  Coarse-grained (CG) models facilitate an efficient exploration of complex
systems by reducing the unnecessary degrees of freedom of the fine-grained (FG)
system while recapitulating major structural correlations. Unlike structural
properties, assessing dynamic properties in CG modeling is often unfeasible due
to the accelerated dynamics of the CG models, which allows for more efficient
structural sampling. Therefore, the ultimate goal of the present series of
articles is to establish a better correspondence between the FG and CG
dynamics. To assess and compare dynamical properties in the FG and the
corresponding CG models, we utilize the excess entropy scaling relationship.
For Paper I of this series, we provide evidence that the FG and the
corresponding CG counterpart follow the same universal scaling relationship. By
carefully reviewing and examining the literature, we develop a new theory to
calculate excess entropies for the FG and CG systems while accounting for
entropy representability. We demonstrate that the excess entropy scaling idea
can be readily applied to liquid water and methanol systems at both the FG and
CG resolutions. For both liquids, we reveal that the scaling exponents remain
unchanged from the coarse-graining process, indicating that the scaling
behavior is universal for the same underlying molecular systems. Combining this
finding with the concept of mapping entropy in CG models, we show that the
missing entropy plays an important role in accelerating the CG dynamics.
