Manuscript Em01-E@2 · Em01-E-v2.tex · proposition · 54f8f8c2

Proposition: Exact translation and scale profiles

Three sufficient verification routes › Route C: an exact translation-origin certificate · lines 1017–1063 · proof included or linked

Em01-E@2#prop:translation, Em01-E@2#eq:translation, Em01-E@2#eq:scaleprofile

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The compact distribution $Ω_{Φ}$ exists, and

t_{δ} (Φ) = Ω_{Φ} (ρ_{δ}) .

For

θ (h) = - 8 ρ (h) - h \cdot \nabla ρ (h),

one has

\int θ = 0, δ \partial_{δ} t_{δ} (Φ) = Ω_{Φ} (θ_{δ}) .

These are complete-family identities and respect reconstruction.

Proof. Transposing the derivatives in (eq:vertextest) gives

⟨ i V (K, H), Φ ⟩ = I, μ, ν \sum ⟨ Λ_{I, μν} [Φ], H_{I}^{μν} ⟩ .

The kinetic sign becomes $+ i$ , the trace-kinetic sign $- i /2$ , and the mass sign remains $- i m^{2} /2$ . Symmetry accounts for the first-line symmetrization.

Substitute $H = ρ_{δ} * (χ C)$ . The tensor pairing is against $ρ_{δ} (x - v)$ . Both tensor factors are compactly supported, so the pushforward in (eq:Omega) is defined and yields (eq:translation).

Differentiation gives $δ \partial_{δ} ρ_{δ} = θ_{δ}$ . Integration of $- div (h ρ)$ gives $\int θ = 0$ . Proposition prop:positive justifies the scale identity. Every construction is linear in the actual internal distribution.

Displayed formulas in this passage

Em01-E@2#eq:translation · line 1020
$t_{δ} (Φ) = Ω_{Φ} (ρ_{δ}) .$
Em01-E@2#eq:scaleprofile · line 1029
$\int θ = 0, δ \partial_{δ} t_{δ} (Φ) = Ω_{Φ} (θ_{δ}) .$
unlabeled · line 1025
$θ (h) = - 8 ρ (h) - h \cdot \nabla ρ (h),$
unlabeled · line 1041
$⟨ i V (K, H), Φ ⟩ = I, μ, ν \sum ⟨ Λ_{I, μν} [Φ], H_{I}^{μν} ⟩ .$

Proposition: Exact translation and scale profiles

Displayed formulas in this passage

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