Hierarchical Riesz Bases of Exponentials for Partitions into Five Intervals

We prove that every partition of \( [0,1) \) into five intervals admits a partition of \( \mathbb Z \) whose nonempty unions are Riesz spectra for the corresponding unions of intervals. No arithmetic assumption on the endpoints is required. The proof combines folding along the primes with local constructions in the rational affine space determined by the lengths. The rational-rank \( 3 \) case is based on a computer-assisted verification of a finite family of Fourier minors by exact integer arithmetic.