Paper 6 of this series derived the linearised rotating exterior perturbation of the four-dimensional scale space framework — the scale-Kerr solution to first order in angular momentum — but left three items unresolved: the separation constant k (which determines the spatial scale of the off-diagonal field), the amplitude normalisation (stated heuristically as−2GJ/(ρc^2)), and the correct value of the scale separation ∆s for real binary pulsars. We resolve all three. First, we derive the interior scale-velocity distribution of a rigidly rotating neutron star: ˙s(r) = 2GM(r)/(rc) = ˙ssurf(r/R)^2 for a uniform sphere, with surface value ˙ssurf = 2GM/(Rc) and volume average (3/5) ˙ssurf. This distribution is the source for the interior off-diagonal field h^int ϕs. Second, we perform the interior–exterior matching at the stellar surface ρ= R. The matching condition selects the DC mode (k→0), corresponding to the slowly rotating limit in which the off-diagonal field has no spatial oscillation and decays as 1/ρ for ρ>R. This is the exact scale-space analogue of the Hartle (1967) slowly rotating star result in GR. Third, the DC-mode matching fixes the amplitude: the Paper 6 heuristic −2GJ/(ρc^2) is corrected by the factor 5/7, yielding −(5/7)·2GJ/(ρc^2) e^(λexact(s−sM)) for the exterior DC-mode field. This value is exact for a uniform-density sphere; realistic neutron star equations of state shift the factor within the range∼4/7–8/7, leaving the order of magnitude robust. The correction is derived from the l = 1 Hartle–Thorne Green’s function integral and is verified symbolically by SymPy. Fourth, we resolve the scale separation of binary pulsars. The scale coordinate s is the logarithm of a physical length, not an accumulated displacement; for two neutron stars with the same radius, ∆s = ln(MA/MB), confirming the Paper 2 estimate from a first-principles derivation that avoids the dimensional confusion between ˙ s (metres per second) and ds/dt (nats per second). For PSR J0737−3039, ∆s= ln(1.338/1.249) = 0.0688 nats, and all Paper 7 numerical results are confirmed.

The mathematical development in this paper was produced in dialogue with Claude.ai (Anthropic) in Spring 2026, directed by the author. Use of AI assistance is acknowledged in accordance with standard scholarly practice.