Manuscript III@2 · III.tex · section · 82dad3e5

Compact adjoint repair and initial surfaces

Sign, inverse, boundary, and kernel audit › Compact adjoint repair and initial surfaces · lines 3156–3174

Both identities

\int h J_{ψ} = \int ψ L_{h} Φ, \int (P ψ) φ = \int ψ P φ

use compact smooth spacetime $ψ$ .

Replacing $ψ$ by a time-truncated response kernel can introduce initial-surface distributions and changes the argument. The zero-repair theorem does not make that replacement.

The positive Ward pair avoids it differently: its coefficient is not $P ψ$ for any compact smooth $ψ$ , as shown by its positive pairing with a free homogeneous solution.

Displayed formulas in this passage

unlabeled · line 3159
$\int h J_{ψ} = \int ψ L_{h} Φ, \int (P ψ) φ = \int ψ P φ$