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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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Aug 22, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: site-confirmed. Publication: preprint. AI contribution: ai-discovered. Imported under CC BY 4.0.

Canonical aliases: Haglund's Zero-Trajectory Conjecture for the First Riemann Xi Approximant · Haglund Conjecture 4, k=1

Confidence: Not scored

Registry verification: site confirmed · preprint · candidate

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Attribution

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model · ai model contributor · OpenAI

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Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

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

specialist registry · passed

Site-confirmed: the computational certificate was replayed here on 26 August 2026 and it passes. Cloned mbaccaro-dev/mathematical-proofs, installed python-flint 0.9.0 into a venv per requirements.txt, ran its own reproduce.py in full (4 min 17 s), and got STATUS=PASS with claim_ceiling=Haglund_Conjecture_4_for_k_equals_1_only - via MANIFEST_PASS at 289 files, S1_STRUCTURE, NONREAL_JOIN, GLOBAL_LIFT, OUTER_REGION, and crucially S2_SCIENTIFIC_PASS at patches=238, source_calls=60930, the paper's own figures for the certified first-quadrant proposition, plus S3 at 323 patches and 46,514 calls. What that does not settle: the script recomputes its own certificates against its own manifest, so it shows the computation replays and is internally consistent, not that the interval arithmetic implies the theorem (argued in prose) nor any analytic step around it. The Lean is unchanged from first review - no sorry, admit or declared axiom in the Solution closure, but it verifies one abstract collision theorem, and the repository still says the atlas, the incomplete-gamma estimates and the assembly are not consequences of it. End-to-end Lean remains pending by the author's account; the paper is unrefereed. One packaging defect found while replaying: on a fresh clone reproduce.py aborts at the manifest gate. Of 289 hashes, 12 match only as CRLF, all Windows console receipts, and with no .gitattributes no checkout satisfies both sets; the run above needed those 12 converted first.

Source ↗

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

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Manuscript source, Lean project and verification certificate

code · pending

code link supplied by the VibeMathed registry.