Manuscript XVI-P@p23 · XVI-P.tex · section · 9b0037a7

Fixed scalar extension u μ R (o70-sync)

Fixed scalar extension u μ R (o70-sync) · lines 1784–1809

(only after the holomorphy premise). Gate 0 gives $D_{F} = - A (Q - i 0)^{- 1}$ with $A = 1/ (4 π^{2})$ and $(- □) D_{F} = i δ$ , hence

□ (Q - i 0)^{- 1} = 4 π^{2} i δ .

The second recurrence

□ ϕ_{- 1 + ζ} = 4 ζ (- 1 + ζ) ϕ_{- 2 + ζ}

and the holomorphy of $ϕ_{- 1 + ζ}$ yield

⟨ ϕ_{- 2 + ζ}, f ⟩ = \frac{r f ( 0 )}{ζ} + FP + O (ζ), r = - i π^{2} .

Scale $μ^{2 ζ}$ affects only the finite part. Operational Laurent identity:

⟨ u_{ζ}, f ⟩ = \frac{r f ( 0 )}{ζ} + ⟨ u_{μ}^{R}, f ⟩ + O (ζ) .

Euclidean radial integral ( $∣ S^{3} ∣ = 2 π^{2}$ ) is an independent sign check, not the sole justification. Not momentum-space MS; not XVI-J's value-jet extension of $d (z)^{2}$ .

Displayed formulas in this passage

unlabeled · line 1789
$□ (Q - i 0)^{- 1} = 4 π^{2} i δ .$
unlabeled · line 1793
$□ ϕ_{- 1 + ζ} = 4 ζ (- 1 + ζ) ϕ_{- 2 + ζ}$
unlabeled · line 1797
$⟨ ϕ_{- 2 + ζ}, f ⟩ = \frac{r f ( 0 )}{ζ} + FP + O (ζ), r = - i π^{2} .$
unlabeled · line 1802
$⟨ u_{ζ}, f ⟩ = \frac{r f ( 0 )}{ζ} + ⟨ u_{μ}^{R}, f ⟩ + O (ζ) .$

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section: Fixed scalar extension u μ R (o70-sync)