Theorem: Finite-order interacting classical BV

The Hamiltonian (eq:newaction) satisfies

It is a classical deformation with no quantum term. Its field rules remain (eq:sr)–(eq:snonminimal), while its scalar antifield equation changes by

Every other fundamental generator has zero change from the additional vertex.

Proof. The vertex has no antifields, so $(V_a,V_a)=0$. Since $s_0\varphi =0$,

All other cross brackets with the added ${\kappa }^3{V}_a$ have order at least four. Lemma lem:referenceCME therefore gives the first identity in (eq:newCME).

The odd Jacobi identity yields

giving the second identity.

An antifield-independent vertex changes no field rule. Its only nonzero fundamental antifield bracket is its Euler derivative with respect to $\varphi$, namely $a{\varphi }^3/6$.