\shortmid}}\partial^{7}\mathbf{\ {}^{\shortmid}}h^{8}}d\mathbf{\
{}^{\shortmid}}x^{i_{3}}\\}^{2}+$ $\displaystyle\mathbf{\
{}^{\shortmid}}h^{8}\\{dE+[\ _{1}^{\shortmid}n_{k_{3}}+\
_{2}^{\shortmid}n_{k_{3}}\int dp_{7}\frac{(\mathbf{\
{}^{\shortmid}}\partial^{7}\mathbf{\ {}^{\shortmid}}h^{8})^{2}}{|\int
dp_{7}\mathbf{\ {}^{\shortmid}}\partial^{7}[\ \
_{3}^{\shortmid}\Im^{\star}\mathbf{\ {}^{\shortmid}}h^{8}]|\ (\mathbf{\
{}^{\shortmid}}h^{8})^{5/2}}]d\mathbf{\ {}^{\shortmid}}x^{k_{3}}\\}.$

The nonlinear symmetries of such s-metrics for generic off-diagonal flows on
nonassociative phase spaces are included in Table 3. They allow to allow to
re-define the generating functions and generating sources and relate them to
conventional $\tau$-running cosmological constants $\
{}_{s}\Lambda^{\star}(\tau).$

####  A.2.4 Nonassociative quasi-stationary evolution and gravitational
polarizations

Quasi-stationary $\tau$-running solutions can be re-defined in terms of
gravitational $\eta$-polarizations (A.17). We can study off-diagonal
deformations of a family of prescribed prime s-metrics $\
{}_{s}^{\shortmid}\mathbf{\mathring{g}}(\tau)$ into other families of target
ones,

$\ {}_{s}^{\shortmid}\mathbf{\mathring{g}}(\tau)=[\
^{\shortmid}\mathring{g}_{\alpha_{s}}(\tau),\
^{\shortmid}\mathring{N}_{i_{s-1}}^{a_{s}}(\tau)]\rightarrow\
_{s}^{\shortmid}\mathbf{g}(\tau)=[\ ^{\shortmid}\eta_{\alpha_{s}}(\tau)\
^{\shortmid}\mathring{g}_{\alpha_{s}}(\tau),\
^{\shortmid}\eta_{i_{s-1}}^{a_{s}}(\tau)\
^{\shortmid}\mathring{N}_{i_{s-1}}^{a_{s}}(\tau)].$

The s-adapted coefficients of $\
{}_{s}^{\shortmid}\mathbf{\mathring{g}}(\tau)$ can be arbitrary ones or chosen
to define certain physically important solutions of some nonholonomic
geometric flows, Ricci solitons, or (modified) Einstein equations, for
instance of BH or BE types. The $\tau$-running generating functions can be
parameterize in the form:

$\psi(\tau)\simeq\psi(\hbar,\kappa;\tau,x^{k_{1}}),\eta_{4}(\tau)\
\simeq\eta_{4}(\tau,x^{k_{1}},y^{3}),\ ^{\shortmid}\eta^{6}(\tau)\simeq\
^{\shortmid}\eta^{6}(\tau,x^{i_{2}},p_{5}),\ ^{\shortmid}\eta^{8}(\tau)\simeq\
^{\shortmid}\eta^{8}(\tau,x^{i_{2}},p_{5},p_{7}),$ (A.45)

when the target s-metrics $\ {}_{s}^{\shortmid}\mathbf{g}(\tau)$ are solutions
of type (A.2.3). Such quasi-stationary $\tau$-parametric s-metrics define
nonlinear quadratic elements303030proofs are similar to those presented in
[36, 19] but for different types of effective generating sources encoding
nonassociative star product R-flux data

$\displaystyle d\ \ ^{\shortmid}\widehat{s}^{2}(\tau)$ $\displaystyle=$ $\ \
{}^{\shortmid}g_{\alpha_{s}\beta_{s}}(\hbar,\kappa,\tau,x^{k},y^{3},p_{a_{3}},p_{a_{4}};\
^{\shortmid}\mathring{g}_{\alpha_{s}}(\tau);\eta_{4}(\tau),\
^{\shortmid}\eta^{6}(\tau),\ ^{\shortmid}\eta^{8}(\tau),\
_{s}\Lambda^{\star}(\tau);\ \ _{s}^{\shortmid}\Im^{\star}(\tau))d~\
{}^{\shortmid}u^{\alpha_{s}}d~\ {}^{\shortmid}u^{\beta_{s}}$ $\displaystyle=$
$\displaystyle e^{\psi(\tau,\
_{1}\Lambda^{\star}(\tau))}[(dx^{1})^{2}+(dx^{2})^{2}]-$
$\displaystyle\frac{[\partial_{3}(\eta_{4}(\tau)\
\mathring{g}_{4}(\tau))]^{2}}{|\int dy^{3}\
_{2}{}^{\shortmid}\Im^{\star}(\tau)\partial_{3}(\eta_{4}(\tau)\
\mathring{g}_{4}(\tau))|\
(\eta_{4}(\tau)\mathring{g}_{4}(\tau))}\\{dy^{3}+\frac{\partial_{i_{1}}[\int
dy^{3}\ _{2}\Im(\tau)\ \partial_{3}(\eta_{4}(\tau)\mathring{g}_{4}(\tau))]}{\
{}_{2}^{\shortmid}\Im^{\star}(\tau)\partial_{3}(\eta_{4}(\tau)\mathring{g}_{4}(\tau))}dx^{i_{1}}\\}^{2}+$
$\displaystyle\eta_{4}(\tau)\mathring{g}_{4}(\tau))\\{dt+[\
_{1}n_{k_{1}}(\tau)+\
_{2}n_{k_{1}}(\tau)\int\frac{dy^{3}[\partial_{3}(\eta_{4}(\tau)\mathring{g}_{4}(\tau))]^{2}}{|\int
dy^{3}\
_{2}{}^{\shortmid}\Im^{\star}(\tau)\partial_{3}(\eta_{4}(\tau)\mathring{g}_{4}(\tau))|\
[\eta_{4}(\tau)\mathring{g}_{4}(\tau)]^{5/2}}]dx^{k_{1}}\\}-$
$\displaystyle\frac{[\ ^{\shortmid}\partial^{5}(\ ^{\shortmid}\eta^{6}(\tau)\
^{\shortmid}\mathring{g}^{6}(\tau))]^{2}}{|\int dp_{5}~\
{}_{3}^{\shortmid}\Im^{\star}(\tau)\ \ ^{\shortmid}\partial^{5}(\
^{\shortmid}\eta^{6}(\tau)\ \ ^{\shortmid}\mathring{g}^{6}(\tau))\ |\ (\
^{\shortmid}\eta^{6}(\tau)\
^{\shortmid}\mathring{g}^{6}(\tau))}\\{dp_{5}+\frac{\
{}^{\shortmid}\partial_{i_{2}}[\int dp_{5}\ {}_{3}^{\shortmid}\Im(\tau)\
^{\shortmid}\partial^{5}(\ ^{\shortmid}\eta^{6}(\tau)\
^{\shortmid}\mathring{g}^{6}(\tau))]}{~\ {}_{3}^{\shortmid}\Im^{\star}(\tau)\
^{\shortmid}\partial^{5}(\ ^{\shortmid}\eta^{6}(\tau)\
^{\shortmid}\mathring{g}^{6}(\tau))}dx^{i_{2}}\\}^{2}+$ $\displaystyle(\
^{\shortmid}\eta^{6}(\tau)\ ^{\shortmid}\mathring{g}^{6}(\tau))\\{dp_{6}+[\
_{1}^{\shortmid}n_{k_{2}}(\tau)+\
_{2}^{\shortmid}n_{k_{2}}(\tau)\int\frac{dp_{5}[\ ^{\shortmid}\partial^{5}(\
^{\shortmid}\eta^{6}(\tau)\ ^{\shortmid}\mathring{g}^{6}(\tau))]^{2}}{|\int
dp_{5}\ {}_{3}^{\shortmid}\Im^{\star}(\tau)\ \partial^{5}(\
^{\shortmid}\eta^{6}(\tau)\ ^{\shortmid}\mathring{g}^{6}(\tau))|\ [\
^{\shortmid}\eta^{6}(\tau)\
^{\shortmid}\mathring{g}^{6}(\tau)]^{5/2}}]dx^{k_{2}}\\}-$
$\displaystyle\frac{[\ ^{\shortmid}\partial^{7}(\ ^{\shortmid}\eta^{8}(\tau)\
^{\shortmid}\mathring{g}^{8}(\tau))]^{2}}{|\int dp_{7}\
{}_{4}^{\shortmid}\Im^{\star}(\tau)\ ^{\shortmid}\partial^{8}(\
^{\shortmid}\eta^{7}(\tau)\ ^{\shortmid}\mathring{g}^{7}(\tau))\ |\ (\
^{\shortmid}\eta^{7}(\tau)\
^{\shortmid}\mathring{g}^{7}(\tau))}\\{dp_{7}+\frac{\
{}^{\shortmid}\partial_{i_{3}}[\int dp_{7}\
{}_{4}^{\shortmid}\Im^{\star}(\tau)\ ^{\shortmid}\partial^{7}(\
^{\shortmid}\eta^{8}(\tau)\ \ ^{\shortmid}\mathring{g}^{8}(\tau))]}{\
{}_{4}^{\shortmid}\Im^{\star}(\tau)\ ^{\shortmid}\partial^{7}(\
^{\shortmid}\eta^{8}(\tau)\ ^{\shortmid}\mathring{g}^{8}(\tau))}d\
^{\shortmid}x^{i_{3}}\\}^{2}+$ $\displaystyle(\ ^{\shortmid}\eta^{8}(\tau)\
^{\shortmid}\mathring{g}^{8}(\tau))\\{dE+[\ _{1}n_{k_{3}}(\tau)+\
_{2}n_{k_{3}}(\tau)\int\frac{dp_{7}[\ ^{\shortmid}\partial^{7}(\
^{\shortmid}\eta^{8}(\tau)\ ^{\shortmid}\mathring{g}^{8}(\tau))]^{2}}{|\int
dp_{7}\ {}_{4}^{\shortmid}\Im^{\star}(\tau)[\ ^{\shortmid}\partial^{7}(\
^{\shortmid}\eta^{8}(\tau)\ ^{\shortmid}\mathring{g}^{8}(\tau))]|\ [\
^{\shortmid}\eta^{8}(\tau)\ ^{\shortmid}\mathring{g}^{8}(\tau)]^{5/2}}]d\
^{\shortmid}x^{k_{3}}\\}.$

String parametric deformations are modeled for $\
{}_{s}^{\shortmid}\eta(\tau)\ ^{\shortmid}\mathring{g}_{\alpha_{s}}(\tau)\sim\
^{\shortmid}\zeta_{\alpha_{s}}(\tau)(1+\kappa\
^{\shortmid}\chi_{\alpha_{s}}(\tau))\ \
^{\shortmid}\mathring{g}_{\alpha_{s}}(\tau).$

####  A.2.5 Nonassociative s-metrics with small parametric quasi-stationary
evolution of double phase space wormholes

We provide explicit formulas for explicit small parametric $\tau$\- and
$\kappa$-parametric decompositions of solutions (4.1.2) using
$\chi$-polarizations as we explain in subsection (4.1.3):

$\displaystyle d\ \widehat{s}^{2}(\tau)$ $\displaystyle=$
$\displaystyle\widehat{g}_{\alpha\beta}(\hbar,\kappa,\tau,l,\theta,\varphi,p_{l},p_{\varphi};\
^{\shortmid}\mathring{g}_{\alpha_{s}};\eta_{4}(\tau),\
^{\shortmid}\eta^{6}(\tau),\
^{\shortmid}\eta^{8}(\tau),\Lambda_{[h]}^{\star}(\tau);\
_{s}^{\shortmid}\Im^{\star}(\tau),\
^{\shortmid}\Lambda_{[c]}^{\star}(\tau))du^{\alpha}du^{\beta}$
$\displaystyle=$ $\displaystyle e^{\psi_{0}(\hbar,\tau,l,\theta)}[1+\kappa\
^{\psi(\hbar,\tau,l,\theta)}\chi(l,\theta)][(dx^{1}(l,\theta))^{2}+(dx^{2}(l,\theta))^{2}]$
$\displaystyle-\\{\frac{4[\partial_{\varphi}(|\zeta_{4}\
\breve{g}_{4}|^{1/2})]^{2}}{\breve{g}_{3}|\int
d\varphi\\{\Lambda_{[h]}^{\star}(\tau)\partial_{\varphi}(\zeta_{4}\
\breve{g}_{4})\\}|}-\kappa[\frac{\partial_{\varphi}(\chi_{4}|\zeta_{4}\breve{g}_{4}|^{1/2})}{4\partial_{\varphi}(|\zeta_{4}\
\breve{g}_{4}|^{1/2})}-\frac{\int d\varphi\\{\
\Lambda_{[h]}^{\star}(\tau)\partial_{\varphi}[(\zeta_{4}\
\breve{g}_{4})\chi_{4}]\\}}{\int
d\varphi\\{\Lambda_{[h]}^{\star}(\tau)\partial_{\varphi}(\zeta_{4}\
\breve{g}_{4})\\}}]\\}\ \breve{g}_{3}$
$\displaystyle\\{d\varphi+[\frac{\partial_{i_{1}}\ \int d\varphi\
\Lambda_{[h]}^{\star}(\tau)\
\partial_{\varphi}\zeta_{4}}{(\check{N}_{i_{1}}^{3})\
\Lambda_{[h]}^{\star}(\tau)\partial_{\varphi}\zeta_{4}}+\kappa(\frac{\partial_{i_{1}}[\int
d\varphi\ \Lambda_{[h]}^{\star}(\tau)\
\partial_{\varphi}(\zeta_{4}\chi_{4})]}{\partial_{i_{1}}\ [\int d\varphi\
\Lambda_{[h]}^{\star}(\tau)\partial_{\varphi}\zeta_{4}]}-\frac{\partial_{\varphi}(\zeta_{4}\chi_{4})}{\partial_{\varphi}\zeta_{4}})]\check{N}_{i_{1}}^{3}dx^{i_{1}}\\}^{2}$
$\displaystyle+\zeta_{4}(1+\kappa\ \chi_{4})\
\breve{g}_{4}\\{dt+[(\check{N}_{k_{1}}^{4})^{-1}[\ _{1}n_{k}+16\
_{2}n_{k_{1}}[\int d\varphi\frac{\left(\partial_{\varphi}[(\zeta_{4}\
\breve{g}_{4})^{-1/4}]\right)^{2}}{|\int
d\varphi\partial_{\varphi}[\Lambda_{[h]}^{\star}(\tau)(\zeta_{4}\
\breve{g}_{4})]|}]$ $\displaystyle+\kappa\frac{16\ _{2}n_{k_{1}}\int
d\varphi\frac{\left(\partial_{\varphi}[(\zeta_{4}\
\breve{g}_{4})^{-1/4}]\right)^{2}}{|\int d\varphi\partial_{\varphi}[\
\Lambda_{[h]}^{\star}(\tau)(\zeta_{4}\
\breve{g}_{4})]|}(\frac{\partial_{\varphi}[(\zeta_{4}\
\breve{g}_{4})^{-1/4}\chi_{4})]}{2\partial_{\varphi}[(\zeta_{4}\
^{cy}g)^{-1/4}]}+\frac{\int
d\varphi\partial_{\varphi}[\Lambda_{[h]}^{\star}(\tau)(\zeta_{4}\chi_{4}\
\breve{g}_{4})]}{\int d\varphi\partial_{\varphi}[(\zeta_{4}\
\breve{g}_{4})]})}{\ {}_{1}n_{k_{1}}+16\ _{2}n_{k_{1}}[\int
d\varphi\frac{\left(\partial_{\varphi}[(\zeta_{4}\
\breve{g}_{4})^{-1/4}]\right)^{2}}{|\int
d\varphi\partial_{\varphi}[\Lambda_{[h]}^{\star}(\tau)(\zeta_{4}\
\breve{g}_{4})]|}]}]\check{N}_{k_{1}}^{4}dx^{k_{1}}\\}^{2}$ (A.47)
$\displaystyle-\\{\frac{4[\ ^{\shortmid}\partial^{5}(|\ ^{\shortmid}\zeta^{6}\
\ {}^{\shortmid}\breve{g}^{6}|^{1/2})]^{2}}{\ {}^{\shortmid}\breve{g}^{5}|\int
dp_{5}\\{\ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\ ^{\shortmid}\partial^{5}(\
^{\shortmid}\zeta^{6}\ \ {}^{\shortmid}\breve{g}^{6})\\}|}-\kappa[\frac{\
{}^{\shortmid}\partial^{5}(\ ^{\shortmid}\chi^{6}|\ ^{\shortmid}\zeta^{6}\ \
{}^{\shortmid}\breve{g}^{6}|^{1/2})}{4\ ^{\shortmid}\partial^{5}(|\
^{\shortmid}\zeta^{6}\ \ {}^{\shortmid}\breve{g}^{6}|^{1/2})}-\frac{\int
dp_{5}\\{\ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\
^{\shortmid}\partial^{5}[(\ ^{\shortmid}\zeta^{6}\ \
{}^{\shortmid}\breve{g}^{6})\ ^{\shortmid}\chi^{6}]\\}}{\int dp_{5}\\{\
^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\ ^{\shortmid}\partial^{5}(\
^{\shortmid}\zeta^{6}\ \ {}^{\shortmid}\breve{g}^{6})\\}}]\\}\
^{\shortmid}\breve{g}^{5}$ $\displaystyle\\{dp_{5}+[\frac{\
{}^{\shortmid}\partial_{i_{2}}\ \int dp_{5}\ \
^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\ ^{\shortmid}\partial^{5}\
{}^{\shortmid}\zeta^{6}}{(\ ^{\shortmid}\check{N}_{5i_{2}})\ \
^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\ ^{\shortmid}\partial^{5}\
{}^{\shortmid}\zeta^{6}}+\kappa(\frac{\ {}^{\shortmid}\partial_{i_{2}}[\int
dp_{5}\ \ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\ \
^{\shortmid}\partial^{5}(\ ^{\shortmid}\zeta^{6}\ {}^{\shortmid}\chi^{6})]}{\
{}^{\shortmid}\partial_{i_{2}}\ [\int dp_{5}\ \
^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\ ^{\shortmid}\partial^{5}\
{}^{\shortmid}\zeta^{6}]}-\frac{\ {}^{\shortmid}\partial^{5}(\
^{\shortmid}\zeta^{6}\ {}^{\shortmid}\chi^{6})}{\ {}^{\shortmid}\partial^{5}\
{}^{\shortmid}\zeta^{6}})]\ ^{\shortmid}\check{N}_{5i_{2}}dx^{i_{2}}\\}^{2}$
$\displaystyle+\ ^{\shortmid}\zeta^{6}(1+\kappa\ \ ^{\shortmid}\chi^{6})\ \
^{\shortmid}\breve{g}^{6}\\{dt+[(\ ^{\shortmid}\check{N}_{5i_{2}})^{-1}[\
_{1}n_{k_{2}}+16\ _{2}n_{k_{2}}[\int dp_{5}\frac{\left(\
{}^{\shortmid}\partial^{5}[(\ ^{\shortmid}\zeta^{6}\ \
{}^{\shortmid}\breve{g}^{6})^{-1/4}]\right)^{2}}{|\int dp_{5}\
^{\shortmid}\partial^{5}[\ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)(\
^{\shortmid}\zeta^{6}\ \ {}^{\shortmid}\breve{g}^{6})]|}]$
$\displaystyle+\kappa\frac{16\ _{2}n_{k_{2}}\int dp_{5}\frac{\left(\
{}^{\shortmid}\partial^{5}[(\ ^{\shortmid}\zeta^{6}\ \
{}^{\shortmid}\breve{g}^{6})^{-1/4}]\right)^{2}}{|\int dp_{5}\
^{\shortmid}\partial^{5}[\ \ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)(\
^{\shortmid}\zeta^{6}\ \ {}^{\shortmid}\breve{g}^{6})]|}(\frac{\
{}^{\shortmid}\partial^{5}[(\ ^{\shortmid}\zeta^{6}\ \
{}^{\shortmid}\breve{g}^{6})^{-1/4}\ {}^{\shortmid}\chi^{6})]}{2\
^{\shortmid}\partial^{5}[(\ ^{\shortmid}\zeta^{6}\ \
{}^{\shortmid}\breve{g}^{6})^{-1/4}]}+\frac{\int dp_{5}\
^{\shortmid}\partial^{5}[\ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)(\ \
^{\shortmid}\chi^{6}\ \ {}^{\shortmid}\zeta^{6}\ \
{}^{\shortmid}\breve{g}^{6})]}{\int dp_{5}\ ^{\shortmid}\partial^{5}[(\
^{\shortmid}\zeta^{6}\ \ {}^{\shortmid}\breve{g}^{6})]})}{\
{}_{1}n_{k_{2}}+16\ _{2}n_{k_{2}}[\int dp_{5}\frac{\left(\
{}^{\shortmid}\partial^{5}[(\ ^{\shortmid}\zeta^{6}\ \
{}^{\shortmid}\breve{g}^{6})^{-1/4}]\right)^{2}}{|\int dp_{5}\
^{\shortmid}\partial^{5}[\ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)(\
^{\shortmid}\zeta^{6}\ \ {}^{\shortmid}\breve{g}^{6})]|}]}]$ $\
{}^{\shortmid}\check{N}_{6k_{2}}dx^{k_{2}}\\}^{2}$ $\displaystyle-\\{\frac{4[\
^{\shortmid}\partial^{7}(|\ ^{\shortmid}\zeta^{8}\ \
{}^{\shortmid}\breve{g}^{8}|^{1/2})]^{2}}{\ {}^{\shortmid}\breve{g}^{7}|\int
dp_{7}\\{\ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\ ^{\shortmid}\partial^{7}(\
^{\shortmid}\zeta^{8}\ \ {}^{\shortmid}\breve{g}^{8})\\}|}-\kappa[\frac{\
{}^{\shortmid}\partial^{7}(\ ^{\shortmid}\chi^{8}|\ ^{\shortmid}\zeta^{8}\ \
{}^{\shortmid}\breve{g}^{8}|^{1/2})}{4\ ^{\shortmid}\partial^{7}(|\
^{\shortmid}\zeta^{8}\ \ {}^{\shortmid}\breve{g}^{8}|^{1/2})}-\frac{\int
dp_{7}\\{\ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\
^{\shortmid}\partial^{7}[(\ ^{\shortmid}\zeta^{8}\ \
{}^{\shortmid}\breve{g}^{8})\ ^{\shortmid}\chi^{8}]\\}}{\int dp_{7}\\{\
^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\ ^{\shortmid}\partial^{7}(\
^{\shortmid}\zeta^{8}\ \ {}^{\shortmid}\breve{g}^{8})\\}}]\\}\
^{\shortmid}\breve{g}^{7}$ $\displaystyle\\{dp_{7}+[\frac{\
{}^{\shortmid}\partial_{i_{3}}\ \int dp_{7}\ \
^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\ ^{\shortmid}\partial^{7}\
{}^{\shortmid}\zeta^{8}}{(\ ^{\shortmid}\check{N}_{7i_{2}})\ \
^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\ ^{\shortmid}\partial^{7}\
{}^{\shortmid}\zeta^{8}}+\kappa(\frac{\ {}^{\shortmid}\partial_{i_{3}}[\int
dp_{7}\ \ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\ \
^{\shortmid}\partial^{7}(\ ^{\shortmid}\zeta^{8}\ {}^{\shortmid}\chi^{8})]}{\
{}^{\shortmid}\partial_{i_{3}}\ [\int dp_{7}\ \
^{\shortmid}\Lambda_{[c]}^{\star}(\tau)\ ^{\shortmid}\partial^{7}\
{}^{\shortmid}\zeta^{8}]}-\frac{\ {}^{\shortmid}\partial^{7}(\
^{\shortmid}\zeta^{8}\ {}^{\shortmid}\chi^{8})}{\ {}^{\shortmid}\partial^{7}\
{}^{\shortmid}\zeta^{8}})]\ ^{\shortmid}\check{N}_{8i_{3}}dx^{i_{3}}\\}^{2}$
$\displaystyle+\ ^{\shortmid}\zeta^{8}(1+\kappa\ \ ^{\shortmid}\chi^{8})\ \
^{\shortmid}\breve{g}^{8}\\{dE+[(\ ^{\shortmid}\check{N}_{7i_{3}})^{-1}[\
_{1}n_{k_{3}}+16\ _{2}n_{k_{3}}[\int dp_{7}\frac{\left(\
{}^{\shortmid}\partial^{7}[(\ ^{\shortmid}\zeta^{8}\ \
{}^{\shortmid}\breve{g}^{8})^{-1/4}]\right)^{2}}{|\int dp_{7}\
^{\shortmid}\partial^{7}[\ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)(\
^{\shortmid}\zeta^{8}\ \ {}^{\shortmid}\breve{g}^{8})]|}]$
$\displaystyle+\kappa\frac{16\ _{2}n_{k_{3}}\int dp_{7}\frac{\left(\
{}^{\shortmid}\partial^{7}[(\ ^{\shortmid}\zeta^{8}\ \
{}^{\shortmid}\breve{g}^{8})^{-1/4}]\right)^{2}}{|\int dp_{7}\
^{\shortmid}\partial^{7}[\ \ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)(\
^{\shortmid}\zeta^{8}\ \ {}^{\shortmid}\breve{g}^{8})]|}(\frac{\
{}^{\shortmid}\partial^{7}[(\ ^{\shortmid}\zeta^{8}\ \
{}^{\shortmid}\breve{g}^{8})^{-1/4}\ {}^{\shortmid}\chi^{8})]}{2\
^{\shortmid}\partial^{7}[(\ ^{\shortmid}\zeta^{8}\ \
{}^{\shortmid}\breve{g}^{8})^{-1/4}]}+\frac{\int dp_{7}\
^{\shortmid}\partial^{7}[\ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)(\ \
^{\shortmid}\chi^{8}\ \ {}^{\shortmid}\zeta^{8}\ \
{}^{\shortmid}\breve{g}^{8})]}{\int dp_{7}\ ^{\shortmid}\partial^{7}[(\
^{\shortmid}\zeta^{8}\ \ {}^{\shortmid}\breve{g}^{8})]})}{\
{}_{1}n_{k_{3}}+16\ _{2}n_{k_{3}}[\int dp_{7}\frac{\left(\
{}^{\shortmid}\partial^{7}[(\ ^{\shortmid}\zeta^{8}\ \
{}^{\shortmid}\breve{g}^{8})^{-1/4}]\right)^{2}}{|\int dp_{7}\
^{\shortmid}\partial^{7}[\ ^{\shortmid}\Lambda_{[c]}^{\star}(\tau)(\
^{\shortmid}\zeta^{8}\ \ {}^{\shortmid}\breve{g}^{8})]|}]}]\times$ $\
{}^{\shortmid}\check{N}_{8k_{8}}dx^{k_{8}}\\}^{2}.$

Details on proofs for $\tau$-families of such nonassociative s-metrics can be
found, for instance, for BH and BE configurations in partner works [36, 37,
18, 19]. In abstract symbolic geometric forms, we can change BH coordinates
into respective wormhole ones, then the prime s-metrics are taken for
wormholes.

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