Theorem: Em01-A release

Under the explicit common-setting assumptions of this paper:

1. a compact symmetric tensor distribution with a smooth compact wave-shell repair has a smooth compact conserved representative on a vacuum comparison space if and only if it is conserved;

2. the possible difference between that conserved representative and the wave-shell representative on the full vacuum space is exactly $-⟨r,\square h⟩$;

3. smooth compact wave-shell representability alone does not imply conserved-current representability, even on the de Donder wave shell;

4. partial witness contraction is noninjective on smooth compact kernels;

5. a presentation-level linear candidate defines a map on reconstructed kernels exactly when it annihilates every null presentation;

6. extension within a fixed supported normal-order class is equivalent to extension of its unique coefficient distributions, and normal order alone does not make the supported ambiguity finite-dimensional.

Proof. Assertions (i) and (ii) are Theorem thm:compatibility. Assertion (iii) is Proposition prop:counterexample. Assertion (iv) is Proposition prop:partial-contraction. Assertion (v) is Theorem thm:descent. Assertion (vi) follows from Theorem thm:jetextension and Proposition prop:infinite-ambiguity.