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

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Haglund's Zero-Trajectory Conjecture for the First Riemann Xi Approximant parent of this event

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. parent of this event

Manuscript source, Lean project and verification certificate evidence for this event

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

Verification archive: reproduce.py, source manifest and interval certificates evidence for this event

Ahn, On the Zeros of Component Functions of the Riemann Xi Approximates (Penn MA thesis, 2012) evidence for this event

Some conjectures on the zeros of approximates to the Riemann Ξ-function and incomplete gamma functions evidence for this event

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