= Solution
Choose a null <four-momentum> with $E>0$ and orient the third axis so that $2\sigma^\mu p_\mu=\operatorname{diag}(4E,0)$. The <Super-Poincaré algebra> gives
$$
\{Q_1^A,(Q_1^B)^\dagger\}=4E\delta^{AB},\qquad
\{Q_2^A,(Q_2^B)^\dagger\}=0.
$$
For every state $|v\rangle$ in a <unitary representation>, the second <anticommutator> is the sum $\|Q_2^A v\|^2+\|(Q_2^A)^\dagger v\|^2$. Thus both operators act trivially. A <central charge in supersymmetry> also vanishes in this <massless supermultiplet>: its mixed-spinor <anticommutator> contains the now-trivial $Q_2$. The remaining $\mathcal N$ pairs, normalized by $2\sqrt E$, satisfy the <canonical anticommutation relations>
$$
\{a_A,a_B^\dagger\}=\delta_{AB},\qquad
\{a_A,a_B\}=\{a_A^\dagger,a_B^\dagger\}=0.
$$
Label a <Clifford vacuum> by its highest <helicity> $\lambda$ and choose the lowering convention $[J_3,a_A^\dagger]=-a_A^\dagger/2$. Acting with distinct <fermionic creation operators> produces the <fermionic Fock space>
$$
a_{A_1}^\dagger\cdots a_{A_k}^\dagger|p,\lambda\rangle,
\qquad
\boxed{h_k=\lambda-\frac{k}{2},\quad n_k=\binom{\mathcal N}{k},\quad k=0,\ldots,\mathcal N}.
$$
This proves the <helicity spectrum of a massless supermultiplet> and the $2^{\mathcal N}$ state count. If the internal charges or <helicities> do not already make the spectrum invariant under the <CPT theorem>, add its conjugate using the <CPT completion of a supermultiplet>.
The <spin bound for massless supermultiplets> follows directly from the width $\mathcal N/2$ of this <helicity> interval. Requiring $|h|\le1$ gives \b[$\mathcal N\le4$], achieved by the <four-dimensional N=4 super Yang-Mills theory> <vector multiplet> with multiplicities $1,4,6,4,1$ at <helicities> $1,1/2,0,-1/2,-1$. These <helicity> bounds are necessary spin restrictions for the usual interacting <renormalizable quantum field theory>; they do not by themselves establish renormalizability for an arbitrary action. Requiring $|h|\le2$ gives \b[$\mathcal N\le8$]. The <four-dimensional N=8 supergravity> <supergravity multiplet> attains this bound with highest <helicity> $2$ and multiplicities $1,8,28,56,70,56,28,8,1$. There is one $+2$ state and one $-2$ state, the two polarizations of a single <graviton>.
The stronger restriction relevant to matter is the <chirality constraint on extended supersymmetry>. An $\mathcal N=1$ <chiral superfield> can contain a left-handed <Weyl spinor> in a complex <gauge group representation> $R$. By contrast, a full $\mathcal N=2$ <hypermultiplet> consists of $\mathcal N=1$ matter in $R\oplus\overline R$, and the <gauginos> of a <vector multiplet> are in the real adjoint <group representation>. Higher unbroken <extended supersymmetry> retains this pairing. A <pseudoreal representation> may allow a half-<hypermultiplet>, but it does not yield net chirality in a genuinely complex <gauge group representation>. Consequently the <chiral gauge spectrum> of the <Standard Model> permits \b[at most unbroken $\mathcal N=1$ supersymmetry] in its four-dimensional low-energy matter sector.
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