Proposition: Point splitting still requires subtraction

The unrenormalized diagonal pullback of $W_{\beta }$ fails the ordinary wavefront criterion. In contrast, the local remainder $H_{\beta }$ in (eq:localHadamard) has well-defined diagonal jets of every finite order. Consequently the Wick square, stress tensor, and current prescriptions of the regular-part companions remain well-defined at every point of $M$.

Proof. The diagonal wavefront contains

These covectors are normal to the diagonal embedding. Thus one cannot justify the unrenormalized pullback by the ordinary criterion. The subtracted remainder is smooth in a neighbourhood of each regular diagonal point, so all its required local diagonal derivatives exist.

The apex front cannot meet the spacetime diagonal: at $X=X^{\prime }$, $F_{\mathrm{ap}}=4r^2>0$. The additional global fronts therefore do not alter this local subtraction argument.