The residue translation functors give the integrable Fock-space action

Flags and the $K_0$ action. By [F1] the tensoring and block-projection functors are exact and biadjoint, their generalized-standard flags are multiplicity-free and given by the valid single-entry moves, and the standard-class map is an isomorphism of the block Grothendieck group with the integral Fock module. Checking the induced operators on the standard basis gives the full integrable ${\mathfrak{g}\mathfrak{l}}_{I^{+}}$ Chevalley action and its integral Chevalley/Serre relations; this is a statement about $K_0$, not yet about heads and socles.

The ground state. For the ground multiset the column-strict admitted label is unique: successive strict entries are at least their row ground values and the minimal total weight forces equality, so the ground label is the only one of its central character. In this one-label highest-weight block, [F3] gives $[M(A_{\mathrm{\Lambda }}):L(A_{\mathrm{\Lambda }})]=1$ and $(P(A_{\mathrm{\Lambda }}):M(A_{\mathrm{\Lambda }}))=1$ by reciprocity. Hence its standard, simple and projective coincide; the entire highest constituent $V(\mathrm{\Lambda })$ is generated by its ground class, and the tensor Fock module is represented by all standard classes, so the ground class generates exactly that multiplicity-one constituent. This need not be the entire tensor Fock module; at level $l=1$, however, the defining tensor product has the single irreducible factor $F(\mathrm{\Lambda })=V({\mathrm{\Lambda }}_{m_1})$, so the ground class does generate the whole module.

Head and socle. Apply the local lemma [F2] to the positive biadjoint pair $E=f_i$, $F=e_i$ (and to the swapped root orientation for the dual direction). Its categorical string argument makes every nonzero translate of a simple a self-dual indecomposable with simple equal head and socle. Its label comparison uses precisely the full source upper chart of [F4]: the adapted kernel heights and the nonzero associated-string coefficients identify the unique maximal-height factor as $L({\tilde{f}}_{i}A)$, respectively $L({\tilde{e}}_{i}A)$, and absence of the edge is exactly vanishing. Thus these labels use the stated $-+$ signature, not the original-order lower tensor chart. Restricted duality commutes with the functors by [F3], so it preserves the head/socle conclusions.

Conclusion. The residue functors have multiplicity-free generalized-standard flags, induce the full integral Chevalley Fock action on $K_0$, vanish on a simple exactly when the crystal edge is missing, and otherwise produce a self-dual indecomposable with simple equal head and socle at the stated crystal label; the ground-state class generates exactly its multiplicity-one highest constituent.