VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
Koizumi and Liu conjectured that for every real hyperplane arrangement $\mathcal A$, the coefficients of
$$
\operatorname{Mag}(\mathcal A;-t)
$$
are eventually nonnegative, equivalently that the coefficients of $\operatorname{Mag}(\mathcal A;q)$ eventually alternate in sign.
The conjecture is false. There exists a rank-$6$ real hyperplane arrangement $\mathcal A$ such that
$$
(-1)^\ell[q^\ell]\operatorname{Mag}(\mathcal A;q)<0
$$
for infinitely many $\ell$.
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
GPT-5.6 Sol; GPT-6 Astra
The author reports extensive mathematical interaction with GPT-5.6 Sol and GPT-6 Astra. AI assistance included developing mathematical arguments, performing computations, and drafting and revising the manuscript. The paper does not separately attribute the eventual-sign-alternation counterexample or any particular main theorem to one model. Junnosuke Koizumi states that he critically evaluated the generated material, independently verified the mathematical arguments and references, and takes responsibility for the paper.
The paper contains a complete conventional proof, and the author states that he independently checked the AI-assisted mathematical arguments and references. No independent external expert review, referee report, or proof-assistant formalization is reported. The arXiv submission is a fresh preprint marked “Comments welcome!”
Koizumi constructs a representable simple rank-$6$ matroid $M$ for which
$$
\operatorname{Mag}(M;q)
$$
has a pole of order $4$ at $q=-1$ but a pole of order $5$ at $q=i$.
After writing
$$
A_M(t)=\operatorname{Mag}(M;-t),
$$
the higher-order pole at $t=i$ forces the Taylor coefficients of $A_M(t)$ to fail eventual nonnegativity. Consequently,
$$
(-1)^\ell[q^\ell]\operatorname{Mag}(M;q)<0
$$
for infinitely many $\ell$.
The matroid is representable by twelve vectors in $\mathbb R^6$, so it yields an actual real central hyperplane arrangement and therefore directly disproves Koizumi–Liu's conjecture for real arrangements.
Known method families
construction (source-reported)
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
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.