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

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.

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.