Proposition

Contact moments and the local correction are

Odd moments vanish by evenness. Then ${\Pi }_w+L={\Pi }_{\mathrm{tr}}$, and $\partial ({\Pi }_w+L)=0$ on both pairs. That identity is kernel_two_point_Ward. It is not functional ST4b. The distribution $L$ has order at most $4$. The kernel-divergence contact bound on $\partial L$ is five. The pairing is a formal transpose, not an $L^2$ adjoint. Lorentz covariance of $L$ is not claimed. The counterterm density $C_L=\frac{1}{2}\int hLh$ has ghost number $0$. The variation $s_0{C}_L$ is not evaluated. A kernel counterterm is not bulk BV restoration. A missing one-point moment D12 stays Missing and is never defaulted to $0$. Under the stated massless dimensional prescription the recorded value is zero; that record is not a default.