The Apparent Velocity Gradient: From Train Windows to Galactic Arms — A Unified Observation under F = ∇(1/N)
Abstract
Everyone has observed it from a moving vehicle: nearby objects rush past in a blur, while distant mountains barely seem to move. This paper proposes that this everyday phenomenon — the apparent velocity gradient — is not merely a perceptual artifact but a direct, observable expression of a unified physical principle: F = ∇(1/N), where N is the layer density of the medium between observer and object. We demonstrate that the same mathematical structure governing the view from a train window also governs the rotation curves of spiral galaxies, the Hubble redshift of distant galaxies, and the compression logic of the human heart. The apparent velocity gradient is thus a universal signature of density-gradient dynamics, observable at every scale from the human eye to the cosmological horizon. Pre-registration chain: DOI 10.5281/zenodo.20539183