Cyclic type-A KLR regular Hom norm gives the total dimension

Rank decomposition. For fixed $d$, the corner idempotents $e(\beta ,i)$ over all root components $\beta$ of height $d-1$ and colors $i$ are orthogonal and sum to $1_{B_d}$: each word has exactly one last color and one prefix root, so it lies in exactly one corner. By [F1] the multiplication map ${⨁}_{\beta ,i}{B}_{d}e(\beta ,i){\otimes }_{R^{\mathrm{\Lambda }}(\beta )}{R}^{\mathrm{\Lambda }}(\beta )\to B_d$ is an isomorphism; spelling out the left action by the basic induction functor gives $P_d=D{P}_{d-1}=D^d{P}_0$. The zero-corner case gives $B_{d}e(\beta ,i)=0$ and is included since its summand is zero.

The mixed commutators at $v=1$. Forgetting the grading shifts in the same-color categorical decomposition gives $[E_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}},F_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}]=h_i(\mathrm{\Lambda }-\beta )\mathrm{Id}$ on the root component $\beta$; its positive identity summands contribute $h_i$ copies and its negative identity summands contribute the signed counterpart, so the integral $[h_i{]}_v$ specializes at $v=1$ to the integer $h_i$ with no division by $v-v^{-1}$. The unequal-color mixed relation gives $[E_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}},F_j^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}]=0$ for $i\mathrm{\ne }j$. Hence $[E,D]={\sum }_i[E_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}},F_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}]=({\sum }_i{h}_i(\mathrm{\Lambda }-\beta ))\mathrm{Id}$ on root component $\beta$. For affine type A every Cartan column sums to zero, including the double-edge matrix at $e=2$, so ${\sum }_i{h}_i(\mathrm{\Lambda }-\beta )={\sum }_i{\lambda }_i-{\sum }_j{\beta }_j{\sum }_i{a}_{ij}=\mathrm{\ell }$ independently of $\beta$. Therefore $[E,D]=\mathrm{\ell }\cdot \mathrm{Id}$, and since rank-zero restriction is zero, $E{P}_0=0$; expanding $[E,D^d]=d\mathrm{\ell }{D}^{d-1}$ gives $E{P}_d=d\mathrm{\ell }{P}_{d-1}$.

The Hom-norm recurrence. For finite graded projectives define $h$ as in [F4]; it is additive in both variables, unchanged by shifts, and descends to projective $K_0$ modulo $v-1$. Pairing distinct root blocks by zero and summing over a fixed rank, the ordinary first adjunction $F_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}⊣E_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}$ gives $h(F_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}P,Q)=h(P,E_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}Q)$, hence $h(DP,Q)=h(P,EQ)$; all restricted modules remain finite projectives by [F3], so the recurrence stays inside the declared group. Combining with step 1.2, $h(P_d,P_d)=h(D{P}_{d-1},P_d)=h(P_{d-1},E{P}_d)=d\mathrm{\ell }h(P_{d-1},P_{d-1})\mathrm{.}$

Conclusion. The initial value is $h(P_0,P_0)={\dim }_F{\mathrm{End}}_F(F)=1$; since ${\mathrm{End}}_{B_d}(B_d)\cong B_d^{\mathrm{o}\mathrm{p}}$ by right multiplication, $h(P_d,P_d)={\dim }_F{B}_d$, and induction on $d$ gives ${\dim }_F{B}_d={\mathrm{\ell }}^{d}d!$ as an ordinary integer. Nothing reduces the integer dimension or the factor $d\mathrm{\ell }$ inside $F$, so the argument covers every field characteristic; for $\mathrm{\ell }=0$ and $d>0$ the recurrence gives ${\dim }_F{B}_d=0$; and for $d=0$, independently of $\mathrm{\ell }$, $B_0=F$ has the singleton empty monomial as basis. No Frobenius theorem, tensor-form perfectness, highest-weight or global-basis isometry, Hecke basis or choice principle is used.