Assumption: Frozen convolution data

Fix the source-specific convolution operator $S_{\epsilon }$, switch and Cauchy prescription, compact real observation test $f_C$, and real-linear supported gauge algebra ${\mathfrak{g}}_C$ of X-A–Convolution. Retain the following explicit inputs:

1. for the source's operators,

   $${\mathsf{E}}_0\mathsf{C}{T}_{g,\epsilon }(K\xi )=S_{\epsilon }\xi ,\xi \in {\mathfrak{g}}_C;$$

2. an admitted witness ${\xi }_{\ast }\in {\mathfrak{g}}_C$ has the finite real coefficient

   $$c_{\epsilon }={\int }_{\hat{M}}{f}_C(x)({\xi }_{\ast }-S_{\epsilon }{\xi }_{\ast }{)}^{\mu }(x){\partial }_{\mu }{x}^1{\mathrm{d}}^{4}x>0,0<\epsilon \le \frac{1}{8};$$

3. every real multiple $\lambda {\xi }_{\ast }$ remains in ${\mathfrak{g}}_C$, and the evaluated first-order coefficient has the orbit

   $$a(\lambda )=\lambda c_{\epsilon },\lambda \in \mathbb{R}.$$

The operators, normalization, width range, and witness in these statements retain their source meanings XConv,XEnter.