= Solution
In <extended supersymmetry> there are $\mathcal N$ Weyl-spinor charges $Q_\alpha^A$, $A=1,\ldots,\mathcal N$. A positive Hermitian mixed bracket can be diagonalized and normalized in this internal index. The ordinary four-dimensional particle algebra is
$$
\boxed{\begin{aligned}
\{Q_\alpha^A,\bar Q_{\dot\beta B}\}&=2\delta^A{}_B\sigma^\mu_{\alpha\dot\beta}P_\mu,\\
\{Q_\alpha^A,Q_\beta^B\}&=\epsilon_{\alpha\beta}Z^{AB},\qquad Z^{AB}=-Z^{BA},\\
\{\bar Q_{\dot\alpha A},\bar Q_{\dot\beta B}\}&=\epsilon_{\dot\alpha\dot\beta}(Z^{AB})^*.
\end{aligned}}
$$
All supercharges commute with translations. The new <central charges in supersymmetry> are Lorentz scalars commuting with momentum, Lorentz generators and supercharges. Their antisymmetry follows because the full anticommutator is symmetric under interchange of the two paired spinor/internal labels, while $\epsilon_{\alpha\beta}$ is antisymmetric. They can be nonzero only for $\mathcal N>1$. Internal <R-symmetry> automorphisms rotate the supercharge index; the algebra without specified central charges admits unitary rotations, and a chosen central-charge matrix restricts the subgroup preserving it. Central charges allow <BPS states> and shortened massive representations, with the numerical BPS normalization depending on whether a factor two is included in the definition of $Z$.
For a massless unitary representation at the momentum used above, every $Q_1^A$ has zero anticommutator with its adjoint and hence vanishes. Its mixed same-chirality bracket with $Q_2^B$ then forces $Z^{AB}=0$ on that representation. The surviving $\mathcal N$ charges give $\mathcal N$ independent <fermionic creation operators>, each lowering helicity by one half. Starting from helicity $\lambda$, their antisymmetric products give
$$
h_k=\lambda-\frac k2,\qquad\text{multiplicity }\binom{\mathcal N}{k},\qquad k=0,\ldots,\mathcal N.
$$
Thus the <helicity spectrum of a massless supermultiplet> occupies an interval of width $\mathcal N/2$ and has $2^{\mathcal N}$ states before any separate CPT completion.
The assumption excluding helicity above two also excludes helicity below minus two in a CPT-complete physical spectrum. Therefore both ends of the interval must obey
$$
\lambda\le2,\qquad\lambda-\mathcal N/2\ge-2,
$$
and the <spin bound for massless supermultiplets> gives
$$
\boxed{\mathcal N\le8.}
$$
This bound is attained: for $\mathcal N=8$ and highest helicity two, the helicities are $2,3/2,1,1/2,0,-1/2,-1,-3/2,-2$ with multiplicities $1,8,28,56,70,56,28,8,1$. They form the CPT-self-conjugate multiplet of <four-dimensional N=8 supergravity>, with $128$ bosonic and $128$ fermionic states. Its bosonic graviton, $28$ vectors and $70$ scalars agree with the toroidal reduction counted above. Here $\mathcal N=8$ means eight Weyl-spinor supersymmetries, or $32$ real supercharge components, not eight real components.
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