One number governs entanglement, positivity, and sensing in qubit networks

For any closed network of N qubits sharing a single excitation, closed-form propagators are derived for any subsystem of arbitrary size. Remarkably, one scalar amplitude determines whether each subsystem's dynamics preserves positivity, how entanglement is distributed, and how much Fisher information is available for estimating global parameters. Complete positivity and ordinary positivity always coincide and reduce to a simple condition: the propagator contracts the subsystem toward its fixed point. The Fisher information splits cleanly into state and process contributions, with the process term growing without bound while the state term saturates.