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

Proposition: Exact seven-row starting value

The exact common pairing · lines 464–498 · proof included or linked

Em02-A@6#prop:starting, Em02-A@6#eq:exact-raw, Em02-A@6#eq:exact-derivative

Status: Proved for displayed mathematical data.

The raw and differentiated complete pairings are

t_{δ}^{ζ} (Φ) = - i I, J \sum [t_{δ} ˙^{ζ} (Φ) = - i I, J \sum [e = 47, 5 a, 5 z \sum ⟨ μ_{e}, H_{e, c_{δ}}^{I, J} [Φ]⟩ + e = 6, 8, 9 d, 9 m \sum I_{resp} (H_{e, c_{δ}}^{I, J} [Φ])], e = 47, 5 a, 5 z \sum ⟨ μ_{e}, H_{e, c ˙_{δ}}^{I, J} [Φ]⟩ + e = 6, 8, 9 d, 9 m \sum I_{resp} (H_{e, c ˙_{δ}}^{I, J} [Φ])] .

Every fixed-test pairing is smooth for $δ > 0$ .

Proof. Apply the adjoint identities of Theorem thm:adjoints. Distributional convolution may be differentiated on compact positive-scale intervals because the profiles form smooth families with bounded test seminorms. Linearity of the vertex then gives (eq:exact-derivative).

Displayed formulas in this passage

Em02-A@6#eq:exact-raw · line 468
$t_{δ}^{ζ} (Φ) = - i I, J \sum [t_{δ} ˙^{ζ} (Φ) = - i I, J \sum [e = 47, 5 a, 5 z \sum ⟨ μ_{e}, H_{e, c_{δ}}^{I, J} [Φ]⟩ + e = 6, 8, 9 d, 9 m \sum I_{resp} (H_{e, c_{δ}}^{I, J} [Φ])], e = 47, 5 a, 5 z \sum ⟨ μ_{e}, H_{e, c ˙_{δ}}^{I, J} [Φ]⟩ + e = 6, 8, 9 d, 9 m \sum I_{resp} (H_{e, c ˙_{δ}}^{I, J} [Φ])] .$
Em02-A@6#eq:exact-derivative · line 468
$t_{δ}^{ζ} (Φ) = - i I, J \sum [t_{δ} ˙^{ζ} (Φ) = - i I, J \sum [e = 47, 5 a, 5 z \sum ⟨ μ_{e}, H_{e, c_{δ}}^{I, J} [Φ]⟩ + e = 6, 8, 9 d, 9 m \sum I_{resp} (H_{e, c_{δ}}^{I, J} [Φ])], e = 47, 5 a, 5 z \sum ⟨ μ_{e}, H_{e, c ˙_{δ}}^{I, J} [Φ]⟩ + e = 6, 8, 9 d, 9 m \sum I_{resp} (H_{e, c ˙_{δ}}^{I, J} [Φ])] .$

Nearby in this section

section: The exact common pairing