An explicit free-shell witness

The theorem supplies explicit allowed tests; it does not assert nonvanishing for every possible fixed $f$. For a particular test, (eq:sB) is the necessary condition to examine. A smearing can annihilate a nonzero field.

Noncompact smooth ghost data are legitimate here, just as in XI-B. If compact ghost Cauchy data are desired, choose them equal to the affine data on a ball containing the initial causal shadow of $\mathrm{supp}f$. Finite propagation makes the resulting ghost identical to the displayed one throughout the dependence region, leaving (eq:witness) unchanged.