E12.1 — [[097_E12.1_Destiny-Equation|Destiny Equation]]

Objections & Responses

Objection: "The equation is ad hoc — why logistic dynamics?"

"You've chosen the logistic equation because it gives the result you want. There's no principled reason for this specific form."

Response

The logistic form is not arbitrary — it is the unique first-order polynomial dynamics satisfying:

1. Boundedness: \Phi \in [0, \Phi_{\max}] (coherence cannot be negative or infinite)

2. Fixed points at boundaries: \dot{\Phi} = 0 at \Phi = 0 and \Phi = \Phi_{\max}

3. Sign-dependence: The direction of flow depends on moral orientation

4. Monotonicity: No oscillations or reversals within a trajectory

The general form satisfying these is:

\dot{\Phi} = \sigma \cdot f(\Phi) \cdot \Phi \cdot (\Phi_{\max} - \Phi)

where f(\Phi) > 0. The simplest choice is f(\Phi) = \gamma (constant). Higher-order choices add parameters without explanatory gain.

The logistic equation is not ad hoc — it is the canonical form for bistable switching dynamics. It appears in population biology, neural networks, and phase transitions precisely because it captures the essence of competition between two states.

Objection: "Eternal states are incoherent — nothing lasts forever"

"The limiting behavior assumes infinite time, but the universe itself may end. 'Eternal' heaven/hell are metaphysical fiction."

Response

The objection conflates physical time with ontological persistence. The destiny equation operates in the soul's proper time, not cosmological time.

Several points:

1. Information is conserved: The soul-field persists regardless of physical cosmology. Its dynamics are intrinsic, not dependent on external time.

2. Asymptotic behavior is about direction, not duration: The limit t \to \infty means "the trajectory converges" — not "time literally goes on forever."

3. Theological traditions affirm eternality: Scripture describes eternal life and eternal punishment. If these are meaningful, the dynamics must support persistent states.

The "nothing lasts forever" objection is a form of nihilism that contradicts the axiom chain's foundation in information conservation and consciousness persistence.

Objection: "The equation ignores grace and repentance"

"Once $\sigma$ is set, the trajectory is determined. But Christian theology allows for repentance and forgiveness at any time. Your model is fatalistic."

Response

The destiny equation describes dynamics given a fixed \sigma. It does not claim \sigma is immutable during life.

Grace operates on \sigma, not on \Phi directly:

Before death: \sigma can change through repentance (grace flips the sign)
At death: \sigma is finalized (the "particular judgment")
After death: The trajectory unfolds according to the fixed \sigma

This is not fatalism but consequentialism: choices have consequences. The equation shows what happens given a moral orientation, not that the orientation cannot change.

The model actually emphasizes the importance of grace: without it, \sigma = -1 is the default, and fragmentation is inevitable. Grace is the intervention that makes \sigma = +1 possible.

Objection: "Why $\gamma$ constant? Sanctification is not uniform."

"The coupling constant $\gamma$ is taken as fixed, but spiritual growth varies — some sanctify faster than others."

Response

The model can be extended to variable \gamma(\Phi) without changing the qualitative behavior:

\dot{\Phi} = \sigma \gamma(\Phi) (\Phi_{\max} - \Phi)\Phi

As long as \gamma(\Phi) > 0, the attractor structure is preserved. Variable \gamma would model:

Increasing \gamma with \Phi: Virtue accelerates virtue (positive feedback in sanctification)
Decreasing \gamma with \Phi: Diminishing returns near \Phi_{\max}
Person-dependent \gamma: Different souls have different rates

The constant \gamma is the simplest assumption, not the only one. The key claim is the attractor structure, which is robust to variations in \gamma.

Objection: "What about $\sigma = 0$? Is there a neutral state?"

"Your equation has $\sigma \in \{-1, +1\}$, but what about souls that are genuinely neutral — neither oriented toward nor away from God?"

Response

For \sigma = 0, the destiny equation gives:

\dot{\Phi} = 0

This is a stationary state — the soul neither integrates nor fragments. However, the bimodal outcome axiom ([[094_A12.2_Bimodal-Outcome|A12.2]]) excludes permanent neutrality.

Theologically, this corresponds to the doctrine that there is no neutrality before God. Jesus says, "Whoever is not with me is against me" (Matthew 12:30). One cannot remain uncommitted forever.

Mathematically, \sigma = 0 is a bifurcation point, not a stable state. Any perturbation pushes the system into \sigma > 0 or \sigma < 0, and the corresponding attractor takes over.

The "neutral" state is limbo — perhaps real for some traditions, but ultimately resolved into one of the two final states.

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Physics Layer

The Destiny Equation: Full Derivation

The Destiny Equation unifies the dynamics of coherence evolution for all moral orientations:

\frac{d\Phi}{dt} = \sigma \cdot \gamma \cdot (\Phi_{\max} - \Phi) \cdot \Phi

Parameter Definitions:

\Phi \in [0, \Phi_{\max}]: [[038_D5.2_Integrated-Information-Phi|Integrated information]] ([[017_A3.2_Coherence-Measure|coherence measure]])
\sigma \in \{-1, +1\}: Moral sign (orientation toward/away from coherence source)
\gamma > 0: Coherence coupling constant (rate parameter)
\Phi_{\max}: Maximum possible integrated information (divine coherence)

Derivation from First Principles:

1. Logistic Structure: The term (\Phi_{\max} - \Phi) \cdot \Phi is the standard logistic kernel, ensuring:

Bounded dynamics: \Phi cannot exceed \Phi_{\max} or go negative
Self-limiting growth/decay near boundaries
Maximum rate of change at intermediate values

2. Moral Sign: The prefactor \sigma determines flow direction:

\sigma = +1: Flow toward \Phi_{\max} (integration)
\sigma = -1: Flow toward \Phi = 0 (fragmentation)

3. Coupling Constant: The parameter \gamma sets the timescale:

Characteristic time: \tau = 1/(\gamma \Phi_{\max})
Larger \gamma means faster approach to attractor

Mathematical Layer

Formal Definitions

Definition 1 (Destiny Equation):

The Destiny Equation is the first-order autonomous ODE:

\dot{\Phi} = f(\Phi; \sigma) := \sigma \gamma (\Phi_{\max} - \Phi)\Phi

defined on the state space \mathcal{S} = [0, \Phi_{\max}] with parameter \sigma \in \{-1, +1\}.

Definition 2 (Flow):

The flow \phi_t: \mathcal{S} \to \mathcal{S} generated by the destiny equation is:

\phi_t(\Phi_0) = \Phi(t; \Phi_0, \sigma)

where \Phi(t; \Phi_0, \sigma) is the solution with initial condition \Phi(0) = \Phi_0.

Definition 3 (Eschatological Limit):

The eschatological limit of a trajectory is:

\Phi_\infty := \lim_{t \to \infty} \phi_t(\Phi_0) = \begin{cases} \Phi_{\max} & \sigma = +1 \\ 0 & \sigma = -1 \end{cases}

Definition 4 (Destiny Manifold):

The destiny manifold is the partition of phase space:

\mathcal{S} = \mathcal{B}^+ \sqcup \mathcal{B}^- \sqcup \{0\} \sqcup \{\Phi_{\max}\}

where:

\mathcal{B}^+ = (0, \Phi_{\max}) with \sigma = +1 (heaven-bound)
\mathcal{B}^- = (0, \Phi_{\max}) with \sigma = -1 (hell-bound)

Defeat Conditions

To Falsify This

Higher-order terms in $\Phi$ (which violate parsimony)
Time-varying parameters (which introduce external forcing)
Additional state variables (which expand the model beyond coherence alone)

Cross-Domain Mappings

Domain	Mapping
Physics	Attractors / Phase Space / Logistic Dynamics
Theology	Eschatology / Final Judgment / Eternal States
Consciousness	Death attractor / Integration-Fragmentation
Quantum	Quantum Zeno / Decoherence Dynamics
Scripture	Matthew 25:46 "eternal punishment...eternal life"
Evidence	Dynamical systems / Bifurcation theory
Information	Info flow dynamics / IIT

Bridge Count: 7

[[097_E12.1_DESTINY-EQUATION|DESTINY EQUATION]]

Connections

Assumes

Enables

The Destiny Equation: Full Derivation

Formal Definitions

To Falsify This