Common-Source Echo Limit

Abstract

Messages derived from one common source cannot resolve a target blind to that source.

Theorem 1.1 (Common-source repetition cannot resolve a blind target).

$$\forall I\in Type,X\in Type,B\in Type,Y\in Type,BM\in I\to Type,S\in X\to B,M\in \left(\forall i\in I,X\to BM\left(i\right)\right),T\in X\to Y,\left(\neg \mathrm{Refines}\left(\mathrm{canonicalTargetReadout}\left(T\right),S\right)\right)\Rightarrow \left(\left(\left(\forall i\in I,\mathrm{Refines}\left(M\left(i\right),S\right)\right)\Rightarrow \left(\neg \mathrm{Refines}\left(\mathrm{canonicalTargetReadout}\left(T\right),\mathrm{jointReadout}\left(M\right)\right)\right)\right)\wedge \left(\mathrm{Refines}\left(\mathrm{canonicalTargetReadout}\left(T\right),\mathrm{jointReadout}\left(M\right)\right)\Rightarrow \left(\exists i\in I,\neg \mathrm{Refines}\left(M\left(i\right),S\right)\right)\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Discussion/CommonSourceEchoLimit.common_source_repetition_cannot_resolve_blind_target (✓ std3). ∎

Source. Repository-derived.

Commentary.

Take an arbitrary indexed family of messages. If every message factors through the same source readout, their canonical joint readout also factors through that source and cannot determine a target that the source does not determine.

Consequently, if the joint message readout does determine the target, at least one component message must introduce a distinction that does not factor through the common source.

References

• Truth anchor: D5/S3/ConceptDynamics/Discussion/CommonSourceEchoLimit.common_source_repetition_cannot_resolve_blind_target
• Dependency: D5/S3/ConceptDynamics/Communication/IndexedCommonSourceUpperBound
• Dependency: D5/S3/ConceptDynamics/Refinement/RefinementTransitivity
• Dependency: D5/S3/ConceptDynamics/Sufficiency/UniversalSufficiencyFactorization