Combinatorics·OpenAI Math·Result 168·Lean formalisation listed
Combinatorial invariance of Kazhdan–Lusztig polynomials
One of the 372 results in OpenAI's public mathematics catalogue, filed under combinatorics: 1 preprint, the latest dated 24 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a result titled ‘Combinatorial invariance of Kazhdan–Lusztig polynomials’ (combinatorics) in a preprint dated 24 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More, is on YouTube.
Chapter 21 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 6:56 · watch from there on YouTube →
What OpenAI claims
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. (Lean)
Paper
Combinatorial invariance of Kazhdan–Lusztig polynomials
OpenAI's abstract 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.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- 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. (Lean)
- Where is the proof?
- In the paper, Combinatorial invariance of Kazhdan–Lusztig polynomials (PDF).
- Is there a video?
- Yes, on YouTube (29:10, chapter 21).
Also in
- Chapter 6 of 10 in Borsuk's Conjecture Fails in Dimension 9, Sidorenko's Conjecture and 8 More · starts at 1:45 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · Compilation · 29:10 · chapter 21 from 6:56 · description, sources and chapters →
Related results in Combinatorics
- 169Shareshian–Wachs elementary positivityVideoLean
- 170Sharp logarithmic exponents for off-diagonal Ramsey numbersVideoLean
- 171The hypercube Ramsey conjectureVideo
- 172Classification of finite Euclidean Ramsey configurationsVideoLean
- 173Seymour’s second-neighborhood conjectureVideoLean
- 174Deterministic construction of strong thin spanning treesVideoLean
