Unconditional hyperbolicity of the critical Laguerre family

For every integer d ≥ 1, all 2d complex zeros of the polynomial Ψ_d are real. Here Ψ_d is an explicit deformation of the Laguerre polynomial L_d^(−1/2) at its critical scaling: each coefficient is multiplied by a moment M_k built from factorials and harmonic numbers. These moments turn out to be the moment sequence of a concrete compound-Poisson random variable, and the theorem says that averaging the Laguerre polynomial against that randomness never pushes a zero off the real line - at any degree.