= Solution
{c}
Write $X\simeq Y$ when the two densities differ by a <total derivative>. Under the <boundary conditions> used in the <action>, they have the same integral. The following identities establish the <integration-by-parts reduction of scalar operators>:
$$
\begin{aligned}
\partial_\mu(\phi^3\partial^\mu\phi)&=3\phi^2(\partial\phi)^2+\phi^3\Box\phi,
&\mathcal O_2&\simeq-\frac13\mathcal O_3,\\
\partial_\mu(\phi\partial^\mu\Box\phi)&=\partial_\mu\phi\partial^\mu\Box\phi+\phi\Box^2\phi,
&\mathcal O_4&\simeq-\mathcal O_5,\\
\partial_\mu(\partial^\mu\phi\Box\phi)&=(\Box\phi)^2+\partial_\mu\phi\partial^\mu\Box\phi,
&\mathcal O_5&\simeq-\mathcal O_6,\\
\partial_\mu(\partial_\nu\phi\partial^\mu\partial^\nu\phi)&=
(\partial_\mu\partial_\nu\phi)^2+\partial_\nu\phi\partial^\nu\Box\phi,
&\mathcal O_7&\simeq-\mathcal O_5.
\end{aligned}
$$
Consequently
$$
\boxed{\{\phi^6,\ \phi^3\Box\phi,\ (\Box\phi)^2\}}
$$
spans the <operator basis in effective field theory> before <equation-of-motion reduction of scalar operators>. These four independent relations reduce the seven representatives to three; the three sectors have different field or derivative content, so <integration by parts> cannot identify them further.
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