Manuscript Em01-C@3 · Em01-C-v3.tex · theorem · f0953702

Theorem: Exact complete-sum translation representation

The complete contraction as a translation distribution › The translation distribution · lines 1186–1236 · proof included or linked

Em01-C@3#thm:translation, Em01-C@3#eq:origin-evaluation, Em01-C@3#eq:Omega-order

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For $K = K_{ret}$ ,

t_{δ} (Φ) = Ω_{Φ} (ρ_{δ}) .

Every internal row and every vertex term is retained.

If $χ C$ has compact distributional order at most $p$ , then

∣ Ω_{Φ} (ψ) ∣ \leq C_{L} ∥Φ ∥_{5} ∥ ψ ∥_{p + 5}, supp Φ \subset L,

on a fixed compact neighborhood of the translation support.

Every equality of actual internal distributions is respected. In particular, if $Rec A = 0$ , then the corresponding $Ω_{Φ}$ and all regularized values are zero.

Proof. Insert $H = C_{δ}$ into (eq:transpose). The result is

I, μ, ν \sum ⟨ Λ_{I, μν} [Φ] (x) \otimes χ C_{I}^{μν} (v), ρ_{δ} (x - v) ⟩,

which is the pushforward definition of (eq:origin-evaluation).

Linearity gives

Ω_{Φ} = b \in B \sum Ω_{b, Φ}, B = {47, 5 +, 5 -, 6, 8, 9 der, 9 mass} .

Each row uses all terms of (eq:Lambda).

The tensor pairing requires at most five $x$ -derivatives and at most $p$ $v$ -derivatives. On $ψ (x - v)$ their combined order is at most $p + 5$ . The compact-support cutoffs have fixed derivative bounds. This proves (eq:Omega-order).

The construction begins with the actual $K$ , so a zero internal distribution remains zero at every stage.

Displayed formulas in this passage

Em01-C@3#eq:origin-evaluation · line 1189
$t_{δ} (Φ) = Ω_{Φ} (ρ_{δ}) .$
Em01-C@3#eq:Omega-order · line 1196
$∣ Ω_{Φ} (ψ) ∣ \leq C_{L} ∥Φ ∥_{5} ∥ ψ ∥_{p + 5}, supp Φ \subset L,$
unlabeled · line 1211
$I, μ, ν \sum ⟨ Λ_{I, μν} [Φ] (x) \otimes χ C_{I}^{μν} (v), ρ_{δ} (x - v) ⟩,$
unlabeled · line 1222
$Ω_{Φ} = b \in B \sum Ω_{b, Φ}, B = {47, 5 +, 5 -, 6, 8, 9 der, 9 mass} .$

Theorem: Exact complete-sum translation representation

Displayed formulas in this passage

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