Manuscript XV-F@3 · XV-F-v3.tex · section · d4e74afa

The K 8 and K 9 kernels

Source manifest (Gate F0) › The K 8 and K 9 kernels · lines 1291–1335

XV-F@3#eq:Q, XV-F@3#eq:T, XV-F@3#eq:K8, XV-F@3#eq:K9, XV-F@3#eq:Lresp

XIV-I's input statement Iresponse, copied from XIV-H, is as follows. The differentiated source kernel is

Q^{L, I, J} (v; a, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q^{L, I, J}_{α β} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a),

supported on $v = a = z$ , with coefficients taken from the density-weighted source and not reproduced; the order bound is XIV-H's Hq. The response kernel is

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

With $B = R * R$ as in (eq:RR) and $F = E^{A}_{m} f$ ,

K_{8}^{I, J} (a, b, z) K_{9}^{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], = - \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 source states that 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,

with zero $y$ covector. The symbols $P$ , $b$ , $g$ are not defined in the record; $q_{α β}$ is paraphrased; $E^{A}_{m}$ is a candidate. XII-B's weighted Euler identity XIIB shows that any density-weighted cubic source carries terms in $\partial σ_{ε}$ , consistent with XIV-H's “including ramp derivatives” and with the weights $w_{Γ}$ of Definition def:Cpart. This is a reading remark, not a datum.

Displayed formulas in this passage

XV-F@3#eq:Q · line 1295
$Q^{L, I, J} (v; a, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q^{L, I, J}_{α β} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a),$
XV-F@3#eq:T · line 1304
$T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v .$
XV-F@3#eq:K8 · line 1309
$K_{8}^{I, J} (a, b, z) K_{9}^{I, J} (a, b, z) = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
XV-F@3#eq:K9 · line 1309
$K_{8}^{I, J} (a, b, z) K_{9}^{I, J} (a, b, z) = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
XV-F@3#eq:Lresp · line 1323
$Λ_{resp} = N^{*} {(b - a)^{2} = 0, a = z} ∖ 0 \subset T^{*} W,$