Paper 1: A Geometric Model of Logarithmic Scale as an Extra Dimension: Anti-de Sitter Geometry, Emergent Gravity, and RG Flow

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 Poincar´e 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.