Abstract

For the insertion-kernel regularization fixed in VII-C, we prove existence of the strong distributional finite part of its numerical reflection-triangle coefficient. The proof applies to arbitrary compact joint tests. An explicit normalization ledger identifies the leading reflected contribution with the prescribed light-ray residue, including its density and every Fourier factor. A joint spatial Fourier argument gives a regulator-independent ultraviolet decomposition: the radial coefficient differs from that residue by $O(r^{-1})$ in a fixed finite-order distribution seminorm, and the remaining regulator-placement error is $O(\epsilon (1+|\log \epsilon |))$. The estimates retain the full phase and are uniform through the intersection of reflected collinearity with the first–third observation diagonal. For two admissible normalized radial mollifiers, the complete conversion is $c_{ab}r$, with an explicit real coefficient, as a corollary of the ultraviolet decomposition. The result is a numerical subtraction prescription, not an algebra construction.