From Exact Stationary Kerr--Newman Inner-Horizon Geometry to a Compact APS Collar

This preprint establishes a geometric-to-analytic bridge from the exact stationary subextremal Kerr--Newman inner-horizon geometry to a compact Atiyah--Patodi--Singer collar model for a specified intrinsic elliptic reduction. The paper works in the nondegenerate Kerr--Newman parameter regime\[0<a^2+Q^2<M^2,\]so that the inner horizon \(r=r_-\) is simple and the Schwarzschild degeneracy \(a=Q=0\) is excluded. On a spacelike horizon-penetrating Eddington--Finkelstein time slice, the induced Riemannian metric extends smoothly to the inner-horizon two-sphere. Passing to Gaussian collar coordinates yields a smooth compact control collar and its intrinsic spin Dirac operator. The main construction proves that, after controlled productization, the productized Dirac operator has the boundary normal form\[Dprod=σ(∂_u+B)+R,\]where \(B\) is the tangential boundary spin Dirac operator on the horizon metric and \(R\) is an order-zero term. In the exact untwisted intrinsic spin-Dirac model treated here, \(B\) is identified with the intrinsic spin Dirac operator of the horizon two-sphere. Since every metric on \(S^2\) is conformal to the round metric and the two-dimensional spin Dirac kernel is conformally invariant, \(B\) is invertible. Compact subextremal parameter families away from \(a=Q=0\) therefore admit a uniform spectral gap, continuous APS projections, and a uniform Weyl shell bound. Thus the paper supplies the local exact-model bridge connecting Kerr--Newman inner-horizon geometry with the analytic collar framework of Paper~A and the boundary spectral framework of Paper~B. Paper~A proves APS/Fredholmness and index stability on compact collars. Paper~B proves dyadic shell trace-norm cutoff removal under an independent shell-damping hypothesis. The present Paper~C verifies, for the exact stationary intrinsic model, the compact collar, tangential operator, spectral-gap, APS-projection, and Weyl-shell inputs needed to connect those analytic results to the stationary Kerr--Newman inner-horizon setting. The result is local and model-specific. It does not claim a theorem on full nonlinear Einstein--Maxwell evolution, nonlinear stability of the Cauchy horizon, mass-inflation resolution, strong cosmic censorship, black-hole evaporation, Page-curve behavior, observational comparison, or a full black-hole information theorem.