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

Haglund's Conjecture 4 reads: for $k\ge1$, the imaginary part of each non-real zero of $\Xi_k(z)+t\Phi_{k+1}(z)$ decreases monotonically as $t$ goes from 0 to 1, where the $\Phi_n$ are the incomplete-gamma summands of Riemann's series for $\Xi$ and $\Xi_k=\sum_{n\le k}\Phi_n$. This work proves the case $k=1$, the pencil $\Phi_1+t\Phi_2$: every non-real zero in the closed first quadrant is simple and the imaginary part of its analytic branch strictly decreases. It adds two statements Conjecture 4 does not itself assert - no non-real branch escapes to infinity on a bounded forward parameter interval, and at a real collision of any finite multiplicity the full local Weierstrass-Puiseux multiset stays real to the right. The cases $k\ge2$ remain open, and nothing is claimed about the zeros of $\Xi$ or the Riemann hypothesis.

Proves Haglund's Conjecture 4 for $k=1$: every non-real first-quadrant zero of $\Phi_1+t\Phi_2$ is simple with strictly decreasing imaginary part, no branch escapes forward, and every finite-multiplicity real collision stays real afterwards. The cases $k\ge2$ remain open. Two readings worth separating: Conjecture 4 asserts the monotone descent alone, so the no-escape and stays-real statements are this paper's own additions rather than Haglund's text, and they are the stronger part of the theorem. The descent itself, part (i), is the part that rests on the unavailable interval-arithmetic certificate.