Abstract

We construct a supported local adaptation of the IX-A admission domain for the exact off-shell harmonic insertion

on a prescribed $3+1$-dimensional conical spacetime. A causally saturated regular developing presentation contains the entire truncated Cauchy past of each admitted observation. This makes the local zero-Cauchy response independent of auxiliary field extensions without asserting a global Green operator on the non-globally-hyperbolic regular cone.

The resulting space consists of actual compact-distribution polynomial functionals. A canonical trace on their linear metric descendants, taken modulo microcausal linear kernels, defines a Hausdorff locally convex topology. The carrier contains the bulk polynomials, exact sharp insertions, coordinate descendants, and arbitrary finite classical pointwise words. Directional differentiation and conjugation are continuous, and pointwise multiplication is separately continuous. For every nonzero real observation test, an explicit positive advanced-Cauchy moment produces a normalized continuous separator: it equals one on the sharp insertion and zero on every unchanged regular approximant. Thus faithful admission does not identify the sharp element with that regularization limit.

The additional contribution is a supported transfer theorem and a complete finite operational interface around these classical objects. One destructive clock selects installed evaluation tables; their outputs and an arbitrary reference-entangled payload admit exact causal transport through a compact regular laboratory. Finite records can certify the displayed separator but cannot determine the full infinite-dimensional admission topology or the quantum residual. This combination provides a controlled benchmark for later deformation and acquisition proposals while keeping classical admission, statistical readout, and quantum multiplication distinct. The admission construction itself is attributed to IX-A; no priority is claimed for its underlying distributional methods. No renormalized quantum product, quantum associativity, Einstein dynamics, or spacetime emergence is constructed.