Why the raw root needs a collar

Thus the independent source pairing of XII-C now becomes a derived equality. It was not renamed as the insertion. The raw-root support requirement has been corrected explicitly, without adding a contact term or altering the source.

The set $O_{f,g}$ controls the scalar slot. It does not control the entire functional support: the metric slots can occur in its causal past. This distinction prevents a false support claim for a nonlocal response functional.