Conjecture 6.9: Structural Stability

VALIDATED Converged systems recover equilibrium within \( O(10) \) adaptation cycles after perturbation.

Statement

A composed adaptive system at equilibrium, when subjected to a bounded perturbation \( \|\delta\| \leq \epsilon \), will return to a neighbourhood of the original equilibrium within \( O(10) \) adaptation cycles, provided the perturbation does not violate the contraction conditions.

Status: Validated

Recovery is fast and robust across all perturbation types tested. The \( O(10) \) bound holds consistently, with most systems recovering in 6–12 cycles.

Evidence Summary

The experiments apply various perturbation types to converged systems:

Parameter perturbation (random noise on weights): recovery in 8–10 cycles
Objective perturbation (modified loss function): recovery in 10–12 cycles
Structural perturbation (removed subsystem, then restored): recovery in 6–8 cycles
Large perturbation (\( \epsilon = 0.3 \), ~30% displacement): recovery in 11–14 cycles

The key insight is that the contraction mapping property guarantees exponential return to equilibrium. The \( O(10) \) timescale reflects the typical contraction rate \( \beta \approx 0.7 \) — after 10 steps, the displacement is reduced by a factor of \( 0.7^{10} \approx 0.028 \), bringing the system within 3% of equilibrium.

Relevant Experiments

exp_memory_dynamics.sx — perturbation recovery dynamics
exp_sensitivity.sx — sensitivity to perturbation magnitude (3 OOM stable)
exp_contraction.sx — contraction rates that determine recovery speed

What This Means

Structural stability is the practical guarantee that converged systems are robust to transient disruptions. A system that has converged to equilibrium will not be permanently derailed by noise, temporary changes in the environment, or brief component failures. The \( O(10) \) recovery time is fast enough for most practical applications — the system self-heals within a small number of adaptation cycles without external intervention.