A12.1 — Asymptotic Behavior

Mathematical Layer

Axiom ([[093_A12.1_Asymptotic-Behavior|A12.1]]): For all soul states ψ ∈ H_soul with dynamics governed by the destiny equation:

\exists L \in \{0, \Phi_{max}\}: \lim_{t \to \infty} \Phi(\psi(t)) = L

Every trajectory converges to a limiting value, and that value is one of the two fixed points.

Defeat Conditions

**Demonstrate eternal oscillation** — Show that soul trajectories can oscillate forever without converging. This would require a perpetual energy source for oscillation (violates thermodynamics) and a mechanism that prevents damping (contradicts the dissipative nature of the universe).
**Prove trajectory divergence** — Show that Φ → ∞ or Φ → -∞ without bound. But Φ is bounded: 0 ≤ Φ ≤ Φ_max (physical limits on coherence). Unbounded trajectories are impossible.
**Find strange attractors in soul dynamics** — Demonstrate chaotic, aperiodic behavior that never settles. While possible in principle, the soul's dynamics (governed by the [[097_E12.1_Destiny-Equation|destiny equation]]) produce simple fixed-point attractors, not chaos.
**Show eternal limbo is possible** — Prove that trajectories can remain at intermediate Φ values forever without converging to an attractor. This contradicts the gradient structure of the dynamics—the "potential well" has no flat regions at intermediate values.

Cross-Domain Mappings

Bridge Count: 7

ASYMPTOTIC BEHAVIOR