A nonsplit logarithmic Heisenberg module obstructs rationality

The space $M_0=P\otimes \mathbb{C}{e}_0$ is stable under every current mode: multiplication and differentiation act only on $P$, while $a_0$ acts as the identity on $\mathbb{C}{e}_0$. Each state field of $V$ is a normal-ordered product of divided derivatives of this current, so every coefficient of every state field also preserves $M_0$. Hence $M_0$ is a $V$-submodule. The state modes descend to $F\mathrm{/}{M}_0$ and its module identities descend from those on $F$. The induced grading and the restriction grading on $M_0$ satisfy the same admissible mode-shift rule in [F2], so both are admissible.

The module $M_0$ is simple. Let $0\mathrm{\ne }f\otimes e_0\in M_0$, and choose a monomial $x^{\beta }={\prod }_r{x}_r^{{\beta }_r}$ of maximal ordinary total degree among the monomials of $f$ with nonzero coefficient. Applying ${\prod }_r{a}_r^{{\beta }_r}$ differentiates by ${\prod }_r(r{\mathrm{\partial }}_{x_r}{)}^{{\beta }_r}$. Every term of lower total degree is killed; among terms of the same total degree, only $x^{\beta }$ survives, since a distinct exponent vector of the same total degree cannot dominate $\beta$ coordinatewise. The result is a nonzero multiple of $1\otimes e_0$. The negative modes $a_{-r}$ then generate every monomial times $e_0$, so every nonzero submodule is all of $M_0$.

If $V$ were rational, write $F={⨁}_{i\in I}{S}_i$ as a direct sum of simple admissible submodules. A nonzero vector of $M_0$ has nonzero projection to some $S_j$, because every vector in a direct sum has finite support. The coordinate projection ${\pi }_j:F\to S_j$ is a graded module homomorphism. Its restriction to $M_0$ is nonzero; simplicity of $M_0$ makes that restriction injective, and simplicity of $S_j$ makes its image all of $S_j$. Thus ${\pi }_j{\mathrm{\mid }}_{M_0}$ is an isomorphism. The map $r=({\pi }_j{\mathrm{\mid }}_{M_0}{)}^{-1}{\pi }_j:F\to M_0$ is a graded module retraction, so $U=\ker r$ is a graded module complement and $F=M_0\oplus U$.

By [F2], the generalized $L_F(0)$-eigenspace of weight $\frac{1}{2}$ in $F$ is exactly $E=P_0\otimes E$, and in $F\mathrm{/}{M}_0$ it is $E\mathrm{/}\mathbb{C}{e}_0$, which is one dimensional. Since $U$ is a module complement, the quotient map restricts to an $L_F(0)$-intertwining isomorphism from $U$ onto $F\mathrm{/}{M}_0$; it therefore maps the generalized weight-$\frac{1}{2}$ subspace of $U$ isomorphically onto $E\mathrm{/}\mathbb{C}{e}_0$. Thus $U\cap E$ would be a one-dimensional line complementary to $\mathbb{C}{e}_0$.

Write a nonzero vector on that line as $v=c{e}_0+d{e}_1$ with $d\mathrm{\ne }0$. Since $U$ is a $V$-submodule, it is stable under the current zero mode, but $a_{0}v=(I+N)v=v+d{e}_0$ is not in $\mathbb{C}v$: membership would imply $e_0\in \mathbb{C}v$, contrary to $d\mathrm{\ne }0$. This contradicts stability of $U\cap E$. Therefore the sequence in the Statement has no admissible module complement.

The contradiction in step 5.1 rules out the rationality assumption used in step 3.1. By the definition [F4], $V$ is not rational. This conclusion is a mathematical deduction from the pinned formal VOA/module definitions; it does not assert a physical realization.