The highest weight Fock submodule carries the quantum-group global bases

The lower family and the pairing. Use the based inclusions, rectification and adjoint projection proved in [F1]. The normalized highest injection preserves intrinsic divided strings, the divided-power integral generators and the normalized bar. On a finite cut the balanced uniqueness of [F2] and the supported finite tensor construction of [F4] therefore identify the image of each lower global lift with the corrected tensor lower lift. At $t=s^{-1}$ this is $T_{A^{\uparrow }}$ in the original order, and ${\tilde{T}}_A$ in reverse order; the rectification proved in that supplier gives the indicated matching of labels. Write $j_{\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}}$ for its normalized reverse-order highest injection. The supplier constructs its projection by $C_0(v,\pi y)=C(j_{\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}}v,y)$: nondegeneracy of $C_0$, the locally proved $E\mathrm{/}F$ contravariance of $C$ and highest normalization make this a normalized module projection. Thus it is the $\pi$ of its Statement, rather than an assumed adjoint for arbitrary module maps. Taking $v=G_{\mathrm{l}\mathrm{o}\mathrm{w}}(A)$ and $y=L_B$, the reverse delta pairing $C({\tilde{T}}_A,L_B)={\delta }_{AB}$ of [F3] gives $C_0(T_{A^{\uparrow }},D_B)={\delta }_{AB}$.

The weight recursion $B_8=s^{-a}{C}_0$. The generator identities of the dictionary determine the library contravariant form $B_8$ and the normalized source form $C_0$ compatibly; on weight $\mathrm{\Lambda }-\alpha$ the semilinear normalization gives $B_8=s^{-a(\alpha )}{C}_0$. Indeed the dictionary sends the library $\sigma$ at $t=s^{-1}$ to the source anti-involution, whose action on a lowered vector of weight $\mu$ contributes $s^{-⟨\mu ,h_i⟩-1}$, and the exponent identity $a(\alpha )-a(\alpha -{\alpha }_i)=⟨\mu ,h_i⟩+1$ with highest normalization and induction on lowering monomials proves the equality on every weight space.

Identification of the upper family. Applying step 1.2 to step 1.1 gives $B_8(T_{A^{\uparrow }},s^a{D}_B)={\delta }_{AB}$; by the dual-family uniqueness in [F2], the vectors $G_{\mathrm{u}\mathrm{p}}(A)=s^a{D}_A$ are the actual library form-dual upper vectors, fixed by the form-adjoint upper bar with the genuine dual lattices and charts.

The quasi-canonical family on the constituent. The source form satisfies $⟨x,y⟩=s^a{C}_0(\psi x,y)$ on the constituent. To verify the actual recurrence, put $K_i=D_i{D}_{i+1}^{-1}$. The coefficient Gram formula ${\mathrm{\Theta }}^{-1}$ and the minuscule generator transpose identities give $⟨E_{i}x,y⟩=⟨x,s{F}_i{K}_i^{-1}y⟩$, $⟨F_{i}x,y⟩=⟨x,s^{-1}{K}_i{E}_{i}y⟩$ and the toral adjoint $D_a^{-1}$; the finite coproduct intertwiner transports these identities through each tensor prefix. For example, on input weight $\mu$ the first adjoint has scalar $s^{1-\mu (h_i)}$. This is precisely the ratio $s^{a(\alpha )-a(\alpha +{\alpha }_i)}$, since $a(\alpha +{\alpha }_i)-a(\alpha )=\mu (h_i)-1$. Together with $C_0(E_{i}x,y)=C_0(x,F_{i}y)$, the ordinary bar and highest value one, induction on divided lowering monomials proves the displayed identity. These comparisons are made over $\mathbb{Q}(s)$, with the specified coefficient conjugation on complex scalar extension. The left annihilator of $\ker \pi$ under this form is $U$-stable by its anti-involution, contains the ground highest vector, and has dimension $\dim V(\mathrm{\Lambda })$; hence it is exactly the highest constituent. The delta evaluations of the quasi-canonical vectors show that $P_A$ lies in that annihilator for $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$; on the constituent the candidate $s^a{T}_{A^{\uparrow }}$ has the same delta evaluations against the $D$-basis by step 1.2 and annihilates the off-standard $L$-vectors by [F1], so whole-Fock dual uniqueness gives $P_A=s^a{T}_{A^{\uparrow }}$ on $V(\mathrm{\Lambda })$.

Integrality, cuts and limits. The units $s^{\pm a(\alpha )}$ are Laurent monomials, so the $D$-to-library-upper and $P$-to-library-lower comparisons of steps 1.1-3.1 are separately weightwise integral; their charts are the corresponding actual rescaled or dual charts. These comparisons do not decide equality of the two embedded lattices. For the single fundamental factor with $I=\{0\}$, the lower basis is $v_0,v_1=F_0{v}_0$; distinct weights are orthogonal and $B_8(v_1,v_1)=B_8(v_0,t^{-1}{K}_0{E}_0{v}_1)=1$, so the Gram matrix is the identity and its integral form-dual equals the lower lattice. In contrast, the root-highest-weight-two string has $B_8(Fv,Fv)=t[2{]}_t=1+t^2$, a nonunit, so its dual lattice differs. The finite double-cut compatibility of the Fock bases and of the library global bases, together with uniqueness of the normalized highest maps, proves the full direct-limit statement, and only the actual integral and specialization clauses of [F4] are used.

Conclusion. The library lower global vector is $G_{\mathrm{l}\mathrm{o}\mathrm{w}}(A)=s^{-a}{P}_A$, the actual form-dual upper vector is $G_{\mathrm{u}\mathrm{p}}(A)=s^a{D}_A$, and the source quasi-canonical family satisfies $P_A=s^a{T}_{A^{\uparrow }}$ on the highest constituent; the source $D$ image lattice compares with the library form-dual upper lattice by $s^a$, while the source $P$ lattice compares with the library lower divided-power lattice by $s^{-a}$. The embedded upper and lower lattices can coincide but need not, and no ordinary-bar fixedness is claimed for the unqualified library upper family.