==== TABLE OF CONTENTS ====
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//appendices/constraints.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//appendices/far-energy.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//appendices/far-sources.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//appendices/triangular.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//glyph-mappings.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//macros.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//main.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/01-introduction.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/02-category.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/03-geometry.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/04-stationary.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/05-compact.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/06-far.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/07-interior.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/08-comparison.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/09-packet.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/history-fragment.tex
Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//references.bib
===========================
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//appendices/constraints.tex ====
\section{Vacuum constraint correction outside a protected ball}
\label{pk:constraints}
Localized deformation of the vacuum constraints was developed by
Corvino, Corvino--Schoen, and Chru\'sciel--Delay
\cite{Corvino2000,CorvinoSchoen2006,ChruscielDelay2003}; conical localization
was established by Carlotto--Schoen \cite{CarlottoSchoen2016}. We require a
correction that fixes an inner ball and allows a decaying tail. The
explicit inverse below provides this support property with full
derivative gain. A nonlinear solve at one small base norm then gives
smooth exact constraints, even when higher data norms are large.
Section~\ref{pk2:constraints} applies the result to the rescaled beam
seed.
We use the vacuum constraint map
\[
\Phi(g,k)=
\bigl(R_g+(\tr_g k)^2-|k|_g^2,\,
\nabla^j k_{ij}-\nabla_i\tr_g k\bigr).
\]
\paragraph{The exterior correction problem.}
For a nonnegative integer \(m\) and a decay exponent \(j\), put
\begin{equation}
\|v\|_{H^m_j}
=\|v\|_{H^m(B_2)}
+\sup_{L\geq1} L^j
\|v(L\,\cdot)\|_{H^m(\{1<|y|<2\})}.
\label{pk:constraint-spaces}
\end{equation}
Componentwise versions of these norms are used for tensors. For an
integer \(s\geq3\), let
\[
Y^s=H^s_4(\R^3;\R\oplus T^*\R^3),\qquad
X^s=H^{s+2}_1(\R^3;\operatorname{Sym}^2)
\oplus H^{s+1}_2(\R^3;\operatorname{Sym}^2).
\]
The Euclidean linearization of \(\Phi\), denoted only in this subsection
by \(P\), is
\begin{equation}
P(q,p)=
\bigl(\partial_i\partial_jq_{ij}-\Delta q_{ii},\,
\partial_jp_{ij}-\partial_i p_{jj}\bigr).
\label{pk:constraint-P}
\end{equation}
\begin{proposition}[Exact correction outside a protected ball]
\label{pk:constraint-nonlinear}
Fix an integer $s\ge3$. There are constants $\epsilon_s,C_s>0$
with the following property. Let $d_B=(\delta+a_B,b_B)$ be smooth
metric and second fundamental form data on $\R^3$, with
$\delta+a_B$ positive definite and, for every multi-index $\alpha$,
\[
|\partial^\alpha a_B(y)|\le C_\alpha\la y\ra^{-1-|\alpha|},
\qquad
|\partial^\alpha b_B(y)|\le C_\alpha\la y\ra^{-2-|\alpha|}.
\]
Let $\phi$ be a smooth compactly supported scalar/covector pair such that
\[
\supp\phi\subset\{|y|\ge1\},\qquad
\Phi(d_B)=\phi\quad\hbox{on }|y|\ge1.
\]
Define the two base sizes
\[
\beta_s=\|a_B\|_{H^{s+2}_1}+\|b_B\|_{H^{s+1}_2},
\qquad \eta_s=\|\phi\|_{Y^s}.
\]
If $\beta_s+\eta_s<\epsilon_s$, there is a smooth tensor pair
$v_c=(q_c,p_c)$ such that
\[
\supp v_c\subset\{|y|\ge1\},\qquad
\Phi(d_B+v_c)=0\quad\hbox{on }|y|\ge1,
\qquad \|v_c\|_{X^s}\le C_s\eta_s.
\]
The corrected metric remains positive definite. The correction is smooth
across $|y|=1$ and satisfies symbol bounds of metric weight one and
second fundamental form weight two at every order. No smallness is
required of these higher bounds. No moment condition is imposed on
$\phi$, and the correction need not be compactly supported; its tail
may change the asymptotic charges. No constraint equation is asserted
for the artificially supplied datum on $|y|<1$.
\end{proposition}
The proof first constructs an exterior right inverse with full derivative
gain. Its definition on all inputs, not only supported ones, will permit
smoothness to be proved by rotations and dilations at the same base order.
\begin{lemma}[Exterior right inverse for the flat constraints]
\label{pk:constraint-linear}
There is a linear operator \(\mathcal K\), defined on every \(Y^s\) and
bounded \(Y^s\to X^s\) for each integer \(s\geq3\), such that
\begin{equation}
\supp z\subset\{|y|\geq1\}
\quad\Longrightarrow\quad
P\mathcal Kz=z,\qquad
\supp\mathcal Kz\subset\{|y|\geq1\}.
\label{pk:constraint-right-inverse}
\end{equation}
The same construction is used at all orders. Its conjugates by rotations
and positive dilations of the argument are smooth in the operator norm
from \(Y^s\) to \(X^s\).
\end{lemma}
\begin{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.
\paragraph{The Newton tail and its cutoff error.}
First construct an inverse without a support restriction. For
\(z=(f,F)\), let
\[
-2\Delta u=f,\qquad \Delta W_i=F_i
\]
be the Newton potentials, and set
\begin{equation}
q_{0,ij}=u\delta_{ij},\qquad
p_{0,ij}=\partial_iW_j+\partial_jW_i
-\tfrac12\delta_{ij}\partial_kW_k.
\label{pk:constraint-Newton}
\end{equation}
Then the scalar component of \(P(q_0,p_0)\) is \(-2\Delta u\).
Also \(p_{0,jj}=\tfrac12\partial_jW_j\), so the gradient terms in the
momentum component cancel and that component is \(\Delta W_i\).
Hence \(P(q_0,p_0)=z\).
Scaled Sobolev embedding gives
\(|z(y)|\leq C_s\la y\ra^{-4}\|z\|_{Y^s}\). Splitting the Newton
integral into a neighborhood of the origin, a neighborhood of \(y\),
and their complement yields
\[
|u(y)|+|W(y)|\leq C_s\la y\ra^{-1}\|z\|_{Y^s}.
\]
For completeness, on \(|v|<|y|/2\) the Newton kernel is \(O(|y|^{-1})\)
and the source has bounded total integral; on
\(|v-y|<|y|/2\), its integral against the source is \(O(|y|^{-2})\);
the remaining region has the same or a smaller bound. Local estimates
on a fixed ball handle bounded \(y\).
On an annulus of radius \(L\), the rescaled right-hand side of the
potential equation has \(H^s\) norm \(O(L^{-2}\|z\|_{Y^s})\).
Interior elliptic estimates on a finite cover by slightly larger
rescaled balls therefore give
\[
\|u(L\,\cdot)\|_{H^{s+2}}
+\|W(L\,\cdot)\|_{H^{s+2}}
\leq C_sL^{-1}\|z\|_{Y^s}.
\]
One physical derivative of \(W\) introduces an additional factor
\(L^{-1}\). Consequently
\begin{equation}
\|(q_0,p_0)\|_{X^s}\leq C_s\|z\|_{Y^s}.
\label{pk:constraint-Newton-bound}
\end{equation}
Choose a smooth radial \(\chi\) which is zero on \(|y|\leq1.2\)
and one on \(|y|\geq1.8\). With
\[
G=P\bigl((\chi-1)(q_0,p_0)\bigr)
\]
we have \(P(\chi(q_0,p_0))=z+G\). The pair
\((\chi-1)(q_0,p_0)\) defining \(G\) is compactly supported.
Integration by parts against affine
functions and Euclidean Killing fields gives
\begin{equation}
\int G_{\rm sc}=0,\quad \int y_aG_{\rm sc}=0,\quad
\int G_{{\rm vec},a}=0,\quad
\int(y_aG_{{\rm vec},b}-y_bG_{{\rm vec},a})=0.
\label{pk:constraint-moments}
\end{equation}
Indeed two derivatives annihilate an affine scalar; one derivative
annihilates a translation, while pairing the divergence of a symmetric
tensor with a rotation gives zero. If \(z\) is supported in
\(\{|y|\geq1\}\), then \(G=-z=0\) in \(|y|<1\), and \(G=0\) in
\(|y|>1.8\). It is therefore supported in
\[
A=\{1\leq|y|\leq2\}.
\]
We next give a compactly supported primitive for every pair on \(A\)
satisfying \eqref{pk:constraint-moments}.
\smallskip
\emph{A divergence primitive on the annulus.}
We use a capwise version of Bogovskii's divergence primitive
\cite{Bogovskii1980}, whose support and regularity properties are treated
by Costabel--McIntosh \cite{CostabelMcIntosh2010}. The finite mass
transfers below adapt the construction to the annulus.
Fix a unit-mass bump \(\zeta_0\) supported strictly inside \(A\).
There is a linear operator \(\mathcal D\), bounded from \(H^m\) to
\(H^{m+1}\) on inputs localized to a fixed compact set for each integer
\(m\geq0\), such that, for every scalar \(e\) supported in \(A\),
\begin{equation}
\partial_i(\mathcal De)_i=e-\zeta_0\int e,
\qquad \supp\mathcal De\subset A.
\label{pk:constraint-divergence}
\end{equation}
Here is the construction and the derivative estimate.
Cover \(A\) by sufficiently narrow angular caps and take smooth
compactly supported cutoffs \(\psi_a\) whose sum is one on \(A\).
They can be chosen so that their sum is a fixed cutoff \(\eta\)
equal to one on \(A\), with support in \(1/2<|y|<3\).
For each cap choose a unit-mass bump \(\zeta_a\) near radius \(3/2\)
in its central direction. Make the cap and bump small enough that
\[
b\cdot\widehat v>1,\qquad |b|<2
\]
whenever \(v\in A\cap\supp\psi_a\) and \(b\in\supp\zeta_a\).
Every point on the segment from \(v\) to \(b\) then has projection
at least one on \(\widehat v\), and lies in the radius-two ball by
convexity. The whole segment is thus contained in \(A\).
For a smooth compactly supported scalar \(e\), define
\begin{equation}
(\mathcal D_\zeta e)(x)
=\int e(v)(x-v)
\int_1^\infty \zeta\bigl(v+t(x-v)\bigr)t^2\,\dd t\,\dd v.
\label{pk:constraint-Bogovskii}
\end{equation}
Its support is contained in the union of the segments joining
\(\supp e\) to \(\supp\zeta\): if the integrand is nonzero, then
\(x=(1-t^{-1})v+t^{-1}b\) for some \(b\in\supp\zeta\).
Testing against a gradient and changing variables
\(b=v+t(x-v)\) gives
\[
\int \mathcal D_\zeta e\cdot\nabla\varphi
=\int e(v)\int\zeta(b)
\bigl(\varphi(b)-\varphi(v)\bigr)\,\dd b\,\dd v.
\]
In this calculation, the substitution \(\tau=t^{-1}\) reduces the
inner integral to the integral of the derivative of
\(\varphi(v+\tau(b-v))\) for \(0\leq\tau\leq1\).
It follows that
\(\operatorname{div}\mathcal D_\zeta e=e-\zeta\int e\).
The bumps may be chosen as translates of one small bump. Join their
centers to that of \(\zeta_0\) along paths at radius \(3/2\), and choose
the common bump small enough that every translated support remains
inside the open annulus. If \(\gamma_a\) is such a path from the
center of \(\zeta_a\) to the center of \(\zeta_0\), then
\[
V_a(x)=\int_0^1\gamma_a'(\tau)
\zeta_{\rm base}(x-\gamma_a(\tau))\,\dd\tau
\]
is smooth, compactly supported in the open annulus, and satisfies
\(\operatorname{div}V_a=\zeta_a-\zeta_0\). Define
\[
\mathcal De=
\sum_a\mathcal D_{\zeta_a}(\psi_a e)
+\sum_a V_a\int\psi_a e.
\]
For annulus-supported \(e\) this proves
\eqref{pk:constraint-divergence}. The same formula defines
\(\mathcal D\) on arbitrary inputs after the fixed localization by
the \(\psi_a\). Its values are always supported in a fixed compact
set, although exactness and annulus support are only needed and
asserted on the indicated inputs.
To prove the one-derivative bound, write \(r=|x-v|\) and
\(\omega=(x-v)/r\). The kernel in
\eqref{pk:constraint-Bogovskii} is
\[
\omega r^{-2}\int_r^\infty\zeta(v+\tau\omega)\tau^2\,\dd\tau
=\omega r^{-2}a(v,\omega)+E(x,v),
\qquad
a(v,\omega)=\int_0^\infty\zeta(v+\tau\omega)\tau^2\,\dd\tau .
\]
On fixed compact sets \(a\) is smooth, and
\[
E(x,v)
=-(x-v)\int_0^1\zeta\bigl(v+t(x-v)\bigr)t^2\,\dd t.
\]
This remainder is jointly smooth on fixed compact sets, with bounds
controlled by smooth seminorms of the bump. The leading kernel has
degree \(-2\) in \(x-v\). Its distributional \(x\) derivative is a
bounded multiplication term plus a principal-value kernel of degree
\(-3\), whose angular mean is zero. To see the cancellation, integrate
the ordinary derivative over a concentric annulus: the two boundary
fluxes agree by degree \(-2\) homogeneity, so the coefficient of the
radial logarithm is zero. Expansion of its smooth angular dependence
in spherical harmonics writes it as a summable series of mean-zero
angular convolution kernels acting on the input multiplied by smooth
coefficient functions of \(v\). These coefficient functions have
rapidly decreasing bounds in the harmonic degree. The classical
\(L^2\) estimate for a truncated homogeneous mean-zero singular
integral, with norm controlled by finitely many angular seminorms,
therefore bounds the differentiated kernel on \(L^2\).
The undifferentiated weak singularity is integrable, and the portions
away from the diagonal are smooth on bounded sets. This proves
\[
\|\mathcal D e\|_{H^1}\leq C\|e\|_{L^2}
\]
for the fixed localization.
Higher orders follow without loss by simultaneous translations.
For the unlocalized formula,
\[
\partial_j\mathcal D_\zeta e
=\mathcal D_\zeta(\partial_j e)
+\mathcal D_{\partial_j\zeta}e.
\]
The same kernel estimate applies to derivatives of the bump, without
a unit-mass requirement. Iterating this identity and differentiating
the fixed cutoffs and finite-rank terms gives
\begin{equation}
\|\mathcal D e\|_{H^{m+1}}\leq C_m\|e\|_{H^m}
\quad (m\geq0)
\label{pk:constraint-divergence-bound}
\end{equation}
for the localized operator. Approximation extends the formula and
its support assertions from smooth inputs to supported Sobolev inputs.
\smallskip
\emph{Symmetric divergence and the scalar primitive.}
Let \(F'\) be supported in \(A\), with zero translation and rotation
moments. Set
\[
T_{ij}=(\mathcal DF'_i)_j.
\]
Then \(\partial_jT_{ij}=F'_i\), and integration by parts gives
\[
\int(T_{ij}-T_{ji})
=\int(y_iF'_j-y_jF'_i)=0.
\]
Define
\begin{equation}
D_{ijk}=\bigl(\mathcal D(T_{ij}-T_{ji})\bigr)_k,\qquad
H_{ijk}=\tfrac12(D_{ijk}-D_{jki}+D_{kij}),\qquad
S_{ij}=T_{ij}-\partial_kH_{ijk}.
\label{pk:constraint-symmetric-divergence}
\end{equation}
Since \(D_{ijk}=-D_{jik}\), direct expansion gives
\[
H_{ijk}=-H_{ikj},\qquad H_{ijk}-H_{jik}=D_{ijk}.
\]
Thus \(S\) is symmetric, and
\[
\partial_jS_{ij}
=F'_i-\partial_j\partial_kH_{ijk}=F'_i.
\]
All tensors remain supported in \(A\). The two applications of
\(\mathcal D\) and the one derivative in
\eqref{pk:constraint-symmetric-divergence} show that this symmetric
divergence primitive maps \(H^m\) to \(H^{m+1}\).
The inverse trace reversal
\[
p_{ij}=S_{ij}-\tfrac12\delta_{ij}S_{kk}
\]
satisfies \(p_{ij}-\delta_{ij}p_{kk}=S_{ij}\), and is the desired
momentum primitive.
Now let \(f'\) be supported in \(A\), with zero constant and linear
moments. Put \(Y=\mathcal Df'\). Then
\[
\partial_iY_i=f',\qquad
\int Y_a=-\int y_af'=0.
\]
Set
\[
M_{ab}=\int(y_aY_b-y_bY_a),\qquad
C_{ab}=-\tfrac12M_{ab},\qquad
\widehat Y_i=Y_i+\partial_j(C_{ij}\zeta_0).
\]
The skew-symmetry of \(C\) implies
\[
\partial_i\widehat Y_i=f',\qquad \int\widehat Y_i=0.
\]
Moreover,
\[
\int\bigl(y_a\partial_j(C_{bj}\zeta_0)
-y_b\partial_j(C_{aj}\zeta_0)\bigr)=2C_{ab},
\]
so every rotation moment of \(\widehat Y\) vanishes.
Apply the symmetric divergence primitive to \(\widehat Y\), obtaining
\(S\) with \(\partial_jS_{ij}=\widehat Y_i\), and put
\(q_{ij}=S_{ij}-\tfrac12\delta_{ij}S_{kk}\). Then
\[
\partial_i\partial_jq_{ij}-\Delta q_{ii}
=\partial_i\partial_jS_{ij}=f'.
\]
This gains two derivatives and preserves support in \(A\).
Together these constructions give an operator \(\mathcal B\) satisfying
\begin{equation}
P\mathcal B(f',F')=(f',F'),\qquad
\|\mathcal B(f',F')\|_{H^{s+2}\oplus H^{s+1}}
\leq C_s\|(f',F')\|_{H^s\oplus H^s}
\label{pk:constraint-compact-primitive}
\end{equation}
on the supported moment-free pairs, with output supported in \(A\).
They also define a bounded operator on arbitrary compact inputs:
use the fixed localization in every occurrence of \(\mathcal D\),
and symmetrize the final tensors. On moment-free annulus-supported
inputs this final symmetrization changes nothing. Off that subspace
no exactness is asserted or required.
We may now set
\begin{equation}
\mathcal Kz=\chi(q_0,p_0)-\mathcal BG.
\label{pk:constraint-K}
\end{equation}
The estimates already proved give its boundedness on all \(Y^s\).
For exterior-supported inputs, both the support assertion and
\(P\mathcal Kz=z\) follow from
\eqref{pk:constraint-compact-primitive} and \(P(\chi(q_0,p_0))=z+G\).
Finally, all the operators have smooth conjugates under the stated
group actions. For the argument dilation
\((T_\lambda e)(y)=e(\lambda y)\), substitution in
\eqref{pk:constraint-Bogovskii} gives
\begin{equation}
T_\lambda\mathcal D_\zeta T_\lambda^{-1}
=\lambda\mathcal D_{\zeta_\lambda},\qquad
\zeta_\lambda(y)=\lambda^3\zeta(\lambda y).
\label{pk:constraint-conjugation}
\end{equation}
The transformed cutoffs, path fields, and moment functionals likewise
depend smoothly on \(\lambda\). Rotations transform the bumps and
cutoffs smoothly and multiply tensor components by rotation matrices.
The kernel bounds above are controlled by finitely many smooth
seminorms on fixed compact sets, so Taylor's formula for these
parameter-dependent coefficients proves smoothness in the operator
norm with the full derivative gain. Newton inversion conjugates by
the factor \(\lambda^2\), and its first derivative by the corresponding
factor \(\lambda\); it is equivariant under rotations.
Composition in \eqref{pk:constraint-K} proves the final assertion.
\end{proof}
\begin{proof}[Proof of \Cref{pk:constraint-nonlinear}]
The same fixed point will supply both the base estimate and the
all-order regularity.
\paragraph{A contraction at one base order.}
Use the operator $\mathcal K$ from \Cref{pk:constraint-linear}.
Define
\begin{equation}
N(q,p)=\Phi(d_B+(q,p))-\Phi(d_B)-P(q,p).
\label{pk:constraint-nonlinearity}
\end{equation}
For small \(\beta_s\) and on a sufficiently small \(X^s\) ball this
is a smooth map to \(Y^s\). We record the weights and derivatives
to justify this assertion. Metric perturbations have weight one,
their first and second derivatives have weights two and three,
and second fundamental form perturbations and their first derivatives have weights
two and three. After subtraction of \(P\), the scalar terms are
linear combinations, with smooth bounded inverse-metric factors,
of terms with the budgets
\[
a\,\partial^2a:\ 1+3=4,\qquad
(\partial a)^2:\ 2+2=4,\qquad
b^2:\ 2+2=4.
\]
The corresponding momentum budgets are
\[
a\,\partial b:\ 1+3=4,\qquad
(\partial a)b:\ 2+2=4.
\]
Here \(a,b\) may be background or correction factors, with the
background-only terms canceled in
\eqref{pk:constraint-nonlinearity}. In particular, a coefficient in
front of a highest derivative differs from its Euclidean value by
a weight-one factor. Applying the Sobolev product estimates on
the unit ball and on the rescaled annuli, using \(s\geq3\), yields
\begin{equation}
\|N(v)-N(w)\|_{Y^s}
\leq C_s(\beta_s+R)\|v-w\|_{X^s},
\qquad \|v\|_{X^s},\|w\|_{X^s}\leq R.
\label{pk:constraint-Lipschitz}
\end{equation}
The same products after differentiation with respect to \(v\) prove
smoothness. Inverse-metric factors are smooth because the metrics
in this ball remain uniformly positive. Notice that the assertion
concerns \(N\); the uncanceled Euclidean leading terms of \(P\) have
weight three on a general element of \(X^s\).
Let \(C_{\mathcal K,s}\) be the norm of \(\mathcal K:Y^s\to X^s\).
For sufficiently small \(\beta_s,\eta_s\), the equation
\begin{equation}
z_c=-\phi-N(\mathcal Kz_c)
\label{pk:constraint-fixed-point}
\end{equation}
is a contraction on \(\|z_c\|_{Y^s}\leq2\eta_s\). Indeed take
\(R=2C_{\mathcal K,s}\eta_s\) in
\eqref{pk:constraint-Lipschitz} and require
\(C_sC_{\mathcal K,s}(\beta_s+R)<1/2\).
Starting the iteration at zero preserves support in \(\{|y|\geq1\}\):
\(\phi\) has that support,
\Cref{pk:constraint-linear} gives it for every correction, and
\(N(v)=0\) pointwise wherever \(v\) vanishes on an open set.
The limit therefore has the same support. With \(v_c=\mathcal Kz_c\),
\begin{equation}
\|v_c\|_{X^s}\leq2C_{\mathcal K,s}\eta_s,\qquad
\Phi(d_B+v_c)=0\quad\hbox{on } |y|\geq1.
\label{pk:constraint-correction-bound}
\end{equation}
The equation holds there because \(Pv_c=z_c\) and
\(\Phi(d_B)=\phi\) there. On the inner side \(v_c=0\).
\paragraph{Higher smoothness from rotations and dilations.}
It remains to establish smoothness at every order, without imposing
smallness of all high Sobolev norms. We do this at the single base
order \(s\). Let \(U_g\) and \(V_g\) be the actions of a rotation and
a positive argument dilation on tensor pairs and on scalar/covector
pairs respectively. Rotations act on components as well as arguments;
dilations here act on the argument alone. These actions are uniformly
bounded on \(X^s,Y^s\) for \(g\) near the identity. Set
\[
\mathcal K_g=U_g\mathcal K V_g^{-1},\qquad
\phi_g=V_g\phi,\qquad
N_g(w)=V_gN(U_g^{-1}w).
\]
The operator family \(\mathcal K_g\) is norm-smooth by
\Cref{pk:constraint-linear}. All background coefficients have the assumed smooth symbol bounds,
and the source is smooth and compactly supported. Thus \(g\mapsto\phi_g\) is smooth in \(Y^s\).
Writing \(N_g\) using the weighted products displayed above shows
that it is jointly smooth in \(g,w\): parameter derivatives fall on
the smooth background coefficients or introduce the explicit scale
and component factors in a transformed derivative. They do not
differentiate the unknown \(w\) beyond the derivatives already
allowed in \(X^s\).
Consider the equation on the full space \(X^s\),
\begin{equation}
w+\mathcal K_g\phi_g+\mathcal K_gN_g(w)=0.
\label{pk:constraint-parameter-equation}
\end{equation}
At \(g\) equal to the identity its solution is \(v_c\).
The derivative in \(w\) is
\(\Id+\mathcal K_gD N_g(w)\), whose inverse is a convergent Neumann
series on a sufficiently small common ball by
\eqref{pk:constraint-Lipschitz}. The implicit function theorem
therefore gives a smooth solution of
\eqref{pk:constraint-parameter-equation} in the group parameters.
We specify the uniqueness argument because argument dilations move
the support boundary. First choose a fixed small ball on which
the contraction estimates, and their nearby transformed versions,
hold. Taking the defect sufficiently small puts \(v_c\) strictly
inside a smaller ball. The uniform boundedness of \(U_g\) then puts
every \(U_gv_c\), for \(g\) near the identity, in the common contraction
ball. It satisfies \eqref{pk:constraint-parameter-equation} by
conjugation. Uniqueness in that ball identifies it with the smooth
implicit-function branch. This uses the all-input definition of
\(\mathcal K\), so no fixed support subspace has to be invariant
under dilation. Nor does it presuppose norm continuity of the
group orbit of the unknown solution.
Distributional differentiation of this smooth orbit shows that all
iterated rotations
\(\Omega_{ij}=y_i\partial_j-y_j\partial_i\) and dilations
\(E=y\cdot\nabla\), applied to \(v_c\), lie in \(X^s\).
The rotation of tensor components contributes only constant
zeroth-order matrices, which may be subtracted in identifying these
derivatives.
On every rescaled exterior annulus these fields span the scaled
coordinate derivatives. More explicitly, with \(r=|y|\),
\[
r^2\partial_i=y_iE+\sum_j y_j\Omega_{ji}.
\]
Their iterates therefore control every scaled spatial derivative.
The same identity is nondegenerate on a neighborhood of \(|y|=1\).
It follows that the correction is smooth across its support
boundary as well as on the exterior, and has symbol bounds of all
orders with metric weight one and second fundamental form weight two.
These higher constants may depend on the higher symbol bounds of
$d_B$ and on the smooth source $\phi$; their smallness is not required.
The small base norm also keeps the corrected metric positive definite.
\end{proof}
\paragraph{Application to the beam seed.}
The rescaling, artificial interior extension, and verification of the
source hypotheses are carried out in Section~\ref{pk2:constraints},
using \eqref{pk:constraint-rescaling}--\eqref{pk:constraint-background}.
That application retains the complete correction tail and proves
approximation in any prescribed finite list of the original data
seminorms. Higher symbol bounds ensure smooth membership of each
corrected datum; they need not be uniformly small or give convergence
at all orders for one fixed frequency exponent.
==== END Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//appendices/constraints.tex ====
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//appendices/far-energy.tex ====
\section{Far-region geometry and energy estimates}\label{fe:section}
We prove the energy estimate used in \Cref{sec:far} for a smooth
finite slab $0\le t\le t_{\rm end}$. The structural inputs are
the lab equation and differentiated gauge in Lemma~\ref{far:structure},
and the initial and starting-annulus bounds in
Lemma~\ref{far:boundary-input}. We derive the point estimates needed
for the coefficients, then prove the two multiplier estimates conditional
on explicit source budgets. Appendix~\ref{fs:section} supplies those
budgets; the simultaneous exterior closure is proved in
Proposition~\ref{far:closure}. The radial weighted
currents follow the method of Dafermos--Rodnianski
\cite[Sections~3--4]{DafermosRodnianski2010}, adapted here to the moving emitter
coordinates and the coupled source estimates.
Throughout, smallness of the finite-slab norms is a bootstrap hypothesis;
their finiteness follows from finite-time smooth existence and symbol
decay. Constants on superlinear terms may depend on all fixed scales.
Linear constants acting \emph{on other far quantities} in absorption or
finite hierarchies are independent of both large matching radii. Take
$R_0$ beyond all spatial kernels and the transition where target smoothing
reaches its prescribed far form. Comparable smoothed radii use the same
positive acceleration majorant $\widetilde a$ on a slightly enlarged
sampling interval at that scale.
\subsection{Emitter coordinates and the three energy levels}\label{fe:coordinates}
For the calculations in both far appendices we abbreviate
$T_l,C_e,s_e,\ell_n,\delta_f,\tau_f,l_f,m_f,M_f$ by
$T,C,s,\ell,\delta,\tau,l,m,M$, respectively, and write
$X,\bar X$ for $X_g,\bar X_g$. These are the same
coordinates, fields and weights as in \Cref{sec:far}.
In the lab chart put
\begin{equation*}
C(q)=-\lambda(q),\quad s(q)=C'(q),\quad t=q+r,\quad z=C(q)+rn,\quad |n|=1,\quad k=1-s(q)\cdot n .
\end{equation*}
These are smooth coordinates for large emitter radius \(r\), by the small speed and the implicit function theorem; the radii \(|x|,|y|\) of the reduction are comparable to \(r\) (the position differences on time intervals of length \(O(r)\) are small times \(r\); use also the contraction defining \(x\)). Work with
\begin{equation*}
T=\partial_{t,{\rm lab}},\quad L=\partial_{r|q,n}=T+n\cdot\partial_z,\quad
Z=(S=rL,\Omega),\quad s_s=-v_s,\quad D=T+s_s\cdot\partial_z ,
\end{equation*}
where \(\Omega\) are the spherical rotations at fixed \(q,r\). Angular differentiations below can equivalently use finite smooth frames. Components of tensor fields in all the far wave differentiations are \emph{lab} components. Write \(H^{ij}=g^{ij}-\eta^{ij}\), \(\ell_i=(1,-n)\), and subscript \(\ell\ell\) for contraction of upper indices with this pair. The lab null contravariant vector corresponding to \(-\eta^{ij}\ell_j\) is \(L\). Derivatives have the identities
\begin{equation*}
\partial_i=\ell_i T+r^{-1}b_i Z
=\ell_i k^{-1}\partial_q+r^{-1}(b_i-\ell_i k^{-1}s^d b_d)Z ;
\end{equation*}
\(b_t=0\), \(b_dZ=n_d S+\) a tangential spherical derivative for spatial indices. In particular the \(q\)-derivative of the second good-frame coefficient (before \(r^{-1}\)) is proportional to \(\ell_i\) and of size \(O(|a(q)|)\). All filtered unsmoothed acceleration jets needed below are bounded and decay with at least \(\langle q\rangle^{-A/2}\) on positive times, by the compact budgets and \eqref{co2:sliced-acceleration}, and are zero at negative times. The enlarged average has
\begin{equation*}
r\widetilde a\lesssim \mathsf N(1+t/r)^{-A/2}
\end{equation*}
by \(L^1\) for \(t\lesssim r\) and weighted Cauchy--Schwarz for \(t\gg r\). Here and in low point estimates \(\mathsf N\) can include the data norm, and fixed compact-reaching constants multiplying small quantities are allowed when such bounds are used \emph{nonlinearly}. Derivatives of \(s_s\) of positive order \(d\) are \(O(r^{1-d}\widetilde a)\) in the lab chart; \(B-\eta\) has symbol size \(O(r^{-1})\) there in all the spacetime derivatives. With ordered commutations (\(T^j\) first, then \(Z^I\)) these retain their respective bounds (\(T\) a lab derivative, \(Z\) scaled first order); arbitrary re-expansions differentiating emitter frame factors need not gain powers for additional \(T\)'s. For example \(T^m n=O(1/r)\) for finite \(m\ge1\), with \(Z\)-derivatives allowed.
Use three levels labeled \(0,H,P\). Their unscaled fields \(U_i\), number of commutations \(N_i\), time exponents \(A_i\), and outgoing powers \(p_i\) are
\begin{equation*}
\begin{array}{c|c|c|c|c}
i&U_i&N_i&A_i&p_i\cr\hline
0&h&9&0&.64\cr
H&h&8&1-\eta_*&1-\eta_*/6\cr
P&\psi=D h&7&A=1.36&1.62 .
\end{array}
\end{equation*}
Here \(\eta_*>0\) is sufficiently small, say at most \(10^{-8}\). Put \(\delta=\eta_*/100,\ \nu=10\delta,\ e=.1\delta,\ d_*=200\eta_*\).
The strict margins serve different purposes and will be kept throughout:
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Margin & Use \\
\midrule
$p_0,p_H<1$ & Initial energy for arbitrary $r^{-1}$ metric tails \\
$p_i>A_i$, $p_i+.1A_i<2$ & Positive outgoing bulk and Hardy control \\
$A>1$ & Time integrability of the acceleration \\
$2A_H>A$ & Weak-field products in the strong equation \\
$p_P-A=.26$ & Radius-uniform pairing with adjoint zero modes \\
\bottomrule
\end{tabular}
\end{center}
Take
\begin{equation*}
\mu=\langle q/r^\delta\rangle,\quad l=\langle q/r^\nu\rangle,\quad
m=(r^{-1}+l^{-1})^{-1},\quad W_i=r^{p_i-A_i}(t+10r)^{A_i}.
\end{equation*}
Use time-energy weights, for commutations with \(j\) lab times,
\begin{equation*}
\begin{split}
w_H&=r^{-e}\mu^{A_H},\\
w_{P,j}&=w_H l^{A-A_H}m^{-(7-j)d_*},\\
w_{0,j}&=r^{-.02}\,m_-^{j d_*}m_+^{-(9-j)d_*}.
\end{split}
\end{equation*}
Here \(m_-\) is a smooth positive monotone weight in \(-q/r^\nu\) comparable to \(1+(-q/r^\nu)_+\) (constant on the nonpositive half-line, bounded logarithmic-symbol derivatives); \(m_+\) is the analogous weight in \(q/r^\nu\), capped smoothly by harmonic addition at \(r\) as for \(m\). The comparability in this text always allows fixed constants. Put \(\tau=1+t/r\). Thus \(W_i\asymp r^{p_i}\tau^{A_i}\); \(W_i/w_{i,j}\ll r^{2-.3}\), and in particular \(w_{P,j}\gtrsim \langle t\rangle^A r^{-.025},\ w_H\gtrsim \langle t\rangle^{A_H} r^{-e-\delta A_H}\) for \(t\gg r\).
Our scalar wave unknowns (componentwise) are
\begin{equation*}
\chi=Z^I(rT^j U_i),\qquad n_*=|I|+j\le N_i .
\end{equation*}
We write \(B_1\chi\) for its full first derivatives (size comparable to \(|\partial_q\chi|+r^{-1}|Z\chi|\)), and \(G_1\chi=r^{-1}Z\chi\). In formulas indicating bounds, lists of fields/derivatives denote their Euclidean size, with sums over finite components or lower commutations as specified. The main measured quantities, before specifying a current correction, are
\begin{equation}\label{fe:F1}
\begin{array}{ll}
L^2(dt\,dr\,d\omega/r): & W_i^{1/2}(G_1\chi,\chi/r),\quad
w_{i,j}^{1/2}B_1\chi ,\\
L^\infty_t L^2(dr\,d\omega):& W_i^{1/2} G_1\chi,\quad
w_{i,j}^{1/2}B_1\chi
\end{array}
\end{equation}
on \(r\ge R_0\). Here \(d\omega\) is round measure of \(n\). In particular the \emph{flat} good derivatives are included. The total size \(\mathsf F\) sums these and the extra current norms below; quantities on a starting overlap annulus a fixed factor farther in, when needed in estimates, can instead use \(C(R_0)\mathsf K_R+C_{\rm fix}(e_{\rm in}+\mathsf N^2)\), by the co/lab comparisons on bounded annuli. Indeed changing \(u\) to \(\psi\) costs only field-derivative products with strong coefficient differences such as \(v(t)-v_s\) there. No supremum of characteristic flux in this overlap is required from compact estimates. All a priori integrations are for smooth solutions with finite symbol decay on bounded intervals; or one first truncates at infinity where this decay gives the limiting estimates. The time-zero energies have the stated orders by the initial symbol controls (\(T^j\) costs at least \(r^{-j}\) on undifferentiated initial jets, startup flat). For \(h\), even with no lab-time commutation, \(p_i<1\); for \(D h\) the extra decay easily gives finiteness. Constants on the input may depend on fixed scales.
\subsection{The wave-gauge constraint and weak-null structure}\label{fe:weak-null}
The weak-null organization of the Einstein equations in wave coordinates
is due to Lindblad--Rodnianski \cite[Sections~3--5]{LindbladRodnianski2003}; see also
\cite{LindbladRodnianski2010}.
The component calculation below records the precise polarization and
weighted source structure required here.
The lab gauge identity on the end gives an improved differentiated constraint. With \(\Gamma_\mu(g)=g^{ab}\Gamma(g)_{\mu;ab}\) (semicolon here separates the lowered first Christoffel index), it is
\begin{equation*}
\Gamma_\mu(g)=g_{\mu\sigma}g^{ab}\Gamma(B)^\sigma_{ab}+\Theta_\mu .
\end{equation*}
In particular its size is \(O(r^{-2})\). Subtracting the first derivatives of \(B\), the flat linear gauge of \(h\) costs terms with smooth coefficients of sizes
\begin{equation}\label{fe:F2}
O((r^{-1}+|h|)\partial h)+O(r^{-2}h)+\Theta .
\end{equation}
These are structural product bounds: the first coefficient is smooth vanishing in \(B-\eta,h\); linear-size fixed coefficients obey lab symbol differentiation, including the enlarged-average bounds for \(\Theta\); differentiation in \(h\) obeys the chain rule. Also the reduced wave equation has the form
\begin{equation*}
g^{ij}\partial_{ij}h=f_h ;
\end{equation*}
its pure-background term is of symbol size \(O(r^{-2}\widetilde a)\), with \(D\) derivative \(O(r^{-3}\widetilde a)\), and the remaining terms have sizes/structure
\begin{equation}\label{fe:F3}
O(r^{-2}\partial h+r^{-3}h)+Q_\eta(\partial h,\partial h)
+O((r^{-1}+|h|)|\partial h|^2).
\end{equation}
Here smooth bounded multiplicative coefficients, lab symbol background coefficients as above, and smooth functions of \(h\) are understood in the \(O\)'s. \(Q_\eta\) is a constant flat quadratic polynomial. On replacing \(\partial_i h\) by \(\ell_i p\) with \(\ell^\mu(p_{\mu\nu}-\frac12\eta_{\mu\nu}{\rm tr}_\eta p)=0\), it is a constant multiple of
\begin{equation*}
\ell\otimes\ell\ |\Pi(n)p|^2 ,
\end{equation*}
where \(\Pi\) takes the two-dimensional tracefree tangential part on the spatial screen. Thus errors of this replacement with \(p=T h\) use either a good derivative or a factor of the gauge error \eqref{fe:F2}. To check the structure, write Ricci as \(-\tfrac12 g^{ab}\partial_{ab}g_{\mu\nu}+\partial_{(\nu}\Gamma_{\mu)}(g)\) plus the products
\begin{equation*}
(\partial_a g^{ab})\Gamma(g)_{b;\mu\nu}
-(\partial_{(\nu}g^{ab})\partial_a g_{\mu)b}
+\Gamma^a_{a d}\Gamma^d_{\mu\nu}-\Gamma^a_{\nu d}\Gamma^d_{a\mu}.
\end{equation*}
Substitute the gauge identity. Pure terms at \(h=0\) are Ricci of the target plus differentiated compensator terms. In the flat product evaluation the first two displayed products cancel on \(\ell p\) as above. Writing the Christoffel matrices at fixed \(\mu\) as \((\ell_\mu p^\sharp+p_\mu^\sharp\otimes\ell-\ell^\sharp\otimes p_\mu)/2\), the last two products then give \((({\rm tr}\,p)^2/8-{ \rm tr}(p^2)/4)\ell_\mu\ell_\nu\), as asserted by the constrained-component relation in the flat null frame. This calculation also works polarized. One can differentiate in lab coordinates first and project afterwards, or retain \eqref{fe:F2} as an exact error identity; when differentiating projected formulas each lab differentiation hitting the frame has a factor \(O(1/r)\) with bounded emitter jets. Here the tangent \(p^\sharp\) notation uses flat raising.
For completeness, rotate the screen so that $n=(1,0,0)$ and use
$\eta=\diag(-1,1,1,1)$, $\ell=(1,-1,0,0)$. The four gauge relations
are solved by the six-parameter matrix
\begin{equation*}
p=\begin{pmatrix}
a&b&c&d\\
b&-a-2b&-c&-d\\
c&-c&e&f\\
d&-d&f&-e
\end{pmatrix}.
\end{equation*}
Consequently $\tr_\eta p=-2(a+b)$,
$\tr((\eta^{-1}p)^2)=2(a+b)^2+2(e^2+f^2)$, and
\begin{equation*}
\frac18(\tr_\eta p)^2-\frac14\tr((\eta^{-1}p)^2)
=-\frac12(e^2+f^2)=-\frac14|\Pi p|^2.
\end{equation*}
Polarization gives the same identity with $|\Pi p|^2$ replaced by the
screen pairing of two constrained tensors. Thus the quadratic restriction
and the projection cancellation remain valid when the two factors carry
different ordered commutations.
We will also use the corresponding projection onto the constrained plane.
Define
\begin{equation}\label{fe:gauge-projection}
\begin{aligned}
\mathcal G_np&=\ell^\mu
(p_{\mu\nu}-\eta_{\mu\nu}\tr_\eta p/2),
&\breve p&=p-J_n\mathcal G_np,\\
\mathcal G_nJ_n&=\Id,
&\Pi J_n&=0.
\end{aligned}
\end{equation}
The matrix calculation shows that $\mathcal G_n$ has rank four and that
every screen polarization occurs in its kernel. Its restriction to
$\ker\Pi$ is therefore surjective; taking the Euclidean right inverse
on this restriction gives a smooth bounded $J_n$. Consequently
$\mathcal G_n\breve p=0$ and $\Pi\breve p=\Pi p$. The four gauge
errors measure exactly the $LL$, $L$--screen, and screen-trace components.
\subsection{The scalar emitter operator}\label{fe:emitter-operator}
For a scalar, the emitter operator is
\begin{equation*}
P_g\chi:=kr g^{ij}\partial_{ij}(\chi/r),\qquad
P_\eta\chi=-2\partial_q\partial_r\chi+r^{-2}\mathcal A_s\chi ,
\end{equation*}
\begin{equation}\label{fe:F4}
\mathcal A_s=(1+s\cdot n)(S^2-S)-2(s\cdot n)(S-1)
+2s^A\nabla_A(S-1)+ k\Delta_{\mathbb S^2}.
\end{equation}
Here is a direct computation that also records the absence of acceleration
terms in the flat operator. Put $\sigma=s\cdot n$ and let $\gamma$ be the
round screen metric. The nonzero covariant entries are
\begin{equation*}
\eta_{qq}=-1+|s|^2,\quad \eta_{qr}=-k,\quad
\eta_{qA}=rs_A,\quad \eta_{AB}=r^2\gamma_{AB}.
\end{equation*}
Their inverse and density are
\begin{equation*}
\eta^{qr}=-k^{-1},\quad \eta^{rr}=(1+\sigma)/k,\quad
\eta^{rA}=s^A/(kr),\quad \eta^{AB}=r^{-2}\gamma^{AB},\quad
\sqrt{|\det\eta|}=kr^2\sqrt{\det\gamma},
\end{equation*}
with $\eta^{qq}=\eta^{qA}=0$.
In the divergence formula the only coefficient differentiated in $q$ is
$kr^2\eta^{qr}=-r^2$, which is independent of $q$.
Using $\nabla_As^A=-2\sigma$ and $\nabla_Ak=-s_A$ gives
\begin{equation*}
P_\eta\chi=-2\chi_{qr}+(1+\sigma)\chi_{rr}
-\frac{2\sigma}{r}\chi_r+\frac{2\sigma}{r^2}\chi
+\frac{2s^A}{r}\nabla_A\chi_r
-\frac{2s^A}{r^2}\nabla_A\chi+\frac{k}{r^2}\Delta_\gamma\chi.
\end{equation*}
Replacing $r\partial_r$ by $S$ proves \eqref{fe:F4}. More generally for an upper-index coefficient \(J^{ij}\), the additional expression \(P[J]\chi=kr J^{ij}\partial_{ij}(\chi/r)\) consists of
\begin{equation}\label{fe:F5}
J_{\ell\ell}\partial_q(k^{-1}\partial_q\chi)
+ O\left(
\frac{J}{r}(1,Z)\partial_q\chi+
\frac{J}{r^2}(1,Z,Z^2)\chi+
\frac{a(q)}{r}J_{\ell\ell}(1,Z)\chi\right),
\end{equation}
where the other coefficients are smooth bounded functions of \(n,s(q)\). This follows from the derivative formula above (including differentiating \(1/r\)); a derivative of a good-frame coefficient in \(q\) has the extra \(\ell\) contraction. The term with \(a(q)\) allows contractions of its components. These identities apply \emph{after} lab times have been commuted.
\subsection{Spacelike anchors and point estimates}\label{fe:anchors}
The following estimates are implications of the finite-time bootstrap norms, \eqref{fe:F2}, and \eqref{fe:F3}; they do not assume the uniform estimate that will close the bootstrap.
The starting scale can and will always be taken so large that even the inner band radii of the following auxiliary construction (a fixed constant times \(Q\gtrsim R_0^{.45}\)) are in the lab symbol regime beyond all spatial kernels and all fixed smoothing transitions. No comparison with compact estimates at radii of this size is used as input for the spacelike observation.
\begin{lemma}[Spacelike anchors]\label{fe:spacelike-anchors}
For \(r\asymp R_f,\ q\le-R_f^\theta/10\), \(\theta=.45\), the center is zero. On a band \(-q\asymp Q\), we have
\begin{equation}\label{fe:F6}
\|Z^I(rT^j h)\|_{L^2_\omega}\lesssim (e_{\rm in}+\mathsf S_a)(r/Q)^{1/2}
\quad (|I|+j\le9),
\end{equation}
allowing fixed input constants; the constant here on \(\mathsf S_a\) is uniform far out. Also the ordinary unweighted \(L^2_z\) energy on such a band at fixed \(t\) of Minkowski commutations through nine is \(\lesssim (e_{\rm in}+\mathsf S_a)Q^{-1/2}\).
\end{lemma}
\begin{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 \eqref{fe:F3} 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 \eqref{fe:F6}, uniformly also if the lifespan truncates the band.
\end{proof}
In low point estimates on products below we abbreviate a constant times small bootstrap size by \(N\). This can include fixed input and fixed compact-to-far point constants. When \eqref{fe:F6} is used on a \emph{linear} energy error we keep its specified independent bounds instead. Endowing point bounds with \(L^2_\omega\) means they hold on each sphere.
\begin{lemma}[Radial point estimates]\label{fe:point-estimates}
The finite-time norms \eqref{fe:F1} imply
\begin{equation}\label{fe:F7}
\begin{array}{ll}
\|\chi(h)\|_{L^2_\omega}\lesssim N r^{2\eta_*}\tau^{-A_H/2}
& (n_*\le8),\\
\|\chi(h)\|_{L^2_\omega}\lesssim N r^{.52}
& (n_*\le9),\\
\|B_1\chi(h)\|_{L^2_\omega}\lesssim N r^{e/2}\mu^{-A_H/2}
& (n_*\le7).
\end{array}
\end{equation}
\end{lemma}
\begin{proof}
For \(t\lesssim r\) integrate to an anchor with \(r'\asymp r\) large enough compared with \(t\), using \eqref{fe:F6}; 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 \eqref{fe:F7}; 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 \(n_*\le7\),
\begin{equation}\label{fe:F8}
\|\chi(\psi)\|_{L^2_\omega}
\lesssim N r^{e/2}\mu^{-A_H/2}.
\end{equation}
Use \(D=(1-s_s\cdot n)T+r^{-1}s_s^d b_dZ\), with lab-time commutators of \(s_s\) taken \emph{before} expansion, and \eqref{fe:F7}; \(rZ^IT^j\) 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.
\end{proof}
\subsection{Improved null projections}\label{fe:projections}
\begin{lemma}[Differentiated null projection]\label{fe:null-projection}
In any bound with a \(T^j\) on an upper-index tensor, contract with \(\ell\ell\) \emph{after} those derivatives and then apply the indicated \(Z\)'s to the full contraction.
\begin{equation}\label{fe:F9}
\begin{array}{ll}
\|Z^I H_{\ell\ell}\|_{L^2_\omega}
\lesssim r^{-1}+N r^{-1+2\eta_*}\tau^{-A_H/2}&(|I|\le8),\\
\|Z^I (T^j H)_{\ell\ell}\|_{L^2_\omega}
\lesssim r^{-2}+N r^{-2+8\eta_*}\tau^{-A_H/2}
&(j\ge1,\ |I|+j\le8),\\
\|\partial_q Z^I (T^j H)_{\ell\ell}\|_{L^2_\omega}
\lesssim r^{-2}+N r^{-2+8\eta_*}\tau^{-A_H/2}
&(|I|+j\le7).
\end{array}
\end{equation}
\end{lemma}
\begin{proof}
The first estimate follows by smooth inversion of $B+h$, the symbol
bound on $B-\eta$, and \eqref{fe:F7} through order eight.
For the last two, \eqref{fe:F2} after ordered differentiation controls the bad derivative of $h_{LL}$, the $L$--screen components, and the spherical trace of $h$: \eqref{fe:F7} 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 \eqref{fe:F2} and \eqref{fe:gauge-projection}, 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 \(H\).
\end{proof}
\begin{lemma}[Tight cone projection and top-order control]\label{fe:tight-projection}
Set $b=1+\min(|q|,r)$. Then
\begin{equation}\label{fe:F10}
\begin{aligned}
r\|Z^I H_{\ell\ell}\|_{L^2_\omega} &\lesssim 1
&&(|I|\le8,\ |q|\le 10 r^\theta),\\
r\|Z^I (T^j H)_{\ell\ell}\|_{L^2_\omega} &\lesssim r^{-.46}
&&(j\ge1,\ |I|+j\le9),\\
r\|Z^9 H_{\ell\ell}\|_{L^2_\omega}
&\lesssim \log r + b^{1/2}r^{-p_0/2}+b r^{-1+8\eta_*}.
\end{aligned}
\end{equation}
Constants here can be bounded independently of \(R_0\) by choosing smallness of the nonlinear quantities after the radii; equivalently retain background plus \(O(C_{\rm fix}\mathsf N)\) in them.
\end{lemma}
\begin{proof}
The second follows exactly as in \eqref{fe:F9}, 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 \(O(r^{-.46})\).
\smallskip\noindent\emph{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 \eqref{fe:F2} 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.
\emph{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 \eqref{fe:F6}. 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 \eqref{fe:F6}, and in products with a high field the bad derivative now costs \(1/Q\) only on a low factor.
\emph{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 \eqref{fe:F7} 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.
\emph{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 \eqref{fe:F6}. 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 \eqref{fe:F10}.
\end{proof}
\subsection{A uniform bound for the lowest radiative jets}\label{fe:news}
\begin{lemma}[Uniform news bound]
On every finite smooth bootstrap interval, the Cartesian lab-time derivatives satisfy
\begin{equation}\label{fe:F11}
|r\Pi T h|+|r\Pi T^2h|+|r\Pi T\psi|\lesssim N.
\end{equation}
The constant uses only the previously stated low bootstrap, initial, and compact overlap controls.
\end{lemma}
\begin{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
\begin{equation}\label{fe:news-principal}
-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.
\end{equation}
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 \Cref{fs:section}.
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 \eqref{fe:F7}, 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 \eqref{fe:F5} 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 \eqref{fe:F10} 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 \eqref{fe:F9} 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 \eqref{fe:F5}; 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}\tr_\eta\breve p_i\bigr)=0,
\qquad \Pi\breve p_i=\Pi T^{i+1}h.
\end{equation*}
Take $\breve p_i$ from \eqref{fe:gauge-projection} with
$p=T^{i+1}h$. The good derivative terms and
\eqref{fe:F2}, 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
\eqref{fe:F3}, 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
\begin{equation}\label{fe:news-transport}
|(-2k\partial_r+H_{\ell\ell}\partial_q)\mathcal Y_j|
\lesssim Nr^{-2+.04}.
\end{equation}
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 \eqref{fe:F11}, 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 \eqref{fe:F11}.
\end{proof}
\subsection{True null directions, spacelike caps, and corrected currents}\label{fe:currents}
We now construct the energy currents and specify the source norms that they require. Let \(X,\bar X\) in this calculation denote the \emph{true} outward/inward future null normals to the emitter spheres, normalized by \(dt=1\), and \(G=-g(X,\bar X)>0\). These are algebraic smooth functions of \(g,n\) near flat, with components in the lab chart, independent of \(s(q)\) given those arguments. They are \(O(r^{-1}+|h|)\) away from the corresponding flat normals. With \(z_X=n\cdot X^z\), one has \(Xq=(1-z_X)/k\), bounded in size by a constant times
\begin{equation*}
d_g=|H_{\ell\ell}|+|H|^2.
\end{equation*}
Indeed the linearization of the radial speed on the outgoing root costs just the \(L L\) covariant variation. Also \(Xr=1-Xq\). The estimates above imply \(d_g\lesssim r^{-1}\) for \(|q|\le 2r^\theta\), \(d_g\lesssim r^{-1+2\eta_*}\) elsewhere. Thus we can use global spacelike caps \(\mathfrak u=u\), where for example
\begin{equation*}
\mathfrak u=q+\int_{R_0/2}^r
[\rho^{\delta-1}+\rho^{20\eta_*-1}j(|q|/\rho^\theta)]\,d\rho ,
\end{equation*}
with \(j=1\) above 2 and zero below 1, a smooth transition. One has \(\partial_q\mathfrak u=1+o(1)\), \(X\mathfrak u>0,\ \bar X\mathfrak u>c_0>0\) for \(R_0\) large, and \(|\mathfrak u-q|\lesssim r^\delta+|q|^{20\eta_*/\theta}\). The induced changes between \(r^\delta+|q|\) and \(r^\delta+|u|\) are comparable. No angular component of \(d\mathfrak u\) on the spheres occurs. Fix
\begin{equation*}
M=r^{.72}\mu^{-A_H}j_0(q/r),\qquad K=1+r^{-1}\int_q^\infty M(q',r)\,dq',
\end{equation*}
with \(j_0\) nonnegative, one on \([-1,1]\), zero outside \([-2,2]\), smooth. Then \(K=1+o(1)\), \(r|XK|=o(1)\), \(-r\bar X K\gtrsim M-o(1)\). Indeed the integral costs \(o(r)\) with the same bounds after \(r\partial_{r|q}\), and \(M Xq=o(1)\).
For each equation a specified scalar current correction \(\mathcal P\) will be used (possibly zero), with
\begin{equation*}
\partial_q=\alpha\bar X+\text{a combination of }X,\ e_A,\quad
c=\frac{G\alpha}{2k},\qquad
\vartheta=-g(\bar X,\cdot)/G,\qquad
\xi=d\chi+c\mathcal P\vartheta .
\end{equation*}
Here \(e_A\) is a \(g\)-orthonormal screen, the indicated coefficients are bounded, and \(\alpha=(z_X-s\cdot n)/(z_X-z_{\bar X})\). Complete the definition of \(\mathsf F\) by including
\begin{equation}\label{fe:F12}
\|(W_i(1+M))^{1/2}\xi_X\|_{L^2(dt\,dr\,d\omega/r)}
+\sup_u\|W_i^{1/2}\xi_X\|_{L^2(dr\,d\omega;\ \mathfrak u=u)}
\end{equation}
out to the current time. Only the directional component, not \(M^{1/2}\) times the uncorrected good frame, is required with the extra \(M\).
Use the notation \(\|\cdot\|_{\rm b}\) here just for \(L^2(dt\,dr\,d\omega/r)\). The source splitting in the outgoing multiplier may be written
\begin{equation}\label{fe:F13}
P_g\chi=\lambda_1 L\chi/r+ f_{\rm flat}
+\partial_q\mathcal P+f_{\rm r}+f_{\rm mix},
\end{equation}
where \(\lambda_1\ge0\) is bounded and the flat triangular terms are specified below. In the lab-time multiplier we keep the whole raw source \(f_{\rm raw}\) (after the first two terms). The budgets to estimate for each level are
\begin{equation}\label{fe:F14}
\begin{split}
&\|W_i^{1/2}(\mathcal P,Z\mathcal P,r d_g\,\partial_q\mathcal P)\|_{\rm b}
+\sup_t\|W_i^{1/2}\mathcal P\|_{L^2_{r,\omega}},\\
&\| (W_i/(1+M))^{1/2}r f_{\rm r}\|_{\rm b}
+\|w_{i,j}^{1/2}r f_{\rm raw}\|_{\rm b}.
\end{split}
\end{equation}
For \(f_{\rm mix}\), supported where \(t\lesssim r\), an alternative charge on each dyadic shell \(r\asymp R_f\) is as follows. Suppose
\begin{equation*}
\|r f_{\rm mix}\|_{L^2_\omega}
\le l_{R_f}(\mathfrak u)\,\mathcal T(t,r),
\qquad \sup_t\|\mathcal T(t,\cdot)\|_{L^2_r}
\le E_{\rm infield}.
\end{equation*}
The corresponding mixed budget is
\begin{equation}\label{fs:mixed-charge}
\sum_{R_f}R_f^{(p_i-1)/2}
\|l_{R_f}\|_{L^2(du)}E_{\rm infield}.
\end{equation}
To see the shell power, pair the source with $W_i\xi_X$ using measure
comparable to $dt\,dr\,d\omega$. Cauchy--Schwarz on each outgoing cap
and then in $u$ gives
\begin{equation*}
C R_f^{p_i/2-1}
\sup_u\|W_i^{1/2}\xi_X\|_{L^2(\mathfrak u=u)}
\|l_{R_f}\|_{L^2(du)}
\left(\int_{t\lesssim R_f}\!\!\int\mathcal T(t,r)^2\,dr\,dt\right)^{1/2}.
\end{equation*}
The last factor is at most $C R_f^{1/2}E_{\rm infield}$, producing
exactly the stated charge. Positive sums of such charges are allowed.
\begin{proposition}[Far energy reduction]\label{fe:energy-reduction}
Assume that each equation has the splitting \eqref{fe:F13}, and that its correction, raw, remainder, and mixed budgets satisfy, in the triangular order specified below, bounds of the form
\begin{equation}\label{fe:source-hypothesis}
o_{R_0\to\infty}(1)(\mathsf F+\mathsf S_a)
+C\sum_{\rm prior}\mathsf F_{\rm prior}
+C(R_0)\mathsf K_R+C_{\rm fix}(e_{\rm in}+\mathsf N^{1+c_1})
\end{equation}
for fixed \(c_1>0\). Assume also that the corrections have initial weighted energy and averaged
starting-annulus costs bounded by the initial and overlap terms in
\eqref{fe:source-hypothesis}; the annular norm is the compact weighted
$L^2$ norm. Then the two energy multipliers give
\begin{equation}\label{fe:far-conclusion}
\mathsf F\ \le\ o(1)(\mathsf F+\mathsf S_a)
+C(R_0)\mathsf K_R+C_{\rm fix}(e_{\rm in}+\mathsf N^{1+c_2})
\end{equation}
(with \(c_2>0\), allowing analogous small input bootstrap errors). It suffices also to take tunably small uniform coefficients on \(\mathsf F\) in this argument. A convenient prior order for linear terms is: level \(P\) before the \(h\) hierarchies, within a level lower total orders first, more lab times first at fixed total order, and then fewer \(S\)'s first. Small radius gains and perturbative nonlinear effects may cross this order. Norms aggregated at equal indices are treated together.
\end{proposition}
\begin{proof}
\par\medskip\noindent\textbf{Claim 1: the corrected divergence identity.}\quad
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 \eqref{fe:F4}, \eqref{fe:F5} on a constant and is small times \(r^{-2}\), by \eqref{fe:F9}, \eqref{fe:F10} 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 \eqref{fe:F7} 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
\begin{equation}\label{fe:F16}
\frac{W_i}{r}\big((1+M)|\xi_X|^2+|\xi_{\rm scr}|^2\big)
\end{equation}
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 \eqref{fe:F9}, 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 \eqref{fe:F16}. 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
\eqref{fe:F12}, 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 \eqref{fe:F1} follows within this argument: \({\rm div}' X=O(r^{-2+.04})\), so integrate \({\rm div}'(W_i X|\chi|^2/r^2)\), using the \emph{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 \eqref{fe:F14} 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 \eqref{fe:F14}.
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 \eqref{fe:F14}, not claimed small by sign). The remaining wave source terms use exactly \eqref{fe:F14}, \eqref{fe:F16} and the mixed entry.
\par\medskip\noindent\textbf{Claim 2: the lab-time multiplier controls the bad derivatives.}\quad
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 \eqref{fe:F9}; 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 \eqref{fe:F14} 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 \(t\asymp R_f\), \eqref{fe:F6} 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 \eqref{fe:F14}; the signed leading term in \eqref{fe:F13} just uses the flat good/bad weight gap here.
\par\medskip\noindent\textbf{Claim 3: the flat commutators are triangular.}\quad
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 \eqref{fe:F13}. 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.
\par\medskip\noindent\textbf{Claim 4: the boundary costs use the stated inputs.}\quad
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 \(\mathsf S_a\), \emph{not} a large overlap coefficient times \(\mathsf S_a\). The required direct source estimate is proved in \Cref{fs:section}. 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.
\par\medskip\noindent\textbf{Claim 5: finite hierarchical absorption.}\quad
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_{ll}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 \eqref{fe:far-conclusion}. 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 \Cref{fs:section}, yielding \eqref{fs:F}. This proves the stated energy reduction.
\end{proof}
==== END Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//appendices/far-energy.tex ====
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//appendices/far-sources.tex ====
\section{Far-source estimates and uniform coupling}
\label{fs:section}
We verify all source and current-correction hypotheses of
\Cref{fe:energy-reduction} for the lab equation of
Lemma~\ref{far:structure} and its differentiated $D$ equation, on an
arbitrary smooth finite slab. In particular the background coefficients
have the stated symbol and $D$ gains, the exact lab gauge gives the
differentiated weak-null polarization, and
Lemma~\ref{far:boundary-input} supplies the initial and overlap bounds.
The quantities from \Cref{fe:section}, including the news estimate
\eqref{fe:F11}, are consequences of the same small bootstrap used here.
They are not assumptions about an already global evolution.
We write $N$ for a fixed multiple of the total bootstrap size, allowing
also the input size in this coefficient. The three target levels are
$(U_i,N_i)=(h,9),(h,8),(\psi,7)$ for $i=0,H,P$ respectively.
The coefficients, source splittings and current corrections below are
always expressed in Cartesian laboratory components before conversion
to the emitter frame.
The source budgets are \eqref{fe:F14} and the mixed charge
\eqref{fs:mixed-charge}, with the bulk measure
$\|\cdot\|_{\mathrm b}=L^2(dt\,dr\,d\omega/r)$ from
\Cref{fe:currents}. For each equation, $\mathcal P$ is the sum of its
corrections with their actual signs. The mixed charge uses only the
corrected $\xi_X$ cap in \eqref{fe:F12}; it requires no $M$-weighted
bound on an arbitrary uncorrected good derivative.
\subsection{Ordered derivatives and weight comparisons}
For a target level $i$ and $|I|+j\le N_i$, write
$\chi^i_{Ij}=Z^I(rT^jU_i)$ for the corresponding unknown of
\Cref{fe:coordinates}. Positive sums over spatial-emitter words up to
length $b$, with \emph{fixed} lab-time count $j$, will be used; define
lists, or sizes of lists, by
\begin{equation}
\label{fs:lists}
\begin{aligned}
V^U_{b j}&=|Z^{\le b}T^j U|,
&G^U_{b j}&=V^U_{b+1,j},\\
B^U_{b j}&=G^U_{b j}+r V^U_{b,j+1},
&&\qquad U=h,\psi .
\end{aligned}
\end{equation}
A superscript can equally be the level used. These letters with derivative indices do not denote the target or its velocity. In bulk \(W_i^{1/2}G^i_{b j},\ w_{i,j}^{1/2}B^i_{b j}\) are controlled by \eqref{fe:F1} with spatial counts \(\le b\). In uses of the time-slice cap, \(B^i_{b j}\) is bounded by \(C\sum_{|I|\le b}|B_1\chi_{I j}|\) plus \(C V^i_{b j}\); we will charge the latter separately there. Similar statements hold for \(G\) with \(G_1\). These facts follow by commuting \(Z\) at fixed \(\partial_q\), using the first derivative formulas. Point entries on spheres, in \(L^2_\omega\), include
\begin{equation}
\label{fs:G1}
\begin{array}{ll}
V^h_{b j}\lesssim N r^{-1+2\eta_*}\tau^{-A_H/2}&(b+j\le8),\\
V^h_{b j}\lesssim N r^{-.48}&(b+j\le9),\\
B^h_{b j},\ r V^\psi_{b j}\lesssim N w_H^{-1/2}&(b+j\le7),\\
B^\psi_{b j}\lesssim N w_H^{-1/2}&(b+j\le6).
\end{array}
\end{equation}
Use \eqref{fe:F7}, \eqref{fe:F8} (including expanding one more lab \(T\) before the \(Z\)'s for the last bound). Two additional rotations give sup versions. In particular low \(G^h\) has the bound of the first line, and low bad entries of either field have size \(\lesssim N\sqrt{1+M}\).
Some comparisons used below, for large \(r\), are
\begin{equation}
\label{fs:G2}
\begin{aligned}
\frac{W_i}{w_{k,j}}&\lesssim r^{p_i+.021}\tau^{A_i-A_k},
&w_{i,j}&\lesssim W_i,\\
\frac{w_{P,j}}{w_H}&\lesssim l^{A-A_H},
&\frac{w_H}{w_{P,j}}&\lesssim l^{-(A-A_H)+7d_*},\\
\frac{w_{0,j}}{w_H}&\lesssim r^{-.018}\mu^{-A_H}.
\end{aligned}
\end{equation}
Write \(m_{\rm cone}\) for the weight \(m=(r^{-1}+l^{-1})^{-1}\) of \Cref{fe:section}, to distinguish it from derivative counts denoted by \(m\) below. For \(i=0,P\), \(\sqrt{w_{i,j}/w_{i,j'}}\lesssim m_{\rm cone}^{9d_*/2}\), and for \(j'=j+1\) the bound is \(C m_{\rm cone}^{-d_*/2}\). Indeed \(q\ge-r\), \(m_-m_+\asymp m_{\rm cone}\). On \(|q|\le C r^\nu\) weights within each level are equivalent in \(j\), and \(w_{P,j}\asymp w_H\). Outside \(|q|\le r^\nu\),
\[
w_H^{-1/2}\lesssim r^{(e-(\nu-\delta)A_H)/2}l^{-A_H/2}.
\]
We use without further comment \(r\widetilde a\lesssim N\tau^{-A/2}\), \(|a(q)|\lesssim N\langle q\rangle^{-A/2}\) as coefficient bounds. When \(|q|\gtrsim r^\theta\), \(\theta=.45\), both \(m_{\rm cone}^{-d_*/2}\) and \(l^{(-(A-A_H)+7d_*)/2}\) gain more than \(r^{-3\eta_*}\).
Two useful bad-commutation comparisons, with \(c_0\) any bounded smooth coefficient of \(n,s(q)\), are
\begin{equation}
\label{fs:G3}
\begin{split}
|\partial_q(c_0\partial_q\chi^i_{I j})| \lesssim&
B^i_{b,j+1}+r^{-1}B^i_{b+1,j}+|a(q)|B^i_{b j},\qquad b=|I|,\\
B^h_{b,j+1}\lesssim&
B^\psi_{b j}+r^{-1}
\sum_{i'\le j,\ b'+i'\le b+j+1} B^h_{b' i'} .
\end{split}
\end{equation}
Indeed \(\partial_q(rT^j U)=krT^{j+1}U+s^d b_d Z T^j U\) and \(Z\) commutes with \(\partial_q\). For the second line express
\begin{equation}
\label{fs:G4}
\begin{aligned}
T^{j+1} h=(1-s_s\cdot n)^{-1}
\Bigl(&T^j\psi-r^{-1}s_s^d b_d Z T^j h\\
&-\sum_{i'0$, by
\begin{equation}
\label{fs:F15}
o_{R_0\to\infty}(1)(\mathsf F+\mathsf S_a)
+C\sum_{\mathrm{prior}}\mathsf F_{\mathrm{prior}}
+C(R_0)\mathsf K_R
+C_{\mathrm{fix}}(e_{\mathrm{in}}+\mathsf N^{1+c_1}).
\end{equation}
The finite prior order is: level $P$ before the two $h$ hierarchies;
within each level lower total order first, then more laboratory times
at equal total order, then fewer dilations $S$. Equal-index norms are
aggregated. The constants on prior far quantities are independent of
both matching radii. Small radial gains and nonlinear bootstrap
products may cross this order. After fixed finite hierarchical
weighting, \Cref{fe:energy-reduction} therefore gives, for some $c_2>0$,
\begin{equation}
\label{fs:F}
\mathsf F\le o_{R_0\to\infty}(1)(\mathsf F+\mathsf S_a)
+C(R_0)\mathsf K_R
+C_{\mathrm{fix}}(e_{\mathrm{in}}+\mathsf N^{1+c_2}).
\end{equation}
The initial and starting-annulus correction costs satisfy the same
bound. In particular the pure acceleration defect on the starting
annulus has its direct small radial coefficient on $\mathsf S_a$.
\end{proposition}
The following ledger identifies the origin and destination of every term.
The numbered claims retain the coefficient placements, corrections, and
endpoint estimates needed for each row.
\begin{center}
\small
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{.20\linewidth}
>{\raggedright\arraybackslash}p{.21\linewidth}
>{\raggedright\arraybackslash}p{.24\linewidth}
>{\raggedright\arraybackslash}p{.25\linewidth}@{}}
\toprule
Source & Derivative placement & Outgoing treatment & Raw and boundary checks \\
\midrule
Principal commutators & Coefficient counts $\le5$, $\ge6$;
exceptional count $9$ & Currents, direct gain, or mixed caps according
to the placement & \eqref{fs:G3}--\eqref{fs:G6}; Claim 1 \\[3pt]
Additional $D$ commutators & Seven target counts; $B^h$ through eight
& Projected gain and acceleration currents
& \eqref{fs:G7}; Claim 2 \\[3pt]
Pure background defect & Symbol derivatives at every base level
& Direct averaging gain, also on the starting annulus
& \eqref{fs:defect-shell-margins}; Claim 3 \\[3pt]
Linear and suppressed products & Nine counts for $h$; eight $h$ counts
or seven mixed counts for the strong equation
& Radial suppression in both bulk weights
& \eqref{fs:G9}; Claim 3 \\[3pt]
Leading screen quadratic & $h$ levels through $N_i$; strong correction
through eight $B^h$ counts
& Mixed cap for $h$; current \eqref{fs:G12} for $\psi$
& \eqref{fs:G10}--\eqref{fs:G12}; Claims 4--5 \\[3pt]
Correction boundaries & Same time weights; strong products use $2A_H>A$
& Sum signed corrections before forming $\xi$
& Initial symbols and off-trap overlap; Claim 6 \\
\bottomrule
\end{tabular}
\end{center}
\begin{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.
\par\medskip\noindent\textbf{Claim 1: common principal commutators.}\label{fs:claim-principal}\quad
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.
\smallskip\noindent\emph{Source forms and sphere placements.}
After the flat recurrence in Claim 3 of the proof of
Proposition~\ref{fe:energy-reduction}, every such term is as in \eqref{fe:F5} 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 \eqref{fe:F5}); 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'1$. The margin needed for the mixed shell charge is
\begin{equation}\label{fs:principal-shell-margin}
e+\delta A_H+\nu(1-A_H)
=.011\eta_*+.09\eta_*^2<.013\eta_*<1-p_H.
\end{equation}
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 $b_0\le8$.
\emph{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.
\par\medskip\noindent\textbf{Claim 2: the additional strong-equation terms.}\label{fs:claim-strong}\quad
Before treating \(f_h,D f_h\), the additional strong equation commutators arise already at order zero:
\begin{equation}
\label{fs:G7}
\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}
\end{equation}
In \eqref{fe:F5} 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 \eqref{fs:G4} but solved for \(T^m\psi\), and the projected estimate on \(T^{m+1}h\) in the proof of \eqref{fe:F9}, i.e. without a stationary background term since this is \(h\) itself; \(\Theta\) there costs \(O(r^{-1}\widetilde a)\). Quadratic inverse errors use \eqref{fs:G1}. The full \(r\)-gain on the projected coefficient therefore applies even through all the commutations of \eqref{fs:G7}. After multiplying by \(r\), the bad, mixed/good and \(a(q)\) terms of \eqref{fe:F5} now cost an acceptable amount directly in both raw weights: use \eqref{fs:G3} on \(h\) through total order eight for \(B\), weighted by \(w_H\); \eqref{fs:G2} 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 \eqref{fs:G7} costs \(O(N r^{-2+2\eta_*}\tau^{-A/2})\) in \emph{all} components through these coefficient orders, hence works directly. The last term of \eqref{fs:G7}, 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 \eqref{fs:G5}, \eqref{fs:G6} for the input \(h\) can equal \(n_*\le7\). In the time multiplier use \eqref{fs:G3} in full for \eqref{fs:G5}: 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 \eqref{fs:G2}, as do the ordinary mixed/good terms of \eqref{fs:G6}. In the outgoing multiplier use current correction \(C c_0\partial_q\chi^h_\ell\) as for \eqref{fs:G5}. Its \eqref{fe:F14} costs all use \eqref{fs:G2} and \(w_H\), with up to order eight by \eqref{fs:G3}, \(r d_g\lesssim r^{2\eta_*}\); 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 \eqref{fe:F5} (not the one inside \eqref{fs:G5}). 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 \eqref{fe:F5}). 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 \eqref{fs:G2}. 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 \eqref{fe:F14}. For its time-slice budget use \(G_1\chi_\ell\) with \(W_H\); any undifferentiated \(\chi_\ell/r\) term uses \eqref{fs:G1} directly with \(\widetilde a\), giving summability in \(L^2(dr)\) even with \(W_P^{1/2}\). This proves sufficiency for \eqref{fs:G7} except \(D f_h\).
\par\medskip\noindent\textbf{Claim 3: background, linear, and suppressed products.}\label{fs:claim-suppressed}\quad
We finish with the semilinear sources, i.e. linear combinations (from the flat recurrence, thus with constant scalars out front) of
\begin{equation}
\label{fs:G8}
Z^{c_3}(kr T^j f_h),\qquad Z^{c_3}(kr T^j D f_h),
\qquad c_3+j\le n_* .
\end{equation}
All product estimates allow indices \emph{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 \eqref{fe:F14}; 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
\begin{equation}\label{fs:acceleration-average}
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).
\end{equation}
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:
\begin{equation}\label{fs:average-bounds}
\|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.
\end{equation}
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{equation}\label{fs:defect-shell-margins}
\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}
\end{equation}
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 \eqref{fe:F1}, \eqref{fs:G2} 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 \(h\), \eqref{fs:G1} on the extra factors, and \eqref{fs:G2} 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 \eqref{fe:F3}; it does not require \(D r\) for the emitter radius to vanish.
Here are reusable quadratic product placements. The last term of \eqref{fe:F3} 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{equation}
\label{fs:G9}
\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}
\end{equation}
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 \eqref{fs:G1}, 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 \eqref{fs:G2}, \eqref{fs:G1}, \eqref{fs:G9} fits both weights with strict radial slack also in the order-nine coefficient case.
For strong estimates a similar sufficient suppressed product is \eqref{fs:G9} 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 \eqref{fs:G2} 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 \eqref{fs:G1}). 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 $B^\psi$.
\par\medskip\noindent\textbf{Claim 4: the leading weak-null source and its corrections.}\label{fs:claim-quadratic}\quad
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 \eqref{fe:gauge-projection} 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 \eqref{fe:F2} with bounded sphere coefficients. Here the lab differentiations in \eqref{fe:F2} 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
\begin{equation}
\label{fs:G10}
T^j Q_\eta
=r^{-2}\ell\otimes\ell\sum_{i+l=j}\mathcal C_{il}(F_i,F_l)
+{\rm Err}_j,
\end{equation}
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 \eqref{fs:G9} or explicit \(O(r^{-1})\) suppressions of that type, \(\widetilde a\,B^h\), or
\begin{equation}
\label{fs:G11}
G^h_{b_1 i_1}\,B^h_{b_2 i_2},\qquad
b_1+i_1+b_2+i_2\le n_* .
\end{equation}
Smooth bounded compositions are allowed on suppressed terms. These assertions follow from \eqref{fe:F2}, including \(r\Theta=O(\widetilde a)\). Terms with \(\widetilde a\) fit directly by \eqref{fs:G2}. For \eqref{fs:G11} 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 \eqref{fs:G10} 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 \eqref{fs:G11}, 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 \eqref{fs:G11} 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 \eqref{fs:G2}. Low placements here have at most four derivatives before the two spare rotations. This proves acceptability of all the remainders of \eqref{fs:G10}.
Consider the outgoing budget of the first term of \eqref{fs:G10} 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 \eqref{fs:G11}. 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 \eqref{fs:G1}. Both costs give shell summability with the indicated \(p_{i_{\rm lev}}\).
The squared mixed-shell exponents just obtained are
\begin{equation}
\label{fs:quadratic-shell-margins}
\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}
\end{equation}
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 \eqref{fs:G10}, use the correction (one for each term with its actual \(\mathcal C_{il}\))
\begin{equation}
\label{fs:G12}
\mathcal P_*= Z^{c_3}\!\left[
(1-s_s\cdot n)r^{-1}\ell\otimes\ell\,\mathcal C_{il}(F_i,F_l)
\right].
\end{equation}
Indeed \(kD=(1-s_s\cdot n)\partial_q+r^{-1}d' Z\), \(k\) here being the one in \eqref{fs:G8}; \(\partial_q\) 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 \eqref{fe:F14}, 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 \eqref{fs:G12} 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 \eqref{fs:G1} 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 \(L^2(dr)\).
\par\medskip\noindent\textbf{Claim 5: raw leading products on the finite slab.}\label{fs:claim-raw}\quad
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 \eqref{fe:F11}: \(w_{i_{\rm lev},j}/w_{i_{\rm lev},j'}\lesssim l^{9d_*}\) 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 \eqref{fs:G1}, \eqref{fs:G2}. 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
\begin{equation}\label{fs:tile-measure}
t=q+e^\rho,\qquad \frac{dt\,dr}{r}=dq\,d\rho.
\end{equation}
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 \eqref{fe:F11}.
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.
\par\medskip\noindent\textbf{Claim 6: initial and starting-annulus costs.}\label{fs:claim-boundary}\quad
All corrections are added before forming \(\xi\) for each equation; the bounds concern \eqref{fe:F1} 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: \eqref{fs:G5} 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 \eqref{fs:G7} smooth-acceleration corrections the coefficient carries \(\widetilde a\). The products in \eqref{fs:G12} 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 \eqref{fs:F15} and therefore \eqref{fs:F}, with every positive-radius-gain term summable and the prior linear costs taken with independent far constants as specified there.
\end{proof}
\subsection{Simultaneous exterior closure}
The compact, far, and acceleration estimates concern the same finite
smooth slab. Write $K=\mathsf K_R$, $F=\mathsf F$, and
$a_*=\mathsf S_a$, and collect the fixed data and superlinear
remainders in $E$. The three estimates \Cref{co2:compact-high}, \eqref{fs:F},
and \Cref{co2:sliced-acceleration} have the form
\begin{equation}\label{fs:coupled-inequalities}
\begin{split}
K&\le\varepsilon_R(K+F)+E,\\
F&\le\varepsilon_0(F+a_*)+B_0K+E,\\
a_*&\le D_0K+C_0F+E .
\end{split}
\end{equation}
Here $\varepsilon_0$ becomes small by increasing $R_0$, whereas
$\varepsilon_R$ is made small later by the compact matching radius
and plateau choices. To choose them in this order, we need $C_0$
to stay bounded as $R_0$ grows. The next lemma proves exactly that
fact and the compatible treatment of the pure acceleration defect.
\begin{lemma}[Radius-uniform coupling]
\label{fs:uniform-coupling}
The coefficient of $\mathsf F$ in \Cref{co2:sliced-acceleration} may be taken independent
of $R_0$. The comparison of the background defect $f_B$ with $Q(0)a$
may be split at a radius $Y$ fixed before $R_0$: its compact part costs
$C(Y)\mathsf K_R$ and fixed data/nonlinear terms, and its tail costs
$C Y^{-1+\epsilon}\mathsf S_a$, with any sufficiently small fixed
$\epsilon>0$. Conversely, the pure far defect, including the starting
annulus, has an $o_{R_0\to\infty}(1)$ coefficient on $\mathsf S_a$ in
\eqref{fs:F}.
\end{lemma}
\begin{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{equation}
\label{fs:slice-jacobian}
\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}
\end{equation}
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
\Cref{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 \eqref{fs:slice-jacobian} 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 \eqref{fe:F1} 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
\Cref{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 \eqref{fs:acceleration-average}.
On the cutoff $\chi_t$, $t$ is a sufficiently large fixed multiple
of $\rho$, so \eqref{fs:average-bounds} 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 \Cref{fs:source-estimate} 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.
\end{proof}
\begin{corollary}[Global exterior estimates from the base neighborhood]
\label{fs:closure}
There is a neighborhood specified solely by sufficiently small
$p_{10}$ difference from the Kerr bridge such that the two reduced
computing-cylinder evolutions are global and smooth. They obey
\Cref{co2:low-bound,co2:sliced-acceleration,co2:sliced-static,co2:compact-high} and \eqref{fs:F}, with arbitrarily
small total base bounds after the neighborhood is sufficiently
shrunk. The higher smooth norms on growing cylinders have the
subexponential bounds of \Cref{co2:higher}; their constants may depend
on the fixed smooth datum and the order. No smallness of those
higher initial norms is required.
\end{corollary}
\begin{proof}
Proposition~\ref{far:closure} 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~\ref{co2:higher},
whose induction uses finite higher initial seminorms without shrinking
the common base neighborhood.
\end{proof}
==== END Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//appendices/far-sources.tex ====
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//appendices/triangular.tex ====
\section{A triangular comparison with one common rate}
\label{app:triangular}
\begin{lemma}[A finite triangular system with one common rate]
\label{pk:triangular-comparison}
Fix an integer $k\ge0$ and $C_0>0$. For each $T\ge1$, let
$00\) and \(0<\mathfrak a0\) and \(0<\mathfrak a0\) such that the data in
\[
\mathcal U_\epsilon
=\{d\in\mathcal D:p_{10}(d-(h_*,K_*))<\epsilon\}
\]
whose full maximal globally hyperbolic vacuum developments admit a
\(C^2\) future extension form a meagre subset of \(\mathcal U_\epsilon\).
Equivalently, the future-inextendible data contain a countable intersection
of relatively open dense subsets of \(\mathcal U_\epsilon\).
\end{theorem}
Thus \Cref{thm:main} resolves positively the stated local \(C^2\)
strong cosmic censorship problem at every strictly subextremal rotating
Kerr bridge. It is not a global assertion for all vacuum initial data,
nor a claim at a lower extension regularity. Constants and
\(\epsilon\) may depend on the fixed center; no estimate uniform as
\(\mathfrak a\to0\) or \(\mathfrak a\to M\) is claimed. Although
all higher seminorms are finite, none beyond \(p_{10}\) must be small.
\subsection{History and relation to earlier work}
\input{sections/history-fragment}
\subsection{From regular exits to finite observations}
The proof turns a possible extension into a finite condition that can
be violated by a small vacuum perturbation. A \(C^2\) extension supplies
an open family of timelike geodesics whose proper durations and curvature
components in parallel frames are bounded. A countable basis of initial
geodesic states and integer bounds give countably many such tests.
Each test is closed: a strict violation occurs on a compact geodesic
segment and persists under a sufficiently small change of the data.
Consequently, to prove meagreness, it suffices to violate the relevant
test arbitrarily close to every datum admitting a regular exit.
\Cref{sec:category} makes this reduction precise. Its decisive feature
is that one large tidal measurement at a finite point suffices; the
perturbation need not establish an asymptotic radiation law for the
starting datum.
We obtain that measurement from the blueshift of a small oscillatory
signal approaching an interior branch. In the optical coordinates used
below, a branch means a limiting side where one coordinate remains finite
and the other tends to infinity; at a corner both tend to infinity.
These are coordinate-limit regimes of the comparison region.
The main analytic work has two
parts. First we construct a comparison development and control its
geometry throughout one fixed neighborhood of Kerr data. Then, for a
particular datum and a prescribed finite approximation order, we place
a late signal in that geometry and realize it in the full development
of nearby exact vacuum data. The distinction matters: the time and
frequency of the experiment may depend on the chosen finite order,
whereas the original \(p_{10}\) neighborhood must serve all tests.
\paragraph{Exterior control and the ergoregion.}
The exterior construction combines stationary estimates in a bounded
region with energy estimates on the asymptotically flat ends.
Stationary time translation need not be timelike near all trapped rays
at arbitrary spin. We instead separate directions in the cotangent
bundle. \Cref{sec:geometry} constructs a horizon-crossing spacelike
slicing for which the spatial projections of the trapped null
covectors lie in a cone where the zero-temporal-frequency stationary
symbol is elliptic. This directional ellipticity persists even when
the base point lies in the ergoregion. \Cref{sec:stationary} uses the
cone to localize the additional gauge terms and calibrate tensor
transport while preserving the stationary modes and their
time-translation chains. The stationary analysis of
\cite{HHV2025} and the normally hyperbolic propagation estimates of
\cite{Dyatlov2016,Hintz2017} enter at the points specified there.
The compact estimate in \Cref{sec:compact} couples the two-derivative
elliptic gain on that cone to a one-derivative propagation estimate,
with one additional source derivative, away from the projected trapped
sets. On the ends, the weak-null cancellations and weighted energy
currents control the arbitrary symbol-bounded tails in
\eqref{intro:seminorms}. \Cref{sec:far} closes these far estimates
simultaneously with the compact estimates and the evolution of the
Kerr parameters. The detailed far energy and source arguments are in
\Cref{fe:section,fs:section}. All of these are estimates on an arbitrary
finite interval of smooth existence, with bounds that permit
continuation; a global nonlinear stability statement for this data
class is not assumed. Once the low-order bounds close, each fixed
higher derivative order has a subexponential bound with constants
depending on the smooth datum.
\paragraph{The interior and every possible exit.}
\Cref{sec:interior} prepares the data entering the black hole and
constructs a double-null comparison region. Exact Kerr optical
profiles, with their two ends normalized to a common positive inner
rate, organize the lapse, angular metric, and transport equations.
The low-order estimates control the geometry; the higher estimates
control the finite derivatives needed later by the oscillatory
construction. \Cref{sec:comparison} joins this region to the exterior
and proves that the entire original bridge is Cauchy. Geometric
uniqueness therefore embeds the resulting comparison development in
the full maximal development of the same data.
We then follow the geodesic family supplied by a putative extension.
Exterior completeness and the causal geometry force its relevant
first exit from the comparison region to occur at an interior branch
or in the corner regime. In the corner case, a small
change of initial geodesic state within the same open family produces
runout at a noncorner branch of the comparison region. Thus the original
closed curvature test still applies
to the selected geodesic. This step connects the local interior
coordinates to an arbitrary future extension of the full development;
it does not require the comparison region to exhaust that development.
\paragraph{A small exact vacuum perturbation with a large tidal effect.}
\Cref{sec:packet} constructs a high-frequency signal near a late point
of the selected branch. The branch geodesic measures an exponentially
increasing optical frequency. Its contraction with the oscillatory
curvature therefore yields a large tidal component even when the
signal is very small at entry. A return-ray calculation and a complex
Gaussian beam carry the required finite entry derivatives backwards
to a distant initial annulus, with controlled losses. The subsequent
forward comparison uses ordinary local wave-map reduction with the
actual background metric as target.
An approximate signal does not yet give vacuum initial data.
The exterior solve proved in \Cref{pk:constraints} and applied in
\Cref{pk2:constraints} supplies an exact correction that vanishes on a
protected inner ball and retains a decaying tail at infinity. The tail
allows the asymptotic charges to change and is included in the complete
corrected bridge. A causal estimate shows that this correction cannot
alter the earlier entry segment on which the comparison relies.
Finally, the finite Cauchy attachment in \Cref{sec:packet} places the
tidal observation, its frame, and its geodesic back to the original
initial surface in the full development of the corrected data.
The resulting strict violation makes the corresponding extendible
set nowhere dense. The countable union from \Cref{sec:category}
completes the proof.
Figure~\ref{fig:proof} displays the two constructions and the point
at which they meet. The fixed background neighborhood is chosen before
the test or approximation order. The finite derivative orders and
their losses are chosen before the small frequency exponent; the
formal accuracy and late time are then chosen to achieve the specified
tolerance. In particular, the construction requires neither smallness
of all higher seminorms nor one perturbation sequence converging in
every seminorm at once.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
box/.style={draw,align=center,text width=4.8cm,inner sep=7pt,font=\small},
arr/.style={-{Stealth[length=2mm]},thick},node distance=1.05cm and 1.3cm]
\node[box] (base) {One small \(p_{10}\) neighborhood\\all higher seminorms finite};
\node[box,below=of base] (comparison) {Background comparison\\development\\complete original bridge\\Cauchy};
\node[box,below=of comparison] (branch) {Branch seed in the original open family\\closed test bounds parallel curvature};
\node[box,right=of branch] (signal) {Late finite tidal experiment\\parameters depend on a finite order};
\node[box,above=of signal] (corrected) {Exact corrected vacuum data\\arbitrarily close in that order};
\node[box,above=of corrected] (mg) {Finite Cauchy attachment\\observation in the new full development};
\draw[arr] (base)--(comparison);
\draw[arr] (comparison)--(branch);
\draw[arr] (branch)--(signal);
\draw[arr] (signal)--(corrected);
\draw[arr] (corrected)--(mg);
\end{tikzpicture}
\caption{The comparison region locates a possible regular exit. The
separate finite experiment violates its closed curvature test in the
full development of nearby data. Neither an exhaustive description
of that development nor smallness at every derivative order is needed.}
\label{fig:proof}
\end{figure}
The intermediate results also separate the evolution problem from
the regularity of a proposed extension. The complete comparison
development, the entry and normalized double-null estimates, the
linear beam preparation, the exterior constraint inverse, and the
whole-data attachment can be used when the final obstruction is
holonomy rather than curvature. For \(C^1\) extensions and for
continuous extensions with locally square-integrable connection,
one must additionally control signed or averaged parallel transport.
Those uses require their own nonlinear amplitude estimates: the
tidal normalization in this paper does not supply the larger-amplitude
comparisons needed for these weaker regularities.
These regularity refinements are established in
\cite[Theorem~2.3]{KerrCOne} and
\cite[Theorem~1.2]{KerrWeakConnection}, respectively, as local residual
statements near each fixed rotating subextremal Kerr bridge.
\subsection{Scaling and conventions}
It suffices to prove the theorem with \(M=1\). Under the constant
rescaling \(g\mapsto M^{-2}g\), the induced covariant data become
\((M^{-2}h,M^{-1}K)\). After identifying the bridge by dimensionless
Kerr coordinates, this is a homeomorphism of the constraint spaces;
the background connections agree and the tensor norms and radial
weights change by fixed-order comparable factors. Completeness,
vacuum maximality and the extension property are preserved. An
original \(p_{10}\) ball can therefore be chosen inside the inverse
image of the rescaled neighborhood, and relative meagreness restricts
to that open ball. We retain \(M\) in geometric identities when doing
so clarifies their dependence on the background.
We write \(\mathfrak a\) for Kerr rotation when another local quantity
is called \(a\), notably a parameter derivative in exterior estimates
or a null lapse coefficient in the interior. All spacetime and
angular Sobolev norms are computed in the specified fixed charts or
frames; on fixed supports, smooth changes give equivalent norms.
Supported and extendible spaces at an outflow boundary are dual in
the usual way. Symbol bounds on an end include all differentiated
bounds, with an additional inverse radial power for each unscaled
spatial derivative. A bound \(e^{o(T)}\) means that for every
\(\sigma>0\) the quantity is at most \(C_\sigma e^{\sigma T}\);
the constant may depend on the smooth datum and on a fixed derivative
order. These conventions will always be used at fixed subextremal
parameters.
==== END Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/01-introduction.tex ====
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/02-category.tex ====
\section{Closed tests and the finite-observation reduction}\label{sec:category}
We first separate the topological argument from the evolution estimates.
The facts in this section are independent of the Kerr parameters. We
include their short proofs to make precise why a finite tidal
observation suffices, and why no regular endpoint tangent is part of
the extension hypothesis.
\begin{lemma}[The data topology]\label{cat:baire}
The space \(\mathcal D\) is completely metrizable and
\(\mathcal U_\epsilon\) is Baire. Every neighborhood of a datum contains
a ball \(p_m(d'-d)<\eta\) relative to \(\mathcal D\), for some finite
\(m\) and \(\eta>0\).
\end{lemma}
\begin{proof}
A sequence Cauchy in every \(p_m\) has compatible smooth limits on
compact coordinate sets. Passing to the pointwise limit in its weighted
uniform Cauchy bounds proves convergence to that limit in every
\(p_m\), including the finiteness of each limiting seminorm. Thus
\(E\) is a Fr\'echet space. For example its topology is defined by
the complete metric
\(\sum_{m\geq0}2^{-m-1}\min\{1,p_m(d-d')\}\).
Any positive metric in the affine space is uniformly comparable to
\(h_*\): this follows from the decaying difference on the ends and
compactness on their complement. Hence it is complete, and positivity
is open already in the \(p_0\) topology. The constraint equations are
closed under local smooth convergence within this positive open set.
An open subset of a completely metrizable space is completely
metrizable, as is a relatively closed subset thereof. Consequently
\(\mathcal D\), and then its open subset \(\mathcal U_\epsilon\),
are completely metrizable and Baire. Finally the seminorms increase
with \(m\), so one sufficiently small ball in the largest of finitely
many orders lies inside any basic neighborhood.
\end{proof}
\subsection{Seeds and curvature measurements}
A seed \(w\in T\Sigma\) includes its base point. For datum
\(d=(h,K)\), its initial future unit velocity is
\[
U_d(w)=\sqrt{1+h(w,w)}\,n+w,
\]
where \(n\) is the future unit normal. Let
\(\gamma_{d,w}:[0,T_d(w))\to M_d\) be the maximal future geodesic
with this initial velocity, parametrized by proper time. Along it
parallel transport the positive inner product \(1\oplus h\) on
\(\mathbb R n\oplus T\Sigma\), and call the result \(k_{d,w}\).
Set
\begin{equation}\label{cat:Q}
Q_d(w,\tau)=|\operatorname{Riem}(g_d)|_{k_{d,w}(\tau)}.
\end{equation}
The curvature tensor is covariant and the norm is its positive
Hilbert--Schmidt norm, independently of any initial orthonormal frame.
Choose a countable relatively compact open basis \(\mathscr O\) of
\(T\Sigma\). For \(O\in\mathscr O\) and an integer \(N\geq1\), define
\begin{equation}\label{cat:test}
A(O,N)=\left\{d\in\mathcal D:
\begin{array}{ll}
T_d(w)\leq N &\text{for every }w\in O,\\
Q_d(w,\tau)\leq N &\text{for every }w\in O,
\ 0\leq\tauN\), use a
geodesic segment extending beyond \(N\). If a curvature bound fails,
use a segment through a point \(\tauN\). 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.
\end{proof}
\begin{lemma}[Tidal contractions]\label{cat:tidal}
If \(U=\dot\gamma_{d,w}\), \(g_d(X,X)=1\), and \(g_d(U,X)=0\), then
\begin{equation}\label{cat:tidal-bound}
|\operatorname{Riem}(U,X,U,X)|
\leq (1+2h(w,w))^2 Q_d(w,\tau).
\end{equation}
The factor is uniformly bounded for data and seeds near a fixed pair.
\end{lemma}
\begin{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.
\end{proof}
\subsection{From an arbitrary extension to regular exits}
We will use an intrinsic compactness fact. It avoids imposing causal
convexity on the image in an extension.
\begin{lemma}[Fixed-endpoint homotopy]\label{cat:homotopy}
Let \(P\) be a globally hyperbolic open subspacetime of a Hausdorff
Lorentzian manifold. In a continuous family of future causal paths
with fixed endpoints \(p,q\in P\), if the first path lies in \(P\),
then all paths lie in \(P\).
\end{lemma}
\begin{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.
\end{proof}
\begin{proposition}[Regular-exit covering]\label{cat:regular-exit}
If \(d\in\mathcal U_\epsilon\) admits a \(C^2\) future extension,
there exist \(O\in\mathscr O\) and \(N\geq1\) such that
\(d\in A(O,N)\) and every seed in \(O\) reaches its finite MGHD
endpoint on a regular ambient geodesic arc. Here regular means that
position and unit tangent continue to and through the endpoint.
\end{proposition}
\begin{proof}
Starting from \Cref{intro:extension}, 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.
\Cref{cat:homotopy} 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
\eqref{cat:Q} 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.
\end{proof}
Let \(B_\epsilon\) denote the extendible data in
\(\mathcal U_\epsilon\). Define \(\mathcal B(O,N)\) to consist of
the data in \(B_\epsilon\cap A(O,N)\) for which at least one seed
in \(O\) has a regular exit in some extension. Then
\begin{equation}\label{cat:cover}
B_\epsilon=\bigcup_{O\in\mathscr O,\,N\geq1}\mathcal B(O,N),
\qquad
\overline{\mathcal B(O,N)}^{\,\mathcal U_\epsilon}
\subset A(O,N)\cap\mathcal U_\epsilon.
\end{equation}
The pieces \(\mathcal B(O,N)\) themselves need not be closed.
\subsection{The analytic task}
\begin{proposition}[Finite-observation criterion]\label{cat:destruction}
Suppose a single \(\epsilon>0\) has the following property. For every
\(O\in\mathscr O\), \(N\geq1\), \(d\in\mathcal B(O,N)\),
integer \(m\geq10\) and \(\eta>0\), there exist
\(d'\in\mathcal U_\epsilon\), \(w'\in O\), and
\(0\leq\tauN.
\]
Then \Cref{thm:main} holds for this \(\epsilon\).
\end{proposition}
\begin{proof}
If the relative closure of \(\mathcal B(O,N)\) had nonempty interior,
that interior would meet the piece and, by \eqref{cat:cover}, 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 \eqref{cat:cover}.
\end{proof}
All choices needed to fix the neighborhood in
\Cref{cat:destruction} will precede the choice of
\(O,N,m,\eta\). The exterior and interior estimates construct the
background comparison region at this common smallness threshold.
Higher constants may then depend on the fixed smooth datum. Only
afterwards do we choose the late finite perturbation that violates
the selected test.
==== END Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/02-category.tex ====
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/03-geometry.tex ====
\section{Kerr geometry and spatial microlocal separation}
\label{sec:geometry}
The exterior construction requires a spacelike stationary slicing, a
description of all null rays that can remain in a bounded cylinder, and a
spatial conic neighborhood of trapping on which the stationary operator is
elliptic at zero frequency. We establish these properties for every fixed
subextremal Kerr metric. Although the mass has been normalized to $M=1$ in
the main argument, we retain $M$ in the identities below to display their
parameter dependence. Throughout this section,
\[
M>0,\qquad 0<\mathfrak a0$ and both numbers $\kappa_\pm$ are positive; at the
inner horizon we use the absolute value of the surface gravity.
The exterior computing cylinder will extend to an inner face $r=r_b$ with
$r_-r_+$,
one can choose a stationary time
\begin{equation}
t=T+r_*-G(r)
\label{geo:time}
\end{equation}
with the following properties. One has $G'=1$ on the prescribed region,
$1\leq G'\leq r_*'$ in a subsequent exterior interpolation, and $G=r_*$
near infinity. The function $t$ is regular at the future event horizon,
its level hypersurfaces are spacelike, and $r=r_b$ is strictly future
outflow. The delay $G-r_*$ at each fixed depth above $r_-$ can be made
arbitrarily large by moving the exterior bend outward.
\end{lemma}
\begin{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
\begin{equation}
\Sigma_K b^{-1}(dt,dt)
=-2(r^2+\mathfrak a^2)G'+\Delta(G')^2
+\mathfrak a^2\sin^2\theta.
\label{geo:time-norm}
\end{equation}
For $G'=1$, the right-hand side is
\begin{equation}
-\Sigma_K-2Mr<0.
\label{geo:time-norm-one}
\end{equation}
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.
\end{proof}
As in \Cref{cmp:bent-time}, the delays place the $t=0$ computing slices on
or to the future of the bridge and separate their trapped-block portions
by the required finite margins. The order of choices is important:
first choose the entry depth and these margins, then choose $r_b$, and
finally move the bend far enough outward. Lemma~\ref{geo:slicing} permits
this order without altering the slicing on the prescribed interval.
After proving compactness of the trapping radii below, we take $R_1$
larger than all of them.
Near infinity $t=T$, so all components and inverse components have the
ordinary stationary Kerr symbol expansions. In particular,
\begin{equation}
\begin{gathered}
b-\eta=O(r^{-1}),\qquad b^{ti}=O(r^{-2}),\qquad
\Gamma(b)=O(r^{-2}),\\
r_*=r+2M\log r+O(1),\\
b_{tt}=-1+\frac{2M}{r}+O(r^{-2}),\qquad
b_{ij}=\delta_{ij}+\frac{2M y_i y_j}{r^3}+O(r^{-2}).
\end{gathered}
\label{geo:end-expansions}
\end{equation}
Here $\eta$ is the Minkowski metric, and every remainder includes its
differentiated symbol bounds. The terms of order $r^{-1}$ are rotation
invariant. The leading term of $b_{ti}$ is a nonzero constant multiple of
$(\mathbf J\times y)_i/r^3$, where
$\mathbf J=M\boldsymbol{\mathfrak a}$ and
$\boldsymbol{\mathfrak a}$ is the oriented spin vector in the chosen
axes. These assertions follow by expanding the Boyer--Lindquist metric;
the angular expressions and their rotations are smooth on the sphere.
\subsection{Radial dynamics and the horizon source}
We now classify the null rays in a large computing cylinder. A future
null covector means a nonzero null covector $k$ whose metric dual
$k^\sharp$ is future directed. Set
\begin{equation}
\begin{gathered}
E=-k(\partial_t)=-k(\partial_T),\qquad
L_z=k(\partial_\varphi),\\
D=k_\theta^2+
\left(\frac{L_z}{\sin\theta}
-\mathfrak a E\sin\theta\right)^2,\qquad
P=(r^2+\mathfrak a^2)E-\mathfrak a L_z.
\end{gathered}
\label{geo:constants}
\end{equation}
The angular expression is understood intrinsically at the poles: its
expansion is the smooth round cotangent norm, minus
$2\mathfrak a E L_z$, plus $\mathfrak a^2E^2\sin^2\theta$.
The separated principal symbol in Boyer--Lindquist variables is
\begin{equation}
q_E:=\Sigma_K b^{-1}(k,k)
=\Delta k_r^2+D-\frac{P^2}{\Delta}.
\label{geo:separated-symbol}
\end{equation}
Stationarity, axial symmetry, and separation show that $E,L_z,D$ are
constant along null rays. With a dot denoting the affine geodesic
parameter, the corresponding equations are
\begin{equation}
\begin{aligned}
(\Sigma_K\dot r)^2&=P^2-\Delta D,\\
\Sigma_K\dot T&=
\frac{(r^2+\mathfrak a^2)P}{\Delta}
+\mathfrak a(L_z-\mathfrak a E\sin^2\theta).
\end{aligned}
\label{geo:ray-equations}
\end{equation}
These are Carter's separated geodesic equations
\cite{Carter1968}, in the conventions used for the characteristic-flow
calculation below.
\begin{lemma}[Radial alternatives]
\label{geo:radial}
For the fixed parameters above, future null rays in $r>r_+$ have $P>0$.
If $E\leq0$, their allowed radial set has at most one simple turning
point, and there is no photon trapping. If $E>0$, every trapped ray is a
spherical photon ray, with constant radius satisfying
\begin{equation}
F=\frac{2r\Delta}{r-M},\qquad
\frac{D}{E^2}=\frac{4r^2\Delta}{(r-M)^2},\qquad
F:=\frac PE=r^2+\mathfrak a^2-\mathfrak a\frac{L_z}{E}.
\label{geo:double-root}
\end{equation}
The energy-normalized trapped directions form a compact set separated
from $r=r_+$ and from infinity. The relevant radial critical point is a
unique nondegenerate minimum. A simple radial turning point reverses
toward the same end of its allowed interval, and there is no homoclinic
orbit from a double radial root back to that root.
\end{lemma}
\begin{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
\begin{equation}
\left(\frac{F}{\sqrt\Delta}\right)'
=\frac{N}{\Delta^{3/2}},\qquad
N=2r\Delta-(r-M)F,
\label{geo:radial-derivative}
\end{equation}
and
\begin{equation}
N'=2\Delta+2r(r-M)-F,\qquad N''=6(r-M)>0.
\label{geo:radial-convexity}
\end{equation}
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~\eqref{geo:radial-convexity} 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 \eqref{geo:double-root}. Moreover,
\begin{equation}
\left|\frac{L_z}{E}\right|-\mathfrak a
\leq\frac{\sqrt D}{E}.
\label{geo:impact-bound}
\end{equation}
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 \eqref{geo:impact-bound}. They cannot approach infinity
either: \eqref{geo:double-root} 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.
\end{proof}
\begin{lemma}[The outer-horizon radial source]
\label{geo:horizon-source}
The outer-horizon generators are a strict radial source for the null
ray flow parameterized by increasing regular time $t$ on spatial
cosphere directions. More precisely, let $H$ be the horizon Killing
generator, normalized by its Killing parameter, so that
\[
\nabla_HH=\kappa_+H,\qquad H^t>0.
\]
There is a smooth positive degree-one frequency scale $b_1$, equal to
$|\rho|$ on $k=\rho H^\flat$, such that, for the increasing-time ray
field $V$,
\begin{equation}
V\log b_1=-\frac{\kappa_+}{H^t}
\quad\hbox{on the generators}.
\label{geo:source-frequency}
\end{equation}
On null directions near the generators, the degree-zero function
\begin{equation}
f_H=\frac{E^2+D}{b_1^2}
\label{geo:source-function}
\end{equation}
vanishes precisely at the generators and satisfies
$Vf_H\geq c_H f_H$ for some $c_H>0$ after shrinking the neighborhood.
\end{lemma}
\begin{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 \eqref{geo:source-frequency}. 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.
\end{proof}
Together, Lemmas~\ref{geo:radial} and \ref{geo:horizon-source} give the
escape alternatives used below. Between the horizons $dr$ is timelike,
so every future ray there moves toward the inner outflow face. In a
sufficiently large exterior cylinder, a ray away from trapping and the
generators escapes in at least one time direction, or approaches the
generator source in backward regular time. An orbit approaching
$r=r_+$ without a transverse crossing has a normalized invariant limit
set tangent to that horizon, and hence consisting of generators.
The radial alternatives exclude any additional recurrent set.
\subsection{Normal hyperbolicity and tangent dynamics}
The radial classification identifies the only exterior obstruction to
escape. The smooth symplectic structure and normal hyperbolicity of
the subextremal Kerr trapped set are established in
\cite[Section~II]{DyatlovZworski2013}. For the estimates below we give
explicit defining functions for its stable and unstable surfaces and
verify that their exponential rates dominate every fixed-order
tangent derivative of the trapped flow.
\begin{lemma}[Transverse defining functions]
\label{geo:normal-functions}
Near the energy-normalized trapped set on either temporal sheet, the
critical radius $r_c=r_c(L_z/E)$ is smooth. There are smooth homogeneous
degree-zero functions
\begin{equation}
\begin{aligned}
z_1&=\frac{\Sigma_K\dot r}{E\sqrt\Delta},\\
z_2&=\operatorname{sgn}(r-r_c)
\sqrt{\frac{F(r)^2}{\Delta(r)}
-\frac{F(r_c)^2}{\Delta(r_c)}}
\end{aligned}
\label{geo:normal-variables}
\end{equation}
whose common zero set is trapping. For the increasing-time ray field,
\begin{equation}
Vz_2=\lambda z_1,\qquad Vz_1=\lambda z_2,
\qquad \lambda>0.
\label{geo:normal-system}
\end{equation}
Consequently $\varphi_\pm=z_1\pm z_2$ satisfy
$V\varphi_\pm=\pm\lambda\varphi_\pm$. Their reduced spatial
symplectic bracket on either halfwave is nonzero on the trapped set.
The transverse rates have a uniform positive lower bound on each
fixed normalized trapped set.
\end{lemma}
\begin{proof}
The strict minimum established in Lemma~\ref{geo:radial} 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, \eqref{geo:ray-equations} gives
\begin{equation}
z_1^2-z_2^2=U_c-\frac{D}{E^2},
\label{geo:normal-invariant}
\end{equation}
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 \eqref{geo:normal-invariant} gives the second equation
of \eqref{geo:normal-system} 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,
\begin{equation}
\{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,
\label{geo:root-bracket}
\end{equation}
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 \eqref{geo:normal-system}. Thus the nonzero
fixed-energy bracket survives on each halfwave. Since
$\{\varphi_+,\varphi_-\}=\pm2\{z_1,z_2\}$, the assertion follows.
\end{proof}
\begin{lemma}[Polynomial growth of tangent derivatives]
\label{geo:tangent-growth}
Fix $E\neq0$. On a neighborhood of the null trapped set in the radial
critical manifold
\[
r=r_c(L_z/E),\qquad k_r=0,
\]
the flow of the separated fixed-frequency symbol has polynomial
growth of every fixed-order tangent derivative. The bounds are
uniform on compact invariant subsets, include variation through a
sufficiently small interval of nearby symbol levels, and persist
under positive smooth time changes with speed bounded above and
away from zero. In particular, they hold in the regular stationary
time $t$, on either sheet with its increasing-time orientation.
\end{lemma}
\begin{proof}
We first establish a strict angular margin on the null trapped set.
There $D=P^2/\Delta$ and $\Delta\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}
\label{geo:angular-margin}
\end{equation}
Thus
\begin{equation}
c_1=D+2\mathfrak a E L_z-2\mathfrak a^2E^2>0.
\label{geo:oscillator-coefficient}
\end{equation}
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 \eqref{geo:separated-symbol} 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{equation}
\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}
\label{geo:oscillator}
\end{equation}
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
\begin{equation}
d_1^2=\frac{2Q}{c_1+\sqrt{c_1^2+4AQ}}.
\label{geo:amplitude}
\end{equation}
This formula is smooth even at $Q=0$. Substituting
$x_1=d_1\sin u$ gives the positive invariant period
\begin{equation}
T_{\mathrm{ang}}(D,L_z)
=\int_0^{2\pi}
\frac{du}{\sqrt{c_1+Ad_1^2(1+\sin^2u)}}.
\label{geo:angular-period}
\end{equation}
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
\begin{equation}
\Phi^{T_{\mathrm{ang}}(z)}z=R_{\alpha(z)}z.
\label{geo:angular-return}
\end{equation}
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
\begin{equation}
\Phi^s z=R_{\alpha(z)^n}
\Phi^{s-nT_{\mathrm{ang}}(z)}z.
\label{geo:angular-iterate}
\end{equation}
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
\begin{equation}
\|D_z^j\Phi^s\|\leq C_j(1+|s|)^{N_j}.
\label{geo:polynomial-tangent}
\end{equation}
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 00$, allowing $D$ to vary. On a nearby level of
\eqref{geo:separated-symbol}, 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.
\end{proof}
\begin{proposition}[Normally hyperbolic trapping with escape]
\label{geo:normal-hyperbolicity}
For each fixed subextremal Kerr metric, the energy-normalized photon
trap is symplectically transverse of codimension two and has smooth
local stable and unstable surfaces. Their normal expansion and
contraction rates are bounded away from zero. All fixed-order
tangent derivatives grow subexponentially, so the normal rates
dominate the tangent dynamics at every fixed order. These
properties hold on a sufficiently small interval of nearby levels
of the separated fixed-frequency symbol. A departing radial tail
escapes the radial trapping neighborhood and cannot return along a
homoclinic orbit.
\end{proposition}
\begin{proof}
The functions $\varphi_\pm$ of
Lemma~\ref{geo:normal-functions} 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~\ref{geo:tangent-growth} bounds
every fixed-order tangent derivative polynomially, which proves
the claimed domination.
For nearby levels, use the symbol $q_E$ in
\eqref{geo:separated-symbol} 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
\begin{equation}
z_1^2-z_2^2=\frac{q_E}{E^2}+U_c-\frac{D}{E^2}.
\label{geo:off-null-invariant}
\end{equation}
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~\ref{geo:radial} 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.
\end{proof}
This is the trapping geometry used by the local normally
hyperbolic microlocal estimate below
\cite{Dyatlov2016,Hintz2017}. The argument requires neither a
pinching-dependent resonance expansion nor a stationary Killing
field that is timelike over the whole trapping base. The local
stable and unstable tails are the real surfaces just constructed;
outside their neighborhood the radial alternatives permit
propagation to microlocal absorption or to the ordinary ends.
All of these trapping and horizon-source descriptions are invariant
under stationary chart changes. They also persist, with uniform
positive margins after shrinking the parameter neighborhood, for
small changes of the exact Kerr parameters.
\subsection{Ellipticity after spatial projection}
The remaining issue is specific to arbitrary spin. A trapping base
point can lie in the ergoregion, so one cannot require the stationary
operator to be elliptic on an entire base neighborhood. The spatial
projection of a trapped covector nevertheless lies in an elliptic
cone for the slicing of Lemma~\ref{geo:slicing}.
\begin{proposition}[An elliptic cone containing the projected trap]
\label{geo:projected-ellipticity}
Choose $G'=1$ throughout the photon region. The union of the
spatial projections of both trapped null sheets is contained, with
a positive margin, in a spatial conic neighborhood $\mathcal T$
with compact base support separated from the horizon and from
both spatial ends of the estimates. On this neighborhood,
\begin{equation}
b^{ij}(y)\eta_i\eta_j\geq c|\eta|^2
\label{geo:conic-ellipticity}
\end{equation}
for some $c>0$ and any fixed auxiliary cotangent norm on the
compact base support. Both temporal roots there are nonzero and
have absolute values comparable to $|\eta|$. One can choose nested
smaller and larger conic neighborhoods with these properties, and
the same choices work after sufficiently small changes of exact
Kerr parameters and stationary charts.
\end{proposition}
\begin{proof}
At a trapped null covector, $\dot r=0$, so $\dot t=\dot T$.
Substituting \eqref{geo:double-root} into
\eqref{geo:ray-equations} gives
\begin{equation}
\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}.
\label{geo:trapped-time-speed}
\end{equation}
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, \eqref{geo:time-norm-one}, and
\eqref{geo:trapped-time-speed}, we obtain the exact identity
\begin{equation}
\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}
\label{geo:projected-symbol}
\end{equation}
The last inequality is strict for every $r>r_+>M$.
The normalized trap is compact by Lemma~\ref{geo:radial}.
Its spatial projection is nonzero, since a nonzero multiple of
$dt$ is timelike and cannot be a null covector. Consequently
\eqref{geo:projected-symbol} 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
\begin{equation}
b^{tt}E^2-2b^{ti}\eta_i E+b^{ij}\eta_i\eta_j=0.
\label{geo:temporal-roots}
\end{equation}
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.
\end{proof}
There is no assertion of base-space stationary ellipticity in
Proposition~\ref{geo:projected-ellipticity}. Elsewhere the temporal
root gap is still comparable to $|\eta|$ on each fixed compact
spatial set: this follows from the timelikeness of $dt$ and the
positive definite inverse metric induced on its slices. Thus
spatial halfwave diagonalization remains available even where one
temporal root vanishes. Together with strict outflow and the
escape alternatives, this permits the full-order positive energy
sign away from trapping. Near infinity the entire zero-frequency
spatial symbol is elliptic.
\begin{lemma}[Zero-energy directions at spatial fiber infinity]
\label{geo:zero-energy}
The zero-temporal-energy characteristic set has no photon trapping
away from the horizon radial set. Strictly outside the horizon,
its spatial Hamilton flow is real principal and nonradial,
including at a covector dual to the stationary Killing vector on
the stationary limit surface. Consequently the extra
characteristics at spatial fiber infinity in an ergoregion do not
introduce a further stationary trapped set at bounded temporal
frequency.
\end{lemma}
\begin{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
\begin{equation}
R(r)=\mathfrak a^2L_z^2-\Delta D,\qquad
R'(r)=-2(r-M)D<0.
\label{geo:zero-energy-radial}
\end{equation}
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 \eqref{geo:zero-energy-radial} 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~\ref{geo:radial}
and \ref{geo:horizon-source} 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.
\end{proof}
After the standard horizon-source propagation, with the supported
reverse estimate for the transpose problem, these zero-energy
directions use ordinary real-principal propagation. More
generally, fixed-energy spatial propagation follows the projected
null flow, oriented by increasing regular time or reversed for
the opposite estimate. The ray alternatives connect the trap,
the horizon radial region, the far end, and inner outflow without
requiring a positive Killing-current norm in the ergoregion.
Stationary affine-in-time coordinate changes preserve this
description. These are the geometric inputs for the stationary
and compact estimates in the next sections.
==== END Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/03-geometry.tex ====
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/04-stationary.tex ====
\section{Stationary gauges and causal modulation}\label{sec:stationary}
Fix one of the horizon-crossing computing cylinders constructed above,
with stationary coordinates $(t,y)$ and exact Kerr metric $b$.
The purpose of this section is to construct a reduced equation whose
stationary zero modes can be controlled by causal parameter feedback.
We first modify the gauge at high spatial frequency, retaining the
stationary spectral information needed at low frequency. We then
construct the parameter families, the causal target, and the tested
inverse that will supply the compact low-frequency estimate.
All constructions are made separately on each end. Constants may
depend on the fixed subextremal Kerr parameters and prescribed cylinder.
\subsection{The local spectral input and conventions}
We use the Laplace convention $\partial_t=\zeta$, with
$\operatorname{Re}\zeta\geq0$. Choose a smooth stationary function
$\phi$ equal to the Kerr tortoise coordinate $r_*$ near infinity and
zero on a fixed interior collar. Outgoing fields have the factor
$e^{-\zeta\phi}$. At infinity our convention is
\begin{equation}\label{st:b-weights}
H_{\mathrm b}^{s,\ell}
=r^{-\ell-3/2}H_{\mathrm b}^{s}(dr\,d\omega/r),
\end{equation}
where the derivatives in $H_{\mathrm b}^{s}$ are $r\partial_r$ and
unscaled angular derivatives. Interior spaces are extendible at the
strict inner outflow boundary; their dual spaces are supported across
that boundary. We denote these conventions by a bar and a dot,
respectively. The notation $O(r^{-j+})$ permits an arbitrarily small
loss in the exponent, chosen within the relevant indicial interval.
\begin{proposition}[Local spectral input]\label{st:hhv}
On the fixed subextremal Kerr cylinder there are arbitrarily small,
stationary, smooth, compactly based bundle-map modifications of
harmonic gauge and of its symmetric-gradient partner with these
properties.
\begin{enumerate}
\item The stationary gauge-potential operator, obtained by applying
the modified gauge to a pure Lie derivative, is invertible at decay
strictly between the powers $1$ and $r^{-1}$.
\item The modified linearized reduced vacuum operator has outgoing
inversion at nonzero frequency in the closed physical half-plane.
Its bounded-frequency estimates, with a weaker compact error, are
uniform down to zero in the weight interval $-3/2<\ell<-1/2$.
Its zero-frequency realization has index zero and a seven-dimensional
kernel: four gauge-corrected Kerr variations and three translations.
\item In outgoing time, a polynomial mode with coefficients of
order $O(r^{-1+})$ has translation leading coefficient if its
degree is one. There are no such modes of degree at least two.
\end{enumerate}
For tensors one may take $s>5/2$, with strict margin, also sufficient
for gauge potentials. The uniform near-zero equation norm is
$H_{\mathrm b}^{s-1,\ell+2}$. At nonzero frequency the outgoing
conditions are $\ell<-1/2$ and $s+\ell>-1/2$.
\end{proposition}
\begin{proof}
The gauge-potential assertion is \cite[Theorem~5.1]{HHV2025};
the discussion following that theorem permits arbitrarily small local
damping parameters. The bounded-frequency estimates and stationary
index are \cite[Theorem~5.2]{HHV2025}, and nonzero-frequency inversion
and the seven-dimensional kernel are \cite[Theorem~6.6]{HHV2025}.
The polynomial-mode assertions are
\cite[Proposition~6.8 and Lemma~6.9]{HHV2025}.
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.
\end{proof}
Normalize the reduced equation by
\begin{equation}\label{st:reduced-normalization}
\operatorname{Ric}(g)+(\mathcal D_g+C)\Upsilon=0,\qquad
(\mathcal D_g\omega)_{\mu\nu}=\nabla^g_{(\mu}\omega_{\nu)}.
\end{equation}
For a target metric $B$, the differential wave-map gauge is
\begin{equation}\label{st:wave-gauge}
(\Upsilon_{\mathrm{wave}}(g;B))_\mu
=-g_{\mu\kappa}g^{\alpha\beta}
\bigl(\Gamma(g)^\kappa_{\alpha\beta}
-\Gamma(B)^\kappa_{\alpha\beta}\bigr).
\end{equation}
Write $E_{\mathrm g}$ for the addition to this gauge and $C$ for
the addition to its partner, including the local terms just obtained.
At $g=B=b$ write
\[
\mathcal B=D\Upsilon_{\mathrm{wave}}|_{b;b}+E_{\mathrm g},
\qquad
L=\operatorname{Ric}'_b+(\mathcal D_b+C)\mathcal B.
\]
The notation $L(\zeta)$ means substitution of $\partial_t=\zeta$.
We write $L_{\mathrm h}$ for the local operator before the
high-spatial-frequency changes made next.
\subsection{Gauge transport on the spatial elliptic cone}
The geometric analysis supplies a compactly based spatial conic
neighborhood $\mathcal T$ of both projected trapped sets, disjoint
from the horizons and ends of the cylinder, on which the
zero-frequency spatial principal symbol is strictly elliptic.
We use this cone instead of a timelike neighborhood in the spatial
base. The two temporal characteristic roots remain separated because
the $t$-slices are spacelike.
For this construction let $\bar g$ be an exact nearby stationary Kerr
metric in a chart of the same type, and let $B_0$ be a close stationary
target. We calibrate the transport of
\begin{equation}\label{st:frozen-general}
\operatorname{Ric}'_{\bar g}
+(\mathcal D_{\bar g}+C)
\bigl(D\Upsilon_{\mathrm{wave}}|_{\bar g;B_0}
+E_{\mathrm g}\bigr).
\end{equation}
At equal metrics without additions, this is ordinary tensor parallel
transport after removal of scalar wave-amplitude transport.
Changing $B_0$ inserts only an algebraic term $K_{\mathrm{alg}}$
in the linearized gauge. We use these wave-gauge identities in the
following construction.
\begin{lemma}[Conic gauge calibration]\label{st:conic-gauge}
The additional gauge and partner terms can be chosen as real spatial
pseudodifferential operators $A_0+A_1\partial_t$, with respective
spatial orders $0,-1$, such that:
\begin{enumerate}
\item their kernels are compactly supported in both variables away
from horizons and boundaries, and their high-frequency microsupport
lies in $\mathcal T$;
\item at sufficiently high spatial frequency their transport near
each trapped set is exactly compatible with a smooth positive
homogeneous bundle metric;
\item at the base metric the unsaturated transport and cutoff
transition have imaginary subprincipal defect smaller than any
prescribed fixed fraction of the minimum normal expansion rate;
\item the additional temporal-acceleration coefficients have spatial
order at most $-1$ and are small on any prescribed finite Sobolev
list when the high-frequency cutoff $\Lambda$ is sufficiently large.
\end{enumerate}
The prescription is smooth in the stationary metric and target.
The symbol bounds are uniform for large $\Lambda$ and for a
sufficiently small fixed parameter neighborhood.
\end{lemma}
\begin{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
\begin{equation}\label{st:root-interpolation}
A_1=\frac{E_+^{\mathrm{add}}-E_-^{\mathrm{add}}}
{-i(P_+-P_-)},\qquad
A_0=E_+^{\mathrm{add}}+iP_+A_1.
\end{equation}
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.
\end{proof}
\subsection{Stationary estimates after the conic modification}
We separate bounded frequency, high frequency, and the outgoing
spatial transpose. The last is the bilinear \emph{spatial}
transpose, whose outgoing exponential has the same sign as the
primal exponential. Fix a small weight slack $0<\iota\ll1$ in the
weighted scattering estimates below.
\begin{proposition}[Stationary estimates]\label{st:stationary-estimates}
For sufficiently large $\Lambda$, the conically modified base
operator has the following estimates.
\begin{enumerate}
\item Bounded-frequency estimates with weaker compact errors persist
uniformly as $\Lambda\to\infty$, at the orders and zero-frequency
weights of Proposition~\ref{st:hhv}. The stationary index is zero,
and kernel and cokernel are upper semicontinuous. On closed bounded
bands excluding zero, the physical estimates persist without compact
error, up to $\operatorname{Re}\zeta\downarrow0$.
\item For every fixed strip $0\leq\operatorname{Re}\zeta\leq D_*$
and fixed admissible order, there are polynomial high-frequency
bounds without kernel, with threshold independent of sufficiently
large $\Lambda$. These include the compact $H^s$ estimate from
$H^{s-1}$, for $s=5$ or a fixed higher admissible order, the supported
dual solution estimate of order $1-s$, and
\[
\langle r\rangle^{-1/2-\iota}(v,\partial_yv)\in L^2
\]
for the physical transpose with a compact smooth source, with
polynomial frequency cost.
\item Zero-frequency estimates, including those for gauge
potentials, persist uniformly for nearby exact Kerr principal
metrics in smooth stationary, asymptotically affine charts of
the prescribed type. Fixed smooth finite-dimensional supplements
remove their kernel and cokernel locally in parameters.
\end{enumerate}
The same-sign outgoing transpose has the a priori estimate and
index stated in Lemma~\ref{st:transpose}.
\end{proposition}
\begin{proof}
\emph{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 $K'$,
\begin{equation}\label{st:elliptic-smallness}
\|(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')}.
\end{equation}
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~\ref{st:hhv}. 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
\begin{equation}\label{st:outgoing-symbol}
(\partial_{r_*}-2\zeta)\partial_{r_*}w
=O(r^{-2})\bigl(\partial_\omega^{\leq2}w,
\partial_\omega^{\leq1}\partial_{r_*}w\bigr).
\end{equation}
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~\ref{st:hhv} 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 \eqref{st:elliptic-smallness} 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.
\emph{High frequency.}
We apply the local outgoing normally hyperbolic estimate of
\cite[Theorem~1]{Dyatlov2016}, with the scalar-principal bundle
formulation and positive pseudodifferential metrics of
\cite[Condition~(2.5), Proposition~3.12]{Hintz2017}. 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~\ref{st:conic-gauge}
verify these hypotheses for each nonzero-energy sign with margin.
To use the separated symbol $q_E=\Sigma_Kb^{-1}(k,k)$ of
\eqref{geo:separated-symbol}, 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.
\emph{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.
\end{proof}
The transpose argument uses three realizations of the stationary
operator. The physical equations act on the original fields through
$L(\zeta)$ and its supported spatial transpose $L(\zeta)^t$.
Removing the outgoing exponential gives, respectively,
\[
P_{\rm out}(\zeta)=e^{\zeta\phi}L(\zeta)e^{-\zeta\phi},
\qquad
P^\sharp(\zeta)=e^{\zeta\phi}L(\zeta)^t e^{-\zeta\phi}.
\]
The second is a same-sign outgoing conjugation of the physical
transpose. In particular,
\[
P_{\rm out}(\zeta)^t=e^{-\zeta\phi}L(\zeta)^t e^{\zeta\phi}.
\]
The conjugating factors are reversed when the outgoing primal is
transposed. We first estimate $P^\sharp$ near zero and then compute
its index at nonzero frequency, using different exterior equation
spaces for these two purposes.
\begin{lemma}[The outgoing spatial transpose]\label{st:transpose}
Fix $\ell_{\mathrm{wt}}=-1/2-\iota$, with $0<\iota\ll1$, and
a sufficiently large fixed exterior derivative order $j$.
Let $\mathcal A$ have supported interior order $-4$ and exterior
space $H_{\mathrm b}^{j,\ell_{\mathrm{wt}}}$; let $\mathcal S$
have supported interior order $-5$ and exterior space
$H_{\mathrm b}^{j-1,\ell_{\mathrm{wt}}+2}$.
For
\[
P^\sharp(\zeta)=e^{\zeta\phi}L(\zeta)^t e^{-\zeta\phi}
\]
there is an estimate from $\mathcal S$ to $\mathcal A$ with a
weaker compact error, uniform near zero.
At each fixed small nonzero $\zeta$ with
$\operatorname{Re}\zeta>0$, the graph using instead the exterior
equation space $H_{\mathrm b}^{j,\ell_{\mathrm{wt}}+1}$
has index zero, also after compact smooth finite-rank augmentation.
\end{lemma}
\begin{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~\ref{st:hhv} 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.
\end{proof}
\subsection{Stationary profiles and exact parameter families}
We next retain the physical zero modes while installing the conic
gauge, and integrate the relevant directions into exact Kerr
families. The indicial calculation below identifies the required
parameter directions at the fixed center.
\begin{lemma}[Profiles and mode-preserving corrections]\label{st:profiles}
There are smooth stationary tensors
$H_i$ for $1\leq i\leq4$ and $J_i,H_i^1$ for $1\leq i\leq3$,
with stationary vector fields $V_i,W_i$, such that
\begin{gather}
J_i=\mathcal L_{V_i}b,\qquad
H_i^1=\mathcal L_{W_i}b+2dt\odot V_i^\flat,
\label{st:profile-definition}\\
V_i=e_i+O(r^{-1}),\qquad
W_i=y_i\partial_t+O(r^{0+}),
\label{st:vector-profiles}\\
\mathcal BH_i=\mathcal BJ_i
=\mathcal B(tJ_i+H_i^1)=0,
\qquad
L(0)H_i=L(0)J_i=0,\qquad
L(0)H_i^1=-[L,t]J_i.
\label{st:profile-identities}
\end{gather}
Here $e_i$ are coordinate translations near infinity and $H_i$
are gauge-corrected Kerr variations. In symbol orders,
\begin{equation}\label{st:profile-tails}
H_i,H_i^1=r^{-1}(\text{smooth angular term})+O(r^{-2+}),
\qquad
J_i=r^{-2}(\text{smooth angular term})+O(r^{-3+}).
\end{equation}
The additional high-frequency gauge can be changed by
$O(\Lambda^{-N})$ finite-rank smoothing terms, for every fixed
$N$, so that these identities remain exact. Its decaying
stationary kernel then consists precisely of the seven
independent $H_i,J_i$.
\end{lemma}
\begin{proof}
For the local gauge, the starting profiles are the Kerr, translation,
and boost profiles of \cite[Lemmas~6.1--6.3, Proposition~6.5,
and~(6.6)]{HHV2025}. 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~\ref{st:hhv}.
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
\eqref{st:profile-identities} for the local gauge.
The flat-Laplacian indicial expansion gives
\eqref{st:profile-tails}, 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~\ref{st:stationary-estimates} 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.
\end{proof}
The decaying stationary modes are not the whole kernel when a field
is allowed slight growth at infinity. A constant symmetric covariant
two-tensor in four dimensions has ten independent coefficients.
The next proposition realizes these constant asymptotes by additional
homogeneous modes and uses them to choose an exact stationary reference.
Their constant leading terms arise from affine coordinate changes;
their normalization may also add decaying Kerr and gauge directions.
They therefore do not represent ten additional physical Kerr parameters.
\begin{proposition}[Exact stationary families and supplements]
\label{st:families}
For small $p=(m,v)\in\mathbb R^4\times\mathbb R^3$ there is
a smooth family $G_p$ of exact diffeomorphic stationary Kerr
metrics with $G_0=b$, tangent directions $H,\mathcal L_Wb$,
and symbol asymptote
\begin{equation}\label{st:affine-asymptote}
G_p-\eta_v=O(r^{-1}),\qquad
\eta_v=-(1-|v|^2)dt^2-2v_i\,dy_i\,dt+|dy|^2.
\end{equation}
Let $L_p(0)$ denote the stationary linear reduced operator at
metric and target $G_p$. It has seven independent decaying
modes and ten further independent stationary homogeneous
metric modes with constant tensor asymptotes. There are no
additional dimensions at any growth power strictly between
$0$ and $1$.
There exist a fixed compact smooth seven-vector functional
$\ell$, smooth ten-vector functionals $\ell_C^p$, and a smooth
list $H_C(p)$ of the ten additional modes, satisfying
\[
\ell(H,J)=I,\qquad
\ell H_C(p)=0,\qquad
\ell_C^p H_C(p)=I,
\]
with $\ell_C^p$ annihilating the seven decaying modes.
Their supports can be fixed outside a physical trapping
neighborhood. There is a second smooth exact stationary Kerr
family $\overline G_{p,c}$, $c\in\mathbb R^{10}$ small, with
\begin{equation}\label{st:second-family}
\overline G_{p,0}=G_p,\qquad
\partial_c\overline G_{p,0}=H_C(p).
\end{equation}
The gauge calibration extends smoothly to both families.
The frozen gauge of $\overline G_{p,c}$ relative to $G_p$
is $O(c^2)$ on fixed compact sets.
\end{proposition}
\begin{proof}
Integrate the stationary correction flows in
Lemma~\ref{st:profiles}. For the velocity directions use
the affine leading chart
\begin{equation}\label{st:affine-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.
\end{equation}
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
\eqref{st:affine-asymptote}. 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~\ref{st:profiles}.
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.
\end{proof}
The two families will serve different parts of the finite-time
argument. Causal modulation will control the seven decaying modes,
while the ten measured coefficients $c$ select the instantaneous
reference $\overline G_{p,c}$ in the sliced estimate. The difference
from this reference, rather than the entire perturbation, will have
the integrable low norms needed in the energy comparison.
For later energy estimates the frozen transport operator is now
unambiguously
\begin{equation}\label{st:frozen-operator}
\mathbf L_{p,c}
=\operatorname{Ric}'_{\overline G_{p,c}}
+(\mathcal D_{\overline G_{p,c}}+C(p,c))
\bigl(D\Upsilon_{\mathrm{wave}}|_{\overline G_{p,c};G_p}
+E_{\mathrm g}(p,c)\bigr).
\end{equation}
It does not include the derivative of $\mathcal D_g$ applied
to the residual gauge. That omitted coefficient is
$O(c^2)$ compactly and will be treated in the nonlinear
comparison. The construction order has been $b$, then $G_p$,
then $\overline G_{p,c}$.
\begin{lemma}[Adjoint tails]\label{st:adjoint-tails}
Let $\mathcal K_p^*$ be the supported bilinear-spatial-transpose
kernel dual to the decaying stationary problem. It is
seven-dimensional, locally of order $\dot H^{-4}$, and
conormal $O(r^{-2+})$ at infinity. It has locally uniform
weak bases with uniform symbol bounds sufficiently far out.
If $H_0$ is a smooth extendible stationary tensor of symbol
growth $O(r^{0+})$, then
\begin{equation}\label{st:adjoint-pairing}
\langle v,L_p(0)H_0\rangle=0,\qquad v\in\mathcal K_p^*.
\end{equation}
\end{lemma}
\begin{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
\Cref{st:families} 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 \eqref{st:adjoint-pairing}; the extra
adjoint tail power, rather than dual Fredholm information at
one weight alone, is what permits these pairings.
\end{proof}
\subsection{The causal target and reduced evolution}\label{st:reduction}
We now construct the time-dependent target and causal feedback that
will control the stationary zero modes. The radius-dependent averages
preserve the asymptotic symbol class; the filters control the parameter
jets; and the compensator restores the desired stationary linearization.
After proving their defect bounds, we solve the reduced evolution and
verify propagation of the vacuum constraints. Write $p(t)=(m(t),v(t))$,
let $a(t)=p'(t)$, and let $\lambda'(t)=v(t)$.
Extend $p$ and $\lambda$ by zero for $t\leq0$.
The co-moving coordinates are $(t,y)$ and the laboratory
coordinates are $(t,z)$, where
\[
z=y-\lambda(t).
\]
The symbol $a$ in this section denotes the parameter derivative,
not the fixed Kerr rotation parameter.
Choose a smooth function $d(x)\geq0$, zero on a large fixed
ball containing both variables of every compact kernel and all
probes, and equal to $|x|$ near infinity. Choose a real smooth
signed kernel $k_0$ compactly supported in $(0,1/10)$ with
\[
\int k_0(\sigma)\,d\sigma=1,\qquad
\int \sigma k_0(\sigma)\,d\sigma=0.
\]
For a zero-past function $f$ define its radius-dependent causal
average by
\begin{equation}\label{st:averaging}
f_s(t,x)=\int k_0(\sigma)f(t-\sigma d(x))\,d\sigma .
\end{equation}
The implicit equation
\begin{equation}\label{st:implicit-position}
x=z+\lambda_s(t,x)
\end{equation}
is a contraction for sufficiently small $\sup|v|$.
It gives $x=y$ in the unsmoothed region, and $1+|x|$ is
comparable to $1+|y|$.
Define the target by evaluating the family coefficients in the
formal coframe
\begin{equation}\label{st:causal-target}
B=G_{p_s(t,x)}(x)
\quad\hbox{in the coframe }(dt,dz+v_s(t,x)\,dt).
\end{equation}
The zero first moment has a useful exact consequence. If $p$ is
constant and $\lambda$ is affine, with $\lambda'=v$, throughout the
interval from the earliest sampled time to the current time $t$, then
$p_s=p(t)$, $\lambda_s=\lambda(t)$, and
$x=y$. On a region where these identities hold, $B$ is exactly $G_p$
in the co-moving chart, expressed in the laboratory chart by the
coframe $dy=dz+v\,dt$. Thus the construction changes an exact
stationary metric only where the parameter history changes; the
following defect estimates quantify that change.
For the evolving metric $g$, set
\begin{equation}\label{st:unknowns}
h=g-B,\qquad
u=\partial_t h_{\mathrm{co}},\qquad
D=\partial_{t,\mathrm{lab}}-v_s\cdot\partial_z,\qquad
\psi=D h_{\mathrm{lab}}.
\end{equation}
Subscripts specify the coordinate components and derivatives.
Fix a smooth startup cutoff $\chi_{\mathrm{st}}$, zero near
zero and one after time one. The feedback is
\begin{equation}\label{st:feedback}
\begin{split}
c&=\ell_C^p h_{\mathrm{co}},\\
a&=F(\partial_t)(\chi_{\mathrm{st}}\ell h_{\mathrm{co}}),
\qquad
F=\operatorname{diag}(F_m I_4,F_v I_3),\\
F_m(\zeta)&=\frac{\sigma_0}{(1+\zeta)^P},\qquad
F_v(\zeta)=\frac{2\sigma_0\zeta+\sigma_0^2}{(1+\zeta)^P}.
\end{split}
\end{equation}
Here $P\geq5$ is a fixed integer, increased below to dominate
a fixed finite list of frequency losses, and $\sigma_0>0$
is sufficiently small. Both filters are retarded.
\begin{lemma}[Filter regularity]\label{st:filter}
For each $j\geq0$, the $j$th derivative of $a$ is controlled
by derivatives of its input through order
$\max(0,j-P+1)$ in polynomially weighted $L^2_t$ and
$L^\infty_t$, also on finite intervals. For $j\geq1$ one
may first differentiate the input and use $\ell u$, with
the resulting startup term charged to initial data.
\end{lemma}
\begin{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 $\lambda$.
\end{proof}
Use
\begin{equation}\label{st:full-gauge}
\Upsilon=\Upsilon_{\mathrm{wave}}(g;B)
+E_{\mathrm g}(p,c)h+\Theta
\end{equation}
in \eqref{st:reduced-normalization}, with partner $C(p,c)$.
The spatial additions act in the co-moving chart.
To define the compensator, contract the profile indices and put
\begin{equation}\label{st:compensator}
Y=H m_s+J\lambda_s+H^1v_s
+2(V-e)^\flat\odot d_x\lambda_s .
\end{equation}
On the unsmoothed region, containing all kernels with a buffer,
set $\Theta=\mathcal B Y$, with the base coefficients of
$\mathcal B$. Outside, compute the same local expression in
independent coordinates $(t,x)$, then evaluate at the implicit
$x$ and pull its components into the formal coframe of
\eqref{st:causal-target}. The definitions coincide where $d=0$.
\begin{lemma}[Joint linearization and column bounds]\label{st:columns}
Linearizing the joint reduced equation at zero parameters and
$g=b$ gives
\begin{equation}\label{st:joint-linearization}
Lh+Q(\partial_t)a.
\end{equation}
For
\[
K_d(y,\zeta)=\int k_0(\sigma)e^{-\zeta\sigma d(y)}\,d\sigma
\]
the seven columns of $Q$ are
\begin{equation}\label{st:Q-definition}
\begin{split}
Y_m&=H K_d,\\
Y_v&=(J+\zeta H^1)K_d
+2(V-e)^\flat\odot d_yK_d,\\
Q(\zeta)&=
\left(\frac{L(\zeta)Y_m}{\zeta},
\frac{L(\zeta)Y_v}{\zeta^2}\right).
\end{split}
\end{equation}
The apparent poles are removable. On every fixed compact set,
$Q(\partial_t)$ has only bounded-length causal memories,
including distributional derivatives, and its fixed spatial
derivative costs are polynomial in frequency. For every
fixed $N$,
\begin{equation}\label{st:Q-tail}
Q(\zeta)=O_N\bigl(r_y^{-2}(1+|\zeta|r_y)^{-N}\bigr),
\qquad r_y=|y|,\quad \operatorname{Re}\zeta\geq0,
\end{equation}
with the corresponding symbol derivatives near infinity.
\end{lemma}
\begin{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 \eqref{st:joint-linearization}, with the columns
\eqref{st:Q-definition}.
The stationary identities and the chain identity
\eqref{st:profile-identities} 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 \eqref{st:Q-tail}. Spatial derivatives
on a fixed annulus give only fixed polynomial frequency costs.
\end{proof}
\begin{lemma}[Nonlinear target defect]\label{st:target-defect}
Suppose $\int|a|$ and $\sup|v|$ are sufficiently small.
For comparable large radii $r\sim|x|$, set
\[
\widetilde a(t,r)
=Cr^{-1}\int_{t-Cr}^{t}|a(\tau)|\,d\tau,
\]
with the positive majorant enlarged as needed at comparable
radii. In laboratory components,
\begin{equation}\label{st:target-defect-orders}
\Theta=O(r^{-1}\widetilde a),\qquad
f_B:=\operatorname{Ric}(B)
+(\mathcal D_B+C(p,c))\Theta
=O(r^{-2}\widetilde a),\qquad
Df_B=O(r^{-3}\widetilde a).
\end{equation}
These hold with each fixed number of scaled laboratory
derivatives. Also $DB=O(r^{-1}\widetilde a)$ with symbol
derivatives. On fixed transition and inner sets the analogous
defects use bounded-length parameter jets and vanish at
exact freezing with affine position.
\end{lemma}
\begin{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{equation}\label{st:smoothed-jets}
\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}
\end{equation}
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 \((t,x)\),
\begin{equation}\label{st:target-frame}
\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}.
\end{equation}
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 \(r\widetilde a\ll1\).
\paragraph{The metric defect.}
The formal coframe substitution cancels the affine flat asymptote exactly: substituting \(dy=dz+v\,dt\) in \(\eta_v\) gives \(\eta\). Hence in lab components
\[
B=\eta+\mathfrak h(p_s,x),\qquad
\partial_x^\alpha\partial_p^\beta\mathfrak h=O(r^{-1-|\alpha|}).
\]
Here \(\mathfrak h\) is the smooth family of lab-component metric remainders. At a point freeze \(p_s,v_s\), and use the constant-parameter affine-worldline Kerr metric with position \(x\). Its first and second lab derivatives differ from those of \(B\) by
\(O(r^{-1}\widetilde a)\) and \(O(r^{-2}\widetilde a)\), respectively. Indeed the second derivative expansion consists of
\[
\mathfrak h_{xx}CC,\quad \mathfrak h_x\partial C,\quad
\mathfrak h_{xp}C\partial p_s,\quad \mathfrak h_{pp}(\partial p_s)^2,\quad
\mathfrak h_p\partial^2p_s,
\]
with both mixed terms understood, where \(C\) is \(S_x\) or \(v_sS_x\). The coordinate-coefficient difference is \(O(r\widetilde a)\), \(\partial C=O(\widetilde a)\), and \(\partial^2p_s=O(r^{-1}\widetilde a)\). These give the claimed orders. The frozen metric is vacuum, so smooth expansion of Ricci gives \(\operatorname{Ric}(B)=O(r^{-2}\widetilde a)\).
For the additional \(D\) derivative, \(DB=O(r^{-1}\widetilde a)\) with two further lab symbol derivatives by \eqref{st:smoothed-jets}. For any two lab coordinate derivatives, the exact commutator identity is
\[
\begin{split}
D\partial_\alpha\partial_\beta B
={}&\partial_\alpha\partial_\beta(DB)
+(\partial_\alpha v_s^j)\partial_j\partial_\beta B
+(\partial_\beta v_s^j)\partial_\alpha\partial_j B\\
&+(\partial_\alpha\partial_\beta v_s^j)\partial_jB.
\end{split}
\]
Here \(\partial v_s=O(\widetilde a)\), \(\partial^2v_s=O(r^{-1}\widetilde a)\), \(\partial B=O(r^{-2})\), and \(\partial^2B=O(r^{-3})\). Every displayed term is therefore \(O(r^{-3}\widetilde a)\). Differentiating the coefficients in Ricci adds only terms of that order or smaller. Thus \(D\operatorname{Ric}(B)=O(r^{-3}\widetilde a)\).
\paragraph{The compensator and its derivative.}
At infinity the gauge is local; write it as
\(\mathcal B=\beta^\mu(x)\partial_\mu+\beta_0(x)\), and put \(U_i=V_i-e_i\). The identities
\[
\mathcal BH=\mathcal BJ=0,\qquad
\mathcal BH^1=-[\mathcal B,t]J,\qquad
\partial_t\lambda_s=v_s
\]
reduce \(\mathcal BY\) to sums of
\[
[\mathcal B,m_s]H,\qquad
\beta^iJ\,\partial_{x_i}\lambda_s+[\mathcal B,v_s]H^1,
\qquad
\mathcal B(2U^\flat\odot d_x\lambda_s).
\]
In particular no undifferentiated position or velocity remains. The profile orders
\(H,H^1,U=O(r^{-1})\), \(J=O(r^{-2})\) and
\eqref{st:smoothed-jets} bound each expression by
\(O(r^{-1}\widetilde a)\). A \(D=\partial_t|_x\) hit on a memory gains one inverse radius; for example it sends
\(\partial_x\lambda_s=O(r\widetilde a)\) to
\(\partial_xv_s=O(\widetilde a)\). The implicit-coordinate and coframe changes contribute only \(S_x,v_s\) and their controlled derivatives. A derivative on a coframe coefficient produces
\(r^{-1}\widetilde a^2\lesssim r^{-2}\widetilde a\) at the \(\Theta\) level and
\(r^{-2}\widetilde a^2\lesssim r^{-3}\widetilde a\) after one further derivative. Together with \eqref{st:target-frame}, this proves
\[
\Theta=O(r^{-1}\widetilde a),\quad
\partial\Theta,D\Theta=O(r^{-2}\widetilde a),\quad
D\partial\Theta=O(r^{-3}\widetilde a).
\]
Christoffel terms in \(\mathcal D_B\Theta\) have smaller orders, and \(C\) vanishes here. We obtain the required defect bounds
\begin{equation}\label{st:full-defect-identity}
\Theta=O(r^{-1}\widetilde a),\qquad
f_B:=\operatorname{Ric}(B)+(\mathcal D_B+C)\Theta
=O(r^{-2}\widetilde a),\qquad
Df_B=O(r^{-3}\widetilde a).
\end{equation}
The calculation allows every fixed number of scaled lab derivatives, with constants depending on that number. Each use may enlarge the comparable averaging interval. At the far-current application, lab-time derivatives are taken before the outgoing commutations; their coefficients contain only smoothed parameters. No unsmoothed acceleration is created by the calculation above.
On the fixed transition annulus \(d\) may vanish, so an inverse-\(d\) estimate is neither needed nor asserted. There the defect is a smooth function of finitely many parameter jets and bounded-length memories. In the unsmoothed region it also contains the fixed compact pseudodifferential operations. The exact identities above make it vanish whenever the relevant accelerations vanish. The filter lemma supplies every fixed required jet list, and the compact estimates treat these transition terms with fixed-radius constants.
\end{proof}
The target, feedback, and compensator now have the spatial decay and
finite-jet control needed by the evolution. We next show that their
retarded dependence closes the local Cauchy problem and that vacuum
initialization gives the vacuum equation throughout the resulting slab.
For this local construction, the base wave-pair norm is
\[
\mathcal E_{10}(t)=\|h_{\rm co}(t)\|_{H^{10}_{\rm uloc}}
+\|\partial_t h_{\rm co}(t)\|_{H^9_{\rm uloc}}.
\]
Here the uniformly local norms use a fixed buffered region containing
all compact kernels and regular end charts of fixed size. The contraction
norm is the wave-pair norm one order lower, $H^9_{\rm uloc}\times
H^8_{\rm uloc}$. These are sufficient fixed orders, not a minimal
regularity assertion.
\begin{proposition}[Local evolution and vacuum initialization]
\label{st:local-evolution}
The reduced equation
\eqref{st:reduced-normalization},
\eqref{st:full-gauge}, with retarded feedback
\eqref{st:feedback}, has smooth local evolution for smooth
initial wave data small in $\mathcal E_{10}$.
It continues on finite time intervals while $\mathcal E_{10}$ remains
bounded, the parameters stay in their chosen small range, the normalized
acceleration and implicit-coordinate inverses stay uniformly controlled,
and the spacelike and strict outflow margins remain positive.
Smooth higher orders propagate on bounded time intervals;
these higher seminorms need only be finite.
For smooth vacuum constraint data, the initial lapse and shift
derivatives can be chosen so that $\Upsilon$ and its first
normal derivative vanish initially. The resulting evolution
is vacuum.
\end{proposition}
\begin{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.
\end{proof}
\subsection{Augmented inversion and retarded tests}
The unaugmented operator has stationary modes. The feedback removes
their obstruction without assuming a uniform unaugmented inverse
at zero. For a large radius $R$, choose a smooth scaled cutoff
$\chi_R(y)$, equal to one for $r_y\leq R$ and zero for
$r_y\geq2R$, and define
\begin{equation}\label{st:augmented-operator}
M_R(\zeta)=L(\zeta)+\chi_R Q(\zeta)F(\zeta)\ell.
\end{equation}
We first identify the zero-frequency coupling.
\begin{lemma}[Zero-frequency coupling]\label{st:zero-coupling}
For sufficiently large $\Lambda$, the map
\[
Q(0)^t:\mathcal K_0^*\longrightarrow\mathbb C^7
\]
has full rank.
\end{lemma}
\begin{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~\ref{st:adjoint-tails}. 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~\ref{st:hhv} is applied. The residuals at the relevant
orders are exactly the equation errors of
\begin{equation}\label{st:polynomial-tests}
t_*H,\qquad
\tfrac12t_*^2J+t_*(H^1+\phi J).
\end{equation}
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 \eqref{st:polynomial-tests}.
Every coefficient of the resulting outgoing-time polynomial
then lies in the decaying class of Proposition~\ref{st:hhv};
none of the discarded $O(r^{0+})$ coefficients occurs in it.
If its quadratic coefficient were
nonzero, \eqref{st:polynomial-tests} 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~\ref{st:hhv}.
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 $\Lambda$.
\end{proof}
\begin{proposition}[Retarded stationary tests]\label{st:retarded-tests}
Fix a compact test support and a finite number of required
angular derivatives. There are a sufficiently high fixed
test Sobolev order $q_*$ and a finite polynomial degree $N_*$
with the following property. Choose the filter order $P$
sufficiently large and $\sigma_0>0$ sufficiently small, then
take $\Lambda,R$ sufficiently large in the order specified
below. For $0<\operatorname{Re}\zeta\leq D_*$ on that strip
and a compactly supported test $f$ of order $q_*$, the spatial
transpose equation
\[
M_R(\zeta)^t v_*=f
\]
has a solution satisfying
\begin{align}
\|v_*\|_{\dot H^{-4}(K)}
&\leq C R^{3\iota}\langle\zeta\rangle^{N_*}
\|f\|_{H^{q_*}},
\label{st:test-compact}\\
\|(v_*,r_y(\partial_{r_y}+\zeta)v_*)\|_{L^2_\omega}
&\leq C r_y^{-1+\iota}R^{3\iota}
\langle\zeta\rangle^{N_*}\|f\|_{H^{q_*}}
\qquad(r_y\gg1).
\label{st:test-radiation}
\end{align}
Here $K$ is any prescribed fixed compact reach, and the second
bound includes the prescribed angular derivatives.
The bounds are uniform down to
$\operatorname{Re}\zeta\downarrow0$ on a fixed strip reaching
a forward Laplace line. The slack $\iota>0$ can be arbitrarily
small without making $N_*$ diverge. The associated tested
operators are retarded.
\end{proposition}
\begin{proof}
We give the zero, intermediate, and high frequency arguments
separately.
\emph{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~\ref{st:transpose}, 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
\begin{equation}\label{st:near-zero-comparison}
\|e^{\zeta\phi}v_*-v_0\|_{\mathcal A}+|\delta_f|
\leq C\bigl(\|f\|_{H^{q_*}}+|\zeta||v_0|\bigr).
\end{equation}
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~\ref{st:zero-coupling} 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~\ref{st:transpose} 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 \eqref{st:test-radiation} in this regime, using
$r_y(1-\phi')=O(1)$.
\emph{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
\begin{equation}\label{st:filter-matrix}
I+[I_H,I_J+\zeta\,\ell H^1]\,
\operatorname{diag}(F_m/\zeta,F_v/\zeta^2).
\end{equation}
Here $[I_H,I_J]$ is the identity divided into its four and
three mode columns. Formula \eqref{st:filter-matrix}
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.
\emph{High frequency and radiation.}
The polynomial transpose losses from
Proposition~\ref{st:stationary-estimates}, 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
\begin{equation}\label{st:transpose-radial}
(\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).
\end{equation}
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 \eqref{st:transpose-radial}. 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 \eqref{st:test-radiation}, with all compact
modifications interior to the computation.
\emph{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
\eqref{st:test-compact}--\eqref{st:test-radiation}, gives
the retarded tested operators down to the boundary of the
physical half-plane.
\end{proof}
\begin{remark}[Fixed base choices and higher regularity]
\label{st:order-quantifiers}
The choice of $\Lambda$ requires smallness on a finite base
Sobolev list, not on all orders simultaneously. Once the
negative-order acceleration correction is invertible at a low
order, its derivative gain yields estimates at every fixed
higher order, with order-dependent constants. The saturated
transport calibration is exact and independent of the energy
order. For nearby $p$, the seven-dimensional kernel is
reconstructed using the invertible gauge-potential problem:
four physical variations and three translations give the
lower bound, while stationary compactness gives the matching
upper bound. Preservation of the base translation chains is
needed for the base columns and coupling, not asserted as
an unchanged chain at every nearby parameter.
Finally the worst filter order is $1-P$, so
$a^{(s+3)}$ uses at most
$\max(0,s+4-P)\leq s-1$ projection derivatives when $P\geq5$.
This count is uniform in $s$. Thus the fixed base construction
does not introduce a requirement of smallness at arbitrarily
high data orders. The subsequent energy argument can use the
order-dependent stationary constants when proving smooth
higher bounds for each fixed datum.
\end{remark}
==== END Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/04-stationary.tex ====
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/05-compact.tex ====
\section{Compact estimates and smooth propagation}\label{sec:compact}
We next pass from the stationary estimates to finite-time estimates for the
nonlinear reduced equation. The essential new point is directional: the
stationary operator is elliptic on the spatial cone containing the projected
trapped covectors, not necessarily over their entire spatial base. We therefore
retain a full-order bulk estimate for first jets outside that cone. Combined
with a conic elliptic correction, this gives the same integrable static defect
that is needed in the nonlinear energy comparison. The interaction of
elliptic and integrated wave estimates has a methodological parallel in
Dafermos--Holzegel--Rodnianski--Taylor
\cite{DafermosHolzegelRodnianskiTaylor2024}; the conic estimates for the
reduced Einstein system used here are established in this section.
Throughout this section the solution is smooth on an arbitrary finite bootstrap
slab $[0,T]$. No constant in an estimate below depends on $T$. We use the
modulation, targets, reduced equation, stationary family and causal filters
constructed in the preceding section. In particular,
$u=\partial_t h_{\rm co}$, $\psi$ is the differentiated field in laboratory
components, $p'=a$, $c=\ell_C^p h_{\rm co}$, and
$\bar g=\overline G_{p,c}$ is the instantaneous stationary reference.
All compact pseudodifferential operators act spatially at the current time and
have kernel support strictly inside the computing cylinder.
\subsection{Finite-slab norms and their hierarchy}
For a Banach space $X$, write
\[
\|V\|_{L^\nu_\alpha X}
=\|\langle t\rangle^{\alpha/2}V\|_{L^\nu([0,T];X)}.
\]
We may replace $\langle t\rangle$ by $\tau=T_0+t$. Here $T_0$ is fixed
after all spatial radii, and differences on a fixed initial interval are charged
to the input. The three levels are
\begin{equation}\label{co2:levels}
\begin{gathered}
\begin{array}{c|c|c|c}
U&U_l&s&A_i\\ \hline
h_{\rm co}&h_{\rm lab}&10&0\\
h_{\rm co}&h_{\rm lab}&9&A_H\\
u&\psi&8&A
\end{array}\\[3pt]
A_H=1-\eta_*,\qquad A=1.36,\qquad A_S=1.06.
\end{gathered}
\end{equation}
where $\eta_*>0$ is sufficiently small. In particular $2A_H>A$.
Fix a compact physical neighborhood of trapping, disjoint from the horizon,
the inner face, the spatial end and all coefficient probes. Also choose a
scalar spatial operator
\begin{equation}\label{co2:directional-cutoff}
B_{\rm nt}=1-\Psi_{\rm tr}
\end{equation}
such that $\Psi_{\rm tr}$ has compact interior kernel support and microsupport
in the stationary-elliptic cone $\mathcal T$. At high frequency $B_{\rm nt}$
vanishes in a neighborhood of the projections of \emph{both} temporal trapped
sets, with a margin valid for nearby frozen parameters. Slightly nested
versions will be used without changing notation. The distinction between
this directional cutoff and the physical trapping neighborhood is important.
For every level in \eqref{co2:levels}, the compact budget $\mathsf K_R$ includes
on fixed enlarged compact sets
\begin{equation}\label{co2:compact-norms}
\partial_t^bU\in
L^\infty_{A_i}H^{s-b}\cap L^2_{A_i}H^{s-b-1/2},
\qquad 0\le b\le s.
\end{equation}
It includes full order $L^2_{A_i}H^{s-b}$ outside the physical trapping
neighborhood, and additionally
\begin{equation}\label{co2:directional-norms}
B_{\rm nt}\partial_t^bU\in L^2_{A_i}H^{s-b},
\qquad b=0,1,
\end{equation}
on fixed compact reaches. Full endpoint order is always retained. We do
\emph{not} require \eqref{co2:directional-norms} for all higher time jets in the
physical trapping neighborhood.
All compact norm sums use a finite collection of nested reaches and cutoffs.
The collection contains the kernel and probe supports, the error supports in
the estimates below, and the fixed starting overlaps with the far region.
It may be enlarged after the far starting radius $R_0$ is chosen, but is fixed
before the upper matching radius $R$. Define $\mathsf K_R$ as the sum of the
listed norms on this collection, with fixed positive coefficients, together
with the differential-end norms described next. Constants on a fixed reach
may depend on it, but not on $R$. The first-jet norms extend to a
nonincreasing upper cutoff equal to one through $3R$ and zero
after $4R$. In the differential exterior region they are the energy, bulk and
Hardy norms of
\[
\widetilde r\,\partial_y^\mu U,\qquad |\mu|\le s-1,
\]
with bulk weight $\widetilde r^{-1-\delta}$ and time weight $\tau^{A_i}$ in
the radial-round measure; $\widetilde r$ is the radius of the constant affine
asymptote of $\bar g$, and $\delta>0$ is arbitrarily small. We will construct
the corresponding positive energy below. Mixed norms on other bounded annuli
up to the matching radius may be added with sufficiently small positive
$R$-dependent normalizations. Their unnormalized estimates are used only in
products or in the choice of $T_0$, never as uniform linearly contracting
controls. Those normalizations are fixed after $R$ and before the bootstrap
and data smallness. This convention avoids asserting radius-independent
equivalence of all unscaled mixed norms through the upper shell.
We use a far budget $\mathsf F$ beginning at a fixed large radius $R_0$, chosen
before $R$. Its complete definition and estimate are given in the next
section; the following components suffice here. In emitter coordinates
\[
t=q+r,\qquad z=-\lambda(q)+rn,\qquad |n|=1,
\]
let $T_l=\partial_{t,\rm lab}$, $X=T_l+n\cdot\partial_z$, and
$Z=(rX,\Omega_{q,r})$, with $\Omega_{q,r}$ the sphere rotations. The good
bulk norms of $(1,Z)Z^I T_l^bU_l$, for $|I|+b\le s-1$ at each
level, in $dt\,dr\,d\omega/r$, have squared weight
\begin{equation}\label{co2:far-weights}
W_i=r^{p_i-A_i}(t+10r)^{A_i},
\qquad (p_0,p_H,p_P)=(0.64,1-\eta_*/6,1.62).
\end{equation}
The bad first-derivative norms of $r\partial Z^I T_l^bU_l$ have the weights
of \Cref{fe:F1}; on $t\ge Cr$ the low orders needed in the sliced estimate
are bounded below by $r^{-0.03}\langle t\rangle^{A_i}$. On this timelike
region we also use
\begin{equation}\label{co2:far-pointwise}
|h_{\rm lab}|\lesssim\mathsf N r^{-1+0.01}(t/r)^{-A_H/2},
\qquad
|\partial h_{\rm lab}|\lesssim\mathsf N r^{-1+0.03}t^{-A_H/2}.
\end{equation}
The relevant co-moving and emitter radii are uniformly comparable.
Set $\mathsf S_a=\|a\|_{L^2_A}$. Since $A>1$, this controls
$\int|a|$ and hence the change of $p$. Filtered feedback from the $u$ budget
controls the required positive acceleration derivatives in $L^2_A$, and then
their finite list of endpoint norms. The total bootstrap size $\mathsf N$
includes $\mathsf K_R,\mathsf F,\mathsf S_a$ and a static budget.
To define it, put $k_{\rm stat}=g_{\rm co}-\overline G_{p,c}$ and let
$K$ run over the fixed compact reaches, with $K_{\rm off}$ the fixed
sets outside the physical trapping neighborhood. Sum the quantities
\[
\sum_{b=0}^{8}\|\partial_t^bk_{\rm stat}\|_{L^2_{A_S}H^{8-b}(K)}
+\sum_{b=0}^{8}\|\partial_t^{b+1}k_{\rm stat}
\|_{L^2_{A_S}H^{8-b}(K_{\rm off})}.
\]
These are assumed finite-slab budgets at this stage; the sliced estimate
will bound them. All reaches are fixed before $R$. Write $e_{\rm in}$ for the input through the base orders.
Constants denoted $C_{\rm fix}$ may depend on all scales fixed before the
final data smallness, but never on the lifespan. These are the norm conventions
of \eqref{co2:compact-norms}, with the additional directional control
\eqref{co2:directional-norms}.
The role of the directional control is particularly concrete in the sliced
stationary equation. On each fixed compact reach its source will have
spatial order six in all directions and order seven after $B_{\rm nt}$.
An order-$-2$ inverse on the complementary
elliptic cone supplies an $H^8$ correction; subtracting it leaves an $H^7$
source for the ordinary one-order stationary estimate. This recovers an
$H^8$ static residual whose $L^2_{A_S}$ bound, with $A_S>1$, makes its
low coefficient norms integrable in time. The argument in
\Cref{co2:sliced} uses only the directional first jets specified above.
The low-frequency estimate, proved first, has a different role: it makes
lower compact errors and the linear parameter forcing small enough to absorb
in the final compact energy estimate.
\subsection{The compact low estimate}
Choose a sufficiently large fixed $N_-$ and let $\mathsf l$ be the sum, on
the compact reaches used below, of the $L^2_{A_H}H^{-N_-}$ norms of
$h_{\rm co},\partial_t h_{\rm co}$ and the $L^2_AH^{-N_-}$ norms of
$u,\partial_tu$. Enlarging $N_-$ merely weakens this norm.
\begin{proposition}[Low-frequency transfer]\label{co2:low-transfer}
Under the simultaneous finite-slab budgets above, there is $\kappa_0>0$ such
that, after choosing $R$ and then $T_0$ sufficiently large,
\begin{equation}\label{co2:low-bound}
\mathsf l\le C R^{-\kappa_0}(\mathsf K_R+\mathsf F)
+C_{\rm fix}(e_{\rm in}+\mathsf N^2).
\end{equation}
The exponent can be fixed before enlarging any prior compact reach. Such
enlargements change constants but not the available radius gain.
\end{proposition}
\begin{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 \eqref{co2:compact-norms}. 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
\begin{equation}\label{co2:wronskian}
\int \chi_R' b^{rr}
(v_*\partial_rU_*-U_*\partial_rv_*)r^2\,dr\,d\omega,
\qquad r=r_y.
\end{equation}
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
\eqref{co2:wronskian} 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
\begin{equation}\label{co2:low-shell}
C R^{-(p_i-A_i)/2}\mathsf F
+C_{\rm fix}(e_{\rm in}+\mathsf N^2).
\end{equation}
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 \eqref{co2:low-bound}. 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.
\end{proof}
\subsection{Frozen time-local energies}
We now construct the energy supplying the compact budget. The construction
has two separate roles: a compatible frozen transport gives a half-order loss
at trapping, while strict sign away from both trapped sheets gives the extra
directional first-jet estimate. Subsequently we will compare the actual
equation with this frozen energy; we will not assume that the actual evolving
metric has a stationary trapped set.
\begin{proposition}[Frozen energy]\label{co2:stationary-energy}
Fix a smooth nearby stationary parameter pair $(p,c)$ and an integer $s\ge8$.
The frozen operator $\mathbf L_{p,c}$ admits a positive time-local first-jet
energy, uniformly for nearby parameters, with full endpoint orders
$H^s\oplus H^{s-1}$. On each fixed interior reach its favorable derivative
controls
\begin{equation}\label{co2:frozen-dissipation}
\|\mathcal Y\|_{H^{s-3/2}}^2
+\|B_{\rm nt}\mathcal Y\|_{H^{s-1}}^2
-C_s\|\mathcal Y\|_{H^{s-2}({\rm enlarged})}^2,
\qquad \mathcal Y=(\langle D_y\rangle U,D_tU),
\quad D_t=-i\partial_t.
\end{equation}
It also controls full mixed physical-nontrapping orders after equation
recovery. At the inner and outer differential regions the energy is an
ordinary wave current with strict bulk sign and an exiting inner flux. All
remaining compact derivative errors have total field order at most $s-1$.
The compact additions to the differential currents have strictly interior
spatial kernel support.
\end{proposition}
\begin{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.
\paragraph{First-jet reduction.}
With $(p,c)$ fixed, the time-acceleration coefficient is an invertible
multiplication operator plus a compact interior operator of spatial order
$-1$, small at a base order. Its inverse is multiplication plus negative
order at every smooth order, by the spatial parametrix and derivative gain.
After solving this coefficient and making the elliptic first-jet rescaling
in \eqref{co2:frozen-dissipation}, the evolution has two smooth blocks with
separated scalar real principal roots. Separation follows from spacelike
slicing, not from timelikeness of the stationary Killing vector. An
eigenbasis conjugation, followed by the order-by-order off-diagonal equations
dividing only by the root gap, diagonalizes to any required lower order on
compact interior supports.
Write $m_s=2s-2$ for the order of the jet quadratic form. On each block its
diagonal principal form, expressed on $U$ polarizations, is a positive
homogeneous Hermitian form times the wave action
\[
|\det\bar g|^{1/2}|\bar g^{t\alpha}k_\alpha|,
\]
of total order $2s$. Its diagonal energy derivative is ray transport of the
extra Hermitian form, including the nonscalar amplitude transport. The
scalar geometric-optics divergence is absorbed by wave action and spatial
pushforward. Equivalently, this is the symmetrized derivative formula in
stationary half-density coordinates for the first-jet system.
Here is the operative compact estimate that this construction will establish.
Let $K$ lie inside a fixed buffered interior reach $K'$, and let $\mathcal Y$
be a smooth first-jet test field supported in $K'$. Write its normalized
first-jet equation as $\partial_t\mathcal Y=\mathcal G_{p,c}\mathcal Y
+\mathcal F$, where $\mathcal F$ is the full residual, and denote the
constructed positive quadratic form by
$\mathcal E_s=\langle\mathcal A_s\mathcal Y,\mathcal Y\rangle$.
The operator $\mathcal A_s$ has order $m_s$, and
$\mathcal E_s\asymp\|\mathcal Y\|_{H^{s-1}}^2$ on these supports.
For fixed $(p,c)$ the estimate is
\begin{equation}\label{co2:compact-energy-form}
\begin{split}
\frac{d}{dt}\mathcal E_s
&+c_s\|\mathcal Y\|_{H^{s-3/2}(K)}^2
+c_s\|B_{\rm nt}\mathcal Y\|_{H^{s-1}(K)}^2\\
&\le C_s\|\mathcal Y\|_{H^{s-2}(K')}^2
+2\operatorname{Re}\langle\mathcal A_s\mathcal Y,\mathcal F\rangle.
\end{split}
\end{equation}
No support preservation by the normal-coefficient inverse is assumed:
$\mathcal F$ includes the entire residual. This formula describes the
buffered compact part; the differential end currents below retain their own
bulk norms and boundary flux. Higher mixed time jets are subsequently
recovered from the equation and the corresponding source jets.
\paragraph{The half-order loss at trapping.}
Use the compatible fiber metric constructed in the calibrated stationary
family. Multiply it by
\begin{equation}\label{co2:trap-weight}
|k(\partial_t)|^{2s-1}b_C,
\qquad b_C=\exp\bigl(-C(\varphi_+^2-\varphi_-^2)\bigr),
\qquad \varphi_\pm=z_1\pm z_2.
\end{equation}
Here $C>0$ is a scalar constant, unrelated to the gauge operator. The normal
variables for the exact Kerr reference satisfy
$V\varphi_\pm=\pm\lambda\varphi_\pm$, with $\lambda>0$. Exact
transport compatibility therefore gives a negative principal derivative of
relative size
$2C\lambda(\varphi_+^2+\varphi_-^2)$. Gauge cutoffs below saturation
contribute only fixed lower errors. This applies to both roots, including
where the spatial base intersects the ergoregion.
The order immediately below this vanishing principal derivative needs care.
In a stationary half-density Weyl chart on a diagonalized block, start the
energy with $e_{m_s}=b_C e^0_{m_s}$ and take its next symbol to be $b_C$
times a $C$-independent choice. Since the order-one evolution has real scalar
principal symbol $p_{\rm hw}$, its even second Weyl bracket cancels in the
degree-$(m_s-1)$ derivative. Only first derivatives of $b_C$ can enter from
$e_{m_s}$. A stationary frame change adds to the next symbol terms linear
in $db_C$, with $C$-independent coefficients. In this frame the next
symbol has the form
\begin{equation}\label{co2:frame-subprincipal}
e'_{m_s-1}=b_C r_0+\sum_\alpha(\partial_\alpha b_C)r_\alpha,
\end{equation}
where the normalized symbols $r_0,r_\alpha$ are independent of $C$.
On the trapped set,
\[
db_C=0,\qquad H_{p_{\rm hw}}(db_C)=0,
\]
the second identity following because $H_{p_{\rm hw}}b_C$ vanishes
quadratically. More explicitly, for $V=H_{p_{\rm hw}}$,
$V(\partial_jb_C)=\partial_j(Vb_C)-(\partial_jV^k)\partial_kb_C$;
both terms vanish there. Thus the critical derivative on trapping is bounded
independently of $C$ relative to the unweighted scales. Off-diagonal errors
are handled by the preceding diagonalization, chosen before $C$.
Factor $C$ from the principal dissipation. On a normalized symbol patch
let $g$ be the positive Hermitian square root of
$2\lambda b_Ce^0_{m_s}$ and set
$G_\pm=\operatorname{Op}^{\rm w}(g\varphi_\pm)$, chosen self-adjoint.
Both factors have order $m_s/2$ and use the same matrix weight. Their
squares have principal sum
$2\lambda b_Ce^0_{m_s}(\varphi_+^2+\varphi_-^2)$.
Their degree-$(m_s-1)$ square errors vanish at trapping: at the zero of a
scalar $\varphi$, the first derivatives of $g\varphi$ are $g$ times
scalar derivatives. For a smooth test section $z$, the elementary inequality
\begin{equation}\label{co2:trap-uncertainty}
\|G_+z\|^2+\|G_-z\|^2
\ge \pm\langle i[G_+,G_-]z,z\rangle
\end{equation}
follows from the nonnegativity of
$\|(G_+\mp iG_-)z\|^2$. On trapping the commutator has, up to the
fixed sign convention, symbol $g^2\{\varphi_+,\varphi_-\}$ of order
$m_s-1$. The bracket is nonzero with uniform normalized size, and $g^2$
on trapping is positive and independent of $C$.
Retain a fixed fraction of the original $C$-scaled squares and apply
\eqref{co2:trap-uncertainty} to the rest, choosing the sign on each patch.
The resulting positive order-$(m_s-1)$ form dominates the $C$-independent
critical derivative once $C$ is large. Fix $C$ first and then shrink the
neighborhood on which that form is strict. Outside it, the retained squares
supply strict principal sign. Operator-square partitions and stationary
frame changes have on trapping only the critical errors just controlled.
Spatial G\aa rding leaves order $m_s-2$, with constants now allowed to
depend on the fixed $C$.
Consequently the bulk controls $U$ in $H^{s-1/2}$ and $U_t$ in
$H^{s-3/2}$ modulo total field order $s-1$, while the endpoint orders stay
full. This argument uses symplectic transversality and exact compatible
transport, not stationary ellipticity over a base annulus.
\paragraph{The horizon and compact propagation region.}
At the horizon radial set take action times $b_1^{2s-1}$ and a positive
tensor metric adapted to the horizon-generator null flag. In generator
parameter, parallel transport scales the null line at rate $-\kappa_+$,
the opposite quotient at rate $\kappa_+$, and rotates the screen quotient.
Flag scaling makes bounded null rotations arbitrarily small. The rank-two
amplitude norm consequently grows at most at rate $2\kappa_++\epsilon'$.
The local HHV additions are chosen sufficiently small at the center; the
additional high-frequency gauges miss the horizon, and nearby target changes
are small. The frequency rate is $-\kappa_+$ in the same parametrization,
so
\[
(2s-1)\kappa_+>4\kappa_+
\]
gives a strict margin, preserved for nearby parameters. Increasing $s$ only
improves it.
Use the source-neighborhood function and the escape alternatives proved
earlier to extend the principal forms over a large fixed cylinder. The
remaining compact set, outside smaller trapping and horizon neighborhoods
where the signs have
already been obtained, is covered by finitely many flow tubes. The ends of
each central bump lie outside this remaining set: they are in exit regions
or in collars with an already strict principal sign. A stable or unstable
tail may end in the trapping collar; it need not reach a cylinder boundary.
Integrating along the tube gives the required positive derivative, with any
unfavorable endpoint bump confined to those controlled collars. Their
existing strict signs absorb these endpoint terms after fixed weight choices;
in the horizon collar one uses the increasing source function.
The correction is constant near the exact trapped set, so it does not alter
the preceding critical-order calculation. Multiplication of the metric by
its decreasing exponential supplies the missing strict signs. Smooth exact-Kerr models and
finitely many tubes make these choices uniform nearby. This is a separate
construction on each root block; it does not use positivity of a stationary
Killing current throughout the exterior.
\paragraph{Literal differential currents near the boundaries.}
Outside all compact kernels, the stationary metric has a constant affine
Minkowski leading model $(\widetilde t,\widetilde y)$, with
$\widetilde y=A_{\rm aff}y$ and $\partial_t$ parallel to
$\partial_{\widetilde t}$. Let $\widetilde r=|\widetilde y|$ and use the
wave stress-energy currents, summed on components, for
\begin{equation}\label{co2:outer-current}
\chi_\mu=\widetilde r\,\partial_y^\mu U,\quad |\mu|\le s-1,
\qquad
N=\partial_{\widetilde t}
+(f_\infty-\widetilde r^{-\delta})\partial_{\widetilde r},
\quad 0.90$ 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
\begin{equation}\label{co2:sliced-equation}
L_p(0)(\chi_t h_{\rm co})
=\chi_t(\mathcal R_h-f_B)+[L_p(0),\chi_t]h_{\rm co}.
\end{equation}
The defect $\mathcal R_h$ subtracts the nonstationary reduced equation and
compares all field-dependent terms with their stationary linearized action at
$G_p$.
\paragraph{Compact source orders.}
On a fixed reach the order-four estimate is
\begin{equation}\label{co2:compact-defect}
\|\mathcal R_h\|_{L^2_A\overline H^4}
+\|f_B-Q(0)a\|_{L^2_A\overline H^4}
\le C_{\rm loc}\mathsf K_R+C_{\rm fix}(e_{\rm in}+\mathsf N^2).
\end{equation}
The same source has order six globally, and order seven after $B_{\rm nt}$.
To verify the last, potentially limiting assertion, subtract first at the
frozen target $G_p$ and frozen additions $E_{\rm g}(p,0),C(p,0)$, then
linearize in the field. The linear time-deletion terms have order at
most two and at least one time derivative.
Changed coefficients cost $h,c$, their jets, or acceleration jets and bounded
memories, including $B-G_p$. True derivatives of the family and its projections
minus frozen derivatives have the same allocation. Ricci and wave-gauge terms
are quasilinear of order two, with smooth undifferentiated coefficients.
The extra gauges use order-zero or lower spatial kernels and at most one
time derivative apiece. Thus at most two ordinary field differentiations
precede the spatial source norm, including within compositions.
On the target, the first variation is $Q(\partial_t)a$ with bounded-length
memory. Its difference from $Q(0)a$ starts with $a'$, controlled by the
$u$ feedback and startup. The nonlinear difference contains a strong factor
by exact freezing. A sufficiently large fixed filter order $P$ supplies every
fixed spatial order here. The coefficient probes are smooth and physically
nontrapped, so their full-order bounds are available.
Order six is below the global half-loss bulk orders: use strong order eight
for $u$, weak order nine for $h$, and $2A_H>A$. At source order seven the
only limiting factors are
\[
\partial_y^2 h,\qquad\partial_yu,\qquad\partial_tu.
\]
They have bulk regularity $H^{6.5}$ globally and $H^7$ after $B_{\rm nt}$.
Their coefficient factors are undifferentiated and have endpoint $H^9$.
For a purely spatial second derivative of $h$, its coefficient difference
also has a weak or strong weighted small factor: the exact stationary linear
part has already been subtracted. We perform this calculation in the reduced
equation \emph{before} inverting its acceleration coefficient. Hence an
unsmoothed second field derivative has only undifferentiated metric or target
coefficients. A second-ordinary-derivative placement from an added order-zero
gauge instead carries a time-differentiating kernel of spatial order at most
$-1$; differentiating that kernel in a parameter does not change this spatial
order. All other connections and inverse-coefficient derivatives distribute
at most first derivatives on individual field factors. In particular, the
argument never commutes a first-field-derivative coefficient past an unsmoothed
second derivative.
The needed cutoff estimate in three dimensions is
\begin{equation}\label{co2:cutoff-commutator}
\|[B_{\rm nt},f]v\|_{H^7}
\le C\|f\|_{H^9}\|v\|_{H^{6.5}}.
\end{equation}
It holds also for matrix-valued $f$, since $B_{\rm nt}$ is scalar.
For completeness, in a compact symbol chart Fourier transform the symbol
difference $b(x,\eta+\ell)-b(x,\eta)$ in $x$. It decays rapidly in the
dual $x$ variable and is bounded there by a symbol seminorm times
$\min(1,|\ell|/\langle\eta\rangle)$. After the input Sobolev weight is
removed, the output multiplier is bounded by
\[
\langle\eta+\ell\rangle^7\langle\eta\rangle^{-6.5}
\min(1,|\ell|/\langle\eta\rangle)
\le C\langle\ell\rangle^7.
\]
Indeed, for $|\ell|\le|\eta|/2$ it is at most
$C|\ell|\langle\eta\rangle^{-1/2}$; on the complementary region use
$\langle\eta+\ell\rangle\le C\langle\ell\rangle$.
The smooth $x$-frequency convolution adds only a fixed polynomial weight.
The weighted Fourier
$L^1$ norm of $f$ is bounded by $\|f\|_{H^9}$ because
$9-7>3/2$. Young's inequality proves \eqref{co2:cutoff-commutator}; smoothing
remainders are harmless. Commuting the scalar cutoff through a fixed smooth
order-zero matrix kernel similarly gains one order; negative-order kernels
have additional room. Slightly enlarged supports therefore allow the cutoff
to be placed on the limiting factor in each composition. Spatially constant
time-dependent coefficients multiply this argument; no time-dependent cutoff
commutator is involved. This proves the asserted directional source gain.
\paragraph{Far source orders.}
Put $h_l=h_{\rm lab}$ and, on the cut,
\[
\Delta_a=C\int_{t-C'r_y}^t|a(\sigma)|\,d\sigma
\lesssim\mathsf S_a t^{-A/2}r_y^{1/2}.
\]
In addition to spatially constant leading combinations of
$T_l\psi$ and $\partial_z\psi$, the far sliced defect is bounded by
\begin{align}\label{co2:sliced-far-source}
&(r_y^{-1}+\Delta_a)|\partial\psi|+r_y^{-2}|\psi|\notag\\
&\quad+\Delta_a\bigl(|\partial_y^2h_l|
+r_y^{-1}|\partial h_l|+r_y^{-2}|h_l|\bigr)\notag\\
&\quad+|h_l||\partial^2h_l|+|\partial h_l|^2
+r_y^{-1}|h_l||\partial h_l|+r_y^{-2}|h_l|^2.
\end{align}
This proves the stated source bounds. To check its applicability,
freeze the affine worldline at the current slice. Stationary time deletion
then uses $T_l=v(t)\partial_y$, whereas the actual field satisfies
$T_lh_l=\psi+v_s\partial_yh_l$. The frozen co/lab basis change is spatially
constant. Comparing in the actual lab equation avoids unsmoothed $v'$ terms.
The first-moment cancellation gives $x-y=O(r_y\Delta_a)$, and the target
jets of order $b\le2$ differ from frozen jets by
$O(\Delta_a r_y^{-1-b})$. The affine flat leading term cancels exactly;
the Kerr corrections, coefficient derivatives and nonlinear wave-gauge terms
give \eqref{co2:sliced-far-source}. The cutoff commutator contributes
$O(r_y^{-1}|\partial_yh_l|+r_y^{-2}|h_l|)$ on $r_y\sim c_1t$.
\paragraph{Acceleration pairing.}
Pair \eqref{co2:sliced-equation} with the stationary adjoint kernel
$\mathcal K_p^*$. The stationary operator term vanishes, and its pairing
with $Q(0)$ has full rank. Fix a sufficiently large $Y$ before $R_0$.
Replacing $\chi_t f_B$ by $Q(0)a$ on $r_y\le Y$ costs the compact
comparison above. Beyond $Y$ it costs
$CY^{-1+\epsilon_1}\mathsf S_a$, which is absorbed by the choice of $Y$:
the adjoint kernels are $O(r_y^{-2+\epsilon_1})$,
$\chi_t f_B=O(r_y^{-2}\chi_t\widetilde a)$, and enlarged averaging over
comparable radii is uniformly bounded in the strong weighted norm on the cut.
Choose $0<\epsilon_1<\zeta_1<0.13$, with $\zeta_1$ small.
For leading terms beyond $CR_0$, write $T_l=X-n\cdot\partial_z$ and
integrate the remaining spatial derivative once on the time slice, as in
\Cref{fs:uniform-coupling}. The test coefficient is
$O(r_y^{-2+\epsilon_1})$ and gains an inverse radius upon differentiation.
This includes derivatives of the cutoffs and $n$, since the emitter-coordinate
first derivatives require only bounded velocities. It therefore suffices to
control
\[
\|r^{\zeta_1}t^{A/2}(rX\psi,\psi)\|_{L^2(dt\,dr\,d\omega/r)}
\le C\mathsf F.
\]
Here $C$ is independent of $R_0$, precisely because $p_P-A=0.26$.
For the other far terms $S'$ in \eqref{co2:sliced-far-source}, use directly
the norm of $r_y^{1+\zeta_1}t^{A/2}S'$ in this measure. The bad norm of
$r_y\partial\psi$ loses at most $0.015$ powers of radius, compensated by
$r_y^{-1}+\Delta_a$. Using tangential $r^{-1}Z$, $X$, and
$T_lh_l=\psi+v_s\partial_yh_l$, with $v_s$ small, gives
\[
|\partial_y^2h_l|
\lesssim |\partial\psi|+r_y^{-1}|Z^{\le1}\psi|
+r_y^{-2}|Z^{\le2}h_l|.
\]
The differentiated coordinate coefficients are $O(r_y^{-1})$, including
$\partial v_s$. In weak terms use
$t^{A/2}\Delta_a\lesssim\mathsf S_a r_y^{1/2}$. For quadratic terms
combine \eqref{co2:far-pointwise} with weak bulk norms of
$h_l,r_y\partial^{1,2}h_l$, losing at most $0.015$ powers. For example the
extra radius factor for $h_l\partial^2h_l$ is bounded by
\[
C\mathsf N r_y^{\zeta_1+0.015-1+0.01+(A-A_H)/2},
\]
a negative power. The cutoff commutator has the required extra inverse
radius. This proves \eqref{co2:sliced-acceleration}; dependence on $R_0$
occurs only in the fixed compact and starting regions.
\paragraph{The conic elliptic correction.}
Apply stationary Fredholm regularity at growth power $\gamma=0.96$ to
$\chi_t h_{\rm co}-H_C(p)c$, with the seven decaying-mode measurements and
the ten constant-asymptote measurements from the preceding section. The
remaining measurement costs
$C_{\rm loc}(\mathsf S_a+\mathsf K_R+e_{\rm in})$: only $\ell h_{\rm co}$
remains, controlled because $F(0)$ is invertible and
$(F(\zeta)-F(0))/\zeta$ is stable on
$\partial_t(\chi_{\rm st}\ell h_{\rm co})$.
At infinity the source needs only
$r_y^{2-\gamma}t^{A_S/2}S'$ in $L^2(dt\,dr_y\,d\omega/r_y)$, with
equation derivative order zero. The direct bounds just proved supply it,
including $\chi_t f_B$ by strong averaging. Even the unsubtracted leading
terms fit, since
\[
1-\gamma+0.015<(A-A_S)/2.
\]
On a fixed enlarged compact reach let $f$ denote the full sliced source.
We have established $f\in H^6$ and $B_{\rm nt}f\in H^7$, with the
claimed time-weighted norms. Choose a properly supported order-$-2$
parametrix $\mathcal Q_p$ for $L_p(0)$ on an enlarged microsupport of
$\Psi_{\rm tr}$, so that
\[
L_p(0)\mathcal Q_p\Psi_{\rm tr}
=\Psi_{\rm tr}+\mathcal R_p,
\]
where $\mathcal R_p$ is smoothing to any prescribed finite order. Include
the compact pseudodifferential lower terms in the full symbol construction;
finite-rank terms and off-cone leakage are smoothing. The correction
$\mathcal Q_p\Psi_{\rm tr}f$ is $H^8$, and its fixed coefficient
measurements have the same bound. Subtracting it leaves the $H^7$ source
$B_{\rm nt}f-\mathcal R_pf$. The ordinary stationary estimate with its
one-order gain now gives $H^8$ for the remainder.
The stationary application therefore uses two spatial regularity regimes:
source order six with the additional $B_{\rm nt}$ order seven on the fixed
compact reaches, and only the weighted source order zero specified above
near infinity. Switch between them across a fixed exterior annulus where
the \emph{full} spatial symbol is elliptic, retaining a margin of high source
regularity through the switch. The stationary propagation and radial
estimates inside, and $b$-ellipticity with the same indicial gap outside, are exactly the
variable-order observation in \Cref{st:stationary-estimates}. Lowering far
derivative orders introduces no new kernel ambiguity. This argument needs
conic ellipticity near projected trapping and ordinary ellipticity far out;
it does not assume ellipticity on an entire trapping base annulus.
Finally, replacing $G_p+H_C(p)c$ by $\overline G_{p,c}$ costs quadratic
$c$ terms; replacing $B$ by $G_p$ costs bounded acceleration memories.
Mixed jets with a time derivative are obtained directly from $u$, not by
differentiating the parametrix: they use order seven globally and full order
eight outside physical trapping. Positive derivatives of the reference and
projections use strong parameter and field jets through order nine for $c$.
At order nine the leading term is $\ell_C^p\partial_t^8u$ at the physically
nontrapped probes; the highest strong order is needed only in bulk. This
proves \eqref{co2:sliced-static} and its extra-time-derivative assertion.
\end{proof}
\subsection{Comparison with the actual principal part}
The frozen energy alone does not estimate the nonlinear equation. Its
principal wave coefficients must be replaced by the actual ones without
losing a derivative. The sliced estimate makes the lower coefficient
differences integrable; the differential principal difference is instead
absorbed algebraically into the energy.
\begin{proposition}[Actual-principal comparison]\label{co2:actual-comparison}
For each level $(U,s,A_i)$ in \eqref{co2:levels}, let
$d_M^{\alpha\beta}$ be the change from $\bar g$ to $g$ of the differential
order-two wave coefficient in stationary normalization. On fixed compact
reaches the reduced equation has the form
\begin{equation}\label{co2:source-ledger}
\mathbf L_{p(t),c(t)}U+d_M^{\alpha\beta}\partial_{\alpha\beta}U
=S_i^{\rm lin}+S_i^{\rm err},\qquad
S_i^{\rm lin}=\pm Q(\partial_t)b_i,\qquad
b_i=a,a,a',
\end{equation}
up to common normalizations, where
\begin{equation}\label{co2:error-source-space}
S_i^{\rm err}\in L^1_{A_i}H^{s-1}+L^2_{A_i}H^{s-1/2},
\qquad
\|S_i^{\rm err}\|\le C_{\rm fix}\mathsf N^2
+C_{\rm fix}e_{\rm in}.
\end{equation}
On larger bounded differential annuli only $L^2_{A_i}H^{s-1}$ is required.
There is an actual-principal correction of the frozen energy whose additional
integrated derivative errors are bounded in squared estimates by
$C_{\rm fix}\mathsf N^3$, up to input terms. The frozen half-loss bulk and
directional improvement remain on the left side of that estimate.
\end{proposition}
\begin{proof}
We compare the frozen and evolving equations, retaining the actual
principal coefficients and checking every source derivative count.
\paragraph{Integrable low differences.}
Acceleration and $\dot c$ have strong $A$-weighted bulk and endpoint bounds
at the required differentiated orders; $c$ has weak $A_H$ bounds and
unweighted high bounds. In detail, derivatives of $c$ of orders one through
nine use the full strong $u$ jets at the probes. The unweighted top $h$
control also supplies $c^{(10)}$, but no strong weighted estimate for this
jet is used. The mixed recovery below differentiates the $h$ equation at
most eight times and the differentiated $u$ equation at most six times;
its parameter placements require at most $c^{(9)}$. Derivatives of $p$ use
the fixed smooth filter and startup bounds. On compact sets $B-G_p$ and true derivatives
minus affine-position frozen derivatives are bounded strong acceleration
memories, with no position growth. Put
\begin{equation}\label{co2:recentered-fields}
\xi=\bar g-G_p,\qquad k_h=h_{\rm co}-\xi,\qquad X_0=B+\xi.
\end{equation}
The reference increment $\xi=O(c)$ has fixed smooth spatial orders.
By Proposition~\ref{co2:sliced}, $k_h$ and $g_{\rm co}-\bar g$ have
integrable low norms of size $C_{\rm fix}\mathsf N$, apart from initial
terms. Indeed $A_S>1$ and Cauchy--Schwarz turn their weighted $L^2$ mixed
order-eight bounds into $L^1_t$ bounds. These control spatial Lipschitz norms
and the first true time derivative of low principal differences. Likewise
$|a|+|\dot c|$ and the true-minus-frozen reference derivatives are integrable.
High placements are controlled by their ordinary field and coefficient
budgets, not by an asserted integrable high norm. On larger bounded annuli
the full physical-nontrapping and far overlap controls suffice with
$R$-dependent constants in products, including the strong comparison of $u$
and $\psi$.
\paragraph{The nonlinear source ledger.}
For $U=h_{\rm co}$, subtract the reduced expression at $g=B$, namely $f_B$.
Treat target, gauge additions and compensator as fixed coefficients and
inputs, also within kernel applications. Taylor-expand both
$X_0+k_h$ and $X_0-\xi$ at the \emph{same} $X_0$. For the reduced
differential and kernel expression $\mathcal D_{\rm red}$ the field-dependent
part is
\begin{equation}\label{co2:taylor-recentering}
\mathcal D_{\rm red}'(X_0)h_{\rm co}
+\operatorname{Rem}_{X_0}(k_h)
-\operatorname{Rem}_{X_0}(-\xi).
\end{equation}
Each remainder is at least quadratic in its displayed increment and jets,
with smooth coefficient multiplicities. This common expansion point avoids
an unabsorbed high first jet multiplied only by a single low slowly decaying
$\xi$. By stationary tangency, the low residual gauge of $X_0$ at freezing
is $O(c^2)$; away from freezing it also has integrable acceleration memories
and parameter derivatives, including the compensator. The linear term in
\eqref{co2:taylor-recentering} is $\mathbf L_{p,c}$, up to the differential
principal change and smooth integrable lower coefficient errors. In
particular the omitted derivative of the wave-gauge operator acting on the
gauge of $X_0$ is of this class, since $H_C$ solves the stationary linear
gauge equation.
The smooth $-\xi$ remainder is quadratic. In the other remainder count the
changed leading coefficient on $\partial^2k_h$ in
$d_M\partial^2h_{\rm co}$; its leftover action on $\partial^2\xi$ needs
at most order-zero $h$ factors. Remaining ordinary products have at most
one derivative on each $k_h$ factor, or a high undifferentiated field times
fixed smooth reference derivatives. A low $k_h$ factor through order two,
with a fixed Sobolev buffer, is integrable. Thus high first derivatives fit
$L^1_{A_i}H^{s-1}$ using endpoints, and high undifferentiated factors fit the
half-loss bulk-source space. Smooth quadratic $\xi$ products use
$2A_H>A$.
For $U=u$, differentiate the full $h$ equation. Its field derivative is
at $g$, with the principal difference as in \eqref{co2:source-ledger}.
Lower coefficient differences, including the residual-gauge terms, remain
integrable at low order. A high order-two $h$ jet multiplied by $u$ or a
differentiated coefficient drift uses unweighted $H^{7.5}$ bulk from the top
mixed budget and a strong low endpoint. A high first derivative of $u$
uses an integrable low difference. A high differentiated $h$ factor and a
low differentiated $u$ factor multiply their unweighted and strong $L^2_t$
budgets into $L^1_{A_i}H^{s-1}$. Derivatives of $B,E_{\rm g},C,\Theta$
relative to the zero-field expression contain strong coefficient factors.
The spatial tame product estimates in dimension three therefore give the
claimed source orders. The following representative borderline placements
record the distinct uses of endpoints and bulk norms. Each low factor uses
the fixed spatial multiplication buffer in its compact budget; integrable
coefficient differences use the sliced bound.
Here $K_{-1}$ denotes a kernel of spatial order $-1$, and $d_1$ an
undifferentiated coefficient difference.
\begin{center}
\small
\begin{tabular}{@{}p{0.21\linewidth}p{0.43\linewidth}p{0.25\linewidth}@{}}
\toprule
Placement & Factors used & Source space \\
\midrule
High first derivative of $k_h$ times a low derivative, at the $h$ levels
& Weighted endpoint $H^{s-1}$ times a low $L^1_t$ factor
& $L^1_{A_i}H^{s-1}$ \\[3pt]
$u\,\partial^2h$ at $s=8$
& Unweighted $H^{7.5}$ bulk for $\partial^2h$ times a strong low endpoint
& $L^2_AH^{7.5}$ \\[3pt]
$K_{-1}(d_1\partial^2u)$ at $s=8$
& Input in $L^1_AH^6$: low integrable $d_1$ times a weighted endpoint,
or unweighted high bulk $d_1$ times strong low bulk
& $L^1_AH^7$ after the spatial gain \\
\bottomrule
\end{tabular}
\end{center}
Opposite high/low placements and smooth coefficient multiplicities obey the same tame product estimates.
The nonlocal terms use the spatial gain in the last row. The added
operators have spatial orders zero and $-1$; a time hit carries order $-1$.
They occur in compositions of at most two first-order gauge factors with
smooth wave coefficients. Any changed coefficient on a remaining order-two
field term beyond the differential principal change therefore has a spatial
gain. For example, an order-$-1$ kernel on $d_1\partial^2u$, where $d_1$
is an undifferentiated coefficient difference, uses either integrable low
$d_1$ and the weighted endpoint $H^{s-2}$ norm of $\partial^2u$, or
unweighted bulk $H^{s-2}$ for $d_1$ and strong low bulk for
$\partial^2u$. The output is $L^1_{A_i}H^{s-1}$. For $h$ split
$k_h+\xi$ and use half-loss bulk for a high coefficient multiplying a
slowly varying spatially smooth factor. The same argument applies when the
negative-order kernel precedes multiplication, using a fixed low Sobolev
buffer, or appears on both gauge factors. Hits on kernel parameters use the
weighted parameter derivatives already counted. Finally, after its linear
part is removed, $f_B$ and $\partial_t f_B$ contain a strong acceleration
factor, jet or memory on $r_y\le4R$, by exact affine-position freezing.
This proves \eqref{co2:error-source-space}.
\paragraph{The separate linear source.}
It must not be treated as a nonlinear error. On prior fixed sets its
required $L^2_{A_i}$ spatial orders are bounded by
$C\mathsf l+C_{\rm fix}e_{\rm in}$ through filtered feedback, from
$\ell h,\ell h,\ell u$ respectively. Enough fixed filter gain also gives
the required endpoint orders. The differential exterior estimate uses
\begin{equation}\label{co2:outer-linear-source}
\left(\int^{O(R)}\widetilde r^{1+\delta}
\|\widetilde r\,\partial^{\le s-1}Q(\partial_t)b_i
\|_{L^2_{A_i}L^2_\omega}^{2}\,d\widetilde r\right)^{1/2}
\lesssim C R^{\delta/2}\mathsf l+C_{\rm fix}e_{\rm in}.
\end{equation}
Explicitly, $Q$ and its required derivatives cost
$r_y^{-2}$ times bounded averaging of filtered jets. Taking $T_0\gg R$
makes the plateau weights comparable across those memories; slowly varying
affine measures are uniformly comparable as well.
\paragraph{Absorption of the principal difference.}
Use the actual wave metric in the differential currents. On fixed compact
sets the principal commutator differences obey the tame split
\begin{equation}\label{co2:principal-commutator}
C\|\nabla_y d_M\|_\infty\|\partial^2U\|_{H^{s-2}}
+C\|d_M\|_{H^{s-1}}\|\partial^2U\|_{\rm low}.
\end{equation}
The first term is integrable low times endpoint. For a high coefficient
with $h$, split $k_h+\xi$ as above, allowing the extra half order on the
smooth $\partial^2\xi$ placement; with $u$, multiply unweighted and strong
bulk norms. Derivatives of the principal-current coefficients cost current
order times integrable low. On larger bounded annuli use full mixed bulk at
$R$-dependent constants. The strict inner outflow remains exiting, including
when a cutoff differential current by itself has no sign in the interior.
On buffered compact kernel supports, the solved first-jet evolution is the
complete frozen evolution plus $iT_{\Delta P}\mathcal Y$, modulo the source
classes already listed. Here $T_{\Delta P}$ is the spatial paradifferential
order-one change of the differential principal symbol. To see this, first
normalize actual and frozen equations by their differential time coefficients.
Their inverse difference times stationary lower terms belongs to the same
source classes; the extra stationary acceleration term has spatial gain.
Inverting its normalized $I+P_{-1}$ contributes only an order-$-1$
correction on the differential principal difference. The non-paralow
remainders cost high coefficient $H^{s-1}$ times low $\partial^{\le2}U$,
with the smooth $\xi$ placement permitted at order $s-1/2$. These are the
preceding product estimates. Stationary cutoff errors stay within the
stationary calculation, so no nonlocal side condition is introduced.
Conjugate the first jets by $\langle D_y\rangle^{s-1}$ on these supports
and put $z=\langle D_y\rangle^{s-1}\mathcal Y$. Write the actual
differential evolution symbol as
\[
p_d=S_d\operatorname{diag}(\lambda_+\mathrm{Id},\lambda_-\mathrm{Id})S_d^{-1},
\]
where $S_d$ depends algebraically and smoothly on the wave coefficients.
Let $e_\pm$ be the stationary diagonal principal forms for the compact
addition to the base differential currents; they need not be positive
separately. Correct this addition at spatial paradifferential principal
order by
\begin{equation}\label{co2:algebraic-pullback}
A_d=S_d^{-*}\operatorname{diag}(e_+,e_-)S_d^{-1}.
\end{equation}
Keep the lower stationary prescriptions. Since the roots are real scalar
roots, the algebraic identity
\begin{equation}\label{co2:principal-symmetrization}
A_dp_d=p_d^*A_d
\end{equation}
cancels the order-one part of $i(A_dP_d-P_d^*A_d)$ exactly. The
differential wave currents separately symmetrize their actual and frozen
symbols, so the same cancellation holds for the total corrected energy.
After this exact cancellation, only one-order symbolic accuracy is needed.
For the symbolic calculus, see Bony
\cite[\S 3, Theorems~3.2--3.3]{Bony1981}; a quantitative formulation
of the one-spatial-Lipschitz-derivative composition and adjoint bounds is
\cite[Definition~2.1 and Theorem~2.6(ii)--(iii), $\rho=1$]{AlazardBurqZuily2014}.
The first-order spatial Bony composition and adjoint estimates
\cite[Theorems~2.4.11--2.4.12]{Metivier2003} leave an order-zero
difference from the stationary derivative. After the source
pairings already estimated, its quadratic form is bounded by
\begin{equation}\label{co2:integrable-energy-error}
C_s\bigl(\|d_M\|_{W^{1,\infty}}+\|\partial_td_M\|_\infty
+|a|+|\dot c|\bigr)\|z\|_{L^2}^2.
\end{equation}
The fixed low regularity buffer from mixed $H^8$ control supplies the
spatial coefficient and smoothing-remainder bounds in these rules.
Normalized frequency derivatives use finitely many smooth angular symbol
seminorms, with the same low coefficient bounds; their number does not grow
with the energy order. The algebraic pullback has one true time derivative,
acting on parameter values and low variation. Stationary subprincipal terms
and compact kernel-cutoff interactions with the paradifferential change have
at most current order. The coefficients of $b_C$ have already been fixed.
Smallness preserves endpoint positivity, and integration of
\eqref{co2:integrable-energy-error} against $\tau^{A_i}$ costs
$C_{\rm fix}\mathsf N^3$ in squared estimates, up to input.
Thus the corrected energy retains the frozen dissipation, including the
$B_{\rm nt}$ first-jet term in \eqref{co2:frozen-dissipation}, with only an
integrable endpoint error. It requires neither a trapping sign for the
actual metric nor full directional estimates on higher mixed time jets.
The scalar cutoff has not been commuted through the changing order-two wave
equation, and every compact kernel remains inside its buffered support,
away from the outflow trace.
\end{proof}
\subsection{The integrated compact estimate}
\begin{proposition}[Compact high estimate]\label{co2:compact-high}
Under the simultaneous finite-slab budgets, for some $\kappa>0$,
\begin{equation}\label{co2:high-bound}
\mathsf K_R\le C R^{-\kappa}(\mathsf F+\mathsf K_R)
+C_{\rm fix}\bigl(e_{\rm in}+\mathsf N^{3/2}
+(e_{\rm in}\mathsf N)^{1/2}\bigr).
\end{equation}
The acceleration and sliced static bounds enter this estimate only as
coefficient smallness and nonlinear errors, not as additional linear forcing
terms. The choices are the stationary energy parameters, then a sufficiently
large $R$, then the plateau $T_0$, and finally the bootstrap and data
smallness.
\end{proposition}
\begin{proof}
Apply Proposition~\ref{co2:actual-comparison} to the frozen energies at the
three levels \eqref{co2:levels}. 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
\begin{equation}\label{co2:high-shell-flux}
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})}.
\end{equation}
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 \eqref{co2:high-shell-flux}.
The square root of \eqref{co2:high-shell-flux} is at most
\begin{equation}\label{co2:high-shell-bound}
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\}.
\end{equation}
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 \eqref{co2:far-weights}; 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
\begin{equation}\label{co2:lower-interpolation}
R^{-\iota_0}\mathsf K_R+C R^{K\iota_0}\mathsf l,
\qquad K\iota_0<\kappa_0/2,
\end{equation}
where $K$ is a sufficiently large fixed interpolation constant and
$\kappa_0$ is from Proposition~\ref{co2:low-transfer}. Very-low-order
coercivity supplements fit the same estimate. After substituting
\eqref{co2:low-bound}, the data and quadratic pieces have only
$C_{\rm fix}$ costs. The linear forcing pairing from
\eqref{co2:outer-linear-source} 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,
\begin{equation}\label{co2:plateau-interpolation}
\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}}.
\end{equation}
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 \eqref{co2:high-shell-bound},
\eqref{co2:lower-interpolation} and the cubic comparison errors, proves
\eqref{co2:high-bound} 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 \eqref{co2:high-bound}.
\end{proof}
\subsection{Higher orders after base closure}
The next statement is used only after the compact and far estimates have been
closed at the fixed base orders in the next section. It is important that
it impose no smallness on higher data seminorms.
\begin{proposition}[Subexponential smooth bounds]\label{co2:higher}
Assume the global base bounds resulting from the compact-far closure. For
every fixed $C_0>0$, every fixed smooth order, and every $\sigma>0$, all
metric, coefficient and parameter norms at that order on
\[
0\le t\le T',\qquad r_y\le C_0(1+T')
\]
are bounded by $C_\sigma e^{\sigma T'}$. The constant may depend on the
smooth datum, the order and $C_0$, but the base neighborhood and fixed filter
do not. We denote these bounds by $e^{o(T')}$.
\end{proposition}
\begin{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,
\begin{equation}\label{co2:low-decay}
h=o(1),\qquad g_{\rm co}-\bar g=o(1),\qquad
c,\dot c,a=o(1),\qquad t\longrightarrow\infty.
\end{equation}
On fixed compact sets use weighted endpoints; at large radius use the far
point estimates, including \Cref{fe:F7} and \Cref{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
\begin{equation}\label{co2:higher-cylinder}
C_0(1+T')+C_1(T'-t),
\end{equation}
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 \eqref{co2:low-decay}.
At each $s\ge8$, Proposition~\ref{co2:stationary-energy} 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 \eqref{co2:higher-cylinder}, with no
matching-shell loss. Its top-order energy equivalence constants are
independent of the terminal radius, since
\begin{equation}\label{co2:affine-density}
\|\widetilde r G\|_{L^2(d\widetilde r\,d\omega)}^2
=|\det A_{\rm aff}|\|G\|_{L^2(d^3y)}^2,
\end{equation}
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
\begin{equation}\label{co2:filter-count}
\max(0,s+4-P)\le s-1
\end{equation}
derivatives of $\ell h$ once the \emph{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 \eqref{co2:affine-density}.
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 H\"older 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{equation}\label{co2:higher-gronwall}
\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}
\end{equation}
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
\eqref{co2:filter-count}. 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.
\end{proof}
The remaining task in the exterior analysis is to prove the far estimate and
close its coupling with \eqref{co2:sliced-acceleration} and
\eqref{co2:high-bound}. The higher-order proposition will then apply to the
resulting solution in the same single base neighborhood.
==== END Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/05-compact.tex ====
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/06-far.tex ====
\section{The far region and simultaneous closure}\label{sec:far}
The compact estimates leave two quantities to control: radiation at large
radius and the acceleration of the modulated Kerr parameters. We now prove
the far estimate with a radius-small coefficient on both quantities, and
then close it simultaneously with the compact and sliced estimates.
The far equation has no conic pseudodifferential modification. We first
verify its target symbols, differentiated weak-null structure, and
initial and overlap bounds. The energy and product estimates in
Appendices~\ref{fe:section} and~\ref{fs:section} then apply to this
equation. Their role is to turn the structural bounds into a
radius-small far estimate; the final subsection here closes its
coupling to the compact region.
Throughout we work on an arbitrary smooth finite bootstrap slab. All
constants are independent of its terminal time. The compact norm
$\mathsf K_R$, the acceleration norm
$\mathsf S_a=\|a\|_{L^2_A}$, the total bootstrap size $\mathsf N$, and the
base input size $e_{\mathrm{in}}$ have the meanings fixed in the compact
argument. Here $a$ in $\mathsf S_a$ denotes the modulation acceleration,
not the fixed Kerr rotation parameter. A constant $C_{\mathrm{fix}}$ may
depend on every scale chosen before the final data smallness, but not on
the lifespan.
\subsection{Emitter coordinates and the three far norms}
Let $\eta$ denote the Minkowski metric in the lab coordinates $(t,z)$.
Write
\[
\begin{gathered}
C_e(q)=-\lambda(q),\qquad s_e=C_e',\qquad s_s=-v_s(t,x),\\
t=q+r,\qquad z=C_e(q)+rn,\qquad |n|=1,
\end{gathered}
\]
and set $k=1-s_e\cdot n$. In this section $r$ is emitter radius; the
small speed bound makes $k$ uniformly close to one and makes $r$, $|x|$,
and the co-moving radius $|y|$ comparable. Put
\[
\begin{gathered}
T_l=\partial_{t,\mathrm{lab}},\qquad
L=T_l+n\cdot\partial_z=\partial_{r|q,n},\\
S=rL,\qquad Z=(S,\Omega),\qquad D=T_l+s_s\cdot\partial_z,
\end{gathered}
\]
where $\Omega$ denotes the emitter sphere rotations. All tensor
components in this section are lab components, and we abbreviate
\[
h=h_{\mathrm{lab}},\qquad \psi=Dh,\qquad H^{ij}=g^{ij}-\eta^{ij}.
\]
The covector $\ell_n=(1,-n)$ is used only for null contractions. Thus
$H_{\ell_n\ell_n}=H^{ij}\ell_{n,i}\ell_{n,j}$. Let $\Pi(n)$ be the
projection of a covariant symmetric tensor onto its tracefree spatial
screen part orthogonal to $n$.
The elementary derivative conversion is
\begin{equation}\label{far:derivative-conversion}
\partial_i=\ell_{n,i}T_l+r^{-1}b_iZ
=\ell_{n,i}k^{-1}\partial_q
+r^{-1}(b_i-\ell_{n,i}k^{-1}s_e^db_d)Z.
\end{equation}
Here $b_t=0$, while $b_dZ$ is $n_dS$ plus the corresponding tangential
rotation combination. We always commute lab time derivatives before
the $Z$ derivatives. Choose the starting radius $R_0$ beyond all fixed
transitions and all compact kernels. The fixed-factor starting overlap
is then physically nontrapping. We require, in addition, that radii as
small as $cR_0^{0.45}$ lie in the lab differential and symbol regime;
these smaller radii occur only in the spacelike-anchor argument below.
We use the three levels
\begin{equation}\label{far:levels}
\begin{array}{c|ccc}
i&0&H&P\\ \hline
U_i&h&h&\psi\\
N_i&9&8&7\\
A_i&0&A_H=1-\eta_*&A=1.36\\
p_i&0.64&1-\eta_*/6&1.62
\end{array}
\end{equation}
with a fixed $0<\eta_*\le 10^{-8}$. For the far weights define
\[
\delta_f=\eta_*/100,\qquad \nu=10\delta_f,
\qquad e=0.1\delta_f,\qquad d_*=200\eta_*.
\]
These quantities are distinct from the slack in the compact
differential-current construction. With $\langle s\rangle=(1+s^2)^{1/2}$,
put
\begin{equation}\label{far:weights}
\begin{gathered}
\tau_f=1+t/r,\qquad \mu=\langle q/r^{\delta_f}\rangle,
\qquad l_f=\langle q/r^\nu\rangle,\\
m_f=(r^{-1}+l_f^{-1})^{-1},\qquad
W_i=r^{p_i-A_i}(t+10r)^{A_i},\\
w_{H,j}=w_H=r^{-e}\mu^{A_H},\qquad
w_{P,j}=w_Hl_f^{A-A_H}m_f^{-(7-j)d_*},\\
w_{0,j}=r^{-0.02}m_-^{jd_*}m_+^{-(9-j)d_*}.
\end{gathered}
\end{equation}
Here $m_-$ is a smooth monotone function of $-q/r^\nu$, comparable to
$1+(-q/r^\nu)_+$, constant when its argument is nonpositive, and with
bounded logarithmic-symbol derivatives. The weight $m_+$ is the
positive-$q$ analogue capped at $r$ by harmonic addition.
For each word $I$ in $Z$ and each $j$ with $|I|+j\le N_i$, define
$\chi=Z^I(rT_l^jU_i)$. Write $G_1\chi=r^{-1}Z\chi$ and let
$B_1\chi$ denote its full first derivative. The bulk measure will always
be $dt\,dr\,d\omega/r$. The preliminary far norm is the sum, over these
indices and tensor components, of
\begin{equation}\label{far:basic-norm}
\begin{split}
&\|W_i^{1/2}(G_1\chi,\chi/r)\|_{\mathrm{bulk}}
+\|w_{i,j}^{1/2}B_1\chi\|_{\mathrm{bulk}}\\
&\hspace{8mm}
+\sup_t\|(W_i^{1/2}G_1\chi,w_{i,j}^{1/2}B_1\chi)
\|_{L^2(dr\,d\omega)}.
\end{split}
\end{equation}
We complete this norm by corrected-current terms in
\eqref{far:corrected-norm} below and denote the result by $\mathsf F$.
The weights are those of \Cref{fe:F1}. On $t\gg r$ they satisfy
\begin{equation}\label{far:timelike-weights}
w_{i,j}\gtrsim\langle t\rangle^{A_i}r^{-0.03},
\qquad W_i\gtrsim t^{A_i}r^{p_i-A_i}.
\end{equation}
Differentiating $\chi$ supplies the bad-derivative controls used in the
sliced argument; the extra lower-order terms are controlled by the good
entries. Thus \eqref{far:basic-norm} contains those earlier far inputs,
with any necessary smaller positive radial slack.
\subsection{Verification of the lab equation}
The target metric is denoted by $B$, its wave compensator by $\Theta$,
and its pure equation defect by $f_B$. Enlarge the positive acceleration
majorant at comparable radii when necessary, and write
\[
\widetilde a(t,r)=Cr^{-1}\int_{t-Cr}^t|a(\tau)|\,d\tau.
\]
The histories are zero before startup. The particular constant $C$ in
this notation can be increased finitely many times.
\begin{lemma}[Far symbol and weak-null structure]\label{far:structure}
For the arbitrary fixed subextremal Kerr family constructed above, the
lab target satisfies
\begin{equation}\label{far:target-symbols}
\begin{gathered}
B-\eta=O(r^{-1}),\qquad DB=O(r^{-1}\widetilde a),\qquad
\partial^ds_s=O(r^{1-d}\widetilde a)\quad(d\ge1),\\
\Theta=O(r^{-1}\widetilde a),\qquad
f_B=O(r^{-2}\widetilde a),\qquad Df_B=O(r^{-3}\widetilde a),\\
\qquad r\widetilde a\lesssim\mathsf N\tau_f^{-A/2}.
\end{gathered}
\end{equation}
These are symbol estimates at every fixed order needed here, with one
inverse power per additional lab derivative. Their constants may depend
on the fixed family but not on either matching radius. The equation is
\begin{equation}\label{far:wave-structure}
\begin{gathered}
g^{ij}\partial_{ij}h=f_h,\\
\begin{aligned}
f_h={}&f_B+O(r^{-2}\partial h+r^{-3}h)
+Q_\eta(\partial h,\partial h)\\
&+O((r^{-1}+|h|)|\partial h|^2),
\end{aligned}
\end{gathered}
\end{equation}
up to fixed harmless changes in $f_B$. The first- and zeroth-order
background coefficients have a relative $O(\widetilde a)$ gain under
$D$. The flat linear wave-gauge contraction of $h$ is
\begin{equation}\label{far:flat-gauge}
O((r^{-1}+|h|)\partial h)+O(r^{-2}h)
+O(r^{-1}\widetilde a).
\end{equation}
The estimates allow smooth composition in the fields and distributed
product differentiation. The quadratic form $Q_\eta$ has the constrained
weak-null polarization described below.
\end{lemma}
\begin{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 \eqref{far:target-symbols}.
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 \eqref{far:flat-gauge}. Expanding vacuum Ricci in this
gauge gives \eqref{far:wave-structure}. 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 \eqref{far:target-symbols} and $[D,\partial]$.
For a symmetric polarization $p$ satisfying
\begin{equation}\label{far:polarization-constraint}
\ell_n^\alpha\bigl(p_{\alpha\beta}
-\tfrac12\eta_{\alpha\beta}\operatorname{tr}_\eta p
\bigr)=0,
\end{equation}
the flat Christoffel identity of \Cref{fe:weak-null} 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
\eqref{far:polarization-constraint} 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
\Cref{fe:gauge-projection}. Apply the differentiated identity
\eqref{far:flat-gauge} 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.
\end{proof}
\begin{lemma}[Initial and overlap inputs]\label{far:boundary-input}
The initial, current-correction, and field overlap costs at the
fixed-factor starting annulus in the far energy estimates are bounded by
\begin{equation}\label{far:overlap-bound}
C(R_0)\mathsf K_R+C_{\mathrm{fix}}(e_{\mathrm{in}}+\mathsf N^2).
\end{equation}
The initial input requires only the base metric $H^{10}$ and second-form
$H^9$ bounds and their order $-1,-2$ end-symbol bounds. It imposes no
condition on leading tail coefficients and no smallness on higher
seminorms. Pure acceleration forcing is estimated separately, including
on this annulus, by the direct averaging estimate
\eqref{fs:average-bounds} in Appendix~\ref{fs:section}.
\end{lemma}
\begin{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 \eqref{far:overlap-bound} and
verifies the boundary hypotheses of \Cref{fs:claim-boundary}.
\end{proof}
\subsection{Point bounds and corrected energy currents}
We record the precise point information entering the source estimates.
It also supplies the timelike point bounds used in the sliced comparison.
For the following assertions, $N\le C_{\mathrm{fix}}\mathsf N$ denotes
the small factor used in product estimates, not a new independent norm.
The finite-slab implications of \Cref{fe:spacelike-anchors},
\Cref{fe:point-estimates}, \Cref{fe:null-projection},
\Cref{fe:tight-projection}, and \Cref{fe:F11} give
\begin{align}
\|\chi(h)\|_{L^2_\omega}
&\lesssim Nr^{2\eta_*}\tau_f^{-A_H/2}
&&\text{through order eight},\label{far:point-h}\\
\|\chi(h)\|_{L^2_\omega}
&\lesssim Nr^{0.52}&&\text{at order nine},\notag\\
\|B_1\chi(h)\|_{L^2_\omega}+\|\chi(\psi)\|_{L^2_\omega}
&\lesssim Nr^{e/2}\mu^{-A_H/2}
&&\text{through order seven}.\label{far:point-first}
\end{align}
The first-derivative version for $\psi$ holds one order lower. Two spare
rotations give the corresponding sphere supremum bounds. Contract lab
time jets before applying $Z$ to $H_{\ell_n\ell_n}$. Through order eight
their angular $L^2$ bounds are
\begin{equation}\label{far:projected-bounds}
O(r^{-1}+Nr^{-1+2\eta_*}\tau_f^{-A_H/2}),
\qquad O(r^{-2}+Nr^{-2+8\eta_*}\tau_f^{-A_H/2})
\end{equation}
in general and with at least one lab time, respectively. The second
bound also holds after $\partial_q$ at orders through seven. Without a
lab time, $r$ times the coefficient is $O(1)$ through order eight on
$|q|\lesssim r^{0.45}$. At order nine it is bounded by
\begin{equation}\label{far:top-projected}
O\bigl(\log r+b^{1/2}r^{-p_0/2}+br^{-1+8\eta_*}\bigr),
\qquad b=1+\min(|q|,r);
\end{equation}
with a lab time the bound for $r$ times the order-nine coefficient
improves to $O(r^{-0.46})$. Finally, the lowest radiative jets satisfy
\begin{equation}\label{far:news}
|r\Pi T_lh|+|r\Pi T_l^2h|+|r\Pi T_l\psi|\lesssim N.
\end{equation}
There is a radius-uniform point in applying these implications.
The negative-$q$ spacelike anchor is proved on an auxiliary exterior
\begin{equation}\label{far:anchor-domain}
\begin{gathered}
|z|\ge t+cQ+C_1\int_0^t((u+Q)Q)^{-1/2}\,du,\\
0\le t\lesssim R_f,\qquad Q\gtrsim R_f^{0.45}.
\end{gathered}
\end{equation}
Here $R_f$ is a dyadic radius. This domain has $q<0$; zero startup
histories give $\lambda(q)=0$ and hence $r=|z|$. Its lower boundary may
lie below $R_0$, but our
choice of $R_0$ places all of it in the differential lab end. Hence the
same equation and initial symbol bounds apply there. Regard the
histories as fixed in this domain estimate. The pure source after up to
nine Minkowski commutations costs
$O(\mathsf S_a|z|^{-3})$ by the same acceleration majorant.
Thus the moving-boundary energy bootstrap of
\Cref{fe:spacelike-anchors} uses no radius-dependent replacement by
$\mathsf K_R$. Constants acting linearly on far quantities remain
bounded independently of $R_0$; nonlinear smallness can be imposed after
the radii are fixed. The news estimate \eqref{far:news} follows from the
emitter equations on $rT_l^jh$, $j=0,1$: projection kills the leading
quadratic by Lemma~\ref{far:structure}. No point smallness in a deep
compact region is used, except on the prescribed overlaps.
To describe the currents, set
\[
P_g\chi=krg^{ij}\partial_{ij}(\chi/r),\qquad
P[J]\chi=krJ^{ij}\partial_{ij}(\chi/r).
\]
The exact flat emitter operator and the principal perturbation identity are
\begin{align}
P_\eta={}&-2\partial_q\partial_r+r^{-2}\bigl[
(1+s_e\cdot n)(S^2-S)-2(s_e\cdot n)(S-1)\notag\\
&\hspace{40mm}+2s_e^A\nabla_A(S-1)+k\Delta_{\mathbb S^2}\bigr],
\label{far:flat-operator}\\
P[J]\chi={}&J_{\ell_n\ell_n}\partial_q(k^{-1}\partial_q\chi)
+O\bigl((J/r)(1,Z)\partial_q\chi
+(J/r^2)(1,Z,Z^2)\chi\notag\\
&\hspace{42mm}+(a(q)/r)J_{\ell_n\ell_n}(1,Z)\chi\bigr).
\label{far:perturbation}
\end{align}
The $O$ notation here denotes finite sums with bounded smooth
emitter-frame coefficients. These identities are \Cref{fe:F4} and
\Cref{fe:F5}; they are unchanged by the choice of Kerr spin.
Let $X_g,\bar X_g$ be the true future outgoing and incoming null normals,
normalized by $dt=1$, let $G=-g(X_g,\bar X_g)$, and let $\alpha$ be the
$\bar X_g$ coefficient of $\partial_q$ modulo $X_g$ and the screen.
Define
\begin{align}
d_g&=|H_{\ell_n\ell_n}|+|H|^2,\qquad
M_f=r^{0.72}\mu^{-A_H}j_0(q/r),\notag\\
\mathfrak u&=q+\int_{R_0/2}^r
\left[\rho^{\delta_f-1}
+\rho^{20\eta_*-1}j(|q|/\rho^{0.45})\right]d\rho.
\label{far:caps}
\end{align}
The smooth cutoffs satisfy $j_0=1$ on $[-1,1]$, $j_0=0$ off $[-2,2]$,
$j=0$ below one, and $j=1$ above two. The tight projection bounds make
the caps $\mathfrak u=u_0$ spacelike when $R_0$ is large.
For each commuted equation the outgoing source is split as
\begin{equation}\label{far:source-split}
P_g\chi=\lambda_1L\chi/r+f_{\mathrm{flat}}
+\partial_q\mathcal P+f_{\mathrm r}+f_{\mathrm{mix}}.
\end{equation}
Here $\lambda_1$ is bounded and nonnegative; it and the triangular
$f_{\mathrm{flat}}$ arise from flat dilation and rotation commutators.
Write $f_{\mathrm{raw}}$ for the unsplit remaining source after these
flat terms. The correction $\mathcal P$ is formed term by term in the commuted
lab equation, keeping each algebraic sign. In the following list,
$\chi'=Z^{I'}(rT_l^{j'}U)$ is the commuted input field,
$U\in\{h,\psi\}$; a coefficient $Z^dT_l^mH$ has count $d+m$.
For the smooth-acceleration terms, $C$ denotes the twice-null
contraction of an ordered derivative of $2\eta^{ij}\partial_i s_s^d$,
the coefficient of $\partial_{jd}h$ in the strong equation.
There are three types:
\begin{enumerate}[label=(\roman*)]
\item A double-bad principal term
$C_h\partial_q(c_0\partial_q\chi')$ contributes
$C_hc_0\partial_q\chi'$ when the coefficient count is at most five,
or at every coefficient count at level $P$.
Here $C_h=Z^d(T_l^mH)_{\ell_n\ell_n}$ and
$c_0=c_0(n,s_e(q))$ is the smooth emitter coefficient.
The smooth-acceleration double-bad terms in the $D$ equation use the
same rule with their coefficient in place of $C_h$.
\item A frame term $C(\partial_qc_2)D_Z\chi'/r$ in that equation
contributes $Cc_2D_Z\chi'/r$, where $c_2=c_2(n,s_e(q))$ is smooth
and bounded, and $D_Z$ has $Z$ order at most one and is independent of $q$.
\item Put $F_m=r\Pi T_l^{m+1}h$. Each leading strong quadratic
pairing contributes
\begin{equation}\label{far:leading-correction}
\mathcal P_*=Z^{b'}\!\left[(1-s_s\cdot n)r^{-1}\ell_n\otimes\ell_n
\mathcal C_{mm'}(F_m,F_{m'})\right],
\end{equation}
where $\mathcal C_{mm'}$ is its actual polarized screen pairing.
\end{enumerate}
The ordered expansions and these choices are given in
\eqref{fs:G5}, \eqref{fs:G7}, and \eqref{fs:G10}--\eqref{fs:G12}.
The raw source keeps all three types unsplit for the lab-time multiplier.
With their signed sum $\mathcal P$, define
\[
\xi_{\mathrm{curr}}
=d\chi+\frac{G\alpha}{2k}\mathcal P
\frac{-g(\bar X_g,\cdot)}{G}.
\]
We now complete $\mathsf F$ by adding, for each equation,
\begin{equation}\label{far:corrected-norm}
\|(W_i(1+M_f))^{1/2}\xi_{\mathrm{curr},X_g}\|_{\mathrm{bulk}}
+\sup_{u_0}\|W_i^{1/2}\xi_{\mathrm{curr},X_g}
\|_{L^2(dr\,d\omega;\,\mathfrak u=u_0)}.
\end{equation}
The source costs of \Cref{fe:energy-reduction} are
\begin{equation}\label{far:source-costs}
\begin{split}
&\|W_i^{1/2}(\mathcal P,Z\mathcal P,rd_g\partial_q\mathcal P)
\|_{\mathrm{bulk}}
+\sup_t\|W_i^{1/2}\mathcal P\|_{L^2_{r,\omega}}\\
&\quad+\|(W_i/(1+M_f))^{1/2}rf_{\mathrm r}\|_{\mathrm{bulk}}
+\|w_{i,j}^{1/2}rf_{\mathrm{raw}}\|_{\mathrm{bulk}}.
\end{split}
\end{equation}
There is also a mixed cap charge: on $t\lesssim r$, $r\sim R_f$, a bound
$\|rf_{\mathrm{mix}}\|_{L^2_\omega}
\le l_{R_f}(\mathfrak u)\mathcal T(t,r)$ is charged by
\begin{equation}\label{far:mixed-charge}
\sum_{R_f}R_f^{(p_i-1)/2}\|l_{R_f}\|_{L^2(du_0)}
\sup_t\|\mathcal T(t,\cdot)\|_{L^2_r}.
\end{equation}
The geometric assumptions of the energy reduction follow from the
point bounds already recorded:
\[
X_gq=O(d_g),\qquad
d_g\lesssim r^{-1}\ \text{if }|q|\le2r^{0.45},\qquad
d_g\lesssim r^{-1+2\eta_*}\ \text{otherwise}.
\]
Good derivatives of lab metric components, and an unrestricted first
derivative when both metric slots are the outgoing flat vector
$L=(1,n)$, cost $O(r^{-2+0.04})$. This uses
\eqref{far:point-first}, \eqref{far:projected-bounds}, and smooth inverse
composition. The bad component of $X_g-L$ is $O(d_g)$, and the screen
is tangent to the sphere, so true outgoing and screen derivatives are
good derivatives for these estimates. The exact lab gauge gives the
lowered wave contraction $O(r^{-2})$. Thus the divergence comparison
for $r^{-2}d\operatorname{vol}_g$ and the deformation bounds, including
\Cref{fe:F16}, have their stated inputs.
\subsection{The far estimate}
The point bounds and the corrections above have two distinct uses.
The outgoing multiplier controls good derivatives and the corrected
cap flux; the laboratory-time multiplier controls bad derivatives.
Appendix~\ref{fe:section} proves both estimates, with explicit source
costs \eqref{far:source-costs} and \eqref{far:mixed-charge}.
Appendix~\ref{fs:section} estimates those costs by the ordered
commutations of the equation just verified.
\begin{proposition}[Far estimate]\label{far:estimate}
After giving the finitely many commuted energies fixed hierarchy weights,
there is $c_1>0$ such that
\begin{equation}\label{far:far-bound}
\mathsf F\le o_{R_0}(1)(\mathsf F+\mathsf S_a)
+C(R_0)\mathsf K_R
+C_{\mathrm{fix}}\bigl(\text{input errors}
+\mathsf N^{1+c_1}\bigr).
\end{equation}
The coefficient denoted by $o_{R_0}(1)$ tends to zero as $R_0\to\infty$,
independently of the upper compact matching radius $R$. The input errors
vanish with $e_{\mathrm{in}}$, uniformly on bounded small bootstrap
ranges.
\end{proposition}
\begin{proof}
Apply the complete source estimate, Proposition~\ref{fs:source-estimate},
to the lab equation of Lemma~\ref{far:structure} 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
\eqref{far:point-h}--\eqref{far:news} supply the product estimates.
Lemma~\ref{far:boundary-input} 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
\begin{equation}\label{far:strong-equation}
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.
\end{equation}
Its seven commutations use $B^h$ indices through eight in the notation of
\eqref{fs:lists}: this means at most nine physical derivatives of $h$,
available at level $H$. Its twice-null coefficient has the extra inverse radius because
\begin{equation}\label{far:strong-projection-identity}
\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_{j0$, $\varepsilon_0(R_0)\to0$, and, crucially, $C_*$ is
independent of $R_0$. The first inequality is \eqref{co2:high-bound}, the compact estimate with
mixed-order recovery. Its acceleration and static quantities occur
only as small coefficient differences or products, not as linear
forcing. The last inequality is \eqref{co2:sliced-acceleration}; its
radius-uniform far coefficient is proved in
Lemma~\ref{fs:uniform-coupling} by a single spatial adjoint pairing
and the strict margin $p_P-A=0.26$. The compact-tail defect
comparison uses a cut $Y$ fixed before $R_0$.
\begin{proposition}[Simultaneous closure]\label{far:closure}
For each fixed strictly subextremal center and fixed computing cylinders,
there is a base smallness threshold depending only on the finite
$p_{10}$ input such that the compact, far, static, and modulation bounds
hold on every finite slab. They yield global smooth vacuum evolution
on both computing cylinders. Every fixed higher-order bound of
Proposition~\ref{co2:higher} then holds with datum- and order-dependent
constants; those higher seminorms need only be finite.
\end{proposition}
\begin{proof}
The common finite-slab norm includes the three compact levels with
mixed jets, the far levels including \eqref{far:corrected-norm},
$\mathsf S_a$, and the static norms of \eqref{co2:sliced-static}
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
\eqref{far:sliced-coupling} into \eqref{far:far-coupling} 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 \eqref{far:compact-coupling} 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 \Cref{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 \eqref{far:anchor-domain} 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 $c_2>0$,
\begin{equation}\label{far:total-closure}
\mathsf N\le C_{\mathrm{fix}}
\bigl(\mathcal R(e_{\mathrm{in}},\mathsf N)
+\mathsf N^{1+c_2}\bigr),
\end{equation}
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
\eqref{far:compact-coupling}.
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
\eqref{far:caps}, as in the construction of \Cref{fe:currents}.
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~\ref{st:local-evolution} 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 \eqref{far:levels} 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.
\end{proof}
==== END Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/06-far.tex ====
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/07-interior.tex ====
\section{The two-ended double-null interior}\label{sec:interior}
The exterior estimates supply data on the two late ends of a fixed,
deep spacelike cylinder. Its bounded middle is supplied by finite-time
evolution of the bridge. We construct the future double-null development
of these cylinder data, with uniform low-order control and arbitrarily
small exponential losses at every fixed higher order. The latter bounds
will be used on finite regions whose optical length tends to infinity.
The double-null framework follows the interior-stability method of
Dafermos--Luk \cite{DafermosLuk2017}; the entry and wedge estimates
required here are proved below. Throughout, the physical Kerr parameters
lie in a sufficiently small compact neighborhood of the fixed pair
$0<\mathfrak a0$, it is at
most $C_\delta e^{\delta T}$ for $T\ge1$. The constant may depend
on the fixed smooth datum, derivative order, and fixed domain
parameters. No uniformity in derivative order is asserted.
\subsection{Gauge and exact profiles}
Let $y_0,y_1$ be future-increasing optical functions and set
\begin{equation}\label{in:gauge}
\begin{gathered}
s=\frac{y_0+y_1}{2},\qquad x=\frac{y_1-y_0}{2},\qquad
\mathcal S=\{s=0\},\qquad q=q_0e^{-2\kappa s},\\
g=-2a\,dy_0dy_1+
\gamma_{AB}(d\theta^A-b^A dy_0)(d\theta^B-b^Bdy_0),
\qquad a=qA.
\end{gathered}
\end{equation}
Here $\kappa>0$ is fixed, for example equal to the background inner
surface gravity in absolute value; $q_0>0$ will be chosen below;
$A>0$; $\gamma$ is a Riemannian metric on $\mathbb S^2$; and $b$ is
a vector field on $\mathbb S^2$. Angular coordinates are understood in
a finite smooth atlas. The symbols $a,A,q$ in this section denote the
null factor, its rescaling, and its exponential scale, respectively.
The rate $\kappa$ here is unrelated to the exterior radius-decay
exponent denoted by the same letter.
For distinct $i,j\in\{0,1\}$, write
\[
B_0=b,\qquad B_1=0,\qquad
L_i=\partial_{y_i}+B_i,\qquad
\mathscr L_i=\partial_{y_i}+\mathcal L_{B_i}.
\]
The last operator acts on angular tensors. Define
\begin{equation}\label{in:variables}
\begin{gathered}
\chi_i=\tfrac12\mathscr L_i\gamma,\qquad
\tau_i=\operatorname{tr}_\gamma\chi_i,\qquad
\ell_i=L_i\log a,\\
\xi_0+\xi_1=d\log a,\qquad
\xi_0-\xi_1=-a^{-1}\gamma\,\partial_{y_1}b,\\
\beta_i=\operatorname{div}\chi_i-d\tau_i,\qquad
c_i^{\mathrm{ang}}=\operatorname{curl}\xi_j,\qquad
Q_i=(K,c_i^{\mathrm{ang}}),\qquad
\mu_i=K-\operatorname{div}\xi_j.
\end{gathered}
\end{equation}
In this section $K$ is the Gaussian curvature of $\gamma$.
All angular differentials, contractions, divergences, and curls use
$\gamma$ unless explicitly indicated otherwise.
\begin{lemma}[Exact profiles with a common rate]\label{in:profiles}
For physical Kerr parameters in a sufficiently small compact
neighborhood of the fixed subextremal parameters, there are exact
profiles in the gauge \eqref{in:gauge}, depending smoothly only on
$(q,\theta)$ and the parameters and invariant under axial rotations,
such that $A,\gamma$ and their inverses extend smoothly to $q=0$,
$b=O(q)$, and
\begin{equation}\label{in:profile-orders}
\chi_i,\ \ell_i+\kappa,\ \beta_i=O(q),\qquad
\xi_i,\ K,\ c_i^{\mathrm{ang}},\ \mu_i=O(1).
\end{equation}
These estimates hold after every fixed angular or physical-parameter
derivative. Their constants can be fixed before $q_0$ is chosen.
The same statement holds with either optical direction chosen to have
zero angular shift.
\end{lemma}
\begin{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{equation}\label{in:BL-metric}
\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}
\end{equation}
Temporal and axial conventions may be reversed simultaneously when
passing between the two ends.
Solve the axisymmetric Hamilton--Jacobi equation
\begin{equation}\label{in:HJ}
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,
\end{equation}
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 \eqref{in:BL-metric} 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 \eqref{in:HJ} is precisely the null equation.
Define
\begin{equation}\label{in:exact-coordinates}
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.
\end{equation}
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 \eqref{in:BL-metric}.
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 \eqref{in:gauge}, with $q=Q$. Since $L_iq=-\kappa q$,
the definitions \eqref{in:variables} imply
\eqref{in:profile-orders}, with all asserted differentiated bounds.
The same construction eliminates the other shift after exchanging the
optical directions.
\end{proof}
Choose $q_0$ sufficiently small for the characteristic absorptions
below, uniformly for both angular-transport conventions. We also impose
the exact-profile depth condition used when returning the later packet
to the exterior. Since $\partial_\vartheta F=O(r-r_-)=O(q_0)$ at
entry and the spin stays bounded away from zero in the fixed
parameter neighborhood, we require the uniform strict inequality
\begin{equation}\label{in:profile-depth}
\left(|\partial_\vartheta F|^2+
\mathfrak a^2\sin^2\vartheta\right)_{\mathrm{entry}}
<4\mathfrak a^2.
\end{equation}
This supplies a strict positive margin for the radial potential
$(r^2+\mathfrak a^2)^2-\Delta
(|\partial_\vartheta F|^2+\mathfrak a^2\sin^2\vartheta)_{\mathrm{entry}}$
for $r\ge r_+$, since $\Delta0$.
The low smallness depends only on the common base-order neighborhood;
constants may depend on the already fixed $q_0$ and coordinate choices.
Higher-order constants may depend on the individual smooth datum.
\end{proposition}
\begin{proof}
We derive the entry trace from the mixed exterior estimates.
On the fixed graph supports, Proposition~\ref{co2:sliced} 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~\ref{in:moving-coordinates}, replacing the
instantaneous parameter by \eqref{in:parameter-profile} 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
\begin{equation}\label{in:trace}
\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.
\end{equation}
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
\eqref{in:entry-orders} 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 \eqref{in:trace} 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.
\end{proof}
\subsection{The null equations and uniform low estimates}
The entry estimate supplies summable errors along the initial cylinder.
We now propagate those errors through the wedge using only the vacuum
equations in \eqref{in:gauge}, the exact profile orders, and the entry
bounds. The sums along each optical direction will control transport
coefficients on arbitrarily long null segments; their vanishing tails
will later give the small exponential rates at higher orders.
With curvature convention $R(X,Y)=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$,
the vacuum identities in this gauge are
\begin{equation}\label{in:vacuum}
\begin{aligned}
L_i\tau_i&=\ell_i\tau_i-|\chi_i|^2,\\
\mathscr L_i\xi_j&=\beta_i+\tau_i\xi_i,\\
\mathscr L_j\chi_i
&=a\operatorname{Sym}\nabla\xi_j-\tfrac a2K\gamma
+a\xi_j^{\otimes2}
-\tfrac12(\tau_i\chi_j+\tau_j\chi_i)
+\chi_i\chi_j+\chi_j\chi_i,\\
L_j\ell_i
&=aK-\chi_i\mathbin{\cdot}\chi_j+\tau_i\tau_j
+a(2\xi_i\mathbin{\cdot}\xi_j-|\xi_i|^2).
\end{aligned}
\end{equation}
\paragraph{Derivation of the null equations.}
For angular vectors $E,H$, Koszul's formula and
$[L_i,L_j]=a(\xi_i-\xi_j)^\sharp$ give
\begin{equation}\label{in:connection-identities}
\begin{aligned}
\nabla^g_{L_i}L_i&=\ell_iL_i,&
\nabla^g_E L_i&=\chi_i(E)^\sharp+\xi_i(E)L_i,\\
\nabla^g_{L_i}L_j&=a\xi_i^\sharp,&
\nabla^g_EH&=\nabla_EH+
a^{-1}\{\chi_i(E,H)L_j+\chi_j(E,H)L_i\}.
\end{aligned}
\end{equation}
The same-index null Ricci contraction is
$-L_i\tau_i+\ell_i\tau_i-|\chi_i|^2$, giving Raychaudhuri.
Direct calculation also gives
\[
g(R(L_i,E)L_j,H)
=(\mathscr L_i\chi_j-\chi_j\chi_i
-a\nabla\xi_i-a\xi_i^{\otimes2})(E,H).
\]
The angular Gauss contraction is
\[
K\gamma+a^{-1}
\{\tau_i\chi_j+\tau_j\chi_i-\chi_i\chi_j-\chi_j\chi_i\}.
\]
Thus $\Ric_{EH}=0$ gives the sum of the two cross equations for
$\chi$. Their difference follows from
\[
\mathscr L_j\chi_i-\mathscr L_i\chi_j
=\tfrac12\mathcal L_{[L_j,L_i]}\gamma
=a\,\operatorname{Sym}\nabla(\xi_j-\xi_i)
+a(\xi_j^{\otimes2}-\xi_i^{\otimes2}),
\]
where $da=a(\xi_i+\xi_j)$ was used. Half the sum and difference give
the cross-shape equation of \eqref{in:vacuum}.
In $\Ric_{iE}$ the angular and null contractions are, respectively,
$\beta_i-\chi_i\xi_i+\tau_i\xi_i$ and
$-\mathscr L_i\xi_j+\chi_i\xi_i$. This proves the torsion equation.
In $\Ric_{ji}$ the angular contraction, after use of the cross
equation, is $aK-\chi_i\cdot\chi_j+\tau_i\tau_j$, while the remaining
contraction is
$a(2\xi_i\cdot\xi_j-|\xi_i|^2)-L_j\ell_i$. This proves the lapse
equation.
The corresponding differentiated angular systems have the factor structure
\begin{equation}\label{in:angular-system}
\begin{aligned}
\mathscr L_j\beta_i
&={}\tfrac a2(dK+Jd c_i^{\mathrm{ang}})
+a(K*\xi+\xi*\nabla\xi+\xi^3)
+\chi_i*\nabla\chi_j+\chi_j*\nabla\chi_i,\\
L_i(Q_i,\mu_i)
&={}((\operatorname{div}\beta_i,
\operatorname{curl}\beta_i),0)
+\chi_i*(K,\nabla\xi)+\nabla\chi_i*\xi.
\end{aligned}
\end{equation}
Here $Jd$ is the rotated angular differential and $*$ denotes the
indicated smooth angular metric contractions; every differentiated
factor type is retained. Also
\begin{equation}\label{in:bracket-shift}
[L_i,L_j]=a(\xi_i-\xi_j)^\sharp,\qquad
\partial_{y_1}b=a\gamma^{-1}(\xi_1-\xi_0).
\end{equation}
To verify this, apply divergence minus trace differential to the cross
equation. On $\operatorname{Sym}\nabla\xi_j$ the second-derivative
part is $\tfrac12Jd\,\operatorname{curl}\xi_j$; the curvature
commutators give $K*\xi$. Differentiating $a$ uses
$da=a(\xi_i+\xi_j)$, and varying the angular connection uses
$\mathscr L_j\gamma=2\chi_j$. These give exactly the displayed remaining
types. Surface curvature variation gives
\begin{equation}\label{in:curvature-variation}
L_iK=\operatorname{div}\beta_i-\tau_iK .
\end{equation}
For a one-form $\zeta$,
\[
L_i\operatorname{div}\zeta
=\operatorname{div}(\mathscr L_i\zeta)
-2\chi_i^{AB}\nabla_A\zeta_B
-(2\operatorname{div}\chi_i-d\tau_i)\cdot\zeta .
\]
Together with the torsion equation this cancels
$\operatorname{div}\beta_i$ in $L_i\mu_i$. Variation of the area form
similarly gives
$L_i\operatorname{curl}\xi_j
=\operatorname{curl}\beta_i+
\operatorname{curl}(\tau_i\xi_i)-\tau_i\operatorname{curl}\xi_j$.
This proves \eqref{in:angular-system}.
Frozen hats satisfy these equations at fixed parameters. In differences,
parameter defects in direction $i$ contain only $\sigma_i$.
We first estimate the solution on finite past diamonds
\[
\mathcal D_{C_0,C_1,s_*}
=\{0\le s\le s_*,\ y_0\le C_0,\ y_1\le C_1\}
\times\mathbb S^2,
\qquad C_0+C_1>0,\quad s_*>0.
\]
At fixed $y_j$, the past coordinate segment ends at $y_i=-y_j$
on $\mathcal S$ and remains in this diamond. The upper null sides and
upper $s$ cuts are future boundaries for the energy currents. All norms
below are restricted to this domain. The slightly tilted enlargements
used for continuation will retain these outgoing flux signs.
Fix $0<\eta\ll\kappa$ and let
$w_{i,n}=e^{-\eta|y_i-n|}$, $n\in\mathbb Z$.
For nonnegative angular norm functions $Y,Z$, define
\begin{equation}\label{in:flux-norms}
\begin{aligned}
[Y]_{i,n}
&=\sup_{y_j}\|w_{i,n}Y\|_{L^2(dy_i)}
+\sup_s\|w_{i,n}Y\|_{L^2(dx)},\\
\{Z\}_{i,n}
&=\sup_{y_i}\|w_{i,n}\sqrt q\,Z\|_{L^2(dy_j)}
+\sup_s\|w_{i,n}\sqrt q\,Z\|_{L^2(dx)}
+\|w_{i,n}\sqrt q\,Z\|_{L^2(dy_0dy_1)}.
\end{aligned}
\end{equation}
Every integral and supremum carries the domain restriction. Angular
$H^k$ norms in fixed controlled charts are written $|\cdot|_k$.
For actual-minus-hat differences, denoted by $\delta$, set
\begin{equation}\label{in:low-variables}
\begin{gathered}
Y_i=|\delta\beta_i|_5,\qquad
Z_i=|\delta(Q_i,\mu_i)|_5,\qquad
R_i=|\delta\chi_i|_6,\qquad
X_j=|\delta\xi_j|_6,\qquad d_i=|\delta B_i|_6,\\
E=\sum_{i,n}([Y_i]_{i,n}+\{Z_i\}_{i,n}),\qquad
R=\sum_{i,n}[R_i]_{i,n},\qquad
X=\sum_{i,n}\{X_j\}_{i,n},\\
D_b=\sum_{i,n}
\|w_{i,n}\sup_{y_j}d_i\|_{L^2(dy_i)}.
\end{gathered}
\end{equation}
The indexing of $X_j$ in the $i$ norm is deliberate: $j=1-i$.
For the remaining low quantities put
\begin{equation}\label{in:low-point-budgets}
\begin{gathered}
p_*=|\delta(\gamma,\log A)|_6+|\delta\xi|_5,\qquad
P=\sup p_*+\sum_n\sup e^{-8\eta s}w_{0,n}p_*,\\
g_i(y_i)=\sup_{y_j}
\bigl(|\delta(\chi_i,\ell_i)|_4+|\delta B_i|_5\bigr),
\qquad S=\sum_{i,n}\sup w_{i,n}g_i.
\end{gathered}
\end{equation}
The summations over cells are unsquared. They are needed both for
integrable transport coefficients and for their vanishing tails.
The paired angular equations estimate $E$. The remaining budgets
recover the shape and torsion fields ($R,X$), the shift ($D_b$),
the metric and lower torsion coefficients ($P$), and the integrable
directional envelopes ($S$) of $\delta\chi_i,\delta\ell_i,\delta B_i$.
These recoveries are coupled: the proof first reduces
their combined size to $E$ and the entry error, then closes the small
feedback into $E$.
\begin{proposition}[Uniform low characteristic estimates]
\label{in:low}
Choose $q_0$ sufficiently small in terms of the uniform profile
constants, and then choose $e_{\mathcal S}$ sufficiently small.
On every smooth finite past diamond satisfying the entry assumptions,
\begin{equation}\label{in:low-conclusion}
E+R+X+D_b+P+S\le C(q_0)e_{\mathcal S}.
\end{equation}
The metrics $A,\gamma$ and their inverses remain uniformly bounded
and nondegenerate at the indicated low orders. The constants are
independent of the location and size of the diamond.
\end{proposition}
\begin{proof}
Bootstrap the budgets small and the metrics in a uniformly
nondegenerate neighborhood. We first derive the Hodge and transport
estimates for the auxiliary budgets. Trace damping supplies the
shape estimate needed in this group. We then estimate the paired
angular currents and perform the simultaneous absorption.
The calculation uses only the displayed profile and equation
hypotheses. The actual radiative factors at low orders have
size $C(q+g_i)$; nonradiative coefficients are bounded and their
differences cost $Cp_*$. An orientation change obeys
\begin{equation}\label{in:weight-switch}
w_{i,n}\le e^{2\eta s}w_{j,-n}.
\end{equation}
Unit-cell convolution bounds the localized $L^2_x$ norm of $p_*$ by
$Ce^{10\eta s}$ times an $\ell^1$ envelope of total size $P$.
Positive powers of $q$ absorb this growth when $\eta$ is small.
The div--curl system for $\xi_j$ and the trace-free divergence
system for $\chi_i$ give
\begin{equation}\label{in:low-hodge}
X\le C(E+\sqrt{q_0}P),\qquad
R_i\le C\bigl(Y_i+|\delta(\tau_i/A)|_6
+|\delta\chi_i|_0+qp_*\bigr).
\end{equation}
Their constants use only the low angular ellipticity and coefficient
bounds. Coefficient differences acting on hats cost $p_*$, with the
additional $q$ in the shape estimate. Integrating the shift equation
in \eqref{in:bracket-shift}, with Cauchy--Schwarz and
\eqref{in:weight-switch}, yields
\begin{equation}\label{in:low-shift}
D_b\le Ce_{\mathcal S}+Cq_0^c(X+P)
\end{equation}
for a fixed $c>0$. The cross transports for $\chi_i,\ell_i$ have
homogeneous coefficient $C(q+g_j)$ and remaining difference source
$Cq(p_*+g_j+\sigma_j)$. Their quadratic terms pair opposite
radiative orientations. The lower shift estimate has source $Cqp_*$.
Consequently
\begin{equation}\label{in:low-characteristic}
S\le Ce_{\mathcal S}+Cq_0^c(P+S).
\end{equation}
The metric and torsion transports, using
$\partial_{y_1}\gamma=2\chi_1$,
$\partial_{y_1}\log A=\ell_1+\kappa$, and
$d\log A=\xi_0+\xi_1$, give
\begin{equation}\label{in:low-metric}
P\le C(e_{\mathcal S}+E+R+D_b+S).
\end{equation}
The unweighted free sources integrate by the cell sums. For the
localized part, the target-to-source ratio of
$e^{-8\eta s}w_{0,n}$ is bounded along either past coordinate path;
the remaining weight change and the square root of the path length
are absorbed by this exponential factor, proving
\eqref{in:low-metric} with the current profile constants.
Raychaudhuri has the exact rescaled form
\begin{equation}\label{in:trace-equation}
(L_i+\kappa)(\tau_i/A)=-A^{-1}|\chi_i|^2.
\end{equation}
The differentiated difference has source
$C((q+g_i)R_i+q^2p_*+q(\sigma_i+d_i))$,
and the tame transport commutators additionally cost
\[
C(q+g_i)|\delta(\tau_i/A)|_6
+C d_i(g_i+qp_*).
\]
The first term is absorbed in the trace damping under the small
bootstrap. The second uses the small low envelopes and the shift
budget; its contribution is included in the
$\varepsilon_{\mathrm{boot}}(P+D_b+R)$ term below.
Exponential convolution
in $y_i$ preserves the localized flux norms: a displacement $h$
changes the weight by at most $e^{\eta h}$ and samples the earlier
$s-h/2$ slice. Combining this damping with
\eqref{in:low-hodge} gives
\begin{equation}\label{in:low-shape}
R\le C(E+e_{\mathcal S}+S)
+C(q_0^c+\varepsilon_{\mathrm{boot}})(P+D_b+R).
\end{equation}
It remains to close the paired angular energies. Commute
\eqref{in:angular-system} through angular order five and subtract
hats, retaining the actual principal operators. The source norms are
bounded by constants times
\begin{equation}\label{in:paired-sources}
\begin{aligned}
G_i={}&q(X_0+X_1+Z_i+p_*+\sigma_j)
+(q+g_j)(Y_i+R_i)+(q+g_i)(R_j+d_j),\\
H_i={}&Y_i+R_i+d_i+\sigma_i+qp_*
+(q+g_i)(X_0+X_1+Z_i).
\end{aligned}
\end{equation}
Indeed, angular principal commutators cost $qZ_i$ or $Y_i$;
coefficient differences on hats have the stated profile gain; and
unweighted shape products pair opposite orientations. Shift
commutators use the low radiative bounds. Parameter defects occur
only in their own direction and inherit the $q$ factor whenever the
hatted field is radiative. This verifies the source list \eqref{in:paired-sources}.
Use squared currents with weights $w_{i,n}^2$ on $Y_i$, flowing
along $j$, and $w_{i,n}^2a/2$ on $(Q_i,\mu_i)$, flowing along $i$.
Angular integration by parts cancels the principal pair. The latter
currents have damping because
\[
L_i\log(aw_{i,n}^2)
\le-\kappa+C(q+g_i)+2\eta.
\]
All future null and $s$ fluxes are positive. The squared flux is
bounded by the initial norm plus
\begin{equation}\label{in:paired-current}
C\int w_{i,n}^2(Y_iG_i+qZ_iH_i)\,dy_i\,dy_j.
\end{equation}
Terms retaining a positive power of $q$ use slice
Cauchy--Schwarz and \eqref{in:weight-switch}. Same-orientation
$g$ terms use a null flux and $\int g_i$. The remaining
opposite-orientation terms obey \Cref{in:cross-flux}:
\begin{equation}\label{in:cross-flux}
\begin{aligned}
\int w_{i,n}^2Y_i g_iR_j
&\le C[Y_i]_{i,n}\|w_{i,n}g_i\|_{L^2(dy_i)}
\sum_m[R_j]_{j,m},\\
\int q w_{i,n}^2Z_i g_iX_i
&\le C\{Z_i\}_{i,n}\|w_{i,n}g_i\|_{L^2(dy_i)}
\sum_m\{X_i\}_{j,m}.
\end{aligned}
\end{equation}
For completeness, partition $y_j$ into unit cells. Two-dimensional
Cauchy--Schwarz uses the fixed-$y_j$ flux for the first energy factor
and the fixed-$y_i$ flux for the opposite factor; in the second line,
the spacetime $\sqrt q\,X_i$ norm on a cell is controlled by its
$j$-localized norm. Partial cells and past-diamond restrictions remain
inside the integrals. The same argument applies to $d_j$ in place of
$R_j$. Summing the unsquared budgets gives
\begin{equation}\label{in:low-energy}
E\le Ce_{\mathcal S}
+C(q_0^c+S)(E+R+D_b+P+X).
\end{equation}
To close the estimates, write
$\mathcal A_{\mathrm{aux}}=P+R+X+D_b+S$.
Substitute the Hodge bound for $X$, the shift and cross-transport
bounds for $D_b,S$, and the damped shape bound for $R$ into the
metric estimate for $P$. For some fixed $c'>0$, this gives
\[
\mathcal A_{\mathrm{aux}}
\le C(E+e_{\mathcal S})
+C(q_0^{c'}+\varepsilon_{\mathrm{boot}})
\mathcal A_{\mathrm{aux}}.
\]
Here $c'$ may be decreased to include the $\sqrt{q_0}$ term in
\eqref{in:low-hodge}. Absorbing the last term yields
$\mathcal A_{\mathrm{aux}}\le C(E+e_{\mathcal S})$.
The paired-current estimate \eqref{in:low-energy}, using the
bootstrap bound for $S$, now becomes
\[
E\le Ce_{\mathcal S}
+C(q_0^{c'}+\varepsilon_{\mathrm{boot}})
(E+e_{\mathcal S}).
\]
Choose $q_0$ first using only the profile and energy constants,
then the bootstrap size, to absorb this feedback. Finally choose
the entry size so that the resulting bounds strictly improve the
bootstrap, including nondegeneracy. This proves
\eqref{in:low-conclusion}.
\end{proof}
\subsection{The first pure longitudinal derivatives}
The low flux bounds control the cross derivatives in the null
equations. Normalized curvature also contains pure derivatives such
as $\mathscr L_i\chi_i$, which those equations do not directly bound
pointwise. Differentiating the cross equations gives transport
equations for these missing jets along the opposite direction, with
no additional small data norm.
\begin{proposition}[Pure jets and flank limits]\label{in:pure-jets}
Under the assumptions of Proposition~\ref{in:low}, the sum of
unit-$y_i$-cell suprema of
\begin{equation}\label{in:pure-budget}
\sup_{y_j}\left(
|\mathscr L_i\chi_i-\widehat{\mathscr L_i\chi_i}|_2
+|L_i\ell_i-\widehat{L_i\ell_i}|_2\right)
\end{equation}
is at most $Ce_{\mathcal S}$. For $i=0$ the same holds for
$\partial_{y_0}b-\partial_{y_0}^{\mathrm{fr}}\widehat b$
in angular $H^2$. On a full wedge obtained by exhaustion,
$A^{\pm1},\gamma^{\pm1}$ are uniformly bounded in $H^6$ and
$\xi_i,a^{-1}\partial_{y_1}b$ in $H^5$, and
\begin{equation}\label{in:flank}
\begin{gathered}
|b|_5+\sum_i|(\chi_i,\ell_i+\kappa)|_4\longrightarrow0,\\
\sum_{k=0}^1\left(
|\partial_{y_k}b|_2+
\sum_i|\partial_{y_k}(\chi_i,\ell_i)|_2
+|\partial_{y_k}\gamma|_4
+|\partial_{y_k}\log a+\kappa|_4\right)\longrightarrow0
\end{gathered}
\end{equation}
as $s\to\infty$ and $|y_0|,|y_1|\to\infty$.
All displayed quantities are uniformly bounded elsewhere.
No smallness of the limiting angular metric or rescaled torsion is
asserted.
\end{proposition}
\begin{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{equation}\label{in:pure-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}
\end{equation}
reduce derivatives of torsion, curvature, and angular connections to
already controlled fields. Differentiating the cross equations in
direction $i$ gives \Cref{in:pure-top}:
\begin{equation}\label{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.
\end{equation}
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
\eqref{in:entry-orders} and the known cross equations.
Localized Gronwall along $j$, with the same orientation and cell
estimates as above, proves \eqref{in:pure-budget}.
For the shift write $\zeta_\xi=\xi_1-\xi_0$ and
$C_b=a\gamma^{-1}\zeta_\xi=\partial_{y_1}b$.
The identity \Cref{in:pure-shift} is
\begin{equation}\label{in:pure-shift}
\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}
\end{equation}
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 \eqref{in:profile-orders}, the point budget therefore
gives the first line of \eqref{in:flank}. The cross equations give
the opposite first derivatives, while \eqref{in:pure-budget} and
\eqref{in:pure-shift} 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.
\end{proof}
\subsection{Higher estimates on finite and expanding regions}
We next distinguish two uses of higher regularity. On a fixed finite
diamond, arbitrary finite bounds suffice to restart the optical
development. On expanding diamonds, the packet construction requires
arbitrarily small exponential rates. The first estimate uses no
global tails; the second is applied only after the low estimates have
been exhausted to the full wedge.
\begin{proposition}[Higher estimates]\label{in:higher}
Assume the entry bounds of Proposition~\ref{in:entry} and the low
estimates on smooth finite past diamonds.
\begin{enumerate}
\item On every fixed finite diamond, all fixed ordinary coordinate
derivatives of $\gamma^{\pm1},\log A,b,a^{-1}\partial_{y_1}b$
have finite bounds depending only on the corresponding smooth initial
norms, the low bounds, and that diamond.
\item If the solution and its uniformly summed low estimates are
defined on the full wedge, fix $U\in\mathbb R$, $C>0$, and either
labeling $\{d,l\}=\{0,1\}$. On
\begin{equation}\label{in:high-domain}
s\ge0,\qquad y_d\le U,\qquad y_l\le CT,
\end{equation}
every fixed ordinary coordinate derivative of
$\gamma^{\pm1},\log A,b,a^{-1}\partial_{y_1}b$ is bounded by
$e^{o(T)}$. The same bounds hold on all restrictions and coordinate-path
subsegments of this domain.
\end{enumerate}
Neither conclusion requires any higher-order initial norm to be small.
\end{proposition}
\begin{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{equation}\label{in:high-variables}
\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}
\end{equation}
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
$d\log A=\xi_0+\xi_1$.
\paragraph{Angular currents.}
The tame sphere Hodge estimates give
\begin{equation}\label{in:high-hodge}
R_i\le C_m(Y_i+z_i+h_iP_*),\qquad
|\xi_j|_{m+1}\le C_m(Z_i/\sqrt q+P_*).
\end{equation}
These constants do not contain high coefficient norms. Indeed,
identify trace-free tensors using smooth $\gamma$-dependent bases,
freeze coefficients in sufficiently small charts, and commute the
first-order angular system. A highest coefficient placement costs
$C_m|\gamma|_{m+1}\|f\|_{C^1}$; interpolation absorbs the
penultimate field norm with a constant depending on $m$ and the low
bounds. In the shape estimate its low factor is $O(h_i)$.
Use the squared paired currents $Y_i^2$ in direction $j$ and
$Z_i^2$ in direction $i$, keeping their multipliers equal.
Add $z_i^2$ and $P_i^2$ in direction $i$, and $D^2,P_\gamma^2$
in direction $1$. Apart from favorable damping, their absolute
interaction costs are bounded by the following list:
\begin{equation}\label{in:high-costs}
\begin{aligned}
C_mY_i\big[&\rho_q(Z_0+Z_1+P_*)
+h_j(Y_i+R_i)+h_i(R_j+d_j^*)+h_i h_jP_*\big],\\
C_mZ_i\big[&\rho_q(Y_i+R_i+d_i^*+P_*)
+h_i(Z_0+Z_1)\big],\\
C_mz_i h_i(&z_i+R_i+d_i^*+h_iP_*),\\
C_mD\big[&\rho_q(Z_0+Z_1+\rho_qP_*)+h_1D\big],\\
C_mP_\gamma(&R_1+h_1P_\gamma),\\
C_mP_i(&Y_i+R_i+d_i^*+h_iP_*).
\end{aligned}
\end{equation}
We derive these costs from the actual equations already displayed.
Terms with $a$ retain
$\rho_q$ after pairing, by \eqref{in:high-hodge}; high angular
connection placements in $\chi*\nabla\chi$ cost $h_i h_jP_*$.
Shift commutators on $\beta_i$ cost $h_jY_i+h_id_j^*$, while
those on the weighted fields retain $\rho_q$. Metric and area
derivatives have coefficient $h_j$ on $Y_i^2$ and $h_i$ on
$Z_i^2$. The identity $L_i\log a=-\kappa+O(h_i)$ supplies the
favorable lapse damping. Trace and shift currents follow from
\eqref{in:trace-equation} and \eqref{in:bracket-shift}; the remaining
two currents follow from the metric and torsion transports.
Thus this actual-field system uses no special values of a Kerr profile.
\paragraph{Fixed regions.}
On a diamond $0\le s\le s_*$, substitute
\eqref{in:high-hodge} into \eqref{in:high-costs}. With all current
multipliers one, the resulting quadratic form is bounded by $C_m$
times their total size. The common weight
\begin{equation}\label{in:finite-weight}
e^{-2K_{\mathrm{fin}}s},\qquad
-L_i\log(e^{-2K_{\mathrm{fin}}s})=K_{\mathrm{fin}},
\qquad K_{\mathrm{fin}}>C_m,
\end{equation}
absorbs the cost in both directions. Removing the weight costs only
the fixed factor $e^{2K_{\mathrm{fin}}s_*}$ on squared energies.
The recovery arguments below then give finite bounds for every
coordinate derivative. For this use extend each scalar envelope
$g_i$ by zero outside the coordinate projection of the diamond; no
spacetime field is extended. In particular, no tail assumption enters
this fixed-region argument.
\paragraph{The two limiting interactions.}
For the sharper estimate now assume the full wedge and global
summable envelopes. When $\rho_q=h_0=h_1=0$, the only unsuppressed
arrows in \eqref{in:high-costs}, after substitution of
\eqref{in:high-hodge}, are
\[
Y_i,z_i\longrightarrow P_i,\qquad
D\longrightarrow P_0,\qquad Y_1,z_1\longrightarrow P_\gamma.
\]
This first limiting system is not sufficient throughout
\eqref{in:high-domain}: the coordinate $y_d$ may remain bounded
while $y_l$ tends to infinity, so $h_d$ need not tend to zero there.
We will instead give the currents flowing along $d$ a large damping
on a fixed interval $-B\le y_d\le U$. Removing this part of the
weight then costs a fixed factor, independent of $T$. To identify
what the remaining currents must control, consider a second limiting
system in which the $d$-flowing currents are strongly damped.
Those currents comprise
\[
\mathcal D_d=\{Y_l,Z_d,z_d,P_d\}
\quad\text{and also }\{D,P_\gamma\}\text{ if }d=1.
\]
The currents flowing along $l$ comprise
\[
\mathcal L_l=\{Y_d,Z_l,z_l,P_l\}
\quad\text{and also }\{D,P_\gamma\}\text{ if }l=1.
\]
Delete products containing a variable of $\mathcal D_d$ and set
$\rho_q=h_l=0$, allowing $h_d$ to remain bounded. The only
remaining arrows are
\begin{equation}\label{in:asymmetric-arrows}
z_l\longrightarrow Y_d,P_l,
\qquad z_l\longrightarrow P_\gamma\quad\text{if }l=1.
\end{equation}
Indeed the possible term $h_dd_l^*$ either vanishes or contains
$D$ flowing along $d$; if $D$ remains in $\mathcal L_l$, its cost
vanishes. The retained $P_l$ source contains $Y_l$, which belongs
to $\mathcal D_d$, and the retained trace cost has factor $h_l$.
There is consequently no unsuppressed return to $z_l$.
Both limiting systems are triangular: their surviving interactions
can be made as small as desired by rescaling the auxiliary currents.
The actual equations retain all their terms; these two comparisons
only determine the multipliers and the directional damping.
Fix an arbitrarily small $\delta_h>0$. Keep multiplier one on each
$Y_i,Z_i$ pair and on $D^2$, multiply $z_i^2$ by $\Lambda^2$
with $\Lambda$ large, and multiply the $P$ currents by
$\epsilon_P^2$ with $\epsilon_P$ small. The resulting size is
\begin{equation}\label{in:rescaled-energy}
\mathcal E_m=\sum_i(Y_i^2+Z_i^2)+D^2
+\Lambda^2\sum_i z_i^2
+\epsilon_P^2(P_\gamma^2+P_0^2+P_1^2).
\end{equation}
In the scaled variables, the first limiting comparison has coefficients
$O(\epsilon_P)$ or $O(\epsilon_P/\Lambda)$, and
\eqref{in:asymmetric-arrows} has coefficients
$O(h_d/\Lambda)$ or $O(\epsilon_P/\Lambda)$.
Choose the rescaling so that both limiting costs are at most
$\delta_h\mathcal E_m/4$ in their respective retained groups.
The principal paired angular cancellation is unchanged.
Continuity of the remaining finite quadratic forms gives a threshold
$\epsilon_*>0$ such that the full cost is small when
$\rho_q,h_0,h_1\le\epsilon_*$, and the cost involving only
$\mathcal L_l$ is small when $\rho_q,h_l\le\epsilon_*$.
This threshold is chosen before the large damping constant below.
\paragraph{Directional damping on expanding regions.}
Use the common weight
\begin{equation}\label{in:high-weight}
\mathfrak w=\exp\{-2\delta_h(y_0+y_1)
-2K_0(y_d+B)_+-2K_0\min(s,S_0)\}.
\end{equation}
Choose $B,S_0$ large, independently of $T$, so that for
$s>S_0,y_d<-B$, both $h_i$ and $\rho_q$ are below
$\epsilon_*$. For $s>S_0,-B\le y_d\le U$, make
$h_l,\rho_q$ small; this is possible because
$y_l=2s-y_d\ge2S_0-U$. Almost everywhere the damping in direction
$k$ is
\[
-L_k\log\mathfrak w
=2\delta_h+2K_0\mathbf1_{\{k=d,\ y_d>-B\}}
+K_0\mathbf1_{\{s4C_m''$ absorbs the two integrals. Hence
$P_*,z_i\le e^{o(T)}$, and \eqref{in:high-hodge} makes $R_i$
integrable along $i$ at that cost. On fixed diamonds the same argument
uses the finite flux bound from \eqref{in:finite-weight} and the
zero-extended scalar envelopes, giving finite constants.
The cross transports for $\chi_i,\ell_i+\kappa$ now give
pointwise angular bounds, initially through $H^{m-1}$.
Their homogeneous coefficient is $h_j$; the terms $qP_*$ have the
necessary spare angular derivatives, and the remaining terms
$h_i(R_j+d_j^*)+h_i h_jP_*$ integrate by the bounds just obtained.
The shift equation gives pointwise angular control through order $m$.
Since $m$ was arbitrary, every fixed angular order is available.
\paragraph{Longitudinal recovery.}
Induct on the number of longitudinal derivatives, retaining arbitrary
fixed angular orders. Proposition~\ref{in:pure-jets} estimated the first
pure derivatives at low angular order. To start the present induction,
commute their equations \eqref{in:pure-top} and \eqref{in:pure-shift}
at each fixed angular order. Their inhomogeneous terms now have the
angular bounds just proved, and their initial jets follow from the
entry tangential derivatives and the cross equations. The top transport
coefficient, including the first angular derivative of the shift, is
bounded by $C_mh_j$. Higher angular derivatives of coefficients multiply
lower angular derivatives of the pure jet. Induction in angular order therefore bounds the first
pure shape jets, then the pure lapse jets through their triangular
equation, and the pure shift jets. This starts the longitudinal
induction at every fixed angular order.
At the next level, differentiation of the cross equations gives
transports for $\mathscr L_i^k\chi_i,L_i^k\ell_i$ along $j$.
The only new pure jet on the right is the shape jet, with homogeneous
coefficient $Ch_j$; all other terms have lower longitudinal counts.
This follows from \eqref{in:pure-identities}: a derivative of either
torsion uses the torsion equation and $d\ell_i$, and curvature
variation uses $\beta_i$. Angular connection variations are expressed
by $\chi_i$ and its angular derivatives. Differentiated brackets
are angular and use only previous longitudinal orders. Derivatives of
$a$ retain its factor.
The initial pure jets follow from the noncharacteristic initial
equations, or equivalently from tangential $x$ derivatives and the
cross equations. Pointwise transport Gronwall uses only $Ch_j$,
including the first angular derivative of the shift. A subsidiary
angular induction assigns higher coefficient derivatives to already
controlled factors. Integration over length $O(T)$ of an $e^{o(T)}$
source is still subexponential. Mixed longitudinal derivatives reduce
to the cross equations and the angular commutators.
Finally recover ordinary derivatives of the shift from
\eqref{in:bracket-shift}. Pure $\partial_{y_0}^k b$ is integrated
in $y_1$; after converting Lie derivatives to coordinate derivatives,
its source contains the shift only at preceding longitudinal orders.
The other derivatives are direct, and
$a^{-1}\partial_{y_1}b=\gamma^{-1}(\xi_1-\xi_0)$ is algebraic.
The same recovery with finite bounds proves the first assertion, and
with \eqref{in:weight-range} proves the second.
\end{proof}
\subsection{Continuation and exchange of the null labels}
We now construct the full wedge. Only the fixed-region part of
Proposition~\ref{in:higher} is used for continuation. Once the wedge
exists, its global low envelopes supply the tails required for the
subexponential conclusion.
\begin{proposition}[Continuation from the full cylinder]
\label{in:continuation}
The smooth vacuum cylinder data of Proposition~\ref{in:entry} have
a smooth development on the full product wedge
$\{s\ge0\}\times\mathbb R_x\times\mathbb S^2$ in the gauge
\eqref{in:gauge}. It satisfies Propositions~\ref{in:low},
\ref{in:pure-jets}, and both conclusions of
Proposition~\ref{in:higher}. With a sufficiently small past collar,
$\mathcal S$ is a Cauchy surface for this development. The construction
does not identify this region with the whole maximal globally
hyperbolic development.
\end{proposition}
\begin{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
\eqref{in:finite-weight}, followed by the recovery part of
Proposition~\ref{in:higher}, 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 \eqref{in:high-weight}; this proves the expanding
subexponential bounds without circularity.
Finally $s$ is temporal. Causality in \eqref{in:gauge} 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.
\end{proof}
\begin{lemma}[The same wedge with exchanged labels]\label{in:label-swap}
The two optical labels may be exchanged while retaining the same
initial cylinder and the same product spacetime. In the corresponding
one-shift angular gauge, all conclusions of
Propositions~\ref{in:entry}--\ref{in:continuation} remain valid,
with the same common $\kappa$ and with a frozen physical profile
depending only on the new $y_0$.
\end{lemma}
\begin{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~\ref{in:entry} 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.
\end{proof}
==== END Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/07-interior.tex ====
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/08-comparison.tex ====
\section{The comparison development and finite geodesic runout}
\label{sec:comparison}
The preceding estimates describe two exterior evolutions and a
double-null interior evolution. We now place them in a single vacuum
development containing the complete initial bridge. We then prove two
geometric facts needed for the perturbation argument: a geodesic of
finite future proper duration first leaves this comparison development
through an interior branch or corner, and an open family containing a
regular exiting geodesic contains a branch geodesic. The comparison
development will be an open subspacetime of the full maximal globally
hyperbolic development; we never assert that it exhausts that development.
Throughout the section the Kerr center is fixed, with
$0<\mathfrak a0.
\]
In Cartesian coordinates at either end the reference bridge has
$h_*-\delta=O(r^{-1})$ and connection coefficients $O(r^{-2})$, with
the corresponding differentiated symbol bounds. Consequently, at every
fixed order, its covariant weighted seminorms are equivalent to the
ordinary component symbol seminorms. On compact sets they control the
usual local norms. No asymptotic coefficient of the actual datum is
fixed by this identification.
Fix first the profile depth $q_0$ and common inner rate $\kappa>0$ from
Section~\ref{sec:interior}. The exact initial interior cylinder
$\mathcal S_* =\{Q=q_0\}=\{s=0\}$ is a graph strictly above $r_-$.
Choose the computing boundary $r_b$ strictly below this graph in radial
value, with margins on both sides and $r_b>r_-$. The metric dual of
$ds$ is past timelike; the future side of $\mathcal S_*$ is its
smaller-radius side.
At each end use the horizon-crossing time
\begin{equation}\label{cmp:bent-time}
t_e=T_e+r_*-G(r).
\end{equation}
Here $G'=1$ through the fixed interior and trapping ranges,
$1\leq G'\leq r_*'$ in an exterior transition, and $G=r_*$ near
infinity. Ordinary ingoing angular coordinates are used at the
corresponding future outer horizon. The temporal sign was established
in the geometric construction; explicitly,
\begin{equation}\label{cmp:temporal-sign}
\bigl(r^2+\mathfrak a^2\cos^2\vartheta\bigr)
g_K^{-1}(dt_e,dt_e)
=-2(r^2+\mathfrak a^2)G'+\Delta(G')^2
+\mathfrak a^2\sin^2\vartheta<0.
\end{equation}
For $G'=1$ the right side is
$-r^2-2Mr-\mathfrak a^2\cos^2\vartheta$; convexity gives the sign in
the exterior transition.
Moving that transition outward makes $G-r_*$ arbitrarily positive at
each fixed inner radius, since the exterior integral of $r_*'-1$
diverges logarithmically. Moreover $G-r_*\geq0$ outside, and
$(G-r_*)'=1-r_*'>0$ between the horizons where $G'=1$. Thus the two
partial slices $t_e=0$ lie to the future of the bridge, or on it, and
agree with it near their respective infinities. Their portions through
the chosen depth are separated with a fixed margin in $T_K$, including
their subsequent nonnegative-time sides before that depth. Choose the
delay so that the bounded matching bands required at the two ends of
$\mathcal S_*$ are disjoint.
\begin{lemma}[Finite bridge evolution]\label{cmp:finite-range}
Fix the depths and delayed slices just described, together with bounded
forward matching bands on them. After enlarging these fixed regions,
there is a finite vacuum Cauchy development of every sufficiently
$p_{10}$-small datum on the complete bridge which contains the two
delayed slices and the matching bands, as well as a past collar of the
whole bridge. On the delayed slices the induced data are small at
metric order $H^{10}$ and second-form order $H^9$ on compact sets, and
retain the prescribed symbol bounds at the ends. The artificial future
cuts are strict future exits. Higher fixed smooth norms are finite,
without an additional smallness hypothesis.
\end{lemma}
\begin{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
\begin{equation}\label{cmp:bl-cone}
\left|\frac{dT_K}{dr}\right|
\leq
\frac{\sqrt{(r^2+\mathfrak a^2)^2
-\Delta\mathfrak a^2\sin^2\vartheta}}
{|\Delta|}.
\end{equation}
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--S{\'a}nchez \cite[Theorem~1.1]{BernalSanchez2005}. 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.
\end{proof}
\begin{proposition}[The comparison development]\label{cmp:development}
There is a single sufficiently small $p_{10}$ neighborhood such that,
for each datum $d$ in it, the preceding exterior and interior solutions
assemble with the finite bridge evolution into a smooth time-oriented
vacuum development $\mathcal P_d$ with the complete bridge $\Sigma$
as a Cauchy surface. It contains an entire spacelike cylinder
$\mathcal S\simeq\mathbb R\times\mathbb S^2$ carrying the entry data
of Section~\ref{sec:interior}, followed by its full product double-null
wedge. There is a time-orientation-preserving open isometric embedding
$\mathcal P_d\hookrightarrow M_d$ into the full MGHD of $d$.
\end{proposition}
\begin{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~\ref{cmp:finite-range}. 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~\ref{cmp:finite-range}. 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.
\end{proof}
The construction has now placed all of our estimates inside $M_d$.
The next objective is to locate the first departure of a finite-duration
geodesic from $\mathcal P_d$. This requires proper-time control, since
bounded spacetime coefficients alone do not bound a geodesic's boost.
\subsection{Compact exterior excursions}
Consider one exterior end, with time $t=t_e$ and co-moving spatial
coordinates $y$. On any fixed compact cylinder containing the matching
interface, the mixed exterior estimates give a majorant
\begin{equation}\label{cmp:compact-integrable}
\|g-\overline G_{p(t),c(t)}\|_{C^1}
+|p'(t)|+|c'(t)|\leq\rho_c(t),
\qquad \rho_c\in L^1([0,\infty)).
\end{equation}
The low errors are uniformly small and the parameters remain small.
The integrability follows by mixed Sobolev embedding and
Cauchy--Schwarz with the static weight $A_S>1$. Constants in
\eqref{cmp:compact-integrable} are independent of the starting time
and duration of an excursion.
Let $U$ be a future unit geodesic tangent. Put
$\dot t=dt(U)>0$, $V=U/\dot t$, and use a prime for differentiation
along the track with respect to $t$. Normalized causal velocities $V$
are bounded on fixed compact cylinders. Define
\[
T_{\mathrm{co}}=\partial_t|_y,
\qquad E=-g(T_{\mathrm{co}},U),
\qquad P=g(\Phi_{p,c},U),
\]
where $\Phi_{p,c}$ is the axial Killing field of the instantaneous
exact Kerr reference. The deformation identity and frozen stationarity
and axisymmetry yield
\begin{equation}\label{cmp:energy-momentum}
|E'|+|P'|\leq C\rho_c(t)\dot t.
\end{equation}
The time derivative of $\Phi_{p,c}$ is included in the parameter cost.
\begin{lemma}[Compact excursions]\label{cmp:compact-excursions}
For a future unit timelike geodesic which is inextendible in
$\mathcal P_d$ and never crosses $\mathcal S$, every portion
$[t_1,t_2]$ inside a fixed sufficiently
large co-moving radius satisfies
\begin{equation}\label{cmp:compact-speed}
\dot t(t)\leq C\dot t(t_1),\qquad t_1\leq t\leq t_2,
\end{equation}
with $C$ independent of its duration and of the number of visits to the
outer-horizon strip. If the portion starts and ends at an interface
where $T_{\mathrm{co}}$ is uniformly future timelike, then
\begin{equation}\label{cmp:return-energy}
E(t_2)\leq E(t_1)
\exp\left(C\int_{t_1}^{t_2}\rho_c(t)\,dt\right).
\end{equation}
In particular, there is no fixed multiplicative loss at each return.
\end{lemma}
\begin{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
\begin{equation}\label{cmp:redshift}
(\mathcal L_{N_H}b)(V,V)>c_*>0.
\end{equation}
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
\begin{equation}\label{cmp:momentum-control}
\dot t\leq C(|E|+|P|).
\end{equation}
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, \eqref{cmp:energy-momentum} and
\eqref{cmp:momentum-control} 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
\eqref{cmp:compact-speed}, 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 \eqref{cmp:return-energy}.
\end{proof}
\subsection{Integrability along far portions}
We next bound the geodesic energy between compact excursions. In the
far lab coordinates $(t,z)$ recall the emitter variables
\[
t=q+r,\qquad z=C_e(q)+rn,\qquad |n|=1,
\qquad k=1-C_e'(q)\cdot n,
\]
and the field
\[
T_l=\partial_t|_z,\qquad D=T_l+s_s\cdot\partial_z,
\qquad g=B+h,\qquad \psi=Dh.
\]
In this subsection $q$ is retarded time and $r$ is emitter radius;
neither is an interior optical variable. The emitter speed is small,
and $k$ is bounded above and away from zero. Interpolate from
$T_{\mathrm{co}}$ near the compact interface through a fixed overlap
to $D$ far out, obtaining a uniformly future timelike field $Z_K$
outside the interface. Its difference from either prescribed field on
the overlap, with first derivatives, has an integrable parameter cost.
Define $E_K=-g(Z_K,U)\asymp\dot t$. The deformation identity gives
\begin{equation}\label{cmp:outer-energy}
|(\log E_K)'|
\leq C|(\mathcal L_{Z_K}g)(V,V)|.
\end{equation}
\begin{lemma}[Outer track integrability]\label{cmp:outer-integrability}
Along any one future unit timelike geodesic which is inextendible in
$\mathcal P_d$, eventually proceeds in this end, and never crosses
$\mathcal S$, the right side of
\eqref{cmp:outer-energy} is integrable over all its outer portions,
including arbitrarily many returns to the compact interface.
\end{lemma}
\begin{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
\begin{equation}\label{cmp:averaged-cost}
|DB|+|\partial s_s|
\leq\frac{C}{|x_{\mathrm{av}}|}
\int_{t-|x_{\mathrm{av}}|/10}^{t}|p'(v)|\,dv.
\end{equation}
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 \eqref{cmp:averaged-cost}
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,
\label{cmp:trace-field}\\
\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.
\label{cmp:trace-derivative}
\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
\begin{equation}\label{cmp:trace-supremum}
\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.
\end{equation}
The shell sum converges. Cauchy--Schwarz in time, using $A>1$, yields
\begin{equation}\label{cmp:timelike-cone-integral}
\int_1^\infty
\sup_{R_0\leq r\leq t/C_1,\omega}|\psi(t,r,\omega)|\,dt
<\infty.
\end{equation}
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{equation}\label{cmp:far-point}
\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}
\end{equation}
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{equation}\label{cmp:retarded-variation}
\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}
\end{equation}
Decomposing $V-L$ into radial and screen parts, the improved
contractions in \eqref{cmp:far-point} imply
\begin{equation}\label{cmp:field-contraction}
|\psi(V,V)|\leq Cr^{-2+c_3}
+C|\psi|(|q'|+r^{-1+c_3}).
\end{equation}
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 \eqref{cmp:retarded-variation} its total negative variation is
$O(R^{c_3})$, while its range has length $O(R^\nu)$. Hence
\begin{equation}\label{cmp:clipped-variation}
\int_{\substack{R\leq t\leq2R\text{ on outer portions}\\
|q|\leq r^\nu}}|q'|\,dt
\leq C(R^\nu+R^{c_3}).
\end{equation}
Multiplying by $CR^{-1+e/2}$ is summable over dyads, because
$\nu,c_3<1-e/2$. The other two terms in
\eqref{cmp:field-contraction} are integrable directly.
Together with \eqref{cmp:timelike-cone-integral},
\eqref{cmp:averaged-cost}, and the fixed-overlap estimate, this proves
integrability over all outer portions of the track, including all
returns to the compact interface.
\end{proof}
\subsection{Proper time and the first-runout alternatives}
\begin{proposition}[No finite exterior runout]\label{cmp:completeness}
A future unit timelike geodesic from $\Sigma$, inextendible as a
geodesic in $\mathcal P_d$, which never crosses $\mathcal S$ has
infinite future proper duration.
\end{proposition}
\begin{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 \eqref{cmp:return-energy} bounds each increment from above by
$C\int_{a_j}^{b_j}\rho_c$. On the complement,
Lemma~\ref{cmp:outer-integrability} 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
\eqref{cmp:compact-speed} 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 $t\to\infty$.
\end{proof}
In the interior wedge we call the following limiting coordinate
behaviors its two \emph{branches} and its \emph{corner}:
\begin{equation}\label{cmp:runout-types}
\begin{array}{lll}
\mathcal B_+:&y_0\longrightarrow y_0^*<\infty,
&y_1\longrightarrow+\infty,\\
\mathcal B_-:&y_1\longrightarrow y_1^*<\infty,
&y_0\longrightarrow+\infty,\\
\mathcal K^+:&y_0\longrightarrow+\infty,
&y_1\longrightarrow+\infty.
\end{array}
\end{equation}
These are runout regimes, not points adjoined to the manifold.
\begin{corollary}[First-runout classification]\label{cmp:classification}
Let a future unit timelike geodesic from $\Sigma$ have finite maximal
future proper duration in $M_d$. Follow its initial connected portion
in $\mathcal P_d$ up to first runout. It crosses $\mathcal S$, and
its first runout has exactly one of the behaviors
\eqref{cmp:runout-types}.
\end{corollary}
\begin{proof}
The first-runout time is no later than its finite maximal time in
$M_d$. Proposition~\ref{cmp:completeness} 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.
\end{proof}
First runout can precede the geodesic's maximal time in $M_d$; in that
case its endpoint is an ordinary point of $M_d$. This distinction will
be important in the next argument, which uses an ambient regular chart
but makes no assertion about the whole inner boundary of $M_d$.
\subsection{From a regular corner to a branch seed}
We recall exactly the interior bounds used here. On
$y_0+y_1\geq0$ the metric is
\begin{equation}\label{cmp:wedge-metric}
\begin{aligned}
g&=-2a\,dy_0dy_1+
\gamma_{AB}(d\theta^A-b^A dy_0)(d\theta^B-b^B dy_0),\\
L_0&=\partial_{y_0}+b^A\partial_{\theta^A},\qquad
L_1=\partial_{y_1},\qquad
a=qA,\qquad q=q_0e^{-\kappa(y_0+y_1)}.
\end{aligned}
\end{equation}
Here $a$ is the positive null factor, not the Kerr rotation parameter;
$q$ now denotes the interior exponential scale. The metric
$\gamma$ and its inverse are uniformly comparable with a fixed round
metric, and $A,A^{-1}$ are uniformly bounded. Set
$\chi_i=\tfrac12\mathcal L_{L_i}\gamma$ on angular tensors and
$\ell_i=L_i\log a$. The angular one-forms $\xi_i$ are characterized by
\[
\xi_0+\xi_1=d_\theta\log a,
\qquad \xi_0-\xi_1=-a^{-1}\gamma\,\partial_{y_1}b.
\]
Section~\ref{sec:interior} gives bounded $\xi_i$ and nonnegative
functions $g_i\in L^1(\mathbb R)$ such that
\begin{equation}\label{cmp:corner-low-input}
|\chi_i|_\gamma+|\ell_i+\kappa|
\leq C(q+g_i(y_i)),\qquad
|b|_{\mathrm{round}}\leq C(q+g_0(y_0)).
\end{equation}
All angular statements below are intrinsic to the sphere. We use only
these low bounds and the connection identities
\begin{equation}\label{cmp:null-connections}
\begin{aligned}
\nabla^g_{L_i}L_i&=\ell_iL_i,&
\nabla^g_E L_i&=\chi_i(E)^\sharp+\xi_i(E)L_i,\\
\nabla^g_{L_i}L_j&=a\xi_i^\sharp
&&(i\ne j),
\end{aligned}
\end{equation}
for angular $E$, obtained by Koszul's formula.
The next lemma compares the positive optical variables
$Y_i=e^{-\kappa y_i}$, both of which vanish at a corner.
Near a corner, their speed ratio changes
by a small multiplicative factor that is independent of the initial
unit timelike tangent. A reciprocal change of the two null velocity
components can therefore prevent the variables from vanishing at
the same proper time. This will select a branch while preserving
unit normalization.
\begin{lemma}[Corner ratios]\label{cmp:corner-ratios}
Consider a future unit timelike geodesic starting at a fixed point of
the double-null wedge, and put $Y_i=e^{-\kappa y_i}$. Its maximal
remaining wedge track has finite proper duration, has an angular limit,
and ends in a branch or corner. Each $|\dot Y_i|$ is comparable above
and below to its positive initial value, and its screen velocity is
bounded on that fixed track. Moreover there is
$\delta_c(y_0^{\mathrm{in}},y_1^{\mathrm{in}})$, independent of the
starting unit timelike tangent, with $\delta_c\to0$ as both initial
coordinates tend to $+\infty$, for which
\begin{equation}\label{cmp:ratio-control}
e^{-2\delta_c}
\frac{|\dot Y_0^{\mathrm{in}}|}{|\dot Y_1^{\mathrm{in}}|}
\leq\frac{|\dot Y_0|}{|\dot Y_1|}
\leq e^{2\delta_c}
\frac{|\dot Y_0^{\mathrm{in}}|}{|\dot Y_1^{\mathrm{in}}|}.
\end{equation}
If the track ends at the corner, then necessarily
\begin{equation}\label{cmp:corner-necessary}
e^{-2\delta_c}
\frac{|\dot Y_0^{\mathrm{in}}|}{|\dot Y_1^{\mathrm{in}}|}
\leq\frac{Y_0^{\mathrm{in}}}{Y_1^{\mathrm{in}}}
\leq e^{2\delta_c}
\frac{|\dot Y_0^{\mathrm{in}}|}{|\dot Y_1^{\mathrm{in}}|}.
\end{equation}
For a general fixed starting point the same conclusions hold with a
finite, not necessarily small, logarithmic-variation constant.
\end{lemma}
\begin{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
\eqref{cmp:null-connections} give
\begin{align}
1+|\Xi|_\gamma^2&=2a\dot y_0\dot y_1,
\label{cmp:unit-normalization}\\
\frac{\dot p_i}{p_i}
&=\ell_i\dot y_i+(\xi_i-\xi_j)(\Xi)
-\frac{\chi_i(\Xi,\Xi)}{p_i}.
\label{cmp:momentum-equation}
\end{align}
Since $|\Xi|_\gamma^2/p_i\leq2\dot y_i$, we obtain
\begin{equation}\label{cmp:log-momentum}
\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}
\end{equation}
Both $Y_i$ decrease, and \eqref{cmp:unit-normalization} 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
\begin{equation}\label{cmp:angular-length}
\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}}}.
\end{equation}
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 \eqref{cmp:angular-length} and a fixed
constant factor, is a choice of $\delta_c$ in
\eqref{cmp:log-momentum}. 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
\eqref{cmp:ratio-control}; the bounds for $A^{\pm1}$ give individual
speed comparability. The normalization then bounds $\Xi$ on the
fixed track. Also \eqref{cmp:corner-low-input} 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
\eqref{cmp:ratio-control}, using
$\int |\dot Y_i|\,d\tau=Y_i^{\mathrm{in}}$, gives
\eqref{cmp:corner-necessary}. This proves the assertions, including
the finite-variation version starting at a general fixed point.
\end{proof}
\begin{proposition}[A branch seed in the original open family]
\label{cmp:branch-seed}
Let $d$ belong to the fixed base neighborhood and let
$O\subset T\Sigma$ be open. Suppose $w\in O$ has finite future
proper duration in $M_d$ and its endpoint is reached on a regular
geodesic arc in a $C^2$ extension. Then some seed $w'\in O$ first
runs out of $\mathcal P_d$ at one of its noncorner branches.
The new seed need not have a regular exit in that extension.
\end{proposition}
\begin{proof}
Corollary~\ref{cmp:classification} 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
\begin{equation}\label{cmp:reciprocal-change}
U_\sigma=(1+\sigma)\dot y_0L_0
+(1+\sigma)^{-1}\dot y_1L_1+\Xi,
\qquad \sigma>0.
\end{equation}
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
\eqref{cmp:corner-necessary}; a corner for the changed track would
satisfy the same necessary condition with this new ratio. The two
allowable intervals are disjoint once
\begin{equation}\label{cmp:corner-exclusion}
4\delta_c<2\log(1+\sigma).
\end{equation}
For each fixed $\sigma>0$, a sufficiently late change therefore
produces a branch by Lemma~\ref{cmp:corner-ratios}.
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
\begin{equation}\label{cmp:rapidity}
-g(U,U_\sigma)
=1+(1+|\Xi|_\gamma^2)
\left(\frac{c+c^{-1}}2-1\right).
\end{equation}
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 $t_00$ for these
tolerances; finally choose $z_n$ so late that
\eqref{cmp:corner-exclusion} holds. The last choice forces a branch
without changing any preceding tolerance.
\end{proof}
In particular, if $d\in\mathcal B(O,N)$ is a datum in a regular-exit
piece of the closed-test reduction, the preceding proposition provides
a branch seed in the same $O$. Although regular exit need not persist
for that seed, every duration and curvature bound of $A(O,N)$ still
holds along it. The following finite-signal construction will use only
this branch and those closed-test bounds.
==== END Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/08-comparison.tex ====
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/09-packet.tex ====
\section{Finite tidal observations and the category conclusion}
\label{sec:packet}
The comparison development constructed above supplies a branch along
which a bounded-curvature geodesic experiences an exponentially large
optical frequency. We now use that frequency amplification to make a
finite tidal observation arbitrarily large by an arbitrarily small
change in any prescribed finite list of data seminorms. The construction
has three distinct objects: an approximate interior metric perturbation
$h^p$, an approximate exterior perturbation $h^B$ transferring its entry
jets to the original initial end, and exact vacuum data $d_T$ obtained
by correcting the constraints of the exterior seed. A separate Cauchy
attachment places the observation in the full development of $d_T$.
All wave comparisons
in this section use ordinary wave-map reduction with the fixed vacuum
background $g=g_d$ as target. They are local hyperbolic comparisons,
independent of the exterior modulation gauge.
Explicitly, for the unknown metric $\widetilde g$ we use
\begin{equation}
\begin{split}
\operatorname{Ric}(\widetilde g)_{\mu\nu}
+\nabla^{\widetilde g}_{(\mu}
\Upsilon_{\nu)}(\widetilde g;g)&=0,\\
\Upsilon_\mu(\widetilde g;g)
&=-\widetilde g_{\mu\lambda}\widetilde g^{\alpha\beta}
\bigl(\Gamma(\widetilde g)^\lambda_{\alpha\beta}
-\Gamma(g)^\lambda_{\alpha\beta}\bigr).
\end{split}
\label{pk2:local-reduction}
\end{equation}
The target $g$ is a fixed smooth coefficient field in this equation.
There are no spatially nonlocal gauge additions or modulation feedback
variables in \eqref{pk2:local-reduction}. Its exact wave-map Cauchy
initialization and the vacuum constraints give homogeneous gauge
propagation and ordinary finite domain of dependence.
\begin{proposition}[A finite tidal observation]
\label{pk2:finite-signal}
Let $d\in\mathcal U_\epsilon$, and let $w\in T\Sigma$ be a seed whose future
unit timelike geodesic has a noncorner branch as its first runout from
the comparison development $\mathcal P_d$. Suppose its curvature is
bounded in the parallel positive metric up to that runout. For every
integer $m\ge10$, every $\eta>0$, and every $J>0$, there are smooth
complete vacuum data $d_T\in\mathcal D$ and seeds $w_T\to w$, for
arbitrarily large finite $T$, such that
\[
p_m(d_T-d)\longrightarrow0,
\]
and the unit geodesic of $w_T$ in the full MGHD of $d_T$ has a finite
point with tangent $U_T$ and a unit spacelike vector $X_T\perp U_T$
for which
\[
\bigl|\operatorname{Riem}(g_{d_T})(U_T,X_T,U_T,X_T)\bigr|>J.
\]
In particular $p_m(d_T-d)<\eta$ for sufficiently large $T$, and the
data may be required to remain in any prescribed relatively open
neighborhood of $d$.
\end{proposition}
We prove the proposition in the following subsections. The orders of
all comparison norms are fixed before choosing the positive frequency
exponent; formal accuracy is chosen afterwards, and $T$ is chosen last.
The notation $e^{o(T)}$ denotes a bound by $C_\sigma e^{\sigma T}$ for
every $\sigma>0$, at each fixed derivative and formal expansion order.
Constants and thresholds may depend on those orders and on $d$.
For a positive quantity, ``two-sided $e^{o(T)}$'' means that the same
convention applies to its reciprocal. No uniformity in an unbounded
sequence of derivative orders is asserted.
\subsection{The branch state and the normalized coefficients}
\label{pk2:branch}
Arrange the double-null labels so that
\[
u=y_0,\qquad v=y_1,\qquad
s=\frac{u+v}{2},\qquad x=\frac{v-u}{2},
\qquad v\longrightarrow\infty,\quad u\longrightarrow u_*<\infty
\]
on the chosen branch. The label-interchange construction proved above
provides these conventions also for the other branch, on the same
surface $\mathcal S=\{s=0\}$ and the same product wedge. Its new angular
labels are obtained by finite sphere flow at every finite point; their
estimates are then rederived from the prepared entry data. Thus no
uniform derivative bound for an unestimated long angular flow is used.
Throughout the experiment these are fixed background coordinates.
Write
\[
g=-2a\,du\,dv+
\gamma_{AB}(d\theta^A-b^Adu)(d\theta^B-b^Bdu),
\qquad L_0=\partial_u+b^A\partial_A,
\quad L_1=\partial_v,
\quad a=q_0e^{-\kappa(u+v)}A.
\]
Let $P_T$ be the background geodesic point with $v=T$, and let $U$ be
its unit tangent. Set
\[
\Xi=\dot\theta-b\dot u,\qquad p=a\dot v>0,
\qquad P=\gamma\Xi,\qquad \pi_u=-p-b\cdot P.
\]
The dot here denotes proper-time differentiation; $P$ is the angular
covector momentum, whereas $P_T$ is a spacetime point.
\begin{lemma}[Branch normalization]
\label{pk2:branch-normalization}
Along the branch, $p\asymp1$, and $P$, $\Xi$, and $\dot u$ remain
bounded. In particular, with $Y_T=\dot v(P_T)$,
\begin{equation}
Y_T\asymp e^{\kappa T}.
\label{pk2:blueshift}
\end{equation}
\end{lemma}
\begin{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
\begin{equation}
-\pi_v=\frac{a(1+|P|_{\gamma^{-1}}^2)}{2p},
\qquad
\frac{du}{dv}=\frac{a(1+|P|_{\gamma^{-1}}^2)}{2p^2}.
\label{pk2:branch-hamiltonian}
\end{equation}
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
\eqref{pk2:blueshift}.
\end{proof}
Fix $u_2>u_*$, $c_2>0$, and $c_0>0$ small relative to these margins.
The finite comparison domain is
\begin{equation}
0\le s\le\frac{T+u_*}{2}+c_0,
\qquad u+\frac{c_0s}{T}\le u_2,
\qquad v+\frac{c_0s}{T}\le T+c_2.
\tag{P1}\label{pk2:domain}
\end{equation}
Its slice lengths in $x$ have a positive lower bound. All constructions
are first made with enlarged domains and margins. For auxiliary
transports fix $u_f>u_2$. The strip $-v\le u\le u_f$, with $v$ in a
fixed neighborhood of $T$, is contained in the expanding region of
the higher double-null estimate, with bounded-direction endpoint
$u_f$. This observation includes the moving initial foot $u=-v$.
Use normalized vector slots and derivative fields
\[
\hat e_i=a^{-1/2}L_i,\qquad \hat e_A=\partial_A,
\qquad Z_\alpha=\sqrt a\,\hat e_\alpha,
\qquad W_t=L_0+L_1,\quad W_x=L_1-L_0.
\]
Thus $Z_i=L_i$. For a covariant two-tensor entry $n$, let $n_i$ count
its radial $i$-slots and put $c_n=\kappa(n_0-n_1)/2$.
In the coefficient estimates below, $Db$ denotes the collection of
first ordinary coordinate derivatives of $b$ in the fixed
$(u,v,\theta)$ charts.
\begin{lemma}[Coefficient inputs for the finite comparison]
\label{pk2:coefficients}
The matrices $V_\alpha$ of $\nabla_{Z_\alpha}$ on the normalized
vector basis are bounded. In the bulk limit
$\min(v,-u,s)\to\infty$,
\[
V_0\longrightarrow
D_0=\operatorname{diag}(-\kappa/2,\kappa/2,0,0),
\qquad V_1\longrightarrow-D_0,
\qquad V_A\longrightarrow0.
\]
The contracted differentiated connection matrices in $a\square_g$
and every normalized entry of $a\operatorname{Riem}(g)$ are bounded
and small in that limit. The undifferentiated connection products in
the tensor wave operator retain the radial diagonal constants.
Moreover
\[
b,\ Db,\ W_t\gamma,\ W_t\log a+2\kappa\longrightarrow0
\]
there, while first ordinary derivatives of $\log a$ and $\gamma$ are
bounded. Every fixed ordinary derivative of the normalized arrays
has size $e^{o(T)}$ on the required domains. Derivatives of $a$,
$\sqrt a$, and $\partial_vb$ retain, respectively, the factors
$a$, $\sqrt a$, and $a$.
\end{lemma}
\begin{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,
\label{pk2:radial-connection}\\
\nabla_{L_i}\partial_A
&=(\chi_{i,A}{}^B-\partial_AB_i^B)\partial_B
+\sqrt a\,\xi_{i,A}\hat e_i,
\notag\\
\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.
\notag
\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.
\end{proof}
\subsection{A constrained oscillatory packet in the deep region}
\label{pk2:deep-packet}
Let $\mathcal P$ be the covariant tensor wave operator with curvature
potential obtained by trace reversal of the linearized wave-map reduced
vacuum equation at $g$, and let $\mathcal C=\operatorname{div}_g$ on
these tensors. Linearized Bianchi gives
\[
\mathcal C\mathcal P=\mathcal P_1\mathcal C,
\]
where $\mathcal P_1$ is the one-form wave operator. Put
\[
k_v=g^{-1}dv=-a^{-1}L_0,
\qquad K_vh=h(k_v,\cdot),
\qquad \mathcal T=2\nabla_{k_v}+\square_gv.
\]
For a sufficiently large finite $L$, solve the transport and gauge
hierarchy
\begin{equation}
i\mathcal Tp_j+\mathcal Pp_{j-1}=0,
\qquad iK_vp_j+\mathcal Cp_{j-1}=0,
\qquad p_{-1}=0,
\label{pk2:hierarchy}
\end{equation}
on $-v\le u\le u_f$, with $v$ near $T$.
Multiplication of the first equation by $-a/2$ gives connection
transport along $L_0$ plus area expansion, with a known source
supplied by $a\mathcal Pp_{j-1}$. At fixed $j$, the unknown normalized
entries of $p_j$ form triangular blocks: the block of weight $c_n$
has same-weight transport $L_0+c_n$, with bounded additional
coefficients small in the bulk, and couples only to higher-weight
blocks. The boundary values for each $p_j$ are assigned as follows:
\[
\begin{array}{c|c|l}
\text{entries}&c_n&\text{boundary value}\\ \hline
00&\kappa&\text{gauge equation at }u=-v\\
0A&\kappa/2&\text{gauge equation at }u=-v\\
01&0&\text{gauge equation at }u=-v\\
AB&0&\text{free angular data at }u=-v\\
1A&-\kappa/2&\text{zero at }u=u_f\\
11&-\kappa&\text{zero at }u=u_f
\end{array}
\]
In this table $A,B$ are angular slots. At $j=0$ the free angular
data are a compact smooth bump times an angular trace-free
polarization of unit norm at the foot of the $v=T$ ray through
$P_T$, multiplied by a constant unit phase to be fixed below.
They vanish for $j>0$. Solve the blocks in descending weight,
retaining the coupled angular transport within each block.
Nonnegative-weight blocks are integrated from $u=-v$, and
negative-weight blocks from $u_f$ backwards; the constant diagonal
part has the favorable sign in each chosen direction.
The gauge contraction uses precisely the entries with a $0$-slot,
so the terminal choices do not prescribe extra gauge data. To check
that the initial gauge equations persist, set
$q_j=iK_vp_j+\mathcal Cp_{j-1}$. Conjugated divergence compatibility
gives
\[
i\mathcal T_1q_j+\mathcal P_1q_{j-1}=0,\qquad q_{-1}=0,
\]
where $\mathcal T_1$ is the corresponding one-form transport. Induction
and the zero initial gauge jets give $q_j=0$ throughout the strip.
All amplitudes and fixed ordinary derivatives in normalized entries
have size $e^{o(T)}$. The one-flank interior estimate applies to the
whole strip: choose its bounded coordinate to be $y_0=u\le u_f$
and its other coordinate $y_1=v\le CT$, with any fixed $C>1$.
This includes both $u=-v$ at entry and $u=u_f$ at the terminal edge.
The angular vector field $b$ and its first angular derivatives have
arbitrarily small average outside bounded end portions. Differentiated angular flow
and inverse flow therefore have subexponential fixed jets; higher
coefficient derivatives enter only lower induction orders. The
normalized sources $a\mathcal Pp_{j-1}$ and the prescribed gauge
components $\sqrt a\,\mathcal Cp_{j-1}$ obey the same bounds.
Locality of each transport, including the backwards integrations,
keeps support in the flow tube of the initial bump. The initial
patch can have fixed small size in uniform charts on $\mathcal S$,
although its center may vary with $T$. The leading tensor is
trace-free and annihilates $k_v$. Its screen polarization at $P_T$
has two-sided $e^{o(T)}$ norm, since it is parallel transported times
an area factor with small-average logarithmic derivative.
Choose $X\perp U$ with $dv(X)=0$ and $g(X,X)=1$, using an angular
unit vector and a bounded multiple of $L_0$. Choose its direction so
that the diagonal screen polarization has absolute value
$\sigma_T$, of two-sided $e^{o(T)}$ size. Fix the constant phase
to make that contraction fully real at $P_T$. Let $C_w$ bound the
background unit tidal contraction on this seed; it is finite by its
bounded parallel curvature and its fixed initial boost. Choose
$J_{\mathrm{amp}}>0$ so that $2J_{\mathrm{amp}}>J+C_w+2$, and set
\begin{equation}
\lambda_p=e^{\zeta T},\qquad
A_T=\frac{4J_{\mathrm{amp}}}{Y_T^2\lambda_p^2\sigma_T},\qquad
h^p=A_T\operatorname{tr.rev.}_g
\operatorname{Re}\left(e^{i\lambda_pv}
\sum_{j=0}^L\lambda_p^{-j}p_j\right).
\label{pk2:packet-amplitude}
\end{equation}
Here $\zeta>0$ will be chosen after all base norm orders. The normalized
reduced-equation residual at $g+h^p$, multiplied by $a$, is
\begin{equation}
e^{o(T)}\bigl(A_T\lambda_p^{-M_1}
+A_T^2\lambda_p^{P_*}\bigr)
\label{pk2:deep-residual}
\end{equation}
at those orders. Increasing the finite hierarchy makes $M_1$ arbitrarily
large, whereas $P_*$ depends only on the fixed base orders. The linear
phase and transport errors cancel successively,
and subtracting the linearization leaves quadratic relative-field
products in scaled covariant derivatives and smooth inverse-array
coefficients, including algebraic $a\operatorname{Riem}(g)$ terms.
There is no inverse-lapse loss in these normalized residual arrays.
\subsection{The exact deep comparison and its observation}
\label{pk2:deep-comparison}
The next proposition specifies exactly how accurately the entry jets
must be transferred from the original initial end. We first define
its mixed error norms using the background fields. For a positive
mass parameter $\varepsilon_0$, a normalized entry $f$ of weight
$c_n$ has reference slice density
\begin{equation}\label{pk2:reference-density}
\frac12\left(|W_tf|^2+|(W_x-2c_n)f|^2
+2a|d_\theta f|_\gamma^2+\varepsilon_0^2|f|^2\right)
dx\,d\operatorname{vol}_\gamma.
\end{equation}
For a normalized tensor array $F$, let $E_j(F;s)$ be the square root
of the sum of these integrals over the $s$-slice of
\eqref{pk2:domain}, with $f=D^\alpha F_n$, over all entries $n$
and ordinary multiindices $|\alpha|\le j$. Angular derivatives and
localization use the fixed finite sphere atlas. These are the
through-$j$ mixed energies below; $\varepsilon_0$ will be fixed
sufficiently small after the desired growth-rate tolerance.
\begin{proposition}[Deep comparison]
\label{pk2:deep-proposition}
Fix a mixed Sobolev comparison order $k\ge10$. For every sufficiently
small fixed $\delta_2>0$, one may fix $\varepsilon_0>0$ and then
$\nu_0>0$ sufficiently small as follows. Suppose the exact
reduced-equation Cauchy jets on $s=0$ satisfy the vacuum constraints
and wave-map gauge, agree with $g$ for $v<(1-\nu_0)T$, and differ
from the jets of $g+h^p$ by at most
\begin{equation}
e^{-7\kappa T/2}
\quad\hbox{in }H^{k+1}\times H^k(dx\,d\theta).
\label{pk2:entry-accuracy}
\end{equation}
Then these data have an exact vacuum development $g'$ on
\eqref{pk2:domain}, with normalized error
$F=g'-g-h^p$ bounded in the through-$k$ mixed energies by
\begin{equation}
C e^{(\kappa+\delta_2)T}
\left(e^{-7\kappa T/2}+A_T\lambda_p^{-M_1}
+A_T^2\lambda_p^{P_*}\right).
\tag{P7}\label{pk2:deep-error}
\end{equation}
The conclusion holds when this bound has a fixed exponential margin
below $A_T$ and the required base packet norms are sufficiently
exponentially small. The same estimate, with fixed buffers, controls
normalized $C^2$ error.
\end{proposition}
\begin{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 \eqref{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~\ref{pk2:coefficients} 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
\eqref{pk2:reference-density}. 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 $s(1-\nu_0)T-B-o(1)$. 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
\[
E_j'\le r_TE_j+b_T\sum_{\ellJ.
\]
The $g'$-geodesic with state $(P_T,U')$, traced backwards to
$\mathcal S$, has state converging to the background geodesic's
transverse entry state and stays within the available side margins.
\end{lemma}
\begin{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 \eqref{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 \eqref{pk2:branch-hamiltonian} 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.
\end{proof}
\subsection{Return rays to the original initial end}
\label{pk2:return-rays}
The interior packet is useful only if its entry jets can be produced
by data on the original bridge. We first verify the real-ray geometry
needed for the exterior beam construction, including its dependence
on the fixed spin.
Use the pertinent exterior time $t_e$, co-coordinate $y$, and lab
coordinate $z=y+C_e(t_e)$. Orient the aligned Boyer--Lindquist time
$T_K$ on this end so that the preparation parameter $I$ satisfies
\[
I'=\lambda_-\varsigma>0,
\qquad 00$, after choosing a sufficiently
large fixed radius and then taking $T$ sufficiently large, all their
arrival points satisfy
\begin{equation}
(1-\alpha)t_T<|z-z_*|<(1+\alpha)t_T,
\qquad z_*=C_e(t_T).
\tag{E1}\label{pk2:arrival-shell}
\end{equation}
Their entry crossing is unique and transverse within the computing
region, with separation from the entry level outside a uniform
collar. All fixed jets of the phase-space flow on all subsegments,
including launch-center derivatives, have bounds $e^{o(T)}$.
Tensor parallel transport has bounded norm and inverse along these
rays. The assertions hold with short past and future launch margins,
including passage through the deeper inner exit.
\end{proposition}
\begin{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
\begin{equation}
(r_K^2+\mathfrak a^2)^2
-\Delta\left(|\partial_\vartheta F|^2
+\mathfrak a^2\sin^2\vartheta\right)_{\rm init}.
\label{pk2:radial-potential}
\end{equation}
The bracket is at most $\mathfrak a^2+O(q_0^2)$, since
$F(r_-)=0$ and the profile is smooth. For $r_K\ge r_+$,
\[
\Delta0$, choosing the original exact-profile
depth sufficiently small over a small physical parameter range
makes \eqref{pk2:radial-potential} 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{equation}
\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}
\label{pk2:far-ray-coefficients}
\end{equation}
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 \eqref{pk2:far-ray-coefficients} 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
\eqref{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 \eqref{pk2:far-ray-coefficients}; 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.
\end{proof}
\subsection{Constrained complex beams and matching}
\label{pk2:beams}
We now transfer the interior packet to the bridge using complex
Gaussian beams. Their complex phases provide transverse localization
through real caustics, while superpositions recover oscillatory Cauchy
data \cite{Ralston1982,Tanushev2008,LiuRunborgTanushev2013}.
We retain the phase-space and matching calculations because we also need
bounds on growing time intervals and Einstein wave-gauge compatibility.
\begin{proposition}[Exterior beam approximation]
\label{pk2:beam-proposition}
For every fixed base-order buffer $N_b$, every prescribed inverse
accuracy $M_1$, and sufficiently high finite phase and amplitude
orders, there is a real exterior perturbation $h^B$ such that
\begin{equation}
\begin{split}
\|h^B\|_{C^{N_b}}&\le A_T\lambda_p^{N_b+3/2}e^{o(T)},\\
\text{reduced-wave and gauge residuals}
&\le e^{o(T)}
\left(A_T\lambda_p^{-M_1}+A_T^2\lambda_p^{P_0}\right),
\qquad P_0=2N_b+7,
\end{split}
\tag{E2}\label{pk2:beam-estimates}
\end{equation}
after enlarging $N_b$ for the required trace and derivative buffers.
Its jets match those of $h^p$ at $\mathcal S$ with linear error
$A_T\lambda_p^{-M_1}e^{o(T)}$. The positive exponent $P_0$ depends
only on the base orders, not on later formal accuracy. On the
original initial slice its support lies in
\eqref{pk2:arrival-shell}, up to arbitrarily thin tube margins.
\end{proposition}
\begin{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.
\paragraph{Phase and central amplitude transport.}
Using the ray, subsegment, and collar bounds just established,
parameterize centers $Y$
by a fixed patch of the entry slice. With spatial entry coordinates
$w_s=(x,\theta)$, start the phase at
\[
v(0,w_s)+\frac{i}{2}|w_s-Y|^2,
\]
choosing its temporal root to agree with $dv$ at the center.
Evolve the phase jets along the real rays in lab time by the
complex gradient-graph construction. Transverse projection from
the entry slice gives a positive imaginary Hessian on the lab
slice; its spacetime null direction is the ray direction.
For the position and momentum differential blocks $P_c,Q_c$,
the preserved symplectic identity is
\[
P_c^*Q_c-Q_c^*P_c=2i\operatorname{Im}M_c,
\]
where $M_c$ is the initial complex Hessian. The all-subsegment
bounds imply $P_c^{-1}=e^{o(T)}$ in bound. The propagated Hessian
has positive imaginary part with inverse bounded by $e^{o(T)}$.
Taylor composition and reversion use only finite smooth jets at
real positions, so no spatial analyticity is required.
The central amplitude propagator is tensor parallel transport
times the inverse square root of the density ratio formed from
\[
\sqrt{|\det g|}\,k^{t_e}\det P_c,
\qquad k=g^{-1}d\varphi.
\]
It and its inverse have subexponential all-subsegment bounds.
Higher transport jets have the same homogeneous propagator with
lower-jet subexponential sources.
\paragraph{Compatibility of the wave and gauge jets.}
Impose both wave transports and gauge equations in the finite
Taylor quotient, taking the eikonal orders sufficiently high.
For a complex phase $\varphi$ set
\[
\mathfrak h=g^{-1}(d\varphi,d\varphi),\quad
\mathcal P_\lambda=\mathcal P+i\lambda\mathcal T-\lambda^2\mathfrak h,
\quad\mathcal C_\lambda=\mathcal C+i\lambda K,
\quad\mathcal P_{1,\lambda}=\mathcal P_1+i\lambda\mathcal T_1-\lambda^2\mathfrak h.
\]
Here $K$ contracts by $g^{-1}d\varphi$ and $\mathcal T,\mathcal T_1$
are the tensor and one-form transports. Conjugation gives the exact
identity $\mathcal C_\lambda\mathcal P_\lambda
=\mathcal P_{1,\lambda}\mathcal C_\lambda$. In the finite jet quotient
where all required eikonal jets vanish, put
\[
W_j=i\mathcal T r_{j,Y}+\mathcal P r_{j-1,Y},\qquad
q_j=iKr_{j,Y}+\mathcal C r_{j-1,Y},\qquad r_{-1,Y}=0.
\]
The coefficient of $\lambda^{2-j}$ is
\begin{equation}\label{pk2:beam-defect-identity}
iKW_j+\mathcal C W_{j-1}=i\mathcal T_1q_j+\mathcal P_1q_{j-1},
\qquad W_{-1}=q_{-1}=0.
\end{equation}
The extra phase degrees eliminate every differentiated $\mathfrak h$
term in these quotients. Once lower defects vanish, solving $W_j=0$
makes $q_j$ satisfy homogeneous one-form transport.
At launch the contraction by $k$ has the right inverse
\[
f\longmapsto l\otimes f+f\otimes l-f(k)l\otimes l,
\qquad l(k)=1.
\]
The nonzero pairing is inverted also in formal jets. This fixes
the forced gauge components while leaving free projected kernel
jets for matching.
\paragraph{Triangular matching at the entry slice.}
Integrate the beams with measure
$(\lambda_p/2\pi)^{3/2}dY$. In a short entry collar, center
positions project diffeomorphically, with the stated low and
subexponential high bounds. At a center, the phase difference
from $v$ starts quadratically and has positive imaginary Hessian.
Changing the center variable to its position on the current
entry slice and evaluating Gaussian moments gives an expansion
\[
e^{i\lambda_pv}\sum_j\lambda_p^{-j}\mathfrak p_j
\]
in integer inverse powers. On the launch surface, set
$a_j(Y)=r_{j,Y}(0,Y)$. The initial phase and normalization give,
after removing $e^{i\lambda_pv}$,
\[
\left(\frac{\lambda_p}{2\pi}\right)^{3/2}
\int e^{-\lambda_p|w_s-Y|^2/2}r_{j,Y}(0,w_s)\,dY
=a_j(w_s)+O(\lambda_p^{-1})e^{o(T)}.
\]
This holds in each required fixed finite spatial norm once
sufficient jets have been supplied; tube cutoffs contribute only
Gaussian-small errors. The integrand includes the chosen off-center
amplitude jets. At order $j$, the integrated coefficient is therefore
the new central value $a_j$ plus terms already fixed by earlier jets.
Derivatives of that new central amplitude first enter one inverse
power later, since odd Gaussian moments vanish.
For each fixed $T$, let an auxiliary frequency tend
to infinity in the differentiated wave and gauge residuals, after
supplying sufficiently many finite jets. Comparison of coefficients
shows that the integrated coefficients satisfy both real-phase
hierarchies; still unmatched higher jets can be completed subject
to their gauge conditions without changing a fixed coefficient.
Once lower coefficients match $p_i$, $i0$ small for the specified finite order; the
complement is Gaussian-small. The Taylor remainder uses just
the supplied smooth subexponential jets.
There are no other entry crossings by
Proposition~\ref{pk2:escape}, and the far escape is disjoint from
the prepared surfaces. Each new undifferentiated formal jet costs
only $e^{o(T)}$, irrespective of the chosen finite degree.
Absolute center integration costs $\lambda_p^{3/2}$, giving the
first bound in \eqref{pk2:beam-estimates}. Nonlinear differences
cost at worst the square of that base bound with two extra
derivatives, giving $P_0=2N_b+7$ in
\eqref{pk2:beam-estimates}.
Increasing formal accuracy adds only finitely many Taylor and
transport coefficients, each with subexponential fixed jets and
its assigned nonpositive frequency power. At a fixed base
derivative order it does not increase the differentiation loss
$\lambda_p^{N_b}$ or the integration loss $\lambda_p^{3/2}$.
The same observation in the interior hierarchy fixes $P_*$ from
its comparison orders before $M_1$ is increased. Constants and
the eventual threshold for $T$ may increase with formal accuracy.
Take real parts, undo trace reversal, and multiply by $A_T$ to
obtain $h^B$. Enlarged domains through $t_T+O(1)$ and radius
$C(1+T)$ contain all tubes and their buffers. At $t_e=0$ the tubes
lie in the shell \eqref{pk2:arrival-shell}, where the slice is the
original bridge. Gauss--Codazzi and the differentiated wave and
gauge residuals show that its induced data, set equal to $d$
elsewhere, have a compactly supported constraint defect with
the residual bound \eqref{pk2:beam-estimates}.
\end{proof}
\subsection{Exact constraints outside a protected ball}
\label{pk2:constraints}
The beam seed is only approximately constrained. The following
precise correction result permits exact vacuum data without fixing
the asymptotic charges or disturbing a large interior region.
For a tensor in Cartesian components define
\[
\|f\|_{H_i^j}
=\|f\|_{H^j(B_2)}
+\sup_{L\ge1}L^i
\|f(L\,\cdot)\|_{H^j(1<|\cdot|<2)},
\qquad
X^s=H_1^{s+2}\oplus H_2^{s+1}.
\]
Let $\Phi$ denote the vacuum scalar and momentum constraint map.
\begin{proposition}[Protected exterior correction]
\label{pk2:correction-theorem}
Fix an integer $s\ge3$. Suppose $d_B=(\delta+a_B,b_B)$ are smooth data on
$\mathbb R^3$, the metric is positive, and all derivatives of
$a_B,b_B$ have symbol orders $-1,-2$, respectively. Suppose a
smooth compact source $\phi$, supported in $\{|y'|\ge1\}$,
agrees with $\Phi(d_B)$ there. If
\[
\|a_B\|_{H_1^{s+2}}+\|b_B\|_{H_2^{s+1}}
+\|\phi\|_{H_4^s}
\]
is sufficiently small, there is a smooth correction $v_c$ supported
in $\{|y'|\ge1\}$ such that
\[
\Phi(d_B+v_c)=0\quad\text{there},
\qquad \|v_c\|_{X^s}\le C_s\|\phi\|_{H_4^s}.
\]
The correction is smooth across the support boundary, preserves
positivity, and satisfies symbol bounds at every order. There is
no higher-order smallness assumption and no condition on the
artificial interior defect. No moment condition is imposed on
$\phi$; the correction may have a noncompact tail changing the
asymptotic coefficients.
\end{proposition}
\begin{proof}
Apply \Cref{pk:constraint-nonlinear} 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.
\end{proof}
Apply this proposition with
\begin{equation}
\rho_T=(1-2\alpha)t_T,
\qquad y'=\frac{z-z_*}{\rho_T}.
\label{pk2:rescaling}
\end{equation}
Write the unperturbed datum on this end as $d=(h,K)$.
This rescaling keeps the Cartesian metric component functions and
multiplies the second-form component functions by $\rho_T$:
\begin{equation}\label{pk:constraint-rescaling}
\widetilde h(y')=h(z_*+\rho_Ty'),\qquad
\widetilde K(y')=\rho_TK(z_*+\rho_Ty').
\end{equation}
Both constraint components then scale homogeneously by $\rho_T^2$.
Small fixed
drift and small $\alpha$ give $|z|\asymp\rho_T|y'|$ for
$|y'|\ge1/2$. More explicitly, the original symbol bounds imply,
for every fixed derivative order $j$,
\begin{equation}\label{pk:constraint-background}
|\partial_{y'}^j(\widetilde h-\delta)|
\le C_j\rho_T^{-1}|y'|^{-1-j},\qquad
|\partial_{y'}^j\widetilde K|
\le C_j\rho_T^{-1}|y'|^{-2-j}.
\end{equation}
The unperturbed scaled pair therefore has $X^s$ size
$O_s(\rho_T^{-1})$. Cut these perturbations off smoothly to flat
data between $|y'|=3/4$ and $1/2$; no constraint condition is
needed on this artificial interior.
By \eqref{pk2:arrival-shell}, the scaled beam seed lies strictly
outside the unit ball. Its size at the chosen solve order tends
to zero after $\zeta$ is made sufficiently small, with only
polynomial rescaling costs. The exterior constraint defect,
extended by zero, is smooth and compactly supported because
the seed vanishes near the unit sphere and the remaining data
there are exactly vacuum. Its norm has the residual bound in
\eqref{pk2:beam-estimates}, with polynomial factors included in
$e^{o(T)}$. Thus every hypothesis of
Proposition~\ref{pk2:correction-theorem} holds for large $T$.
Undo the scaling and retain $d$ on the protected interior and
the rest of the bridge. The correction is smooth and identically
zero on the protected side. The resulting data $d_T$ are smooth,
vacuum, positive, and complete; outside a compact set their metric
is uniformly comparable to the original complete metric. They
belong to $\mathcal D$ at every order, agree with $d$ on
$|z-z_*|<\rho_T$ and off the selected end, and may change the
permitted asymptotic coefficients. Here the Cartesian symbol norms
on the end are equivalent, at every fixed order, to the
fixed-background covariant norms defining $\mathcal D$.
For the prescribed $m\ge10$, choose $s\ge m+3$, increasing it also
for the comparison and trace buffers. Scaled annular Sobolev
embedding and unscaling give
\begin{equation}
p_m(d_T-d)
\le e^{o(T)}\left(
A_T\lambda_p^{m+5/2}
+A_T\lambda_p^{-M_1}+A_T^2\lambda_p^{P_0}\right).
\label{pk2:data-approximation}
\end{equation}
One extra beam derivative accounts for the second form. To see the
unscaling cost for the correction, write its scaled components as
$(q_c,p_c)$. Its physical components are $q_c(y')$ and
$\rho_T^{-1}p_c(y')$. Thus a derivative of order $j$, multiplied
by the respective physical weights $|z|^{1+j}$ and $|z|^{2+j}$,
costs at most $C\rho_T$ times the corresponding scaled weighted
bound in both cases. These polynomial factors are harmless.
No compact-support assertion
about the final exact change at infinity is used.
\subsection{Ordinary exterior comparison and causal protection}
\label{pk2:exterior-comparison}
The corrected data are now exact vacuum data. We next show that their
ordinary wave-map evolution supplies the entry jets in
\eqref{pk2:entry-accuracy}. Give $d_T$ to the partial exterior problem
on $t_e=0$. Its bent compact portion lies in the dependence region
of unchanged data for large $T$. Choose the initial lapse and shift
as for $g+h^B$, and determine the normal metric derivatives from
the actual corrected second form and the exact wave-map condition.
Spacelikeness makes this algebraic system invertible. The resulting
metric and first jets differ from $g+h^B$ by the residual bound
in \eqref{pk2:beam-estimates}, by the correction and gauge-defect
estimates, and agree exactly with $g$ in the protected region.
Take the exterior comparison through $t_e=t_T+C_2$, with the fixed
deeper inner exit and shrinking outer cutoff
\begin{equation}
|y|0$, the localization parameters
and far threshold can be chosen so that the exact ordinary
wave-map solution exists on this finite exterior domain and its
error from $g+h^B$, in the required fixed mixed norms, is bounded by
\begin{equation}
C e^{\delta_3T}
\left(A_T\lambda_p^{-M_1}+A_T^2\lambda_p^{P_0}\right).
\tag{E3}\label{pk2:exterior-error}
\end{equation}
It is exactly $g$ on the earlier needed compact-in-radius region
up to $t_T-\alpha'T$, where $\alpha'>0$ can be prescribed
sufficiently small. In particular, with the choices below, its
entry jets agree exactly with $g$ for $v<(1-\nu_0)T$ and otherwise
meet \eqref{pk2:entry-accuracy}.
\end{proposition}
\begin{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
\begin{equation}
(1-2\alpha)t_T
\le(1+\delta_R)\tau+s_*(t_T-\tau)+O(R).
\label{pk2:first-hit}
\end{equation}
Equivalently,
\begin{equation}
t_T-\tau\le
\frac{2\alpha+\delta_R}{1+\delta_R-s_*}\,t_T+O_R(1).
\label{pk2:late-first-hit}
\end{equation}
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
\eqref{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 \eqref{pk2:late-first-hit}. 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 \eqref{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
\eqref{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~\ref{pk2:beam-proposition}. The choices below make
the total smaller than \eqref{pk2:entry-accuracy}; the already
proved protected-region statement supplies its exact early part.
All estimates allow fixed small enlargements for the final gluing.
\end{proof}
\subsection{The order of the parameter choices}
\label{pk2:choices}
We verify simultaneously the comparison margins and approximation in
the prescribed data topology. First fix the finite data order $m$,
the correction order $s\ge m+3$, the interior and exterior comparison
orders, and every Sobolev, normal-jet, and trace buffer. Fix $N_b$
large enough for these requirements and then fix $P_*,P_0$ for the
quadratic residuals. None of these choices depends on formal accuracy.
Choose $\delta_2,\delta_3>0$ small relative to $\kappa$. The deep
mass, bulk threshold, and $\nu_0$ supply the rate in
\eqref{pk2:deep-error}; the far radius and localization parameters
$\alpha,\alpha'$ supply \eqref{pk2:exterior-error} and the exact
early agreement. The far radius may be enlarged later for
\eqref{pk2:arrival-shell}: the compact top rate is uniformly
bounded independently of it. These choices use fixed base bounds
and qualitative tails, with thresholds allowed to depend on the
smooth background datum.
Next take $\zeta>0$ small enough for every positive loss in the
chosen base norms, retaining more than $\kappa T/2$ of exponential
absorption for the perturbative deep products, and satisfying
\begin{align}
\delta_3+(P_0-4)\zeta&<\kappa/2,&
\delta_2+2\zeta&<\kappa/2,
\label{pk2:positive-choices}\\
\delta_2+(P_*-2)\zeta&<\kappa,&
(m+\tfrac12)\zeta&<\kappa.
\notag
\end{align}
Choose $0\tfrac32\kappa+\delta_3,
\qquad M_1\zeta>\kappa+\delta_2,
\label{pk2:inverse-choices}
\end{equation}
and take sufficiently large finite transport, phase-Taylor, and
matching orders to achieve this accuracy. Finally take $T$ large
depending on all choices. Every subexponential factor can be
assigned a share of the strict margins.
By \eqref{pk2:packet-amplitude} and \eqref{pk2:blueshift},
$A_T=\exp((-2\kappa-2\zeta)T+o(T))$ with two-sided size.
The linear and quadratic exponents in the exterior error are
\[
-2\kappa-(M_1+2)\zeta+\delta_3,
\qquad -4\kappa+(P_0-4)\zeta+\delta_3.
\]
Both are strictly below $-7\kappa/2$ by
\eqref{pk2:positive-choices}--\eqref{pk2:inverse-choices}.
After division by $A_T$, the three terms of the deep error have
exponent bounds
\[
-\tfrac12\kappa+\delta_2+2\zeta,
\qquad \kappa+\delta_2-M_1\zeta,
\qquad -\kappa+\delta_2+(P_*-2)\zeta,
\]
all strictly negative. This proves the entry condition and both
comparison improvements, and makes the observation errors
$o(Y_T^{-2})$. The direct beam contribution to the data seminorm
has exponent
\[
-2\kappa+(m+\tfrac12)\zeta<0;
\]
the two correction contributions in
\eqref{pk2:data-approximation} also tend to zero. Thus
$p_m(d_T-d)\to0$ for this prescribed finite $m$. Smooth membership
of every $d_T$ was already obtained from the correction theorem;
no single frequency exponent has to give convergence at all orders.
\subsection{Attachment to the whole corrected bridge}
\label{pk2:attachment}
The finite exterior and deep solutions have now been constructed
with exact constraints, matching jets, and strict margins. It remains
to identify their observation with a point of the full MGHD of the
whole corrected datum. We attach these finite solutions to a lower
development containing every radius of the corrected bridge, then
use the outflow cuts to prove that the bridge remains Cauchy.
\begin{proposition}[Full-bridge Cauchy attachment]
\label{pk2:attachment-proposition}
For sufficiently large finite $T$, the exact data $d_T$ have a smooth
globally hyperbolic vacuum development containing the entire initial
bridge and the finite observation from
Lemma~\ref{pk2:observation}. Its adaptive backward geodesic reaches
$\Sigma$ with a seed $w_T\to w$. The development embeds openly
in the full MGHD of $d_T$, retaining the entire constraint-correction
tail on $\Sigma$.
\end{proposition}
\begin{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
\eqref{pk2:outer-cutoff} 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
\begin{equation}
\{t_e0.
\]
Thus the strict sublevels, their union, and their intersection
in \eqref{pk2:partial-union} 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 \eqref{pk2:domain}.
More explicitly its active future cuts are
\[
s0.
\]
The signal point has $v=T$, $u=u_T0$ 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~\ref{pk2:choices} at this enlarged
finite order. The constrained beam correction
gives smooth complete exact vacuum data $d_T$ satisfying
\eqref{pk2:data-approximation}. The ordinary exterior comparison
supplies the exact entry jets required by the deep comparison.
Lemma~\ref{pk2:observation} gives the strict finite tidal
observation and convergent backward entry states.
Proposition~\ref{pk2:attachment-proposition} 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.
\end{proof}
\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}]
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~\ref{pk2:finite-signal} 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~\ref{sec:category}. 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 a0\), and a faster-decaying
remainder; it does not directly include the arbitrary order-\(r^{-1}\)
symbol tails permitted here. We therefore prove the compact and far
estimates for our data class together. Their wave-coordinate
cancellation is related to the weak-null structure identified and
used by Lindblad--Rodnianski
\cite{LindbladRodnianski2003,LindbladRodnianski2010}.
The use of elliptic control at selected frequencies together with
frequency-localized physical-space energy currents also has a close methodological
parallel in the full-subextremal quasilinear-wave analysis of
Dafermos--Holzegel--Rodnianski--Taylor
\cite{DafermosHolzegelRodnianskiTaylor2024}. The tensorial gauge
construction and its estimates are established here for the
Einstein system.
\paragraph{Exact data and finite signals.}
Localized deformation of the constraints was developed by Corvino,
Corvino--Schoen, and Chru\'sciel--Delay
\cite{Corvino2000,CorvinoSchoen2006,ChruscielDelay2003};
Carlotto--Schoen \cite{CarlottoSchoen2016} proved a further conical
localization of asymptotically flat data. Our correction fixes an inner
region while allowing an asymptotic tail, so it can change the charges.
The explicit inverse uses a localized divergence primitive in the
tradition of Bogovskii \cite{Bogovskii1980}, whose support and regularity
properties are treated by Costabel--McIntosh
\cite{CostabelMcIntosh2010}. We prove the required inverse here because its
support, derivative gain, and nonlinear dependence are needed in the
finite experiment. Similarly, complex Gaussian beams and their
superpositions provide the linear antecedents of the oscillatory
construction \cite{Ralston1982,Tanushev2008,LiuRunborgTanushev2013}.
The matching of Einstein wave-gauge data and the bounds on time
intervals growing with the experiment parameter are proved below.
==== END Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//sections/history-fragment.tex ====
==== BEGIN Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility//references.bib ====
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