Level zero integrable affine highest weight simples are one dimensional

By F2, every label $m_i=\lambda (h_i)$ is a nonnegative integer. The zero-level hypothesis and F1 give $0={\sum }_i{b}_i{m}_i$. Since every coefficient is positive, a positive label would make this sum strictly positive. Hence every $m_i=0$. The terminal assertion in F2 now gives $f_{i}v=0$ for every $i$. Each $e_{i}v=0$ by the highest-vector condition, and every $h\in \mathfrak{h}$ acts on $v$ by $\lambda (h)$.

Thus the nonzero line $\mathbb{C}v$ is invariant under all generators $e_i,f_i,\mathfrak{h}$ and therefore under their Lie algebra and enveloping algebra. As $v$ generates the module, the module equals that line and has dimension one. On a one-dimensional representation every action is scalar, so for any $x,y$, the commutator acts by $\rho (x)\rho (y)-\rho (y)\rho (x)=0$. All of the derived algebra therefore acts trivially. The argument kills the simple-coroot labels, but it imposes no vanishing on values of $\lambda$ outside their span; these remain the scalar actions of complementary Cartan directions.

This proves the asserted irreducible case and the stronger cyclic case without using irreducibility. The zero-module exception is excluded by $v\mathrm{\ne }0$; the zero-level endpoint was used essentially in 1.1 and no positive-level conclusion is inferred here. There is no choice of bases or infinite family of witnesses: the single line is specified by $v$, and all label comparisons are finite. No AC or character formula enters.