Abstract

We solve an exact finite contact-membership problem for a fixed two-species Pauli–Fierz tree. On an explicit three-label family, the single-longitudinal reference response is $m_S=(0,16/3,16/3{)}^T$. A complete redundant catalogue of $15120$ local, parity-even, two-derivative quartic contacts has evaluated rank $2$, equal to the rank of its target augmentation. Two original raw columns are $(0,8,0{)}^T$ and $(0,0,8{)}^T$; their minor is $64$, and the constant coefficients $2/3,2/3$ fit the target simultaneously on all three labels. The complete left kernel is $\mathrm{span}\{(1,0,0)\}$, so no separator exists on this family against the declared class. We derive the required Einstein cubic contraction, prove the analytic controls, and provide an integer encoding of every contact-matrix entry together with an optional exact checker. The status is $\mathrm{SEP}$-finite: the finite-family problem is closed, while global membership or separation remains undecided here. We distinguish action representatives, response classes, affine fitting freedom, class enlargement, and weighted response prescriptions. Companion results are cited only for comparison; none supplies a contact column or a premise of the certificate.