Lemma A: exact phase and complete error budget

. Lemma A holds on every compact of $X$, with test order $64$, independently of the regulator. The proof covers the comparable central sector and all noncomparable shifted tails, uniformly in the momentum direction. It uses only upper bounds on $a,b,|d|$. Thus it applies unchanged on neighborhoods of $\mathrm{Rel}\cap {\Delta }_{13}$.

The estimate is an equality and bound on all joint tests. In particular, an additional order-$r^0$ contribution proportional to $\delta (d)$, a normal derivative of the residue, or any other nonzero distribution could not be hidden in its error: it would contradict (eq:Ar) on a test detecting that distribution.