The Fock pairing matches the projective-simple and standard-costandard pairings

The general delta identities. Substituting the inverse transition matrices of [F1] into the explicitly bar-twisted pairings gives the general identities $⟨P_A,L_B⟩={\delta }_{AB}$, $C({\tilde{L}}_A,T_B)={\delta }_{AB}$ and the complementary ones; the source sesquilinear form has semilinear $\tau$-adjunction, while the reverse-factor bilinear form has the linear $\mathcal{E}\mathcal{F}$-swap adjunction with the torus fixed, checked on the raw generator matrices and their coefficient conjugates.

The highest-constituent form $C_0$. On irreducible highest constituents the normalized nondegenerate restriction of the reverse-factor bilinear form is $C_0$; the dictionary sends the library $\sigma$ with $\sigma (F_i)=t^{-1}{K}_i{E}_i$ to the source anti-involution acting by $s{D}^{-1}E$, which contributes $s^{-⟨\mu ,h_i⟩-1}$ on a target vector of weight $\mu$, and the exponent identity $a(\alpha )-a(\alpha -{\alpha }_i)=⟨\mu ,h_i⟩+1$ with highest normalization and induction on lowering monomials proves $B_8=s^{-a}{C}_0$ on every weight space of the constituent.

The specialization at $s=1$ and the Hom pairing. At $s=1$ the ordinary bar becomes the identity because the normalized Hecke generators are bar-fixed there, so the labelled monomial Gram matrices specialize to the identity and the delta identities of step 1.1 give $⟨P_A,L_B⟩={\delta }_{AB}$. For a projective $P$ and finite-length $N$, exactness and additivity of $\mathrm{Hom}(P,-)$ on a composition series, together with the projective/simple pairing, give $⟨[P],[N]⟩=\dim \mathrm{Hom}(P,N)$.

The standard/costandard pairing. The pairing $\mathrm{Hom}(M(A),\mathrm{\nabla }(B))={\delta }_{AB}\mathbb{C}$ is the actual generalized-standard Hom theorem of the block structure [F2], transported by restricted duality and the typed opposite filtered equivalence; it is not a restriction of the ordinary Verma lemma, and it is a separate pairing from the source Fock forms. The types original/reverse/dual and ordinary/adjoint bars remain explicit: the library upper family uses the form-adjoint bar and the rescaling of [F3], not the bare source forms.

Conclusion. The source Fock forms pair projectives with simples and reverse-canonical with canonical vectors by the stated deltas, the standard/costandard pairing is the separate generalized Hom delta, and the library polarization is $B_8=s^{-a}{C}_0$ on the highest constituent; the source forms, the standard/costandard form and the library form are not conflated.