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Large systoles in every genus: the constant 4/3

For g≥2g\geq 2 let Sys(g)\mathrm{Sys}(g) be the supremum of the systoles of closed oriented hyperbolic surfaces of genus gg. Determine the asymptotic behaviour of Sys(g)/log⁡g\mathrm{Sys}(g)/\log g; in particular, what is the best constant cc with lim inf⁡g→∞Sys(g)/log⁡g≥c\liminf_{g\to\infty}\mathrm{Sys}(g)/\log g\geq c?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Construction
Field
Hyperbolic geometry
Posed by
—
Year posed
—
Years open
—
Solved
2026-10
Model
ChatGPT 5.6 (construction); ChatGPT 6.1 Sol and Astra 6.0 via Codex (formalization)
Vendor
OpenAI
Collaborators
Nihar Gargava, Yifei Cai, Mingkun Liu
Verification
Lean-checked, statement unaudited
Publication
Announced
Significance
22 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

lim inf⁡g→∞Sys(g)/log⁡g≥4/3\liminf_{g\to\infty}\mathrm{Sys}(g)/\log g\geq 4/3. This raises the every-genus constant from 11 (Cai, August 2026, the entry linked below) to 4/34/3, the value Buser-Sarnak reached only along a subsequence of genera. The optimal constant is still open: the area bound gives at most 22.

What the AI did

The README says: "The autoformalization was done with the help of Codex running various version of ChatGPT. Most of the code was written by ChaGPT 6.1 Sol with various proof outlines written by ChatGPT Astra 6.0. The proof itself was based on a candidate construction given by ChatGPT 5.6." formalization.yaml adds that Nihar Gargava directed the mathematical and implementation goals and Codex agents developed the Lean proofs.

As the submitter describes it, the model saw that Buser-Sarnak's arithmetic surfaces can be glued together with Katz-Sabourau's techniques, using an analytic number theory input of Cook-Magyar on prime cubes, to reach the constant 4/3 in every large genus.

Verification

Read by this site on 3 November 2026 at commit ddf43a1, not rebuilt. The headline statement in Challenge.lean was read against the problem: closed oriented hyperbolic surfaces of genus g are discrete, freely acting subgroups of SL(2,R) with compact quotient of the upper half-plane and an isomorphism to the standard genus-g surface group; the systole is the infimum of displacements by nontrivial deck transformations; Sys(g) is the supremum; the claim is that 4/3 is at most the liminf of Sys(g)/log g. Lean's junk values for an unbounded supremum or liminf would make the claim harder, not easier. comparator.json asks Comparator to match Solution.lean against it with only propext, Classical.choice and Quot.sound. A grep of the 1,088 Lean files found no sorry outside Challenge.lean and no admit, native_decide, unsafe or implemented_by; the only axiom lines are examples inside a vendored docstring. The statement was written by the author's own agents (formalization.yaml: review self-assessed), and a Palomar listing is a precondition for, not an independent audit of, statement fidelity, so the entry is Lean-checked. Comparator was not run here. There is no paper yet.

Sources

Submitted by SwiftOsprey610 on

Changelog2 changes

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