Radu nontrapping global regularity package

The Radu NT global regularity package is the GR data of arXiv:2609.23872v1 §1.1: a connected oriented time-oriented smooth four-spacetime with $\mathrm{Ric}(g)=0$, a smooth stationary Killing field $T$, and an embedded spacelike acausal edgeless partial Cauchy hypersurface ${\mathrm{\Sigma }}^0$. Supply a diffeomorphism ${\mathrm{\Phi }}_0:E_{1\mathrm{/}2}\to {\mathrm{\Sigma }}^0$, that is, a bijective smooth map with smooth inverse, where $E_a=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>a\}$. Put ${\mathrm{\Sigma }}_1={\mathrm{\Phi }}_0(E_1)$. There is a smooth end chart $\mathrm{\Psi }:\mathbb{R}\times E_{R_0}\to M^{\mathrm{e}\mathrm{n}\mathrm{d}}$, with $T={\mathrm{\Psi }}_{\ast }{\mathrm{\partial }}_t$, in which ${\mathrm{\Sigma }}^0\cap M^{\mathrm{e}\mathrm{n}\mathrm{d}}=\mathrm{\Psi }(\{0\}\times E_{R_0})$. Thus the end chart and ${\mathrm{\Phi }}_0$ parametrize the same slice on their overlap; their spatial coordinates need not be identical. For $M_0>0$ and $S\in {\mathbb{R}}^3$ with $\mathrm{\mid }S\mathrm{\mid }<M_0^2$, require $g_{00}=-1+2M_0\mathrm{/}r+O_6(r^{-2}),g_{ij}={\delta }_{ij}+O_6(r^{-1}),g_{0i}=-2{ϵ}_{ijk}{S}^j{x}^k{r}^{-3}+O_6(r^{-3})\mathrm{.}$ Here $r=\mathrm{\mid }x\mathrm{\mid }$, ${ϵ}_{123}=1$, and derivatives are spatial in this stationary chart. The exterior $E=I_M^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})\cap I_M^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$ is globally hyperbolic, ${\mathrm{\Sigma }}^0\cap I_M^{\pm }(M^{\mathrm{e}\mathrm{n}\mathrm{d}})={\mathrm{\Sigma }}_1$, and its stationary orbits are complete and meet ${\mathrm{\Sigma }}_1$. These are source hypotheses, not consequences of a generic black-hole definition. No maximality, quantitative perturbation bound or smallness threshold for $\mathrm{\mid }S\mathrm{\mid }$ is included beyond the explicit strict subextremal parameter range.

All causal predicates for ${\mathrm{\Sigma }}^0$ are relative to $M$. A set $A$ is achronal if no future timelike curve has both endpoints in $A$, and acausal if no nonconstant future causal curve has both endpoints in $A$, including coincident endpoints. For achronal $A$, its edge consists of the points $q\in {\bar{A}}^M$ such that every open neighborhood $U\ni q$ contains points $p\in I_U^{-}(q)$ and $s\in I_U^{+}(q)$ joined by a future timelike curve contained in $U\setminus A$. Here $I_U^{\pm }(q)$ uses curves entirely within $U$. Edgeless means that this edge is empty; partial Cauchy hypersurface means acausal and edgeless. Smooth embeddedness and spacelikeness are separately imposed above.

The supplied diffeomorphisms determine the slice parametrization, its distinguished subset ${\mathrm{\Sigma }}_1$ and the end coordinates. Their overlap is a smooth change of spatial coordinates on the same zero-time slice, without a prescribed identity transition. The metric and time orientation determine the causal curve classes. Applying the chronological definitions to each open $U$ makes the edge predicate unambiguous; its closure is taken in $M$. The stated causal relations for the slice use $M$, while global hyperbolicity is tested on $E$. The asymptotic bounds, strict parameter range and complete-orbit/section condition are separate supplied predicates. This is a mathematical definition of admissible data, not a derivation of the vacuum equations from observations or an assertion that such data exist. It inserts neither a maximal section, a horizon-smallness condition nor a smooth structure on an orbit quotient.

Radu's version-1 section 1.1 GR passage supplies the source-specific package, including equation (1.1), the diffeomorphism ${\mathrm{\Phi }}_0$, complete exterior orbits, chronological intersections and the zero-time end slice. The four original local definitions supply vocabulary, not this particular list of hypotheses.

Minguzzi's Definitions 2.19, 2.95, 2.130 and 3.35 supply the causal predicates written explicitly in the Definition. No theorem about the resulting hypersurface or its domain of dependence is imported.