Perfect Observational Correlation from a Shared Source

Abstract

Two identity observations of one fair Boolean source have conditional success probability one after observing true and zero after observing false.

Theorem 1.1 (A fair shared source gives perfect observational correlation).

$$\begin{array}{r}\frac{\mathrm{conceptLaw}\left(u\mapsto \frac{1}{2},u\mapsto (u,u),(true,true)\right)}{\mathrm{conceptLaw}\left(u\mapsto \frac{1}{2},u\mapsto u,true\right)}=1,\\ \frac{\mathrm{conceptLaw}\left(u\mapsto \frac{1}{2},u\mapsto (u,u),(false,true)\right)}{\mathrm{conceptLaw}\left(u\mapsto \frac{1}{2},u\mapsto u,false\right)}=0.\end{array}$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Information/SharedSourcePerfectCorrelation.fair_shared_source_perfect_observational_correlation (✓ std3). ∎

Source. Repository-derived.

Commentary.

The source law assigns mass one half to each Boolean value, and both X and Y are the identity readout of that same source. The joint event X = true, Y = true therefore has mass one half, equal to the X = true marginal, so their ratio is one.

The joint event X = false, Y = true is impossible under the shared identity readout, while the X = false marginal is one half. Its conditional ratio is therefore zero.

References

• Truth anchor: D5/S3/ConceptDynamics/Information/SharedSourcePerfectCorrelation.fair_shared_source_perfect_observational_correlation
• Dependency: D5/S3/ConceptDynamics/Information/SharedSourceObservationDependence