Over every characteristic-zero field, the \((1,1)\) entry of a product of n independent \(n\times n\) variable matrices requires \(n^{\Theta(\sqrt n)}\) gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables. (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 Homogeneous depth-five lower bounds for iterated matrix multiplication
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
There exist absolute constants c>0 and n_0 such that, for every integer n\geq n_0, every syntactically homogeneous \Sigma\Pi\Sigma\Pi\Sigma circuit over \C computing \IMM_{n,n} has size at least
One may take c=1/400.
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.
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DepthFiveThe paper states
In Homogeneous depth-five lower bounds for iterated matrix multiplication
There exist absolute constants c>0 and n_0 such that, for every integer n\geq n_0, every syntactically homogeneous \Sigma\Pi\Sigma\Pi\Sigma circuit over \C computing \IMM_{n,n} has size at least
One may take c=1/400.
The words they share: computing, homogeneous, least, syntactically.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
DepthFive.lean, the release lists it as a supporting statement.
theorem main : (∃ threshold : ℕ, ∀ size : ℕ, threshold ≤ size → ∀ circuit : Depth5Circuit ℂ size, circuitValue circuit = imm ℂ size → (size : ℝ) ^ (Real.sqrt (size : ℝ) / 400) ≤ (circuitSize circuit : ℝ)) ∧ (∃ threshold : ℕ, ∀ (FieldType : Type u) [Field FieldType] [CharZero FieldType], ∀ size : ℕ, threshold ≤ size → ∀ circuit : Depth5Circuit FieldType size, circuitValue circuit = imm FieldType size → (size : ℝ) ^ (Real.sqrt (size : ℝ) / 400) ≤ (circuitSize circuit : ℝ)) ∧ (∀ (FieldType : Type u) [Field FieldType] (size : ℕ), 2 ≤ size → ∃ circuit : Depth5Circuit FieldType size, circuitValue circuit = imm FieldType size ∧ circuitSize circuit ≤ upperGateBound size ∧ (upperGateBound size : ℝ) ≤ (size : ℝ) ^ (Real.sqrt (size : ℝ) + 4)) := 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.
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 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.
Definitions the group asks readers to compare with the paper: Depth5Circuit is a bespoke structure (leaf/bottom/lower/middle/upper/output arrays with bottomHomogeneous, middleHomogeneous, outputHomogeneous degree-equality fields) and circuitSize counts all nodes; homogeneity is syntactic and the model charges nodes, not wires
“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
Let $\mathrm{IMM}_{n,n}$ be the $(1,1)$ entry of the product of $n$ independent $n\times n$ variable matrices. The formalization proves that, over every characteristic-zero field, syntactically homogeneous depth-five $\Sigma\Pi\Sigma\Pi\Sigma$ circuits computing this polynomial require at least $n^{\sqrt n/400}$ gates for all sufficiently large $n$, with a threshold independent of the field. Arbitrary bottom support, finite fan-in and fan-out, and sharing are allowed.
It also gives, over every field and for $n\ge2$, circuits with at most $n^{\sqrt n+4}$ gates. These are the lower and upper bounds selected from the paper.
Homogeneous depth-five bounds for iterated matrix multiplication DepthFive
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem with the explicit exponent √n/400 over every field of characteristic zero, together with the upper bound n^{√n+4}.
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.
DepthFiveTwo 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 |
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Why this passage is shown
Theorem 1.1
In Homogeneous depth-five lower bounds for iterated matrix multiplication
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, DepthFive
Proof-verification report
MathVet, DepthFive
Lean statement
DepthFive