Universal irreducible tensor squares for symmetric groups
Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.
Theorem selected for verification:OAI.UniversalTensorSquares.universal_tensor_square
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Definitions needed to read it
Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.
defwordRep (n d : ℕ) : Representation ℂ (Equiv.Perm (Fin n)) (WordSpace n d) where toFun g := { toFun := fun f a => f (a ∘ g) map_add' := by intros; rfl map_smul' := by intros; rfl } map_one' := by ext f a; rfl map_mul' g h := by ext f a; rfl
defcyclic {G : Type uG} {V : Type uV} [Group G] [AddCommGroup V] [Module ℂ V] (ρ : Representation ℂ G V) (v : V) : Subrepresentation ρ where
Show the remaining 13 lines
toSubmodule := Submodule.span ℂ (Set.range fun g : G => ρ g v) apply_mem_toSubmodule g := by intro x hx induction hx using Submodule.span_induction with | mem x hx =>obtain ⟨h, rfl⟩ := hx apply Submodule.subset_span exact ⟨g * h, by simp only [map_mul, Module.End.mul_apply]⟩ | zero => simp | add x y hx hy ihx ihy => simpa only [map_add] using Submodule.add_mem _ ihx ihy | smul a x hx ih => simpa only [map_smul] using Submodule.smul_mem _ a ih
defcolumnGroup {n : ℕ} {μ : YoungDiagram} (t : Tableau n μ) : Subgroup (Equiv.Perm (Fin n)) where
Show the remaining 9 lines
carrier := {g | ∀ i, (t (g i)).val.2 = (t i).val.2} one_mem' := by intro i; rfl mul_mem' := by intro g h hg hh i exact (hg (h i)).trans (hh i) inv_mem' := by intro g hg ihave h := hg (g⁻¹ i) simpa using h.symm
defkronecker {n : ℕ} {α β μ : YoungDiagram} (a : Tableau n α) (b : Tableau n β) (t : Tableau n μ) : ℕ := Module.finrank ℂ (Representation.IntertwiningMap (spechtRep t) ((spechtRep a).tprod (spechtRep b)))
Possible paper match, with no recorded review
Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Words of the Lean statement, its names split into words, that also appear in the theorem: one, tensor, representation, square, integer and irreducible. Found in the paper’s title, which its main theorem states: universal and groups. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: notin, positive and containing.
Statements in Universal Tensor Squares for Symmetric Groups come first because the release’s name for this Lean statement shares words with that paper’s title.
Theorem 1.1 of Universal Tensor Squares for Symmetric Groups
Theorem
For every positive integer n\notin\{2,4,9\} there is a partition \lambda\vdash n such that
g(\lambda,\lambda,\nu)>0\qquad\text{for every }\nu\vdash n.
Thus one irreducible representation has a tensor square containing every irreducible representation of S_n.
For every positive integer $n\notin\{2,4,9\}$ there is a partition
$\lambda\vdash n$ such that
\[
g(\lambda,\lambda,\nu)>0\qquad\text{for every }\nu\vdash n.
\]
Thus one irreducible representation has a tensor square containing every
irreducible representation of $S_n$.
Other possible paper matches, closest wording first
Corollary 1.2 of Universal Tensor Squares for Symmetric Groups. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, range, tensor, squares, complex, integer and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: notin, positive and contains.
Theorem 1.1 of A Cyclic Polytabloid Proof of Saxl’s Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: saxls, representation, complex, integer and irreducible. Found in the paper’s title, which its main theorem states: polytabloid and cyclic. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equivalently and contains.
Theorem 2.4 of Universal Tensor Squares for Symmetric Groups. Words of the Lean statement, its names split into words, that also appear in the theorem: eq, saxl, cyclic, module and irreducible.
Proposition 4.1 of Universal Tensor Squares for Symmetric Groups. Words of the Lean statement, its names split into words, that also appear in the theorem: zeros, column, row and integers.
Proposition 5.8 of Universal Tensor Squares for Symmetric Groups. Words of the Lean statement, its names split into words, that also appear in the theorem: group, row and module. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: contains.
Proposition 4.2 of A Cyclic Polytabloid Proof of Saxl’s Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: map, module and irreducible.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
Atlas has no report of an independent group using Comparator to verify OpenAI’s proof of this Lean statement at the version shown here. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…
No verdict reachedTheir term: “REFUSED”.
Re-run through Comparator did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)).
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 205’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: Saxl.saxl_conjecture : ∀ m ≥ 1, ∀ μ with μ.card = (staircase m).card, 0 < kronecker (staircase m) (staircase m) μ and universal_tensor_square (n) (hn : 0 < n) (h2 : n ≠ 2) (h4 : n ≠ 4) (h9 : n ≠ 9) : ∃ λ, (∀ ν, 0 < kronecker λ λ ν) ∧ IsIrreducible (spechtRep λ) ∧ ∀ irreducible ρ, ∃ injective intertwiner ρ → S^λ ⊗ S^λ. Both summary claims are stated.
Definitions the group asks readers to compare with the paper: Specht modules are hand-built: spechtSub t := cyclic (wordRep n d) (polytabloid t) as the orbit span of the column-antisymmetrized row-word inside (Fin n → Fin d) → ℂ, and kronecker := finrank ℂ (IntertwiningMap (spechtRep t) ((spechtRep a).tprod (spechtRep b))). This is the standard polytabloid construction (and IsIrreducible of the chosen Specht module is part of the universal statement), but it is not Mathlib's notion, so Saxl's statement is only as faithful as this construction.
OpenAI’s own note on what this Lean statement covers
“For every positive integer $n\notin\{2,4,9\}$, it constructs an irreducible complex representation of $S_n$ whose tensor square contains every irreducible complex representation of $S_n$. Equivalently, all corresponding Kronecker coefficients are positive. The statement also supplies injective intertwining maps for arbitrary finite-dimensional irreducible representations.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
110 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 35 to its own declarations, 63 to Mathlib’s documentation. 13 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
For every positive integer $n\notin\{2,4,9\}$ there is a partition
$\lambda\vdash n$ such that
\[
g(\lambda,\lambda,\nu)>0\qquad\text{for every }\nu\vdash n.
\]
Thus one irreducible representation has a tensor square containing every
irreducible representation of $S_n$.
Corollary(Unipotent tensor squares)
For each positive integer n\notin\{2,4,9\}, choose a partition \lambda_n supplied by Theorem thm:main, including its finite-range witnesses. For every prime power q, let U_q^\lambda denote the unipotent irreducible complex character of \GL_n(\mathbb F_q) indexed by \lambda\vdash n. Then U_q^{\lambda_n}\otimes U_q^{\lambda_n} contains U_q^\nu for every \nu\vdash n.
For each positive integer $n\notin\{2,4,9\}$, choose a partition
$\lambda_n$ supplied by Theorem~\ref{thm:main}, including its
finite-range witnesses. For every prime power $q$, let $U_q^\lambda$
denote the unipotent irreducible complex character of $\GL_n(\mathbb F_q)$
indexed by $\lambda\vdash n$. Then
$U_q^{\lambda_n}\otimes U_q^{\lambda_n}$ contains $U_q^\nu$
for every $\nu\vdash n$.
For every integer $m\ge1$ and every partition $\lambda\vdash N_m$,
\[
g(\rho_m,\rho_m,\lambda)>0.
\]
Equivalently, $S^{\rho_m}\otimes S^{\rho_m}$ contains every irreducible
complex representation of $S_{N_m}$.
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.