Cyclic Laguerre Residue Projectors and Real-Zero Rigidity

Let p be a real polynomial and let L_n[p](x) denote the generalized Laguerre coefficients in the expansion of |p(x+iρ)|^2. For an integer m ≥ 2, we consider the symmetric products p(x+z_j)p(x-z_j) on the m unoriented axes z_j = iρ exp(π i j/m) and take a discrete Fourier transform in the axis index. The resulting real-valued channels isolate the residue classes of the Laguerre hierarchy modulo m: P_{m,k}[p](x,ρ) = Σ_{n≡k (mod m)} L_n[p](x) ρ^(2n). After removing the L_0=p^2 term from the zero residue class and applying the natural power normalizations, we study global nonnegativity of all normalized channels. For every m ≥ 3, this condition forces every zero of p to be real. The key step is a restart lemma: any upper-half-plane zero forces all Fourier channels and all rotated symmetric products to vanish, and the first oblique axis produces another upper-half-plane zero whose height is multiplied by cos(π/m). Iterating this step produces infinitely many distinct zeros. The case m=2 is qualitatively different because the first propagated height is zero; q(z)=z(z^4-1) is an explicit nonreal-rooted polynomial satisfying both global channel inequalities. We also prove a one-way extension to arbitrary nonzero real entire functions, multiplicative closure of the cyclic-positive classes, and a divisibility refinement law. The construction is compared explicitly with the closest generalized-Laguerre and complex shift-product criteria; the classical Laguerre coefficients and roots-of-unity shift geometry are treated as prior art.