Triangle lower tail: Zhao’s conjecture settled with GPT-6 Astra and Claude
Awaiting review
The takeaway
Dong Huang and Shuangping Li solve the graphon variational problem behind the lower tail of triangle counts in G(n,p) for n^(-1/2) ≪ p ≪ 1. Minimizers are constant or balanced two-block graphons, with a first-order phase transition at η* = 0.009217…. The authors say GPT-6 Astra helped find key steps and Claude Opus 5.5 helped build the computer-assisted proof. The arguments are formalized in Lean 4.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- 3 of 5
- Last verified
- 26 October 2026
Your AI and this story
- GPT-6 Astra176 days after its cutoff
- Claude Opus 5.5115 days after its cutoff
- Gemini 4 Argon206 days after its cutoff
- Grok 4.7145 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 115 days before it.
Key facts
- Authors: Dong Huang and Shuangping Li (affiliations listed in the paper: Tsinghua University, Department of Statistics and Data Science; Yale University, Department of Statistics and Data Science)
- Result: complete description of the lower tail of triangle counts for n^(-1/2) ≪ p ≪ 1; conditioned graph is close to Erdős–Rényi or to a symmetric two-block stochastic block model; threshold η* = 0.009217…
- AI statement (verbatim, excerpt): ‘Discussions with GPT-6 Astra helped the authors identify how the penalized value theorem of Kwan and Pham [50], applied with penalty coefficient γ = µr/3, could determine the optimal value for 0 < r ≤ r− = 0.03965 . . .. These discussions also helped the authors find the scaling argument … With assistance from Claude Opus 5.5, the authors developed the computer-assisted proof at r∗. Further discussions with GPT-6 Astra helped develop the continuation argument, together with a family of computer-assisted certificates … Parts of the verification programs and of the Lean 4 formalization were written by GPT-6 Astra and GPT-6.1 Sol.’
- Formal verification: ‘We have also formalized the arguments of this paper in Lean 4 with mathlib’; validated-numerics programs at github.com/fifalsp/sparse-triangle-lower-tail-verification
- Background: earlier work reduced both the rate and the structure of the lower tail to a graphon variational problem; Zhao conjectured the constant/two-block answer
What happened
How unlikely is it that a random graph has far fewer triangles than expected, and what does such a graph look like? For sparse random graphs, earlier work turned both questions into a minimisation problem over graphons. Huang and Li solve that problem for every target fraction η. A graph conditioned to have few triangles looks either like a thinner uniform random graph or, below a threshold of about 0.92% of the expected count, like a balanced two-community graph. The switch between the two is abrupt.
The paper says the expected answer came from Zhao’s work. The hard part was finding inequalities that are tight exactly at the two-block graphons. Its AI statement credits GPT-6 Astra with suggesting how to apply Kwan–Pham’s penalized value theorem and the continuation argument, and Claude Opus 5.5 with the computer-assisted proof at the critical point.
Why it matters
It settles a named problem in random-graph large deviations. The proof mixes computer-assisted certificates with a Lean formalization, and the paper says which model contributed which step.
What is disputed or not yet verified
| Verification | Unrefereed arXiv preprint with validated numerics and a Lean 4 formalization |
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Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 2 primary
Primary
Changes
- Filed (arXiv PDF read)