Hecke intertwiners produce the local KLR generators

The cyclotomic and dot-sliding relations. Multiply the first-variable cyclotomic relation [F2] by $e(i)$. For every $j\mathrm{\ne }{i}_1$, the factor $x_1-j$ on $e(i)$ in DG has nonzero constant $i_1-j$ plus a nilpotent, and the factor $X_1-q^j$ in NG has nonzero constant $q^{i_1}-q^j$ plus a nilpotent. Each is therefore invertible in that word corner by the finite geometric series. Cancelling these other-residue factors leaves $(x_1-i_1{)}^{{\lambda }_{i_1}}e(i)=0$ or $(X_1-q^{i_1}{)}^{{\lambda }_{i_1}}e(i)=0$, respectively. Since these remaining factors are $y_1$ and $-q^{i_1}{y}_1$, this is exactly $y_1^{{\lambda }_{i_1}}e(i)=0$. If ${\lambda }_{i_1}=0$ it says $e(i)=0$. No invertibility on the complementary summands is needed. The dot and idempotent relations are immediate from the definitions of the idempotents $e(i)$ and the dots $y_r$, and the dot-sliding relations follow from the dot-sliding intertwiner identities, the definitions of $p_r(i),P_r(i)$ and the equal-colour case of the normalization.

Quadratic and all braid patterns. Work first in the faithful formal affine-Hecke calculation of the normalization supplier, then evaluate its regular identities in the sector. On equal colours the normalized intertwiner is $D=(S-1)\mathrm{/}(u-v)$, so $D^2=0$ and $Dv-uD=1=vD-Du$. On unequal colours it is $S{c}_{ij}(u,v)$, with $c=(1-p)\mathrm{/}N$ in DG or $(1-P)\mathrm{/}N$ in NG; its square is $c_{ij}(u,v)c_{ji}(v,u)=K_{ij}(u,v)$, giving respectively $1,v-u,u-v,(v-u)(u-v)$. For the braid, put $W=S_1{S}_2{S}_1$ and dots $u,v,w$. Distinct colours give both words $W{c}_{ij}(u,v)c_{ik}(u,w)c_{jk}(v,w)$. For $(i,i,k)$ both words are $(W-S_1{S}_2)c_{ik}(u,w)c_{ik}(v,w)\mathrm{/}(v-w)$; for $(i,k,k)$ they are $(W-S_2{S}_1)c_{ik}(u,v)c_{ik}(u,w)\mathrm{/}(u-v)$, with denominators on the left and coefficient products on the right. The products are symmetric in the two equal-colour variables, so their divided differences vanish. For $(i,j,i)$ the longest terms cancel and the identity coefficients differ by $(K_{ij}(w,v)-K_{ij}(u,v))\mathrm{/}(u-w)$; direct substitution gives $0,1,-1,u+w-2v$. On all equal colours the common value is $-(1-S_1-S_2+S_1{S}_2+S_2{S}_1-W)\mathrm{/}((u-v)(u-w)(v-w))$. These are the supplier's explicit local computations, using only affine-Hecke relations and scalar normalizers, not the opposite relation-map lemma. The double-edge correction and its sign remain valid in characteristic two. Clearing temporary differences in the faithful formal calculation gives regular relations; only those relations, and unit coefficients, are evaluated on nilpotent dots. Hence each of the BK quadratic and braid equations holds in $H_{\alpha }^{\mathrm{\Lambda }}$.

The homomorphism. The verified relations are exactly the presentation of $R_{\alpha }^{\mathrm{\Lambda }}$ recalled in [F2] and in The type-A cyclotomic specialization of the QG-KLR algebras, so the universal property of that presentation gives a unique $F$-algebra homomorphism $\rho \text{\hspace{0.17em}⁣}:R_{\alpha }^{\mathrm{\Lambda }}\to H_{\alpha }^{\mathrm{\Lambda }}$ sending the generators to the displayed elements. No choice principle is used.

Conclusion. Steps 1.1 and 1.2 verify all relations of the sign-modified cyclotomic KLR presentation for $\{e(i)\}\cup \{y_r\}\cup \{{\psi }_r\}$, and step 2.1 invokes the universal property to produce $\rho$. The zero sector is included: all constructions are supported on the projectors $e(i)$, which may vanish.