  Massive stars are linked with diverse astronomical processes and objects
including star formation, supernovae and their remnants, cosmic rays,
interstellar media, and galaxy evolution. Understanding their properties is of
primary importance for modern astronomy, and finding simple rules that
characterize them is especially useful. However, theoretical simulations have
not yet realized such relations, instead finding that the late evolutionary
phases are significantly affected by a complicated interplay between nuclear
reactions, chemical mixing, and neutrino radiation, leading to non-monotonic
initial mass dependencies of the iron core mass and the compactness parameter.
We conduct a set of stellar evolution simulations, in which evolutions of He
star models are followed until their central densities uniformly reach
10$^{10}$ g cm$^{-3}$, and analyze their final structures as well as their
evolutionary properties including the lifetime, surface radius change, and
presumable fates after core collapse. Based on the homogeneous data set, we
have found that monotonicity is inherent in the cores of massive stars. We show
that not only the density, entropy, and chemical distributions, but also their
lifetimes and explosion properties such as the proto-neutron-star mass and the
explosion energy can be simultaneously ordered into a monotonic sequence. This
monotonicity can be regarded as an empirical principle that characterizes the
cores of massive stars.
