= Solution
{c}
The <action> is dimensionless, so in four dimensions the <Lagrangian density> has <mass dimension> four. Its kinetic term fixes $[\phi]=1$ and $[\partial_\mu]=1$. An interaction $g_D\mathcal O_D$ with $[\mathcal O_D]=D$ has $[g_D]=4-D$. Therefore
$$
\boxed{D\leq4\ \text{is renormalizable by power counting};\qquad D>4\ \text{has a negative-dimensional coupling}.}
$$
In an <effective field theory> with heavy scale $\Lambda$, a higher-dimensional <local operator> enters as $c_D\mathcal O_D/\Lambda^{D-4}$. For characteristic scales $E,m\ll\Lambda$ and coefficients compatible with this expansion, its effects are suppressed by powers of the small ratios. Omitting these terms defines the leading approximation and its <truncation error in effective field theory>. It is not a fundamental prohibition: such terms must be retained if that accuracy is required. A continuum <renormalizable quantum field theory> can instead be defined by the closed set of <counterterms> of dimension at most four.
Up to <total derivatives>, all <Lorentz scalar> terms of dimension at most four built from one <real scalar field> are
$$
1,\quad\phi,\quad\phi^2,\quad\phi^3,\quad\phi^4,\quad
\partial_\mu\phi\partial^\mu\phi.
$$
The term $\phi\Box\phi$ is equivalent to the kinetic term by <integration by parts>; $\Box\phi$ and $\Box(\phi^2)$ contribute only at the boundary. The displayed <Lagrangian density> omits the linear and cubic interactions and a field-independent constant. The exact <Z2 symmetry> $\phi\mapsto-\phi$ forbids both odd interactions, and <radiative corrections> preserving this <symmetry> cannot generate them. The constant shifts the vacuum <energy> and the normalization of $Z$ and can be dropped when gravity and absolute vacuum <energy> are not being studied. A constant vacuum <counterterm> may still be used when renormalizing unnormalized vacuum diagrams. The kinetic coefficient has already been fixed by field normalization.
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