An observer-centered scale–motion relativity principle: Causal transformations and exact second-order cosmography

We propose a scale–motion relativity principle to address the parameter-dependence of standard cosmology, offering an alternative foundation where an observer's position, motion, and measurement scale change interactively under observer transformations. The core physical intuition arises from defining motion as spatial change per unit time and deriving it via central projection geometry. This projection scales apparent transverse sizes by a factor s = R0/(R0 + X), and restoring this reference scale uniquely fixes the radial comoving distance to χ = X/s. Embedding this scale-governed distance into a transparent, spatially flat FLRW metric allows pure geometric scaling to determine the expansion rate, proper time, and light propagation without invoking dark matter fractions or a cosmological constant. Explaining late-time acceleration through this metric, the derived luminosity distance achieves an exact analytical match with flat ΛCDM at Ωm = 1/3 through second order in redshift without any fitted shape parameters. The model yields a distinct, testable deviation at the third order via a fixed cubic term -z^3/12 in H0DL/c, providing definitive observational signatures that can be rigorously evaluated against standard cosmology using redshift drift, the jerk parameter, fast-radio-burst dispersion, and strong gravitational lensing.