  We present experimental evidence of the generation of distinct types of
genuine multipartite entanglement between three degrees of freedom, two
internal and one external degree of freedom, within single-neutron quantum
systems. This is achieved via the development of new spin manipulation
apparatuses for neutron interferometry and the genuine multipartite
entanglement is detected via appropriately designed and optimized non-linear
witnesses. Via the applied criteria we reveal even finer properties of the type
of the genuine multipartite entanglement that is produced in the experiment. In
a subsequent analysis we show the extraordinarily high fidelity of genuine
tripartite entangled states, proving the outstanding level of controllability
of these systems.
