Fourier quasicrystals with unit masses are sections of Lee–Yang varieties

We characterize unit-mass Fourier quasicrystals as real sections Λ(X,L) = {x : exp(2πiLx) ∈ X} of Lee–Yang varieties, where the matrix L has positive maximal minors and every intersection has local degree one. For Delone sets the variety can be chosen strict. Every unit-mass Fourier quasicrystal is the common complex zero set of finitely many real-valued trigonometric polynomials. These results answer Questions (1), (2), and (4) of Alon, Kummer, Kurasov, and Vinzant. The proof recovers a finite-dimensional torus from the spectrum, proves that the resulting analytic set is algebraic, and reconstructs its complex branches from their moments. A support-height theorem for holonomic sequences gives the required linear spectral bound. Singular Lee–Yang sections are Fourier quasicrystal measures with their natural local intersection degrees; discarding those degrees can destroy the Fourier quasicrystal property. Version 2: revised exposition; the main results are unchanged. This version corrects several slips in the arguments, adds a splitting lemma, examples, and Theorem F (a special case of the algebraicity theorem), and reorganizes several proofs.