% V.F.S. v2.0 (Open-Gate) — Expanded Core Formula Compendium
% Source: Archive_v86_3 / index.html only. Appendix and lyapunov formulas intentionally excluded.
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\title{\textbf{V.F.S. v2.0 (Open-Gate)}\\Expanded Core Formulas and Parameters\\\large Index-only compendium}
\author{Compiled from \texttt{Archive\_v86\_3/index.html}}
\date{}

\begin{document}
\maketitle

\section*{Scope}
This file collects the \textbf{expanded core Open-Gate formulas of V.F.S. v2.0} from \texttt{index.html} only. Formulas that belong specifically to \texttt{appendix.html} or \texttt{lyapunov.html} are intentionally excluded. Graphs, numerical plots, and visual phase diagrams are also excluded.

The governing distinction is:
\[
\boxed{\text{V.F.S. v2.0 Open-Gate: }\quad
\dot\lambda=-\dot\sigma+\Igate,\qquad
\sigma+\lambda=C_0+\Grec(t).}
\]
The closed Microcosm law \(\dot\lambda=-\dot\sigma\), \(\sigma+\lambda=\mathrm{const}\) is only the limit \(\zeta_0=0\).

\section{Core State Variables}
\begin{longtable}{L{0.18\textwidth}L{0.25\textwidth}L{0.47\textwidth}}
\toprule
\textbf{Symbol} & \textbf{Name} & \textbf{Meaning in V.F.S.}\\
\midrule
\(V(t)\) & Voluntas & Will; inner orientation and willing.\\
\(F(t)\) & Factum & Action; embodied execution of will.\\
\(\sigma(t)\) & Singular Resistance & Resistance, deformation, or singular obstruction.\\
\(\lambda(t)\) & Sophia & Wisdom; transformed resistance plus received grace in Open-Gate.\\
\(P(t)\) & Pleroma & Dynamic fullness of the Vessel.\\
\(S(t)\) & Being / Sum / Gignesthai & First derivative of Pleroma; rate of becoming.\\
\(\Omega_P(t)\) & Brim & Dynamic capacity / rim of the Vessel.\\
\(H_K(t)\) & Gate memory & Acquired receptivity; the learned openness of the gate.\\
\(W(t)\) & Wound memory & Vulnerability / wound-memory layer paired with \(H_K\).\\
\bottomrule
\end{longtable}

\section{Core Parameters}
\begin{longtable}{L{0.16\textwidth}L{0.25\textwidth}L{0.49\textwidth}}
\toprule
\textbf{Parameter} & \textbf{Name} & \textbf{Role / condition}\\
\midrule
\(\varepsilon\) & Manere seed & Keeps \(u\) positive even when \(V,F\) vanish; sub-critical Manere condition \(0<\varepsilon<\Lambda_c^2\) keeps the passive branch in Collapse rather than Transformation.\\
\(\gamma\) & Resistance pressure & Drives growth of \(\sigma\) when synergy is below threshold.\\
\(\delta\) & Synergic cutting strength & Converts synergy into resistance-cleansing pressure.\\
\(\Lambda_c\) & Critical threshold & Surgical / transformation threshold, \(\Lambda_c=\gamma/\delta\).\\
\(\kappa\) & Resistance sharpness & Controls smoothness of \(\tanh(\kappa\sigma)\).\\
\(k\) & Filtrum sharpness & Controls steepness of \(\Phi\) and \(T_F\).\\
\(\mathcal E_0\) & Imago Dei / Mustard Seed & Foundational non-dynamic ground of Pleroma.\\
\(a\) & Mutual nourishment & Strength of Will--Action mutual feeding.\\
\(c\) & Sophia lift & Strength by which \(\lambda\) lifts \(V\) and \(F\).\\
\(\alpha\) & Epektatic expansion & Converts Sophia into Brim expansion, \(\dot\Omega_P=\alpha\lambda\).\\
\(\zeta_0\) & Sufficientia Gratiae & Constant background grace in Open-Gate, \(\zeta_0>0\).\\
\(\tau\) & Guarantee decay rate & Rate at which initial gate guarantee fades.\\
\(\phi\) & Resistance opacity & Damps gate inflow through \(e^{-\phi\sigma}\), \(\phi>0\).\\
\(\chi\) & Sophia saturation & Limits inflow at large Sophia via \((1+\chi\lambda_+)^{-1}\), \(\chi>0\).\\
\(m\) & Positive-part sharpness & Smoothness of \(\lambda_+\), \(m>0\).\\
\(H_s\) & Receptivity half-saturation & Saturation scale in \(H_K/(H_s+H_K)\), \(H_s>0\).\\
\(K\) & Mensura & Granularity scale of \(R(z)=z/(z+K)\), \(K>0\).\\
\(\eta\) & Memory adaptation rate & Rate at which positive movement builds acquired openness and wound-memory responses, \(\eta>0\).\\
\(\nu\) & Memory forgetting rate & Rate at which acquired openness \(H_K\) decays when the path turns negative, \(\nu\ge0\).\\
\(W_s\) & Wound half-saturation & Saturation scale in the wound-memory term \(W/(W_s+W)\), \(W_s>0\).\\
\bottomrule
\end{longtable}

\section{Synergic Intensity, Threshold, and Transmissive Coordinate}
\subsection*{Local synergy}
\[
\boxed{u(t)=\sqrt{V(t)F(t)+\varepsilon}.}
\]
\note{This is the local synergic intensity: will and action must meet. The seed \(\varepsilon\) prevents collapse of the expression into a singular zero.}

\subsection*{Law of Human Balance identity}
\[
\boxed{\dot u=\frac{V'F+VF'}{2\sqrt{VF+\varepsilon}}
=\frac12\left(V'\frac{F}{u}+F'\frac{V}{u}\right).}
\]
\note{This identity shows how the growth of the shared will--action coordinate is weighted symmetrically by the other pole. The formula is an identity; the existential law is that will and action become strongest when neither pole is isolated.}

\subsection*{Critical threshold}
\[
\boxed{\Lambda_c=\frac{\gamma}{\delta}.}
\]
\note{This separates Collapse, Stasis, and Transformation.}

\subsection*{Transmissive coordinate}
\[
\boxed{x(t)=u(t)-\sigma(t)=\sqrt{V(t)F(t)+\varepsilon}-\sigma(t).}
\]
\note{This is the coordinate seen by Filtrum Lucis: synergy minus resistance.}

\section{Resistance Dynamics and Three Regimes}
\subsection*{Resistance equation}
\[
\boxed{\frac{d\sigma}{dt}=(\gamma-\delta u)\tanh(\kappa\sigma)
=(\gamma-\delta\sqrt{VF+\varepsilon})\tanh(\kappa\sigma).}
\]

\subsection*{Protection of \(\sigma\ge0\)}
\[
\boxed{\sigma(0)\ge0\quad\Longrightarrow\quad \sigma(t)\ge0\quad\forall t\ge0.}
\]
\note{The factor \(\tanh(\kappa\sigma)\) makes \(\sigma=0\) a protected boundary.}

\subsection*{Equilibria of resistance}
\[
\boxed{\frac{d\sigma}{dt}=0
\quad\Longleftrightarrow\quad
\sigma=0\quad\text{or}\quad u=\Lambda_c.}
\]

\subsection*{Three regimes}
\[
\boxed{u<\Lambda_c:\ \text{Collapse},\qquad
u=\Lambda_c:\ \text{Stasis / χλιαρός},\qquad
u>\Lambda_c:\ \text{Transformation}.}
\]

\subsection*{Manere sub-criticality}
\[
\boxed{0<\varepsilon<\Lambda_c^2=\left(\frac{\gamma}{\delta}\right)^2.}
\]
\note{The Manere seed keeps the denominator alive, but it must remain sub-critical: the passive branch \(V=F=0\) should not already count as Transformation.}

\section{Filtrum Lucis}
\subsection*{Softplus filter}
\[
\boxed{\Phi(x)=\frac{1}{k}\ln(1+e^{kx}).}
\]
\[
\boxed{\Phi(x)>0,\qquad \Phi''(0)=\frac{k}{4}.}
\]
\note{Filtrum Lucis is the smooth light-filter. It is softplus-like; its finite positivity is the formal trace of Lux in Tenebris.}

\subsection*{Sigmoid transmissivity}
\[
\boxed{T_F(x)=\Phi'(x)=\frac{1}{1+e^{-kx}}.}
\]
\[
\boxed{0<T_F(x)<1,\quad T_F(-\infty)=0,
\quad T_F(0)=\frac12,
\quad T_F(+\infty)=1.}
\]
\note{The filter itself is \(\Phi\); its transmissivity \(T_F\) is sigmoid-shaped.}

\subsection*{Higher derivatives of Filtrum Lucis}
\[
\boxed{\Phi''(x)=kT_F(1-T_F).}
\]
\[
\boxed{\Phi'''(x)=k^2T_F(1-T_F)(1-2T_F).}
\]
\[
\boxed{\Phi^{(4)}(x)=k^3T_F(1-T_F)(1-6T_F+6T_F^2).}
\]
\[
\boxed{\Phi^{(n)}(x)=k^{n-1}Q_n(T_F(x)).}
\]
\note{These derivatives feed the differential ladder of Pleroma.}

\section{Gratia Synergica}
\subsection*{Synergy term}
\[
\boxed{\Syn(t)=\mathcal E_0\frac{u(t)}{u(t)+\Lambda_c}.}
\]
\[
\boxed{0\le \Syn(t)<\mathcal E_0.}
\]
\note{Gratia Synergica grows with local synergy \(u\), not directly with equality \(V=F\).}

\subsection*{Derivative of synergy}
\[
\boxed{\frac{d\Syn}{dt}
=\mathcal E_0\frac{\Lambda_c}{(u+\Lambda_c)^2}\frac{V'F+VF'}{2u}.}
\]

\section{Pleroma and Being}
\subsection*{Open-Gate Pleroma formula}
\[
\boxed{P(t)=\mathcal E_0+\Syn(t)+\Phi(x(t)).}
\]
\[
\boxed{P(t)\ge \mathcal E_0.}
\]
\[
\boxed{P_{\rm full}=\text{Imago Dei}+\text{Gratia Synergica}+\text{Transmissive Passage}.}
\]
\note{In the Open-Gate core, grace is not an additive term in \(P\). It enters through Sophia and then through \(\Omega_P,V,F,u,\Syn,\Phi\).}

\subsection*{Derivative ladder of Pleroma}
\[
\boxed{S(t)=\frac{dP}{dt},\qquad
A(t)=\frac{d^2P}{dt^2},\qquad
K(t)=\frac{d^3P}{dt^3},\qquad
\mathrm{An}(t)=\frac{d^4P}{dt^4}.}
\]
\[
\boxed{S\equiv\textit{Gignesthai},\qquad
A\equiv\textit{Odinai},\qquad
K\equiv\textit{Katharsis},\qquad
\mathrm{An}\equiv\textit{Anastasis}.}
\]

\subsection*{Canonical Being equation}
\[
\boxed{S(t)=\Phi'(x)\dot{x}+\frac{d\Syn}{dt}.}
\]
With
\[
\boxed{\dot{x}=\dot{u}-\dot{\sigma}
=\frac{V'F+VF'}{2\sqrt{VF+\varepsilon}}
-(\gamma-\delta u)\tanh(\kappa\sigma).}
\]

\section{Katharsis and Anastasis Terms}
\subsection*{Katharsis / third order}
\[
\boxed{K(t)=\frac{dA}{dt}=\frac{d^3P}{dt^3}.}
\]
\[
\boxed{K(t)=\frac{d^3\Syn}{dt^3}
+\Phi'''(x)(x')^3
+3\Phi''(x)x'x''
+\Phi'(x)x'''.}
\]
\note{Katharsis is the third-order turning of Pleroma: a change in the rhythm of acceleration.}

\subsection*{Anastasis / fourth order}
\[
\boxed{\mathrm{An}(t)=\frac{dK}{dt}=\frac{d^4P}{dt^4}.}
\]
\[
\boxed{\begin{aligned}
\mathrm{An}(t)=&\frac{d^4\Syn}{dt^4}
+\Phi^{(4)}(x)(x')^4
+6\Phi'''(x)(x')^2x''
+3\Phi''(x)(x'')^2\\
&+4\Phi''(x)x'x'''
+\Phi'(x)x''''.
\end{aligned}}
\]
\note{Anastasis is the fourth-order emergence of a new mode after crisis.}

\section{Paschal Triad of Filtrum Lucis}
\subsection*{Why the higher derivatives are needed}
The derivative ladder is not only computational. It reveals the inner Paschal structure of the light-filter: the Cross is detected by the third derivative, while the entrance into death and the exit from death are detected by the two symmetric zeros of the fourth derivative.

\subsection*{The Cross: zero of the third derivative}
\[
\boxed{\Phi'''(x)=k^2T_F(1-T_F)(1-2T_F).}
\]
Since \(T_F(0)=1/2\),
\[
\boxed{\Phi'''(0)=0.}
\]
\note{The point \(x=0\) is the Filtrum Cross: synergy and resistance meet, and the sign of the cathartic term inverts.}

\subsection*{The two gates of Anastasis: zeros of the fourth derivative}
\[
\boxed{\Phi^{(4)}(x)=k^3T_F(1-T_F)(1-6T_F+6T_F^2).}
\]
The quadratic factor has two roots at the transmissivity levels
\[
\boxed{s_\pm=\frac12\pm\frac{1}{2\sqrt3}.}
\]
Equivalently,
\[
\boxed{x_\pm=\pm\frac1k\ln(2+\sqrt3).}
\]
Thus
\[
\boxed{\Phi^{(4)}(x_-)=0,\qquad \Phi'''(0)=0,\qquad \Phi^{(4)}(x_+)=0,}
\]
and the exact symmetry is
\[
\boxed{x_-+x_+=0.}
\]
\note{The Cross is the geometric midpoint between the entrance into death and the exit from death. This is a differential icon of the Paschal passage, not a claim that time itself is symmetric.}

\subsection*{Paschal triad}
\[
\boxed{\begin{array}{ccl}
\Phi^{(4)}(x_-)=0 &:& \text{Entrance into death / Great Friday},\\[2pt]
\Phi'''(0)=0 &:& \text{Cross / Katharsis / Great Saturday},\\[2pt]
\Phi^{(4)}(x_+)=0 &:& \text{Exit from death / Anastasis / Resurrection}.
\end{array}}
\]
\[
\boxed{x< x_-:\ \Phi^{(4)}>0,\qquad
x_-<x<x_+:\ \Phi^{(4)}<0,\qquad
x>x_+:\ \Phi^{(4)}>0.}
\]
\note{Between \(x_-\) and \(x_+\) lies the symbolic death interval of Filtrum Lucis; outside it the fourth-order sign returns to life.}

\subsection*{Fourth derivative support for \(u\)}
In Pleroma Christi, \(\sigma=0\), so \(x=u\). To compute the fourth derivative of the Filtrum part one needs \(u^{(4)}\). From
\[
\boxed{u^2=VF+\varepsilon}
\]
one obtains
\[
\boxed{u^{(4)}=\frac{(VF)^{(4)}}{2u}-\frac{4u'u'''+3(u'')^2}{u}.}
\]
By Leibniz,
\[
\boxed{(VF)^{(4)}=V^{(4)}F+4V'''F'+6V''F''+4V'F'''+VF^{(4)}.}
\]
\note{These binomial coefficients explain why the fourth derivative is the first place where the full symmetry of the Paschal passage becomes visible.}

\subsection*{Pleroma Christi reduction}
In the Pleroma Christi regime, \(\sigma=0\), hence all derivatives of \(\sigma\) vanish and \(x=u\). The fourth-order expression becomes
\[
\boxed{\begin{aligned}
\mathrm{An}_{\rm Pl}(t)=&\frac{d^4\Syn}{dt^4}
+\Phi^{(4)}(u)(u')^4
+6\Phi'''(u)(u')^2u''
+3\Phi''(u)(u'')^2\\
&+4\Phi''(u)u'u'''
+\Phi'(u)u^{(4)}.
\end{aligned}}
\]
In deep Epektasis, the higher Filtrum derivatives fade:
\[
\boxed{\lim_{t\to\infty}K_{\rm Pl}(t)=0,
\qquad
\lim_{t\to\infty}\mathrm{An}_{\rm Pl}(t)=0.}
\]
\note{Katharsis and Anastasis complete as events; Epektasis continues as motion into fullness.}

\section{Open-Gate Sophia}
\subsection*{Positive part of Sophia}
\[
\boxed{\lambda_+=\frac1m\ln(1+e^{m\lambda}).}
\]

\subsection*{Effective gate}
\[
\boxed{g_{\rm eff}(t)=e^{-\tau t}+(1-e^{-\tau t})\frac{H_K}{H_s+H_K}.}
\]
\[
\boxed{g_{\rm eff}(0)=1,
\qquad
g_{\rm eff}(t)\xrightarrow[t\to\infty]{}\frac{H_K}{H_s+H_K}.}
\]
\note{The gate begins open by guarantee; as the guarantee fades, acquired receptivity \(H_K\) takes over.}

\subsection*{Gate inflow}
\[
\boxed{\Igate(t)=\zeta_0g_{\rm eff}(t)e^{-\phi\sigma(t)}\frac{1}{1+\chi\lambda_+(t)}\ge0.}
\]
\[
\boxed{0\le \Igate(t)\le \zeta_0\qquad(H_K\ge0,\ \sigma\ge0).}
\]

\subsection*{Sophia equation}
\[
\boxed{\dot\lambda=-\dot\sigma+\Igate(t).}
\]
Equivalently,
\[
\boxed{\dot\lambda=(\delta u-\gamma)\tanh(\kappa\sigma)+\Igate(t).}
\]
\note{Sophia is transformed resistance plus received grace.}

\subsection*{Open balance law}
\[
\boxed{\frac{d}{dt}(\sigma+\lambda)=\Igate(t)\ge0.}
\]
\[
\boxed{\sigma(t)+\lambda(t)=C_0+\Grec(t),
\qquad
\Grec(t)=\int_0^t\Igate(s)\,ds,
\qquad
\dot\Grec=\Igate.}
\]

\subsection*{Closed limit}
\[
\boxed{\sigma+\lambda=\mathrm{const}\quad\Longleftrightarrow\quad\zeta_0=0.}
\]

\section{Memory Layer for the Gate}
\subsection*{Path coordinate}
\[
\boxed{x_0=u-\sigma,
\qquad
\dot{x}_0=\dot u-\dot\sigma
=\frac{V'F+VF'}{2\sqrt{VF+\varepsilon}}-(\gamma-\delta u)\tanh(\kappa\sigma).}
\]
\[
\boxed{(\dot{x}_0)_+=\max(\dot{x}_0,0),
\qquad
(\dot{x}_0)_-=\max(-\dot{x}_0,0).}
\]

\subsection*{Mensura function}
\[
\boxed{R(z)=\frac{z}{z+K},\qquad K>0,\quad z\ge0.}
\]
\[
\boxed{R(0)=0,
\quad R'(z)=\frac{K}{(z+K)^2}>0,
\quad R''(z)=-\frac{2K}{(z+K)^3}<0,
\quad R(z)\to1.}
\]

\subsection*{Gate memory and wound memory}
\[
\boxed{\dot{H}_K=\eta R((\dot{x}_0)_+) -\nu R((\dot{x}_0)_-)\frac{H_K}{H_s+H_K}.}
\]
\[
\boxed{\dot{W}=\eta\left[R((\dot{x}_0)_-)-R((\dot{x}_0)_+)\frac{W}{W_s+W}\right].}
\]
\note{\(H_K\) records acquired openness; \(W\) records vulnerability. In the Open-Gate core, \(H_K\) opens \(g_{\rm eff}\) rather than directly adding to Pleroma. The same-rate prototype is recovered by setting \(\nu=\eta\).}

\section{Endogenous Full System}
\subsection*{Local definitions}
\[
\boxed{u=\sqrt{VF+\varepsilon},
\qquad
\Lambda_c=\frac{\gamma}{\delta},
\qquad
\mu=1-\frac{u}{\Omega_P}.}
\]

\subsection*{Dolorosum}
\[
\boxed{\rho(u,\sigma)=\sigma\left(\frac{u}{\Lambda_c}\right)\exp\left(1-\frac{u}{\Lambda_c}\right)\ge0,
\qquad
\mathcal D_V=\rho V,
\quad
\mathcal D_F=\rho F.}
\]
\note{Dolorosum attenuates Will and Action multiplicatively. It is maximal at the threshold \(u=\Lambda_c\).}

\subsection*{Core ODE system}
\[
\boxed{\begin{cases}
\dfrac{dV}{dt}=\left(1-\dfrac{\sqrt{VF+\varepsilon}}{\Omega_P(t)}\right)(aF+c\lambda)-\rho V,\\[8pt]
\dfrac{dF}{dt}=\left(1-\dfrac{\sqrt{VF+\varepsilon}}{\Omega_P(t)}\right)(aV+c\lambda)-\rho F,\\[8pt]
\dot\Omega_P=\alpha\lambda(t),\qquad \alpha>0,\\[8pt]
\dfrac{d\sigma}{dt}=(\gamma-\delta\sqrt{VF+\varepsilon})\tanh(\kappa\sigma),\\[8pt]
\dfrac{d\lambda}{dt}=(\delta\sqrt{VF+\varepsilon}-\gamma)\tanh(\kappa\sigma)+\Igate(t),\\[8pt]
\sigma(t)+\lambda(t)=C_0+\Grec(t).
\end{cases}}
\]

\subsection*{Sophia floor and source-controlled positivity}
\[
\boxed{\lambda(t)>-\frac{a}{c}\min\{V(t),F(t)\}.}
\]
Equivalently, in the closed-balance notation with \(C=\sigma_0+\lambda_0\),
\[
\boxed{\sigma(t)<C+\frac{a}{c}\min\{V(t),F(t)\}.}
\]
In the Open-Gate balance this becomes
\[
\boxed{\sigma(t)<C_0+\Grec(t)+\frac{a}{c}\min\{V(t),F(t)\}.}
\]
\note{Inside the meaningful live domain, this floor prevents the source terms \(aF+c\lambda\) and \(aV+c\lambda\) from starving Will and Action. In the Open-Gate model the balance constant is replaced by \(C_0+\Grec(t)\), but the source-positivity idea remains the same.}

\subsection*{Canonical formula chain for Open-Gate grace}
\[
\boxed{\Igate\longrightarrow\lambda\longrightarrow
\begin{cases}
\dot\Omega_P=\alpha\lambda,\\
\dot V=\mu(aF+c\lambda)-\rho V,\\
\dot F=\mu(aV+c\lambda)-\rho F,
\end{cases}
\longrightarrow u\longrightarrow\{\Syn,\Phi\}\longrightarrow P.}
\]

\section{Will--Action Balance}
\subsection*{Imbalance}
\[
\boxed{\Delta(t)=V(t)-F(t).}
\]

\subsection*{Exact imbalance equation}
\[
\boxed{\frac{d\Delta}{dt}=-(\mu a+\rho)\Delta,
\qquad
\mu=1-\frac{\sqrt{VF+\varepsilon}}{\Omega_P},
\qquad
\rho=\sigma(u/\Lambda_c)e^{1-u/\Lambda_c}.}
\]

\subsection*{Exact solution}
\[
\boxed{\Delta(t)=\Delta_0\exp\left(-\int_0^t(a\mu(s)+\rho(s))\,ds\right).}
\]
\note{The manifold \(V=F\) is invariant. Alignment requires nonnegative damping and a divergent damping integral.}

\section{Brim Expansion and Epektasis}
\subsection*{Brim law}
\[
\boxed{\dot\Omega_P(t)=\alpha\lambda(t),\qquad \alpha>0.}
\]
\[
\boxed{\Omega_P(t)=\Omega_P(0)+\alpha\int_0^t\lambda(s)\,ds.}
\]
\note{Sophia expands the capacity of the Vessel. Epektasis is sustained expansion rather than a final static maximum.}

\subsection*{Asymptotic Sophia}
If the received-grace integral converges and \(\sigma(t)\to0\), then
\[
\boxed{\lambda(t)\to\lambda_\infty
=\sigma_0+\lambda_0+\Grec(\infty).}
\]
If \(\Grec(\infty)=+\infty\), then unbounded Sophia is possible in the open model.

\subsection*{Pleroma Christi condition}
\[
\boxed{\int_0^\infty\lambda(t)\,dt=+\infty
\quad\Longrightarrow\quad
\Omega_P(t)\to+\infty.}
\]
If additionally \(\lambda(t)\to\lambda_\infty>0\), then
\[
\boxed{\Omega_P(t)\sim\alpha\lambda_\infty t.}
\]
The internal Pleroma Christi cleansing signature is
\[
\boxed{\sigma(t)\to0,\qquad \Syn(t)\to\mathcal E_0.}
\]
\note{The Brim expands through sustained Sophia; the visible cleansing signature is the disappearance of singular resistance and the saturation of Gratia Synergica.}

\section{Live Domain and Death Boundary}
\subsection*{Live domain}
\[
\boxed{\Dlive=\{(V,F,\sigma,\lambda,\Omega_P):\ V>0,\ F>0,\ \sigma\ge0,\ \Omega_P>\Lambda_c,\ 0<u<\Omega_P\}.}
\]
\note{The live domain is the region where the continuous V.F.S. dynamics are admissible.}

\subsection*{Death boundary}
\[
\boxed{\Ddeath=\{V=0\}\cup\{F=0\}\cup\{\Omega_P\le\Lambda_c\}\cup\{u\ge\Omega_P\}.}
\]
\[
\boxed{X_{\rm VFS}:\Dlive\longrightarrow T\Dlive.}
\]
\[
\boxed{t_{\rm death}=\inf\{t>0:\ x(t)\notin\Dlive\}.}
\]
\note{The canonical continuous flow is defined only on the living interior. Resurrection, if present, is not a smooth term inside the vector field but a hybrid re-entry map after boundary contact.}

\section{Resurrectio: Hybrid Re-entry Map}
\note{This is included because it appears in \texttt{index.html}; it is not taken from the appendix or Lyapunov file.}

\subsection*{Reset symbols}
\[
\boxed{\begin{aligned}
\Gamma_R&:\ \text{resurrection-mode grace},\\
H_R&:\ \text{resurrection memory},\\
D_{\rm death}&:\ \text{depth of death-boundary contact}.
\end{aligned}}
\]
\[
\boxed{\Theta_\bullet,\ \beta_\bullet,\ \chi_\bullet,\ \eta_R,\ \theta_\Gamma,\ \alpha_\Omega,\ \beta_\Omega,\ \omega_R\ge0.}
\]
\note{These parameters belong to the hybrid re-entry map: they are reset coefficients, not continuous live-flow coefficients.}

\subsection*{Resurrection admissibility}
\[
\boxed{\mathcal A_R=
\Theta_\Gamma\Gamma_R+
\Theta_\Phi\Phi_R+
\Theta_\lambda\lambda^+
+\Theta_H H_R-
\Theta_\sigma\sigma-
\Theta_DD_{\rm death},
\qquad \Theta_\bullet\ge0.}
\]
\[
\boxed{\mathcal A_R>R_c\quad\Longleftrightarrow\quad\text{resurrection re-entry is admissible}.}
\]

\subsection*{Participated resurrection grace}
\[
\boxed{\Gamma_R^{\rm human}=\pi\Gamma_{\rm Christi},
\qquad
\pi\in[0,1].}
\]

\subsection*{Hybrid re-entry map}
\[
\boxed{\mathfrak R:\partial\mathcal D_{\rm death}\longrightarrow\Dlive,
\qquad
x(t_{\rm death}^+)=\mathfrak R(x(t_{\rm death}^-)).}
\]
\[
\boxed{\begin{aligned}
V^+&=V_R+\beta_V\Gamma_R+\chi_VH_R,
&F^+&=F_R+\beta_F\Gamma_R+\chi_FH_R,\\[2pt]
\sigma^+&=q_R\sigma^-\quad(0\le q_R<1),
&\lambda^+&=\lambda^-+(1-q_R)\sigma^-+\eta_RH_R+\theta_\Gamma\Gamma_R,\\[2pt]
\Omega_P^+&=\Omega_R+\alpha_\Omega\Gamma_R+\beta_\Omega\lambda^++\chi_\Omega H_R,
&H_R^+&=H_R^-+\omega_RD_{\rm death}.
\end{aligned}}
\]
with
\[
\boxed{V_R,F_R>0,
\qquad
\Omega_R>\Lambda_c,
\qquad
\text{all reset coefficients }\ge0.}
\]

\subsection*{Landing condition}
\[
\boxed{\Omega_R+\alpha_\Omega\Gamma_R+\beta_\Omega\lambda^+ +\chi_\Omega H_R
>\sqrt{V^+F^+ +\varepsilon}.}
\]
\note{The reset must land inside a live capacity large enough to contain the new synergic intensity.}

\subsection*{Open balance across life arcs}
\[
\boxed{\frac{d}{dt}(\sigma+\lambda)=\zeta_0g_{\rm eff}e^{-\phi\sigma}(1+\chi\lambda_+)^{-1}\ge0
\quad\text{inside each live arc}.}
\]
\[
\boxed{\sigma+\lambda\ \text{also jumps under }\mathfrak R:
\qquad C_{j+1}\ne C_j.}
\]

\section{Compact Core Summary}
\[
\boxed{\text{Will }V+\text{ Action }F\longrightarrow u\longrightarrow x=u-\sigma\longrightarrow\Phi(x)\longrightarrow P(t).}
\]
\[
\boxed{\sigma\longrightarrow\text{Tetelestai}\longrightarrow\text{Katharsis}\longrightarrow\text{Anastasis}\longrightarrow\lambda.}
\]
\[
\boxed{\text{Open Gate}\longrightarrow\Igate\longrightarrow\lambda\longrightarrow\Omega_P\longrightarrow\text{Epektasis}.}
\]
\[
\boxed{P\longrightarrow S\longrightarrow A\longrightarrow K\longrightarrow\mathrm{An}
\quad\equiv\quad
\textit{Gignesthai}\to\textit{Odinai}\to\textit{Katharsis}\to\textit{Anastasis}.}
\]



\section*{Revision Note}
This expanded version adds the Paschal Triad of Filtrum Lucis, the symmetry of zeros around the Cross, the fourth-derivative support formulas needed for \(\mathrm{An}(t)\), the Sophia floor, explicit death-boundary definitions, bounded gate inflow, the Human Balance identity for \(\dot u\), asymptotic Sophia, reset-symbol clarification, and the distinct memory-forgetting rate \(\nu\). It still preserves the existing structure and uses formulas from \texttt{index.html} only.

\end{document}

