Assumption: Supported carrier input

The carrier is IX-A-ST's adapted $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$, with the following explicitly used properties IXAST.

1. Elements are actual off-shell polynomial functionals with compact distributional configuration kernels. Equality means equality on all smooth configurations, not equality modulo equations of motion.

2. Smooth configurations and compactly supported smooth variations are unrestricted in the retained chart.

3. Classical pointwise multiplication, conjugation, and fixed smooth directional differentiation preserve the carrier. Multiplication by a fixed classical factor is continuous.

4. Every coefficient must satisfy the inherited metric admission condition

   $$({\mathrm{D}}_{\mathbf{\eta }}F{)}_h^{\prime }(\Psi )\in {\mathcal{S}}_1={\mathcal{M}}_1+{\mathcal{Q}}_{\Sigma }$$

   for every configuration and required finite direction list. Existence of a distributional pairing alone is not admission.

5. The supplied local response $L={\mathsf{E}}_{0}C$ has the retained support property: a metric variation outside $P_F$ cannot change $Lh$ on $F$. The IX-A-ST saturation statement makes its retained response independent of auxiliary extensions. No global conical inverse is supplied.

6. The sharp metric adjoint, on its supplied source domain, retains the actual identity

   $$L^{\ast }a=\theta (t)C^{\ast }b+\delta (t)\mathcal{B}(b{|}_{\Sigma }),b={\mathsf{E}}_0^{\ast }a.$$

   The source domain and the actual adjoint must be matched before a formula for $b{|}_{\Sigma }$ is imported.

7. Unchanged regular approximants do not converge:

   $$B_{f,\epsilon }/⟶B_f\text{in }{\tau }_{\Sigma }(0\ne f\text{ real}).$$

The boundary-trace certificate (TC) of Part IV is not included in this assumption.