= Solution
Use a positive-energy <unitary representation> with finite <helicity> states, excluding <continuous-spin representations>. In the specified frame the <Pauli matrices> give
$$
2\sigma^\mu p_\mu=2E(\mathbf1+\sigma^3)
=4E\begin{pmatrix}1&0\\0&0\end{pmatrix}.
$$
For every $A$ and every state $|\chi\rangle$,
$$
0=\langle\chi|\{Q_2^A,(Q_2^A)^\dagger\}|\chi\rangle
=\|Q_2^A|\chi\rangle\|^2+\|(Q_2^A)^\dagger|\chi\rangle\|^2.
$$
Positivity therefore gives $Q_2^A=\bar Q_{\dot2 A}=0$ on this <massless supermultiplet>. The equal-<chirality> <anticommutator> $\{Q_1^A,Q_2^B\}=\epsilon_{12}Z^{AB}$ then shows that every <central charge in supersymmetry> vanishes on this <superalgebra representation>.
Normalize the remaining <supercharges> as <fermionic raising and lowering operators>:
$$
a_A=\frac{Q_1^A}{\sqrt{4E}},\qquad a_A^\dagger=\frac{\bar Q_{\dot1 A}}{\sqrt{4E}}.
$$
They obey the <canonical anticommutation relations>
$$
\{a_A,a_B^\dagger\}=\delta_{AB},\qquad
\{a_A,a_B\}=\{a_A^\dagger,a_B^\dagger\}=0.
$$
Choose a <Clifford vacuum> $|\lambda\rangle$ of <helicity> $\lambda\in\tfrac12\mathbb Z$ annihilated by every $a_A$. Such a vector exists because the commuting <fermion number operators> $a_A^\dagger a_A$ have eigenvalues $0$ and $1$, and lowering every occupied mode produces a nonzero empty state. In an <irreducible representation> this empty space has dimension one; multiple empty states produce a <direct sum> of copies. The states are
$$
|A_1\cdots A_k;\lambda\rangle
=a_{A_1}^\dagger\cdots a_{A_k}^\dagger|\lambda\rangle,
\qquad 1\leq A_1<\cdots<A_k\leq\mathcal N,
\qquad 0\leq k\leq\mathcal N.
$$
The <canonical anticommutation relations> make these states orthonormal for a normalized <Clifford vacuum>. In the <helicity> convention $[J_3,Q_1^A]=\tfrac12Q_1^A$, its adjoint has $[J_3,a_A^\dagger]=-\tfrac12a_A^\dagger$. Thus the <Clifford vacuum> is the highest-<helicity> state and
$$
\boxed{h_k=\lambda-\frac{k}{2},\qquad d(h_k)=\binom{\mathcal N}{k}.}
$$
Reversing the rotation convention reverses the displayed ordering and leaves all counts unchanged. The level-$k$ states form the <exterior power> $\Lambda^k\mathbb C^{\mathcal N}$, which explains the <binomial coefficient>. Summing the <binomial coefficients> gives the <helicity spectrum of a massless supermultiplet>:
$$
\boxed{\dim\mathcal H_p=\sum_{k=0}^{\mathcal N}\binom{\mathcal N}{k}=2^{\mathcal N},
\qquad h_{\max}-h_{\min}=\frac{\mathcal N}{2}.}
$$
Each application of a <fermionic raising and lowering operator> reverses <fermion parity>. The even and odd levels each contain $2^{\mathcal N-1}$ states, so a physical <massless supermultiplet> has $2^{\mathcal N-1}$ <boson> and $2^{\mathcal N-1}$ <fermion> states.
These formulas describe one <irreducible representation> of the massless <Super-Poincaré algebra> at fixed <four-momentum>. A reducible <superalgebra representation> can contain several such <supermultiplets>. Also, the <CPT theorem> sends <helicity> $h$ to $-h$ and conjugates internal charges. <CPT completion of a supermultiplet> may therefore require a second $2^{\mathcal N}$-state <supermultiplet>, with highest <helicity> $\mathcal N/2-\lambda$. A necessary condition for the original <helicity spectrum of a massless supermultiplet> to be self-conjugate is $\lambda=\mathcal N/4$; internal charges must also admit the conjugation in a realization compatible with the <Spin-statistics theorem>. Thus this <helicity> condition alone is not sufficient for <CPT completion of a supermultiplet> without doubling. In particular, the original irreducible count is not automatically the count after <CPT completion of a supermultiplet>.
For the conventional bounds on <extended supersymmetry>, suppose all massless states satisfy $|h|\leq s_{\max}$. The <helicity spectrum of a massless supermultiplet> occupies an interval of width $\mathcal N/2$, whereas the available interval has width $2s_{\max}$. Hence
$$
\boxed{\mathcal N\leq4s_{\max}.}
$$
An ordinary four-dimensional <renormalizable quantum field theory> without gravity uses interacting massless fields of <spin angular momentum> at most one. Massless <spin angular momentum> $3/2$ requires <supergravity>, whose gravitational coupling has negative <mass dimension> and is a <nonrenormalizable interaction> by <power counting in quantum field theory>. Setting $s_{\max}=1$ gives \b[$\mathcal N\leq4$ for conventional renormalizable theories]. The bound is attained by <four-dimensional N=4 super Yang-Mills theory>: choosing $\lambda=1$ gives
$$
\begin{array}{c|ccccc}
h&1&\tfrac12&0&-\tfrac12&-1\\\hline
d(h)&1&4&6&4&1
\end{array}
$$
for its <supersymmetric vector multiplet>.
Allowing gravity permits $s_{\max}=2$, so \b[$\mathcal N\leq8$ for conventional theories including gravity]. The <four-dimensional N=8 supergravity> <supergravity multiplet> attains the bound with $\lambda=2$:
$$
\begin{array}{c|ccccccccc}
h&2&\tfrac32&1&\tfrac12&0&-\tfrac12&-1&-\tfrac32&-2\\\hline
d(h)&1&8&28&56&70&56&28&8&1
\end{array}
$$
It has $256$ states, $128$ <boson> and $128$ <fermion> states. The conventional claim about a maximum “in general” assumes local interacting four-dimensional theories in <Minkowski spacetime> with a finite spectrum of fields and no massless <spin angular momentum> above two. It is not a theorem excluding arbitrary free higher-<spin angular momentum> constructions or all theories in other spacetime settings.
Finally, a <chiral gauge spectrum> means that the independent left-handed <Weyl spinors> occur in complex <gauge group representations> without obligatorily paired conjugate left-handed <Weyl spinors>. In <four-dimensional N=1 supersymmetry>, an irreducible matter <supermultiplet> with <helicities> $(0,-\tfrac12)$ can carry a complex <gauge group representation> $R$. Its <CPT completion of a supermultiplet> supplies the right-handed <antiparticle> in $\bar R$, as every left-handed <Weyl spinor> requires; it does not supply a second independent left-handed <Weyl spinor> in $\bar R$. Consequently a <chiral gauge spectrum> is possible.
For $\mathcal N=2$, the matter <hypermultiplet> has <helicities> $(\tfrac12,0,0,-\tfrac12)$ before internal <charge conjugation>. If it carries a complex <gauge group representation> $R$, its <CPT completion of a supermultiplet> adds the conjugate <hypermultiplet>. Equivalently, the resulting full <hypermultiplet> contains two $\mathcal N=1$ <chiral superfields>, one in $R$ and one in $\bar R$. These give a <vectorlike gauge spectrum>. Half-<hypermultiplets> can occur in <pseudoreal representations>, which also do not produce genuinely complex <chiral gauge spectra>. A <supersymmetric vector multiplet> has <Adjoint representation>, hence real gauge <quantum numbers>. Since every $\mathcal N\geq2$ algebra contains an $\mathcal N=2$ subalgebra, the same obstruction applies to greater <extended supersymmetry>.
Therefore \b[a conventional chiral gauge spectrum is possible only for $\mathcal N=0$ or $\mathcal N=1$]. Here $\mathcal N=0$ means no <supersymmetry>; among theories with actual <supersymmetry>, only $\mathcal N=1$ qualifies. These are necessary structural conditions, not a guarantee that a chosen <chiral gauge spectrum> satisfies <anomaly cancellation>.
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