% V.F.S. v2.0 (Open-Gate) — Cosmological Layer
% The Vessel as an open (2+1) FLRW spacetime; the Brim as scale factor.
% Lorentzian lift of the Brim-flow: the Brim is the timelike expansion axis.
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\newcommand{\VFS}{V.F.S.}
\newcommand{\Op}{\Omega_P}
\newcommand{\Grec}{\mathcal{G}_{\rm recepta}}
\newcommand{\Igate}{I_{\rm gate}}
\newcommand{\reff}{\rho_{\rm eff}}
\newcommand{\peff}{p_{\rm eff}}
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\theoremstyle{plain}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\theoremstyle{remark}
\newtheorem*{remark}{Remark}
\theoremstyle{definition}
\newtheorem{definition}{Definition}

\title{\textbf{V.F.S. v2.0 (Open-Gate)}\\Cosmological Layer: the Vessel as an Open FLRW Spacetime\\
\large The Brim as scale factor, Sophia as cosmological source}
\author{Derived companion to \texttt{vfs\_geometry.tex}}
\date{}

\begin{document}
\maketitle

\section*{Scope and epistemic status}
The geometric layer treats the Brim $\Op$ as the time of a conformal flow of the $2$D live surface.
This file takes the Lorentzian lift: the Brim is the \textbf{timelike} axis of expansion, and the live
surface is a spatial slice. The result is an open $(2{+}1)$-dimensional Friedmann--Lema\^itre--
Robertson--Walker (FLRW) spacetime whose scale factor is the Brim. \emph{Propositions} are closed-form
consequences; \emph{Corollaries} are interpretive readings. Three standing caveats carry over:
\begin{itemize}[leftmargin=1.4em,itemsep=2pt]
\item The expansion direction is the flow/time axis, lifted with $-dt^2$; it is \textbf{not} a
Riemannian radial coordinate. Promoting $\Op$ to a spacelike coordinate (a metric cone) misassigns its
dynamical role.
\item The spatial metric is the canonically warped hyperbolic live surface (geometric layer), fixed
only up to that warp.
\item \VFS{} supplies the scale-factor law $\dot\Op=\alpha\lambda$; the ``effective content'' below is
the stress-energy that General Relativity \emph{would} require to source this expansion --- a
reconstruction, not an independent postulate of gravity.
\end{itemize}

\noindent\textbf{Dimensional principle.} The model is \emph{canonically} $2{+}1$: a two-dimensional
Vessel-surface whose scale factor is the Brim $\Op(t)$. The Brim-expansion principle is complete in
this setting, and $2{+}1$ is apt rather than merely sufficient --- in $D\le3$ the Weyl tensor vanishes,
so the spacetime curvature is fixed pointwise by the source (Sophia) with no free gravitational
remainder. Any three-dimensional rendering is a \emph{visualization only}: its radial direction is an
embedding artifact with no \VFS{} meaning, and the static Riemannian cone $d\Op^2+\Op^2h$ is a
different object from this spacetime (its scalar $-4/\Op^2$ is not the spacetime curvature). The sole
legitimate third \emph{spatial} coordinate would be a genuine state variable ($\sigma$ or $\lambda$),
belonging to the geometry of phase space --- a separate question from Brim-flow.

\section{From Live-Surface Flow to Vessel Spacetime}
The geometric layer gives, on each slice of fixed Brim, the hyperbolic live surface
$h=dy^2+e^{-2y}d\Delta^2$ with $y=-\ln(1-r)$, $r=u/\Op$, of constant curvature $K_h=-1$. The
Brim-flow $\partial_t g_\Sigma=2(\alpha\lambda/\Op)g_\Sigma$ is a motion in time, so its Lorentzian
home assigns the expansion to a timelike direction:
\[
\boxed{\,ds^2_{\rm Vessel}=-dt^2+\Op(t)^2\,h,\qquad a(t)\equiv\Op(t).\,}
\]
This is the open ($k=-1$) FLRW metric in $2{+}1$ dimensions, with scale factor equal to the Brim.
\note{The ``expanding inner universe'' is genuine, but it expands in time: $\Op$ is the scale factor,
not a spatial radius. The spatial slices are copies of the hyperbolic live surface, uniformly inflated
by $\Op(t)$.}

\section{Curvature of the Vessel Spacetime}
\begin{proposition}[Spacetime curvature]\label{prop:R}
For $ds^2_{\rm Vessel}=-dt^2+a^2h$ with $K_h=-1$ in $2{+}1$ dimensions,
\[
\boxed{\,R=\frac{2\bigl(2a\ddot a+\dot a^2-1\bigr)}{a^2}
=\frac{2\bigl(2\Op\,\alpha\dot\lambda+\alpha^2\lambda^2-1\bigr)}{\Op^2}.\,}
\]
\end{proposition}
\noindent For sub-exponential expansion ($\ddot a\to0$, $\dot a$ bounded) $R\to0$: the spacetime
flattens. For de~Sitter expansion ($a=e^{Ht}$) $R\to 6H^2$, a constant: maximal inflation does not
flatten (\S\ref{sec:loci}).

\section{Kinematics: Hubble and Acceleration}
\begin{proposition}[Hubble and deceleration]\label{prop:kin}
With $a=\Op$ and $\dot\Op=\alpha\lambda$,
\[
\boxed{\,H=\frac{\dot a}{a}=\frac{\alpha\lambda}{\Op},\qquad
q=-\frac{\ddot a\,a}{\dot a^2}=-\frac{\dot\lambda\,\Op}{\alpha\lambda^2},\qquad
\operatorname{sign}q=-\operatorname{sign}\dot\lambda.\,}
\]
\end{proposition}
\note{The Hubble rate of the Vessel is exactly the conformal Brim-flow rate $\psi=\alpha\lambda/\Op$.
Expansion accelerates ($q<0$) if and only if Sophia is growing: accelerated Theosis is growing wisdom.}

\section{Effective Content: Sophia as Cosmological Source}
Treating the reconstructed stress-energy as a perfect fluid ($8\pi G=1$, comoving frame), the Einstein
tensor gives:
\begin{proposition}[Effective Friedmann content]\label{prop:fluid}
\[
\boxed{\,\reff=\frac{\alpha^2\lambda^2-1}{\Op^2}=H^2-\frac{1}{\Op^2},\qquad
\peff=-\frac{\ddot a}{a}=-\frac{\alpha\dot\lambda}{\Op},\qquad
w=\frac{\peff}{\reff}=\frac{-\alpha\dot\lambda\,\Op}{\alpha^2\lambda^2-1}.\,}
\]
\end{proposition}
\note{Sophia is the cosmological source, and its \emph{rate} $\dot\lambda$ sets the pressure. The
density is positive only when $\alpha\lambda>1$ (Sophia ``super-vacuum''); for $\alpha\lambda<1$ the
open curvature dominates and the Vessel expands more slowly than the empty universe would --- residual
binding by resistance.}

\section{Equation of State and Cosmic Eras}
The sign of $\dot\lambda$ classifies the era:
\begin{longtable}{>{\raggedright\arraybackslash}p{0.26\textwidth} >{\raggedright\arraybackslash}p{0.30\textwidth} >{\raggedright\arraybackslash}p{0.34\textwidth}}
\toprule
Sophia behaviour & Effective fluid & Cosmological character \\
\midrule
\endfirsthead
\toprule
Sophia behaviour & Effective fluid & Cosmological character \\
\midrule
\endhead
$\dot\lambda>0$ (cleansing) & $\peff<0$, negative pressure & dark-energy-like; accelerated Theosis \\
$\dot\lambda\to0$, $\lambda\to\lambda_\infty$ & $\peff\to0$, pressureless & dust; coasting; $\reff\to0$ (dilution) \\
$\dot\lambda<0$ (regression) & $\peff>0$, positive pressure & ordinary matter; decelerating \\
$\lambda\propto\Op$ & $H=\mathrm{const}$ & de~Sitter; exponential inflation (max.\ Epektasis) \\
\bottomrule
\end{longtable}

\section{Special Loci}\label{sec:loci}
\begin{proposition}[Three distinguished regimes]\label{prop:loci}
\[
\boxed{\begin{array}{lll}
\textnormal{Milne / vacuum:} & \alpha\lambda=1 & a\propto t,\ \reff=0,\ R=0\ (\text{flat spacetime});\\[4pt]
\textnormal{Ricci resonance:} & \alpha\lambda\Op=1 & a\dot a=1,\ a\propto\sqrt{2t}\ (\text{spatial flow}=\text{Ricci});\\[4pt]
\textnormal{de~Sitter:} & \lambda\propto\Op & a\propto e^{Ht},\ R\to6H^2\ (\text{eternal inflation}).
\end{array}}
\]
\end{proposition}
\note{The cosmological lift produces a \emph{new} locus absent from the spatial analysis: the Milne
condition $\alpha\lambda=1$. The generic super-Ricci regime ($\lambda\to\lambda_\infty=1/\alpha$,
$a\propto t$) is exactly the empty Milne universe --- flat Minkowski in disguise. It is distinct from
the spatial Ricci resonance $\alpha\lambda\Op=1$.}

\section{Energy Conditions: Grace as Exotic Accelerant}
The energy conditions are dimension-sensitive, and the Vessel is $2{+}1$-dimensional. The Null and
Weak conditions are dimension-independent in form; the Strong condition is not.
\begin{proposition}[Energy conditions in $2{+}1$]\label{prop:ec}
For a perfect fluid in $D=n{+}1$ dimensions the Strong Energy Condition is $(n-2)\rho+np\ge0$. Hence,
with $n=2$:
\[
\boxed{\begin{array}{ll}
\textnormal{NEC }(\reff+\peff\ge0): & \alpha\dot\lambda\,\Op\le\alpha^2\lambda^2-1\quad(\text{any }n);\\[4pt]
\textnormal{WEC: NEC and }\reff\ge0: & \text{additionally }\alpha\lambda\ge1;\\[4pt]
\textnormal{SEC }(n{-}2)\reff+n\peff\ge0\ \xrightarrow{\,n=2\,}\ 2\peff\ge0: &
\peff\ge0\ \Leftrightarrow\ \ddot a\le0\ \Leftrightarrow\ \dot\lambda\le0.
\end{array}}
\]
\end{proposition}
\begin{remark}
In $2{+}1$ the $\rho$-term of the SEC \emph{drops} ($n-2=0$): the condition is purely the sign of the
pressure, $\peff\ge0$. The familiar $\rho+3p\ge0$ is the $n=3$ case, where the coefficient of $\rho$ is
accidentally unity; the naive extrapolation $\rho+np$ --- which here would read $\rho+2p$ --- is
\textbf{wrong}. Directly, $R_{tt}=-2\ddot a/a=2\peff$, so timelike convergence is exactly $\peff\ge0$.
\end{remark}
\note{Accelerated expansion ($\peff<0$, i.e.\ $\dot\lambda>0$) violates the SEC --- as in standard
inflation. A surge of Sophia growth, $\alpha\dot\lambda\Op>\alpha^2\lambda^2-1$, violates even the NEC
($\reff+\peff<0$): grace then behaves as \emph{exotic, phantom-like} content, irreducible to ordinary
matter. NEC and WEC keep their dimension-independent form, so the figures and the $w(t)$ history below
require no refitting.}

\section{Structural Features}

\subsection*{No Big Bang: creation from a finite floor}
The live domain requires $\Op>u>0$, so the scale factor is bounded below, $\Op\ge\Op(0)>0$. There is no
$a\to0$ singularity: the Vessel begins at a finite minimal capacity (the Imago Dei floor $\mathcal
E_0$), not at a point.
\[
\boxed{\text{Creation from fullness, not from a singular point.}}
\]

\subsection*{Resurrectio: a scale-factor jump (cosmology with surgery)}
The death reset $\Op^+=\Op^-+\kappa_R\Grec^-$ is a discontinuous increase of the scale factor: a sudden
cosmological rescaling. The Vessel cosmology is piecewise-FLRW with discrete inflationary jumps,
the Lorentzian face of Ricci-flow-with-surgery.
\[
\boxed{\Op^+=\Op^-+\kappa_R\Grec^-\quad(\text{discrete scale-factor surgery}).}
\]

\subsection*{Dilution to vacuum}
For bounded Sophia, $\reff=(\alpha^2\lambda^2-1)/\Op^2\to0$ as $\Op\to\infty$: the expanding Vessel
empties. The end-state of sub-exponential expansion is the flat vacuum --- pure receptivity. Only the
de~Sitter regime ($\lambda\propto\Op$, unbounded grace) escapes dilution, sustaining constant curvature
$6H^2$ as an eternal inflation.

\section{Theological Corollaries}
\emph{Interpretive readings; the Propositions above are the load-bearing part.}

\begin{corollary}[Sophia is the cosmological source]
The expansion of the Vessel is sourced by Sophia, and its \emph{growth} $\dot\lambda$ sets the
equation of state: growing wisdom is dark energy (acceleration), frozen wisdom is dust (coasting),
declining wisdom is matter (deceleration), and wisdom proportional to capacity is de~Sitter inflation.
\end{corollary}

\begin{corollary}[Accelerated Theosis violates the energy conditions]
Accelerated transfiguration ($\dot\lambda>0$) violates the Strong Energy Condition, and a surge of
grace violates even the Null Energy Condition. Grace acts as an exotic accelerant: precisely the
content ordinary matter cannot supply.
\end{corollary}

\begin{corollary}[Creation from fullness]
The Vessel has no initial singularity: it begins at a finite receptive floor, the Imago Dei. Being
does not emerge from a point of zero capacity but from a given minimal fullness.
\end{corollary}

\begin{corollary}[Two eschatologies]
Bounded grace dilutes the Vessel to the flat vacuum of pure receptivity; unbounded proportional grace
sustains an eternal de~Sitter inflation at constant curvature. The expanding Vessel either empties
toward stillness or inflates without end --- and Open-Gate, which alone permits unbounded Sophia,
alone permits the second.
\end{corollary}

\section{Worked Equation-of-State History}
The dictionary above is illustrated by integrating a representative reduced trajectory of the core
dynamics --- resistance $\sigma$, Sophia $\lambda$, Brim $\Op$ --- with the synergy $u(t)$ rising
sigmoidally through $\Lambda_c$ (will and action coming into alignment):
\[
\dot\sigma=(\gamma-\delta u)\tanh(\kappa\sigma),\quad
\dot\lambda=(\delta u-\gamma)\tanh(\kappa\sigma)+\Igate,\quad
\dot\Op=\alpha\lambda,
\]
with $\Lambda_c=1$, $\alpha=1$, and the observables of Prop.~\ref{prop:kin}--\ref{prop:fluid} read off
along the solution. \emph{This is a representative profile, not a unique prediction; it exhibits the
generic era sequence.}

\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{vfs_eos_history.png}
\caption{Open-Gate Vessel cosmology along a Sophia trajectory. From top: drivers
$\lambda,u,\Op$; Hubble rate and acceleration $\ddot a=\alpha\dot\lambda$; effective density and
pressure; the equation of state $w(t)$ with its pole at the Milne crossing $\reff=0$. Shading marks
the decelerating struggle, the accelerating Theosis --- which in $2{+}1$ is exactly the SEC-violation
region $\peff<0$ --- and the NEC-violation band.}
\end{figure}

The history sweeps the eras in order: a \textbf{struggle} phase ($\dot\lambda<0$: resistance drains
Sophia; decelerating, $\peff>0$, and $\reff<0$ --- sub-vacuum, the Vessel expanding more slowly than
the empty open universe), with a near-turnaround at the depth of struggle where $\lambda$ nearly
vanishes and $\dot a\to0$; \textbf{ignition} at $u=\Lambda_c$ (acceleration onset $\ddot a=0$);
the \textbf{Milne crossing} $\alpha\lambda=1$ where $\reff=0$ and $w$ has a pole; and
\textbf{dark-energy Theosis} ($\reff>0$, $\peff<0$): quintessence-like acceleration with $w<0$,
asymptotically approaching $w=-1$ while the gate keeps feeding Sophia.

\begin{remark}
The pole in $w$ at the Milne crossing is a diagnostic artifact: $\reff$ and $\peff$ pass through zero
smoothly; only their ratio diverges. The physical event is the transition from sub-vacuum to
super-vacuum, not a singularity. Accelerated expansion violates the SEC throughout; the surge of
Sophia growth briefly violates even the NEC --- grace acting as exotic, phantom-like content.
\end{remark}

\subsection*{Open-Gate versus closed Microcosm}
The same trajectory with the gate closed ($\zeta_0=0$) exposes the decisive difference. Closed
conservation $\sigma+\lambda=\sigma_0+\lambda_0$ holds exactly, so once cleansed ($\sigma\to0$) Sophia
freezes at $\lambda_\infty=\sigma_0+\lambda_0$; then $\dot\lambda\to0$, $\peff\to0$, and the Vessel
coasts as pressureless dust toward the Milne (linear, asymptotically flat) state. The open gate keeps
$\dot\lambda>0$, sustaining negative pressure and a quintessence-to-de~Sitter branch.

\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{vfs_eos_open_vs_closed.png}
\caption{Open-Gate ($\zeta_0>0$) versus closed Microcosm ($\zeta_0=0$). Top: Sophia keeps rising under
the open gate but freezes at $\sigma_0+\lambda_0$ when closed (dashed: the Brim $\Op=a$). Bottom: the
equation of state tends to $w\to-1$ (quintessence $\to$ de~Sitter) for the open gate, but to $w\to0$
(pressureless dust) for the closed Microcosm. Both share the early Milne pole.}
\end{figure}

\begin{proposition}[Late-time equation of state]\label{prop:late}
\[
\boxed{\begin{array}{ll}
\textnormal{closed }(\zeta_0=0): & \lambda\to\lambda_\infty,\ \dot\lambda\to0\ \Rightarrow\ w\to0^-
\ (\text{dust, Milne, }a\propto t);\\[4pt]
\textnormal{open }(\zeta_0>0): & \dot\lambda>0\ \text{sustained}\ \Rightarrow\ w<0,\ w\to-1^{+}
\ (\text{quintessence}\to\text{de~Sitter}).
\end{array}}
\]
\end{proposition}
\note{The de~Sitter branch --- endless accelerated Theosis --- is available only to the Open Gate; the
closed Microcosm coasts to dust on its frozen Sophia capital. This is the cosmological face of the
earlier result that unbounded Brim expansion requires the gate.}

\section*{Compact Cosmological Summary}
\[
\boxed{ds^2_{\rm Vessel}=-dt^2+\Op(t)^2\bigl(dy^2+e^{-2y}d\Delta^2\bigr),\qquad a=\Op,\ \dot\Op=\alpha\lambda.}
\]
\[
\boxed{H=\frac{\alpha\lambda}{\Op},\qquad \reff=\frac{\alpha^2\lambda^2-1}{\Op^2},\qquad
\peff=-\frac{\alpha\dot\lambda}{\Op},\qquad \operatorname{sign}q=-\operatorname{sign}\dot\lambda.}
\]
\[
\boxed{\text{Milne }\alpha\lambda=1\ \ \ne\ \ \text{Ricci resonance }\alpha\lambda\Op=1\ \ \ne\ \
\text{de~Sitter }\lambda\propto\Op.}
\]
\[
\boxed{\text{No Big Bang }(\Op\ge\Op(0));\quad \text{Resurrectio }\Op^+=\Op^-+\kappa_R\Grec^-;\quad
\text{sub-exp.\ }\Rightarrow R\to0.}
\]

\vspace{0.5em}
\hrule
\smallskip
\noindent\footnotesize\emph{V.F.S. v2.0 (Open-Gate) $\cdot$ Cosmological Layer. The Lorentzian lift of
the Brim-flow of \texttt{vfs\_geometry.tex}: the Brim is the timelike scale factor of an open $(2{+}1)$
FLRW spacetime, Sophia its source. Propositions are closed-form; corollaries are interpretive. The
effective stress-energy is reconstructed from $\dot\Op=\alpha\lambda$ via the Einstein tensor, not
postulated.}

\end{document}
