Theorem: Main result: a specified C-obstruction

For the fixed matrix (eq:Hhh)–(eq:Hscalar), there exist real nonnegative $f,f^{\prime },g\in {C_c}^{\infty }(M_{+})$, with pairwise disjoint supports, and regular polynomials $B_{f,\epsilon },B_{f^{\prime },\epsilon }$ such that:

1. their mixed kernels converge to the exact kernels of $B_f,B_{f^{\prime }}$ in the normal-topology stages specified in Section sec:topology;

2. with $W_g=\int g(y)h_{00}(y{)}^2{\mathrm{d}}^{4}y$,

   $${\mathrm{ev}}_0[{\hslash }^3]\left(B_{f,\epsilon }{\star }_H{W}_g{\star }_H{B}_{f^{\prime },\epsilon }\right)=\mathcal{C}(f,g,f^{\prime })\log ({\epsilon }^{-1})+O(1),$$

   where $\mathcal{C}(f,g,f^{\prime })>0$;

3. the corresponding fully contracted coefficient has no sequentially continuous extension at the pair of sharp kernels that agrees with the old Wick product on regular kernels.

In particular, no algebra extension satisfying Assumption ass:extension exists.