Exact Density Profiles of 1D Quantum Fluids in the Thomas-Fermi Limit: Geometric Hierarchy to the Tonks-Girardeau Gas

We present a geometric framework for 1D quantum fluids across interaction regimes in the Thomas-Fermi limit. Based on the Linearization Principle via the $q$-logarithm, macroscopic density profiles form a discrete hierarchy: the ideal Bose gas ($q=1$), the mean-field Gross-Pitaevskii condensate ($q=-1$), and the strongly correlated Tonks-Girardeau gas ($q=-3$). We further derive a universal sound velocity scaling, $c\propto {\rho }^{(1-q)\mathrm{/}4}$, valid in the interacting regimes ($q\le -1$). This establishes a non-perturbative link between static geometry and dynamical excitations in many-body systems.