SS-GEO: Logarithmic Scale as a Geometric Coordinate: Hyperbolic Structure and Finite-Profile Projection

Physical scale is represented by the dimensionless logarithmic coordinate u = ln(ℓ/ℓ₀). Under the explicit postulate that ordinary spatial intervals scale as eᵘ and that the scale direction has constant metric coefficient, the configuration metric is dΣ² = e^(2u)dx² + a²du². It is four-dimensional Riemannian hyperbolic space with invariant curvature radius a. We derive its connection, curvature, geodesics, combined scale-translation/spatial-dilation isometry, invariant measure and scalar Laplace–Beltrami operator. The decaying scalar Green function is an exact reciprocal function of hyperbolic distance and has the local 1/(4π²d²) singularity. A finite-profile projection theorem shows that this local scalar kernel gives a leading 1/R term when the two normalized scale profiles have continuous nonzero same-scale overlap. The result is geometric and kinematic. It does not determine Newton's constant, a universal gravitational source law, propagating gravitational degrees of freedom or phenomenological bounds. Those questions are assigned to the later dynamics, matter and phenomenology papers. The present successor also records the correction history of its Paper 01 predecessor: normalization of the scale coordinate, correction of the geodesic sign, and withdrawal of the earlier interpretation of a reduced one-sided scale kernel as fundamental gravity.