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Precise statement
We show that for every sufficiently large genus $g$, there exists a closed hyperbolic surface $S_g$ with systole $\mathrm{sys}(S_g)\geq \log g-12\log\log g$. In particular,
$$
\liminf_{g\to \infty}\frac{\max\{\mathrm{sys}(S):S\in \mathcal{M}_g\}}{\log g}\geq 1,
$$
improving the previously known bound $2/9$. This note is a continuation of our previous work on the diameter of finite covers arXiv:2608.12887, using the same framework of constant-twist pants decomposition to study systoles.
The proof was developed by GPT-5.6 Sol through an extended discussion with the author.
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
Disclosed twice: in the abstract, and in a dedicated section 1.3 "Declaration on the use of AI", which reads in full: "Starting from the constant-twist pants decomposition approach described in this note, GPT-5.6 Sol (OpenAI) developed the first complete proof of the main theorem through an extended discussion with the author. The proof in this manuscript is checked, simplified and reorganized by the author. The author takes full responsibility for the content and correctness of this manuscript."
AI-discovered on that wording: the model produced the proof and the human verified and wrote it up, which is what the tier means. The framework it started from was not the model's - the constant-twist pants decomposition comes from Cai and Luo's earlier work on the diameter of finite covers (arXiv:2608.12887), and the note is explicitly a continuation of it. So the human set the approach and the model built the proof inside it.
Unreviewed: an arXiv preprint one day old (v1, 27 August 2026, math.GT), unrefereed, with no formalization and no computational certificate, so there was nothing mechanical to re-run and no mathematics was checked here. What was verified on 28 August 2026: the paper exists at arXiv:2608.26660 with this title and author; the theorem and the $\liminf\ge1$ corollary are its abstract and Theorem 1; the AI declaration is section 1.3, quoted in the AI-role note; and every prior-work claim is as the introduction states - Katz-Sabourau's $19/120$, Liu-Petri's $2/9$, Petri-Walker's constant $1$ along a subsequence, Brooks and Buser-Sarnak's $\limsup\ge4/3$, and the area bound $\max\mathrm{sys}\le2\log(4g-2)$.
The theorem improves the best lower bound valid in *every* sufficiently large genus from asymptotic constant $2/9$ to $1$. The every-genus ladder it climbs is Katz-Sabourau's $19/120$ and then Liu-Petri's $2/9$, the latter also by a random construction. Constant $1$ was already reached by Petri-Walker along a subsequence of genera, following Erdos-Sachs, so the new contribution is achieving it uniformly rather than the constant itself.
The asymptotic problem stays open, and the remaining gap is wide: Brooks and Buser-Sarnak give $\limsup\ge4/3$, while the elementary area bound is $\max\mathrm{sys}(S)\le2\log(4g-2)$, asymptotically $2\log g$. So this closes much of the liminf gap and determines no optimal constant.
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.