Smooth rigidity from a quantitatively small horizon field

In the four-dimensional classical vacuum Einstein model with zero cosmological constant, impose the exact AIK quantitative stationary package, including GR, SBS and the asymptotically flat maximal section. Once $A$ is fixed there is ${ϵ}_{\ast }(A)>0$ such that ${\sup }_{S_0}\mathrm{\mid }g(T,T)\mathrm{\mid }<{ϵ}_{\ast }^2$ implies that the exterior $E$ is isometric to a subextremal Kerr exterior, with the source stationary and axial symmetries. Here the horizon bound is equivalently ${\sup }_{S_0}\mathrm{\mid }T{\mathrm{\mid }}_h<{ϵ}_{\ast }$, using the induced positive metric on $S_0$. The source SBS hypothesis retains $T{\mathrm{\mid }}_{S_0}\equiv ̸0$. The non-timelike subset of the maximal section is then contained in the local rotational-rigidity neighborhood. Smallness of ADM angular momentum is not substituted for this horizon-field condition.

Import AIK Theorem 1.1 in its complete GR/SBS/maximal and quantitative setting. The source condition is $\mathrm{\parallel }g(T,T){\mathrm{\parallel }}_{L^{\mathrm{\infty }}(S_0)}<{ϵ}^2$, with $ϵ$ sufficiently small depending only on $A$ from (1.10). The source abstract identifies the isometry conclusion as one about the domains of outer communications. This contract imports the full rigidity theorem, including the global extension and classification it invokes; it does not assert a locally reconstructed rigidity proof.

The exact AIK package retains its own stronger coordinate decay, its GR/SBS assumptions and its nonzero-horizon-field clause; it is separate from a generic black-hole regularity package. AIK uses the non-strict set $g(T,T)\ge 0$ in Proposition 1.3, while some terminology calls only $g(T,T)>0$ the ergoregion; the explicit inequality here avoids that convention ambiguity. No source defect or independent mathematical acceptance is asserted.