What has not been supplied by the algebraic sum

The kernels $K_j$ are defined distributions, but a wavefront inclusion alone does not assign them a common scalar principal symbol for the composition (eq:retainedfull).

Some kernels include the observable metric diagonal $a=z$. Others arise from tree compositions with additional internal vertices. To use a common microlocal symbol calculation one must specify the corresponding conormal or multiphase representations, their composition maps, and their orders. One must then control the complete grouped amplitudes and the remaining terms. The distribution product and pushforward criteria do not supply this information merely from an incidence count Hormander.

In particular, the following replacements are not made:

1. evaluating $K_j$ at a singular point;

2. setting every hidden internal vertex equal to one of the retained three vertices;

3. adding differential numerators from inequivalent unidentified phases as if they were one wavefront section;

4. calling a cancellation of highest differential factors a proof that all lower-order singularities disappear;

5. expanding the grouped line difference into intermediate products whose existence is unknown.

The first missing analytic step is a controlled common symbol/composition description of the complete germ (eq:retainedfull). No equality to zero and no nonzero full symbol section is proved here.