Version v2 is a substantial technical and expository revision of v1. The principal mathematical statement and the deliberately local scope of the paper are retained, but the Fredholm proof, the deformation hypotheses, and the varying-domain argumenthave been reformulated to make every analytic input explicit. The main changes from v1 are: Precise APS terminology. The condition \ (₀ (B) =0\) is now consistently called the Atiyah-Patodi-Singer boundary condition. It is identified more precisely as the \ (a=0\) member of the generalized APS family in the terminology of Bär-Ballmann. Elliptic estimate for the actual operator. Proposition 4. 4 has been rewritten as an APS ellipticity and graph-norm estimate for the Dirac-type operator \ (D\) itself: \ \|\|₇℉ C (\|D\|₋ℂ+\|\|₋ℂ). \ In v1, the corresponding discussion was formulated through an auxiliaryproduct-type operator \ (D₀= (ᵤ+B) \). Direct Fredholm proof. The proof of the APS Fredholm theorem now invokes standard first-order elliptic boundary-value theory directly for \ (D\) on the compact manifold with boundary. The earlier compact-perturbation decomposition\ (D=D₀+R\) has been removed from the proof. This avoids treating the local boundary product expression as though it defined a global product operator on the entire collar. Independent closedness argument. Closedness of the APS realization is now proved separately and directly from the elliptic estimate and the closedness of the APS domain in \ (H¹\). Local product structure stated explicitly. A new clarification records that only the product structure in a boundary collar is used; no globally defined operator \ ( (ᵤ+B) \) on all of \ (C\) is introduced. Stronger and more transparent deformation hypotheses. Condition (D4) now explicitly assumes operator-norm continuity of the APS projections both on \ (L²\) and on \ (H^1/2\). It is stated as an independent hypothesis, not as aconsequence of (D1) - (D3), continuity of \ (Bₛ\), or the spectral gap alone. A concrete sufficient class is also recorded: continuous or smooth families of self-adjoint first-order elliptic tangential operators on the fixed boundary, with a uniform open spectral gap. Proposition 7. 15 of Booß-Bavnbek-Lesch-Zhu then gives the required continuity on \ (Hʳ\) for\ (|r| 1/2\). Uniform spectral gap stated as a hypothesis. The gap condition is now given directly by the existence of \ (>0\) such that \ ( (-, ) (Bₛ) =\) for every \ (s0, 1\). This replaces the stronger inference made in v1 from pointwise invertibility and family continuity. Stable spectral-cut lemma recalibrated. The lemma now proves the fixed-parameter functional-calculus identity \ Pₛ=12 (Id+sgn (Bₛ) ) \ and Sobolev boundedness of each \ (Pₛ\), while explicitly stating that continuity of the family \ (s Pₛ\) is supplied by (D4) and is not derived from the lemma. Completed local trivialization of the APS domains. The construction using the close-projections intertwiner \ (Wₛ\), a bounded trace right inverse \ (E\), and the maps \ (Uₛ\) and \ (Vₛ\) is written out explicitly, including the inverseidentities \ (VₛUₛ=Id\) and \ (UₛVₛ=Id\). Fixed-domain Fredholm family. Proposition 6. 7 now defines \ (Tₛ=Dₛ^APSUₛ\) on a fixed Banach domain and proves that it is a norm-continuous family of bounded Fredholm operators with\ (ind (Tₛ) =ind (Dₛ^APS) \). Sharper index-stability proof. The index theorem is now obtained by applying the standard local constancy of the Fredholm index to the norm-continuous fixed-domain family \ (Tₛ\). This replaces the less explicit graph-topologyformulation used in v1. Scope and exposition aligned with the hypotheses. The abstract, introduction, theorem statements, and final scope section now state explicitly that Sobolev-scale projector continuity is assumed. The introduction additionallyexplains the specific Kerr-Newman motivation: in horizon-penetrating coordinates the Dirac equation separates into radial and angular first-order systems, with controlled Cauchy-horizon asymptotics. The paper carefully states that the compactcollar retains only this local radial-tangential pattern after renormalization and is not presented as a direct derivation from the Lorentzian spacetime. It continues to make no claim about global nonlinear Kerr-Newman dynamics, mass inflation, strong cosmic censorship, evaporation, or observational consequences. Editorial cleanup. Cross-references and theorem labels have been corrected, the bibliography has been streamlined, and the source has been recompiled with all references resolved and no LaTeX warnings.
Tosho Lazarov Karadzhov (Thu,) studied this question.