= Solution
The <Wigner classification> labels a massive one-particle state by $|p;j,m_s\rangle$, where $p^2=m^2>0$, $p^0>0$, $j$ is the spin representation of the $SO(3)$ <little group>, and $m_s=-j,\ldots,j$. A finite-spin massless state is $|p;\lambda\rangle$, where $p^2=0$, $p^0>0$, and the <helicity> $\lambda$ labels a one-dimensional representation of the rotational part of the $ISO(2)$ little group. A parity-invariant theory pairs nonzero helicities $\lambda$ and $-\lambda$.
The physical polarization counts are
$$
\begin{array}{c|cccc}
\text{spin}&0&\tfrac12&1&2\\ \hline
\text{massless}&1&1\text{ per helicity}&2&2\\
\text{massive}&1&2&3&5.
\end{array}
$$
A scalar obeys the <Klein-Gordon equation>. A spinor obeys the <Dirac equation>, and a massless irreducible spinor additionally has a fixed <chirality>. A massive vector obeys the <Proca equation>, whose divergence gives $\partial_\mu A^\mu=0$ and leaves three polarizations. A massless vector instead has the <gauge redundancy> $A_\mu\sim A_\mu+\partial_\mu\alpha$, leaving two transverse polarizations. A massive symmetric tensor obeys the <Fierz-Pauli equations>: symmetry, transversality, and tracelessness leave five components. A massless tensor has linearized diffeomorphism redundancy $h_{\mu\nu}\sim h_{\mu\nu}+\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$, leaving helicities $\pm2$. A redundancy identifies field configurations representing the same physical state; <gauge invariance> is invariance under that identification.
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