Paper 15: The Scale-Integration Reduction: Effective Four-Dimensional Einstein Equations from the Five-Dimensional Scale-Space Field Theory

Paper 14 of this series derived Newton’s law from the linearised 5D Einstein equations with a point mass source, establishing the kinematic Poisson equation of Paper 1 as a dynamical theorem. It identified a residual open problem: the coupling constant κ5 = 2G/[(L + 2)c^2L] derived in that paper depends on the compactness parameter L = Rc^2/(GM) of the source body, making κ5 body-dependent rather than universal. The present paper performs the s-integration reduction of the 5D field equations, showing how effective 4D Einstein equations emerge and how the body-dependence is resolved.

We prove four results. First, the spatial and temporal projections of the 5D equations onto a fixed scale surface s = s0 reduce exactly to the standard 4D Einstein equations in the weak-field limit, recovering GR at a single scale. Second, the matter-weighted s-integration — averaging the 5D equations with the physical volume element W(s) = e^(3s/L) over the support of the matter source — produces effective 4D equations with coupling κeff depending on L. Third, the universal background stress-energy T^(5) bg,tt identified in Paper 11, together with the identity Ls˙ = 2c from Paper 12, provides the normalisation condition that fixes κeff independently of any particular body: κeff = 8πG/c^4 = κ4. Fourth, the effective 4D cosmological constant is Λeff = 3/(c^2L^2), which in the physical limit L → Luniverse ∼ 10^60 nats gives Λeff ∼ 10^(−120) in Planck units — consistent in order of magnitude with the observed value. General relativity is thereby recovered as the effective 4D theory of the scale-space framework, valid when scale structure is averaged over a matter source’s characteristic scale range.