= Solution
{c}
For <dimension-six operators of a real scalar field>, let $n$ count fields and $r$ count <partial derivatives>. <Mass dimension> gives $n+r=6$. <Lorentz invariance> requires even $r$. Discard derivatives acting on the entire expression, but \b[do not yet identify different expressions by integration by parts]. A complete list of seven representatives is
$$
\boxed{\begin{aligned}
\mathcal O_1&=\phi^6,\\
\mathcal O_2&=\phi^2\partial_\mu\phi\partial^\mu\phi,&
\mathcal O_3&=\phi^3\Box\phi,\\
\mathcal O_4&=\phi\Box^2\phi,&
\mathcal O_5&=\partial_\mu\phi\,\partial^\mu\Box\phi,\\
\mathcal O_6&=(\Box\phi)^2,&
\mathcal O_7&=(\partial_\mu\partial_\nu\phi)(\partial^\mu\partial^\nu\phi).
\end{aligned}}
$$
For $(n,r)=(6,0)$ there is one choice; for $(4,2)$ the derivative distribution is either $2$ or $1+1$; for $(2,4)$ it is $4$, $3+1$, or $2+2$, the last allowing two distinct metric contractions. Commuting <partial derivatives> identifies alternative orderings. Contractions with a <Levi-Civita symbol> vanish because an antisymmetric index pair necessarily contracts symmetric derivatives. The remaining case $(0,6)$ gives no field-dependent expression. No independent odd-$r$ <Lorentz scalar> exists.
The source's seven means this unreduced list. If “ignoring total derivative operators” were interpreted as already taking the full quotient by <total derivatives>, seven would be incorrect: the next part gives three representatives of that quotient. Dimensionful factors multiplying lower-dimensional terms do not count as new canonical-dimension-six field monomials.
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