  It has recently been shown that the M theory lift of a IIA 4D BPS Calabi-Yau
black hole is a 5D BPS black hole spinning at the center of a Taub-NUT-flux
geometries, and a certain linear relation between 4D and 5D BPS partition
functions was accordingly proposed. In the present work we fortify and enrich
this proposal by showing that the M-theory lift of the general 4D multi-black
hole geometry are 5D black rings in a Taub-NUT-flux geometry.
