Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in \(O(n^8)\) time for every graphical labeled degree sequence. The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time. (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.
The two independent assessments disagree on how much of the result described above is expressed by the Lean statements. Read the assessments
In Polynomial mixing of the switch chain for every graphical degree sequence, 25 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
For every n\ge4 and every graphical degree vector d,
If |\Omega_d|>1, the spectral gap of P_d is at least [24n^2\binom n4]^{-1}.
In Polynomial mixing of the switch chain for every graphical degree sequence, 25 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
There is a uniform randomized algorithm that, given any finite graphical labeled degree vector d=(d_1,\ldots,d_n), uses unbiased random bits, terminates almost surely, and outputs a graph chosen exactly uniformly from the simple undirected graphs on \{1,\ldots,n\} with labeled degree vector d. Its expected bit running time is bounded by a fixed polynomial in the binary encoding length of d.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
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.
SwitchChainThe paper states
In Polynomial mixing of the switch chain for every graphical degree sequence, 25 September 2026
For every n\ge4 and every graphical degree vector d,
If |\Omega_d|>1, the spectral gap of P_d is at least [24n^2\binom n4]^{-1}.
The words they share: spectral, mix, gap, least, graphical, degree, chain, switch, binom.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
SwitchChain.lean, the release lists it as a supporting statement.
theorem switch_chain_main : ∀ (n : Nat) (d : Fin n → Nat), 4 ≤ n → (∀ v, d v ≤ n - 1) → Graphical n d → ((∃ h : (∃ t : Nat, MixedAt n d t), mixingTime n d h ≤ 2 * n ^ 8) ∧ (stateCount n d = 1 → (MixedAt n d 0 ∧ ∀ h : (∃ t : Nat, MixedAt n d t), mixingTime n d h = 0)) ∧ (1 < stateCount n d → hasSpectralGapAtLeast n d ((1 : ℝ) / (24 * (n : ℝ) ^ 2 * (Nat.choose n 4 : ℝ))))) := 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.
The file states two more theorems: switch_, switch_.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
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: Mixing time <= 2 n^8 (TV 1/4, worst-case start), spectral gap, TV bound and connectivity are stated; the exact uniform sampler (expected polynomial bit time) is not.
Definitions the group asks readers to compare with the paper: Mixing is defined by hand: kernelPow (matrix powers via sums over allGraphStates), tvDistanceFrom := (1/2) ∑ |P^t(G,H) - 1/|Ω||, MixedAt, mixingTime := Nat.find h (the proof argument h is part of the statement), hasSpectralGapAtLeast via Dirichlet form; standard in content, no Mathlib Markov-chain objects
“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 a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
The gap they name: The exactly uniform sampler is not stated (also n ≥ 4, a harmless restriction).
“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.
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 Kannan–Tetali–Vempala conjecture asks for polynomial mixing of the switch chain for every graphical degree sequence. The formalization proves the simple undirected case: for $n\ge4$, the lazy chain that proposes switches on four vertices has total-variation mixing time at distance $1/4$ at most $2n^8$. It also proves switch connectivity, the stated exponential total-variation bound, and a spectral gap of at least $1/(24n^2\binom n4)$ when more than one graph realizes the degree sequence.
The paper’s exactly uniform sampling algorithm is outside these selected chain estimates.
Polynomial mixing and connectivity for every graphical degree sequence SwitchChain
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the switch chain mixes in at most 2n⁸ steps for every graphical degree sequence, with connectivity and the spectral-gap bound. The exact sampling corollary is outside the statement.
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.
SwitchChainTwo 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 |
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.
No paper elsewhere in the release cites this family’s papers.
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
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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.
Author not stated. Source
tparker, Philip Weiss, Will Sawin, Nik Weaver, Pace Nielsen, mathworker21, Joshua P. Swanson, Timothy Chow, Daniel Soltesz, MathOverflow: how many of the released results went against the conventional wisdom?, 10 October 2026. Source
A researcher who worked on switch chain mixing says the team was convinced it mixes rapidly and is glad the release proved it.
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 Polynomial mixing of the switch chain for every graphical degree sequence, 25 September 2026
Why this passage is shown
Corollary 1.2 (Exact uniform sampling)
In Polynomial mixing of the switch chain for every graphical degree sequence, 25 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, SwitchChain
Proof-verification report
MathVet, SwitchChain
Lean statement
SwitchChain