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%  V.F.S. — Volo. Facio. Sum.   Editio LXXXVIII (Open-Gate)
%  Hierarchy of Fundamental Conclusions (English)
%  + Theological Corollaries (from lyapunov & index §11)
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\newcommand{\E}{\mathcal{E}_0}
\newcommand{\Lc}{\Lambda_c}
\newcommand{\Grec}{\mathcal{G}_{\mathrm{recepta}}}
\newcommand{\Op}{\Omega_P}
\newcommand{\Lvfs}{\mathcal{L}_{\mathrm{VFS}}}
\newcommand{\DA}{\mathcal{D}_A}
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\newcommand{\concl}[2]{\smallskip\noindent\textbf{#1.}\ \emph{#2}\par\nobreak\smallskip}
% scripture epigraph (kept short)
\newcommand{\scrip}[2]{\par\nopagebreak\smallskip{\small\itshape ``#1''\quad---\ #2}\par\smallskip}

\title{\textbf{V.F.S.} \;---\; \emph{Volo. Facio. Sum.}\\[2pt]
\large Hierarchy of Fundamental Conclusions \;\&\; Theological Corollaries\\[2pt]
\normalsize Editio LXXXVIII \;$\cdot$\; \emph{Hierarchia Conclusionum \& Corollaria Theologica} \;$\cdot$\; English}
\author{}
\date{}

\begin{document}
\maketitle
\vspace{-2.4em}

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\section*{Part I.\quad Hierarchy of Fundamental Conclusions}
\textit{Hierarchia Conclusionum} $\cdot$ \textit{Editio Integrata} (thirteen conclusions).

\concl{I. Being has no final attractor state}{the governing principle of the architecture.}
\emph{Pleroma Christi} is not a static point of completion. The very motion toward Fullness
expands the metric of Fullness itself, $\dot{\Op}=\alpha\,\lambda(t)$. Saturation never finally
completes; the horizon of attainable being recedes continuously; \emph{Theosis} is an endless
asymptotic expansion.

\concl{II. Being is a flow, not a state}{the primary reality is the rate, not the level.}
The principal reality is not $P$ but $S=dP/dt$. Existence is defined by the rate of filling
toward \emph{Pleroma Christi}; Being is a trajectory, not a fixed identity.

\concl{III. Salvation is a geometric transformation, not annihilation}{resistance is transmuted, not destroyed.}
The system does not destroy $\sigma$; it transmutes it into Sophia,
$\sigma(t)+\lambda(t)=C_0+\Grec(t)$ (closed limit $\sigma+\lambda=\text{const}$) --- the
\textbf{Law of Open Balance}: an exact symmetry at $\zeta_0=0$, a balance law \emph{with a source}
in the open generalisation.

\concl{IV. Sophia ($\lambda$) is free memory, not eternal accumulation}{a two-sided process, by essence.}
Sophia obeys a two-sided functional
\[ \frac{d\lambda}{dt}=\bigl(\delta\sqrt{VF+\varepsilon}-\gamma\bigr)\tanh(\kappa\sigma)
   +\zeta_0\,g_{\mathrm{eff}}\,e^{-\phi\sigma}\,\frac{1}{1+\chi\lambda_+}, \]
whose sign is set against the surgical threshold $\Lc=\gamma/\delta$: for
$\sqrt{VF+\varepsilon}\ge\Lc$ Sophia accumulates ($\sigma\!\to\!\lambda$); for
$\sqrt{VF+\varepsilon}<\Lc$ it may temporarily regress (Wisdom can become Resistance again, ready
for new transmutation). The two-sidedness is the essence, not a defect.

\concl{V. Katharsis is a phase transition of the geometry flow}{a passage, not an emotion.}
Katharsis is not emotional purification but a passage through the critical surface
$x=\sqrt{VF+\varepsilon}-\sigma=0$, where the old geometry flow loses stability, surgery ceases to
dominate, and a luminous asymptotic regime appears.
\smallskip\par\noindent\emph{Va. Critical slowing near the Cross.} At $x=0$ the system undergoes
critical slowing --- a classic mark of phase transitions. The curvature of the Filtrum Lucis is
maximal there, $\Phi''(0)=k/4$, yet the relaxation is slowest,
$\tau_{\mathrm{relax}}^{-1}\sim|\omega_{\mathrm{lin}}(x)|$ ($\omega_{\mathrm{lin}}$ the smallest
linearised eigenvalue). \emph{Metastable Cathartic Layer:} outer stagnation may conceal an inner
accumulation of Sophia.

\concl{VI. Hysteresis of Salvation}{the system remembers what it has overcome.}
The path of falling and the path of recovery are not symmetric. Once $\sigma$ is transmuted into
$\lambda$, the system cannot return to its former geometry --- the trajectory carries the trace of
what it has crossed, and that trace reshapes its future response. Beyond a passive record, lived
struggle (especially in the zone $x_0>0$ where synergy exceeds resistance) leaves an \emph{active}
path-memory of effort --- \emph{Memoria Hysteretica Salutis} --- which enters not as a new member
of the Pleroma but as a modification of the synergistic intensity itself. The system remembers not
only what it \emph{is}, but also what it has \emph{overcome}.

\concl{VII. Surgery is not the final geometry}{pain is a transitional operator.}
Surgery exists only while singularities do: $\tanh(\kappa\sigma)\to0$ as $\sigma\to0$. Salvation is
not an eternal operation; pain is not the final geometry.
\smallskip\par\noindent\emph{VIIa. After Katharsis $V$ and $F$ are freed, not abolished.} They
remain the fundamental engines of Being's trajectory, ceasing only to be instruments of struggle
against $\sigma$. Before: $\dot V=\mu(aF+c\lambda)-\rho V$ with Dolorosum
$\rho=\sigma(u/\Lc)e^{1-u/\Lc}$ dominant near the threshold; after, in Transformation $u>\Lc$:
$\dot V\approx\mu(aF+c\lambda)$ --- pure participation-driven dynamics. $\Syn$ does not replace
$V$ and $F$.

\concl{VIII. Light cannot be fully absorbed}{no absolute collapse.}
The Filtrum Lucis guarantees $\Phi(x)>0\ \forall x$ (since $\Phi(0)=\ln2/k>0$ and $\Phi$ is
monotone increasing). Even in the deepest darkness a minimal permeability of light remains.

\concl{IX. Grace does not replace will}{it initiates, it does not substitute.}
The initial impulse decays exponentially, $\mathcal{P}(t)=\tau G_{\max}e^{-\tau t}\to0$. Grace
initiates the trajectory but does not move the system in the person's stead; once it fades, the
dynamics rest on $V,F,\lambda$ --- on human participation. (In the open gate this background grace
re-enters continuously as $I_{\mathrm{gate}}$; cf.\ Conclusion of the Theological Corollaries.)

\concl{X. Will and Action are fundamentally symmetric}{imbalance is structurally penalised.}
Mutual nourishment $a$ enters symmetrically ($V\!\leftrightarrow\!F$). Exactly,
$\dot\Delta=-\mu(t)\,a\,\Delta$ with $\Delta=V-F$; balance comes through the \emph{symmetry} of
nourishment, not the intensity of synergy. With Epektasis $\int_0^\infty\mu\,ds=+\infty\Rightarrow
\Delta_\infty=0$. Will without action degenerates; action without will exhausts itself.
Convergence is slow and polynomial, $\Delta(t)\sim\Delta_0/t$ (universal exponent $\beta=1$).

\concl{XI. Man is never fully cut off}{hope is structurally inscribed.}
\emph{Manere}: $\varepsilon>0$ guarantees a nonzero minimum of connection --- no absolute
ontological severance. At total exhaustion ($V=F=0$) the synergy does not vanish algebraically,
$\sqrt{0\cdot0+\varepsilon}=\sqrt{\varepsilon}>0$. Whether the trajectory recovers from this
boundary depends on $\lambda,\sigma$ and the regime (Lemma on the domain, \S6): hope is inscribed
in the model, though its realisation lies beyond its guarantees.

\concl{XII. Being is an anti-collapse architecture}{the Imago-Dei core is indestructible.}
\emph{Imago Dei}: $\E>0$ is the indestructible invariant core, so $P(t)\ge\E$ always --- even at a
full collapse of $V,F,\sigma,\lambda$ the algebraic core of the Pleroma persists. Algebraic
collapse of the Pleroma is impossible; ontological collapse of the trajectory ($V,F\to0$) is
possible and exits the live domain $\mathcal{D}_{\mathrm{live}}$ (\S6), yet $\E$ as Imago Dei
remains even then.

\concl{XIII. V.F.S.\ is a self-renormalizing geometric flow}{the structure of Theosis.}
The architecture is a conservation-driven renormalizing flow: surgery of singularities generates
Sophia ($\sigma\to\lambda$), Sophia expands the geometry of attainable being
($\dot{\Op}=\alpha\lambda$), and the moving boundary of Fullness continuously renormalizes the
trajectory. Transmuted resistance enlarges the space of participation; the enlarged space permits
new participation; new participation breeds new challenges, again transmuted into new Sophia. The
cycle is endless --- this \emph{is} the structure of Theosis.
\smallskip\par\noindent\emph{The Paschal Triad as the crown of the hierarchy.} The three zeros of
the Filtrum Lucis derivatives resonate with the full Paschal cycle: $\Phi^{(4)}(x_-)=0$ --- Good
Friday (entry into death); $\Phi'''(0)=0$ --- Holy Saturday, the Cross, Katharsis;
$\Phi^{(4)}(x_+)=0$ --- Resurrection Sunday, Anastasis. By the symmetry $x_-+x_+=0$ the Cross is the
geometric centre of the triad. The three-day repose in the tomb (Matt 12:40) is reflected as a
\emph{differential icon}, not a chronological formula: invariant is the $x$-interval
$|x_+-x_-|=\tfrac{2}{k}\ln(2+\sqrt3)$ (depending on $k$ alone), while the temporal duration depends
on the dynamics (\S8).
\scrip{Christ is risen from the dead, the firstfruits of them that slept.}{1 Corinthians 15:20}

\concl{Closing note}{}
\!\!The model is stable within its domain of correctness (Lemma \S6) and yields philosophically
valid results in all its asymptotic regimes: \emph{Pleroma Christi}, the lukewarm
(\emph{chliaros}), Collapse, and the Dark Night of the Soul.

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\section*{Part II.\quad Theological Fruit of the Lyapunov Stability (index \S11)}
The Lyapunov analysis of the active domain bears a theological fruit --- concise and honest, not
an automatic victory but a \emph{stable dynamics of transfiguration in the synergistic domain}:
\begin{quote}
grace does not destroy freedom; freedom does not save itself; and resistance does not have the
last word.
\end{quote}
In the active synergistic domain will and action come into harmony, resistance fades
asymptotically ($\sigma\to0$), and the resistance one has lived through can be transmuted into
Sophia \emph{without destroying personal continuity}. Compactly,
\[ \text{explicit admissibility}\Rightarrow\DA\text{ positively invariant}
   \Rightarrow\dot{\Lvfs}\le C-C_1\Lvfs\Rightarrow\sigma(t)\to0,\ D_\Delta(t)\to0. \]
Thus V.F.S.\ describes neither mechanical salvation nor self-salvation, but the stability of the
synergistic path in a domain where grace, will, action and purification already stand in the
right dynamic order.

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\section*{Part III.\quad Theological Corollaries (lyapunov \S12)}
\emph{Corollaria Theologica.} The functional must not be read as a proof of automatic or global
salvation from an arbitrary state; its content is the internal dissipative structure of the
positively-invariant active domain.

\begin{enumerate}[leftmargin=1.8em,itemsep=4pt]

\item \textbf{Resistance is not a final ontological reality.} In the active domain $\sigma(t)\to0$
--- supporting \emph{privatio boni}: evil/sin/resistance has no eternal substantial standing of
its own. It is real existentially (it shapes the dynamics) but not final ontologically: real as a
disease, not final as being.
\scrip{And the light shineth in darkness; and the darkness comprehended it not.}{John 1:5}

\item \textbf{Will and action must converge.} The imbalance $D_\Delta=\tfrac12(h_1-h_2)^2\to0$,
i.e.\ $h_1-h_2\to0$ (\emph{volo}$\leftrightarrow$\emph{facio}). Holiness is neither pure inner
intention nor mechanical moral activism, but the harmonisation of willing and embodied action;
will without action and action without will are unstable.
\scrip{Not everyone who says `Lord, Lord' will enter, but he who does the Father's will.}{cf.\ Matthew 7:21}

\item \textbf{Grace does not abolish structure but stabilises it.}
$\text{gratia}+\text{voluntas}+\text{factum}\Rightarrow\text{stabilitas transformationis}$.
Grace is a \emph{bounded source} (it raises $C$) while the dissipation coefficient $C_1$ is left
untouched --- the structure is not cancelled but precisely stabilised.
\scrip{Without me ye can do nothing.}{John 15:5}

\item \textbf{Spiritual life has a living domain.} The result is domain-based, not global:
$x(0)\in\DA\Rightarrow x(t)\in\DA\ \forall t\ge0$. Not every state purifies automatically; but once
the system enters such a domain, its internal dynamics already point toward transfiguration.
\scrip{The kingdom of God is within you.}{Luke 17:21}

\item \textbf{Manere does not replace living synergy.} Under $r_*^2>\varepsilon/\Omega_{\min}^2$,
the regulariser $\varepsilon$ (Manere) is the minimal holding of being --- but stability cannot
rest on that residue alone; real synergy $VF>0$ is required. Survival is not transfiguration.
\scrip{Thou hast a name that thou livest, and art dead.}{Revelation 3:1}

\item \textbf{Sophia is not an elitist precondition.} The proof needs only $q\ge0$ ($q$-free), not
an artificial $q\ge q_0>0$. Sophia may be the \emph{fruit} of transmuting resistance, not a
precondition of access; the concave Mensura kernel favours many small attempts over one grand
gesture --- the advantage of the poor.
\scrip{Those who are well need no physician, but the sick; I came to call sinners.}{cf.\ Mark 2:17}

\item \textbf{Resistance can be transmuted into Sophia.} In the closed limit
($\sigma+\lambda=\text{const}$), if $\sigma\to0$ resistance is not merely removed --- it is
transmuted, $\text{resistance}\to\text{Sophia}$. \emph{(Open generalisation: the pair is balanced
not conservative, $\sigma+\lambda=C_0+\Grec(t)$; the Pearl is transmuted resistance \emph{plus}
received gift, not a zero sum.)} Struggle, falls and trauma can become the material of wisdom.
\scrip{The stone the builders rejected has become the cornerstone.}{cf.\ Matthew 21:42}

\item \textbf{Non-Zeno has spiritual meaning.} The dwell-time bound $\Delta t_j\ge\Delta t_*>0$:
the hybrid system makes no infinitely many transitions in finite time. Purification keeps rhythm,
duration and sequence, preserving personal continuity --- it is not a chaotic infinite crisis.
\scrip{In your patience possess ye your souls.}{Luke 21:19}

\item \textbf{Pleroma is not reached by a violent leap.} The decay $\sigma(t)\to0$ is asymptotic,
not finite-time zeroing --- consistent with \emph{epektasis}: the motion toward fullness is real,
stable and directed, but not reducible to a technical completion button. Epektasis acts as a
universal regulator (dilutes the source, holds the $q$-ceiling, pushes back the death-wall).
\scrip{First the blade, then the ear, after that the full grain in the ear.}{cf.\ Mark 4:28}

\item \textbf{Grace is one, but acts in two modes.} All grace is a single background $\zeta_0$
entering through the receptivity gate. \emph{Modus I} (in life): a continuous inflow with a
guaranteed regime $e^{-\tau t}$ and a synergistic regime $H_K/(H_s+H_K)$. \emph{Modus II} (in
death): where the gate has collapsed, the same grace acts by a discrete re-creation $\mathfrak{R}$,
participating in the archetype $\mathfrak{R}_{\mathrm{Christi}}$ --- unmerited, precisely because
receptivity has fallen to zero. The modes are complementary: where one fades, the other engages.
\end{enumerate}

\bigskip
\hrule
\smallskip
\noindent\footnotesize\emph{V.F.S. $\cdot$ Volo. Facio. Sum. $\cdot$ Editio LXXXVIII $\cdot$ Open-Gate Sophia.
Part~I translated from the \texttt{index} section \emph{Hierarchy of Fundamental Conclusions};
Part~II from \texttt{index} \S11 (\emph{Lyapunov Stability of the Active Domain});
Part~III from \texttt{lyapunov} \S12 (\emph{Theological Corollaries}). Scripture epigraphs as in
the source.}
\end{document}
