Theorem: Exact adjoints and existence

Status: Proved for displayed mathematical data.

For compact ambient $\Psi$,

Their compact-local distributional orders are at most $4,4,4,2,3,3,2$, respectively. The response rows are supported in $\{a=z,b-a\in \partial J^{+}\}$.

Proof. For $5a$, $s=b-a$, $t=a-z$ give

The internal signs cancel. Transposing the outer divergence and three seed derivatives gives (eq:P5a). The $5z$ proof is identical.

For $47$, substitute

The bracket has three terms with signs $+,-,-$, containing respectively three, two, and three seed derivatives. The outer negative divergence transposes positively; the remaining signs are $-,-,+$. The outer ambient $b$ derivative becomes ${\mathcal{B}}_{\mu }$, proving (eq:P47), including the derivative-of-$f$ term.

For $6$, transpose the outer divergence and the separate $-{\partial }_{s}R$ factor without differentiating the incidence delta. This gives (eq:P6).

For the response rows, source-transposition signs cancel those in (eq:response-definition). Outer negative divergences transpose positively. Lemma lem:B gives the $s_{\kappa }R/2$ factor in $8$. This proves the remaining formulas, including both root terms.

Every integral is against a locally finite seed measure with a compact smooth differential test amplitude. Counting its test derivatives proves existence, continuity, and the orders. Differentiation preserves the stated support.