Proves the sixth case of Foulkes’ conjecture: \(\mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^bV)\) embeds equivariantly in \(\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6V)\) for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map \(\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^aV)\to\mathop{\mathrm{Sym}}\nolimits ^a(\mathop{\mathrm{Sym}}\nolimits ^bV)\) is surjective for a ≥ 2 and \(b\ge a(a-1)\), giving dimension-independent quadratic stabilization. (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 independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments
In Quadratic stabilization of the canonical Foulkes–Howe map, 25 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 integer a\ge2, every integer b\ge a(a-1), and every finite-dimensional complex vector space V, the map \mu_{a,b,V} in (2) is surjective. Equivalently, A_b=R_b.
In Foulkes’ conjecture for the sixth symmetric power, 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 integer b\ge6 and every finite-dimensional complex vector space V, there is a \GL(V)-equivariant injection
Equivalently, h_b[h_6]-h_6[h_b] is Schur-positive for every b\ge6.
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 closer match to the description’s wording first. 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.
FoulkesHoweThe paper states
In Quadratic stabilization of the canonical Foulkes–Howe map, 25 September 2026
For every integer a\ge2, every integer b\ge a(a-1), and every finite-dimensional complex vector space V, the map \mu_{a,b,V} in (2) is surjective. Equivalently, A_b=R_b.
The words they share: map, surjective, finite, dimensional, complex, integer, canonical, howe, foulkes, space, vector.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
FoulkesHowe.lean, the release lists it as a main result.
theorem canonical_foulkes_howe_surjective : ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧ ∀ ν : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V ν → ν = μ := 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
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.
IsFoulkesMap Model.lean, line 36IsPlethysmAction Model.lean, line 42IsGLEquivariantEmbedding Model.lean, line 52IsSymmetricMultilinearForm Model.lean, line 69Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
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 part of the advertised result (their term: “partial”).
Their note: Quadratic stabilization is stated: canonical_foulkes_howe_surjective : ∀ V f.d., ∀ a b, 2 ≤ a → a * (a - 1) ≤ b → ∃ μ, IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧ (unique). The headline sixth case of Foulkes’ conjecture for all b >= 6 is only reached for b >= 30 = 6*5 (docs: 'The sixth-power specialization is covered for b ≥ 30; the companion’s full range b ≥ 6 is not included'); the equivariant embedding is only implicit (dual of surjectivity).
Definitions the group asks readers to compare with the paper: foulkesFormula a b V v := (a!)^(-b) • Σ_{σ : Fin b → Perm (Fin a)} sym_a(fun i => sym_b(fun j => v j (σ j i))) and IsFoulkesMap define the ‘canonical’ map by hand on SymPow n V := span of degree-n products in SymmetricAlgebra ℂ V; the theorem also asserts uniqueness, which pins the definition.
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 sixth Foulkes case for 6 ≤ b < 30 is not stated (only b ≥ 30 follows), and the equivariant embedding itself is not stated even for b ≥ 30.
“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 formalized result proves surjectivity of the canonical averaged Foulkes–Howe map $\mathrm{Sym}^b(\mathrm{Sym}^a\,V)\to\mathrm{Sym}^a(\mathrm{Sym}^b\,V)$ for every finite-dimensional complex vector space, $a\ge2$, and $b\ge a(a-1)$. It also covers the bijective $a=1$ case, the vanishing-on-products consequence, and the corresponding equivariant embedding in the reverse direction. The sixth-power specialization is covered for $b\ge30$; the companion’s full range $b\ge6$ is not included.
Quadratic stabilization of the canonical Foulkes–Howe map FoulkesHowe
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the canonical Foulkes–Howe map is surjective for a ≥ 2 and b ≥ a(a − 1). The companion paper’s sixth-power result for the full range b ≥ 6 is not included.
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.
FoulkesHoweTwo 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 |
Other people’s accounts of running proof-verification software, with links to their reports.
realopjt, realopjt/foulkes-sixth-power-check: independent check of the base range in the Foulkes sixth-power paper, 7 October 2026. Source
Finite base-range computation reproduced independently in Rust.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
Foulkes’ conjecture for the sixth symmetric power, 25 September 2026
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.
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 “A strong professor’s paper of the year”. The page says that nothing on it is verified.
Author not stated. Source
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
Only surjectivity is checked; further results the doc calls formalised are proved in solution files but not named in the Comparator configuration.
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
In Quadratic stabilization of the canonical Foulkes--Howe map, 25 September 2026
Why this passage is shown
Theorem 1.1
In Foulkes’ conjecture for the sixth symmetric power, 25 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, FoulkesHowe
Proof-verification report
MathVet, FoulkesHowe
Report using another proof or calculation method
Foulkes’ conjecture for the sixth symmetric power, 25 September 2026
Exact calculations for finite claims, cert-machine
9814e4cLean statement
FoulkesHowe