The Mechanical Earth: Differential Rotation as a Density-Gradient Clockwork under F = ∇(1/N)

Consider a mechanical clockwork in which inner gears spin faster than outer gears. Scale this model to planetary size: an inner core rotating at maximum speed, a mantle at intermediate speed, a crust at minimum speed — a 'Sliced Earth'. This paper proposes that Earth's observed differential rotation is a direct structural consequence of the principle F = ∇(1/N), where N is the layer density. Near the center, N is maximum and rotational velocity is maximum. The layers do not fly apart because N forms a continuous gradient — a seamless density field that acts as the medium of spacetime itself. We argue that spacetime curvature is the geometric expression of the N-gradient, and that the minimal design parameter of Earth is N: set N, and time, force, gravity, and Earth's architecture follow automatically. Observational support: Zhang et al. (2023, Nature Geoscience) confirmed inner-core super-rotation of 0.3–0.5 degrees/year. Chain DOI: 10.5281/zenodo.20539183