Proof. Every strict failure has a compact witness. If \(T_d(w)>N\), use a geodesic segment extending beyond \(N\). If a curvature bound fails, use a segment through a point \(\tau<T_d(w)\) with \(Q_d(w,\tau)>N\). Smooth local Cauchy stability and geometric uniqueness place a neighborhood of this fixed compact segment in a common gauge for all sufficiently close data. For the compact segment \(K\), its causal shadow \(J^-(K)\cap\Sigma\) is compact; otherwise a limit of past causal curves to escaping initial points would contradict Cauchyness. Finite local Cauchy constructions therefore suffice. Geodesic and parallel-transport ODE stability preserve the strict witness. This proves openness of the complement, without asserting continuity of the maximal duration. ◻

Proof. In the transported Lorentz orthonormal frame write \(U=\alpha e_0+v\), \(X=\beta e_0+z\). Orthogonality and the unit conditions give \(\alpha^2=1+|v|^2\), \(|z|^2=1+\beta^2\), and \(\alpha\beta=v\cdot z\); hence \(\beta^2\leq|v|^2\). Therefore \(|X|_k^2\leq|U|_k^2=1+2h(w,w)\), the latter identity being constant by parallel transport. Tensor Cauchy–Schwarz proves the assertion. ◻

Proof. Parametrize the family by a connected compact interval. The set of parameters for which the whole path lies in \(P\) is open. Every such path lies in the intrinsic compact diamond \(J_P^+(p)\cap J_P^-(q)\), which is closed in the ambient Hausdorff manifold. Continuity makes the parameter set closed as well. It is therefore the whole interval. ◻

Proof. Starting from 1, identify the development with its image and write \(z=c(1)\). The curve is future-inextendible within the development: an interior endpoint would equal \(z\) by the Hausdorff property. Extend it maximally to the past inside the development. Cauchyness then puts a sufficiently late point \(p=c(s_0)\) to the intrinsic future of \(\Sigma\). In a small temporal chart at \(z\), the local timelike relation from \(p\) to \(z\) gives a short timelike geodesic continuing past \(z\). Only this local relation is needed, not a continuing unit tangent of \(c\). For a \(C^2\) metric the Christoffel symbols are \(C^1\), so the geodesic flow has the usual local uniqueness and continuous dependence.

Consider sufficiently close geodesic arcs with starting positions on the same temporal slice inside the image, and close unit directions. They remain in a common compact chart and phase-space neighborhood. Parametrize these arcs by chart time on a common interval \([t_0,t_1]\), with the reference arc passing through \(z\) at \(t_z\in(t_0,t_1)\). Each nearby arc must exit the image. To see this, suppose an arc \((t,x_\gamma(t))\) stayed inside. Choose a fixed smooth bump \(\beta\), supported in \((t_0,t_1)\), with \(\beta(t_z)=1\). The paths \[\bigl(t,x_\gamma(t)+\lambda\beta(t) (x(z)-x_\gamma(t_z))\bigr),\qquad 0\le\lambda\le1,\] form a fixed-endpoint timelike homotopy: after shrinking the arc family, the spatial change is uniformly small in \(C^1\), and all paths stay in the chart with its uniform timelike margin. Under the supposition that the original arc stayed inside, both endpoints belong to the image. 4 would then keep every path inside, although the path at \(\lambda=1\) passes through \(z\), a contradiction. Thus every nearby arc has a first exit, with position and tangent regular there. By geodesic uniqueness this first exit is its maximal intrinsic endpoint, regardless of possible later reentry.

The reference arc continued backwards inside the development meets \(\Sigma\). To see that a maximal timelike geodesic cannot end at an interior position limit, use a precompact temporal chart \((t,x)\). Causal slopes \(V=(1,\mathrm dx/\mathrm dt)\) are bounded and the geodesic equation gives \[\frac{\mathrm d}{\mathrm dt}\log\frac{\mathrm dt}{\mathrm d\tau} =-\Gamma^0_{\alpha\beta}V^\alpha V^\beta.\] Over a finite chart-time tail this bounds the unit tangent and gives its limit, so the geodesic continues. Cauchyness now gives the claimed intersection. Flow dependence through that fixed earlier compact segment yields an open seed family whose arcs have the exits just constructed.

Shrink the seed family to a relatively compact one. The earlier tube has uniformly bounded duration, parallel frames and curvature. In the extension chart, unit tangents stay in a fixed compact phase-space set. The parallel-transport ODE has bounded coefficients on a bounded chart-time interval, so its transport matrices and their inverses are bounded; curvature is continuous and bounded as well. Duration and the norm (3) are consequently uniformly bounded up to first exit. Choose a basis set \(O\) within the seed family and an integer \(N\) larger than both bounds. ◻

Proof. If the relative closure of \(\mathcal B(O,N)\) had nonempty interior, that interior would meet the piece and, by (6), be contained in \(A(O,N)\). At an intersection point, choose a finite seminorm ball within the interior, increasing the order to at least ten if necessary. The asserted perturbation is in that ball but fails the test, a contradiction. Hence each closure is nowhere dense. The countable intersection of their relatively open dense complements contains no extendible datum, by (6). ◻

Proof. Use the regular ingoing azimuth near the horizon, \[d\widetilde\varphi=d\varphi+\frac{\mathfrak a}{\Delta}\,dr,\] and return smoothly to the Boyer–Lindquist azimuth toward infinity. These coordinates, together with \(r\) and local sphere coordinates, give stationary spatial coordinates \(y\); near infinity we use their Boyer–Lindquist Cartesian version. Let \(b\) denote the Kerr metric in \((t,y)\). Direct substitution in the Boyer–Lindquist inverse metric gives \[ \Sigma_K b^{-1}(dt,dt) =-2(r^2+\mathfrak a^2)G'+\Delta(G')^2 +\mathfrak a^2\sin^2\theta. \tag{9}\] For \(G'=1\), the right-hand side is \[ -\Sigma_K-2Mr<0. \tag{10}\] In the exterior it is a convex quadratic in \(G'\). Its value at \(G'=r_*'=(r^2+\mathfrak a^2)/\Delta\) is \[-\frac{(r^2+\mathfrak a^2)^2}{\Delta} +\mathfrak a^2\sin^2\theta<0.\] Both endpoint values are negative, so every indicated interpolation is spacelike. Near the horizon, \(t\) is the regular ingoing time minus the smooth function \(G\). Between the horizons \(b^{-1}(dr,dr)=\Delta/\Sigma_K<0\); with the chosen future orientation, \(r\) decreases along future causal curves. This proves strict outflow at \(r_b\).

Choose the additive constant of \(G\) by imposing \(G=r_*\) after the interpolation. Extending the interval where \(G'=1\) changes the delay at any fixed smaller radius by the integral of \(r_*'-1\) over the added interval. Since \[r_*'-1=\frac{2Mr}{\Delta}=\frac{2M}{r}+O(r^{-2}),\] this integral tends to positive infinity as the bend moves outward. Thus the delay can satisfy all fixed-depth margins at once, while \(G'=1\) continues to hold throughout the prescribed interval. ◻

Proof. In the exterior, \[X=(r^2+\mathfrak a^2)\partial_T+\mathfrak a\partial_\varphi\] is future timelike: direct evaluation gives \(b(X,X)=-\Sigma_K\Delta\). Since \(k(X)=-P\), pairing with the future null vector \(k^\sharp\) gives \(P>0\).

Suppose first that \(E\leq0\). Then \(P'=2rE\leq0\), while \(\sqrt{\Delta D}\) is strictly increasing when \(D>0\). The case \(D=0\) cannot occur: it would force \(k_\theta=0\) and \(L_z=\mathfrak a E\sin^2\theta\), and hence \(P=E\Sigma_K\leq0\). Thus the inequality \(P^2\geq\Delta D\), on its positive-\(P\) branch, has at most one boundary point, and that point is simple. It cannot contain either a double root or a bounded allowed island.

For \(E>0\), restrict to the interval where \(F>0\). Differentiation gives \[ \left(\frac{F}{\sqrt\Delta}\right)' =\frac{N}{\Delta^{3/2}},\qquad N=2r\Delta-(r-M)F, \tag{16}\] and \[ N'=2\Delta+2r(r-M)-F,\qquad N''=6(r-M)>0. \tag{17}\] If \(F(r_+)>0\), then \(N(r_+)<0\). The strictly convex function \(N\) tends to positive infinity and has exactly one zero beyond \(r_+\); its derivative at that zero is strictly positive. Consequently \(F/\sqrt\Delta\) has a unique nondegenerate minimum. If instead \(F(r_+)\leq0\), let \(r_0\geq r_+\) be its first zero. Since \(F'=2r>0\), \[N(r_0)=2r_0\Delta(r_0)\geq0,\qquad N'(r_0)=2\Delta(r_0)+2r_0(r_0-M)>0.\] Equation (17) then gives \(N>0\) wherever \(F>0\). In this second case there is no radial critical point.

A double root of \(P^2-\Delta D\) must therefore occur at the unique minimum in the first case. Setting \(N=0\) and \(F^2=\Delta D/E^2\) gives (15). Moreover, \[ \left|\frac{L_z}{E}\right|-\mathfrak a \leq\frac{\sqrt D}{E}. \tag{18}\] If double roots approached \(r_+\), their identities would force \[\frac{L_z}{E}\longrightarrow \frac{r_+^2+\mathfrak a^2}{\mathfrak a}>\mathfrak a, \qquad \frac{D}{E^2}\longrightarrow0,\] contrary to (18). They cannot approach infinity either: (15) makes \(|L_z/E|\) quadratic in \(r\) to leading order, whereas \(\sqrt D/E\) is only linear. The radial range, the two impact quantities, and the normalized angular covectors are therefore bounded. Smoothness of the angular expression at the poles gives compactness in the intrinsic cotangent bundle.

Finally, the same one-minimum description classifies the allowed radial intervals. A simple endpoint is a reflection toward the same end of its interval. At a double root, \(F^2/\Delta\) increases strictly in either radial direction away from the minimum, so a departing branch cannot turn back to it. There are no further radially bounded trajectories. Covector reversal gives the corresponding description of the past sheet. ◻

Proof. The only null tangencies to a null horizon are its generators. Along an affine generator, write \(k^\sharp=\rho H\) and use a Killing-invariant normalization of \(H\). The affine geodesic equation and \(\nabla_HH=\kappa_+H\) give \[H(\log|\rho|)=-\kappa_+.\] Dividing by \(H^t\) proves (19). Extend \(|\rho|\) smoothly and homogeneously to a positive frequency scale in a conic neighborhood. Since \(E\) and \(D\) are conserved, \[Vf_H=-2(V\log b_1)f_H.\] The coefficient is strictly positive on the generators and remains bounded below nearby. On the null set, \(E=D=0\) forces \(L_z=k_\theta=0\); near the horizon the remaining null direction is exactly its generator covector. This proves the asserted zero set and the strict source inequality. ◻

Proof. The strict minimum established in Lemma 6 and the implicit function theorem give the smooth function \(r_c\). Set \(U=F^2/\Delta\) and \(U_c=U(r_c)\). Near the minimum, \[U(r)-U_c=(r-r_c)^2 a_0(r,L_z/E),\qquad a_0>0\] with \(a_0\) smooth. Thus \(z_2=(r-r_c)\sqrt{a_0}\) is smooth and \(\partial_rz_2>0\) after restricting the neighborhood. On the null set, (14) gives \[ z_1^2-z_2^2=U_c-\frac{D}{E^2}, \tag{23}\] which is conserved. Since \(E\) and \(L_z\) are constant, \[Vz_2=(\partial_rz_2)\frac{\dot r}{\dot t}=\lambda z_1, \qquad \lambda:=(\partial_rz_2)\frac{E\sqrt\Delta}{\Sigma_K\dot t}.\] On a trapped future ray \(E>0\), while \(\dot t>0\) because the slicing is spacelike. Both signs reverse on the opposite sheet, so \(E/\dot t>0\) on both sheets. Hence \(\lambda>0\) there and nearby. Differentiating (23) gives the second equation of (22) where \(z_1\neq0\); smoothness extends it across \(z_1=0\). Compactness gives the positive lower bound.

For symplectic transversality first hold \(E\) fixed in the Boyer–Lindquist spatial phase space. Then \(z_1=\sqrt\Delta\,k_r/E\), and the radial contribution to \(\{z_1,z_2\}\) is, up to the bracket convention, \[\frac{\sqrt\Delta}{E}\,\partial_rz_2\neq0.\] There is no axial contribution because the functions are independent of the axial coordinate, and no polar contribution. At fixed energy, a stationary change of time adds an exact one-form times \(E\) to the spatial covector, while a spatial coordinate change acts by cotangent pullback. The reduced bracket is therefore unchanged.

Finally pass from fixed \(E\) to a temporal root \(E=e(y,\eta)\) of the characteristic equation. Write \(f|_e=f(y,\eta,e(y,\eta))\), and use the bracket convention \(H_f g=\{f,g\}\). Implicit differentiation gives, on the characteristic set, \[ \{f|_e,g|_e\} =\{f,g\} +\frac{g_E\{q_E,f\}-f_E\{q_E,g\}}{\partial_E q_E}, \qquad \partial_Eq_E=-2\Sigma_K k^\sharp(dt)\neq0, \tag{24}\] where the symbol is expressed in the regular stationary coordinates, the brackets on the right hold \(E\) fixed, and \(\partial_E\) holds their spatial covector \(\eta\) fixed. The denominator is nonzero because \(dt\) is timelike. For \(f=z_1\) and \(g=z_2\), both characteristic derivatives in the numerator vanish on the invariant trapped set by (22). Thus the nonzero fixed-energy bracket survives on each halfwave. Since \(\{\varphi_+,\varphi_-\}=\pm2\{z_1,z_2\}\), the assertion follows. ◻

Proof. We first establish a strict angular margin on the null trapped set. There \(D=P^2/\Delta\) and \(\Delta<r^2\). Expanding \(P\) gives \[ \begin{aligned} D+2\mathfrak a E L_z &>\frac{P^2}{r^2}+2\mathfrak a E L_z\\ &=(r^2+2\mathfrak a^2)E^2 +\frac{\mathfrak a^2(\mathfrak a E-L_z)^2}{r^2} \geq(r^2+2\mathfrak a^2)E^2. \end{aligned} \tag{25}\] Thus \[ c_1=D+2\mathfrak a E L_z-2\mathfrak a^2E^2>0. \tag{26}\] Compactness makes this inequality uniform, and it persists on nearby levels. Also \[|p_{\mathbb S^2}|^2 =D+2\mathfrak a E L_z-\mathfrak a^2E^2\sin^2\theta >\mathfrak a^2E^2,\] so the angular momentum covector never vanishes.

Use half the Hamiltonian (13) to normalize the separated time. On the radial critical manifold its angular field is \(H_{D/2}+\omega(L_z)H_{L_z}\), where \(\omega\) is smooth on the relevant impact interval. The second term is an axial rotation and preserves \(D,L_z\). Put \[A=\mathfrak a^2E^2,\qquad C=D+2\mathfrak a E L_z,\qquad Q=C-L_z^2-A=D-(L_z-\mathfrak a E)^2,\] and let \(x_1=\cos\theta\). The globally smooth function \(x_1\) satisfies \[ \begin{aligned} (x_1')^2&=(1-x_1^2)C-L_z^2-A(1-x_1^2)^2,\\ (x_1')^2+c_1x_1^2+Ax_1^4&=Q,\qquad x_1''=-c_1x_1-2Ax_1^3. \end{aligned} \tag{27}\] The second-order equation follows by differentiation away from the turning points and extends to them by smoothness. It is a confining oscillator because \(c_1>0\).

In particular \(Q\geq0\), and its squared amplitude is \[ d_1^2=\frac{2Q}{c_1+\sqrt{c_1^2+4AQ}}. \tag{28}\] This formula is smooth even at \(Q=0\). Substituting \(x_1=d_1\sin u\) gives the positive invariant period \[ T_{\mathrm{ang}}(D,L_z) =\int_0^{2\pi} \frac{du}{\sqrt{c_1+Ad_1^2(1+\sin^2u)}}. \tag{29}\] It depends smoothly on the conserved quantities, including at the constant equatorial orbit where its value is \(2\pi/\sqrt{c_1}\). It also remains smooth when the orbit passes through a pole.

Let \(\Phi^s\) denote this angular flow, and let \(R_\zeta\) be axial rotation by \(\zeta\in\mathbb S^1\). The circle action is free on the angular phase states under consideration. Away from the poles it is free on the base; at a pole a nonzero covector has trivial rotational stabilizer. The data \(D,L_z,x_1,x_1'\) determine the angular phase state modulo this action. Away from the poles this follows from \(x_1'=-\sin\theta\,p_\theta\); at a pole \(D\) determines the nonzero covector’s norm, and rotations act transitively on covectors of that norm. Therefore there is a unique \(\alpha(z)\in\mathbb S^1\) such that \[ \Phi^{T_{\mathrm{ang}}(z)}z=R_{\alpha(z)}z. \tag{30}\] To justify smoothness without treating the listed invariants as coordinates, let \(\mathcal U\) be the angular phase region with nonzero covectors. Its free circle action is proper, and the map \[\mathbb S^1\times\mathcal U\longrightarrow \mathcal U\times_{\mathcal U/\mathbb S^1}\mathcal U, \qquad (\zeta,z)\longmapsto(z,R_\zeta z)\] is a diffeomorphism. The smooth pair \(z\mapsto(z,\Phi^{T_{\mathrm{ang}}(z)}z)\) lies in this fiber product by the preceding orbit classification. Its inverse gives the smooth \(\alpha\), including at poles and the equatorial limit. Invariance of \(T_{\mathrm{ang}}\) and equivariance of the flow under axial rotations show that \(\alpha\) is invariant along the flow.

For every fixed integer \(n\), it follows that \[ \Phi^s z=R_{\alpha(z)^n} \Phi^{s-nT_{\mathrm{ang}}(z)}z. \tag{31}\] At each evaluation point choose \(n\) so the residual time is bounded, and keep this integer fixed while differentiating the identity. On a compact invariant subset the period is bounded above and away from zero, so \(|n|=O(1+|s|)\). Every fixed derivative of the circle power map and of \(s-nT_{\mathrm{ang}}(z)\) costs only a polynomial factor in \(n\), while bounded-time flow derivatives are uniformly bounded. Hence for each fixed integer \(j\geq1\) there are constants \(C_j,N_j\) such that \[ \|D_z^j\Phi^s\|\leq C_j(1+|s|)^{N_j}. \tag{32}\] This argument uses the circle-valued return rotation and requires no global real-valued lift of its angle.

For a positive time change, write \[t=\int_0^s\tau(\Phi^u z)\,du,\qquad 0<c\leq\tau\leq C.\] The bounds just proved give polynomial bounds for every fixed derivative of this integral. Its \(s\)-derivative is \(\tau(\Phi^s z)\geq c\); repeated implicit differentiation gives polynomial bounds for its inverse \(s=s(t,z)\), and \(|s|\) is comparable to \(|t|\). Composition with \(\Phi^s\) preserves (32). The time changes between separated time and a regular stationary Kerr time have precisely these properties on the compact trapped set; reverse the separated orientation on the past sheet when parameterizing by increasing \(t\).

Finally, all formulas hold on an angular neighborhood where \(c_1>0\), allowing \(D\) to vary. On a nearby level of (13), the radial critical equations are unchanged and only the allowed value of \(D\) is shifted. The smooth embedding \(r=r_c(L_z/E)\), \(k_r=0\) transfers the estimates to every tangent jet of the restricted flow on this critical manifold. ◻

Proof. The functions \(\varphi_\pm\) of Lemma 8 give the two invariant surfaces; their common zero set is the trap, and their nonzero bracket proves symplectic transversality. Their differential equations give the normal exponential rates. Lemma 9 bounds every fixed-order tangent derivative polynomially, which proves the claimed domination.

For nearby levels, use the symbol \(q_E\) in (13) and the spatial field \(X=H_{q_E}/2\). Continue \(z_1\) algebraically as \(z_1=\sqrt\Delta\,k_r/E\). On every nearby level one has the exact identity \[ z_1^2-z_2^2=\frac{q_E}{E^2}+U_c-\frac{D}{E^2}. \tag{33}\] The quantities \(q_E,D,E,L_z\) are constant for \(X\), and \(Xr=E\sqrt\Delta\,z_1\). Therefore \[Xz_2=(\partial_rz_2)E\sqrt\Delta\,z_1,\qquad Xz_1=(\partial_rz_2)E\sqrt\Delta\,z_2.\] For the full stationary field, differentiation at fixed Boyer–Lindquist spatial covector gives \(X^T=-\partial_Eq_E/2\). In the regular time \(t=T+r_*-G\), its speed is instead \[X^t=X^T+(r_*'-G')Xr.\] On the trap \(Xr=0\), so \(E/X^t>0\) there and in a sufficiently small neighborhood on both sheets. Dividing by \(X^t\) gives the same positive \(\lambda\) for both equations, with increasing-time orientation. Thus these are invariant hypersurfaces in a full fixed-frequency neighborhood, not merely in its null level. Their bracket remains nonzero by continuity. The positive angular margin also persists, as proved above.

For a trapped impact parameter, Lemma 6 gives \(F(r_+)>0\). Along a departing tail \(L_z/E\) and \(D/E^2\) remain fixed, and \(F^2/\Delta\) increases strictly as \(r\) moves away from \(r_c\) on either side. Thus the departure is genuine escape, with no return to the double root. ◻

Proof. At a trapped null covector, \(\dot r=0\), so \(\dot t=\dot T\). Substituting (15) into (14) gives \[ \frac{\Sigma_K k^\sharp(dt)}{E} =\frac{(r^2+\mathfrak a^2)F}{\Delta} +\mathfrak a\left(\frac{L_z}{E} -\mathfrak a\sin^2\theta\right) =\Sigma_K+\frac{4Mr^2}{r-M}. \tag{35}\] Write \(k=-E\,dt+\eta\) in the stationary splitting. The zero-temporal-component lift of its spatial projection is \[\eta^\uparrow=k+E\,dt.\] Using nullness, (10), and (35), we obtain the exact identity \[ \begin{aligned} \frac{\Sigma_K}{E^2} b^{-1}(\eta^\uparrow,\eta^\uparrow) &=\frac{2\Sigma_K k^\sharp(dt)}{E} +\Sigma_K b^{-1}(dt,dt)\\ &=\Sigma_K+\frac{8Mr^2}{r-M}-2Mr>0. \end{aligned} \tag{36}\] The last inequality is strict for every \(r>r_+>M\).

The normalized trap is compact by Lemma 6. Its spatial projection is nonzero, since a nonzero multiple of \(dt\) is timelike and cannot be a null covector. Consequently (36) gives a positive lower bound relative to \(|\eta|^2\) on that projection. Covector reversal supplies the same bound on the other sheet. By continuity, a neighborhood in the spatial cosphere has the asserted inequality. Its base support may be confined to a fixed compact radial interval separated from the horizon and the spatial ends, and nested neighborhoods may be chosen inside this open elliptic set.

For completeness, the temporal characteristic equation is \[ b^{tt}E^2-2b^{ti}\eta_i E+b^{ij}\eta_i\eta_j=0. \tag{37}\] Here \(b^{tt}<0\), while the constant term is positive on \(\mathcal T\). The two real roots therefore have opposite signs and neither vanishes. Homogeneity and compactness give the comparability of their absolute values with \(|\eta|\). The radial description places the trap of every nearby exact Kerr metric in the same compact neighborhood, after shrinking the parameter range. Smooth dependence of the charts, the symbol, and the trapped set preserves all the strict margins. ◻

Proof. For a future null covector with \(E=0\) and \(r>r_+\), positivity of \(P=-\mathfrak a L_z\) implies \(L_z\neq0\) and hence \(D>0\). The radial polynomial and its derivative are \[ R(r)=\mathfrak a^2L_z^2-\Delta D,\qquad R'(r)=-2(r-M)D<0. \tag{38}\] Thus there is at most one radial turn, and it is simple.

The spatial fixed-energy Hamilton field is the projection of the full null Hamilton field. If its spatial base component vanishes, the full metric dual must be proportional to \(\partial_T\). Nullness then places the base point on the stationary limit surface. At such a point (38) still has a simple root, so the spatial Hamilton field is nonzero. It cannot be a nonzero pure fiber dilation: axial momentum \(L_z\neq0\) is conserved, whereas a nonzero dilation would change it. This proves the nonradial real-principal assertion even at these apparently stationary covectors.

Between the horizons, the radial component of a null ray is nonzero because \(dr\) is timelike. At the horizon, all crossings other than the generators are transverse. Lemmas 6 and 7 therefore exhaust the possible non-escaping behavior. At bounded temporal frequency, spatial fiber infinity corresponds to normalized temporal energy zero, so it contributes exactly the directions just analyzed and no additional trapping. ◻

Proof. The gauge-potential assertion is (Häfner et al. 2025, Theorem 5.1); the discussion following that theorem permits arbitrarily small local damping parameters. The bounded-frequency estimates and stationary index are (Häfner et al. 2025, Theorem 5.2), and nonzero-frequency inversion and the seven-dimensional kernel are (Häfner et al. 2025, Theorem 6.6). The polynomial-mode assertions are (Häfner et al. 2025, Proposition 6.8 and Lemma 6.9). We verify the geometric conventions for applying these results here. The stationary time is horizon-regular, with outgoing time \(t-\phi\) at infinity. In Boyer–Lindquist coordinates there, \[r_*=r+2M\log r+O(1),\quad b^{ti}=O(r^{-2}),\quad \partial_y(\phi-r)=O(r^{-1}),\quad b^{-1}\bigl(d(t-\phi),d(t-\phi)\bigr)=O(r^{-1}),\] with symbol derivatives. The logarithmic coefficient is independent of spin. Bounded smooth order-zero offsets in \(\phi\) preserve the outgoing graphs and polynomial-mode statements. Stationary regular chart changes on compact slabs act by smooth component maps and bounded time shifts, with only polynomial losses on fixed strips.

The inner edge may be extended to the chosen \(r_b\) between the horizons. Between any two strict depths, \(dr\) is temporal, so ordinary hyperbolic propagation applies. On a strip with bounded \(\operatorname{Re}\zeta\), its high-frequency cost is polynomial: \(\operatorname{Im}\zeta\) is a real tangential Fourier parameter in the radial-time Cauchy estimate, and the real Laplace part is a bounded lower-order weight. We choose the local damping sufficiently small for the finitely many strict thresholds below. This input concerns the local modified operator; it invokes neither nonlinear stability nor inversion after spatially nonlocal gauge changes. ◻

Proof. For a null covector \(k\) define \[Q_kH=H(k^\sharp,\cdot) -\tfrac12(\operatorname{tr}_{\bar g}H)k, \qquad d_ke=k\odot e,\] where \(\odot\) is averaged symmetric product. The map \(d_k\) is injective and \(Q_k\) is surjective, since trace reversal is invertible. Nullity gives \(\operatorname{im}d_k\subset\ker Q_k\). Parallel transport preserves this flag, with quotient the traceless symmetric tensors of the Euclidean screen. Up to a common scalar normalization the available changes are \(d_kE_{\mathrm g}(k)\) and \(C(k)Q_k\).

Normalize \(k\) by its conserved nonzero stationary energy: \(n=k^{\sharp_{\bar g}}/E\) is parallel along its null geodesic. Choose a stationary future \(\bar g\)-unit timelike field \(T\) and set \[\alpha=-\bar g(n,T)>0,\qquad l=\frac{T}{\alpha}-\frac{n}{2\alpha^2}.\] Thus \(\bar g(l,l)=0\) and \(\bar g(n,l)=-1\). In the screen \(\{n,l\}^{\perp}\) choose local orthonormal vectors \(e_A\). Writing \(\nabla\) for transport along the geodesic, preservation of these inner products gives \[\nabla n=0,\qquad \nabla l=w,\qquad \nabla e_A=\sum_B\omega_{AB}e_B+\bar g(w,e_A)n, \qquad \omega_{AB}=-\omega_{BA},\] where \(w\) is a screen vector. In particular there is no diagonal boost term: conservation of \(E\) has made \(n\) parallel, not merely parallel up to scale. Give \(n\) positive length \(\epsilon\), \(l\) length \(\epsilon^{-1}\), and the screen its ordinary metric. The null-rotation terms have norm \(O(\epsilon)\) in this metric, whereas the screen rotation is skew. The construction patches without a global screen frame and gives a positive homogeneous tensor metric. Its non-skew transport is arbitrarily small, also on the fixed tensor power.

Choose metric transport preserving the orthogonal flag splitting, with the same induced quotient transport. Its difference \(A\) from the original transport maps \(\ker Q_k\) into \(\operatorname{im}d_k\) and is otherwise unrestricted. It is therefore realized by the available changes: for a smooth right inverse \(R_k\) of \(Q_k\), put \(P_k=1-R_kQ_k\) and take \[E_{\mathrm g}(k)=d_k^{-1}AP_k,\qquad C(k)=AR_k.\] Then \(d_kE_{\mathrm g}(k)+C(k)Q_k=A\). Compensate also \(K_{\mathrm{alg}}\) and the local additions in prescribing the total transport. Smooth projections and right inverses give smooth symbols. The normalized-ray construction can be chosen real and invariant under simultaneous covector reversal. Transport before time normalization is odd of degree one, and the gauge-factor symbols are even of degree zero.

At \(b\) choose the local additions small after the flag scaling. Convex interpolation to compatible transport preserves the strict subprincipal margin through the high-frequency transition. Write the roots as \(k=(-P_\pm(y,\eta),\eta)\) and let \(E_\pm^{\mathrm{add}}\) be the extra value prescribed for either gauge factor on the two roots. Set \[ A_1=\frac{E_+^{\mathrm{add}}-E_-^{\mathrm{add}}} {-i(P_+-P_-)},\qquad A_0=E_+^{\mathrm{add}}+iP_+A_1. \tag{43}\] Since \(|P_+-P_-|\gtrsim|\eta|\), these have orders \(-1,0\). Reversal interchanges the sheets; thus \(A_0\) is real-even, \(A_1\) imaginary-odd, and \(A_0+A_1\partial_t\) is real. Extend and patch the values on each sheet before the single spatial interpolation. Overlap of their projections creates no conflict, including over the ergoregion.

Use an even conic cutoff in \(\mathcal T\), equal to one near both projected trapped sets, and a cutoff \(|\eta|\gtrsim\Lambda\). Quantize with compact kernel cutoffs in both variables. Products using both order-zero gauge factors have no remaining first derivative and do not contribute to transport order. Thus the common scalar cutoff really interpolates the first-order transport; symbol derivatives in compositions are lower order. Kernel localization preserves its homogeneous symbol. Outside a slightly enlarged cone, separated-support composition or integration by parts away from the spatial diagonal gives smoothing leakage \(O(\Lambda^{-N})\) in every prescribed fixed seminorm, for every fixed \(N\). The full symbols are uniformly bounded in their orders. Each extra coefficient of a second time derivative has a spatial gain of at least one, hence is small on any fixed finite Sobolev list. The mode-preserving finite-rank corrections below are smoothing and do not alter the calibration. Repeating the prescription at each frozen metric makes compatibility exact near its trap. ◻

Proof. Bounded frequency. The additional operator has spatial order at most one and compact elliptic microsupport. Let \(\beta\) be an order-zero elliptically supported cutoff equal to one on an enlarged neighborhood of that microsupport. On fixed enlarged compact sets \(K'\), \[ \|(L-L_{\mathrm h})(\zeta)u\|_{H^{s-1}} \leq C\Lambda^{-1} \|\operatorname{Op}(\beta)u\|_{H^{s+1}} +O(\Lambda^{-N})\|u\|_{H^s(K')}. \tag{44}\] Nested cutoffs and the high-frequency cutoff give this estimate, including arbitrary-order separated-symbol remainders. An elliptic parametrix for the operator without additions controls the stronger norm. The addition is already bounded \(H^s\to H^{s-1}\), so the bounded-frequency interior graph domains agree. The same argument applies to supported dual orders, since reversed covectors remain elliptic, and to gauge potentials. It proves persistence of compact-error estimates, index, and semicontinuity. Smoothing subtractions of size \(O(\Lambda^{-N})\) do not change this reasoning.

We recall why physical homogeneous limiting-absorption states on a closed nonzero band belong to the outgoing class in Proposition 5. The passage from physical limiting absorption to outgoing conormality uses the unchanged Kerr end and the interior propagation verified above. The physical solution and source weights are \(\langle y\rangle^{-1/2-\iota}\) and \(\langle y\rangle^{1/2+\iota}\). The incoming radial estimate gives the outgoing microlocal condition for a homogeneous real-frequency limit. Exterior rotation commutators are short-range because the principal \(1/r\) correction is rotation invariant. Divided rotation differences, exterior propagation, and spatial ellipticity for unscaled derivatives and fixed-annulus errors give the same scattering estimate after any fixed number of rotations, with positive slack. Thus, for a limit \(h_{\mathrm{lim}}\) of the local operator, \(rh_{\mathrm{lim}}\) and its radial derivative lie in \(L^2(r^{-1-2\iota}dr\,d\omega)\) with those rotations.

Divide the radial equation by its exact radial coefficient. Its leading factorization in tortoise coordinate is \((\partial_{r_*}-\zeta)(\partial_{r_*}+\zeta)(rh_{\mathrm{lim}})\); the remaining symbol terms are \(O(r^{-2})\) with at most two angular derivatives or one radial and one angular derivative. Integrating the first factor from infinity, with the incoming homogeneous term excluded, gives \(O(r^{-1+\iota})\) for the second factor by weighted Cauchy–Schwarz. On the imaginary axis put \(w=e^{\zeta r_*}rh_{\mathrm{lim}}\). It has size \(O(r^\iota)\) and the first symbol-derivative estimate, and \[ (\partial_{r_*}-2\zeta)\partial_{r_*}w =O(r^{-2})\bigl(\partial_\omega^{\leq2}w, \partial_\omega^{\leq1}\partial_{r_*}w\bigr). \tag{45}\] Here is the derivative induction explicitly. Write the right side as \(F=r^{-2}(A_2w+A_1w')\), with symbol-bounded angular operators \(A_2,A_1\) of orders at most two and one. Initially \(w=O(r^\iota)\) and \(w'=O(r^{-1+\iota})\). The equation first gives the raw bound \(w''=2\zeta w'+F=O(r^{-1+\iota})\). It follows that \(F'=O(r^{-3+\iota})\), including the term \(r^{-2}A_1w''\). Differentiating the equation and excluding its nondecaying homogeneous solution gives \[w''(r_*)=-\int_{r_*}^{\infty} e^{2\zeta(r_*-s)}F'(s)\,ds=O(r^{-2+\iota}).\] In general, symbol bounds through \(w^{(j-1)}\) first give the raw bound \(w^{(j)}=O(r^{-j+1+\iota})\). Then \(F^{(j-1)}=O(r^{-j-1+\iota})\), including its highest derivative term, and integration of \((\partial_{r_*}-2\zeta)w^{(j)}=F^{(j-1)}\) gives \(w^{(j)}=O(r^{-j+\iota})\). The raw decay excludes the homogeneous term also when \(\operatorname{Re}\zeta=0\). Angular derivatives are included throughout, and initial slack is chosen smaller than the intended final slack. This gives outgoing b-regularity, and radial propagation upgrades interior regularity above threshold. Proposition 5 excludes the resulting homogeneous limit. In the open half-plane one may instead use its outgoing primal solvability and the physical index argument below. The microlocal smallness (44) now retains the closed-band estimates for \(L\). Dual estimates follow from the open half-plane. The contour argument will use bounds up to the imaginary axis, not an ordinary unweighted inverse there.

High frequency. We apply the local outgoing normally hyperbolic estimate of (Dyatlov 2016, Theorem 1), with the scalar-principal bundle formulation and positive pseudodifferential metrics of (Hintz 2017, Condition (2.5), Proposition 3.12). Its hypotheses are a symplectically transverse codimension-two trapped set, smooth stable and unstable surfaces, and imaginary subprincipal transport defect strictly below half the minimum normal rate in a positive metric, including its transport and the half-density convention. The geometric defining functions \(z_1\pm z_2\) and Lemma 11 verify these hypotheses for each nonzero-energy sign with margin.

To use the separated symbol \(q_E=\Sigma_Kb^{-1}(k,k)\) of (13), an operator initially written with principal symbol \(b^{-1}(k,k)\) can be normalized microlocally by symmetric sandwiching with the smooth elliptic multiplier \(\sqrt{\Sigma_K}\). In half-density Weyl calculus the two first-order bracket terms in this symmetric sandwich cancel, so its subprincipal symbol is \(\Sigma_K\) times the original one. On the characteristic trap \(H_{\Sigma_K p}=\Sigma_KH_p\); the derivative of a chosen bundle metric scales by the same factor. Consequently the pointwise relative transport margin is preserved. For a theorem stated with the supremum of the defect and the minimum normal rate, fix this normalization first and choose the defect sufficiently small using the positive minimum and finite maximum of the scaling factor on the compact trapped neighborhood. The gauge calibration permits arbitrarily small defect before saturation and zero defect at saturation, so this choice imposes no additional hypothesis.

For \(k=-E\,dt+\eta\), \[\partial_E\bigl(b^{-1}(k,k)\bigr)=-2k^\sharp(dt).\] Its sign is separated on each characteristic sheet because \(dt\) is timelike, also after rescaling at spatial fiber infinity. Orient projected propagation by increasing regular \(t\), dividing the Hamilton field by its \(t\)-speed. At trapping \(k^\sharp(dt)/E>0\); the frequency and time normalizations differ by a bounded positive factor after fixing orientations. The subprincipal margin is selected after these normalizations. More precisely, if \(f\) is the positive time-change factor, \(f_{\max}\) its maximum on the compact microlocal neighborhood, and \(\nu_{\min}^{\mathrm{new}}>0\) the minimum normal rate for the normalized flow, choose the arbitrarily small flag defect \(\delta\) so that \(f_{\max}\delta<\nu_{\min}^{\mathrm{new}}/2\) with fixed surplus. Choose the local additions smaller than that surplus. Compatible saturated transport has zero defect, and the convex cutoff transition retains the margin. Merely scaling rates and defects pointwise would not suffice for this global supremum–infimum comparison. Scalar half-density transport is scalar wave-amplitude transport with stationary density; its remaining bundle transport is the calibrated one. The bounded positive real Laplace part therefore supplies damping on either trapped sheet. The horizon radial operator is unchanged and has positive redshift.

The ray alternatives give strict outflow and no returns from infinity or homoclinic returns. Incoming propagation from infinity, ordinary real-principal propagation, radial estimates, and the trap estimate control the characteristic set modulo lower remainders. For the transpose use the opposite propagation direction. A fixed Killing-energy sign can encounter ergoregion rays with a different time orientation. They cannot be trapped, since the double-root identities with future \(P>0\) force future \(E>0\). They use only escape or radial propagation. At spatial fiber infinity with normalized temporal energy zero, the new symbols miss all nonelliptic directions and the horizon radial geometry; the zero-energy alternatives give ordinary nonradial propagation. The positive metric extends to the required microlocal neighborhood. This uses a local outgoing theorem, not global Kerr–de Sitter inversion or a positive Killing-current norm in the ergoregion.

We check uniformity in \(\Lambda\). On characteristic added microsupport, \(c|\zeta|\leq|\eta|\leq C|\zeta|\). Put \(h_f=|\zeta|^{-1}\) and \(\xi=h_f\eta\). On \(|\eta|\geq c'|\zeta|\), the additional terms in \(h_f^2L\) are uniformly \(O(h_f)\) of maximal spatial order one, by the ordinary \(S^j_{1,0}\) bounds on coefficients of orders \(1,0,-1\) multiplying zero, one, and two powers of \(\zeta\). Outside an enlarged elliptic cone they are \(O(h_f^\infty)\) smoothing. Where the cutoff changes at the trap, \(\Lambda/|\zeta|\) is bounded above and below, so its rescaled derivatives are uniformly bounded. Its first transport is the controlled scalar interpolation; symbol derivatives in gauge composition are lower order. Substitution of the bounded real Laplace part in the extra time-derivative factors also changes only lower orders. At large relative spatial frequency the added microsupport is itself elliptic, and the additions vanish at the radial set.

For \(|\eta|\leq c'|\zeta|\), with \(c'\) sufficiently small, the wave symbol has size \(\gtrsim|\zeta|^2\). Write the perturbation as \[\zeta^2K_{-1}+\zeta K_0+K_1,\] with subscripts denoting maximal spatial orders. Relative to elliptic inversion its first term costs \(O(\Lambda^{-1})\) and the others \(O(|\zeta|^{-1})\). Separated localizations from this sector to the characteristic set have rapid frequency decay, including for fixed-rank smoothing terms. No uniform smooth semiclassical symbol at \(\xi=0\) is asserted. These errors are absorbed after propagation with a threshold independent of large \(\Lambda\). Large positive Laplace lines also admit forward wave energy after the small acceleration correction is solved. This proves the polynomial strip bounds and the exterior scattering bounds.

Nearby stationary parameters. At zero frequency the horizon inequalities remain strict after pullback to nearby exact Kerr principal models. Lower-order terms vary smoothly, high additions remain elliptic and away from the horizon, and the normal operator at infinity is a positive constant-coefficient Laplacian after a constant linear spatial change, with unchanged indicial gaps. Add a fixed smooth finite-dimensional supplement removing the center’s kernel and cokernel. Primal and dual compact-error estimates, compactness, and equation convergence keep the supplemented problem invertible: otherwise a normalized contradicting sequence yields a nonzero limiting nullstate. This avoids demanding smallness of a changing principal operator on a fixed one-derivative graph. Gauge potentials obey the same argument. When high spatial regularity is needed only on a compact set, the b-elliptic end permits lower solution and equation derivative orders after a switch across fixed full elliptic annuli, with the higher source norm maintained slightly farther out. ◻

Proof. Combine the supported interior stationary dual estimate with the primal outgoing estimate cut to infinity. In Cartesian trivializations the difference of the conjugated transpose and comparison primal operator is \[O(r^{-2})\partial_y+O(r^{-3}) +(\zeta+\zeta^2)O(r^{-2}).\] The symbol bounds, notably \(b^{ti}=O(r^{-2})\) and connection coefficients of that order, give this expression. It is small between the indicated spaces sufficiently far out and near zero. Switch orders across fixed full elliptic annuli beyond all compact kernels. This proves the a priori estimate.

For the index, first the unconjugated physical primal and supported transpose have index zero for each \(\operatorname{Re}\zeta>0\). Use interior solution/equation orders \(s,s-1\) with \(s=5\) and \(1-s,-s\), respectively, and ordinary unweighted fixed-frequency graphs outside. The Fourier normal operator at infinity is a multiple of the flat spatial Laplacian minus \(\zeta^2\), hence invertible. At high interior spatial frequency the normalized temporal energy is zero; ordinary propagation, non-adverse redshift, their supported dual estimates, and strict outflow give compact-error estimates in both directions. The additions are lower order over ellipticity or smoothing with compact kernels. Thus the graph domains are locally constant in the open half-plane and both problems are Fredholm.

On a sufficiently large forward line, wave energy and past integration give primal inversion and uniqueness after solving the small order-\((-1)\) normal-acceleration correction. Begin with natural unweighted orders, then use exterior ellipticity. Duality gives transpose solvability; testing against primal solutions at slightly raised compact orders gives its uniqueness. Continuation establishes the physical indices, which compact smooth finite-rank terms do not change.

The comparison outgoing primal problem uses \(L_{\mathrm h}\) with extendible interior orders and exactly the exterior graph proposed for the outgoing transpose. Proposition 5 gives solvability on compact smooth data, trivial kernel, and an estimate with compact error. Altering interior orders uses the one-order propagation estimate, homogeneous regularity above threshold, and annular ellipticity. This comparison also has index zero. Exchange its end with the physical supported transpose end in their direct sum, by a smooth signed exchange on an elliptic annulus outside all kernels. Domain and range spaces identify, and collar errors are Rellich compact.

On the conjugated end the fixed-frequency difference is \(O(r^{-2})\partial_y+O(r^{-2})\). Its graph additionally controls \(H_{\mathrm b}^{j+1,\ell_{\mathrm{wt}}-1}\). Indeed, if \(V\in H_{\mathrm b}^{j,\ell_{\mathrm{wt}}}\) and its equation \(PV\in H_{\mathrm b}^{j,\ell_{\mathrm{wt}}+1}\), write \[r^2P=B_2+rB_1+B_0,\] where \(B_2\) is b-elliptic of order two and the remaining terms have order at most one, with uniformly bounded symbol coefficients at this fixed nonzero frequency. Both \(r^2PV\) and \((rB_1+B_0)V\) lie in \(H_{\mathrm b}^{j-1,\ell_{\mathrm{wt}}-1}\); elliptic regularity gives the asserted extra derivative. Since \(r^{-2}\partial_y\) is \(r^{-3}\) times a b-vector field, the first-order difference maps this improved graph regularity to \(H_{\mathrm b}^{j,\ell_{\mathrm{wt}}+2}\). The zeroth-order difference maps the original \(V\) to the same space. Thus the difference gains one tail weight relative to the equation space \(H_{\mathrm b}^{j,\ell_{\mathrm{wt}}+1}\), with norm \(O(Y^{-1})\) beyond radius \(Y\). On each finite annulus ordinary graph ellipticity gives two local derivatives over the equation order, so Rellich compactness handles the remaining compact part. The other exchanged summand is a physical primal problem of index zero. Hence the desired outgoing transpose graph has index zero. All high additions remain on their original interior pieces in this argument.

This gives an index, not automatically an inverse. If a compactly forced outgoing conjugated transpose problem has trivial kernel, index zero supplies its inverse. Its solution \(V\) gives the physical test \(v=e^{-\zeta\phi}V\), exponentially decaying at infinity. Supported–extendible integration by parts against these tests kills the physical primal kernel, and physical index zero yields inversion. The stronger near-zero \(\mathcal S\) estimate is used on graph states with that extra source weight. This never identifies \(P^\sharp(\zeta)\) with the transpose of the conjugated primal. ◻

Proof. For the local gauge, the starting profiles are the Kerr, translation, and boost profiles of (Häfner et al. 2025, Lemmas 6.1–6.3, Proposition 6.5, and (6.6)). We record the end estimates needed here, including the sharper translation-potential bound, and then preserve their identities under the conic modification.

Begin with the local gauge of Proposition 5. Correct cut-off Kerr, translation, and boost generators successively with its stationary gauge-potential inverse. The errors requiring nondecaying correction, for Kerr and the stationary boost part, are \(O(r^{-2})\). Shifting to a slight growth weight \(O(r^{0+})\) is surjective by indicial Fredholm regularity: its cokernel is already zero at decay.

The translation correction has an \(O(r^{-3})\) leading error which is a constant linear combination of second Cartesian derivatives of degree-\((-1)\) symbol-leading coefficients. Each has zero spherical mean, as follows by integration over shells and surface scaling. Hence the decaying vector correction has an \(r^{-1}\) angular leading term without a leading logarithm. The identity \(\mathcal L_{te_i+y_i\partial_t}\eta_{\mathrm{Mink}}=0\) and the gauge correction give \(tJ_i+H_i^1=\mathcal L_{tV_i+W_i}b\) and (48) for the local gauge. The flat-Laplacian indicial expansion gives (49), using \(b^{ti}=O(r^{-2})\) in the equation for \(H_i^1\). The sublinear correction vectors for Kerr and for \(W_i-y_i\partial_t\) have first derivatives \(O(r^{-1})\) with symbol bounds. Indeed their flat symmetric gradients have that order by the tensor profile expansion. Differentiating and permuting indices gives second derivatives of order \(r^{-2}\); integrating along rays from infinity, where first derivatives vanish, gives the stated bound. These arguments use smooth Kerr symbols, indicial regularity at infinity, and compact propagation above threshold, without a restriction on spin.

The seven stationary profiles are independent already on a sufficiently far exterior region, away from the trapping base. A stationary sublinear-symbol gauge preserves the \(1/r\) coefficient of the stationary Killing-time norm and the leading curl of \(b_{ti}\,dy_i/b_{tt}\). A stationary time displacement adds an exact form to the latter, and its spatial Lie derivative is lower order. The Kerr expansions therefore detect the mass and angular momentum vector, with fixed normalized Cartesian axes. A nonzero translation combination cannot have zero Lie derivative there, since it commutes with \(\partial_t\) and its action on the leading radial coefficient of \(b(\partial_t,\partial_t)\) is nonzero. Thus compact smooth functionals away from a physical trapping neighborhood separate the modes; no elliptic unique continuation across that neighborhood is required.

At the base point, subtract finite-rank spatial smoothing terms from the high addition to \(E_{\mathrm g}\) so that its \(A_0\) part annihilates the span of \(H,J,H^1\) and its \(A_1\) part annihilates \(J\). To be explicit, choose a compact cutoff \(\chi_{\mathrm{in}}\) equal to one on all input supports. Let \(q_\alpha\) be a basis of the actual finite-dimensional span of \(\chi_{\mathrm{in}}H,\chi_{\mathrm{in}}J, \chi_{\mathrm{in}}H^1\), and choose compact smooth tests \(\lambda_\alpha\) with \(\lambda_\alpha(q_\beta)=\delta_{\alpha\beta}\). For the high-cutoff operator \(A_{0,\Lambda}\) subtract \[S_0f=\sum_\alpha (A_{0,\Lambda}q_\alpha) \lambda_\alpha(\chi_{\mathrm{in}}f).\] The identity \(A_{0,\Lambda}=A_{0,\Lambda}\chi_{\mathrm{in}}\) shows that \(A_{0,\Lambda}-S_0\) annihilates every original profile, including when their restrictions are dependent. Use the same construction on the span of \(\chi_{\mathrm{in}}J\) for \(A_{1,\Lambda}\). Each \(q_\alpha\) is fixed, smooth, and compactly supported. Consequently \(A_{0,\Lambda}q_\alpha=O(\Lambda^{-N})\) in each fixed smooth seminorm for every fixed \(N\), and the finite-rank kernel formula gives the same statement for \(S_0\) and \(S_1\) in every fixed smoothing norm. The constants may depend on that norm and on \(N\): only the finite base list is made small by the single choice of \(\Lambda\). Hence the profile identities remain exact. The seven modes persist, and Proposition 6 excludes any enlargement of the decaying kernel. Gauge-potential invertibility persists by the same elliptic smallness. Index and weak compactness give the corresponding cokernel continuity towards the local-gauge limit. Extend these smoothing subtractions smoothly to nearby parameters. ◻

Proof. Integrate the stationary correction flows in Lemma 13. For the velocity directions use the affine leading chart \[ t_K=\sqrt{1-|v|^2}\,t+ \frac{v\cdot y}{\sqrt{1-|v|^2}}, \qquad y_K=\left(I+\frac{vv^t}{1-|v|^2}\right)^{1/2}y. \tag{52}\] Pullback of the flat metric is \(\eta_v\). Nearby raw Kerr metrics are first expressed in smooth ingoing coordinates at their outer horizons, bent on a larger domain, and corrected to Boyer–Lindquist conventions towards infinity before restriction. Since the center has nonzero spin, its direction can be rotated smoothly relative to the fixed Cartesian axes. The correcting maps have stationary spatial components and time component equal to a nonzero constant times \(t\) plus a spatial offset. Their vector-field derivatives retain the tight \(r^{-1}\) bounds. This constructs \(G_p\) and (50). Apply the smooth calibration at \(G_p\).

The supplemented stationary and gauge-potential estimates give seven gauge-corrected decaying modes and no more. Physical variations are taken with the leading affine chart held fixed. At vector growth just above constants the correction equations are surjective, as in Lemma 13. Ten independent constant tensor asymptotes are obtained from pure affine gauge at infinity, stationary as metric variations, including a scaling of the stationary generator; correct them by stationary \(O(r^{0+})\) vectors. The flat indicial gap, unchanged by the constant anisotropic chart, shows that these are exactly the ten extra dimensions below growth power \(1\). Uniform supplemented estimates at each fixed high order and difference quotients of the differentiated stationary equations prove smooth parameter dependence, with only finitely many extra orders at each step.

Normalize their representatives by \(\ell H_C(p)=0\). For the probes, choose fixed tests separating the limiting spans on a far exterior region. The stationary estimates with compact error, the dimension counts, and equation convergence prevent any normalized nearby mode from being annihilated by all of them. Smooth parameter-dependent linear combinations therefore produce \(\ell_C^p\) with fixed supports. These supports will lie in the unsmoothed region of the causal construction. Integrating the Kerr and diffeomorphism directions that define \(H_C(p)\) gives \(\overline G_{p,c}\). Its slightly looser symbol decay after its affine asymptote is harmless. Continue the calibration at this family. Since its tangent satisfies the linear stationary gauge, its frozen gauge residual is quadratic in \(c\). All horizon, slicing, trapping, and conic-support conditions retain strict margins after shrinking the parameter range. ◻

Proof. Index and kernel dimension give the dimension. Apply Green’s identity to the constant-coefficient flat, possibly anisotropic, normal operator. Indicial regularity initially permits order-\((-1)\) fundamental solutions. Pairing with the ten independent constant-asymptote homogeneous modes of 7 kills their leading coefficients. The supported convention supplies no inner boundary contribution, so the remaining adjoint tail is \(O(r^{-2+})\). The uniform stationary estimates and exterior indicial regularity give the asserted weak bases and symbol bounds.

For \(v=O(r^{-2+\epsilon'})\) and \(H_0=O(r^\epsilon)\) the outer Green boundary terms are \(O(r^{-1+\epsilon+\epsilon'})\). Choose \(\epsilon+\epsilon'<1\) and let the sphere tend to infinity. The inner term vanishes by supported–extendible duality. This proves (54); the extra adjoint tail power, rather than dual Fredholm information at one weight alone, is what permits these pairings. ◻

Proof. Put \(d_n=\max(0,n-P+1)\) and let \(q\) be the smooth zero-past input. Let \(G_0=\delta_0\) and, for \(j\ge1\), let \[G_j(t)={\bf1}_{t\ge0}\,e^{-t}\frac{t^{j-1}}{(j-1)!}.\] Write \(K_F\) for the diagonal convolution kernel of \(F\). Its mass/spin kernel is \(\sigma_0G_P\), and its velocity kernel is \[2\sigma_0G_{P-1}+(\sigma_0^2-2\sigma_0)G_P,\] since \(\zeta(1+\zeta)^{-P}=(1+\zeta)^{-(P-1)}-(1+\zeta)^{-P}\). The distributional identity \(\partial_tG_j=G_{j-1}-G_j\) shows that every derivative through order \(P-1\) of these kernels is a finite measure, including its possible endpoint delta. All its polynomial moments are finite. Put \(r_n=\min\{n,P-1\}\). Differentiating the convolution gives \[a^{(n)}=(\partial_t^{r_n}K_F)*q^{(d_n)}.\] There are no startup delta terms from \(q\), since its zero extension is smooth. For \(0\le u\le t\), \[\frac{(T_0+t)^{\alpha/2}}{(T_0+t-u)^{\alpha/2}} \le (1+u)^{\alpha/2}.\] Weighted Young’s inequality for this finite measure proves the asserted weighted bounds, on finite intervals by causality. For the input \(q=\chi_{\mathrm{st}}\ell h_{\mathrm{co}}\), differentiating once uses \(\ell u\) and a fixed startup term. The primitive relation \(p^{(j)}=a^{(j-1)}\) gives no strong weight for \(p\) or \(\lambda\). ◻

Proof. At zero parameters the variation of \(B\) in the fixed co-moving chart, plus the Lie derivative along \(e\lambda+(V-e)\lambda_s\), is precisely \(Y\). The location variation supplies \(\lambda_s-\lambda\) and the formal-coframe variation supplies \(v_s-v\). Since \(b\) is vacuum, adding this Lie derivative does not alter the linearized Ricci tensor. The definition of \(\Theta\) therefore gives (62), with the columns (63).

The stationary identities and the chain identity (48) cancel the zeroth and first Taylor terms required by the displayed denominators. The zeroth and first moments of \(k_0\) give \(K_d=1+O(|\zeta|^2r_y^2)\) at small scaled frequency. Thus both poles are removable; in the time domain their antiderivatives have bounded memory on bounded spatial sets. At large scaled frequency integrate by parts in \(\sigma\). Its compact support is strictly at positive delays, so the bound is uniform for \(\operatorname{Re}\zeta\geq0\). Together with the tight profile orders \(H,H^1,V-e=O(r_y^{-1})\) and \(J=O(r_y^{-2})\), these calculations prove (64). Spatial derivatives on a fixed annulus give only fixed polynomial frequency costs. ◻

Proof. Work first where \(d(x)=r=|x|\), with the enlarged positive majorant \(\widetilde a=Cr^{-1}\int_{t-Cr}^t|a(\tau)|\,d\tau\). Assume the finite-slab bootstrap \(\int|a|\) is small, so \(r\widetilde a\ll1\). Integration by parts in the smooth positive-delay kernel, with no endpoint terms, gives \[ \begin{aligned} |\partial_t^j\partial_x^\alpha p_s| &\le C_{j,\alpha}r^{1-j-|\alpha|}\widetilde a, &&j+|\alpha|\ge1,\\ |\partial_t^j\partial_x^\alpha J_s| &\le C_{j,\alpha}r^{1-j-|\alpha|}\widetilde a, &&j,|\alpha|\ge0,\qquad J_s=\partial_x\lambda_s. \end{aligned} \tag{66}\] For the second line the zero first moment removes the constant-velocity term: \(J_s=O(r\widetilde a)\), while \(\partial_tJ_s=\partial_xv_s=O(\widetilde a)\). Each differentiated kernel is estimated separately by the positive majorant; the argument never differentiates \(\widetilde a\).

Put \(S_x=(I-J_s)^{-1}\). Implicit differentiation yields, in the independent variables \((t,x)\), \[ \partial_{z_i}=(S_x)^j{}_i\partial_{x_j},\qquad D:=\partial_{t,\mathrm{lab}}-v_s\cdot\partial_z=\partial_t|_x, \qquad [D,\partial_{z_i}] =(\partial_{z_i}v_s^j)\partial_{z_j}. \tag{67}\] Thus \(S_x-I=O(r\widetilde a)\), \(DS_x=O(\widetilde a)\), and \(\partial_xS_x=O(\widetilde a)\), with one inverse radius for each further ordinary derivative. Products of acceleration averages obey the same bounds since \(r\widetilde a\ll1\).

Proof. After pointwise normalization of the wave principal coefficient, the extra normal-acceleration coefficient has compact interior spatial order at most \(-1\). Choose it small on a low norm and on the finite base Sobolev list needed for iteration. Its identity-plus-correction inverse then extends to every smooth order by the derivative gain and Sobolev multiplication. The identity-minus-inverse error gains one spatial derivative; apply it to \(\partial_y^2h\) and \(\partial_y\partial_th\). The solved equation therefore retains its differential strictly hyperbolic principal part, and every nonlocal correction is at most first order on the wave jets, with tame bounds.

The compact kernels are bounded in a high spatial wave-pair energy and are Lipschitz one energy order lower on bounded sets. Metric multiplication inside their negative-order gains uses ordinary spatial Sobolev multiplication. Before spatial commutation, the equation contains at most \(\partial_tc\) from the parameter-dependent additions, a projection of the two available wave jets rather than a new unsolved acceleration. At any fixed base iteration order choose \(P\) with surplus so that spatial differentiation of bounded-radius histories uses only available projection jets. The implicit-position inverses use \(I-\partial_x\lambda_s\) uniformly close to the identity, and the filters are stable zero-past Volterra variables, equivalently a finite system of retarded ordinary differential equations.

On a short slab integrate the high-energy estimate and the one-order-lower contraction estimate, using suprema of earlier wave energies for the history terms. Work on regions containing the compact kernels with buffers and use uniformly local wave estimates outside. The inner boundary has strictly exiting flux. Conic localization preserves the spatial mapping properties and the time-local or retarded dependence. Both variables of every compact spatial kernel stay within the buffered interaction region, away from the outflow boundaries. This does not assert metric finite propagation inside that region; later geometric attachment uses ordinary local wave-map equations. At higher orders, spatial commutation through source order \(s-1\) uses at most \(a^{(s+3)}\) in targets and compensators. The filter gain charges these to at most \(s-1\) total-order projection jets when \(P\geq5\). Successive normal solves and wave-energy induction, or smooth regularization with these tame estimates, propagate smoothness without demanding smallness of the negative-order coefficient on every high Sobolev space. Outside the compact interaction region, evolution for fixed histories is local. Annular rescaling, together with the fact that sufficiently distant memories sample times before startup, propagates all symbol orders on bounded time slabs.

For constraint data, imposing \(\Upsilon=0\) determines the initial lapse and shift derivatives by the same small normal coefficient solve. The normal Einstein constraints in the reduced equation then imply \(\partial_t\Upsilon=0\) initially. The contracted Bianchi identity gives a homogeneous wave system for \(\Upsilon\); the compact terms from \(C\) have the same lower-order type. Uniqueness yields \(\Upsilon=0\) and \(\operatorname{Ric}(g)=0\). The startup cutoff on feedback ensures that the initial wave jets through the finite base budget are determined by the corresponding geometric data orders. None of these arguments requires stationary time to be timelike on the kernel supports. ◻

Proof. Expand the columns in outgoing time \(t_*=t-\phi\), equivalently multiplying the profiles by \(e^{\zeta\phi}\). In pairing their Taylor coefficients with \(\mathcal K_0^*\), discard terms \(L(0)H_0\) with \(H_0=O(r_y^{0+})\) by Lemma 14. This operation is essential for the coefficient class of the eventual polynomial mode. For a Kerr column the discarded coefficient includes \(\phi H\). For a translation column, the constant and linear coefficients of the conjugated profile are \(J\) and \(Z_1=H^1+\phi J=O(r^{-1+})\); its quadratic coefficient includes \(\phi H^1+\phi^2J/2\), together with the averaging and coordinate profile terms. The latter coefficients are \(O(r^{0+})\) and contribute only exact stationary terms \(L(0)H_0\) at the order under consideration. They are discarded in the adjoint pairing, not retained as constant coefficients of a polynomial to which Proposition 5 is applied. The residuals at the relevant orders are exactly the equation errors of \[ t_*H,\qquad \tfrac12t_*^2J+t_*(H^1+\phi J). \tag{70}\] These errors are stationary and \(O(r_y^{-3+})\). To see the extra decay, the outgoing linear Taylor coefficient on the end is a constant multiple of \(\partial_r+1/r\), modulo \(O(r^{-1})\partial_y+O(r^{-2})\), and annihilates the leading \(r^{-1}\) power. The outgoing quadratic coefficient is \(O(r^{-1})\) or better. These are the flat and Kerr asymptotic computations using precisely the orders of \(b\), \(\phi\), and the profiles established above.

At the local-gauge limit, if the resulting seven-dimensional coupling were singular, a nonzero combination of these residuals would pair to zero with the entire adjoint kernel. The improved \(O(r_y^{-3+})\) residual could then be solved away in a decaying \(O(r_y^{-1+})\) space. Add only this decaying stationary solution to the corresponding combination in (70). Every coefficient of the resulting outgoing-time polynomial then lies in the decaying class of Proposition 5; none of the discarded \(O(r^{0+})\) coefficients occurs in it. If its quadratic coefficient were nonzero, (70) would give a forbidden degree-two outgoing polynomial mode. Otherwise a nonzero linear combination of the four Kerr directions would give a degree-one mode with nontranslation leading coefficient. Both contradict Proposition 5. It is essential here to discard exact stationary terms before invoking solvability for the improved residual.

The added gauge preserves all identities used in these Taylor expansions. Uniform stationary compactness and the \(O(r_y^{-2+})\) adjoint tails give continuity of this finite coupling towards the local-gauge limit. Thus its rank persists for sufficiently large \(\Lambda\). ◻

Proof. We give the zero, intermediate, and high frequency arguments separately.

Near zero. Put \[\delta_f=F(\zeta)^tQ(\zeta)^t\chi_Rv_*, \qquad L(\zeta)^t v_*=f-\ell^t\delta_f.\] Use the spaces \(\mathcal A,\mathcal S\) of Lemma 12, and supplement their compact-error estimate by a compact measurement \(\Pi\) that separates \(\mathcal K_0^*\). It controls both \(e^{\zeta\phi}v_*\) and \(\delta_f\) with the extra forcing \(\ell^t\delta_f\) placed on the left. Indeed in a normalized contradicting sequence, the limiting stationary equation paired with \(H,J\) gives \(\delta_f=0\), because \(\ell(H,J)=I\). The measurement \(\Pi\) then kills its remaining stationary kernel element.

Choose \(v_0\in\mathcal K_0^*\) matching \(\Pi(e^{\zeta\phi}v_*)\). Its conjugated equation residual is \(O(|\zeta|)|v_0|\) in \(\mathcal S\), by the adjoint \(r_y^{-2+}\) tail with smaller slack and the stationary end bounds; here \(|v_0|\) denotes its finite-dimensional coefficient size. The supplemented estimate gives \[ \|e^{\zeta\phi}v_*-v_0\|_{\mathcal A}+|\delta_f| \leq C\bigl(\|f\|_{H^{q_*}}+|\zeta||v_0|\bigr). \tag{73}\] By the column bounds, pairing the physical difference \(e^{-\zeta\phi}(e^{\zeta\phi}v_*-v_0)\) against \(\chi_RQ\) costs at most \[C_\iota\min(R,|\zeta|^{-1})^{2\iota} \bigl(\|f\|_{H^{q_*}}+|\zeta||v_0|\bigr).\] On normalized \(v_0\), the pairing of \(e^{-\zeta\phi}v_0\) with \(\chi_RQ(\zeta)\) converges uniformly to \(Q(0)^t v_0\) as \(R\to\infty\) and \(\zeta\to0\). Lemma 18 and nonsingularity of \(F(0)\) therefore absorb the \(O(|\zeta|^{1-2\iota})|v_0|\) term after choosing \(R\) large and a sufficiently small fixed zero-frequency neighborhood. This gives the claimed compact bound and trivial augmented transpose kernel there. Lemma 12 supplies index zero and the return to physical inversion. Its stronger near-zero source estimate is applied only to these graph states, whose forcing is compactly supported. At infinity b-Sobolev regularity for \(e^{\zeta\phi}v_*\) gives (72) in this regime, using \(r_y(1-\phi')=O(1)\).

Closed nonzero bands. Use the estimates for \(L^t\) first with the full column \(Q\), whose tail rapidly decreases, and then treat the removed tail \((1-\chi_R)Q\) as small. Spatial integration by parts reduces the full finite-dimensional condition to \[ I+[I_H,I_J+\zeta\,\ell H^1]\, \operatorname{diag}(F_m/\zeta,F_v/\zeta^2). \tag{74}\] Here \([I_H,I_J]\) is the identity divided into its four and three mode columns. Formula (74) uses \(d=0\) on the probe supports and the exact stationary and chain identities.

This matrix is invertible in the closed right half-plane for sufficiently small \(\sigma_0\). It is a perturbation of the identity when \(|\zeta|/\sigma_0\) is large. When \(w=\zeta/\sigma_0\) stays bounded, multiply its four Kerr columns by \(w\) and its three translation columns by \(w^2\). The limiting diagonal blocks are \(1+w\) and \((1+w)^2\), which have no zeros for \(\operatorname{Re}w\geq0\). The zero-frequency neighborhood has already been handled; these two regimes cover the remaining closed bounded bands. Taking \(R\) large absorbs the omitted tail.

High frequency and radiation. The polynomial transpose losses from Proposition 6, the polynomial column bounds on compact sets, and their rapid tail decrease are absorbed by taking \(P\) large. Away from zero, \(L^t v_*\) is supported in a fixed compact set. At the end we obtain the radiation estimate as follows. Commute rotations in Cartesian components, with cutoff in an elliptic annulus. Their order-two coefficient errors are \(O(r_y^{-2})\) by rotation invariance of the mass correction. The scattering estimate and exterior spatial ellipticity control these unscaled derivatives and fixed-annulus errors with polynomial cost. For \(w_*=r_yv_*\) this gives, with the requisite rotations, \[(w_*,\partial_{r_y}w_*)\in L^2(r_y^{-1-2\iota}dr_y\,d\omega)\] with the asserted right side. In tortoise coordinate the source-free radial equation is \[ (\partial_{r_*}-\zeta)(\partial_{r_*}+\zeta)w_* =O(r_y^{-2})\langle\zeta\rangle^2 O\bigl(\partial_\omega^{\leq2}w_*, \partial_\omega^{\leq1}\partial_{r_*}w_*\bigr). \tag{75}\] This follows by dividing by the radial second-derivative coefficient in Boyer–Lindquist Kerr. For positive real part, integrate the first factor from infinity; Cauchy–Schwarz with the scattering weight costs \(O(r_y^{-1+\iota})\). To see the exact radial gain, if \(w_*=r_yv_*\), the scattering weight gives \[\int_r^\infty \rho^{-2}|w_*|\,d\rho \le \left(\int_r^\infty\rho^{-1-2\iota}|w_*|^2\,d\rho\right)^{1/2} \left(\int_r^\infty\rho^{-3+2\iota}\,d\rho\right)^{1/2} \lesssim r^{-1+\iota}.\] The same estimate holds for the finitely many angular and radial jets on the right side of (75). The first integration has a kernel bounded by one; the second costs only a constant proportional to \(1/\iota\). Reducing \(\iota\) changes constants but not the polynomial frequency degree, because all differential orders were fixed beforehand. Integrate the second factor outwards. Returning to the ordinary good derivative uses \(r_*'=1+O(r_y^{-1})\). This proves (72), with all compact modifications interior to the computation.

Choice of constants and causality. First fix the finite testing and differentiation orders and their polynomial losses. Choose \(P\) and then \(\sigma_0\). A forward energy line can be fixed uniformly before finalizing \(\Lambda,R\): the acceleration coefficients are small, the compact symbols are uniformly bounded, and \(Q\) acting on the filtered parameters is a bounded energy forcing by its memory estimates. The high and near-zero thresholds are uniform for sufficiently large \(\Lambda\). Enlarge \(\Lambda\) for the intervening band, and then take \(R\) sufficiently large. This is an augmented argument using column identities, rank, and transpose graphs; it does not require persistence of unaugmented inversion uniformly to zero.

On the open strip, the physical graph domains are fixed locally in \(\zeta\): the end is parameter elliptic and compact first-order changes are bounded on the graph. The inverse is analytic there. Starting on the forward line, wave and Volterra uniqueness identify the tested inverse as causal. Contour displacement within the strip, using (71)–(72), gives the retarded tested operators down to the boundary of the physical half-plane. ◻

Proof. Apply the augmented retarded operator \(M_R(\partial_t)\) to \(\chi_R U\); for \(U=u\) differentiate the reduced equation first. The first variation at the base is exactly \(Lh+Q(\partial_t)a\). After this linear part is removed, the compact forcing other than \([L,\chi_R]U\) and startup terms has norm \(C_{\rm fix}\mathsf N^2\) in \(L^2_{A_i}\overline H^4\); the bar means extendibility at the outflow boundary. We spell out why the arbitrary-spin localization does not alter this fact. A varying coefficient on a field jet costs \(p,c,h\), their derivatives, or their bounded memories, and never an uncontrolled position \(\lambda\). At zero field the exact affine-position freezing identity removes the target defect; its nonlinear remainder therefore contains a strong acceleration factor. After one time derivative a changed coefficient on an undifferentiated-in-time field jet contains \(u,\dot c\), or an acceleration jet. A product of two weak factors also suffices, since \(2A_H>A\). There are at most two ordinary differentiations before the four spatial source derivatives. These products lie strictly below the half-loss bulk orders in (78). The compact kernels preserve the stated orders, and their distant bounded reaches use the compact-far overlap norms. Startup terms and filtered startup tails are bounded by finite-time stability in terms of \(e_{\rm in}\).

Commuting \(\tau^{A_i/2}\) through bounded memories and the stable convolution filter costs \(C_R T_0^{-1}\) times the matched budgets, together with initial data terms. Apply also the causal smoothing filter \[\left(\frac{c_*}{c_*+\partial_t}\right)^{M_*},\] where \(M_*\) exceeds the fixed polynomial test losses, including those of a once differentiated output. For the shell forcing, the integration by parts gives the leading Wronskian \[ \int \chi_R' b^{rr} (v_*\partial_rU_*-U_*\partial_rv_*)r^2\,dr\,d\omega, \qquad r=r_y. \tag{83}\] Here \(v_*\) is the transpose test and \(U_*\) is the weighted temporal transform. Replacing both radial derivatives by \(\partial_r+\zeta\) leaves the Wronskian unchanged. Angular, density and short-range terms have no worse orders by the Boyer–Lindquist asymptotics; compact additions produce no shell term. The transpose radiation estimates and Plancherel thus bound (83) by \(C c_*^{N'}R^{4\iota}\) times the shell norm of \[(1,\Omega_{\rm co},r_y(\partial_{r_y}+\partial_t))U\] in \(L^2(\langle t\rangle^{A_i}dt\,dr_y\,d\omega/r_y)\), up to the plateau errors already listed.

This shell norm is bounded by \[ C R^{-(p_i-A_i)/2}\mathsf F +C_{\rm fix}(e_{\rm in}+\mathsf N^2). \tag{84}\] Indeed \(y=rn+\lambda(t)-\lambda(q)\), so the two good-direction systems differ by small parameter coefficients times full first derivatives. The component transformation uses \(v(t)\), and the difference between \(u\) and the transform of \(\psi\) is \(O(v_s-v)\partial h_{\rm lab}+O(a)h_{\rm lab}\). On this fixed shell the coefficient differences, also after the required differentiation, have bounded-memory strong weighted bounds. Bad derivatives therefore occur only in products, with allowable \(R\)-dependent constants.

All tested multipliers are causal: identification on a forward Laplace line and strip displacement apply separately to the two test kernels produced by the Wronskian rearrangement. We can consequently stop forcing at \(T\) when estimating earlier outputs without paying final-time data. Initial jumps are absorbed by the smoothing filter. Taking duals of sufficiently high-order compact test bounds gives the negative spatial norm defining \(\mathsf l\). Removing the auxiliary filter costs \(C c_*^{-1}\) times adjacent local time-jet budgets and startup transients. Choose \[c_*=R^{\kappa'},\qquad \kappa'N'+4\iota<\frac12\min_i(p_i-A_i),\] and then take \(T_0\) sufficiently large after \(R\). All \(p_i-A_i\) are positive, so this gives (82). The powers arise only from the fixed test loss and far weights, not from constants on compact reaches. Interpolation is consequently available at every order strictly below the corresponding bulk order. No full-order bulk estimate on a base-space stationary-timelike trapping region was used. ◻

Proof. We first construct dissipation near trapping and the horizon, and extend it through the compact propagation region. We then join these forms to literal differential currents near the boundaries. Finally we recover the original first jets and record how the energy changes with the stationary parameters.

Proof. We first derive the source regularity, then estimate the acceleration by adjoint pairing, and finally recover the static field. At sufficiently late time choose \(\chi_t=1\) for \(r_y\le c_1t\) and \(\chi_t=0\) for \(r_y\ge2c_1t\), with \(c_1>0\) small enough that its support lies in \(t\ge Cr_y\). The fixed earlier interval costs input. With \(p=p(t)\) the sliced equation is \[ L_p(0)(\chi_t h_{\rm co}) =\chi_t(\mathcal R_h-f_B)+[L_p(0),\chi_t]h_{\rm co}. \tag{93}\] The defect \(\mathcal R_h\) subtracts the nonstationary reduced equation and compares all field-dependent terms with their stationary linearized action at \(G_p\).

Proof. We compare the frozen and evolving equations, retaining the actual principal coefficients and checking every source derivative count.

Proof. Apply Proposition 13 to the frozen energies at the three levels (76). Put a nonincreasing cutoff, with derivative \(O(R^{-1})\), on the outer differential current. Its unfavorable squared flux is bounded, up to finite-time costs, by \[ C\sum_{|\mu|\le s-1} \|\tau^{A_i/2}(1,\Omega,rX)\partial_y^\mu U \|^2_{L^2(dt\,dr\,d\omega/r;\,\mathrm{shell})}. \tag{107}\] On a pure outgoing flat covector the flux exits strictly because \(f_\infty<1\). After the matching currents are fixed, the affine, cutoff normal and cone changes are small for large \(R\) and a small bootstrap. Finite-dimensional definiteness therefore bounds every unfavorable component by the good derivatives of \(\chi_\mu\); differentiating its radius factor accounts for the order-zero term in (107).

The square root of (107) is at most \[ C R^{-\kappa_b}\mathsf F+C_{\rm fix}(e_{\rm in}+\mathsf N^2), \qquad 0<\kappa_b<\min_i\bigl\{(p_i-A_i)/2,0.98\bigr\}. \tag{108}\] Indeed \[\partial_{z_d}=-n_dT_l+r^{-1}(n_drX+\Omega_d)\] expands a spatial derivative string into \(r^{-b}c(n)Z^IT_l^j\), with \(j+|I|\le|\mu|\) and \(b\ge|I|\) at the flat level. Order the lab time derivatives before \(Z\) derivatives. Perturbed emitter and co-frame coefficients add only small coefficient products controlled by the fixed-annulus overlap norms. The \(I=0\) terms use the good-weight gain in (80); every \(I\ne0\) term has an inverse radius, and the bad weights \(\gtrsim t^{A_i}r^{-0.04}\) then suffice. At the strong level the difference between \(u\) and the transformed \(\psi\) contains strong bounded memories such as \(v(t)-v_s,a\), and at most nine derivatives of \(h_{\rm lab}\). At the \(h\) levels any time derivative of the component change carries strong acceleration and lower fields. Thus full good and bad first-jet far controls through \(j+|I|\le s-1\) suffice. The earlier fixed interval, allowed to depend on \(R,T_0\), costs input by finite-order Cauchy stability.

All fixed compact lower derivative errors have total spatial or first-time order at most \(s-1\), strictly below the half-loss bulk. At matched time weights interpolate them by \[ R^{-\iota_0}\mathsf K_R+C R^{K\iota_0}\mathsf l, \qquad K\iota_0<\kappa_0/2, \tag{109}\] where \(K\) is a sufficiently large fixed interpolation constant and \(\kappa_0\) is from Proposition 10. Very-low-order coercivity supplements fit the same estimate. After substituting (82), the data and quadratic pieces have only \(C_{\rm fix}\) costs. The linear forcing pairing from (101) is at most \[C R^{\delta/2}\mathsf l\,\mathsf K_R +C_{\rm fix}e_{\rm in}\mathsf N.\] Choose \(0<\delta<\kappa_0\) sufficiently small. The low-transfer exponent is already fixed before these enlarged compact reaches: their constants and the outer current’s slack power do not worsen its test loss or far-weight gain.

Differentiating the time weight uses the unweighted top bulk for the extra half derivative. On a fixed compact set, for example, \[ \tau^{A_i-1}\|U\|_{H^s}^2 \lesssim\tau^{A_i/2-1} \bigl(\tau^{A_i/2}\|U\|_{H^{s-1/2}}\bigr) \|U\|_{H^{s+1/2}}. \tag{110}\] The analogous first-time-jet estimate shifts the spatial order by one. For the strong level it uses \(u_t=\partial_t^2h\) in unweighted \(H^{7.5}\) bulk. On farther bounded annuli the permitted radius-dependent constants and mixed top orders suffice. Since \(A_H,A<2\), taking \(T_0\) large after \(R\) makes these coefficients as small as required. Integration up to arbitrary endpoints, using (108), (109) and the cubic comparison errors, proves (106) first for spatial and first-time jets, with the full mixed budget still on its right.

It remains to recover that mixed budget. Successively solve the equations with the stationary normal coefficient inverse, including its order-\(-1\) addition, after \(j\) time derivatives, for \(0\le j\le s-2\). Lower time jets in the stationary homogeneous part have already been bounded; parameter derivatives on coefficients are products or input. A nonlinear term, including a principal difference on the current time jet, needs only bulk order \[r'=s-j-2-\theta, \qquad\theta=1/2\text{ globally},\quad \theta=0\text{ in a full-order region},\] and endpoint order \(s-j-2\). Its norm is \(C_{\rm fix}\mathsf N^2\) by the bootstrapped mixed estimates. Explicitly, if \(k\) time derivatives hit an undifferentiated field coefficient on an order-two jet, put that coefficient in endpoint \(H^{s-k}\) and the other factor in bulk \(H^{r'+k}\). The product estimates have the margins \[r'\ge-1/2,\qquad s-k\ge r'+2,\qquad (s-k)+(r'+k)-r'=s.\] Two first-order placements and further smooth coefficient factors have the same margin. Place the matched time weight on one factor and use two endpoints for endpoint output. Products containing \(u\) use unweighted top \(h\) and strong \(u\); drifts involving \(h\) alone carry strong differentiated parameters. Kernels preserve or improve spatial order and have at most two time derivatives before this mixed differentiation. The preceding parameter counts cover all cases. The linear source is still filtered low measurement with fixed compact costs.

For full mixed bulk outside physical trapping, use nested smooth spatial cutoffs with margins. Place the cutoff on the bulk factor in products; commuting it through stationary kernels gains a full spatial order, more than the global half-order loss. No full \(B_{\rm nt}\)-cut estimate for \(j\ge2\) is needed inside the remaining physical neighborhood. Additional mixed norms on larger annuli follow at \(R\)-dependent cost and are included with the small normalizations specified at the start; they occur elsewhere only in products or plateau choices. Thus the left side becomes the full \(\mathsf K_R\), proving (106). ◻

Proof. We use induction in derivative order, with the conic energy and its parameter counts explicit. The closed base estimates imply, uniformly on the slice in the low unscaled supremum norms needed below, through at least two field derivatives, \[ h=o(1),\qquad g_{\rm co}-\bar g=o(1),\qquad c,\dot c,a=o(1),\qquad t\longrightarrow\infty. \tag{111}\] On fixed compact sets use weighted endpoints; at large radius use the far point estimates, including [fe:F7] and [fe:F8], with their spare rotations. They give uniform decay of low unscaled lab derivatives as \(r\to\infty\), while the co transformation adds bounded velocities and differentiated parameters. Fixed annuli have decay in \(t\). Also \(B-G_p=o(1)\): compactly this is a bounded memory, and near infinity both leading co asymptotes are exactly the affine flat metric \(\eta_{v(t)}\), with symbol-decaying differences and bounded differentiated frame changes. True low reference derivatives minus their formal frozen counterparts are \(o(1)\). The affine changes and symbol remainders in \(\bar g-G_p\) use \(c=o(1)\). No convergence of the coordinates is required, although \(p\) does converge because \(a\in L^1\).

Fix a terminal time \(T'\) and work on shrinking cylinders with outer radius \[ C_0(1+T')+C_1(T'-t), \tag{112}\] where \(C_1\) is large enough for strict exit and all compact kernels have a buffer inside. Induct on total order, using a larger such cylinder for the previous order if needed. Orders through seven start the induction: bulk bounds through order eight convert to unscaled \(L^2\) bounds with at most polynomial losses in \(T'\) from the measures and radial weights on these growing balls. Conversion of derivatives uses lab time, \(r^{-1}Z\), bounded velocities and the available filtered acceleration jets. Enlarged spatial regions and backward unit slabs give order-seven endpoints; startup uses fixed finite-time bounds. The spare rotations in the far point estimates ensure the uniform low decay used in (111).

At each \(s\ge8\), Proposition 11 is available near the limiting stationary parameters. Exact compatible saturation at trapping is independent of \(s\), and the horizon inequality becomes stronger. For this one order, start at a sufficiently late time that the actual-principal correction preserves positivity; the preceding fixed interval has smooth symbol-tail estimates by finite-time propagation. Continue the differential current to the exiting boundary (112), with no matching-shell loss. Its top-order energy equivalence constants are independent of the terminal radius, since \[ \|\widetilde r G\|_{L^2(d\widetilde r\,d\omega)}^2 =|\det A_{\rm aff}|\|G\|_{L^2(d^3y)}^2, \tag{113}\] and \(f_\infty<1\) is now fixed. Lower terms from differentiating \(\chi_\mu\) use the preceding induction. The parameter derivatives of \(\widetilde r\) relative to \(\widetilde r\), of the affine angles, and of the density ratio are uniformly bounded, so differentiating current coefficients incurs no terminal-radius factor. The frozen derivative is favorable modulo the fixed compact lower order already estimated.

In the principal comparison use \(o(1)\) instead of integrability for the low top coefficients. Products of finitely many lower-order norms and polynomial losses are \(e^{o(T')}\). The parameter and pure target jets in spatial source order \(s-1\) have this property. The smoothing, implicit position map and compensator may require acceleration jets through \(a^{(s+3)}\), accounting for spatial derivatives of shifts and the two-order wave expression. The filter has worst differential order \(1-P\); hence these jets require at most \[ \max(0,s+4-P)\le s-1 \tag{114}\] derivatives of \(\ell h\) once the single fixed \(P\ge5\) is chosen. The larger fixed \(P\) needed by the base resolvent tests only improves this count. Smooth probes, position integrations, exponential or bounded/reach memories cost lower products and polynomials. The original spatial equation uses only low \(c\) jets in its kernel families.

Solving for \(\partial_t^2h\) in \(H^{s-2}\) in a commutator gives \(C_s\sqrt{\mathcal E_s}+e^{o(T')}\), where \(\mathcal E_s\) is the positive first-jet energy. Besides the differential pointwise solve, the order-\(-1\) addition uses only its preceding induction order and coefficient products. For example \(K_{-1}\partial_t^2h\) needs \(H^{s-3}\) on that jet and coefficient algebra orders at most \(s-2\). Consequently, after exact absorption of the differential principal part, the source difference at \(H^{s-1}\) costs \[o(1)\sqrt{\mathcal E_s}+e^{o(T')}.\] Placements at orders already covered by induction give the additive term. Top first derivatives multiply low vanishing differences. Top second derivatives in coefficient commutators or lower acceleration terms have the one-order spatial gain and an \(o(1)\) Lipschitz or coefficient difference. Interior paraproduct pullbacks and differential currents also use only low differences in front of top squared energy. The outer products are estimated in uniformly local unscaled coordinate patches of bounded overlap, and the affine source factor changes density as in (113). Every top placement thus has a coefficient bounded, or vanishing, independently of the terminal radius; high coefficients of order at most \(s-1\) multiply low arguments and belong to the additive induction bound. If an interior kernel or paraproduct acts on a low placement, the base estimates supply a fixed compact Sobolev or Hölder buffer. No polynomial in \(T'\) multiplies the top incremental coefficient.

It follows that \(X_s=(1+\mathcal E_s)^{1/2}\) satisfies on the late interval \[ \begin{gathered} \dot X_s\le\alpha_s(t)X_s+B_{s-1}(T'), \qquad B_{s-1}(T')=e^{o(T')},\\ \alpha_s(t)\longrightarrow0 \quad\text{independently of the terminal radius}. \end{gathered} \tag{115}\] For any prescribed \(\sigma>0\), make \(\alpha_s\le\sigma/3\) by a later fixed starting time, use the induction with a rate smaller than \(\sigma/3\) on \(B_{s-1}\), and absorb the polynomial integration factor. Gronwall gives \(X_s\le C_\sigma e^{\sigma T'}\). This proves the spatial and first-time orders.

Recover the higher time jets successively. The normalized actual \(I+K_{-1}\) is uniformly invertible on spatial \(L^2\) by low-order smallness. At \(H^{s-j}\) for \(\partial_t^jh\), \(2\le j\le s\), iterate its tame negative-order gain; coefficient derivatives through at most \[\max(s-j+1,4)\le s-1\] and fixed symbol seminorms suffice. These have the induction bound. The right side uses \(c^{(j-1)}\) or lower, already recovered in earlier time counts, while \(p\) derivatives have the filter buffer (114). Full mixed endpoints are therefore subexponential without smallness of any high Sobolev norm of the gauge or datum. Sobolev embedding on enlarged regions and finite products give the stated smooth bounds. Every estimate was on a common finite slab; no limiting inner horizon has entered the argument. ◻

Proof. The far background in the formal lab coframe has the form \(\eta+\mathfrak h(p_s(t,x),x)\), where, at independent family argument, \[\partial_p^b\partial_x^\alpha\mathfrak h =O(|x|^{-1-|\alpha|}).\] This statement includes the coframe change and holds for every member of the chosen fixed Kerr parameter family. Implicit differentiation of the smoothed coordinates gives \[\partial_zx=S_x=(1-\partial_x\lambda_s)^{-1},\qquad T_lx=S_xv_s,\qquad S_x-I=O(r\widetilde a).\] The first derivatives of \(S_x\) and \(v_s\) are \(O(\widetilde a)\), with the corresponding differentiated symbol bounds. In particular \(Dx=0\). Positive-delay smoothing and its signed moments imply \(\partial_x\lambda_s=O(r\widetilde a)\). The affine flat leading term cancels exactly in lab components. Comparing at a point with an exact constant-parameter Kerr metric whose affine worldline agrees there gives the first- and second-jet errors \(O(r^{-1}\widetilde a)\) and \(O(r^{-2}\widetilde a)\). Commuting \(D\) past a lab derivative introduces a derivative of \(s_s\); this proves the differentiated bounds for \(B\) and its defect.

For clarity, the compensator uses the stationary profiles with orders \(H_{\mathrm{mode}},H^1_{\mathrm{mode}},V_i-e_i=O(r^{-1})\) and \(J=O(r^{-2})\) from the stationary construction. We append the subscript here only to distinguish the mode profiles from \(H^{ij}\). Its local gauge image consists of first-order commutators against the smoothed mass and velocity parameters, terms containing \(J\) and \(\partial_x\lambda_s\), and the gauge derivative of \((V_i-e_i)^\flat\odot d_x\lambda_s^i\). When only one derivative falls on \(\lambda_s\) in the last term, a derivative of the profile, or a zeroth-order connection coefficient, supplies the extra inverse radius. The chain identities cancel the free position and velocity terms. Thus neither an unsmoothed acceleration nor a growing free position occurs in a lab symbol coefficient. Filtered unsmoothed accelerations and the finitely many low jets needed here are bounded by \(C_{\mathrm{fix}}\mathsf N\langle q\rangle^{-A/2}\) and vanish before startup. These observations prove (121).

Constraint propagation gives the exact lab identity \[\Gamma_\mu(g)=g_{\mu\sigma}g^{ab}\Gamma(B)^\sigma_{ab}+\Theta_\mu,\] where \(\Gamma_\mu(g)\) is the lowered wave contraction. Expanding this identity gives (123). Expanding vacuum Ricci in this gauge gives (122). The added nonlocal gauge terms vanish in the far region; only the differential wave compensator remains. The coefficients of the linear terms use first and second jets of \(B\), their products, and compensator jets, so their claimed \(D\) gain follows from (121) and \([D,\partial]\).

For a symmetric polarization \(p\) satisfying \[ \ell_n^\alpha\bigl(p_{\alpha\beta} -\tfrac12\eta_{\alpha\beta}\operatorname{tr}_\eta p \bigr)=0, \tag{124}\] the flat Christoffel identity of 10.2 says that \(Q_\eta(\ell_n\otimes p,\ell_n\otimes p)\) is a constant multiple of \(\ell_n\otimes\ell_n|\Pi(n)p|^2\); its polarized version holds as well. Indeed the constraint eliminates \(p_{LL}\), \(p_{LA}\), and the screen trace in a flat null frame. The linear map to the four contractions in (124) is surjective on \(\ker\Pi\). Consequently there is a uniformly smooth projection to the constrained polarizations that leaves \(\Pi p\) fixed and changes \(p\) by bounded multiples of those four contractions; this is [fe:gauge-projection]. Apply the differentiated identity (123) to \(p=T_l^{j+1}h\) before taking this projection. After multiplying a first jet by \(r\), every error contains a good derivative, a vanishing coefficient times an ordinary scaled jet, a symbol-suppressed field, or \(O(\widetilde a)\). Distributed lab-time differentiations retain exactly these placements. This verifies the differentiable weak-null structure, not just its undifferentiated algebraic identity. ◻

Proof. Initially the lapse and shift have background values and their derivatives are fixed by the gauge. At infinity startup has zero parameters and no kernel terms. The wave equation supplies mixed jets through order ten, with one inverse radius per derivative. The incoming weighted integrals converge because \(p_0,p_H<1\) for \(\chi(h)\), whereas \(\psi\) has an additional inverse power. Reduced local propagation gives finiteness of all higher symbol orders on bounded time intervals without making their constants small.

On a fixed-factor overlap at \(R_0\), all unscaled mixed lab jets of \(U_i\) through order \(N_i+1\) have full compact bulk and endpoint control, with squared time weight \(\langle t\rangle^{A_i}\). The component transformation from \(h_{\mathrm{co}}\) to \(h_{\mathrm{lab}}\) depends pointwise on \(v(t)\). Under this transformation \(\psi\) equals \(u\) plus terms of the forms \((v(t)-v_s)\partial h_{\mathrm{co}}\) and \(v'h_{\mathrm{co}}\) with smooth bounded component factors. Their coefficients carry the strong time weight by the fixed-interval acceleration averages and parameter jet bounds. Inverse-coordinate and differentiated-frame costs are bounded on this fixed overlap, and \(\partial^{\le9}h_{\mathrm{co}}\) has at least the weak weighted bound. These terms therefore contribute products, not a radius-growing linear coefficient on \(\mathsf S_a\). The emitter commutations have bounded conversion coefficients there. The weights \(W_i,w_{i,j}\), and the factor \(1+M_f\) defined below cost at most the indicated time weights, up to \(R_0\)-dependent constants; for \(1+M_f\) the cost is bounded. This proves (125) and verifies the boundary hypotheses of [fs:claim-boundary]. ◻

Proof. Apply the complete source estimate, Proposition 34, to the lab equation of Lemma 19 and its \(D\) derivative. The lemma supplies its background symbol bounds, the additional \(D\) gain on the defect and linear coefficients, and the differentiated weak-null polarization. The point bounds in (126)–(130) supply the product estimates. Lemma 20 supplies the full mixed-order, physically nontrapping overlap and the initial symbol input. These are the actual inputs of the six source estimates, rather than bounds on an already global solution.

Two features explain the choice of levels. A common principal commutator places the differentiated coefficient on a lower field. An extra bad laboratory-time derivative on \(h\) can be replaced by \(\psi\) and good derivatives using \(Dh=\psi\); the raw \(h\) estimates therefore use the prior strong level. Conversely the differentiated equation is \[ g^{ij}\partial_{ij}\psi =Df_h-(Dg^{ij})\partial_{ij}h +2g^{ij}(\partial_is_s^d)\partial_{jd}h +g^{ij}(\partial_{ij}s_s^d)\partial_dh. \tag{141}\] Its seven commutations use \(B^h\) indices through eight in the notation of (271): this means at most nine physical derivatives of \(h\), available at level \(H\). Its twice-null coefficient has the extra inverse radius because \[ \begin{split} T_l^m\psi={}&(1-s_s\cdot n)T_l^{m+1}h +r^{-1}s_s^db_dZT_l^mh\\ &+\sum_{j<m}c_{jm}(T_l^{m-j}s_s^d)\partial_dT_l^jh. \end{split} \tag{142}\] The differentiated gauge controls the first term without a stationary background offset, since it is a derivative of \(h\) itself. The other terms carry an inverse radius or an acceleration symbol. The full coefficient placements and transferred angular derivatives are checked in Claims 1 and 2 of Proposition 34.

The pure defect uses a different estimate from field products. After the \(r^2\) source scaling it is \(O(\widetilde a)\) for \(h\) and \(O(r^{-1}\widetilde a)\) for \(\psi\). The averaging estimates (284) give squared shell powers \(p_i-1,p_i-A\) at the two \(h\) levels and two additional inverse powers at level \(P\). All are negative, as recorded in (285). In particular its starting-annulus cost has a small coefficient on \(\mathsf S_a\), not a growing compact coefficient. The remaining source terms are handled by the strict radial and time-weight margins in Claims 3–5; the leading strong weak-null term uses the third current correction above.

Thus every correction, remainder, raw and mixed source budget has the bound (276). The size-one terms respect the finite order: level \(P\) before the \(h\) levels, lower total count, more lab times at fixed count, and then fewer dilations. The flat recurrence and its nonnegative diagonal term are proved in Claim 3 of Proposition 33. Its Claim 5 chooses fixed hierarchy weights before increasing \(R_0\); the source constants on prior far energies are independent of both matching radii. The two corrected multiplier estimates therefore give (140). ◻

Proof. The common finite-slab norm includes the three compact levels with mixed jets, the far levels including (137), \(\mathsf S_a\), and the static norms of (92) on fixed reaches containing all compact error supports and deep matching regions. Include its extra nontrapped time orders and the low norm \(\mathsf l\) when needed. Filtered feedback from the strong \(u\) level supplies the base parameter jets. The static estimate supplies the integrable coefficient difference near trapping; in the compact high estimate it multiplies another bootstrap factor and therefore creates no additional linear feedback term.

Choose \(R_0\) so large that \(\varepsilon_0(R_0)(1+C_*)\) is as small as needed. Substituting (145) into (144) and absorbing the resulting \(\mathsf F\) term bounds \(\mathsf F+\mathsf S_a\) by a fixed \(C(R_0)\mathsf K_R\) plus input and superlinear errors. Now enlarge \(R\) so that substitution in (143) absorbs its compact feedback. Finally enlarge the plateau time \(T_0\) after \(R\) to obtain the plateau-small terms used in that compact estimate. This is the order of [fs:coupled-inequalities], and it has no radius cycle.

We spell out which earlier choices this order permits. The inner profile depth and finite Cauchy-join margins are fixed first. The slightly enlarged spacelike and outflow cylinders are therefore fixed before their stationary gauges. Measurement and gauge supports, the high spatial cutoff \(\Lambda\), the feedback filter, the small exponent in the compact low transfer, and then the outer-current slack can all be fixed independently of \(R_0,R\). Their polynomial frequency degrees require only fixed differentiation, propagation, and testing orders. Compact stationary energy gluing to the differential end, and the cut \(Y\) in the first sliced tail, are fixed before \(R_0\) as well. After choosing \(R_0\), prior fixed compact reaches may include its starting overlaps, with constants independent of \(R\) and unchanged exponents. The original choice of \(R_0\) also places the auxiliary anchor exterior (131) in the prescribed lab end. Every constant needed for a radius-uniform absorption was fixed before \(R_0\); all other linear far costs were explicitly put on \(C(R_0)\mathsf K_R\).

The low norm and the second sliced norms cost only fixed linear constants times the three absorbed norms, plus data and superlinear errors. Thus for some \(c_2>0\), \[ \mathsf N\le C_{\mathrm{fix}} \bigl(\mathcal R(e_{\mathrm{in}},\mathsf N) +\mathsf N^{1+c_2}\bigr), \tag{146}\] where \(\mathcal R(e_{\mathrm{in}},\mathsf N)\to0\) as \(e_{\mathrm{in}}\to0\), uniformly on bounded small bootstrap ranges. This notation includes the mixed positive input powers in (143). Choose a bootstrap cap so small that the superlinear term is less than one quarter of that cap, then choose the base input small enough that the other term is less than one quarter. The resulting strict improvement is uniform in the terminal time.

For smooth finite-time solutions these norms vary continuously with the slab endpoint. For the cap suprema use the spacelike transversality of (134), as in the construction of 10.7. Finite-time symbol bounds and strict radial convergence margins control the tails. Startup and earlier bounded intervals have ordinary finite-time wave stability. Since \(A>1\), the acceleration bound keeps the parameters small, and the spacelike, outflow, and wave-coefficient continuation conditions persist. Proposition 8 now supplies causal local continuation and constraint propagation, hence global smooth vacuum evolution on both cylinders. Its finite-time tame propagation of higher smooth orders requires finiteness, not smallness, of those orders; no higher-order trapping energy has been invoked to reach this conclusion.

All small initial norms used here are supplied by \(p_{10}\) for the fixed cylinder-slice construction. The order-nine, eight, and seven far commutations in (117) carry one additional energy derivative, exactly matching the compact field orders ten, nine, and eight for \(h,h,u\), respectively. The higher-order induction is applied only after this base closure. Its later datum- and order-dependent starting times and constants impose no further smallness on the common neighborhood. ◻

Proof. Align the spin axes smoothly in the central trapped block. Let \(T_K\) be signed Boyer–Lindquist Killing time, and let \((\vartheta,\varphi)\) denote the corresponding sphere variables. For the current exact physical parameters, put \[\Sigma_K=r^2+\mathfrak a^2\cos^2\vartheta,\quad \Delta=(r-r_-)(r-r_+),\quad r_*'=\frac{r^2+\mathfrak a^2}{\Delta},\quad \kappa_-=\frac{r_+-r_-}{2(r_-^2+\mathfrak a^2)}.\] The Boyer–Lindquist metric is \[ \begin{split} g_K={}&-\frac{\Delta}{\Sigma_K} (dT_K-\mathfrak a\sin^2\vartheta\,d\varphi)^2 +\frac{\Sigma_K}{\Delta}\,dr^2+\Sigma_K\,d\vartheta^2\\ &+\frac{\sin^2\vartheta}{\Sigma_K} ((r^2+\mathfrak a^2)d\varphi-\mathfrak a\,dT_K)^2. \end{split} \tag{150}\] Temporal and axial conventions may be reversed simultaneously when passing between the two ends.

Solve the axisymmetric Hamilton–Jacobi equation \[ 2(r^2+\mathfrak a^2)F_r+\Delta F_r^2 +|d_\omega F|_{\mathrm{rd}}^2 +\mathfrak a^2\sin^2\vartheta=0, \qquad F(r_-)=0, \tag{151}\] where the norm is that of the round sphere. The finite root is noncharacteristic at the initial sphere, since its coefficient of \(F_r\) is \(2(r_-^2+\mathfrak a^2)>0\). Smooth Hamilton–Jacobi evolution therefore gives a parameter-dependent solution on a short radial interval, uniformly over the chosen compact parameter set. The equation is smooth on the whole sphere, and uniqueness preserves axisymmetry. The inverse of (150) gives \[\begin{split} \Sigma_K g_K^{-1} \bigl(d(r_*+F\pm T_K),d(r_*+F\pm T_K)\bigr) ={}&\Delta\left(\frac{r^2+\mathfrak a^2}{\Delta}+F_r\right)^2 +|d_\omega F|_{\mathrm{rd}}^2\\ &-\frac{(r^2+\mathfrak a^2)^2}{\Delta} +\mathfrak a^2\sin^2\vartheta. \end{split}\] Thus (151) is precisely the null equation.

Define \[ Q=e^{-2\kappa_-(r_*+F)},\qquad s=-\frac{\log(Q/q_0)}{2\kappa},\qquad \lambda_- =\frac{\kappa_-}{\kappa},\qquad x=\lambda_-T_K. \tag{152}\] Both \(s-x\) and \(s+x\) increase to the future near the inner horizon. Indeed their sum has temporal gradient, with \(s\) increasing as \(r\) decreases. The parameter-smooth normalization \[r_*=-\frac{1}{2\kappa_-}\log(r-r_-)+\text{smooth}\] shows that \(Q\) is \((r-r_-)\) times a smooth strictly positive factor. The factor \(\lambda_-\) in both optical directions is what makes \(\kappa\) common to every profile.

Corotate the sphere by \(-\Omega_-T_K\), where \(\Omega_- =\mathfrak a/(r_-^2+\mathfrak a^2)\). In the resulting \((s,x,\omega)\) coordinates, coefficients are independent of \(x\) and invariant under axial rotations. To see their regularity, the pairings of \(\partial_{T_K}+\Omega_-\partial_\varphi\) with itself and with angular vectors are \(O(Q)\) by (150). The simple pole of the radial coefficient is canceled by \(r_s,\partial_\vartheta r|_s=O(Q)\), and \(r_s/Q\) is smoothly nonzero. Hence the angular cross coefficients with \(ds,dx\) are \(O(Q)\), while the angular metric extends smoothly and positively to \(Q=0\). The absolute spacetime volume divided by \(Q\) is smooth and positive, as follows from the Boyer–Lindquist density \(\Sigma_K\) relative to round sphere measure. These are smooth sphere-tensor statements, including at the poles; here \(r_->0\) is essential.

The two-shift null decomposition consequently has shifts \(O(Q)\) and positive null factor \(Q\) times a smooth positive function. Eliminate the second shift by angular labels \(H(Q,\omega)\), with \(H(0,\cdot)=\operatorname{Id}\). After division by \(Q\), their transport equation has the form \(-\kappa\partial_Q\) plus a smooth sphere vector field. It yields axis-equivariant sphere diffeomorphisms and the one-shift gauge (147), with \(q=Q\). Since \(L_iq=-\kappa q\), the definitions (148) imply (149), with all asserted differentiated bounds. The same construction eliminates the other shift after exchanging the optical directions. ◻

Proof. This is the preparation in [in:phases]; we check its dependence on the present exact family. At a time \(t_0\), compare with the map frozen at \(\pi(t_0)\) and give its Killing-time and rotation products the constant offsets \[I(t_0)-(\lambda_-\varsigma)(\pi(t_0))t_0, \qquad I_{\mathrm{rot}}(t_0)-(\Omega_-\varsigma)(\pi(t_0))t_0.\] The actual and frozen maps then agree in value. The offsets change neither the stationary nor the axisymmetric prepared profile. Their differentiated discrepancies contain only positive parameter jets with bounded smooth coefficients. Differentiating a phase gives a bounded frequency or a parameter jet, never the product of an elapsed time and a parameter derivative. Axis alignment precedes rotation, so variations of the axis also have uniformly bounded coefficients. The same comparison applies to the metric, with the additional residual \(g_{\mathrm{co}}-\overline G_{p,c}\).

For fixed \((s,x)\), first solve the \(s\) equation for radius over a stationary sphere. It is a uniformly small graph perturbation. Along that level, the derivative of \(x\) in time is \(I'+O(|\pi'|)\), bounded away from zero in absolute value. The graph times therefore differ from \(I^{-1}(x)\) by a bounded amount. At fixed \(x\), angular differentiation of the total rotating phase costs \(O(|\pi'|)\), by \(dx=0\). The angular map is consequently a uniformly nonsingular perturbation of an aligned rotation composed with \(H\). It is a sphere diffeomorphism; equivalently, finitely many uniform graph charts give the same estimates. ◻

Proof. We derive the entry trace from the mixed exterior estimates. On the fixed graph supports, Proposition 12 controls mixed derivatives of \(g_{\mathrm{co}}-\overline G_{p(t),c(t)}\) through order eight, and one further time derivative of that order, in local spacetime \(L^2\) with squared time weight \(\langle t\rangle^{A_S}\), where \(A_S>1\). These supports are outside physical trapping. Positive parameter derivatives through order eight have enlarged-cell supremum bounds supplied by their through-order-nine weighted \(L^2\) controls, again with exponent greater than one. The one-dimensional \(H^1\) inequality and weighted Cauchy–Schwarz make their cell sums small. Bounded graph-time offsets cost the corresponding local parameter variations.

The prepared frozen fields are the same smooth functions of intrinsic parameters and \((q,\theta)\) at every time. After the value-matching comparison of Lemma 22, replacing the instantaneous parameter by (155) changes only that parameter argument, with positive parameter jets charged when differentiated. In particular it does not rotate a phase by \(t\) times a parameter variation.

For a residual derivative \(f=D^{\alpha} (g_{\mathrm{co}}-\overline G_{p,c})\), \(|\alpha|\le7\), on an enlarged fixed \((t,r)\) rectangle, the iterated one-dimensional Sobolev inequality gives \[ \int_{\mathbb S^2}\sup_{t,r}|f|^2\,d\omega \le C\sum_{\epsilon,\epsilon'\in\{0,1\}} \|\partial_t^\epsilon\partial_r^{\epsilon'}f \|_{L^2(t,r,\omega)}^2. \tag{157}\] Even the mixed time-radial term lies in the stated budget: it uses the full mixed order-eight estimate for the time derivative of the metric residual, not a ninth-order pointwise metric trace. The graph Jacobians and the bounded overlap of enlarged cells transfer this estimate to the prepared spheres. If \(a_n\) denotes the resulting cell norm, then \[\sum_n a_n \le\left(\sum_n\langle n\rangle^{-A_S}\right)^{1/2} \left(\sum_n\langle n\rangle^{A_S}a_n^2\right)^{1/2}.\] The first factor is finite, and the second is small by the sliced estimate.

The first and second initial jets of the actual optical and angular chart are smooth functions of the auxiliary metric through order one: solve the noncharacteristic eikonal and angular-transport equations for normal derivatives. At exact freezing these jets agree with the auxiliary ones. Further initial jets require only ordinary derivatives of these identities. The geometric fields in (156) therefore use metric derivatives through order seven: \(A,\gamma,b\) use order zero and the one-derivative geometric fields use order one before angular differentiation. The coordinate transformations through this order use parameter jets through order eight. Sphere Sobolev products, with spare angular suprema on low residual factors, and (157) give exactly the displayed norms. Strict optical transversality requires only low suprema after \(q_0\) and the coordinate constants have been fixed.

In the bounded middle and enlarged joins, finite-time spacetime estimates through order nine supply the same trace. Use common identified coordinates for the lower computations; any order loss in the identifying map is assigned to the pulled-back metric bounds in that finite-time matching. Smooth fixed coordinate prescriptions and small parameter-dependent changes suffice to interpolate the preparation there. At higher orders the identical argument uses the subexponential exterior estimates on expanding intervals and finite smooth constants in the middle. This proves every assertion. ◻

Proof. We use the pure transport equations. Set \(U_i=\mathscr L_i\chi_i\) and \(V_i=L_i\ell_i\). The identities \[ \begin{gathered} \mathscr L_i\xi_j=\beta_i+\tau_i\xi_i,\qquad \mathscr L_i\xi_i=d\ell_i-\beta_i-\tau_i\xi_i,\\ L_iK=\operatorname{div}\beta_i-\tau_iK,\qquad [\mathscr L_j,\mathscr L_i] =\mathcal L_{a(\xi_j-\xi_i)^\sharp} \end{gathered} \tag{180}\] reduce derivatives of torsion, curvature, and angular connections to already controlled fields. Differentiating the cross equations in direction \(i\) gives [in:pure-top]: \[ \mathscr L_jU_i=-\tfrac12\tau_jU_i +U_i\chi_j+\chi_jU_i+\mathcal F_i, \qquad L_jV_i=-U_i\mathbin{\cdot}\chi_j+\mathcal G_i. \tag{181}\] Neither remainder contains a new pure jet. Differentiating \(a\) retains \(a\); a differentiated angular connection uses \(\chi_i\) and its angular derivatives; and the bracket is an angular operator with an \(a\) factor. Differentiating unweighted shape products introduces either this factor or a cubic product containing both orientations. The strongest angular term is \(\nabla\beta_i\), controlled in \(H^2\) by the available \(H^3\) bound for \(\beta_i\).

After subtraction of frozen differentiated fields, the inhomogeneous sources in \(H^2\) cost \[C\{q(p_*+g_i+g_j+\sigma_j)+g_i g_j\},\] and the homogeneous coefficient along \(j\) is \(C(q+g_j)\). Only a top shape jet times \(\chi_j\) recurs without a \(q\) factor. The initial pure jets are obtained from the tangential derivatives in (156) and the known cross equations. Localized Gronwall along \(j\), with the same orientation and cell estimates as above, proves (178).

For the shift write \(\zeta_\xi=\xi_1-\xi_0\) and \(C_b=a\gamma^{-1}\zeta_\xi=\partial_{y_1}b\). The identity [in:pure-shift] is \[ \begin{split} \partial_{y_1}\partial_{y_0}b ={}&a\{\ell_0\gamma^{-1}\zeta_\xi -2\gamma^{-1}\chi_0\gamma^{-1}\zeta_\xi +\gamma^{-1}(2\beta_0+2\tau_0\xi_0-d\ell_0)\} -[b,C_b]. \end{split} \tag{182}\] Every term retains \(a\), and no new pure shift derivative appears on the right. Its difference source costs \(Cq(p_*+g_0)\) in \(H^2\). Since the profile depends only on \(y_0\), differentiating its frozen pure shift in direction \(1\) adds no parameter defect. Integration from its tangential/cross initial value proves the shift assertion.

Summable cell suprema vanish in the corresponding infinite tails. Together with (149), the point budget therefore gives the first line of (179). The cross equations give the opposite first derivatives, while (178) and (182) give the pure ones. Converting Lie derivatives to coordinate derivatives uses already controlled \(b,d_\theta b\). Finally, \[\partial_{y_i}\gamma=2\chi_i-\mathcal L_{B_i}\gamma, \qquad \partial_{y_i}\log a+\kappa =\ell_i+\kappa-B_i\mathbin{\cdot}d\log a.\] These identities give the remaining \(H^4\) limits. The uniform metric, lapse, and torsion bounds follow from \(P\) and smooth inversion, and \(a^{-1}\partial_{y_1}b=\gamma^{-1}(\xi_1-\xi_0)\) has the asserted \(H^5\) bound. ◻

Proof. We derive high-order currents and recover the unweighted derivatives from their directional estimates. All high variables below are actual fields, not differences from profiles. Put \[h_i=C_0(q+g_i),\qquad \rho_q=\sqrt q,\] where \(C_0\) bounds all required low radiative norms. The functions \(h_i\) are bounded and have uniformly bounded directional integrals. On a full wedge their tails become arbitrarily small when \(s,|y_i|\to\infty\).

Fix an angular order \(m\ge5\). In the rest of the proof redefine \[ \begin{gathered} Y_i=|\beta_i|_m,\qquad Z_i=\sqrt q\,|(Q_i,\mu_i)|_m,\qquad R_i=|\chi_i|_{m+1},\qquad z_i=|\tau_i/A|_{m+1},\\ d_0^*=D=|b|_{m+1},\qquad d_1^*=0,\qquad P_\gamma=1+|\gamma|_{m+1},\qquad P_i=|\xi_j|_m,\qquad P_*=P_\gamma+P_0+P_1. \end{gathered} \tag{184}\] In particular, \(Z_i\) already includes the lapse weight. The quantity \(P_*\) controls \(\gamma^{-1}\) and \(A^{\pm1}\) at the corresponding angular orders, using the low bounds and \(d\log A=\xi_0+\xi_1\).

Proof. Start with smooth vacuum Cauchy evolution near \(\mathcal S\) and solve the noncharacteristic optical and angular transport equations there. We continue by exhaustion and finite restarts. For a target compact subdiamond work on a slightly larger region \[0\le s\le s_*,\qquad y_i\le C_i-\alpha s,\] where \(\alpha>0\) is small and the constants \(C_i\) leave positive slice length and a fixed margin around the target. The tilted sides are strictly exiting, so all previous past-cap energies retain their flux signs.

Bootstrap the uniform low estimates. The common finite weight (187), followed by the recovery part of Proposition 21, bounds every fixed smooth norm up to a putative terminal time. Since \(s\) is bounded here, \(a=q_0e^{-2\kappa s}A\) has a positive lower bound; the lapse and spatial metrics remain uniformly nondegenerate.

Restart from a slice before the alleged terminal time. Extend its induced metric and second fundamental form smoothly across the interval endpoints, preserving positivity and bounded requisite finite Sobolev norms on an enlarged margin. Constraints are unnecessary outside that margin. A reduced wave Einstein equation, initialized in wave gauge, has a uniform short existence time from these bounds. Solve the optical and angular transport equations with their inherited initial values. Their transversality and invertibility on a smaller margin also persist for a uniform short time. Higher smooth norms propagate separately.

Restrict to the exiting region. Future causal vectors obey \(dy_i\ge0\) in this gauge, so data outside the margin cannot enter the region under consideration. Constraint and gauge propagation give vacuum there. Geometric uniqueness, followed by uniqueness of the optical functions and angular transport, identifies the restarted and previous solutions. Slightly larger diamonds supply the endpoint jets and margins for each restart, patchwise on the sphere. Thus no finite terminal time occurs on a compact target diamond.

Exhaustion gives the full smooth product wedge. Increasing-domain suprema and their cell sums preserve the uniform low bounds and define global summable envelopes \(g_i\). Only now use their tails in the \(B,S_0\) choices of (190); this proves the expanding subexponential bounds without circularity.

Finally \(s\) is temporal. Causality in (147) implies \(|dx/ds|\le1\), and angular speeds are bounded on compact \((s,x)\) ranges. A past causal curve from the wedge therefore reaches \(\mathcal S\) or has an ordinary limit where it continues; it cannot escape at finite decreasing \(s\). Choose a local past collar of possibly varying smooth thickness in \(x\), with sufficiently small Lipschitz slope within the local existence region. Future causal curves in that collar reach the cylinder before leaving it. The cylinder is consequently Cauchy for this development, which can be attached along all of \(\mathcal S\) by geometric Cauchy uniqueness. ◻

Proof. Negate the prepared \(x\) on \(\mathcal S\), including its bounded joins. The old optical functions exchanged have the required eikonal initial data, so optical uniqueness identifies them with the new pair. For the angular labels use the exact preparation that eliminates the new second shift. In the existing geometry, transport these labels along the old \(L_0\). At every finite point this is a finite-duration smooth flow on a compact sphere over a finite coordinate interval, hence a sphere diffeomorphism with smooth inverse. It defines the new angular chart on the same full wedge before any uniform estimate on that flow is assumed.

To verify the initial comparison, reverse \(T_K,\varphi\) simultaneously at both ends, keep \(Q\), and use \(-I,-I_{\mathrm{rot}}\) with the new transport map \(H\) and the corresponding bounded angular joins. Thus the surface \(s=0\) is unchanged and its prepared \(x\) is exactly negated. Use the reversed \(I\) to define a single physical parameter profile along the new \(y_0\). The exact profile estimates and the trace proof of Proposition 18 apply with this other frozen angular map. Apply the finite-diamond a priori estimates to the new fields and exhaust. This reestablishes all uniform low and subexponential high bounds on the same spacetime, without requiring an a priori uniform estimate for the long angular coordinate change. ◻

Proof. We first construct a suitable exact Kerr range. Starting at the bridge, cut in the future at a radius strictly smaller than \(r_b\) but larger than \(r_-\), and at \(t_e=L\) on both ends, where the fixed number \(L\) is large enough to leave all required bands and margins. In the trapped region impose both time inequalities. At either future outer horizon its own time coordinate is regular and the opposite time inequality is automatic. Every active cut is a spacelike future exit. We construct on a slightly larger range and restrict only afterward.

For completeness, the causal control of this exact finite range is as follows. In the trapped block, \(r\) increases strictly to the past. The Boyer–Lindquist cone inequality gives \[ \left|\frac{dT_K}{dr}\right| \leq \frac{\sqrt{(r^2+\mathfrak a^2)^2 -\Delta\mathfrak a^2\sin^2\vartheta}} {|\Delta|}. \tag{198}\] This follows by Cauchy–Schwarz on the positive directions tangent to an \(r\) level. Near \(r_+\) the right side is \(|r_*'|+O(1)\), so the derivatives of both \(r_*+T_K\) and \(r_*-T_K\) are bounded above in the past-directed radial parameter. The positive interior Kruskal variables \[U_+=e^{\kappa_+(r_*+T_K)},\qquad V_+=e^{\kappa_+(r_*-T_K)}\] therefore have finite nonnegative limits. Indeed each logarithm becomes monotone after subtracting a bounded multiple of \(r\) and hence has a limit in \([-\infty,\infty)\).

The angular squares in the Kerr metric control \(d\vartheta\) and \(\sin\vartheta((r^2+\mathfrak a^2)d\varphi-\mathfrak a\,dT_K)\). Replacing \(\mathfrak a/(r^2+\mathfrak a^2)\) by its horizon value shows that the round speed of the corotating angle is \(O(|\Delta|^{-1/2})\) in the radial parameter, an integrable bound. Together with \(U_+,V_+\) this gives a finite point in regular outer horizon coordinates. To check regularity directly in the Boyer–Lindquist expression, \(r-r_+\) is a signed product \(U_+V_+\) times a smooth nonvanishing factor; after corotation, the mixed \(dT_K\)–angular coefficients are \(O(\Delta)\), and the sum of the \(dT_K^2\) and \(dr_*^2\) coefficients is \(O(\Delta^2)\). Substitution therefore gives the smooth nondegenerate horizon metric, up to a smooth angular-coordinate adjustment.

A past causal continuation at this horizon either follows its generator toward the bifurcation sphere or enters the corresponding exterior. There \(T_e\) decreases toward its bridge value zero. The exterior radial causal-speed bound is \(O(r-r_+)\) near the horizon and is bounded at infinity. A curve starting strictly outside cannot reach the horizon or spatial infinity in its bounded remaining \(T_e\) interval. Ordinary continuation at a finite interior endpoint then carries it to the bridge. The same argument applies in bounded interior radial ranges.

At bounded spatial radius, the two time cuts give compact closure in regular coordinates. Compact subsets have compact causal shadows toward the bridge; far out only bounded-time propagation with bounded speed is involved. Adding a short open past collar gives an exact Cauchy range with all the stipulated margins. The boundary signs require only fixed subextremality.

Apply finite-time vacuum Cauchy stability in wave-map coordinates on an enlargement of this range. For compact portions include their compact causal shadows. The \(p_{10}\) bound supplies small \(H^{10}\times H^9\) geometric data there. The positive wave-energy flux across a fixed uniformly spacelike bent surface, after commutation through order nine, controls tangential metric derivatives through ten and normal first-jet derivatives through nine. Thus these Cauchy orders are obtained by a spacelike flux estimate, without a graph-restriction loss at the highest derivative. Forming the induced metric and second fundamental form uses exactly those orders. At infinity the bent slices are the original ends; uniformly local end charts, bounded propagation speeds, and uniqueness on overlaps give the corresponding finite-time control there. All higher symbol seminorms remain finite.

The spacelike signs persist by low-order closeness. Past curves cannot leave through the artificial future cuts. The smooth Cauchy temporal function on the exact globally hyperbolic development is supplied by Bernal–Sánchez (Bernal and Sánchez 2005, Theorem 1.1). On a fixed bounded-radius enlargement it remains temporal with a uniform margin for the perturbed metric. In the far part the exterior times are temporal, speeds are bounded, and the time to the bridge is bounded. An excursion from a fixed bounded radius can therefore travel only a fixed additional distance. Enlarge the compact temporal-function region beyond that distance and handle curves entirely far out in exterior time. The resulting coordinate-length bounds preclude an inextendible endpoint inside these bounded ranges. The short past collar is handled by the same bounded-geometry construction: every future-inextendible causal curve there crosses the bridge. These are all fixed finite-time choices, independent of any later entry or experiment time. ◻

Proof. Evolve the two exterior problems from the geometric data on \(t_e=0\). Choose a fixed matching time \(t_a>1\) inside the finite range of Lemma 24. An enlarged band about \(t_e=t_a\) reaches slightly deeper than every cylinder intersection that will be used. Both evolutions are vacuum on the overlapping domains.

We describe the identification because its regularity is needed at \(\mathcal S\). Solve a wave map from the exterior evolution into the finite evolution with the initial slice identification and matching unit normals. On the fixed earlier slab and matching band it is close to the identity, with a strict deeper outflow margin and uniform control at infinity. Pulling the target metric back by this map gives matching initial first jets: the metric values agree, the second fundamental form fixes the normal evolution of the spatial components, and the relative wave-map condition fixes the remaining normal derivatives. The two metrics then solve the reduced wave equations with a common target. Ordinary uniqueness on the enlarged domain, with the exiting boundary signs, proves that the map is an isometry on the required smaller region.

The fixed-time map has \(H^{10}\) wave control on compact sets. After differentiating its equation, the Christoffel terms need derivatives through nine. Composing the target coefficients with the map costs their through-nine spacetime norms on enlarged sets times bounded composition constants under the \(H^{10}\) bootstrap. This follows from Leibniz and interpolation estimates, bounded low derivatives and inverse Jacobian, and change of variables. Smooth background compositions cost the map difference. The resulting spacetime \(L^2\) terms have finite-time \(L^1\) bounds; current-order terms are treated by Gronwall. The metric order-ten estimates therefore give the required smallness for the map and, after straightening the finite coordinates near the actual matching face, at least spacetime \(H^9\) closeness through the join. Cut off this small coordinate change farther back. Uniformly local end estimates give the same low diffeomorphism control near infinity. An embedding on the smaller region follows from \(C^1\) closeness and the fixed margins. Only a fixed finite interval is involved in this gauge comparison.

In these identified charts join the two prepared end portions of \(\mathcal S=\{s=0\}\) to the central preparation within the bounded overlaps. Use small smooth interpolations, implemented by small vector fields for angular maps. The radial positions of the instantaneous \(Q=q_0\) graphs stay uniformly close to the exact graph even at late times; the instantaneous profile comparison does not introduce an accumulated angular error in this assertion. In the bounded joining regions the integrated phases, with their fixed constants, are close as well. The timelike conormal and sphere-transversality margins persist.

Retain the finite future piece below \(\mathcal S\), with both cuts \(t_e\leq t_a\), and attach each late exterior piece \(t_e\geq t_a\) below \(\mathcal S\). The deeper computing boundaries are discarded; all joins lie strictly before them. The agreement bands contain the necessary cylinder neighborhoods. We identify the same hypersurfaces in common straightened coordinates, not unequal coordinate-time levels. This constructs the whole cylinder with the entry estimates already proved. Across it, geometric Cauchy uniqueness identifies the double-null solution with matching induced data, using optical coordinates in local collars.

The resulting topology retains disjoint interiors and identifies common closed matching faces, with the common collars providing charts. We do not retain two open future continuations of the same face. The faces are closed on their respective matching sides and have no extra finite accumulation points, since late end parameters have unbounded end times. Each identification is a homeomorphism. At a corner of the joining prescription, first identify the two lower charts and then cross \(\mathcal S\). These product neighborhoods give a Hausdorff smooth manifold, together with the original charts and the open past collar.

It remains to check the Cauchy property for the complete bridge. In the wedge, write \[s=\frac{y_0+y_1}{2},\qquad x=\frac{y_1-y_0}{2}.\] Along past causal curves \(s\) decreases and \(|dx/ds|\leq1\); angular speeds are bounded on bounded coordinate ranges. Such a curve reaches \(\mathcal S\) or has an ordinary interior continuation before doing so. In a late exterior part, \(t_e\) decreases with finite-time speed bounds. The sign of \(\mathcal S\) excludes a past exit there, so the curve proceeds into the finite block. Its remaining passage to \(\Sigma\) is controlled by Lemma 24. Curves in the past collar reach \(\Sigma\) to the future, and a crossing of \(\Sigma\) has only the prescribed orientation. Thus every inextendible causal curve meets \(\Sigma\) exactly once. The vacuum MGHD theorem supplies the stated open embedding, which we use henceforth as an identification. ◻

Proof. We first control the horizon strip. Let \(H\) be the future outer-horizon generator of the base metric \(b\), normalized by its Killing parameter, so \(\nabla_HH=\kappa_+H\). Choose a stationary axially invariant future timelike field \(N_0\) there. Invariance gives \[b(\nabla_HN_0,H)=-\kappa_+b(N_0,H)>0.\] At the horizon, \(dr_K(V)\leq0\) for normalized future causal vectors, with equality only on the generator; here \(r_K\) denotes the reference Kerr radius. Multiplying \(N_0\) by \(e^{K_0(r_K-r_+)}\) adds \[2K_0e^{K_0(r_K-r_+)}dr_K(V)b(N_0,V)\] to its deformation form. This is positive off the generator. Compactness of normalized causal directions, followed by continuity, gives a fixed strip and a stationary future timelike field \(N_H\) on it for which \[ (\mathcal L_{N_H}b)(V,V)>c_*>0. \tag{203}\] The strict sign and timelikeness persist for \(g\). Hence \(-g(N_H,U)\asymp\dot t\) is nonincreasing during each visit to this strip.

Below a smaller interior-side level, \(dr_K(U)/\dot t<-c\) all the way to \(\mathcal S\). A geodesic continuing while avoiding \(\mathcal S\) cannot enter this region: it cannot return through the level and would reach \(\mathcal S\) in bounded \(t\). Outside the strip on the exterior side, stationary energy together with angular momentum controls the normalized velocity. Indeed in exact Kerr \[b\left(\partial_T+ \frac{\mathfrak a}{r^2+\mathfrak a^2}\partial_\varphi, \partial_T+ \frac{\mathfrak a}{r^2+\mathfrak a^2}\partial_\varphi\right) =-\frac{\Delta(r^2+\mathfrak a^2\cos^2\vartheta)} {(r^2+\mathfrak a^2)^2}<0.\] On the fixed exterior compact range the coefficients are bounded and the timelike margin is uniform, also at the axis. Transferring this combination to each nearby frozen chart and then perturbing gives \[ \dot t\leq C(|E|+|P|). \tag{204}\] Thus no timelike-Killing-field assumption is made inside the ergoregion.

Set \(X_1=|E(t_1)|+|P(t_1)|\) and \[B(t)=X_1+C\int_{t_1}^t\rho_c(v)\dot t(v)\,dv.\] If the portion starts in the strip, include its initial \(N_H\) energy in \(X_1\). Outside the strip, (200) and (204) give \(\dot t\leq CB\). Within the strip, its nonincreasing \(N_H\) energy bounds \(\dot t\) by the value at the initial point or latest strip entry. Using a fixed overlap of the two control regions and the monotonicity of \(B\), the same bound holds there. Consequently \(B'\leq C\rho_c B\), and Gronwall proves (201), with one majorant for all strip visits.

At the large interface \(X_1\leq CE(t_1)\). Integrating the \(E\) equation itself now gives \[E(t_2)\leq E(t_1) \left[1+C_0\left(e^{C\int_{t_1}^{t_2}\rho_c}-1\right)\right] \leq E(t_1)e^{C'\int_{t_1}^{t_2}\rho_c}.\] This proves the return estimate (202). ◻

Proof. We prove separately the integrability of the averaged target terms and of the field contraction. Besides \(\psi(V,V)\) and the already integrable overlap errors, the deformation cost is bounded by \(C(|DB|+|\partial s_s|)\). Write \(x_{\mathrm{av}}\) for the implicit averaging position, defined by \(x_{\mathrm{av}}=z+\lambda_s(t,x_{\mathrm{av}})\). The target construction gives \(Dx_{\mathrm{av}}=0\), averaging scale \(d(x_{\mathrm{av}})=|x_{\mathrm{av}}|\), and delays in \((0,1/10)\). Consequently \[ |DB|+|\partial s_s| \leq\frac{C}{|x_{\mathrm{av}}|} \int_{t-|x_{\mathrm{av}}|/10}^{t}|p'(v)|\,dv. \tag{206}\]

Along every late outer portion of this fixed track, \(|x_{\mathrm{av}}|\leq2t\) and \(r\leq Ct\), uniformly over repeated visits. Indeed the far lab speed is at most \(1+\) a small constant. At entry from bounded co-radius one has \(|z|\leq O(1)+\epsilon't\), and past sampling gives \(x_{\mathrm{av}}-z=O(\epsilon't)\) for small drift \(\epsilon'\). Starting from any fixed finite point changes only a bounded initial time interval. The averaging interval in (206) therefore lies at times comparable to \(t\). If \(\mathcal M\) is the one-dimensional maximal operator, extending \(p'\) by zero before startup gives the bound \[C\langle t\rangle^{-A/2} \mathcal M\bigl(\langle\cdot\rangle^{A/2}|p'|\bigr)(t).\] The acceleration belongs to weighted \(L^2_A\), and \(A=1.36>1\). The \(L^2\) maximal inequality and Cauchy–Schwarz prove integrability.

For the field contraction, first consider \(t\geq C_1r\) with \(C_1\) large and fixed. Here even the spatial supremum of \(\psi\) is integrable. We include the trace argument to justify evaluation on a geodesic. On a radial shell \(R\leq r\leq2R\), use \(\varrho=\log r\). The strong bulk bounds through two angular rotations, with the comparisons \(W_P\geq cR^{0.26}t^A\) and \(w_{P,j}\geq cR^{-0.03}t^A\), imply \[\begin{align*} \int_{C_1R}^{\infty}t^A \|\psi\|_{L^2_{\varrho}H^2_\omega}^2\,dt &\leq CR^{-0.26}\mathsf F^2, \tag{207}\\ \int_{C_1R}^{\infty}t^A \|\partial_{\varrho}\psi\|_{L^2_{\varrho}H^2_\omega}^2\,dt &\leq C(R^{0.03}+R^{-0.26})\mathsf F^2. \tag{208}\end{align*}\] The norms use a slightly enlarged shell, and \(C_1\) is increased if necessary to put all of it in the same timelike region. To see the derivative count, with \(S=r(\partial_t+n\cdot\partial_z)\) one has \(r\partial_{r|t,n}=S-r\partial_q\). Good and full derivatives of \(r\psi\) control the two terms, while the field term accounts for differentiating \(r\). Two rotations and one derivative lie strictly within the strong commutation budget.

The Hilbert-valued multiplicative one-dimensional trace inequality and \(H^2(\mathbb S^2)\hookrightarrow L^\infty(\mathbb S^2)\) now give \[ \int_{C_1R}^{\infty}t^A \sup_{R\leq r\leq2R,\omega}|\psi|^2\,dt \leq C\left(R^{-0.26}+R^{-(0.26-0.03)/2}\right)\mathsf F^2 \leq CR^{-0.115}\mathsf F^2. \tag{209}\] The shell sum converges. Cauchy–Schwarz in time, using \(A>1\), yields \[ \int_1^\infty \sup_{R_0\leq r\leq t/C_1,\omega}|\psi(t,r,\omega)|\,dt <\infty. \tag{210}\] The radial loss \(0.03\) still leaves a strictly negative trace power, which is the margin needed for the shell sum.

On the remaining late outer portions \(r\asymp t\). Set \(c_3=0.01\). The point estimates and differentiated gauge identity give \[ \begin{aligned} |\psi|&\leq Cr^{-1+e/2} \langle q/r^{\delta_f}\rangle^{-A_H/2},\qquad |g-\eta|\leq Cr^{-1+c_3},\\ |\psi(L,L)|+|\psi(L,e_{\mathrm{scr}})|&\leq Cr^{-2+c_3}, \end{aligned} \tag{211}\] where \(L=(1,n)\), \(e_{\mathrm{scr}}\) is a spatial unit screen vector orthogonal to \(n\), and the far exponents satisfy \(\nu=10\delta_f\), \(e=0.1\delta_f\). We recall why the improved projections apply to \(\psi\). For \(p=T_lh\) and the flat null covector \(\ell_n=(1,-n)\), put \[\mathcal C_\beta(p) =\ell_n^\alpha \left(p_{\alpha\beta} -\tfrac12\eta_{\alpha\beta}\operatorname{tr}_\eta p\right).\] Since \(\ell_n^\sharp=-L\), contraction against \(L\) or a screen vector has no trace term and equals respectively \(-p_{LL}\) or \(-p_{L\,\mathrm{scr}}\). The far gauge estimate bounds these by good derivatives, coefficient-suppressed full derivatives, and smaller field and target terms. Good derivatives cost \(O(r^{-2+2\eta_*})\); a product of \(h\) with a full derivative can cost \(O(r^{-2+2\eta_*+e/2})\). The fixed choice \(c_3=0.01\) absorbs these losses, giving the stated \(O(r^{-2+c_3})\) bound. Passing from \(T_lh\) to \(\psi\) uses \[\psi=(1-s_s\cdot n)T_lh+r^{-1}s_s^d b_dZh,\] with the bounded angular/radial decomposition coefficients \(b_d\) of the far construction. It adds only good derivatives. Thus no derivative of an unsmoothed velocity is used in this conversion.

Write \(V=(1,\mathbf v)\) on the track. Differentiating the emitter identity and using causality gives \[ \begin{aligned} kq'&=1-\mathbf v\cdot n,\\ q'&\geq-Cr^{-1+c_3},\\ |\mathbf v-(\mathbf v\cdot n)n|^2 &\leq C(|q'|+r^{-1+c_3}),\qquad |q'|\leq C. \end{aligned} \tag{212}\] Decomposing \(V-L\) into radial and screen parts, the improved contractions in (211) imply \[ |\psi(V,V)|\leq Cr^{-2+c_3} +C|\psi|(|q'|+r^{-1+c_3}). \tag{213}\] Where \(|q|\geq r^\nu\), the field itself is integrable, since \[\frac{A_H}{2}(\nu-\delta_f)-\frac e2>0.\] It remains to control the \(|q'|\) term near the cone.

On a late time dyad \(R\leq t\leq2R\), clip \(q\) to \([-C_4R^\nu,C_4R^\nu]\), with \(C_4\) containing all values in \(|q|\leq r^\nu\). Extend the clipped function by its upper constant during compact excursions. Near the interface \(r\) is bounded and \(q=t-r\gg C_4R^\nu\), so this extension is continuous. It is uniformly Lipschitz, even for countably many accumulating excursions: on each outer component the derivative is bounded, and it matches the same constant on the complement. Its nonsaturated part has \(r\asymp R\). By (212) its total negative variation is \(O(R^{c_3})\), while its range has length \(O(R^\nu)\). Hence \[ \int_{\substack{R\leq t\leq2R\text{ on outer portions}\\ |q|\leq r^\nu}}|q'|\,dt \leq C(R^\nu+R^{c_3}). \tag{214}\] Multiplying by \(CR^{-1+e/2}\) is summable over dyads, because \(\nu,c_3<1-e/2\). The other two terms in (213) are integrable directly. Together with (210), (206), and the fixed-overlap estimate, this proves integrability over all outer portions of the track, including all returns to the compact interface. ◻

Proof. Such a geodesic must eventually proceed in one end with \(t_e\to\infty\). In the finite piece its far excursions have bounded time and bounded speed; the remaining bounded-radius range, including the matching faces, has compact closure in enlarged regular charts. Global hyperbolicity excludes total causal imprisonment there. There is also no finite proper-time tangent blowup at an interior position limit: in a regular temporal chart the causal slopes are bounded, the tail has finite coordinate length, and the parallel-tangent equation \[\frac{dU^\alpha}{dt} =-\Gamma^\alpha_{\beta\gamma} \frac{dx^\beta}{dt}U^\gamma\] is a linear equation with bounded coefficients. It bounds the tangent and gives ordinary continuation. The future sign of an end matching face prevents return to the finite block before crossing \(\mathcal S\). At bounded \(t_e\), finite propagation and the same local argument prevent runout in the late end. This proves \(t_e\to\infty\).

Extend \(Z_K\) to a smooth future timelike field \(X\) throughout this end, agreeing with \(Z_K\) outside and near the compact interface. A partition of unity gives such an extension by convexity of future timelike cones. On any fixed finite track interval \([t_1,t_2]\) with outer endpoints, \(F(t)=\log[-g(X,U)]\) is absolutely continuous. Write the open set of times strictly inside the interface as \(\mathcal O=\bigsqcup_j(a_j,b_j)\). Then \[F(t_2)-F(t_1) =\int_{[t_1,t_2]\setminus\mathcal O}F'(t)\,dt +\sum_j\bigl(F(b_j)-F(a_j)\bigr).\] On this finite interval the series converges absolutely, bounded by \(\int|F'|\). At excursion endpoints \(X=Z_K=T_{\mathrm{co}}\), so (202) bounds each increment from above by \(C\int_{a_j}^{b_j}\rho_c\). On the complement, Lemma 26 bounds the absolute variation. The resulting upper bound for \(F(t_2)-F(t_1)\) is independent of \(t_2\) and the number of excursions. No deformation bound for the arbitrary continuation \(X\) inside the interface has been used.

Thus \(E_K\) remains bounded on all outer portions, and (201) bounds \(\dot t\) on every intervening compact portion by its entering value. An initial or final portion remaining entirely compact uses that same supremum estimate. Consequently \(\dot t\) is bounded on the entire late track, and \(\int dt/\dot t=\infty\) as \(t\to\infty\). ◻

Proof. The first-runout time is no later than its finite maximal time in \(M_d\). Proposition 24 rules out a track avoiding \(\mathcal S\). In the wedge both \(y_i\) increase strictly along future timelike curves. If both had finite limits, then \(s,x\) would be bounded. Causal slopes in regular temporal charts and compactness of the sphere would give a position endpoint; the parallel-tangent argument just used would give ordinary geodesic continuation. Thus at least one limit is infinite, giving exactly the three listed alternatives. ◻

Proof. Dots denote proper-time derivatives. Decompose \[U=\dot y_0L_0+\dot y_1L_1+\Xi, \qquad \Xi=\dot\theta-b\dot y_0.\] Future timelikeness gives \(\dot y_i>0\). For \(j=1-i\) define the null momentum \(p_i=-g(U,L_i)=a\dot y_j>0\). Unit normalization and (218) give \[\begin{align*} 1+|\Xi|_\gamma^2&=2a\dot y_0\dot y_1, \tag{221}\\ \frac{\dot p_i}{p_i} &=\ell_i\dot y_i+(\xi_i-\xi_j)(\Xi) -\frac{\chi_i(\Xi,\Xi)}{p_i}. \tag{222}\end{align*}\] Since \(|\Xi|_\gamma^2/p_i\leq2\dot y_i\), we obtain \[ \begin{aligned} \left|\frac{d}{d\tau}\log(e^{\kappa y_i}p_i)\right| &\leq C(q+g_i(y_i))\dot y_i+C|\Xi|_\gamma,\\ e^{\kappa y_i}p_i&=\frac{q_0}{\kappa}A|\dot Y_j|. \end{aligned} \tag{223}\] Both \(Y_i\) decrease, and (221) becomes \[1+|\Xi|_\gamma^2 =\frac{2q_0A}{\kappa^2}|\dot Y_0|\,|\dot Y_1|.\] On any initial portion of the remaining track, Cauchy–Schwarz and monotonicity imply \[ \int|\Xi|_\gamma\,d\tau \leq C\left(\int|\dot Y_0|\,d\tau \int|\dot Y_1|\,d\tau\right)^{1/2} \leq C\sqrt{Y_0^{\mathrm{in}}Y_1^{\mathrm{in}}}. \tag{224}\] For each \(i\) there are also the bounds \[\int q\dot y_i\,d\tau \leq\frac{q_0}{\kappa}Y_0^{\mathrm{in}}Y_1^{\mathrm{in}}, \qquad \int g_i(y_i)\dot y_i\,d\tau \leq\int_{y_i^{\mathrm{in}}}^{\infty}g_i(v)\,dv.\] Their sum, together with (224) and a fixed constant factor, is a choice of \(\delta_c\) in (223). It is independent of the initial tangent, finite at any fixed starting point, and small when both initial coordinates are large.

The logarithmic variation of each \(A|\dot Y_i|\) is at most \(\delta_c\). Taking their ratio cancels \(A\), giving (219); the bounds for \(A^{\pm1}\) give individual speed comparability. The normalization then bounds \(\Xi\) on the fixed track. Also (217) implies \(\int |b|_{\mathrm{round}}\dot y_0\,d\tau<\infty\). Since \(\dot\theta=\Xi+b\dot y_0\), the angular track has finite round length and an angular limit.

An individual \(|\dot Y_i|\) has a positive lower bound, so the remaining proper duration is finite. Both \(Y_i\) have nonnegative limits. If both were positive, the optical coordinates would be bounded; the speed bounds and angular limit would give ordinary continuation in a regular chart, contradicting maximality of the wedge track. Thus one or both limits vanish. In the corner case integration of (219), using \(\int |\dot Y_i|\,d\tau=Y_i^{\mathrm{in}}\), gives (220). This proves the assertions, including the finite-variation version starting at a general fixed point. ◻

Proof. Corollary 1 applies to the original geodesic. If its first runout is a branch, take \(w'=w\). Suppose it is a corner. At a late point change its tangent to \[ U_\sigma=(1+\sigma)\dot y_0L_0 +(1+\sigma)^{-1}\dot y_1L_1+\Xi, \qquad \sigma>0. \tag{225}\] The product of its null coefficients and its screen velocity are unchanged, so it is future unit timelike. Its initial \(Y\)-speed ratio is multiplied by \((1+\sigma)^2\). The original corner satisfies (220); a corner for the changed track would satisfy the same necessary condition with this new ratio. The two allowable intervals are disjoint once \[ 4\delta_c<2\log(1+\sigma). \tag{226}\] For each fixed \(\sigma>0\), a sufficiently late change therefore produces a branch by Lemma 27.

It remains to make one choice of \(\sigma\) which returns the changed geodesic to \(O\), uniformly as the change point approaches runout. This is a statement in regular ambient coordinates, not in the degenerating optical coordinates. On the fixed original tail \(\Xi\) is bounded. Writing \(c=1+\sigma\), direct calculation gives \[ -g(U,U_\sigma) =1+(1+|\Xi|_\gamma^2) \left(\frac{c+c^{-1}}2-1\right). \tag{227}\] The relative rapidity is therefore \(O(\sigma)\) uniformly late.

Let \(\tau_P\) be first runout from \(\mathcal P_d\), and let \(\tau_M=T_d(w)\). If \(\tau_P<\tau_M\), the runout state is an ordinary smooth state in \(M_d\). If \(\tau_P=\tau_M\), the hypothesis provides a regular geodesic state in the \(C^2\) extension. In either case choose one precompact product temporal chart \((t,x)\) around that ambient state, with a convex spatial coordinate ball containing a tail of the original arc. The original unit tangent is bounded there. Shrink the chart so that one fixed convex ball of spatial slopes around the reference slopes consists of future timelike slopes at every point of the chart.

Choose fixed times \(t_0<t_1\) before the runout time such that the original segment \([t_0,t_1]\) has a compact tube contained in \(\mathcal P_d\). At change points \(z_n\) after \(t_1\) tending to runout, (227) and the bounded original tangent give an \(O(\sigma)\) change of ambient coordinate phase state. The geodesic vector field is \(C^1\) even for a \(C^2\) ambient metric. On a compact phase-space tube its ODE constants are uniform. Backward integration to \(t=t_0\) therefore exists and changes the state by \(O(\sigma)\), with one constant independent of \(n\). Its endpoint \(p_n'\) lies in the fixed interior tube for sufficiently small \(\sigma\).

We must also show that this entire backward arc lies in \(\mathcal P_d\). Write \(x_0(t)\) for the original spatial path and choose \(\chi\in C^\infty([t_0,t_1])\) equal to one at \(t_0\) and zero near \(t_1\). Form a reference path on the early interval by \[x_0(t)+\chi(t)\bigl(x(p_n')-x_0(t_0)\bigr),\] and then follow \(x_0(t)\) to \(z_n\). The early part lies in the fixed compact tube, the rest in the original track, and the whole path is timelike for small \(\sigma\). It has exactly the endpoints of the backward-integrated geodesic. Both paths have slopes in the common convex timelike ball. Spatial linear interpolation at the same chart time is thus a timelike fixed-endpoint homotopy within the ambient box.

Apply the intrinsic-diamond argument of 4 to the globally hyperbolic open subspacetime \(\mathcal P_d\). For clarity, the parameters for which the full path lies inside are open, since the path is compact. They are closed because such paths lie in the intrinsic compact causal diamond of their two fixed endpoints. Hence every path of the homotopy, including the changed arc, lies in \(\mathcal P_d\). No causal convexity in the ambient extension is required.

The changed state at \(t=t_0\) is uniformly close to the original one. Ordinary geodesic flow through the fixed earlier compact segment, including its transverse crossing of \(\Sigma\), then gives a seed in \(O\). The choices have the following order: fix the ambient chart, the two early slices, their compact tube, and an earlier-flow neighborhood mapping into \(O\); choose one sufficiently small \(\sigma>0\) for these tolerances; finally choose \(z_n\) so late that (226) holds. The last choice forces a branch without changing any preceding tolerance. ◻

Proof. Unit normalization gives \(1+|\Xi|_\gamma^2=2p\dot u\). The geodesic equation used in the corner-ratio argument implies \[|\dot{\log p}|\le C(\dot u+|\Xi|_\gamma).\] The integral of \(\dot u\) is finite. Also, by the same normalization and Cauchy–Schwarz, \[\int|\Xi|_\gamma\,d\tau \le C\left(\int du\right)^{1/2} \left(\int a\,dv\right)^{1/2}<\infty.\] Thus \(p\asymp1\). Near the angular limit the unit-shell Hamiltonian with \(v\) as time is \[ -\pi_v=\frac{a(1+|P|_{\gamma^{-1}}^2)}{2p}, \qquad \frac{du}{dv}=\frac{a(1+|P|_{\gamma^{-1}}^2)}{2p^2}. \tag{230}\] The low coefficient bounds give \(|dP/dv|\le C(p+|P|)\,du/dv\). Since the total variation of \(u\) is finite, Gronwall bounds \(P\) and hence \(\Xi\) and \(\dot u\). Finally \(a\asymp e^{-\kappa T}\) at \(P_T\), proving (229). ◻

Proof. Write \(B_0=b\), \(B_1=0\), and \(j=1-i\). The complete connection formulas include \[\begin{align*} \nabla_{L_i}\hat e_i&=\tfrac12\ell_i\hat e_i,& \nabla_{L_i}\hat e_j&=-\tfrac12\ell_i\hat e_j +\sqrt a\,\xi_i^A\partial_A, \tag{231}\\ \nabla_{L_i}\partial_A &=(\chi_{i,A}{}^B-\partial_AB_i^B)\partial_B +\sqrt a\,\xi_{i,A}\hat e_i, \\ \nabla_{Z_A}\hat e_i &=\chi_{i,A}{}^B\partial_B +\tfrac12\sqrt a(\xi_i-\xi_j)_A\hat e_i,& \nabla_{Z_A}\partial_B &=\sqrt a\,\Gamma_{AB}^C\partial_C +\chi_{0,AB}\hat e_1+\chi_{1,AB}\hat e_0. \end{align*}\] The low estimates and pure-jet limits give the asserted limiting matrices. In the mixed trace the radial derivatives are cross derivatives; in particular \(L_1\partial_Ab=\partial_A\partial_vb\) retains \(a\). For curvature use the exact matrix identity \[aR(\hat e_\alpha,\hat e_\beta) =Z_\alpha V_\beta-Z_\beta V_\alpha +[V_\alpha,V_\beta]-V_{[Z_\alpha,Z_\beta]}.\] The brackets satisfy \[[L_0,L_1]=\sqrt a(\xi_0-\xi_1)^AZ_A, \qquad [L_i,Z_A]=\tfrac12\ell_iZ_A-(\partial_AB_i^B)Z_B.\] The new unsuppressed pure longitudinal curvature term is \(L_i\chi_i\), covered by the pure-jet estimate. Longitudinal torsion derivatives reduce to \(\beta_i\), \(d\ell_i\), and \(\tau_i\xi_i\) by the double-null identities. The actual low bounds on \(\ell_i+\kappa\) and \(\chi_i\) in angular \(H^4\) give \(d\ell_i\) and \(\beta_i\) in \(H^3\); the pure jets are in \(H^2\), which embeds in \(C^0\) on the spheres. Thus the required limits do not differentiate an uncontrolled moving profile. The factor \(\sqrt a\) in the angular connection and the limiting commutation relation do not create a constant curvature term. The only constant tensor-wave contribution is \[-(L_0+C)(L_1-C)-(L_1-C)(L_0+C), \qquad C=\operatorname{diag}(c_n),\] whose \(n\)th entry is the radial part of \(e^{2c_nx}a\square_g^{\mathrm{scal}}e^{-2c_nx}\). The higher double-null bounds, on the strip including \(u_f\), give all remaining fixed derivative estimates with the stated factors. ◻

Proof. Bootstrap the error energies by \(e^{-(2\kappa+c')T}\) for a small fixed \(c'>0\). The relative cone change is exponentially small. Therefore the side inequalities in [pk2:domain] are strict outflow, since \(du/ds,dv/ds\ge-\)exponentially small on future causal vectors. Finite propagation keeps the exact change supported in \(v\ge(1-\nu_0)T-o(1)\).

For \(f=D^jF_n\), where \(D\) denotes ordinary derivatives through order \(k\), the scalar operator retaining the diagonal drift is \[e^{2c_nx}a\square_{g'}^{\mathrm{scal}}(e^{-2c_nx}f).\] Lemma 29 gives bounded top-order residual coefficients small in the bulk, with \(e^{o(T)}\) coefficients only on lower commuted energies. The potentially unscaled angular commutator has the form \[(Db)\partial_v(\partial_AD^{j-1}F)\] when one derivative hits \(b\partial_A\partial_v\). The angular derivative is inside the ordinary order-\(j\) field, so its coefficient is the small \(Db\). With two or more coefficient hits the field is at a lower energy level. A first hit on \(a\gamma^{-1}\partial_\theta^2\) retains at least \(\sqrt a\) against the scaled angular derivative; higher hits again enter lower levels. The factor \(\partial_vb\) retains \(a\), and angular partitions cost \(O(|b|+\sqrt a)\) at top order.

After retaining the exact principal coefficients, each perturbative commutation or lower product contains an additional packet-small or bootstrap-small factor at an error placement. The highest error order \(j+1\) can lose \(O(a^{-1/2})\le Ce^{\kappa T/2}\); further error factors through half the total order \(j+2\) are controlled by the mixed \(k\)-order bootstrap and slice Sobolev embedding. Taking \(\zeta\) sufficiently small leaves an exponential margin to absorb these factors. Smooth higher background derivatives cost only \(e^{o(T)}\).

Apply massive scalar stress energy to \(e^{-2c_nx}f\) with mass squared \(\varepsilon_0^2/(2a)\) and multiplier \(e^{4c_nx}W_t\). Up to an exponentially small relative metric change, its slice density is (235). Thus the exact-principal stress energy is uniformly equivalent to the reference mixed norm just defined. The limiting nonsource bulk terms are \[-4c_nW_tf(W_x-2c_n)f -2\kappa a|d_\theta f|_\gamma^2 +\varepsilon_0^2fW_tf.\] Here \(W_tx=0\), \(W_xx=1\), \(W_t\log a\to-2\kappa\), and \(W_t\gamma\to0\). The brackets of \(W_t\) with \(W_x,Z_A\) are small in the scaled basis except for the limiting term \(-\kappa Z_A\). All exits have favorable flux. Since \(|c_n|\le\kappa\), fixing \(\varepsilon_0\) small and then a bulk threshold gives square-root energy rate \(2\kappa+\)arbitrarily small in the bulk, and a bounded rate elsewhere. The part of the support not entirely in that bulk has time length \(O(1)+O(\nu_0T)\), whereas the whole interval has length \(T/2+O(1)\). More explicitly, for a fixed bulk threshold \(B\) the bulk is \(\min(v,-u,s)\ge B\). The support has \(v>B\) for large \(T\), the region \(s<B\) has bounded length, and \(-u<B\) implies \(2s=u+v>(1-\nu_0)T-B-o(1)\). Its possible remaining time length is at most \(\nu_0T/2+O(B+1)\). A common nonnegative majorant therefore integrates to \((\kappa+\delta_2)T\) after slightly reserving its strict margin.

For clarity, if \(E_j\) are the successive error norms, their inequalities have the form \[E_j'\le r_TE_j+b_T\sum_{\ell<j}E_\ell+S_j, \qquad \int b_T\le e^{o(T)}.\] Factoring out \(\exp(\int r_T)\) and integrating successively in \(j\) costs only a fixed polynomial in \(\int b_T\). This is 37, and is still \(e^{o(T)}\) at every fixed \(k\). The initial mixed jets follow from the reduced equation with the same source and subexponential costs: \(s=0\) is uniformly spacelike in its nondegenerate entry coordinates. This proves [pk2:deep-error].

Its strict exponential improvement closes the bootstrap. Ordinary hyperbolic restarts on enlarged domains within the outflow margins give continuation, since for each finite \(T\) all coordinates are nondegenerate. Homogeneous wave-constraint propagation gives vacuum. Adding fixed Sobolev buffers gives the final \(C^2\) assertion. ◻

Proof. Put \(H=U/Y_T\). In the normalized primal basis, \(H\) has size \(O(\sqrt a)\) and \(X\) has size \(O(1)\), while \(dv(H)=1\) and \(dv(X)=0\). The phase-phase leading term in \(\operatorname{Riem}(g')(H,X,H,X)\) has absolute value \[\tfrac12 A_T\lambda_p^2\sigma_T.\] All other packet terms cost \(e^{o(T)}(A_T\lambda_p+A_T^2\lambda_p^{P_*})\); comparison terms cost [pk2:deep-error] times \(e^{o(T)}\), enlarging the fixed \(P_*\) if necessary. Indeed express the connection difference using first \(\nabla^g\) derivatives of \(g'-g\), then use its covariant derivative, connection products, and inverse-metric products for curvature. Multiplication by \(a\) allows both differentiated slots to use \(Z=\sqrt a\,\hat e\) in normalized entries, with only subexponential conversion costs, including hits on the scaling. Evaluation on two \(H\) and two \(X\) slots cancels this curvature scaling. The single order-zero amplitude supplies the displayed leading term; trace reversal does not alter it, and the ordinary second-metric-derivative formula gives the factor \(1/2\).

All remaining terms are \(o(Y_T^{-2})\) under the choices made below. A normalized near-identity frame isometry sends \(U,X\) to a \(g'\)-orthogonal future/unit and spacelike/unit pair \(U',X'\), with relative frame error \(O(A_Te^{o(T)})\). Its curvature slot error is still \(o(Y_T^{-2})\) at the \(H\) scale. Multiplying by \(Y_T^2\) and using the choice of \(J_{\mathrm{amp}}\) proves the strict tidal bound.

For the backward state, use (230) on a compact \(u,\theta,\pi_u,P\) state tube around the background tail, with \(p=-\pi_u-b\cdot P\) bounded below. Its background Lipschitz cost is \(ae^{o(v)}\), integrable as \(v\to\infty\). The perturbed unit shell is solved on the same branch by putting \(\pi_v=az'\): the background equation for bounded \(z'\) has derivative bounded away from zero. Normalized covector arguments have size \(O(a^{-1/2})\), but the differentiated relative metric change is exponentially smaller than \(a\) on this bounded-\(u\) path. The Hamiltonian and its first state derivatives therefore change by exponentially small absolute amounts. The terminal errors are \[\delta(\pi_u,P)=O(A_Te^{o(T)}/\sqrt a), \qquad \delta(\pi_v/a)=O(A_Te^{o(T)}/a).\] The second estimate uses \(\partial_v=\sqrt a\,\hat e_1\) and also shows that the chosen future tangent lies on the continued root. Differentiating the shell equation in state variables costs at most the same inverse \(a\) on normalized coefficient differences, with subexponential background factors. Integrating the exponentially small vector-field error over length \(O(T)\) and applying Gronwall with the integrable background Lipschitz majorant proves convergence through the transverse hit of \(s=0\). Fixed earlier portions use ordinary compact geodesic flow. Uniform state and side margins keep the perturbed paths in the stated domain. ◻

Proof. On compact spatial subsets of the co cylinder, \(g\to G_{p_\infty}\) in the required low differentiated orders. Indeed \(p\to p_\infty\) follows from the integrable modulation derivative, and \(p',c,c'\) and further required low jets tend to zero by the positive-weight endpoint and feedback bounds. The compact \(A_H\)-weighted endpoint estimates control \(h_{\mathrm{co}}\), while the target memories have bounded length there. Higher compact and far estimates supply fixed jets of size \(e^{o(T)}\) for time and radius \(O(T)\).

The entry surface is strictly spacelike and optically transverse. A small uniform collar obtained by local eikonal and angular flow therefore has bounded low transition jets and subexponential high jets. The covectors \(dv\) approach the frozen covectors \[\lambda_-d(r_*+F+T_K)\] in aligned Kerr variables. This follows from the prepared-coordinate Jacobian comparison, the decay of the parameter derivatives, and the noncharacteristic null root. Accumulated phase offsets do not affect this assertion. Here \(F\) is the exact axisymmetric optical-profile function, not the metric comparison error.

The frozen covector has zero angular Killing momentum. By Kerr separation its radial potential divided by squared Killing energy is \[ (r_K^2+\mathfrak a^2)^2 -\Delta\left(|\partial_\vartheta F|^2 +\mathfrak a^2\sin^2\vartheta\right)_{\rm init}. \tag{237}\] The bracket is at most \(\mathfrak a^2+O(q_0^2)\), since \(F(r_-)=0\) and the profile is smooth. For \(r_K\ge r_+\), \[\Delta<r_K^2, \qquad \frac{(r_K^2+\mathfrak a^2)^2}{r_K^2}\ge4\mathfrak a^2.\] For every fixed \(\mathfrak a>0\), choosing the original exact-profile depth sufficiently small over a small physical parameter range makes (237) strictly positive. It is also positive between the horizons, where \(\Delta<0\), with uniform margins on compact crossings. This depth choice was made before the exterior constants and packet parameters.

In the frozen chart \(g^{-1}dv\) is past-directed with Killing energy \(-\lambda_-\). Its radial direction is increasing, since initially \[\Sigma_K\,dr_K(g^{-1}dv) =\lambda_-\bigl(r_K^2+\mathfrak a^2+\Delta F_r\bigr)>0\] by the finite Hamilton–Jacobi root. At the outer horizon its radial covector component has leading coefficient \(\lambda_-(r_K^2+\mathfrak a^2)/\Delta\) with bounded remainder. The coordinate \(T_K+r_*\) and the ingoing azimuth are regular, and the angular momentum stays bounded, with zero azimuthal coefficient. Thus the ray crosses the future event horizon transversely and spends bounded regular \(t_e\)-time through every fixed compact crossing. The same holds forwards from entry to the deeper exit. No timelike stationary Killing field is needed in an ergoregion.

Near entry, \(Q=(r_K-r_-)f\) with \(f>0\) smooth and axisymmetric. Separation and the strict radial potential bound give bounded \(|d\vartheta/dr_K|\). Hence \(dQ/dr_K\) has a positive lower bound in a sufficiently small profile neighborhood of \(r_-\). Singular azimuthal winding does not differentiate \(Q\); round colatitude length or smooth sphere charts give the same conclusion at poles. Outside this neighborhood the used compact crossing portions are separated from the entry level. Compact convergence of the metric and prepared level graphs therefore gives a unique transverse entry crossing, separated from all other used portions by a uniform collar. Only times \(t_T+O_R(1)\) are needed until a fixed large radius \(R\) is reached. This comparison does not require the launching azimuthal phase to converge.

The frozen branch is asymptotically radial. The limiting co-chart is asymptotically affine in differential order, with small changes of axes, shear, and boost. At a sufficiently large fixed co-radius \(R\), followed by sufficiently large \(T\), the past-directed lab velocity \(\mathbf w=dz/d(-t_e)\) has length \(1+o_{R\to\infty}(1)\) and makes a small angle with \(y/|y|\). There \(z=z_*+O_R(1)\) and \(t_e=t_T+O_R(1)\), with uniformly nondegenerate momentum and velocity normalizations on the whole launch patch. Small drift and shear use only the fixed base smallness, not frequency-dependent smallness.

In the far lab region, with emitter radius \(r\asymp|y|\) and \(z=C_e(q)+rn\), \(t_e=r+q\), the far estimates give, uniformly in time, \[ \begin{split} |g-\eta|&\lesssim r^{-1+.01},\\ |\partial g|&\lesssim r^{-2+.01} +r^{-1+e/2}\left\langle q/r^{\delta_f}\right\rangle^{-A_H/2}, \qquad |\partial^2g|=o_{r\to\infty}(1). \end{split} \tag{238}\] These are ordinary lab derivatives. To obtain them, use two spare rotations in the point bounds and \(\partial_i=\ell_{n,i}T_l+r^{-1}b_iZ\). Second derivatives expand with lab times commuted innermost; angular and \(Z\) terms have an extra \(r^{-1}\), and the remaining low unrestricted jets are at worst a bounded coefficient times the displayed bad-field size. The differentiated emitter conversion coefficients remain bounded. The smoothed target’s differentiated symbol terms also tend to zero.

While \(\mathbf w\) remains close to its crossing value, \(dy/d(-t_e)=\mathbf w+s_e(t_e)\) lies in an outward cone, \(n\) is close to \(y/|y|\), and \[\frac{dr}{d(-t_e)} =\frac{n\cdot(\mathbf w+s_e(q))}{1-n\cdot s_e(q)}\] has positive lower and upper bounds. Thus \(q\) decreases at a rate bounded below. On each large radial dyad \(r\sim R_1\), integration in \(q\) over an interval of length \(O(R_1)\) bounds the second term of (238) by \[O\left(R_1^{(e-(1-\delta_f)A_H)/2}\right).\] Here \(A_H<2\), and the exponent is strictly negative for the fixed far parameters. The first term is summable as well. The lab geodesic equation in time parameter bounds both the velocity derivative and logarithmic frequency-scale derivative by \(C|\partial g|\); spacelike temporal hyperbolicity bounds \(|\mathbf w|\). Choosing \(R\) large makes the integrated cost small, closes the outward-cone argument, and keeps momentum nondegenerate until \(t_e=0\). The ray then remains in the far portion on which \(t_e=0\) is the original bridge. Integrating its velocity proves [pk2:arrival-shell].

Finally consider the time-dependent null Hamiltonian root in lab coordinates. In its first variation, the diagonal blocks and the position-to-momentum block tend to zero with radius by the first two derivative bounds in (238); the momentum-to-position block is bounded. On the rescaled variables \((\epsilon_1\delta z,\delta\pi_{\rm sp})\), choose \(\epsilon_1>0\) small first and then a sufficiently large fixed radius. The rate is then arbitrarily small on the far portion. The complement has time length \(O_R(1)\). This gives subexponential all-subsegment propagators. Higher jet induction has the same homogeneous propagator and sources made from lower jets and the available subexponential coefficient derivatives. It also covers derivatives in the launch-center parameters. The integrated connection bound gives bounded tensor parallel transport and inverse. ◻

Proof. The construction has four stages. We first propagate complex phase jets and the central tensor amplitude along the return rays. We then impose the wave-gauge equations on every finite jet. At each inverse frequency order, Gaussian superposition leaves a new central amplitude subject to the gauge equation. Choosing its free part to match the interior coefficient makes the real transport hierarchy determine the matching normal jets. Finally we choose finite Taylor orders and cutoffs to obtain the required residual bounds.

Proof. Apply 35 with \(Y^s=H_4^s\) and \(X^s=H_1^{s+2}\oplus H_2^{s+1}\). Its right inverse fixes the unit inner ball and permits the stated asymptotic tail. Its single-base-order nonlinear solve proves both the estimate and higher smoothness without additional smallness assumptions. ◻

Proof. We first prove the exact support assertion; the correction’s tail need not be compact. Write \(s_* =\sup|C_e'|<1\), fixed by the base smallness. Suppose a causal path from the changed initial support first reaches a fixed sufficiently large co-radius \(R\) at time \(\tau\le t_T\). Before this hit its lab speed is at most \(1+\delta_R\), where \(\delta_R\to0\) as \(R\to\infty\), also in the exponentially small comparison bootstrap. Its initial point is outside the radius \(\rho_T\) about \(z_*\), and its hit point is \(C_e(\tau)+O(R)\). Thus \[ (1-2\alpha)t_T \le(1+\delta_R)\tau+s_*(t_T-\tau)+O(R). \tag{245}\] Equivalently, \[ t_T-\tau\le \frac{2\alpha+\delta_R}{1+\delta_R-s_*}\,t_T+O_R(1). \tag{246}\] Since \(t_T\asymp T\), a fixed large \(R\) and a fixed small \(\alpha\) make the right side less than any prescribed small multiple of \(T\) for large \(T\). Only the inner exclusion radius of the change enters this estimate; its outer extent is irrelevant. The beam supports obey the same conclusion by their real centers and exponentially thin tube margins.

Moreover \(I(t_T)=T\) and \(I'\le C_I\) show that the selected entry points with \(v<(1-\nu_0)T\) have \[t_e\le t_T-\frac{\nu_0}{C_I}T+O(1).\] Choose \(\alpha'<\nu_0/(2C_I)\) and make the first-hit allowance smaller still. Ordinary local wave-map uniqueness, from exactly equal metric and gauge Cauchy jets, proves exact equality on that earlier entry interval. On \(s=0\), the interval of [pk2:domain] is \(-u_2\le v\le T+c_2\); its negative part is fixed, and its middle and unselected-end portions are protected by finite-time domain of dependence. The deeper inner face is future outflow, so no past influence can enter from the deep block. There is consequently no additional dependence on an unexamined route through the other end.

For the error estimate use the actual-principal ordinary scalar stress energies on component errors, with lab mixed commutations and a small fixed positive mass. The future slice normal is a uniformly timelike multiplier, and both artificial boundaries have strict outflow fluxes. The first- and second-order lab background derivatives that determine the top linear rate are globally bounded and tend uniformly to zero far out. Indeed the linear tensor-wave lower terms are connections, their derivatives, and curvature, while leading principal commutations use first coefficient derivatives. Fix the mass small first, then choose \(R\) large: the square-root top rate on the far support is arbitrarily small. On its complement that rate is bounded independently of \(R\) and \(T\). Higher background hits multiply lower commuted error norms, with subexponential coefficients.

The non-small-rate part of the support has time length at most \(\alpha'T+O(1)\) by (246). The lower-jet triangular estimate used in the deep comparison costs just an \(e^{o(T)}\) factor. Retaining the exact principal coefficients, quasilinear remainders have an exponentially small packet or bootstrap factor against each remaining error placement, plus the source in [pk2:beam-estimates]. Ordinary Sobolev product estimates on the uniform annuli and inner charts apply without an inverse lapse loss. Initial mixed jets follow from the equation on uniformly spacelike slices. Taking the far rate and \(\alpha'\) sufficiently small therefore gives [pk2:exterior-error].

The parameter inequalities below strictly improve the exponentially small bootstrap. Ordinary continuation preserves the two boundary margins; exact wave-constraint propagation yields vacuum. Transform the resulting entry jets to the fixed background wedge coordinates. The transition and trace costs, with fixed buffers, are only \(e^{o(T)}\). The wave-map condition with target \(g\) is tensorial and exact. Beam matching adds the linear inverse-accuracy bound from Proposition 29. The choices below make the total smaller than (236); the already proved protected-region statement supplies its exact early part. All estimates allow fixed small enlargements for the final gluing. ◻

Proof. The corrected data are complete and converge in \(p_{10}\). They agree with \(d\) off the selected end and inside the protected ball. The center satisfies \(|z_*|\le s_*t_T+O(1)\), with \(s_*<1-2\alpha\), so that this ball contains the fixed-origin ball of radius \[(1-2\alpha-s_*)t_T-O(1)\longrightarrow\infty.\] Every fixed compact causal shadow used in the early bridge construction is therefore unchanged for large \(T\).

Construct the full finite lower bridge development for \(d_T\), including all initial spatial radii and a past collar, by the finite-range Cauchy argument already proved. The bounded central joins, the unselected end, and the bent compact part of the selected \(t_e=0\) slice agree with the original finite solution by domain of dependence. Near infinity the selected horizontal slice is the original initial surface carrying the actual corrected data, including their tail.

Attach the selected ordinary wave-map exterior comparison through an enlarged band about the fixed time \(t_a\), only inside (244) and before its terminal time. Use the finite wave-map uniqueness and straightening from the comparison development construction, with the same new data on both sides. Fixed-time bounds and strict deeper outflow and spatial margins give the identification, uniformly through the finite outer cutoff. The straightening agrees on the used band and is cut off inside its margin; on bounded central portions it is the original identification because those solutions remain unchanged until much later.

To see why the entire initial bridge is retained, put \(f(t_e,y)=|y|-R_T(t_e)\) and let \(t_{\rm end}\) denote the comparison terminal time. Locally the selected-end union is \[ \{t_e<t_{\rm end}\}\cap \bigl(\{t_e<t_a\}\cup\{f<0\}\bigr), \tag{249}\] with the lower \(\mathcal S\) cut imposed where active. The lower set \(t_e<t_a\) includes every spatial radius; only the added future block is truncated. On future causal curves, \[\frac{df}{dt_e} =\frac{d|y|}{dt_e}+V_{\rm out} \ge V_{\rm out}-V_{\rm max}>0.\] Thus the strict sublevels, their union, and their intersection in (249) are preserved to the past. A past curve in the added block retains \(f<0\) until it crosses \(t_a\), then enters the unrestricted lower block. It cannot end at the omitted corner \(t_e=t_a\), \(f=0\). Unmatched future faces are omitted; the original bridge is not truncated or replaced by the old datum outside the computing cutoff.

The used portion of \(\mathcal S\) and a matching collar lie within the available margins. Its large positive endpoint has \(t_e\le t_T+O(1)\) and bounded co-radius, while the middle and unselected portions are unchanged. The fixed background surface is still spacelike for the new metric. Its exact comparison jets and wave-map gauge agree with the deep Cauchy data. In the common background collar, identify the deep vacuum solution across it, retaining only the strict interior of [pk2:domain].

More explicitly its active future cuts are \[s<s_{\rm max},\qquad F_0=u+c_0s/T<u_2,\qquad F_1=v+c_0s/T<T+c_2, \qquad s_{\rm max}=(T+u_*)/2+c_0.\] If \(\varepsilon_T\) is the exponentially small relative cone error, then \(du/ds,dv/ds\ge-C\varepsilon_T\) and hence \[\frac{dF_i}{ds}\ge c_0/T-C\varepsilon_T>0.\] The signal point has \(v=T\), \(u=u_T<u_*\), with terminal margin at least \(c_0\) and strict side margins by the choice of \(c_0\). Past causal paths therefore reach \(s=0\) strictly inside the attached interval. For each finite \(T\) all relevant coordinate ranges are bounded and the metric is nondegenerate, so ordinary continuation handles interior endpoints.

Omit every unused future face and edge from the lower pieces. Their retained neighborhoods at intersections of cuts are finite unions and intersections of the same past-preserved strict sublevels; such corners cannot be past endpoints from the interior. Use the identified collars to make the product gluing of the comparison development, with no duplicate open continuations. This yields a smooth Hausdorff vacuum manifold. Every past-inextendible causal curve from the experiment reaches the entry slice and then the complete corrected bridge. Lower pieces have the finite Cauchy control already proved; bounded speeds exclude escape to infinity in bounded time. The past collar supplies the other direction of the Cauchy property. Thus \(\Sigma\) is Cauchy, and the maximal-development theorem embeds this development into the full MGHD of \(d_T\).

Finally the fixed earlier compact segment of the background geodesic, through its transverse entry and back to \(\Sigma\), lies with a neighborhood in the unchanged region for large \(T\). The backward entry states from Lemma 30 continue through this fixed tube to seeds \(w_T\to w\) by ordinary geodesic flow dependence. This proves both the attachment and the seed assertion. ◻

Proof of Proposition 26. Fix \(m\), \(\eta\), and \(J\). If a relatively open neighborhood \(\mathcal V\) of \(d\) is also prescribed, choose a finite order \(m_{\mathcal V}\) and \(\eta_{\mathcal V}>0\) such that the relative ball \(p_{m_{\mathcal V}}(d'-d)<\eta_{\mathcal V}\) lies in \(\mathcal V\). Replace the construction order by \(\widehat m=\max(m,m_{\mathcal V},10)\) and its tolerance by \(\widehat\eta=\min(\eta,\eta_{\mathcal V})\); without a prescribed neighborhood use \(\widehat m=m\) and \(\widehat\eta=\eta\). Make all choices in Subsection 9.8 at this enlarged finite order. The constrained beam correction gives smooth complete exact vacuum data \(d_T\) satisfying (243). The ordinary exterior comparison supplies the exact entry jets required by the deep comparison. Lemma 30 gives the strict finite tidal observation and convergent backward entry states. Proposition 32 places the point, its frame, and its geodesic back to \(\Sigma\) in the full MGHD of \(d_T\), with \(w_T\to w\). Convergence in \(p_{\widehat m}\) attains both the original tolerance and the prescribed neighborhood for sufficiently large finite \(T\). The frequency exponent may depend on this enlarged order; no convergence in all seminorms for one fixed construction is asserted. ◻

Completion of the proof of Theorem 1. Use the single base neighborhood constructed in the preceding sections. Fix \(O,N\), \(d\in\mathcal B(O,N)\), a finite \(m\ge10\), and a tolerance \(\eta>0\). The first-runout and corner-to-branch arguments give a seed \(w\in O\) with a noncorner first runout. The closed test \(A(O,N)\) supplies its bounded parallel curvature, whether or not this new branch seed itself has a regular exit.

Choose a neighborhood of \((d,w)\) on which the bounded-seed tidal comparison has a common constant \(C_1\): \[|\operatorname{Riem}(U',X',U',X')| \le C_1Q_{d'}(w',\tau) \quad\text{for }g_{d'}(X',X')=1,\quad g_{d'}(X',U')=0.\] Indeed parallel transport back to the initial frame gives the explicit factor \((1+2h'(w',w'))^2\), uniformly bounded near \((d,w)\). Choose \(J>C_1N\) before constructing the late experiment. Apply Proposition 26 with tolerance smaller than \[\min\left\{\eta, \epsilon-p_{10}\bigl(d-(h_*,K_*)\bigr)\right\}.\] The second quantity is positive. For large finite \(T\), one has \(w_T\in O\), \(d_T\in\mathcal U_\epsilon\), \(p_m(d_T-d)<\eta\), and, at the finite point in its full MGHD, \[Q_{d_T}(w_T,\tau_T) \ge C_1^{-1} |\operatorname{Riem}(g_{d_T})(U_T,X_T,U_T,X_T)|>N.\] This is exactly the destruction of the closed test required in Section 2. Each relative closure of \(\mathcal B(O,N)\) is therefore nowhere dense. Their countable union covers the future-extendible data, proving the stated meagreness and the open-dense-intersection assertion.

We recall why the same \(\epsilon\) serves all these tests. The physical profile range and depth, computing depths, ray and horizon margins are fixed first at the given mass and spin. Next come the cylinder gauges, measuring radii, compact/far absorption scales, fixed overlap times and regions, and the base-order continuation thresholds for the exterior, entry, and double-null constructions. All resulting base input conditions on the two ends, bent slices, and finite joins follow from one sufficiently small \(p_{10}\) threshold. Higher smooth estimates for a particular \(d\) have datum- and order-dependent constants, but arbitrarily small exponential rates obtained by later-time or larger-radius choices, without higher-order data smallness. The seed family, \(N\), the finite \(m\), branch location, comparison orders, frequency, formal accuracy, and late time affect only this finite experiment. They do not alter the base neighborhood. All spin gaps and constants need be positive only for the fixed \(0<\mathfrak a<M\), as required. ◻

Proof. These anchors will be proved without a characteristic commutator estimate. Up to time \(\lesssim R_f\) work outside \[\begin{equation*} |z|\ge t+cQ+C_1\int_0^t ((u+Q)Q)^{-1/2}du \end{equation*}\] with \(c\) a sufficiently small fixed band constant, \(C_1\) large fixed. Indeed its ratio to \(Q\) is bounded by \(C\sqrt{R_f}Q^{-3/2}\lesssim R_f^{(1-3\theta)/2}=R_f^{-.175}\), so the correction is \(o(Q)\) uniformly on the band. Bootstrap unweighted slice energies of \(h\) after rotations, boosts, translations and the flat dilation through order nine by a large constant times \((e_{\rm in}+\mathsf S_a) Q^{-1/2}\). The initial norm involves at most ten derivatives of \(h\) and at most nine derivatives of its initial normal jet, all controlled by the small initial seminorm. The squared integral of the leading \(r^{-2}\) first jet outside \(r\asymp Q\) is \(O(Q^{-1})\), which gives the claimed initial energy. Radial integration to infinity gives the commuted fields in \(L^2_\omega\) of size \(O(|z|^{-1/2})\) times the energy, and sphere sup with two spare rotations. The boundary moves at least as fast outwards as the outgoing light speeds including bootstrap errors. The lab time multiplier for the true principal is uniformly timelike and controls full unweighted first derivatives. Here are the energy error details. Commuting the principal leaves, besides constant linear multiples of lower flat wave terms (eliminate recursively), terms \(\mathcal Z^b H^{ij}\partial^2\mathcal Z^d h\), \(b+d\le9,d<9\), for the Minkowski list \(\mathcal Z\), including index mixing by commutators with derivatives. The first factor is bounded on the sphere by \(C/|z|+C_{\rm boot}(e_{\rm in}+\mathsf S_a)(Q|z|)^{-1/2}\); if high, use only \(L^2_\omega\) on it, placing two extra rotations on the low second factor. Smooth composition has the same placements. One derivative of the second factor costs at most \(C/Q\) times one dilation/boost/rotation commutation and lower terms (apply to first derivatives). Derivative-of-principal energy errors and the differentiated quadratic first-jet terms of (252) obey the same product bounds, using the derivative placement with \(1/Q\) on a coefficient. Linear lower terms with symbol-decaying coefficients do also, by energy/Hardy or the commuted field bound. For instance a coefficient \(O(|z|^{-3})\) times a high field costs \(C(t+Q)^{-2}\) times the energy at that order. The time integrals of all multiplier losses just described are \(o(1)\), including \(O(\sqrt{R_f}Q^{-3/2})\). The pure defect commutations cost \(C\mathsf S_a |z|^{-3}\) pointwise and \(\lesssim\mathsf S_a Q^{-1/2}\) in \(L^1_t L^2_z\). This proves the auxiliary bootstrap by the ordinary domain energy inequality. At a spacelike evaluation band, \(Z\) and its iterates can be expanded with bounded coefficients in these commutations (\(r(T+\partial_{|z|})\) uses the sum of a boost and dilation); the corresponding first-jet statements include \(O(1/r)\) coefficient derivatives. This gives (256), uniformly also if the lifespan truncates the band. ◻

Proof. For \(t\lesssim r\) integrate to an anchor with \(r'\asymp r\) large enough compared with \(t\), using (256); the integral of \(w_H^{-1}\) costs \(\lesssim r^{e+\delta A_H+1-A_H}\). For \(t\gg r\) integrate inward to the fixed starting annulus, using a compact weighted endpoint anchor there (an average suffices). The top time weight is everywhere at least a constant times \(r^{-.021}\). For the last estimate take a radial unit trace using the endpoint bad derivatives also at the next order, and (if undifferentiated terms occur) the first bound divided by \(r\). In detail, \(Z,\partial_q\) commute and \(\partial_q(rT^j h)=k r T^{j+1}h+r^{-1}s^d b_dZ(rT^j h)+O(r^{-1})rT^j h\). Thus \(\partial_q^2\) in this trace expands into first derivatives of the next and lower fields with bounded coefficients and possible \(r^{-1}\) times fields through order eight. The harmless field terms satisfy the required weights, and first derivatives against \(w_H\) have a comparable weight throughout the interval. Angular and \(r|q\) derivatives are simpler. This proves (257); the compact overlap or one-sided traces suffice near a radial cutoff. The same convention of two spare rotations gives angular sup versions. In particular for \(n_*\le7\), \[ \|\chi(\psi)\|_{L^2_\omega} \lesssim N r^{e/2}\mu^{-A_H/2}. \tag{258}\] Use \(D=(1-s_s\cdot n)T+r^{-1}s_s^d b_dZ\), with lab-time commutators of \(s_s\) taken before expansion, and (257); \(rZ^IT^j\) or \(Z^I rT^j\) field lists are equivalent. All first-derivative versions at one lower order follow too, up to harmless \(O(1/r)\) lower terms. ◻

Proof. The first estimate follows by smooth inversion of \(B+h\), the symbol bound on \(B-\eta\), and (257) through order eight. For the last two, (251) after ordered differentiation controls the bad derivative of \(h_{LL}\), the \(L\)–screen components, and the spherical trace of \(h\): (257) bounds each good derivative by \[\begin{equation*} C N r^{-2+2\eta_*}\tau^{-A_H/2}, \end{equation*}\] and products such as \(H\,\partial h\) satisfy the bound as well (lab differentiations may be distributed before projecting; the highest differentiated \(\partial h\) is of total order at most eight and its bad part can use a \(T\)-field). This is just solving the flat gauge on the \(\ell\) gradient using (251) and (253), using \(r^{-1}b Z\) for the omitted part, or using \(k^{-1}\partial_q\) for the bad derivative. Derivatives of bounded metric compositions have at most one high sphere factor; inverse contraction differs from the negated flat-raised covariant perturbation by a smooth quadratic. \(\Theta\) costs at most \(C\widetilde a/r\) with ordered derivatives allowed. The \(r^{-2}\) term at this precision is for the stationary background contribution, not a claimed \(\tau\)-decay of all of \(H\). ◻

Proof. The second follows exactly as in (259), using the rough \(r^{.52}\) order-nine field bound. For the others it suffices to prove corresponding covariant \(h_{LL}\) estimates: smooth quadratic inverse errors at order nine after multiplication by \(r\) are even \(O(r^{-.46})\).

The constrained radial integral. Keep \(t,n\) fixed and integrate the equation for the bad derivative of \(Z^d(r h_{LL})\) from negative \(q\) toward the cone. At fixed \(t\) this adds a flat good derivative, since the integration differentiates by \(\partial_q-\partial_r\). By (251) the terms to be integrated have the following sphere sizes, after distributing \(Z\)’s through order \(d\): flat \(G_1\chi(h),\chi(h)/r\); products of \(B_1\chi(h),\chi(h)/r\) with factors from the order \(r^{-1}\) background perturbation or \(r^{-1}\chi(h)\), with further bounded metric compositions allowed; and \(O(\widetilde a)\). Here all \(\chi\)’s need only spatial-emitter commutations (no lab times); this follows using \(\partial_i=(\ell_i/k)\partial_q+\) good and \(Z\) on the full constrained combinations.

Negative-\(q\) anchor bands. When \(t\asymp r\), start at \(-q\asymp r\); radii during integration stay comparable up to a target in \(t\asymp r\). In the spacelike bands \(-q\asymp Q\gtrsim r^\theta\), for \(d\le8\) the integral of good terms costs \(CN (Q/r)^{1/2}\) bandwise by (256). Products there are summable as well: a bad derivative in these products through that order is bounded using the extra Minkowski commutation with loss \(C/Q\), giving e.g. \(O(N/Q)\) for the quadratic band cost. For \(d=9\), use the unweighted spacelike-band slice energy for a highest first derivative, giving \(O(N)\) per band; undifferentiated order-nine fields use (256), and in products with a high field the bad derivative now costs \(1/Q\) only on a low factor.

Near the cone. Up through \(|q|\lesssim r^\theta\) the remaining integral of \(G_1\chi\) costs \(\lesssim N r^{(\theta-p_i)/2}\) at the corresponding \(H,0\) time-slice orders. For \(d=9\), \(\chi_9/r\) with nine \(Z\)’s uses the \(W_H\)-controlled \(G_1\chi_8\) on this slice. Products with high \(B_1\chi_9\) use \(L^2_{q,\omega}\) size \(\lesssim N r^{.011}\) on comparable radii and gain the factor \(r^{-1+2\eta_*}\); products with \(\chi_9/r\) and low \(B_1\chi\) can use the same \(G_1\chi_8\) bound and (257) on the low factor. All lower undifferentiated terms alone cost at most \(CN r^{-1+2\eta_*}\) per unit length. These estimates give tightness for \(d\le8\), at most logarithmic loss for nine.

Longer intervals and timelike evaluation. For evaluation up to length \(b\gtrsim r^\theta\) they cost at most the two additional terms of the last bound, using time-slice Cauchy–Schwarz with length \(b\). Targets with \(t\ll r\) need only (256). If \(t\gg r\), start instead by a length-\(\asymp r\) local average of \(Z^9(rh_{LL})\) on comparable radii (use the \(G_1\chi_8\) time-slice norm), then integrate the same estimates over this length. This proves (260). ◻

Proof. Put \(\chi_j=rT^jh\), \(j=0,1\), and \(\mathcal Y_j=\Pi k^{-1}\partial_q\chi_j\). The projection \(\Pi\) depends on \(n\) only; \(k\) depends on \(q,n\) only. Thus projection of the two leading terms of the scalar emitter operator is exactly \[ -2\Pi\partial_q\partial_r\chi_j +H_{\ell\ell}\Pi\partial_q(k^{-1}\partial_q\chi_j) =(-2k\partial_r+H_{\ell\ell}\partial_q)\mathcal Y_j. \tag{262}\] We show that every remaining term is bounded by \(CNr^{-2+.04}\). This calculation uses only low commutations, so it precedes and supplies the undifferentiated supremum used in the high-order products of 11.

Choose a temporary fixed \(\varepsilon>0\) larger than the finitely many \(\eta_*,e\) losses in the following bounds, with \(3\varepsilon<.04\). The sphere estimates in (257), with two spare rotations, give \[\begin{equation*} |Z^{\le2}\chi_j|+|Z^{\le1}\partial_q\chi_j| +|\partial_q^2\chi_0|\lesssim Nr^{\varepsilon}, \qquad j=0,1. \end{equation*}\] For the last term, use \(\partial_q(rT^jh)=krT^{j+1}h+s^db_dZ(T^jh)\), differentiate once, and commute \(\partial_q\) through \(Z\). This leaves first derivatives of the next and lower fields, \(r^{-1}\) times good derivatives, and \(a(q)\partial_q\chi_0\). The largest field count in the displayed point estimates, including the two rotations, is at most \(j+2+2\le5\); the unit trace for a bad derivative uses one further order, still below the weak cap \(N_H=8\). Also \[\begin{equation*} |H|\lesssim r^{-1}+Nr^{-1+\varepsilon},\qquad |H_{\ell\ell}|\lesssim r^{-1+\varepsilon},\qquad |a(q)|\lesssim N. \end{equation*}\] The angular term \(r^{-2}\Pi\mathcal A_s\chi_j\) is therefore \(O(Nr^{-2+\varepsilon})\). The three remainder types in (255) have bounds \[\begin{equation*} \frac{|H|}{r}|(1,Z)\partial_q\chi_j|,\quad \frac{|H|}{r^2}|(1,Z,Z^2)\chi_j|,\quad \frac{|a(q)H_{\ell\ell}|}{r}|(1,Z)\chi_j| \lesssim Nr^{-2+2\varepsilon}. \end{equation*}\] In particular no uncontrolled radial power accompanies the acceleration. One also has \[\begin{equation*} r|a(q)H_{\ell\ell}|\lesssim1, \end{equation*}\] by (260) near the cone and the decay of \(a(q)\) away from it.

For \(j=1\), commuting the Cartesian derivative \(T\) through the physical wave operator introduces \(-P[TH]\chi_0\). The full coefficient obeys \(TH=O(r^{-2})+O(Nr^{-1+\varepsilon})\), whereas (259) gives \[\begin{equation*} (TH)_{\ell\ell}=O(r^{-2})+O(Nr^{-2+\varepsilon}). \end{equation*}\] Thus its two-bad part, \((TH)_{\ell\ell}\partial_q(k^{-1}\partial_q\chi_0)\), is \(O(Nr^{-2+2\varepsilon})\). The mixed part has the additional \(r^{-1}\) in (255); its good and acceleration parts are smaller. It is the improved double contraction, rather than an improved bound on every component of \(TH\), that is needed.

For the quadratic source, polarize the constant Cartesian polynomial before projecting. In particular \(TQ_\eta(\partial h,\partial h)=2Q_\eta(\partial Th,\partial h)\). For \(h_i=T^ih\), \(i=0,1\), write \[\begin{equation*} \partial_\alpha h_i=\ell_\alpha\breve p_i+E_{\alpha i}, \qquad \ell^\mu\bigl((\breve p_i)_{\mu\nu} -\tfrac12\eta_{\mu\nu}\operatorname{tr}_\eta\breve p_i\bigr)=0, \qquad \Pi\breve p_i=\Pi T^{i+1}h. \end{equation*}\] Take \(\breve p_i\) from (253) with \(p=T^{i+1}h\). The good derivative terms and (251), differentiated once in Cartesian lab time when \(i=1\), give \(|E_i|\lesssim Nr^{-2+2\varepsilon}\). For example the differentiated gauge error contains \((Th)\partial h\), of size \(O(N^2r^{-2+2\varepsilon})\), and not a term of order \(r^{-1}\). The constrained–constrained polarization is a multiple of \(\ell\otimes\ell\) and is annihilated by \(\Pi\). Every other quadratic term has one \(E_i\) and one full first jet of size \(O(Nr^{-1+\varepsilon})\); after the physical-source factor \(r\), it is \(O(N^2r^{-2+3\varepsilon})\). Differentiating an already expanded frame instead gives \(T\ell=O(r^{-1})\) and the same suppression. The vanishing-coefficient quadratic terms in (252), including their single \(T\) derivative, also satisfy this bound. The linear lower terms are \(O(Nr^{-2+\varepsilon})\), and the pure defect costs \(r^{-1}\widetilde a\lesssim Nr^{-2}\). We have proved \[ |(-2k\partial_r+H_{\ell\ell}\partial_q)\mathcal Y_j| \lesssim Nr^{-2+.04}. \tag{263}\]

At fixed \(n\), this transport direction has strictly decreasing lab time, with \(dr/d\sigma=-2k<-c\) and \(dt/dr=1-H_{\ell\ell}/(2k)=1+O(r^{-1+2\eta_*})\). Its backward integral curve reaches \(t=0\) or the fixed-radius starting region and stays within the current finite lifespan. Since \(\int_{R_0}^{\infty}r^{-2+.04}\,dr<\infty\), the initial symbol bounds and compact low point bounds control \(\mathcal Y_j\) by \(CN\). The displayed identity for \(\partial_q\chi_j\) converts this to the first two terms of (261), the additional term being good. Finally differentiate \(\psi=(1-s_s\cdot n)Th+r^{-1}s_s^db_dZh\) once in lab time, taking the lab derivative before expanding the frame. It gives \(r\Pi T\psi=(1-s_s\cdot n)r\Pi T^2h\) plus good-frame and \(\widetilde a\) terms bounded by the same low estimates. This proves the last term of (261). ◻

Proof. Claim 1: the corrected divergence identity. Use volume \(r^{-2}d{\rm vol}_g\), comparable to \(dt\,dr\,d\omega\); prime on a divergence indicates this density. The normalized equation for \({\rm div}'(d\chi)^\sharp\) differs from \(P_g\chi/k\) by a first-order coefficient \(O(r^{-2})\) (contracted Christoffels) and a potential of size \[\begin{equation*} O(r^{-2})\,O\big(|s(q)|+|H|+r|a(q)H_{\ell\ell}|\big). \end{equation*}\] The latter follows from (254), (255) on a constant and is small times \(r^{-2}\), by (259), (260) and the acceleration decay. Write \(Q(\xi)=\xi\otimes\xi-g^{-1}(\xi,\xi)g/2\). For a multiplier \(Y\) the primed divergence identity has deformation part \[\begin{equation*} g(\nabla_{\xi^\sharp}Y,\xi^\sharp) -\tfrac12{\rm div}'Y\,g^{-1}(\xi,\xi) \end{equation*}\] and other terms \(\xi_Y\,{\rm div}'\xi^\sharp+d\xi(\xi^\sharp,Y)\). Use \(Y=W_iKX\) and factor out its scalar weight. To calculate the correction, write \(F=c\mathcal P\). Since \(\vartheta^\sharp=-\bar X/G\), we have \[\begin{equation*} \xi^\sharp=-\frac{\xi_{\bar X}}G X-\frac{\xi_X}G\bar X +\xi_{\rm scr}^{\sharp},\qquad \vartheta(X)=1,\quad \vartheta(\xi^\sharp)=-\xi_{\bar X}/G. \end{equation*}\] The differentiated-\(F\) contribution from divergence is \(-G^{-1}\xi_X\bar XF\). In the curl contribution the term \(-G^{-1}\xi_{\bar X}XF\) from \(dF(\xi^\sharp)\) cancels \(-dF(X)\vartheta(\xi^\sharp)\) exactly. The sum is therefore \[\begin{equation*} -2G^{-1}\xi_X\bar XF +\nabla_{\rm scr}F\cdot\xi_{\rm scr}. \end{equation*}\] Since \(2c/G=\alpha/k\), this cancels the indicated source component with its exact sign. When the derivative falls on \(\vartheta\), the divergence term is proportional to \(\xi_X\), while in \(F\,d\vartheta(\xi^\sharp,X)\) the \(\xi_{\bar X}\) part of \(\xi^\sharp\) vanishes by \(d\vartheta(X,X)=0\). Thus no remaining correction term from divergence or curl involves \(\xi_{\bar X}\). They are bounded, without the weight, by \[\begin{equation*} C r^{-1} \big[ |\xi_X|((1+\sqrt M)|\mathcal P|+|Z\mathcal P|+r d_g|\partial_q\mathcal P|) +|\xi_{\rm scr}|(|\mathcal P|+|Z\mathcal P|)\big]. \end{equation*}\] To check coefficients, first derivatives in a good true direction of \(g,n,X,\bar X\) in lab components cost \(O(1/r)\); first unrestricted derivatives cost \(O((1+\sqrt M)/r)\), by (257) and the frame formulas. The same estimates apply to \(\vartheta\), in particular to \(d\vartheta(e_A,X)\) and \(d\vartheta(\bar X,X)\), respectively. Screen derivatives of \(c\) cost \(O(1/r)\); for \(\bar X c\) the extra dependence on \(s(q)\) uses \(\alpha/k=(1+(z_X-1)/k)/(z_X-z_{\bar X})\), hence \(r|a(q)|d_g\lesssim1\). Also \(\partial_q-\alpha\bar X\) differentiates \(\mathcal P\) by true good directions, giving the \(r d_g\) allowance.

The outgoing deformation has a lower bound of a positive constant times \[ \frac{W_i}{r}\big((1+M)|\xi_X|^2+|\xi_{\rm scr}|^2\big) \tag{270}\] up to small errors in the joint estimates. For details, the derivative of \(W_iK\) alone gives \[\begin{equation*} -G^{-1}\bar X(W_iK)|\xi_X|^2-\tfrac12 X(W_iK)|\xi_{\rm scr}|^2. \end{equation*}\] Here \(-r\bar XW_i/W_i\) is positive with fixed margin (each \(p_i>A_i\)), and \(r XW_i/W_i<2\) with fixed margin. The following frame estimates check the remaining deformation: \[\begin{equation*} \begin{split} &g(X,\nabla_Y X)=0,\\ &g(e_B,\nabla_{e_A}X)=\delta_{AB}/r+O(r^{-2+.04}),\\ &g(e_B,\nabla_X X),\ g(\bar X,\nabla_X X)=O(r^{-2+.04}),\\ &{\rm tr}_{\rm scr}\nabla X-2Xr/r=O(r^{-2+.04}). \end{split} \end{equation*}\] Indeed differentiation of the algebraic normals in \(X\) gains \(Xg=O(r^{-2+.04})\), \(Xn=O(r^{-2+.04})\); spherical differentiation gives the ordinary flat expansion, and Christoffels with the displayed slots use good derivatives or the improved first derivative of \(g_{LL}\) from (259), to the precision indicated. The other components of \(\nabla X\), after subtracting the flat emitter ones of order \(1/r\), cost \(O(r^{-2+.04})+O(N\sqrt{1+M}/r)\). Any non-small flat off-expansion components relevant to cross terms between \(\xi_X,\xi_{\rm scr}\) or to \(\xi_X^2\) here cost only \(O(|s|/r)\) (for the flat \(L\), incoming variation of \(n\) is \(O(|s|/r)\)). Consequently the potentially two-bad square vanishes exactly by nullity, mixed terms involving the bad derivative have coefficients \(O(r^{-2+.04})\), and the screen expansion supplies the missing positive \(W_iK|\xi_{\rm scr}|^2/r\). For example the square of a mixed coefficient, relative to the two positive bulk weights, is bounded by \(r^{-2+.08}W_i/w_{i,j}\lesssim r^{-.22}\). We absorb mixed bad terms using \(W_i/w_{i,j}\ll r^{1.7}\); other cross terms use smallness and (270). Here \(\xi_{\bar X}=\bar X\chi\).

The favorable volume sign is positive with the future energy convention (time endpoint energy is subtracted at positive divergence). Integrate on slabs cut off either just by a time endpoint or also by an outgoing cap. All future endpoint stresses in this argument are nonnegative as energies. At a time endpoint the outgoing one controls \(W_i\) times \(|\xi_X|^2+|\xi_{\rm scr}|^2\); at \(\mathfrak u=u\) it controls \(W_i|\xi_X|^2\,dr\,d\omega\). To verify the latter with its measure, \(d\mathfrak u\) has no screen component, so a future normal is \[\begin{equation*} -\nabla\mathfrak u=\frac{\bar X\mathfrak u}{G}X +\frac{X\mathfrak u}{G}\bar X. \end{equation*}\] The flux density after the coarea Jacobian is, up to the uniformly comparable primed volume factor, \[\begin{equation*} \frac{W_iK}{\partial_q\mathfrak u} \left(\frac{\bar X\mathfrak u}{G}|\xi_X|^2 +\frac{X\mathfrak u}{2}|\xi_{\rm scr}|^2\right) \,dr\,d\omega. \end{equation*}\] Here \(\partial_q\mathfrak u\asymp1\), \(\bar X\mathfrak u\ge c_0\) and \(X\mathfrak u>0\). This proves the uniform lower bound required in (264), even when \(X\mathfrak u\) tends to zero at infinity. The starting radial flux costs are estimated by averaging the lower radius in \([R_0/2,R_0]\) and using the stated compact comparisons. The Hardy control of \(\chi/r\) in (250) follows within this argument: \({\rm div}' X=O(r^{-2+.04})\), so integrate \({\rm div}'(W_i X|\chi|^2/r^2)\), using the negative strict leading derivative of \(W_i/r^2\); only past/radial anchors are charged and \(X\chi=\xi_X-c\mathcal P\). In detail, \(X\log(W_i/r^2)\le-\epsilon_i/r\) for some fixed \(\epsilon_i>0\), and hence \[\begin{equation*} \operatorname{div}'(W_iX|\chi|^2/r^2) \le -\frac{\epsilon_i}{2}\frac{W_i|\chi|^2}{r^3} +\frac{2W_i}{r^2}\chi X\chi. \end{equation*}\] Young’s inequality bounds the last term by half the displayed negative bulk plus \(C_iW_i|X\chi|^2/r\). Future boundary terms of this scalar current are nonnegative with the energy convention, so the bulk \(W_i^{1/2}\chi/r\) norm is bounded by the good bulk, the correction budget, and the past and radial anchors. Conversion from true corrected good derivatives to the flat \(G_1\) in bulk and at time endpoints similarly charges (266) and a small amount of bad energy (\(d_g^2 W_i\ll w_{i,j}\)); it is not needed on outgoing caps. The pointwise comparison behind this step is \[\begin{equation*} |G_1\chi|\le C(|\xi_X|+|\xi_{\rm scr}|+|\mathcal P|) +C d_g|B_1\chi|. \end{equation*}\] The screen bases span the same emitter sphere. The coefficient of the incoming derivative in the difference between \(L\) and \(X\) is \(O(d_g)\); all other difference coefficients are \(O(|H|)\). Since \(d_g^2W_i/w_{i,j}\lesssim r^{-.3+4\eta_*}\), the last term is small in both the spacetime and the time-endpoint norms. The correction terms at time endpoints use the supremum in (266). The first-order and potential differences from \(P_g/k\) listed above are absorbable by these estimates. Also \(\xi_X\lambda_1 L\chi/(kr)\) has good sign up to such conversions (its \(\mathcal P\) part costs (266), not claimed small by sign). The remaining wave source terms use exactly (266), (270) and the mixed entry.

Claim 2: the lab-time multiplier controls the bad derivatives. For the lab-time multiplier \(wT\), \(w=w_{i,j}\), use the uncorrected \(Q(d\chi)\). Its endpoint energy controls the full bad-weight cap on time slices. In the deformation, the two-bad coefficient from \(\mathcal L_T g\) is improved by (259); bad-good mixed errors from metric/density differentiation are absorbable since \(w/W_i\) has a strict negative radial power and \(r|\partial g|\lesssim r^{-1}+N r^{e/2}\mu^{-A_H/2}\) at the required first order. More explicitly \(w_H/W_H\lesssim r^{-e-(p_H-A_H)}\), and the gaps at the other two levels are larger. The weight-derivative part is \[\begin{equation*} -G^{-1} (Xw)Q(d\chi)(\bar X,T) -G^{-1}(\bar Xw)Q(d\chi)(X,T). \end{equation*}\] We have \(r Xw/w<-e/2\): differentiate explicitly, using the tight \(Xq=O(1/r)\) near \(|q|\lesssim r^\theta\) and the looser bound elsewhere. Possible adverse log derivatives from \(m^{-d}\) or \(m_+^{-d}\) in the radial powers are at most \(9d_*\nu\); top \(m_-\) is favorable at fixed \(q<0\). Thus we gain the two-bad bulk with a fixed margin. Adverse \(\bar Xw\) at top is at most \(Cw/r\) by \(q\)-monotonicity; at strong level it is at most \(Cw((r^\delta+|q|)^{-1}+r^{-1})\ll W_P/r\). At weak level the same absorption into \(W_H/r\) holds except \(0\le q\le r^{.3}\); indeed \(1-p_H<.3(1-A_H)\). Here and in this multiplier conversions charging (266) are understood.

On the exceptional weak-weight range use \(G_1\chi=r^{-1}Z\chi\), where the undivided \(Z\chi\) is a field of total order at most nine. At \(r\asymp R_f\), uniformly from \(q=-R_f^\theta\) to \(R_f^{.3}\) at fixed \(t\asymp R_f\), (256) and the top \(w_{0,j}\)-endpoint give the squared sphere bound \[\begin{equation*} \|Z\chi\|^2_{L^2_\omega}\lesssim (e_{\rm in}+\mathsf S_a)^2 R_f^{1-\theta} +\mathsf F^2 R_f^{\theta+.021}. \end{equation*}\] On the shell the bad exceptional integral for the flat good derivatives therefore costs at most \[\begin{equation*} C(e_{\rm in}+\mathsf S_a+\mathsf F)^2 R_f^{-1+.3+\max(1-\theta,\theta+.021)}. \end{equation*}\] The true-minus-flat derivative contribution is absorbed directly by \(w_H\) bulk with the extra \(d_g^2\). The exponent is explicitly \(-1+.3+\max(.55,.471)=-.15\). Thus these shell costs are summable with radius gain, explaining why no extra positive-\(q\) weighted bad flux is required at the weak level. All raw wave sources in this multiplier now use (266); the signed leading term in (265) just uses the flat good/bad weight gap here.

Claim 3: the flat commutators are triangular. To specify the flat terms, commute \(Z\) after lab differentiations, eliminating \(P_g\) at lower orders; separate the \(P[H]\) commutations, including any scalar commutator multiples of it, into the nonflat sources below. The flat recurrence is \(P_\eta S=(S+1)P_\eta+r^{-2}\mathcal A_s\); rotation commutator errors are \(r^{-2}O(s)(1,Z,Z^2)\). Thus each \(S\) count contributes \(+(1+s\cdot n)L\chi/r\) from the pure \(S^2\) leading term on the right, giving \(\lambda_1\) in (265). Terms where extra rotations differentiate its \(s\)-dependent coefficient can instead be counted as \(O(s)\). All remaining size-one terms in \(f_{\rm flat}\) cost, after multiplication by \(r\), good derivatives/field-over-radius of lower orders or of the same order with fewer \(S\)’s, with unchanged \(j\); feedback terms at this precision have \(O(s)\) coefficients. In particular a pure \(r^{-2}\chi\) term at the same order from a commuted \(S\) uses \(r^{-1}\chi=L\chi_{\rm prior}\). This proves their triangular acceptability.

Claim 4: the boundary costs use the stated inputs. In this argument entries involving the outer cap supremum need only be estimated beyond \(R_0\). In the averaging annulus of the initial radial cutoff all field bulk/time-slice controls can instead use the compact estimates with constants depending on \(R_0\), which have full mixed orders there. For instance every common principal commutator puts order at most \(N_i+1\) on the differentiated field (one lower commutation count before the two wave derivatives); in the \(D\)-equation an additional order-two term on \(h\) uses at most nine derivatives but carries either a parameter acceleration coefficient or a strong field derivative. Thus field source/current products there can use ordinary time-weighted Sobolev estimates off trapping, not an outgoing cap input on the overlap. On linear pure-defect forcing involving acceleration alone one keeps the direct far \(\widetilde a\) bound with its same radius gain on \(\mathsf S_a\), not a large overlap coefficient times \(\mathsf S_a\). The required direct source estimate is proved in 11. The surface cutoffs themselves can be selected by averaging against the full time annular bounds, irrespective of the outer cap used. Divergence integrations at infinity can also first take a sequence of radial truncations at each bounded time range; ordinary high symbol decay there suffices on the raw jets (with \(p_H<1\), and an extra decay power for the strong field), and the displayed cap/bulk bounds on the specific corrections, or their bounded-time symbol estimates, suffice for correction fluxes.

Claim 5: finite hierarchical absorption. Add the two multiplier estimates with fixed coercive relative factors, use Hardy and the true-to-flat conversions, and take separate suprema over the endpoints. Enumerate the finitely many blocks in the stated prior order as \(1,\ldots,J\) and let \(E_k\) denote their full norms. After Young’s inequality each block has an estimate of the form \[\begin{equation*} E_k^2\le \epsilon(R_0)\sum_{l=1}^JE_l^2 +C\sum_{l<k}E_l^2+B^2, \end{equation*}\] where \(B\) is the nonhierarchical right-hand budget in (268); the same argument applies to tunably small uniform errors. All size-one constants \(C\) and the number \(J\) are independent of the two large matching radii. Choose fixed positive weights \(a_k\) decreasing so rapidly that \(C\sum_{k>l}a_k\le a_l/4\) for every \(l\). Summing \(a_kE_k^2\) absorbs every prior-order term. Only after these weights have been chosen do we enlarge \(R_0\) so that \(\epsilon(R_0)\sum_k a_k\le\frac14\min_k a_k\). The resulting weighted norm is equivalent to the finite sum defining \(\mathsf F\), with constants independent of the matching radii, and its square root gives (269). Small multiples of any same-time right hand budgets in the bootstrap can be kept on the right before combining all levels. Compact boundary/source squares costing \(C(R_0)\mathsf K_R^2\) are harmless with these conventions. The source hypotheses are verified in 11, yielding (277). This proves the stated energy reduction. ◻

Proof. We verify the catalogue in six claims. Coefficients denoted by \(N\) are bootstrap/input factors, so \(N\mathsf F\) is nonlinear. The pure background symbol bound \(B-\eta=O(r^{-1})\) has no such smallness factor. All wave equations and laboratory derivatives are taken before the emitter decomposition.

Claim 1: common principal commutators. We first derive the common source forms, then estimate their raw and outgoing budgets separately. The distinction is necessary because the outgoing correction avoids one bad derivative, whereas the raw budget uses the derivative hierarchy directly.

Source forms and sphere placements. After the flat recurrence in Claim 3 of the proof of Proposition 33, every such term is as in (255) applied at lower indices \(\ell=(b',j')\), with coefficients \[C_h=Z^d (T^m H)_{\ell_{\rm null}\ell_{\rm null}}, \qquad J_h=Z^{\le d} T^m H .\] Here \(\ell_{\rm null}=(1,-n)\) (the same \(\ell\) as in (255)); the contraction is inside the \(Z\)’s. A length written as an exponent allows any word of that length. The indices, for a target total order \(n_*\) and time count \(j\), have \[j'=j-m,\qquad n'+b_0\le n_*,\quad n'<n_*, \qquad n'=b'+j',\quad b_0=d+m.\] More explicitly the bad term has the form \[ C_h\partial_q(c_0\partial_q\chi_\ell) \tag{278}\] where \(c_0\) is as in (274), and spatial counts on \(\chi_\ell\) can be smaller. The other terms, multiplied by \(r\), cost \[ C |J_h|(B^i_{b'+1,j'}+G^i_{b'+1,j'}) + C r |a(q) C_h|\, G^i_{b' j'}. \tag{279}\] These are bounds term by term, allowing sums and bounded coefficients. Indeed lab commutation first gives \(P[T^m H]\) on \(rT^{j'}U\) if \(m\ge1\). In the \(Z\) commutations use \(P_g=P_\eta+P[H]\); differences from the flat recurrence differentiate (255) by Leibniz (including a scalar multiple from each dilation recurrence), with highest second derivatives on the field cancelling. In particular there is no differentiation of the projector outside the full differentiated contraction in (278); differentiations of \(k^{-1}\) by \(Z\) stay inside \(\partial_q\). The mixed and spatial second derivatives and all lower terms obey (279). The \(a(q)\) terms there only differentiate \(s\)-dependent frame factors, with the double contraction of (255).

Products in these formulas and their estimates below use sphere \(L^2_\omega\) on a coefficient if it is high, moving two extra rotations to a low companion; otherwise we use coefficient sup with spare rotations. For \(J_h,C_h\) one may use the latter when \(b_0\le5\); when \(b_0\ge6\) there is room since \(n'+3\le n_*\). The composition sphere bounds follow by the ordinary chain rule with at most one highest factor not in sup. Write \(L_h\) for the bound of \(r|C_h|\) with the indicated placement. For \(b_0\le8\) (without increasing a high index) it is \(O(1+N r^{2\eta_*})\), tight \(O(1)\) on \(|q|\le 10r^\theta\), and \(O(r^{-1+8\eta_*})\) if \(m\ge1\); these use (259), (260), with smallness taken after any fixed bootstrap constants. For \(b_0=9\) use the sphere bounds of (260) exactly. Now (279) costs an acceptable amount in both raw bulk weights of (266), even ignoring the \(1+M\) denominator. Indeed \(J_h\) costs \(r^{-1+2\eta_*}\) to order eight or \(r^{-.48}\) at nine (then the target is level 0). Use (273). Also \(L_h |a(q)|\lesssim N\) by the tightness and decay split, except possibly at nine; there use \(G^H_{b'j'}\) with \(W_H\) on the low field, so \((W_0/W_H)^{1/2}\) and \(a(q)\) decay give the same sufficiency by (260).

The raw time-multiplier budget. Estimate (278) without splitting, namely \(L_h\) times the first line of (274) with target square weight \(w_{i,j}\). For \(m\ge1\) it is small by (259), (260) and (273), including the \(j\)-transfer. It remains to treat \(m=0\). At levels \(P,0\), except \(b_0=9\), the \(B_{b',j+1}\) term costs prior norms: \(L_h m_{\rm cone}^{-d_*/2}\) is bounded by tightness and the outer \(m_{\rm cone}\)-gain. The other two terms are small by \(1/r\) and \(a(q)\).

At level \(H\) instead use both lines of (274); \(B^\psi_{b'j}\) has total order at most seven, and \(L_h(w_H/w_{P,j})^{1/2}\lesssim1\). Angular placements respect this order, and level \(P\) is prior.

The exceptional coefficient count \(b_0=9\) occurs only at level 0 and has a low input. Use the same conversion: (273) gives \(L_h(w_{0,j}/w_{P,j})^{1/2}\lesssim1\), since the logarithm of (260) is offset at small \(|q|\), and the term \((1+\min(|q|,r))^{1/2}r^{-p_0/2}\) is offset too (use \(\mu^{-A_H/2}l^{(-(A-A_H)+7d_*)/2}\) at large \(|q|\)). Similarly \(L_h|a(q)|(w_{0,j}/w_H)^{1/2}\lesssim N\); the \(r^{-1}\) terms are small directly.

The outgoing budget with a current correction. If \(b_0\le5\), or if the target is level \(P\), use the correction \[\mathcal P=C_h c_0\partial_q\chi_\ell\] for (278), with its actual sign/factor. Corrections for different terms are always summed. The remainder differentiates \(C_h\) by \(\partial_q\), so the bound after multiplying by \(r\) uses (259), cost \(r^{-1+8\eta_*}\), against \(B_1\) in the field; this is small in (266) by (273). For the correction norms \(C_h,ZC_h\) cost at most \(O(r^{-1+2\eta_*})\), and when \(Z\) hits the field \(n'+1\le n_*\). In the term \(r d_g\partial_q\mathcal P\) one has \(r d_g\lesssim r^{2\eta_*}\) and uses (274). Thus all its bulk costs are small too; its time cap uses \(B_1\chi_\ell\) alone and is small. Sphere placements for \(b_0\le5\) take \(\partial_q C_h,Z C_h\) with two extra rotations, allowed in (259).

The outgoing budget with a high coefficient. It remains to treat \(b_0\ge6\) at levels \(0,H\), without a current correction. If \(m\ge1\), the same raw bound as before, now using \(W_i/w_{i,j'}\) or \(W_i/w_{i,j'+1}\), works with strict gain by (259), (260).

If \(m=0\), use (274) in full, including the two extra rotations on the low input. The terms with \(r^{-1}\) have strict gain by (273). Thus it suffices to bound \(L_h\) times cap/bulk derivatives \(B_1\) of low \(\psi\) fields, and \(L_h |a(q)|\) times \(B_1\) of low \(h\) fields. Indeed the \(V\) pieces in the cap comparison above can instead use (250) bulk with \(W_P,W_H\) respectively, by (260) (as for (279), and \(L_h(W_i/W_P)^{1/2}\ll 1\)). All low orders used here fit even with two extra rotations.

On the timelike region \(t\gg r\), the bad bulk weights of levels \(P,H\) suffice for the \(B_1\) costs with gain: use \(w_{P,j}\gtrsim t^A r^{-.025}\), \(W_i\asymp r^{p_i-A_i}t^{A_i}\), and \(w_H\gtrsim t^{A_H}r^{-e-\delta A_H}\) with \(|a(q)|\) decaying; the extreme \(b_0=9\) only occurs for \(W_0\).

On the remaining region \(t\lesssim r\), \(r\asymp R_f\), use the mixed entry with \(l_{R_f}\) equal to the bounds in (259), (260) for \(L_h(w_{P,j}^{-1/2}+|a(q)|w_H^{-1/2})\), supremized at fixed \(\mathfrak u\) on the shell. Weights and power thresholds may use \(R_f\) in place of \(r\), with fixed comparable constants.

Counts six through eight. The squared integral of this kernel is \(O(R_f^{.013\eta_*})\). Indeed \[w_{P,j}^{-1}\lesssim r^e\mu^{-A_H}l^{-(A-A_H)+7d_*},\] whose integral is \(O(r^{e+\delta A_H+\nu(1-A_H)})\) because \(A-7d_*>1\). The margin needed for the mixed shell charge is \[ e+\delta A_H+\nu(1-A_H) =.011\eta_*+.09\eta_*^2<.013\eta_*<1-p_H. \tag{280}\] On \(|q|\gtrsim r^\theta\), the tail integral has enough additional power gain to offset \(L_h^2\); the decay of \(a(q)\) handles its entry as well. Supremizing uses only comparability of \(r^\delta+|q|\) and \(r^\delta+|\mathfrak u|\), so the small-\(q\) acceleration bound may be broadened to a band \(O(R_f^\delta)\). This proves summability for \(b_0\le8\).

Count nine. Only weight \(W_0\) occurs. The squared extra term \((1+\min(|q|,r))r^{-p_0}\) in \(L_h^2\), including the mixed shell factor squared, contributes at most \[C R_f^{-1}\int_{|\mathfrak u|\lesssim R_f} (R_f^\delta+|\mathfrak u|) (w_{P,j}^{-1}+|a(q)|^2w_H^{-1})_{\sup}\,du.\] Split the \(w_P^{-1}\) part at \(|\mathfrak u|=R_f^\nu\). It is bounded by the sum of \[R_f^{-1+e+\delta A_H+\nu(2-A_H)},\qquad R_f^{1-A+7d_*+e+\delta A_H+\nu(A-A_H-7d_*)}.\] Both exponents are less than \(-1/4\) for the fixed range of \(\eta_*\). For the acceleration part, use \(|a|\le CN\) on the broadened band \(|\mathfrak u|\le C R_f^\delta\) and \(|a(q)|^2\le CN^2|\mathfrak u|^{-A}\) outside it. Its powers are bounded by \(R_f^{-1+e+2\delta}\) and \(R_f^{-1+e+\delta(2-A)}\). The integral is therefore \(O(R_f^{-c_1})\) for a fixed \(c_1>0\); logarithms and \(R_f^{8\eta_*}\) losses in the ninth-order coefficient leave a strict summable margin. This accounts for every common principal commutator in both multipliers.

Claim 2: the additional strong-equation terms. Before treating \(f_h,D f_h\), the additional strong equation commutators arise already at order zero: \[ \begin{aligned} g^{ij}\partial_{ij}\psi &=D f_h-(Dg^{ij})\partial_{ij}h\\ &\quad+2g^{ij}(\partial_i s_s^d)\partial_{jd}h +g^{ij}(\partial_{ij}s_s^d)\partial_d h . \end{aligned} \tag{281}\] In (255) on the second term, \(J\) (full) and \(C\) (double contraction) after ordered differentiations through \(b+m\le7\) satisfy in sphere \(L^2\) \[|J|\lesssim N r^{-1+8\eta_*}\tau^{-A_H/2},\qquad |C|\lesssim N r^{-2+8\eta_*}\tau^{-A_H/2}.\] Indeed \(D B=O(r^{-1}\widetilde a)\) with symbol bounds (freezing makes this derivative zero), \(D h=\psi\). The improved estimate on the double contraction of \(\psi\) uses \(T^m D T^0 h\) via the same formula as (275) but solved for \(T^m\psi\), and the projected estimate on \(T^{m+1}h\) in the proof of (259), i.e. without a stationary background term since this is \(h\) itself; \(\Theta\) there costs \(O(r^{-1}\widetilde a)\). Quadratic inverse errors use (272). The full \(r\)-gain on the projected coefficient therefore applies even through all the commutations of (281). After multiplying by \(r\), the bad, mixed/good and \(a(q)\) terms of (255) now cost an acceptable amount directly in both raw weights: use (274) on \(h\) through total order eight for \(B\), weighted by \(w_H\); (273) works because \(A<2A_H\). For \(a(q)\) times good entries use \(W_H\). If the coefficient is high use the two rotations instead on a low input (\(b+m\ge6\) suffices). Likewise the piece \(2H^{ij}\partial_i s_s^d\) in the third term of (281) costs \(O(N r^{-2+2\eta_*}\tau^{-A/2})\) in all components through these coefficient orders, hence works directly. The last term of (281), after the \(r^2\) source scaling, costs \(C\widetilde a\, B^h\) with bounded coefficient products and likewise works.

Here the derivative accounting is explicit. If \(c=b+m\) derivatives hit \(Dg\) and \(n'\) hit \(h\), then \(c+n'\le7\). The two physical derivatives on \(h\) require a \(B^h\) index of at most \(n'+1\le8\), available at level \(H\). For \(c\le5\) two additional rotations on the coefficient stay within its order-seven estimate. For \(c\ge6\), place it in \(L^2_\omega\); the input including the two transferred rotations has \(B\) count at most \(7-c+1+2\le4\). For example, comparison of the \(J\) or \(rC\) bound with the high \(w_H\)-weighted bad norm gives the squared factor \[C N^2 r^{p_P-2+16\eta_*+e}\tau^{A-2A_H}.\] The radial exponent is \(-.38+16.001\eta_*<0\) and \(A-2A_H=-.64+2\eta_*<0\). These are strict gains before using the \(1+M\) denominator.

For the smooth \(Y=2\eta^{ij}\partial_i s_s^d\) coefficient of \(\partial_{jd}h\), \(J=Z^bT^m Y\) is \(O(r^{-m}\widetilde a)\); \(\partial_q C\) for its differentiated contraction as before costs an extra \(r^{-1}\), since \(\partial_q,Z\) commute and no derivative of the fixed-\(n\) projector occurs. Here \(n'\) in (278), (279) for the input \(h\) can equal \(n_*\le7\). In the time multiplier use (274) in full for (278): the \(\psi\) term costs \(C r^{1-m}\widetilde a\, (w_{P,j}/w_{P,j-m})^{1/2}\lesssim C N\) times its weighted bad entry, admissible even at the same order. The \(a(q)\) terms on \(B^h,G^h\) use \((w_{P,j}/w_H)^{1/2}\lesssim l^{(A-A_H)/2}\) and acceleration decay. All \(1/r\)-suppressed terms work directly by (273), as do the ordinary mixed/good terms of (279). In the outgoing multiplier use current correction \(C c_0\partial_q\chi^h_\ell\) as for (278). Its (266) costs all use (273) and \(w_H\), with up to order eight by (274), \(r d_g\lesssim r^{2\eta_*}\); the bulk source remainder costs \(r|\partial_q C|\lesssim\widetilde a\) times \(B_1\). The ordinary mixed/good terms are small too.

One further correction here deals with the explicit \(a(q)\) term of (255) (not the one inside (278)). More precisely, with \(d_i Z=(b_i-\ell_i s^d b_d/k)Z\) in the first derivative formula, it comes from \[-Y_{\ell_{\rm null}\ell_{\rm null}}\, \partial_q(s^d/k)\, b_d Z(\chi/r)\] in \(P[Y]\chi\), and the corresponding ordered differentiations (lab times on \(Y\) and the input taken before (255)). It thus has the form \(C\,(\partial_q c_2)\, D_Z\chi_\ell/r\), with sign included in \(D_Z\), an operator of order at most one in \(Z\) with smooth sphere coefficients independent of \(q\); \(c_2\) smooth bounded in \(n,s(q)\). Use correction \(C c_2 D_Z\chi_\ell/r\). The two remainders, multiplied by \(r\), cost \(C'\widetilde a\) times \(B^h,G^h\) entries through order eight, all suppressed by (273). For the bulk correction budget even after \(Z\) it uses \(G^h\) through order eight with the same coefficient bound; \(r d_g\partial_q\) additionally costs \(O(r^{2\eta_*}\widetilde a)\) against \(G^h\) and \(O(r^{-1+2\eta_*}\widetilde a)\) against \(B^h\) (differentiate the formula, including \(|\partial_q c_2|\lesssim |a(q)|\)). These meet (266). For its time-slice budget use \(G_1\chi_\ell\) with \(W_H\); any undifferentiated \(\chi_\ell/r\) term uses (272) directly with \(\widetilde a\), giving summability in \(L^2(dr)\) even with \(W_P^{1/2}\). This proves sufficiency for (281) except \(D f_h\).

Claim 3: background, linear, and suppressed products. We finish with the semilinear sources, i.e. linear combinations (from the flat recurrence, thus with constant scalars out front) of \[ Z^{c_3}(kr T^j f_h),\qquad Z^{c_3}(kr T^j D f_h), \qquad c_3+j\le n_* . \tag{282}\] All product estimates allow indices up to the indicated ones. The estimates for \(r\) times these expressions thus concern \(r^2\)-scaled physical sources. Unless specifying a different treatment, a "both weights" bulk estimate below uses \(w_{i,j}^{1/2}\) and \((W_i/(1+M))^{1/2}\) of (266); for suppressed products it suffices simply to use \(W_i^{1/2}\).

For the pure background source, the \(r^2\) scaling leaves \(\widetilde a\) at levels \(0,H\) and \(r^{-1}\widetilde a\) at level \(P\); the latter gain comes from the \(D\) derivative of the defect. We estimate these terms directly, including on the starting annulus. On a shell \(r\asymp R_f\), an enlarged positive majorant is \[ A_{R_f}(t)=\frac{C_1}{R_f} \int_{t-C_2R_f}^{t}|a(u)|\,du,\qquad a(u)=0\quad(u<0). \tag{283}\] Since \(A>1\), Cauchy–Schwarz gives \(\|a\|_{L^1}\le C_A\mathsf S_a\). The averaging kernel has \(L^2\) norm \(O(R_f^{-1/2})\), hence the first estimate below. For \(t\ge2C_2R_f\), the time weight is comparable throughout the averaging interval; the bounded \(L^1\) norm of the kernel gives the second: \[ \|A_{R_f}\|_{L^2_t}\le C R_f^{-1/2}\mathsf S_a, \qquad \|\langle t\rangle^{A/2}A_{R_f}\|_{L^2(t\ge2C_2R_f)} \le C\mathsf S_a. \tag{284}\] The fixed intermediate time band is included in the early estimate. As \(A_i\le A\), the late contribution satisfies \[R_f^{p_i-A_i} \|\langle t\rangle^{A_i/2}A_{R_f}\|_{L^2(t\ge C R_f)}^2 \le C R_f^{p_i-A}\mathsf S_a^2 .\] The squared early/late shell exponents are \(p_i-1,p_i-A\) for \(i=0,H\), with two further inverse powers for \(i=P\): \[ \begin{array}{c|cc} i&\text{early}&\text{late}\\ \hline 0&-.36&-.72\\ H&-\eta_*/6&-.36-\eta_*/6\\ P&-1.38&-1.74 \end{array} \tag{285}\] and their dyadic sums tend to zero with \(R_0\). This calculation also applies on \([R_0/2,R_0]\): the pure defect is estimated directly there, without a growing compact coefficient on \(\mathsf S_a\).

For the terms of linear jet size \(O(r^{-2}\partial h+r^{-3}h)\), after multiplication by \(r^2\) one uses \(r^{-1} B^h_{b j'}\), times bounded smooth coefficients with their distributed differentiations. The \(h\)-jet factors in those coefficients cost at most \(C\) in \(L^2_\omega\) even through order nine; in a product with a high such factor, transfer two rotations to the bulk field for the angular placement. More precisely one needs this transfer only at high coefficient orders (say seven and up); all its counts then fit the same level. Using \(w_{i,j'}^{1/2} B^i_{b j'}\) from (250), (273) supplies strict radial gain. The leading stationary coefficient constants have symbol bounds independent of the matching radii. In \(D f_h\) these terms give the same estimate for \(\psi\) in place of \(h\), and terms with \(r^{-1}B^h_{b j'}\) accompanied additionally by \(\widetilde a\) or a factor \(V^\psi\) (with distributed indices). For the latter estimates use level \(H\) on \(h\), (272) on the extra factors, and (273) with \(A<2A_H\). In particular \(\mu\gtrsim\tau\). The orders here sum to at most seven; a high extra factor can be put in sphere \(L^2\) with its stated decay, using spare rotations on the others. To justify this differentiation description one can take \(D\) before the lab \(T^j\), still in Cartesian lab formulas before frame and scale expansions. Linear jet coefficients before smooth field composition consist of background first jets (and gauge coefficients of first-jet size) at order \(O(r^{-2})\), second or first-product jets at \(O(r^{-3})\). On these \(D\) gains \(O(\widetilde a)\) relative to the indicated bounds: \(D B=O(r^{-1}\widetilde a)\) with symbols by \(D x=0\) and the parameter/formal-frame differentiation of the order \(-1\) residual from flat in lab components. Commuting lab derivatives costs \(\partial s_s=O(\widetilde a)\) with symbol bounds; the compensator and its derivatives have the stated stronger bounds from the reduction. Smooth \(h\)-coefficients instead differentiate by \(D h=\psi\). This applies directly in the Ricci and gauge substitution formulas giving (252); it does not require \(D r\) for the emitter radius to vanish.

Here are reusable quadratic product placements. The last term of (252) has vanishing coefficient \(\mathcal H=O(r^{-1}+|h|)\) in front of two first jets. After the scaling and differentiations for \(h\), this costs products bounded by \[ \begin{gathered} |Z^b T^m\mathcal H|\, B^h_{b_1 i_1} B^h_{b_2 i_2},\\ b+m+b_1+i_1+b_2+i_2\le n_* . \end{gathered} \tag{286}\] with harmless additional smooth sphere coefficients, sums, and analogous metric compositions. Use \(r^{-1+2\eta_*}\) on the coefficient through order eight, \(r^{-.48}\) at nine, in sphere \(L^2\). This also bounds the corresponding rates in sup with two spare rotations. If \(b+m\le6\) take that sup with the better rate, and otherwise put both \(B\) factors in sphere sup using two more rotations on each. Put a lower-order \(B\) in point control (272), the other in bulk \(w_{i,i_k}^{1/2}\) of its occurrence (here \(i=H,0\) is the level, \(i_k\) the relevant time count). There is enough room, since the lower-order factor needs order at most four before the rotations and a coefficient consuming at least seven leaves both factors low. By (273), (272), (286) fits both weights with strict radial slack also in the order-nine coefficient case.

For strong estimates a similar sufficient suppressed product is (286) with total count at most eight and target \(W_P^{1/2}\); we now put the high \(B^h\) in \(w_H^{1/2}\) and the lower one in point with \(Nw_H^{-1/2}\lesssim N r^{e/2}\tau^{-A_H/2}\). The \(\tau\) loss of (273) is covered. Coefficients through this total order cost at most \(C r^{-1+2\eta_*}\). The placements work also with plain \(O(r^{-1+3\eta_*})\) bounded coefficients, or with one \(B^h\) replaced by \(B^\psi\) if the total count on the two inputs is at most seven (measure a high \(B^\psi\) in its own level; point \(B^\psi\) through its stated orders in (272)). Thus \(T^jD\) on the vanishing-coefficient terms works simply counting \(D\) as up to one extra lab first derivative on the metric/field factors; differentiated \(s_s\) coefficients are bounded. Additional unscaled spatial lab derivatives can always be distributed before switching to \(Z,T\) counts, using \(\partial_d=\ell_d T+r^{-1}b_d Z\) on lab jets with \(T\) still commuting through other lab derivatives. Smooth bounded field compositions with explicit \(r^{-1}\) suppression and the analogous symbol bounds work by the same allocations.

Two explicit comparisons record the restrictive suppressed products. At strong level a coefficient \(O(r^{-1+3\eta_*})\), a high \(B^h\) measured with \(w_H\), and a low \(Nw_H^{-1/2}\) give \[W_Pr^{-2+6\eta_*}w_H^{-2} \lesssim r^{p_P-2+6\eta_*+2e}\tau^{A-2A_H},\] whose radial exponent is \(-.38+6.002\eta_*<0\). At coefficient count nine only level \(0\) occurs; its squared comparison is bounded by \(r^{p_0+.021-.96+e}=r^{-.299+e}\). For coefficient count at least seven the two input counts sum to at most two, so both inputs admit two additional rotations. For a lower coefficient the smaller input has count at most four before those rotations. The mixed strong product has total count seven, and its lower input needs at most five counts after the rotations, within the order-six point bound for \(B^\psi\).

Claim 4: the leading weak-null source and its corrections. We treat \(Q_\eta\) separately. First take \(T^j\) on it, for either equation, giving the polarization sum. Set \(h_i=T^i h,\ p_i^\flat=T^{i+1}h\) (here this decorated letter denotes just the symmetric tensor, not a radial power), and set \[F_i=r\Pi T^{i+1} h,\qquad \widehat F_i=r\Pi T^{i+1}\psi .\] For each polarization pair, use (253) to write \(\partial_\alpha h_i=\ell_\alpha\breve p_i+E_{\alpha i}\) with \(\mathcal G_n\breve p_i=0\) and \(\Pi\breve p_i=\Pi p_i^\flat\). The error \(r E_{\alpha i}\) is made of \(b Z h_i\) good terms and \(r\) times the differentiated gauge error (251) with bounded sphere coefficients. Here the lab differentiations in (251) are taken before the bad/flat-good decomposition. Expand each quadratic polarization sequentially, so each remainder need use just one such error and as its partner a full \(\partial h_l\) or a projected \(\ell\breve p_l\), estimated without any substitution of gauge errors into that partner. This gives \[ T^j Q_\eta =r^{-2}\ell\otimes\ell\sum_{i+l=j}\mathcal C_{il}(F_i,F_l) +{\rm Err}_j, \tag{287}\] where the pairings include constants. One may use ambient spatial tensor components for the screen, the Euclidean pairing being smooth. After \(Z^{c_3}\) the \(r^2\)-scaled error terms have placements (286) or explicit \(O(r^{-1})\) suppressions of that type, \(\widetilde a\,B^h\), or \[ G^h_{b_1 i_1}\,B^h_{b_2 i_2},\qquad b_1+i_1+b_2+i_2\le n_* . \tag{288}\] Smooth bounded compositions are allowed on suppressed terms. These assertions follow from (251), including \(r\Theta=O(\widetilde a)\). Terms with \(\widetilde a\) fit directly by (273). For (288) at levels \(H,0\), if the good input is lower order use its point \(C N r^{-1+2\eta_*}\tau^{-A_H/2}\) and measure the bad input in \(w_{i,i_2}\). If the good input is higher order measure it in \(W_i\), the low bad one in point \(C N w_H^{-1/2}\); this works in both weights since \(w_H^{-1}\lesssim1+M\), and \(w_{i,j}w_H^{-1}\lesssim W_i\).

In \(T^j D Q_\eta\) the commutator terms \((T^m s_s^d)\partial_d T^{j-m}Q_\eta,\ m\ge1\), already have the suppressed strong placement via \(T^m s_s^d=O(\widetilde a)\) with \(Z\)-bounds. Consider thus \(D\) on (287) before the \(Z^{c_3}\) distribution. When it differentiates \({\rm Err}_j\) the suppressed terms are still suppressed, now at total orders up to eight; expressions involving the gauge have the same differentiable product structure, with \(O(1/r)\) when a sphere projector is differentiated by \(D\). The good factor underlying (288), if hit by \(D\), now costs \(C r^{-1} B^h_{b_1+1,i_1}\): indeed \(D=(1-s_s\cdot n)k^{-1}\partial_q+r^{-1}d Z\) with bounded coefficients under spatial-emitter differentiation, and \(\partial_q\) commutes with \(Z\) before expanding on \(h_i\). Differentiation of scale factors costs a relative \(O(1/r)\) too. Thus the remaining unsuppressed placements are (288) with total count up to eight (one extra derivative can now be on the bad input). If good is low the previous point placement and \(w_H\) on the bad input suffice even with \(W_P\) since \(2A_H>A\). If good is high, use \(W_H\) and the low bad point bound; the relevant comparisons are \[(W_P/W_H)w_H^{-1}\lesssim1+M,\qquad (w_{P,j}/W_H)w_H^{-1}\lesssim1.\] The first uses \(p_P-p_H+e<.72\) within \(|q|\le r\); outside, use \(\mu\gtrsim r^{1-\delta}\tau\). The second follows from (273). Low placements here have at most four derivatives before the two spare rotations. This proves acceptability of all the remainders of (287).

Consider the outgoing budget of the first term of (287) for \(i_{\rm lev}=H,0\). On \(t\gg r\) ordinary high-low placement on differentiated \(F_i,F_l\) suffices with \(B^{i_{\rm lev}}\) for the high one, low \(w_H^{-1/2}\) in point for the other: \[(W_{i_{\rm lev}}/w_{i_{\rm lev},i'})w_H^{-1}\lesssim1 \qquad(t\gg r,\ i'\ {\rm in\ that\ level}).\] On the remaining \(t\lesssim r\) use the mixed entry. A high differentiated \(F_i\) is controlled by the corresponding \(B_1\chi\)’s of \(h\) and the \(V^h_{b i}\) term in the cap comparison. The latter term can instead be charged as a high good input in bulk paired with low point \(N w_H^{-1/2}\), as in (288). The \(B_1\)’s have time cap with square weight \(w_{i_{\rm lev},i}\). Thus the mixed kernel is bounded by \[C N\sup(w_{i_{\rm lev},i}w_H)^{-1/2} \quad(\text{sup at fixed }\mathfrak u\text{ in the shell})\] on \(|u|\lesssim R_f\). Its squared integral for \(i_{\rm lev}=H\) costs \(C N^2 R_f^{2e+\delta}\). For \(i_{\rm lev}=0\) use \(w_{0,i}^{-1}\lesssim R_f^{.021}\), giving cost at most \(C N^2 R_f^{.021+e+\delta A_H+1-A_H}\). Here \(r^\delta+|u|\) is comparable to \(r^\delta+|q|\), and the low point placement including angular sup has the orders of (272). Both costs give shell summability with the indicated \(p_{i_{\rm lev}}\).

The squared mixed-shell exponents just obtained are \[ \begin{split} p_H-1+2e+\delta&=(-1/6+.012)\eta_*<0,\\ p_0-1+.021+e+\delta A_H+1-A_H &=-.339+1.011\eta_*-.01\eta_*^2<0 . \end{split} \tag{289}\] Their strict signs justify summing the norm charges themselves over dyadic shells.

In the strong outgoing budget for the first term of \(D\) applied to (287), use the correction (one for each term with its actual \(\mathcal C_{il}\)) \[ \mathcal P_*= Z^{c_3}\!\left[ (1-s_s\cdot n)r^{-1}\ell\otimes\ell\,\mathcal C_{il}(F_i,F_l) \right]. \tag{290}\] Indeed \(kD=(1-s_s\cdot n)\partial_q+r^{-1}d' Z\), \(k\) here being the one in (282); \(\partial_q\) commutes with \(Z\). The outgoing remainder after multiplying by \(r\) costs \(C\widetilde a \sum|Z^{\le c_3}(F_i F_l)|+C r^{-1}\sum|Z^{\le c_3+1}(F_i F_l)|\) with product derivatives allowed individually in the sums. This uses \(\partial_q(1-s_s\cdot n)=O(\widetilde a)\) with spatial-emitter derivatives. These are suppressed products through total order eight. In the bulk correction costs of (266), all terms have at least \(r^{-1}\) in front of differentiated products of \(F_i,F_l\), up to the factor \(r d_g\lesssim r^{2\eta_*}\). Here \(Z^b F_i\) uses \(B^h_{b i}\), and \(\partial_q Z^b F_i\) uses \(B^h_{b,i+1}\) by commuting then expanding on \(T^{i+1}h\). Thus the total input count never exceeds eight in these \(B\) indices either; the \(r^{-1+2\eta_*}\) suppressed estimate with \(W_P^{1/2}\) proves sufficiency. At a time-slice cap the count in (290) itself is at most seven; make the same high-low comparisons with \(w_H\) on the high \(B_1\chi\)’s. A \(V^h\) term missing from their cap control is bounded by (272) directly; after the weights and low point factor this term has sphere \(L^2\) bounded by \(C N^2 r^{p_P/2-2+3\eta_*}\) with favorable \(\tau\) factor, summable in \(L^2(dr)\).

Claim 5: raw leading products on the finite slab. It remains to verify the raw time-multiplier budgets of the leading terms. At \(H,0\) they use, after \(r^2\) scaling, products \(F_i F_l\) with \(i+l=j\); at level \(P\), \(F_i\widehat F_l\) with \(i+l=j\), modulo suppressed terms, all still differentiated by \(Z\) to order \(c_3\). Indeed \(D F_i=\widehat F_i+O(r^{-1})B^h_{0,i}+O(\widetilde a)\sum_{m\le i}B^h_{0,m}\) as a differentiable product bound under \(Z\), by commuting \(D\) with lab times before expanding its spatial-frame coefficients. On \(|q|>r^\nu\), normal high-low placement works without using (261): \(w_{i_{\rm lev},j}/w_{i_{\rm lev},j'}\lesssim l^{9d_*}\) in a level, and \(w_{P,j}/w_H\lesssim l^{A-A_H}\) for a high \(F\) in the strong product. The low \(w_H^{-1/2}\) point factors offset both ratios by (272), (273). High factors use their corresponding \(B\) in weighted bulk.

Outside the near band the squared coefficient needed for the within-level transfer is at most \(l^{9d_*}w_H^{-1}\); for a high \(h\) input in the strong product it is at most \(l^{A-A_H}w_H^{-1}\). They are bounded because \[\begin{split} e-(\nu-\delta)A_H&=-.089\eta_*+.09\eta_*^2<0,\\ 9d_*-A_H&=-1+1801\eta_*<0,\qquad A-2A_H=-.64+2\eta_*<0 . \end{split}\]

On \(|q|\le r^\nu\), a uniform bound on the undifferentiated news, rather than radial losses in every low differentiated supremum, closes the raw product estimate. First replace \(F_i\) by \(T^iF_0\) and \(\widehat F_l\) by \(T^l\widehat F_0\). Since \(T^m(r\Pi)=O(1)\) for \(m\ge1\), with \(Z\) derivatives allowed, each replacement error has \(r^{-1}\) size relative to \(B^h\) or \(B^\psi\) at the allowed indices. The remaining raw jet still contains a lab \(T\) derivative. Products with these errors have the suppressed estimates already proved, for both \(h\) and \(\psi\) inputs.

Tile the near region in \((q,\rho,n)\), with \(\rho=\log r\), using small fixed coordinate boxes and a fixed enlargement of bounded overlap. The measure identity is \[ t=q+e^\rho,\qquad \frac{dt\,dr}{r}=dq\,d\rho. \tag{291}\] On the enlarged boxes, \(|q|\lesssim r^\nu\), the operators \(T,Z\) have uniformly bounded smooth coefficients through the required orders, and the weights are comparable. Within each level all time counts have equivalent weights there; also \(w_{P,j}\asymp w_H\). Angular coordinates change the measure by a uniformly bounded factor.

Ordinary derivatives of \(F_0\) through order \(n_*\) are controlled in the required weighted \(L^2\) norm by \(B^h_{bi}\) with \(b+i\le n_*\); the same statement holds for \(\widehat F_0\) and \(B^\psi\) through the strong target order. To verify this count, commute \(\partial_q\) through \(Z\), expand it on \(T^{i+1}h\) successively as \(kT+r^{-1}s^db_dZ\), and retain the initial lab time. Sphere rotations span the angular derivatives. High \(h\) factors use their own level in an \(h\)-only product and level \(H\) in the strong product.

Write \(\partial^a\) for the array of ordinary tile derivatives of order \(a\). For \(a+b=m>0\), Hölder and Gagliardo–Nirenberg interpolation on an enlarged tile give \[\|\partial^af\,\partial^bg\|_2 \le \|\partial^af\|_{2m/a}\|\partial^bg\|_{2m/b} \le CN\bigl(\|f\|_{H^m}+\|g\|_{H^m}\bigr), \qquad \|f\|_\infty+\|g\|_\infty\le CN.\] An order-zero factor uses the \(L^\infty\) endpoint; the last inequality is weighted arithmetic–geometric mean. Insert cutoffs on the enlarged tiles for inhomogeneous norms. The needed supremum bound for \(f,g\in\{F_0,\widehat F_0\}\) is precisely (261). Local weight comparability, not a uniform bound on \(\mu\) over the whole near region, permits squaring and summing this estimate.

The same argument applies up to the current time endpoint. For large \(R_0\) the initial slice does not meet this band. If the upper endpoint meets a tile, it is the uniformly smooth graph \(\rho=\log(t_{\rm end}-q)\), with the domain below. Extend across it by a finite linear combination of values at negative multiples of the signed graph-height, matching normal derivatives through order \(n_*-1\). Change of variables gives uniform \(H^{n_*}\) and supremum bounds from enlarged interior boxes, and reflected weights remain comparable. Choose the fixed tile sizes small enough that, for boxes meeting \(r\ge R_0\), the required enlargements and reflected points never extend inward beyond the fixed-factor overlap below \(R_0\); near-cutoff boxes may equivalently use compact overlap controls. Source integrals with an additional cap truncation can be bounded on the full time slab. Summing with bounded overlap proves the near-region raw budgets, with only the stated compact-overlap and bootstrap-product costs.

Claim 6: initial and starting-annulus costs. All corrections are added before forming \(\xi\) for each equation; the bounds concern (250) for uncorrected fields, already part of the total norm. On fixed starting overlap annuli their extra current and boundary costs satisfy the same compact convention: (278) corrections use a first derivative of a lower commuted field with bounded coefficients (weighted at the same time level, with acceleration decay where strong and weak inputs couple); in (281) smooth-acceleration corrections the coefficient carries \(\widetilde a\). The products in (290) on such annuli cost quadratically by weak fields with \(2A_H>A\). One can use the off-trap compact controls and sphere Sobolev placements there. Time-zero costs use the same cap estimates or directly the input symbol bounds. All patch/annulus constants on nonlinear terms are allowed fixed before smallness. This verifies (276) and therefore (277), with every positive-radius-gain term summable and the prior linear costs taken with independent far constants as specified there. ◻

Proof. On a fixed \(t\) slice the emitter coordinates satisfy \[y=C(t-r)-C(t)+rn,\qquad \partial_r y=n-s(q),\qquad \partial_Ay=r\partial_A n .\] It follows that \[ \begin{gathered} d^3y=kr^2\,dr\,d\omega,\qquad \partial_{y_i}r=\frac{n_i}{k},\\ \partial_{y_i}n_j= \frac1r\left(\delta_{ij}-\frac{(n_j-s_j)n_i}{k}\right), \qquad \operatorname{div}_y n=\frac2r . \end{gathered} \tag{293}\] All comparison constants depend only on the fixed small speed bound. Indeed \((1-\|s\|_\infty)r\le r_y\le(1+\|s\|_\infty)r\). These first derivatives contain no acceleration. In particular one must perform the single spatial integration by parts below in \(d^3y\) before comparing it to the emitter measure.

Let \(v_p\) be any adjoint zero mode in the fixed normalization of [co2:sliced-acceleration]. Its uniform symbol bounds are \(|v_p|\le C r_y^{-2+\epsilon}\) and \(|\partial_yv_p|\le C r_y^{-3+\epsilon}\). The leading linear source pairing has the form \[I(t)=\int \chi_t\chi_>v_{p(t)} \bigl[A(p(t))T\psi+B^d(p(t))\partial_{y_d}\psi\bigr]\,d^3y .\] The coefficient matrices and the laboratory/component transformation are spatially constant on this slice. The outer cutoff \(\chi_>\) can be chosen so its emitter support begins at \(r\ge cR_0\); a fixed-factor intermediate annulus is charged to \(C(R_0)\mathsf K_R\). Write \(T\psi=L\psi-n^d\partial_{y_d}\psi\). Then \[I(t)=\int\chi_t\chi_>v_p A L\psi\,d^3y -\int\partial_{y_d}\bigl(\chi_t\chi_>v_p(B^d-An^d)\bigr) \psi\,d^3y .\] Derivatives of either cutoff cost \(C/r_y\) on its support. Equation (293) and the adjoint symbol bounds give \[|I(t)|\le C\int_{r\ge cR_0} r^\epsilon\bigl(|rL\psi|+|\psi|\bigr)\,\frac{dr\,d\omega}{r}.\] Choose \(\epsilon<\zeta<(p_P-A)/2=.13\). Cauchy–Schwarz contributes \[\left(\int_{cR_0}^\infty r^{-2(\zeta-\epsilon)}\frac{dr}{r}\right)^{1/2} =\frac{(cR_0)^{-(\zeta-\epsilon)}}{\sqrt{2(\zeta-\epsilon)}}.\] This is uniformly bounded as \(R_0\) grows. Since \(W_P=r^{p_P-A}(t+10r)^A\) and (250) controls both \(\psi=(r\psi)/r\) and \(rL\psi=L(r\psi)-\psi\), we obtain \[\|\langle t\rangle^{A/2}I\|_{L^2_t}\le C\mathsf F ,\] with \(C\) independent of \(R_0\). Bounded initial times are included in the fixed finite-time input. No integration by parts in time, nor any differentiation of \(p(t)\), has been used.

Choose \(Y\) beyond the fixed compact operator supports and the transition where target smoothing reaches its prescribed far form. The compact defect comparison on \(r_y\le2Y\) is given independently by [co2:compact-defect]. On its complement it is enough to bound the two terms separately: \[|f_B|+|Q(0)a|\le C r_y^{-2} \bigl(\widetilde a+|a(t)|\bigr).\] Use the majorant \(A_\rho\) from (283). On the cutoff \(\chi_t\), \(t\) is a sufficiently large fixed multiple of \(\rho\), so (284) gives \(\|\langle t\rangle^{A/2}A_\rho\mathbf1_{\{t\ge2C_2\rho\}} \|_{L^2_t}\le C\mathsf S_a\), uniformly in \(\rho\). Pairing with \(v_p\), converting \(d^3y\), and using Minkowski yields \[C\mathsf S_a\int_Y^\infty \rho^{-2+\epsilon}\,d\rho \le C_\epsilon Y^{-1+\epsilon}\mathsf S_a .\] The omitted \((1-\chi_t)Q(0)a\) tail obeys the same bound directly. Thus \(Y\) is fixed before \(R_0\) without producing a \(C(R_0)\mathsf S_a\) term. Finally the direct shell estimates for the pure defect in the proof of 34 also apply to the single shell \([R_0/2,R_0]\). They prove the last assertion. All convolution bounds remain valid up to an arbitrary finite lifespan by extending \(a\) by zero after the endpoint; causality leaves the values inside the slab unchanged. ◻

Proof. Proposition 17 proves the simultaneous absorption and continuation from these estimates. Its bootstrap cap can be chosen arbitrarily small before the \(p_{10}\) neighborhood is shrunk. This gives the asserted small base bounds on both cylinders. The higher-order conclusion is then Proposition 15, whose induction uses finite higher initial seminorms without shrinking the common base neighborhood. ◻

Proof. We construct the inverse in three steps. Newton potentials solve the flat constraints and supply the permitted tail. Cutting them off on the inner ball creates a compact error whose Euclidean Killing moments vanish automatically. An annulus-supported primitive removes that error without changing the tail. Only this compact error has moment conditions; the original source is unrestricted.

Proof of 35. The same fixed point will supply both the base estimate and the all-order regularity.