Behavior Completion Stability

Abstract

The effective behavior completion is stable under the source update.

Theorem 1.1 (Behavior completion carries the canonical shift dynamics).

$$\begin{array}{c}\forall X,B,\\ F:X\to X,q:X\to B,\\ \mathrm{Surjective}(q)\Rightarrow \\ \exists Fbar:\mathrm{ItineraryRange}(F,q)\to \mathrm{ItineraryRange}(F,q),\\ \mathrm{rangeFactorization}(\mathrm{completeItinerary}(F,q))\circ F=Fbar\circ \mathrm{rangeFactorization}(\mathrm{completeItinerary}(F,q)).\end{array}$$

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

Source. Repository-derived.

Commentary.

Let F update a state space X, and let q be a surjective interface onto its effective output codomain.

The behavior completion is constructed as the realized range of the full future q-itinerary. Its interface map is the canonical factorization through that range.

The induced map is the existing itinerary shift: it drops the current coordinate and advances every remaining future coordinate by one. Because a shifted realized itinerary is realized by F(x), the map stays on the exact effective-image carrier.

The displayed commutation equation is precisely interface stability. No parallel completion or shift definition is introduced.

References

• Truth anchor: D5/S3/ObserverMemory/RefinementClosure/BehaviorCompletionStability.behavior_completion_is_stable
• Dependency: D5/S3/ObserverMemory/Prediction/ItineraryCompletion