The Rail Ledger: Scale-Covariant Log-Domain Dynamics with Saturation Horizons

We present a scale-covariant framework in which all quantities are indexed on the Rail coordinate \(\Rail = \ln(r/r_0)\). The log-contracting nozzle model supplies the reactive closure: continuity and isentropic flow yield \(M^2(\Rail) \approx M_0^2 e^{4\beta \Rail}\) (regime \(M \gg 1\), \(\beta\Rail \gg 1\)), while pressure saturates at a dimensionless floor \(\delta_p \sim 10^{-3}\) (Zayyay floor \(\Xi \approx 0.0041\)). Scale covariance suggests a universal saturation structure across domains, though control parameters differ. Physical time is defined operationally as \(\Delta\tau \propto \Pi^{-1}\) with thermodynamic entropy precipitation \(\Pi = \dd S/\dd\magpress\). The framework provides a unifying description for scale-invariant systems with boundary saturation and is falsifiable via laboratory nozzle experiments and GWTC statistics.