Simples, projectives and tiltings map to the three distinguished Fock bases

Simples. The mixed singular/parabolic coefficient theorem gives $[M(B):L(A)]=d_{A,B}(1)$ including repeated entries; inverting this unitriangular matrix by the standard-class isomorphism of [F1] gives $[L(A)]=L_A$, because $L_A$ is defined as the inverse chart of the monomial basis through the same matrix $d$ as the Fock bases.

Projectives. BGG reciprocity [F2] gives $[P(A)]={\sum }_B[P(A):M(B)][M(B)]={\sum }_B{d}_{A,B}(1)M_B=P_A$, which is exactly the quasi-canonical family of [F1].

Tiltings. The opposite block labels and reversed order were computed by the semi-regular label helper, and Ringel reciprocity [F3] gives $[T(A)]={\sum }_B{\tilde{d}}_{A,B}(1)M_B$; the typed reverse-factor pairing of [F4] has labelled monomial Gram matrix specializing to the identity at $s=1$, and it identifies exactly this latter matrix with the source canonical coefficients, giving $[T(A)]=T_A$. The matrix $\tilde{d}$ is the opposite/reverse-column matrix and is not generally the inverse transpose of the original composition matrix.

Conclusion. At $s=1$ the simple, projective and tilting classes of the block are the dual-canonical, quasi-canonical and canonical Fock vectors respectively, with the stated coefficient matrices; the result is ungraded and asserts neither Koszul grading nor an identification of library upper and lower families.