We propose that physical scale — the logarithm of apparent object size — constitutes a genuine spatial coordinate s alongside the familiar three spatial dimensions (x,y,z). A point (x,y,z,s) has a spatial location and a scale address. We show that the requirement of exponential spatial scaling with s-position uniquely determines the metric to be dσ^2 = e^(2s/L) (dx^2 + dy^2 + dz^2) + α^2 ds^2,where Lis a curvature radius (nats) and α= √Lmetres/nat. This metric is precisely anti-de Sitter space AdS4 in Poincare coordinates. We derive the geodesic equations, the four-dimensional Laplace–Beltrami field equation and its point-mass solution, and the action with its Noether conservation laws. The Newtonian weak-field limit is derived rigorously: the background geodesic coupling term is second order in the slow-motion limit and drops out, and Newton’s inverse-square law is recovered with no free parameters when L= Rc^2/GM. The background geometry is AdS4 with a universal curvature radius L; in the weak-field single-body limit, matching to Newton’s law yields the effective identification Leff = Rc^2/GM for a body of mass M and radius R, playing the role of a solution parameter analogous to the Schwarzschild radius. Evaluated at the event horizon this gives Leff = 2 nats universally for every black hole, independently of mass. The directly derived prediction distinguishing the framework from GR is a scale-dependent correction to the force law F(r,s) = GMm/r^2 · e^(−3∆s/L), measurable in principle near compact objects. Additional predictions — an asymmetric two-body force for bodies of different compactness, and five gravitational wave polarisation states including a pure scale mode invisible to current laser interferometric detectors — are derived in a companion paper (in preparation). This work proposes a geometric kinematic model of scale; a full dynamical gravity theory in which matter sources the scale geometry is left to future work.

Manuscript prepared for submission. This paper was developed in dialogue with Claude.ai (Anthropic) in March 2026. The founding hypothesis was first conceived by the author in 1977. Related publications: Palmer, D.G. (2025a), Advances in Theoretical & Computational Physics, 8(4), 01-09; Palmer, D.G. (2025b), Space Science Journal, 2(4), 01-05.