{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-23-triangle-lower-tail-zhao-conjecture-gpt-6-astra-claude/","md":"https://postcutoff.com/e/2026-10-23-triangle-lower-tail-zhao-conjecture-gpt-6-astra-claude/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-23-triangle-lower-tail-zhao-conjecture-gpt-6-astra-claude","date":"2026-10-23","date_precision":"day","short_title":"Triangle lower tail: Zhao's conjecture settled with GPT-6 Astra and Claude","deck":null,"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.32879: The lower tail of triangle counts in sparse random graphs (Huang, Li)","url":"https://arxiv.org/abs/2610.32879","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GitHub: sparse-triangle-lower-tail-verification","url":"https://github.com/fifalsp/sparse-triangle-lower-tail-verification","type":"code","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-10-26","updated":"2026-10-26","orgs":["Tsinghua University","Yale University","OpenAI","Anthropic"],"title":"Lower tail of triangle counts in sparse random graphs solved, settling Zhao's conjecture and a Janson–Warnke problem, with GPT-6 Astra and Claude Opus 5.5 (Lean formalization)","summary":"arXiv 2610.32879 (23 Oct 2026, 66 pages): for the Erdős–Rényi graph with n^(-1/2) ≪ p ≪ 1 and every η ∈ (0,1), the authors solve the variational problem governing P(T_n ≤ ηE[T_n]). Every minimizer is a constant graphon or a two-block step graphon with blocks of equal size, separated by a first-order phase transition at η* = 0.009217…. This \"resolves a conjecture of Zhao and an open problem of Janson and Warnke on triangles\". The AI statement credits GPT-6 Astra discussions with the key strategy steps, Claude Opus 5.5 with help on the computer-assisted proof at the critical point, and GPT-6 Astra / GPT-6.1 Sol with parts of the verification programs and the Lean 4 formalization.","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"],"key_numbers":[],"tags":["math","probability","random-graphs","large-deviations","gpt-6-astra","claude-opus-5-5","gpt-6-1-sol","lean","computer-assisted-proof","arxiv"],"science":{"field":"mathematics","subfield":"probability / random graphs (large deviations)","problem":"Lower tail of triangle counts in sparse G(n,p); conjecture of Yufei Zhao and open problem of Janson–Warnke","result":"Variational problem solved for every η ∈ (0,1): minimizers are constant or balanced two-block graphons, with a first-order transition at η* = 0.009217…","open_since":"","ai_system":["GPT-6 Astra","Claude Opus 5.5","GPT-6.1 Sol"],"human_role":"AI-assisted: authors expected the answer from Zhao's work; GPT-6 Astra suggested key proof strategies, Claude helped with the computer-assisted proof, models wrote parts of the verification code and Lean","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed arXiv preprint with validated numerics and a Lean 4 formalization"},"status":"pending","shock":""},"body_md":"## What happened\n\nHow unlikely is it that a random graph has far fewer triangles than expected, and what does such a graph look like? For sparse\nrandom graphs, earlier work turned both questions into a minimisation problem over graphons. Huang and Li solve that problem for\nevery target fraction η. A graph conditioned to have few triangles looks either like a thinner uniform random graph or, below a\nthreshold of about 0.92% of the expected count, like a balanced two-community graph. The switch between the two is abrupt.\n\nThe paper says the expected answer came from Zhao's work. The hard part was finding inequalities that are tight exactly at the\ntwo-block graphons. Its AI statement credits GPT-6 Astra with suggesting how to apply Kwan–Pham's penalized value theorem and the\ncontinuation argument, and Claude Opus 5.5 with the computer-assisted proof at the critical point.\n\n## Why it matters\n\nIt settles a named problem in random-graph large deviations. The proof mixes computer-assisted certificates with a Lean\nformalization, and the paper says which model contributed which step.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint with validated numerics and a Lean 4 formalization","sources":[]}],"related":[{"id":"2026-10-26-ai-assisted-results-october-26-listing","url":"https://postcutoff.com/e/2026-10-26-ai-assisted-results-october-26-listing/","date":"2026-10-26","date_precision":"day","short_title":"AI-assisted results in the 26 Oct 2026 sweep","deck":"Claude-written Lean proof of a frame-indifferent rank-one convex function that is not quasiconvex, a sculptor and Claude Code prove Takemura's conjecture, Claude's contribution to deterministic root-finding, and Zenodo records","takeaway":"Monday's arXiv announcement (papers submitted 22–23 Oct) brought four results with their own entries: OpenAI's propeller conjecture reproved by Heilman, Tokareva's conjecture disproved by MiniMax agents, the triangle lower tail with GPT-6 Astra and Claude, and graph-connectivity communication complexity with ChatGPT. This page lists the next tier, from Lean proofs written entirely by Claude to papers where the AI only checked the text.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-26","updated":"2026-10-26","orgs":["Anthropic","OpenAI","various researchers"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-26","type":"filed","text":"Created (arXiv PDF read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-26","run":null,"sources_read":"ArXiv PDF read","updated":"2026-10-26","human_review":null,"version":{"date":"2026-10-27"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":176,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":115,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":206,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":145,"in_training_data":false}],"short_url":null}