Scale-Gauged Cosmological Observation Theory \vspace{0.3cm

Abstract: We establish a cosmological observation theory centered on the duality between the scale gauge (denoted a(t)) and the internal observational metric (c-lock, denoted R(t)). Setting a(t) = R(t)^{-1} and \kappa(t) := a/a = -R/R, cosmological redshift and time dilation unify as Mellin dilation on the energy axis: 1+z = a(t_0)/a(t_e) = R(t_e)/R(t_0), \nu_0 = \nu_e/(1+z), \Delta t_0 = (1+z)\Delta t_e. All readouts are aligned on the ``mother scale'': where \rho is the relative density of sta