Abstract

We calibrate a selected sharp-insertion triangle relative to the actual-functional carrier ${\mathcal{P}}_{\Sigma }$ of IX-A. We classify the eight connected contraction skeletons through three edges and identify the unique fully contracted graph for two sharp insertions and one quadratic metric observable. Using the quantified VII-D finite-part theorem, we prove stability under fixed observation differential operators and smooth spectator multipliers, with test-order loss $64\to 64+d$. We fix the sign and multiplicity of the two-factor output already admitted by IX-A, retain the observation-diagonal obstruction, and prove that the observation-derivative orbit of the residue is linearly independent. The infinite distribution orbit is compatible with the single parent conversion parameter of VII-D. Both regularized parenthesizations have the same calibrated numerical limit, but the normal-topology obstruction of VII-C remains unchanged. The manuscript does not factor the numerical finite part ${u_{\mu }}^{\rho }$ through sharp binary multiplications.