Bloch constant conjecture (Ahlfors–Grunsky, 1937) disproved
Teng Zhang shows B < 0.47186… − 1.3×10⁻⁹, with a Lean proof generated by Codex
Awaiting review
Importance: major (4 of 5)The takeaway
The value Ahlfors and Grunsky conjectured for Bloch’s constant in 1937 is not the true value. The human proof uses a genus-two translation surface; ChatGPT helped explore approaches and Codex generated the Lean formalization.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean proof, not yet reviewed
- Importance
- Major (4 of 5)
- Last verified
- 14 October 2026
Your AI and this story
- GPT-6 Astra165 days after its cutoff
- Claude Opus 5.5104 days after its cutoff
- Gemini 4 Argon195 days after its cutoff
- Grok 4.7134 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 104 days before it.
Key facts
- Paper: ‘The Bloch constant is strictly smaller than the Ahlfors–Grunsky value’, Teng Zhang (Xi’an Jiaotong University), arXiv 2610.16808 [math.CV], 30 pages, 12 Oct 2026
- Conjecture: Ahlfors and Grunsky (Math. Z. 1937) built a function whose inverse Riemann surface is tiled by equilateral triangles, giving B ≤ A_AG = Γ(1/3)Γ(11/12)/(√(1+√3) Γ(1/4)) = 0.471861653…, and conjectured equality
- Result: B < A_AG (Corollary 1.2) and B < A_AG − 1.3 × 10⁻⁹ (Theorem 1.3); the paper says the explicit bound ‘can certainly be improved’
- Method: starts from a genus-two translation surface tiled by equilateral triangles; its developing map gives an Ahlfors–Grunsky function, which is then deformed to lower the supremum of univalent-disk radii
- Lower bounds for context (paper’s Table 1): Ahlfors √3/4; Chen–Gauthier and later improvements up to √3/4 + 0.0153 (Wikström)
- AI use: ‘ChatGPT was used for English-language editing and proofreading, and as an exploratory tool for discussing possible approaches to selected parts of the argument. The mathematical arguments and proofs in the original manuscript were independently developed, checked, and written by the author. The Lean 4 formalization in [19] was generated using OpenAI’s Codex and checked by the Lean kernel.’
- Lean: cited as github.com/zhangteng2000/Bloch-constant; that URL returned 404 on 14 Oct 2026, so the formalization could not be inspected
What happened
On 12 Oct 2026 Teng Zhang posted arXiv 2610.168081. Bloch’s constant B is the largest r such that every holomorphic f on the unit disk with |f′(0)| = 1 maps some subdomain conformally onto a disk of radius r. In 1937 Ahlfors and Grunsky gave the upper bound A_AG ≈ 0.4718617 and conjectured it was sharp. Zhang proves B < A_AG, and B < A_AG − 1.3×10⁻⁹, by deforming an Ahlfors–Grunsky-type function obtained from a genus-two translation surface tiled by equilateral triangles.
The AI statement credits the mathematics to the author. ChatGPT was used for editing and “as an exploratory tool for discussing possible approaches to selected parts of the argument”, and the Lean 4 formalization of Theorem 1.1, Corollary 1.2 and Theorem 1.3 “was generated using OpenAI’s Codex and checked by the Lean kernel”. The GitHub repository the paper cites returned 404 on 14 Oct, so the formal statement could not be compared with the paper.
No expert reactions were found on 14 Oct (web search; X not searched directly).
Why it matters
If it holds up, a famous extremal problem in geometric function theory has a negative answer after almost 90 years. The AI role is modest in the mathematics but large in the checking: the Lean proof, once public, would let anyone confirm the inequality without trusting the 30-page argument.
What is disputed or not yet verified
| Verification | Lean 4 formalization reported (repository not reachable on 14 Oct); unrefereed preprint |
|---|
Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 2 primary
Primary
Changes
- Filed from the arXiv paper; Lean repository not reachable