Continuous Behavior Closure Stability

Abstract

Continuous dynamics preserve the closure of realizable behaviors.

Theorem 1.1 (Continuous actions preserve behavior closure).

$$\begin{array}{r}\forall B:\mathrm{Type},\mathrm{TopologicalSpace}\left(B\right),\\ S:B\to B,I:\mathrm{Set}\left(B\right),\\ \mathrm{Continuous}\left(S\right)\wedge \mathrm{MapsTo}\left(S,I,I\right)\Rightarrow \\ \mathrm{MapsTo}\left(S,\mathrm{closure}\left(I\right),\mathrm{closure}\left(I\right)\right).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/ObserverMemory/RefinementClosure/ContinuousBehaviorClosureStability.continuous_dynamics_preserves_behavior_closure (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let B carry a topology, let I be its set of realizable behaviors, and let S be a continuous self-action on B.

When S maps every realizable behavior back into I, continuity sends every limit of realizable behaviors into the closure of I.

The proof applies the pinned closure mapping theorem directly; no parallel closure or dynamics primitive is introduced.

References

• Truth anchor: D5/S3/ObserverMemory/RefinementClosure/ContinuousBehaviorClosureStability.continuous_dynamics_preserves_behavior_closure