Paper 25: Scale-Extended Matter Sources and the 4D Poisson Equation

The Scale Space Framework (Paper 1) treats every gravitating body as a point source at a single scale address sM = ln(R/ℓ0), fixed by the outer radius R. This is inconsistent with Postulate 5 (scale extent postulate): every massive object has genuine, irreducible extent in the scale direction s. We resolve the inconsistency by replacing the point source Mδ^(3)(x)δ(s−sM) with a 4D matter density ρm(x,s) distributed over the body’s full scale extent [smin,smax] and solving the 4D Poisson equation. Three results follow within the kinematic weak-field approximation. First, the exterior potential recovers Φ ≈−GMeff/r in the far field, where the effective gravitational mass is Meff = F·Mtot and the shape factor 0 < F≤1 encodes the body’s internal scale structure via scale shielding: matter at depth s′ < sM contributes to the exterior field with weight e^(3(s′ −sM )/L) <1. Second, the effective scale curvature parameter Leff emerges as the mass-weighted harmonic mean of the local compactness L(r) = rc^2/[GM(r)] across the body, recovering Lbody when ∆s ≪L. Third, for compact objects where ∆s ∼L — neutron stars in particular — the internal mass distribution becomes the primary driver of F.
An EOS survey over five density profiles (ρc/¯ρ = 1–9) gives an EOS-robust suppression band F ≈0.68–0.85 (15–32% gravitational mass suppression), fully separated from the general relativity prediction F= 1. The Tolman VII profile admits the exact analytic result δs = 8/15 nats, F = 0.773. For bodies with L ≫∆s (Earth, white dwarfs), internal structure is irrelevant and existing results are unaffected.

Epistemic status of all results: Derived-conditional. All results are derived within the kinematic fixed-background weak-field framework of Paper 1. The dynamical self-consistency of the extended-source metric has not been verified against the 5D field equations (Papers 17–20). The corrections to the force law are model predictions, not yet confronted with observation.