Hierarchy-Dependence of the Hubble Parameter (H₀) Under Multi-Interface Closure: A Structural Toy Diagnostic

This technical note presents a minimal structural stress test examining whether the Hubble parameter (H₀) remains invariant under changes in interface hierarchy during multi-source cosmological integration. Using a synthetic multi-interface toy model inspired by contemporary inference architectures (CMB-like, BAO-like, local ladder-like, and lensing-like constraints), the study evaluates the dependence of inferred global values on the order and topology of closure. The results show that the inferred H₀ is not invariant under permutation of interface hierarchy within the toy construction. A sink-like interface (lensing analogue) reduces structural spread but does not eliminate hierarchy dependence. The hierarchy-induced variation is further propagated to a derived functional (cosmic age), which remains band-stable under moderate thresholds despite primary parameter sensitivity. The analysis is purely diagnostic and does not propose new physical models. It formalizes an operational criterion for assessing structural legitimacy of global parameters in multi-interface systems.