= Solution
Varying the <vector field> gives the <Proca equation>,
$$
\partial_\mu F^{\mu\nu}+m_A^2A^\nu=0.
$$
Its divergence uses antisymmetry of $F^{\mu\nu}$ to cancel the first term. Since $m_A\ne0$,
$$
\boxed{\partial_\nu A^\nu=0,\qquad(\Box+m_A^2)A^\nu=0.}
$$
For an on-shell <momentum> this imposes $p_\nu\epsilon^\nu=0$ on the four <polarization> components, leaving three independent solutions. In the rest frame, $p=(m_A,\mathbf0)$ makes $\epsilon^0=0$, leaving precisely the three spatial <polarizations> of a massive spin-one particle.
There is no fourth independent oscillator hidden in $A^0$. Its canonical <momentum> is zero because the <Lagrangian> has no $\dot A^0$, and the temporal equation constrains $A^0$ in terms of the spatial data. Equivalently the primary and secondary <second-class constraints> remove two variables from the eight-dimensional <phase space>, leaving six phase-space variables, or three physical degrees of freedom.
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