Why raw internal contraction is different

The kernel contraction

is not justified merely by (eq:microcausal). At coincidence $K_{f,R}$ has opposite-slot covectors with arbitrary spatial component, including null ones. The Hadamard kernel has opposite-slot null covectors. These can cancel in the product criterion D1.

Similarly, the diagonal pullback of $W$, or of its differentiated entries, meets the conormal of the diagonal. Local Wick subtraction is required.

Thus a compact normal-ordered quadratic element can exist even though its raw internal contraction is undefined by the ordinary product criterion. This distinction is retained in the compact Wick framework HWlocal.