Manuscript XIV-E@1 · XIV-E.tex · theorem · e89fd2a5

Theorem: Exact regrouping of the complete retained expression

E1: exact calculation of the complete precomposition sum › The first calculation: the entire six-branch sum · lines 661–717 · proof included or linked

XIV-E@1#thm:regroup, XIV-E@1#eq:fullregroup

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For all smooth directions for which the inherited slab response formulas are used,

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 .

This identity retains all six branches. It involves no characteristic composition and no regularization.

Proof. Symmetry of $\partial_{μ} \partial_{ν} ψ$ gives

M_{7} = \int f X_{G}^{ν} X_{h}^{μ} \partial_{ν} \partial_{μ} ψ d^{4} x .

Compact integration by parts in the second term of $M_{4}$ yields

\int f X_{G}^{ν} \partial_{ν} X_{h}^{μ} \partial_{μ} ψ = - \int \partial_{ν} (f X_{G}^{ν}) X_{h}^{μ} \partial_{μ} ψ - \int f X_{G}^{ν} X_{h}^{μ} \partial_{ν} \partial_{μ} ψ .

The last integral cancels $M_{7}$ exactly. The first term of $M_{4}$ and the remaining integral are precisely

M_{4} + M_{7} = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} x .

Next, the definition of $U$ expresses $M_{5}$ as the retarded inverse of the divergence of the first two terms in (eq:Btensor). The third term is $M_{6}$ . Since

- X^{μ} [r_{h}] = E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α μ} [r_{h}]),

the fourth term gives $M_{8}$ , with its inherited sign. Linearity of $E_{0}^{R}$ combines these three branches.

Finally $M_{9}$ is unchanged. Adding the three groups proves (eq:fullregroup).

Displayed formulas in this passage

XIV-E@1#eq:fullregroup · line 665
$M_{ret} [G; h, ψ] = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} x + \int f E_{0}^{R} (\partial_{α} (σ_{ε} B^{α μ} [G; h])) \partial_{μ} ψ d^{4} x + \int f p [r_{h}; ψ] d^{4} x .$
unlabeled · line 683
$M_{7} = \int f X_{G}^{ν} X_{h}^{μ} \partial_{ν} \partial_{μ} ψ d^{4} x .$
unlabeled · line 689
$\int f X_{G}^{ν} \partial_{ν} X_{h}^{μ} \partial_{μ} ψ = - \int \partial_{ν} (f X_{G}^{ν}) X_{h}^{μ} \partial_{μ} ψ - \int f X_{G}^{ν} X_{h}^{μ} \partial_{ν} \partial_{μ} ψ .$
unlabeled · line 699
$M_{4} + M_{7} = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} x .$
unlabeled · line 708
$- X^{μ} [r_{h}] = E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α μ} [r_{h}]),$

Theorem: Exact regrouping of the complete retained expression

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