Self-dual indecomposable KLR projectives give the lower global basis (symmetric type)

(The graded algebra comparison.) Set $P_{ij}(u,v)=(v-u{)}^{h_{ij}}$ for $i\mathrm{\ne }j$. These are nonzero, including when $h_{ij}=0$, and $P_{ij}(u,v)P_{ji}(v,u)=(-1{)}^{h_{ij}}(u-v{)}^{h_{ij}+h_{ji}}=Q_{ij}(u,v)$. Thus Faithful polynomial representation of a KLR algebra gives a faithful action of $R(\beta ;Q)$ by projections, dots and unequal crossings $(x_r-x_{r+1}{)}^{h_{ij}}{s}_r$, with equal crossings $(1-s_r)\mathrm{/}(x_r-x_{r+1})$. The actual geometric action of [F3] uses exactly those operators. In both actions give the unit of the $\nu$ polynomial summand degree $-d_{\nu }$: the incidence formula $d_{\nu }={\sum }_i{\beta }_i({\beta }_i-1)\mathrm{/}2+{\sum }_{a>b}{h}_{{\nu }_a{\nu }_b}$ gives $d_{s_r\nu }-d_{\nu }=h_{ij}-h_{ji}$, so the unequal crossing has degree $2h_{ij}+d_{\nu }-d_{s_r\nu }=h_{ij}+h_{ji}$; equal crossings have degree $-2$ and dots degree $2$. Both faithful images are therefore the identical graded subalgebra generated by these operators, yielding $R(\beta ;Q)\cong S_{\mathrm{\Gamma }}(\beta )\cong {\mathrm{Ext}}_G^{\bullet }(L,L)$ for $L={⨁}_{\nu }{L}_{\nu }$. This is also the direct comparison of the identical specialized presentations: all adjacent unequal dot slides and the native mixed and braid signs are present in [F3]. The equal geometric crossing is minus its standard complex Borel–Moore class, so no sign change is implicit.

(The actual normalized complex.) By [F4], the same $L={⨁}_{\nu }R{\pi }_{\nu !}k[d_{\nu }]$ of step 1.1 is a finite direct sum of shifts of simple actual equivariant perverse objects. The normalization $[d_{\nu }]$ has already been included in $L_{\nu }$; no further shift of $L$ is made. This uses the proved semisimplicity clause in the complete loopless multiedge scope, with no semismallness or finite-ADE restriction.

(Graded Yoneda.) Use the shifted opposite-composition product of [F3] and define $Y_T={\mathrm{Ext}}_G^{\bullet }(L,T)$ as a graded left $R$-module: for homogeneous $a:L\to L[n]$ and $y:L\to T[m]$, set $a\cdot y=y[n]\circ a$, so $(a\ast b)\cdot y=a\cdot (b\cdot y)$. On $T=L$ this is the regular left module. A homogeneous map $f:T\to U[t]$ acts by postcomposition $y\mapsto f[m]\circ y$ of degree $t$. For $T=U=L$ every graded $R$-linear map $R\to R$ is right multiplication by its value at $1$, hence is induced uniquely by the corresponding $f$; extend this identification to finite sums and shifts and then to idempotent summands. Since the actual category is idempotent complete and all finite graded projectives are summands of finite sums of shifts of $R$, this gives a fully faithful equivalence from the additive category generated by shifts of $L$ onto finite graded projectives, with ${\mathrm{End}}_R^{\bullet }(Y_T)\cong {\mathrm{Ext}}_G^{\bullet }(T,T)$ using ordinary categorical composition on both endomorphism rings.

(Indecomposables and scalar degree zero.) Each simple perverse summand $T$ in step 2.1 belongs to that additive category up to shift. By [F4] and step 2.2 its projective $Y_T$ has no negative-degree endomorphisms and its degree-zero endomorphism ring is a division algebra, hence it is graded indecomposable. These exhaust the indecomposable projective shift orbits: every projective is a summand of a finite sum of the $Y_T$ and their shifts, and [F5] gives the finite graded Krull–Schmidt decomposition. Distinct simple perverse objects yield distinct orbits, since full faithfulness would otherwise give $T\cong U[t]$, and perverse concentration forces $t=0$. Finally the simple head $S$ of $Y_T$ has ${\mathrm{End}}_R^0(S)=k$ by [F5]. Every degree-zero endomorphism preserves the unique maximal graded submodule, so descent to the head is a unital $k$-algebra homomorphism ${\mathrm{End}}_R^0(Y_T)\to k$; its kernel is zero because its source is a division algebra, and its image contains all scalars, so ${\mathrm{End}}_R^0(Y_T)=k$.

(Self-dual normalization, norm one and signed globalization.) By [F1] every indecomposable projective shift orbit has a unique ${}^{\mathrm{\#}}$ self-dual representative $P_b$; put $x_b={\gamma }^{-1}[P_b]$, which is integral and bar fixed. The exact Hom/tensor comparison for a self-dual $P$ gives $B_L(x_b,x_b)=h(P_b,P_b)$, a nonnegative integer power series in $q$ congruent to $1$ modulo $q$, since the geometric nonnegative-End property is unchanged by grading shifts. Dividing by $C_{\beta }$ (which is $1$ modulo $q$) gives a value $B_K(x_b,x_b)$ congruent to $1$ modulo $q$ with integer coefficients, so the general half norm criterion places $x_b$ in the $q^0$ crystal lattice; being bar fixed, it also lies in the actual bar chart, and the integral balanced globalization writes $x_b$ as a finite $\mathbb{Z}$-linear combination of lower global vectors. Their residue Gram matrix is the identity, so norm residue $1$ forces the sum of coefficient squares to be $1$: $x_b$ is exactly plus or minus one lower global vector. Distinct projective shift orbits are independent and have the same finite weight cardinality as the global basis by the integral $K_0$ isomorphism, so they exhaust this signed set.

(Positivity and the based quotient.) The positive divided-string leading-term supplier [F2] removes every minus sign using its integral LEFT-string ideal equality, its coefficient-$1$ projective adjunction expansion, and the general adapted LEFT global recursion (carried with its inherited Axiom of Choice, The Axiom of Choice); batch 8's finite-ADE canonical-comparison item is not used. Consequently the sign in step 4.1 is $+1$, giving $\gamma (G(b))=[P_b]$ for every lower global vector $G(b)$ and establishing the bijection of the statement. The compatibility with the based quotient is the general zero/survival projection theorem of the QG-5 machinery.