Self-dual simple cyclotomic modules give the upper global basis

The functional ${\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}(S)$. By [F1] the pairing is diagonal in normalized bases, and by [F2] the class $\varphi (P(S))$ is the lower global vector with the same crystal label; therefore ${\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}(S)$ is exactly the weightwise dual lower-basis functional with that label.

Identification with the upper global basis. The integral projective identification [F3] identifies the tensor form on projectives with the actual highest-weight contravariant form; the upper/lower duality theorem identifies these restricted dual functionals with the upper global vectors and preserves crystal labels. This proves the bijection and the normalization statements, using neither a generic negative-half form nor a vector-dual trace shift.

Cover/simple correspondence and the two actions. A simple and its normalized projective cover have the same $B(\mathrm{\Lambda })$ label by the cover/simple correspondence of [F2], not by applying projective Hom duality to a simple. The $G_0$ theorem [F3] proves the canonical action and its defect-rescaled KK version; keeping the two maps distinct preserves both sound quantum actions and the rank-one $\lambda =2$ check, with ${\mathrm{\Psi }}_{\mathrm{K}\mathrm{K}}=q^{{\mathrm{def}}_{\mathrm{\Lambda }}(\beta )}{\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}$ on root $\beta$, so that only ${\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}$ gives the unrescaled upper-basis claim. The Axiom of Choice is inherited through the lower projective and canonical crystal suppliers, the QG-5 global-basis duality and the restricted-dual identifications of [F2] and [F3].