Manuscript XIV-H@1 · XIV-H.tex · input · 695c7c3a

Input statement: XIV-E's operator identity

H0: copied handoffs and their domains › The exact regrouping and complete scalar root · lines 414–469

XIV-H@1#input:E, XIV-H@1#eq:regroup, XIV-H@1#eq:B, XIV-H@1#eq:root

With

X_{G} = X [G], X_{h} = X [h], r_{h} = r_{D}^{'} [G] h = P E_{0}^{R} (J_{D}^{'} [G] h),

the complete retained smooth-slot expression is

M_{ret} [G; h, ψ] = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} x + \int f E_{0}^{R} (\partial_{α} (σ_{ε} B^{α μ} [G; h])) \partial_{μ} ψ d^{4} x + \int f p [r_{h}; ψ] d^{4} x,

where

C_{G}^{μ}_{ν} = f \partial_{ν} X_{G}^{μ} - δ^{μ}_{ν} \partial_{ρ} (f X_{G}^{ρ}),

and

B^{α μ} [G; h] = a_{1}^{α β} [h] \partial_{β} X_{G}^{μ} + a_{1}^{α β} [G] \partial_{β} X_{h}^{μ} + a_{2} ˙^{α μ} [G; h] + a_{1}^{α μ} [r_{h}] .

All six branches are retained.

For $F = E_{m}^{A} f$ , the complete scalar-response root is

\int f p [r_{h}; ψ] = \int g (a_{1}^{μν} [r_{h}] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r_{h}] F ψ) d^{4} x .

The derivative and mass terms are both retained.

For each fixed smooth compact $G$ , the tested attachment kernel is compact and satisfies the balanced attachment bound. This does not give the untested joint symbol or its uniform estimates XIVE.

Displayed formulas in this passage

XIV-H@1#eq:regroup · line 422
$M_{ret} [G; h, ψ] = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} x + \int f E_{0}^{R} (\partial_{α} (σ_{ε} B^{α μ} [G; h])) \partial_{μ} ψ d^{4} x + \int f p [r_{h}; ψ] d^{4} x,$
XIV-H@1#eq:B · line 441
$B^{α μ} [G; h] = a_{1}^{α β} [h] \partial_{β} X_{G}^{μ} + a_{1}^{α β} [G] \partial_{β} X_{h}^{μ} + a_{2} ˙^{α μ} [G; h] + a_{1}^{α μ} [r_{h}] .$
XIV-H@1#eq:root · line 455
$\int f p [r_{h}; ψ] = \int g (a_{1}^{μν} [r_{h}] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r_{h}] F ψ) d^{4} x .$
unlabeled · line 417
$X_{G} = X [G], X_{h} = X [h], r_{h} = r_{D}^{'} [G] h = P E_{0}^{R} (J_{D}^{'} [G] h),$
unlabeled · line 435
$C_{G}^{μ}_{ν} = f \partial_{ν} X_{G}^{μ} - δ^{μ}_{ν} \partial_{ρ} (f X_{G}^{ρ}),$