Canonical variational completion and 4D Gauss–Bonnet gravity

Recently, a proposal to obtain finite contribution of second derivative order the gravitational field equations in \(D = 4\) dimensions from renormalized Gauss-Bonnet term action has received wave attention. It triggered discussion whether employed renormalization procedure yields well-defined theory. One main criticisms is based on fact that resulting cannot be obtained as Euler-Lagrange diffeomorphism invariant action. In this work, we use techniques inverse calculus variations point out truncated any at all (either or not), dimension. Then, employ canonical variational completion, notion Vainberg-Tonti Lagrangian - which consists adding canonically defined correction given system equations, so make them derivable an To apply technique suggested $4$D extend completion algorithm some classes PDE systems for usual integral providing diverges. We discover $D>4$ can variationally completed, choosing either metric its variables; both approaches yield consistently same Lagrangian, whose variation leads fourth equations. $D=4$, completed theory diverges cases.