Finite Spectral Data Do Not Determine a Real-Rooted Entire Function

Abstract

We prove an explicit non-uniqueness theorem for real-rooted entire functions. Starting from an even genus-zero canonical product with simple positive zeros $\gamma_n$ of exponent of convergence one, and fixing arbitrary integers $k,J$, we perturb the infinite tail while compensating the first $J$ reciprocal-zero power sums by moving only $J$ nearby zeros. The implicit-function Jacobian is an exact scaled Vandermonde determinant. The resulting function has only real zeros, shares the first $k$ positive zeros and the even Taylor jet through order $2J$ with the reference function, but is not equal to it. We also prove a companion compactness principle: local divisor convergence and point normalization identify the target under a genuinely uniform order-one growth envelope, whereas separate per-approximant order bounds do not. The results isolate a precise missing compactness ingredient in spectral approximation schemes for the Riemann Xi function; they do not assume or prove the Riemann hypothesis.