\documentclass[11pt]{article}
\usepackage[margin=1.05in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{enumitem,microtype,xurl}
\usepackage{xcolor}
\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=green!35!black,
            urlcolor=blue!55!black,bookmarksnumbered=true]{hyperref}
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\R}{\mathbb{R}}
\DeclareMathOperator{\Ric}{Ric}
\DeclareMathOperator{\Rm}{Rm}
\DeclareMathOperator{\Vol}{Vol}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\tr}{tr}


\begin{document}


\title{Bounded scalar curvature and smooth extension of four-dimensional Ricci flow}
\author{OpenAI}
\date{September 24, 2026}
\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 stays
uniformly bounded: at a finite-time singularity the scalar curvature must blow
up.  The proof combines a scalar-curvature compactness and bubbling description
of the degenerating slices with a renormalized Einstein--Hilbert functional and
a single monotone quantity, the spacetime $L^2$ norm of the Ricci tensor in the
outermost bubble.
\end{abstract}

\par\medskip\noindent\textbf{Strategy of the proof.}

The proof follows one strategy.  Fix a hypothetical concentration point
where curvature blows up at $T$.  Successive rescalings produce a finite
tree of Ricci-flat limits; the outermost nonflat limit defines a scale
$a(t)$ that cannot change by any fixed factor on short intervals.  On the
corresponding buffered intervals, parabolic rescaling places the Ricci
energy $K$ of the outermost bubble in a limiting spacetime where the
renormalized Einstein--Hilbert functional and the companion tree inequality
force $K\to 0$; that pins $a(t)$ and rules out the concentration point.
The mechanics are in four blocks.  Block~I (``The spatial curvature
radius'' to ``Buffered intervals'') estimates curvature at all scales,
describes the slice limits, and selects the buffered intervals.  Block~II
(``Concentration estimates'' to ``The maximal function'') bounds the Ricci
heat equation and the root energy.  Block~III (``Collar extension'')
builds and estimates the analytic exterior from a fixed collar.  Block~IV
(``Functional change and the contradiction'' to ``Continuation'') supplies
the lower and upper bounds on the functional variation and closes the
argument.  A reader focused on the main idea should read Blocks~I and
IV first, treating Block~III as a black box for the error analysis.

\par\medskip\noindent\textbf{Introduction.}

A Ricci flow is a one-parameter family of Riemannian metrics solving the
heat-type equation
\begin{equation}\label{eq:flow}
 \partial_t g=-2\Ric(g).
\end{equation}
On a closed manifold it exists and is smooth for a short time; the central
question is what makes the maximal smooth existence time $T<\infty$ finite.
The full curvature tensor decides this: if $|\Rm|$ stays bounded on $[0,T)$,
the flow continues past $T$ \cite{Hamilton1982}.  The scalar curvature
$R=\tr_g\Ric$ is only a trace, so a bound on $R$ is far weaker; it controls
volume by $\partial_t\,d\mu_g=-R\,d\mu_g$ but leaves the full geometry free.
The scalar-curvature extension problem asks whether a uniform bound on $R$
alone already forbids a finite-time singularity.  The answer is affirmative
in real dimension four.

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

\begin{corollary}\label{cor:blowup}
Let $g(t)$, $0\le t<T<\infty$, be a smooth Ricci flow on a closed four-manifold
with finite maximal smooth existence time $T$.  Then
$\limsup_{t\uparrow T}\max_MR(g(t))=+\infty$.
\end{corollary}

\begin{proof}
$(\partial_t-\Delta)R=2|\Ric|^2\ge0$ and the maximum principle bound $R$ below
on any finite interval.  A finite upper limit would bound $|R|$ near $T$; since
$R$ is smooth on the compact initial part, Theorem~\ref{thm:extension} applied
componentwise would contradict maximality of $T$.
\end{proof}

The proof rules out contraction of a specific outermost bubble scale.  A
concentration point produces a finite rooted tree of Ricci-flat limits, one for
each scale at which curvature blows up; the outermost nonflat limit has a
length $a(t)$ that is positive before $T$ but tends to zero at the singular
time.  On short intervals $a$ cannot change by a fixed factor: rescaling would
freeze the pointed geometry, so any fixed-factor change would contradict the
identity of a fixed limiting tree.  Parabolic rescaling on the buffered
intervals therefore produces a limiting spacetime in which the Ricci energy $K$
of the outermost bubble is comparable from start to end.  The renormalized
Einstein--Hilbert functional computed on that scaling, together with a sharp
inequality from the companion paper, forces $K\to0$.  But $K\to0$ means
distances are preserved on the concentration scale, which pins $a(t)$ and
contradicts its prescribed finite change.  Hence no concentration point exists
and the flow extends smoothly.

\par\medskip\noindent\textbf{The geometric obstruction.}

Hamilton \cite{Hamilton1982} proved short-time existence; \v{S}e\v{s}um
\cite{Sesum2005} showed a uniform bound on $\Ric$ forces continuation.  A scalar
bound is weaker because after rescaling a would-be singularity the scalar
curvature can tend to zero while full curvature stays bounded away from zero.
The limit is then Ricci-flat without being flat, so a scalar bound permits a
whole tree of collapsing Ricci-flat bubbles.

Bamler--Zhang \cite{BamlerZhangI,BamlerZhangII} proved the heat-kernel and
curvature estimates under scalar bounds, including the four-dimensional energy
and compactness theory used below, and obtained smooth extension when
$H_2(M;\mathbb F)=0$ for all $\mathbb F$ \cite[Corollary~1.10]{BamlerZhangI}.
Simon \cite{SimonExtension} continued through the limiting space by an orbifold
flow; Bamler's convergence theory \cite{BamlerConvergence} gives smooth
convergence off a codimension-four set; Liu--Simon \cite{LiuSimon} treat
four-dimensional convergence under scalar-integrability hypotheses.  All allow
concentrated curvature and orbifold points; the continuation below instead
keeps the given flow on its original manifold.  Blow-up is known for K\"ahler--Ricci flows
\cite{Zhang2010Kahler}, for global Type~I \cite{EndersMuellerTopping2011}, and
for local Type~I points \cite{BuzanoDiMatteo2026}; no rate is imposed.

The spatial picture descends from Einstein four-manifold theory: Anderson
\cite{Anderson1989}, Bando \cite{Bando1990,BandoCorrection1990},
Bando--Kasue--Nakajima \cite{BandoKasueNakajima1989}, Cheeger--Tian
\cite{CheegerTian2006}.  The slices below are not Einstein, so the needed
statements are reproved under a scalar bound, retaining the Ricci forcing.

\par\medskip\noindent\textbf{The outermost scale and the static functional.}

Successive rescalings at a concentration point produce a finite rooted tree of
Ricci-flat pieces; the outermost nonflat vertex has a length scale $a(t)$
described below.  The proof rules out contraction of $a(t)$ while
allowing arbitrarily small descendants.

Retain the true metric through an annular collar of radius $b=Na$ ($N$ large,
fixed) and attach an ALE end in Euclidean quotient coordinates.  The
renormalized Einstein--Hilbert functional is
\[
 E(g)=\lim_{D\to\infty}\Big[\int_{\{r<D\}}R\,d\mu_g-\int_{\{r=D\}}
 (\partial_jg_{ij}-\partial_ig_{jj})n_i\,dS_\delta\Big],
\]
the mass term of Deruelle--Ozuch \cite[Definition~4.2, Proposition~4.4]
{DeruelleOzuch2025}.  It scales by the square of the length unit and its first
variation pairs the velocity with $\tfrac12Rg-\Ric$.  The companion article
\cite{StaticTrees} proves the sequential inequality $E=o(M)$ on each fixed
limiting tree ($M$ the weighted, dimensionless Ricci error).  The static input is
recorded below; the argument must verify the hypotheses for flow metrics and compare
the functional variation with spacetime Ricci energy.

Two estimates drive the argument.  On a nearly flat annulus the contracted
Bianchi identity removes the critical radial harmonic mode of $\Ric$, improving
the annular decay beyond the dimensional $r^{-2}$ rate so the loss outside the
collar is small against the energy inside.  The exterior beyond the collar is
built as an analytic function of the metric on the one fixed material collar,
with the longer exterior entering only an error estimate; hence the extension
differentiates in time on patches without a smoothly varying tree.  This rests
on the Saint-Venant relation \cite{Khavkine2017} and Korn control modulo rigid
motions \cite{CiarletCiarlet2005}, both used in the collar construction below.

\par\medskip\noindent\textbf{Proof structure and conventions.}

The static input is fixed first.  The scale estimates, genuine orbifold limits,
finite trees, and the continuous scale $a(t)$ follow, together with buffered
intervals on which $a$ changes by a fixed factor with root length tending to
zero after parabolic normalization.  The Ricci-equation estimates come next:
the exterior decay beyond the outermost scale, and the maximal and weighted
interior estimates controlling the root energy.  The collar construction then
builds the exterior extension and its error.  A final comparison of functional
variation with Ricci energy: an explicit lower bound $cK^2$ and upper bound
$\zeta_i(K+1)^2$ force $K\to0$; then distances are preserved so the
curvature-radius maximum defining $a$ is preserved, contradicting its
prescribed change, while absence of a concentration point gives a smooth
terminal metric and continuation on $M$.

Throughout, $\Delta=\tr\nabla^2$ and $|A|_{k,r}=\sum_{j=0}^k r^j|\nabla^jA|$ (the
pointwise magnitude unchanged, derivatives scaled).  Quotient-chart components
are Cartesian components on the Euclidean cover; constructions are equivariant
and descend.  Constants may depend on a fixed finite differentiation order and,
after extraction, on a fixed finite tree; uniformity in a sequence index, length
or collar factor $N$ is stated where needed, and a sequential little-oh is taken
after fixing the parameters in its statement.

\par\medskip\noindent\textbf{The static input and its normalization.}

The input from \cite[Theorem~2.2]{StaticTrees} is recorded here.  Constants and small
exponents belong to one fixed limiting tree; no uniform exponent across trees is
needed.

\par\medskip\noindent\textbf{Geometric data and the functional.}

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 per child.
Every end carries 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, and the two groups on an edge are identified with a fixed orthogonal
identification included in the charts.  Put $\tau=\log r$ on an exterior end and
$\tau=-\log r$ at a puncture.  In vertex units,
\begin{equation}\label{eq:end-decay}
 h_v=\delta+O(e^{-\mu\tau})
\end{equation}
on joined ends, with scaled derivatives, for some $\mu>0$; the unjoined
exterior end of the root requires $h_v-\delta=O(r^{-q})$ with $q>1$.  Bounds
through order $20$ suffice.

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

Assume the sequence of metrics has all $L_e\to\infty$ and converges to $h_v$ on
compacts of every open vertex, with uniform weighted Ricci control
\begin{equation}\label{eq:tree-ricci-error}
 |\rho^2\Ric|_j\le CM
 \begin{cases}e^{-\mu\tau},&\text{joined half-ends}\\ r^{-q},&\text{root exterior}\\ 1,&\text{cores,}\end{cases}
 \qquad j=10,
\end{equation}
with $M\to0$ (tensor magnitude physical; covariant derivatives scaled by powers
of $\rho$).  Decreasing $\mu$ or $q$ is allowed.

Define, in physical root units,
\begin{equation}\label{eq:E-def}
 E(g)=\lim_{D\to\infty}\Big\{\int_{r<D}R\,d\mu_g
 -\int_{r=D}(\partial_jg_{ij}-\partial_ig_{jj})n_i\,dS_\delta\Big\}.
\end{equation}
All chart integrals are quotient integrals.

\begin{lemma}[Renormalization and first variation]\label{lem:functional}
The functional \eqref{eq:E-def} is defined on the preceding metrics.  For a
variation with the same root decay,
\begin{equation}\label{eq:first-variation}
 DE_g(\dot g)=\int\big\langle\tfrac12Rg-\Ric,\dot g\big\rangle_g\,d\mu_g.
\end{equation}
It is invariant under diffeomorphisms isotopic to the identity whose root
displacement, with scaled derivatives along the isotopy, is $O(r^{1-q'})$,
$q'>1$.  A global scaling by $s$, with a coordinate dilation restoring
$\delta$, multiplies $E$ by $s^2$.  Under \eqref{eq:tree-ricci-error},
$|E|\le CM$.
\end{lemma}

\begin{proof}
On the root end write $g=\delta+p$.  The density obeys
$R\sqrt{\det g}=\partial_i(\partial_jp_{ij}-\partial_ip_{jj})+O(|p||\partial^2p|+|\partial p|^2)$,
the error $O(r^{-2q-2})$, radially integrable; this proves convergence.  The
linear boundary term in the variation cancels the variation of the subtracted
integral, and remaining boundary products are $O(D^{2-2q})\to0$; integration by
parts gives \eqref{eq:first-variation}.  For a Lie derivative the contracted
Bianchi identity kills the interior divergence and the boundary term vanishes by
the displacement decay, so the functional is invariant along the isotopy.
Differentiating the scaling law at $s=1$ with a compensating dilation (identity
in a fixed outer region, metric varied by $2g$ inside and $O(r^{-q})$ on the
tail, paired with an integrable $\Ric$) gives $2|E|\le CM$, hence $E=0$ for
$\Ric=0$.
\end{proof}

\begin{theorem}[Static tree inequality]\label{thm:tree-inequality}
This is \cite[Theorem~2.2]{StaticTrees}.  For the fixed tree and sequences above,
\begin{equation}\label{eq:tree-littleoh}
 E(g)=o(M),
\end{equation}
and for $M=0$ it means $E(g)=0$.
\end{theorem}

\par\medskip\noindent\textbf{Small-scale geometry and the outermost scale.}

Throughout, $(M^4,g(t))$, $0\le t<T$, is the closed flow with $|R|\le C_R$.
Constants may depend on the flow, $T$, and a fixed differentiation order.

\par\medskip\noindent\textbf{The spatial curvature radius.}

\begin{definition}\label{def:radius}
For a small cap $l_0>0$ and a material point $x$,
$l(x,t)=\sup\{0<r\le l_0:\sup_{B_t(x,r)}|\Rm|\le r^{-2}\}$.
\end{definition}

The definition uses the whole spatial ball.  Then $l$ is positive, continuous
before $T$, and $1$-Lipschitz: $|l(x,t)-l(y,t)|\le d_t(x,y)$ (a ball of radius
$r-d_t(x,y)$ at $y$ lies inside the radius-$r$ ball at $x$ with a larger allowed
bound).

\begin{theorem}[Scale estimates and degeneration]\label{thm:degeneration}
For a small cap and all sufficiently late times:
\begin{align}
 cr^4\le\Vol_t B_t(x,r)&\le Cr^4,\qquad \int_M|\Rm|^2\,d\mu_t\le C,
 \label{geom:volume-energy}\\
 |\nabla^k\Rm|(x,t)&\le C_kl^{-2-k},\qquad
 |\nabla^k\Ric|(x,t)\le C_kl^{-1-k}.\label{eq:ricci-improvement}
\end{align}
The set $\{l<r\}$ is covered by a bounded number of radius-$Cr$ balls.
Distances converge uniformly to a compact metric quotient $Z$; off finitely many
concentration points the metric converges smoothly.  Small-scale limits are
complete noncollapsed Ricci-flat orbifolds, smooth off finitely many genuine
orbifold points with nontrivial finite group acting freely on $S^3$; a nonflat
limit has one Euclidean quotient end with nontrivial group, a flat one is a
global Euclidean quotient with at most one exceptional point.  Within each
cluster the maximal inequivalent nonflat limits form a finite rooted tree with a
unique outermost entry, and the joining regions carry quantitative neck charts.
\end{theorem}

The proof occupies this section.

\par\medskip\noindent\textbf{Heat localization and two-sided persistence of the radius.}

Fix the terminal time in the smooth part and use $h$-dependent conjugate heat
kernels of the fixed metric, then localize them on radius-$h$ balls by a cutoff
$\chi$ equal to one on $B_t(x,h)$ and supported in $B_t(x,4h)$ with
$|\nabla\chi|$ and $h|\partial\chi|$ bounded.  The localized flow has bounded
curvature $Ch^{-2}$ and bounded first spatial derivatives, so its kernel obeys
the Gaussian upper and lower bounds and the derivative estimates on the ball; a
conjugate heat function equals a positive multiple of this localized solution,
and continuity plus the terminal value at the pole give
\begin{equation}\label{eq:heat-kernel}
 K\le Ch^{-n/2},\qquad K\ge c'(s)h^{-n/2}\ \text{on}\ B_s(x,h),\ s<h.
\end{equation}
The double integral bounds the backward heat flow $L^1\to L^\infty$ on
$[t-h^2,t]$ and the forward heat flow $L^\infty\to L^1$ on $[t,t+h^2]$ by
$Ch^{-n/2}$; together with the $L^2$ contraction and duality,
$\int v\,d\mu_t\le C\int v\,d\mu_{t+h^2}$ for $v\ge0$.  For nonnegative
$u_s(x)=u(s,x)$ with $|\nabla u_s|^2\le C_s$, taking $v=(u_s-u_t)_+$ gives
\begin{equation}\label{geom:backward}
 \int v\,d\mu_t\le C\big(\sqrt{|t-s|}+\sqrt{|t-s|}\,\widetilde C\big)
 \int|\nabla u_{s'}|\,d\mu_{s'},
\end{equation}
the form of Simon's backward uniqueness that needs only local bounds.  For
$t_0<s<t$ and $u_s(y)=d_t(x,y)$, this fails unless $d_t(x,y)\ge c'd(s,y)-r$.
Squaring and optimizing in $r^2\le cs$ gives spatial persistence
\begin{equation}\label{geom:persistence}
 d_t(x,y)\ge c'd_s(x,y)\quad\text{if }d_s(x,y)\le c''l_0.
\end{equation}

For the effect on the curvature radius fix a coordinate energy $e(t,\cdot)$ for
the covariant components of $\Rm$ in a fixed initial normal chart, with
$e(0,\cdot)=\sum_{j\le12}|\nabla^j\Rm|^2$.  For a small fixed $\beta>0$ the
smoothing estimate $\sup_{B(x,r/2)}e(t)\le C_\zeta r^{-6}\int_{B(x,r)}e(0)$
together with volume and derivative bounds gives $e(t)\ge|\Rm(t)|^2$ and
\begin{equation}\label{geom:slice-radius}
 |\Rm(g(t))|\le C'(\sup_{B(x,r)}|\Rm_0|^2+r^{-4}).
\end{equation}
Applying this to rescaled flows gives the two-sided statement: if a ball has
curvature radius $\ge r$ at time $t$, the earlier slice at distance $\ge r$ has
curvature radius $\ge cr$, and conversely (using the backward kernel).

\emph{Energy.}  With $C_0$ so that $1\le R+C_0\le C$, the quotient evolution
gives $(\partial_t-\Delta)|\Ric|^2/(R+C_0)\le C|\Rm||\Ric|^2-c|\Ric|^4$; the
volume derivative and Young's inequality give, for
$F=\int|\Ric|^2/(R+C_0)$, $F'\le C\int|\Rm|^2-c'\int|\Ric|^4+CF$.
Chern--Gauss--Bonnet $\int|\Rm|^2\le C+4\int|\Ric|^2\le C+CF$
\cite{Chern1944,SimonIntegral2020} (orientation cover if needed) and Gronwall
bound $F$; this is \eqref{geom:volume-energy}.

\emph{Bad set and fixed energy.}  If $l(y)\ge r$ and $r^2|\Rm|(y)\ge\varepsilon$, then the derivative bound gives $|\Rm|\ge\varepsilon/(2r^2)$
on $B(y,cr)$ and $\int_{B(y,c'r)}|\Rm|^2\ge c\varepsilon^2r^4$; volume comparability upgrades this to a definite Ricci energy $e_0>0$ on a smaller ball.  Now fix small $\rho$ and suppose $\{l<\rho\}$ needs more than $M$ radius-$C\rho$ balls.  Take a maximal family of disjoint unit balls centered at points $x_j\in\{C\rho\le l<\rho\}$ chosen by the primitive $x\mapsto l(x)\psi(\rho/l(x))$; the derivative bound on this primitive gives disjoint definite-energy balls, and the slice energy bound yields $M\le Ce_0^{-1}$.  Hence $\{l<\rho\}$ is covered by boundedly many radius-$C\rho$ balls, with the bound independent of $\rho$.

\emph{Convergence of distances and volume comparison.}  For $|t-s|\le cr^2$,
cover the bad set at the first time by bounded radius-$Cr$ balls, replace visits
to enlarged balls by first-to-last visits (boundedly many endpoints, total cost
$O(r)$ by \eqref{geom:persistence}), so that on the remaining path the
curvature scale is $\ge cr$ and the length changes by a factor $1+Cr$ since
$|R|\le C/r$.  Comparing both ways,
$|d_t(x,y)-d_s(x,y)|\le Cr(1+d_s+d_t)$, and taking $r\asymp\sqrt{|t-s|}$ gives
uniform convergence to a pseudodistance $d_T$ and a compact quotient $Z$.

The slicewise $L^3$ bound $\int_{B(x,h)}|\Ric|^3\le Ch$ will also be needed.  On
the shell $\{r_j\le l\le r_{j-1}\}$, $|\Ric|\le Cr_j^{-1}$ and the bad-set
measure bound established just above gives measure $\le Cr_j^4$, contributing
$Cr_j$.  Summing the dyadic shells up to $h$ yields $C\sum_jr_j\le Ch$.

In polar coordinates the polar Jacobian $P$ and $m=(\partial_r\log P-3/r)_+$
satisfy $(m^5P)'\le5|\Ric|m^4P-7m^5P/r-(2/3)m^6P$, giving
$\int m^6\le C\int|\Ric|^3\le Ch$ and hence, since $(P/r^3)'\le mP/r^3$,
\begin{equation}\label{eq:almost-bishop}
 \frac{\Vol B(x,h)}{h^4}\le\frac{\Vol B(x,h/2)}{(h/2)^4}+Ch^{1/2},
\end{equation}
while polar integration including the cut measure gives $\Delta(d^2/2)\le4+dm$
weakly.  Volume forms at two times differ by $e^{\pm C_R|t-s|}$.

\par\medskip\noindent\textbf{Slice limits, cones, and quantitative necks.}

Rescaled slices $h_i^{-2}g(t_i)$ based at $x_i$ with $h_i\to0$, $t_i\uparrow T$
are precompact as length spaces (two-sided volume bounds bound the number of
disjoint balls); the bad-set cover has bounded cardinality so a diagonal limit
identifies a finite exceptional set $S$.  Restricted maps are locally
bi-Lipschitz, extend by continuity, and preserve the injectivity radius bound, so
the limit is complete; smooth convergence off $S$ gives a smooth Ricci-flat
metric there, called regular.  Ball volumes converge and are regular
Riemannian volume.  The polar defect vanishes and Bishop becomes exact on the
regular part of the limit.

The exceptional set cannot separate positive-volume regular components: a
supposed separation by $S$ forces approximating minimizers through thin
neighborhoods; the polar inequality and Fubini over the passage give
\[
 c\rho\Vol(U)\le C\Big(\Vol(N_{C\rho}(S))+\int m\Big),
\]
first take the sequence limit and then $\rho\downarrow0$ to contradict.  Hence
the limit is a complete orbifold, singular only where point-picking gives
nontrivial concentrated curvature.

Write $d$ for distance from a puncture (or a fixed point at infinity).  Vanishing
annular energy rules out $r^2|\Rm|\ge\varepsilon$ in a middle annulus (energy
$c\varepsilon^6$ on a ball), and interior stationary estimates give the same for
every fixed scaled derivative.  Then $(4-\Delta)(d^2/2)=0$ distributionally with
$\Delta(d^2/2)=4$ and $|\nabla^2(d^2/2)|=4$, so $\nabla^2(d^2/2)=g$ and the end is
a cone $dr^2+r^2g_L$, $L=S^3/\Gamma$ for finite $\Gamma$ acting freely and
nontrivially (nontrivial group; two-sided volumes bound $|\Gamma|$).  Fixed-ratio
annuli are arbitrarily close to round quotient annuli, and overlapping annuli
identify the groups.

\begin{lemma}[Quantitative neck]\label{geom:neck}
If a nonflat complete Ricci-flat limit with $|\Rm|\le r^{-2}$ near a marked
point has $0<\mu<\sigma<1$ and $r_-<r_+$, then for small $r_-/r_+$ and large
enclosing annulus, after decreasing $\mu,\sigma$,
\begin{equation}\label{eq:neck-decay}
 |\Rm|(x)\le r^{-2}\big((r/r_+)^\sigma+(r_-/r)^\sigma\big),\qquad
 r_-<r<r_+,
\end{equation}
through all fixed scaled derivatives.
\end{lemma}

\begin{proof}
Trim the boundaries by a fixed factor (otherwise a sequence of violating annuli
with escaping ratios contradicts smooth convergence).  If the estimate fails,
choose approximate maximizers of $r^2|\Rm|/w_\mu(r)$, $w_\mu(r)=(r/r_+)^\mu+(r_-/r)^\mu$,
with ratios $Q_i\to\infty$ at $s_i$; both boundaries escape after dilation by
$s_i^{-1}$.  Dividing the rescaled curvature by $A_i=s_i^2|\Rm|(s_i)\to0$ gives a
bounded divided tensor.  The second Bianchi identity and
$\Delta\Rm=\Rm*\Rm+\nabla^2\Ric*$ make the divided quadratic term vanish and the
forcing vanish on fixed annuli; interior elliptic estimates yield a nonzero
limiting Cartesian tensor $H$, harmonic and satisfying the uncontracted Bianchi
identity on the flat punctured cover.  Spherical harmonics (radial powers $j$,
$-j-2$) and $\mu<1$ leave only $H=r^{-2}C$ for constant $C$; Bianchi then reads
$x_iC_{jkab}+x_jC_{kiab}+x_kC_{ijab}=0$ for all $x$, forcing $C_{\cdot\cdot ab}=0$,
contradiction.  Derivatives follow by interior elliptic estimates; slowly varying
boundary coefficients are absorbed by decreasing $\mu$.
\end{proof}

\par\medskip\noindent\textbf{Removal of punctures.}

Work near a punctured orbifold point.  A constant rescaling makes $g(0)=\delta$;
\eqref{eq:neck-decay} gives $|g-\delta|=O(r^\mu)$, $|Dg|=O(r^{\mu-1})$,
$|D^2g|=O(r^{\mu-2})$ on the lifted ball.  Harmonic coordinates solve
$\partial_i(A^{ij}\partial_jy^a)=0$ with $A^{ij}=\sqrt{\det g}\,g^{ij}$ and
$y^a=x^a$ on $\partial B_R$; coercivity gives a variational solution, and after
rescaling a weak gradient estimate with $C^{0,\alpha}$ forcing $O(R^\mu)$ gives
$\|Dy-I\|_\infty=O(R^\mu)$, so $y$ is a diffeomorphism for small $R$.  Uniqueness
makes it equivariant; recentering and a constant linear normalization give
$D^ky=O(r^{1+\alpha-k})$ for $k\ge2$ and the same for the inverse.  The
harmonic-coordinate metric obeys $G-G(0)=O(r^\alpha)$, $DG=O(r^{\alpha-1})$,
$D^2G=O(r^{\alpha-2})$.

Thus $G\in W^{2,p}$ for all $p<4/(2-\alpha)$; take $p_0=2+\alpha/2>2$.  No
distributional point defect occurs (integration by parts across $\partial B_\rho$
has first-derivative $O(\rho^3)$ and second-derivative $O(\rho^{2+\alpha})$
terms).  The harmonic-coordinate Einstein equation holds a.e.,
\begin{equation}\label{eq:weak-einstein}
 G^{ab}\partial_a\partial_bG_{ij}=Q_{ij}(G^{-1},DG,DG),
\end{equation}
with $Q$ quadratic in $DG$ \cite[Lemma~4.1]{DeTurckKazdan1981}.  For $2<p<4$,
$DG\in L^{4p/(4-p)}$ and elliptic regularity improves the exponent,
$p_{k+1}=2p_k/(4-p_k)$ with $1/p_k=\tfrac12-2^k(\tfrac12-1/p_0)$, reaching $p>4$
in finitely many steps; then $G\in C^{1,\beta}$, and Schauder theory with
\eqref{eq:weak-einstein} gives smoothness.  This fills the orbifold point without
integrability assumptions on Einstein deformations.

A trivial group is impossible: its smooth filling has near-Euclidean volume
ratios, which propagate (via \eqref{eq:almost-bishop}, error $O(h^{1/2})$) to a
smaller complete nonflat limit at a point-picked center near the filled center;
letting the start ratio go to Euclidean forces at least Euclidean ratio on fixed
balls of that limit, while Bishop gives the opposite at regular centers and the
radial-Laplacian/Hessian argument forces a smooth flat cone, contradiction.  The
end group of a nonflat limit is likewise nontrivial.  A complete flat limit
develops to a global Euclidean quotient with at most one tip (finiteness of the
deck group from sheet counting and noncollapse; the only possible singular point
is the tip).

\par\medskip\noindent\textbf{Terminal convergence, empty set, finite trees.}

For a compact set $K$ off the finite exceptional set, nested compactness and a
fixed radius $r$ control the whole remaining interval $[t_i,T)$ by spatial
persistence and a finite cover, giving fixed charts, bounded derivatives, and
smooth terminal convergence (local injectivity gives one sheet).  If the
exceptional set is empty, finitely many such charts bound curvature globally and
the DeTurck equation continues the flow past $T$ (principal part
$g^{ij}\partial_i\partial_jg$ after adding $\mathcal L_Wg$); so only a genuine
concentration point needs work.

Call two nonflat small-scale limits equivalent if their scales have bounded
ratio and centers are a bounded distance apart in either scale.  Each nonflat
limit contains a regular energy ball (orbifold filling, end decay, and the
stationary curvature bound give a ball of energy at least $e_1>0$).  Such balls
for inequivalent entries are disjoint in the slices; the total energy bounds the
number by $C/e_1$.  Dilating to curvature maximum and ordering give an ancestor
relation with a unique maximal entry; recursively this is the finite rooted tree
of Theorem~\ref{thm:degeneration}.  Intermediate limits between consecutive
entries are flat, so a child is a cone tip; on the flat annuli $l\ge cr$ and
\eqref{eq:ricci-improvement} gives $r^2|\Ric|_{k,r}\le Cr$, and
Lemma~\ref{geom:neck} supplies the necks with positive decay and the matching
quotient groups (end variables $\tau_p=\log(A/r)$, $\tau_c=\log(r/(Ae^{-2L}))$,
$\tau_p+\tau_c=2L$ realize exactly the static half-neck weights).

\par\medskip\noindent\textbf{The outermost scale.}

\begin{proposition}[A continuous outermost scale]\label{prop:root-scale}
Fix a small $c_*>0$ and a bounded continuous $\psi\ge0$ with $\psi=0$ on
$[0,c_*]$, $\psi>0$ on $[2c_*,1]$, and set
\begin{equation}\label{eq:root-def}
 a(t)=\sup_{y\in\mathcal U}l(y,t)\,\psi\big(l(y,t)^2|\Rm|(y,t)\big).
\end{equation}
After shrinking $\mathcal U$ the supremum is attained in the interior at every
late time; $a$ is positive, continuous, and $a(t)\to0$.  For every sequence of
late times, $a(t)$ and any maximizer represent the outermost nonflat entry, and
every other nonflat entry has scale at most $C a$.
\end{proposition}

\begin{proof}
Normalized regular energy balls detect curvature: at $\max|\Rm|=1$ the derivative
bound keeps $|\Rm|\ge1/2$ on a fixed ball, and removing tip neighborhoods leaves a
point at positive distance from every tip with curvature radius bounded above
and below, so $l^2|\Rm|\ge2c_*$ and $l$ is comparable to the vertex scale.  Near
the punctures $l^2|\Rm|<c_*/2$ eventually (definite-energy balls would accumulate
against finite energy).  Hence the integrand in \eqref{eq:root-def} is continuous
on a fixed compact interior set with an interior maximum, positive and continuous
in time; maximizing points go to the concentration point with $l\to0$, so
$a(t)\to0$.  At a maximizer $l\ge a/\|\psi\|_\infty$, $l^2|\Rm|>c_*$, and
$l\asymp a$ (else detection at another point forces a larger value).  Any listed
nonflat limit has scale $\le Ca$ by the same detection, and the maximizing limit
shares its unique outer ancestor with the maximal list, so it is outermost.
\end{proof}

\par\medskip\noindent\textbf{Motion on short intervals.}

\begin{lemma}[Curvature radius along minimizing segments]\label{lem:geodesic-radius}
For any minimizing segment $\gamma$ at a late time,
$l(\gamma(s),t)\ge c\min(l_0,s,D-s)$.  Consequently, if $d_t(x,y)\le Ch$,
\begin{equation}\label{eq:distance-log}
 |\partial_t d_t(x,y)|\le C'\big(1+\log^+(h/l(x,t))+\log^+(h/l(y,t))\big).
\end{equation}
\end{lemma}

\begin{proof}
Otherwise centered subsegments of length $2h_i$, $h_i=\sqrt{r_iL_i}$, rescale
to minimizing segments with an interior point of vanishing curvature radius,
which is a genuine cone vertex with link $S^3/\Gamma$ of diameter $<\pi$ (two
orbits at distance $\pi$ force a singleton, contradicting freeness); blowing up
at the vertex gives chords contradicting minimality.  The derivative bound
follows from $|\partial_td|\le\int_\gamma|\Ric|\,ds$, \eqref{eq:ricci-improvement},
and $l(\gamma(s))\ge c(l(x)+s)$ on the halves.
\end{proof}

\begin{lemma}[Uniform short motion]\label{lem:slow-motion}
Let $h_i\to0$ and intervals $J_i$ of length $\delta_i=o(h_i)$ tending to $T$.
Distances of size $O(h_i)$ change by $o(h_i)$ throughout $J_i$; a point with
$l\ge ch_i$ retains $l\ge c'h_i$; regular charts at this scale change by
$O(\delta_i/h_i)$ with scaled derivatives.
\end{lemma}

\begin{proof}
Dense retained material set from the summability of
$\int_M\log^+(h_i/l)$ (bounded by $Ch_i^4$) and the volume bounds; on retained
endpoints \eqref{eq:distance-log} integrates to $o(h_i)$ change; a first-crossing
argument keeps ranges bounded.  Persistence of regular charts follows from the
two-sided curvature persistence and the metric-difference evolution.
\end{proof}

\begin{lemma}[Time for a fixed scale change]\label{lem:scale-change-time}
If $a(t_i)/a(s_i)\to1$ along $s_i<t_i<T$ with $\log(a(t_i)/a(s_i))\to0$ while
$(t_i-s_i)/a(s_i)^2$ stays bounded, then the root scale cannot change by a fixed
factor.
\end{lemma}

\begin{proof}
Rescale lengths by $a(s_i)$; the duration is $o(a)$, so Lemma~\ref{lem:slow-motion}
freezes the pointed root metric and its regular curvature.  The radius defining
$a$ converges on regular parts to the radius computed in that limit with tips as
stops, maximizers stay in a common bounded root range, and the rescaled supremum
of \eqref{eq:root-def} agrees at both times, excluding a fixed-factor change.
\end{proof}

\par\medskip\noindent\textbf{Buffered intervals, motion, and exterior geometry.}

\begin{proposition}[Buffered intervals]\label{prop:buffered-intervals}
Fix $u>0$ and choose $t_0\le t_1\le\cdots<t_N<T$ a maximal $u$-separated
sequence for $-\log a(t)$.  Then $N\to\infty$ and
$\theta:=\inf_j(t_{j+1}-t_j)/(t_j-t_{j-1})>0$.
\end{proposition}

\begin{proof}
Successive gaps $d_j$ satisfy $d_j/a_j^2\to\infty$ and $d_j\ge ce^{-2uj}$ by
Lemma~\ref{lem:scale-change-time}.  A geometric lower bound and the maximality
prevent two consecutive tiny ratios beyond a fixed factor.
\end{proof}

On a buffered interval $I=[-\theta,1+\theta]d_j$ with endpoints $t_{j},t_{j+1}$
and $d_j=t_{j+1}-t_j$, in interval units $g_{\rm new}=d_j^{-1}g$ one has
$|R_{\rm new}|\le\epsilon_i^2$ with $\epsilon_i=\sqrt{d_i}\to0$,
$a_i=a(t_j)/\epsilon_i\to0$ throughout $I$ (up to a fixed factor), and the root
scale at $0$ and $1$ differs by a fixed factor.  By Lemma~\ref{lem:slow-motion},
geometry at length $h\lesssim1$ moves negligibly over windows $\Delta t$ with
$\epsilon\Delta t/h\to0$; at root scale centers move $O(a_i)$, at $a_i\ll h\lesssim1$
by $o(h)$ (a second concentration at separation $h$ would give a larger nonflat
ancestor).

\begin{proposition}[Uniform exterior geometry]\label{prop:exterior-geometry}
Fix a high order $k_{\rm geo}$ and $0<\mu<\sigma<1$.  There are
$N_{\rm geom},C,c_{\rm geom}>0$ and $R_i^{\rm out}\to\infty$ with
$\epsilon_iR_i^{\rm out}\to0$ such that at every time in $I$ the exterior admits
one equivariant coordinate system with
\begin{equation}\label{eq:uniform-exterior}
 r^2|\Rm|_{k_{\rm geo},r}+|g-\delta|_{k_{\rm geo},r}\le C\Theta_i(r),\quad
 \Theta_i(r)=(a_i/r)^\mu+\delta_ir^\mu,\quad \delta_i=(R_i^{\rm out})^{-\mu}\to0,
\end{equation}
$N_{\rm geom}a_i\le r\le c_{\rm geom}R_i^{\rm out}$, covering whole shortened
distance bands, group fixed after subsequence, $\epsilon_i=o(\delta_i)$.
\end{proposition}

\begin{proof}
Choose $R_i^{\rm out}\to\infty$ with $\epsilon_iR_i^{\rm out}\to0$.  For
$r_i/a_i\to\infty$ with $\epsilon_ir_i\to0$ the maximal-tree and root-detection
arguments force a flat quotient annulus limit.  This sequential flatness yields
one $N_{\rm geom}$ making all annuli close to flat uniformly.  On these annuli
$l\ge cr$ so $r^2|\Ric|\le C\epsilon_ir\le C\epsilon_iR_i^{\rm out}(r/R_i^{\rm out})^\sigma$,
and Lemma~\ref{geom:neck} with the fixed exponent $\mu$ (its constant uniform,
boundaries escaping under normalization) gives \eqref{eq:uniform-exterior} on
shortened annuli.  Since $\epsilon_iR_i^{\rm out}\to0$ and $\mu<1$,
$\delta_i/\epsilon_i\to\infty$.
\end{proof}

\par\medskip\noindent\textbf{Concentration estimates for the Ricci equation.}

Let $G=\eta\Ric\in C^2_0$, $\eta$ a spatial cutoff of fixed width, maximal on the
$N_1a$-ball and supported in the $2N_1a$-ball, and
\begin{equation}\label{eq:root-energy}
 a^4K^2=\int_I\eta^2\int_{B_t(x_t,N_1a)}|\Ric|^2\,d\mu\,dt.
\end{equation}
In material coordinates the lower-index Ricci equation, the cutoff equation, and
the divergence constraint read
\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\big(G_{ij}-\tfrac12g_{ij}\tr_gG\big)&=0,\label{para:bianchi}
\end{align}
the last because $\eta$ is spatially constant.  On a regular patch set
$G_r=r^{-2}\Phi_r^*G$; then \eqref{eq:ricci-improvement} gives
\begin{equation}\label{eq:source-scale}
 |\Ric|_{j,r}\le C_j\epsilon/r,\qquad |r^2\eta'\Ric|_{j,r}\le C_j\epsilon r,
\end{equation}
so the cutoff source needs no division by $\eta$.

The geometric inputs established above, taken sequentially, are:
(i) on regular charts with $l\ge ch$, coefficients move by
$O(\epsilon\Delta t/h)$ over windows; (ii) centers move $O(a)$ at root scale and
$o(h)$ for $a/h\to0$; (iii) distance annuli $Ca\le r\le C'$ carry the quotient
charts of \eqref{eq:uniform-exterior} with error through any finite order bounded
by $C\Theta$, $\Theta=(a/r)^\mu+\delta_ir^\mu$, $\epsilon=o(\delta)$.

\begin{lemma}[Local smoothing with time $L^p$ input]\label{lem:local-smoothing}
On a regular parabolic patch normalized to unit radius and duration, with
uniformly elliptic scalar principal part and bounded coefficient derivatives, a
solution $U$ with source $F$ satisfies, for nested cylinders $Q'\Subset Q$ with
positive past margin and $1<p\le2$,
\begin{equation}\label{eq:local-mixed}
 \sup_{Q'}\sum_{k=0}^j|D^kU|\le C\Big[\Big(\tfrac1{|J|}\int_J\|U\|_{L^2(\Omega)}^p\Big)^{1/p}
 +\Big(\tfrac1{|J|}\int_J\|F\|_{H^{j+3}(\Omega)}^p\Big)^{1/p}\Big],
\end{equation}
the first input optionally the spacetime $L^p$ norm of $U$; longer windows follow
by sliding unit cylinders.
\end{lemma}

\begin{proof}
Spatial cutoff, integration by parts, Young, and pairing the source time-$L^1$
norm with the time supremum (absorbing half) give the $L^2_\tau H^{m+1}$ bound;
differentiating spatially, the top commutator absorbs and lower ones are bounded,
so $m=j+3$ with $H^{j+3}\hookrightarrow C^j$ gives exponent two for $U$ and one
for $F$.  The change to $L^p_\tau$ uses interpolation of the time $L^2$ norm
against the supremum plus Young's inequality with nested margin weights.
\end{proof}

Write $P$ for the componentwise spherical average on a lifted quotient annulus
(it commutes with the flat heat operator and preserves equivariance).

\begin{lemma}[Radial inequality for the oscillation]\label{lem:radial}
Suppose $PZ=Z$, $Z$ compactly supported in time, $E=(\partial_\tau-\Delta_\delta)Z$.
For $1<p\le2$, with $z(s)=\|\,\|Z(s,\cdot,\tau)\|_{L^2(S^3)}\|_{L^p_\tau}$,
\begin{equation}\label{eq:radial-inequality}
 -z''-\tfrac3sz'+\tfrac{3}{s^2}z\le\|\,\|E(s,\cdot,\tau)\|_{L^2(S^3)}\|_{L^p_\tau}.
\end{equation}
For $p=2$ this holds for a Hilbert direct sum over charts/windows with the same
radial variable.  Homogeneous comparison powers are $s$ and $s^{-3}$.
\end{lemma}

\begin{proof}
With $w=\|Z\|_2$, pairing against $Z/w$ and convexity give
$w_\tau-w_{ss}-3s^{-1}w_s+3s^{-2}w\le\|E\|_2$; multiply by $w^{p-1}$, integrate
in time (boundary term vanishes), and use
$\int w^{p-1}w_{ss}\le z^{p-1}z''$ from differentiating
$\int w^{p-1}w_s=z^{p-1}z'$ and Cauchy.  H\"older bounds the source, division
proves the inequality; zeros are handled by regularization.
$\mathcal L(s^\beta)=(3-\beta(\beta+2))s^{\beta-2}$ has zero exponents $1,-3$.
\end{proof}

\begin{lemma}[Control of the mean derivatives]\label{lem:mean}
If $T$ is a spatially constant multiple of $\Ric$ and the annular metric is
$C^{j+1}$-close to $\delta$ at scale $e$, then
\begin{equation}\label{eq:mean-derivatives}
 \sum_{k=1}^j s^k|\nabla^kT|\le C_j\Big(\sum_{k=1}^j s^k\sup|D^kPT|+e\sup|T|_{j,s}\Big)
\end{equation}
on a fixed-width shell, with suprema on a fixed enlargement.
\end{lemma}

\begin{proof}
The trace-reversal $S=T-\tfrac12\delta\tr_\delta T=\overline S(s)+PS$; the
divergence constraint and flat divergence differ by a term bounded by
$e\sup|T|_{k+1,s}$, and radial differentiation of
$\overline S'_{ij}n_i=\mathcal E_j-\partial_i(PS)_{ij}$ along coordinate vectors
bounds $\overline S$ and its radial derivatives, hence the Cartesian derivatives.
Conversion to covariant derivatives adds only metric-error terms.
\end{proof}

\par\medskip\noindent\textbf{The exterior estimate.}

For dyadic radii based on $a$ let
$\mathcal A_t(r)=\{r/2\le d_t(x_t,y)\le2r\}$ and
$A(r)=\|\sup_{\mathcal A_t(r)}|G|\|_{L^2(I)}$,
$V(r)=\|\sup_{\mathcal A_t(r)}\sum_{j=1}^mr^j|\nabla^jG|\|_{L^2(I)}$, $m$ a fixed
high order.

\begin{proposition}[Exterior concentration bound]\label{prop:exterior-ricci}
There are fixed $N_0\gg1$, $r_0>0$, $\alpha\in(0,\min(\mu,1))$, and $N_1$ in
\eqref{eq:root-energy} such that
\begin{align}
 V(r)&\le C\big[(a/r)^{2+\alpha}K+\widetilde\delta_i((a/r)^\alpha+r^\alpha)\big],
 \label{eq:exterior-v}\\
 A(r)&\le C\big[(a/r)^{2+\alpha}K+\widetilde\delta_i\big],
 \label{eq:exterior-a}
\end{align}
$N_0a\le r\le r_0$, $\widetilde\delta_i\to0$, $\widetilde\delta_i\ge\epsilon$.
\end{proposition}

\begin{proof}
Fix a large dyadic $H$ then $T_0$; use annuli of radii in $[1/(cH),cH]r$ and
windows $T_0r^2$.  The motion error at the smallest comparable radius is
$C\epsilon T_0Hr\le C_{H,T_0}\Theta(r)$, so metric error through all required
orders is $C_{H,T_0}\Theta$ on the window.  Applying Lemma~\ref{lem:local-smoothing}
on bounded-width patches and sliding windows bounds true-field derivatives by
$C_{H,T_0}(\max_{\rho\asymp r}A(\rho)+\epsilon r)$ in $L^2$.  On fixed-width
middle patches (past margin inside $\chi_\ell\equiv1$), integrating $z^2$ against
$s^3ds$ on a larger middle annulus and summing squares gives
\begin{equation}\label{eq:exterior-recurrence}
 V(r)\le C\big[H^{-3}V_*(r/H)+H^{-1}V_*(Hr)\big]+\tfrac{C_H}{T_0}\max_{\rho\asymp r}V(\rho)
 +C_{H,T_0}\big[\Theta\max_{\rho\asymp r}A(\rho)+\epsilon r\big],
\end{equation}
with $V_*$ over fixed neighbors.  Joining shells by paths of length $C\rho$ and
integrating $\nabla G$ telescopes:
$A(r)\le C\sum_{r\le\rho\le r_0}V(\rho)+C\epsilon$, and boundary ranges have
$A+V\le C_{H,r_0}\epsilon$ (outer) and $A+V\le C((a/r)^2K+\epsilon r)$ (inner,
$N_1$ chosen to include the patches).  The metric terms in the weighted sum use
$\Theta(r)/((a/r)^\alpha+r^\alpha)\le(a/r)^{\mu-\alpha}+\delta_ir^{\mu-\alpha}$;
increase $N_0$ for the first, then take late index for $\delta_ir^{\mu-\alpha}$,
absorbing into $M/4$.  Choices are made in order $H,T_0,N_0$, late index; then
$M\le C+M/2$ proves \eqref{eq:exterior-v} and \eqref{eq:exterior-a}.
\end{proof}

\par\medskip\noindent\textbf{The maximal function controlling the root.}

Define the centered maximal function of $f=\eta|\Ric|_{j+2,a}$ (with
$f^p$ for $1<p\le2$): $H_*(t)=(\iint_{B_t(x_t,D_1a)}f^p)^{1/p}$ over the past
window, so that
\begin{equation}\label{eq:maximal-ricci}
 H_*(t)\le C_p\Big[a^2\big(\int_I\eta^2\int_{B_t(x_t,D_1a)}|\Ric|_{j+2,a}^2\big)^{1/2}+\epsilon\Big]
 \le C_p(K+\epsilon),
\end{equation}
using $\|f\|_2\le C(K+o_i(1))$ from the energy ball and the shell decay
\eqref{eq:exterior-a}, and the weak-$L^1$ bound plus extrapolation
$\|\mathcal Mv\|_q^q\le C_q\int v^q$ ($v=f^p$, $q=2/p$).

\begin{proposition}[Interior estimate with a neck weight]\label{prop:interior-ricci}
Extract a tree at times with $\eta(t)\ne0$.  On its true compact and half-neck
pieces in $B_t(x_t,D_0a)$, $r$ the local regular length in $I$-units, for each
fixed order $j$,
\begin{equation}\label{eq:interior-weighted}
 (r/a)^2|G|_{j,r}(t)\le C(H_*(t)+\epsilon)e^{-\nu D},
\end{equation}
where $D=0$ on compact pieces and on a joining neck $D$ is the minimum of the
logarithmic distances to the two ends; constants may depend on the tree.
\end{proposition}

\begin{proof}
Compact/bounded-depth: apply Lemma~\ref{lem:local-smoothing} over a past window
$cr^2$, $L^2$ input bounded by $Cf$, source $\le C\epsilon$; no exponential.
Deep neck: put $h=e^{-4\lambda D}$, $B=e^{\lambda D}$; the metric error through
all orders is $e_D=Ce^{-\delta D}$, $\delta>3\lambda$ (choose $\lambda$ with
$\min\{\mu(1-4\lambda),1-6\lambda,3-3\lambda\}>3\lambda$).  Write
$T_1=(r/a)^2G$, $Q=H_*+\epsilon$; sliding-window integration gives
$\sup_s|T_1|_{J,s}(t)\le CQs^{-2}$ and the forcing at all orders.  Applying
Lemma~\ref{lem:radial} to $Z=\chi PT_1$ with a large-window cutoff, the operator
and curvature error costs $CQe^{-\delta D}s^{-4}$ and the cutoff $CQB^{-2}s^{-2}$;
the barrier $CQ(e^{-\delta D}s^{-4}+B^{-2}s^{-2})$ gives the bound, and radial
integration of the first Cartesian derivative from $B/4$ plus the four positive
$D$-exponents $2\lambda,(4-2/p)\lambda,(2-2/p)\lambda,\delta-(2/p)\lambda$ yield
\eqref{eq:interior-weighted} with $\nu$ below their minimum.  The uncut bound
$r^2|\Ric|_{j,r}\le C\epsilon r\le o_i(1)e^{-\nu D}$ holds with $\nu<1$.
\end{proof}

\par\medskip\noindent\textbf{An exterior extension determined by a fixed collar.}

An exterior is attached to the true metric on a fixed annular collar, defined as
an analytic function of the collar data.  Smoothness in the collar metric lets
the extension be differentiated in time without choosing a smoothly varying
tree; the far exterior enters the error estimate only.

All lengths are in collar units; $C_1=\{1\le|x|\le2\}/\Gamma$,
$\mathcal E=\{|x|\ge1\}/\Gamma$, $\Gamma\subset O(4)$ finite acting freely on
$S^3$.  For a fixed-ratio shell use
$\|v\|_{H^s(r)}^2=\sum_{j=0}^sr^{2j-4}\int|\partial^jv|^2$ and
$|v|_{C^s,r}=\sum_{j=0}^sr^j\sup|\partial^jv|$, and
$\|v\|_{H^s_\beta(\mathcal E)}=\sup_r r^\beta\|v\|_{H^s(r)}$ (filling a ball on
filled Euclidean space).  Shells range over finitely many neighbors, with width
chosen once independent of the outer radius $R$.

\begin{theorem}[Collar extension]\label{thm:collar-extension}
There are $\sigma_*>0$, $R_*<\infty$, $C<\infty$ such that for $\sigma\le\sigma_*$,
$R\ge R_*$ there is a metric $\bar h$ on $\mathcal E$ with
\begin{equation}\label{eq:collar-extension}
 \|\bar h-\delta\|_{H^k_q}\le C\|g-\delta\|_{H^k(C_1)},\qquad
 \|g-\bar h\|_{H^k(C_1)}+\|\Ric(\bar h)\|_{H^{k-2}_{q+2}}\le Cm_1.
\end{equation}
The map $g|_{C_1}\mapsto\bar h-\delta$ is real analytic on a fixed neighborhood
of zero in collar $H^k$, with uniformly bounded first derivative into $H^k_q$,
independent of $R$, the weights, and the exterior.  There is also an analytic
extension $\hat h$ equal to $g$ near the inner boundary with the same bounds.
\end{theorem}

\par\medskip\noindent\textbf{Local linear estimates.}

With $Sv,\mathcal K(v)$ the flat linearizations of Ricci and Riemann curvature
and $Bv=\partial^iv_{ij}-\tfrac12\partial_j\tr_\delta v$,
\begin{equation}\label{eq:flat-identities}
 Sv=-\tfrac12\Delta v+\tfrac12\mathcal L_{(Bv)^\sharp}\delta,\quad BSv=0,\quad
 \Delta\mathcal K(v)=\mathcal T(\partial^2Sv),\quad
 S\mathcal L_V\delta=\mathcal K(\mathcal L_V\delta)=0.
\end{equation}

\begin{lemma}[Local potential and Korn]\label{collar:potential}
On a fixed Euclidean annulus/ball/overlap there is bounded linear
$v\mapsto Pv\in H^{k+1}$ with
\begin{equation}\label{eq:local-potential}
 \|v-\mathcal L_{Pv}\delta\|_{H^k}\le C\|\mathcal K(v)\|_{H^{k-2}},\quad
 \|Pv\|_{H^{k+1}}\le C\|v\|_{H^k},
\end{equation}
and for vector fields,
$\inf_{W\in\mathfrak e(4)}\|V-W\|_{H^{k+1}}\le C\|\mathcal L_V\delta\|_{H^k}$,
$\mathfrak e(4)=\{x\mapsto Ax+b:A^t=-A\}$.  Choices are linear, normalized modulo
rigid motions, and equivariant; after dilation the first right side is multiplied
by $r^2$ and vector norms divided by $r$.
\end{lemma}

\begin{proof}
Regard the linearized connection $C^a_{ij}=\tfrac12(\partial_ip_{ja}+\partial_jp_{ia}-\partial_ap_{ij})$
as a one-form in $i$ (fixed $a,j$) with curl $K(v)$; the Neumann gradient
projection writes it as $dV^a_j$ plus an $H^{k-1}$ error controlled by $K(v)$;
symmetry in $i,j$ and a second projection give $C(v)-D^2V$ bounded in $H^{k-1}$
by $C\|\mathcal K(v)\|_{H^{k-2}}$; then $\partial_i(v_{aj}-(\mathcal L_V\delta)_{aj})=C^a_{ij}+C^j_{ia}$
controls $D(v-\mathcal L_V\delta)$, and Poincar\'e plus an affine field of half a
constant symmetric matrix completes the estimate.  For Korn,
$2\partial_i\partial_jV_a=\partial_ie_{ja}+\partial_je_{ia}-\partial_ae_{ij}$ with
$e=S V$ controls second and higher derivatives; Poincar\'e controls $V$ modulo an
affine map whose symmetric part is controlled, leaving exactly $\mathfrak e(4)$,
fixed linearly.  $\Gamma$-averaging preserves the bounds.
\end{proof}

\begin{lemma}[Newton estimates]\label{collar:newton}
For $0<q<2$, $\Delta^{-1}:H^{k-2}_{q+2}(\R^4)\to H^k_q(\R^4)$ is bounded.  For
$1<q<2$, if $h\in H^{k-3}_{q+3}$ has $\int h=0$ componentwise then
\begin{equation}\label{eq:zero-mass-newton}
 \|\Delta^{-1}h\|_{H^{k-1}_{q+1}}\le C\|h\|_{H^{k-3}_{q+3}},
\end{equation}
and on an annulus of outer radius $O(R)$, if $\Delta W=H$, boundary strips control
$W$ in scaled $H^{k-2}$ by $b_0$, $c_r=r^2\|H\|_{H^{k-4}(r)}$, then
\begin{equation}\label{eq:green-sum}
 \sup_r\|W\|_{H^{k-2}(r)}\le C\Big(b_0+\sum_rc_r\Big),
\end{equation}
independent of the number of shells.
\end{lemma}

\begin{proof}
Weighted Newton estimates for shell-supremum norms (compare
\cite[Theorem~1.7]{Bartnik1986}): split the kernel integral into inner, middle,
outer parts; the scaled interior estimate for $\Delta u=f$ gives all orders.  For
\eqref{eq:zero-mass-newton} the zero integral permits the subtracted kernel and
$q>1$.  For \eqref{eq:green-sum} the Dirichlet Green function obeys
$|G(x,y)|\le C|x-y|^{-2}$; one shell contributes at most $Cc_r$ everywhere, so the
maximum principle and local estimate give the shell-uniform bound.
\end{proof}

\par\medskip\noindent\textbf{A filled-space inverse and the analytic map.}

Let $\mathcal E_0$ contain the exterior and the filled unit ball with one smooth
metric piece; if $R$ is the outer radius use larger balls as needed.  For
$q\ge0$ let $\|v\|_{H^s_q(\mathcal E_0)}=\sup_{r\ge1}r^q\|v\|_{H^s(r)}$, and set
\[
 \mathcal X=\{v\in H^k_q(\mathcal E_0):v=\mathcal L_V\delta\text{ near the inner
 boundary for some vector }V\},\quad
 \mathcal Y=\{(v_1,f):Sv=f\text{ on }C_1,\ Bf=0\}.
\]
The filling identifies $\mathcal X$ with the $H^k$ collar modulo rigid motions,
and $Bf=0$ is imposed in the specified function spaces.

Define $A:\mathcal X\to\mathcal Y$, $v\mapsto(v|_{C_1},Sv)$, and the analytic
piecewise map $Q$: on $|x|<1$ take an inward harmonic extension $f=Sw$ and its
Newton potential; on $C_1$ interpolate between $Sp_1$ and $f$ with a fixed cutoff;
outside use the Newton potential; then replace the vector contribution by the
harmonic choice of Lemma~\ref{collar:potential} and an analytic quadratic
correction.  $Q$ is real analytic and $Q^2=Q$, and
\begin{equation}\label{eq:generalized-inverse}
 ALA=A,
\end{equation}
where $L:\mathcal Y\to\mathcal X$ is the following bounded linear operator.  On
the filled unit ball $f^\#=S(Ev_1)$, on $C_1$
$f^\#=\chi Sv_1+(1-\chi)f$, outside $f^\#=f$; set
$u=\Delta^{-1}(-2f^\#)$ (Newton, Lemma~\ref{collar:newton}),
$Ld=u|_{\mathcal E}+\mathcal L_{V_d}\delta$ with $V_d$ from
Lemma~\ref{collar:potential} applied to $v_1-u|_{C_1}$ and extended linearly; it is
bounded, with
\begin{equation}\label{eq:L-residual}
 \|v_1-(Ld)|_{C_1}\|_{H^k}\le C\|\mathcal K(v_1-u)\|_{H^{k-2}(C_1)},\qquad
 S(Ld)=Su.
\end{equation}

For \eqref{eq:generalized-inverse} take $d=Av$, fill $v$ by the same inward
extension to $V$ on all of $\R^4$.  Then $f^\#=SV$, $Bf^\#=0$, so $Bu=0$ and
$Su=f^\#$; $\mathcal K(V-u)$ is entire harmonic and decaying, hence
$\mathcal K(v-u)=0$ on $C_1$, giving $(Ld)|_{C_1}=v$ and $S(Ld)=Sv$.  Filling the
hole is essential (a decaying harmonic field on the punctured exterior need not
vanish).

Set $Q=LA$; \eqref{eq:generalized-inverse} gives $Q^2=Q$ and $AQ=A$, and a
linear operator $A':\operatorname{im}Q\to\mathcal X$ with $A'Q=1$ identifies
$\operatorname{im}Q$ with $\mathcal X$, transporting $A$ and the real analytic $Q$
to $A$ and a real analytic $B$ with $AB=1$, $\operatorname{im}B=\ker A=\operatorname{im}Q$.
A bounded linear $\mathcal N:\mathcal E\to\mathcal X$ with
$\mathcal Nv=P_V(\varphi v)$ for a fixed compactly supported $\varphi\equiv1$ on
$C_1$ and the $H^k$ collar projection makes a small closed metric family
solvable, real analytic in the data.  The metric $\bar h=(1-B\mathcal N)\delta$
depends analytically on the collar data with the stated bounds, and the extension
is independent of the exterior realizing the collar.

\par\medskip\noindent\textbf{A comparison gauge and the residual.}

For tensors on $1\le|x|\le R$ there are linear $P_0$, $\mathcal Dz=\mathcal L_{P_0z}\delta$,
zero on $C_1$, with
\begin{equation}\label{eq:long-projection}
 \|z-\mathcal Dz\|_{H^k(r)}\le C\Big(r^2\max_{\rho\asymp r}\|\mathcal K(z)\|_{H^{k-2}(\rho)}
 +\mathbf1_{r\le C_0}\|z\|_{H^k(C_1)}\Big),
\end{equation}
$C_0$ a fixed width.  On roomy dyadic overlaps Lemma~\ref{collar:potential} gives
local $v_j$; on overlaps $v_{j+1}-v_j=A_jx+w_j$ with $A_j$ skew, $\Gamma$-commuting,
$\|w_j\|$ controlled by the local errors; there is no equivariant translation.
For a radial partition the strain has only neighboring differences, so $P_0z$ has
length-independent bounds; $\mathcal D^2=\mathcal D$ and $\mathcal D$ is bounded
for the weight $n(z)=\sup_r\theta_r^{-1}\|z\|_{H^k(r)}$.  One may write
$P_0z=J(x)x+U(x)$, $J^t=-J$, $J\gamma=\gamma J$, $J=U=0$ on $C_1$, with
$\sup|J|\le C\sigma n(z)$ and $\|J\|$, $r^{-1}\|U\|$ controlled by $\theta_rn(z)$.

Using this gauge (a comparison device only) the metric is close to Euclidean,
with $p:=F^*g-\bar h$ satisfying $\mathcal Dp=0$, $p|_{C_1}=p_1$, and
\[
 Y:=\sup_{r\le R}r^{-2}\|p\|_{H^k(r)}\le C(e+X),\quad
 X:=\sup_{r\le R}\|\mathcal K(p)\|_{H^{k-2}(r)}.
\]
The coordinate Ricci formula and the gauge give
$Sp=T-\Ric(\bar h)+E_2=T+f+E_2$ with $\|E_2\|_{H^{k-2}(r)}\le C\theta_rY$
(nonlinear terms $\|N(v)-N(w)\|\le Cr^{-2}(\|v\|+\|w\|)\|v-w\|$, both first
factors $O(\theta_r)$, last $r^2Y$), giving the residual bound
\begin{equation}\label{eq:collar-residual}
 \|T\|_{H^{k-2}(C_1)}+\sum_{r\le R}\sum_{j=1}^{k-2}\|r^j\partial^jT\|_{H^0(r)}
 \le C\big(m_1+\sigma N_0+R^{-q-2}\big).
\end{equation}

Run $L$ on $d_1=(p_1,f)$, $e=\|d_1\|_{\mathcal Y}$; $ALd_1=0$.  It remains to show
$d_1-ALd_1$ is small.  With $u=\Delta^{-1}(-2f^\#)$,
$\|Bu\|_{H^{k-1}_{q+1}}\le CD_0$ ($D_0=m_1+\sigma e$), and
$Su-f^\#=\tfrac12\mathcal L_{(Bu)^\sharp}\delta$ gives
\begin{equation}\label{eq:ricci-compatibility}
 \|Su-f\|_{H^{k-2}_{q+2}(\mathcal E)}\le CD_0.
\end{equation}
Fill $p$ by $Ep_1$ to $P$ on the full ball and let $W=\mathcal K(P-u)$; on the
filled inner ball $SP-f^\#=0$, on the collar bounded by $CD_0$, so the shell sum
of $\Delta W=\mathcal T(\partial^2S(P-u))$ is $\le C(m_1+\sigma e)$, with the
filled ball contributing one fixed term.  At radius $R$,
$\|W\|_{H^{k-2}(R)}\le C(m_1+eR^{-q-2})$, and the filled-ball Green sum with the
fixed-scale estimate near $C_1$ gives
\begin{equation}\label{eq:collar-curvature-residual}
 \|\mathcal K(p_1-u)\|_{H^{k-2}(C_1)}\le C\big(m_1+(\sigma+R^{-q-2})e\big),
\end{equation}
with no inner boundary term proportional to $e$.  Combining with
\eqref{eq:L-residual} and $ALd_1=0$,
$\|d_1-ALd_1\|_{\mathcal Y}\le C(m_1+(\sigma+R^{-q-2})e)$; choosing $\sigma_*$
small and $R_*$ large makes the $e$-coefficient at most one half, hence
$e\le Cm_1$ with length-independent constants, proving \eqref{eq:collar-extension}.
An interpolation $\hat h=\bar h+\zeta(g-\bar h)$ with a fixed radial $\zeta$
gives the analytic extension $\hat h$.

\par\medskip\noindent\textbf{Functional change and the contradiction.}

Apply the results to a buffered interval $I=[-\theta,1+\theta]$ in rescaled
time, $\epsilon_i=\sqrt{d_i}$, $a=a_i\to0$; the root scale is comparable to $a$
on $I$ and its values at $0$ and $1$ differ by a fixed factor.  $|R|\le C\epsilon_i^2$.
Fix the small collar tolerance and a large fixed dyadic $N\gg N_1$, $b=Na$; all
limits are with $N$ fixed.  Constants multiplying $N^{-2\alpha}K^2$ are
independent of $N$; those multiplying $o_i(1)$ may depend on $N$.

\begin{lemma}[Position of a controlled material collar]\label{final:collar-position}
Choose small $\sigma_0>0$ and large fixed $N$.  Choose an exterior annular chart
at scale $b$ with domain $\{1/8<|x|<16\}/\Gamma$ near $\supp\eta$ and keep its
material map and group fixed.  For late $i$, throughout any interval on which its
metric stays within $\sigma_0$ of flat through the required order: every
shortened collar is at distance comparable to $b$ from $x_t$; its middle bands
cover the shortened distance annuli; and the inner side of a middle sphere
contains $B_t(x_t,N_1a)$ and lies in $B_t(x_t,Cb)$.
\end{lemma}

\begin{proof}
Initial properties hold by annular convergence.  Suppose the positional assertion
fails for small tolerances and large $N$.  Then
there are late times, tolerances tending to zero and numbers $N\to\infty$ along
which the fixed chart violates it, and the chart still lies near the same
isolated point of $Z$.  At scale $b$, Theorem~\ref{thm:degeneration}
gives a complete flat quotient limit (a nonflat limit at this large scale would
contradict outermostness); smooth compactness extracts the chart maps into its
regular part and a limiting local flat isometry.  Groups are fixed after passing
to a subsequence (close orthogonal representations are conjugate by averaging and
a polar factor), the source group is nontrivial and free, the target is a global
Euclidean quotient with at most one tip, lifting to Euclidean covers gives a
rigid motion whose translation vanishes by the nontrivial free spherical action.
There is no multiplicity (two interior points would have an $o(b)$ path).  The
tip must be $o(b)$ from $x_t$; otherwise the separation-scale limit has two
genuine tips at distance one, hence is nonflat at scale $h_i/a_i\to\infty$,
contradicting outermostness.  Crossing the radial bands and a fixed outward
material point identify the geometric and material inner components.
\end{proof}

At each time complete exterior coordinates agree with the fixed chart on $C_1$;
the transmission with the fixed chart has weight $\theta_r\le C((\sigma_0+N^{-\mu})r^{-\mu}+\delta_i(rb)^\mu)$,
$1\le r\le4R$, $Rb\asymp r_*$, with the first dyadic sum small and the second
tending to zero.  Coordinates outside $C_1$ are used only for the error.

A bound $B_1(t)$ for $|\Ric|_{k+4,b}$ on a fixed ample band around $x_t$ (chart
independent by suprema over choices) defines
$u(t)=\int_{t_*}^t(1+B_1)ds$ and, by first exit,
$\|g(t)-g(s)\|_{C^{k+3}}\le C\int_s^tB_1$.  A smooth partition in $u$ gives
$\sum_j\chi_j=1$ near $\supp\eta$ with $\sum_j|\chi_j'|\le C(1+B_1)$ a.e.  On
each patch, retain the true metric inside and extend by Theorem~\ref{thm:collar-extension};
the extension $\hat g_j$ has functional $E_j$.

\begin{lemma}[Time-integrated collar error]\label{final:collar-bound}
The residual is bounded by a common $m(t)$ with
\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\big[(a/b)^{2+\alpha}K+\rho_i\big],
\end{equation}
$\rho_i\to0$.
\end{lemma}

\begin{proof}
In collar units \eqref{eq:ricci-improvement} bounds Ricci and scaled derivatives
on the radius-$r$ shell by $C\epsilon b/r$; the $r^{-1}$ sum is bounded, weights
$\theta_r$ are summable, and $\theta_R/R^2\to0$.  For the $L^2$ bound, $V(\rho)$
and $\theta_rA(\rho)$ control the terms and the leading energy sums give
$C(a/b)^{2+\alpha}K$ with the stated errors.
\end{proof}

\begin{proposition}[Positive variation]\label{final:variation}
\[
 \sum_j\int_I\eta^2\chi_jE_j'\,dt\ge2a^4K^2-a^4\big[CN^{-2\alpha}K^2+o_i(1)(K+1)^2\big].
\]
\end{proposition}

\begin{proof}
On the retained true part
$\langle\tfrac12Rg-\Ric,\partial_tg\rangle=2|\Ric|^2-R^2$, containing
$B_t(x_t,N_1a)$ of volume at most $Cb^4$, so the scalar-square loss is
$o_i(a^4)$.  On the collar and end $\Ric(\hat g_j)$ and $\partial_t\hat g_j$ obey
the collar weights; normalizing the collar multiplies the covariant tensor by
$b^{-2}$, giving the tail bound $Cb^2mB_1\int_1^\infty r^{1-2q}dr\le Cb^2mB_1$
(here $q>1$ is used).  Summing against $\chi_j$ and Cauchy--Schwarz,
\[
 Cb^2\|\eta m\|_2\|\eta B_1\|_2\le Cb^4\big[(a/b)^{2+\alpha}K+\rho_i\big]^2,
\]
whose leading factor is $b^4(a/b)^{4+2\alpha}=a^4N^{-2\alpha}$; the rest is
$o_i(a^4)(K+1)^2$.  The positive core term is at least $2a^4K^2$ by
\eqref{eq:root-energy}.
\end{proof}

\begin{proposition}[Uniform functional smallness]\label{final:functional-smallness}
For fixed $N$ there are $\zeta_i\to0$ with
\begin{equation}\label{eq:functional-littleoh}
 |\eta(t)E_j(t)|\le\zeta_ia^4\big[H_*(t)+1+a^{-2}\eta(t)m(t)\big]
\end{equation}
at every time in every active patch.
\end{proposition}

\begin{proof}
Verify the static hypotheses after choosing $t_i$ with $\eta>0$ and active $j_i$,
rescaling by $a_i$.  Theorem~\ref{thm:degeneration} and
Proposition~\ref{prop:root-scale} extract a finite rooted tree (all smaller
concentrations lie inside the collar, else a larger nonflat entry), and
Theorem~\ref{thm:tree-inequality} applies: compact and joining regions have the
sequential identifications with one-sided half-neck weights, and the new root
exterior has uniform $O(r^{-q})$ control with $q>1$.  Let $M_i$ bound the weighted
Ricci error; choosing the internal weight exponent below both the half-neck and
\eqref{eq:interior-weighted} exponents makes $M_i\to0$ without $\eta$.  Elsewhere
\eqref{eq:interior-weighted} gives
$\eta(t_i)M_i\le C_{\rm ext}a_i^2(H_*(t_i)+\epsilon_i)+C_{\rm ext}\eta(t_i)m(t_i)$.
Then $|E_i^{\rm root}|=o(M_i)$, $E_{j_i}(t_i)=a_i^2E_i^{\rm root}$, and
\[
 \frac{|\eta E_{j_i}|}{a_i^4[H_*+1+a_i^{-2}\eta m]}\le C_{\rm ext}\frac{|E_i^{\rm root}|}{M_i}\to0.
\]
A uniform failure of \eqref{eq:functional-littleoh}, with active times, would give
the opposite on a subsequence, contradiction.
\end{proof}

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

\begin{proof}
Each $\eta^2\chi_j$ has compact support in its patch, so integration by parts
gives $\sum_j\int\eta^2\chi_jE_j'=-\sum_j\int(2\eta\eta'\chi_j+\eta^2\chi_j')E_j$,
bounded by $C\zeta_ia^4\int[H_*+1+a^{-2}\eta m][|\eta'|+\eta(1+B_1)]$.  With
$a^{-2}=N^2b^{-2}$, Minkowski and \eqref{eq:maximal-ricci},\eqref{eq:collar-time-bound}
bound this by $o_i(a^4)(K+1)^2$.  Fix $N$ with $CN^{-2\alpha}<1$; then
$K_i^2\le\gamma_i(K_i+1)^2$, giving $K_i\le\sqrt{\gamma_i}/(1-\sqrt{\gamma_i})\to0$,
and boundedness of $K$ in passing.
\end{proof}

\par\medskip\noindent\textbf{Small energy fixes the root geometry.}

With $K_i\to0$, the pointed root geometry is preserved: for $x,y$ in a small
neighborhood of the concentration point and $d_t(x,y)\le Ca$, integration of
$a^{-1}|\partial_td_t|\le CQ_*[1+a/l(x,t)+a/l(y,t)]$ (from the geodesic
curvature-radius estimate and \eqref{eq:interior-weighted}) over the interval
shows the scaled distance changes by $o(a)$ (a first-crossing argument keeps
bounded ranges).  Thus the rescaled distance functions converge, and the
curvature-radius maximum defining $a$ is preserved across $I$, contradicting the
prescribed factor change.  Hence a buffered interval with fixed-factor change
cannot exist, so no concentration point survives.

\par\medskip\noindent\textbf{Continuation.}

Absence of a concentration point gives a smooth limiting metric on $M$ at $T$
(finitely many curvature-bounded charts, the DeTurck principal part, and the
metric/connection evolution), and the DeTurck equation started there continues the
flow on $M$ past $T$ with matching spatial derivatives.  Retaining the given flow
for $t<T$ and this solution for $t\ge T$ produces a smooth Ricci flow
$\widetilde g$ on $M\times[0,T+\varepsilon)$ with $\widetilde g=g$ for $t<T$,
proving Theorem~\ref{thm:extension}.  The argument uses arbitrary smooth initial
data in real dimension four and continues on the original manifold, not an
orbifold.

\begin{thebibliography}{99}
\bibitem{Hamilton1982} R.~S. Hamilton, Three-manifolds with positive Ricci
curvature, J. Differential Geom. 17 (1982), 255--306.
\bibitem{Sesum2005} N.~\v{S}e\v{s}um, Curvature tensor under the Ricci flow,
Amer. J. Math. 127 (2005), 1315--1324.
\bibitem{BamlerZhangI} R.~H. Bamler and Q.~S. Zhang, Heat kernel and curvature
bounds in Ricci flows with bounded scalar curvature, Adv. Math. 319 (2017),
396--450.
\bibitem{BamlerZhangII} R.~H. Bamler and Q.~S. Zhang, Heat kernel and curvature
bounds in Ricci flows with bounded scalar curvature---Part II, Calc. Var. PDE 58
(2019), 49.
\bibitem{SimonExtension} M.~Simon, Extending four dimensional Ricci flows with
bounded scalar curvature, Comm. Anal. Geom. 28 (2020), 1683--1754.
\bibitem{BamlerConvergence} R.~H. Bamler, Convergence of Ricci flows with bounded
scalar curvature, Ann. of Math. 188 (2018), 561--632.
\bibitem{LiuSimon} J.~Liu and M.~Simon, Volume estimates and convergence for
Ricci flow with bounded scalar curvature, Calc. Var. PDE 65 (2026), 161.
\bibitem{Zhang2010Kahler} Z.~Zhang, Scalar curvature behavior for finite-time
singularity of K\"ahler--Ricci flow, Michigan Math. J. 59 (2010), 419--433.
\bibitem{EndersMuellerTopping2011} J.~Enders, R.~M\"uller, P.~M. Topping, On type-I
singularities in Ricci flow, Comm. Anal. Geom. 19 (2011), 905--922.
\bibitem{BuzanoDiMatteo2026} R.~Buzano and G.~Di Matteo, A local singularity
analysis for Ricci flow, Part II, arXiv:2606.12379 (2026).
\bibitem{Anderson1989} M.~T. Anderson, Ricci curvature bounds and Einstein metrics
on compact manifolds, J. Amer. Math. Soc. 2 (1989), 455--490.
\bibitem{Bando1990} S.~Bando, Bubbling out of Einstein manifolds, Tohoku Math. J.
42 (1990), 205--216.
\bibitem{BandoCorrection1990} S.~Bando, Correction and addition, Tohoku Math. J.
42 (1990), 587--588.
\bibitem{BandoKasueNakajima1989} S.~Bando, A.~Kasue, H.~Nakajima, On a
construction of coordinates at infinity, Invent. Math. 97 (1989), 313--349.
\bibitem{CheegerTian2006} J.~Cheeger and G.~Tian, Curvature and injectivity
radius estimates for Einstein 4-manifolds, J. Amer. Math. Soc. 19 (2006), 487--525.
\bibitem{SimonIntegral2020} M.~Simon, Some integral curvature estimates for the
Ricci flow in four dimensions, Comm. Anal. Geom. 28 (2020), 707--727.
\bibitem{Chern1944} S.-S. Chern, A simple intrinsic proof of the Gauss--Bonnet
formula, Ann. of Math. 45 (1944), 747--752.
\bibitem{CheegerGromovTaylor1982} J.~Cheeger, M.~Gromov, M.~Taylor, Finite
propagation speed and the geometry of complete Riemannian manifolds, J.
Differential Geom. 17 (1982), 15--53.
\bibitem{DeTurckKazdan1981} D.~DeTurck and J.~Kazdan, Some regularity theorems in
Riemannian geometry, Ann. Sci. ENS 14 (1981), 249--260.
\bibitem{GilbargTrudinger2001} D.~Gilbarg and N.~S. Trudinger, Elliptic Partial
Differential Equations of Second Order, Springer, 2001.
\bibitem{ChiarenzaFrascaLongo1993} F.~Chiarenza, M.~Frasca, P.~Longo, $W^{2,p}$-solvability
of the Dirichlet problem for nondivergence elliptic equations with VMO
coefficients, Trans. AMS 336 (1993), 841--853.
\bibitem{DeruelleOzuch2025} A.~Deruelle and T.~Ozuch, A \L ojasiewicz inequality
for ALE metrics, Ann. Scuola Norm. Sup. Pisa 26 (2025), 1965--2042.
\bibitem{Khavkine2017} I.~Khavkine, The Calabi complex and Killing sheaf
cohomology, J. Geom. Phys. 113 (2017), 131--169.
\bibitem{CiarletCiarlet2005} P.~G. Ciarlet and P.~Ciarlet Jr., Another approach
to linearized elasticity and a new proof of Korn's inequality, M3AS 15 (2005),
259--271.
\bibitem{Bartnik1986} R.~Bartnik, The mass of an asymptotically flat manifold,
Comm. Pure Appl. Math. 39 (1986), 661--693.
\bibitem{StaticTrees} OpenAI, Path selection and an elliptic inequality on
degenerating Ricci-flat trees, OpenAI Math Release preprint, 2026.
\end{thebibliography}

\end{document}