The string-ideal triples are balanced: the induction step (Kashiwara 7.4.1 / JKK 10.4)

Descending setup and the extension engine. Fix the degree $\lambda -\alpha$ of height $r$ and work by descending induction on $n\ge 1$: for $n$ strictly larger than the height of the degree, $J_{(i,n)}^A=0=I_{(i,n)}^A$ is trivially balanced, so assume $J_{(i,n+1)}^A$ balanced with its adapted basis. The chart-extension engine [F1], in the relative form with $N=J_{(i,n+1)}^A$, will produce the balancedness and basis statements as soon as a finite free module $F$ with the two injective chart images and the generation property is exhibited.

The boundary regime. Put $\mu =\lambda -\alpha$ and $n_0=-\mu (h_i)$. When $n_0\ge 1$, [F2] gives $J_{(i,n_0)}^A{}_{\mu }=V_A(\lambda {)}_{\mu }$. For $1\le n<n_0$, both $J_{(i,n)}^A{}_{\mu }$ and its field string space equal the corresponding objects at $n_0$, since every admissible position at $\mu$ is at least $n_0$. Thus, after proving balancedness at $n_0$ by the next steps, these smaller $n$ inherit it. It suffices to perform the relative induction for $n\ge 1$ with $n+\mu (h_i)\ge 0$, including equality.

Decomposition of the next ideal. From this point assume $n+\mu (h_i)\ge 0$. By definition $J_{(i,n)}^A=F_i^{(n)}{V}_A(\lambda {)}_{\mu +n{\alpha }_i}+J_{(i,n+1)}^A$, and the first summand is generated by the images ${\phi }_b:=F_i^{(n)}{G}_{\mathrm{\infty }}(b)v_{\lambda }$ for those $b\in {\mathcal{B}}_{\mathrm{\infty }}$ of degree $-\alpha +n{\alpha }_i$ with ${\tilde{e}}_{i}b=0$ and ${\pi }_{\lambda }(b)\mathrm{\ne }0$; the remaining generators with ${\tilde{e}}_{i}b\mathrm{\ne }0$ have $G_{\mathrm{\infty }}(b)\in I_{(i,1)}^A$ by [F3], so their $F_i^{(n)}$-multiples are integral Gaussian multiples and lie in $J_{(i,n+1)}^A$ (The Lusztig divided-power integral form $U_A$).

Simultaneous quotient lifts. In the half decomposition the projection to position zero is $P_i=1-{\tilde{f}}_i{\tilde{e}}_i$, so it preserves $\mathcal{L}(\mathrm{\infty })$ by [F4]. For a lower-height global lift $G_{\mathrm{\infty }}(b)$ with ${\tilde{e}}_{i}b=0$, its primitive component $P_i{G}_{\mathrm{\infty }}(b)$ has residue $b$. Therefore $F_i^{(n)}{P}_i{G}_{\mathrm{\infty }}(b)={\tilde{f}}_i^n{P}_i{G}_{\mathrm{\infty }}(b)$ lies in the zero lattice, with residue ${\tilde{f}}_i^{n}b$. The remaining integral string components, after multiplication by $F_i^{(n)}$, lie in $I_{(i,n+1)}^A$ by [F2]. Projection gives ${\phi }_b\in \mathcal{L}(\lambda )+J_{(i,n+1)}^A$. Since ${\phi }_b$ is bar fixed and the next ideal is bar stable, it also belongs to $\psi (\mathcal{L}(\lambda ))+J_{(i,n+1)}^A$. These are quotient-chart memberships; membership in the two charts themselves is not presumed.

The new residue classes and injections. The condition $n+\mu (h_i)\ge 0$ makes the module string at every surviving top class have length at least $n$, so ${\tilde{f}}_i^n{\pi }_{\lambda }(b)$ is nonzero and sits at exactly position $n$; these classes biject onto $({\mathcal{B}}_{\lambda }\cap {\tilde{f}}_i^n{\mathcal{B}}_{\lambda })\setminus {\tilde{f}}_i^{n+1}{\mathcal{B}}_{\lambda }$ by the completed crystal partial inverses and the lower-degree based quotient. By the descending balancedness of $J_{(i,n+1)}^A$, its rational zero-residue has basis the higher positions, so the finite free $\mathbb{Z}$-module $F={⨁}_b\mathbb{Z}{\phi }_b$ maps injectively into $(L_0+J_{(i,n+1)})\mathrm{/}(q{L}_0+\mathbb{Q}{J}_{(i,n+1)})$; the infinity injection is its bar image under step 1.4.

Balancedness of the module ideal. Applying the relative chart-extension criterion [F1] to $M=J_{(i,n)}^A$, $N=J_{(i,n+1)}^A$ and $F$ as in step 2.1 proves integral balancedness of $J_{(i,n)}^A$ in degree $\mu$, the exact integral residue sequence, and the $\mathbb{Z}$-basis consisting of the new $n$-positions together with all higher positions; finite $A$-freeness of the Laurent extension follows as a conclusion. This proves (i) and the module part of (iii).

The half ideal. For the negative half, choose the auxiliary explicit regular dominant datum of [F5] with all coordinates large compared with the height-$r$ degree; take $\lambda (h_j)\ge r$ for every $j$, large enough also for the chart isomorphism of [F5]. The highest-weight kernel is generated by singular vectors of degrees $(\lambda (h_j)+1){\alpha }_j$, whose heights exceed $r$ (the calculation in [F5]); thus its component at $-\alpha$ vanishes. In this degree, ${\pi }_{\lambda }$ identifies the field spaces and sends $(U_A^{-})[-\alpha ]$ onto $V_A(\lambda {)}_{\lambda -\alpha }$ and $(I_{(i,n)}^A)[-\alpha ]$ onto $(J_{(i,n)}^A{)}_{\lambda -\alpha }$ by their defining images. It identifies the rational charts and commutes with bar. Transporting the already-proved balanced module triple and its signed-free residue basis of step 3.1 back along this isomorphism proves the half statements (ii) and the half part of (iii), with the residue $\mathbb{Z}$-span of ${\mathcal{B}}_{\mathrm{\infty }}\cap {\tilde{f}}_i^n{\mathcal{B}}_{\mathrm{\infty }}$ from [F2]. No global half basis and no saturation equality was assumed.

Conclusion. Steps 1.1–1.2 reduce the problem and dispose of the nonpositive-string regime; steps 1.3–3.1 prove the module balancedness, residue basis and freeness; step 4.1 transports the result to the half; thus (i)–(iii) hold in degree $\alpha$ under the stated height-$<r$ hypotheses, and the saturation identification remains a conclusion of the completed construction.