Manuscript XIII-C@1 · XIII-C.tex · section · e6556893

Derivative placement

C1: complete incidence inventory for the selected pair › Derivative placement · lines 572–639

Move each displayed divergence onto its incoming retarded kernel. Leave derivatives of uncontracted fields on those fields. Let $e = 1$ for a kinetic scalar monomial and $e = 0$ for a mass monomial. Write $Γ (a)$ for a metric slot at $a$ and $φ (a)$ for a scalar slot there.

Branch	Three field-slot locations	Internal retarded edges	Maximum total field/line derivative count
$B_{1}$	$Γ (u), Γ (v), φ (v)$	$x - u, u - v$	$4$
$B_{2}$	$Γ (u), Γ (u), φ (u)$	$x - u$	$2$
$B_{3}$	$Γ (u), Γ (v), φ (v)$	$x - u, x - v$	$4$
$B_{4}$	$Γ (u), Γ (v), φ (x)$	$x - u, x - v$	$4$
$B_{5}$	$Γ (u), Γ (v), φ (x)$	$x - u, u - v$	$4$
$B_{6}$	$Γ (u), Γ (u), φ (x)$	$x - u$	$2$
$B_{7}$	$Γ (u), Γ (v), φ (x)$	$x - u, x - v$	$4$
$B_{8}$	$\partial^{α} Γ (v), \partial^{β} Γ (v), φ (x)$	$x - u, u - v$	$4$
$B_{9}$	$\partial^{α} Γ (v), \partial^{β} Γ (v), φ (u)$	$x - u, u - v$	$4$
$B_{10}$	$φ (v), φ (v), φ (x)$	$x - u, u - v$	$4$
$B_{11}$	$φ (v), φ (v), φ (u)$	$x - u, u - v$	$4$

The exact cross-line derivative order is the sum of the derivative orders of its two matched field slots. Thus:

d_{γ γ} = 0 + ∣ α ∣ when a differentiated source metric is selected, d_{ϕϕ} = e_{vertex} + d_{observable scalar} .

Coordinate metric slots outside the source have order zero. Scalar slots in a switched scalar or stress vertex have order $e$ ; observation scalar slots in $B_{4}, B_{5}, B_{6}, B_{8}, B_{10}$ have order one, and that in $B_{7}$ has order two. These rules, together with (eq:branches), specify every derivative placement without imposing a tensor-component truncation.

Displayed formulas in this passage

unlabeled · line 626
$d_{γ γ} = 0 + ∣ α ∣ when a differentiated source metric is selected, d_{ϕϕ} = e_{vertex} + d_{observable scalar} .$