β = (v/c)²: A Two-Level Kinematic Indicator of Galaxy State

Preprint version 2.0.0. We investigate the kinematic parameter β = (v/c)² as a two-level indicator of galaxy state. On a sample of ALFALFA α.40 × SDSS DR7 × galSpecExtra DR8 (N = 9724), we confirm that globally β anti-correlates with gas fraction (r = −0.347, p < 10⁻²⁷²) and positively correlates with stellar mass (r = +0.569, p < 10⁻³⁰⁰). After controlling for stellar mass, the correlation between β and f_gas reverses sign: r_partial = +0.312 (p < 10⁻²¹⁸, 95% CI [+0.296, +0.335]). The effect is present in all 15 mass bins (r = +0.191…+0.439) and is strongest for low- and intermediate-mass galaxies. Independent checks confirm the result: BTFR: β correlates strongly with baryonic mass (r = +0.957); Virgo cluster: β correlates positively with HI mass (r = +0.573, p < 10⁻¹¹); DESI Legacy Survey: red galaxies are truncated from below in β. Robustness is confirmed by bootstrap (1000 iterations), mass-error simulations (100 realizations), and randomization tests. We interpret β as a two-level indicator: globally it reflects mass and evolutionary stage; locally, after mass control, it reflects the ability of a galaxy to retain cold gas. The repository includes the compiled PDF, LaTeX source, figures, tables, and Python code for full reproduction.