Extension family

The finite collection of point-extension constants is finite dimensional before additional relations. Edge-exchange symmetry can be retained by choosing ${\bar{p}}_{\mu \nu }={\bar{p}}_{\nu \mu }$, for example by averaging. That algebraic restriction does not fix the remaining constants.

A subtraction projection can be specified by functions $w_{\gamma }(r)$ satisfying

and by the values of the extension on these functions. No such values, covariance conditions, Ward conditions, or renormalization scale are selected here.

There are two different ambiguity statements:

• The numerical translation-invariant building blocks have the constant-coefficient point ambiguities (eq:ambiguity).

• An unrestricted extension of the full distribution in $(r,w)$ can have coefficients depending on $w$; transverse order alone does not make that larger freedom finite dimensional.

The present construction uses the first, explicitly specified extension family. It does not classify every possible tangential or fully root-attached ambiguity.