Weak original data furnish the one recovery family and pole

Choose the end chart compatible with $o$: if necessary reflect one spatial coordinate. This preserves $r$, the original $t=0$ section, mass and all weak Cartesian decay bounds. The future-normal orientation of the original section then agrees with $d{x}^1\wedge d{x}^2\wedge d{x}^3$ on the end. The original outer-section supplier [F10] gives a compact smooth annulus $B_R$ in the original section for every sufficiently large $R$. The chosen orientation toward infinity on $S$ is opposite to its inner-boundary orientation, so $\mathrm{\partial }{B}_R=S_R\bigsqcup (-S)$ with the outward end-sphere orientation. This verifies the orientation and bounding hypotheses in [F4] without a condition on the transformed section being spacelike.

The regular definition gives the two smooth transverse null chronological end-boundaries through $S$ and the smooth spacelike original section. By [F1] and [F10], its exterior side is exactly $\{\rho >1\}$ and its stationary-flow product is $E$. The adapted topological section coordinates chosen in [F10] give a smooth collar $c(s,\tau )$ corresponding to $(1+\tau )s$, for $s\in S^2$ and $\tau \ge 0$ small; its positive-parameter points belong to $E$. Thus every hypothesis of [F6] holds, with the original section's future unit normal and its unit normal toward increasing $\rho$. It gives $E\cap O=\{b>\mathrm{\mid }a\mathrm{\mid }\}$ in the normal-exponential tube, the precise outward-wedge attachment required by [F2].

Apply the boost assertion in [F7] to the $K$ and normal frame from [F1]. Its hypotheses hold because $K{\mathrm{\mid }}_S=0$ and its normal boost matrix is $\kappa \left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ with $\kappa >0$. A local $g$-isometry preserves the metric-selected connection: transporting the connection preserves torsion freedom and metric compatibility, so uniqueness in [F7] identifies it with the original connection. Geodesic initial-value uniqueness then gives ${\mathrm{\Phi }}_t^K({\exp }_{s}v)={\exp }_s(d{\mathrm{\Phi }}_t^K{\mathrm{\mid }}_{s}v)$ wherever the local maps are defined. In the tube of step 1.2 put $u=b+a$ and $v=b-a$. The boost therefore acts as $(s,u,v)\mapsto (s,e^{\kappa t}u,e^{-\kappa t}v)$. This establishes the constant nonzero normal boost action required by [F2], before any primitive is selected.

First verify the circle-preservation hypothesis of [F2]. By [F11], $T_{s}S=T_s{\mathcal{H}}^{+}\cap T_s{\mathcal{H}}^{-}$, and regularity gives $T_s\in T_{s}S$ for every $s\in S$. If ${\mathrm{\Omega }}_H=0$, the relation $K=T+{\mathrm{\Omega }}_{H}Z$ and $K{\mathrm{\mid }}_S=0$ would give $T{\mathrm{\mid }}_S=0$, contrary to the Statement. Thus ${\mathrm{\Omega }}_H\mathrm{\ne }0$, and $Z{\mathrm{\mid }}_S=-{\mathrm{\Omega }}_H^{-1}T{\mathrm{\mid }}_S$ is tangent to $S$. The compact embedded sphere is closed in the common smooth neighborhood on which $Z$ is defined. Apply [F12] there: every defined $Z$-flow map sends $S$ into $S$, and its inverse flow gives equality. Hence the supplied period-$2\pi$ circle action preserves $S$. Now apply [F2] with $\bar{\mathrm{\Sigma }}=C\cup S$, $\mathrm{\Sigma }=C$ and the actual original exterior section, whose maximality is stipulated in the Statement. The stationary-flow product in [F1] uses this same section. The physical truncations in [F10] are compact and smooth, nested and exhausting, have boundary $S\bigsqcup S_R$, and contain a relative neighborhood of $S$; their physical closed tails are closed and disjoint from $S$. Thus they meet the primitive's end-attachment hypotheses with separate radii $\rho$ and $r$, without any coordinate identification. Step 1.2 verifies its outward-wedge hypothesis, and step 2.1 verifies its constant nonzero normal boost hypothesis independently of $W_0$. By [F9], the original ADM mass-one convention identifies $M_0=1$, so the unchanged original end has exactly the weak mass-one coefficient bounds required by [F2], with $T$ future timelike on a far tail. Together with the commuting fields, circle action, common pole and normalized Ernst potential from [F1], all hypotheses of [F2] now hold. Choose its one $W_0$, smooth across $S$ and locally $K,Z$-invariant, with $d_0=D_0(p)$ and original-section spatial coefficients $O_5(r^{-3})$. Apply the original lapse/product result in [F1] using the assumed maximality of this actual $C$; it gives $f<1$ on $E$. Since $S$ lies in the spacelike original section, $f=-g(T,T)\le 0$ on $S$. Directly from [F3], $\mathrm{Re}{B}_q=2q+(1-q)(1-f)$ is positive on $E$ for $q<1$ and equals $2$ for $q=1$, so $D_q>0$ on $E$. On $S$, $\mathrm{Re}{B}_q\ge 1+q$ and $D_q\ge (1+q{)}^2$. Intersect the preceding common working neighborhood of $S$ with the open set $\{f<1\mathrm{/}2\}$, which contains $S$, and call the result $O$. On this same $O$ for every $q\in [0,1]$, $\mathrm{Re}{B}_q\ge 2q+(1-q)\mathrm{/}2\ge 1\mathrm{/}2$, hence $D_q>0$. At the common pole $T_p=0$, so $d_q=(1+q{)}^2+(1-q{)}^2\psi (p{)}^2>0$. The family normalization in [F3] now gives $D_q+W_q(T)=d_q>0$. All hypotheses of the rank-one result in [F3] hold on $U=E\cup O$: the original stationary Lorentzian data, supplied normalized potential, zero $T_p=0$, smooth stationary $W_q$, contraction identity and positive $D_q$. It therefore makes $g_q$ Lorentzian there for every $q$, with the stated time orientation. This argument uses maximality only on the physical $C$ and does not identify the independent charts.

Fix that $W_0$ and use exactly the formulas in [F3] for all parameters. Their curl and contraction normalization are the local hypotheses of source 0, and the positivity hypotheses were proved in step 3.1. Apply the imported local non-null and compatible-null calculation and its zero-locus extension. This proves the displayed electrovacuum equations on $U$, in the conventions of [F5], without using a global construction theorem.

The coefficients of $W_q$, $B_q$ and $D_q$ are polynomial in $q$ with fixed smooth spacetime coefficients. Since $D_q>0$, the reciprocals and hence $A_q$ and $g_q$ are jointly smooth locally, with local extensions across both parameter endpoints. Set $J_q=-({\bar{B}}_q\mathrm{/}{B}_q)A_q^{\ast }\mathcal{F}$; it too is jointly smooth locally up to both endpoints. Then ${\mathcal{H}}_q=\sqrt{1-q^2}{J}_q$. Every spacetime derivative of $J_q$ is continuous and bounded on compact parameter-chart subsets, so multiplication by the continuous scalar factor proves the asserted $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}$ continuity of $F_q$ and ${\star }_{g_q}{F}_q$ on $[0,1]$. For $q<1$ that scalar is locally smooth as well. At $q=1$, $W_1=0$, $B_1=2$, $D_1=4$, $A_1=I$ and the scalar factor is zero; thus $(g_1,F_1)=(g,0)$.

Work on one connected neighborhood of $S$ where the local fields and the chosen $W_0$ are defined. For $X=K,Z$, the commutator $[X,T]=0$ and ${\mathcal{L}}_{X}g=0$ imply invariance of $T^{\mathrm{♭}}$, $f$, $F=\frac{1}{2}d{T}^{\mathrm{♭}}$ and $\mathcal{F}=F+i{\star }_{g}F$; the local isometry flows preserve the fixed orientation and hence the Hodge star. Therefore $d(X\sigma )={\mathcal{L}}_{X}d\sigma =0$ by the Ernst equation. Both $K$ and $Z$ vanish at the common pole $p$, so $X\sigma (p)=0$ and the constant $X\sigma$ is zero on this connected neighborhood. The chosen primitive is $K,Z$-invariant by step 3.1. Consequently $B_q,D_q,W_q,A_q$, and then $g_q,{\mathcal{H}}_q,F_q$ in [F3], are preserved by these same fields for every $q$. This proves symmetry for the actual weak-end-decay primitive, without identifying it with another existential witness.

By [F9], on the unchanged original end, $\sigma =1-2\mathrm{/}r+O_1(r^{-2})$, the metric has the weak $O_6$ bounds and the spatial coefficients of $W_0$ are $O_5(r^{-3})$. The seed-form and spatial-Hodge estimates in [F4] give exactly the end-restriction hypotheses. For each fixed $0<q<1$, the positive-parameter charge lemma therefore gives end limits $(Q_E,Q_M)=(\sqrt{1-q^2},0)$ with the orientations of step 1.1. The two smooth Maxwell forms in step 4.1 are closed on the original annulus and through its boundary $S$. Applying the compact-cobordism case of the closed-flux lemma to $B_R$ gives equality of their integrals on $S_R$ and $S$. Taking the end limit proves the same flux pair on $S$ and every such $S_R$. The homologous-surface clause transfers it to each $C$ in the Statement.

In the tube and boost coordinates established in steps 1.2 and 2.1, the two horizon axes supplied by [F6] are $u=0$ and $v=0$. Every point on either axis sufficiently near $S$ has a half-orbit staying in the tube and tending to its base point in $S$: shrink the nonzero coordinate exponentially in the appropriate time direction; the resulting explicit curve is an integral curve on that whole half-line. By step 4.3, $W_q(K)$ is constant on that curve, and its limit is zero since $K{\mathrm{\mid }}_S=0$. Thus $W_q(K)=0$ on both branches, for this chosen family. By [F8], $K$ is the original null normal generator; tangency of $T$ to each branch gives $g(K,T)=0$ there.

On the one fixed compact oriented sphere $S$, step 4.2 verifies the continuity hypotheses of the endpoint lemma in [F4]. Its fixed-sphere limit extends the identities from $0<q<1$ to $q=0,1$ on $S$. The closed-flux lemma then transfers these endpoint identities to every surface $C$ specified in the Statement, using the Maxwell equations at that endpoint. In particular the flux pair at $q=0$ is $(1,0)$ on every fixed large end sphere, so it has that same end limit. At $q=1$ it is $(0,0)$, also directly from $F_1=0$. No interchange of parameter and spatial-infinity limits is used.

Expand the rank-one metric in [F3] as $4g_q=D_{q}g+T^{\mathrm{♭}}\otimes W_q+W_q\otimes T^{\mathrm{♭}}-f{D}_q^{-1}{W}_q\otimes W_q$. On either branch, step 5.2 gives $g_q(K,\cdot )=(D_q\mathrm{/}4)g(K,\cdot )$. Hence $K$ remains tangent, null and normal on the unchanged nonzero-generator part. With step 4.3 and the Lorentzian character from step 3.1, these branches are Killing horizons in the sense of [F8]. At $S$, $T\in TS$ makes $A_q(TS)\subset TS$, and invertibility makes $A_q{\mathrm{\mid }}_{TS}$ invertible; thus $g_q{\mathrm{\mid }}_{TS}$ is positive definite. Differentiating $4g_q(K,K)=D_{q}g(K,K)+2g(K,T)W_q(K)-f{D}_q^{-1}{W}_q(K{)}^2$ as a spacetime scalar, the vanishing of $g(K,K),g(K,T),W_q(K)$ on each branch leaves $d(g_q(K,K))=(D_q\mathrm{/}4)d(g(K,K))$. The signed seed equations in [F8] therefore give $d(g_q(K,K))=-2\kappa g_q(K,\cdot )$ on ${\mathcal{H}}^{+}$ and $d(g_q(K,K))=+2\kappa g_q(K,\cdot )$ on ${\mathcal{H}}^{-}$. Their smooth zero extensions at $S$ retain these branch normalizations. The normal flow action, and hence its boost rate, is unchanged. All construction inputs use this one primitive, gauge and pole; the original maximal section was never replaced. Together with steps 4.1, 4.2, 5.1 and 6.1, this proves every Statement contract.