Observer Strategy Factorization

Abstract

Observer strategy factorization is equivalent to reverse kernel inclusion.

Theorem 1.1 (An effective interface implements exactly its fiber-constant policies).

$$\begin{gathered} \forall X:\mathrm{Type},O:\mathrm{Type},Policy:\mathrm{Type}, \\ q:X\to O,Pi:X\to Policy, \\ \mathrm{Surjective}(q)\Rightarrow (\mathrm{Refines}(Pi,q)⟺\ker (q)\subseteq \ker (Pi)). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/RefinementAlgebra/ObserverStrategyFactorization.observer_strategy_factorization (✓ std3). ∎

Source. Repository-derived.

Commentary.

The observer interface is surjective onto its declared coordinate carrier, matching the source convention that interfaces use only realized values. The policy readout need not be surjective.

A factorization makes the policy constant on every interface fiber. Conversely, a section of the effective interface constructs the policy implementation from the kernel-inclusion premise.

The repository’s existing effective-kernel theorem assumes both readouts are surjective, so applying it here would add a premise absent from the source.

References

• Truth anchor: D5/S3/ConceptDynamics/RefinementAlgebra/ObserverStrategyFactorization.observer_strategy_factorization
• Dependency: D5/S3/ConceptDynamics/ConceptJoinUniversal