Quantifiers and parameter gap in the ADM smallness hypothesis

In mass-one Radu ADM data with $J<1$, for any supplied positive threshold function $j(A)$, the quantitative application predicate at fixed $A$ is ${\bar{A}}_{\mathrm{A}\mathrm{D}\mathrm{M}}\le A$ and $J<j(A)$. A theorem asserting existence of a suitable threshold uses the order: first fix $A$, then choose a positive threshold uniformly for data with that bound. The geometric bound implies $A\ge 1$ and $J\le 1-A^{-1}$. Hence it is a simultaneous geometric and angular-momentum condition, not the statement that a universal numerical angular tolerance works for every geometry. These algebraic consequences do not prove a Kerr conclusion. Any Kerr rigidity implication would require a separately supplied theorem under its complete hypotheses, including GR/SBS and maximality; no rigidity theorem is an input to this lemma.

For these comparisons, the horizon scalar is $H={\sup }_{p\in S_0}\mathrm{\mid }{g}_p(T_p,T_p)\mathrm{\mid }$ on the supplied bifurcation sphere. When its own defining Ernst/curvature data are supplied, $\mathcal{W}$ denotes the smooth weighted Mars–Simon tensor. A Kerr conclusion means an isometry of the exterior to the subextremal Kerr exterior with parameters $M=1$ and $a=J$, using the target definition's stated countable-choice convention. No bound for $H$ or $\mathcal{W}$ is asserted here.