Manuscript Em02-A@6 · Em02-A-v6.tex · proposition · cb2f0a9a

Proposition: Exact complete translation bridge

Route A: the complete origin contribution and its cross-check · lines 1177–1210 · proof included or linked

Em02-A@6#prop:translation, Em02-A@6#eq:translation-bridge

Status: Proved for displayed mathematical data.

The translation distribution is compact and satisfies

⟨ Ω_{Φ}^{ζ}, ψ ⟩ = - i e, I, J \sum I_{e} (P_{e}^{I, J} [W_{ψ * C^{χ}}^{I, J} [ζ Φ]]) .

Consequently

t_{δ}^{ζ} (Φ) = ⟨ Ω_{Φ}^{ζ}, ρ_{δ} ⟩, t_{δ} ˙^{ζ} (Φ) = ⟨ Ω_{Φ}^{ζ}, κ_{δ} ⟩ .

Proof. The compact-local internal order-four estimate and the vertex give order at most five for each relative functional. Both tensor factors are compact, so their difference pushforward exists. Tensor evaluation gives $⟨ Ω_{Φ}^{ζ}, ψ ⟩ = Λ_{Φ}^{ζ} (ψ * C^{χ})$ . The adjoints prove (eq:translation-bridge). A detailed order and support estimate appears in Appendix app:translation.

Displayed formulas in this passage

Em02-A@6#eq:translation-bridge · line 1181
$⟨ Ω_{Φ}^{ζ}, ψ ⟩ = - i e, I, J \sum I_{e} (P_{e}^{I, J} [W_{ψ * C^{χ}}^{I, J} [ζ Φ]]) .$
unlabeled · line 1192
$t_{δ}^{ζ} (Φ) = ⟨ Ω_{Φ}^{ζ}, ρ_{δ} ⟩, t_{δ} ˙^{ζ} (Φ) = ⟨ Ω_{Φ}^{ζ}, κ_{δ} ⟩ .$

Proposition: Exact complete translation bridge

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