The combinatorial invariance conjecture for Kazhdan-Lusztig polynomials

Theorem 1.1: for arbitrary Coxeter systems $(W,S)$ and $(W',S')$, including infinite and noncrystallographic ones, and $u\le b$, $u'\le b'$, any poset isomorphism $[u,b]\to[u',b']$ gives $P^W_{u,b}=P^{W'}_{u',b'}$ for the equal-parameter polynomials; restricting to subintervals determines every $P_{x,y}$ and, by reciprocity, every $R$-polynomial on the interval. The proof uses moment-graph sheaves (Braden-MacPherson, Fiebig, Elias-Williamson), an edge schedule transported from a reflection order of the second system, and Dyer's path formula. It gives no procedure computing $P$ from the poset, and it treats equal parameters only.

Theorem 1.1: for arbitrary Coxeter systems $(W,S)$ and $(W',S')$, including infinite and noncrystallographic ones, and $u\le b$, $u'\le b'$, any poset isomorphism $[u,b]\to[u',b']$ gives $P^W_{u,b}=P^{W'}_{u',b'}$ for the equal-parameter polynomials; restricting to subintervals determines every $P_{x,y}$ and, by reciprocity, every $R$-polynomial on the interval. The proof uses moment-graph sheaves (Braden-MacPherson, Fiebig, Elias-Williamson), an edge schedule transported from a reflection order of the second system, and Dyer's path formula. It gives no procedure computing $P$ from the poset, and it treats equal parameters only.