  Sunspots and active regions observed on the solar surface are widely believed
to be manifestations of compact predominantly-toroidal magnetic field
structures (``flux tubes") that emerge by magnetic buoyancy from the deeper
interior of the Sun. Much work has examined the evolution of such magnetic
structures, typically considering them as idealized isolated magnetic entities
and not as more realistic magnetic concentrations in a volume-filling
background magnetic field. Here, we report results that explore the buoyant
rise dynamics of magnetic concentrations in a volume-filling field in the full
three dimensions. Earlier 2.5D work in this series (arXiv:1805.08806,
arXiv:2101.03472, arXiv:2204.13078) established the remarkable fact that the
twist orientation of a flux concentration relative to the background field
affected it's likelihood to rise and emerge, regardless of whether the buoyant
rise took place in the absence or presence of convection. The contrasting
dynamics between structures with differing orientations leads to a selection
mechanism that reproduces characteristics of the ``solar hemispheric helicity
rule(s)" (SHHR) observations strikingly well. Here, we show that this
two-dimensional selection mechanism persists in the face of the added
complexity of three-dimensional dynamics. Arching of the magnetic structure in
the third dimension, as might be expected in the solar application, is
introduced. The role of tension force leading to this selection mechanism is
elucidated and subtle differences that arise due to the three-dimensional
geometry are discussed.
