Connection between galaxy morphology and dark-matter halo structure I: a running threshold for thin discs and size predictors from the dark sector

We study the connection between galaxy morphology and host dark matter (DM) halo structure using cosmological simulations. Introducing a new kinematic decomposition scheme, we robustly separate thin and thick discs and measure halo properties, including cosmic web locations, internal structures, and assembly histories. In the TNG50 simulation, we find that the orbital-circularity threshold for disc differentiation varies systematically with galaxy mass and redshift. Similarly, the energy threshold between stellar halos and inner galaxies depends on mass and redshift, minimizing at sub-Galactic halo mass where the circularity threshold approaches its peak. Revisiting galaxy size predictors, we show that disc sizes in TNG50 correlate with three structural parameters beyond virial mass and redshift: 1) a positive correlation with halo spin $\lambda$ across redshifts -- stronger than previously reported for zoom-in simulations but still weaker than the simple $r_{1/2}/R_{\rm vir} \propto \lambda$ scaling; 2) an anti-correlation with DM concentration $c$; 3) larger discs in more actively accreting haloes. Disc mass fraction is higher in rounder haloes and in cosmic knots and filaments, implying that disc development needs both stable halo conditions and continuous material supply. Our methodology is public and adaptable to other simulations.