The tadpole correction to the free-limit assertion

This is not a removable bookkeeping term. Erasing it for arbitrary $h$ would change the reference action and confuse it with a different source prescription XHST.

Thus the phrase “free-limit recovery” has two distinct, proved meanings here:

1. on fields and the homogeneous free causal carrier, the limit is exactly $s_0$;

2. on the complete BV algebra, exact free recovery holds in the centered background sector.

No unconditional full-BV free-limit claim is made for arbitrary fixed off-shell backgrounds.