Exclusion of regular analytic multiple horizons

Apply the stationary normal-form input on a sufficiently far-out end. The canonical-slice calculation gives $h=(1+2m\mathrm{/}r)\delta +o_2(r^{-1})$ and $k=o_2(r^{-1})$. Apply the ADM calculation in [F1] to this same Cartesian end: its spatial remainder is $o_2(r^{-1})$ and its stronger $k=o_2(r^{-2})$ estimate verifies the stated ADM falloffs. That calculation yields $M_{\mathrm{A}\mathrm{D}\mathrm{M}}=m$, so the spatial expansion has exactly the ADM-mass coefficient required in (H1), as well as its differentiated decay. Cofinal end invariance keeps the same chronological sets, DOC and horizon. The section-deformation conclusion supplied in [F1] supplies an acausal section containing the canonical end and retaining its compact horizon cross-section; equality of chronological sets alone is not used to supply that section. Completeness and timelikeness are the supplied stationary hypotheses, while analyticity and vacuum are unchanged. Thus the exact geometric source hypotheses (H1)–(H3) hold in an allowed end representation.

For this conditional comparison, suppose both the local component data and the componentwise label correspondence required in [F2] are supplied, and consider local case (R). The correspondence gives a rotating component in the source terminology, so use the explicitly stated rotating geometric consequence in [F2]. It supplies global axisymmetry and connectedness for the regular analytic geometry, without an a priori connectedness assumption. On each component degenerate under the matched labels, apply the source’s area equality to its own angular-momentum scalar and supplied smooth compact spacelike cross-section. The arbitrary-scalar implication in the positive-area supplier then shows that this source angular momentum is nonzero. This use does not identify it with a separately normalized local Komar functional and does not require nonzero total angular momentum. The quantitative negative rod defects remain the narrow balance theorem’s stated mechanism in its own map class; here the global geometric consequence is imported directly, rather than treating a bare regular axisymmetric geometry as an already verified map-class input. Hence more than one horizon component is impossible in case (R).

Under the same local-component-data and label-correspondence premises in [F2], local case (N) says every component is non-rotating and nondegenerate. The correspondence makes every component non-rotating and nondegenerate in the source terminology as well. Apply the exact multi-component staticity/Schwarzschild consequence in [F2]. Its geometric source statement already includes the required maximal-slice and static-classification preparation; that substantial input is imported, not replaced by the local single-bifurcation-sphere statement. Its connected Schwarzschild horizon excludes more than one horizon component.

Under the same local-component-data and label-correspondence premises in [F2], local case (D) says all components are non-rotating and some is degenerate. The correspondence supplies a component that is both non-rotating and degenerate in the source terminology. The Main Theorem in [F2] excludes even one such source-labelled component, including in a possibly disconnected horizon. It therefore excludes this case entirely.

The partition lemma makes the local cases (R), (N), (D) exhaustive when the component data are supplied. Under the additional componentwise label correspondence in [F2], steps 2.1–2.3 apply the respective source conclusions, preserving the article’s case-split explanation, including mixed degeneracy. These auxiliary calculations establish neither the existence of local component data nor their correspondence with the source labels. For the unconditional conclusion under the Statement’s hypotheses, apply instead the full geometric connectedness input [F3]: step 1.1 supplies its exact source hypotheses on the same geometry and preserves its horizon. That imported conclusion directly excludes more than one component. No premise about a locally specified Killing generator or surface-gravity normalization, and no smooth nonanalytic result, is used in this application.