\eta_{4}(\hbar,\kappa;\tau,\breve{r},\hat{x}^{2},\hat{x}^{3})\simeq\zeta_{4}(\hbar,\tau,\breve{r},\hat{x}^{2},\hat{x}^{3})(1+\kappa\
\chi_{4}(\hbar,\tau,\breve{r},\hat{x}^{2},\hat{x}^{3})),$ $\
{}^{\shortmid}\eta^{6}(\tau)$ $\displaystyle\simeq$ $\
{}^{\shortmid}\eta^{6}(\hbar,\kappa;\tau,\breve{r},\hat{x}^{2},\hat{x}^{3},\hat{x}^{5})\simeq\
^{\shortmid}\zeta^{6}(\hbar,\kappa;\tau,\breve{r},\hat{x}^{2},\hat{x}^{3},\hat{x}^{5})(1+\kappa\
^{\shortmid}\chi^{6}(\hbar,\kappa;\tau,\breve{r},\hat{x}^{2},\hat{x}^{3},\hat{x}^{5})),$
$\ {}^{\shortmid}\eta^{8}(\tau)$ $\displaystyle\simeq$ $\
{}^{\shortmid}\eta^{8}(\hbar,\kappa;\tau,\breve{r},\hat{x}^{2},\hat{x}^{3},\hat{x}^{5},p_{7})\simeq\
^{\shortmid}\zeta^{8}(\hbar,\kappa;\tau,\breve{r},\hat{x}^{2},\hat{x}^{3},\hat{x}^{5},p_{7})(1+\kappa\
^{\shortmid}\chi^{8}(\hbar,\kappa;\tau,\breve{r},\hat{x}^{2},\hat{x}^{3},\hat{x}^{5},p_{7})).$

Using formulas (94), we can extract solutions with rotoid spacetime
configurations determined by nonassociative star product R-flux deformation
(considering $\chi$-polarizations), or to compute volume forms (5.1) for
$\eta$-polarizations.

####  5.3.3 Bekenstein-Hawking entropy of $\tau$-running phase space RN-AdS
BEs configurations

For a subclass of nonholonomic configurations, the s-metrics (5.3.2) define
higher dimension BH and/or BE configurations with conventional horizons which
can be characterized variables in the framework of generalized Bekenstein-
Hawking thermodynamics [47, 48, 49, 50]. For simplicity, we shall consider
solutions for 6-d $\tau$-running quasi-stationary configurations evolving in a
8-d phase space, for simplicity, with trivial integration functions of type $\
{}_{1}^{\shortmid}n_{k_{s}}=0$ and $\ {}_{2}^{\shortmid}n_{k_{s}}=0.$ The
corresponding nonlinear quadratic elements are parameterized in the form:

$\displaystyle d\ _{\shortmid}^{\chi}s_{[6\subset 8d]}^{2}(\tau)$
$\displaystyle=$ $\displaystyle e^{\psi_{0}}(1+\kappa\ ^{\psi(\tau)}\
{}^{\shortmid}\chi(\tau))[\
\breve{g}_{1}(\breve{r})d\breve{r}^{2}+\breve{g}_{2}(\breve{r})(d\hat{x}^{2})]$
$\displaystyle-\\{\frac{4[\hat{\partial}_{3}(|\zeta_{4}(\tau)\breve{g}_{4}(\breve{r})|^{1/2})]^{2}}{\
\breve{g}_{4}(\breve{r})|\int d\hat{x}^{3}\\{\
_{2}\Im(\tau)\hat{\partial}_{3}(\zeta_{4}(\tau)\
\breve{g}_{4}(\breve{r}))\\}|}-\kappa[\frac{\hat{\partial}_{3}(\chi_{4}(\tau)|\zeta_{4}(\tau)\
\breve{g}_{4}(\breve{r})|^{1/2})}{4\hat{\partial}_{3}(|\zeta_{4}(\tau)\
\breve{g}_{4}(\breve{r})|^{1/2})}$ $\displaystyle-\frac{\int d\hat{x}^{3}\\{\
_{2}\Im(\tau)\hat{\partial}_{3}[(\zeta_{4}(\tau)\
\breve{g}_{4}(\breve{r}))\chi_{4}(\tau)]\\}}{\int d\hat{x}^{3}\\{\
_{2}\Im(\tau)\hat{\partial}_{3}(\zeta_{4}(\tau)\
\breve{g}_{4}(\breve{r}))\\}}]\\}\ \breve{g}_{3}(\mathbf{e}^{3}(\tau))^{2}+\
\zeta_{4}(\tau)(1+\kappa\ \chi_{4}(\tau))\breve{g}_{4}(\breve{r})dt^{2}$
$\displaystyle-\\{\frac{4[\hat{\partial}_{5}(|\ ^{\shortmid}\zeta^{6}(\tau)\ \
\breve{g}^{6}|^{1/2})]^{2}}{~{}\breve{g}_{5}(\breve{r})|\int d\hat{x}^{5}\\{\
_{3}^{\shortmid}\Im(\tau)\ ^{\shortmid}\partial^{7}(\
^{\shortmid}\zeta^{6}(\tau)~{}\breve{g}^{6})\\}|}-\kappa[\frac{\
\hat{\partial}_{5}(\ ^{\shortmid}\chi^{6}(\tau)|\ ^{\shortmid}\zeta^{6}(\tau)\
\breve{g}^{6}|^{1/2})}{4\hat{\partial}_{5}(|\ ^{\shortmid}\zeta^{6}(\tau)\
\breve{g}^{6}|^{1/2})}$ $\displaystyle-$ $\displaystyle\frac{\int
d\hat{x}^{5}\\{\ _{3}^{\shortmid}\Im(\tau)\ \hat{\partial}_{5}[(\
^{\shortmid}\zeta^{6}(\tau)\breve{g}^{6})\
^{\shortmid}\chi^{8}(\tau)]\\}}{\int d\hat{x}^{5}\\{\
_{3}^{\shortmid}\Im(\tau)\ \hat{\partial}_{5}[(\
^{\shortmid}\zeta^{6}(\tau)\breve{g}^{6})]\\}}]\\}\
\breve{g}_{5}(\breve{r})(\mathbf{e}^{5}(\tau))^{2}+\
^{\shortmid}\zeta^{6}(\tau)\ (1+\kappa\
^{\shortmid}\chi^{6}(\tau))(dp_{6})^{2}+(dp_{7})^{2}-dE^{2},$

where

$\mathbf{e}^{3}(\tau)=d\hat{x}^{3}+[\frac{\hat{\partial}_{i_{1}}\int
d\hat{x}^{3}\ _{2}\Im(\tau)\
\hat{\partial}_{3}\zeta_{4}(\tau)}{\breve{N}_{i_{1}}^{3}\
{}_{2}\Im(\tau)\hat{\partial}_{3}\zeta_{4}(\tau)}+\kappa(\frac{\hat{\partial}_{i_{1}}[\int
d\hat{x}^{3}\
_{2}\Im(\tau)\hat{\partial}_{3}(\zeta_{4}(\tau)\chi_{4}(\tau))]}{\hat{\partial}_{i_{1}}\
[\int d\hat{x}^{3}\
_{2}\Im(\tau)\hat{\partial}_{3}\zeta_{4}(\tau)]}-\frac{\hat{\partial}_{3}(\zeta_{4}(\tau)\chi_{4}(\tau))}{\hat{\partial}_{3}\zeta_{4}(\tau)})]\
\breve{N}_{i_{1}}^{3}dx^{i_{1}},$ $\displaystyle\ \mathbf{e}^{5}(\tau)$
$\displaystyle=$ $\displaystyle d\hat{x}^{5}+[\frac{\hat{\partial}_{i_{2}}\
\int d\hat{x}^{5}\ _{3}{}^{\shortmid}\Im(\tau)\ \ \hat{\partial}_{5}(\
^{\shortmid}\zeta^{6}(\tau))}{\ {}^{\shortmid}\breve{N}_{i_{2}}^{5}\
{}_{3}^{\shortmid}\Im(\tau)\ \ \hat{\partial}_{5}(\
^{\shortmid}\zeta^{6}(\tau))}+$
$\displaystyle\kappa(\frac{\hat{\partial}_{i_{2}}[\int d\hat{x}^{5}\
_{3}{}^{\shortmid}\Im(\tau)\ \ \hat{\partial}_{5}(\
^{\shortmid}\zeta^{6}(\tau)\ \ \breve{g}^{6})]}{\hat{\partial}_{i_{2}}\ [\int
d\hat{x}^{5}\ _{3}{}^{\shortmid}\Im(\tau)\ \ \hat{\partial}_{5}(\
^{\shortmid}\zeta^{6}(\tau))]}-\frac{\ \hat{\partial}_{5}(\
^{\shortmid}\zeta^{6}(\tau)\ \ \breve{g}^{6})}{\ \hat{\partial}_{5}(\
^{\shortmid}\zeta^{6}(\tau))})]\ \ ^{\shortmid}\breve{N}_{i_{2}}^{5}d\
^{\shortmid}x^{i_{2}}.$

A subclass of solutions (5.3.3) generates $\tau$-families of rotoid
configurations in coordinates $(\breve{r},\hat{x}^{2},\hat{x}^{3})$ (as
nonholonomic deformations of the phase BH solution (92)) if we chose such
generating functions:

$\ \
\chi_{4}(\tau)=\hat{\chi}_{4}(\tau,\breve{r},\hat{x}^{2},\hat{x}^{3})=2\underline{\chi}(\tau,\breve{r},\hat{x}^{2})\sin(\omega_{0}\hat{x}^{3}+\hat{x}_{0}^{3}),$
(96)

where $\underline{\chi}(\tau,\breve{r},\hat{x}^{2})$ are smooth functions (or
constants), and $(\omega_{0},\hat{x}_{0}^{3})$ is a couple of constants. In a
conventional 5-d phase space on shells $s=1,2,3,$ trivially imbedded into a
8-d phase space posses a distinct ellipsoidal type horizon with respective
eccentricity $\kappa$ a stated by the equations

$\zeta_{4}(\tau)(1+\kappa\ \chi_{4}(\tau))\breve{g}_{4}(\breve{r})=0\mbox{
i.e. }(1+\kappa\
\chi_{4})\breve{f}(\breve{r})=(1-\frac{\hat{m}}{\breve{r}^{2}}-\frac{\Lambda_{[5]}}{6}\breve{r}^{2}+\frac{\hat{q}^{2}}{\breve{r}^{4}}+\kappa\
\chi_{4})=0,$

for $\zeta_{4}\neq 0.$ For small parametric deformations and configurations
with
$-\frac{\Lambda_{[5]}}{6}\breve{r}^{2}+\frac{\hat{q}^{2}}{\breve{r}^{4}}\approx
0,$ we can approximate for a fixed $\tau_{0},$
$\breve{r}\simeq\hat{m}^{1/2}/(1-\frac{\kappa\ }{2}\hat{\chi}_{4}).$ These are
parametric formulas for a rotoid horizon defined by small gravitational R-flux
polarizations. In the limits of zero eccentricity, such e BE configurations
transform into a 5-d BH embedded into nonassociative 8-d phase space.

Extending the concept of Bekenstein-Hawking entropy for phase spaces
determined by quadratic linear elements (92), we can define such thermodynamic
values (computations are similar to those for formulas (8)-(15) in [56] but
with different constants and following our notations):

$\ {}^{0}\breve{S}$ $\displaystyle=$ $\displaystyle\frac{\
{}^{0}\breve{A}}{4G_{[5]}}=\frac{\omega_{[3]}\breve{r}_{h}}{4G_{[5]}}\mbox{
and }\
^{0}\breve{T}=\frac{1}{2\pi\breve{r}_{h}}(\epsilon+2\frac{\breve{r}_{h}^{2}}{l_{[5]}^{2}})-\frac{2G_{[10]}^{2}\hat{Q}^{2}}{3\pi^{9}l_{[5]}^{8}\breve{r}_{h}^{5}},\mbox{
for }$ $\displaystyle\hat{M}$ $\displaystyle=$
$\displaystyle\frac{3\omega_{[3]}\hat{m}}{16\pi
G_{[5]}}(\epsilon\breve{r}_{h}^{2}+\frac{\breve{r}_{h}^{4}}{l_{[5]}^{2}}+\frac{4G_{[5]}\hat{Q}^{2}l_{[5]}^{2}}{3\pi^{2}\breve{r}_{h}^{2}}),$
(97)

where $\breve{r}_{h}$ and ${}^{0}\breve{A}$ are, respectively the horizon and
area of horizon of 5-d BH, $G_{[5]}=G_{[10]}/(\pi^{3}l_{[5]}^{5})$ and
$G_{[10]}=\ell_{p}^{8}.$ Using these formulas for rotoid deformations
$\breve{r}_{h}\rightarrow\hat{m}^{1/2}/(1-\frac{\kappa\ }{2}\hat{\chi}_{4})$
and ${}^{0}\breve{A}$ $\rightarrow\ ^{rot}\breve{A},$ with
$\hat{\chi}_{4}(\tau)$ (96), we compute for respective BE configurations:

$\breve{S}(\tau)=\ ^{0}\breve{S}(1+\frac{\kappa\
}{2}\hat{\chi}_{4}(\tau))\mbox{ and }\breve{T}(\tau)=\
^{0}\breve{T}+\kappa\left(-\frac{\epsilon}{4\pi\breve{r}_{h}}+\frac{\breve{r}_{h}}{2\pi
l_{[5]}^{2}}-\frac{5G_{[10]}^{2}\hat{Q}^{2}}{3\pi^{9}l_{[5]}^{8}\breve{r}_{h}^{5}}\right)\hat{\chi}_{4}(\tau).$
(98)

The modified Hawking temperatures $\breve{T}(\tau)$ and$\ {}^{0}\breve{T}$ are
stated by requiring the absence of the potential conical singularity of the
Euclidean BH at the horizon in the phase space.

####  5.3.4 G. Perelman thermodynamics of nonassociative flows of phase RN-AdS
BHs

We can not apply the Bekenstein-Hawking thermodynamic paradigm in order to
characterize physical properties of general classes of quasi-stationary
solutions of type (5.3.2) and/or (5.3.3) excepting very special cases of
nonassociative deformations, for instance, to BE configurations of type (96).
The G. Perelman approach is more general and allows to define and compute
statistical thermodynamic variables of type (69). Let us sketch how to compute
such values for any data $\ {}^{\shortmid}\mathbf{\breve{g}}_{\alpha_{s}}$
(92), $\ {}_{s}^{\shortmid}\Im(\tau)$ related via nonlinear symmetries (60) to
$\ \Lambda_{[5]},$ and nontrivial (on shells $s=1,2,3,$ see also formulas
(94)). For simplicity, we can consider the same value of cosmological constant
on such shells when

$|\ \Lambda_{[5]}\ \eta_{4}(\tau)\breve{g}_{4})|=|\ \Lambda_{[5]}\
\zeta_{4}(\tau)\breve{g}_{4}|(1-\kappa\chi_{4}(\tau)),|\
\Lambda_{[5]}~{}^{\shortmid}\eta^{6}(\tau)~{}^{\shortmid}\breve{g}^{6}|=|\
\Lambda_{[5]}~{}^{\shortmid}\zeta^{6}(\tau)~{}^{\shortmid}\breve{g}^{6}|(1-\kappa~{}^{\shortmid}\chi^{6}(\tau))$
(99)

for a a subclass of s-metrics of type (5.3.3). We obtain such thermodynamic
functionals:

$\ {}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau)$
$\displaystyle=$
$\displaystyle\int\nolimits_{\tau^{\prime}}^{\tau}\frac{d\tau}{32(\pi\tau)^{4}}\frac{2\tau\Lambda_{[5]}^{2}-1}{\Lambda_{[5]}^{2}}\
_{\eta}^{\shortmid}\mathcal{\breve{V}}(\tau),\ \
_{s}^{\shortmid}\widehat{\mathcal{Z}}_{\kappa}^{\star}(\tau)=\exp\left[\int\nolimits_{\tau^{\prime}}^{\tau}\frac{d\tau}{(2\pi\tau)^{4}}\frac{1}{\Lambda_{[5]}^{2}}\
_{\eta}^{\shortmid}\mathcal{\breve{V}}(\tau)\right],$ (100) $\
{}_{s}^{\shortmid}\widehat{\mathcal{E}}_{\kappa}^{\star}(\tau)$
$\displaystyle=$
$\displaystyle-\int\nolimits_{\tau^{\prime}}^{\tau}\frac{d\tau}{64\pi^{4}\tau^{3}}\frac{\tau\Lambda_{[5]}-1}{\Lambda_{[5]}^{2}}\
_{\eta}^{\shortmid}\mathcal{\breve{V}}(\tau),\
_{s}^{\shortmid}\widehat{\mathcal{S}}_{\kappa}^{\star}(\tau)=-\int\nolimits_{\tau^{\prime}}^{\tau}\frac{d\tau}{64(\pi\tau)^{4}}\frac{\tau\Lambda_{[5]}-2}{\Lambda_{[5]}^{2}}\
_{\eta}^{\shortmid}\mathcal{\breve{V}}(\tau).$

In these formulas, we use the running phase space volume functional

$\ {}_{\eta}^{\shortmid}\mathcal{\breve{V}}(\tau)=\int_{\
{}_{s}^{\shortmid}\widehat{\Xi}}\ ^{\shortmid}\delta\
_{\eta}^{\shortmid}\mathcal{V}(\
_{s}^{\shortmid}\Im(\tau),~{}^{\shortmid}\breve{g}_{\alpha_{s}}),\mbox{ for
}s=1,2,3.$ (101)

Above presented values are determined by prescribed classes of generating
$\eta$-functions (99), effective generating sources $\
{}_{s}^{\shortmid}\Im(\tau),$ coefficients of a prime s-metric $\
{}^{\shortmid}\breve{g}_{\alpha_{s}}$ and nonholonomic distributions defining
the hyper-surface $\ {}_{s}^{\shortmid}\widehat{\Xi},$ when the volume forms
are computed

$\ {}^{\shortmid}\delta\ ^{\shortmid}\mathcal{V}(\tau)=\ ^{\shortmid}\delta\
^{\shortmid}\mathcal{V}(\ \ _{s}^{\shortmid}\Im(\tau),\ \Lambda_{[5]},\
^{\shortmid}\eta_{\alpha_{s}}(\tau)~{}^{\shortmid}\breve{g}_{\alpha_{s}})=\frac{1}{\Lambda_{[5]}^{2}}\
^{\shortmid}\delta\ _{\eta}^{\shortmid}\mathcal{V}(\
_{s}^{\shortmid}\Im(\tau),\ ^{\shortmid}\breve{g}_{\alpha_{s}}),\mbox{ where
}$ $\ {}^{\shortmid}\delta\ _{\eta}^{\shortmid}\mathcal{V}(\
_{s}^{\shortmid}\Im(\tau),~{}^{\shortmid}\breve{g}_{\alpha_{s}})=\
^{\shortmid}\delta\ _{\eta}^{1}\mathcal{V}[_{1}\Im(\tau),\eta_{2}(\tau)\
\breve{g}_{2}]\times\ ^{\shortmid}\delta\
_{\eta}^{2}\mathcal{V}[_{2}\Im(\tau),\eta_{4}(\tau)\ \breve{g}_{4}]\times\
^{\shortmid}\delta\
_{\eta}^{3}\mathcal{V}[_{3}\Im(\tau),~{}^{\shortmid}\eta^{6}(\tau)~{}^{\shortmid}\breve{g}^{6}].$

G. Perelman thermodynamic variables (100) can be computed in explicit form if
we prescribe certain nonholonomic distributions in $\
{}_{\eta}^{\shortmid}\mathcal{\breve{V}}(\tau)$ (101) which results in
physically viable geometric thermodynamic models. For instance, we can study
solutions with more general $\kappa$-parametric deformations than those
determined by rotoid generating functions (96). Prescribing generating sources
$\ {}_{s}^{\shortmid}\Im(\tau)$ and $\eta$-generating functions for $\
{}^{\shortmid}\delta\ _{\eta}^{\shortmid}\mathcal{V},$ we define explicit
thermodynamic models of deformed phase space BH objects propagating on a local
temperature like parameter $\tau$ like, for instance, in the hydrodynamic of
moving media. This analogy is very rough because instead of hydrodynamic flow
equations we consider nonassociative geometric flow equations and certain
special classes of solutions. The physical interpretation of corresponding
entropy and temperature from (100) is different from that for the hyper-
surface variables (98) in the Bekenstein-Hawking approach. Finally we note
that the effective running of thermodynamic variables on shell cosmological
constants in (90) and (100) is very different. We have to analyze such
different implications, for instance, in extending the swampland conjecture
for nonassociative BHs and $\tau$-running quasi-stationary solutions.

##  6 Extending swampland conjectures for nonassociative geometric flows and
nonholonomic BHs deformations

Connections between the swampland conjectures in string theory, QG, exact
solutions in MGTs , effective QFTs and (non) commutative geometric flow
equations, with a number of examples with BH solutions, were studied in a
series of works [22, 23, 24, 25, 26, 27]. We also cite such papers and
references therein for recent developments and applications in modern gravity,
cosmology and astrophysics; and studies of various types of Hawking–Page phase
transitions under Ricci, Yamanabe, Ricci-Bourguignon flows etc. It should be
remembered that the general aim of the swampland program [19, 20, 21, 18] is
to elaborate on mathematical and physical criteria for distinguishing (low-
energy) effective classical and quantum theories, QG and MGTs, which can be
completed in the UV, from other classes of theories which cannot.

The approach to nonassociative geometry and gravity [38, 39] and nonholonomic
generalizations with the AFCDM for constructing exact/parametric solutions in
nonassociative/noncommutative Ricci flow theories and MGTs [12, 13, 40, 41, 6,
7, 8] was developed for star product R-flux deformation models in
string/M-theory. Respective fundamental nonassociative geometric flow and
(modified) vacuum gravitational (Ricci soliton) equations consist very
sophisticate systems of coupled nonlinear PDEs encoding nonassociative data as
effective sources and generic off-diagonal terms of nonholonomic shell adapted
metrics. Such equations can be integrated in very general exact/parametric
forms, for instance, for quasi-stationary configurations running on
$\tau$-flow parameter as we proved in section 4, see also related results
examples in section 4 of [41]. Positively, for certain classes of solutions
with respective nonlinear symmetries and $\tau$-running, or prescribed/fixed,
shell effective cosmological constants, such solutions and respective
nonassociative geometric thermodynamic models can be constructed to be
compatible with certain refined versions of the swampland conjectures (both on
higher dimension spacetimes and/or phase space). Nevertheless, generalized
classes of solutions do not obligatory involve certain effective (running)
cosmological constants, for instance, for nonassociative BH deformations, see
subsections 5.2.4 and 5.3.4. Even for such nonassociative geometric flow
configurations, we can always compute the modified G. Perelman F- and
W-functionals and related thermodynamic variables, the issues of compatibility
or not compatibility with the swampland ideas and conjectures have to be
studied respectively for any explicit class of solutions.

In this section, we analyze and discuss two kinds of nonassociative
generalized swampland and Ricci flow conjectures. First, there are outlined
refined versions on the phase space of typical results obtained for high
dimensional (A)dS spaces, for instance, modifications of the Black Hole
Entropy Distance Conjecture (BHEDC) to the case of nonassociative black
ellipsoid, BE, configurations etc. Such constructions are possible for
nonassociative $\tau$-running quasi-stationary solutions with conventional
hyper-surface horizons and associated modified Bekenstein-Hawking
thermodynamic models studied above in subsections 5.2.3 and 5.3.3. Second, we
elaborate on some main points of this paper when the swampland program is
extended to nonassociative geometric flow and MGTs using the concept of
W-entropy and G. Perelman thermodynamics. There are considered certain well-
defined conditions when positively swampland and nonassociative Ricci flow
conjectures can formulated for such generalizations. We also analyze explicit
examples of $\tau$-running quasi-stationary solutions when it is not clear and
we can’t conclude without an additional analysis if such low-energy effective
(encoding nonassociative data) geometric flow and gravity models can be
completed into QG in the UV and distinguished from those that cannot.

###  6.1 The generalized distance conjecture, nonassociative Ricci flows and
gravity

The goal of this subsection is to show how the swampland, gradient flows and
infinite distance conjectures [22, 23] can be extended on nonassociative phase
spaces for solutions of nonassociative Ricci flow/soliton equations derived in
sections 2.3 and 2.4 and $\kappa$-parameterized in the form (41).

####  6.1.1 Phase space generalized and Weyl distances, and the AdS distance
conjecture

For a phase space with star product R-flux nonassociative deformation,
${}_{s}\mathcal{M}\rightarrow\ _{s}^{\star}\mathcal{M}$, restricted to a
finite region $\ {}_{s}^{\star}\mathcal{U}\subset\ _{s}^{\star}\mathcal{M},$
we consider a $\tau$-family of symmetric s-metrics $\
{}_{s}^{\shortmid}\mathbf{g}(\tau)$ following Convention 2 (26). There are
used $\kappa$-linear parametric decomposition when $\
{}_{\star}^{\shortmid}\mathfrak{\check{g}}_{\alpha_{s}\beta_{s}}^{[0]}(\tau)=\
_{\star}^{\shortmid}\mathbf{g}_{\alpha_{s}\beta_{s}}(\tau)=\
^{\shortmid}\mathbf{g}_{\alpha_{s}\beta_{s}}(\tau)$, for star product flows
$\star_{s}(\tau)$ determined by s-adapted frames $\
{}^{\shortmid}\mathbf{e}_{i_{s}}(\tau)$ in (2.1.2) and canonical data $(\
_{s}^{\shortmid}\mathbf{g}(\tau),\
_{s}^{\shortmid}\widehat{\mathbf{D}}(\tau)).$ We can consider arbitrary
variations of such a s-metric, or to choose a class of solutions of
nonassociative geometric flow equations (41) for the data $[\
_{\star}^{\shortmid}\mathfrak{g}_{\alpha_{s}\beta_{s}}(\tau),\
_{s}^{\shortmid}\widehat{\mathbf{D}}^{\star}(\tau)]$, when the nonsymmetric $\
{}_{\star}^{\shortmid}\mathfrak{g}_{\alpha_{s}\beta_{s}}$ are computed using
$\kappa$-linear parameterizations (20)–(22). On $\ {}_{s}^{\star}\mathcal{U}$,
a path $\gamma(\tau),\tau_{i}\leq$ $\tau\leq\tau_{f},$ is considered (for
which a proper metric distance $\aleph$ is defined by $\
{}^{\shortmid}\mathbf{g}_{\alpha_{s}\beta_{s}}(\tau))$ (1.1).171717In this
work, we write $\aleph$ instead of $\bigtriangleup$ considered, for instance,
in formula (1) from [23] for formulating the general distance conjecture. On
nonassociative phase spaces, we elaborate a different system of notations when
the symbol $\ {}^{\star}\widehat{\bigtriangleup}$ is the Laplace operator
constructed for $\ {}_{s}^{\shortmid}\widehat{\mathbf{D}}^{\star}.$ Here we
also note that in field theories, the parameter $\tau$ can be an arbitrary
one, parameterizing some curves/geodesic etc. In this section, we are
interested in models when $\tau$ can be always related to a geometric flow /
temperature like parameter. We introduce

$\ {}_{g}^{s}\aleph=\aleph[\
^{\shortmid}\mathbf{g}_{\alpha_{s}\beta_{s}}(\tau),\
_{s}^{\shortmid}\Lambda(\tau)]=\mathcal{C}\int\nolimits_{\tau_{i}}^{\tau_{f}}\left(\frac{1}{\mathcal{V}_{\mathcal{U}}}\int_{\mathcal{U}}\
^{\shortmid}\mathbf{g}^{\alpha_{s}\mu_{s}}\
{}^{\shortmid}\mathbf{g}^{\beta_{s}\nu_{s}}\frac{\ \partial\
^{\shortmid}\mathbf{g}_{\alpha_{s}\beta_{s}}}{\partial\tau}\ \ \frac{\partial\
\ ^{\shortmid}\mathbf{g}_{\mu_{s}\nu_{s}}}{\partial\tau}d\
^{\shortmid}\mathcal{V}ol(\tau)\right)^{1/2}d\tau$ (102)

for the volume form $d\ ^{\shortmid}\mathcal{V}ol(\tau)$ (36), where the
constant $\mathcal{C}\sim\mathcal{O}(1)$ depending on dimension of
$\mathcal{U}$ and $\mathcal{V}_{\mathcal{U}}$ is the volume of
$\mathcal{U}\rightarrow\ _{s}^{\star}\mathcal{U}.$ In this work, we consider
that $\ {}_{g}^{s}\aleph$ is determined by a $\tau$-family of
$\kappa$-parametric solutions of geometric flow equations (62) for some
prescribed running shell cosmological constants $\
{}_{s}^{\shortmid}\Lambda(\tau).$

For phase spaces, the generalized distance conjecture states that there must
be an infinite tower of states with an effective mass scale $\
{}^{\shortmid}m(\tau),$ when

$\ {}^{\shortmid}m(\tau_{f})\sim\ ^{\shortmid}m(\tau_{i})e^{-\hat{\alpha}|\
_{g}^{s}\aleph|},\mbox{ where }\hat{\alpha}\sim\mathcal{O}(1).$ (103)

This formula was introduced without labels "$\ {}^{\shortmid}$" for
corresponding notations for higher dimension spacetimes in [22, 23], when

$m\sim M_{p}e^{-\alpha|\ _{g}\aleph|}.$ (104)

In field theories, it is considered that generically the masses $m(\tau_{i})$
at some initial values $\tau_{i}$ are of order of $M_{p}$ (the Planck mass
constant, determined by $\hbar).$ We have to introduce such constants with
additional labels also for phase spaces $\ {}_{s}^{\star}\mathcal{M}$, when we
elaborate on (nonassociative and noncommutative) quantum geometric flow models
and/or QFTs. On 4-d spacetime base with shells $s=1,2,$ the formula $\
{}^{\shortmid}m$ (103) still contains nonassociative star product and R-flux
data encoded into $\ {}_{g}^{s}\aleph.$ Only for very special nonholonomic
configurations, non involving the effects of possible off-diagonal and
generalized effective sources, and omitting $\kappa$-parametric terms, we may
follow the assumptions for $m$ (104). We note that the mass scale $m$ in
associative/commutative classical and quantum theories states a natural cut-
off above which the effective field description is not valid. In such cases,
for large distance variations in the space of metrics, when $|\
_{g}\aleph|\rightarrow\infty,$ we get for QG models a respective massless
tower of states. This breaks down/ invalidates the effective field theory
description. In modern literature [19, 20, 21, 18, 22, 23], it is argued that
for respective conditions such theories belong to the swampland. The
conjectures and related results have to be revised for nonassociative
geometric flow and gravity/matter field theories when exact/parametric generic
off-diagonal solutions of effective nonholonomic Ricci flow and/or Ricci
soliton equations are considered following the AFCDM. We formulate:

The generalized distance conjecture and claim for nonassociative phase spaces,
CCL1:

a)

Conjecture: _For nonassociative geometric flow, gravity, and classical field
theories, and respective QG and QGTs encoding star product and R-flux data
from string/M-theory, there are classes of exact/parametric solutions with
nonlinear phase space symmetries of type ( A.8) connecting effective sources
to effective shell cosmological constants (for $\tau$-running or fixed
$\tau_{0}$ quasi-stationary and/or locally anisotropic cosmological
configurations with a time like coordinate ), when the corresponding effective
geometric/physical models belong to the swampland._

b)

Claim: _In a general context, nonassociative/ noncommutative/ nonholonomic /
generic off-diagonal/ generalized (non) linear connection modifications of
gravity theories contain also large classes of exact/parametric solutions
(involving, or not, nonlinear symmetries of type ( A.8)) defining effective
$\tau$-running, or fixed $\tau_{0})$, configurations which are physically well
defined and characterized by a respective modified G. Perelman thermodynamics
with variables of type (50). We have to analyze additionally for which
conditions such models belong or not to the swampland._

It should be noted here that above a) Conjecture is a nonassociative phase
space version of the generalized distance conjecture for (associative and
commutative) d-dimensional manifolds studied in [22, 23]. In this work, we
show that such a conjecture (and related ones, see next subsections) can be
formulated for respective conditions on nonassociative geometric flows when $\
{}_{g}^{s}\aleph$(102)$\rightarrow\infty$ in the space of $\tau$-running, or
fixed $\tau_{0},$ pase space s-metrics. Nevertheless, using the AFCDM, we can
construct large classes of exact/parametric solutions when $\
{}_{g}^{s}\aleph$ is "freezed" someway, for respective nonholonomic
configurations, and do not result in an infinite tower of effective phase
masses $\ {}^{\shortmid}m$ (103). This reflects a more rich propriety of
respective classes of solutions of corresponding systems of nonlinear PDEs
(describing the geometric flow evolution and/or corresponding field equations)
when generic off-diagonal interactions and evolution scenarios are modeled in
a more general way than in the case when the analysis is performed in the
framework of diagonalizable solutions and ODEs. Such results (we provided very
general classes of quasi-stationary solutions and explicit examples in the
previous section) motivate the b)  Claim. So, a (nonassociative) modified
geometric flow, gravity, and/or classical/quantum field theory may involve
large classes of solutions, and respective effective models which belong, or
not to the swampland. It depends on the type of nonholonomic constraints,
nonlinear symmetries and nonlinear interactions, circumstances of the
considered coupling, prescribed generating functions and effective sources,
and integration functions/constants. In all cases, for various models with
solutions satisfying the conditions of above stated CCL1, we are able to
elaborate on modified G. Perelman thermodynamic models (50), for general
assumptions on nonassociative geometric flows; or (51), for (92),
thermodynamic variables; and (90), or (100), for explicit examples of general
nonassociative BH/BE deformations.

At the next step, we analyze how keeping the conditions of CCL1, with notions
formulated for the (phase) spaces of s-metrics, it is possible to formulate a
respective conjecture and claim involving the so-called Weyl distance [22] but
generalized for nonassociative phase spaces. Specifically, we can consider an
external conformal re-scaling of s-metrics on ${}_{s}\mathcal{M}$,

$\
{}^{\shortmid}\mathbf{\tilde{g}}_{\alpha_{s}\beta_{s}}(\bar{\tau})=e^{2\bar{\tau}}\
{}^{\shortmid}\mathbf{g}_{\alpha_{s}\beta_{s}},$ (105)

where the (re-scaling) parameter $\bar{\tau}$, in general, is different from a
geometric flow/temperature like parameter $\tau.$ Introducing (105) in (102),
we compute $\
{}_{g}^{s}\bar{\aleph}\simeq\bar{\tau}_{f}-\bar{\tau}_{i}\simeq\bar{\tau}.$
Following a) Conjecture, there is a corresponding tower of phase space states
$\tilde{m}$ (103) with a scale of effective masses as $\tilde{m}\sim
e^{-\hat{\alpha}\bar{\tau}}.$ We obtain light masses in respective towers if
$\ {}_{g}^{s}\bar{\aleph}\simeq\bar{\tau}\rightarrow\infty$ (we can consider
also an opposite large distance limit, $\
{}_{g}^{s}\bar{\aleph}\simeq\bar{\tau}\rightarrow-\infty$, when light states
are with masses $\tilde{m}^{\prime}\sim e^{\hat{\alpha}\bar{\tau}}$). For a
Weyl re-scaling, we can consider variations of the cosmological constant, $\
{}^{\shortmid}\bar{\Lambda}_{i}\leq\
^{\shortmid}\bar{\Lambda}(\bar{\tau})\leq\ ^{\shortmid}\bar{\Lambda}_{f}$ for
corresponding families of (A)dS phase space vacua. We can express the
$\bar{\tau}$-transforms of the cosmological constant for an arbitrary
dimension $\check{d}$ of ${}_{s}\mathcal{M}$ ( in this work, typical
constructions are performed for $\check{d}$ =8),

$\
{}^{\shortmid}\bar{\Lambda}=-\frac{1}{2}(\check{d}-1)(\check{d}-2)M_{p}^{2}e^{-2\bar{\tau}}.$
(106)

The phase space metric distance between any initial, $\
{}^{\shortmid}\Lambda_{i},$ and finial, $\ {}^{\shortmid}\Lambda_{f},$ values
of such running cosmological constant can be approximated

$\ {}_{g}^{s}\bar{\aleph}\simeq\bar{\tau}_{f}-\bar{\tau}_{i}\simeq\log(\
^{\shortmid}\Lambda_{i}/\ ^{\shortmid}\Lambda_{f}),$ (107)

which states that the limit $\ {}^{\shortmid}\Lambda_{f}\rightarrow 0$ is at
infinite distance with respect to the Weyl re-scaling (105) of an (A)dS phase
s-metric. This corresponds to the limit $\bar{\tau}\rightarrow\infty.$

Combining the behaviour determined by the Weyl re-scaling formulas (105) -
(107) and CCL1, we extend the AdS Distance conjecture from [22, 23] in such a
form:

The AdS distance conjecture and claim for nonassociative phase spaces, CCL2:

a)

Conjecture: _For models of (non) associative phase space geometric flow and
gravity theories, and QG on a $\check{d}$-dimensional phase space
${}_{s}\mathcal{M}$ with cosmological constant $\ {}^{\shortmid}\bar{\Lambda}$
(106), there exists an infinite tower of phase space states with mass scale $\
{}^{\shortmid}m$ which, as $\ {}^{\shortmid}\Lambda\rightarrow 0,$ behave as
$\ {}^{\shortmid}m$ $\sim|\ ^{\shortmid}\Lambda|^{\bar{\alpha}},$ where
$\bar{\alpha}$ is a positive order-one constant. Such a behavior can be
described by phase space s-metrics (92) and their nonassociative quasi-
stationary deformations encoding star product and R-flux data from
string/M-theory with nonlinear phase space symmetries of type (A.8) connecting
effective sources to effective shell cosmological constants $\
{}_{s}^{\shortmid}\Lambda(\tau)=\ ^{\shortmid}\bar{\Lambda}.$ _

b)

Claim: _In general, in QG models with nonassociative/ noncommutative/
nonholonomic / generic off-diagonal/ generalized (non) linear connection
modifications, there are large classes of exact/parametric solutions
(involving, or not, nonlinear symmetries of type ( A.8) when the approximation
(107) is not valid. Nevertheless, corresponding classes of solutions defining
effective $\tau$-running, or fixed $\tau_{0}$, configurations, are physically
well defined and characterized by a respective modified G. Perelman
thermodynamics with variables (50)_.

We formulate the b)  Claims  in CCL2 because for (non) associative geometric
flow theories and gravity, and their generic off-diagonal solutions, the
conformal symmetry of s-metric is not a general property. In some models, one
speculates on possible duality between AdS and dS configurations (see footnote
3 in [23] on necessary exchange of the mass-towers and powers of cosmological
constants). Such duality properties and re-scaling behaviour are not important
for the procedure of generating exact/parametric solutions using the AFCDM.

Finally, it should be noted that we can also consider a "dual" infinite
distance limit when the associated curvatures of phase space/ spacetime and
(effective) cosmological constants became very large for
$\bar{\tau}\rightarrow-\infty.$ Such conditions for the string/M-theory and
various coupling constants are analyzed in section III of [23]. In general, it
is not clear if a tower of light states may appear for such models large
curvature/ cosmological constants. Similar conclusions can be drawn for
nonassociative phase models determined, for instance, by diagonal s-metrics of
type (92) and certain $\kappa$-parametric deformations. For more general
classes of solutions, limits of type $\bar{\tau}\rightarrow-\infty$ may be not
obligatory correlated to certain large curvature/cosmological constants values
because of generic off-diagonal interactions and imposed nonholonomic
constraints on the geometric flow evolution and/or gravitational and matter
field dynamics. Such issues have to be investigated for any class of well-
defined physical solutions determined by respective generating functions and
effective sources, integration functions and prescribed (non) linear
symmetries.

####  6.1.2 Nonassociative Ricci flows and infinite phase space distance
conjectures and claims

For associative and commutative Riemannian metrics, the geometric flow
equations imply the properties that for positive Ricci curvature the manifold
is contracting and for regions of negative curvature the manifold is
extending. Such properties exist for space like configurations in pseudo-
Riemannian geometry and nonassociative generalizations when, in general, the
evolution scenarios depend on the type of solutions for respective systems of
nonlinear PDEs. Let us begin with some properties of the associative and
commutative R. Hamilton equations for the LC-connection $\nabla,$

$\frac{\partial\mathbf{g}(\tau)}{\partial\tau}=-2\
\mathcal{R}ic[\nabla](\tau).$ (108)

The Ricci flow ends in certain fixed points $\tau_{\odot},$ when

$\frac{\partial\mathbf{g}(\tau)}{\partial\tau}\mid_{\tau=\tau_{\odot}}=0\mbox{
and }\mathcal{R}ic(\tau_{\odot})=0,$

i.e. a fixed point defines a flat spacetime with vanishing curvature and zero
cosmological constant, $\Lambda=0$. Because a metric with vanishing
cosmological constant is at infinite Weyl distance in the space of all AdS
metrics, it was conjectured [22, 23] that in QG on a family of background
metrics $\mathbf{g}(\tau)$ under Ricci flows "there exists an infinite tower
of states which become massless when following the flow towards a fixed point
$\mathbf{g}_{\odot}=\mathbf{g}(\tau_{\odot})$ at infinite distance".

In nonassociative geometric flow theory, the equations (108) can be
generalized in the form (2) for a family of nonassociative metrics $\
{}^{\shortmid}\mathfrak{g}^{\star}(\tau)$ and respective family of
nonassociative LC-connections $\ {}^{\shortmid}\nabla^{\star}(\tau)$ on phase
space $\ {}^{\star}\mathcal{M}.$ Such nonassociative Ricci flow equations
involve $\kappa$-parametric decompositions of geometric objects. Corresponding
decompositions of nonsymmetric metric, canonical s-connection, and Ricci
s-tensor structures are given, for instance, in formulas (20), (29) and
(2.1.2); for $\ {}^{\shortmid}\nabla,$ such formulas were proven in abstract
and coordinate forms for LC-configurations in [39] and generalized in
nonholonomic s-adapted forms in [12, 13]. The issue on definition of fixed
points for nonassociative Ricci flows is more sophisticate and requests a
rigorous analysis of ceratin families of exact/parametric solutions of
nonlinear systems of PDEs of type (41) and their $\kappa$-linear parametric
decompositions (46). For $\tau$-families of configurations with nonlinear
symmetries (A.8) and respectively prescribed running/ fixed shell cosmological
constants $\ {}_{s}^{\shortmid}\Lambda(\tau),$ we can analyze fixed point
properties of $\kappa$-parametric of geometric flow equations (62), when

$\frac{\partial\
_{s}^{\shortmid}\mathbf{g}(\tau)}{\partial\tau}\mid_{\tau=\tau_{\odot}}=0\mbox{
and }\ ^{\shortmid}\widehat{\mathbf{R}}_{\ \
\gamma_{s}}^{\beta_{s}}(\tau_{\odot},\ _{s}\Phi(\tau_{\odot}),\
_{s}^{\shortmid}\Im(\tau_{\odot}))={\delta}_{\ \ \gamma_{s}}^{\beta_{s}}\
{}_{s}^{\shortmid}\Lambda(\tau_{\odot})=0.$ (109)

Here we note that fixed points determined by such equations for the canonical
s-connection do not define, in general, fixed points for nonassociative
geometric flows with $\ {}^{\shortmid}\nabla^{\star}(\tau).$ Nevertheless, we
can extract and study properties of LC-configurations considering additional
nonholonomic constraints (25),

$\ {}_{\star s}^{\shortmid}\widehat{\mathbf{Z}}(\tau_{\odot})=0,\mbox{ which
is equivalent to }\ _{s}^{\shortmid}\widehat{\mathbf{D}}_{\mid\
_{s}^{\shortmid}\widehat{\mathbf{T}}(\tau_{\odot})=0}^{\star}(\tau_{\odot})=\
\ _{\star}^{\shortmid}\nabla(\tau_{\odot}).$ (110)

Above formulas and observations lead to:

The fixed points of nonassociative geometric flows conjecture and claim, CCL3:

a)

Conjecture: _Consider a QG model on a family of background s-metric $\
{}_{s}^{\shortmid}\mathbf{g}(\tau)$ satisfying the $\kappa$-parametric
nonassociative geometric flow equations (62). There exists an infinite tower
of phase space states with zero effective masses when following the
nonassociative geometric flow evolution toward a fixed point $\
{}_{s}^{\shortmid}\mathbf{g}_{\odot}=\
_{s}^{\shortmid}\mathbf{g}(\tau_{\odot})$ at infinite distance. _

b)

Claim: _For QG models with nonassociative star product and R-flux
modifications, there are large classes of exact/parametric solutions
(involving, or not, nonlinear symmetries of type ( A.8)) when conditions of
type (109) do not hold, for instance, for LC-configurations (110). The
conditions of existence at fixed points, of infinite towers of phase spaces
and behaviour of corresponding effective masses must be analyzed
correspondingly for any class of solutions of $\kappa$-parametric of geometric
flow equations (62) or certain their equivalents. _

We can compute the metric distance (102) along the nonassociative Ricci flows
between a $\tau$ and a fixed point $\tau_{\odot}$ as function
${}_{g}^{s}\aleph(\tau,\tau_{\odot})$ determined by a background solution $\
{}_{s}^{\shortmid}\mathbf{g}(\tau)$ (we can compute respective nonsymmetric
and symmetric components of the nonassociative s-metric) in formula (102). For
such configurations, possible associated towers of phase space states as in a)
Conjecture in CCL3  scale as

$\ {}^{\shortmid}m(\tau_{\odot})\sim\ ^{\shortmid}m(\tau)e^{-\hat{\alpha}|\
_{g}^{s}\aleph(\tau,\tau_{\odot})|}.$

For many examples, a fixed point will be of the form $\tau_{\odot}=\pm\infty.$
The tower of nonholonomic states can only appear when the nonassociative Ricci
flow evolves to a fixed point. In general, a model/ off-diagonal solution is
not necessary characterized by an infinite tower of effective massless states
in phase space. If ${}_{g}^{s}\aleph(\tau,\tau_{\odot})\rightarrow\pm\infty,$
a fixed point/ solution can never be reached. This means the transition
between (non) associative geometric flow theories/ Ricci solitons/ gravity
and/or respective classes of solutions along the geometric flows to the fixed
point is discontinuous.

The a) Conjecture in CCL3 was checked in section 2.1 of [23] (see Conjecture A
in that work) for simple cases of associative and commutative Ricci flows,
simple cases of Einstein spaces (for instance, for (A) dS with non-zero
cosmological constant) and string theory and QG realizations. Similar behavior
can be stated by phase space s-metrics (92) and their nonassociative quasi-
stationary deformations encoding star product and R-flux data when nonlinear
phase space symmetries (A.8) are considered for connecting effective sources
to effective shell cosmological constants $\ {}_{s}^{\shortmid}\Lambda(\tau),$
in particular, we can approximate $\ {}_{s}^{\shortmid}\Lambda(\tau)=\
^{\shortmid}\bar{\Lambda}.$

####  6.1.3 Gibbons-Hawking entropy, scalar curvatures, and generalized
distances for
nonassociative geometric flows

Let us consider an example of nonassociative Ricci flow equations (62) defined
with nonlinear symmetries (A.8) resulting in a conventional phase space
cosmological constant $\ {}_{s}^{\shortmid}\Lambda(\tau)=\
^{\shortmid}\Lambda_{0}.$ Considering classes of solutions involving
conventional phase space horizons, we can apply the concept of Gibbons-
Hawking, GB, entropy [57] and write $\ {}^{\shortmid}\mathcal{S}_{GH}=(\
^{\shortmid}\Lambda_{0})^{-1}$. For quantum models, $\
{}^{\shortmid}\mathcal{S}_{GH}$ can be related to the dimension of the Hilbert
space of $\ {}^{\shortmid}\mathcal{H}$ for observer’s causal domain [58, 59],
when

$\dim\ ^{\shortmid}\mathcal{H}=e^{1/\
^{\shortmid}\Lambda_{0}}\rightarrow\infty\mbox{ for }\
^{\shortmid}\Lambda_{0}\rightarrow 0.$

This property was used in [60] to relate the modified de Sitter conjecture to
the GB entropy of tower of massless states (for infinite distance at $\
{}^{\shortmid}\Lambda_{0}\rightarrow 0$). This property can be extended to
nonassociative phase spaces $\ {}^{\star}\mathcal{M}$ for respective
conditions with $\ {}_{s}^{\shortmid}\Lambda(\tau)=\ ^{\shortmid}\Lambda_{0}$
and, for instance, for s-metrics of type (92). As explained above, see formula
(107), we can write $\ {}_{g}^{s}\bar{\aleph}\simeq\log|\
^{\shortmid}\Lambda_{0}|,$ which for de Sitter configurations with positive
cosmological constant results in $\ {}_{g}^{s}\bar{\aleph}\simeq\log\
^{\shortmid}\mathcal{S}_{GH}.$ We conclude that in the large distance limit
the GB entropy becomes large and this leads to a large tower of light states.

The concepts of Gibbons-Hawking and/or Bekenstein-Hawking entropies can be
applied only for very special classes of solutions (with conventional
horizons) in geometric flow and gravity theories. In next subsections we shall
consider modified G. Perelman functionals in order characterize more general
classes of generic off-diagonal solutions.

Here, we analyze another type of geometric properties of the nonassociative
phase space geometric distance. To work directly with formula $\
{}_{g}^{s}\aleph$ (102) is quite sophisticate and not completely clear how to
formulate certain consistent geometric criteria to decide which families of
background s-metrics can be used as backgrounds in a consistent QG theory not
involving an additional tower of light states, with or not generic off-
diagonal terms, in the infinite distance limit. That why we define in this
work alternative, more general and more abstract distance measures, which can
encode nonassociative star product R-flux contributions.

Let us consider nonassociative Ricci flows for a family of geometric data $[\
_{\star}^{\shortmid}\mathfrak{g}_{\alpha_{s}\beta_{s}}(\tau),\
_{s}^{\shortmid}\widehat{\mathbf{D}}^{\star}(\tau)]$ subjected to the
condition that they are determined by a solution of nonassociative R. Hamilton
equations (41). Such equations involve complex like variables and canonical
nonholonomic structures modeling generic off-diagonal evolution processes and,
for fixed Ricci soliton configurations, gravitational interactions subjected
to non-integrable constraints. Important geometric and possible physical
properties are encoded in the corresponding families of canonical
nonassociative Ricci scalars $\
{}_{s}^{\shortmid}\widehat{\mathbf{R}}sc^{\star}(\tau)$ (31). In partner works
[12, 13], it is elaborated a procedure of parametric decompositions of related
fundamental geometric s-objects $\
{}^{\shortmid}\widehat{\mathbf{R}}_{\quad\alpha_{s}\beta_{s}\gamma_{s}}^{\star\mu_{s}}$
(28), $\ {}^{\shortmid}\widehat{\mathbf{R}}ic_{\alpha_{s}\beta_{s}}^{\star}$
(2.1.2) and $\ {}_{s}^{\shortmid}\widehat{\mathbf{R}}sc^{\star}$ (31) with
$[01,10,11]:=\left\lceil\hbar,\kappa\right\rceil$ components, using the
parametric form of the canonical s-connections (29). This procedure is
summarized in by formulas (28) - (31) in Appendix.

The Conjecture B1 from [23] states that the distance $\ \aleph_{R}$ in the
field space of the background Riemannian metrics along the Ricci flows on a
$\check{d}$-dimensional Riemannian manifold is determined by the scalar
curvature of the LC-connection, $R(\tau):=\mathcal{R}sc[\nabla](\tau),$ when
at $R=0$, there is an infinite tower of additional massless states in QG. That
Conjecture do not have a straightforward extension for nonassociative
geometric flows and related gravity theories because of such reasons:

  1. 1.

Nonassociative phase spaces $\ {}_{s}^{\star}\mathcal{M}$ can be endowed with
different classes of (non) linear connections and respective curvature and
Ricci s-tensors, and respective scalars. There are necessary additional
assumptions and physical motivations on how, for instance, the canonical
geometric objects are related to similar ones for LC-configurations.

  2. 2.

The generalized distance functional $\ {}_{g}^{s}\aleph$ (102) and, for
instance, the canonical scalar curvature $\
{}_{s}^{\shortmid}\widehat{\mathbf{R}}sc^{\star}(\tau)$ (31), and their
restrictions for LC-configurations considering additional nonholonomic
constraints (25), involve complex coordinates and geometric variables. It is
not clear, in general, how to find a physical interpretation for such non-
quantum models even the complex variables are important in QG.

We can solve in general forms the problems stated above in paragraphs 1 and 2
if we work with canonical geometric data when the terms with complex variables
are transformed into almost complex and almost symplectic ones. If necessary,
such nonholonomic geometric objects can be considered as certain canonical
distortions of certain LC-models. Respective $\kappa$-linear decompositions to
models of parametric (real and effective associative and commutative, but
nevertheless nonholonomic) geometric flows, with nonholonomic R. Hamilton
equations (46), also allow to elaborate on physical viable theories with
corresponding Conjectures and Claims:

The canonical distance - scalar curvature conjecture and claim for
nonassociative geometric flows, CCL4:

a)

Conjecture: _Consider a model of nonassociative geometric flows determined by
$\kappa$-parametric R. Hamilton equations (46). The generalized distance $\
{}_{g}^{s}\aleph$ (102) can be defined in a form $\
{}_{R}^{s}\widehat{\aleph}$ determined by the canonical scalar curvature $\
{}_{s}^{\shortmid}\widehat{\mathbf{R}}sc[\ _{s}^{\shortmid}\mathbf{g}(\tau),\
_{s}^{\shortmid}\widehat{\mathbf{D}}(\tau),\
^{\shortmid}\Im_{\alpha_{s}\beta_{s}}(\tau)]$ for certain canonical data for
the equivalent system of PDEs (53). For QG models at $\
{}_{s}^{\shortmid}\widehat{\mathbf{R}}sc=0,$ there exists a canonical infinite
tower of phase space states with zero effective masses._

b)

Claim: _For configurations with nonlinear symmetries ( 60) and transforms of
generating functions and generating sources, $[\ _{s}\Psi(\tau),\
_{s}^{\shortmid}\Im(\tau)]\rightarrow[\ _{s}\Phi(\tau),\
_{s}^{\shortmid}\Lambda(\tau)],$ we can construct exact/parametric solutions
of $\tau$-running $\kappa$-linear modified Einstein equations (62) satisfying
the conditions stated by above conjecture. LC-configurations can be extracted
if additional nonholonomic constraints (25) are imposed. _

Generalizing for nonassociative canonical geometric flows on $\
{}^{\star}\mathcal{M}$ the formula (51) from [23], we consider an alternative
definition of (102) when

$\ {}_{R}^{s}\widehat{\aleph}\simeq\log(\
_{s}^{\shortmid}\widehat{\mathbf{R}}sc(\tau_{i})/\
_{s}^{\shortmid}\widehat{\mathbf{R}}sc(\tau_{f})).$

For some classes of solutions, we can consider flow evolution models with
$\tau_{f}<\tau_{i}.$ If in the vicinity of a fixed point $\
{}_{s}^{\shortmid}\widehat{\mathbf{R}}sc(\tau_{f})=0$ but $\
{}_{s}^{\shortmid}\widehat{\mathbf{R}}sc(\tau_{i})\neq 0,$ we can approximate

$\ {}_{R}^{s}\widehat{\aleph}(\tau)\simeq\log(\
_{s}^{\shortmid}\widehat{\mathbf{R}}sc(\tau))\simeq\log(\sum\nolimits_{s}\
{}_{s}^{\shortmid}\Lambda(\tau)),$ (111)

for configurations with $\tau$-running shell cosmological constants. The
nonassociative off-diagonal geometric flux evolution and gravitational
interactions define in the infinite canonical distance limit (when $\
{}_{R}^{s}\widehat{\aleph}\rightarrow\infty$ with $\
{}_{s}^{\shortmid}\widehat{\mathbf{R}}sc\rightarrow 0)$ a canonical tower of
effective massless states

$\ {}^{\shortmid}m\sim M_{p}e^{-\hat{\alpha}|\ \
_{R}^{s}\widehat{\aleph}|}\simeq(\
_{s}^{\shortmid}\widehat{\mathbf{R}}sc)^{\hat{\alpha}}\simeq(\sum\nolimits_{s}\
{}_{s}^{\shortmid}\Lambda)^{\hat{\alpha}}.$ (112)

The statements of CCL4 can be reformulated for LC-configurations. Here we note
that, in general, the condition $\ {}_{s}^{\shortmid}Rsc[\
^{\shortmid}\nabla]=0$ is characterized by $\
{}_{s}^{\shortmid}\widehat{\mathbf{R}}sc\neq 0$ which reflects the
nonholonomic, locally anisotropic and off-diagonal characters of effective
masses $\ {}^{\shortmid}m$ (103) and/or (112). We prefer to work with
canonical s-adapted geometric variables because this allows us to apply the
AFCDM and construct general classes of solutions. The problem to consider $\
{}^{\shortmid}\nabla$ as a more fundamental than the (auxiliary) canonical
s-connection, $\ {}_{s}^{\shortmid}\widehat{\mathbf{D}},$ or other not/or
s-adapted linear connections, has to be analysed using certain additional
theoretical arguments and experimental/observational data. For pseudo-
Riemannian spacetimes, the associated entropy $\mathcal{S}\simeq\exp(\
_{R}^{s}\widehat{\aleph}(\tau))$ become infinite in the flat spacetime limit.
This was also pointed out in [61] for the BH entropy considerations using the
Bekenstein-Hawking paradigm. Restricting above formulas for base spacetime LC-
configurations, we conclude that the flat space limit should be accompanied by
an infinite number of massless stated and that, for instance, the Minikowki
spacetime is infinitely far away from the curved manifolds along the geometric
flow. This means that the flat space never be reached, and that the transition
to flat space is discontinuous. Nevertheless, such results have to be revised,
for instance, if we consider contributions from nonassociative star products
and R-fluxes and work with more general classes of generic off-diagonal
solutions.

####  6.1.4 Generalized distances and nonassociative geometric flow G.
Perelman functionals

We have to introduce and study more complicate distance functionals and test
respective claims and conjectures for nonassociative geometric flows and
related classical gravity and QG models. Let us begin with original
considerations for geometric flows on $\check{d}$-dimensional Riemannian
manifolds, see Conjecture B2 in [23] for the F-functional
$F(\tau)=F(g(\tau),f(\tau))$ (9) and (in our notations) respective distance
functional $\aleph_{F}.$ It was conjectured that in the background field space
along the combined dilaton-metric flow $\ \aleph_{F}$ is determined by
F-functional when in QG models related to such Ricci flows there is an
infinite tower of additional massless states at $F=0.$ Here we note that
$f(\tau)$ is not obligatory a dilaton type field and originally it was
considered as a normalizing function [1], see also the footnote 10.

In the case of nonassociative geometric flow, we can elaborate on physical
viable theories with:

The canonical distance - F-functional conjecture and claim for nonassociative
geometric flows, CCL5:

a)

Conjecture: _Consider a model of nonassociative geometric flows determined by
a $\kappa$-parametric functional $\
{}_{s}^{\shortmid}\widehat{\mathcal{F}}_{\kappa}^{\star}(\tau)$ (44). The
generalized distance $\ {}_{g}^{s}\aleph$ (102) can be defined and computed in
a form $\ {}_{F}^{s}\widehat{\aleph}$ determined by the canonical scalar
curvature for certain canonical data $[\ _{s}^{\shortmid}\mathbf{g}(\tau),\
_{s}^{\shortmid}\widehat{\mathbf{D}}(\tau),\
^{\shortmid}\Im_{\alpha_{s}\beta_{s}}(\tau)]$ as solutions of nonassociative
geometric flow equations (53). For QG models at
${}_{s}^{\shortmid}\widehat{\mathcal{F}}_{\kappa}^{\star}=0,$ there exists a
canonical infinite tower of phase space states with zero effective masses._

b)

Claim: _For nonassociative geometric flow models and gravitational
configurations with nonlinear symmetries ( 60), we can construct exact/
parametric solutions of $\tau$-parametric running $\kappa$-linear modified
Einstein equations (62) satisfying, or not, the conditions stated by above
conjecture. We can extract LC-configurations for additional nonholonomic
constraints (25). _

We can define and compute a generalized distance functional if

$\ {}_{F}^{s}\widehat{\aleph}\simeq[\
_{s}^{\shortmid}\widehat{\mathcal{F}}_{\kappa}^{\star}(\tau_{i})/\
_{s}^{\shortmid}\widehat{\mathcal{F}}_{\kappa}^{\star}(\tau_{f})].$

In the case when $\
{}_{s}^{\shortmid}\widehat{\mathcal{F}}_{\kappa}^{\star}(\tau_{f})$ is a fixed
point of the nonassociative flow equations (53), the canonical tower of
effective massless stated scale for $\
{}_{F}^{s}\widehat{\aleph}\rightarrow\infty$ as

$\ {}^{\shortmid}m\sim M_{p}e^{-\hat{\alpha}|\
_{F}^{s}\widehat{\aleph}|}\simeq\left(\
{}_{s}^{\shortmid}\widehat{\mathcal{F}}_{\kappa}^{\star}(\tau)\right)^{\hat{\alpha}}\simeq\left(\
{}_{s}^{\shortmid}\widehat{\mathbf{R}}sc(\tau)+\
_{s}^{\shortmid}\widehat{\mathbf{K}}sc(\tau)\right)^{\hat{\alpha}}.$

A normalizing function $\widehat{f}(\tau)$ can be prescribed in a form when
the distortion $\ {}_{s}^{\shortmid}\widehat{\mathbf{K}}sc(\tau)$ is absorbed
in such a form that the nonassociative geometric flow is determined by $\
{}_{F}^{s}\widehat{\aleph}\simeq\ _{R}^{s}\widehat{\aleph}\simeq\log(\
_{s}^{\shortmid}\widehat{\mathbf{R}}sc)$ as in (111). For other type models,
for instance, with $f$ treated as a dilaton field, and considering only a base
spacetime flow of dilaton, we can approximate $\
{}_{F}^{s}\widehat{\aleph}\simeq\log g_{s},$ where $g_{s}$ is the string
coupling (see formula (65) and footnote 8 in [23]). Nonassociative geometric
flow configurations with nonlinear symmetries (60) can be characterized by
${}_{F}^{s}\widehat{\aleph}\simeq-\log\left|\sum\nolimits_{s}\
{}_{s}^{\shortmid}\Lambda(\tau)\right|,$ which is compatible with the
conditions of previous CCL4.

G. Perelman [1] defined also another important functional, the W-functional,
which is a "minus entropy" and can be used for treating the case of collapsing
cycles of the Ricci flow theory of Riemannian metrics. In connection to the
generalized distance functional, the Conjecture B3 from [23] states that for
geometric flows of Riemannian metrics we can introduce a generalized distance
$\ {}_{W}^{s}\widehat{\aleph}$ in the background field space along the Ricci
flow evolution which is determined by the W-entropy. For QG models in the
points where this variable vanishes, there is an infinite tower of massless
states. Such constructions involve more rich nonholonomic geometric structures
for nonassociative geometric flows, and results in physical models with local
anisotropy.

For $\kappa$-parametric decompositions, we can introduce a re-scaling factor
$\chi(\tau)$ in the definition of $\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau)$ (45), when

$\ {}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}[\
_{s}^{\shortmid}\mathbf{g}(\tau),\
_{s}^{\shortmid}\widehat{\mathbf{D}}(\tau),\chi(\tau);\
^{\shortmid}\Im_{\alpha_{s}\beta_{s}}(\tau)]=$ (113) $\displaystyle\int_{\
{}_{s}^{\shortmid}\widehat{\Xi}}\delta\
^{\shortmid}u^{\gamma_{s}}(\chi)\sqrt{|\
^{\shortmid}\mathbf{g}_{\alpha_{s}\beta_{s}}\ (\chi)|}\left[\chi\left(\
{}_{s}^{\shortmid}\widehat{\mathbf{R}}sc(\chi)+(\
_{s}^{\shortmid}\widehat{\mathbf{D}}(\chi)\ \
_{s}^{\shortmid}\widehat{f}(\chi))^{2}\right)+\ \
_{s}^{\shortmid}\widehat{f}(\chi)-8\right]\frac{e^{-\
_{s}^{\shortmid}\widehat{f}(\chi)}}{(4\pi\chi)^{4}},$

under constraint $\int_{\ {}_{s}^{\shortmid}\widehat{\Xi}}\delta\
^{\shortmid}u^{\gamma_{s}}(\chi)\sqrt{|\
^{\shortmid}\mathbf{g}_{\alpha_{s}\beta_{s}}\ (\chi)|}e^{-\
_{s}^{\shortmid}\widehat{f}(\chi)}/(4\pi\chi)^{4}=1.$ The normalizing function
$\ {}_{s}^{\shortmid}\widehat{f}(\chi)$ and re-scaling factor $\chi(\tau)$ can
be chosen in such a form that the resulting variational geometric flow
equations involve only the canonical Ricci s-tensor and respective scalar
curvature. As a result, the nonassociative geometric flow equations (46) can
be written equivalently as

$\displaystyle\partial_{\tau}\
^{\shortmid}\mathbf{g}_{\alpha_{s}\beta_{s}}(\tau)$ $\displaystyle=$
$\displaystyle-2\ ^{\shortmid}\widehat{\mathbf{R}}_{\
\alpha_{s}\beta_{s}}(\tau),$ (114) $\displaystyle\partial_{\tau}\ \
{}_{s}^{\shortmid}\widehat{f}(\tau)$ $\displaystyle=$ $\
{}_{s}^{\shortmid}\widehat{\mathbf{R}}sc(\tau)-\widehat{\bigtriangleup}(\tau)\
_{s}^{\shortmid}\widehat{f}(\tau)+(\
_{s}^{\shortmid}\widehat{\mathbf{D}}(\tau)\
_{s}^{\shortmid}\widehat{f}(\tau))^{2}(\tau)+4/\chi(\tau),$
$\displaystyle\partial_{\tau}\ \chi(\tau)$ $\displaystyle=$ $\displaystyle-1.$

Here we note that the nonassociative star product and R-flux contributions of
order $\left\lceil\hbar,\kappa\right\rceil$ are encoded into generating
functions, generating sources and nonlinear symmetries (60) of s-metrics $\
{}^{\shortmid}\mathbf{g}_{\alpha_{s}\beta_{s}}(\tau).$ Such $\tau$\- and
$\chi$-families of s-metrics can be found as exact/parametric solutions (62)
with prescribed effective running cosmological constants $\
{}_{s}^{\shortmid}\Lambda(\tau).$ Finding a solution of the nonlinear system
of PDEs (114), we can introduce it in the formula (113) and compute the
W-entropy of associated thermodynamic system. This functional is important for
a further analysis if such a nonassociative model belongs, or not, to
swampland.

Summarizing above considerations, for nonassociative geometric flows, we
formulate:

The canonical distance - W-functional conjecture and claim for nonassociative
geometric flows, CCL6:

a)

Conjecture: _For models of nonassociative geometric flows determined by
$\kappa$ -parametric functional $\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau)$ (113), the
generalized distance $\ {}_{g}^{s}\aleph$ (102) can be defined in a form $\
{}_{W}^{s}\widehat{\aleph}$ determined by such a canonical W-entropy
functional defined and computed for corresponding canonical data and solutions
of nonassociative geometric flow equation (46). For QG models at $\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}=0,$ there exists a
canonical infinite tower of phase space states with zero effective masses._

b)

Claim: _If certain nonassociative models of geometric flows / Ricci solitons
are constructed to posses nonlinear symmetries ( 60) to running cosmological
constants $\ {}_{s}^{\shortmid}\Lambda(\tau)$, we can generate exact/
parametric solutions of $\tau$-parametric running $\kappa$-linear modified
Einstein equations (62) satisfying, or not, the conditions stated by above
conjecture. LC-configurations can be extracted for additional nonholonomic
constraints (25). The formulas for $\ {}_{W}^{s}\widehat{\aleph}$ can be
expressed as functionals on $\ {}_{s}^{\shortmid}\Lambda(\tau)$ and respective
volume forms and the conclusion that such a nonassociative geometric and
thermodynamic model belongs, or not, to swampland depend on the type of
prescribed generating functions and running effective cosmological constants._

In analogy to CCL4 and CCL5, we define and compute for CCL6 such a generalized
distance functional:

$\ {}_{W}^{s}\widehat{\aleph}\simeq\log(\
_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau_{i})/\
_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau_{f})).$ (115)

If $\ {}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau_{f})$ is a
fixed point of the nonassociative flow equations (114), the canonical tower of
effective massless stated scale for $\
{}_{W}^{s}\widehat{\aleph}\rightarrow\infty$ as

$\ {}^{\shortmid}m\sim M_{p}e^{-\hat{\alpha}|\ \
_{W}^{s}\widehat{\aleph}|}\simeq\left(\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau)\right)^{\hat{\alpha}}.$
(116)

In [23] (see formulas (88)-(95) in that work), there are provided explicit
computations for functionals (115) and (116) in some cases of associative and
commutative geometric flows, for instance, for high dimensional Einstein
spaces and 2-d spacetime with $SO(2)$ symmetries. Another example was studied
for modified gravity with F(R) Ricci flows (section 4, in that work). Here we
note that more general examples and various applications for $R^{2}$-gravity
with locally anisotropic cosmological and BH solutions were studied in [62].
We shall provide explicit proofs of W-entropy and related formulas for
generalized distance and effective massless stated scales for explicit
examples of nonassociative BH solutions in next subsections.

####  6.1.5 Generalized nonassociative phase space swampland conjectures for
effective (A) dS phase
space configurations and the Bekenstein–Hawking entropy

In [63, 26], there were studied possible physical implications of the Black
Hole Entropy Distance Conjecture, BHEDC. It was postulated in the framework of
a research elucidating a close connection between the infinite distance
conjecture in the context of swampland and area law for the Bekenstein-Hawking
entropy, $S_{BH}.$ That conjecture sates that for BH spacetime configurations
in QG, the formulas $\ {}_{g}^{s}\aleph$ (102) and $m$ (104) can be written,
respectively, in the form

$\ {}_{BH}^{s}\aleph\sim\log S_{BH}\mbox{ and }m\sim
S_{BH}^{-\hat{\alpha}},\mbox{ where }\hat{\alpha}\sim\mathcal{O}(1).$ (117)

For such considerations, the limit $S_{BH}\rightarrow\infty$ is identified
with the Minkowski spacetime. The BHEDC was tested and generalized for
concrete string setups with AdS configurations; for metrics belonging the
family of RNdS solutions; multi-horizon spacetimes; for BH evaporation etc.

Nonassociative star product and R-flux deformations result in different types
of nonlinear systems of PDEs characterized by more general classes of generic
off-diagonal solutions, nonholonomic structures and (non) linear connections.
Such solutions are generated for corresponding nonlinear symmetries and for
coefficients of the fundamental geometric objects depending on all phase space
coordinates and flow evolution parameter. Nevertheless, we can construct also
some very special and physically important classes of solutions with
conventional horizons when the nonassociative contributions are encoded into
respective classes of generating functions (for instance, rotoid
deformations). Our strategy for generalizing the BHEDC on nonassociative phase
spaces $\ {}^{\star}\mathcal{M}$ is stated as follow:

The phase space black hole entropy distance conjecture and claim for
nonassociative geometric flows, CCL7:

a)

Conjecture: _Exact/ parametric solutions with conventional phase space hyper-
surface horizons, hh, defining models of nonassociative geometric flows and/or
gravity theories with $\kappa$-parametric modified R. Hamilton (46) / Ricci
soliton (43) equations, and describing $\tau$-families of star product R-flux
deformed BHs. The QG models for such quasi-stationary solutions are
characterized by a Bekenstein-Hawking entropy variable,
${}_{s}^{\shortmid}S_{hh}$ defined on corresponding phase spaces when the
generalized distance $\ {}_{hh}^{s}\aleph\sim\log(\ _{s}^{\shortmid}S_{hh})$
and effective mass scale for a respective tower of states is $\
{}_{s}^{\shortmid}m\sim\ _{s}^{\shortmid}S_{hh}^{-\hat{\alpha}},$ for
$\hat{\alpha}\sim O(1).$_

b)

Claim: _For nonassociative geometric flow models and gravitational
configurations with nonlinear symmetries ( 60) for running cosmological
constants $\ {}_{s}^{\shortmid}\Lambda(\tau)$, we can construct
exact/parametric solutions of $\tau$-parametric running $\kappa$-linear
modified Einstein equations (62) satisfying, or not, the conditions stated by
the phase space BHEDC. We can extract BE and BH solutions for LC-
configurations for additional nonholonomic constraints (25). _

Additionally, in this subsection, we provide a list of four parametric
solutions for $\tau$-families of nonassociative BE and BH solutions which are
characterized by the conditions CCL7 and can be investigated as models of
nonassociative geometric flows for which the paradigm of Bekenstein–Hawking
thermodynamics holds true.

| Type of solution |  | Nonlinear
---
quadratic
element
Generalized Bekenstein-Hawking entropy, ${}_{s}^{\shortmid}S_{hh}$
1. |  |  Nonassociative $\tau$ -deformed
---
double RNdS BHs, dissipation
to BEs & Schwarzschild BHs,
| ${d\ }_{\shortmid}^{\flat}{s}_{[8d]}^{2}{(\tau),}$
---
see (5.2.3);
| ${}_{s}^{\shortmid}S_{hh}=S_{0}(\tau,r,\theta,\varphi)+\
_{\shortmid}S_{0}(\tau,p,p_{\theta},p_{\varphi}),$
---
$S_{0}=\frac{\Omega_{2}\times(r_{s})^{4}}{4}[1+\frac{4\kappa}{3}\underline{\chi}{(\tau,r,\theta)}\sin{(\omega}_{0}{\varphi+\varphi}_{0})],$
$\
{}_{\shortmid}S_{0}=\frac{\Omega_{2}\times(p_{s})^{4}}{4}[1+\frac{4\kappa}{3}\overline{\chi}{(\tau,p,p}_{\theta})\sin(\
_{\shortmid}^{p}{\omega}_{0}{p}_{\varphi}+p_{\varphi}^{0})$
see (85).
2. |  |  Nonassociative $\tau$-running
---
couples of Schwarzschild-AdS
BHs and BEs deformations,
| $d\ _{\shortmid}^{\epsilon}{s}_{[8d]}^{2}{(\tau),}$
---
see (5.2.3);
| $\ {}_{s}^{\shortmid}S_{hh}=\ ^{\epsilon\chi}S(\tau,r,\theta,\varphi)+\
_{\shortmid}^{\epsilon\chi}S(\tau,p,p_{\theta},p_{\varphi}),$
---
$\ {}^{\epsilon\chi}S=\
^{\epsilon}S_{0}(\tau)[1+\frac{4\kappa}{3}\underline{\chi}(\tau,r,\theta)\sin(\omega_{0}\varphi+\varphi_{0})],$
$\ {}_{\shortmid}^{\epsilon\chi}S=\
_{\shortmid}^{\epsilon}S_{0}(\tau)[1+4\kappa\overline{\chi}(\tau,p,p_{\theta})\sin(\
_{\shortmid}^{p}\omega_{0}\ p_{\varphi}+p_{\varphi}^{0})]$
see (89) and (88).
3. |  |  $d=5$ RN AdS metric
---
embedded into a
8-d phase space $\ {}_{s}M,$
| $d\ \breve{s}_{[5+3]}^{2},$
---
see (92);
$\ {}^{0}\breve{S}=\frac{\
{}^{0}\breve{A}}{4G_{[5]}}=\frac{\omega_{[3]}\breve{r}_{h}}{4G_{[5]}},$ see
(97).
4. |  |  $\tau$-running phase space
---
RN-AdS BEs
| ${d\ }_{\shortmid}^{\chi}{s}_{[6\subset 8d]}^{2}{(\tau),}$
---
see (5.3.3);
${\breve{S}(\tau)=\ }^{0}{\breve{S}(1+}\frac{\kappa\
}{2}{\hat{\chi}}_{4}{(\tau))},$ see (98).

There are possible different scenarios of nonassociative geometric flow
evolution defined by above outlined solutions. They depend on the type of
generating functions and prescribed $\tau$-running/fixed cosmological
constants, and respective nonlinear symmetries (60). In the case 1, the off-
diagonal interactions and flow evolution may transform a phase space double
RNdS BH configuration into a phase system of BEs and Schwarzschild BHs, when
the thermodynamic variables are computed as rotoid type locally anisotropic
deformations of respective variables in Bekenstein-Hawking thermodynamics. For
certain classes of nonholonomic constraints, the corresponding ellipsoidal
deformations can be stable for a fixed value $\tau_{0}$ (i.e. there are
defined stable nonassociative phase space Ricci solitons) and describe
$\tau$-evolution preserving the prime configuration. Such models are described
by solutions of type 2. Similar scenarios can be described by solutions of
type 3 and 4 but with different thermodynamic models and physical
interpretations which is typical for the phase space higher dimension BEs and
BHs.

###  6.2 Nonassociative geometric flows, swampland conjectures, and G.
Perelman thermodynamics

A class of solutions for nonassociative geometric flows and/or nonassociative
gravitational equations can be always characterized by an associated G.
Perelman thermodynamic model, when, for instance, the conditions of CCL6 are
satisfied but those stated in CCL7 are not applicable. We have to investigate
explicitly (for a class of $\tau$-quasi-stationary solutions) if and for which
the conditions and statements of CCL1-CCL6 are true, not true, or
undetermined. In this subsection, we study a certain important examples how
the swampland conjectures have to be extended to nonassociative Ricci flows.
In explicit form, we show how such constructions are related to CCL6 when the
W-entropy is used for defining and computing the generalized canonical
distance functional and respective tower of effective mass stated scaling, see
respective formulas (115) and (116). Similar analysis on applicability and
testing of CCL1-CCL5 are not presented in this work because, the formulas with
canonical scalar curvature, F-functionals etc. are more cumbersome and not
related directly to computation of variables of G. Perelman thermodynamics.

####  6.2.1 The canonical distance - W-functional conjecture and claim, CCL6,
for $\tau$-running
nonassociative generating sources and quasi-stationary solutions

For any class of solutions of geometric flow equations (53) with
$\tau$-running effective sources $\ {}_{s}^{\shortmid}\Im(\tau)$ (52),
represented in a form (4.1), or (4.4), or (4.5), we can compute the star
product deformed W-functional $\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}^{\star}(\tau)$ (39). Let us consider a
s-metric $\ {}_{s}^{\shortmid}\mathbf{g}[\hbar,\kappa,\tau,\psi(\tau),\
_{s}\Psi(\tau),\ _{s}^{\shortmid}\Im(\tau)]$ for quasi-stationary solutions of
type (A.10). We can compute the W-entropy and S-entropy functions as in
formulas (69) but for corresponding generating functions and effective
sources, and volume form ${}^{\shortmid}\delta\ ^{\shortmid}\mathcal{V}(\tau)$
(49),

$\ {}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau)$
$\displaystyle=$
$\displaystyle\int\nolimits_{\tau^{\prime}}^{\tau}\frac{d\tau}{(4\pi\tau)^{4}}\int_{\
{}_{s}^{\shortmid}\widehat{\Xi}}\left(\tau[\sum\nolimits_{s}\
{}_{s}^{\shortmid}\Im(\tau)]^{2}-8\right)\ ^{\shortmid}\delta\
^{\shortmid}\mathcal{V}(\tau),$ (118) $\
{}_{s}^{\shortmid}\widehat{\mathcal{S}}_{\kappa}^{\star}(\tau)$
$\displaystyle=$
$\displaystyle-\int\nolimits_{\tau^{\prime}}^{\tau}\frac{d\tau}{(4\pi\tau)^{4}}\int_{\
{}_{s}^{\shortmid}\widehat{\Xi}}\left(\tau[\sum\nolimits_{s}\
{}_{s}^{\shortmid}\Im(\tau)]-8\right)\ ^{\shortmid}\delta\
^{\shortmid}\mathcal{V}(\tau).$ (119)

So, nonassociative geometric flows can be characterized by two entropy type
variables: the "standard" statistical thermodynamics entropy (similar to
constructions in hydrodynamic models of moving media but in terms of curvature
scalars, metrics, connections) , $\
{}_{s}^{\shortmid}\widehat{\mathcal{S}}_{\kappa}^{\star}(\tau),$ with
$[\sum\nolimits_{s}\ {}_{s}^{\shortmid}\Im(\tau)];$ and the "minus" entropy,
i.e. W-entropy, $\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau),$ with
$[\sum\nolimits_{s}\ {}_{s}^{\shortmid}\Im(\tau)]^{2}.$

Using (119), we can modify the formulas (117) for BHEDC and the conditions of
CCL7,

$\ {}_{\mathcal{S}_{\kappa}^{\star}}^{s}\widehat{\aleph}\ \sim\log(\
_{s}^{\shortmid}\widehat{\mathcal{S}}_{\kappa}^{\star}(\tau))\mbox{ and }\
_{s}^{\shortmid}m\sim(\
_{s}^{\shortmid}\widehat{\mathcal{S}}_{\kappa}^{\star}(\tau))\
^{-\hat{\alpha}\ }\mbox{ for }\ \hat{\alpha}\sim O(1).$

Alternatively, we can follow the CCL6 and compute (115) and (116) for (118).
Respectively, we obtain a generalized distance functional:

$\
{}_{W}^{s}\widehat{\aleph}\simeq\int\nolimits_{\tau_{f}}^{\tau_{i}}\frac{d\tau}{(4\pi\tau)^{4}}\int_{\
{}_{s}^{\shortmid}\widehat{\Xi}}\left(\tau[\sum\nolimits_{s}\
{}_{s}^{\shortmid}\Im(\tau)]^{2}-8\right)\ ^{\shortmid}\delta\
^{\shortmid}\mathcal{V}(\tau),$ (120)

when the canonical tower of effective massless stated scale for $\
{}_{R}^{s}\widehat{\aleph}\rightarrow\infty$ as

$\ {}^{\shortmid}m\sim M_{p}e^{-\hat{\alpha}|\ \
_{W}^{s}\widehat{\aleph}|}\simeq\left(\
\int\nolimits_{\tau_{i}}^{\tau_{f}}\frac{d\tau}{(4\pi\tau)^{4}}\int_{\
{}_{s}^{\shortmid}\widehat{\Xi}}\left(\tau[\sum\nolimits_{s}\
{}_{s}^{\shortmid}\Im(\tau)]^{2}-8\right)\ ^{\shortmid}\delta\
^{\shortmid}\mathcal{V}(\tau)\right)^{\hat{\alpha}}.$ (121)

In general, it is not clear if such nonassociative geometric flow models
belong, or not, to swampland. It depends of the data for generating functions
encoded in $\ {}^{\shortmid}\delta\ ^{\shortmid}\mathcal{V}(\tau)$ and
effective sources $\ {}_{s}^{\shortmid}\Im(\tau).$ We have to analyze such
issues for explicit classes of solutions of (53).

####  6.2.2 CCL6 for $\tau$-running effective shell cosmological constants and
quasi-stationary solutions

Using nonlinear symmetries (60), with $\
{}_{s}^{\shortmid}\mathbf{g}[\hbar,\kappa,\tau,\psi(\tau),\ _{s}\Psi(\tau),\
_{s}^{\shortmid}\Im(\tau)]\rightarrow\
_{s}^{\shortmid}\mathbf{g}[\hbar,\kappa,\tau,\psi(\tau),\ _{s}\Phi(\tau),\
_{s}^{\shortmid}\Lambda(\tau)],$ we can study the conditions of CCL5 involving
quasi-stationary solutions of type (A.10). This allows us to simplify the
formulas for explicit computation of W-entropy $\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau)$ in terms of
running/fixed effective cosmological constants $\
{}_{s}^{\shortmid}\Lambda(\tau),$ see (69) with $\ {}^{\shortmid}\delta\
^{\shortmid}\mathcal{V}(\tau)$ (5.1). Introducing such values, respectively,
in (115) and (116), we obtain:

$\
{}_{W}^{s}\widehat{\aleph}\simeq\int\nolimits_{\tau_{f}}^{\tau_{i}}\frac{d\tau}{(4\pi\tau)^{4}}\int_{\
{}_{s}^{\shortmid}\widehat{\Xi}}\left(\tau[\sum\nolimits_{s}\
{}_{s}^{\shortmid}\Lambda(\tau)]^{2}-8\right)\ ^{\shortmid}\delta\
^{\shortmid}\mathcal{V}(\tau),$ (122)

when the canonical tower of effective massless stated scale for $\
{}_{R}^{s}\widehat{\aleph}\rightarrow\infty$ is computed as

$\ {}^{\shortmid}m\sim M_{p}e^{-\hat{\alpha}|\ \
_{W}^{s}\widehat{\aleph}|}\simeq\left(\
\int\nolimits_{\tau_{i}}^{\tau_{f}}\frac{d\tau}{(4\pi\tau)^{4}}\int_{\
{}_{s}^{\shortmid}\widehat{\Xi}}\left(\tau[\sum\nolimits_{s}\ \
{}_{s}^{\shortmid}\Lambda(\tau)]^{2}-8\right)\ ^{\shortmid}\delta\
^{\shortmid}\mathcal{V}(\tau)\right)^{\hat{\alpha}}.$ (123)

Prescribing data for $\ {}_{s}^{\shortmid}\Lambda(\tau)$ and $\
{}^{\shortmid}\delta\ ^{\shortmid}\mathcal{V}(\tau),$ we are able to generate
nonassociative geometric flow scenarios which belong, or not, two swampland.
Such an analysis is more simple than that for (120) and (121) because
respective formulas (122) and (123) contain effective cosmological constants
which can be prescribed in certain forms which allow generating viable and
important physical models.

Above formulas can be computed in explicit $\kappa$-parametric form when $\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau)=\
_{s}^{\shortmid}\widehat{\mathcal{W}}_{0}+\kappa\
_{s}^{\shortmid}\widehat{\mathcal{W}}_{1}^{\star}(\tau),\ $ for $\
{}^{\shortmid}\delta\ ^{\shortmid}\mathcal{V}=\ ^{\shortmid}\delta\
^{\shortmid}\mathcal{V}_{0}+\kappa\ ^{\shortmid}\delta\
^{\shortmid}\mathcal{V}_{1}$ (5.1). We omit in this work such cumbersome
computations and incremental formulas with $\kappa$–linear decomposition as in
(76), which can be considered for solutions of type (5.2.3), with
$\chi$-polarization functions.

####  6.2.3 CCL6 for nonassociative flows of phase space deformed (double) RN-
dS BHs

The W-entropy from (90) was computed for target s-metrics (80) with
$\eta$-polarization functions defining general quasi-stationary star-product
R-flux deformations of double RN-dS BHs in nonassociative phase spaces. Such
configurations can be characterized by effective cosmological constants $\
{}_{1}^{\shortmid}\Lambda_{0}=\
_{2}^{\shortmid}\Lambda_{0}=\check{\Lambda}\geq 0$ and $\
{}_{3}^{\shortmid}\Lambda_{0}=\ _{4}^{\shortmid}\Lambda_{0}=\
^{\shortmid}\check{\Lambda}\geq 0.$ This class of solutions can be extended
for running cosmological constants $\
{}_{s}^{\shortmid}\Lambda(\tau)=[\check{\Lambda}(\tau),\
^{\shortmid}\check{\Lambda}(\tau)]\geq 0;$ when respective effective sources
$\ {}_{s}^{\shortmid}\Im(\tau)$ are related via nonlinear symmetries (60) to
such $\ {}_{s}^{\shortmid}\Lambda(\tau).$ In general, such nonassociative
geometric flow deformations do not describe BH configurations and their
general physical interpretation is not clear. For $\kappa$-parametric
decompositions with $\chi$-generating functions (5.2.3), we can model stable
BE and BH configurations embedded self-consistently into nonassociative phase
space backgrounds.

Using (90) and introducing $\
{}_{s}^{\shortmid}\Lambda(\tau)=[\check{\Lambda}(\tau),\
^{\shortmid}\check{\Lambda}(\tau)]$ in formulas (122) and (123), we compute

$\
{}_{W}^{s}\widehat{\aleph}\simeq\int\nolimits_{\tau_{f}}^{\tau_{i}}\frac{d\tau}{64(\pi\tau)^{4}}\frac{\tau[\check{\Lambda}(\tau)+\
^{\shortmid}\check{\Lambda}(\tau)]^{2}-2}{|\check{\Lambda}(\tau)\
^{\shortmid}\check{\Lambda}(\tau)|}\
_{\eta}^{\shortmid\flat}\mathcal{V}(\tau),$

when the canonical tower of effective massless stated scale for $\
{}_{R}^{s}\widehat{\aleph}\rightarrow\infty$ as

$\ {}^{\shortmid}m\sim M_{p}e^{-\hat{\alpha}|\ \
_{W}^{s}\widehat{\aleph}|}\simeq\left(\
\int\nolimits_{\tau_{i}}^{\tau_{f}}\frac{d\tau}{64(\pi\tau)^{4}}\frac{\tau[\check{\Lambda}(\tau)+\
^{\shortmid}\check{\Lambda}(\tau)]^{2}-2}{|\check{\Lambda}(\tau)\
^{\shortmid}\check{\Lambda}(\tau)|}\
_{\eta}^{\shortmid\flat}\mathcal{V}(\tau)\right)^{\hat{\alpha}}.$

Prescribing data for $\check{\Lambda}(\tau)$ and$\
{}^{\shortmid}\check{\Lambda}(\tau)$ and $\
{}_{\eta}^{\shortmid\flat}\mathcal{V}(\tau),$ we are able to generate
nonassociative geometric flow scenarios for deforming double BH solutions in
phase space, which belong, or not, to swampland. This depends on behaviour of
$\ {}_{W}^{s}\widehat{\aleph}$ (if there are limits to $\inf$, or zero) for
some types of chosen functions under the integral on $\tau$.

####  6.2.4 CCL6 for nonassociative flows of phase space nonholonomic deformed
RN-AdS BHs

The conditions of CC6 can be tested for another class of exact/parametric
solutions (5.3.2) and (5.3.3) with W-entropy (100) with volume form (101). In
such a target metric, there is an effective cosmological constant
${}_{s}\Lambda(\tau)=\Lambda_{[5]}<0,$ for $s=1,2,3,4;$ when respective
effective sources $\ {}_{s}^{\shortmid}\Im(\tau)$ are related via nonlinear
symmetries (60) to $\ \Lambda_{[5]}.$ Using (100), (122) and (123), we write:

$\
{}_{W}^{s}\widehat{\aleph}\simeq\int\nolimits_{\tau_{f}}^{\tau_{i}}\frac{d\tau}{32(\pi\tau)^{4}}\frac{2\tau\Lambda_{[5]}^{2}-1}{\Lambda_{[5]}^{2}}\
_{\eta}^{\shortmid}\mathcal{\breve{V}}(\tau),$

when the canonical tower of effective massless is characterized by such a
scale behaviour for $\ {}_{R}^{s}\widehat{\aleph}\rightarrow\infty:$

$\ {}^{\shortmid}m\sim M_{p}e^{-\hat{\alpha}|\ \
_{W}^{s}\widehat{\aleph}|}\simeq\left(\ \int\nolimits_{\tau_{i}}^{\tau_{f}}\
\frac{d\tau}{32(\pi\tau)^{4}}\frac{2\tau\Lambda_{[5]}^{2}-1}{\Lambda_{[5]}^{2}}\
_{\eta}^{\shortmid}\mathcal{\breve{V}}(\tau)\right)^{\hat{\alpha}}.$

Such formulas can be generalized for a $\tau$-running cosmological constant
$\Lambda_{[5]}(\tau)<0$ with the same value for all shells. Effective volume
flows $\ {}_{\eta}^{\shortmid}\mathcal{\breve{V}}(\tau)$ can be adapted to
nonholonomic s-distributions which allow to conclude if such nonassociative
geometric flow deformed phase space RN-AdS BHs belong, or not, to swampland.

##  7 Discussion and conclusions

This is the fifth partner work in a series of articles devoted to the theory
of nonassociative geometric and information flows, modified gravity, and
applications in astrophysics and cosmology [12, 13, 40, 41]. Such a research
program is motivated by important results on nonassociative geometry and
physics involving theories with star product R-flux deformations in string and
M-theory [34, 35, 36, 37, 38, 39]. In this paper, we postulate the main
functionals and derive the geometric evolution equations for such a theory of
nonassociative Ricci flows and gravity; state the necessary conditions for a
general decoupling and integrability of necessary systems of nonlinear PDEs;
and construct new classes of exact/parametric solutions describing
nonassociative flux evolution on quasi-stationary and BH configurations. There
are computed and analyzed physical properties of nonassociative phase space
thermodynamic variables developing the paradigms defined (for subclasses of
solutions with conventional hyper-surface horizons) by the Bekenstein-Hawking
entropy, and/or following the G. Perelman W-entropy and geometric
thermodynamic approach which is applicable to general geometric evolution
models and generic off-diagonal solutions. Finally, we study key issues
(formulated as conjectures and claims) related to nonassociative extensions
and revision of the swampland program [19, 20, 21, 18, 60, 22, 23, 24, 25, 26,
27].

###  7.1 Summary and discussion of main results

Let us discuss how the results of this paper fit in with the Objectives (Aims
1-5) stated in section 1.2. We analyze the most important ideas and innovative
methods, consider possible interpretations, and present the key points of
novelty and originality:

  1. 1.

Using nonassociative star product, $\star$, deformations, we define in section
2 (first objective) generalized G. Perelman F- and W-functionals, $\
{}_{s}^{\shortmid}\widehat{\mathcal{F}}^{\star}(\tau)$ and $\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}^{\star}(\tau),$ see formulas (38) and
(39); and effective functionals $\
{}_{s}^{\shortmid}\widehat{\mathcal{F}}_{\kappa}^{\star}(\tau)$ (44) and $\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau)$ (45), for
computing $\kappa$-parametric decompositions (on string constant $\kappa$,
when $\tau$ is a temperature like parameter). Such functionals are used for
deriving nonassociative versions of R. Hamilton equations describing
nonassociative and noncommutative geometric flow evolution models. For self-
similar configurations, they define nonassociative Ricci soliton geometries
and, in particular, nonassociative vacuum gravity theories with running
cosmological constants. We note that nonassociative geometric flow evolution
theories and applications have not been studied in modern mathematics and
physics. There were elaborated only approaches related to noncommutative
geometry (a lá A. Connes) [15], geometric and quantum information flow
theories and modified gravity [62, 6, 7, 8]; and, in associative/commutative
form, it is performed a recent research on swampland conjectures and Ricci
flows [23, 24, 26, 27]. Our nonassociative geometric models and solutions are
self-consistent and well-defined in a sense that they are nonholonomic
deformations with 2+2+2+… and/or (3+1)+(3+1)+…. decompositions of dimensions,
encoding nonassociative geometric data, and generalizing the fundamental
geometric objects and evolution equations in standard Ricci flow theory [4, 5,
1, 2, 3, 9, 10, 11]. Such geometric constructions can be performed in
relativistic forms for Lorentz manifolds and their (co) tangent bundles, when
the nonassociativeand noncommutative structures are determined by star product
and R-flux deformations in string and M-theory. In [38, 39], two similar
models of nonassociative gravity theories were elaborated up to defining and
computing the nonassociative Ricci tensor for the nonassociative star deformed
Levi-Civita, LC, connection, involving (non) symmetric metric structures. The
originality of nonassociative geometric methods elaborated for the
nonholonomic shell oriented structures in [12, 13] consists in developing a
formalism which allows us to apply the anholonomic frame and connection
method, AFCDM [16, 17, 62] for constructing exact and parametric solutions in
modified gravity theories, MGTs; see a brief review in Appendix A. Using the
AFCDM we can prove a general decoupling and integration property of of
nonassociative R. Hamilton equations (41), see section 4 and [40, 41], for
related details for nonasscociative vacuum Einstein equations. Modern
mathematics states explicit limitations, in the sense of topology and
geometric analysis methods, for elaborating a general nonassociative and/or
noncommutative Ricci flow theory. This is because an infinite number of models
of nonassociative/ noncommutative geometries can be elaborated (some of them
present interest in modern mathematical physics and information theory) and it
was not formulated at least an example of nonassociative/ noncommutative /
relativistic generalization of the Poincaré-Thurston conjecture. Nevertheless,
nonassociative geometric flow models with star product R-flux deformations are
well defined (at least in $\kappa$-parametric form, in the framework of
string/ M-theory, and corresponding phase space generalizations of the
Einstein gravity). Applying the AFCDM, we can construct physically important
solutions of respective nonlinear systems of PDEs.

  2. 2.

Theories of nonassociative geometric flows and modified gravity are
characterized by respective statistical thermodynamic models defined by
W-entropy $\ {}_{s}^{\shortmid}\widehat{\mathcal{W}}^{\star}(\tau)$ and
related statistical generating functional $\
{}_{s}^{\shortmid}\widehat{\mathcal{Z}}^{\star}(\tau)$. The novelty of the
second objective  (for section 3) consists in formulating a nonassociative
geometric thermodynamic theory determined by variables (50) encoding
nonassociative geometric data in the $\kappa$-linear thermodynamic variables
$[\ _{s}^{\shortmid}\widehat{\mathcal{E}}_{\kappa}^{\star}(\tau),\
_{s}^{\shortmid}\widehat{\mathcal{S}}_{\kappa}^{\star}(\tau),\
_{s}^{\shortmid}\widehat{\sigma}_{\kappa}^{\star}]$ (51). Such statistical
thermodynamic variables are defined using fundamental (non) associative/
commutative geometric objects (the geometric measure, (non) linear
derivatives, respective nonassociative Ricci tensors and scalar curvatures).
They can be computed, at least in $\kappa$-parametric forms, for all classes
of solutions of corresponding geometric flow/ gravitational field equations.
Former results were for associative and commutative theories [23, 24, 26, 27,
63] and involving special classes of solutions with conventional horizons when
the concept of Bekenstein-Hawking entropy [47, 48, 49, 50] is well-defined.
Following a nonassociative generalized G. Perelman functional formalism [1, 9,
10, 11, 7], we can elaborate on a new statistical/ geometric thermodynamic
paradigm for classical and quantum gravity and information theories, and study
nonlinear models of quasi-stationary and locally anisotropic cosmological
evolution in MGTs, see details in [62, 6, 7, 8, 17, 62].

  3. 3.

In our approach, the nonassociative geometric and gravity theories with star
product R-flux deformations are defined on nonholonomic phase spaces modelled
as cotangent Lorentz bundles, $\ {}_{s}^{\star}\mathcal{M}=T_{s}^{\ast}V,$
with nonholonomic shell, s, dyadic splitting of dimensions, 2+2+2+2. Following
the third objective of this work, in section 4, we show how the AFCDM
developed for nonassociative gravity theories in [13, 40, 41] can be applied
for generating $\tau$-evolving classes of quasi-stationary solutions of
nonassociative geometric flow equations written in $\kappa$-parametric form
(46). An original and important result consists from the proof that, for well-
defined nonholonomic geometric and relativistic physical assumptions, such
equations can be written in two equivalent forms:

     * •

1] as $\tau$-running canonically s-deformed Einstein equations (53), with
effective sources $\ {}^{\shortmid}\Im_{\alpha_{s}\beta_{s}}(\tau)$ encoding
nonassociative geometric data; and

     * •

2] as equivalent families of modified vacuum gravitational equation (62), with
effective $\tau$-running shell cosmological constants $\
{}_{s}^{\shortmid}\Lambda(\tau).$

Such functional representations of $\kappa$-parametric nonassociative R.
Hamilton equations are important because they allow us to use directly the
AFCDM (as for the nonholonomic Einstein equations but with additional
$\tau$-dependence) and generate various classes of generic off-diagonal
solutions. The method simplifies substantially if there are considered, for
instance, some general classes of quasi-stationary solutions involving
nonlinear symmetries of type (60) (for generating locally anisotropic
cosmological solutions, we have to consider certain dual nonlinear
symmetries). We constructed in general off-diagonal forms such
exact/parametric nonassociative geometric flow solutions for $\tau$-evolving
quasi-stationary configurations: with effective sources, (4.1); with running
cosmological constants (4.3); and for various types of s-metric generating and
polarizations functions, see (4.4), or (4.5). The coefficients of such (non)
symmetric metrics and (non) linear connections depend, in general, on all
conventional space like, co-fiber coordinates, for certain Killing type phase
space symmetries, being determined by corresponding $\tau$-families of
generating and integration functions, generating effective sources and running
cosmological constants as in (4.1). The AFCDM is an innovative geometric and
analytic method for constructing exact/parametric solutions of physically
important systems of nonlinear PDEs in (non) associative/ commutative MGTs and
geometric flow generalizations. It allows to generate vacuum and non-vacuum
metrics for very general off-diagonal ansatz not using special assumptions for
diagonalizable metrics resulting into some associated systems of nonlinear
ODEs (such details, for GR, are presented in [43, 44, 45, 46]). Another
important difference from other analytic and numeric methods is that using the
AFCDM we work with certain canonical s-adapted connections which allow to
decouple necessary types of nonlinear systems of equations. We have to impose
additional nonholonomic constraints of type (25) in order to extract LC-
configurations and model their nonassociative geometric flow evolution. It
should be noted that the AFCDM can be applied for constructing
exact/parametric solutions of nonassociative vacuum gravitational equations
with coefficients proportional to the complex unity. Such $\kappa$-parametric
decompositions are for nonassociative Ricci tensors are considered, for
instance, in [38, 39], when only the real coefficients are taken for analyzing
possible consequences for classical models. It is not clear what physical
importance may have such complex solutions in classical theories but,
positively, they can be used to study quasi-classical approximations in QG and
string/ M-theory. In our approach, we can redefine all (non) associative/
geometric constructions using almost complex/symplectic structures and related
real nonholonomic geometric models [12, 13, 40].

  4. 4.

The fourth objective of this work, in section 5, is twofold on its novelty of
results and methods: 1] There are constructed and analyzed most important
physical properties of two new classes of $\tau$-running nonassociative phase
space black hole, BH, black ellipsoid, BE, and other types of more general
quasi-stationary deformations. Such generic off-diagonal solutions are
classified with respect to two different types of primary phase space
s-metrics defined by double Schwarzschild - AdS BHs and, in the second case,
phase space generalizations of Reisner-Norström BHs, when the generated target
s-metrics have different interpretations. For certain types of nonholonomic
constraints and generating functions, there are generated $\tau$-evolving BE
type solutions with conventional phase space hyper-surfaces, when the concept
of Bekenstein–Hawking is applicable, and we compute respective thermodynamic
variables in sections 5.2.3 and 5.3.3. 2] Nevertheless, general models with
nonassociative and generic off-diagonal Ricci flow evolution of physically
important quasi-stationary solutions do not contain hyper-surfaces and we have
to apply more general concepts of nonassociative generalized G. Perelman
thermodynamics. The formulas for respective thermodynamic variables (73) are
defined by effective running shell cosmological constants, $\
{}_{s}^{\shortmid}\Lambda(\tau)$ and certain volume forms depending on the
prescribed nonholonomic structure. Such values are computed in explicit forms
for corresponding classes of nonassociative Ricci flow deformed BHs solutions,
see subsections 5.2.4 and 5.3.4. They functionally depend on generating and
integration functions, which can be chosen in such forms when the
configurations are well-defined as effective relativistic thermodynamic
variables, preserving certain physical important properties under
$\tau$-evolution, to belong, or not to swampland models etc.

  5. 5.

We study nonassociative modifications of the swampland program [19, 20, 21,
18, 60], which consists the fifth objective of this paper, see section 6. As
motivated in modern literature on high energy physics and gravity [15, 6, 17,
7, 8, 22, 23, 24, 25, 26, 27], the Ricci flow is closely related to the RG
flow with respect to the energy scale in the underlying two-dimensional non-
linear sigma-models and various modified gravity and string theories. This
provides a connection between RG, Ricci flows, and the swampland idea. In this
work, we formulate a theory of nonassociative geometric flows using
generalizations of G. Perelman functionals and related thermodynamic models
and develop the AFCDM of constructing new classes of $\tau$-evolving
nonassociative quasi-stationary star-deformed BH and BE solutions. In general,
the thermodynamic physical properties of such nonassociative geometric flow
and gravity generic off-diagonal solutions can not be investigated following
only the Bekenstein-Hawking entropy paradigm. We analyze how the a series of
important swampland and Ricci flow conjectures can be extended to
nonassociative geometric flows and respective extensions to the modified G.
Perelman thermodynamic models. Then, we argue that we can prescribe certain
subclasses of nonholonomic constraints, generating and integration functions
when corresponding models satisfy the conditions of swampland conjectures
studied in [23, 24, 25, 26, 27]. Corresponding phase space generalizations and
modifications involving respective generalized distance functionals and
effective mass scale of towers of states are stated respectively in
subsections 6.1 and 6.2 and numbered as a) Conjectures 1-7. We provide and
analyze four examples of $\tau$-running quasi-stationary phase space solutions
for nonassociative BH configurations with conventional hyper-surface horizons
when the concept of generalized Bekenstein–Hawking entropy is applicable, see
the end of subsection 6.1. This proves that for certain defined geometric
conditions the swampland conjectures can be extended to conventional phase
space and generalized for such nonassociative geometric flow and gravitational
configurations. For more general nonassociative $\kappa$-parametric off-
diagonal solutions, characterized by respective W-entropy functionals, we need
a more tedious analysis in order to conclude if a corresponding nonassociative
geometric flow model belongs, or not, to swampland following b) Claims 1-7. We
formulate such criteria in the form of Conjectures and Claims, CCL, 1-7. In
subsection 6.2, we provide explicit formulas for computing nonassociative and
off–diagonal modifications of general distances and towers of effective mass
states for various classes of $\tau$-running nonassociative quasi-stationary
equations following the CCL5 and effective cosmological constants $\
{}_{s}^{\shortmid}\Lambda(\tau).$

###  7.2 Conclusions, validity of claims and results, and perspectives

Discussing Objectives 1-5 for nonassociative Ricci flow theory and
applications, we have advocated for modifications of the Swampland Program
summarized in a series of Conjectures and Claims, CCL:

  * •

CCL1: The generalized distance conjecture and claim for nonassociative phase
space. We conclude that for very special nonholonomic/ diagonal configurations
and additional assumptions on the type of nonlinear symmetries for the
$\kappa$-parametric distance there are limits of the general distance
functional $\ {}_{g}^{s}\aleph$ (102)$\rightarrow\infty$ resulting in infinite
towers of effective phase masses $\ {}^{\shortmid}m$ (103). This extends for
nonassociative phase space the generalized distance conjecture studied in [22,
23] for (associative and commutative) d-dimensional manifolds. In this work,
we show that using the AFCDM, we can construct new classes of exact/parametric
solutions when, for nonassociative configurations, when $\ {}_{g}^{s}\aleph$
and $\ {}^{\shortmid}m$ may have a different behaviour because generic off-
diagonal interactions and a more rich nonholonomic structure of phase spaces
and nonassociative geometric flux evolution. This claims additional
investigations for any class of off-diagonal solutions in order to understand
if they result in effective models which belong, or not, to swampland.

  * •

CCL2: The AdS distance conjecture and claim for nonassociative phase spaces.
Another important conclusion of this work is that combining the conditions of
CCL1 with the notion of (conformal) Weyl re-scaling of nonassociative phase
space metrics, we can extend the AdS Distance conjecture in a form including a
part a) as a conjecture for a subclass of phase space quasi-stationary
s-metrics with $\kappa$\- and $\tau$ parametric nonlinear symmetries (60)
connecting effective sources to effective running cosmological constants. A
part b) Claim in CCL2 is motivated because the conformal transforms and
symmetries are relevant only to very special classes of solutions encoding
nonassociative data and generic off-diagonal geometric flow evolution
scenarios and/or nonassociative gravitational interactions. In general, we
need an additional analysis in order to determine if a class of parametric
solutions for nonassociative Ricci flows belong, or not, to swampland.

  * •

CCL3: The fixed points of nonassociative geometric flows conjecture and claim.
This concludes that in the nonassociative geometric flow theory elaborated in
this work there exist parametric solutions defining infinite towers with
effective zero masses points for flow evolution toward a fixed point at
infinite distance. The a) Conjecture in CCL3 can be checked on nonassociative
phase spaces, for instance, for s-metrics (92) in a form similar to that for
associative and commutative Ricci flows in [23]. For more general classes of
nonassociative quasi-stationary off-diagonal solutions, we need a more
rigorous analysis for chosen classes of generating functions and effective
cosmological constants. Different nonholonomic configurations may result, or
not, in well defined effective classical and QG models.

  * •

CCL4: The canonical distance - scalar curvature conjecture and claim for
nonassociative geometric flows. The AFCDM allows to decouple (modified) Ricci
flow and gravitational equations and construct exact/parametric solutions if
the canonical s-connection is used instead of the LC-connection. An original
result of the subsection 6.1.3 consists in deriving formulas for the
generalized canonical distance and related tower of effective mass states in
terms of the canonical scalar curvatures and running shell cosmological
constants (see constructions related to (111) and (112)). So, in principle, we
can check if some classes of parametric solutions belong or not to swampland
if we formulate all results in terms of the canonical s-connections. We can
extract LC-configurations imposing additional nonholonomic constraints but the
corresponding geometric constructions and analysis of possible physical
implications depend on the type of generated families of quasi-stationary
solutions. This imposes certain limitations for testing in general LC-form the
CCL4. Nevertheless, such formulas involving the canonical s-curvature allow to
express the necessary results in terms of $\ {}_{s}^{\shortmid}\Lambda(\tau)$
which is important for testing, for instance, the CCL6 and CCL7 (see below).

  * •

CCL5: The canonical distance - F-functional conjecture and claim for
nonassociative geometric flows. The idea to define the general distance
functional in terms of F- and W-functionals was elaborated for associative and
commutative geometric flows, see Conjectures B2 and B3 in [23]. The subsection
6.1.4 consists an original extension of those constructions for nonassociative
geometric flows determined by respective $\kappa$-parametric generalizations
of G. Perelman’s functionals. Using $\
{}_{s}^{\shortmid}\widehat{\mathcal{F}}_{\kappa}^{\star}(\tau)$ (44), we can
compute the generalized distance and state the conditions of existence of
canonical infinite tower of phase space states with zero effective masse as in
a) Conjecture of CCL5. The part b) Claim states that in general, solving
respective $\tau$\- and $\kappa$-parametric equations we can express necessary
formulas in terms of respective F-functionals and decide if a class of
solutions belong or not two swampland and speculate on how LC-configurations
can be extracted. Nevertheless, we conclude that the constructions with
generalized F-functionals are quite sophisticate when we study the statistic
and geometric thermodynamic properties of modified (nonassociative and/or
other types) geometric flows and Ricci soliton configurations and, in
particular of quasi-stationary and nonholonomic BH solutions.

  * •

CCL6: The canonical distance - W-functional conjecture and claim for
nonassociative geometric flows. The a) Conjecture and b) Claim parts are
similar to those stated for CCL5 but using the star deformed W-functional. We
derived the formulas $\ {}_{W}^{s}\widehat{\aleph}\simeq\log(\
_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau_{i})/\
_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau_{f}))$ (115) and
$\ {}^{\shortmid}m\sim M_{p}e^{-\hat{\alpha}|\ \
_{W}^{s}\widehat{\aleph}|}\simeq\left(\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau)\right)^{\hat{\alpha}}$
(116), for $\ {}_{W}^{s}\widehat{\aleph}\rightarrow\infty$, which are very
important and simplify the procedure for analyzing if certain classes of
exact/ parametric solutions belong, or not, to swampland. Using nonlinear
symmetries, we can express $\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau_{i})$ as a
functional of running cosmological constants and volume forms and, then, we
can speculate on classes of generating functions which drive a class of
solutions to swampland, or inversely, keep it with some well-defined
properties. For associative and commutative Ricci flows, such formulas were
conjectured in [23].

  * •

CCL7: The phase space black hole entropy distance conjecture and claim for
nonassociative geometric flows. An original approach to swampland and BH
physics was elaborated in [63, 26] exploiting possible connections between the
distance conjecture and area low, involving Ricci flow conjectures, for the
Bekenstein-Hawking entropy. The statements of the Black Hole Entropy Distance
Conjecture, BHEDC, can be extended to certain classes of nonassociative phase
solutions for geometric flows/ Ricci solitons if the s-metrics contain certain
hyper-surface configurations, as we concluded in a) Conjecture of CCL7.
Nevertheless, even for such type of quasi-stationary $\kappa$\- and
$\tau$-running solutions related via nonlinear symmetries to running
cosmological constants, we may need an additional analysis (the b) Claim of
CCL7) in order to decide if such a solution belong to swampland or not. We
list a table with 4 examples of such nonlinear quadratic elements and
respective formulas for phase generalized Bekenstein-Hawking entropy computed,
for instance, for rotoid configurations.

Summarizing CCL1-CCL7, we conclude that, in general, the solutions of
nonassociative geometric flow and Ricci soliton equations can be characterized
by a corresponding modified G. Perelman thermodynamic model, when the
conditions of CCL6 can be analyzed with respect to the purposes of the
Swampland Program, even the statements of CCL7 are not applicable. Here we
note that we can define and compute two types of entropy variables: the
"minus" entropy, i.e. W-entropy, $\
{}_{s}^{\shortmid}\widehat{\mathcal{W}}_{\kappa}^{\star}(\tau)$ and $\
{}_{s}^{\shortmid}\widehat{\mathcal{S}}_{\kappa}^{\star}(\tau)$, with
effective shell sources $\sum\nolimits_{s}\ {}_{s}^{\shortmid}\Im(\tau)$
(119). Using nonlinear symmetries / transforms (60), such formulas allow us to
modify respectively the statements of CCL6 in certain forms which are
applicable to compute the generalized distance functional and the effective
mass stated scale in terms of effective running cosmological constants. We
present such important formulas for quasi-stationary solutions, see (122) and
(123). Together with CCL6, the modified CCL7 is tested by explicit examples of
nonassociative quasi-stationary geometric flow deformations of phase space
double nonholonomic deformed RN-dS BHs and nonholonomic deformed RN-AdS BHs,
see respective formulas in subsections 6.2.3 and 6.2.4.

Let us discuss the proofs and validity of results and claims of this article
stated as Issues 1-5, Is1-5:

  * •

Is1: Postulating nonassociative geometric flow and vacuum gravitational
equations. It is not possible to formulate a general theory of nonassociative
or noncommutative geometric flows or MGTs involving different types of non-
Riemannian metric and (non) linear connection structures. For a subclass of
theories determined by R-flux deformations [12, 13, 40, 41], it is possible to
define star variants of Riemannian and Ricci tensors and perform corresponding
$\kappa$–linear decompositions of geometric objects (generalizing in
nonholonomic form the constructions from [38, 39]). The geometric
constructions are very different for other models involving octonionic or
supersymmetric variables, spectral triples etc. [15, 33, 34, 35, 36, 37, 70].
Using $\ {}_{s}^{\shortmid}\widehat{\mathcal{R}}ic^{\star}$ and $\
{}_{s}^{\shortmid}\widehat{\mathcal{R}}sc^{\star}$ from (27), we can postulate
the nonassociative geometric flow equations (2), or (41), describing the
evolution of $\tau$-families $\left(\
{}_{\star}^{\shortmid}\mathfrak{g}_{\alpha_{s}\beta_{s}}(\tau),\
_{s}^{\shortmid}\widehat{\mathbf{D}}^{\star}(\tau)\right)$ (as we considered
in subsection 2.3.1). For some subclasses of nonassociative Ricci solitons, we
obtain nonassociative generalizations of the Einstein equations (43), or (32),
which can be formulated independently in abstract symbolic geometric form as
in [43] but for nonassociative geometric s-objects.

  * •

Is2: Postulating nonassociative Perelman’s functionals and thermodynamic
variables. Considering $\ {}_{s}^{\shortmid}\widehat{\mathcal{R}}ic^{\star}$
and $\ {}_{s}^{\shortmid}\widehat{\mathcal{R}}sc^{\star}$, and respective
normalizing functions and volume forms, we can define in abstract geometric
form the F- and W-functionals (38) and (39). Such Lyapunov type functionals
consist nonholonomic relativistic generalizations and star product
deformations of standard Perelman functionals (9) for Ricci flows of
Riemannian metrics. The nonassociative W-functional (39) is similar to a
"minus" nonassociative entropy, which allowed us to define geometrically
(generalizing the constructions from [1]) nonassociative versions of
statistical/ geometric thermodynamic variables (50). Such nonassociative
geometric thermodynamic models can be defined in abstract symbolic geometric
form and corresponding physical motivations are provided at the end of
subsection 2.3.2, see also below the paragraphs Is3 - Is5.

  * •

Is3: Problems and a cure for formulating variational procedures for deriving
nonassociative geometric flow and nonassociative gravitational field
equations. One of the fundamental G. Perelman’s result [1] consisted in a
proof that for Riemannian geometries the R. Hamilton equations (108) can be
derived using variational procedures for F- or W-functionals (9). Those
geometric constructions were used for elaborating rigorous proofs of the
Poincaré-Thurston conjecture [1], see details in monographs [9, 10, 11].
Unfortunately, it is not possible to formulate a self-consistent variational
principle for general nonassociative deformations determined by a general
twist product and in other types of nonassociative and noncommutative
theories. Here, we note that one can be defined an infinite number of
nonassociative and noncommutative differential and integral calculi which is
different from the commutative (pseudo) Riemannian geometry. So, it is not
possible to formulate some general forms of nonassociative the Poincaré-
Thurston conjecture and use "twisted" variational constructions for
corresponding proofs. Nevertheless, this does not prohibit us to elaborate on
physically important and well-defined models of nonassociative geometric
flows. We can postulate in symbolic geometric form nonassociative geometric
flow equations (2), or (41), and then to study their thermodynamic properties
using associated nonassociative F-and W-functionals. For such nonassociative
geometric constructions, we can perform always parametric decompositions and
compute all terms for $\kappa$-deformed R. Hamilton equations (46) and
Perelman’s functionals (44) and (45). Working with $\kappa$–linear deformed
theories, we can always elaborate self-consistent variational procedures, when
corresponding nonassociative geometric flow/ gravitational equations are
characterized by $\kappa$-linear thermodynamic variables (51).

  * •

Is4: Physically important nonvariational theories. There are various physical
arguments (similarly to those for models of hydrodynamics with parametric
turbulence, or for thermodynamic models with parametric phase transitions,
multi-diffusion and branch decomposition and evolution) to study
nonassociative geometric flow theories. In such cases, the definition of self-
consistent variational procedures is not uniquely defined but depends on some
prescribed (non) linear / associative / commutative symmetries, on the order
of parametric decompositions and corresponding nonholonomic s-distributions.
We emphasize that the variational procedure described in subsection 2.3.2 can
be performed in a quite general form if we begin with a corresponding
commutative canonical s-adapted variational nonholonomic configuration and
subject the constructions to $\kappa$-parametric deformations following the
Convention 2 (26). Such a variational procedure can be performed recurrently
for higher orders on $\kappa$. Unfortunately, to prove some general
convergence conditions summarizing all terms on powers of $\kappa$ is an
unsolved mathematical problem, which is typical for nonlinear functional
analysis. Theoretically, we can consider an inverse situation and begin with
general nonassociative formulas for flow equations (2) and formal functionals
(38) and (39). Then, we can perform adapted $\kappa$-parametric decompositions
involving imaginary and real terms. Unfortunately, in such cases, we are not
able to elaborate a general method for constructing exact/parametric solutions
but following the inverse procedure we can always apply the AFCDM as we proved
in previous sections (for quasi-stationary solutions) and in partner works
[13, 40, 41]. This is a generic property for any nonassociative/
noncommutative differential and integral calculus.

  * •

Is5:  Validity through exact and parametric solutions of physically important
nonlinear systems of PDE encoding nonassociative data. So, haven defined
corresponding fundamental nonassociative geometric s-objects like a star
deformed metric, a canonical s-connection and respective Ricci s-tensor and
canonical scalar, we can always postulate geometrically certain nonassociative
geometric flow equations and F- and W-functionals. Such values are well-
defined in all orders on $\kappa$ but for twisted theories there is not a
general nonassociative variational proof for nonassociative geometric flow/
gravitational equations from some generalized Perelman/ action functionals.
For applications in modern physics and quantum information theory, we can
consider only $\kappa$-linear constructions, apply the AFCDM and construct
physically important solutions (star-deformed black holes, wormholes, locally
anisotropic cosmological s-metrics), and speculate on verifiable models
encoding nonassociative data. Recurrently, we can construct nonassociative and
noncommutative solutions with higher orders on $\kappa$ and $\hbar$ but the
technically the resulting formulas are much cumbersome. For any such higher
order parametric configurations, we can define and compute effective F- and
W-functionals, and respective nonassociative thermodynamic variables. Re-
defining the measures (for respective normalizing s-functions and nonholonomic
s-distributions), we suppose that a corresponding variational procedure can be
formulated for any stated polynomial order on $\kappa$ and $\hbar$ (like we
considered in subsection 2.3.2). But this do not provide us a general well-
defined nonassociative variational theory with twist product. We can construct
and study physical properties of realistic nonassociative theories considering
physically important solutions encoding $\kappa$-linear data.

Finally, with respect to above Is1-5, we note that (in certain similar forms)
such problems exist in all classical gravity and QG theories. For instance,
the classical Einstein equations are linear on $\varkappa=8\pi G/c^{4},$ were
$G$ is the Newtonian constant of gravitation and $c$ is the light velocity
constant. We do not have yet a well-defined theory of QG, but it is aways
possible to elaborate on physically important effects proportional to
$\varkappa$ and $\hbar$ and consider higher orders on such constants involving
(square) curvature terms etc. Such effective models are commutative or
noncommutative. R-flux deformations from string theory result in
nonassociative geometric configurations determined by twist products. This
results in ambiguities for constructing general self-consistent principles,
involves complex terms for curvatures and Ricci tensors for some parametric
decompositions on $\kappa$ and $\hbar$ like in [38, 39, 12] etc. This reflects
our not complete knowledge about string theory and QG. The priority of
nonholonomic s-adapted definition of star product (2.1.2) is that using the
Convention 2 (26) we can define and compute recurrently parametric R-flux
deformations of some (associative/commutative) variatonal equations and apply
the AFCDM in order to construct physically important solutions of
nonassociative geometric flows and vacuum gravitational equations. In general
form, such nonassociative solutions are characterized by respective modified
Perelman (thermodynamic) functionals and variables. This states a new paradigm
for formulating and investigating possible physical implications of
nonassociative theories and provides a general computational method when
nonassociative effects can be computed and the validity of certain claims can
be verified at least (for simplicity) in the linear approximation on $\kappa$.

Above conclusions and theoretical/ computational tests of CCL1-CCL7 support
the The Main Hypothesis, MH,  of this work (formulated at the end of
subsection 1.2) _that the Swampland Program has to be revised/ modified in
order to elaborate explicit criteria how to include nonassociative and
noncommutative geometric flows, QM, QFTs, and MGTs in elaborating QG theories.
The corresponding Conjectures and Claims allows us to select self-consistent
nonassociative geometric and physical models (defined by generic off-diagonal
solutions) encoding at least in parametric form nonassociative star product
and R-flux data for string / M-theory. In low-energy limits, such
configurations can be completed into QG in the UV forms and distinguished from
another classes of theories/models/ solutions which do not have such
properties._

The MH modifies a series of future research purposes stated in our partner
works [12, 13, 40, 41]. We list and speculate on five perspective directions
of research involving nonassociative geometric flow methods, classical and
quantum gravity, and quantum information models181818for instance, Q1a is used
for modifications involving nonassociative Ricci flow of the query Q1 from
[41]; we also state a new query Q5.:

  * •

Q1a: Nonassociative Einstein-Yang-Mills-Higgs systems. A full investigation of
nonassociative geometric flow and gravity theories should involve flux
evolution and field equations with nontrivial matter sources and other types
of nonassociative/ noncommutative structures (for instance, octonions etc.)
which are not necessarily determined by R-flux deformations but based on other
types of star product and/or nonassociative algebraic structures. One of the
next steps is to formulate and study models with nonassociative nonholonomic
deformations and geometric flow evolution of Einstein-Yang-Mills-Higgs systems
resulting in nonassociative gravity and matter field theories with
nonsymmetric metrics, generalized connections and nonzero sources. Such
nonassociative models should generalize in nonassociative form the Einstein-
Eisenhart-Moffat theories [64, 65, 66, 66, 68], see respective nonholonomic
and phase space Ricci flow constructions in [69]. References [12, 13, 40, 41]
provide a series of new ideas on further nonassociative/ nonsymmetric metric /
nonholonomic developments when the AFCDM can be applied for constructing
various classes of exact/parametric solutions (for instance, defining locally
anisotropic wormholes, BHs and BEs, solitonc hierarchies, nonassociative
quasi-periodic cosmological structures etc.).

  * •

Q2a: Nonassociative Finsler-Lagrange-Hamilton geometric flows/gravity, their
almost symplectic models, and deformation quantization. Nonassociative
geometric flow and gravity theories determined by a star product of type
(2.1.2) on tensor products of cotangent bundles consist of a class of
nonassociative generalizations of the Finsler-Lagrange-Hamilton geometry which
have generalized metrics/connections depending on velocity/ momentum-like
coordinates. In relativistic/ noncommutative and commutative/ supersymmetric /
fractional etc. variants, there are status reports and reviews of results [70,
8]. The importance of Finsler-like geometric objects and nonholonomic
variables (they can be defined even on Lorentz manifold enabled with a
nonolonomic fibered structure) is that they can be reformulated as some
equivalent almost Kaehler/ symplectic and almost complex variables. This
allows us to elaborate on realistic classical models for (non) associative/
commutative geometric flow and gravity theories (to apply complex variables
for quantum models and almost complex variables for classical models) and
apply/develop methods of deformation and/or geometric quantization. A more
sophisticate program is to formulate and study models of QG involving
nonassociative/ noncommutative Lagrange-Hamilton structures and/or almost
symplectic variables.

  * •

Q3a: Elaborating Bekenstein-Hawking and G. Perelman thermodynamic and locally
anisotropic kinetic models for physically important solutions in
nonassociative classical and quantum theories. The sections 2 and 3 of this
work provide a solution of query Q3 stated at the end of [41]. Here we
remember that the nonassociative gravity with star product and R-flux
deformations was formulated up to defining and computing the nonassociative
Ricci tensor in [38, 39]. This was enough to develop models of nonassociative/
noncommutative Ricci flows but those abstract geometric and/or coordinate
frame constructions based on LC-connection do not allow to motivate respective
generalizations of R. Hamilton equations, prove decoupling and integration
properties of such geometric flow equations and find exact/ parametric
solutions. To study possible physical implications we had to elaborate a
respective nonholonomic formalism with dyadic shell structures, see details in
[12, 13] and Appendices to this work. Such constructions were motivated by
finding physically important exact/parametric solutions and computing
respective Bekenstein-Hawking and/or G. Perelman thermodynamic variables in
section 4 of [41] and section 5 in this paper.

  * •

Q4a: The program on nonassociative geometric and quantum information flow
theories. We point again to perspectives and importance of a research program
to extend in nonassociative forms the geometric flow information theory [6, 7]
to theories with nonassociative qubits and entanglement, conditional entropies
etc. Such new directions in modern QFT, strings and gravity, theory of quantum
computers have deep roots in nonassociative quantum mechanics and gauge models
[30, 31, 33, 37] and motivations from noncommutative geometry, string and
M-theory [15, 34, 35, 36, 38, 39].

  * •

Q5: Cosmological models and dark energy and dark matter physics encodding
quasi periodic structure and data for nonassociative theories. In this work,
we constructed and studied the main properties of a series of new classes of
nonassociative exact/ parametric solutions for $\tau$-running quasi-stationary
physically important solutions defining star product and R-flux deformed BHs,
BEs etc. The AFCDM can be also developed and applied in certain dual forms
when there are generated locally anisotropic solutions [72, 17, 71]. Extending
such geometric and analytic methods, we plan to elaborate on a series of works
on phase space and spacetime nonassociative quasi-crystal configurations and
related web filaments, nonassociative quasi-periodic geometric flow evolution
and pattern forming structures in accelerating and inflationary cosmology and
dark energy and dark matter physics. Such models are generic off-diagonal (for
certain classes of holonomic configurations, the s-metrics can be
diagonalized) encoding nonassociative star product deformations. They are
characterized by respective generalized G. Perelman thermodynamic variables
and put their imprints and requests for modifications of the Swampland
Program.

We shall develop on above directions and report on progress for queries
Q1a-Q4a,Q5 in future works.

Acknowledgments: This is the second work together with [41] supported by a
Fulbright senior fellowship for SV and hosted by the physics department at
California State University at Fresno, USA. Some results are related also to a
research program proposed for the PNRR-Initiative 8 of the Romanian Ministry
of Research, Innovation and Digitization and for a visiting program to be
decided by CASLMU München. It is also a partner work to [12, 13, 40] extending
to nonassociative geometric flows former research programs and fellowships at
CERN (Geneva, Switzerland) and Max Planck Institut für Physik / Werner
Heisenberg Institut, München (Germany). SV is also grateful to professors D.
Lüst, N. Mavromatos, J. Moffat, Yu. A. Seti, P. Stavrinos, and M. V. Tkach for
hosting visits and collaborations. Authors thank the referee for very
important critics and requests to provide substantial modifications and proofs
on s-adapted variational calculus for nonassociative geometric flow equations
performed in section 2.3.

##  Appendix A The anholonomic frame and connection deformation method, AFCDM

We outline the AFCDM for constructing exact and parametric solutions for
nonlinear systems of PDEs (16). Details with proofs and methods in commutative
gravity theories are provided in [7, 8] and, for recent nonassociative
generalizations, [12, 13, 40, 41]. In this section, we consider effective
sources parameterized in the form (35) encoding nonassociative star product
and R-flux deformations. Such PDEs can’t be integrated in some general off-
diagonal forms if we follow standard methods from GR [46, 43, 45, 44] when
solutions are found for some special diagonal ansatz of metrics transforming
the (modified) Einstein equations into certain systems of nonlinear ordinary
differential equations, ODE. For diagonalizable metrics, there are imposed
some higher order symmetries (spherical, cylindrical type etc.) which allow to
integrate the resulting ODEs in very general forms. The solutions are
classified by respective integrations constants. This is used, for instance,
for constructing black hole, BH, solutions. The AFCDM is a more genera
geometric method for generating exact and parametric solutions, when a
necessary type of auxiliary connection (for instance, the canonical
s-connection $\ {}_{s}^{\shortmid}\widehat{\mathbf{D}}$) is used. This allows
us to decouple and integrate systems of (modified) geometric flow and
gravitational field equations defined by generic off-diagonal metrics, (non-)
Riemannian connections etc. with coefficients depending, in principle, on all
spacetime and phase space coordinates. Different classes of such exact/
parametric solutions are determined by respective classes of generating
functions, generating sources, integrating functions and constants, and/or
decompositions certain small physical parameters (for instance, on string and
Plank constants) etc.

To apply the AFCDM the quadratic linear element for generating quasi-
stationary solutions is parameterized by such an off-diagonal ansatz for
s-metrics:

$d\widehat{s}^{2}=g_{i_{1}}(x^{k_{1}})(dx^{i_{1}})^{2}+g_{a_{2}}(x^{i_{1}},y^{3})(\mathbf{e}^{a_{2}})^{2}+\
^{\shortmid}g^{a_{3}}(x^{i_{2}},p_{5})(\ ^{\shortmid}\mathbf{e}_{a_{3}})^{2}+\
^{\shortmid}g^{a_{4}}(\ ^{\shortmid}x^{i_{3}},E)(\
^{\shortmid}\mathbf{e}_{a_{4}})^{2},$ (A.1)

for
$i_{1},j_{1},k_{1}...=1,2;a_{2}=3,4;i_{2}=(i_{1},b_{2}),b_{2}=3,4;i_{3}=(i_{2},b_{3}),b_{3}=5,6;b_{4}=7,8$
etc.; and coordinates
$x^{i_{1}}=(x^{1},x^{2});u^{i_{2}}=x^{i_{2}}=(x^{i_{1}},y^{b_{2}}),y^{4}=t;\
^{\shortmid}u^{i_{3}}=\ ^{\shortmid}x^{i_{3}}=(x^{i_{2}},p_{b_{3}});\
^{\shortmid}u^{i_{4}}=(\ ^{\shortmid}x^{i_{3}},p_{b_{4}}),p_{8}=E$. The dual
bases in (A.1),

$\mathbf{e}^{a_{2}}=dy^{a_{2}}+N_{k_{1}}^{a_{2}}(x^{i_{1}},y^{2})dx^{k_{1}},\
^{\shortmid}\mathbf{e}_{a_{3}}=dp_{a_{3}}+\ ^{\shortmid}N_{a_{3}k_{2}}^{\
}(x^{i_{2}},p_{6})dx^{k_{2}},\ ^{\shortmid}\mathbf{e}_{a_{4}}=dp_{a_{4}}+\
^{\shortmid}N_{a_{4}k_{3}}^{\ }(\ ^{\shortmid}x^{i_{3}},E)d\
^{\shortmid}x^{k_{3}},$

are determined by respective N-connection coefficients,

$\displaystyle N_{k_{1}}^{3}$ $\displaystyle=$ $\displaystyle
w_{k_{1}}(x^{i_{1}},y^{3}),N_{k_{1}}^{4}=n_{k_{1}}(x^{i_{1}},y^{3});$ (A.2) $\
{}^{\shortmid}N_{5k_{2}}^{\ }$ $\displaystyle=$ $\displaystyle
n_{k_{2}}(x^{i_{2}},p_{5}),\ ^{\shortmid}N_{6k_{2}}^{\
}=w_{k_{2}}(x^{i_{2}},p_{5});\ ^{\shortmid}N_{7k_{3}}^{\
}=n_{k_{3}}(x^{i_{3}},E),\ ^{\shortmid}N_{8k_{3}}^{\
}=w_{k_{3}}(x^{i_{3}},E).$

We emphasize that above coefficients defined with respect to s-adapted frames
do not depend on $u^{4}=y^{4}=x^{4}=t,$ i.e. there is a Killing vector
$\partial_{t}$ on shell $s=2;$ do not depend on $\ {}^{\shortmid}u^{6}=p_{6},$
i.e. there is a Killing symmetry on $\ {}^{\shortmid}\partial^{6}$ on shell
$s=3;$ and do not depend on $\ {}^{\shortmid}u^{7}=p_{7},$ i.e. there is a
Killing symmetry on $\ {}^{\shortmid}\partial^{7}$ on shell $s=4.$ In similar
forms, changing parameterizations of coordinates and coefficients, we can
consider ansatz with Killing symmetry on $\partial_{3}$ instead of
$\partial_{4};$ on $\ {}^{\shortmid}\partial^{5}$ instead of $\
{}^{\shortmid}\partial^{6};$ on $\ {}^{\shortmid}\partial^{8}$ instead of $\
{}^{\shortmid}\partial^{7}.$ Such parameterizations allow to decouple and
integrate in explicit form various classes of vacuum and non-vacuum modified
gravitational equations.

###  A.1 Off-diagonal quasi-stationary solutions with effective sources

Tedious computations provided in [13] (re-defined in real momentum coordinates
$\ {}^{\shortmid}u^{\alpha}$) prove that quasi-stationary solutions of the
nonassociative parametric vacuum gravitational equations (16) with effective
sources (35) are defined by respective s-metric (A.1) and N-connection (A.2)
coefficients:

$\displaystyle g_{1}(x^{i_{1}})$ $\displaystyle=$ $\displaystyle
g_{2}(x^{i_{1}})=e^{\psi(\hbar,\kappa;x^{k_{1}})},$ (A.3) $\displaystyle
g_{3}(x^{i_{1}},y^{3})$ $\displaystyle=$ $\displaystyle\frac{[\partial_{3}(\