Haglund's Zero-Trajectory Conjecture for the First Riemann Xi Approximant

Abstract

This paper proves the first case of Haglund's zero-trajectory conjecture for incomplete-gamma approximants to the Riemann Xi function. Every nonreal zero in the first interpolation is simple and moves strictly toward the real axis as the parameter increases. The zero branches cannot escape forward. After a real collision of any finite multiplicity, the complete local multiset remains real on the forward side. The proof combines exact analytic identities with reproducible interval-arithmetic verification. The result covers only the first interpolation. Higher cases remain open, and no theorem about the zeros of the Xi function or the Riemann hypothesis is claimed.