The QG-8 categorification transports across the blockwise Hecke-KLR bridge

Transport of the projective identification. By [F1] the blockwise graded isomorphisms ${\rho }_{\alpha }$ induce an $A$-linear isomorphism $K_0^{\mathrm{g}\mathrm{r}}(R^{\mathrm{\Lambda }})\to K_0^{\mathrm{g}\mathrm{r}}(H^{\mathrm{\Lambda }})$ which intertwines shifts, Cartan maps and forgetful maps, block by block and then on the restricted direct sums. Composing the inverse of this transport with the KLR-side isomorphism $\varphi$ of [F4] gives ${\varphi }^{\mathrm{H}\mathrm{e}\mathrm{c}\mathrm{k}\mathrm{e}}$; the vacuum class is preserved because restriction of scalars keeps the underlying graded vector spaces and the rank-zero algebra isomorphism fixes the vacuum sector, and the divided powers are intertwined because the Hecke-side $E_i,F_i,K_i$ are defined to be the transported KLR functors [F1, F2, F4]. This proves (1).

Transport of the module identification. Applying the same transport to the exact functor on finite-dimensional graded modules gives an $A$-linear isomorphism $G_0^{\mathrm{g}\mathrm{r}}(R^{\mathrm{\Lambda }})\to G_0^{\mathrm{g}\mathrm{r}}(H^{\mathrm{\Lambda }})$ preserving internal shifts, composition multiplicities and the diagram dualities, by [F1]. Composing the inverse of this transport with ${\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}$ of [F4] gives ${\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}^{\mathrm{H}\mathrm{e}\mathrm{c}\mathrm{k}\mathrm{e}}$ with the same restricted-weight-dual target, the same canonical module action transported through the functor, and normalized simple classes mapping to the form-dual projective basis; this proves (2).

Functors and pairings. The Hecke-side functors $E_i,F_i$ are by definition the images of the KLR functors under the blockwise equivalences, so the transported identifications intertwine them with the quantum action of [F4] [F1, F2]. The graded Hom pairing is identified by [F1], and the identity $h(P,M)=T(P^{\mathrm{\#}},M)$ is the native comparison recorded in [F4] between the Hom pairing and the tensor pairing defining ${\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}$; both are weightwise perfect on the normalized cover and simple bases because on those bases $P^{\mathrm{\#}}=P$ and the actual unscaled cover/simple pairing clause of the module $G_0$ supplier makes the two pairings evaluate to the same delta, while the first-bar rule on shifts is retained on arbitrary shifted first arguments rather than identified with the bilinear form [F1, F2, F4]. This proves (3).

Clause (4) and the type-A hypothesis check. By [F3] the type-$A$ specialization has $d_i=1$, a symmetric Cartan matrix and the finite loopless cyclic quiver; for $e\ge 3$ the ordered adjacent multiplicities are $(1,0)$ or $(0,1)$ and for $e=2$ they are $(1,1)$, giving exactly $Q_{ij}^A=(-1{)}^{h_{ij}}(u-v{)}^{h_{ij}+h_{ji}}$ in both cases. Branch (C0) supplies characteristic zero, so all geometric-parameter hypotheses of the two global-basis theorems hold; transporting their conclusions along the blockwise equivalences identifies the normalized self-dual indecomposable projectives with the lower global basis and the normalized self-dual simples with the upper global basis [F1, F3, F4]. Since the transported functor preserves shifts and dualities, the crystals are the $B({\mathrm{\Lambda }}_0)$-crystals of [F4] and each class carries its label in the restricted set $RP$ [F3, F4].

Both branches and Choice. In an arbitrary branch of the regime the normalized finite-module duality representatives and their right crystal labels exist by the duality and crystal suppliers of [F4] alone, so clauses (1)-(3) hold in both branches and clause (4) only adds the characteristic-zero global-basis correspondence [F4]. No new choice is made: [C1] is inherited through the global-basis suppliers, and all transport verifications are finite substitutions.