Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension \(2n\ge6\). Finitely many closed symplectic balls of capacities \(R_1,\ldots,R_k\) embed disjointly into an open ball of capacity R exactly when \(\sum_iR_i^n\lt R^n\) and \(R_i+R_j\lt R\) for every distinct pair i, j. (OpenAI’s Lean note)
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Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments
The paper states
In Symplectic Ball Packings in Higher Dimensions
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
Let n\ge3 and k\ge1 be integers. For positive real numbers R,R_1,\ldots,R_k, there exists a symplectic embedding
if and only if
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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.
BallPackingThe paper states
Let n\ge3 and k\ge1 be integers. For positive real numbers R,R_1,\ldots,R_k, there exists a symplectic embedding
if and only if
The words they share: integers, real, symplectic, embedding, packings, ball, positive.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
BallPacking.lean, the release lists it as a supporting statement.
theorem main_theorem (n k : ℕ) (hn : 3 ≤ n) (hk : 1 ≤ k) (R : ℝ) (r : Fin k → ℝ) (hR : 0 < R) (hr : ∀ i, 0 < r i) : HasPacking n k R r ↔ PackingInequalities n k R 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.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
BallPackingNecessityThe paper states
Let n\ge3 and k\ge1 be integers. For positive real numbers R,R_1,\ldots,R_k, there exists a symplectic embedding
if and only if
The words they share: integers, real, symplectic, embedding, packings, ball.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
BallPackingNecessity.lean, the release lists it as a supporting statement.
theorem packing_necessity (n k : ℕ) (hn : 3 ≤ n) (hk : 1 ≤ k) (R : ℝ) (r : Fin k → ℝ) (hR : 0 < R) (hr : ∀ i, 0 < r i) (hp : HasPacking n k R r) : PackingInequalities n k R 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.
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: main_theorem (n ≥ 3, k ≥ 1, R, r_i > 0): HasPacking n k R r ↔ (∑ r_i^n < R^n ∧ ∀ i≠j, r_i + r_j < R), where a packing is k disjointly-imaged smooth symplectic embeddings (defined on open neighbourhoods of the closed capacity-r_i balls in ℂⁿ with the standard form, capacity π|z|²) into the open ball of capacity R; BallPackingNecessity repeats the forward direction.
“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 Siegel and Yao’s ball-packing criterion in every symplectic dimension $2n$ with $n\ge3$. For $k\ge1$ closed standard balls of positive capacities $R_1,\ldots,R_k$, disjoint symplectic embeddings into the interior of a ball of capacity $R$ exist exactly when $\sum_i R_i^n<R^n$ and $R_i+R_j<R$ for all $i\ne j$. Capacity is $\pi$ times squared Euclidean radius, and each embedding is defined on a neighborhood of its closed source ball. The linked statements include both the full equivalence and the necessity direction.
Exact criterion for higher-dimensional symplectic ball packings BallPacking
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: in dimension 2n ≥ 6, balls pack symplectically into a ball exactly when the volume and pairwise capacity inequalities hold.
Necessity of the volume and pairwise capacity inequalities BallPackingNecessity
Unreviewed AI draft: states part of a paper’s theorem. States only the necessity direction of the packing criterion (a packing forces the volume and pairwise inequalities); the sufficiency direction is in BallPacking.
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.
BallPackingTwo 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 |
BallPackingNecessityTwo 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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Offer to write an explanationCompare one Lean statement with its paper: BallPacking, BallPackingNecessitySuggest 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 “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
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Why this passage is shown
Theorem 1.1
In Symplectic Ball Packings in Higher Dimensions
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, BallPacking
Proof-verification report
MathVet, BallPacking
Proof-verification report
cert-machine, BallPackingNecessity
Proof-verification report
MathVet, BallPackingNecessity
Lean statement
BallPacking
Lean statement
BallPackingNecessity