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Manuscript XVI-L@1 · XVI-L.tex · section · 35081099

Normalization gate

L.2 Complete graph decomposition › Normalization gate · lines 613–635

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ledger (mandatory fields).

Datum	Record
Source summand	$A \in {C_{2}, M_{2}, J_{3}}$ ; $P_{A}$ from (eq:jet).
Source placements $a, b$	$C_{2}$ : $(0, 0)$ only; $M_{2}$ : $∣ a ∣ + ∣ b ∣ = 1$ ; $J_{3}$ : $∣ a ∣ + ∣ b ∣ = 2$ (Theorem thm:budget).
Vertex $p, q$	$∣ p ∣ + ∣ q ∣ \leq 2$ ; no derivative on $χ_{I}$ (XVI-K, prop:hess).
Dual slots	Source tensors $- P B^{i} = - (G^{- 1})^{ii} P B_{i}$ , $G = diag (1, - 2, - 2, - 2, 1, 2, 2, 1, 2, 1)$ .
Vertex-end signs	$(- 1)^{∣ p ∣ + ∣ q ∣}$ from Lemma lem:Lsign; sits in $a_{ab, pq}$ , not in $q^{\circ}$ .
Preparation jet	In $P_{A} (y)$ : $\partial\partial σ_{ε}$ ( $C_{2}$ ), $\partial σ_{ε}$ ( $M_{2}$ ), $σ_{ε}$ ( $J_{3}$ ).
Residual metric jet	In $L^{pq}_{ij} (w; h)$ , order $2 - ∣ p ∣ - ∣ q ∣$ .
Switch $χ_{I}$	$χ_{I} (w)$ , undifferentiated, inside $a_{ab, pq}$ .
Ordered vs unordered	Ordered: $(A; a, b; p, q)$ . Unordered numerical $λ = {a + p, b + q}$ ; coefficients accumulate over ordered contributions (no universal factor $2$ ).
Combinatorial factor	$\frac{1}{2 !} \cdot 2 = 1$ (Proposition prop:Lnorm).
Coupling/loop weights	Undetermined in L.2; separated from the graph factor.

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