Assumption: Compact root and joint root kernel

There is a real linear compact smooth root $a_{f,g}[\varphi ]$, with the inherited collar and normalization, such that

for the real smooth inputs considered here.

In a fixed source-component frame, indexed by $A$, the root has a real distributional kernel $B_A(y,x)$ satisfying

The joint kernel is compactly supported, with

The root-theorem domain includes the real cosine and sine witness inputs and their required sums, with the same $f,g,\sigma ,m,\vartheta$ used in revised XVI-E/F and current XVI-G.