Proposition: Why the XII-D kernel is the matching kernel

Under Inputs input:kernel and input:cones, the kernel to which the supplied XII-D admission statement applies is exactly $t_{\mathrm{D}}$ used here.

Proof. The diagonal insertion is the same ${\mathcal{C}}_{3,\mathrm{D}}={\mathcal{A}}_1(u_{\mathrm{D}};\cdot )$. Its polarization has the same factor $1/2$. The source polarization reproduces the same ${\mathcal{J}}_{\mathrm{D}}(a,c)$, including the ramp-derivative terms. The root kernel and derivative-transfer signs are those of (eq:t-test).

Thus both constructions act on general tests by (eq:t-test). They define the same distribution. There is no change of normalization, tensor species, preparation, or real domain in this matching.