Reversibility and current recognition for the exact essential image of the twisted quantum Frenkel–Kac functor

Let L be a root lattice of Dynkin type An (n ≥ 1), Dn (n ≥ 4), E6, E7, or E8 (ADE type) and let μ be a finite-order Dynkin diagram automorphism. The lattice construction associates a level-one restricted module of the corresponding twisted quantum affine algebra with each equivariant ϕ-coordinated module. We determine its range and prove that the construction is reversible there. At the algebra level, Lℏ,τ1(g)≃VL[[ℏ]]η. At the module level, the induced functor is an equivalence onto the full subcategory annihilated by the integrability ideal; at level one this is the standard restricted integrable category defined by local nilpotence of all Drinfeld current modes. Membership in this subcategory is characterized by the vanishing of the (−1)-st ϕ-product squares of the normalized simple-root currents. The resulting obstruction fields have a constant-term expression in the Drinfeld currents, and their common zero locus is independent of normalization. Quotient factorization then gives an explicit inverse functor. The algebra isomorphism is obtained by identifying the quotient vertex bialgebra, recovering the mixed and Cartan parameters from the root parameter, and gauging away the regular symmetric part of the comparison kernel. Its antisymmetric normal form is Ωμ. We also obtain a canonical family of ℏ-simple modules with a Hom-space formula and describe the associated based deformation space. Whether the simple family is exhaustive remains open.