Standard tableau monomials form a basis of the highest weight Fock submodule

The normalized highest inclusion. The explicitly constructed fundamental factors make the ground tensor integrable and give it the unique top weight $\mathrm{\Lambda }$. The universal highest-module map from the actual existence-and-uniqueness theorem sends the normalized highest vector to this ground tensor and has nonzero image. Ordinary simplicity of that highest module makes the map $j:V(\mathrm{\Lambda })\to F(\mathrm{\Lambda })$ injective. A module injection preserves each highest kernel and the unique divided-string decomposition, hence commutes with the intrinsic lower string shifts. It also sends divided-power generators to the actual integral tensor lattice. The normalized projection will be constructed by the finite nondegenerate pairing below, rather than assumed from a general semisimplicity assertion. Its uniqueness follows from the one-dimensional top weight: a nonzero map to the simple highest module is surjective, hence cannot vanish on the top weight.

The actual balanced tensor chart. Let $L_0^{\mathrm{t}\mathrm{e}\mathrm{n}}$ and $\mathrm{\Psi }{L}_0^{\mathrm{t}\mathrm{e}\mathrm{n}}$ be the actual two charts of the finite-product theorem, not the raw tensor-word chart. The normalized ground tensor is a nonzero ground crystal residue. Every generator of $L_0(\mathrm{\Lambda })$ is a divided-string word on its highest vector; the naturality established in step 1.1 and the actual tensor crystal rule put its image in $L_0^{\mathrm{t}\mathrm{e}\mathrm{n}}$ with the corresponding nonzero ground-component residue, with nonvanishing ensured by the partial-inverse rule. Divided-power generation gives $j{V}_{\mathcal{A}}(\mathrm{\Lambda })\subseteq F_{\mathcal{A}}$. Both bar composites agree on the highest vector and on every bar-fixed divided lowering monomial, so $j\overline{L_0(\mathrm{\Lambda })}\subseteq \mathrm{\Psi }{L}_0^{\mathrm{t}\mathrm{e}\mathrm{n}}$. Hence $jG(b)$ is integral, lies in both actual charts, is bar fixed and has the required residue. Balanced uniqueness identifies it with the corrected tensor global vector. Since $t=s^{-1}$, this library zero chart is the source infinity chart: the original factor order yields $T_A$ on $\mathrm{R}\mathrm{e}\mathrm{v}$, and the reverse-factor order yields ${\tilde{T}}_A$ on $\mathrm{S}\mathrm{t}\mathrm{d}$. This proves based inclusion rather than assuming every module map is based.

The ground components. On a strict column the crystal signature rule has $+$ for entry $i$ and $-$ for entry $i+1$ with simultaneous occurrences cancelling; concatenating down-column words, cancelling $-+$ pairs, the rightmost surviving $+$ and leftmost surviving $-$ determine the source upper operators. Semistandard rows are stable under this rule: a changed entry violating a row inequality has an equal neighbouring entry which is either a later surviving $+$, an earlier surviving $-$ or its cancelling mate, and column strictness forces the last matching in the same row strip. The only highest tableau of the closed set is the ground state: the rightmost column is consecutive ground, and inducting to the left, rows overlapping the already-ground right columns are forced to be ground while a gap above the overlap produces an unmatched minus; repeated raising decreases the sum of entries, bounded below on a finite interval. Thus the reverse-factor lower ground component, equivalently the source original-order upper signature component, is exactly ${\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$; the original-order lower component is the reverse-standard set from step 2.1.

Rectification. Reversing the tensor order gives the reverse reading of the same signature rule; aligning column tops by the prescribed vertical shifts makes the reverse-standard inequalities stable, and the explicit row-insertion rectification (bumping the smallest larger letter, with the elementary Knuth interchanges preserving the cancelling signs) is a weight-preserving crystal isomorphism between the original lower ground component and the reverse-factor lower ground component ${\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$, with inverse reverse insertion. This is the label dictionary of [F1].

The projection. Define $\pi$ by $C_0(v,\pi y)=C(j_{\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}}v,y)$, where $j_{\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}}$ is the based reverse-factor highest inclusion. The required highest form is constructed from the actual library polarization $B_8$: at weight $\mathrm{\Lambda }-\alpha$ put $C_0=s^{a(\alpha )}{B}_8$, $a(\alpha )=(2\mathrm{\Lambda }-\alpha ,\alpha )\mathrm{/}2$. Its nondegeneracy and highest value one follow from $B_8$. The translated adjoint is $\sigma (F_i)=s{K}_i^{-1}{E}_i$, $K_i=D_i{D}_{i+1}^{-1}$. For input weight $\mu$ this contributes $s^{1-\mu (h_i)}$, canceled by $a(\alpha +{\alpha }_i)-a(\alpha )=\mu (h_i)-1$. Thus $C_0(F_{i}x,y)=C_0(x,E_{i}y)$, and similarly $C_0(E_{i}x,y)=C_0(x,F_{i}y)$; toral operators remain self-adjoint. This constructs, rather than assumes, the normalized nondegenerate $\mathcal{E}\mathcal{F}$-contravariant form. Here is the needed typed contravariance locally. Let $B_0$ be the coefficient dot pairing of reversed and original tensor words, and write $\mathrm{\Psi }=\mathrm{\Theta }{\overline{(\cdot )}}_0$ on a finite product, with ${\overline{(\cdot )}}_0$ the factorwise coefficient bar. Since $\mathrm{\Psi }{M}_B=\mathrm{\Theta }{M}_B$, the normalization $C({\tilde{M}}_A,\mathrm{\Psi }{M}_B)={\delta }_{AB}$ gives $C(x,y)=B_0(x,{\mathrm{\Theta }}^{-1}y)$. On each minuscule factor the matrices of $E_i$ and $F_i$ are transposes. Reversing the factors in the explicit coproduct therefore gives $B_0(E_{i}x,z)=B_0(x,\bar{\mathrm{\Delta }}(F_i)z)$, $B_0(F_{i}x,z)=B_0(x,\bar{\mathrm{\Delta }}(E_i)z)$, while toral $D_a$ is self-adjoint. The finite quasi-$R$ intertwiner satisfies $\mathrm{\Theta }\bar{\mathrm{\Delta }}(u)=\mathrm{\Delta }(\bar{u})\mathrm{\Theta }$; for $E_i,F_i$ their bars equal themselves. Substitution proves $C(E_{i}x,y)=C(x,F_{i}y)$, $C(F_{i}x,y)=C(x,E_{i}y)$ and $C(D_{a}x,y)=C(x,D_{a}y)$, first on two factors and then by coproduct association on each finite product. The required complementary delta pairing can also be checked here. The paired-root Casimir terms of the finite quasi-$R$ expansion, under factor reversal and the minuscule $E\mathrm{/}F$ transpose, give ${\mathrm{\Theta }}_{\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}}^{\mathsf{T}}={\bar{\mathrm{\Theta }}}^{-1}$; the inverse/factorization identities extend this from two factors to every finite product. Consequently $C$ commutes with both bars in the sense $C(\tilde{\mathrm{\Psi }}x,\mathrm{\Psi }y)=\overline{C(x,y)}$. Since $L_B=\mathrm{\Theta }{\overline{L_B}}_0$, one has $C({\tilde{T}}_A,L_B)=B_0({\tilde{T}}_A,{\overline{L_B}}_0)$. The two coefficient families on the right have coefficients in $\mathbb{Z}[s^{-1}]$, with constant matrix the identity and every off-diagonal term in $s^{-1}\mathbb{Z}[s^{-1}]$. Their value is bar fixed by the preceding identity, hence belongs to $\mathbb{Z}[s^{-1}]\cap \mathbb{Z}[s]=\mathbb{Z}$, forcing $C({\tilde{T}}_A,L_B)={\delta }_{AB}$. The triangular bar-fixed families used here are obtained locally in finite weight spaces: at each lower label, involutivity makes the residual coefficient $r(s)$ satisfy $r(s)+r(s^{-1})=0$; its positive-power part, or negative-power part, gives the unique required correction, and induction on the finite lower ideal constructs the complementary charts. Thus this use requires neither an unproved Gram identity nor a presumed based embedding. These identities are over $\mathbb{Q}(s)$; any stated complex coefficient conjugation is restored on scalar extension. Thus the normalized highest form $C_0$ has precisely the required $E\mathrm{/}F$ adjunction and toral symmetry, and $\pi$ is $U$-linear; its highest coefficient is $1$, so its composite with the original highest inclusion is the identity on the simple highest module. It is therefore a genuine normalized projection, and its uniqueness is the top-weight argument of step 1.1.

Vanishing and the dual-canonical basis. Since $j_{\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}}$ sends the lower global basis to ${\tilde{T}}_A$ with $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$, the delta pairing $C({\tilde{T}}_A,L_B)={\delta }_{AB}$ of [F1] gives $\pi L_B=0$ for $B\mathrm{\notin }{\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$, while for $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$ the vectors $D_A=\pi L_A$ form the $C_0$-dual basis of the constituent. This proves the vanishing statement (2) and that the $D_A$ are the source dual-canonical basis.

Straightening and the $S$-basis. Expanding $M_B={\sum }_A{d}_{A,B}(s)L_A$ by the triangular chart of [F1] and applying $\pi$ gives $\pi M_B={\sum }_{A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}}{d}_{A,B}(s)D_A,$ the restriction to ${\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$ of a unitriangular matrix with diagonal $1$; hence the projected monomials $S_A=\pi M_A$, $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$, form a basis of the constituent and the displayed formula is the explicit straightening of every projected monomial, standard or not. No projected-monomial vanishing is claimed.

The integral kernel and the source upper lattice. By step 6.1 the kernel of $\pi$ is spanned by the $L_A$ with $A\mathrm{\notin }{\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$; being spanned by a subset of a basis it is a saturated direct summand of the integral Fock lattice. The image $\pi (F_{\mathcal{A}})$ is spanned by the $D_A$, hence is the source upper/costandard lattice; it is not identified with the lower divided-power lattice by an $A$-unit Gram determinant, since the relevant monomial Gram matrices are the inverse ordinary-bar matrix rather than the identity.

Cuts and the direct limit. All constructions and all coefficients of steps 1.1-8.1 are stable under the upper-cut and common-tail embeddings by the declared truncation compatibility theorem, and every vector lies in some finite double cut; the normalized highest maps are unique, so the maps agree on overlaps. The compatible direct limit gives the full bounded or semi-infinite assertion, and the comparison with the library form-dual upper family is the explicit weight rescaling proved in the global-basis comparison.

Conclusion. The highest Fock constituent $V(\mathrm{\Lambda })$ has the standard monomial basis $S_A$ and the source dual-canonical basis $D_A=[\pi L_A]$ over ${\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$, with the explicit straightening formula and the saturated integral kernel off ${\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$; the library comparison uses the weight rescaling only.