Remark: Scope of the uniqueness statement

Theorem thm:unique is uniqueness relative to the classical normalization on ${\mathcal{L}}_{\mathrm{off}}$ and to the target ${{\mathfrak{A}}_0}^{(1)}$. It replaces the sentence of X-B “makes no uniqueness claim among arbitrary additional contact normalizations” by a precise one: every modification of $R_1$ with values in ${{\mathfrak{A}}_0}^{(1)}$ either changes a classical evaluation or is zero in ${\mathfrak{A}}_0$. Two remarks delimit this.

(1) Under the usual grading in which a term with $j$ contractions has field degree lowered by $2j$ and carries ${\hslash }^j$ DF,FR, a finite renormalization of a would-be $T_2(V,F_c)$ by a local term respecting that grading has field degree $3-2j$ with $j\ge 1$, hence $j=1$ and degree one; after multiplication by $i/\hslash$ it is an element of ${{\mathfrak{A}}_0}^{(1)}$ of the form $\hat{\varphi }(Nc)$ with $N$ local. All such terms are covered by Corollary cor:contact. This remark presupposes a $T_2$, which the sheet does not define (Theorem thm:nonextension); it explains why no additional freedom is hidden at this order, and it is not a theorem about a family.

(2) Uniqueness relative to the axioms (T1)–(T4) of a family is not addressed, because no family is defined.