Abstract

We represent the real polarization of the stated single-metric prepared completion by a compact distributional kernel with two symmetric-tensor slots and one scalar slot. The construction is relative to explicit inherited source and compact-root inputs. It is implemented by transferring derivatives from diagonal source deltas onto general test functions and then pairing with the root kernel. This gives a distribution without invoking an unverified product of arbitrary distributions.

The compact kernel yields joint continuity on the compact-set smooth topology, with finite but unspecified derivative orders. We identify the polarization conventions with those of revised XVI-E and XVI-F and import the current XVI-G witness theorem. The resulting complexified kernel, partially evaluated on the specified tensor witness pair, is exactly the compact smooth scalar density $j_{\lambda }$. This closes the former witness compatibility gap without identifying the scalar density with the full kernel or determining a free-shell class from one witness family.

We also give a concise conditional audit of the inherited past-to-future switch representative. No wavefront set of the full kernel or its trisolution is calculated. No cubic admission, BRST/BV observability, or quantum-product construction is established. The regulator width is fixed throughout.