Haglund's Zero-Trajectory Conjecture for the First Riemann Xi Approximant
candidateconfidence 50%
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Precise statement
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.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
ChatGPT and Codex
The disclosure was rewritten on 25 August, and now reads in full: "OpenAI ChatGPT and Codex models performed most of the proof search, mathematical derivation, computational implementation, Lean proof development, literature discovery, organization, and drafting. The author supplied the research methodology, problem framing, constraints, evaluation criteria, and iterative oversight. The author remains responsible for every mathematical statement, proof, citation, and submission decision." That names the systems and attributes the proof search itself to them, which is why this is filed as AI-discovered. It replaces the earlier version - "AI systems were used extensively in mathematical derivation, Lean proof development, literature discovery, organization, and typesetting" - which named no system and attributed no step, and on which this entry was first filed a rung lower as co-developed. The manuscript still gives no version for either system, so the model field records only what it says.
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.
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.
Known method families
computation (source-reported)
Source-reported tools: computation.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
The source status is candidate and must not be represented as solved.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.