A registry of objects in their order of construction: read from top to bottom, each row uses only what has already been defined above. The culmination is \(P(t)\); after that come the derivatives and the engine that closes the loop.
Columns: Object · Explanation / formula (inputs are visible in the formula) · What it affects. Red marks the target \(P\); grey marks the legacy closed model \(\zeta_0=0\).
Construction spine (within-instant DAG + loop)
givens constants + primitives + state {V, F, σ, λ, ΩP, HK}
V, F, ε → u=√(VF+ε) → x=u−σ → Φ ⌍
u, Λc=γ/δ ────────────→ Syn ⌎→ P = ℰ₀ + Syn + Φ → S=dP/dt → A → K → An
engine: HK, σ, λ → geff, Igate → σ̇, λ̇, Ω̇P, V̇, Ḟ ↺ updates the state (next instant)
A.Givens — constants and primitives
Declared once; they do not evolve. This is the reference layer on which the whole recipe rests.
| Object | Explanation / formula | What it affects |
|---|
| \(\mathcal E_0\)Imago Dei | The mustard seed — the indestructible floor of being (Mt 13:31). \(\mathcal E_0>0\). | → Syn, P |
| \(k\)Lux in Tenebris | The depth of light through darkness (Jn 1:5): \(\Phi=\tfrac1k\ln(1+e^{kx})\); finite \(k\Rightarrow\Phi>0\). | → Φ |
| \(\kappa\)Filtrum Tetelestai | Sharpness of the boundary \(\tanh(\kappa\sigma)\) near \(\sigma=0\). | → \(\dot\sigma\) |
| \(\varepsilon\)Manere | Regularizer in \(u=\sqrt{VF+\varepsilon}\); prevents \(u\) from vanishing. | → u |
| \(\gamma,\ \delta\)Entropic Pressure / Surgical Efficiency | Accumulation of resistance \(\gamma\) versus surgical efficiency \(\delta\). Together they define the threshold \(\Lambda_c=\gamma/\delta\). | → \(\Lambda_c\), \(\dot\sigma\) |
| \(a,\ c\)Mutual Nourishment / Elevation | Mutual nourishment \(V\leftrightarrow F\) through \(a\), and the elevation term \(c\lambda\) from Wisdom. | → \(\dot V,\dot F\) |
| \(\alpha,\ \tau\)Brim rate / Gate decay | Expansion of the Vessel \(\dot\Omega_P=\alpha\lambda\); fading of guaranteed openness \(e^{-\tau t}\). | → \(\Omega_P\), geff |
| \(\zeta_0,\phi,\chi\)Open-gate trio | Background grace \(\zeta_0\); opacity of resistance \(e^{-\phi\sigma}\); saturation \((1+\chi\lambda_+)^{-1}\). | → Igate |
| \(m,H_s,K,\eta\)Memory / Mensura primitives | Smoothness of \(\lambda_+\) through \(m\); half-saturation \(H_s\); granularity \(R(z)=z/(z+K)\); memory rate \(\eta\). | → geff, HK |
B.State at time \(t\)
Integrated variables carried from the previous instant; they hold the system's history. Inputs to recipe C.
| Object | Explanation / formula | What it affects |
|---|
| \(V,\ F\)Voluntas / Factum | Will and Action — intention and its embodiment. \(V,F>0\). | → u |
| \(\sigma\)Singular Resistance | Accumulated resistance / sin. \(\sigma\ge0\). | → x; Igate |
| \(\lambda\)Sophia (Pearl) | Accumulated Wisdom: resistance transmuted plus received grace. | → \(\dot\Omega_P\), \(\dot V,\dot F\), Igate |
| \(\Omega_P\)Brim of the Vessel | The ontological capacity of the Vessel. \(\Omega_P>\Lambda_c\). | → \(S_q=I_{\rm gate}/\Omega_P\) |
| \(H_K\)Acquired openness | Acquired openness as memory of the path. | → geff |
C.The recipe for Pleroma \(P\) at time \(t\)
A strict DAG: everything needed for instantaneous \(P\), and nothing extra. The gate does NOT enter here; it belongs to the engine E.
| Step · Object | Explanation / formula | Built from / affects |
|---|
| 1\(\Lambda_c\)Limen Chirurgiae | \(\Lambda_c=\gamma/\delta\) — the threshold or “eye” of the filter. | from \(\gamma,\delta\) → Syn, \(\dot\sigma\) |
| 2\(u\)Transmissive coordinate | \(u=\sqrt{VF+\varepsilon}\) — the shared magnitude of will-and-action. | from \(V,F,\varepsilon\) → x, Syn |
| 3\(x\)Input of the filter (Cross at \(x=0\)) | \(x=u-\sigma\) — intensity relative to resistance. | from \(u,\sigma\) → Φ |
| 4\(\Phi\)Filtrum Lucis | \(\Phi=\tfrac1k\ln(1+e^{kx})\), with \(\Phi>0\) always; \(\Phi''(0)=k/4\). | from \(k,x\) → P |
| 5\(\mathcal{Syn}\)Gratia Synergica | \(\mathcal{Syn}=\mathcal E_0\cdot\dfrac{u}{u+\Lambda_c}\), \(0\le\mathcal{Syn}<\mathcal E_0\). | from \(\mathcal E_0,u,\Lambda_c\) → P |
| 6\(P(t)\)Pleroma — TARGET | \(P=\mathcal E_0+\mathcal{Syn}+\Phi\). \(P\ge\mathcal E_0\). | from \(\mathcal E_0,\mathcal{Syn},\Phi\) → ladder D |
D.The ladder — derivatives of \(P\)
Once \(P(t)\) exists, it is differentiated in time: each rung is a phase of fullness in motion.
| Object | Explanation / formula | Meaning |
|---|
| \(S(t)\)Being / Sum / Gignesthai | \(S=dP/dt=\Phi'(x)\dot x+\mathcal{Syn}'\). | the speed of fullness becoming lived being |
| \(A(t)\)Odinai | \(A=d^2P/dt^2\). | birth-pang acceleration of becoming |
| \(K(t)\)Katharsis | \(K=d^3P/dt^3\); the Filtrum core has \(\Phi'''(0)=0\) at the Cross. | the cathartic turn / sign-change of transition |
| \(\mathrm{An}(t)\)Anastasis | \(\mathrm{An}=d^4P/dt^4\); the Filtrum core has two symmetric fourth-derivative zeros around the Cross. | resurrectional emergence after the transition |
The Paschal Triad is the signature of this ladder: \(A\) / Odinai enters the pressure of becoming, \(K\) / Katharsis marks the Cross-point of purification, and \(\mathrm{An}\) / Anastasis marks the emergence into a new mode of being.
E.The engine of state evolution
The gate machinery + ODEs that move state B into the next instant. This closes the loop back to B.
| Object | Explanation / formula | Built from / affects |
|---|
| \(\lambda_+\)Positive part | \(\lambda_+=\tfrac1m\ln(1+e^{m\lambda})\). | from \(m,\lambda\) → Igate |
| \(g_{\rm eff}\)Receptivity gate | \(g_{\rm eff}=e^{-\tau t}+(1-e^{-\tau t})\dfrac{H_K}{H_s+H_K}\). | from \(\tau,H_K,H_s\) → Igate |
| \(I_{\rm gate}\)Gate inflow (Modus I) | \(I_{\rm gate}=\zeta_0 g_{\rm eff}e^{-\phi\sigma}(1+\chi\lambda_+)^{-1}\ge0\). | from \(\zeta_0,g_{\rm eff},\sigma,\lambda_+\) → \(\dot\lambda\) |
| \(\mathcal G_{\rm recepta}\)Received grace | \(\mathcal G_{\rm recepta}=\int_0^t I_{\rm gate}\,ds\), monotone. | from \(I_{\rm gate}\) → balance, \(\lambda_\infty\) |
| \(\dot\sigma\)Resistance core | \(\dot\sigma=(\gamma-\delta u)\tanh(\kappa\sigma)\) — the dissipative core. | from \(\gamma,\delta,u,\kappa,\sigma\) → updates \(\sigma\) |
| \(\dot\lambda\)Sophia | \(\dot\lambda=-\dot\sigma+I_{\rm gate}\); regression occurs only when \(I_{\rm gate}<(\gamma-\delta u)\tanh(\kappa\sigma)\). | from \(\dot\sigma,I_{\rm gate}\) → updates \(\lambda\) |
| \(\dot\Omega_P\)Brim | \(\dot\Omega_P=\alpha\lambda\). | from \(\alpha,\lambda\) → updates \(\Omega_P\) |
| \(\dot V,\dot F\)Will–Action | Coupled growth: nourishment \(a\), Dolorosum \(\rho\), and elevation \(c\lambda\). The balanced optimum is \(V=F\). | from \(a,c,\lambda,\rho\) → updates \(V,F\) |
| \(\dot H_K\)Memory | \(\dot H_K=\eta R((\dot x_0)_+)-\nu R((-\dot x_0)_+)\tfrac{H_K}{H_s+H_K}\). | from \(\eta,\nu,x,H_K\) → updates \(H_K\) |
| BalanceLex Aequilibrii Aperti | \(\sigma+\lambda=C_0+\mathcal G_{\rm recepta}\), \(\tfrac{d}{dt}(\sigma+\lambda)=I_{\rm gate}\ge0\). | accounting law (closed case: \(=C_0\)) |
F.Long time — regimes, operators, boundaries
Asymptotics and events after many turns of the loop. Closed-model legacy is marked in grey.
| Object | Explanation / formula | When / what for |
|---|
| \(\mathfrak R\)Resurrection reset (Modus II) | Reset at the point of death; participation continues after closure, with \(\Gamma_{\rm Christi}\). | terminal closure → turn of the cycle |
| KatharsisPurification | Smooth passage through the Cross \(x=0\) (\(\Phi'''(0)=0\)). | metastable cathartic layer |
| EpektasisInfinite expansion | \(\Omega_P\to\infty \Leftrightarrow \int_0^\infty\lambda\,dt=\infty\). | unbounded growth of the Vessel |
| \(\lambda_\infty\)Asymptotic Sophia | \(\lambda_\infty=\sigma_0+\lambda_0+\mathcal G_{\rm recepta}(\infty)\). | \(\Omega_P\sim\alpha\lambda_\infty t\) |
| Pleroma ChristiAbsolute Fullness | Asymptotic regime: \(\sigma\to0,\ \mathcal{Syn}\to\mathcal E_0\). | telos of the movement |
| \(\mathcal G,\mathcal P\)Gratia / Impulse · closed | [Only \(\zeta_0=0\)] The old additive grace / impulse. In v2.0, grace enters through Igate. | not active in LXXXVII |
| \(\lambda_{\max},\ \sigma{+}\lambda{=}C_0\)closed invariants | [Only \(\zeta_0=0\)] \(\lambda_{\max}=\sigma_0+\lambda_0\); exact conservation. | valid only in Microcosm |
The order is topological in construction: A–B provide inputs, C builds \(P\) as a strict DAG (the gate is intentionally outside C because it is not needed for instantaneous \(P\)), D differentiates, and E moves the state and closes the loop ↺ back to B. Circularity is temporal, not within a single instant.