Theorem: Equality of regularized parenthesizations

For every positive cutoff allowed by the supported prescription, the field-independent order-${\hslash }^3$ coefficients of

are equal and have common value

After the single common subtraction their numerical limit is

The corresponding statement on arbitrary compact joint tests is the strong distributional limit in Assumption ass:VIID.

Proof. Use independent fields at the three vertices. The cross-contraction operators ${\mathcal{P}}_{12},{\mathcal{P}}_{13},{\mathcal{P}}_{23}$ have field-independent coefficients and commute on the positive-cutoff old-domain expressions. Both parenthesizations consequently give the finite polynomial expansion of

followed by field identification. The old products in this calculation are defined under Assumption ass:tensor. Proposition prop:unique identifies their scalar order-three graph and multiplicity. The asserted limit is the parent input. Product-test density identifies the same joint numerical distribution; no independent graph is assigned to one parenthesization.