Universal affine vertex algebras exist at every complex level

Let $\mathcal{S}=\{J^x(z):x\in \mathfrak{g}\}$. By [F2], each basis monomial of $V^k(\mathfrak{g})$ is obtained from $\mathbf{1}$ by successively applying negative modes of fields in $\mathcal{S}$; hence their negative modes span the module. By [F3] these are fields, and [F5] supplies their creation, translation covariance and pairwise locality.

The hypotheses of [F6] now hold for $V^k(\mathfrak{g})$, $\mathbf{1}$, $T$, and $\mathcal{S}$, giving a state-field map with the prescribed currents and the axioms in [F7]. For any competing structure with translation $T^{\mathrm{\prime }}$, [F7] gives $T^{\mathrm{\prime }}\mathbf{1}=0$ and $[T^{\mathrm{\prime }},J^x(z)]={\mathrm{\partial }}_z{J}^x(z)$, hence $[T^{\mathrm{\prime }},x_n]=-n{x}_{n-1}$. By [F4], $T^{\mathrm{\prime }}-T$ kills $\mathbf{1}$ and commutes with every current mode, so it vanishes on the spanning monomials of [F2]. Thus $T^{\mathrm{\prime }}=T$, and uniqueness for this fixed translation in [F6] proves uniqueness of the entire structure.

Every construction and verification above uses only the affine bracket, the vacuum character at level $k$, and formal field operations. No step divides by a function of $k$ or assumes a grading, positivity, or a Sugawara operator. Therefore the same vertex-algebra construction applies to every $k\in \mathbb{C}$, with the claimed limits on its conclusion.