Stabilization of complete spectrum spaces on the real line

Let Ωj and Ω have the same positive finite measure, and suppose that 1Ωj converges to 1Ω in L1. If Ω is spectral and every complete spectrum of Ω is a spectrum of every Ωj, then the complete spectrum spaces are equal for all sufficiently large j. The proof uses convergence of a circle-valued phase together with uniform rigidity of nearby spectra of each source. As an application, we use the finite rationality theorem of Fu and Song to show that sufficiently large centered lattice folds of a fixed finite-measure spectral set preserve its entire spectrum space.