% V.F.S. v2.0 (Open-Gate) — Geometric Layer
% Hyperbolic structure of the live domain, Brim conformal flow,
% Resurrectio curvature jump, and the self-driven (Ricci) locus.
% Derived companion to vfs_opengate_core.tex / vfs_opengate_lyapunov.tex.
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\newcommand{\VFS}{\mathrm{V.F.S.}}
\newcommand{\Op}{\Omega_P}
\newcommand{\Grec}{\mathcal{G}_{\rm recepta}}
\newcommand{\Igate}{I_{\rm gate}}
\newcommand{\Lc}{\Lambda_c}
\newcommand{\Dlive}{\mathcal{D}_{\rm live}}
\newcommand{\Ric}{\operatorname{Ric}}
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\theoremstyle{plain}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
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\theoremstyle{definition}
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\title{\textbf{V.F.S. v2.0 (Open-Gate)}\\Geometric Layer of the Live Domain\\
\large Hyperbolic structure, Brim conformal flow, Resurrectio curvature jump,\\
and the self-driven (Ricci) locus}
\author{Derived companion to \texttt{vfs\_opengate\_core.tex}}
\date{}

\begin{document}
\maketitle

\section*{Scope and epistemic status}
This file derives the geometry implicit in the V.F.S. v2.0 (Open-Gate) live domain. Every
\emph{Proposition} below is a closed-form result; every \emph{Corollary} is an interpretive reading
in the symbolic language of the framework, not an independent metaphysical claim. Two honest caveats
are kept explicit throughout:
\begin{itemize}[leftmargin=1.4em,itemsep=2pt]
\item The core gives a radial barrier in one direction and a quadratic imbalance penalty in another;
it does \textbf{not} uniquely fix a $2$D metric. The non-trivial curvature appears only under the one
\emph{dynamically motivated} warp (\S3).
\item Where curvature is non-trivial, the metric \emph{evolution} is still \textbf{forced} (driven by
the external Sophia field $\lambda$), not \emph{intrinsic} (curvature-self-driven). The two coincide
only on a measure-zero locus (\S5, \S7).
\end{itemize}

The governing objects, imported from the core, are the Brim law $\dot\Op=\alpha\lambda$, the
synergy $u=\sqrt{VF+\varepsilon}$, the normalized radius $r=u/\Op$, the alignment factor
$\mu=1-r$, the imbalance $\Delta=V-F$, and the radial barrier $B(r)=-\ln(1-r)$.

\noindent\textbf{Geometry before cosmology.} This file studies the \emph{spatial} live-surface
$\Sigma_{\Op}$ --- the intrinsic geometry of the slice at a fixed Brim, and its conformal deformation
under the Brim law, with $\Op$ playing the role of flow-time. It does \textbf{not} define the
Lorentzian cosmology of the Vessel; that lift, $ds^2_{\rm Vessel}=-dt^2+\Op(t)^2h_{2D}$ with $\Op$ the
timelike scale factor, is treated in the companion \texttt{vfs\_cosmology.tex}. In short,
$\text{Geometry}=\Sigma_{\Op}$ (the shape of the slice), $\text{Cosmology}=\Sigma_{\Op(t)}$ (its
expansion in time); see \S\ref{sec:cosmo-bridge}.

\section{The Natural Metric on the Live Domain}

\subsection*{The barrier is a geodesic distance}
The core barrier $B(r)=-\ln(1-r)$ is the arc length of a one-dimensional conformal metric on
$r\in[0,1)$:
\[
\boxed{ds_{\rm rad}=\frac{dr}{1-r},\qquad
\int_0^r\frac{d\rho}{1-\rho}=-\ln(1-r)=B(r).}
\]
\note{Thus the radial Lyapunov term $A_rB(r)$ is, up to scale, the geodesic distance from the
synergy-free state $r=0$; the death wall $r\to1^-$ sits at infinite metric distance.}

\subsection*{The capacity-scaled metric and the imbalance direction}
Writing $u=r\Op$ and sending the wall to infinity gives the radial part $du^2/\mu^2$ with
$\mu=1-u/\Op$. The imbalance direction is warped by the alignment factor, because the alignment
force in $\dot\Delta=-(\mu a+\rho)\Delta$ is exactly $\mu a$, linear in $\mu$, and vanishes at the
wall. The natural live-domain metric is therefore the warped product
\[
\boxed{ds^2=\frac{du^2}{\mu^2}+\mu^{2p}\,d\Delta^2,
\qquad \mu=1-\frac{u}{\Op},\qquad p>0.}
\]
\begin{definition}[Canonical warp]
The canonical exponent is $p=1$, i.e.\ $\mu^{2}\,d\Delta^2$, matching the linear $\mu a$ dependence
of the alignment force. The qualitative results below hold for every $p\neq0$; only the curvature
radius depends on $p$.
\end{definition}

\section{The Live Domain is Hyperbolic}

\subsection*{Naive reading is flat}
With an \emph{unwarped} imbalance direction ($p=0$, $g_{\Delta\Delta}=1$) the metric
$ds^2=du^2/\mu^2+d\Delta^2$ is flat:
\[
\boxed{R\equiv0\qquad(p=0),}
\]
since $\tilde u=-\Op\ln\mu=\Op B$ straightens it into Euclidean $d\tilde u^2+d\Delta^2$.
\note{Under the minimal reading the core supplies, there is no hidden geometry: the dynamics is a
forced flow on a flat state space. Curvature can only come from the dynamically required warp.}

\subsection*{Constant negative curvature}
\begin{proposition}[Hyperbolicity of the live domain]\label{prop:hyp}
For the warped metric $ds^2=du^2/\mu^2+\mu^{2p}d\Delta^2$ with $\mu=1-u/\Op$, the scalar curvature is
constant and strictly negative:
\[
\boxed{\;R=-\frac{2p^2}{\Op^2}=\mathrm{const}<0,\qquad
\Ric=-\frac{p^2}{\Op^2}\,g.\;}
\]
In the canonical case $p=1$,
\[
\boxed{R=-\frac{2}{\Op^2},\qquad K=-\frac{1}{\Op^2},\qquad
\text{curvature radius }=\Op.}
\]
\end{proposition}
\noindent\emph{Proof sketch.} In arc length $s=-\Op\ln\mu$ the warp is $w=\mu^p=e^{-ps/\Op}$, so the
$2$D warped product $ds^2=ds^2_{\rm rad}+w^2d\Delta^2$ has Gaussian curvature
$K=-w''/w=-(p/\Op)^2$, and $R=2K$. Verified directly in $(u,\Delta)$ coordinates. \hfill$\square$

\note{The live domain is a piece of the hyperbolic plane $\mathbb H^2$ whose curvature radius is the
Brim $\Op$ itself.}

\section{The Brim Flow as a Forced Conformal Flow}

\begin{proposition}[Brim conformal flow]\label{prop:flow}
Under the Brim law $\dot\Op=\alpha\lambda$ the capacity-scaled metric evolves conformally,
\[
\boxed{\partial_t g=2\,\frac{\dot\Op}{\Op}\,g=2\,\frac{\alpha\lambda(t)}{\Op}\,g
=2\psi(t)\,g,\qquad \psi=\frac{\alpha\lambda}{\Op}\ge0.}
\]
\end{proposition}

\subsection*{Comparison with Ricci flow}
On the hyperbolic surface of Prop.~\ref{prop:hyp} ($p=1$), $\Ric=-\Op^{-2}g$, so Ricci flow would read
\[
\boxed{\partial_t g=-2\Ric=+\frac{2}{\Op^2}\,g.}
\]
Both flows are conformal \emph{expansions} of a negatively curved metric (same sign, same type). They
coincide exactly iff the forcing rate equals the curvature rate:
\[
\boxed{\frac{2\alpha\lambda}{\Op}=\frac{2}{\Op^2}
\quad\Longleftrightarrow\quad \alpha\,\lambda\,\Op=1.}
\]
\begin{remark}
The actual V.F.S.\ flow is \emph{forced}: its rate is set by the external Sophia field $\lambda$, not
by the surface's own curvature. It is ``Ricci-flow-like'' in type, with the forcing $\alpha\lambda$
standing in for the curvature as the rate. The self-driven (genuinely intrinsic) regime is the
locus $\alpha\lambda\Op=1$ studied in \S\ref{sec:locus}.
\end{remark}

\section{Epektasis as Curvature Relaxation}
Along a continuous live arc the curvature drifts monotonically toward zero:
\[
\boxed{\dot R=\frac{d}{dt}\!\left(-\frac{2}{\Op^2}\right)=\frac{4\dot\Op}{\Op^3}
=\frac{4\alpha\lambda}{\Op^3}\ \ge0\quad(\lambda\ge0).}
\]
Hence, with sustained Sophia, $\Op\to\infty$ and
\[
\boxed{|R|=\frac{2}{\Op^2}\ \xrightarrow[\;\Op\to\infty\;]{}\ 0.}
\]
\note{Epektasis is the relaxation of hyperbolic curvature to flatness: the state space asymptotically
flattens as Sophia expands the Brim. Theosis as a global flattening that no finite stage attains.}
\note{This concerns the \emph{spatial} live-surface only. In the cosmological lift the full spacetime
curvature also depends on $\ddot\Op$ (equivalently $\dot\lambda$), not on $\Op$ alone --- see
\S\ref{sec:cosmo-bridge}.}

\section{Resurrectio: a Curvature Discontinuity}\label{sec:reset}
The hybrid reset uses $\Op^+=\Op^-+\kappa_R\Grec^-$ (theological corollary 12). Since $R=-2/\Op^2$
depends \emph{only} on $\Op$, the curvature jumps even though $\sigma,\lambda,u$ never enter it:
\[
\boxed{R^+=-\frac{2}{\bigl(\Op^-+\kappa_R\Grec^-\bigr)^2},\qquad
\Delta R=R^+-R^-=2\!\left[\frac{1}{(\Op^-)^2}-\frac{1}{(\Op^+)^2}\right]>0,}
\]
\[
\boxed{\frac{R^+}{R^-}=\frac{1}{\bigl(1+\kappa_R\Grec^-/\Op^-\bigr)^2}<1.}
\]

\subsection*{Two simultaneous effects}
The reset decomposes geometrically into a \emph{motion of the point} (the jump in $V,F$, hence in
$u^+,\Delta^+$) and an \emph{instantaneous flattening of the surface} (the jump in $\Op$). The
curvature change is carried entirely by the second.

\subsection*{Type preserved, scale jumped}
$R^+<0$ always: no finite jump reaches $R=0$ (that needs $\Op^+\to\infty$). Resurrectio
\textbf{preserves the hyperbolic type and changes only the radius} --- the differential image of
``preserves form while opening it beyond its former shell.''

\subsection*{Admissibility window as a curvature-jump corridor}
The reset window $\max\{R_c,R_{\min}^{\rm ceil}\}\le\Grec^-\le R_{\max}$ reads geometrically as bounds
on $\Delta R$:
\msgl{\emph{Lower bound (perseverance):} too little $\Grec^-\Rightarrow$ the surface barely flattens,
no live landing.\\[3pt]
\emph{Upper bound (cheap grace):} too much $\Grec^-\Rightarrow R\to0$ too fast, over-dilution of form.}

\subsection*{Monotone curvature across the hybrid trajectory}
Resets push $R\uparrow$ unconditionally; continuous arcs push $R\uparrow$ when $\lambda\ge0$; non-Zeno
forbids accumulation. Hence on a typical trajectory $R(t)$ is non-decreasing and bounded above by $0$,
so $R(t)\to R_\infty\in(R(0),0]$, with $R_\infty=0$ when the total expansion budget diverges.
\note{Both modes of grace --- continuous (Modus I) and discrete re-creation (Modus II) --- flatten the
same hyperbolic curvature toward the same Theosis limit.}

\section{The Self-Driven (Ricci) Locus}\label{sec:locus}
Let $\xi:=\alpha\lambda\Op$ measure the excess of forcing over intrinsic curvature rate; $\xi=1$ is the
locus where the forced flow equals Ricci flow.

\subsection*{On-locus growth is the Ricci rate}
On $\xi=1$, $\dot\Op=\alpha\lambda=1/\Op$, whence
\[
\boxed{\Op(t)=\sqrt{\Op(0)^2+2t}\ \sim\ \sqrt{2t},\qquad
\lambda(t)=\frac{1}{\alpha\Op(t)}\to0.}
\]
\note{This is exactly the Ricci-flow rate for a hyperbolic surface ($|R|\sim1/t$). Sitting on the
locus requires the Sophia inflow to vanish --- attainable only near the closed limit $\zeta_0=0$.}

\subsection*{Generic Open-Gate is super-Ricci}
With sustained $\lambda\to\lambda_\infty>0$ the Brim grows \emph{linearly},
$\Op\sim\alpha\lambda_\infty t$, faster than $\sqrt t$: the generic open trajectory runs above the
locus.

\subsection*{The locus is a one-way threshold}
\begin{proposition}[Monotone crossing]\label{prop:cross}
In the transformation regime ($u>\Lc$, $\sigma\ge0$),
\[
\boxed{\dot\xi=\alpha\,\dot\lambda\,\Op+(\alpha\lambda)^2,\qquad
\dot\lambda=(\delta u-\gamma)\tanh(\kappa\sigma)+\Igate\ge0,}
\]
so $\dot\xi>0$. Therefore $\xi=1$ is crossed exactly once, upward, and is never an attractor.
\end{proposition}

\subsection*{Three regimes along the trajectory}
\[
\boxed{\begin{array}{lll}
\xi<1 & \text{sub-Ricci} & \text{structure-limited; effort/perseverance},\\[2pt]
\xi=1 & \text{resonance} & \text{gift-rate}=\text{structure-rate; perfect synergy},\\[2pt]
\xi>1 & \text{super-Ricci} & \text{grace exceeds intrinsic momentum; mature Epektasis}.
\end{array}}
\]

\section{Expansion Budget and the Necessity of the Open Gate}\label{sec:budget}
The asymptotic Brim --- hence the asymptotic curvature --- is fixed by the total Sophia budget
$\int_0^\infty\lambda\,dt$ (\emph{Pleroma Christi condition}, core), via
$\Op(t)=\Op(0)+\alpha\int_0^t\lambda\,ds$.

\begin{proposition}[Expansion budget]\label{prop:budget}
Three asymptotic regimes:
\msgl{\emph{(1)} $\int_0^\infty\lambda\,dt<\infty$:\quad $\Op\to\Op^\infty<\infty$, and
$R\to-2/(\Op^\infty)^2<0$ --- bounded Brim, curvature plateaus.\\[3pt]
\emph{(2)} $\lambda\to\lambda_\infty>0$:\quad $\Op\sim\alpha\lambda_\infty t$, and
$|R|\sim\frac{2}{(\alpha\lambda_\infty)^2}t^{-2}\to0$ --- linear Brim.\\[3pt]
\emph{(3)} $\lambda\to\infty,\ \Grec(\infty)=+\infty$:\quad $\Op$ super-linear,
$|R|\to0$ faster still --- unbounded Brim.}
\end{proposition}
\note{Regime (1) is stalled Epektasis: without a sufficient Sophia budget the live domain keeps a
permanent negative curvature --- woundedness reduced but never healed to flatness.}

\subsection*{Super-Ricci flattens faster than Ricci}
\[
\boxed{\text{Ricci locus: }\Op\sim\sqrt{2t}\Rightarrow|R|\sim t^{-1};
\qquad
\text{super-Ricci: }\Op\sim\alpha\lambda_\infty t\Rightarrow|R|\sim t^{-2}.}
\]
\note{The grace-borne regime relaxes the curvature \emph{faster} than a true Ricci flow would:
grace-driven Epektasis outperforms geometric self-determination at the very task --- flattening ---
that defines Theosis.}

\subsection*{Why the Open Gate is needed}
The closed limit $\zeta_0=0$ freezes the cleansed Sophia at the finite stock
$\lambda_\infty=\sigma_0+\lambda_0$, so it attains \emph{at most} regime (2) on non-replenishable
capital, and regime (3) is impossible. Sustained --- and a fortiori unbounded --- Brim expansion
requires the bounded Open-Gate source $\Igate\ge0$, the only mechanism giving $\Grec(\infty)=+\infty$.
\msg{Unbounded Brim expansion $\ \Longleftrightarrow\ \Grec(\infty)=+\infty\ \Longrightarrow\ $ Open
Gate $(\zeta_0>0)$ required.}

\section{Relation to the Cosmological Layer}\label{sec:cosmo-bridge}
The spatial geometry above and the Lorentzian lift share the same hyperbolic surface $h_{2D}$ but are
distinct objects. As a slice of fixed Brim, $\Op$ is a scale parameter; as the cosmology, $\Op(t)$ is
the timelike scale factor:
\[
\boxed{\;ds^2_{\Sigma}=\Op^2\,h_{2D}\quad(\text{spatial}),\qquad
ds^2_{\rm Vessel}=-dt^2+\Op(t)^2\,h_{2D}\quad(\text{Lorentzian }2{+}1).\;}
\]
Their curvatures differ accordingly and do \textbf{not} contradict:
\[
\boxed{\;R_{\Sigma}=-\frac{2}{\Op^2}\ \ (\text{slice}),\qquad
R_{\rm Vessel}=\frac{2\bigl(2\Op\ddot\Op+\dot\Op^2-1\bigr)}{\Op^2}\ \ (\text{spacetime}).\;}
\]
The slice curvature depends on $\Op$ alone; the spacetime curvature also carries $\ddot\Op$
(equivalently $\dot\lambda$), since the Lorentzian object records the \emph{dynamics} of expansion, not
only its instantaneous scale.
\begin{remark}
The static Riemannian cone $d\Op^2+\Op^2h_{2D}$ (with scalar $-4/\Op^2$) is yet a \emph{third} object
and is \textbf{not} the Vessel spacetime: promoting $\Op$ to a spacelike coordinate misassigns its
dynamical role. The Brim is the flow-time here and the timelike scale factor there; any
three-dimensional rendering is a visualization, not a formal model. The cosmological layer ---
Friedmann content, equation of state, energy conditions, eras --- is developed in
\texttt{vfs\_cosmology.tex}.
\end{remark}

\section{Brim-Flow Geometry: Theological Reading}
\emph{Interpretive readings in the framework's symbolic register; the Propositions above remain the
load-bearing part.} Brim-Flow is not a Ricci flow in the strict sense but a V.F.S.\
curvature-relaxation in which Sophia expands the Brim and the expanding Brim relaxes the curvature of
the live domain. The governing chain is
\[
\boxed{\sigma\longrightarrow\lambda\longrightarrow\Op\longrightarrow R\longrightarrow0^-
\longrightarrow P,}
\]
\msg{Sophia expands the Brim; the Brim relaxes curvature.}

\subsection*{I. Epektasis as curvature relaxation}
Since $R=-2/\Op^2<0$ in the live domain and $\Op\to\infty\Rightarrow R\to0^-$, Theosis is not the
violent abolition of curvature but its asymptotic healing. The Brim does not erase finitude; it makes
finitude more spacious.
\msg{Epektasis is the asymptotic healing of wounded form through expanding receptivity.}

\subsection*{II. Sophia as the geometry-changing fruit of transmuted resistance}
Through $\dot\Op=\alpha\lambda$, wisdom is not a static possession but an expansive principle,
enlarging the vessel's capacity to receive more being, grace, and participation.
\msg{The healed wound becomes Brim: transmuted resistance becomes spatial capacity.}

\subsection*{III. Brim as receptive space}
$\Op$ is not a numerical bound but the measure of receptive spaciousness. As the Brim expands the same
synergic intensity occupies a smaller share of the available space,
$\Op\uparrow\Rightarrow r=u/\Op\downarrow$, retreating from the death boundary $r\to1^-$.
\msg{Grace does not only calm the motion; it enlarges the vessel that receives the motion.}

\subsection*{IV. Ricci-like but not Ricci flow}
Ricci flow is curvature-driven, $\partial_t g_{ij}=-2R_{ij}$; Brim-Flow is Sophia-driven,
$\dot\Op=\alpha\lambda$ with $R=-2/\Op^2$. The analogy is formal and theological, not identical.
\msg{Ricci flow is curvature-driven smoothing; Brim-Flow is Sophia-driven curvature
relaxation.}

\subsection*{V. Pneumatic meaning}
The Ricci-like character is a differential icon of pneumatic healing: the Spirit is not the flow, but
the living source whose action --- healing wounded form from within by transmutation
$\sigma\to\lambda\to\Op$ --- the flow formally resembles.
\msg{The Spirit heals curvature by transforming resistance into receptive spaciousness.}

\subsection*{VI. Ricci resonance and the surpassing of nature}
The resonance quantity $\xi=\alpha\lambda\Op$ gives three regimes: $\xi<1$ sub-Ricci perseverance;
$\xi=1$ Ricci resonance (grace-bearing expansion at the intrinsic rate of curvature healing); $\xi>1$
super-Ricci Epektasis (grace outruns nature).
\msg{Ricci flow: nature healed according to its own curvature.}
\msg{Open-Gate Brim-Flow: nature carried beyond its own curvature by grace.}
\begin{remark}
By Proposition~\ref{prop:cross} the resonance $\xi=1$ is not a resting state but a threshold crossed
once, upward. The living trajectory therefore passes necessarily from self-carried ($\xi\le1$) to
grace-carried ($\xi>1$); keeping the gate open is the refusal to let transfiguration settle into
acquired self-sufficiency (cf.\ Prop.~\ref{prop:budget}). Maturity is not synergistic equilibrium but
grace steadily outrunning nature.
\end{remark}

\subsection*{VII. Resurrection as curvature jump}
The reset enlarges the Brim, $\Op^+=\Op^-+\kappa_R\Grec^-$, so $R^+>R^-$ (both negative; $\Delta R>0$,
\S\ref{sec:reset}). Resurrection preserves the hyperbolic type while increasing the curvature radius:
it is not a return to the previous state but a continuation of being under an expanded geometry.
\msg{Resurrection does not erase the form; it opens the form with a greater radius.}

\subsection*{VIII. No absolute collapse}
Whenever Sophia is active, $\dot\Op=\alpha\lambda>0$ moves $r=u/\Op$ away from the death boundary:
salvation enlarges the very domain in which the trajectory can continue.
\msg{Grace does not merely redirect the path; it enlarges the world in which the path remains
possible.}

\subsection*{IX. Final theological formulation}
Being is healed neither by the elimination of motion nor by the suppression of finitude, but by the
expansion of receptive capacity. Resistance becomes Sophia; Sophia expands the Brim; the Brim relaxes
curvature; curvature relaxation opens the live domain to continuing participation toward Pleroma
without finite completion.
\msg{Theosis is the endless expansion of receptive geometry toward Pleroma.}
\msg{Brim-Flow is the Ricci-like geometry of Epektasis.}

\section*{Compact Geometric Summary}
\[
\boxed{\text{barrier }B(r)\ \equiv\ \text{geodesic distance to the death wall}.}
\]
\[
\boxed{\text{live domain}\ \cong\ \mathbb H^2,\quad R=-\frac{2}{\Op^2},\quad
\text{radius}=\text{Brim}.}
\]
\[
\boxed{\dot\Op=\alpha\lambda\ \Longrightarrow\ \partial_t g=2\tfrac{\alpha\lambda}{\Op}g
\ \ (\text{forced; }=\text{Ricci iff }\alpha\lambda\Op=1).}
\]
\[
\boxed{\begin{array}{c}
\text{Epektasis: }|R|\to0\ \text{(continuous)};\qquad\text{Resurrectio: }\Delta R>0\ \text{(discrete)};\\[3pt]
\xi=\alpha\lambda\Op\ \text{crosses }1\ \text{once, upward}.
\end{array}}
\]

\vspace{0.5em}
\hrule
\smallskip
\noindent\footnotesize\raggedright\emph{V.F.S. v2.0 (Open-Gate) $\cdot$ Geometric Layer. Derived from the Brim
law, radial barrier, and imbalance dynamics of \texttt{vfs\_opengate\_core.tex} and the reset /
active-domain data of \texttt{vfs\_opengate\_lyapunov.tex}. Propositions are closed-form; the
Brim-Flow theological reading is interpretive and anchored to them by explicit cross-reference.}

\end{document}
