Operational diagnostics from embedded Liouvillian Jordan geometry in a four-atom collision model

We study a non-Markovian two-qubit collision model through its Markovian
four-atom embedding and argue that, in this setting, defectivity is most naturally formulated
at the level of the embedded GKSL Liouvillian. In the relevant invariant sector the
generator separates into explicit relaxation and coherence blocks that contain both defective
coherence channels and a Jordan-3 relaxation subchannel. After projection onto the stable
subspace, the associated Lyapunov equation remains well posed for positive probes and
yields finite but strongly anisotropic diagnostics; numerical benchmarks on documented
near-critical paths then show that unconstrained optimization is coherence dominated,
whereas a balanced objective selects a compact mixed probe. A Hermitian surrogate retains
most of this optimized response and supports experimentally readable finite-window
drive/readout protocols, so the paper offers a controlled, model-specific route from embedded
Jordan geometry to probe design and readout construction rather than a general
theory of non-Markovian exceptional points.