Paper 8: Interior Matching for the Linearised Rotating Exterior Perturbation of the Four-Dimensional Scale Space Framework: Mode Selection, Amplitude Normalisation, and the Scale Separation of Binary Pulsars

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/ (ρc²) ), 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 hⁱnt 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/ (ρc²) is corrected by the factor 5/7, yielding − (5/7) ·2GJ/ (ρc²) 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.