Theorem: Inherited observation-diagonal limitation

There are nonnegative $f,g,f^{\prime }\in {C_c}^{\infty }(M_{+})$ with pairwise disjoint supports such that

No family of counterterms supported on ${\Delta }_{12}\cup {\Delta }_{23}\cup {\Delta }_{13}$ removes the logarithmic divergence on all joint tests. The same limitation holds for observation boundary faces avoided by these compact supports.

Proof. Use the VII-C separated-support witness VIIC (lem:separated). Its centers are

with positive pairwise distinct $S,T,T^{\prime }$. The source supplies nonnegative bumps with pairwise disjoint supports and $\mathcal{C}(f,g,f^{\prime })>0$. Equation (eq:C) identifies this number with the residue pairing.

For any fixed joint test, Assumption ass:VIID gives

For the selected product test, the leading coefficient is therefore exactly $\mathcal{C}(f,g,f^{\prime })$. For fixed $\mu$, the contribution $-(\log \mu )\mathcal{C}$ is bounded, so this agrees with the VII-C form $\mathcal{C}\log ({\epsilon }^{-1})+O(1)$.

The joint support avoids all three diagonals. Any distribution supported on their union pairs to zero with this test, even if its coefficients depend on $\epsilon$. Such a subtraction cannot remove the nonzero logarithm. This is the observation-diagonal limitation already recorded in VII-C's counterterm discussion.