Combinatorial invariance of Kazhdan–Lusztig polynomials
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Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan–Lusztig polynomials. Thus the abstract order of the interval determines the polynomial, even across different Coxeter systems. (OpenAI’s Lean note)
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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.
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).
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
We prove that an isomorphism of Bruhat intervals in arbitrary Coxeter systems preserves their equal-parameter Kazhdan–Lusztig polynomials. This resolves the full combinatorial invariance conjecture positively.
One Lean statement in this family
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KLInvariance.lean, the release lists it as a supporting statement.
theoremcombinatorial_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' := bysorry
This 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
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Independent comparisons and proof-verification reports
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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: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.
“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.
The combinatorial invariance conjecture asks whether a Kazhdan–Lusztig polynomial depends only on its Bruhat interval as an ordered set. 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.
AI-drafted comparisons, with no recorded human review
Combinatorial invariance of Kazhdan–Lusztig polynomialsKLInvariance
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: an order isomorphism between Bruhat intervals, possibly in different Coxeter systems, preserves the Kazhdan–Lusztig polynomial.
Who ran the proof-verification program?
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Combinatorial invariance of Kazhdan–Lusztig polynomialsKLInvariance
Three 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.
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An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
tparker, Philip Weiss, Will Sawin, Nik Weaver, Pace Nielsen, mathworker21, Joshua P. Swanson, Timothy Chow, Daniel Soltesz, MathOverflow: how many of the released results went against the conventional wisdom?, 10 October 2026. Source
An algebraic combinatorialist says the proof of combinatorial invariance for Kazhdan-Lusztig polynomials agrees with the majority expectation.
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Address a question raised in an independent assessment
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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
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; 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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 178 s; peak memory 6.5 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
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.
Combinatorial invariance of Kazhdan–Lusztig polynomials
Role
the release lists it as a supporting statement
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.