The Smile of Cheshire Cat At High Density

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.