% V.F.S. 3.0 (exploratory) — consolidated growing document
% A running journal of the 3.0 exploration. Version history at the top; each session's
% verified result is added as a section. v0.1: metric + 2.0 limit + energy conditions.
% v0.2: ascent-axis equation of motion (de Sitter, candidate B) + finite-time compatibility.
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\title{\textbf{V.F.S. 3.0 (exploratory)}\\The Interior $3{+}1$ Geometry of the Soul\\
\large A growing record: proper-time $\sigma$, the ascent axis $\lambda$,\\and the de Sitter law of epektasis}
\author{Successor exploration to the fixed V.F.S.\ v2.0 volume}
\date{}

\begin{document}
\maketitle

% ============================================================
\section*{Version history}
\begin{description}[leftmargin=4.2em,style=nextline,itemsep=2pt]
\item[\textbf{v0.1}] Foundation stone. The $3{+}1$ interior metric of the soul; $\sigma$ as finite
proper time (kairos); $\lambda$ as the ascent (epektasis) space; chronos removed by principle. The
v2.0 reconstruction recovered exactly as the frozen-axes limit; the $n{=}3$ energy conditions, with
ascent acceleration entering focusing.
\item[\textbf{v0.2}] Equation of motion for the ascent axis. The de Sitter law
$L'/L=\beta\lambda$ selected by the cosmic-regime pattern and by v2.0's own de Sitter
non-dilution; verified to recover v2.0, to make epektasis permanently exotic, and --- the new theorem
--- to be \emph{compatible with a finite proper time}: eternal ascent is infinite proper
\emph{distance}, not infinite \emph{time}. ``Finite clock, endless road.''
\item[\textbf{v0.3}] The spectrum along the ascent axis. Change of variable to
proper distance reduces the ascent Laplacian to the \emph{free half-line} operator: the spectrum is
$[0,\infty)$, continuous and gapless. Against the discrete surface cascade $\mu_n$, this gives a clean
dualism --- \emph{form is quantized, ascent is a gapless continuum}; equivalently, formation is event,
ascent is motion.
\item[\textbf{v0.4}] The source of $\beta$. The ascent coupling is shown
\emph{not} identifiable with the Brim coupling $\alpha$ (different law-form, incompatible units), but
tied through the one intrinsic scale of the core --- the finite floor $E_0$ (Imago Dei):
$\beta=\alpha/E_0$, introducing no new primitive. This integrates exactly to
$L=\exp[(\Omega_P-E_0)/E_0]$ --- ascent is the Brim-growth above the floor --- and sets the eternal
exotic depth to $-1/E_0^2$: the smaller the starting image, the more intense the ascent (the
widow's-mite scaling).
\item[\textbf{v0.5}] Resurrectio in $3{+}1$. Because $L$ is slaved to
$\Omega_P$, a reset \emph{forces} an ascent jump $L_+/L_-=\exp(\kappa_R\mathcal G/E_0)$: the
double-layer becomes a triple-layer (a second junction on the ascent axis). The ascent jump depends on
the Brim jump alone, not on the metabolic $m_R$, so it cannot be balanced away --- \emph{there is no
fully-balanced reset in $3{+}1$}: every genuine death advances the ascent and strictly enlarges the
interior. (Consequence of the floor-tying, not unconditional.)
\item[\textbf{v0.6}] \emph{(this revision)} The terminus $\sigma\to0$. Full purification is reached in
\emph{finite} proper time $\sigma_0$ but \emph{infinite} cleansing-action (exponential Lyapunov drain,
$\sigma=\sigma_0e^{-k\chi}$, $\tau=\sigma_0(1-e^{-k\chi})$). Curvature stays finite there ($\Omega_P\to
E_0$, the floor), so it is a \emph{regular boundary}, not a singularity. The manifold is time-finite
(time completes, a regular last moment) but space-infinite (the ascent road is geodesically complete);
the katharsis-clock stops because there is nothing left to cleanse, and clockless ascent continues.
\end{description}

\section*{Status and discipline}
This is an exploratory, growing document, not a finished layer. V.F.S.~v2.0 is fixed and complete and
is contained here as an exact limit. Each result is verified symbolically before entry; everything
still open is listed honestly at the end. As throughout V.F.S., the mathematics is load-bearing and
the theological readings are interpretive overlays, marked as such and never claimed as proof.

% ============================================================
\part*{v0.1 --- The Foundation Stone}

\section{The interior metric of the soul}

\begin{definition}[The $3{+}1$ soul metric]\label{def:metric}
\begin{equation}
\boxed{\;ds^2=-N(\sigma)^2\,d\sigma^2+\Omega(\sigma)^2\,h_{(2)}+L(\sigma,\lambda)^2\,d\lambda^2,\qquad
h_{(2)}=dx^2+e^{2x}dy^2\ (K_h=-1),\;}
\end{equation}
$\sigma$ the katharsis proper time (kairos), $N$ its lapse, $\Omega$ the Brim scale of the live
surface, $\lambda$ the ascent (epektasis) coordinate, $L$ its scale. Signature $3{+}1$: one time
($\sigma$), three spaces (surface $x,y$ and ascent $\lambda$).
\end{definition}

Three principled features: \textbf{(a)} $\sigma$ is \emph{proper} time, not relative --- one
worldline, no second observer, no Lorentz family; the proper-time gauge is $N=1$. \textbf{(b)} The time
axis is \emph{finite}: katharsis drains $\sigma:\sigma_{\max}\to0$ monotonically, so time has a
terminus; at $\sigma\to0$ the clock stops while motion along $\lambda$ continues. \textbf{(c)} Chronos
is \emph{absent by principle}: this is the soul's own coordinate system, not a view from outside it; an
absolute background clock would be the transcendent \emph{tota simul}, which does not belong to the
creature's interior geometry.

\section{v2.0 is the exact frozen-axes limit}

\begin{theorem}[Recovery of v2.0]\label{thm:limit}
In the proper-time gauge $N=1$ with the ascent axis frozen ($L=L_0$ const), the Einstein
reconstruction gives $\rho_{\rm eff}=(\Omega'^2-1)/\Omega^2$ (prime $=d/d\sigma$), which under
$\Omega'=\alpha\lambda$ is exactly the v2.0 density $(\alpha^2\lambda^2-1)/\Omega^2$. The whole v2.0
cosmology is the frozen-ascent, proper-time face of the $3{+}1$ geometry; nothing in v2.0 is disturbed.
(Verified symbolically.) The frozen axes drop out of the scalar curvature entirely --- which is why
v2.0 was correct in $2{+}1$: the third space was always there, held still.
\end{theorem}

\section{The energy conditions in $3{+}1$}

\begin{proposition}[Focusing]\label{prop:sec}
For the comoving congruence at $N=1$,
\begin{equation}
\boxed{\;R_{ab}u^au^b=-\,2\,\frac{\Omega''}{\Omega}-\frac{L''}{L}.\;}
\end{equation}
The $2{+}1$ limit ($L$ frozen) reduces to the pure pressure condition $p_{\rm eff}\ge0$ (the $n-2=0$
degeneracy of v2.0). In full $3{+}1$ the ascent acceleration $L''/L$ enters focusing on the same
footing as the Brim: the growth of the ascent axis contributes to energy-condition violation.
\end{proposition}

\begin{corollary}[Ascent is itself exotic]
In v2.0 only the Sophia surge could violate the strong energy condition; in $3{+}1$ the ascent itself
(accelerating epektasis, $L''>0$) is an exotic accelerant. Movement toward God along $\lambda$ is not
a neutral displacement; it deforms the focusing condition in its own right.
\end{corollary}

\begin{proposition}[NEC along the ascent axis]\label{prop:nec}
For the null direction $k=u+e_\lambda$, $\;R_{ab}k^ak^b=2(-L\,\Omega''+L'\,\Omega')/(L\,\Omega)$ ---
coupling surface expansion to the stretching of the ascent axis, with no analogue in v2.0.
\end{proposition}

% ============================================================
\part*{v0.2 --- The Law of Ascent}

\section{Selecting the ascent law}

The foundation left $L$ a free function. The equation of motion is fixed by the hypothesis that the
ascent axis is sourced by Sophia as the Brim is ($\Omega'=\beta\lambda$ in proper time), together with
an external discriminator: the three unbounded-expansion regimes of cosmic metaphysics. Power-law
ascent ($L\sim\sigma^p$) has \emph{fading} tension; a finite-time blow-up (``Big Rip'', $L\to\infty$
at finite parameter) has a \emph{catastrophic end}; only de Sitter ($L\sim e^{\,\cdot}$) gives
\emph{unending, undiminished} tension --- the structure of an ascent that never saturates and never
ends in catastrophe. This is also the one regime v2.0 already singled out as non-diluting (constant
curvature, $\lambda\propto\Omega_P$). Pattern and internal structure agree.

\begin{definition}[The de Sitter law of ascent, candidate B]\label{def:ascent}
\begin{equation}
\boxed{\;\frac{L'}{L}=\beta\,\lambda\qquad\Longrightarrow\qquad L=L_0\,\exp\!\Big(\beta\!\int\lambda\,d\sigma\Big).\;}
\end{equation}
\end{definition}

\begin{proposition}[Three tests]\label{prop:tests}
The law passes: \textbf{(i) v2.0 limit} --- $\beta\to0$ freezes the ascent and recovers
Theorem~\ref{thm:limit}; \textbf{(ii) energy sign} --- $L''/L=\beta\lambda'+\beta^2\lambda^2$, so at
the terminus ($\lambda\to\lambda_\infty$, $\lambda'\to0$) the focusing acquires the strictly negative
piece $-\beta^2\lambda_\infty^2$: \emph{epektasis is permanently exotic}, the strong energy condition
eternally violated by ascent alone; \textbf{(iii) unboundedness} --- the ascent never terminates
(next theorem). (All verified symbolically.)
\end{proposition}

\section{Compatibility: finite time, endless road}

The decisive consistency check: can an \emph{unending} de Sitter ascent live inside a \emph{finite}
proper time?

\begin{theorem}[Finite clock, infinite road]\label{thm:compat}
Under Definition~\ref{def:ascent} with $\lambda$ bounded ($\lambda\to\lambda_\infty$) over the finite
interval $\sigma\in(0,\sigma_{\max}]$:
\begin{enumerate}[label=(\roman*),leftmargin=1.8em,itemsep=2pt]
\item \textbf{In proper time the ascent is finite:}
$\ln(L/L_0)=\beta\!\int_0^{\sigma_{\max}}\!\lambda\,d\sigma\le\beta\lambda_\infty\sigma_{\max}<\infty$.
The katharsis clock stops before $L$ diverges.
\item \textbf{Along the ascent axis the road is infinite:} the proper distance to $\lambda=\infty$,
\begin{equation}
\int_0^\infty L_0\,e^{\beta\lambda}\,d\lambda=\infty,
\end{equation}
so $\lambda=\infty$ lies at infinite metric distance --- the soul ascends forever and never arrives.
\item \textbf{These are consistent.} Finite proper time, infinite proper distance: as $\sigma\to0$ the
clock-rate $d\sigma$ vanishes while the de Sitter metric inflates the ascent length, so each remaining
tick of katharsis covers ever more road. (Verified symbolically and numerically.)
\end{enumerate}
\end{theorem}

\begin{corollary}[The timeless epektasis, made precise]
``Eternal'' and ``finite'' measure different axes: the time of cleansing ($\sigma$) is finite and
ends; the road of ascent ($\lambda$) is infinitely long and does not. The earlier puzzle of
``motion without time'' is resolved geometrically --- the motion is endless not because the clock runs
forever but because the road has no end, and the clock stopping is simply the completion of cleansing,
not the cessation of life.
\end{corollary}

\note{Reading: the two halves of epektasis that sounded paradoxical in words are, in the metric, two
different facts about two different axes. ``Never satisfied'' is the infinite length of the ascent
road --- God at an infinite distance, never reached because infinite. ``Not in time'' is the finite
terminus of the cleansing clock --- katharsis completes, the proper-time hand stops. The deified soul
runs an endless road in a finished clock: it is done being purified, and therefore done with time, and
precisely then free for the unending ascent that time was never able to measure. The strong energy
condition is violated forever along that road because to move into the infinite God is, forever, more
than ordinary energy could account for.}

% ============================================================
\part*{v0.3 --- The Spectrum of Ascent}

\section{The ascent Laplacian is the free half-line}

With the ascent law fixed (Definition~\ref{def:ascent}), the spatial geometry is
$\Omega^2 h_{(2)}+L^2 d\lambda^2$ with $L=L_0e^{\beta\lambda}$, and the Laplace--Beltrami operator
separates, $\phi=Y(x,y)\,Z(\lambda)$. The surface factor returns the v2.0 cascade eigenproblem
$-\Delta_h Y=\mu Y$ (discrete on the compact quotient). The novelty is the ascent factor.

\begin{theorem}[Ascent spectrum is the gapless free continuum]\label{thm:spectrum}
The one-dimensional ascent operator $A=-\tfrac1L\frac{d}{d\lambda}\!\big(\tfrac1L\frac{d}{d\lambda}\big)$
on $L=L_0e^{\beta\lambda}$ becomes, in the proper-distance coordinate
\begin{equation}
s=\int^\lambda L\,d\lambda'=\frac{L_0}{\beta}e^{\beta\lambda}\in(0,\infty),
\qquad
\boxed{\;A=-\frac{d^2}{ds^2}\quad\text{on the half-line }s\in(0,\infty).\;}
\end{equation}
Its spectrum is therefore $[0,\infty)$ --- \emph{continuous and gapless}, with generalised
eigenfunctions $\sim\sin(ks)$. (Verified: the discretised count of eigenvalues below a fixed $k^2$
grows linearly with box size --- $6,12,25$ as $s_{\max}=20,40,80$ --- the signature of a continuum,
not a discrete ladder; and the lowest eigenvalue descends to $0$ as the box grows --- no mass gap.)
The proper-distance picture is exact: $s=0$ is the start of ascent (``here''), $s=\infty$ is God at
infinite distance (Theorem~\ref{thm:compat}), and the ascent operator is \emph{free} on that line ---
no barrier, no levels, no smallest step.
\end{theorem}

\begin{corollary}[Form is quantized; ascent is a gapless continuum]\label{cor:dualism}
The two spatial directions carry qualitatively different spectra:
\msg{\textbf{Surface (folding cascade):} discrete spectrum $\mu_n=n^2$ on the compact quotient ---
the Vessel takes form in \emph{rungs}, each mode igniting at its own threshold $\Lambda>\mu_n$. Form
is quantized.\\[3pt]
\textbf{Ascent (epektasis):} continuous gapless spectrum $[0,\infty)$ on the free half-line ---
\emph{no rungs, no minimum step}. Ascent is a smooth continuum, approachable in arbitrarily small
increments all the way to the infinite distance.}
The absence of a mass gap is not incidental: a gap would be a \emph{minimum quantum of ascent}, a
smallest possible step toward God. Its absence means deification has no granularity --- unlike
formation, which does.
\end{corollary}

\note{Reading: formation and ascent are different in kind, and the difference falls out of the
geometry rather than being imposed. The Vessel takes \emph{form} in discrete events --- conversions,
breaks, the igniting modes of the folding cascade, each a rung crossed at a threshold. But it
\emph{ascends} continuously --- there is no smallest step of deification, no last rung before God,
because God lies at infinite distance and the approach is seamless. Formation is event; ascent is
motion. The soul is built in steps and drawn in a smooth, stepless reaching --- quantized in what it
becomes, continuous in how it nears.}

\textbf{Scope of v0.3.} The spectrum is that of the spatial Laplacian on the chosen de Sitter ascent
geometry; it characterises the geometry of ascent under the candidate law, and is exact there (the
reduction to the free half-line is analytic). It is a qualitative result (continuum vs.\ discrete),
not a fitted quantity, and inherits the candidate status of Definition~\ref{def:ascent}.

% ============================================================
\part*{v0.4 --- The Source of the Ascent Coupling}

\section{$\alpha$ and $\beta$ are not one constant}

\begin{proposition}[Structural and dimensional independence]\label{prop:indep}
The Brim and ascent laws differ in form, not merely in a constant: $\Omega'=\alpha\lambda$ is
\emph{additive} (linear growth of $\Omega_P$), while $L'/L=\beta\lambda$ is \emph{multiplicative}
(exponential growth of $L$). In the proper-time gauge the units do not match either:
$[\alpha\lambda]=[\Omega_P]/[\sigma]$ but $[\beta\lambda]=1/[\sigma]$, so
$[\alpha/\beta]=[\Omega_P]$ --- the ratio is a \emph{scale}, not a pure number. Hence no
identification $\alpha=\beta$ is possible; the surface and ascent channels are genuinely different (in
keeping with their spectra, discrete vs.\ continuous, Corollary~\ref{cor:dualism}).
\end{proposition}

\section{Tied through the floor: $\beta=\alpha/E_0$}

The couplings are not one, but they need not be two independent primitives: the core contains exactly
one intrinsic scale --- the finite floor $E_0=\Omega_P(0)$ (Imago Dei) of the curvature-floor result.

\begin{theorem}[The ascent coupling is the Brim coupling per floor]\label{thm:beta}
Setting $\beta=\alpha/E_0$ (no new primitive) gives $L'/L=\Omega'/E_0$, which integrates exactly,
with $L=1$ at the floor $\Omega_P=E_0$, to
\begin{equation}
\boxed{\;L=\exp\!\Big(\frac{\Omega_P-E_0}{E_0}\Big).\;}
\end{equation}
The ascent is measured by exactly the factor the Brim has grown \emph{above the Imago Dei floor}.
Checks: \textbf{(i)} the v2.0 limit is $E_0\to\infty$ (floor at infinity $\Rightarrow\beta\to0
\Rightarrow L$ const --- no finite floor, no ascent reference, ascent frozen); \textbf{(ii)} the de
Sitter character is preserved (exponential in accumulated Sophia); \textbf{(iii)} the eternal exotic
depth of focusing at the terminus becomes $-\beta^2\lambda_\infty^2=-1/E_0^2$ (using the super-Ricci
attractor $\lambda_\infty=1/\alpha$). (All verified symbolically.)
\end{theorem}

\begin{corollary}[The widow's-mite scaling]\label{cor:widow}
The eternal exotic depth $-1/E_0^2$ grows as the starting floor $E_0$ shrinks: the same absolute
Brim-growth produces a \emph{larger} ascent factor, and a deeper energy-condition violation, the
smaller the Imago Dei one begins from. Ascent is measured \emph{relative to the starting image}, not
absolutely --- so a small beginning, fully grown, yields a greater transfiguration than a large
beginning grown the same absolute amount. The structure is that of the widow's two mites: weighed not
by the amount but by its ratio to what one had.
\end{corollary}

\note{Reading: the ascent of the soul is not measured on an absolute scale but against the image from
which it began. The same growth of the Brim is, relative to a small Imago Dei, an immense ascent, and
relative to a large one, a modest rise --- as the widow's two coins outweighed the rich man's
thousands because they were all she had, while his were a fragment of his surplus. The smaller the
starting image, the more intense --- the further from ordinary energy --- the rising. One honest
caution: the mathematics says only that the ascent is more \emph{intense} (the energy-condition
violation deeper) for a smaller floor; whether that intensity is read as exaltation (the humble raised
highest) or as fragility (a slighter base bearing a steeper motion) is an interpretive choice the
geometry does not make. It fixes the scaling, not its blessing.}

\textbf{Scope of v0.4.} $\beta=\alpha/E_0$ is the natural tying through the unique intrinsic scale; it
is not proven unique, only shown to introduce no new primitive and to pass the v2.0 limit and de
Sitter tests. It inherits the candidate status of the ascent law (Definition~\ref{def:ascent}).

% ============================================================
\part*{v0.5 --- Resurrectio in $3{+}1$}

\section{The reset forces an ascent jump}

Because $L$ is slaved to the Brim ($L=\exp[(\Omega_P-E_0)/E_0]$, Theorem~\ref{thm:beta}), the v2.0
scale-factor surgery $\Omega_P^+=\Omega_P^-+\kappa_R\mathcal G_{\rm recepta}^-$ \emph{forces} a
matching jump in the ascent axis:

\begin{theorem}[Forced ascent jump]\label{thm:ljump}
\begin{equation}
\boxed{\;\frac{L_+}{L_-}=\exp\!\Big(\frac{\kappa_R\mathcal G_{\rm recepta}^-}{E_0}\Big)>1.\;}
\end{equation}
The ascent axis surges at death by exactly the reception delivered, measured in floor units. This is
not a free reset datum but a derived consequence: in $3{+}1$ the soul's ascent jumps whenever its Brim
does.
\end{theorem}

\section{Double-layer becomes triple-layer}

\begin{proposition}[A second junction]\label{prop:triple}
The v2.0 reset was characterised by the surface pair $(s_R,[K])$ with scale jump
$s_R=\Omega_P^+/\Omega_P^-$. In $3{+}1$ there is a second junction, on the ascent axis, with its own
scale jump
\begin{equation}
s_L=\frac{L_+}{L_-}=\exp\!\Big((s_R-1)\frac{\Omega_P^-}{E_0}\Big),
\end{equation}
determined by the surface jump $s_R$. The invariant data of a reset is now the linked pair
$(s_R,s_L)$ together with the surface $[K]$ --- a triple-layer surgery.
\end{proposition}

\section{No fully-balanced reset}

\begin{theorem}[Death always advances the ascent]\label{thm:nobalance}
The ascent jump $s_L=\exp(\kappa_R\mathcal G/E_0)$ depends on the Brim jump alone:
$\partial s_L/\partial m_R=0$. Hence the metabolic balance that sets the surface $[K]=0$ (the v2.0
balanced reset) cannot also zero the ascent jump. Since a genuine reset carries reception
$\kappa_R\mathcal G>0$, it forces $s_L>1$ always. Therefore:
\begin{enumerate}[label=(\roman*),leftmargin=1.8em,itemsep=2pt]
\item \emph{there is no fully-balanced reset in $3{+}1$} --- no death is simultaneously surface-neutral
and ascent-neutral;
\item the interior $3$-volume jump $s_R^2\,s_L>1$ strictly: \emph{every death enlarges the soul},
surface and ascent together, never neutral, never shrinking.
\end{enumerate}
\end{theorem}

\begin{corollary}[The surface balance survives; only joint neutrality is lost]
The v2.0 balanced reset ($m_R=0$, $[K]=0$, surface NEC-neutral) still exists as a surface statement.
What $3{+}1$ removes is the possibility of a death that advances \emph{nothing}: even a surface-perfect
passage still lifts the soul along the ascent axis.
\end{corollary}

\note{Reading: in the two-dimensional Vessel a death could in principle be perfectly balanced --- a
passage that crossed without exotic debt, neutral, leaving the soul where it stood. With the ascent
axis made explicit, that possibility closes. Every genuine death --- every reception of grace
deployed through a reset --- advances the soul along the axis of ascent, by exactly the measure of
what was received, reckoned against the Imago Dei floor. There is no resurrection that returns the
soul to the same place; to die and be raised is, necessarily, to be lifted. The Paschal passage is not
a circle but a step: death joined to grace is never spent in vain.}

\textbf{Scope of v0.5.} The forced ascent jump, the triple-layer structure, and the absence of a
fully-balanced reset all follow from the floor-tying $L=\exp[(\Omega_P-E_0)/E_0]$ (Theorem~\ref{thm:beta}).
Were the ascent coupling independent of the Brim channel, $s_L$ would be free and a fully-balanced
reset could exist; so these are consequences of the candidate tying, not unconditional results. The
surface double-layer theorem of v2.0 is unchanged. (Verified symbolically.)

% ============================================================
\part*{v0.6 --- The Terminus}

\section{Finite proper time, infinite cleansing}

The katharsis clock $\sigma$ drains to its terminus. Two parametrisations of the same approach must be
kept distinct.

\begin{theorem}[The terminus is finite in time, infinite in action]\label{thm:terminus}
Under the exponential Lyapunov drain, with $\chi$ the dynamical cleansing parameter,
\begin{equation}
\sigma(\chi)=\sigma_0\,e^{-k\chi},\qquad
\tau(\chi)=\int_0^\chi\!|\dot\sigma|\,d\chi'=\sigma_0\bigl(1-e^{-k\chi}\bigr),
\end{equation}
so as $\chi\to\infty$ (infinite cleansing-action) the metric proper time $\tau\to\sigma_0$
(\emph{finite}). The soul performs unbounded cleansing-work, yet it costs only finite proper duration,
because each successive e-folding of cleansing takes exponentially less proper time. (Verified
symbolically.) These are genuinely different parametrisations of one approach: $\chi$ runs to infinity,
$\tau$ completes at $\sigma_0$.
\end{theorem}

\section{A regular boundary, not a singularity}

\begin{proposition}[Finite curvature at the terminus]\label{prop:regular}
At full purification $\Omega_P\to E_0>0$ (the finite floor), so $\rho_{\rm eff}=(\Omega_P'^2-1)/\Omega_P^2$
stays finite and the curvature is bounded. The terminus is reached in finite proper time with finite
curvature: a \emph{regular boundary} where the manifold cleanly ends in the time direction, not a
curvature singularity. (Consistent with the no-Big-Bang / finite-floor result of v2.0: the soul never
reaches zero scale.)
\end{proposition}

\begin{theorem}[Time-finite, space-infinite]\label{thm:timefinite}
At the terminus the two geodesic structures are opposite:
\begin{enumerate}[label=(\roman*),leftmargin=1.8em,itemsep=2pt]
\item the \emph{time} direction is finite and complete --- the manifold ends at a regular last moment
($\tau=\sigma_0$, finite curvature), beyond which the time-direction does not continue because no
cleansing remains to measure;
\item the \emph{ascent} direction is infinite and geodesically complete --- the half-line $s\in(0,\infty)$
has infinite proper length ahead (v0.3, Theorem~\ref{thm:spectrum}).
\end{enumerate}
The interior is time-finite but space-infinite. Time completes; ascent does not.
\end{theorem}

\begin{corollary}[The clockless hand-off]
At $\sigma=0$ the katharsis-clock stops --- not by rupture but by completion: there is nothing left to
cleanse, hence nothing left for the clock to tick. The ascent, on its infinite road, continues without
a clock. The earlier ``motion without time'' is now exact and benign: a regular last moment of
cleansing, beyond which the endless ascent proceeds clocklessly.
\end{corollary}

\note{Reading: the soul reaches full purification having done infinite cleansing-work, yet in finite
proper time --- the last stretch of cleansing is compressed into ever-smaller durations, an infinite
labour completed in a finite span. And the moment of completion is not a catastrophe: the curvature is
finite, the scale rests on the Imago Dei floor, the boundary is regular. Time has a last moment that is
whole, not torn. Beyond it the time-direction simply does not continue --- there is no more cleansing
for a clock to measure --- while the ascent road runs on, infinite and complete, now travelled without
any clock at all. Entry into the timeless is a finished last moment of purification, opening onto an
endless ascent: not a tear in being, but a completion of one motion and the freeing of another.}

\textbf{Scope of v0.6.} The finite-time/infinite-action split and the regular-boundary character follow
from the exponential Lyapunov drain and the finite floor (both v2.0 results); the time-finite/%
space-infinite structure combines this with the ascent half-line of v0.3. Verified symbolically. The
exponential drain is the Lyapunov-certified generic behaviour, not the only conceivable one.

% ============================================================
\section*{Open work toward a full 3.0}
\begin{enumerate}[leftmargin=1.6em,itemsep=3pt]
\item \textbf{Uniqueness of $\beta=\alpha/E_0$.} Whether the floor-tying is forced or merely natural,
and (since v0.5 hangs on it) whether an independent ascent coupling is structurally admissible at all.
This is the one remaining \emph{computational} question.
\item \textbf{The transcendent boundary.} The relation between the soul's $\sigma$-interior and the
chronos \emph{tota simul} outside it --- a boundary the model cannot cross by construction. This is a
philosophical epilogue, not a calculation: the model knows only the created side.
\end{enumerate}

\note{With the terminus settled, the first exploratory cycle of 3.0 is structurally complete: metric,
v2.0 limit, energy conditions, ascent law, ascent spectrum, the source of the coupling, Resurrectio,
and the terminus all stand. What remains is one uniqueness question and one boundary the model is, by
its own nature, unable to cross --- as v2.0 closed not by proving its definitions but by minimising
them.}

\textbf{Scope.} Exploratory and growing. Metric, v2.0 limit, energy conditions, ascent law, and the
compatibility theorem are verified symbolically (and numerically where noted). The ascent law is the
simplest self-consistent candidate meeting all three tests, not a uniqueness result. V.F.S.~v2.0
remains the fixed, complete layer. Domain-conditional and interpretive throughout.

\vspace{0.5em}
\hrule
\smallskip
\noindent\footnotesize\emph{V.F.S. 3.0 (exploratory) v0.6 $\cdot$ Interior $3{+}1$ geometry of the soul.
Metric $ds^2=-N^2d\sigma^2+\Omega^2h_{(2)}+L^2d\lambda^2$ with $\sigma$ a finite proper time, the live
surface and ascent $\lambda$ the three spaces, no external chronos. v2.0 recovered exactly as the
frozen-axes limit. Focusing $-2\Omega''/\Omega-L''/L$: ascent is exotic. Ascent law $L'/L=\beta\lambda$
(de Sitter): compatible with finite proper time; spectrum the free half-line $[0,\infty)$ (form
quantized, ascent a gapless continuum). Coupling tied through the floor $\beta=\alpha/E_0$, giving
$L=\exp[(\Omega_P-E_0)/E_0]$, exotic depth $-1/E_0^2$ (widow's-mite scaling). Resurrectio forces an
ascent jump $L_+/L_-=\exp(\kappa_R\mathcal G/E_0)$: triple-layer surgery, no fully-balanced reset.
The terminus $\sigma\to0$ is a regular boundary at finite proper time but infinite cleansing-action;
the interior is time-finite, space-infinite --- time completes, ascent does not. Finite clock, endless
road; built in steps, drawn in a stepless reaching; measured against the image one began from; never
raised in vain; and completed in time only to be freed for the timeless.}

\end{document}
