A Unified Empirical Timescale for Galaxy Dynamics from Kinematic Scale Competition

We define a unified, empirically constructed dynamical observable τ, expressed as a signed characteristic timescale (in Myr), derived from two structural scales extracted directly from kinematic data: a dynamical coherence length ξ_GL and a baryonic confinement scale r_core. These scales are obtained consistently from TDGL-inspired fits to galaxy rotation curves and velocity-dispersion profiles, without imposing morphology priors or assuming a specific microphysical mechanism. Using 165 rotation-supported galaxies from SPARC and 50 pressure-supported ellipticals from ATLAS³D with resolved kinematics, we measure τ across a broad range of galaxy types. We find that τ exhibits a strongly bimodal distribution: rotation-supported galaxies overwhelmingly occupy the positive regime (97.0% with τ > 0) while pressure-supported ellipticals predominantly occupy the negative regime (96.0% τ < 0) with No rotation-supported galaxy exhibits τ < 0; instead, a small boundary population lies near τ ≈ 0. Lenticular (S0) galaxies cluster narrowly around this boundary. A Kolmogorov–Smirnov test rejects the hypothesis that rotation-supported and pressure-supported systems are drawn from the same parent τ distribution at p < 10⁻¹⁴. We further show that τ correlates strongly with long-term baryonic evolution, decreasing with stellar population age and increasing with gas fraction, while remaining independent of instantaneous star-formation rate and environmental metrics. These results establish τ as a robust, empirically measurable structural indicator of galaxy dynamical state, directly derived from kinematics. The observed bimodality and the sharp structural boundary near τ ≈ 0 are consistent with an interpretive framework in which galaxy-scale dynamics reflect competition between coherence and confinement. Subsequent papers examine τ's orthogonality to instantaneous star formation and extend the analysis deeper into the pressure-supported regime.