Partial witness contraction and recovery of the density

The cutoffs in this proof only implement pairing with a compact distribution. They do not replace the witness by a different metric configuration.

limitation of the matching. Equation (eq:q-equals-j) identifies a partial evaluation of $t_{\mathrm{D}}$, not the full kernel with $j_{\lambda }$. It is not a reconstruction or uniqueness theorem for a three-slot kernel from one witness family. It is not an all-free-triples comparison.

The full kernel representation is established by Theorem thm:trep; the witness identities are used only to identify its specified partial contraction.