Proposition: Extension for this insertion pair

Under the stated normalization, the numerical coefficients have unique extensions across $x=y$. They are the distributions in (eq:W) and their first derivatives. No independent diagonal parameter occurs for this pair.

Proof. Existence is supplied by the displayed full-space distributions. For uniqueness, a difference of extensions is supported at the relative origin. Such a distribution is a finite sum of derivatives of a delta distribution. Every nonzero term has scaling degree at least four, and cannot be added to a coefficient of degree at most three. This is the subcritical case of the extension criterion in BF.