The sharp terminal leave in random triangle removal
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Starting from the complete graph on n vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to \(n^{3/2}/(2\sqrt2)\), with mean-square convergence after normalization by n3/2. This proves the triangle case of the Joos–Kühn sharp-constant conjecture. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
The release does not say which paper theorem its catalogue describes. The statement above shares the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
Starting from the complete graph on n vertices, repeatedly remove the three edges of a uniformly chosen remaining triangle. We prove that the number of edges left at termination, divided by n3/2, converges in L2 to \(1/(2\sqrt2)\). This proves the triangle case of the sharp-constant conjecture of Joos and Kühn. In particular, the same limit holds in probability and for the normalized expectation.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
TriangleRemoval.lean, the release lists it as a supporting statement.
theoremsharp_terminal_leave : Tendsto (fun n => expectation n (fun G => (normalizedLeave n G - sharpConstant) ^ 2)) atTop (𝓝 0) ∧ (∀ ε : ℝ, 0 < ε → Tendsto (fun n => probability n (fun G => ε < |normalizedLeave n G - sharpConstant|)) atTop (𝓝 0)) ∧ Tendsto (fun n => expectation n (fun G => (G.card : ℝ)) / normalization n) atTop (𝓝 sharpConstant) := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note:sharp_terminal_leave states, for the Mathlib-PMF Markov chain evolve (completeGraph n) (n.choose 2) that deletes a uniformly chosen remaining triangle (PMF.uniformOfFinset (triangles G)), (i) E[(|G|/n^(3/2) - 1/(2√2))^2] → 0, (ii) convergence in probability, (iii) E|G|/n^(3/2) → 1/(2√2). This is the summary’s mean-square convergence; the process is absorbing at triangle-free graphs so running C(n,2) steps gives the terminal law.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
Start with the complete graph on $n$ vertices and repeatedly delete the three edges of a uniformly chosen remaining triangle. The formalization proves that the number of edges at termination, divided by $n^{3/2}$, converges in $L^2$ to $1/(2\sqrt2)$. It also states convergence in probability and convergence of the normalized expectation to the same constant. This is the triangle case of the sharp terminal-leave conjecture of Joos and Kühn.
AI-drafted comparisons, with no recorded human review
Sharp terminal edge count in random triangle removalTriangleRemoval
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the terminal edge count of random triangle removal, divided by n^{3/2}, converges in L² to 1/(2√2).
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Sharp terminal edge count in random triangle removalTriangleRemoval
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
These reports examine claims in the papers using HOL Light, a different proof assistant, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
q = (1+2Dt)^(−1/2) solves q′ = −Dq³ with q(0) = 1, and the leave constant tends to 1/(2√2)
HOL Light card 188
Reported by
Ben Knill, openai-math-hol-cards
Method
Proof of a selected step in HOL Light
Their verdict
“PASS” The report records a proof of a supporting fact.
What it covers
The heuristic’s arithmetic only; nothing about the random process
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 188No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded.Open the form
Compare the proposed paper and Lean matches
Family 188Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In The sharp terminal leave in random triangle removal, 25 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
starting, triangle
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Independent assessment
MathVet
Their term
“full”
What their term means
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state the headline as OpenAI wrote it.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.