Manuscript XIV-I@1 · XIV-I.tex · input · 29df501d

Input statement: Source incidence and gravitational legs

I0: copied inputs and the exact predecessor stop › The untested source and two response maps · lines 339–403

XIV-I@1#input:Hresponse, XIV-I@1#eq:Q, XIV-I@1#eq:T, XIV-I@1#eq:K8, XIV-I@1#eq:K9

The full differentiated source has the finite local form

(J_{D}^{'} [G] h)^{L} (v) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial^{α} G_{J} (v) \partial^{β} h_{I} (v),

with every coefficient taken from the actual density-weighted source. Before residual testing its kernel is

Q^{L, I, J} (v; a, z) = α, β \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a) .

This is a defined regular incidence supported on $v = a = z$ .

Writing the response kernel with all three species indices,

T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v,

the exact coordinate-leg map on the retained patch is

K_{8}^{I, J} (a, b, z) = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], B = R * R .

The complete scalar-leg root, with $F = E_{m}^{A} f$ , gives

K_{9}^{I, J} (a, b, z) = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .

The mass term is retained.

On the separated chronological patch, both kernels have regular conormal representations of upper order $3/2$ on

Λ_{resp} = N^{*} {(b - a)^{2} = 0, a = z} ∖ 0 \subset T^{*} W .

The $y$ covector is zero XIVH.

Displayed formulas in this passage

XIV-I@1#eq:Q · line 350
$Q^{L, I, J} (v; a, z) = α, β \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a) .$
XIV-I@1#eq:T · line 361
$T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v,$
XIV-I@1#eq:K8 · line 368
$K_{8}^{I, J} (a, b, z) = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], B = R * R .$
XIV-I@1#eq:K9 · line 381
$K_{9}^{I, J} (a, b, z) = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
unlabeled · line 342
$(J_{D}^{'} [G] h)^{L} (v) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial^{α} G_{J} (v) \partial^{β} h_{I} (v),$
unlabeled · line 397
$Λ_{resp} = N^{*} {(b - a)^{2} = 0, a = z} ∖ 0 \subset T^{*} W .$