Manuscript XIV-E@1 · XIV-E.tex · section · e89fd2a5
What would be a joint symbol
The precise E1 stop › What would be a joint symbol · lines 1096–1160
To use the notation
for (eq:target), a successor must supply a common microlocal representation and composition theorem for the grouped coefficient. In particular, it must identify:
the relevant canonical or characteristic relation in the chosen ambient space;
the amplitude orders and the common symbol bundle;
the maps accounting for additional internal vertices in ;
the observable attachment diagonal and its derivatives;
the complete tensor coefficients and the prescribed grouping of the cross-line difference;
a remainder estimate that controls every term not represented by the proposed leading section.
The inherited wavefront inclusions do not themselves supply these data. The ordinary principal-symbol construction requires more than a wavefront-set inclusion Wunsch (Section 8).
Theorem thm:regroup supplies an exact common operator expression before characteristic composition. Theorem thm:nonclean proves that simply adding the selected line phases does not supply the missing calculus.
Neither theorem assigns the requested joint principal symbol.
| Branch | Established contribution | Joint-symbol status |
|---|---|---|
| Included in the exact coefficient | No common characteristic symbol assigned. | |
| The first two terms of | Internal composition and order remain to be controlled. | |
| and the copied diagnostic | Not a symbol of the full sum. | |
| Its explicit second-scalar-derivative term cancels as proved | Remaining terms retained; no microlocal order claim. | |
| in the complete coordinate response | Full differentiated source retained. | |
| The exact root formula (eq:B9root) | Derivative and mass terms both retained. |
A vanishing numerator in a particular presentation would have to be recorded with its remaining terms. It would not authorize deleting a branch from this table.
Displayed formulas in this passage
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