Lean statement

Combinatorial invariance of Kazhdan–Lusztig polynomials

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.KLInvariance.combinatorial_invariance

Lean statement file: KLInvariance.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 168 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem combinatorial_invariance    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')    {u b : W} {u' b' : W'}    (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')    (ι : Interval cs u b ≃o Interval cs' u' b') :    klPolynomial cs u b = klPolynomial cs' u' b' := by  sorry

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.

definition OAI.KLInvariance.BruhatStep line 14

def BruhatStep (cs : CoxeterSystem M W) (x y : W) : Prop :=  cs.length x < cs.length y ∧ ∃ t : W, cs.IsReflection t ∧ y = t * x

definition OAI.KLInvariance.BruhatLE line 18

def BruhatLE (cs : CoxeterSystem M W) : W → W → Prop :=  Relation.ReflTransGen (BruhatStep cs)

definition OAI.KLInvariance.Interval line 22

def Interval (cs : CoxeterSystem M W) (u b : W) :=  {x : W // BruhatLE cs u x ∧ BruhatLE cs x b}

definition OAI.KLInvariance.rankDifference line 45

noncomputable def rankDifference (cs : CoxeterSystem M W) (x y : W) : ℕ :=  cs.length y - cs.length x

abbreviation OAI.KLInvariance.PolynomialFamilies line 48

abbrev PolynomialFamilies (W : Type v) := (W → W → ℤ[X]) × (W → W → ℤ[X])

structure OAI.KLInvariance.NormalizedKL line 55

structure NormalizedKL (cs : CoxeterSystem M W) (RP : PolynomialFamilies W) : Prop where

Show the remaining 16 lines

  R_diagonal : ∀ x, RP.1 x x = 1  R_zero : ∀ x y, ¬ BruhatLE cs x y → RP.1 x y = 0  R_recursion : ∀ x y i, cs.length (cs.simple i * y) < cs.length y →    RP.1 x y =      if cs.length (cs.simple i * x) < cs.length x then        RP.1 (cs.simple i * x) (cs.simple i * y)      else        (X - 1) * RP.1 x (cs.simple i * y) +          X * RP.1 (cs.simple i * x) (cs.simple i * y)  P_diagonal : ∀ x, RP.2 x x = 1  P_zero : ∀ x y, ¬ BruhatLE cs x y → RP.2 x y = 0  P_degree : ∀ x y, BruhatLE cs x y → x ≠ y →    2 * (RP.2 x y).natDegree < rankDifference cs x y  reciprocity : ∀ x y, BruhatLE cs x y →    reflect (rankDifference cs x y) (RP.2 x y) =      ∑ᶠ z : Interval cs x y, RP.1 x z.val * RP.2 z.val y

definition OAI.KLInvariance.klFamilies line 74

noncomputable def klFamilies (cs : CoxeterSystem M W) : PolynomialFamilies W :=  Classical.epsilon (NormalizedKL cs)

definition OAI.KLInvariance.klPolynomial line 77

noncomputable def klPolynomial (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=  (klFamilies cs).2 x y

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: invariance, equal, parameter, combinatorial, coxeter, systems, bruhat, polynomials and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: arbitrary, kazhdan, lusztig and isomorphism.

Theorem 1.1 of Combinatorial invariance of Kazhdan–Lusztig polynomials

Theorem (Combinatorial invariance)

Let (W,S) and (W',S') be arbitrary Coxeter systems, and let u\le b in W and u'\le b' in W'. If

\iota:[u,b]\longrightarrow[u',b']

is an isomorphism of abstract Bruhat interval posets, then their equal-parameter Kazhdan–Lusztig polynomials satisfy

P^W_{u,b}(q)=P^{W'}_{u',b'}(q).

Section 1, Introduction Read it in the paper TeX source, line 22

Show the TeX

Let $(W,S)$ and $(W',S')$ be arbitrary Coxeter systems, and let
$u\le b$ in $W$ and $u'\le b'$ in $W'$.
If
\[
 \iota:[u,b]\longrightarrow[u',b']
\]
is an isomorphism of abstract Bruhat interval posets, then their
equal-parameter Kazhdan--Lusztig polynomials satisfy
\[
 P^W_{u,b}(q)=P^{W'}_{u',b'}(q).
\]

Other possible paper matches, closest wording first

Theorem 3.4 of Combinatorial invariance of Kazhdan–Lusztig polynomials. Words of the Lean statement, its names split into words, that also appear in the theorem: graph, coxeter, systems, reflection, bruhat and intervals. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: arbitrary, isomorphism and preserves.
Proposition 3.5 of Combinatorial invariance of Kazhdan–Lusztig polynomials. Words of the Lean statement, its names split into words, that also appear in the theorem: reflects, rank, differences, order, reflection and intervals. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: preserves.
Corollary 6.1 of Combinatorial invariance of Kazhdan–Lusztig polynomials. Words of the Lean statement, its names split into words, that also appear in the theorem: polynomials. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: corresponding, kazhdan, lusztig, isomorphism and preserves.
Theorem 2.2 of Combinatorial invariance of Kazhdan–Lusztig polynomials. Words of the Lean statement, its names split into words, that also appear in the theorem: order, reflection, polynomial and bruhat.
Proposition 5.2 of Combinatorial invariance of Kazhdan–Lusztig polynomials. Words of the Lean statement, its names split into words, that also appear in the theorem: bound, edge and upward.
Proposition 2.4 of Combinatorial invariance of Kazhdan–Lusztig polynomials. Words of the Lean statement, its names split into words, that also appear in the theorem: degree and normalization.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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.

Carlos Toledo’s cert-machine

How they ran it, in their words

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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 178 s; peak memory 6.5 GiB).
Reported 11 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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.

Allowed axioms: propext, Quot.sound, Classical.choice

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 168’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.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: combinatorial_invariance takes arbitrary Coxeter systems cs, cs' (different W, W' allowed) and ι : Interval cs u b ≃o Interval cs' u' b' and concludes klPolynomial cs u b = klPolynomial cs' u' b'; BruhatLE is the genuine strong Bruhat order (ReflTransGen of length-increasing reflection steps) and the R/P normalization is the standard Kazhdan-Lusztig characterization.
Definitions the group asks readers to compare with the paper: noncomputable def klFamilies (cs) := Classical.epsilon (NormalizedKL cs) and klPolynomial cs x y := (klFamilies cs).2 x y: KL polynomials are defined as an arbitrary solution of the normalization axioms (existence/uniqueness not part of the statement); this is the standard characterization and not trivializing, but it is a non-Mathlib, choice-based definition.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves that every order isomorphism between Bruhat intervals in arbitrary Coxeter systems preserves the corresponding equal-parameter Kazhdan–Lusztig polynomial. The two intervals may come from different Coxeter systems.”
Quoted from OpenAI’s Lean note for this family

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

99 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: 23 to its own declarations, 32 to Mathlib’s documentation. 9 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves combinatorial_invariance exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches 8 of the file’s 9 other declarations; with the theorem itself they span 38 lines. 3 of them are named in the statement itself.

Names in the namespace OAI.KLInvariance are listed without it.

def BruhatStepline 14library
def BruhatLEline 18in the statementlibrary
def Intervalline 22in the statementlibrary
def rankDifferenceline 45library
abbrev PolynomialFamiliesline 48library
structure NormalizedKLline 55library
def klFamiliesline 74library
def klPolynomialline 77in the statementlibrary
theorem combinatorial_invarianceline 88selected theorem

Other Lean rules that may affect the statement 1

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance intervalOrder line 25

import Mathlibnamespace OAInamespace KLInvarianceopen Polynomialuniverse u v u' v'variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}/-- An upward edge in the *strong* Bruhat graph, with left reflection labels. -/def BruhatStep (cs : CoxeterSystem M W) (x y : W) : Prop :=  cs.length x < cs.length y ∧ ∃ t : W, cs.IsReflection t ∧ y = t * x/-- The actual strong Bruhat order, not the weak order. -/def BruhatLE (cs : CoxeterSystem M W) : W → W → Prop :=  Relation.ReflTransGen (BruhatStep cs)/-- The unlabelled interval, including both endpoints. -/def Interval (cs : CoxeterSystem M W) (u b : W) :=  {x : W // BruhatLE cs u x ∧ BruhatLE cs x b}instance intervalOrder (cs : CoxeterSystem M W) (u b : W) :    PartialOrder (Interval cs u b) where  le left right := BruhatLE cs left.val right.val  le_refl _ := Relation.ReflTransGen.refl  le_trans _ _ _ forward backward := forward.trans backward  le_antisymm left right forward backward := by    apply Subtype.ext    have length_mono : ∀ {lower upper : W}, BruhatLE cs lower upper →        cs.length lower ≤ cs.length upper := by      intro lower upper relation      induction relation using Relation.ReflTransGen.head_induction_on with      | refl => exact le_rfl      | head step _ rest => exact le_trans step.1.le rest    by_contra distinct    rcases forward.cases_head with same | ⟨middle, step, rest⟩    · exact distinct same    · exact (not_lt_of_ge (length_mono backward))        (lt_of_lt_of_le step.1 (length_mono rest))/-- `d(x,y)` in the manuscript, used only for comparable endpoints. -/noncomputable def rankDifference (cs : CoxeterSystem M W) (x y : W) : ℕ :=  cs.length y - cs.length xabbrev PolynomialFamilies (W : Type v) := (W → W → ℤ[X]) × (W → W → ℤ[X])/-- Exact equal-parameter normalization in the introduction.The first family is R and the second is P.  `reflect d p` is `q^d p(q⁻¹)`when `p.natDegree ≤ d`, which follows here from the degree bound (or thediagonal normalization). The sum is over the real Bruhat interval;its finiteness is a separate theorem, not a different definition of order. -/structure NormalizedKL (cs : CoxeterSystem M W) (RP : PolynomialFamilies W) : Prop where  R_diagonal : ∀ x, RP.1 x x = 1  R_zero : ∀ x y, ¬ BruhatLE cs x y → RP.1 x y = 0  R_recursion : ∀ x y i, cs.length (cs.simple i * y) < cs.length y →    RP.1 x y =      if cs.length (cs.simple i * x) < cs.length x then        RP.1 (cs.simple i * x) (cs.simple i * y)      else        (X - 1) * RP.1 x (cs.simple i * y) +          X * RP.1 (cs.simple i * x) (cs.simple i * y)  P_diagonal : ∀ x, RP.2 x x = 1  P_zero : ∀ x y, ¬ BruhatLE cs x y → RP.2 x y = 0  P_degree : ∀ x y, BruhatLE cs x y → x ≠ y →    2 * (RP.2 x y).natDegree < rankDifference cs x y  reciprocity : ∀ x y, BruhatLE cs x y →    reflect (rankDifference cs x y) (RP.2 x y) =      ∑ᶠ z : Interval cs x y, RP.1 x z.val * RP.2 z.val y/-- The polynomial families selected by their equal-parameter normalization. -/noncomputable def klFamilies (cs : CoxeterSystem M W) : PolynomialFamilies W :=  Classical.epsilon (NormalizedKL cs)noncomputable def klPolynomial (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=  (klFamilies cs).2 x yend KLInvariancenamespace KLInvarianceuniverse u v u' v'variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}theorem combinatorial_invariance    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')    {u b : W} {u' b' : W'}    (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')    (ι : Interval cs u b ≃o Interval cs' u' b') :    klPolynomial cs u b = klPolynomial cs' u' b' := by  sorryend KLInvarianceend OAI

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The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.