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The sharp threshold for the multiplicative Merino-Welsh inequality

It is proved that $T_M(x_*,0)T_M(0,x_*)\ge T_M(1,1)^2$ for every finite matroid $M$ without loops or coloops, and known counterexamples below $x_*$ make the universal threshold sharp. For $2\le x\le x_*$ , the stronger bound $\frac{T_M(x,0)T_M(0,x)}{T_M(1,1)^2}\ge\left(\frac{x^3}{9(x-1)}\right)^{(m-2\kappa)/2}$ holds, where $m$ is the number of elements and $\kappa$ the number of connected components. Parallel doublings of uniform matroids show the exponential rate is optimal throughout the interval. The proof uses weighted broken circuits, matroid decomposition, and exact checks of rational replacements. The graph conjecture at $x=2$ remains open.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 1, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-co-developed. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The sharp threshold for the multiplicative Merino-Welsh inequality · Merino-Welsh sharp threshold

Confidence: Not scored

Registry verification: unreviewed · preprint · resolved

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VibeMathed
registry · event recorded by

GPT-6 Astra
model · ai model contributor · OpenAI

Claude Opus 5
model · ai model contributor · Anthropic

Mingchang Liu
human · human collaborator

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VibeMathed record: The sharp threshold for the multiplicative Merino-Welsh inequality

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Checked here on 27 September 2026. Csikvári's Conjecture 7.1 was located in arXiv:2502.19196v2, the version the submission links, and the Zenodo record 22911905 of 23 September was read: the threshold is stated as $x_* = 2.2266815969\ldots$ , the largest real root of $x^3 = 9(x-1)$ , with the quantitative bound on $2 \le x \le x_*$ and sharpness of the exponential rate from parallel doublings of uniform matroids. The tools named are a weighted broken-circuit comparison, the Cunningham-Edmonds decomposition and exact polynomial checks of a finite rational replacement table; the table and a checker are in the linked V1.1 release. The mathematics was not checked here, the rational table was not recomputed, and the author's own note is correct that critical AI reviews are not verification. Four days old, no referee.