Smooth invariant Harrison primitive near the horizon and at infinity

In the mass-one stationary vacuum construction under the precise Radu GR/SBS and original maximal-slice hypotheses in [F1], define $F=\frac{1}{2}d{T}^{\mathrm{♭}}$, $f=-g(T,T)$, $\sigma =f+i\psi$, where $\psi$ is the smooth twist potential on the exterior and a neighborhood of the bifurcation sphere satisfying $d\psi =2i_T\star F$ and $\psi \to 0$ as $r\to \mathrm{\infty }$, and put ${\mathrm{\Theta }}_0=4((f-1)F+\psi \star F)$ and $D_0=\mathrm{\mid }1-\sigma {\mathrm{\mid }}^2$. The symbol $\star$ denotes the oriented Lorentzian Hodge star and $T^{\mathrm{♭}}=g(T,\cdot )$. There exists a smooth stationary real one-form $W_0$ on the exterior and a neighborhood of the bifurcation sphere $S$, with $d{W}_0={\mathrm{\Theta }}_0$, locally invariant under the Hawking field $K$ and rotational field $Z$, and satisfying $W_0(T)+D_0=D_0(p)$ at a common zero $p$ of $T,Z$. Here $K=T+{\mathrm{\Omega }}_{H}Z$, $K{\mathrm{\mid }}_S=0$ and $Z$ has period $2\pi$. In the fixed stationary end chart, $W_0=(D_0(p)-D_0)dt+(W_0{)}_{i}d{x}^i$ and $(W_0{)}_i(x)=-{\int }_1^{\mathrm{\infty }}s{x}^j({\mathrm{\Theta }}_0{)}_{ji}(sx)ds=O_5(r^{-3})$. One primitive has both this end gauge and the smooth invariant homotopy gauge near $S$. The local Hawking-field invariance is asserted only where the local field is defined.

The smooth Radu construction in [F1] is imported as an established external input with the complete source statement and hypotheses above. Its spatial decay makes the period on $S_r$ tend to zero: the coefficients are $O(r^{-4})$ and the area scales as $r^2$. Closedness and [F3] therefore give zero period on $S$ and on a generator sphere. The source's cohomology criteria provide the base primitive $\beta$ on $S$ and a global spatial primitive. The first two intermediate lemmas then explain the smooth invariant normal-bundle construction and the forced constant at $p$; the fourth lifts the spatial primitive with exactly this constant. The third proves the radial formula and its five derivative bounds. The fifth explains why exact corrections on the end and the original stationary collar preserve the normalization and leave the Hawking symmetry intact near $S$. No global extension of $K$ or $Z$ is required.

The five intermediate lemmas were fully authored before admitting the residual source inputs. This item retains literature-derived proof provenance and is not a claim that all of Radu's geometric or asymptotic analysis has been reconstructed locally. In particular, the radial lemma proves $C^5$ regularity from the recorded $O_5$ bound; smoothness of the chosen radial end form is included in the imported smooth construction, not inferred solely from five derivative bounds. The collar input is the original $q=1$ collar, avoiding dependence on the Harrison electrovacuum construction. Radu's §3.3 derivation of (3.28) uses the original vacuum end and does not require transformed charge formulas. The separate held vacuum Ernst, local rigidity and Harrison electrovacuum items are not logical suppliers here; their applicable inputs are expressly imported from the independently read source. Source reading does not grant Step-7 judgment or establish a defect in the article under study.