The semi-infinite action on the Fock space is well defined and the Fock space is an integrable $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_e)$-module

Stabilized, normalized coproducts. For a charge-$M$ partition choose a prefix of length $N$ extending beyond its nonvacuum rows and all finitely many node outputs. The stable-prefix computation in [F2] uses $F_i^{[N]}$, $q^{-{\delta }_{i\equiv M-N}}{E}_i^{[N]}$, $q^{{\delta }_{i\equiv M-N}}{K}_i^{[N]}$ and $q^{{\mathrm{\Omega }}_{M-N}(d)}{K}_d^{[N]}$, followed by tail attachment. Here $[N]$ denotes the finite coproduct action. Each has the cutoff-independent node value proved in that supplier; any deeper individual insertion vanishes by [F1]. Thus these normalized expressions stabilize exactly and converge in the separated $q$-adic topology. The unnormalized finite $E_i$ and toral coproducts are not claimed to converge. For charge $M$, the node coefficients are the charge-zero coefficients with colors shifted by $M$, so the mixed commutator and Serre proofs of [F2] apply under that relabelling. The full toral operators are assigned by the displayed label weights, which shift by $\pm {\alpha }_i$ and therefore satisfy every toral-action relation, including the degree direction.

The wedge product formulas. Split a sufficiently deep finite prefix at length $r$, with the right part representing a charge-$(M-r)$ tail $u$. Its artificial boundary is again $M-N$. Write $b={\delta }_{i\equiv M-N}$. The normalized right torus is $K_{i}u=q^b{K}_i^{[N-r]}u$, and its normalized raising operator is $E_{i}u=q^{-b}{E}_i^{[N-r]}u$. Multiplying the finite split coproduct by $q^{-b}$ therefore gives $E_i(v\wedge u)=E_{i}v\wedge K_i^{-1}u+v\wedge E_{i}u$. The finite lowering coproduct needs no scalar correction, giving $F_i(v\wedge u)=F_{i}v\wedge u+K_{i}v\wedge F_{i}u$. For the torus the two labels add, since ${\mathrm{\Omega }}_M={\sum }_{k=1}^r\mathrm{w}\mathrm{t}(u_{M-k+1})+{\mathrm{\Omega }}_{M-r}$ by the vacuum recurrence; the degree normalization agrees at the common cutoff as well. Stabilization from step 1.1 and linearity transport these finite identities through the local wedge quotient, proving quantum-group linearity. On the vacuum the only addable node has residue $M$, all raising operators vanish, and $K_i$ acts by $q^{{\delta }_{i\equiv M}}$.

Integrability and decomposition. For a fixed color, [F2] puts each basis vector in a finite two-vertex tensor cube; a vector is a finite linear combination, so both generators of that color are locally nilpotent on it. The charge-$M$ label is ${\mathrm{\Omega }}_M-{\sum }_i{m}_i(\lambda ){\alpha }_i$, so the support lies in the single downward cone ${\mathrm{\Omega }}_M-Q^{+}$. Since the simple roots of the full affine datum are independent, a weight fixes all residue counts $m_i$ and hence the size ${\sum }_i{m}_i$; only finitely many partitions have that size, proving finite-dimensional label spaces. Thus all the finite-cone and finite-weight hypotheses of [F3] hold, and $F$ is a locally weight-finite direct sum of simple integrable highest weight modules. Choice enters only through [F3]. Characters and multiplicities are computed later from the actual crystal highest vertices.

Conclusion. The stable prefix operators give a well-defined integrable affine action on the charge-$M$ Fock space with the displayed wedge product formulas, $q$-adic convergence and finite stabilisation, and the finite-weight upper cone is a locally weight-finite direct sum of simple integrable highest weight modules, with the Axiom of Choice used only as declared.