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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:Consequentlyspans 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.

Let $a=\lambda /6$. Within terms already suppressed by ${\Lambda }^{-2}$, the leading equation of motion gives $\square \varphi =-m^2\varphi -a{\varphi }^3$. Equality here means equivalence under a perturbative field redefinition, together with integration by parts, rather than a pointwise identity inside arbitrary off shell correlation functions. HenceFor the second relation, factor the difference as $(\square \varphi -m^2\varphi -a{\varphi }^3)(\square \varphi +m^2\varphi +a{\varphi }^3)$, which is explicitly a redundant operator. Ifthe reduced correction isThe first two coefficients shift the already present mass and quartic coupling constant. After these shifts,This is the equation-of-motion reduction of scalar operators through $O({\Lambda }^{-2})$; terms of $O({\Lambda }^{-4})$ are outside that truncation. The lower-dimensional shifts should not be silently dropped when $m\ne 0$.