Local sine-effect access, not coherent access

For $\omega \in {\mathfrak{S}}_C$, exact Weyl relations and Cauchy–Schwarz give

Hence either sine or cosine effect satisfies

For completeness, write $k=l+d$, use the Weyl phase, and bound

Also $|s(k,l)|\le 2\Vert d{\Vert }_{\mathrm{U}}\Vert l{\Vert }_{\mathrm{U}}$.

Choose a measurement time slice strictly later than both source slabs. The compact packet construction of Lemma lem:bridge gives a real compact test $B_L$ with coefficient $l_L$ and error ${\Delta }_L$.

Use the independent massless scalar probe and compact bilinear coupling described explicitly in Appendix app:local. Its final effect is

Its exact induced system effect has the form