Paper 6: A Linearised Rotating Exterior Perturbation of the Four-Dimensional Scale Space Framework: Scale-Kerr to First Order in Angular Momentum

We derive the linearised rotating exterior perturbation of the four-dimensional scale space framework—the “scale-Kerr” solution—to first order in the angular momentum parameter a = J/Mc. The framework treats (x, y, z, s) as a four-dimensional configuration manifold, not a spacetime: s is the physical scale coordinate and time remains an external evolution parameter. The background geometry is AdS4 in Poincar´e coordinates (Paper 1 1) with Λ =−6/L². We write down the 4D linearised Einstein equations, establish the Lichnerowicz operator, impose Lorenz gauge and axisymmetry, and specialise to the rotating off-diagonal sector hϕs in cylindrical coordinates (ρ, ϕ, z, s). The resulting second-order PDE is solved exactly in Poincar´e coordinates ζ= Le^ (−s/L): the ζ-sector is a Bessel equation of order νB = sqrt2 (L+ 1) /L, yielding the separated exterior mode solution hϕs ∝ K1 (kρ) IνB (κζ). In the near zone (kρ≪1) this gives the near-zone asymptotic form of the rotating perturbation with exact scale-decay exponent λexact =−νB/L. The result exhibits off-diagonal gϕs coupling proportional to J/ρ, a proposed scale-ergosphere-analogue condition gss = 0, and exact recovery of the non-rotating solution as a→0. The separation constant k, which sets the spatial scale of the dragging field, is not fixed by the vacuum exterior equations alone and requires interior matching; the paper derives the mode structure and near-zone form, leaving full normalisation to future work. This resolves the linearised exterior rotating sector of the open problem identified in Paper 2 2, and provides the foundation for computing scale-dragging observables in neutron star and black hole binaries.