The whole partition Fock space is not the basic representation

Strictness. By [F1] the full field decomposition contains the summands $V({\mathrm{\Lambda }}_0-k\delta )$ for every $k\ge 0$ with multiplicity $p(k)\ge 1$, and the ${\mathrm{\Lambda }}_0$ summand occurs with multiplicity $p(0)=1$; since $p(k)\ge 1$ for $k\ge 1$ and those constituents are nonzero, $F$ is strictly larger than its ${\mathrm{\Lambda }}_0$ constituent $M$.

Raw-vector exclusion. The shape $1^e$ is not $e$-regular: all its parts equal $1$ and occur $e$ times, so it lies outside the empty component by [F1]; since the residue basis of $F_0\cap M$ is exactly the empty component by [F3], every vector in $F_0\cap M$ has residue a combination of regular labels. The raw vector $\mathrm{\mid }{1}^e⟩$ has nonzero residue in $F_0\mathrm{/}q{F}_0$ equal to the nonregular shape $1^e$, so $\mathrm{\mid }{1}^e⟩\mathrm{\notin }M$; the same argument applies to $G(1^e)$, which is congruent to the same shape modulo $q{F}_0$.

Conclusion. The whole partition Fock space is not the basic representation: it has strictly more constituents than the single $V({\mathrm{\Lambda }}_0)$, and even the explicit raw and lower-lift vectors $\mathrm{\mid }{1}^e⟩$ and $G(1^e)$ lie outside $M$; the nonregular ambient lower vectors are not a basis of a single highest module.