Joint Relational Certification of Articulation and Entanglement from Two Global Observables

Which properties of a quantum state are determined by two global observables and the outcomes of specified measurement sequences? On a calibrated four-level system, the marked observables determine two complementary qubit factor algebras without prior local qubit labels. For each measurement protocol, the paper asks how much entanglement every state compatible with its data must possess. For a single sequential measurement, retaining the complete outcome record gives sharp lower bounds on concurrence and negativity. A second measurement can be recorded either as a parity or as four resolved outcomes. The parity record gives a rank-five measurement space with explicit attainable bounds. Retaining individual population differences within a parity class can strengthen those bounds without further measurement runs. In an explicit example, the guaranteed concurrence rises from zero to one half. The paper identifies which population difference produces that gain and when the resulting certificate is exact. Further results include an exact rank formula for pulse sampling, a four-duration minimum within the specified pulse family, and explicit states for which rank-ten and rank-fourteen protocols certify entanglement in opposite directions. The manuscript gives finite-sample and calibration bounds and distinguishes reconstruction on the assumed system from identification of a physical realisation solely from probabilities. The accompanying archive contains fifteen Python verification scripts, run logs and a SHA-256 manifest. All numerical data are synthetic.