Conditional horizon recovery and Kerr implications

Apply [F1] using (A) to the actual $C={\mathrm{\Sigma }}_0\cap E$. It gives the positive original lapse and stationary product on this geometric section, its compact physical truncations and its adapted topological chart. It does not identify the original specified side of $j$ with $C$. For each selected $U_R$ in (B), two-sided access gives $E\subset I_M^{-}(U_R)\cap I_M^{+}(U_R)$. Conversely $U_R\subset M^{\mathrm{e}\mathrm{n}\mathrm{d}}$ implies $I_M^{-}(U_R)\cap I_M^{+}(U_R)\subset E$ by the defining chronological relations. Thus each such tail has the same actual DOC. This argument does not assert the separate ambient chronological sets or their boundaries are equal.

Apply [F2] under the regular stationary-vacuum hypotheses in [A1]. It supplies an existing ADM limit on this original physical end, equal to its coefficient $m>0$. By [F4] take the normalized potential on $E$ and a neighborhood of $S$, with the fixed orientation and contraction convention. No change of the original section or identification of independent charts is used.

Set $\widehat{g}=m^{-2}g$ and $\widehat{T}=mT$. On the same physical end use $\widehat{t}=t\mathrm{/}m$, $\widehat{x}=x\mathrm{/}m$ and $\widehat{r}=\mathrm{\mid }\widehat{x}\mathrm{\mid }$. Both coordinate differentials acquire a factor $m$, so the components of $\widehat{g}$ in this chart equal the original component functions evaluated at $x=m\widehat{x}$. The chain rule in [F6] gives ${\widehat{g}}_{00}=-1+2\mathrm{/}\widehat{r}+O_6({\widehat{r}}^{-2})$, ${\widehat{g}}_{0i}=O_6({\widehat{r}}^{-2})$ and ${\widehat{g}}_{ij}={\delta }_{ij}+O_6({\widehat{r}}^{-1})$, and $\widehat{T}={\mathrm{\partial }}_{\widehat{t}}$. By [F3], all regular causal and stationary-orbit conditions, the same sphere and nonexpanding branches, nontriviality on $S$, and maximality of the actual original section are preserved. The original topological identification is kept separately; it is not rescaled into the physical coordinates. Access hypothesis (B) is preserved because homothety does not change timelike curves and a hatted physical tail is an original tail of radius multiplied by $m$.

Match the section and potential domains for the local input without a chart-agreement assumption. Use the adapted topological identification from [F1], writing its radius as $s$, with $C=\{s>1\}$ and $S=\{s=1\}$. The potential neighborhood in [F4] contains a full inner collar $1-\eta <s\le 1$ of compact $S$ for some $0<\eta <1\mathrm{/}2$. Restrict the original section to ${\mathrm{\Sigma }}_{\mathrm{A}\mathrm{F}}=\{s>1-\eta \}$, so it consists of $C$, $S$ and that inner collar and lies in the open potential domain $E\cup O$. It is an embedded spacelike hypersurface without manifold boundary. If $y$ is the adapted topological chart, the radial map $y\mapsto (\frac{1}{2}+(s-1+\eta )\mathrm{/}(2\eta ))y\mathrm{/}s$ identifies this restricted section smoothly with $\{\mathrm{\mid }z\mathrm{\mid }>1\mathrm{/}2\}$ and sends $S$ to $\mathrm{\mid }z\mathrm{\mid }=1$. It does not claim agreement with $x$ on the physical end. The hypersurface itself still agrees with physical $t=0$ there, its intersection with $E$ is exactly $C$, and every complete stationary orbit in $E$ meets it at its point on $C$. Its lapse on that intersection is positive by step 1.1. Thus the section, transversality and normalized-potential-domain requirements of the exact AF/SBS interfaces in [F7] are met.

Let $h_{ij}$ denote the original section metric in physical end coordinates. The same section has hatted coordinate components ${\widehat{h}}_{ij}(\widehat{x})=h_{ij}(m\widehat{x})$. In the ADM integral of [F2], differentiation contributes $m$, the Euclidean sphere area contributes $m^{-2}$ relative to the original sphere of radius $m\widehat{R}$, and the radial unit vector is unchanged. Therefore its hatted ADM limit is $m^{-1}$ times the original limit $m$, hence one. This proves the exact mass-one hypothesis on the unchanged maximal section using only a linear rescaling of its physical chart. It invokes neither a compatible global auxiliary chart nor an end-normalized replacement time slice.

The hatted data now satisfy the full regular mass-one, actual-original-section maximality and nontriviality hypotheses of [F4]–[F5]. Apply the local-field and global one-form results to choose their common pole and one invariant smooth primitive, and use that same choice in the regular electrovacuum construction. The original normalized potential remains the hatted normalized potential by [F3]. These are exactly the one-family and pole inputs of [F5]; invoke recovery to obtain $-4{\widehat{\mathcal{F}}}^2=(1-\sigma {)}^4$ on $S$ and the strict reciprocal inequality at that pole. Only the supplied smooth decaying primitive is used, with no assertion that it equals an exact radial integral.

By [F3], ${\widehat{\mathcal{F}}}^2=m^2{\mathcal{F}}^2$ and the normalized scalar is the same $\sigma$. The common zero of $\widehat{T}$ and the selected circle field is also a zero of $T$, since $\widehat{T}=mT$ with $m>0$. Substitute the contraction scaling in step 4.1 to get the asserted original-scale identity and unchanged strict reciprocal inequality at the same pole.

The original $O_6$ metric bounds imply the AF interface's undifferentiated end bounds with the positive mass from step 1.2; $T={\mathrm{\partial }}_t$ and all required orbit conditions hold by regularity and step 2.2. The ambient DOC, its bifurcation sphere and its smooth transverse nonexpanding end-boundaries were not changed. Tangency and $T{\mathrm{\mid }}_S\equiv ̸0$ give the remaining SBS requirements. The potential is normalized to $1$ on this same physical $t=0$ end, with the contraction convention in [F7]. Step 5.1 supplies both technical sphere conditions. Apply the exact imported local criterion: every point of $E$ has the stated local isometry into a Kerr exterior with mass $m$ and $0<a<m$. No global bijection is inferred from these local charts.

For the separate global implication, assume precisely the source-native realization specified in the Statement. Its identified DOC is the unchanged physical $E$, and all GR, SBS and PK predicates in [F8], including the source mass, spin, partial-Cauchy, separate chronological-section, regularity and tensor-bound conditions, are supplied for that realization. Apply [F8] to obtain its single isometry onto the full Kerr DOC of mass $m$ and angular momentum magnitude $J$. Composing with the stipulated identification of its DOC with the original $E$ gives the asserted global isometry and $a=J\mathrm{/}m$. These hypotheses contain no Kerr isometry premise, and step 6.1 has not been used to manufacture missing global hypotheses. This proves the conditional endpoint at its stated scopes.