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Koizumi–Liu Eventual Sign Alternation Conjecture

Koizumi and Liu conjectured that for every real hyperplane arrangement A\mathcal A, the coefficients ofMag⁡(A;−t) \operatorname{Mag}(\mathcal A;-t) are eventually nonnegative, equivalently that the coefficients of Mag⁡(A;q)\operatorname{Mag}(\mathcal A;q) eventually alternate in sign.

The conjecture is false. There exists a rank-66 real hyperplane arrangement A\mathcal A such that(−1)ℓ[qℓ]Mag⁡(A;q)<0 (-1)^\ell[q^\ell]\operatorname{Mag}(\mathcal A;q)<0 for infinitely many ℓ\ell.

Result
Disproved(see note)
Status
Resolved
AI contribution
AI co-developed
Method
Construction
Field
Matroid theory
Posed by
Junnosuke Koizumi and Ye Liu
Year posed
2026
Years open
0y
Solved
2026-09-07
Model
GPT-5.6 Sol; GPT-6 Astra
Vendor
OpenAI
Collaborators
Junnosuke Koizumi
Verification
Unreviewed
Publication
Preprint
Significance
12 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Koizumi constructs a representable simple rank-66 matroid MM for whichMag⁡(M;q) \operatorname{Mag}(M;q) has a pole of order 44 at q=−1q=-1 but a pole of order 55 at q=iq=i.

After writingAM(t)=Mag⁡(M;−t), A_M(t)=\operatorname{Mag}(M;-t), the higher-order pole at t=it=i forces the Taylor coefficients of AM(t)A_M(t) to fail eventual nonnegativity. Consequently,(−1)ℓ[qℓ]Mag⁡(M;q)<0 (-1)^\ell[q^\ell]\operatorname{Mag}(M;q)<0 for infinitely many ℓ\ell.

The matroid is representable by twelve vectors in R6\mathbb R^6, so it yields an actual real central hyperplane arrangement and therefore directly disproves Koizumi–Liu's conjecture for real arrangements.

What the AI did

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.

Verification

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!”

Source

Submitted by VibeGene on

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