Cyclotomic Hecke central sectors are isomorphic to type-A cyclotomic KLR algebras

The forward composite on dots and crossings. On each KLR word the map $\sigma$ kills all idempotents outside content $\alpha$; in particular, since the inverse polynomial generators of $R$ have exactly the eigenword projectors $e(\nu )$ (the nilpotent dots shift distinct scalar eigenvalues), $\sigma (e_{\alpha })=1_R$ and $\sigma$ annihilates every other central sector. The composite $\sigma \rho$ fixes $y_r$ by [F2], and on a crossing $\sigma ({\mathrm{\Phi }}_r{Q}_r(i{)}^{-1}e(i))=({\psi }_r{Q}_r(i)-P_r(i)+P_r(i))Q_r(i{)}^{-1}e(i)={\psi }_{r}e(i),$ using the definitions of the local generating elements in Hecke intertwiners produce the local KLR generators; the degenerate case is the same computation with ${\phi }_r,p_r,q_r$ in place of ${\mathrm{\Phi }}_r,P_r,Q_r$. These are literal ordered expressions, not a commutation of ${\psi }_r$ with its coefficient. Hence $\sigma \rho$ fixes every presentation generator of $R_{\alpha }^{\mathrm{\Lambda }}$.

The reverse composite on generators. By [F2] the composite $\rho \sigma$ fixes the idempotents and dots; on a word, $\rho ({\psi }_r{Q}_r(i)-P_r(i))e(i)={\mathrm{\Phi }}_r{Q}_r(i{)}^{-1}{Q}_r(i)e(i)-P_r(i)e(i)=T_{r}e(i)$ in the nondegenerate case, and the same with $s_r$ in the degenerate case. Again the evaluation is a finite expression in the nilpotent variables. For the polynomial generators the computations are $\rho \sigma (x_r{e}_{\alpha })={\sum }_i((x_r-i_r)+i_r)e(i)=x_r{e}_{\alpha }$ in DG and $\rho \sigma (X_r{e}_{\alpha })={\sum }_i{q}^{i_r}(1-(1-q^{-i_r}{X}_r))e(i)=X_r{e}_{\alpha }$ in NG; their inverses are fixed because homomorphisms preserve inverses. Conversely $\sigma \rho (y_{r}e(i))=(\sigma (x_r)-i_r)e(i)=y_{r}e(i)$ in DG and $(1-q^{-i_r}\sigma (X_r))e(i)=y_{r}e(i)$ in NG. Along with the projector identities of [F1], these verify every presentation generator of both algebras, including the correct inverse NG substitution. Hence $\rho \sigma$ fixes every presentation generator of $H_{\alpha }^{\mathrm{\Lambda }}$.

Mutual inverses and the zero-sector statement. Both composites fix all generators of their source algebras, so $\rho \sigma =\mathrm{id}$ and $\sigma \rho =\mathrm{id}$; in particular the maps are algebra isomorphisms, and on idempotents they are the literal identity. If $H_{\alpha }^{\mathrm{\Lambda }}=0$ then the already constructed $\sigma$ would satisfy $\sigma (e_{\alpha })=1_R$ with $e_{\alpha }=0$, forcing $1_R=0$; conversely the forward identities force $H_{\alpha }^{\mathrm{\Lambda }}=0$ if $R_{\alpha }^{\mathrm{\Lambda }}=0$. Zero rings and rank zero are therefore included without any nonzero-block premise.

Conclusion. Summing the isomorphisms over the finite set of height-$d$ contents and using that the $e_{\alpha }$ are central orthogonal idempotents summing to $1$ by [F3] gives $H_d^{\mathrm{\Lambda }}\cong {⨁}_{\alpha }{R}_{\alpha }^{\mathrm{\Lambda }}$. Neither exact dimension nor primitive terminology has entered the proof; the full colour-case verifications, affine six-coefficient calculation and equal-colour divided-difference computation reside in the normalization supplier and the two preceding local relation lemmas. No choice principle and no algebraic closure are used.