Simple and indecomposable-projective crystals agree with $B(\mathrm{\Lambda })$

The simple graph. Inflate a cyclotomic simple to the generic algebra; by [F1] the one-step restriction has a single simple socle and head once and induction has the unique simple head, giving well-defined operators ${\tilde{e}}_i,{\tilde{f}}_i$ on shift orbits with ${\epsilon }_i$ decreasing by one under ${\tilde{e}}_i$ and increasing by one under ${\tilde{f}}_i$. By [F2] the graph is lowering-generated with strict elementary extractions and positive extraction at every nonvacuum vertex, over every field including positive characteristic. The normalized-label supplier in the Given identifies each simple shift orbit with its unique $D_{\psi }$-self-dual representative. Taking that representative for each nonzero socle and induction head lifts both operators to ${\mathcal{B}}_{\mathrm{\Lambda }}$ and preserves their partial-inverse identities.

Identification with $B(\mathrm{\Lambda })$. Apply the canonical elementary extraction of [F3]: the generic simple graph is canonically $B(\mathrm{\infty })$ and the first-strand cyclotomic survivors are exactly the vertices with $c_i(b)\le {\lambda }_i$ for all $i$, which is the crystal $B(\mathrm{\Lambda })$ with vacuum $u_{\mathrm{\Lambda }}$; the survivor criterion is ${\epsilon }_i^{\mathrm{L}\mathrm{E}\mathrm{F}\mathrm{T}}(S)\le {\lambda }_i$ for every $i$ by the first-dot index of [F1]. The one-step socle/head is one shifted copy, so no quantum-factorial multiplicity enters.

Projective labels. The normalized indecomposable projectives carry the same crystal by their normalized simple heads and covers: the quotient-cover lemma pairs each projective cover with its simple head without scaling and preserves the label under the quotient, while the two Grothendieck identifications of [F3] record the same labels on $K_0$ and $G_0$. The projective graph is therefore explicitly transported from the simple labels, not obtained by applying linear Kashiwara operators to an unspecified projective basis. The assumed Axiom of Choice supplies the inherited hypotheses of the canonical-extraction and Grothendieck-identification suppliers of [F3], including the integral projective theorem and its restricted-dual consequence; transporting normalized cover labels introduces no further choice.