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Large systoles in every sufficiently large genus

We show that for every sufficiently large genus gg, there exists a closed hyperbolic surface SgS_g with systole sys(Sg)≥log⁡g−12log⁡log⁡g\mathrm{sys}(S_g)\geq \log g-12\log\log g. In particular,
lim inf⁡g→∞max⁡{sys(S):S∈Mg}log⁡g≥1, \liminf_{g\to \infty}\frac{\max\{\mathrm{sys}(S):S\in \mathcal{M}_g\}}{\log g}\geq 1,
improving the previously known bound 2/92/9. 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.

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Construction
Field
Hyperbolic geometry
Posed by
—
Year posed
—
Years open
—
Solved
2026-08-27
Model
GPT-5.6 Sol
Vendor
OpenAI
Collaborators
Yifei Cai
Verification
Unreviewed
Publication
Preprint
Significance
22 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

The theorem improves the best lower bound valid in *every* sufficiently large genus from asymptotic constant 2/92/9 to 11. The every-genus ladder it climbs is Katz-Sabourau's 19/12019/120 and then Liu-Petri's 2/92/9, the latter also by a random construction. Constant 11 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 lim sup⁡≥4/3\limsup\ge4/3, while the elementary area bound is max⁡sys(S)≤2log⁡(4g−2)\max\mathrm{sys}(S)\le2\log(4g-2), asymptotically 2log⁡g2\log g. So this closes much of the liminf gap and determines no optimal constant.

What the AI did

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.

Verification

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 lim inf⁡≥1\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 19/12019/120, Liu-Petri's 2/92/9, Petri-Walker's constant 11 along a subsequence, Brooks and Buser-Sarnak's lim sup⁡≥4/3\limsup\ge4/3, and the area bound max⁡sys≤2log⁡(4g−2)\max\mathrm{sys}\le2\log(4g-2).

Source

Submitted by VibeGene on

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