Abstract

We construct an off-shell polynomial admission space for the exact sharp zero-Cauchy harmonic insertion $B_f=-\int f({\mathsf{E}}_{0}Ch{)}^{\mu }{\partial }_{\mu }\varphi$. The construction retains compact distributional kernels and uses actual functionals, rather than independent symbols with unspecified relations. A boundary-trace map on linear metric derivatives is well defined modulo microcausal linear kernels. Requiring this map for every directional derivative gives a domain containing the bulk polynomials, the sharp insertions, their linear coordinate descendants, and arbitrary finite pointwise words. The domain is stable under smooth directional differentiation and complex conjugation.

An explicit Hausdorff locally convex topology combines strong compact-distribution convergence with all these boundary traces. The sharp insertion varies continuously with its observation test, whereas its unchanged smooth-kernel approximants do not converge to it. This failure is proved for every nonzero real observation test by a nonvanishing advanced Cauchy trace. Consequently, the domain does not reproduce the normal-topology admission hypothesis of the mixed-Wick obstruction.

Classical pointwise multiplication is separately continuous. No joint continuity or hypocontinuity is inferred, and no renormalized quantum product is constructed. We prove faithful bulk inclusion, exact off-shell classical normalization, and classical presentation independence. We also identify selected already-defined Wick coefficients that belong to the admission space, without claiming quantum closure. The result closes IX-A only: graph extensions, quantum associativity, quantum presentation independence, interactions, and the switched-gauge gate remain separate tasks.