( (36)(10) \: + \: 162 \right) \: + \: 20 \: = \: 281, \\
n_H - n_V & = & -12 \: \neq \: 244,\end{aligned}$$ and so we see that this cannot satisfy anomaly cancellation, mechanically verifying our previous observation that this theory cannot be consistent.

More generally, any heterotic compactification on a gerbe, in which the bundle is twisted, will be of this same general type, unless the bundle has rank 8 and the trivially-acting group is ${\mathbb Z}_2$. Locally each theory will look like a compactification of a ten-dimensional theory with an altered GSO projection, and except for the case that the GSO projection switches between Spin$(32)/{\mathbb Z}_2$ and $E_8 \times E_8$, the resulting theory cannot be consistent.

For purposes of comparison, and to help illuminate the methods encoded in appendix \[app:spectra\], let us also outline the results in a closely related consistent compactification. If we did not orbifold, if we took a compactification of an $E_8 \times
E_8$ heterotic string on a smooth large-radius $K3$ with a rank 4 vector bundle, then from a similar computation we would find

$h^0(X, {\cal O})=1$ vector multiplets in the adjoint of ${\rm Spin}(10)$,

One vector multiplet in the adjoint of $E_8$,

$h^1(X, {\cal E}) = 16$ half-hypermultiplets in the ${\bf 16}$ of ${\rm Spin}(10)$,

$h^1(X, \Lambda^2 {\cal E}) = 36$ half-hypermultiplets in the ${\bf 10}$ of ${\rm Spin}(10)$,

$h^1(X, \Lambda^3 {\cal E} = {\cal E}^*) = 16$ half-hypermultiplets in the ${\bf 16}$ of ${\rm Spin}(10)$,

20 singlet hypermultiplets for $K3$ moduli,

$h^1({\rm End}\, {\cal E}) = 162$ singlet half-hypermultiplets for bundle moduli,

where the representations of ${\rm Spin}(10)$ are constructed in the same fashion. Altogether, we find that $$\begin{aligned}
n_V & = & 45 \: + \: 248 \: = \: 293, \\
n_H & = & (1/2)\left( (16)(16) \: + \: (36)(10) \: + \: (16)(16)
\: + \: 162 \right) \: + \: 20 \: = \: 537,
\\
n_H - n_V & = & 244,\end{aligned}$$ consistent with anomaly cancellation, in that standard compactification. Unfo