SS-GEO: Logarithmic Scale as a Geometric Coordinate: Exact-Distance Projection and Radial Scale Response

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. We extend the projection calculation to the exact hyperbolic distance and Green function for compactly supported profiles, and distinguish fixed midpoint physical separation from fixed coordinate separation. Their leading 1/R coefficients are generally different, so a measurement convention is part of the statement of the projected scalar interaction. We also compare the scale coordinate to holographic RG and entanglement renormalization geometries without identifying their physical flow laws. On the fixed H4 background, an auxiliary scalar probe admits a radial Hamilton Jacobi response flow with an exact massless Fourier solution. A separate state distance construction gives a conditional, operational criterion for matching an entanglement-renormalization scale path to SS proper distance. 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.