Route A: the complete origin contribution and its cross-check

Choose a translation cutoff $\vartheta =1$ near zero. For sufficiently small $\delta$, both profiles sample only $\vartheta {{\Omega }_{\Phi }}^{\zeta }$ exactly.

The potential row fibers are

where ${\Sigma }_e$ is the corresponding scalar seed support. They are potential contributors only. Classifying their singular, regular, and degenerating pieces requires the actual crossed singular structure. The equation $\pi (w)=v$ alone does not impose $r=0$.