  The dilaton-limit fixed point and the scaling properties of hadrons in the
close vicinity of the fixed point in dense baryonic matter uncovered in hidden
local symmetry implemented with spontaneously broken scale symmetry are shown
to reveal a surprisingly intricate interplay, hitherto unsuspected, between the
origin of the bulk of proton mass and the renormalization-group flow of the
\omega-nuclear interactions. This rends a theoretical support to the previous
(phenomenologically) observed correlation between the dropping nucleon mass and
the behavior of the \omega-nuclear interactions in dense matter described in
terms of half skyrmions that appear at a density denoted $n_{1/2}$ in skyrmion
crystals. The role of the \omega-meson degree of freedom in the source for
nucleon mass observed in this paper is highly reminiscent of its important role
in the skyrmion description of nucleon mass in hidden local symmetric theory.
One of the most notable novel results found in this paper is that the nucleon
mass in dense baryonic medium undergoes a drop roughly linear in density up to
a density (denoted $\tilde{n}$) slightly above nuclear matter density ($n_0$)
and then stays more or less constant up to the dilaton limit fixed point. The
possibility that we entertain is that $\tilde{n}$ coincides with or at least
close to $n_{1/2}$. We note that this feature can be economically captured by
the parity-doublet model for nucleons with the chiral-invariant mass $m_0\sim
(0.7-0.8) m_N$. It is found in one-loop renormalization-group analysis with the
Lagrangian adopted that while the \rho-NN coupling "runs" in density, the
\omega-NN coupling does not scale: it will scale at two-loop or higher-loop
order, but at a slower pace, so it is more appropriate to say it "walks" rather
than runs.
