The actual geometric constant in the small-ADM theorem

For the Radu ADM source choose its chart data $A_1,R_1,{\epsilon }_0$, with $R_1\ge \max (R_0,A_1^4)$ and ${\epsilon }_0>0$. Require $A_1^{-1}\mathrm{\mid }v{\mathrm{\mid }}^2\le h_{ij}{v}^i{v}^j\le A_1\mathrm{\mid }v{\mathrm{\mid }}^2$ on ${\mathrm{\Phi }}_0(E_{3\mathrm{/}4})$, and differentiated coefficients of the three remainders in its end expansion bounded by $A_1$ (at order $m\le 6$ their weights are $r^{m+1}$, $r^{m+2}$ and $r^{m+3}$ for spatial, temporal and mixed remainders respectively). Supply the source extension chart ${\mathrm{\Phi }}_1:(-{\epsilon }_0,{\epsilon }_0)\times E_{1-{\epsilon }_0}\to \tilde{M}$, built by interpolation between the slice unit normal and $T$, agreeing with the stationary chart for $r>2R_1$, and obeying ${\sum }_{i=1}^3\mathrm{\mid }{g}_{0i}\mathrm{\mid }+\mathrm{\mid }{g}_{00}+1\mathrm{\mid }\le A_1\mathrm{/}(R_1+r)\mathrm{.}$ Under $J<M_0^2$ set ${\bar{A}}_{\mathrm{A}\mathrm{D}\mathrm{M}}=\max \{R_1,{\epsilon }_0^{-1},(M_0^2-J{)}^{-1},{\sup }_{\tilde{M}}{\sum }_{m=0}^6({\sum }_{a,b=0}^3\mathrm{\mid }{\mathrm{\partial }}^m{g}_{ab}\mathrm{\mid }+{\sum }_{a=0}^3\mathrm{\mid }{\mathrm{\partial }}^m{T}^a\mathrm{\mid })\}\mathrm{.}$ Here all spacetime coordinate derivatives and components are summed, and the supremum is required finite. The source's quantitative condition is a bound on this actual normalized chart package. It is not a claim that small $J$ alone controls curvature, chart radius, horizon collar width or the distance to extremality.

Every term is computed from the specified source chart, metric, field and positive parameters. The strict parameter inequality makes the inverse gap finite, and the explicit supremum condition makes the last term finite. A maximum of this finite list is a unique finite positive number. The ancillary chart and remainder conditions remain part of the package; merely writing down a maximum on an unrelated chart would not instantiate the source theorem. The chart choices and all their bounds must be supplied for each application.