An APS Fredholm Theorem on a Renormalized Kerr--Newman Inner-Horizon Collar

This preprint proves a local analytic Atiyah--Patodi--Singer/Fredholm theorem for a compact renormalized inner-horizon collar motivated by the subextremal Kerr--Newman geometry. The paper considers a compact collar model \(\mathcal C\) with product-type structure near the boundary and a Dirac-type operator of the form \[ D=\sigma(\partial_u+B)+R, \] where \(B\) is a self-adjoint elliptic tangential boundary operator and \(R\) is an order-zero perturbation. Under explicit compactness, ellipticity, product-structure, and spectral-gap hypotheses, the Atiyah--Patodi--Singer realization \[ D^{\mathrm{APS}}: H^1_{\mathrm{APS}}(\mathcal C;S)\to L^2(\mathcal C;S) \] is proved to be Fredholm. For admissible gap-preserving deformation families \(D_s\), the corresponding APS index \[ \operatorname{ind}(D_s^{\mathrm{APS}}) \] is shown to be locally constant. The Kerr--Newman geometry is used only as geometric motivation for the inner-horizon collar model. The results are local compact-collar analytic theorems and do not claim a theorem on the full nonlinear Kerr--Newman interior, mass inflation, strong cosmic censorship, black-hole evaporation, observational comparison, or the full black-hole information paradox.