Theorem: Equality of the two regularized parenthesizations

For every $\epsilon >0$, the fully contracted order-${\hslash }^3$ coefficients of

agree and equal the single VII-C coefficient

Their common logarithmic subtraction has the numerical limit $⟨{u_{\mu }}^{\rho },f\otimes g\otimes f^{\prime }⟩$. On arbitrary joint tests the corresponding distributional convergence is strong.

Proof. Introduce independent field variables at the three vertices. The cross-contraction operators ${\mathcal{P}}_{ij}$ have field-independent coefficients and commute on the positive-cutoff old-domain expressions. Both parenthesizations therefore expand as

followed by identification of the field variables. The middle Wick symbol is an old bulk polynomial; the two insertion kernels are regular at positive cutoff, so the relevant old products are defined.

Proposition prop:unique identifies the fully contracted graph and its multiplicity. Equations (eq:T)–(eq:G) give the common coefficient, and Assumption ass:VIID gives the limit. Association does not split this one graph divergence into two different logarithms.