  Baryons in finite nuclei, nuclear matter and dense compact-star matter are
described in terms of Cheshire Cat for QCD. A potential conceptual link,
admittedly short in mathematical rigor, between their manifestations is made by
what's called Cheshire Cat Principle. Put in terms taken ``dual" to QCD
variables, going to very high density exposes quantum Hall droplets -- or
pancakes -- at which the dilaton-limit fixed point with $g_A\to 1$, $f_\pi\to
f_\chi$ -- where $f_\pi$ and $f_\chi$ are respectively the pion and dilaton
decay constants -- and the baryon parity-doubling are reached. This scenario
suggests a thus-far totally unexplored structure of dual baryonic matter at
high density which does neither require nor rule out (rapid) first-order phase
transitions from hadrons to quarks in the core of compact stars on the verge of
gravitational collapse.
