Theorem: Finite part for the prescribed VII-C family

Fix the convention-sheet data, a mass $m\ge 0$, and the class of real, nonnegative, even, normalized, Euclidean-radial compact mollifiers. Use one fixed mollifier on both slots of each insertion, and keep the middle observable unsmeared.

Then:

1. For each $\epsilon >0$, the numerical family is the smooth distribution (eq:Texact), with the prescribed product-test pairings.

2. The leading reflected coefficient, including its density, is exactly

   $$r=\frac{1}{64(2\pi {)}^3}{\Psi }_{\ast }\nu .$$

   It is obtained from the exact branch integral by the normalization ledger, not imposed on that integral.

3. For every $\mu >0$, the prescribed subtracted family has a strong limit in ${\mathcal{D}}^{\prime }(M_{+}^3)$, with the fixed-order estimate (eq:finalrate). This includes the full reflected-diagonal intersection.

4. The regulator-independent ultraviolet decomposition is (eq:UV). Its radial and regulator-placement errors satisfy

   $$\begin{array}{rl}⟨A(r)-r,\Phi ⟩& =O_{K,m}(r^{-1})p_{K,64}(\Phi ),\\ ⟨{E_{\epsilon }}^{\rho },\Phi ⟩& =O_{K,\rho ,m}(\epsilon (1+|\log \epsilon |))p_{K,64}(\Phi ).\end{array}$$

5. For any two admissible mollifiers, the complete conversion is (eq:conversion). In particular, ${L_{ab}}^{\perp }=0$ and $c_{ab}\in \mathbb{R}$. The fixed class $\mathfrak{C}=\mathbb{R}r$ suffices.

These conclusions concern only the numerical prescription defined in this paper.

Proof. Proposition prop:positive proves (i). Proposition prop:ledger proves (ii) as an equality on arbitrary joint tests. Lemma lem:S, Lemma A, and Lemma lem:replacement give the ultraviolet decomposition and its uniform error estimates. Theorem thm:finitepart proves (iii), while Section sec:audit records the observation and momentum coverage. Theorem thm:conversion proves (v).