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

An actual cross-check of the transition calculation

Route A: the complete origin contribution and its cross-check › An actual cross-check of the transition calculation · lines 1228–1271

Em02-A@6#eq:transition-translation, Em02-A@6#eq:transition-crosscheck

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For fixed $ϵ$ , define a row functional by inserting $θ_{ϵ}$ after its adjoint, exactly as in $V_{e, δ, ϵ}$ . Construct its translation distribution $Ω_{e, Φ, ϵ}$ by the same component tensor pushforward. Then

V_{e, δ, ϵ} (Φ) = ⟨ Ω_{e, Φ, ϵ}, κ_{δ} ⟩ .

To check the coefficients rather than merely the representation, take an arbitrary compact translation profile $ψ$ and put $c = ψ * C^{χ}$ . Theorem thm:transition yields

⟨ Ω_{5 a, Φ, ϵ}, ψ ⟩ ⟨ Ω_{5 z, Φ, ϵ}, ψ ⟩ ⟨ Ω_{47, Φ, ϵ}, ψ ⟩ = - i c_{R} ϵ^{2} I, J \sum ⟨ μ_{0}, H_{5 a, c}^{I, J} [Φ]⟩ + O (ϵ^{3}) I, J \sum ∥ H_{5 a, c}^{I, J} ∥_{1, K}, = - i c_{R} ϵ^{2} I, J \sum ⟨ μ_{0}, H_{5 z, c}^{I, J} [Φ]⟩ + O (ϵ^{3}) I, J \sum ∥ H_{5 z, c}^{I, J} ∥_{1, K}, = - i c_{47} ϵ^{3} I, J \sum ⟨ ν_{0}, H_{47, c}^{I, J} [Φ]⟩ + O (ϵ^{4}) I, J \sum ∥ H_{47, c}^{I, J} ∥_{1, K} .

Thus the same constants and incidence functionals occur in the seed and translation descriptions for every such $ψ$ , not only the chosen mollifier.

This verifies the calculated internal localization coefficients. It does not supply the missing physical origin-germ expansion.

Displayed formulas in this passage

Em02-A@6#eq:transition-translation · line 1235
$V_{e, δ, ϵ} (Φ) = ⟨ Ω_{e, Φ, ϵ}, κ_{δ} ⟩ .$
Em02-A@6#eq:transition-crosscheck · line 1245
$⟨ Ω_{5 a, Φ, ϵ}, ψ ⟩ ⟨ Ω_{5 z, Φ, ϵ}, ψ ⟩ ⟨ Ω_{47, Φ, ϵ}, ψ ⟩ = - i c_{R} ϵ^{2} I, J \sum ⟨ μ_{0}, H_{5 a, c}^{I, J} [Φ]⟩ + O (ϵ^{3}) I, J \sum ∥ H_{5 a, c}^{I, J} ∥_{1, K}, = - i c_{R} ϵ^{2} I, J \sum ⟨ μ_{0}, H_{5 z, c}^{I, J} [Φ]⟩ + O (ϵ^{3}) I, J \sum ∥ H_{5 z, c}^{I, J} ∥_{1, K}, = - i c_{47} ϵ^{3} I, J \sum ⟨ ν_{0}, H_{47, c}^{I, J} [Φ]⟩ + O (ϵ^{4}) I, J \sum ∥ H_{47, c}^{I, J} ∥_{1, K} .$