Threshold Collapse by Common Provenance

Abstract

Equal formal role counts can conceal radically different capture thresholds.

Theorem 1.1 (Common provenance collapses the capture threshold).

$$\forall n:Nat,0<n\Rightarrow \mathrm{card}\left(\mathrm{Fin}\left(n\right)\right)=n\wedge \mathrm{captureNumber}\left(commonProvenanceReadou{t}_n,commonProvenanceReadou{t}_n\right)=1\wedge \mathrm{captureNumber}\left(independentProvenanceReadou{t}_n,independentProvenanceReadou{t}_n\right)=n.$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/InstitutionalCapture/ThresholdCollapseByCommonProvenance.threshold_collapse_by_common_provenance (✓ std3). ∎

Source. Repository-derived.

Commentary.

The formal roles are the n elements of Fin n, and both constructions use states in Fin n x Bool. The common-provenance readout ignores the state label, so every named role exposes the same Boolean source.

The independent-provenance readout exposes the Boolean value only when the state’s label matches the named source, returning false for all other labels. Consequently, each role has a distinct necessary source.

For every positive n, the two systems therefore have the same formal role cardinality n while their exact capture numbers are one and n. Formal role multiplicity alone does not determine the capture threshold.

References

• Truth anchor: D5/S3/ConceptDynamics/InstitutionalCapture/ThresholdCollapseByCommonProvenance.threshold_collapse_by_common_provenance
• Dependency: D5/S3/ConceptDynamics/InstitutionalCapture/IndependentSourceCaptureLowerBound