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/L^2. 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 = sqrt[2(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.

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.