Positive Rouquier complexes are Delta-split

Induct on the word length. For the empty word $F_e=R$ has just ${\mathrm{\Delta }}_e$; for one letter the exact rank-one sequence is $0\to R_s(-1)\to B_s\stackrel{\mu }{\to }R(1)\to 0$. Its upper $s$ layer is $R_s(-1)={\mathrm{\Delta }}_s$, and its identity layer is the isomorphism $R(1)\to R(1)$, hence contractible. This starts the induction directly for positive complexes; no inverse generator is identified with a positive one.

Choose a Bruhat-compatible enumeration grouped into right-$s$ cosets, listing $y$ before $ys$ when $y<ys$. This exists by ordering the minimal representatives of the finite cosets in Bruhat order and refining each pair; the lifting property from reduced subwords ensures that different coset comparisons respect this order. Every union of complete initial or terminal cosets is right-$s$-stable. For such a set $A$, ${\mathrm{\Gamma }}_A(M{B}_s)={\mathrm{\Gamma }}_A(M)B_s$: the cut and quotient have graph flags, whose free layers split on the right; after exact tensoring, their generic graph supports lie respectively in $A$ and its stable complement. The quotient remains right-free by its flag, so an $A$-supported element of it is generically zero and hence zero. This proves the cut identity and its analogous consecutive-coset quotient identity.

Write $x=zs$ with $\mathrm{\ell }(x)=\mathrm{\ell }(z)+1$. Every term of $F_z$ is supported on subexpressions of a reduced word for $z$, hence on labels at most $z$. In particular its $x$ layer is zero termwise, not merely contractible. Its paired layer on $\{z,x\}$ is therefore its single $z$ layer, homotopy equivalent as a full bimodule complex to ${\mathrm{\Delta }}_z$ by induction. The cut identity of step 1.2 identifies the paired layer of $F_z{F}_s$ with the cone of the natural multiplication map from this paired layer tensored with $B_s$ to its shift $(1)$. A full bimodule homotopy equivalence commutes with this natural counit and induces a homotopy equivalence of the cones. Thus this paired layer is the rank-one graph complex $[R_z(-\mathrm{\ell }(z))B_s\to R_z(-\mathrm{\ell }(z)+1)]$. Its high $x$ layer is the kernel graph $R_x(-\mathrm{\ell }(x))={\mathrm{\Delta }}_x$, and its low $z$ layer is an isomorphism onto $R_z(-\mathrm{\ell }(z)+1)$ and contracts. The two single-layer functors of the paired layer are the original canonical single layers: the support cut and quotient flag, in the chosen enumeration, contain exactly those two labels, so their submodule and quotient are respectively its high and low graph layers. This proves the two required layers with the counit compatibility retained.

On any other pair, both single layers of $F_z$ contract by induction. Their paired sequence becomes termwise split after restriction to right $R^s$, by [F1]. Tensoring with $B_s$ yields a triangle with contractible outside terms, so its paired tensor layer contracts. Applying either single-layer functor preserves its homotopy and gives contractible single layers of $F_z{B}_s$. The same single layers of $F_z(1)$ contract already by induction. Since $F_z{F}_s$ is the cone of the natural multiplication map $F_z{B}_s\to F_z(1)$ with its standard shift, and an additive layer functor commutes with cones, both resulting single layers also contract. This treats every non-top pair without claiming that arbitrary complexes with zero Euler character contract.

The paired enumeration covers every graph; its single layers equal the canonical Bruhat layers by [F1]. Steps 2.1 and 3.1 prove the asserted homotopy isomorphism for the word complex. By [F3] every layer of a removed contractible summand remains contractible, so the same isomorphism holds for its minimal summand. This is exactly the stated $\mathrm{\Delta }$-split property, obtained without any $S$ or Hodge assumption.