Manuscript VII-E@1 · VII-E.tex · proposition · b713cf3c

Proposition: Exact one-dimensional counterterm test

A dictionary for distributional observations › Counterterm-blind tests · lines 1674–1708 · proof included or linked

VII-E@1#prop:blind-tests, VII-E@1#eq:blind-test, VII-E@1#eq:blind-test-criterion

Let $r \neq = 0$ and $L$ belong to $D^{'} (X)$ . Choose $Ψ \in D (X)$ with $⟨ r, Ψ ⟩ = 1$ , and set

Φ^{\circ} = Φ - ⟨ r, Φ ⟩ Ψ.

Then

L \in R r ⟺ {⟨ L, Φ^{\circ} ⟩ = 0 ⟨ L, Ψ ⟩ \in R . for every Φ \in D (X),

Proof. Nonzero $r$ permits a test with nonzero pairing, which can be rescaled to obtain $Ψ$ . By construction, $⟨ r, Φ^{\circ} ⟩ = 0$ .

If $L = cr$ , $c \in R$ , both conditions follow. Conversely, if they hold, then

⟨ L, Φ ⟩ = ⟨ L, Φ^{\circ} ⟩ + ⟨ r, Φ ⟩ ⟨ L, Ψ ⟩ = c ⟨ r, Φ ⟩, c = ⟨ L, Ψ ⟩ \in R .

This holds for every compact test, so $L = cr$ .

Displayed formulas in this passage

VII-E@1#eq:blind-test · line 1678
$Φ^{\circ} = Φ - ⟨ r, Φ ⟩ Ψ.$
VII-E@1#eq:blind-test-criterion · line 1683
$L \in R r ⟺ {⟨ L, Φ^{\circ} ⟩ = 0 ⟨ L, Ψ ⟩ \in R . for every Φ \in D (X),$
unlabeled · line 1701
$⟨ L, Φ ⟩ = ⟨ L, Φ^{\circ} ⟩ + ⟨ r, Φ ⟩ ⟨ L, Ψ ⟩ = c ⟨ r, Φ ⟩, c = ⟨ L, Ψ ⟩ \in R .$