= Solution
Use an unfluxed toroidal zero-mode <dimensional reduction>, with no projection removing fields. A ten-dimensional <gauge field> has $10-2=8$ physical polarizations per <gauge generator>. Its <Majorana-Weyl spinor> partner has sixteen real components before the massless equation, and eight propagating fermionic states. Thus <ten-dimensional super Yang-Mills theory> has sixteen real <supercharges> and eight bosonic plus eight fermionic states per generator.
Split the spacetime index as $M=(\mu,m)$, with $\mu=0,\ldots,3$ and six internal directions. The zero modes of $A_M$ become $A_\mu$ and six real adjoint <scalar fields> $X_m=A_m$. The <spinor representation> decomposes under $\operatorname{Spin}(1,3)\times\operatorname{Spin}(6)$ into four four-dimensional <Weyl spinors>; the Majorana condition supplies their conjugates rather than independent additional spinors. All sixteen <supercharges> survive, giving <four-dimensional N=4 super Yang-Mills theory>. The <supermultiplet> is therefore \b[one gauge vector, four Weyl <gauginos> and six real scalars, all in the <Adjoint representation>], with $2+6=8$ bosonic and $4\times2=8$ fermionic states. The internal $\operatorname{Spin}(6)\simeq SU(4)$ acts as <R-symmetry>, with the scalars in its six-dimensional representation and the Weyl <gauginos> in its four-dimensional representation.
Since the internal derivatives vanish, the internal <Yang-Mills field strength> is $F_{mn}=-ig_4[X_m,X_n]$ in a Hermitian-generator convention. The internal part of the gauge <kinetic term> becomes the <scalar potential>
$$
\boxed{V=\frac{g_4^2}{4}\sum_{m,n,a}(f^{abc}X_m^bX_n^c)^2=\frac{g_4^2}{4}\sum_{m,n}\operatorname{tr}\bigl([X_m,X_n]^\dagger[X_m,X_n]\bigr).}
$$
Here the invariant <trace> is normalized by $\operatorname{tr}(T^aT^b)=\delta^{ab}$ and $[T^a,T^b]=if^{abc}T^c$; the coupling and scalar <kinetic terms> use the same normalization. The sum includes both orders of each pair. For Hermitian $X_m$, the <commutator> is anti-Hermitian, so equivalently $V=-\frac{g_4^2}{4}\sum_{m,n}\operatorname{tr}[X_m,X_n]^2$. It is nonnegative and vanishes exactly when all six adjoint scalars commute. This gives the classical <vacuum moduli space of a supersymmetric gauge theory>.
Interpret $A_{MNP}$ as a massless three-form <gauge field>, with transformation $A_3\mapsto A_3+d\Lambda_2$ and <kinetic term> built from $F_4=dA_3$. The <massless p-form gauge field> count is an on-shell count, rather than the raw $\binom D3$ <tensor> components. For a null plane wave, <gauge transformations> remove the components along the null gauge direction, while the field equation removes the complementary longitudinal components. The independent polarization is consequently an antisymmetric <tensor> on the $D-2$ transverse directions. Thus
$$
\boxed{n_3(D)=\binom{D-2}{3}.}
$$
This argument incorporates the redundancies in the two-form gauge parameter; subtracting $\binom D2$ once from $\binom D3$ would not do so.
Let $n=D-4$ and split indices into four-dimensional and internal ones. The <toroidal reduction of a p-form gauge field> gives one $A_{\mu\nu\rho}$, $n$ two-forms $A_{\mu\nu i}$, $\binom n2$ vectors $A_{\mu ij}$ and $\binom n3$ scalars $A_{ijk}$. A massless three-form in four dimensions has no local propagating state; each massless two-form has one, each vector two and each scalar one. The two-form state can equivalently be represented by a scalar using <massless p-form duality>: its three-form field strength satisfies $H_3=*da$. Consequently
$$
\begin{aligned}
n_{\mathrm{4D}}&=0+n+2\binom n2+\binom n3\\
&=n+n(n-1)+\frac{n(n-1)(n-2)}6\\
&=\frac{n(n+1)(n+2)}6=\binom{D-2}{3}.
\end{aligned}
$$
This verifies the required matching, including $D=6$ where the count is $2+2=4$. The <nondynamical four-dimensional three-form> can carry a constant flux parameter, but that is not an additional propagating <degree of freedom>.
<Eleven-dimensional supergravity> contains the metric $G_{MN}$, a three-form $A_{MNP}$ and a Majorana <gravitino> $\Psi_M$. The massless graviton is a symmetric traceless <tensor> of the nine-dimensional transverse rotation group, giving $9\times10/2-1=44$ states. The three-form gives $\binom93=84$. The <gravitino> is a gamma-traceless transverse vector-spinor: a <spinor representation> of $\operatorname{Spin}(9)$ has sixteen real components, so it gives $9\times16-16=128$ states. Hence the eleven-dimensional balance is $44+84=128$ bosonic states and $128$ fermionic states.
Reduction on a circle yields <type IIA supergravity>, with thirty-two real <supercharges> and opposite ten-dimensional chiralities. The metric decomposes into the ten-dimensional metric, the <Kaluza-Klein> vector $C_\mu=G_{\mu\,11}$ and the radius scalar, or dilaton. Their counts are $35+8+1=44$. The three-form gives a ten-dimensional three-form $C_{\mu\nu\rho}$ and a two-form $B_{\mu\nu}=A_{\mu\nu\,11}$, with $\binom83+\binom82=56+28=84$ states. The eleven-dimensional <Majorana spinor> decomposes into two opposite-chirality <Majorana-Weyl spinors>. The external <gravitino> components give two <gravitini>, each with $8\times8-8=56$ states; the internal component supplies two spin-one-half dilatini, each with eight states. Therefore
$$
\boxed{\text{10D IIA: }35+8+1+56+28=128,\qquad2\times56+2\times8=128.}
$$
The two-form is in the NS--NS sector and the one-form and three-form are the Ramond--Ramond potentials. This reduction is nonchiral IIA rather than IIB.
Reduction on a seven-torus yields <four-dimensional N=8 supergravity>, again preserving thirty-two real <supercharges>. From $G_{MN}$ one gets one graviton, seven vectors $G_{\mu i}$ and $7\times8/2=28$ scalars $G_{ij}$, giving $2+7\times2+28=44$ states. From $A_{MNP}$ one gets a nonpropagating four-dimensional three-form, seven two-forms, $\binom72=21$ vectors and $\binom73=35$ scalars, giving $0+7+21\times2+35=84$ states. Dualizing the seven two-forms to scalars gives the familiar field content
$$
\boxed{\text{4D: one graviton, }28\text{ vectors, }70\text{ real scalars};\qquad2+28\times2+70=128.}
$$
The <gravitino> reduces to eight four-dimensional Majorana <gravitini> and $7\times8=56$ spin-one-half Majorana fermions, after the external <gravitini> are separated from their gamma-trace mixing with the internal components. Their count is $8\times2+56\times2=128$. Together these fields form the single maximal <supergravity multiplet>.
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