The universal affine vacuum module is restricted

We prove by induction on the length of a negative-mode monomial $W\mathbf{1}$ that, for fixed $x\in \mathfrak{g}$, $x_{m}W\mathbf{1}=0$ for all sufficiently large $m$. For the empty monomial this holds for every $m\ge 0$, since $\mathfrak{g}[t]$ kills $\mathbf{1}$.

For the induction step write $W\mathbf{1}=y_{-r}{W}^{\mathrm{\prime }}\mathbf{1}$ with $r\ge 1$. By induction choose bounds for $x_m{W}^{\mathrm{\prime }}\mathbf{1}$ and $[x,y{]}_{m-r}{W}^{\mathrm{\prime }}\mathbf{1}$. Take $m$ larger than both bounds and larger than $r$. Then [F2] and [F3] give $x_m{y}_{-r}{W}^{\mathrm{\prime }}\mathbf{1}=y_{-r}{x}_m{W}^{\mathrm{\prime }}\mathbf{1}+[x,y{]}_{m-r}{W}^{\mathrm{\prime }}\mathbf{1}+mkB(x,y){\delta }_{m,r}{W}^{\mathrm{\prime }}\mathbf{1}=0$. The first two terms vanish by the chosen bounds and the central term vanishes because $m>r$. This proves the induction step.

By [F1], any $v$ is a finite linear combination of such monomials. For each summand step 2.1 supplies a bound; the maximum of these finitely many bounds works for $v$. This is exactly the restrictedness condition in [F4], with no energy grading or unitarity assumed.