The frozen target and the three endings

Use

The slab, all switches, and $\epsilon >0$ remain fixed. The carrier is

The frozen lines satisfy

and metric–scalar blocks vanish.

The interaction remains

The full observable retains the scalar-sourced slot once:

The target is the retained grouped mixed coefficient

The species factor is unchanged. The cross-line difference stays grouped; no separate mixed $H$–${G_{\mathrm{act}}}^{\mathrm{A}}$ product is declared to exist.

The following endings are fixed before the new calculation:

1. Constructed: a common representation of the complete coefficient with a joint principal-symbol section and a remainder controlling every omitted term, yielding the required cancellation; or one named prescription with its ambient space, reference domain, degree, normalization, ambiguity, and wavefront theorem.

2. Proved obstructed: a demonstrated incompatibility excludes every map in a class specified before the test.

3. Unresolved: the common representation, symbol, remainder, or prescription theorem is missing.

These endings apply to (eq:target), not to the success or failure of an individual geometric shortcut.

For the cancellation ending, disappearance of a leading section must be accompanied by the control actually needed for the remaining grouped distribution. Merely lowering a formal order does not define a product that was undefined.