Proves an \(\Omega(n^3)\) lower bound for the border determinantal complexity of the \(n\times n\) permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least \(cn^3\), for an absolute c > 0 and all sufficiently large n; the same bound therefore holds for exact representations. (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 A cubic lower bound for border determinantal complexity of the permanent, 24 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
There exist absolute constants c>0 and m_0 such that, for every integer m\ge m_0,
One may take c=1/(5\,529\,600e) and m_0=1408.
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PermanentCubicThe paper states
In A cubic lower bound for border determinantal complexity of the permanent, 24 September 2026
There exist absolute constants c>0 and m_0 such that, for every integer m\ge m_0,
One may take c=1/(5\,529\,600e) and m_0=1408.
The words they share: integer, cubic, border, lower, permanent, bound.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
PermanentCubic.lean, the release lists it as a supporting statement.
theorem permanent_cubic_lower_bounds : (∀ (m n : ℕ), 1408 ≤ m → HasBorderDeterminant n (permanentPolynomial m) → (m : ℝ)^3 / (5529600 * Real.exp 1) ≤ (n : ℝ)) ∧ (∀ (m n : ℕ), 1408 ≤ m → HasExactDeterminant n (permanentPolynomial m) → (m : ℝ)^3 / (5529600 * Real.exp 1) ≤ (n : 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.
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
SmoothInitialFormThe paper states
No paper theorem is paired with this Lean statement.
no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements it shares words with:
The Lean statement
No paper theorem is paired with it.
SmoothInitialForm.lean, the release lists it as a supporting statement.
theorem smooth_initial_form_lower_bound (d r : ℕ) (g : Poly (Fin d)) (a : Fin d → ℂ) (hd : 2 ≤ d) (hr : 2 ≤ r) (hG : homogeneousComponent r (affineSubstitution a (fun i j : Fin d => if i = j then 1 else 0) g) ≠ 0) (hlow : ∀ j < r, homogeneousComponent j (affineSubstitution a (fun i j : Fin d => if i = j then 1 else 0) g) = 0) (hs : ∀ y : Fin d → ℂ, eval y (homogeneousComponent r (affineSubstitution a (fun i j : Fin d => if i = j then 1 else 0) g)) = 0 → (∀ i, eval y (pderiv i (homogeneousComponent r (affineSubstitution a (fun i j : Fin d => if i = j then 1 else 0) g))) = 0) → y = 0) : ∀ n : ℕ, (HasBorderDeterminant n g → ((r - 1 : ℕ) : ℝ) * ((d - 1 : ℕ) : ℝ) / (4 * Real.exp 1) ≤ (n : ℝ)) ∧ (HasExactDeterminant n g →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.
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.
HasBorderDeterminant Model.lean, line 132HasExactDeterminant Model.lean, line 139MathVet 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: None; the algebraic-branching-program consequences are not stated (docs). Border complexity is encoded as sequential coefficientwise limits (equivalent to Zariski closure here).
“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 formalization proves a cubic lower bound for both exact and border determinantal representations of the complex $m\times m$ permanent. For $m\ge1408$, every affine-linear determinant representation of size $n$, including coefficientwise limits, satisfies $n\ge m^3/(5529600e)$.
A general supporting theorem applies to a polynomial in $d\ge2$ variables whose first nonzero homogeneous Taylor term has degree $r\ge2$ and no nonzero singular zero. Its exact and border determinantal sizes are at least $(r-1)(d-1)/(4e)$. The paper’s algebraic-branching-program consequences are outside these statements.
Cubic lower bounds for permanent determinantal complexity PermanentCubic
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem with its explicit constants: for m ≥ 1408, every exact or border determinantal representation of the m × m permanent has size at least m³/(5529600e).
Determinantal lower bound from a smooth initial form SmoothInitialForm
Unreviewed AI draft: states a supporting result. States the general supporting theorem behind the permanent bound: determinantal size at least (r − 1)(d − 1)/(4e) for a polynomial whose initial form has degree r and no nonzero singular zero.
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.
PermanentCubicThree 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 476 s; peak memory 6.6 GiB). | 11 October 2026 | Original report |
SmoothInitialFormThree 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 186 s; peak memory 6.4 GiB). | 10 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.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
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Why this passage is shown
Theorem 1.1
In A cubic lower bound for border determinantal complexity of the permanent, 24 September 2026
Why this passage is shown
Corollary 9.3 (Smooth initial form)
In A cubic lower bound for border determinantal complexity of the permanent, 24 September 2026
Why this passage is shown
Theorem 2.3 (Lower bound for smooth homogeneous forms)
In A cubic lower bound for border determinantal complexity of the permanent, 24 September 2026
Why this passage is shown
Proposition 3.1 (Coefficient projection)
In A cubic lower bound for border determinantal complexity of the permanent, 24 September 2026
Why this passage is shown
Proposition 7.3 (A smooth form obtained from the permanent)
In A cubic lower bound for border determinantal complexity of the permanent, 24 September 2026
Why this passage is shown
Proposition 7.2 (Smooth linear section)
In A cubic lower bound for border determinantal complexity of the permanent, 24 September 2026
Why this passage is shown
Theorem 1.1
In A cubic lower bound for border determinantal complexity of the permanent, 24 September 2026
Why this passage is shown
Proposition 8.3
In A cubic lower bound for border determinantal complexity of the permanent, 24 September 2026
Why this passage is shown
Proposition 6.5 (Dimension of a matrix fiber)
In A cubic lower bound for border determinantal complexity of the permanent, 24 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, PermanentCubic
Proof-verification report
MathVet, PermanentCubic
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, PermanentCubic
Proof-verification report
cert-machine, SmoothInitialForm
Proof-verification report
MathVet, SmoothInitialForm
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, SmoothInitialForm
Lean statement
PermanentCubic
Lean statement
SmoothInitialForm