A Boundary-Constant Obstruction to Real-Rooted Riemann Incomplete-Gamma Approximants

Let H_N denote the Nth Fourier partial sum obtained from Riemann’s incomplete-gamma expansion, and let c_N = Σn>N(4πn²−1)e^(−πn²) > 0. For every finite N ≥ 1 and every 0 < t ≤ 1, we prove that the even entire function FN,t = H_N + t c_N has a nonreal zero. The direct arithmetic-index truncation is the endpoint FN,1. Indeed, FN,t(x) tends to t c_N at both ends of the real axis, while FN,t(i/2) − t c_N = 1/2 − c_N ≠ 0; moreover, FN,t has order at most one. If all zeros were real, the positive boundary limit would leave only finitely many zeros, and Hadamard factorization together with evenness would force FN,t to be constant. Pólya’s theorem already treats the unshifted endpoint t = 0. The result identifies the obstruction created by the positive constant shift and separates it from Haglund’s next-summand zero-motion conjectures. It has no implication for the Riemann hypothesis itself.