= Solution
In four dimensions, <extended supersymmetry> has $\mathcal N>1$ independent <Weyl spinor> <supercharges>, or $4\mathcal N$ real generators. With the ordinary particle <Super-Poincaré algebra>,
$$
\{Q_\alpha^A,\bar Q_{\dot\beta B}\}=2\delta^A_B\sigma^\mu_{\alpha\dot\beta}P_\mu,
\qquad\{Q_\alpha^A,Q_\beta^B\}=\epsilon_{\alpha\beta}Z^{AB},\qquad Z^{AB}=-Z^{BA}.
$$
The <central charges in supersymmetry> commute with spacetime generators and the <supercharges>; an <R-symmetry> can act on their internal indices. Antisymmetry makes a scalar central charge impossible for $\mathcal N=1$.
At positive timelike momentum, the rest-frame <supercharges> act as fermionic oscillators. Without central charges, a spin-$j$ <Clifford vacuum> generates $(2j+1)2^{2\mathcal N}$ states. For a <massless supermultiplet>, positivity makes half the <supercharges> act trivially. The remaining $\mathcal N$ creation operators generate $2^{\mathcal N}$ states with <helicity> levels $\lambda-k/2$ and multiplicities $\binom{\mathcal N}{k}$. A separate <CPT completion of a supermultiplet> may be necessary. Keeping all helicities between $-1$ and $1$ gives $\mathcal N\leq4$ for a theory without gravity; allowing helicity two gives $\mathcal N\leq8$.
A nonzero <central charge in supersymmetry> changes the oscillator norms. For an $\mathcal N=2$ central-charge block, write $Z^{12}=2z$. Suitable combinations of a charge and the adjoint of the other have <anticommutators> $2(M+|z|)$ and $2(M-|z|)$. Their positivity gives
$$
\boxed{M\geq|z|.}
$$
A <BPS state> saturates this <BPS bound in supersymmetry>. The combinations with zero anticommutator annihilate it, preserving part of <supersymmetry> and giving a shortened <BPS supermultiplet>. In this example a scalar-seeded massive multiplet has four states instead of sixteen, before any required CPT completion. More central-charge blocks allow different fractions of preserved supersymmetry. The mass-charge relation is protected as long as the short state exists; this does not guarantee stability against decay everywhere in parameter space.
The main obstruction to unbroken <extended supersymmetry> at low energies is the <chirality constraint on extended supersymmetry>. An $\mathcal N=2$ <vector multiplet> contains an additional adjoint complex scalar. A full <hypermultiplet> contains two $\mathcal N=1$ matter <chiral superfields> in conjugate gauge representations $R$ and $\overline R$. Its charged fermion spectrum is therefore vectorlike. Half-hypermultiplets in pseudoreal representations do not supply genuinely complex chirality. The $\mathcal N=4$ gauge multiplet is still more restrictive: it has four adjoint <Weyl spinors> and six adjoint real scalars. These spectra and their constrained interactions cannot directly reproduce the chiral electroweak matter and arbitrary Yukawa couplings of the <Standard Model>.
\b[At most unbroken $\mathcal N=1$ supersymmetry can directly describe the observed chiral four-dimensional gauge spectrum.] Extended supersymmetry can be useful in a higher-energy theory, in an internal sector or before compactification, but it must be reduced or broken to obtain that low-energy spectrum. This is a statement about unbroken symmetry of the relevant sector, not a prohibition on extended supersymmetric ultraviolet descriptions.
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