==== TABLE OF CONTENTS ====
Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//01-introduction.tex
Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//02-static-input.tex
Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//04-geometry.tex
Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//05-parabolic.tex
Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//06-collar.tex
Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//07-contradiction.tex
Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//buffered-intervals.tex
Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//overview.tex
Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//paper.tex
Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//preamble.tex
Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//references.bib
===========================

==== BEGIN Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//01-introduction.tex ====
\section{Introduction}\label{sec:introduction}

A Ricci flow is a family of Riemannian metrics satisfying
\(\partial_tg=-2\Ric(g)\). Its scalar curvature \(R=\tr_g\Ric\)
controls the infinitesimal change of volume, whereas its full curvature
tensor controls smooth continuation. The scalar-curvature extension
problem asks whether a uniform scalar bound prevents a finite-time
singularity. We prove its positive resolution on closed manifolds in
real dimension four.

\begin{theorem}[Smooth extension]\label{thm:extension}
Let \(M\) be a closed connected smooth four-manifold, let
\(0<T<\infty\), and let \(g(t)\), \(0\le t<T\), be a smooth Ricci
flow with smooth initial metric. If
\[
 \sup_{M\times[0,T)}|R(g(t))|<\infty,
\]
then there are \(\varepsilon>0\) and a smooth Ricci flow
\(\widetilde g(t)\) on \(M\), \(0\le t<T+\varepsilon\), such that
\(\widetilde g(t)=g(t)\) for every \(t<T\).
\end{theorem}

The same conclusion holds without connectedness. The closed manifold
\(M\) has finitely many connected components, each a closed smooth
four-manifold. The restricted flows satisfy the hypotheses of
Theorem~\ref{thm:extension} at the common time \(T\), so each extends
to \(T+\varepsilon_j\) for some \(\varepsilon_j>0\). Taking
\(\varepsilon=\min_j\varepsilon_j\) and combining these flows gives
the required extension on \(M\).

\begin{corollary}\label{cor:scalar-blowup}
If a Ricci flow on a closed smooth four-manifold has finite maximal
smooth existence time \(T\), then
\[
 \limsup_{t\uparrow T}\max_{x\in M}R(g(t))(x)=+\infty.
\]
\end{corollary}

\begin{proof}
The equation \((\partial_t-\Delta)R=2|\Ric|^2\) and the maximum
principle give a scalar lower bound on a finite time interval. A finite
upper limit in the assertion would therefore give a uniform absolute
bound near \(T\), and smoothness supplies the bound on the remaining
compact time interval. The componentwise extension just described
contradicts maximality.
\end{proof}

\subsection{The extension problem and its geometric obstruction}

Hamilton introduced Ricci flow and established its short-time existence
on closed manifolds \cite{Hamilton1982}. A compact flow continues while
its full curvature remains bounded. \v{S}e\v{s}um showed that a uniform
bound for the Ricci tensor also suffices \cite{Sesum2005}. A scalar bound
is weaker: in a possible curvature blowup, the scalar curvature may
vanish after rescaling while the full curvature stays nonzero. The
limiting geometry can therefore be Ricci-flat without being flat.

Bamler and Zhang developed heat-kernel and curvature estimates under
scalar bounds, including the four-dimensional energy and compactness
theory \cite{BamlerZhangI,BamlerZhangII}. Their work also gives smooth
extension when \(H_2(M;\mathbb F)=0\) for every field \(\mathbb F\)
\cite[arXiv version, Corollary~1.10]{BamlerZhangI}. Simon constructed
continuation from the limiting space by an orbifold Ricci flow
\cite{SimonExtension}. Bamler's convergence theory gives smooth
convergence away from a set of codimension at least four
\cite{BamlerConvergence};
four-dimensional convergence under spatial scalar-integrability
hypotheses is also studied by Liu and Simon \cite{LiuSimon}.
These compactness conclusions permit concentrated curvature and
orbifold points. The assertion here is smooth continuation of the given
flow on its original manifold. The scalar extension problem and its
known special cases are discussed in \cite[Section~2]{LiLiSurvey}.
Finite-time scalar blowup was established for closed K\"ahler--Ricci
flows by Zhang \cite{Zhang2010Kahler} and for compact Ricci flows with
a global Type~I bound, $\sup_M|\Rm(g(t))|\le C/(T-t)$, by
Enders, M\"uller and Topping \cite{EndersMuellerTopping2011}.
More recently, Buzano and Di Matteo proved scalar blowup at local
Type~I singular points without a global Type~I assumption
\cite[Definition~1.4 and Theorem~1.1]{BuzanoDiMatteo2026}.
Here ``local Type~I'' means
\(0<\limsup_{t\uparrow T}(T-t)r_{\Rm}(p,t)^{-2}<\infty\),
where \(r_{\Rm}\) is their parabolic curvature scale at the point \(p\).
That result addresses a specified blowup regime; the argument below
allows unrestricted rates at the hypothetical concentration points.

The spatial degeneration analysis has its origins in the theory of
Einstein metrics. Anderson's compactness theorem produces orbifold
limits with smooth convergence on their regular parts
\cite[Theorem~C]{Anderson1989}. Bando's bubbling analysis, including
its correction, organizes Ricci-flat bubbles and quantitative necks
\cite{Bando1990,BandoCorrection1990}; the work of Bando, Kasue and
Nakajima develops coordinates at infinity under curvature decay and
volume-growth hypotheses \cite{BandoKasueNakajima1989}.
Cheeger and Tian's curvature-energy estimates provide a further
important part of this Einstein four-manifold theory
\cite{CheegerTian2006}. Our slices need not be Einstein. We prove
the required geometric statements below using the scalar-bound
estimates and retaining the Ricci forcing in the neck argument.

At a hypothetical concentration point, successive spatial rescalings
produce a finite tree of Ricci-flat pieces. A vertex represents a
nonflat limit; its singular points may contain smaller vertices. The
outermost nonflat vertex has a distinguished length scale \(a(t)\).
The central problem is to rule out contraction of this outermost scale
while allowing arbitrarily smaller descendants.

\subsection{The static functional and the flow estimates}

Retain the true metric from the cluster through an annular collar of
radius \(b=Na\), with \(N\) large and fixed, and attach an
asymptotically locally Euclidean end. On that end write the metric in
Euclidean quotient coordinates. The renormalized Einstein--Hilbert
functional is
\[
 E(g)=\lim_{D\to\infty}\left[
 \int_{\{r<D\}}R(g)\,d\mu_g-
 \int_{\{r=D\}}(\partial_jg_{ij}-\partial_ig_{jj})n_i\,dS_\delta
 \right].
\]
The boundary subtraction cancels the scalar curvature's linear
divergence term. The functional scales by the square of the length
unit, and its first variation pairs the metric velocity with
\(\tfrac12 Rg-\Ric\).

The companion article \cite{StaticTrees} proves a sequential inequality
\(E=o(M)\) on each fixed limiting tree, where \(M\) is its weighted,
dimensionless Ricci error. Section~\ref{sec:static-input} records the
complete input, including the end weights, derivative orders and
matching quotient groups. Our task is both to verify those hypotheses
for metrics obtained from the flow and to compare their functional
variation with actual spacetime Ricci energy.

\begin{figure}[htbp]
 \centering
 \input{figures/overview}
 \caption{The outermost scale \(a\), its smaller descendants, and the
 material collar at radius \(b=Na\). An edge of logarithmic length
 \(L\) changes the length unit by \(e^{-2L}\). The collar extension
 preserves the inner geometry and supplies an ALE end. Incidence and
 scale are schematic.}
 \label{fig:scales}
\end{figure}

Two flow estimates are decisive. The divergence constraint on Ricci
removes the critical radial harmonic mode on a nearly flat annulus.
This improves annular decay enough that the loss outside the collar
is small compared with the energy inside the root. The exterior is
then constructed as an analytic function of the metric on one fixed
material collar. Longer exterior coordinates enter only an error
estimate. Consequently the extension can be differentiated on time
patches without choosing a smoothly varying bubble tree.
The collar construction uses the classical compatibility between
metric strains and linearized curvature, together with Korn control
modulo rigid motions \cite{Khavkine2017,CiarletCiarlet2005}.
The scale-specific estimates, filled-space inverse and dependence
on the fixed collar are proved in Section~\ref{sec:collar}.

\subsection{Proof structure}

Section~\ref{sec:geometry} establishes the scale estimates, genuine
orbifold limits and finite trees needed here. It defines a continuous
outermost scale and selects buffered time intervals on which that
scale changes by a fixed factor. After parabolic normalization, the
interval has fixed duration and its root length \(a\) tends to zero.

Section~\ref{sec:parabolic} controls Ricci on exterior annuli and on
the root's compact and joining regions. A time maximal function
converts root spacetime energy into the single-time weighted error
required by the static theorem. Section~\ref{sec:collar} constructs
the collar-dependent exterior and proves an error bound independent
of its length.

Section~\ref{sec:contradiction} applies the static theorem separately
to arbitrary sequences of active time patches. If \(K^2\) denotes
the root spacetime Ricci energy divided by \(a^4\), the positive
functional variation and its static smallness imply
\[
 cK^2\le o(1)(K+1)^2.
\]
Hence \(K\to0\), without assuming it was bounded. Dense sets of
material points then preserve distances at scale \(a\) between the
two endpoints. The resulting pointed isometry preserves the
curvature-radius maximum defining the outermost scale. This contradicts
its prescribed factor change. The absence of concentration gives a
smooth terminal metric and continuation on \(M\).

\subsection{Conventions}

We use \(\Delta=\tr\nabla^2\). A tensor norm at scale \(r\)
includes \(r^j\) on its \(j\)th spatial derivative, while leaving
its pointwise magnitude unchanged. Tensor magnitudes use the current
metric unless specified otherwise. Quotient-chart components are
Cartesian components on the Euclidean cover; all constructions are
equivariant and descend to the quotient.

Constants may depend on a fixed finite differentiation order. After
extraction they may also depend on a fixed finite tree. Uniformity
in a sequence index, a length, or the collar factor \(N\) is stated
where needed. A sequential little-oh is taken after fixing the
parameters in its statement.
==== END Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//01-introduction.tex ====

==== BEGIN Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//02-static-input.tex ====
\section{The static input and its normalization}
\label{sec:static-input}

We record the precise input from the companion article
\cite[Theorem~2.2]{StaticTrees}. Its constants and small exponents
belong to one fixed limiting tree. No uniform exponent across different
trees is needed. The elementary functional identities are included here
to fix the normalizations used in the flow calculation.
The remaining static proof, including path selection and the augmented
elliptic construction, is supplied by the complete companion.

\subsection{The geometric data and the functional}
\label{ell:setup}

Let $\mathcal T$ be a finite rooted tree of four-dimensional Ricci-flat
spaces $(V_v,h_v)$.  A vertex has one exterior end and one punctured end
for each child.  Every end has Cartesian coordinates on a Euclidean
annulus modulo a finite subgroup of $O(4)$ acting freely on $S^3$.
The group on a joined end is nontrivial.  The two groups on an edge
are identified; a fixed orthogonal identification is included in the
charts.  Put $\tau=\log r$ on an exterior end and $\tau=-\log r$ at a
puncture.  In these charts, in vertex units,
\begin{equation}
 h_v=\delta+O(e^{-\mu\tau})
 \label{ell:end-decay}
\end{equation}
on joined ends, with derivatives at scale, for some $\mu>0$.
On the unjoined exterior end of the root we require instead
$h_v-\delta=O(r^{-q})$, where $q>1$.
All remaining parts are smooth compact manifolds with the indicated
boundary collars.  Bounds through order $20$ suffice in the data below.

For an edge $e$ from $p$ to $c$, truncate both end coordinates at
$\tau=L_e$ and identify the boundary collars by
\begin{equation}
 x_p=e^{-2L_e}x_c,\qquad a_c=a_p e^{-2L_e},\qquad a_{\rm root}=1.
 \label{ell:scales}
\end{equation}
Thus the metric tensor in vertex units is multiplied by $a_v^2$ in
physical units.  The physical radius $\rho=a_vr$ agrees in both charts,
and $\tau_p+\tau_c=2L_e$.  Gluing just the boundary faces, with their
compatible smooth collars, gives the same manifold.

Consider a sequence of metrics $g$ for which every $L_e\to\infty$,
which converges to $h_v$ on each compact subset of every open vertex.
Assume the same end estimates as in Equation~\eqref{ell:end-decay}
on each joining half, and the stated root estimate, with uniform
finite derivative bounds.  On a compact core put $\rho=a_v$.
Suppose $M\to0$ and
\begin{equation}
 |\rho^2\Ric(g)|_j\le CM
 \begin{cases}
 e^{-\mu\tau},&\text{on joined half-ends},\\
 r^{-q},&\text{on the root exterior end},\\
 1,&\text{on vertex cores},
 \end{cases}
 \qquad j=10.
 \label{eq:tree-ricci-error}
\end{equation}
Here the tensor magnitude is physical, and the subscript includes
covariant derivatives multiplied by the corresponding powers of
$\rho$.  The metric bounds convert these to coordinate estimates.
Decreasing $\mu$ or $q$, while retaining $\mu>0$ and $q>1$, is allowed.

Define, in physical root units,
\begin{equation}
 E(g)=\lim_{D\to\infty}\left\{
 \int_{r<D}R(g)\,d\mu_g-
 \int_{r=D}(\partial_jg_{ij}-\partial_ig_{jj})n_i\,dS_\delta
 \right\}.
 \label{eq:energy-functional}
\end{equation}
All chart integrals are quotient integrals.  The two terms inside the
braces need not converge separately.

The boundary subtraction is the mass term in the renormalized
Einstein--Hilbert expression discussed by Deruelle and Ozuch
\cite[Definition~4.2, Remark~4.3, and Proposition~4.4]{DeruelleOzuch2025}.
Their ALE version of Perelman's functional also minimizes over an
auxiliary potential; the functional $E$ here is the unminimized
scalar-curvature-minus-flux expression.

\begin{lemma}[Renormalization and first variation]
\label{ell:functional}
The functional in Equation~\eqref{eq:energy-functional} is defined on
the preceding metrics.  For a variation with the same root decay,
\begin{equation}
 DE_g(\dot g)=\int\left\langle\tfrac12R(g)g-\Ric(g),\dot g\right\rangle_g
 \,d\mu_g.
 \label{ell:first-variation}
\end{equation}
It is invariant under diffeomorphisms isotopic to the identity whose
root displacement, including scaled derivatives throughout the
isotopy, is $O(r^{1-q'})$ for some $q'>1$.  A global length scaling
by $s$, accompanied by a coordinate dilation at infinity restoring
the leading tensor $\delta$, multiplies $E$ by $s^2$.
Under Equation~\eqref{eq:tree-ricci-error}, $|E(g)|\le CM$.
\end{lemma}

\begin{proof}
Write $g=\delta+p$ on the root end.  The scalar-curvature density is
\[
 R(g)\sqrt{\det g}=
 \partial_i(\partial_jp_{ij}-\partial_ip_{jj})+
 O(|p||\partial^2p|+|\partial p|^2).
\]
The last expression is $O(r^{-2q-2})$ and its radial integral is
bounded by $C\int_{r_0}^\infty r^{1-2q}\,dr$.  This proves convergence.
The linear boundary term in the variation of scalar curvature
cancels the variation of the subtracted integral.  Every remaining
boundary product is $O(D^{2-2q})$; it tends to zero.  Integration by
parts therefore gives Equation~\eqref{ell:first-variation}.
For a Lie derivative, the contracted Bianchi identity makes the
interior divergence vanish.  The remaining boundary term vanishes
with the stated displacement decay.  Integration along the isotopy
gives invariance.  Changing variables under a dilation in
Equation~\eqref{eq:energy-functional} gives the scaling law.

Differentiate this law at $s=1$.  Choose the compensating dilation
to be the identity inside a fixed outer root region.  Its metric
variation equals $2g$ on the interior and decays like $r^{-q}$ on
the root tail.  Its pairing with Ricci on the root is integrable
by $q>1$.  On a punctured half the interior estimate costs
\[
 CM a_p^2\int_{\tau_0}^{L_e}e^{-(2+\mu)\tau}\,d\tau\le CM a_p^2.
\]
On a child exterior half the corresponding quantity is
\[
 CM a_c^2\int_{\tau_0}^{L_e}e^{(2-\mu)\tau}\,d\tau
 \le CM a_p^2(1+L_e)e^{-\min(2+\mu,4)L_e},
\]
with a harmless change of constant for the fixed lower endpoint.
The finite sum of core contributions is bounded as well, since
$a_v\le1$.  Equation~\eqref{ell:first-variation} now gives
$2|E(g)|\le CM$.  In particular $E=0$ when $\Ric=0$.
\end{proof}

\begin{theorem}[Static tree inequality {\cite[Theorem~2.2]{StaticTrees}}]
\label{thm:tree-inequality}
For the fixed tree and the sequences just specified,
\begin{equation}
                         E(g)=o(M).
 \label{ell:little-oh}
\end{equation}
For $M=0$ the assertion means $E(g)=0$.
\end{theorem}

The companion proof constructs a finite-dimensional analytic family with
complex length parameters.  It does not assume that Einstein
deformations are unobstructed, or that the given metrics or the
coefficients of a fixed core metric are spatially analytic.

The theorem is applied in Proposition~\ref{final:functional-smallness}.
There we verify every hypothesis after extracting a fixed tree from an
arbitrary sequence of active times and material collars. The root
exterior has decay exponent greater than one because it is replaced
by the extension of Section~\ref{sec:collar}; the true joining necks
retain their own positive decay exponent.
==== END Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//02-static-input.tex ====

==== BEGIN Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//04-geometry.tex ====
\section{Small-scale geometry and the outermost scale}\label{sec:geometry}

Throughout this section, $(M^4,g(t))$, $0\leq t<T$, is the closed
Ricci flow under consideration and $|R|\leq C_R$. Constants may depend
on this flow, $T$, and a specified finite differentiation order. They are
independent of the small scales subsequently chosen. For a tensor $A$,
write $|A|_{k,r}=\sum_{j=0}^k r^j|\nabla^j A|$; on coordinate annuli
the analogous notation uses Cartesian derivatives. We first establish
the geometric input needed in the later analytic argument. The
bounded-scalar-curvature estimates in
\cite{BamlerZhangI,BamlerZhangII} give background for these estimates;
the scale selection, tree organization, and motion assertions needed
here are included below.

\begin{definition}[Spatial curvature radius]\label{geom:radius}
Fix a small cap $l_0>0$. At a material point $x\in M$ set
\[
 l(x,t)=\sup\{0<r\leq l_0:
       \sup_{B_t(x,r)}|\Rm|\leq r^{-2}\}.
\]
The definition uses the whole spatial ball, not just curvature at its
center. The function $l$ is positive and continuous before $T$, and
$|l(x,t)-l(y,t)|\leq d_t(x,y)$. Indeed a ball of radius $r-d_t(x,y)$
at $y$ is contained in the radius-$r$ ball at $x$, and its allowed
curvature bound is larger. Continuity in time follows by compactness
and smooth dependence of the metric, using radii slightly smaller or
larger than the supremum.
\end{definition}

\begin{theorem}[Scale estimates and degeneration]\label{thm:degeneration}
For a sufficiently small cap and all sufficiently late times, the
following conclusions hold.

\emph{Scale and energy bounds.} One has
\begin{align}
 cr^4\leq\Vol_t B_t(x,r)&\leq Cr^4,
       &\int_M|\Rm|^2\,d\mu_t&\leq C,\label{geom:volume-energy}\\
 |\nabla^k\Rm|(x,t)&\leq C_k l(x,t)^{-2-k},
       &|\nabla^k\Ric|(x,t)&\leq C_k l(x,t)^{-1-k}.
       \label{eq:ricci-improvement}
\end{align}
Here $0<r\leq l_0$ and each fixed $k$ is allowed. The set
$\{l<r\}$ is covered by a uniformly bounded number of balls of radius
$Cr$. Distances converge uniformly to a compact metric quotient $Z$
of the material manifold. Away from finitely many concentration points
the metric convergence is smooth.

\emph{Small-scale limits and genuine singularities.}
At arbitrary sequences of centers and length scales $h_i\to0$, with
times tending to $T$, a subsequence of the rescaled slices converges
to a complete noncollapsed Ricci-flat orbifold, smoothly away from
finitely many genuine orbifold points. Each such point has a nontrivial
finite group acting freely on $S^3$. A nonflat limit has a single
Euclidean quotient end with nontrivial group. A flat limit is a global
Euclidean quotient and has at most one exceptional point.

\emph{Finite trees.} Within each concentrating cluster, maximal
inequivalent nonflat limits form a finite
rooted tree with a unique outermost entry. The joining regions have the
quantitative charts of Lemma~\ref{geom:neck} below.
\end{theorem}

Sections~\ref{geom:local-estimates}--\ref{geom:energy-comparison}
prove the scale and energy estimates and convergence of distances.
Sections~\ref{geom:slice-limits}--\ref{geom:puncture-removal} establish
the small-scale limits and identify their exceptional points as genuine
quotient singularities. Section~\ref{geom:trees-root} completes terminal
regular convergence and constructs the finite trees. The remaining
subsections use these conclusions to compare the outermost scale in time.

\subsection{Heat localization and two-sided persistence of the spatial radius}
\label{geom:local-estimates}

The spatial curvature radius does not by itself control a time cylinder.
Our first goal is to obtain that control in both time directions, and
then to improve the Ricci bound by one power of the spatial radius.
We give the heat estimates first so that the curvature-radius argument
does not assume curvature-dependent cutoff bounds. Set $n=4$ for the
applications, retaining $n$ temporarily in the heat formulas. If
$u=(4\pi\tau)^{-n/2}e^{-f}$ is a positive unit-mass conjugate heat
density, $\partial_\tau=-\partial_t$ and
$\partial_\tau u=\Delta u-Ru$, put
\[
 v=\{\tau(2\Delta f-|\nabla f|^2+R)+f-n\}u.
\]
A direct computation gives
\begin{equation}\label{geom:entropy-identity}
 (\partial_\tau-\Delta+R)v
 =-2\tau|\Ric+\nabla^2f-g/(2\tau)|^2u.
\end{equation}
For completeness, after factoring out $u$ the operator is
$D=\partial_\tau-\Delta+2\nabla f\cdot\nabla$. The identities used are
\begin{align*}
 f_\tau&=\Delta f-|\nabla f|^2+R-n/(2\tau),\\
 D\Delta f&=-2|\nabla^2 f|^2-2\Ric(\nabla f,\nabla f)
                  +\Delta R-2\Ric:\nabla^2f,\\
 D|\nabla f|^2&=2\nabla f\cdot\nabla R
                   -2|\nabla^2 f|^2-4\Ric(\nabla f,\nabla f),\\
 DR&=-2\Delta R-2|\Ric|^2+2\nabla f\cdot\nabla R.
\end{align*}
The metric term in $\partial_\tau\Delta f$ is
$-2\Ric:\nabla^2f$; the contracted Bianchi identity cancels the
first-derivative terms. Substitution gives
Equation~\eqref{geom:entropy-identity}, including its time direction.
This is Perelman's entropy calculation
\cite[Proposition~9.1]{PerelmanEntropy}.

Consequently $\int v\,d\mu_t$ is nondecreasing in forward time.
At time zero it has a lower bound uniform for $0<\tau\leq T+1$:
writing $\phi=\sqrt u$ and $\int\phi^2=1$, its expression is
\[
 \tau\int(4|\nabla\phi|^2+R\phi^2)
 -\int\phi^2\log\phi^2-\frac n2\log(4\pi\tau)-n.
\]
Jensen's inequality followed by the Sobolev inequality of the fixed
initial metric gives this bound, including as $\tau\downarrow0$ by
optimizing the coefficient of $\|\nabla\phi\|_2^2$. Solve the
conjugate equation backwards from an arbitrary smooth positive
terminal density. Monotonicity then gives the same logarithmic Sobolev
inequality at every time for $0<\tau\leq1$. Positive approximation
allows Lipschitz or nonnegative test functions.

Here are two consequences with their normalizations. If a small ball
has arbitrarily small ratio $\Vol(B_r)/r^n$, choose a dyadic descendant
whose inner half has a definite fraction of its volume and whose ratio
is still small. Such a descendant exists: otherwise successive ratios
decrease geometrically to zero, contradicting the Euclidean ratio at a
smooth center. A tent function supported in this descendant, equal to
one on its inner half, has normalized Dirichlet energy $O(r^{-2})$.
Jensen gives $\int\phi^2\log\phi^2\geq-\log\Vol(B_r)$.
The logarithmic Sobolev inequality with $\tau=r^2$ therefore gives a
uniform positive lower volume ratio. Also
$\int\phi^2\log\phi^2\geq-2\log\|\phi\|_1$ for
$\|\phi\|_2=1$. Optimization in $\tau$, using $\tau=1$ when
necessary, gives
\begin{equation}\label{geom:nash}
 \|\phi\|_2^{2+4/n}
 \leq C(\|\nabla\phi\|_2^2+\|\phi\|_2^2)\|\phi\|_1^{4/n}.
\end{equation}
For both forward ordinary heat and backwards conjugate heat, the
$L^1$ norm is bounded by its initial norm times $e^{Ch}$, and the
derivative of the squared $L^2$ norm in the diffusion direction is at
most $-2\|\nabla u\|_2^2+C\|u\|_2^2$. This uses only
$\partial_t d\mu_t=-R\,d\mu_t$. Equation~\eqref{geom:nash} and
integration of the resulting scalar differential inequality give
$\|P_h\|_{1\to2}\leq Ch^{-n/4}$ for $h\leq1$. Duality with the
conjugate equation and composition at the midpoint give the heat
kernel bound $K\leq Ch^{-n/2}$.

For a conjugate kernel with terminal pole $p$,
Equation~\eqref{geom:entropy-identity} also gives $v\leq0$; compare
\cite[Corollaries~9.3--9.4]{PerelmanEntropy}.
One may verify the terminal sign without a uniform curvature bound
near $T$. Fix the terminal time first, when the flow is smooth, and a
positive forward heat test $\zeta$. Integration by parts gives
\[
 \int v\zeta
 =\partial_\tau\left(\tau\int(f-n/2)\zeta u\right)
       -2\tau\int u\Delta\zeta.
\]
The kernel upper bound bounds the integral in parentheses from below
after division by $\tau$. For its upper bound, apply Jensen relative
to $(4\pi\tau)^{-n/2}e^{-d_*^2/(4\tau)}$, where $d_*^2$ is a smooth
function agreeing with squared terminal distance near $p$ and has
$p$ as its unique zero. Its mass tends to $1$, and the heat equation,
tested against $d_*^2\zeta$, gives
$\tau^{-1}\int d_*^2\zeta u\to2n\zeta(p)$. Hence
$\limsup\int(f-n/2)\zeta u\leq0$. Along a sequence approaching zero
the displayed derivative has nonpositive liminf. At this fixed smooth
terminal time, $2\tau\int u\Delta\zeta\to0$, so the same is true of
$\int v\zeta$. Equation~\eqref{geom:entropy-identity} makes
$\int v\zeta$ nonincreasing in $\tau$; hence it is nonpositive at
every earlier time. Arbitrary positive forward heat tests now give
$v\leq0$. Equivalently,
\[
 2\tau f_\tau+\tau|\nabla f|^2-\tau R+f\leq0.
\]
Integration along the constant material path at $p$ gives
$f(p,\tau)\leq C\tau$. The initial term in this integration can
also be obtained from the preceding second moment: some point within
$O(\sqrt\tau)$ of $p$ has bounded $f$; moving from that point to $p$
over a time interval of size $\tau$ costs $O(\sqrt\tau)$ in the
inequality for $\sqrt\tau f$, by completing the gradient square.
Letting the starting time tend to zero removes this cost. Thus
\begin{equation}\label{geom:kernel-diagonal}
 K(p,t;p,s)\geq c(t-s)^{-n/2}.
\end{equation}

For a forward heat solution $|U|\leq1$ over elapsed time $h\leq1$,
the cancellation in the Bochner formula gives
\[
 (\partial_t-\Delta)|\nabla U|^2=-2|\nabla^2U|^2,
 \qquad |\nabla U|\leq Ch^{-1/2}.
\]
The gradient bound follows by applying the maximum principle to
$h|\nabla U|^2+CU^2$. To obtain a time derivative bound observe
$ (\partial_t-\Delta)\Delta U=2\Ric:\nabla^2U$ and
$ (\partial_t-\Delta)R=2|\Ric|^2$. For either sign, apply the
maximum principle to
$Q=h^2(\pm\Delta U+A|\nabla U|^2-AR)$, with fixed large $A$.
The negative terms $-2Ah^2(|\nabla^2U|^2+|\Ric|^2)$ absorb
$2h^2\Ric:\nabla^2U$. The remaining negative Hessian square
absorbs $2h|\Delta U|$; the already proved gradient bound controls
the other elapsed-time terms. At a positive maximum with $Q\geq C h$
the resulting upper bound is $C-cQ^2/h^2$, which rules out a first
crossing of a sufficiently large multiple of $h$. Therefore
\begin{equation}\label{geom:heat-derivatives}
 |\nabla U|\leq Ch^{-1/2},\qquad |\partial_tU|=|\Delta U|\leq C/h.
\end{equation}

Take a forward kernel from $(x,t-r^2)$ and multiply it by $r^n$.
On a short slab about $t$ it is bounded, has value at least $c$ at
$(x,t)$, has spatial gradient $C/r$, time derivative $C/r^2$, and
integral at most $Cr^n$. It is therefore bounded below on a ball of
radius $cr$ at time $t$, proving the upper volume bound. Each
connected component of a fixed positive superlevel has diameter
$O(r)$: a connecting path of greater diameter contains arbitrarily
many disjoint balls of radius $c'r$, where the function is bounded
below; lower volume bounds and the integral estimate bound their
number. The same function, using its time derivative bound and
slightly nested levels, proves
\begin{equation}\label{geom:rough-ball-transfer}
 B_{t_1}(x,cr)\subset B_{t_2}(x,Cr)
       \quad\text{if }|t_1-t_2|\leq c'r^2,
\end{equation}
in either time direction. Indeed every initial radial path in the
smaller ball remains in a positive superlevel at the second time, so
its endpoint remains in the same component as $x$.

In particular, for small $l$ there is a nonnegative Lipschitz cutoff
on $[t-cl^2,t]$, bounded by a fixed constant, with
\begin{equation}\label{geom:heat-cutoff}
 |\nabla\chi|\leq C/l,\quad
 c/l^2\leq\partial_s\chi\leq C/l^2\ \text{ on }\{\chi>0\},
 \quad\chi(x,s)\geq c,\quad\supp\chi\subset B_t(x,l).
\end{equation}
Choose the kernel radius to be a small fixed fraction of $l$, subtract
a suitable level, add $C'(s-t)/r^2$, and take the positive part. Keep
the final superlevel component containing $x$. Choosing $C'$ larger
than the kernel time-derivative constant makes the positive domains
monotone in $s$, so extension by zero outside this component is
legitimate. The component diameter bound puts its support in
$B_t(x,l)$. All constants are independent of the cap $l_0$.

We now bootstrap, rather than assume, the time control needed to use
a spatial curvature radius. Fix a tensor-index convention and
bootstrap
\begin{equation}\label{geom:bootstrap}
 |\partial_t\Rm|+C_n|\Ric||\Rm|\leq A l^{-3}.
\end{equation}
The left side controls the time derivative of the curvature norm.
Assume Equation~\eqref{geom:bootstrap} at preceding times, set
$l=l(x,t)$, and use Equation~\eqref{geom:heat-cutoff}. At time $t$,
$\max\chi^2|\Rm|\leq C l^{-2}$. Choose $B>C$. If this maximum
had been larger than $Bl^{-2}$, take its last crossing down to that
value, and let $y$ be a maximizing point at the crossing. Put
$\chi_y=\chi(y,s)>0$. The maximum and the cutoff gradient bound give,
for $d_s(y,z)\leq c\chi_y l$,
\[
 \chi(z,s)\geq\chi_y/2,\qquad
 |\Rm|(z,s)\leq4B/(\chi_y^2 l^2).
\]
Thus, taking $c_B\leq\min(c,(4B)^{-1/2})$,
\begin{equation}\label{geom:no-circularity}
 l(y,s)\geq c_B\chi_y l.
\end{equation}
This inference uses no curvature derivative estimate. At the crossing
the positive cutoff contribution to $\partial_s(\chi^2|\Rm|)$ is
at least $cB/(\chi_y l^4)$, whereas its possible negative curvature
contribution is at most $C_BA/(\chi_y l^3)$ by the bootstrap.
Choosing $A l_0$ sufficiently small makes the derivative strictly
positive. The upper one-sided derivative of the spatial maximum
cannot then permit the crossing. Approximation of the positive-part
cutoff, or the corresponding one-sided maximum argument on its
positive set, gives the same conclusion. Consequently curvature is
bounded by $Cl^{-2}$ on a definite smaller backward parabolic ball
at $(x,t)$. Its moving spatial balls are contained in the positive
cutoff region by $\chi(x,s)\geq c$ and the gradient bound.

On this parabolic ball the usual differentiated curvature argument
can now be applied without circularity. The connection variation is
\[
 \partial_t\Gamma^k_{ij}
 =-g^{km}(\nabla_i\Ric_{jm}+\nabla_j\Ric_{im}-\nabla_m\Ric_{ij}).
\]
Skew differentiation, the second Bianchi identity, and commuting
derivatives give the heat equation for $\Rm$ and its differentiated
equations, with products
$\nabla^i\Rm*\nabla^{k-i}\Rm$. At unit scale, after bounds through
order $k-1$, the quantity
$(D+|\nabla^{k-1}\Rm|^2)|\nabla^k\Rm|^2$, with large fixed $D$,
satisfies a differential inequality bounded above by a constant minus
a positive multiple of its square. Cauchy's inequality absorbs the
cross-gradient terms. Cutoffs on smaller parabolic balls, whose
distance and Laplacian bounds now follow from the curvature bound,
prove the derivative estimate inductively; distance cutoffs are
interpreted by shortened radial barriers at cut points. This is the
local derivative argument of \cite{Shi1989}.

The coordinate estimates used here have a uniform radius. Normalize
a smaller controlled curvature ball by the length $l$. On a fixed
ball in these units, sectional curvature is bounded above and below,
and every sufficiently small contained ball has volume at least
$c r^4$. The local injectivity estimate of Cheeger--Gromov--Taylor
\cite[Theorem~4.3]{CheegerGromovTaylor1982} therefore gives a positive
uniform injectivity radius on a further smaller ball. To check its
local hypotheses, reserve a fixed outer ball within the curvature
bound and choose all comparison radii below its conjugate-radius
bound. The upper sectional bound supplies that bound; the lower
sectional bound controls the volume of the pulled-back tangent ball
in the theorem's loop estimate, while the lower ball-volume bound
controls its denominator. Closedness supplies compactness of the
smaller closed balls. No curvature bound outside this controlled
region is used. Returning to original units gives normal radius
$c l$. Differentiating radial Jacobi equations then gives the higher
coordinate bounds from the covariant curvature bounds.

At this stage the crude estimates
$|\nabla^k\Ric|\leq C_k l^{-2-k}$ follow by contraction from the
curvature estimates. These already suffice, after dilation to unit
scale, to integrate the metric and connection equations in a fixed
initial normal chart for a time of order $l^2$. Consequently the
chart coefficients and all needed derivatives stay uniformly
controlled, and smooth spacetime cutoffs with the usual scaled
bounds are available before the improved Ricci estimate is proved.

The lower-index Ricci tensor satisfies
\begin{equation}\label{geom:ricci-equation}
 \partial_t\Ric=\Delta\Ric+2\Rm(\Ric)-2\Ric^2,
       \qquad \Rm(g)=\Ric.
\end{equation}
Test the scalar equation against a scaled smooth spacetime cutoff on
the smaller parabolic ball. Such cutoffs are now available in the
controlled coordinates. Integration by parts, $|R|\leq C_R$, and
volume $O(l^4)$ over time $O(l^2)$ give
\begin{equation}\label{geom:scalar-energy}
 \int_{t-cl^2}^{t}\int_{B_s(x,cl)}|\Ric|^2\,d\mu_s\,ds
       \leq Cl^4.
\end{equation}
For example, derivatives of the cutoff cost $l^{-2}$, so its scalar
terms cost $l^{-2}l^4l^2= l^4$; time integration also includes the
bounded $R^2$ measure term. Rescaling lengths by $l^{-1}$ makes the
squared spacetime Ricci norm $O(l^2)$. Equation~\eqref{geom:ricci-equation}
is a homogeneous linear tensor heat equation once its bounded
coefficients are fixed. Interior energy estimates followed by local
Sobolev iteration and differentiated interior estimates give
$|\nabla^k\Ric|\leq C_k l^{-1-k}$ in original units.
This is the local scalar-to-Ricci improvement of
\cite[arXiv version, Lemma~6.1]{BamlerZhangI}; the preceding argument
has verified its bounded-curvature cylinder and cutoff inputs from
the spatial radius. The uniform coefficients used in these interior
estimates are precisely those obtained from the crude curvature bounds above;
the improved Ricci estimates are their conclusion.

Curvature variation now bounds the left side of
Equation~\eqref{geom:bootstrap} by $A_0 l^{-3}$, with $A_0$
independent of $A,l_0$. Choose $A>A_0$, and then $l_0$ small enough
for the crossing argument and for the bootstrap to hold on a fixed
initial compact time interval. A first-failure argument proves it for
all later times. This proves Equation~\eqref{eq:ricci-improvement}.
For forward persistence, use the same heat function with a negative
drift, so that $-C/l^2\leq\partial_s\chi\leq-c/l^2$ on its positive
set and its support stays in the initial radius-$l$ ball. At the
first upward crossing of $\max\chi^2|\Rm|=Bl^{-2}$, the cutoff
gradient again gives $l(y,s)\geq c_B\chi_y l$. The negative cutoff
contribution is at most $-cB/(\chi_y l^4)$ while the curvature
contribution is at most $C_BA/(\chi_y l^3)$. Their sum is negative
for the chosen cap, contradicting the crossing. The cutoff is
positive near the initial center throughout a fixed shorter time
slab, proving the forward bounded-curvature neighborhood. We have
proved both directions of spatial-scale persistence and the derivative
bounds in Equation~\eqref{eq:ricci-improvement}; the two-sided volume
bounds were established by heat localization. The remaining global
input for compactness is the slice energy bound.

\subsection{Energy, small bad sets, and volume comparison}
\label{geom:energy-comparison}

The local estimates now give definite curvature energy at each
concentration scale. To bound how many such regions can occur, we first
control the total slice energy and then derive a summable volume-comparison
error. The latter will distinguish genuine quotient singularities from
smooth points in a limit.
Choose $C_0$ so $1\leq R+C_0\leq C$. Completing the gradient square
in the quotient evolution, as in Simon's integral-curvature argument
\cite[Section~3, equation~(3.2) and Theorems~3.5--3.6]{SimonIntegral2020}, gives
\[
 (\partial_t-\Delta)\frac{|\Ric|^2}{R+C_0}
 \leq C|\Rm||\Ric|^2-c|\Ric|^4.
\]
If $F=\int|\Ric|^2/(R+C_0)$, the volume derivative and Young's
inequality imply
\[
 F'\leq C\int|\Rm|^2-c'\int|\Ric|^4+CF.
\]
In four dimensions the Chern--Gauss--Bonnet formula \cite{Chern1944}
in the full-tensor normalization of
\cite[Equation~(3.6)]{SimonIntegral2020} gives
$\int|\Rm|^2\leq C+4\int|\Ric|^2\leq C+CF$. No orientability
assumption is needed: for a nonorientable $M$ apply the identity to
its orientation double cover with the pulled-back flow and divide
the integral identity by two. Dropping the negative quartic
term and applying Gronwall bounds $F$ and then the slice curvature
energy, proving Equation~\eqref{geom:volume-energy}.

There is a fixed energy $e_0>0$ associated to a point-picked
curvature ball. Indeed maximize curvature times squared distance to
the boundary of a slightly larger ball. At the maximizing point,
curvature $Q$ is bounded by $CQ$ on a ball of radius $cQ^{-1/2}$.
The spatial derivative estimate makes curvature at least $Q/2$ on a
smaller ball; lower volume then gives $\int|\Rm|^2\geq e_0$.
If $l(x)<r$, curvature reaches at least $r^{-2}$ within distance $r$
of $x$ by the definition of $l$. The preceding point selection
therefore produces an energy ball inside $B(x,Cr)$. A maximal
disjoint collection of these larger radius-$Cr$ balls has at most
$C/e_0$ members by the slice energy bound; enlarging them covers
$\{l<r\}$. Thus
\begin{equation}\label{geom:bad-volume}
 \Vol_t\{l<r\}\leq Cr^4.
\end{equation}
On $B(x,h)\cap\{l\geq h\}$, the cubic Ricci integral is at most
$Ch^{-3}h^4$. On $2^{-j-1}h\leq l<2^{-j}h$ it is at most
$C(2^{-j}h)^{-3}(2^{-j}h)^4$. Summing gives
\begin{equation}\label{geom:ricci-L3}
 \int_{B_t(x,h)}|\Ric|^3\,d\mu_t\leq Ch.
\end{equation}

Distances have a uniform terminal limit. For $|t-s|\leq cr^2$,
cover the bad set at the first time by the bounded family of
radius-$Cr$ balls, omitting from a minimizing path its visits to
slightly enlarged balls. Replace first-to-last visits when a ball is
revisited. There are uniformly many replacement endpoints, with
total replacement cost $O(r)$ at the other time by
Equation~\eqref{geom:rough-ball-transfer}. On the remaining path the
curvature scale is $\geq cr$ throughout the interval by forward and
backward persistence. Its length changes by a factor at most $1+Cr$
since $|\Ric|\leq C/r$. Comparing also in reverse gives
\[
 |d_t(x,y)-d_s(x,y)|\leq Cr(1+d_s(x,y)+d_t(x,y)).
\]
First compare with a fixed late time to bound diameter, then take
$r$ a sufficiently large constant times $\sqrt{|t-s|}$. This proves
uniform convergence to a pseudodistance $d_T$ on $M$ and defines its
compact metric quotient $Z$. For scalar-bound distance distortion and
terminal metric convergence, see also
\cite[Theorem~1.1 and Corollary~1.2]{BamlerZhangII}.
Volume forms at two times differ by factors $e^{\pm C_R|t-s|}$.

We will need a comparison estimate stronger than mere volume bounds.
In polar coordinates about a smooth point, before the cut radius,
let $P$ be the polar Jacobian and
$m=(\partial_r\log P-3/r)_+$. The Riccati inequality gives, on
$\{m>0\}$,
\[
 m'+2m/r+m^2/3\leq|\Ric|.
\]
Since $P'=(m+3/r)P$ there,
\[
 (m^5P)'\leq5|\Ric|m^4P-7m^5P/r-(2/3)m^6P.
\]
Integrate to cuts and use Young's inequality on the first term.
The boundary contributions have the correct sign, with $m=0$ at
entrance into a positivity interval. Setting $P=0$ after cuts gives
\begin{equation}\label{geom:polar-defect}
 \int_{B(x,h)}m^6\leq C\int_{B(x,h)}|\Ric|^3\leq Ch.
\end{equation}
Furthermore $(P/r^3)'\leq mP/r^3$. Integrating between comparable
radii yields
\begin{equation}\label{geom:almost-bishop}
 \frac{\Vol B(x,h)}{h^4}
 \leq\frac{\Vol B(x,h/2)}{(h/2)^4}+Ch^{1/2}.
\end{equation}
To see the exponent, the error is bounded by
$Ch^{-3}\int m\leq Ch^{-3}(Ch)^{1/6}(Ch^4)^{5/6}=Ch^{1/2}$.
The same argument holds for any fixed comparable pair of radii.
Errors sum to $Ch^{1/2}$ over all dyadic descendants. Polar
integration including the nonpositive cut measure also gives weakly
\begin{equation}\label{geom:radial-laplacian}
 \Delta(d^2/2)\leq4+dm.
\end{equation}

The scale, energy, and bad-set bounds of Theorem~\ref{thm:degeneration}
are now established, as is uniform convergence of distances. We next
identify the geometry on every shrinking spatial scale; smooth terminal
convergence away from a fixed finite set will then follow by persistence.

\subsection{Slice limits, quotient cones, and quantitative necks}
\label{geom:slice-limits}

We next turn these estimates into a description of every small-scale
slice limit. The essential conclusions are smooth Ricci-flat convergence
off a finite set and quantitative quotient coordinates on the intervening
annuli. Puncture removal and the nontriviality of the quotient groups
will be established after those coordinates are available.
This is the part of the argument corresponding to classical Einstein
orbifold compactness and bubbling \cite{Anderson1989,Bando1990,
BandoCorrection1990}; the estimates below include the Ricci error
present in the flowing slices.
Consider $h_i^{-2}g(t_i)$, based at arbitrary $x_i$, where
$h_i\to0$ and $t_i\uparrow T$. Two-sided volume bounds uniformly
bound the number of disjoint radius-$\rho$ balls in any bounded ball,
giving pointed subsequential compactness as length spaces.
Completeness follows by exhausting with successively larger balls.
The cover of $\{l<\rho h_i\}$ has uniformly bounded cardinality.
Letting $\rho\downarrow0$ diagonally identifies a finite exceptional
set $S$, with $l/h_i$ bounded below on its regular compact complement.
The coordinate and derivative bounds give smooth convergence there.
In the rescaled metrics,
\[
 |\nabla^k\Ric|\leq C_kh_i(l/h_i)^{-1-k}\longrightarrow0
\]
on these compacts, so the limit is Ricci-flat.

Smooth convergence includes embeddings of relatively compact regular
subregions. Take finitely many normal charts with margins and extract
their transition maps together with their metrics. The local inverse
identifications differ smoothly by quantities tending to zero.
In convex limiting balls combine them by a smooth partition and the
unique local squared-distance barycenter. The resulting map is locally
invertible on smaller charts, covers the desired region by the chart
margins, and is injective: two points whose images converge to the same
point have distance tending to zero, hence lie in one chart where
injectivity holds. Construct on a larger compact region before
restricting. This supplies actual embeddings for the later gluing.

There is no volume mass at $S$, since its $\rho$-neighborhood has volume
$O(\rho^4)$. Volume is therefore regular Riemannian volume, and balls
have converging volumes. Indeed at almost every regular point away
from the center, the last regular segment of a minimizing path gives
$|\nabla d|=1$. Distance levels consequently have zero four-dimensional
measure. The regular limit has finite curvature energy by lower
semicontinuity on compact exhaustions. In $h_i^{-1}$ length units,
Equation~\eqref{geom:ricci-L3} gives a cubic Ricci integral
$O(h_i^2)$ on each fixed ball. Thus the normalized polar defect
vanishes, and Equations~\eqref{geom:almost-bishop} and
\eqref{geom:radial-laplacian} give exact Bishop monotonicity and
$\Delta(d^2/2)\leq4$ weakly on the regular part of the limit.

The finite exceptional set cannot separate positive-volume regular
components. Fix a regular initial point and a small regular endpoint
ball $U$ in a supposed other component, both a positive distance from
$S$. All approximating minimizing segments between them must enter
$N_\rho(S)$; otherwise a subsequential limit of such bounded segments
would connect the two components in the regular complement.
Their passage through $N_{C\rho}(S)$ has length at least $c\rho$,
at radii in a fixed compact subinterval of $(0,\infty)$.
The polar inequality gives, for a passage radius $s$ and endpoint
radius $r$,
\[
 P(r)\leq C P(s)+C\int_s^r m(u)P(u)\,du.
\]
Integrating over passage times, endpoint radii, and angles, and using
Fubini on these bounded ranges, gives
\begin{equation}\label{geom:nonseparation}
 c\rho\Vol(U)\leq
 C\left(\Vol(N_{C\rho}(S))+\int m\right).
\end{equation}
Cuts only decrease the polar integrals. First take the sequence
limit, eliminating $\int m$, and then let $\rho\downarrow0$.
The right side is $O(\rho^4)$, a contradiction. This argument also
applies to iterated tangent and end limits: a diagonal choice ensures
that the original physical radii still tend to zero.

Write $d$ for distance from a puncture, or from a fixed point when
studying infinity. At a puncture, finite energy makes the curvature
energy on annuli $\{r/4<d<4r\}$ tend to zero as $r\downarrow0$; at infinity the same
holds as $r\to\infty$. First, point-picking and the fixed energy
$e_0$ rule out unbounded $r^2|\Rm|$ on a smaller annulus. The inherited
local estimates then give $r^3|\nabla\Rm|\le C$ with further fixed
boundary margins. If $r^2|\Rm|\ge\varepsilon>0$ at a point of this
middle annulus, curvature remains at least $\varepsilon/(2r^2)$
on a ball of radius $c\varepsilon r$ contained in the wider annulus.
For each fixed $\varepsilon$, lower volume bounds give energy at
least $c\varepsilon^6$ on that ball, contradicting the vanishing
annular energy. Thus the scaled curvature tends to zero. Interior
stationary estimates, or interpolation with the inherited higher
bounds, give the same conclusion for every fixed number of scaled
derivatives. Tangent and end limits have flat regular annuli and only
the dilation center as a possible incomplete point. Their ball volumes
are exactly $vr^4$: the monotone ratio has a limit at the relevant end,
and dilation freezes that limit at every fixed radius.

Put $u=d^2/2$. The nonnegative distribution $4-\Delta u$ pairs to zero
with every compactly supported radial test by this volume formula
and $|\nabla d|=1$. Dominate an arbitrary nonnegative test by a radial
test on a larger compact annulus to see that the defect vanishes.
Elliptic regularity makes $u$ smooth and $\Delta u=4$. Bochner's formula
and $|\nabla u|^2=2u$ give $|\nabla^2u|^2=4$; since its trace is $4$,
equality in Cauchy's inequality gives $\nabla^2u=g$. Its gradient
curves produce the cone metric $dr^2+r^2g_L$. Flatness makes $L$ a
compact spherical manifold and nonseparation makes it connected.
Thus $L=S^3/\Gamma$ for a finite orthogonal group acting freely.
Two-sided volumes bound $|\Gamma|$. All fixed-ratio annuli sufficiently
far along the original end are therefore arbitrarily close, with
derivatives and radial distances, to round quotient annuli.
Enlarged overlapping annuli identify the groups.

\begin{lemma}[Curvature and chart improvement on necks]\label{geom:neck}
Suppose annular regions have inner and outer radii $r_-\ll r_+$ and
every escaping intermediate annulus converges smoothly to a round flat
quotient annulus. Assume for some $\sigma>0$ and all needed finite
orders that
\[
 r^2|\Ric|_{k+2,r}\leq
 C\big((r/r_+)^\sigma+(r_-/r)^\sigma\big).
\]
Fix $0<\mu<\min(1,\sigma)$. After removing fixed-factor boundary
collars, for this fixed exponent,
\begin{equation}\label{eq:neck-decay}
 r^2|\Rm|_{k,r}\leq
 C\big((r/r_+)^\mu+(r_-/r)^\mu\big).
\end{equation}
A single equivariant coordinate system on a shortened neck gives the
same bound for $g-\delta$, with scaled derivatives. Infinite
half-necks are allowed. One may also use slowly varying coefficients
on the boundary terms, when the forcing and boundary data obey their
corresponding bounds.
\end{lemma}

\begin{proof}
First choose the permitted fixed-factor boundary collars. For a fixed
finite derivative order, a fixed widened annular ratio, and a small
cone-chart tolerance, there are a fixed $K$ and a late index such that
all widened annuli with central radii in $[Kr_-,r_+/K]$ have these
smooth bounds. Otherwise choose increasingly late violating annuli
with radii in $[jr_-,r_+/j]$, $j\to\infty$. Both boundary ratios
escape, contradicting the assumed smooth convergence of the whole
annulus. Enlarge $K$ by a fixed factor to retain boundary margins.
The shortened boundaries $r'_-=Kr_-$ and $r'_+=r_+/K$ therefore have
uniformly bounded scaled curvature and the required derivatives.
For either exponent $\beta=\mu,\sigma$, their weights satisfy
\[
 (r/r'_+)^\beta+(r'_-/r)^\beta
   =K^\beta\big((r/r_+)^\beta+(r_-/r)^\beta\big).
\]
It is enough to prove the estimate on this shortened neck: these
fixed multipliers preserve both the forcing hypothesis and the
claimed conclusion. Relabel its boundary radii as $r_-,r_+$ below.

Write $w_\mu(r)=(r/r_+)^\mu+(r_-/r)^\mu$. If the bound fails, choose
approximate maximizing points of $r^2|\Rm|/w_\mu(r)$ with ratios
$Q_i\to\infty$, and call their radii $s_i$. Both boundaries escape
after dilation by $s_i^{-1}$: near either boundary the weight is bounded
below and scaled curvature is bounded. Put
$A_i=s_i^2|\Rm|(s_i)$. Intermediate annular flatness gives $A_i\to0$.
Divide the rescaled curvature by $A_i$. The maximum property and
\[
 \frac{w_\mu(rs_i)}{w_\mu(s_i)}\leq\max(r^\mu,r^{-\mu})
\]
give the divided tensor the bound
$Cr^{-2}\max(r^\mu,r^{-\mu})$.
The second Bianchi identity and commutation give
$\Delta\Rm=\Rm*\Rm+\nabla^2\Ric*$.
The divided quadratic term vanishes. The divided forcing vanishes on
fixed annuli since $w_\sigma(s_i)/A_i\leq C/Q_i$.
Interior elliptic estimates thus give a nonzero limiting Cartesian
tensor $H$ on the flat punctured cover, componentwise harmonic and
satisfying the uncontracted differential Bianchi identity.

Spherical harmonics have radial powers $j$ and $-j-2$,
$j=0,1,\ldots$. The two bounds with $\mu<1$ leave only
$H=r^{-2}C$ with a constant Cartesian tensor $C$. Bianchi becomes
\[
 x_iC_{jk ab}+x_jC_{ki ab}+x_kC_{ij ab}=0
       \qquad\text{for every }x\in\R^4.
\]
For fixed $a,b$ it says $x\wedge C_{\cdot\cdot ab}=0$ for every
covector $x$, forcing that two-form to vanish. This contradicts the
normalization. Derivatives follow by interior elliptic estimates.
For an infinite end use a remote artificial boundary and pass to its
limit. Slowly varying boundary coefficients obey the same ratio
estimate after decreasing $\mu$ to absorb their change on fixed
logarithmic intervals; the same divided forcing argument applies.

Here are the gauge and gluing details. On a fixed lifted round annulus,
with a larger annulus retained for margins, let
$S(V)=DV+(DV)^t$ and $K(p)$ be linearized curvature at $\delta$.
For $k\geq5$ there is a bounded linear extraction
$E:H^k(\operatorname{Sym}^2)\to H^{k+1}(T)$, modulo Killing fields,
with
\begin{equation}\label{geom:linear-gauge}
 E(SV)=V,\qquad
 \|(1-SE)p\|_{H^k}\leq C\|K(p)\|_{H^{k-2}}.
\end{equation}
Indeed regard the linearized connection
$C^a_{ij}=\tfrac12(\partial_i p_{ja}+\partial_jp_{ia}-\partial_ap_{ij})$
as a one-form in $i$, for each $a,j$. Its curl is $K(p)$.
The Neumann gradient projection writes it as $dU^a_j$ plus an
$H^{k-1}$ error controlled by $K(p)$ in $H^{k-2}$.
Symmetry in $i,j$ makes the curl of $U^a_jdx^j$ controlled; a second
projection writes $U^a=dV^a$ plus a controlled error.
Thus $C(p-SV)$ is controlled, and
$\partial_iq_{ja}=C(q)^a_{ij}+C(q)^j_{ia}$ controls $Dq$.
Absorb the remaining constant symmetric matrix by an affine field.

For clarity, the projection estimate is the div/curl estimate with
zero normal component. Integration by parts makes the boundary
second fundamental form an order-zero term and gives coercivity
modulo $L^2$. Compactness removes this term because a closed remainder
is a gradient on the simply connected annulus and the Neumann
projection makes it orthogonal to gradients. Higher estimates follow
by tangential differentiation in flattened boundary charts, recovering
normal derivatives from div and curl. Fix averages and Killing
ambiguity, or project onto the range of $S$, to obtain the stated
left inverse at all needed orders. Averaging over $\Gamma$ preserves
the estimates and gives equivariance.

For $g=\delta+p$ small in $H^k$, solve
$V=Ep-E((DV)^tDV)$ by contraction. Sobolev multiplication is bounded
since $k>4/2+1$. Put $q=p-SV-(DV)^tDV$, so $Eq=0$, and
$f=(\Id+V)^*\delta$. Since $f$ is flat, the difference of the nonlinear
curvature remainders of $g$ and $f$ has $H^{k-2}$ norm at most
$C\|p\|_{H^k}\|q\|_{H^k}$. Equation~\eqref{geom:linear-gauge} and
absorption give
\[
 \|q\|_{H^k}\leq C\|\Rm(g)\|_{H^{k-2}}.
\]
Changing coordinates by $\Id+V$ gives the local metric estimate.
Higher finite orders and Sobolev embedding give all required scaled
differentiability bounds.

Choose geometric radius steps and charts on much wider annuli whose
images contain whole distance bands with margins. On consecutive
overlaps, the transition satisfies
\[
 (D\Psi)^tD\Psi-I=O(\varepsilon_j),\qquad
 D^2\Psi=O(\varepsilon_j/r_j),\qquad
 \varepsilon_j=Cw_\mu(r_j).
\]
The second estimate follows from the Christoffel transformation law.
Integration gives
$\Psi=P_jx+b_j+O_{k,r_j}(\varepsilon_jr_j)$ for an affine isometry.
Comparing both charts in a larger converging cone chart, or using
nested bands in both directions, identifies their fundamental groups.
Average $P_j$ over this exact deck isomorphism and take its polar
factor to obtain an exact orthogonal intertwiner within the same
error. Project the translation to the invariant subspace; it is zero
when $\Gamma$ is nontrivial and free.

Align consecutive charts by these rigid motions. For the trivial
group, radial positioning bounds translation increments by $o(r_j)$.
Toward a puncture their sum converges; subtract its limit. At infinity
sum outward from an initial chart, giving $o(r_j)$ relative displacement
by geometric summation. Rotations do not affect radial position.
Interpolate the remaining errors on shorter overlaps; derivatives
of the interpolation cutoff cost $r_j^{-m}$ and preserve the scaled
$O(\varepsilon_j)$ bounds. Local invertibility follows from smallness.
For global injectivity, equal images have comparable radii and hence
lie in one common wide annular chart. There the glued map is $C^1$
close to the identity: close points are separated by derivative
bounds and points a fixed fraction of a radius apart by the $C^0$
bound. Radial margins ensure coverage of the shortened neck.
\end{proof}

\subsection{Removal of punctures and intrinsic singular points}
\label{geom:puncture-removal}

The neck estimate supplies coordinates in which a puncture has a
continuous metric completion. We now improve that completion to a smooth
metric on its local cover, and then prove that every actual concentration
point has a nontrivial quotient group. This second assertion is needed
both for organizing trees and for tracking them by their distances.
At a limit puncture Lemma~\ref{geom:neck}, with zero Ricci forcing,
gives on the cover
\begin{equation}\label{geom:puncture-decay}
 |D^k(g-\delta)|\leq C_kr^{\mu-k},\qquad \mu>0.
\end{equation}
Choose $0<\alpha<\min(\mu,1)$ and fill $g(0)=\delta$. Points separated
by a distance comparable to their radii are controlled by the
zeroth-order bound; closer points by the first-derivative bound along
their segment. Thus $g$ and $A^{ij}=\sqrt{\det g}\,g^{ij}$ are
$C^{0,\alpha}$, and after scaling $B_R$ to $B_1$,
$\|A(R\cdot)-I\|_{C^{0,\alpha}}=O(R^\mu)$.

Solve
\[
 \partial_i(A^{ij}\partial_jy^a)=0\quad\text{in }B_R,\qquad
 y^a=x^a\quad\text{on }\partial B_R.
\]
Coercivity gives the unique variational solution. For $w=y-x$,
\[
 \partial_i(A^{ij}\partial_jw^a)
       =-\partial_i(A^{ia}-\delta^{ia}),\qquad
 w^a|_{\partial B_R}=0.
\]
After rescaling to $B_1$, the forcing vector has
$C^{0,\alpha}$ norm $O(R^\mu)$. The weak-solution gradient estimate
\cite[Theorem~8.16 and Corollary~8.35]{GilbargTrudinger2001} gives
$\|Dy-I\|_\infty=O(R^\mu)$ and $y\in C^{1,\alpha}$. This estimate
also follows by absorbing the small coefficient perturbation in the
constant-coefficient divergence estimate. Thus $y$ is a coordinate
diffeomorphism for small $R$. Uniqueness makes it equivariant, and
$y(0)$ is $\Gamma$-fixed, so recentering is permitted. Put $B=Dy(0)$,
and then make a constant linear normalization. Since
$y(x)-y(0)-Bx=O(r^{1+\alpha})$, rescaled interior estimates after
subtracting the affine part give
$D^ky=O(r^{1+\alpha-k})$ for $k\geq2$. The forcing from that affine
part has the stronger coefficient exponent $\mu>\alpha$.
The inverse coordinates obey the same estimates. Consequently the
harmonic-coordinate metric satisfies
\begin{equation}\label{geom:harmonic-decay}
 G-G(0)=O(r^\alpha),\quad DG=O(r^{\alpha-1}),\quad
 D^2G=O(r^{\alpha-2}).
\end{equation}

Thus $G\in W^{2,p}$ across zero for all $p<4/(2-\alpha)$; take
$p_0=2+\alpha/2>2$. There is no distributional point defect:
integration by parts across $\partial B_\rho$ has first-derivative
boundary terms $O(\rho^3)$ and second-derivative terms involving
$DG$ of size $O(\rho^{2+\alpha})$. Both tend to zero.
The harmonic-coordinate Einstein equation holds almost everywhere:
\begin{equation}\label{geom:weak-einstein}
 G^{ab}\partial_a\partial_bG_{ij}=Q_{ij}(G^{-1},DG,DG).
\end{equation}
Here $Q$ is quadratic in $DG$ with smooth coefficients in $G^{-1}$;
this is the Lanczos harmonic-coordinate formula, as recorded in
\cite[Lemma~4.1, equation~(4.3)]{DeTurckKazdan1981}. For $2<p<4$, Sobolev gives
$DG\in L^{4p/(4-p)}$, so elliptic regularity improves the exponent by
\begin{equation}\label{geom:bootstrap-exponents}
 p_{k+1}=\frac{2p_k}{4-p_k}>p_k,\qquad
 \frac1{p_k}=\frac12-2^k\left(\frac12-\frac1{p_0}\right).
\end{equation}
After finitely many steps one can take $p>4$. If a step reaches
exactly $4$, the local embedding $W^{1,4}\subset L^q$ for every
finite $q$ permits the next exponent above $4$.

This gain does not assume the stronger regularity in advance.
On a sufficiently small ball the continuous leading coefficients
are uniformly close to a constant positive matrix. Localize $G$;
its equation has right side in the higher $L^q$ since the cutoff
terms contain only $G,DG$. The constant-coefficient $W^{2,q}$
inverse and a Neumann series for the small coefficient perturbation
give a $W^{2,q}$ solution. Uniqueness in the previously available
$W^{2,p}$ class identifies it with the localized metric.
Alternatively, the global Dirichlet regularity theorem
\cite[Theorem~4.2]{ChiarenzaFrascaLongo1993} gives this upgrade for the
localized metric components, which have zero boundary trace.
Keep the continuous leading coefficients equal to $G^{-1}$ near the
support of the cutoff and extend them to a constant positive matrix
outside a larger compact set. A convex interpolation
preserves ellipticity and gives bounded uniformly continuous, hence
vanishing-mean-oscillation, coefficients.
Once $p>4$, $G\in C^{1,\beta}$.
Equation~\eqref{geom:weak-einstein}, Schauder estimates, and
differentiation give smoothness. This proves the orbifold filling
in changed equivariant coordinates without an assumption about
Einstein deformation integrability.

A concentration point cannot have trivial group. If it did, its
smooth filling would have almost-Euclidean volume ratios on sufficiently
small balls. Smooth convergence on surrounding annuli, negligible
tip volume, and Equation~\eqref{geom:almost-bishop} propagate these
lower ratios to every smaller scale in the approximations: the total
error below original radius $h$ is $O(h^{1/2})$. Concentration and
point-picking give, at a much smaller scale, a complete nonflat
Ricci-flat limit with a regular nonflat region. Point-picked centers
are $o(h)$ from the smoothly filled center. Inclusions between their
radius-$h$ balls and the filled-center balls of radii $h\pm o(h)$
transfer the same lower starting ratios. Let the initial ratio approach
the Euclidean one, choosing the approximating index after each outer
radius; the summable comparison error tends to zero in physical units.
Every fixed ball of the smaller limit has at least
Euclidean volume ratio. At a regular center Bishop comparison gives
the opposite bound. Equality, by the radial Laplacian and Hessian
argument, forces a smooth flat cone from that center, a contradiction.
Thus each exceptional point is a genuine intrinsic singularity with
nontrivial group; in particular a metrically regular point of a limit
cannot conceal a further curvature concentration. The same argument
shows the end group of a nonflat complete limit is nontrivial: a trivial end gives Euclidean
asymptotic volume ratio, which is independent of the chosen center.

A complete flat limit here is a global Euclidean quotient with at
most one tip. Develop the universal cover of its regular locus.
The faithful free linear local holonomy at a puncture ensures that
its local fundamental group injects into that of the regular locus:
the composition of the inclusion with flat holonomy is the faithful
linear action of its quotient group. Hence its lifts contain full
punctured Euclidean balls. Fill their centers.
The resulting flat manifold is complete: a finite-length path
projects to a convergent path downstairs and eventually remains in
one lifted local ball. Adding isolated points preserves simple
connectivity in dimension four. Its development is therefore all
of Euclidean space. The deck group is finite. For any prescribed
finite number of distinct deck transformations, lifts of a regular
expanding ball and their transforms occupy that many sheets in a
comparably enlarged Euclidean ball. Every downstairs point has a
lift within comparable radius by path lifting; paths detour isolated
tips with arbitrarily small excess using their connected links.
Arbitrarily many sheets would force the downstairs Euclidean growth
ratio to zero, contrary to noncollapse. A finite Euclidean isometry
group fixes its barycenter and is orthogonal after translation.
Since singularities are isolated, its only possible singular point
is the origin. This finishes the small-scale orbifold, genuine-tip,
and end assertions of Theorem~\ref{thm:degeneration}. We can now use
the intrinsic singular points to identify the vertices and edges of
a finite tree.

\subsection{Finite bubble trees and the definition of the root scale}
\label{geom:trees-root}

We have shown that a flat limit has at most one genuine tip. This
excludes two separated concentrations without a larger nonflat ancestor.
Together with the slice energy bound it will produce a finite tree and
identify its outermost member by a continuous maximum.
The terminal space $Z$ has only finitely many concentration points.
Fix a sequence $t_i\uparrow T$ and write $\pi:M\to Z$ for the
terminal quotient. For each sufficiently large integer $m$, let
$E_{i,m}=\pi(\{l(\cdot,t_i)\leq2^{-m}\})$. Extract their Hausdorff
limits $F_m$ diagonally in the compact space $Z$, allowing an empty
limit if the sets eventually become empty. They are nested compact
sets. The bad-set cover and uniform convergence of distances show
that each $F_m$ is covered by at most $N$ balls of radius $C2^{-m}$,
where $N,C$ are independent of $m$. Consequently
$S=\bigcap_mF_m$ has at most $N$ points: $N+1$ distinct points
would violate such a cover for large $m$.

Let $K\subset Z\setminus S$ be compact. Nested compactness gives
an $m$ with $K\cap F_m=\varnothing$ and a fixed neighborhood of
$K$ disjoint from $F_m$. For all sufficiently large $i$, every
material point mapping to a slightly smaller neighborhood has
$l(\cdot,t_i)>r=2^{-m}$. Crucially, $r$ is now fixed. Choose one
such $i$ with $T-t_i<c r^2$. Forward spatial persistence on these
radius-$r$ balls controls the entire remaining interval $[t_i,T)$,
and a finite ball cover gives fixed material charts, bounded
derivatives, and smooth terminal convergence over $K$. Local
injectivity gives a single sheet. Thus the possible exceptional
set is fixed and finite throughout late time; this conclusion does
not infer uniformity from shrinking time windows. It completes the
terminal-convergence assertion of Theorem~\ref{thm:degeneration}.

If the exceptional set is empty, finitely many such charts bound
curvature globally. Their derivative estimates, metric evolution,
and compactness give a smooth limiting metric on the original
manifold at $T$. The DeTurck formulation, with
$W^a=g^{ij}(\Gamma^a_{ij}(g)-\Gamma^a_{ij}(g_{\rm fixed}))$,
has principal part $g^{ij}\partial_i\partial_jg$ after adding
$\mathcal L_Wg$. Short-time quasilinear parabolic existence and
pullback by $-W$ continue the flow on the same manifold. The smooth
limit and evolution equation give matching derivatives at $T$;
compare \cite[Theorem 14.1]{Hamilton1982} for the curvature continuation
criterion. Hence only the case of a genuine concentration point
needs further work. Fix a small neighborhood $\mathcal U$ in the
material manifold, pulled back from a $d_T$-ball, whose regular
boundary isolates one such point. Its punctured regular annuli have
finite limiting curvature energy.

For a sequence $t_i\uparrow T$, call two nonflat small-scale limits
in this cluster \emph{equivalent} if their scales have ratio bounded
above and below and their centers remain a bounded distance apart
in either scale. Pass to subsequences so all scale and separation
comparisons converge in the extended sense. Each nonflat limit
contains a regular ball with a uniformly positive energy. Indeed
its orbifold filling and end decay make its maximum curvature finite
and attained. Dilate so that maximum equals $1$. The complete
stationary curvature equation gives a universal derivative bound:
the differentiated-curvature product used above, now stationary,
satisfies $-\Delta Q\leq C-cQ^2$ for
$Q=(D+|\Rm|^2)|\nabla\Rm|^2$. Its maximum is controlled, using decay
at infinity and lifted maximum tests at quotient points.
Thus a definite ball around a curvature maximum has curvature
bounded below. Noncollapse, upper volume bounds, and the uniformly
finite number of tips allow removal of small tip neighborhoods
without exhausting that ball; some smaller regular ball remains
with definite curvature and volume. Its energy is at least a fixed
$e_1>0$.

Such regular energy balls belonging to inequivalent entries are
disjoint in the original slices for large $i$. Separated centers
give disjoint balls directly. If one scale is much smaller and
its center remains bounded on a larger scale, it cannot meet a
fixed regular ball there: smooth regular convergence bounds its
curvature radius from below proportionally to the larger scale.
The smaller entry must approach a tip. Therefore the total energy
bounds the number of inequivalent entries by $C/e_1$.

Construct a maximal list by adjoining, whenever possible on a
further subsequence, a nonflat limit inequivalent to those already
listed. Previous convergences and comparisons persist after
subsequence extraction. The energy bound forces this process to
terminate, including against tests on further subsequences.
The list is nonempty by high-curvature point-picking.
If two entries are separated by a distance much greater than both
scales, their separation-scale limit has two genuine tips. It cannot
be flat, since a flat limit has at most one tip. It is consequently a
new nonflat ancestor unless it is already in the list.
Comparable scales at bounded distance are equivalent; at separated
scales and bounded larger-scale distance the smaller entry approaches
a tip of the larger one. These observations define an ancestor order
on the finite list and show that it has a unique maximal entry.
Applying the same argument to entries at each tip gives a unique
outermost child in that branch, and every tip has a child by
concentration and point-picking. This is the finite rooted tree.

Every intermediate limit between consecutive entries is flat,
since a nonflat one would enlarge the maximal list. The child becomes
its cone tip, and the same conclusion holds between the root and
arbitrarily slowly vanishing outer radii. On these flat annuli
$l\geq cr$, so Equation~\eqref{eq:ricci-improvement} gives
$r^2|\Ric|_{k,r}\leq Cr$ in physical units. Lemma~\ref{geom:neck}
therefore supplies the finite necks and positive decay exponents.
This completes the tree assertion of Theorem~\ref{thm:degeneration}.

These convergences identify actual slice regions with the core/neck
domains of Section~\ref{sec:static-input}. Use compact regular embeddings
on truncated vertices and the single annular coordinates on each
joining region. Dividing physical coordinates by the corresponding
vertex scale gives its end coordinates. If the parent and child
scales are $A$ and $Ae^{-2L}$, their end variables satisfy
\[
 \tau_p=\log(A/r),\qquad
 \tau_c=\log(r/(Ae^{-2L})),\qquad \tau_p+\tau_c=2L.
\]
On the parent half $\tau_p\leq L$,
$e^{-\mu\tau_p}+e^{-\mu\tau_c}\leq2e^{-\mu\tau_p}$;
the child half has the analogous bound. Thus the two-sided neck
estimate supplies exactly the one-sided half-end weights used there.
Fixed initial end collars are absorbed by enlarging constants.

On fixed overlaps the annular charts converge, after a subsequence,
with all required derivatives; they prescribe end charts for the
limiting vertices by diagonal extraction. Adjust the regular
embeddings by the small resulting overlap differences. Regular
convergence covers each vertex except holes about its tips;
annular convergence covers the intervening distances to its children,
and at a leaf a bounded rescaled core is entirely regular.
The same construction passes through a fixed compact root collar if
the exterior is later replaced. It is a sequential construction:
there is no assertion that one tree or one family of identifications
is smooth in time. It gives only a positive joining exponent.
The stronger exponent greater than one on the unjoined exterior
will be supplied by Theorem~\ref{thm:collar-extension} before
Theorem~\ref{thm:tree-inequality} is applied.

\begin{proposition}[A continuous outermost scale]\label{prop:root-scale}
There is a fixed small $c_*>0$ such that the following definition
detects the unique outermost entry. Choose a bounded continuous
$\psi:[0,1]\to[0,\infty)$, zero on $[0,c_*]$ and strictly positive
on $[2c_*,1]$, and set
\begin{equation}\label{geom:root-definition}
 a(t)=\sup_{y\in\mathcal U}
 l(y,t)\psi\big(l(y,t)^2|\Rm|(y,t)\big).
\end{equation}
After shrinking $\mathcal U$, this supremum is attained in its
interior for every late time; it is positive and continuous, and
$a(t)\to0$. For every sequence of late times, scale $a(t)$ and any
maximizing point represent the outermost nonflat entry. Every other
nonflat entry has scale bounded by a constant times $a(t)$.
The constants may be fixed after extracting the sequential tree.
\end{proposition}

\begin{proof}
The normalized regular energy balls just constructed provide uniform
detection. In the normalization $\max|\Rm|=1$, the derivative bound
keeps $|\Rm|\ge1/2$ on a ball of fixed radius. Removing small
neighborhoods of the uniformly bounded number of tips leaves a point
a fixed positive distance from every tip, by the volume bounds.
Its curvature radius is bounded below: a sufficiently small fixed
ball is regular and $|\Rm|\le1$ there. It is bounded above by the
nonzero curvature at its center. Hence at this point
$l^2|\Rm|\geq2c_*$ for a fixed small $c_*$, and $l$ is comparable
to the vertex scale. This assertion transfers to approximating slices.
Here curvature radius in a limit is stopped by its exceptional
points: a ball containing such a point has unbounded approximating
curvature. On compact regular balls, smooth convergence verifies
the opposite inequality in the radius definition.

On the punctured terminal neighborhood the quantity $l^2|\Rm|$
tends uniformly below $c_*/2$ toward the puncture. Otherwise regular
points approaching it with $l^2|\Rm|\geq c_*/2$ yield disjoint
definite-energy balls, using curvature derivatives and noncollapse.
Their radii are at most a constant times distance to the puncture,
since concentration stops the curvature radius. Taking the
terminal limit first keeps these balls regular; their definite
energies contradict finite energy arbitrarily near the puncture.
Shrink $\mathcal U$ within this region. On every compact subset of
its punctured terminal regular part, $l^2|\Rm|<c_*$ eventually.
In particular the integrand in Equation~\eqref{geom:root-definition}
vanishes near its boundary. It is continuous on a fixed compact
interior set, so its maximum is attained and depends continuously
on time. Detection proves positivity. Every sequence of maximizing
points converges to the concentration point and has $l\to0$;
hence $a(t)\to0$.

At a maximizer, $l\geq a/\|\psi\|_\infty$ and
$l^2|\Rm|>c_*$. Its $l$-scale limit is therefore nonflat. If
$l/a\to\infty$, detection at another regular point of that same limit
would give a value of $l\psi$ bounded below by a positive multiple
of $l$, contradicting the definition of $a$. Thus $l\asymp a$ at
maximizers. Conversely detection on any listed nonflat limit bounds
its scale by $Ca$. The maximal list contains the maximizing
$a$-scale limit, and its unique outer ancestor cannot have larger
order than $a$. It is consequently the same entry. This is precisely
the outermost scale; a deepest bubble or the largest curvature may
occur at a much smaller scale.
\end{proof}

\subsection{Motion on short intervals}

The scale just defined must be compared at different times on the same
material manifold. We first use the genuine quotient tips to control
curvature radii along minimizing segments, then construct dense sets of
material points whose distance changes are small.
\begin{lemma}[Curvature radius along minimizing segments]\label{geom:geodesic-radius}
For any minimizing segment $\gamma:[0,D]\to M$ at a late time,
parametrized by arclength,
\[
 l(\gamma(s),t)\geq c\min(l_0,s,D-s),\qquad 0<s<D.
\]
Consequently, whenever $d_t(x,y)\leq Ch$ with $h$ small, its upper
and lower time derivatives satisfy
\begin{equation}\label{geom:distance-log}
 |\partial_t d_t(x,y)|\leq
 C'\left(1+\log^+(h/l(x,t))+\log^+(h/l(y,t))\right).
\end{equation}
Time derivatives can be interpreted almost everywhere or by barriers.
\end{lemma}

\begin{proof}
If the bound fails, choose minimizing segments and interior points
$z_i=\gamma_i(s_i)$ such that, with
\[
 r_i=l(z_i,t_i),\qquad L_i=\min(l_0,s_i,D_i-s_i),
 \qquad r_i/L_i\longrightarrow0,
\]
the scale $h_i=\sqrt{r_iL_i}$ satisfies
$h_i\to0$, $r_i/h_i\to0$, and $h_i\le L_i$.
Restrict to the centered subsegments of length $2h_i$ and rescale
by $h_i^{-1}$. Their limits are minimizing segments with endpoints
at distance one on both sides of the basepoint. Since $l(z_i,t_i)/h_i\to0$,
that basepoint is a genuine exceptional point of the slice limit.
Its tangent cone has link $S^3/\Gamma$ with $\Gamma$ nontrivial and free.
This link has diameter strictly less than $\pi$: if two orbits were
distance $\pi$ apart, every translate of one representative would
have to be the antipode of the other, forcing its orbit to be a
singleton, contrary to freeness. Blowing up the limiting segment at
this point therefore gives a minimizing segment through the cone
vertex with positive length on each side. Joining nearby points by
the shorter cone chord contradicts minimality.

For the derivative bound use first variation
$|\partial_t d|\leq\int_\gamma|\Ric|\,ds$ and
Equation~\eqref{eq:ricci-improvement}. Near $x$ the spatial
Lipschitz property also gives $l(\gamma(s))\geq l(x)-s$.
Combining this with the interior bound gives
$l(\gamma(s))\geq c(l(x)+s)$ on the first half of the segment,
up to the harmless cap, and the analogous expression on the other
half. Integration of their reciprocals gives
Equation~\eqref{geom:distance-log}.
\end{proof}

\begin{lemma}[Uniform short motion]\label{geom:slow-motion}
Let $h_i\to0$ and let $J_i$ be time intervals of length
$\delta_i=o(h_i)$ tending to $T$. Distances initially of order
$O(h_i)$ change by $o(h_i)$ throughout $J_i$. If a material point
has $l\geq c h_i$ at one time, it has $l\geq c' h_i$ throughout
(after extraction for a sequential assertion). Regular compact
charts at this scale persist, and their scaled metric coefficients,
with any fixed finite number of scaled derivatives, change by
$O(\delta_i/h_i)$ in fixed material coordinates.
\end{lemma}

\begin{proof}
Put $q_i=\delta_i/h_i\to0$ and, on material points, define
\[
 I_i(x)=\int_{J_i}\log^+(h_i/l(x,t))\,dt.
\]
The distribution formula and Equation~\eqref{geom:bad-volume} give
\[
 \int_M\log^+(h_i/l)\,d\mu_t
 =\int_0^\infty\Vol_t\{l<h_ie^{-v}\}\,dv\leq Ch_i^4.
\]
Volume forms are comparable across $J_i$, so Fubini gives
$\int I_i\leq C\delta_i h_i^4$. Discard material points with
$I_i>h_i\sqrt{q_i}$. Their volume at every time is at most
$Ch_i^4\sqrt{q_i}$. Lower ball volumes therefore make the retained
set $O(h_iq_i^{1/8})$-dense simultaneously at all times.
For two retained endpoints in a bounded rescaled distance range,
Equation~\eqref{geom:distance-log} integrates to
\[
 |d_t(x,y)-d_s(x,y)|
 \leq C(\delta_i+h_i\sqrt{q_i})=o(h_i).
\]
A first-crossing argument keeps initially bounded distance ranges
bounded. Its reverse version prevents initially remote retained
points from entering a fixed bounded range.

Choose finite increasingly dense labeled nets of retained material
points on bounded balls. Their pairwise distance matrices remain
unchanged in the limit, uniformly in time. Every temporal
subsequence thus has the same pointed metric limit under these
labels. Exceptional points in all such limits are intrinsic genuine
singularities, and each metrically regular compact set has smooth
regular convergence, by the exclusion of trivial-group concentration.

Now take a material point regular at one time, but whose minimum
$l/h_i$ over $J_i$ supposedly tends to zero. By reversing time
direction in the distance comparison if needed, stop at its first
crossing of $l/h_i=\eta_i$, where $\eta_i\downarrow0$ is chosen
above that attained minimum and sufficiently slowly that
$q_i\log(1/\eta_i)\to0$. Until the stopping time its integrated
endpoint logarithm is at most
$\delta_i\log(1/\eta_i)=o(h_i)$. Its distances to retained anchors
therefore stay unchanged to $o(h_i)$. At the crossing it is a
singular point in the common pointed limit with the same anchor
distance vector as its initially regular point, a contradiction.
This proves the asserted uniform positive lower radius.

After taking a subsequence, the other alternative for a tracked point
is $\sup_{J_i}l/h_i\to0$. Such a point initially near a good anchor
must stay near the same singular point. Choose a large anchor radius
different from the finitely many singular-point radii in the common
limit. Escape to that sphere would give a singular point there by
subsequential extraction, which is impossible. In the resulting
bounded ball choose finitely many good labels whose distance vectors
distinguish the singular points. At arbitrary intermediate times,
the persistently bad point's vector is uniformly close to one of
these finitely many distinct vectors; otherwise take a contradicting
temporal subsequence. Its vector is continuous in time, so it stays
close to the same vector throughout. Increasing the finite anchor
nets identifies all its limiting positions with the same tip.

These alternatives prove tracking for every point: any sequence
violating it has either a uniformly regular time subsequence, handled
by the first argument, or a persistently bad subsequence, handled by
the second. Two tracked points can be placed in a common bounded
good-anchor range, so their distance changes by $o(h_i)$.
Finally, on persisting regular charts,
$|\nabla^k\Ric|\leq C_k h_i^{-1-k}$. Integrating the metric and
connection evolution in the fixed initial chart gives
$O(\delta_i/h_i)$ change of the scaled metric and its derivatives.
The coordinate conversion is an induction on derivative order,
using the already controlled lower-order connection terms.
\end{proof}

\begin{lemma}[Time required for a fixed scale change]
\label{geom:scale-change-time}
Fix $u>0$. Let $s_i<t_i<T$ tend to $T$, where $t_i$ is the first time
after $s_i$ for which $|\log(a(t_i)/a(s_i))|=u$. Then
\[
 \frac{t_i-s_i}{a(s_i)^2}\longrightarrow\infty.
\]
\end{lemma}

\begin{proof}
Otherwise a subsequence of these intervals has duration $O(a^2)$,
where $a=a(s_i)$. Rescale lengths by $a$. Their duration is $o(a)$
in physical units, so
Lemma~\ref{geom:slow-motion} freezes the pointed root metric and its
regular curvature in the limit. The radius in the definition of
$a$ converges on regular parts to the radius computed in that
limit with its tips as stops. For the lower bound use smooth
convergence on a slightly smaller admissible ball, whose curvature
threshold has a strict margin. For the upper
bound a ball exceeding the limiting radius either contains a
curvature violation on a regular compact set or encounters a
genuine concentration tip; in either case it is inadmissible in
the approximations.

Maximizers at both times are regular on this scale by
Proposition~\ref{prop:root-scale}. They remain in a common bounded
root range: a nonflat patch at the first time persists at the other,
and uniqueness makes all comparable nonflat scales the same tree
entry. Thus the rescaled supremum in
Equation~\eqref{geom:root-definition} is computed on regular compact
regions in one common pointed limit and has the same value at both
times. A fixed-factor change is impossible.
\end{proof}

\begin{proposition}[Buffered scale-change intervals]\label{prop:buffered-intervals}
If a concentration point exists, there are parabolically rescaled
intervals $I=[-\theta,1+\theta]$, for one fixed $\theta>0$, with
physical squared length units $d_i\to0$, such that the root scale
throughout $I$ is comparable to a number $a_i\to0$ and changes by a
fixed factor between $0$ and $1$. Put $\epsilon_i=\sqrt{d_i}$.
In these units
\[
 |R|\leq C\epsilon_i^2,\qquad
 |\Ric|_{k,r}\leq C_k\epsilon_i/r
\]
at regularity scales $r$ in a bounded range. For $a_i\ll r\lesssim1$
the cluster has flat quotient annular limits with nontrivial group.
At any fixed time, allowable root centers differ by $O(a_i)$.
\end{proposition}

\begin{proof}
Fix $u>0$ and choose successive late times $t_j$ at which
$\log a$ first changes by $u$ in absolute value from its value at
$t_{j-1}$. Positivity, continuity, and $a(t)\to0$ ensure infinitely
many such times with $t_j\uparrow T$. Put
$d_j=t_{j+1}-t_j$ and $a_j=a(t_j)$.
Lemma~\ref{geom:scale-change-time} gives $d_j/a_j^2\to\infty$.
Also $a_j\geq a_0e^{-uj}$, so eventually
\[
 d_j\geq c e^{-2uj}.
\]
Choose $0<\theta<\min(1,e^{-2u})$. Arbitrarily late indices satisfy
$d_{j-1},d_{j+1}\geq\theta d_j$. Otherwise every sufficiently late
index has a neighbor smaller by a factor $\theta$. Starting so late
that its duration is smaller than all durations in the finite
excluded initial segment, follow such neighbors. This chain cannot
enter that initial segment and cannot reverse direction: reversal
would return to the immediately preceding, strictly larger term.
It must run rightward forever. It then gives
$d_{j+k}\leq\theta^k d_j$, contradicting the displayed exponential
lower bound because $\theta<e^{-2u}$.

Use these central steps as the unit time interval. Their neighboring
steps provide buffers of size $\theta$. The first-change definition
makes $a(t)$ comparable to $a_j$ on the central step and both
buffers, with constants at most fixed powers of $e^u$.
In length units $\sqrt{d_j}$ its value is
$a_i=a_j/\sqrt{d_j}\to0$. The central endpoint factor is $e^{u}$
or $e^{-u}$; pass to a subsequence if its sign matters.
The curvature estimates rescale to the claimed bounds since
$|R_{\rm new}|=d_j|R|$ and
$|\Ric|_{k,r,\rm new}\leq C_k\sqrt{d_j}/r$.
Intermediate annuli are flat by maximality of the bubble list.
Their tip is genuine, so their quotient group is nontrivial.
The center bound follows because all maximizing points represent
the unique outermost entry.
\end{proof}

\begin{figure}[htbp]
 \centering
 \input{figures/buffered-intervals}
 \caption{Parabolic normalization of a selected buffered interval.
Write $j=j_i$ for the selected indices and
$d_i=t_{j_i+1}-t_{j_i}$.  The neighboring intervals each have duration
at least $\theta d_i$; only the shortened buffers are drawn.
The root scale is comparable to $a_i$ throughout the normalized
interval, and its values at $0$ and $1$ differ by the fixed factor
$e^u$ or $e^{-u}$.  The diagram is schematic and does not assert
monotonicity of the root scale.}
 \label{fig:buffered-intervals}
\end{figure}

\begin{corollary}[Motion in interval units]\label{geom:rescaled-motion}
On the intervals of Proposition~\ref{prop:buffered-intervals},
geometry at length $h\lesssim1$ moves negligibly over windows
$\Delta t$ satisfying $\epsilon_i\Delta t/h\to0$.
Regular chart coefficients change by
$O(\epsilon_i\Delta t/h)$ with every fixed number of scaled
derivatives. At $h\asymp a_i$, old and new centers are $O(a_i)$
apart; at $a_i\ll h\lesssim1$ they are $o(h)$ apart.
\end{corollary}

\begin{proof}
The corresponding physical duration divided by physical length is
$d_i\Delta t/(\sqrt{d_i}h)=\epsilon_i\Delta t/h$, so
Lemma~\ref{geom:slow-motion} applies. At root scale a nonflat regular
patch persists, and uniqueness identifies the corresponding entry
at both times, giving bounded center displacement. At larger
intermediate scale the root is the intrinsic cone tip and its
location persists by the bad-point tracking argument. Another
concentration at distance comparable to $h$ in the same shrinking
terminal cluster would give two tips at their separation scale,
hence a nonflat ancestor larger than the outermost entry, a
contradiction. This proves the $o(h)$ displacement.
\end{proof}

\begin{proposition}[Uniform exterior geometry]
\label{geom:exterior-geometry}
Fix a sufficiently high finite derivative order $k_{\mathrm{geo}}$
and numbers $0<\mu<\sigma<1$. There are fixed constants
$N_{\mathrm{geom}},C,c_{\mathrm{geom}}>0$ and outer radii
$R_i^{\mathrm{out}}\to\infty$ in interval units, with
$\epsilon_iR_i^{\mathrm{out}}\to0$, with the following property.
At every time in the buffered interval the exterior region admits
one equivariant coordinate system whose annuli satisfy
\begin{equation}\label{geom:uniform-exterior}
 r^2|\Rm|_{k_{\mathrm{geo}},r}+|g-\delta|_{k_{\mathrm{geo}},r}
 \le C\Theta_i(r),\qquad
 \Theta_i(r)=(a_i/r)^\mu+\delta_i r^\mu,\qquad
 \delta_i=(R_i^{\mathrm{out}})^{-\mu}\longrightarrow0
\end{equation}
for $N_{\mathrm{geom}}a_i\le r\le c_{\mathrm{geom}}R_i^{\mathrm{out}}$.
The charts cover whole shortened distance bands with boundary margins;
their nontrivial quotient group is fixed after subsequence extraction.
Moreover $\epsilon_i=o(\delta_i)$. The constant $C$ is uniform in
time and radius and is unchanged when the lower bound is restricted
to $Na_i$ for any larger fixed $N$.
\end{proposition}

\begin{proof}
Choose actual outer radii $R_i^{\mathrm{out}}\to\infty$ in interval
units with $\epsilon_iR_i^{\mathrm{out}}\to0$, slowing their growth
if necessary. For arbitrary choices $t_i\in I$ and
$r_i/a_i\to\infty$ with $\epsilon_i r_i\to0$, the maximal-tree
and root-detection arguments force the annular limit to be a flat
quotient cone. Otherwise a nonflat limit at scale $r_i$ would, by
detection on a regular patch, give $a_i\geq c r_i$. This sequential
assertion implies that, for any fixed small cone-chart tolerance,
one fixed $N_{\mathrm{geom}}$ makes all widened annuli
$N_{\mathrm{geom}}a_i\leq r\leq R_i^{\mathrm{out}}$ sufficiently close to flat,
uniformly in $t\in I$ for large $i$. Failure would choose
$N_i\to\infty$ and violating times and radii, contradicting the
same assertion.

On these annuli $l\geq cr$, and hence
\[
 r^2|\Ric|_{k_{\mathrm{geo}}+2,r}\leq C\epsilon_i r
 \leq C\epsilon_iR_i^{\mathrm{out}}
             (r/R_i^{\mathrm{out}})^\sigma.
\]
Apply the ordinary two-boundary proof of Lemma~\ref{geom:neck}
with $r_-=N_{\mathrm{geom}}a_i$, $r_+=R_i^{\mathrm{out}}$, and the fixed exponent
$\mu$. Its constant is uniform in time: a violation chooses
maximizing pairs $(t_i,r_i)$; both boundary radii escape under
normalization, the annular metric tends to the flat cone, and the
divided Ricci forcing is at most $C/Q_i$. The harmonic-tensor and
Bianchi contradiction uses only $\mu<1$ and the flat cover,
independently of an extracted nonflat tree. This proves
Equation~\eqref{geom:uniform-exterior} on shortened
annuli, with all required derivatives. Fixed boundary factors are
included in $N_{\mathrm{geom}}$ and $c_{\mathrm{geom}}$.
Since $\epsilon_iR_i^{\mathrm{out}}\to0$ and $\mu<1$, one has
$\delta_i/\epsilon_i\to\infty$. Fixed wider collars merely
enlarge $N_{\mathrm{geom}}$ and $C$ once. These geometric choices
are fixed before the later analytical collar parameter $N$ is chosen.
Increasing $N\geq N_{\mathrm{geom}}$ only restricts this existing
estimate to $r\geq Na_i$ and preserves its constant; it does not
reconstruct the neck estimate with a new constant depending on $N$.
At scale one the regular charts persist
through the entire buffered interval by
Corollary~\ref{geom:rescaled-motion}; their quotient group is
therefore fixed after subsequence extraction. Widened overlapping
annuli identify that group and preserve the distance-band margins
in the chart construction at each time. This proves the exterior
estimate uniformly in all times and radii. Only the joining-end
exponents of individual extracted trees elsewhere in the argument
may be decreased after extraction.
\end{proof}
==== END Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//04-geometry.tex ====

==== BEGIN Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//05-parabolic.tex ====
\section{Concentration estimates for the Ricci equation}
\label{sec:parabolic}

We work on the sequence of buffered, parabolically rescaled intervals supplied
by Proposition~\ref{prop:buffered-intervals}.  Write
\[
 I=[-\theta,1+\theta],\qquad \epsilon=\sqrt d\longrightarrow0.
\]
All distances, curvatures, and time variables in this Section refer to these
units.  The number $a\to0$ is constant on each interval and comparable to the
outermost scale at every time.  Sequence indices will usually be suppressed.
Choose a center $x_t$ of the isolated cluster and a nonnegative
$\eta\in C_c^\infty(\operatorname{int}I)$ with $\inf_{[0,1]}\eta>0$.  Set
\begin{equation}
 G=\eta\Ric,\qquad
 K^2=a^{-4}\int_I\int_{B_t(x_t,N_1a)}|G|^2\,d\mu_{g_t}\,dt.
 \label{eq:root-energy}
\end{equation}
The fixed large number $N_1$ will be chosen after the annular parameters below.
All constants may depend on $\eta$.

These integrals can use measurable choices of exact maximizing centers in
Proposition~\ref{prop:root-scale}.  Indeed their nonempty compact sets, in a
fixed closed neighborhood contained in $\mathcal U$, are upper
semicontinuous in time.  To obtain a selection, successively choose the first
member of a finite ordered cover by closed balls of mesh tending to zero that
meets the previously retained compact set, and intersect with that ball.
The resulting compact sets are nested, have diameter tending to zero, and
have a unique common point.  Intersection tests with fixed compact sets are
measurable, so the limiting point is measurable.  All estimates below hold
for every such choice.

\subsection{Geometric inputs and local estimates}
\label{para:inputs}

We specify the consequences of Section~\ref{sec:geometry} that enter the
argument.  They are sequential statements: they apply to arbitrary choices
of times, radii, and windows after passage to a subsequence.
\begin{enumerate}[label=(\roman*)]
\item If $h$ is bounded above and $\epsilon\Delta t/h\to0$, the geometry
at length $h$ moves negligibly on a window of duration $\Delta t$.
On regular charts with $l\ge c h$, its coefficients in fixed material
coordinates change by $O(\epsilon\Delta t/h)$, with any fixed number of
spatial derivatives measured at scale $h$.
\item For $h\asymp a$ on these windows, the old and new centers have
distance $O(a)$.  For $a/h\to0$, they have distance $o(h)$.
For the latter assertion the old cone tip persists as an intrinsic
singularity.  If a concentration representing it were separated from the
current center by a definite fraction of $h$, the limit at their separation
scale would have two distinct genuine exceptional points.  The tree
construction would then give a nonflat ancestor larger than $a$,
contrary to Proposition~\ref{prop:root-scale}.  The cluster itself is fixed
by convergence of the original distances to the isolated point of $Z$.
\item For sufficiently large fixed $C$ and every fixed $C'>0$, distance
annuli with $Ca\le r\le C'$ admit the quotient charts of
Equation~\eqref{geom:uniform-exterior}, with error, through any prescribed finite
order, bounded by a constant times
\begin{equation}
 \Theta(r)=(a/r)^\mu+\delta_i r^\mu,
 \qquad \delta_i\longrightarrow0,
 \qquad \epsilon=o(\delta_i).
 \label{para:theta}
\end{equation}
We decrease $\mu$ when necessary and assume $0<\mu<1$.
The charts cover whole shortened distance annuli, their radial positioning
has arbitrarily small relative error when $C$ is sufficiently large, and
their finite deck groups are nontrivial.
\end{enumerate}
The quantitative estimate in Item (iii) is
Equation~\eqref{geom:uniform-exterior}, with the prescribed finite derivative
order fixed in advance and a fixed positive exterior exponent and constants
uniform in time and radius.  It covers every fixed upper radius $C'$ for
late sequence indices, since $R_i^{\mathrm{out}}\to\infty$.
Whole-band coverage and radial positioning are part of the quotient-chart
construction there.  Positioning is only required with small relative
error, not at the rate $\Theta$.

For a tensor $U$, write
\[
 |U|_{j,r}=\sum_{k=0}^j r^k|\nabla^kU|.
\]
In material coordinates the lower-index Ricci equation and its constraint
are
\begin{align}
 (\partial_t-\Delta_g)\Ric_{ij}
   &=2R_{ikjl}\Ric^{kl}-2\Ric_i{}^k\Ric_{kj},
   \label{para:ricci-equation}\\
 (\partial_t-\Delta_g)G_{ij}
   &=2R_{ikjl}G^{kl}-2\Ric_i{}^kG_{kj}+\eta'\Ric_{ij},
   \label{para:cutoff-equation}\\
 \nabla^i\bigl(G_{ij}-\tfrac12 g_{ij}\tr_gG\bigr)&=0.
   \label{para:bianchi}
\end{align}
The last identity uses that $\eta$ is spatially constant.  On a regular
patch of radius $r$, use time $\tau=(t-t_0)/r^2$ and tensor components whose
size records the original $I$-unit magnitude.  More precisely, for the
spatial chart $\Phi_r$ set
\[
 g_r=r^{-2}\Phi_r^*g,\qquad G_r=r^{-2}\Phi_r^*G,\qquad
 \Ric_r=r^{-2}\Phi_r^*\Ric.
\]
Thus $|G_r|_{g_r}=|G|_g\circ\Phi_r$; the symbol $\Ric_r$ records
the original magnitude and is $r^{-2}\Ric(g_r)$.
We suppress these pullbacks in norm estimates.
Equation~\eqref{eq:ricci-improvement}
gives
\begin{equation}
 |\Ric|_{j,r}\le C_j\epsilon/r,
 \qquad |r^2\eta'\Ric|_{j,r}\le C_j\epsilon r.
 \label{para:source-scale}
\end{equation}
In particular the cutoff source requires no division by $\eta$.

\begin{lemma}[Local smoothing with time $L^p$ input]
\label{para:local-smoothing}
Consider a linear system on a regular parabolic patch, normalized to spatial
radius and temporal duration of order one, with scalar uniformly elliptic
principal part.  Suppose its spatial coefficient derivatives through a
sufficiently high fixed order are bounded.  For nested cylinders
$Q'\Subset Q$ whose evaluation times have a positive past margin, a
solution $U$ with source $F$ satisfies, for $1<p\le2$,
\begin{equation}
 \sup_{Q'}\sum_{k=0}^j|D^kU|
 \le C\left[
 \left(\frac1{|J|}\int_J\|U(\cdot,\tau)\|_{L^2(\Omega)}^p\,d\tau\right)^{1/p}
 +\left(\frac1{|J|}\int_J\|F(\cdot,\tau)\|_{H^{j+3}(\Omega)}^p\,d\tau\right)^{1/p}
 \right],
 \label{para:local-mixed}
\end{equation}
after enlarging $\Omega\times J$ inside $Q$ if necessary.  The first input
can instead be the normalized spacetime $L^p$ norm of $U$.  Constants
depend on the coefficient bounds and margins.  Corresponding time
$L^p$ estimates over longer windows follow by integrating sliding
cylinders of unit duration.
\end{lemma}

\begin{proof}
We give the energy argument, including the change of time exponent.
With a spatial cutoff $\zeta$ and a time cutoff that vanishes on the
past boundary, multiply the equation by $\zeta^2U$ and integrate in
space.  Uniform ellipticity, integration by parts, and Young's inequality
give, on two nested cylinders,
\[
 \sup_\tau\|\zeta U(\tau)\|_2^2
 +\int\|\zeta DU\|_2^2\,d\tau
 \le C_m\int_Q|U|^2
   +C\left(\int_J\|F(\tau)\|_2\,d\tau\right)^2.
\]
Here $C_m$ is bounded by a fixed power of the inverse spatial and temporal
margins.  For the source, pair its time $L^1$ norm with the left-hand
time supremum and absorb half the latter; no time $L^2$ bound on $F$
is used.  First-order and zeroth-order coefficients produce terms
bounded by a small multiple of the gradient energy plus $C\int|U|^2$.

Differentiate spatially.  The top commutator term has the form
$Da\,D^{k+1}U$, paired with $D^kU$, and is absorbed into gradient
energy.  All remaining commutators contain derivatives of $U$ of order
at most $k$ with bounded coefficients.  Applying the preceding estimate
successively on smaller cylinders controls
$\sup_\tau\|U\|_{H^m}$ and $\|U\|_{L^2_\tau H^{m+1}}$ by
$\|U\|_{L^2(Q)}+\|F\|_{L^1_\tau H^m}$.  At each step the spacetime
norm of the next derivative was supplied by the previous gradient
energy.  Taking $m=j+3$ and using the four-dimensional Sobolev embedding
$H^{j+3}\hookrightarrow C^j$ proves the assertion with time exponent
two for $U$ and exponent one for $F$.

For completeness, let $X(\rho)$ denote the resulting spatial and temporal
supremum on a member of a nested family of cylinders.  For
$\rho<\sigma$, interpolate the time $L^2$ spatial $L^2$ norm against
the larger supremum to obtain
\[
 X(\rho)\le C(\sigma-\rho)^{-M}
   X(\sigma)^{1-p/2}\|U\|_{L^p_\tau L^2_x}^{p/2}
   +C(\sigma-\rho)^{-M}\|F\|_{L^p_\tau H^{j+3}_x}.
\]
Normalized and unnormalized time norms are comparable on these fixed
cylinders.  Young's inequality makes the first right-hand term at most
$\kappa X(\sigma)+C_\kappa(\sigma-\rho)^{-M'}\|U\|_{L^p_\tau L^2_x}$.
Take margins proportional to $2^{-n}$ and $\kappa<2^{-M'-1}$.
Iteration sums a convergent geometric series; the terminal term tends
to zero since the smooth solution has finite supremum on the enclosing
compact cylinder.  This proves Equation~\eqref{para:local-mixed}.
Interpolating $\|U\|_{L^2(Q)}$ against the larger spacetime supremum
in exactly this displayed Young inequality, with $\|U\|_{L^p(Q)}$
in place of the mixed norm, proves the spacetime variant.
Finally integrate the $p$th power of the local estimate over the
evaluation time.  Fubini's theorem bounds the integral of the sliding
averages by a constant times the input integral on the enlarged window.
\end{proof}

\subsection{The spherical projection and the missing radial mode}
\label{para:radial-subsection}

Lift a quotient annulus to its Euclidean cover and record tensor
components in a Cartesian frame.  Let $P$ subtract the componentwise
spherical average at every radius.  This operation preserves
equivariance and commutes with the componentwise flat heat operator.

\begin{lemma}[A radial inequality for the oscillation]
\label{para:radial-lemma}
Suppose $PZ=Z$, $Z$ is compactly supported in time, and
$E=(\partial_\tau-\Delta_\delta)Z$.  For $1<p\le2$, set
\[
 z(s)=\left\|\,\|Z(s,\cdot,\tau)\|_{L^2(S^3)}\,\right\|_{L^p_\tau}.
\]
With either ordinary or constant-normalized time measure,
\begin{equation}
 \mathcal Lz:=-z''-\frac3s z'+\frac3{s^2}z
 \le\left\|\,\|E(s,\cdot,\tau)\|_{L^2(S^3)}\,\right\|_{L^p_\tau}.
 \label{para:radial-inequality}
\end{equation}
For $p=2$ the same conclusion holds for a Hilbert direct sum of fields
in different charts or time windows with the same radial variable.
The homogeneous comparison powers are $s$ and $s^{-3}$.
\end{lemma}

\begin{proof}
Put $w(s,\tau)=\|Z(s,\cdot,\tau)\|_2$.  Where $w>0$, pair the
equation with $Z/w$.  Convexity of the Hilbert norm gives
$\langle Z,Z_{ss}\rangle/w\le w_{ss}$, and the zero spherical
mean gives
$\|\nabla_{S^3}Z\|_2^2\ge3\|Z\|_2^2$.  Consequently
\[
 w_\tau-w_{ss}-3s^{-1}w_s+3s^{-2}w\le\|E\|_2.
\]
Multiply by $w^{p-1}$ and integrate in time.  The time term vanishes
by compact support.  If $z=(\int w^p)^{1/p}$, then
\[
 \int w^{p-1}w_s=z^{p-1}z',\qquad
 \int w^{p-1}w_{ss}\le z^{p-1}z''.
\]
For the second inequality, differentiate the first identity and use
$|z'|^2\le z^{2-p}\int w^{p-2}w_s^2$, which is Cauchy's inequality.
H\"older's inequality bounds the integrated source by
$z^{p-1}\|\|E\|_2\|_p$.  Division proves
Equation~\eqref{para:radial-inequality}.  At zeros, work on a finite
time interval containing the support and replace each norm by its
square plus a positive constant under a square root.  The regularized
time boundary values agree, so their derivative integral still
vanishes.  Perform the calculation and let the constant tend to zero
on compact subannuli; the norm and spectral-gap regularization errors
tend to zero there.  The resulting inequalities hold distributionally.
The direct-sum assertion uses the same Hilbert norm computation.
Finally
\[
 \mathcal L(s^\beta)=[3-\beta(\beta+2)]s^{\beta-2},
\]
whose two zero exponents are $1$ and $-3$.
\end{proof}

\begin{lemma}[Control of the spherical mean derivatives]
\label{para:mean-lemma}
Let $T$ be a spatially constant multiple of $\Ric$ on an annulus whose
metric has scaled $C^{j+1}$ distance at most $e$ from $\delta$.
On a fixed-width shell of radius $s$ in its Cartesian coordinates,
\begin{equation}
 \sum_{k=1}^j s^k|\nabla^kT|
 \le C_j\left(\sum_{k=1}^j s^k\sup|D^kPT|
              +e\sup|T|_{j,s}\right),
 \label{para:mean-derivatives}
\end{equation}
where the suprema on the right use a fixed enlargement of the shell.
\end{lemma}

\begin{proof}
Take the flat trace reversal
$S=T-\tfrac12\delta\tr_\delta T$ and write
$S=\overline S(s)+PS$.  Comparing the true divergence constraint
with flat divergence gives
\[
 \partial_iS_{ij}=\mathcal E_j,\qquad
 s^{k+1}|D^k\mathcal E|
 \le C_j e\,\sup|T|_{k+1,s}\quad(0\le k<j).
\]
This follows directly by expanding $g^{ij}-\delta^{ij}$,
$\Gamma(g)$, and the difference between the two trace reversals;
each term contains either a metric error times $DT$ or a metric
derivative times $T$.
For every unit radial vector $n$, the equation at $x=sn$ reads
\[
 \overline S'_{ij}(s)n_i
    =\mathcal E_j(sn)-\partial_i(PS)_{ij}(sn).
\]
Taking $n$ to be each coordinate unit vector bounds every entry
of $\overline S'$.  Radially differentiating this identity at fixed
$n$ bounds the successive derivatives of $\overline S$ by the
displayed error derivatives and derivatives of $PS$.  Cartesian
derivatives of a radial matrix are combinations of these radial
derivatives with powers of $s^{-1}$, so the scaled bounds follow.
In dimension four the trace-reversal map is an involution, since
$\tr_\delta S=-\tr_\delta T$; it is therefore invertible.
Conversion from Cartesian to covariant derivatives adds only the
stated metric-error terms.
\end{proof}

The two Lemmas explain why a degree-zero critical mode cannot obstruct
the estimates.  Although $s^{-2}A$ is componentwise harmonic for a
constant symmetric matrix $A$, its flat trace-reversed divergence is
\[
 -2s^{-3}\bigl(A-\tfrac12\delta\tr_\delta A\bigr)n.
\]
It vanishes for every $n$ only if $A=0$.  Spatially constant tensors
can survive, but have no positive spatial derivatives; their values
will be fixed by an outer boundary estimate.
The trace-reversal calculation is closely related to the improvement
of ALE metric decay in the proof of Deruelle--Kr\"oncke's
Theorem~2.7 in the arXiv version \cite{DeruelleKroncke2021}.
Their field is a stationary
metric perturbation in a chosen gauge. Here the contracted Bianchi
identity constrains the evolving Ricci tensor itself, and the two
lemmas provide the quantitative derivative bounds needed below.

\subsection{The exterior estimate}
\label{para:exterior-subsection}

We apply the missing-mode argument on successive annuli. The goal is
decay faster than the dimensional $r^{-2}$ rate for the part of Ricci
controlled by the root energy. The spatially constant part is instead
controlled from a fixed outer annulus; separating these two contributions
prevents a loss proportional to the number of shells.
Take dyadic radii based on $a$, and define
\[
 \mathcal A_t(r)=\{y:r/2\le d_t(x_t,y)\le2r\},\quad
 A(r)=\left\|\sup_{\mathcal A_t(r)}|G|\right\|_{L^2(I)},\quad
 V(r)=\left\|\sup_{\mathcal A_t(r)}
                   \sum_{j=1}^m r^j|\nabla^jG|\right\|_{L^2(I)}.
\]
Here $m$ is a fixed sufficiently high order, for example $60$.
We may replace widths by fixed neighboring dyadic widths; a star
will indicate a maximum over a fixed number of such neighbors.

\begin{proposition}[Exterior concentration bound]
\label{prop:exterior-ricci}
There are fixed $N_0\gg1$, $r_0>0$, and
$0<\alpha<\min(\mu,1)$, and a choice of $N_1$ in
Equation~\eqref{eq:root-energy}, such that
\begin{align}
 V(r)&\le C\left[(a/r)^{2+\alpha}K
             +\widetilde\delta_i\bigl((a/r)^\alpha+r^\alpha\bigr)\right],
 \label{para:exterior-v}\\
 A(r)&\le C\left[(a/r)^{2+\alpha}K+\widetilde\delta_i\right]
 \label{para:exterior-a}
\end{align}
for $N_0a\le r\le r_0$, where
$\widetilde\delta_i\to0$ and $\widetilde\delta_i\ge\epsilon$.
The constants are independent of $i$ and $r$.
\end{proposition}

\begin{proof}
\emph{Fixed charts and operator errors.}
Fix a large dyadic $H$, to be chosen first, and then a large $T_0$.
Around an intermediate radius $r$ use annuli
$1/(cH)\le s\le cH$ in $r$-units and time windows of length
$T_0r^2$, where $c$ is a fixed ample constant.  Keep each annular
chart fixed in material points.  The motion bound at the smallest
comparable radius is
\[
 C\epsilon T_0Hr\le C_{H,T_0}\Theta(r),
\]
by $\mu<1$ and $\epsilon=o(\delta_i)$.  Thus the metric error through
all required spatial orders is $C_{H,T_0}\Theta(r)$ throughout the
window.  The center and distance comparisons in
Section~\ref{para:inputs} ensure coverage of the same shortened
distance annuli at each time.  Uniform coverage follows by the
sequential criterion, first increasing $N_0$ after $H,T_0$ and then
taking late indices.  Choose $r_0$ small enough that the enlarged
windows near $\supp\eta$ remain in $I$.

Let $\mathcal R(r)$ be the dyadic radii in
$[r/(c'H),c'Hr]$, for another fixed ample $c'$.  Applying
Lemma~\ref{para:local-smoothing} on bounded-width regular patches
and integrating sliding windows bounds true-field derivatives
through any prescribed fixed order by
\begin{equation}
 C_{H,T_0}\left(\max_{\rho\in\mathcal R(r)}A(\rho)+\epsilon r\right)
 \quad\hbox{in }L^2(I).
 \label{para:true-field-bound}
\end{equation}
Regions disjoint from the time support contribute zero.

\emph{Projection and the time cutoff.}
Cover $\supp\eta$ by smaller windows and choose smooth cutoffs
$\chi_\ell$ equal to one there, supported on the respective larger
windows, with uniformly bounded overlap and
$|\partial_t\chi_\ell|\le C/(T_0r^2)$.
Their smaller windows have an additional time margin of order $r^2$.
In the Hilbert direct sum of Lemma~\ref{para:radial-lemma}, use
$Z_\ell=\chi_\ell PG$ in each chart.  Time norms can be measured
in $I$-time: its constant factor relative to $r^2$-time multiplies
both sides of the radial inequality.
Let $\mathcal B_r$ be the dimensionless zeroth-order operator obtained
from the first two terms of Equation~\eqref{para:cutoff-equation}:
\[
 \mathcal B_rU=2\Rm(g_r)(U)-2\Ric(g_r)\circ U.
\]
Its coefficients and scaled spatial derivatives are bounded by
$C_{H,T_0}\Theta(r)$ on the annulus.  With the explicit rescaled
notation above, the flat-equation forcing is the sum of
\[
 \chi_\ell P\bigl[(\Delta_{g_r}-\Delta_\delta)G_r
       +\mathcal B_rG_r+r^2\eta'\Ric_r\bigr]
 \quad\hbox{and}\quad (\partial_\tau\chi_\ell)PG_r,
\]
where the first operator difference includes the connection terms
appropriate to tensor components.
Equation~\eqref{para:true-field-bound} and flatness bound the norm
of the first part, uniformly for $H^{-1}\le s\le H$, by
$C_{H,T_0}(\Theta(r)\max_{\mathcal R(r)}A+\epsilon r)$.
For the second part use the angular Poincar\'e estimate at its
\emph{own} radius:
\[
 \|PG(s)\|_{L^2(S^3)}
 \le Csr\sup|\nabla G|+C_{H,T_0}\Theta(r)\sup|G|.
\]
Bounded overlap is independent of $T_0$.  The cutoff contribution
therefore has the separate bound
\begin{equation}
 \frac{C_H}{T_0}\max_{\rho\in\mathcal R(r)}V(\rho)
   +C_{H,T_0}\Theta(r)\max_{\rho\in\mathcal R(r)}A(\rho).
 \label{para:time-cutoff-term}
\end{equation}
The leading coefficient $C_H/T_0$ contains no chart-persistence
constant.  In particular it does not multiply the uncontrolled
spherical mean of $G$.

At $s=H^{-1}$ and $s=H$ the same angular inequality gives
boundary values at most $C V_*(r/H)$ and $C V_*(Hr)$,
respectively, plus $C_{H,T_0}\Theta\max A$.
The first constant $C$ is independent of $H,T_0$, since each
angular derivative is normalized by the boundary's own radius.
Let $E_0$ denote the sum of the forcing bounds just obtained.
The nonnegative comparison function
\[
 C V_*(r/H)H^{-3}s^{-3}
  +C V_*(Hr)H^{-1}s+\tfrac13H^2E_0
\]
dominates the boundary data after adding the boundary metric errors
to $E_0$, and its $\mathcal L$ is at least $E_0$ on
$[H^{-1},H]$.  The one-dimensional maximum principle, valid also
for the weak inequality, bounds $z$ by this function.  On any fixed
middle annulus this gives the two leading terms with factors
$H^{-3}$ and $H^{-1}$; all inhomogeneous constants may depend on $H$.

Apply flat local smoothing on fixed-width middle patches.  The extra
past margin ensures that each entire unit-size smoothing cylinder lies
where $\chi_\ell\equiv1$.  Thus the equation differentiated in that
cylinder contains only the true operator errors and the $\eta'$ source;
it contains no $\chi_\ell'PG$ term.  Integrating $z^2$ against $s^3\,ds$ on a slightly
larger fixed middle annulus supplies its spacetime input.  Integrate
the resulting local estimates over the smaller time windows and sum
their squares.  The time margins and spatial margins here are of
unit order, so the leading constants are independent of $H,T_0$.
Higher derivatives of the forcing are bounded by
Equation~\eqref{para:true-field-bound} at higher fixed orders.
Finally Lemma~\ref{para:mean-lemma} supplies the positive derivatives
of the spherical mean.  We have proved
\begin{equation}
 \begin{split}
 V(r)\le{}&C\bigl[H^{-3}V_*(r/H)+H^{-1}V_*(Hr)\bigr]
       +\frac{C_H}{T_0}\max_{\rho\in\mathcal R(r)}V(\rho)\\
       &+C_{H,T_0}\left[\Theta(r)
                     \max_{\rho\in\mathcal R(r)}A(\rho)+\epsilon r\right].
 \end{split}
 \label{para:exterior-recurrence}
\end{equation}
It is used only when its enlarged widths stay in the prescribed range.

\emph{Magnitude and boundary estimates.}
At each time join a point on a shell of radius $\rho$ to the next
outgoing shell by a path of length $C\rho$, and join points within
that shell at the same cost.  Connected quotient annuli and whole-band
coverage provide these paths.  Integrating the covariant derivative
of $G$ along the paths, or the derivative of its norm, gives a
pointwise-in-time telescoping bound.  Minkowski's inequality gives
\begin{equation}
 A(r)\le C\sum_{\substack{\rho\text{ dyadic}\\r\le\rho\le r_0}}
                 V(\rho)+C\epsilon.
 \label{para:magnitude-sum}
\end{equation}
A fixed number of neighbors below $r$ may be included to accommodate
the chosen widths.  The outer boundary annuli have $A+V\le
C_{H,r_0}\epsilon$ by Equation~\eqref{para:source-scale}.
The inner boundary annuli have radii comparable to $a$ with fixed,
possibly large, factors.  Choose $N_1$ to include all enlarged
regular patches there at their respective times; this uses the
$O(a)$ center-motion bound.  Local smoothing, followed by integration
of sliding windows, yields
\begin{equation}
 A(r)+V(r)\le C\bigl[(a/r)^2K+\epsilon r\bigr]
 \quad\hbox{on the inner boundary range}.
 \label{para:inner-boundary}
\end{equation}
The factor $r^{-2}$ is the square root of normalized four-dimensional
volume.  Time integration cancels the sliding-window time average,
so it introduces no additional negative power of $r$.

\emph{Closing the recurrence, in the required order.}
Fix $0<\alpha<\min(\mu,1)$, and take
$\widetilde\delta_i\ge\epsilon$ positive and tending to zero
sufficiently slowly.  Put
\[
 W_K(r)=(a/r)^{2+\alpha}K,\quad
 W_0(r)=\widetilde\delta_i\bigl[(a/r)^\alpha+r^\alpha\bigr],\quad
 M=\max_r\frac{V(r)}{W_K(r)+W_0(r)}.
\]
For each index this maximum over finitely many radii is finite.
The geometric series in Equation~\eqref{para:magnitude-sum} imply
\begin{equation}
 A(r)\le C M\bigl[W_K(r)+\widetilde\delta_i\bigr]+C\epsilon.
 \label{para:sum-weight}
\end{equation}
Indeed the three sums involve the powers $-2-\alpha$, $-\alpha$,
and $\alpha$, and are bounded respectively by their initial value,
a constant, and a constant.  Thus no number-of-shells factor occurs.

For each power $r^\beta$ in $W_K+W_0$,
$-2-\alpha\le\beta\le\alpha$.  The two leading terms in
Equation~\eqref{para:exterior-recurrence}, divided by this weight,
are bounded by
\[
 C\bigl(H^{-3-\beta}+H^{-1+\beta}\bigr)M
 \le C' H^{-1+\alpha}M.
\]
Fixed neighboring widths change only $C'$.  Choose $H$ so this
is at most $M/8$.  For this fixed $H$ there is a finite constant
$L_H$ with $\max_{\mathcal R(r)}W/W(r)\le L_H$; choose $T_0$
so $(C_H/T_0)L_H\le1/8$.

For the metric terms in Equation~\eqref{para:sum-weight}, uniform
smallness of $\Theta$ controls the $W_K$ part.  For the constant
$\widetilde\delta_i$ part use the sharper inequality
\begin{equation}
 \frac{\Theta(r)}{(a/r)^\alpha+r^\alpha}
 \le (a/r)^{\mu-\alpha}+\delta_i r^{\mu-\alpha}.
 \label{para:metric-ratio}
\end{equation}
Now increase $N_0$ to make the first term uniformly small on all
needed enlarged radii.  With $r_0$ already chosen for the buffers,
fix $N_1$ large enough for Equation~\eqref{para:inner-boundary}.
Finally take the sequence index sufficiently late to make the second
term small and ensure the stated chart coverage.  This absorbs the
metric contribution into at most $M/4$.  Thus the smallness choices
are made in the order $H$, $T_0$, $N_0$, and the late sequence index;
all constants may depend on the parameters already fixed.
The remaining source is bounded relative to $W_0$ because
$\epsilon r\le\widetilde\delta_i r^\alpha$ for $r<1$.
The two boundary ranges have bounded ratios by
Equation~\eqref{para:inner-boundary} and the outer $O(\epsilon)$
bound.  Their constants may depend on the fixed parameters.
Consequently $M\le C+M/2$.  This proves
Equation~\eqref{para:exterior-v}, and
Equation~\eqref{para:sum-weight} proves Equation~\eqref{para:exterior-a}.
\end{proof}

\subsection{The maximal function controlling the root}
\label{para:maximal-subsection}

Fix a sufficiently large $D_0$ and consider the tree pieces inside
$B_t(x_t,D_0a)$.  Choose an ample fixed $D_1>D_0$, enlarging it to
include the short-time comparison patches.  Extend functions of time
by zero outside $I$, fix $1<p<2$, and set
\begin{equation}
 f(t)=a^{-2}\|G(\cdot,t)\|_{L^2(B_t(x_t,D_1a))},\qquad
 H_*(t)=\bigl(\mathcal M(f^p)(t)\bigr)^{1/p},
 \label{para:maximal-definition}
\end{equation}
where $\mathcal M$ is the centered Hardy--Littlewood maximal operator
on the time line \cite{HardyLittlewood1930}. We include its exact
boundedness argument at the exponent needed below.

\begin{lemma}
\label{para:maximal-lemma}
For every fixed $D_1$,
\begin{equation}
 \|H_*\|_{L^2(\R)}\le C_{D_1,p}(K+o_i(1)).
 \label{eq:maximal-ricci}
\end{equation}
\end{lemma}

\begin{proof}
The energy ball bounds the corresponding part of $\|f\|_2$ by
$K$.  Its complement in the larger ball uses finitely many dyadic
shells, with number and radius factors depending only on $D_1,N_1$.
Volume upper bounds and Proposition~\ref{prop:exterior-ricci} bound
each shell's contribution by
\[
 C(r/a)^2A(r)\le C(a/r)^\alpha K
                       +C(r/a)^2\widetilde\delta_i.
\]
Hence $\|f\|_2\le C_{D_1}(K+o_i(1))$.

We recall the maximal estimate at the exact exponent needed here.
The interval covering argument chooses disjoint witnessing intervals
whose triples cover a compact subset of a maximal level set.  It
gives $|\{\mathcal Mv>\lambda\}|\le C\|v\|_1/\lambda$.
Apply this to $v\mathbf1_{\{v>\lambda/2\}}$, since the discarded
part has every average at most $\lambda/2$.  For $q>1$, integrating
the distribution function then gives
\[
 \begin{split}
 \|\mathcal Mv\|_q^q
 &\le Cq\int_0^\infty\lambda^{q-2}
                     \int_{\{v>\lambda/2\}}v(t)\,dt\,d\lambda\\
 &\le C_q\int_\R v^q.
 \end{split}
\]
Take $v=f^p$ and $q=2/p>1$.  Thus
$\|H_*\|_2\le C_p\|f\|_2$, proving the claim.
\end{proof}

\subsection{A weighted estimate at a single time}
\label{para:interior-subsection}

The exterior estimate controls the collar error in time-integrated
norms. The static input also needs a bound at a single time on every
smaller descendant and joining neck. The maximal function just defined
provides the time control; we now obtain the additional exponential
weight along a deep neck.
\begin{proposition}[Interior estimate with a neck weight]
\label{prop:interior-ricci}
Take any sequence of times with $\eta(t)\ne0$ and extract a tree as in
Theorem~\ref{thm:degeneration}.  On its true compact and half-neck
pieces in $B_t(x_t,D_0a)$, let $r$ be the physical length of the local
regular patch, in $I$-units.  For every fixed required derivative order
$j$ there are $C,\nu>0$ such that
\begin{equation}
 (r/a)^2|G|_{j,r}(t)
       \le C\bigl(H_*(t)+\epsilon\bigr)e^{-\nu D}.
 \label{eq:interior-weighted}
\end{equation}
Here $D=0$ on compact pieces, including truncated outer compact
regions.  On a joining neck $D$, up to bounded shifts, is the minimum
of the logarithmic distances from the two ends.  It is the half-neck
coordinate used in Section~\ref{sec:static-input}.  Constants may depend
on the extracted tree, and the exponent may be decreased.
\end{proposition}

\begin{proof}
For a regular patch of radius $r\lesssim a$, apply
Lemma~\ref{para:local-smoothing} over a past window of length $cr^2$.
The patch lies in the larger ball defining $f$ at its respective
times, by Section~\ref{para:inputs}; shorten its fixed size if needed.
Its normalized spatial $L^2$ input, after multiplying the field by
$(r/a)^2$, is bounded by $Cf$.  The centered maximal average at the
specified time controls its time $L^p$ average.  The source is bounded
by $C\epsilon r(r/a)^2\le C\epsilon$ for late indices.  We obtain
Equation~\eqref{eq:interior-weighted} without its exponential factor.
This proves the compact case and every bounded-depth neck case.

It remains to gain decay as $D\to\infty$.  On such a neck
$r\le Ca e^{-D}$.  Choose $\lambda>0$ sufficiently small and put
\begin{equation}
 h=e^{-4\lambda D},\qquad B=e^{\lambda D}.
 \label{para:neck-window}
\end{equation}
Use the annulus $h\le s\le B$ in $r$-units, with fixed-factor
enlargements, and a time window of length $CB^2r^2$ centered at
the specified time.  By Equation~\eqref{eq:neck-decay}, the initial
spatial error through every required order is bounded by
$Ce^{-\mu(1-4\lambda)D}$.  The worst motion error occurs at the
innermost radius and is at most
\[
 C\epsilon B^2r/h
     \le C\epsilon a e^{-(1-6\lambda)D}.
\]
Choose $\lambda$ so small that
$\min\{\mu(1-4\lambda),1-6\lambda,3-3\lambda\}>3\lambda$.
After decreasing an exponent, the fixed material coordinates
therefore satisfy a uniform scaled error bound
\begin{equation}
 e_D=Ce^{-\delta D},\qquad \delta>3\lambda.
 \label{para:deep-metric-error}
\end{equation}
Regular flat annular limits make the local persistence constants
uniform across these shells.  The entire window stays in $I$,
since $Br\le Ca e^{-(1-\lambda)D}$, and the enlarged spatial patches
remain in the $D_1a$ ball by the same motion comparisons.

Write $T_1=(r/a)^2G$ in components recording unscaled magnitude,
and $Q=H_*(t)+\epsilon$.  For every shell of radius $s$, its
normalized spatial $L^2$ input obeys
\begin{equation}
 (r/a)^2(sr)^{-2}\|G(\cdot,u)\|_{L^2(B_u(x_u,D_1a))}
       =s^{-2}f(u).
 \label{para:interior-normalization}
\end{equation}
Integrate the local estimate on sliding windows of length $c s^2r^2$
inside the larger window.  Fubini's theorem cancels their time
normalization, and the remaining centered window average is bounded
by $H_*(t)$.  Thus, for any fixed high order $J$, both the time
$L^p$ average over a window $J'$ of length comparable to $B^2r^2$ and
the value at time $t$ satisfy
\begin{equation}
 \left(\frac1{|J'|}\int_{J'}\sup_{\text{shell }s}|T_1|_{J,s}^p\,du\right)^{1/p}
      \le CQs^{-2},\qquad
 \sup_{\text{shell }s}|T_1|_{J,s}(t)\le CQs^{-2}.
 \label{para:rough-neck}
\end{equation}
Here derivatives in $|T_1|_{J,s}$ are in $r$-coordinates, so they
represent physical scale $sr$.  To verify the source estimate used
in these applications, its size in shell parabolic units is
\[
 C\epsilon(r/a)^2sr.
\]
After division by $Qs^{-2}$ this is at most
$C(r^3/a^2)s^3\le Ca e^{-(3-3\lambda)D}$.
It is absorbed in Equation~\eqref{para:deep-metric-error}, after
decreasing $\delta$.  This also proves the forcing estimate at
all fixed spatial derivative orders.

Take a time cutoff $\chi$ supported in the large window, equal to
one on an ample middle window, with
$|\partial_\tau\chi|\le CB^{-2}$ in $r^2$-time.  Apply
Lemma~\ref{para:radial-lemma} to $Z=\chi PT_1$, using time measure
normalized by a constant times $B^2$.  The metric and curvature
operator errors cost $CQe^{-\delta D}s^{-4}$: there are two
inverse shell radii from the operator and $Qs^{-2}$ from
Equation~\eqref{para:rough-neck}.  The cutoff costs
$CQB^{-2}s^{-2}$.  Therefore
\[
 \mathcal Lz\le CQ\bigl(e^{-\delta D}s^{-4}+B^{-2}s^{-2}\bigr),\quad
 z(h)\le CQh^{-2},\quad z(B)\le CQB^{-2}.
\]
The four terms in the following barrier have explicit roles:
\begin{equation}
 z(s)\le CQ\left(hs^{-3}+B^{-3}s+B^{-2}
                                  +e^{-\delta D}s^{-2}\right).
 \label{para:interior-barrier}
\end{equation}
The first two are $\mathcal L$-harmonic and dominate the inner and
outer data.  The last two supply the forcing, since
$\mathcal L(B^{-2})=3B^{-2}s^{-2}$ and
$\mathcal L(e^{-\delta D}s^{-2})=3e^{-\delta D}s^{-4}$.
The maximum principle proves Equation~\eqref{para:interior-barrier}.

We next pass from the long time average to the one specified time.
For $1\le s\le B/4$, choose a parabolic patch of temporal size
$cs^2$ where $\chi=1$.  Restricting a normalized $L^p$ time norm
from duration $CB^2$ to this patch costs exactly the bound
\begin{equation}
 C(B/s)^{2/p}.
 \label{para:time-average-loss}
\end{equation}
Angular $L^2$ controls angular $L^p$ because $p\le2$.
Integrating Equation~\eqref{para:interior-barrier} over a fixed-width
radial shell therefore supplies the normalized spacetime $L^p$
input for the flat local estimate in Lemma~\ref{para:local-smoothing}.
The operator errors and their spatial derivatives are controlled
by Equation~\eqref{para:rough-neck}.  Lemma~\ref{para:mean-lemma}
then restores the positive derivatives of the mean, with additional
error $CQe^{-\delta D}s^{-2}$, already bounded by the expression
below.  Since $B^{-3}s\le B^{-2}$ on this range, the result is
\begin{equation}
 \sup_{\text{shell }s}\sum_{k=1}^j s^k|D^kT_1|(t)
 \le CQ(B/s)^{2/p}
        \left(hs^{-3}+B^{-2}+e^{-\delta D}s^{-2}\right).
 \label{para:single-time-derivative}
\end{equation}
Covariant derivatives obey the same bound after the metric-error
terms are included.  The factor in Equation~\eqref{para:time-average-loss}
has been retained in every term.

At radius $B/4$, Equation~\eqref{para:rough-neck} bounds the value
of $T_1(t)$ by $CQB^{-2}$.  Integrate the Cartesian first derivative
along each radial ray down to radius one.  A scaled first derivative
at radius $s$ contributes its bound times $ds/s$; thus
\[
 \begin{split}
 |T_1|_{j,1}(t)
 &\le CQ\left[B^{-2}
   +B^{2/p}\int_1^{B/4}
       \left(hs^{-3-2/p}+B^{-2}s^{-2/p}
                      +e^{-\delta D}s^{-2-2/p}\right)\frac{ds}{s}\right]\\
 &\le CQ\left[B^{-2}+B^{2/p}h+B^{-2+2/p}
                                      +B^{2/p}e^{-\delta D}\right].
 \end{split}
\]
Higher derivatives at radius one are already covered by
Equation~\eqref{para:single-time-derivative}.  The four decay
exponents in $D$ are respectively
\[
 2\lambda,\qquad(4-2/p)\lambda,\qquad
 (2-2/p)\lambda,\qquad\delta-(2/p)\lambda.
\]
They are positive because $1<p<2$ and $\delta>3\lambda$.
Choose $\nu$ smaller than their minimum.  This proves
Equation~\eqref{eq:interior-weighted}.
\end{proof}

There is also an uncut qualitative estimate on these true pieces:
\begin{equation}
 r^2|\Ric|_{j,r}\le C\epsilon r
       \le o_i(1)e^{-\nu D},
 \label{para:uncut-weight}
\end{equation}
with $\nu$ decreased below one, using $r\le Ca e^{-D}$ on deep
necks and $r\le Ca$ on compact pieces.
Equation~\eqref{eq:interior-weighted} supplies the stronger estimate
relative to the energy and cutoff.  In the root units used in
Section~\ref{sec:static-input}, its precise form is
\[
 \eta\,r^2|\Ric|_{j,r}
       \le Ca^2(H_*+\epsilon)e^{-\nu D}.
\]
Together with Equation~\eqref{eq:maximal-ricci}, this is the interface
used for the functional estimate in Section~\ref{sec:contradiction}.
==== END Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//05-parabolic.tex ====

==== BEGIN Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//06-collar.tex ====
\section{An exterior extension determined by a fixed collar}
\label{sec:collar}

A nearly flat exterior can carry a Ricci-flat field whose size is much
larger than its Ricci tensor.  Cutting the metric off directly to the
Euclidean metric would lose this distinction.  We construct an extension
which retains the field detected by a fixed collar.  The extension is an
analytic function of that collar metric.  The longer exterior is used
only to estimate the error of this function; neither its coordinates nor
any comparison gauge will have to vary differentiably in time.

\subsection{Norms and the extension statement}
\label{collar:statement}

All lengths in this Section are in collar units.  Let
\(\Gamma\subset O(4)\) be a nontrivial finite group acting freely on
\(S^3\), and put
\[
 C_1=\{1\le |x|\le2\}/\Gamma,\qquad
 \mathcal E=\{|x|\ge1\}/\Gamma.
\]
Calculations are made with equivariant Cartesian components on the
Euclidean cover.  For a fixed-ratio shell \(\Omega_r\), use the norm
\begin{equation}
 \|v\|_{H^s(r)}^2
 =\sum_{j=0}^s r^{2j-4}\int_{\widetilde\Omega_r}|\partial^jv|^2\,dx,
 \qquad
 |v|_{C^s,r}=\sum_{j=0}^s r^j\sup_{\widetilde\Omega_r}|\partial^jv|.
 \label{collar:scaled-norms}
\end{equation}
Here \(\widetilde\Omega_r\) is the lifted shell.  Thus derivatives, but
not the sizes of tensor components, are rescaled.  In particular,
\(\|\mathcal K(v)\|_{H^{s-2}(r)}\) is bounded by
\(Cr^{-2}\|v\|_{H^s(r)}\).  Equivalent fixed enlargements of the shells
will be used without changing notation.  At a boundary they are
one-sided annuli of a fixed positive relative width.  Define
\[
 \|v\|_{H^s_\beta(\mathcal E)}
   =\sup_{r\in\{1,2,4,\ldots\}}r^\beta\|v\|_{H^s(r)}.
\]
On filled Euclidean space the corresponding norm also includes an
ordinary Sobolev norm on the ball of radius two.  These conventions
give constants independent of \(\Gamma\): all estimates are on the
fixed cover and finite-group averaging has norm at most one.
Every relation \(\rho\asymp r\) below ranges over a fixed finite
number of neighboring dyadic shells.  Its comparison width is
chosen once to include the initial shells affected by the collar
correction and is independent of the outer radius \(R\).

Fix \(k=30\) and \(1<q<2\).  Let \(R\) be a large dyadic radius;
\(R\) in this Section does not denote scalar curvature.  Suppose a
metric \(g\), expressed in these coordinates on
\(1\le |x|\le4R\), satisfies
\begin{equation}
 |g-\delta|_{C^{k+2},r}\le\theta_r,
 \qquad \sum_{1\le r\le R}\theta_r\le\sigma,
 \qquad
 \theta_r\ge \|g-\delta\|_{H^k(C_1)}r^{-q}.
 \label{collar:metric-smallness}
\end{equation}
The weights are positive and comparable on neighboring shells, with
fixed comparison constants; the hypotheses include the fixed
enlargements needed below.  A fixed multiplicative constant in the
last inequality can be incorporated into the weights.  Set
\begin{equation}
\begin{split}
 m_1={}&\sup_{C_1}|\Ric(g)|
 +\sum_{1\le r\le R}\sup_{\Omega_r^+}
 \left(\theta_r|\Ric(g)|+
       \sum_{j=1}^{k-2}r^j|\nabla_g^j\Ric(g)|\right)
 +\frac{\theta_R}{R^2},
\end{split}
\label{eq:collar-residual}
\end{equation}
where \(\Omega_r^+\) denotes the specified ample enlargement.  Curvature
norms on the right are intrinsic; since the metrics are uniformly
close to \(\delta\), changing to Cartesian component magnitudes costs
only a fixed factor.

\begin{theorem}[Collar extension]
\label{thm:collar-extension}
There are \(\sigma_*>0\), \(R_*<\infty\), and \(C<\infty\), depending
only on \(q,k\), the fixed shell choices, and the weight comparison
constants, with the following property.  If
\(\sigma\le\sigma_*\) and \(R\ge R_*\), there is a metric \(\bar h\)
on \(\mathcal E\) such that
\begin{equation}
\begin{split}
 \|\bar h-\delta\|_{H^k_q}
   &\le C\|g-\delta\|_{H^k(C_1)},\\
 \|g-\bar h\|_{H^k(C_1)}
    +\|\Ric(\bar h)\|_{H^{k-2}_{q+2}}
   &\le C m_1.
\end{split}
\label{eq:collar-extension}
\end{equation}
The map \(g|_{C_1}\mapsto\bar h-\delta\) is real analytic on a fixed
neighborhood of zero in collar \(H^k\), and its first derivative into
\(H^k_q\) has uniformly bounded operator norm.  It is independent of
\(R\), the weights, and the choice of an exterior realizing the collar
data.  There is also such an analytic extension \(\hat h\), equal to
\(g\) near the inner boundary, for which
\[
 \|\hat h-\delta\|_{H^k_q}\le C\|g-\delta\|_{H^k(C_1)},
 \qquad
 \|\Ric(\hat h)\|_{H^{k-2}_{q+2}}\le Cm_1,
\]
and whose first derivative has the same bound.
\end{theorem}

The last term in Equation~\eqref{eq:collar-residual} cannot simply be
discarded.  A regular vacuum curvature field on a finite annulus is
detected at its outer boundary even when its Ricci tensor vanishes.
The term \(\theta_R/R^2\) retains this information.

\subsection{Two local linear estimates}
\label{collar:local-linear}

The flat compatibility used here is the Saint-Venant relation between
metric strain and linearized curvature. Its place in the Calabi complex
is described in \cite[Section~2.2]{Khavkine2017}. The relation between
compatibility, potentials and Korn's inequality is also developed in
\cite{CiarletCiarlet2005} in dimension three. We give the required
four-dimensional, equivariant estimates with their actual derivative
orders and kernel below.

Write \(Sv\) and \(\mathcal K(v)\) for the flat linearizations of
Ricci and Riemann curvature.  With plus-contraction divergence, set
\(Bv=\partial^iv_{ij}-\tfrac12\partial_j\tr_\delta v\).  Directly
differentiating the coordinate formulas for curvature gives
\begin{equation}
 Sv=-\tfrac12\Delta v+\tfrac12\mathcal L_{(Bv)^\sharp}\delta,
 \qquad BSv=0,
 \qquad
 \Delta\mathcal K(v)=\mathcal T(\partial^2Sv).
 \label{collar:flat-identities}
\end{equation}
Here \(\mathcal T\) is a fixed linear combination of components,
independent of the annulus.  These identities hold distributionally
at the Sobolev orders used below.  Also
\(S\mathcal L_V\delta=\mathcal K(\mathcal L_V\delta)=0\).

\begin{lemma}[Local potential and Korn estimates]
\label{collar:potential}
On a fixed smooth Euclidean annulus, ball, or fixed-width annular
overlap in dimension four, there is a bounded linear extraction
\(v\mapsto P v\in H^{k+1}\) with
\begin{equation}
 \|v-\mathcal L_{Pv}\delta\|_{H^k}
       \le C\|\mathcal K(v)\|_{H^{k-2}},
 \qquad \|Pv\|_{H^{k+1}}\le C\|v\|_{H^k}.
 \label{collar:local-potential}
\end{equation}
For vector fields on these domains,
\begin{equation}
 \inf_{W\in\mathfrak e(4)}\|V-W\|_{H^{k+1}}
           \le C\|\mathcal L_V\delta\|_{H^k},
 \label{collar:korn}
\end{equation}
where \(\mathfrak e(4)=\{x\mapsto Ax+b:A^t=-A\}\).
All choices can be linear, normalized modulo this rigid-motion
space, and equivariant.  After dilation, the right side of the first
inequality in Equation~\eqref{collar:local-potential} is multiplied
by \(r^2\), and vector norms in both inequalities are divided by \(r\).
\end{lemma}

\begin{proof}
We give the potential construction to specify its order and its
kernel.  On any of the stated domains, which are simply connected,
the Neumann gradient projection of a one-form \(\alpha\) writes
\(\alpha=dv+w\), where \(\operatorname{div}w=0\), \(w(\nu)=0\), and
\[
 \|w\|_{H^{s+1}}\le C\|d\alpha\|_{H^s}.
\]
The constant in \(v\) is fixed by its mean.  To recall why there is
no additional term, the div--curl integration-by-parts identity
first gives this inequality with an \(L^2\) remainder; its boundary
term is of order zero.  Compactness removes that remainder on the
orthogonal complement of the kernel.  A kernel element is closed,
hence exact on the stated domain, and its divergence and normal
conditions make its potential a constant Neumann harmonic
function.  The kernel is therefore zero.  Differentiation in smooth
boundary charts, followed by recovery of normal derivatives from
divergence and curl, gives the displayed estimate in every fixed
integer order.  This also makes the projection bounded and linear.

For a symmetric tensor \(v\), form its linearized connection
\[
 C^a_{ij}(v)=\tfrac12
       (\partial_i v_{aj}+\partial_j v_{ai}-\partial_a v_{ij}).
\]
For fixed \(a,j\), its curl in the index \(i\) is a component of
\(\mathcal K(v)\).  Apply the gradient projection with \(s=k-2\)
to obtain
\(C^a_{ij}=\partial_i w^a_j+e^a_{ij}\), with
\(\|e\|_{H^{k-1}}\le C\|\mathcal K(v)\|_{H^{k-2}}\).
Symmetry of \(C^a_{ij}\) in \(i,j\) bounds the curl of \(w^a_j\)
in \(H^{k-1}\) by the same quantity.  A second projection, now
with \(s=k-1\), writes \(w^a_j=\partial_j V^a+e'^a_j\), with
\(\|e'\|_{H^k}\le C\|\mathcal K(v)\|_{H^{k-2}}\).
Consequently \(C(v)-D^2V\) has that bound in \(H^{k-1}\).
Since \(C(\mathcal L_V\delta)=D^2V\) and
\(\partial_i v_{aj}=C^a_{ij}(v)+C^j_{ia}(v)\), the derivative
of \(v-\mathcal L_V\delta\) has that bound in \(H^{k-1}\).
Poincare's inequality controls the tensor modulo a constant symmetric
matrix.  Add the affine vector field with half this matrix as its
derivative.  The result proves the first inequality in
Equation~\eqref{collar:local-potential}.  Each projection has a
bounded linear normalization, and the construction also gives its
second inequality.

For Equation~\eqref{collar:korn}, if
\(e_{ij}=\partial_iV_j+\partial_jV_i\), the identity
\[
 2\partial_i\partial_jV_a
  =\partial_i e_{ja}+\partial_j e_{ia}-\partial_a e_{ij}
\]
controls second and higher derivatives.  Poincare's inequality
then controls \(V\) modulo an affine map, and the symmetric part
of that affine map is controlled by \(e\).  Its remaining ambiguity
is precisely \(\mathfrak e(4)\).  Projecting onto that finite
dimensional space fixes it linearly.  Averaging all constructions
over \(\Gamma\) preserves the inequalities.  Dilation proves the
scaled assertions.
\end{proof}

\begin{lemma}[Newton estimates used below]
\label{collar:newton}
Let \(G(x)=c_4|x|^{-2}\) be the fundamental solution for the chosen
sign of the Euclidean Laplacian.
For \(0<q<2\), its Newton operator is bounded
\[
 \Delta^{-1}:H^{k-2}_{q+2}(\R^4)\longrightarrow H^k_q(\R^4).
\]
For \(1<q<2\), if \(h\in H^{k-3}_{q+3}(\R^4)\) and
\(\int_{\R^4}h=0\) componentwise, then
\begin{equation}
 \|\Delta^{-1}h\|_{H^{k-1}_{q+1}}
             \le C\|h\|_{H^{k-3}_{q+3}}.
 \label{collar:zero-mass-newton}
\end{equation}
Finally, on an annulus or ball of outer radius comparable to \(R\),
if \(\Delta W=H\), boundary strips control \(W\) in scaled
\(H^{k-2}\) by \(b_0\), and
\(c_r=r^2\|H\|_{H^{k-4}(r)}\), then
\begin{equation}
 \sup_r\|W\|_{H^{k-2}(r)}\le C\left(b_0+\sum_rc_r\right).
 \label{collar:green-sum}
\end{equation}
The constant is independent of the number of shells.  A filled ball
has no inner boundary term; its innermost shell is replaced by a
fixed ball.
\end{lemma}

\begin{proof}
These are elementary weighted Newton estimates; compare the weighted
elliptic theory and Newton-kernel argument of
\cite[Theorem~1.7]{Bartnik1986}. We prove them for the shell-supremum
norms used here, including the zero-mass improvement.
Sobolev embedding on unit shells bounds a source in the first
assertion by \(D(1+|y|)^{-q-2}\).  At \(|x|=\rho\ge2\), split
the integral into \(|y|<\rho/2\),
\(\rho/2\le |y|\le2\rho\), and \(|y|>2\rho\).
The first part is bounded by
\(CD\rho^{-2}\int_0^\rho(1+s)^{1-q}\,ds\le CD\rho^{-q}\);
the middle part uses
\(\int_{|z|\le3\rho}|z|^{-2}\,dz\le C\rho^2\);
and the last part is bounded by
\(CD\int_{2\rho}^\infty s^{-q-1}\,ds\).
This proves the required size bound.  The scaled interior estimate
for \(\Delta u=f\),
\[
 \|u\|_{H^s(r)}\le
 C\bigl(\|u\|_{L^\infty(\Omega_r^+)}
          +r^2\|f\|_{H^{s-2}(\Omega_r^+)}\bigr),
\]
gives all the stated Sobolev orders.  The ball near the origin is
handled by the same local estimate.  These local estimates follow
by a cutoff and the Fourier multiplier for \(\Delta\), or by the
usual interior energy estimate and its derivatives.

For the second assertion \(h\) is integrable because \(q>1\), and
zero integral permits the identity
\[
 (G*h)(x)=\int_{\R^4}(G(x-y)-G(x))h(y)\,dy.
\]
On \(|y|<\rho/2\) the kernel difference is at most
\(C\rho^{-3}|y|\), so this part is bounded by
\[
 CD\rho^{-3}\int_0^\rho s^4(1+s)^{-q-3}\,ds
       \le CD\rho^{-q-1}.
\]
On the remaining region, the integral of \(G(x-y)h(y)\) is bounded
as in the first assertion, now by \(CD\rho^{-q-1}\).  The
subtracted term there is bounded by
\(C\rho^{-2}\int_{|y|\ge\rho/2}|h(y)|\,dy
 \le CD\rho^{-q-1}\).
Local elliptic estimates give Equation~\eqref{collar:zero-mass-newton}.

For the last assertion, the Dirichlet Green function in a Euclidean
domain has absolute value at most \(C|x-y|^{-2}\), as follows by
subtracting its boundary harmonic correction from the fundamental
solution and applying the maximum principle.  One shell of forcing
with supremum at most \(Cc_rr^{-2}\) contributes at most \(Cc_r\)
at every point: if \(x\) is near that shell, integrate the kernel
over a ball of radius \(Cr\); if it is far away, use the separation
and the shell volume.  Sobolev embedding supplies the forcing
supremum, since \(k-4>2\).  Componentwise maximum principle and
addition of the shell contributions bound \(\|W\|_\infty\) by
\(C(b_0+\sum c_r)\).  The displayed local elliptic estimate gives
the higher norms on interior shells.  The given boundary strips
supply the remaining norms.  This proof also applies by weak
approximation at the stated Sobolev regularity.
\end{proof}

\subsection{A filled-space generalized inverse and the analytic map}
\label{collar:analytic-map}

Let
\[
 \mathcal X=H^k_q(\mathcal E;\operatorname{Sym}^2),\qquad
 \mathcal Y=H^k(C_1;\operatorname{Sym}^2)
       \mathbin{\times}H^{k-2}_{q+2}(\mathcal E;\operatorname{Sym}^2),
 \qquad Av=(v|_{C_1},Sv).
\]
The pair \((g|_{C_1}-\delta,0)\) need not lie in the range of
\(A\): prescribed collar data need not admit a linearized Ricci-flat
extension.  We first solve a projected nonlinear equation.  The
supplied metric on the finite annulus will then control the residual
left by the projection.

We construct a bounded linear \(L:\mathcal Y\to\mathcal X\) such that
\begin{equation}
 ALA=A.
 \label{collar:generalized-inverse}
\end{equation}
Fix a bounded linear extension \(E\) from collar \(H^k\) to
\(H^k(B_2)\), equal to its input on the entire collar.  Such an
extension is obtained by reflection across the inner smooth
boundary in finitely many boundary charts, with the usual finite
linear combination of reflections matching derivatives through
order \(k\), followed by a partition of unity and an interior
cutoff.  Average it over \(\Gamma\).  Fix also a radial cutoff
\(\chi\) equal to one near radius one and zero before radius two.
For \(d=(v_1,f)\), define
\begin{equation}
 f^\#=
 \begin{cases}
 S(Ev_1),& |x|<1,\\
 \chi Sv_1+(1-\chi)f,&1\le |x|\le2,\\
 f,&|x|>2.
 \end{cases}
 \qquad u=\Delta^{-1}(-2f^\#).
 \label{collar:filled-source}
\end{equation}
The source lies in filled \(H^{k-2}_{q+2}\), with norm at most
\(C\|d\|_{\mathcal Y}\); Lemma~\ref{collar:newton} gives
\(\|u\|_{H^k_q}\le C\|d\|_{\mathcal Y}\).
Apply Lemma~\ref{collar:potential} to \(v_1-u|_{C_1}\), extend
its extracted vector linearly to the exterior, and cut that vector
off outside a fixed annulus.  Write the resulting vector as
\(V_d\), and set
\[
 Ld=u|_{\mathcal E}+\mathcal L_{V_d}\delta.
\]
This operator is bounded.  Its collar residual satisfies
\begin{equation}
 \|v_1-(Ld)|_{C_1}\|_{H^k}
       \le C\|\mathcal K(v_1-u)\|_{H^{k-2}(C_1)},
 \qquad S(Ld)=Su.
 \label{collar:L-residual}
\end{equation}

To prove Equation~\eqref{collar:generalized-inverse}, take \(d=Av\)
and fill \(v\) itself by the same extension \(E(v|_{C_1})\), obtaining
\(V\) on all of \(\R^4\).  The construction gives \(f^\#=SV\)
everywhere, including the interpolation region.  Equation~\eqref{collar:flat-identities}
gives \(Bf^\#=0\).  Thus \(Bu=0\): either commute the constant
coefficient derivative with the Newton integral in distributions,
or observe that \(Bu\) is an entire harmonic field decaying at
infinity and apply the mean-value property.  Consequently
\(Su=f^\#\).  The last identity in Equation~\eqref{collar:flat-identities}
shows that \(\mathcal K(V-u)\) is harmonic on the whole space.
It decays at infinity and hence vanishes by the same mean-value
argument.  The local extraction in Equation~\eqref{collar:L-residual}
is now exact.  We obtain \((Ld)|_{C_1}=v|_{C_1}\) and \(S(Ld)=Sv\),
which proves Equation~\eqref{collar:generalized-inverse}.  Filling
the hole is essential here: a decaying harmonic field on the
punctured exterior need not vanish.

Put \(Q=LA\).  Equation~\eqref{collar:generalized-inverse} implies
\[
 Q^2=Q,\qquad AQ=A.
\]
In particular \(\mathcal X_0=\operatorname{im}Q\) is a closed
Banach subspace.  For small \(h\in\mathcal X_0\), define
\[
 \Phi(h)=QL\bigl(h|_{C_1},\Ric(\delta+h)\bigr).
\]
This map is analytic.  Indeed the inverse metric is given by its
convergent Neumann series, and the coordinate expression for Ricci
is a sum of inverse-metric multiples of \(D^2h\) and products of
\(Dh\).  On every rescaled shell, \(H^{k-2}\) is a multiplication
algebra; inserting weights gives an analytic map
\(H^k_q\to H^{k-2}_{q+2}\).  Its differential at zero is \(S\).
For \(h\in\mathcal X_0\),
\[
 D\Phi(0)h=QLAh=Q^2h=h.
\]
The analytic inverse function theorem on this Banach space
therefore solves
\begin{equation}
 \Phi(\bar h-\delta)=QL(g|_{C_1}-\delta,0)
 \label{collar:analytic-equation}
\end{equation}
in a fixed neighborhood.  It yields the first estimate in
Equation~\eqref{eq:collar-extension} and the asserted bounded
first derivative.  Everything in this definition is fixed by the
collar, \(q\), and the choices of linear operators.

It remains to estimate the error.  Set
\begin{equation}
 d_1=(p_1,f)
    =(g|_{C_1}-\bar h|_{C_1},-\Ric(\bar h)),
 \qquad e=\|d_1\|_{\mathcal Y}.
 \label{collar:error-data}
\end{equation}
Equation~\eqref{collar:analytic-equation} says \(QLd_1=0\).
Applying \(A\), and using \(AQ=A\), gives
\begin{equation}
 ALd_1=0.
 \label{collar:annihilated-data}
\end{equation}
No identity \(LAL=L\) is needed.  We will show that
\(d_1-ALd_1\) is small, and then use
Equation~\eqref{collar:annihilated-data}.

\subsection{A comparison gauge on the long finite annulus}
\label{collar:comparison-gauge}

The analytic extension has already been defined from the collar alone.
To estimate its error, we are free to compare it with the supplied
metric in a separate gauge on the longer annulus. The construction
below must retain constants independent of that annulus's length,
even when its coordinate rotations accumulate.
We first construct a linear projection which removes coordinate
strain without a constant depending on \(R\).  For tensors on
\(1\le |x|\le R\), there are linear operators
\(P_0\) and \(\mathcal D z=\mathcal L_{P_0z}\delta\), with
\(P_0z=0\) on all of \(C_1\), such that
\begin{equation}
 \|z-\mathcal Dz\|_{H^k(r)}
 \le C\left(r^2\max_{\rho\asymp r}
       \|\mathcal K(z)\|_{H^{k-2}(\rho)}
       +\mathbf1_{r\le C_0}\|z\|_{H^k(C_1)}\right).
 \label{collar:long-projection}
\end{equation}
Here and below \(C_0\) is a fixed width, not a growing end length.

Here is the construction.  Take a sequence of roomy dyadic annuli
\(\Omega_j\) and smaller neighboring overlaps with fixed relative
widths.  Lemma~\ref{collar:potential} supplies local vectors \(v_j\)
with
\[
 r_j^{-1}\|v_j\|_{H^{k+1}(r_j)}\le C\|z\|_{H^k(r_j)},
 \quad
 \|z-\mathcal L_{v_j}\delta\|_{H^k(r_j)}\le e_j,
 \quad
 e_j=Cr_j^2\|\mathcal K(z)\|_{H^{k-2}(\Omega_j)}.
\]
On each overlap, Equation~\eqref{collar:korn} writes
\[
 v_{j+1}-v_j=A_jx+w_j,
 \qquad r_j^{-1}\|w_j\|_{H^{k+1}}
                  \le C(e_j+e_{j+1}),
\]
where \(A_j\) is skew and commutes with \(\Gamma\).
There is no equivariant translation: a nonzero invariant vector
would be fixed by each nonidentity group element, contrary to its
free action on the sphere.  The rigid projection can be chosen
orthogonal, so its coefficient also obeys
\[
 |A_j|\le C\bigl(\|z\|_{H^k(\Omega_j)}
                       +\|z\|_{H^k(\Omega_{j+1})}\bigr).
\]
Choose \(B_{j+1}=B_j-A_j\), and write \(W_j=v_j+B_jx\).
Then \(W_{j+1}-W_j=w_j\) on the overlap.  With a radial partition
\(\sum_j\psi_j=1\), let \(W=\sum_j\psi_jW_j\).
The part of its strain coming from partition derivatives is
\[
 \sum_j\bigl(d\psi_j\otimes W_j^\flat
                  +W_j^\flat\otimes d\psi_j\bigr).
\]
On a shell subtract any one active \(W_i\) inside this sum;
\(\sum d\psi_j=0\) cancels it exactly.  Only differences of
neighboring \(W_j\)'s remain, whose norms are controlled by the
nearby \(e_j\)'s.  Thus the residual estimate in
Equation~\eqref{collar:long-projection} holds for \(W\) without its
collar term.  In particular a cumulative rotation \(B_j\) never
appears in a residual bound.

Normalize the global rigid ambiguity by projecting \(W|_{C_1}\)
off the rigid space.  By Equation~\eqref{collar:korn}, its remaining
collar norm is bounded by \(C\|z\|_{H^k(C_1)}\) plus the nearby
curvature residuals.  Subtract a fixed bounded extension of this
entire collar vector, supported in \(r\le C_0\) and equal to it on
all of \(C_1\).  The resulting vector is \(P_0z\); this proves
Equation~\eqref{collar:long-projection}.  If
\(z=\mathcal L_V\delta\) with \(V=0\) on \(C_1\), both terms on
the right vanish.  Thus \(\mathcal Dz=z\) on such strains.
Moreover \(P_0z-V\) is a Euclidean Killing field vanishing on a
whole collar, so it is zero.  Since every \(\mathcal Dz\) is itself
such a strain, \(\mathcal D^2=\mathcal D\).

For the nonlinear step, define
\[
 n(z)=\sup_{r\le R}\theta_r^{-1}\|z\|_{H^k(r)}.
\]
The preceding construction gives more information than
Equation~\eqref{collar:long-projection}.  Namely the local potentials
have size \(Cr\theta_r n(z)\), successive correcting rotations
differ by \(C\theta_r n(z)\), and the first correcting rotation has
size \(C\theta_1n(z)\).  Consequently \(\mathcal D\) is bounded
for \(n\), independently of \(R\).  Radially interpolate the
correcting rotations, with value zero on \(C_1\), to write linearly
\begin{equation}
 P_0z=J(x)x+U(x),\qquad J(x)^t=-J(x),\qquad
 J\gamma=\gamma J\quad(\gamma\in\Gamma),
 \label{collar:rotation-decomposition}
\end{equation}
where \(J=U=0\) on \(C_1\), and
\begin{equation}
 \sup|J|\le C\sigma n(z),\qquad
 \sum_{d=1}^{k+1}\|r^d\partial^dJ\|_{L^\infty(\Omega_r)}
       +r^{-1}\|U\|_{H^{k+1}(r)}\le C\theta_r n(z).
 \label{collar:rotation-bounds}
\end{equation}
To see these bounds explicitly, before the collar correction use
\(J=\sum\psi_jB_j\) and \(U=\sum\psi_jv_j\).
For a derivative of \(J\), subtract a neighboring \(B_i\) using
the differentiated partition identity.  The resulting coefficient
is bounded by a fixed sum of \(|B_j-B_i|\), hence by
\(C\theta_rn(z)\).  The undifferentiated bound is the telescoping
sum \(\sum\theta_rn(z)\).  The collar subtraction and a cutoff of
\(J\) there change these estimates only on finitely many comparable
shells.  Bounds for arbitrarily many fixed derivatives of \(J\)
follow from the smooth fixed partition.  This justifies the
supremum version used in Equation~\eqref{collar:rotation-bounds}.

For \(z\in\operatorname{im}\mathcal D\), define
\begin{equation}
 F_z(x)=e^{J(x)}(x+U(x)).
 \label{collar:orthogonal-map}
\end{equation}
Fix a sufficiently large but fixed ball \(n(z)\le M_0\).
If \(\sigma\) is small in terms of \(M_0\), the map is the identity
on \(C_1\), is equivariant, and is a coordinate map into
\(1\le|x|\le4R\).  Indeed Sobolev embedding and
Equation~\eqref{collar:rotation-bounds} give
\(\|DF_z-\Id\|_\infty\le C M_0\sigma\) and
\(|F_z(x)-x|\le C M_0\sigma|x|\).
Extend it by the identity to the inner ball.  The derivative bound
on this convex ball of radius \(R\) proves injectivity by
integrating along segments.  Its image cannot enter the unit ball
from the exterior, since it already fixes that ball; the upper
radius is less than \(4R\).  This also verifies the boundary and
domain margins needed for composition.

The use of an orthogonal exponential in
Equation~\eqref{collar:orthogonal-map} gives
\begin{equation}
 F_z^*g-\bar h=g-\bar h+z+\mathcal E(z),\qquad
 \mathcal E(0)=0,
 \qquad
 n(\mathcal E(z)-\mathcal E(z'))\le C\sigma n(z-z')
 \label{collar:nonlinear-gauge-error}
\end{equation}
on this fixed ball.  We verify the relative error bound.  Write
\(O=e^J\) and \(\Omega_i=O^{-1}\partial_iO\).  Then
\[
 O^{-1}\partial_iF_z=e_i+\partial_iU+\Omega_i(x+U),
 \qquad \Omega_i=\partial_iJ+O(|J|\,|\partial_iJ|).
\]
The undifferentiated \(O\) cancels exactly from the Euclidean
pullback.  Its linear metric term is the symmetric derivative of
\(U+Jx\): the extra undifferentiated \(J\) in \(D(Jx)\) has zero
symmetric part.  The remaining products have a shell factor
\(\theta_r\) and a second factor bounded by \(C\sigma\).
The same statement for differences follows from linearity of
\(J,U\), the integral formula for a difference of matrix
exponentials, and Sobolev multiplication on rescaled shells.
For the part involving \(g-\delta\), the maps and their
differences have relative \(H^{k+1}\) size at most
\(C\sigma\) and \(C\sigma n(z-z')\).  The mean-value formula
for \((g-\delta)\circ F_z-(g-\delta)\circ F_{z'}\), together
with the \(C^{k+2}\) bound in
Equation~\eqref{collar:metric-smallness}, gives the factor
\(C\theta_r\sigma n(z-z')\).  Differentiating the pullback
formula through order \(k\) produces products with the same
bound; the highest derivative of a map occurs linearly and is
controlled in \(L^2\), while the other factors are controlled by
Sobolev multiplication.  Neighboring-shell comparability covers
the displaced points.  This proves
Equation~\eqref{collar:nonlinear-gauge-error}.  In particular an
undifferentiated accumulated rotation produces no Euclidean metric
error of size \(|J|^2\) on a remote shell.

Equation~\eqref{collar:metric-smallness} and the first estimate for
\(\bar h\) imply \(n(g-\bar h)\le C\).  The equation
\[
 z=-\mathcal D(g-\bar h)-\mathcal D\mathcal E(z)
\]
is a contraction on a sufficiently large fixed \(n\)-ball in
\(\operatorname{im}\mathcal D\), once \(\sigma\) is small.
For its solution set \(F=F_z\) and \(p=F^*g-\bar h\).
Then \(\mathcal Dp=0\), \(p|_{C_1}=p_1\), and
Equation~\eqref{collar:long-projection} gives
\begin{equation}
 \|F^*g-\delta\|_{H^k(r)}\le C\theta_r,
 \qquad
 Y:=\sup_{r\le R}r^{-2}\|p\|_{H^k(r)}\le C(e+X),
 \quad
 X:=\sup_{r\le R}\|\mathcal K(p)\|_{H^{k-2}(r)}.
 \label{collar:gauge-bound}
\end{equation}
This gauge is only a comparison device for the present estimate.
It is not used to define the analytic map in
Equation~\eqref{collar:analytic-equation}.

\subsection{Curvature control and the compatibility residual}
\label{collar:residual-proof}

The comparison gauge controls metric differences by linearized
curvature. We now estimate that curvature and use the filled-space
inverse again to bound the incompatibility of the collar and Ricci
data. Filling the inner ball is what removes an otherwise uncontrolled
inner boundary term.
Put \(T=\Ric(F^*g)\).  The coordinate Ricci formula and
Equation~\eqref{collar:gauge-bound} imply
\begin{equation}
 Sp=T-\Ric(\bar h)+E_2=T+f+E_2,
 \qquad \|E_2\|_{H^{k-2}(r)}\le C\theta_rY.
 \label{collar:quadratic-error}
\end{equation}
Indeed, writing \(\Ric(\delta+v)=Sv+N(v)\), its nonlinear
terms satisfy on a shell
\[
 \|N(v)-N(w)\|_{H^{k-2}(r)}
 \le Cr^{-2}(\|v\|_{H^k(r)}+\|w\|_{H^k(r)})
                     \|v-w\|_{H^k(r)}.
\]
Both first factors are \(O(\theta_r)\); the last factor is at
most \(r^2Y\).  On the collar the sharper bound
\(\|E_2\|_{H^{k-2}(C_1)}\le C\sigma e\) holds because
\(p=p_1\) there.

We will repeatedly use the consequence of
Equation~\eqref{eq:collar-residual}
\begin{equation}
 \|T\|_{H^{k-2}(C_1)}
 +\sum_{r\le R}\sum_{j=1}^{k-2}
       \|r^j\partial^jT\|_{H^0(r)}
 \le Cm_1.
 \label{collar:positive-ricci-derivatives}
\end{equation}
Here no sum of unweighted zeroth-order exterior values is being
claimed.  For completeness, covariant derivatives commute with
pullback.  The first derivative of \(F\) is bounded, so the
intrinsic terms \(F^*(\nabla_g^j\Ric(g))\) are bounded by the
suprema in Equation~\eqref{eq:collar-residual} on the enlarged
shells.  To pass from these tensors to Cartesian derivatives,
expand iteratively \(\partial T=\nabla_{F^*g}T+\Gamma(F^*g)*T\).
A term involving no positive covariant derivative of \(T\)
contains a metric derivative and hence a factor \(\theta_r\);
the other terms are bounded by the positive covariant derivatives
already present in the data.  Products through order \(k-2\)
are controlled in scaled \(L^2\) by the \(H^k\) bound for
\(F^*g\).  This proves the shell sum.  On \(C_1\), the
unweighted zeroth-order term is supplied separately in
Equation~\eqref{eq:collar-residual}.  Smooth approximation of
\(F\) on the roomy shells proves the identities first for smooth
maps and then in these finite Sobolev orders; their bounds require
no uncontrolled higher derivatives of \(F\).

Apply the last identity in Equation~\eqref{collar:flat-identities}
to \(p\).  Its forcing satisfies
\begin{equation}
 \sum_{r\le R}r^2
 \|\mathcal T(\partial^2Sp)\|_{H^{k-4}(r)}
             \le C(m_1+e+\sigma Y).
 \label{collar:first-forcing}
\end{equation}
The contribution of \(T\) is controlled by
Equation~\eqref{collar:positive-ricci-derivatives}; the contribution
of \(f\) is at most \(Ce\sum r^{-q-2}\); and that of \(E_2\)
is at most \(CY\sum\theta_r\).  The inner boundary strips have
\(\mathcal K(p)\)-norm at most \(Ce\), because \(p=p_1\) on
the collar.  Outer boundary strips have norm at most
\(C\theta_R/R^2\) by Equation~\eqref{collar:gauge-bound} and
the lower bound for \(\theta_R\) in
Equation~\eqref{collar:metric-smallness}.  Lemma~\ref{collar:newton}
therefore yields
\[
 X\le C(m_1+e+\sigma Y).
\]
Together with Equation~\eqref{collar:gauge-bound}, and then
absorption of \(C\sigma(X+Y)\), this gives
\begin{equation}
 X+Y\le C(m_1+e).
 \label{collar:first-absorption}
\end{equation}
The two derivatives of \(Sp\) are essential: a constant Ricci
component is killed, so it does not accumulate once per shell.

We now run the fixed linear construction \(L\) on the data
\(d_1=(p_1,f)\) from Equation~\eqref{collar:error-data}.  In
particular \(f^\#\) and \(u\) always mean precisely the filled
source and Newton potential in Equation~\eqref{collar:filled-source}.
Let \(D_0=m_1+\sigma e\).  We claim
\begin{equation}
 \|Bf^\#\|_{H^{k-3}_{q+3}(\R^4)}\le CD_0,
 \qquad \int_{\R^4}Bf^\#\,dx=0.
 \label{collar:bianchi-source}
\end{equation}
Inside radius one, \(Bf^\#=BS(Ep_1)=0\).
On \(C_1\), Equation~\eqref{collar:quadratic-error} and
Equation~\eqref{collar:positive-ricci-derivatives} give
\[
 \|Sp_1-f\|_{H^{k-2}(C_1)}
       \le C(m_1+\sigma e)=CD_0.
\]
Thus derivatives of the fixed blending cutoff cost at most \(CD_0\).
On the exterior, the contracted Bianchi identity for \(\bar h\)
gives \(B_{\bar h}f=0\).  Comparing it with flat \(B\) yields
terms of the form
\((\bar h^{-1}-\delta^{-1})\partial f\) and
\(\partial\bar h*f\), including the corresponding trace terms.
Their weighted \(H^{k-3}_{q+3}\) norm is at most \(C\sigma e\)
by Sobolev multiplication.  This proves the first assertion.
For the second, integrate the divergence expression for each
component of \(Bf^\#\) over a ball.  Its boundary flux is at most
\(Ce\rho^3\rho^{-q-2}=Ce\rho^{1-q}\), which tends to zero
because \(q>1\).  Integrability follows from the first assertion.
There is no inner boundary in this computation.

Commutation with the Newton operator and
Equation~\eqref{collar:zero-mass-newton} now give
\[
 Bu=\Delta^{-1}(-2Bf^\#),\qquad
 \|Bu\|_{H^{k-1}_{q+1}}\le CD_0.
\]
If desired, the commutation identity follows by comparing the
two distributional solutions: their difference is entire
harmonic and decays, hence is zero.  By
Equation~\eqref{collar:flat-identities},
\(Su-f^\#=\tfrac12\mathcal L_{(Bu)^\sharp}\delta\).
On the collar \(f^\#-f=\chi(Sp_1-f)\), while outside it
\(f^\#=f\).  Hence
\begin{equation}
 \|Su-f\|_{H^{k-2}_{q+2}(\mathcal E)}\le CD_0.
 \label{collar:ricci-compatibility}
\end{equation}

To control the other component of the residual, fill \(p\) by
\(Ep_1\) in the unit ball, obtaining \(P\) on the full ball of
radius \(R\).  Consider \(W=\mathcal K(P-u)\).  On the filled
inner ball, \(SP-f^\#=0\); on the collar,
\(SP-f^\#=(1-\chi)(Sp_1-f)\).  Therefore
\(S(P-u)\) on this whole fixed ball is bounded in
\(H^{k-2}\) by \(CD_0\).  Farther out it is
\[
 S(P-u)=T+E_2+(f-Su).
\]
The shell sum of the forcing
\(\Delta W=\mathcal T(\partial^2S(P-u))\) is consequently
bounded by
\[
 C\bigl(m_1+\sigma Y+D_0\bigr)
     \le C(m_1+\sigma e),
\]
where Equation~\eqref{collar:first-absorption} was used in the
last step.  The filled ball contributes one ordinary fixed-scale
term to this sum.  At its only boundary, near radius \(R\),
\begin{equation}
 \|W\|_{H^{k-2}(R)}
 \le C\left(\frac{\theta_R}{R^2}+eR^{-q-2}\right)
 \le C(m_1+eR^{-q-2}).
 \label{collar:outer-curvature}
\end{equation}
Here \(\|u\|_{H^k_q}\le Ce\) follows from the bounded linear
construction of \(L\), and \(\mathcal K(p)\) is controlled on
the outer strip by Equation~\eqref{collar:gauge-bound}.
The filled-ball version of Equation~\eqref{collar:green-sum},
followed by the fixed-scale interior estimate around \(C_1\), gives
\begin{equation}
 \|\mathcal K(p_1-u)\|_{H^{k-2}(C_1)}
       \le C\bigl(m_1+(\sigma+R^{-q-2})e\bigr).
 \label{collar:collar-curvature-residual}
\end{equation}
There is no inner boundary term proportional to \(e\); that is
the purpose of using the same inward extension a second time.

Equations~\eqref{collar:L-residual},
\eqref{collar:ricci-compatibility}, and
\eqref{collar:collar-curvature-residual} imply
\[
 \|d_1-ALd_1\|_{\mathcal Y}
       \le C\bigl(m_1+(\sigma+R^{-q-2})e\bigr).
\]
Since \(ALd_1=0\), first choosing \(\sigma_*\) small and then
\(R_*\) large makes the coefficient of \(e\) at most one half.
Thus \(e\le Cm_1\), proving the second estimate in
Equation~\eqref{eq:collar-extension}.  All absorptions have
constants independent of the exterior length.

Finally choose a fixed smooth radial \(\zeta\), equal to one
near radius one and zero near radius two, and define
\[
 \hat h=\bar h+\zeta(g-\bar h)\quad\hbox{on }C_1,
 \qquad \hat h=\bar h\quad\hbox{for }|x|\ge2.
\]
It is a positive metric for the same smallness choice.  The
coordinate Ricci formula and bounded cutoff derivatives give
\[
 \|\Ric(\hat h)\|_{H^{k-2}(C_1)}
 \le C\bigl(\|\Ric(\bar h)\|_{H^{k-2}(C_1)}
                  +\|g-\bar h\|_{H^k(C_1)}\bigr)\le Cm_1.
\]
Outside the collar the estimate is already proved.  The
interpolation is linear in the collar input and \(\bar h\), so
analyticity and the bounded derivative persist.  This completes
the proof of Theorem~\ref{thm:collar-extension}.

\begin{remark}[Scaling and the time-dependent interface]
\label{collar:scaling-interface}
If the physical collar radius is \(b\), write the normalized
metric as \(g_b=b^{-2}\Phi_b^*g\), where \(\Phi_b(x)=bx\) in
the chosen coordinates.  Covariant Ricci components of \(g_b\)
are \(b^2\) times the physical Cartesian components; intrinsic
Ricci magnitudes and their scaled derivatives consequently gain
the same \(b^2\) factor.  The residual term in physical units is
\(b^{-2}\theta_R/R^2=\theta_R/(Rb)^2\).
For a differentiable family on a fixed material collar, the
bounded derivative of the analytic extension gives
\[
 |\partial_t\hat g|_{0,r}\le CB_1(t)r^{-q},\qquad
 |\Ric(\hat g)|_{0,r}\le Cb^{-2}m(t)r^{-q-2},
\]
when the collar velocity is bounded by \(B_1(t)\) in the
required fixed \(H^k\) norm and \(m(t)\) bounds
Equation~\eqref{eq:collar-residual} in collar units.
The physical volume factor is \(b^4\), so the tail pairing
of these two tensors is bounded by
\(Cb^2m(t)B_1(t)\int_1^\infty r^{1-2q}\,dr\), which is finite
precisely for \(q>1\).  No derivative of the comparison map
\(F\), the far exterior coordinates, the weights, or \(R\) occurs.
All constructions use only the finite orders specified above.
For \(k=30\), scaled Sobolev embedding in dimension four gives
\(C^{27}\) metric bounds and \(C^{25}\) Ricci bounds from
\(H^{30}\) and \(H^{28}\), respectively.  These supply the finite
\(C^{20}\) metric and \(C^{10}\) Ricci bounds used in
Section~\ref{sec:static-input}.
\end{remark}
==== END Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//06-collar.tex ====

==== BEGIN Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//07-contradiction.tex ====
\section{Functional change and the contradiction}\label{sec:contradiction}

We apply the preceding results to the buffered intervals of
Proposition~\ref{prop:buffered-intervals}.  The interval is
$I=[-\theta,1+\theta]$ in its rescaled time coordinate.  The central
interval $[0,1]$ has original duration $d_i\to0$, whereas the full
buffered interval has original duration $(1+2\theta)d_i$.
Put $\epsilon_i=\sqrt{d_i}$.  We suppress the index $i$ when no limit
is being taken.  Thus $|R|\le C\epsilon_i^2$,
the reference root length $a=a_i$ tends to zero, and the root scale
defined in Proposition~\ref{prop:root-scale} is comparable to $a$
throughout $I$.  Its values at $0$ and $1$ differ by a fixed factor.
The centers $x_t$, cutoff $\eta$, and energy $K$ are those of
Equation~\eqref{eq:root-energy}; in particular
\begin{equation}\label{final:energy-recall}
 a^4 K^2=\int_I\eta(t)^2
       \int_{B_t(x_t,N_1a)}|\Ric|^2\,d\mu_{g(t)}\,dt,
 \qquad \min_{[0,1]}\eta>0.
\end{equation}
The maximal function $H_*$ may use a larger fixed ball than the one in
Equation~\eqref{final:energy-recall}, as permitted in
Equation~\eqref{eq:maximal-ricci}.

First fix the small collar tolerance.  The estimates below hold for
every sufficiently large fixed dyadic number $N\gg N_1$, with
$b=Na$.  All limits $i\to\infty$ are taken with $N$ fixed.
The constants multiplying $N^{-2\alpha}K^2$ will be independent of
large $N$; constants multiplying a quantity denoted $o_i(1)$ may depend
on $N$.  At the final absorption step we choose one such $N$ large
enough to absorb that explicit loss, then take the sequence limit.
No subsequent limit in $N$ is needed.

\subsection{Material collars and time patches}

\begin{lemma}[Position of a controlled material collar]
\label{final:collar-position}
Choose sufficiently small $\sigma_0>0$ and sufficiently large fixed
$N$.  At a time near $\supp\eta$, choose an exterior annular chart at
scale $b$ with domain, in collar units,
$\{1/8<|x|<16\}/\Gamma$, retaining slightly larger boundary margins
when necessary.  Keep its map into the material manifold and its group
fixed.  For all sufficiently late $i$, throughout any interval on
which its metric stays within $\sigma_0$ of the flat metric through the
required fixed derivative order, the following assertions hold.
Every closed shortened collar lies at distance comparable to $b$
from the current center $x_t$; its middle bands cover the corresponding
shortened distance annuli.  The component on the designated inner side
of a middle sphere contains $B_t(x_t,N_1a)$ and is contained in
$B_t(x_t,Cb)$, with fixed constants.
\end{lemma}

\begin{proof}
The initial chart has these properties by the exterior annular
convergence in Proposition~\ref{prop:exterior-ricci}.  In particular,
a middle sphere separates: every path from a sufficiently small ball
to outside a sufficiently large comparable ball crosses the chart's
radial band and hence the sphere.  Its two sides belong to different
components.  A connected two-sided embedded closed hypersurface in a
connected manifold has at most two complementary components: every
component reaches a tubular neighborhood of the hypersurface, and
each of its two connected sides meets only one component.  Thus the
designated inner component is unambiguous and is fixed as a material
subset while the chart is kept fixed.

We justify the uniform positional assertion; closeness of the metric
in a chart alone would not imply it.  If the assertion failed for
arbitrarily small tolerances and arbitrarily large $N$, choose a
countersequence with $\sigma_0\to0$, $N\to\infty$, and indices as late
as necessary for each choice.  The chart lies in the material
neighborhood of the same isolated point of the limiting space $Z$.
At scale $b$, based at an interior chart point, Theorem~\ref{thm:degeneration}
gives a complete flat quotient limit: a nonflat limit at this scale,
which is much larger than $a$, would contradict the outermost-scale
property in Proposition~\ref{prop:root-scale}.  On each compact
subannulus the controlled chart supplies curvature bounds on balls
with a definite margin.  Indeed a path leaving the chart must first
traverse that margin in its controlled metric.  Smooth compactness
therefore extracts the chart maps locally into the regular part of
the flat limit; metric and connection transformation formulas give
convergence of the maps and a limiting local flat isometry.

The groups can be fixed after passage to a subsequence.  Their orders
are bounded by noncollapse, there are finitely many abstract groups
of bounded order, and their orthogonal representations have convergent
subsequences.  Close orthogonal representations of a fixed finite
group are conjugate: averaging an approximate intertwiner over the
group makes it an exact invertible intertwiner, and its polar factor
is orthogonal and still intertwines.  The source group is nontrivial
and acts freely on the sphere.  The target flat limit is a global
Euclidean quotient with at most one exceptional point, by
Theorem~\ref{thm:degeneration}.  Lift the local isometry to Euclidean
covers of regular annuli.  It is the restriction of a rigid motion
$x\mapsto Ax+c$, since its differential is parallel.  Equivariance
conjugates the source group into the target group.  Its translation
part fixes $c$ under that nontrivial free spherical action, so $c=0$
after centering the target quotient.  In particular the target group
is nontrivial and the image is a complete centered open annulus.

There is no multiplicity in that image.  Otherwise two distinct
interior source points would approach one target point.  In the
approximations they could then be joined by a path of length $o(b)$.
A path contained in the chart cannot do this by its controlled
metric, and a path leaving it costs a fixed positive multiple of $b$
before reaching the boundary.  Both alternatives are impossible.
The target tip is represented by points $z_i$ with $l(z_i)/b_i\to0$.
It must lie at distance $o(b_i)$ from the current cluster center.
Otherwise, on a subsequence, put $h_i=d_t(z_i,x_t)\ge cb_i$; the
case $h_i/b_i\to\infty$ is allowed.  The chart's material points
approach the same point of $Z$ by their initial positioning and
uniform convergence of the original distances.  The same holds
for $z_i$, which is within $O(b_i)$ of an interior chart point,
and for $x_t$.  Thus the original length corresponding to $h_i$
tends to zero.  At this separation scale,
\[
 \frac{l(z_i)}{h_i}\longrightarrow0,\qquad
 \frac{l(x_t)}{h_i}\le C\frac{a_i}{h_i}
                    \le\frac{C}{cN_i}\longrightarrow0.
\]
Here the maximizing-center detection in
Proposition~\ref{prop:root-scale} gives $l(x_t)\asymp a_i$;
this placement countersequence has $N_i\to\infty$.
The separation-scale limit has two genuine tips at distance one.
It is therefore nonflat, since a flat limit has at most one tip.
Its scale satisfies $h_i/a_i\ge cN_i\to\infty$, contradicting
the outermost-scale property in Proposition~\ref{prop:root-scale}.

These facts imply coverage in the approximations.  A point whose
limit belongs to the interior image has distance $o(b)$ from an image
point with a fixed chart margin; a path realizing that distance
cannot leave the image, so the point itself lies in it.  Boundary
positions follow by approaching the boundary from interior points
and using the controlled lengths.  Consequently all shortened
radial bands have their claimed position and coverage, to any
prescribed fixed accuracy.  Crossing such bands shows that the
geometric inward component contains the small ball and is contained
in the larger $Cb$ ball.  To identify it with the designated material
component, fix a regular material point $p$ at positive $d_T$-distance
from this isolated cluster.  At the chart's initial time $p$ belongs
to the outward component.  Uniform convergence of original distances
keeps $p$ outside the current $Cb$ ball at every time under consideration.
Thus it still belongs to the geometric outward component.  The
embedded sphere and its two complementary material components were
kept fixed, so the other component remains the designated inward one.
This contradicts the countersequence and proves the lemma with fixed
choices of tolerance and $N$, followed by sufficiently late indices.
\end{proof}

At any individual time in such a patch, complete exterior coordinates
can be chosen to agree with the fixed chart on $C_1=\{1\le|x|\le2\}/\Gamma$.
Here and below an outward fixed rescaling of the chosen middle collar
is harmless.  To see the agreement, take the single neck coordinates
from Equation~\eqref{eq:neck-decay} out to a sufficiently small fixed
radius $r_*<r_0$ in interval units.  On the roomy overlap their
transition with the fixed chart has metric distortion
$O(\sigma_0+N^{-\mu})$ and controlled connection difference.  Lifting
the overlap, integration of the connection transformation makes
this transition close to a rigid equivariant map, through all needed
orders.  Its linear part can be adjusted to be orthogonal by polar
factorization.  The group and whole-band identifications are those
proved in Lemma~\ref{final:collar-position}.  After that rigid
alignment the maps are close to identity, so a radial cutoff
interpolation on a shorter overlap has invertible derivative and
patches the coordinate systems.  This is precisely the overlap
construction of the neck charts, now performed only in a regular
fixed-width band.  Decreasing $\mu>0$ if needed, the shell weights in
Theorem~\ref{thm:collar-extension} can consequently be chosen as
\begin{equation}\label{final:collar-weights}
 \theta_r\le C\bigl[(\sigma_0+N^{-\mu})r^{-\mu}
                    +\delta_i(rb)^\mu\bigr],
 \qquad 1\le r\le4R,\qquad Rb\asymp r_*.
\end{equation}
They include the collar norm times $r^{-q}$ after decreasing $\mu<q$.
The dyadic sum of their first term is small with the fixed choices;
the second sum tends to zero because $Rb\asymp r_*$.
These coordinates outside $C_1$ are used only to estimate the error.
They are not differentiated in time.

Let $B_1(t)$ be an upper bound for $|\Ric|_{k+4,b}$ on a fixed ample
band of true distance radii $[c'b,C'b]$ around $x_t$, in interval
units.  Suprema over all admissible center choices can be used to
make this bound independent of a choice of chart.  It is measurable
and locally bounded for every $i$.  Define an increasing absolutely
continuous time coordinate by
\begin{equation}\label{final:time-coordinate}
 u(t)=\int_{t_*}^{t}(1+B_1(s))\,ds .
\end{equation}
Start each chart with a tolerance strictly better than $\sigma_0$,
as is possible by first taking $N$ large.  As long as the chart
remains within $\sigma_0$, Lemma~\ref{final:collar-position} places
its entire closed domain in the band defining $B_1$.  The flow equation
$\partial_tg=-2\Ric$, together with conversion between covariant and
coordinate derivatives in that controlled chart, then gives
\[
 \|g(t)-g(s)\|_{C^{k+3}(\text{fixed collar})}
       \le C\int_s^t B_1(v)\,dv .
\]
A first-exit argument proves validity for a fixed sufficiently small
$u$-width.  This holds in both time directions, using the absolute
value of the integral.

On an interval with room around $\supp\eta$, choose regularly spaced
centers in the $u$ coordinate, subordinate smooth nonnegative bumps
of that fixed width, and normalize their sum.  Pulling them back gives
an absolutely continuous partition $\{\chi_j\}$ with
\begin{equation}\label{final:partition}
 \sum_j\chi_j=1\quad\hbox{near }\supp\eta,\qquad
 \sum_j|\chi_j'(t)|\le C(1+B_1(t))\quad\hbox{a.e.}
\end{equation}
Its overlap is bounded by a fixed number.  For each $i$ it is finite
on the compact interval in use; no bound on the number of members
uniform in $i$ is asserted or needed.

On patch $j$, retain the true metric on the material inner component
and apply Theorem~\ref{thm:collar-extension} in its fixed collar.
Interpolate as in that theorem.  Denote the resulting metric on the
fixed manifold with one end by $\hat g_j(t)$, and its functional in
interval length units by $E_j(t)$, using
Equation~\eqref{eq:energy-functional}.  Different patches are allowed
to have different extensions and different functional values.

\subsection{The lower bound for functional variation}

\begin{lemma}[The time-integrated collar error]
\label{final:collar-bound}
The residual in Equation~\eqref{eq:collar-residual} for every active
patch is bounded by a common nonnegative function $m(t)$ such that
\begin{equation}\label{eq:collar-time-bound}
 \sup_{\supp\eta}m=o_i(1),\qquad
 b^{-2}\|\eta m\|_{L^2(I)}+\|\eta B_1\|_{L^2(I)}
 \le C\bigl[(a/b)^{2+\alpha}K+\rho_i\bigr],
 \quad \rho_i\longrightarrow0.
\end{equation}
The leading constant is independent of large $N$; $\rho_i$ may depend
on fixed $N$.
\end{lemma}

\begin{proof}
Use the wider true bands in every supremum in the residual.  In collar
units the uncut regularity estimate
Equation~\eqref{eq:ricci-improvement} bounds Ricci and its scaled
derivatives on the shell of radius $r$ by $C\epsilon_i b/r$.
The dyadic sum of $r^{-1}$ is bounded, the zero-order terms carry the
summable weights $\theta_r$, and
$\theta_R/R^2\to0$.  Thus $m=o_i(1)$ uniformly.  This argument uses
the uncut estimate and is valid even where $\eta$ is small.

For the integrated assertion multiply the residual by $\eta$ and
convert a collar-unit Ricci magnitude to interval units by $b^{-2}$.
Writing $\rho=br$ for a shell's interval radius, its positive-order
terms are bounded in time $L^2$ by $V(\rho)$ from
Proposition~\ref{prop:exterior-ricci}; its zero-order term is bounded
by $\theta_r A(\rho)$.  The initial collar term is $A(b)$ up to
fixed-width changes.  Consequently it suffices to sum
\[
 A(b)+\sum_{r}\bigl[V(br)+\theta_r A(br)\bigr]
             +C\frac{\theta_R}{(Rb)^2}.
\]
The leading energy terms sum to $C(a/b)^{2+\alpha}K$.
The errors in $V$ are summable in both directions:
\[
 \sum_{br\in[cb,Cr_*]}
       \bigl[(a/(br))^\alpha+(br)^\alpha\bigr]
       \le C\bigl[(a/b)^\alpha+r_*^\alpha\bigr].
\]
The errors in $A$ are summed only with $\theta_r$, whose sum is
uniformly bounded.  In particular no constant error is multiplied
by the number of shells.  Finally
$\theta_R/(Rb)^2\to0$, because $Rb\asymp r_*>0$ and
Equation~\eqref{final:collar-weights} makes $\theta_R\to0$.
The same bounds for the fixed ample band give the assertion for
$B_1$.  Minkowski's inequality proves
Equation~\eqref{eq:collar-time-bound}.
\end{proof}

\begin{proposition}[Positive variation supplied by the true core]
\label{final:variation}
For the preceding extensions and partition,
\begin{equation}\label{eq:variation-lower}
 \sum_j\int_I\eta^2\chi_j E_j'\,dt
 \ge 2a^4K^2-a^4\bigl[CN^{-2\alpha}K^2
                              +o_i(1)(K+1)^2\bigr].
\end{equation}
\end{proposition}

\begin{proof}
The analytic collar map and its bounded derivative make $E_j$
differentiable on its patch.  The available fixed Sobolev orders
give all spatial derivatives required by the first-variation formula
in Equation~\eqref{ell:first-variation}.  On the retained true part,
\[
 \left\langle \tfrac12 Rg-\Ric,\partial_tg\right\rangle
           =2|\Ric|^2-R^2.
\]
That part contains $B_t(x_t,N_1a)$ by
Lemma~\ref{final:collar-position}, and its volume is at most $Cb^4$.
The scalar-square loss integrated in time is therefore at most
$C\epsilon_i^4b^4=o_i(a^4)$ for fixed $N$.

On the interpolated collar and attached end, use collar radius
$r\ge1$.  Equation~\eqref{eq:collar-extension}, Sobolev embedding,
and the derivative bound for the extension map imply
\[
 |\Ric(\hat g_j)|\le Cb^{-2}m(t)r^{-2-q},\qquad
 |\partial_t\hat g_j|_{\hat g_j}\le CB_1(t)r^{-q},\qquad
 d\mu_{\hat g_j}\le Cb^4r^3\,dr\,d\omega.
\]
There is no extra power of $b$ in the second estimate: normalizing
the collar metric multiplies its covariant tensor by $b^{-2}$,
whereas pulling back the original covariant Ricci tensor contributes
$b^2$ relative to its interval-unit magnitude.  The two factors
cancel.  Since $|R|\le2|\Ric|$ in dimension four, the absolute value
of the tail contribution to $E_j'$ is bounded by
\begin{equation}\label{final:tail-variation}
 Cb^2m(t)B_1(t)\int_1^\infty r^{1-2q}\,dr
       \le Cb^2m(t)B_1(t).
\end{equation}
This is the specific use of $q>1$ in the time-variation estimate.
Summing against $\chi_j$ costs no multiplicity, since
$\sum_j\chi_j=1$.  By Equation~\eqref{eq:collar-time-bound} and
Cauchy--Schwarz the integrated tail loss is at most
\[
 Cb^2\|\eta m\|_2\|\eta B_1\|_2
 \le Cb^4\bigl[(a/b)^{2+\alpha}K+\rho_i\bigr]^2.
\]
Its leading factor is exactly
\begin{equation}\label{final:tail-scaling}
 b^4(a/b)^{4+2\alpha}=a^4N^{-2\alpha}.
\end{equation}
For fixed $N$ the remaining terms are
$o_i(a^4)(K+1)^2$.  The positive core term sums to at least
$2a^4K^2$ by Equation~\eqref{final:energy-recall}.  This proves the
proposition.
\end{proof}

\subsection{Applying the static inequality to every time patch}

The lower variation bound is expressed in terms of the true root
energy. We now obtain the opposing upper bound. The only required
uniformity is over the ratios of functional value to weighted Ricci
error: a new fixed tree may be extracted from each proposed failure
of that uniformity.
\begin{proposition}[Uniform functional smallness]
\label{final:functional-smallness}
For fixed $N$ there are numbers $\zeta_i\to0$ such that, at every
time in every active patch,
\begin{equation}\label{eq:functional-littleoh}
 |\eta(t)E_j(t)|\le
 \zeta_i a^4\bigl[H_*(t)+1+a^{-2}\eta(t)m(t)\bigr].
\end{equation}
Here $H_*$ uses a fixed enlargement depending on $N$.
\end{proposition}

\begin{proof}
We first verify exactly which static hypotheses hold after an
arbitrary choice of indices, times $t_i$ with $\eta(t_i)>0$, and
active patches $j_i$.  Rescale their extensions by the length $a_i$.
Theorem~\ref{thm:degeneration} and Proposition~\ref{prop:root-scale}
extract one finite rooted tree, with the outermost nonflat entry
represented at unit scale.  Every smaller concentration is contained
inside the collar: if a concentration escaped to distance much
greater than $a$, the distance-scale limit with the root would have
two genuine tips and hence a larger nonflat entry.  For large fixed
$N$, the replacement collar and all its overlaps are uniformly
regular and leave a detecting nonflat root patch in the retained
body.

For clarity, the hypotheses of Theorem~\ref{thm:tree-inequality}
are supplied as follows.  The original compact pieces and joining
regions have the sequential identifications from
Theorem~\ref{thm:degeneration}.  On an edge with parent scale $A$
and child scale $Ae^{-2L}$, put
\[
 \tau_p=\log(A/\rho),\qquad
 \tau_c=\log(\rho/(Ae^{-2L})),\qquad \tau_p+\tau_c=2L.
\]
The physical radius is $\rho=Ae^{-\tau_p}$ on the parent half.
On the child half, $\tau_c\le L$, so
$\rho=Ae^{-2L+\tau_c}\le Ae^{-\tau_c}$.
Every parent scale is bounded by a fixed multiple of $a$.
Consequently $\rho\le C_Nae^{-D}$ on both halves, where
$D=\min(\tau_p,\tau_c)$ up to fixed end-coordinate offsets.
Equation~\eqref{eq:neck-decay} bounds the metric error by a constant
times $e^{-\mu\tau_p}+e^{-\mu\tau_c}$.  On the parent half
$\tau_p\le L$ this is at most $2e^{-\mu\tau_p}$, and on the child
half it is at most $2e^{-\mu\tau_c}$.  Thus these are exactly the
one-sided half-neck weights of the static theorem, with a possibly
smaller fixed positive exponent.  The joined groups are nontrivial
and are identified by those same annular charts.

On the new root exterior, Equation~\eqref{eq:collar-extension}
gives uniform $O(r^{-q})$ control with $1<q<2$ and sufficiently many
scaled derivatives.  Expressing it in root units changes its
constant by a factor depending on $N$, which is fixed.  Local Sobolev
compactness extracts convergence through the finite differentiability
order needed by the static theorem.  The limit is Ricci-flat on
the attached portion because $m(t_i)\to0$.  Local harmonic
coordinates and elliptic regularity make this Ricci-flat limit smooth
on its regular part; the weighted finite-order bounds are retained
in the original end coordinates.  The collar embeddings themselves
extract on smaller regular overlaps: metric and
connection bounds control their transitions with the interior
convergence charts.  Adjusting the latter by these limiting
transitions identifies the two limits on the overlaps.  Hence the
substitution changes only a regular portion and the unjoined end
of the root vertex.  It does not alter a joining end or remove the
retained nonflat patch.  This gives one fixed limiting Ricci-flat
root with the required decay $q>1$, while every other vertex is the
one furnished by the true slice convergence.  A fixed tree is
obtained only on this extracted sequence; no tree through time is
being chosen.

Let $M_i$ be a bound for the weighted, scale-neutralized Ricci error
in Equation~\eqref{eq:tree-ricci-error} on this sequence.  Choose its
internal weight exponent smaller than both the half-neck metric
exponent and the exponent in Equation~\eqref{eq:interior-weighted}.
We can take $M_i\to0$ without using $\eta$.  Indeed on a true piece
of physical radius $\rho$ in interval units the uncut bound is
\[
 \rho^2|\Ric|_{j,\rho}\le C\epsilon_i\rho.
\]
On compact pieces $\rho\le C_Na$.  At logarithmic depth $D$ in a
joining neck one has $\rho\le C_Nae^{-D}$, so multiplication by a
sufficiently small positive exponential weight still leaves a
quantity tending to zero.  The substituted region contributes at
most $C_Nm(t_i)$ with the required root-end decay.  These statements
also hold for all derivatives required in the definition of $M_i$.

Independently, Equation~\eqref{eq:interior-weighted} supplies the
stronger bound relative to the energy:
\begin{equation}\label{final:weighted-error}
 \eta(t_i)M_i
 \le C_{\mathrm{ext}}a_i^2\bigl[H_*(t_i)+\epsilon_i\bigr]
                         +C_{\mathrm{ext}}\eta(t_i)m(t_i).
\end{equation}
Here the constant may depend on this extraction and on fixed $N$.
To check the power of $a$, multiply that equation's
$(\rho/a)^2|\eta\Ric|_{j,\rho}$ by $a^2$; the result is the
scale-neutralized tensor $\rho^2|\eta\Ric|_{j,\rho}$ required in
Equation~\eqref{eq:tree-ricci-error}.  Compact retained regions are
handled by shorter regular patches, and the energy ball for $H_*$
is enlarged once to contain their short-time comparisons.  On the
collar and new end the error is $C_{\mathrm{ext}}\eta m$ by
Equation~\eqref{eq:collar-extension}.  This proves
Equation~\eqref{final:weighted-error} for the same $M_i$ obtained as
the maximum of these weighted errors.  Its uncut smallness and its
cutoff energy bound are separate assertions; neither was obtained
by division by $\eta$.

Write $E_i^{\mathrm{root}}$ for the functional in root units.
Theorem~\ref{thm:tree-inequality} now gives
$|E_i^{\mathrm{root}}|=o(M_i)$ on this extraction, and the scaling
law of Equation~\eqref{eq:energy-functional} gives
$E_{j_i}(t_i)=a_i^2E_i^{\mathrm{root}}$.  If $M_i=0$, the
zero-error scaling argument in that theorem gives
$E_i^{\mathrm{root}}=0$ directly.  Otherwise, using
Equation~\eqref{final:weighted-error} and $\epsilon_i\le1$,
\begin{equation}\label{final:ratio-test}
 \frac{|\eta(t_i)E_{j_i}(t_i)|}
 {a_i^4[H_*(t_i)+1+a_i^{-2}\eta(t_i)m(t_i)]}
 \le C_{\mathrm{ext}}\frac{|E_i^{\mathrm{root}}|}{M_i}
 \longrightarrow0.
\end{equation}

If Equation~\eqref{eq:functional-littleoh} failed uniformly, its
nonnegative ratio would be bounded below by some fixed number
along a sequence of active patches and times.  Times with $\eta=0$
have zero numerator, so that sequence has $\eta>0$.  The extraction
just performed yields Equation~\eqref{final:ratio-test} on a
subsequence, a contradiction.  The extraction-dependent constant
is fixed on that subsequence and multiplies a quantity tending to
zero.  This proves the uniform assertion without a uniform tree,
a uniform analytic family, or a uniform power-law exponent.
\end{proof}

\begin{corollary}\label{final:energy-vanishes}
Along the buffered intervals, $K_i\to0$.
\end{corollary}

\begin{proof}
Each product $\eta^2\chi_j$ has compact support in the patch where
$E_j$ is defined.  Integration by parts there gives
\[
 \sum_j\int_I\eta^2\chi_j E_j'\,dt
 =-\sum_j\int_I
       (2\eta\eta'\chi_j+\eta^2\chi_j')E_j\,dt.
\]
Absolute continuity of the partition is sufficient for this identity.
No values of different $E_j$ are equated at a switch.  By
Equations~\eqref{final:partition} and \eqref{eq:functional-littleoh},
the absolute value is at most
\begin{equation}\label{final:ibp-bound}
 C\zeta_i a^4\int_I
 [H_*+1+a^{-2}\eta m]\,[|\eta'|+\eta(1+B_1)]\,dt.
\end{equation}
For fixed $N$, Equations~\eqref{eq:maximal-ricci} and
\eqref{eq:collar-time-bound} show
\[
 \|H_*+1+a^{-2}\eta m\|_2\le C_N(K+1),\qquad
 \||\eta'|+\eta(1+B_1)\|_2\le C_N(K+1).
\]
For the first inequality use $a^{-2}=N^2b^{-2}$; fixed powers of
$N$ are allowed here.  Cauchy--Schwarz in
Equation~\eqref{final:ibp-bound} therefore bounds it by
$o_i(a^4)(K+1)^2$.

Choose and now fix $N$ so large that the leading constant in
Equation~\eqref{eq:variation-lower} satisfies $CN^{-2\alpha}<1$.
Combining the upper and lower bounds and then taking late indices
gives
\[
 K_i^2\le\gamma_i(K_i+1)^2,\qquad\gamma_i\longrightarrow0.
\]
This also proves boundedness if it was not known previously:
$K_i/(K_i+1)\le\sqrt{\gamma_i}$, and, for
$\sqrt{\gamma_i}<1$, one has
$K_i\le\sqrt{\gamma_i}/(1-\sqrt{\gamma_i})\to0$.
\end{proof}

\subsection{Small energy fixes the pointed root geometry}

The vanishing energy is an integral statement. To contradict the
prescribed change of scale, it remains to translate it into distance
preservation on a single dense set of material points and then into
preservation of the curvature-radius maximum.
\begin{lemma}[A common time-dependent Ricci bound]
\label{final:ricci-envelope}
On a slightly smaller material neighborhood of the isolated cluster,
for $0\le t\le1$,
\begin{equation}\label{eq:final-ricci-bound}
 |\Ric|(y,t)\le Q_*(t)\bigl[1+(a/l(y,t))^2\bigr],
 \qquad\|Q_*\|_{L^2([0,1])}\longrightarrow0.
\end{equation}
\end{lemma}

\begin{proof}
Inside a fixed large multiple of $a$, use the ordinary local
parabolic estimate underlying Proposition~\ref{prop:interior-ricci}
on a patch of scale a small fixed multiple of $\min(a,l(y,t))$.
The enlarged energy balls contain these patches and their prior
windows by the short-motion comparison.  Since $\eta$ is bounded
below on $[0,1]$, the estimate and
Equation~\eqref{eq:maximal-ricci} give an envelope bounded by
$C(H_*+\epsilon_i)$ times $1+(a/l)^2$.  Its time $L^2$ norm tends
to zero by Corollary~\ref{final:energy-vanishes}.

On the exterior annular range, an envelope for the magnitude on all
shells is obtained from a fixed outer shell and the sum of the
positive first derivative suprema on the intervening dyadic shells.
Indeed integrate $\nabla\Ric$ along the radial coordinate curves,
using parallel transport for tensor norms.  The controlled neck
charts give length $O(r)$ on each shell, so each shell costs its
scaled first derivative supremum.  Norms on a fixed shell vary by
the same bound along its angular paths.  Dividing the estimates for
$G=\eta\Ric$ by the fixed positive lower bound of $\eta$ is
legitimate on this interval.  Minkowski's inequality and
Proposition~\ref{prop:exterior-ricci} bound the time $L^2$ norm of
that envelope by
\[
 C\left[K+\widetilde\delta_i+
 \widetilde\delta_i\sum_{r\in[N_0a,r_0]}
                 \bigl((a/r)^\alpha+r^\alpha\bigr)\right]
 \longrightarrow0.
\]
The outer-shell value tends to zero by the uncut Ricci improvement
and its regular geometry.  Only the positive-derivative errors are
summed over all shells; the constant error in their zero-order
estimate is not summed.

Outside this range but within the smaller isolated neighborhood,
$l$ is bounded below in interval units.  Otherwise choose a sequence
with $l\to0$ there.  If its images remain in a compact regular part
of the original limiting space, smooth convergence is a
contradiction.  If they approach the isolated point, their
distance from the cluster is bounded below in interval units and
is much larger than $a$.  At that distance scale the vanishing
curvature radius and the root give two genuine tips, forcing a
nonflat entry larger than $a$.  The same argument applies when
that distance tends to infinity in interval units but tends to
zero originally, by rescaling with the distance itself.
Proposition~\ref{prop:root-scale} excludes this alternative.
Equation~\eqref{eq:ricci-improvement} thus bounds Ricci in the
remaining region by $C\epsilon_i$.  Adding these three envelopes
proves the lemma.
\end{proof}

\begin{lemma}[Good material points]
\label{final:good-points}
There is a single set of material points $\mathcal G_i$, independent
of time, whose complement has volume $o(a_i^4)$, which is $o(a_i)$-dense
at every $t\in[0,1]$ in a smaller neighborhood of the cluster.
For points $x,y\in\mathcal G_i$ in that neighborhood whose distance
at either endpoint is at most $Ca_i$,
\begin{equation}\label{final:good-distance}
 |d_1(x,y)-d_0(x,y)|=o(a_i),
\end{equation}
uniformly for each fixed $C$.
\end{lemma}

\begin{proof}
Take nested material neighborhoods compactly contained in the region
of Equation~\eqref{eq:final-ricci-bound}.  Uniform convergence of the
original distances to $d_T$ gives a fixed positive original distance
between the smaller neighborhood and the complement of the larger
one.  Thus any minimizing segment of interval length $O(a)$ with
endpoints in the smaller neighborhood remains in the region where
that equation applies.

For such a segment $\gamma:[0,L]\to M$ parametrized by arclength,
the geodesic curvature-radius estimate of
Lemma~\ref{geom:geodesic-radius} and spatial $1$-Lipschitzness of $l$
imply, on its first half,
\[
 l(\gamma(s))\ge \max\{l(x)-s,cs\}
                  \ge c'\bigl(l(x)+s\bigr).
\]
The fixed cap in that geodesic estimate is irrelevant at these
vanishing original lengths.  The second half has the analogous
bound from $y$.  Integration gives
\begin{equation}\label{final:geodesic-integral}
 \int_\gamma l^{-2}\,ds
          \le C\bigl(l(x,t)^{-1}+l(y,t)^{-1}\bigr).
\end{equation}
The length variation formula under Ricci flow, applied to minimizing
segments in both time directions, therefore yields, almost everywhere
while $d_t(x,y)\le Ca$,
\begin{equation}\label{final:distance-derivative}
 a^{-1}|\partial_t d_t(x,y)|
 \le C_C Q_*(t)
         \bigl[1+a/l(x,t)+a/l(y,t)\bigr].
\end{equation}
One may equivalently first use upper and lower one-sided length
variations; for each fixed smooth flow the distance is locally
Lipschitz in time, so these imply the almost-everywhere inequality.

The scale-sublevel covering in Theorem~\ref{thm:degeneration} gives
$\Vol_t\{l<s\}\le Cs^4$ at these scales.  Decomposing $\{l<a\}$
into the sets $2^{-k-1}a\le l<2^{-k}a$ proves
\begin{equation}\label{final:endpoint-integral}
 \int_{\{l<a\}}\frac a l\,d\mu_{g(t)}
 \le\sum_{k=0}^\infty 2^{k+1}C(2^{-k}a)^4
 \le Ca^4.
\end{equation}
Fix the material measure $d\mu_0=d\mu_{g(0)}$.  Since
$\partial_t d\mu_g=-R\,d\mu_g$ and $|R|\le C\epsilon_i^2$,
all these time-slice measures are uniformly comparable with $d\mu_0$.
Put $q_i=\|Q_*\|_{L^1([0,1])}\to0$ and define
\[
 F_i(x)=\int_0^1 Q_*(t)\frac a{l(x,t)}
                          \mathbf1_{\{l(x,t)<a\}}\,dt.
\]
Fubini and Equation~\eqref{final:endpoint-integral} give
$\int F_i\,d\mu_0\le Ca^4q_i$.  For $q_i>0$, discard the material
set $\mathcal B_i=\{F_i>\sqrt{q_i}\}$.  Then
\[
 \mu_0(\mathcal B_i)\le Ca^4\sqrt{q_i},\qquad
 \int_0^1 Q_*(t)(1+a/l(x,t))\,dt
                    \le2q_i+\sqrt{q_i}\quad(x\notin\mathcal B_i).
\]
If $q_i=0$, take the good set to be the whole manifold, up to a
null set, and the same conclusions hold.  The lower ball-volume
bound $\Vol_t B_t(x,r)\ge cr^4$ shows that every ball of radius
$C'aq_i^{1/8}$ in the smaller neighborhood meets the good set when
$C'$ is large enough.  Indeed such a ball has greater volume than
the entire bad set, uniformly in time.  Taking the complement of
$\mathcal B_i$ for $\mathcal G_i$ proves simultaneous $o(a)$ density.

For good endpoints, the time integral of the right side of
Equation~\eqref{final:distance-derivative} is $o(1)$ with a constant
depending only on a fixed distance range.  If the initial distance
is at most $Ca$, apply that inequality up to the first time it
reaches $(C+1)a$.  Its integrated variation is smaller than $a/2$
for late indices, so such a first crossing cannot occur.  The same
argument backwards in time treats a pair initially specified at
$t=1$.  This proves Equation~\eqref{final:good-distance}, and also
prevents points outside a larger bounded scaled ball from entering
a smaller one during the comparison.
\end{proof}

\begin{proposition}[The root maximum cannot change]
\label{final:root-fixed}
Let $\mathfrak a_i(t)$ denote the root scale of
Proposition~\ref{prop:root-scale}, computed in interval units.  Then
\[
 a_i^{-1}\bigl|\mathfrak a_i(1)-\mathfrak a_i(0)\bigr|
                      \longrightarrow0.
\]
\end{proposition}

\begin{proof}
Choose a good material anchor within $o(a)$ of a maximizing point
at time $0$.  Extract pointed limits at the two endpoints with
length unit $a$.  Lemma~\ref{final:good-points} identifies these
limits isometrically.  Here is the metric argument in detail.
On a fixed bounded ball choose a finite increasingly dense net of
good material labels at time $0$.  Their distance matrices at time
$1$ differ by $o(1)$ in scaled units.  The same is true of nets
chosen at time $1$, and the first-crossing argument keeps every
such label in a bounded ball at the other time.  Apply compactness
to their union and then diagonalize in the radius and the net
mesh.  The limiting correspondence preserves all pairwise
distances and is dense and onto; completing it gives a pointed
isometry.  All bounded balls used here lie near the same isolated
point of $Z$ in original units.

Every exceptional point of either limit is a genuine nontrivial
orbifold point by Theorem~\ref{thm:degeneration}, so the regular
part is determined by the metric space itself.  The isometry is
smooth there and preserves its Riemannian metric: in a sufficiently
small regular convex neighborhood, squared distances to suitable
nearby points give smooth coordinates with independent differentials;
the distance-preserving map has the corresponding smooth coordinate
expressions, as does its inverse.  Smooth convergence on these
regular regions is available at both endpoints.  In particular the
initial nonflat root patch persists in the other endpoint limit.

We next check that the curvature radius itself, rather than only
the distance metric, is preserved.  For a regular point $z$ of the
common nonflat limit $X$ define its stopped radius by
\begin{equation}\label{final:stopped-radius}
\begin{split}
 \lambda_X(z)=\sup\{r>0:\;&B_X(z,r)\text{ contains no exceptional point}\\
                       &\text{and }|\Rm_X|\le r^{-2}\text{ on }B_X(z,r)\}.
\end{split}
\end{equation}
It is finite: a regular point of nonzero curvature exists at finite
distance, so sufficiently large balls fail the curvature test.
For any regular convergent sequence $z_i\to z$ at either endpoint,
\begin{equation}\label{final:radius-convergence}
                         l(z_i,t)/a_i\longrightarrow\lambda_X(z).
\end{equation}
For the lower bound, take $r<s<\lambda_X(z)$ with $s$ admissible.
The closed $r$-ball has compact regular geometry, and its curvature
test has the strict margin $r^2|\Rm_X|\le(r/s)^2<1$.
Smooth convergence and convergence of distances imply admissibility
of every slightly smaller $r$-ball in the approximations.  Let
$r\uparrow\lambda_X(z)$.  For the upper bound, take
$r>\lambda_X(z)$ and then an intermediate radius between them.
If its ball meets an exceptional point, genuine concentration
approaches that point in the approximations and violates the
$r$-curvature test.  If it does not, its inadmissibility gives a
regular point whose curvature strictly exceeds $r^{-2}$; this
violation persists under smooth convergence.  Thus the $r$-ball
is eventually inadmissible.  The original fixed cap for $l$, divided
by the original length corresponding to $a_i$, tends to infinity,
so it imposes no further restriction here.  This proves
Equation~\eqref{final:radius-convergence}, including possible
equality cases at the limiting threshold.

The defining maximum is
\[
 \frac{\mathfrak a_i(t)}{a_i}
 =\max_{y\in\mathcal U}
   \frac{l(y,t)}{a_i}
       \psi\bigl(l(y,t)^2|\Rm|(y,t)\bigr).
\]
At a maximizer, $l/a_i$ is bounded below by a positive constant,
because $\mathfrak a_i(t)/a_i$ is bounded below and $\psi$ is bounded.
It is also bounded above by the detection and outermost-scale
property of Proposition~\ref{prop:root-scale}.  Maximizers at time
$1$ stay at bounded scaled distance from the common anchor.  Indeed
the persistent nonflat root patch supplies a nonflat entry of scale
comparable to $a_i$ at that time; if a maximizing nonflat entry
escaped from it, their distance-scale limit would have two genuine
tips and produce an entry larger than the outermost scale.  The
same bounded-range statement at time $0$ follows from the choice
of the anchor.

Consequently maximizers at either endpoint have subsequences
converging to regular points of $X$.  At any such point,
Equation~\eqref{final:radius-convergence}, smooth curvature
convergence, and continuity of $\psi$ show that their values tend
to
\[
 \lambda_X(z)\,
       \psi\bigl(\lambda_X(z)^2|\Rm_X|(z)\bigr).
\]
That same regular point can be approximated in the other endpoint
sequence, giving the same limit value there.  A subsequence
realizing any proposed strict difference between the two maxima
is therefore impossible: its maximizing point at the larger end
provides a competing point at the smaller end.  Applying this in
both directions proves the proposition.
\end{proof}

\subsection{Smooth continuation on the original manifold}

Proposition~\ref{final:root-fixed} contradicts the definite factor
change of the comparable positive root scales at the endpoints
of the intervals in Proposition~\ref{prop:buffered-intervals}.
Hence a scalar-bounded flow cannot have a concentration point.
Theorem~\ref{thm:degeneration} then gives a uniform full-curvature
bound up to the original finite time $T$.

For completeness, this yields the precise extension required in
Theorem~\ref{thm:extension}.  Curvature differentiation on a closed
manifold gives bounds for every covariant derivative of curvature
on any interval $[t_0,T)$ with $t_0>0$.  Bounded Ricci curvature and
$\partial_tg=-2\Ric$ first make $g(t)$ uniformly equivalent to
$g(t_0)$.  Integrating the connection variation and its successive
derivatives, or working in the resulting fixed controlled charts,
then gives uniform bounds and convergence for all coordinate
derivatives of $g(t)$.  Thus $g(t)$ converges smoothly to a positive
definite smooth metric $g_T$ on the original manifold $M$.
Short-time existence from $g_T$ follows from the Ricci--DeTurck
equation on this same closed smooth manifold, followed by its
diffeomorphism pullback.  The flow equation determines all time
derivatives from the spatial derivatives, so the old and new
solutions join smoothly at $T$.  Retaining the given flow for
$t<T$ and this solution for $t\ge T$ gives a smooth Ricci flow
$\widetilde g$ on $M\times[0,T+\varepsilon)$, for some
$\varepsilon>0$, with $\widetilde g(t)=g(t)$ for every $t<T$.
The argument uses arbitrary smooth initial data in real dimension
four; no change of manifold or orbifold continuation is involved.
==== END Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//07-contradiction.tex ====

==== BEGIN Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//buffered-intervals.tex ====
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  \node at (5.5,-3.05)
    {$g_i(s)=d_i^{-1}g(t_{j_i}+d_i s),\qquad
      \epsilon_i=\sqrt{d_i},\qquad
      a_i=\dfrac{a(t_{j_i})}{\epsilon_i}\longrightarrow0$};
\end{tikzpicture}
==== END Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//buffered-intervals.tex ====

==== BEGIN Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//overview.tex ====
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  x=1cm,y=1cm,>=Stealth,
  piece/.style={draw=blue!55!black,fill=blue!5,rounded corners=3pt,
                minimum width=2.1cm,minimum height=.65cm,align=center},
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  \node[piece] (root) at (0,0) {outermost piece\\scale $a$};
  \node[piece] (left) at (-2.6,-1.65) {descendant\\$a e^{-2L_1}$};
  \node[piece] (right) at (2.6,-1.65) {descendant\\$a e^{-2L_2}$};
  \node[piece] (deep) at (-2.6,-3.2)
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==== END Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//overview.tex ====

==== BEGIN Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//paper.tex ====
\documentclass[11pt]{article}
\input{preamble}

\title{Bounded scalar curvature and smooth extension\\
       of four-dimensional Ricci flow}
\author{OpenAI}
\date{September 24, 2026}

\begin{document}
\maketitle
\begin{abstract}
We prove that a smooth Ricci flow on a closed real four-manifold extends on
the same manifold through every finite time at which its scalar curvature
remains uniformly bounded. This resolves the scalar-curvature extension
problem in dimension four.
\end{abstract}
\tableofcontents

\input{sections/01-introduction}
\input{sections/02-static-input}
\input{sections/04-geometry}
\input{sections/05-parabolic}
\input{sections/06-collar}
\input{sections/07-contradiction}

\bibliographystyle{plain}
\bibliography{sources/references}
\end{document}
==== END Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//paper.tex ====

==== BEGIN Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//preamble.tex ====
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\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{enumitem,booktabs,array}
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\hypersetup{pdftitle={Bounded scalar curvature and smooth extension of four-dimensional Ricci flow},
            pdfauthor={OpenAI},
            pdfsubject={Bounded scalar curvature, Ricci flow, and Ricci-flat bubble trees}}
==== END Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//preamble.tex ====

==== BEGIN Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow//references.bib ====

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