= Solution
A massless <graviton> in $D$ dimensions has a symmetric traceless transverse polarization tensor. Its state count is
$$
\frac{(D-2)(D-1)}2-1=\frac{D(D-3)}2.
$$
Thus the metric of <eleven-dimensional supergravity> has $44$ polarizations, while its three-form has $\binom93=84$. The total bosonic count is $128$.
For circle reduction to ten dimensions, split $M=(\mu,10)$. The metric supplies $g_{\mu\nu}$, a <Kaluza-Klein> vector from $G_{\mu,10}$, and a scalar from $G_{10,10}$, conventionally the <dilaton>. The three-form supplies $C_{\mu\nu\rho}=A_{\mu\nu\rho}$ and $B_{\mu\nu}=A_{\mu\nu,10}$. Their ten-dimensional counts are
$$
\begin{array}{c|c|c}
\text{eleven-dimensional field}&\text{ten-dimensional fields}&\text{polarizations}\\\hline
G_{MN}&g_{\mu\nu},\ C_\mu,\ \phi&35+8+1=44\\
A_{MNP}&C_{\mu\nu\rho},\ B_{\mu\nu}&\binom83+\binom82=56+28=84.
\end{array}
$$
This is the bosonic content of <type IIA supergravity>: $g$, $B_2$ and the dilaton together with $C_1$ and $C_3$. Consequently
$$
\boxed{128=35+8+1+56+28.}
$$
For flat seven-torus reduction directly to four dimensions, split $M=(\mu,i)$, $i=1,\ldots,7$. The metric yields one four-dimensional metric, seven vectors $G_{\mu i}$ and $7\cdot8/2=28$ scalars $G_{ij}$. Their polarizations are $2+7\cdot2+28=44$. The three-form gives
$$
\begin{array}{c|c|c}
\text{component}&\text{multiplicity and rank}&\text{polarizations}\\\hline
A_{\mu\nu\rho}&1\text{ three-form}&0\\
A_{\mu\nu i}&7\text{ two-forms}&7\\
A_{\mu ij}&\binom72=21\text{ vectors}&42\\
A_{ijk}&\binom73=35\text{ scalars}&35.
\end{array}
$$
The three-form count is $0+7+42+35=84$, as required by <toroidal reduction of a p-form gauge field>. Applying <two-form scalar duality> to the seven two-forms, the propagating four-dimensional spectrum is
$$
\boxed{1\text{ graviton},\quad28\text{ vectors},\quad70\text{ real scalars};\qquad2+28\cdot2+70=128.}
$$
This is the bosonic content of <four-dimensional N=8 supergravity>. The remaining four-dimensional three-form has no local polarization and can be omitted from the propagating spectrum, though a flux choice can matter in other backgrounds. The <toroidal reduction of eleven-dimensional supergravity> here retains all zero modes; fixing moduli, introducing fluxes or using an orbifold projection would change the spectrum and would not be this unprojected counting problem.
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