# Bruhat interval polynomial caching library

Family 168: Combinatorial invariance of Kazhdan–Lusztig polynomials. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Symbolic-algebra teams could reuse Kazhdan–Lusztig polynomial results across isomorphic abstract intervals.

## Applicability and commercial boundary

Order-isomorphism testing and polynomial construction may be costly; invariance alone does not give an efficient algorithm or unrestricted unequal-parameter result.

## Initial business decision

Research or conditional engineering only until an effective implementation and a recurring buyer problem are identified. No commercial demand or profitability is established. Where a direct product bridge is weak, the legitimate use is a research reference or evidence adapter, rather than a new standalone company.

## Next verification action

Extract canonical interval encoding and compare with an existing CAS workflow.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and selected formal statement have not yet been compared in depth for this family. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Combinatorial invariance of Kazhdan–Lusztig polynomials](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026/paper.pdf).