Proposition: Point-supported changes

For each label $\mathsf{k}$, any other extension ${\overline{q^{\prime }}}_{\mathsf{k}}$ with $\mathrm{sd}({\overline{q^{\prime }}}_{\mathsf{k}})\le 4+N_{\mathsf{k}}$ satisfies

If the alternative choices are required to define a coherent linear map ${\mathfrak{R}}_{\mathrm{K}}^{\prime }$ on $\mathcal{V}$, these constants are constrained by the constant-coefficient relations among the kernels. They are not independent parameters for all twenty-five labels.

Proof. The difference is supported at zero. The scaling bound and the point-support structure theorem allow only derivatives of $\delta$ through order $N_{\mathsf{k}}$, as in BF (Theorem 5.3). This argument also covers a zero punctured kernel when the declared upper bound is used.

For a relation ${\sum }_{\mathsf{k}}{c}_{\mathsf{k}}{q^{\circ }}_{\mathsf{k}}=0$, coherence requires both extension maps to annihilate that relation. Consequently

Independence of the derivatives of $\delta$ gives the corresponding constraints. Equivalently, ${\mathfrak{R}}_{\mathrm{K}}^{\prime }-{\mathfrak{R}}_{\mathrm{K}}$ is a linear point-supported map on the finite graded span, with grade-$k$ order bound $k-4$.