Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in \(G(n,p)\) is at most \(C p_{\mathrm E}(n,H)(1+\log_2 h)\), with universal C. Here \(p_{\mathrm E}\) is the least density at which every subgraph of H has expected copy count at least 1/2. (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 paper states
In The second Kahn–Kalai conjecture, 24 September 2026
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
For every n\ge2 and every finite simple graph H with h=e(H)\ge1 edges and at most n vertices,
In particular,
Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.
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.
SecondKahnKalaiThe paper states
In The second Kahn–Kalai conjecture, 24 September 2026
For every n\ge2 and every finite simple graph H with h=e(H)\ge1 edges and at most n vertices,
In particular,
The words they share: vertices, graph, edges, simple, finite, min, log, kahn, kalai, second.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
SecondKahnKalai.lean, the release lists it as a main result.
theorem secondKahnKalaiBounds (n : ℕ) (V : Type u) [Fintype V] (H : SimpleGraph V) (hn : 2 ≤ n) (hcard : Fintype.card V ≤ n) (hedge : 1 ≤ edgeCount H) : SecondKahnKalaiBounds n H := by sorryThis 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
SecondKahnKalaiBounds Model.lean, line 58MathVet 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: secondKahnKalaiBounds (n) (V) (H) (hn : 2 ≤ n) (hcard : Fintype.card V ≤ n) (hedge : 1 ≤ edgeCount H) : criticalThreshold n H ≤ min 1 (2048 * exp 50 * expectationThreshold n H * (1 + logTwo (edgeCount H))) ∧ criticalThreshold n H ≤ 6144 * exp 50 * expectationThreshold n H * logTwo n; criticalThreshold = sInf{p | P(G(n,p) contains a copy) ≥ 1/2} using Mathlib copyCount, expectationThreshold = sInf{p | ∀ F : H.Subgraph, E[copies of F] ≥ 1/2}. Matches the summary’s p_E definition with explicit universal C.
“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.
cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: 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.
OpenAI’s own note on what its Lean covers
The second Kahn–Kalai conjecture compares the threshold for a random graph to contain a fixed graph with its expectation threshold. The formalized result proves the comparison up to a universal factor times $1+\log_2|E(H)|$, and hence up to a universal factor times $\log_2 n$, for every graph $H$ with at least one edge and at most $n$ vertices, $n\ge2$. The bounds use the actual containment and expectation thresholds.
Second Kahn–Kalai threshold bounds SecondKahnKalai
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem with its explicit constants: p_c(n, H) ≤ min{1, 2048e^50 p_E(n, H)(1 + log₂ e(H))} and the log₂ n form.
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.
SecondKahnKalaiThree 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. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 179 s; peak memory 3.3 GiB). | 11 October 2026 | Original report |
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.
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
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.
Offer to write an explanationCompare the Lean statement with the paperSuggest a source attributionSuggest a change
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 “Major math journal”. The page says that nothing on it is verified.
Author not stated. Source
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Why this passage is shown
Theorem 1.1
In The second Kahn–Kalai conjecture, 24 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, SecondKahnKalai
Proof-verification report
MathVet, SecondKahnKalai
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, SecondKahnKalai
Lean statement
SecondKahnKalai