==== TABLE OF CONTENTS ====
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//01-introduction.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//02-framework.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//03-kernel.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//04-construction.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//05-cold.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//06-early.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//07-global.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//08-compactness.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//09-dilation.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//10-nonconcentration.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//11-branching.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//12-separation.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//jet-geometry.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//main.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//preamble.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//scales.tex
Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//references.bib
===========================

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//01-introduction.tex ====
\section{Introduction}\label{sec:introduction}

The Boltzmann equation describes the transport and binary collisions of a
dilute gas. Its collision law conserves mass, momentum, and kinetic energy
at each spatial point, while its entropy decreases. A weak solution theory
must retain enough of these properties to distinguish the equation from
a balance law with an unidentified collision defect. This paper constructs
two global solutions with the same initial datum for which all five local
collision-invariant balances hold exactly.

The connection between binary collisions and entropy dissipation goes
back to Boltzmann's kinetic theory~\cite{Boltzmann1872}. For the
spatially inhomogeneous Cauchy problem, DiPerna and Lions established
the large-data theory of global existence and weak stability for
renormalized solutions~\cite{DiPernaLions1989}. On periodic space the
initial assumptions are finite mass, energy, and entropy, and the
solutions obey the entropy-production inequality
\cite[Theorem~1 and Remark~2]{DiPernaLions1991}.
Renormalization addresses a basic difficulty: these bounds alone need
not make the quadratic collision term locally integrable.

Exact conservation is an additional issue at this level of regularity.
The periodic formulation in
\cite[Theorem~4.1]{LevermoreMasmoudi2010} gives local mass conservation,
but allows a nonnegative matrix-valued defect in the momentum balance
and a corresponding defect in total energy. It does not automatically
give exact local momentum and energy conservation. The present result
concerns solutions that obey both of these local identities, with the
energy flux integrable and with the prescribed initial traces.
Their separate collision gains and losses are integrable with every
polynomial velocity weight on each finite time interval.

Uniqueness is available under stronger control. Kaniel and Shinbrot
proved local existence and uniqueness in classes bounded by a multiple
of a Maxwellian~\cite{KanielShinbrot1978}. Ukai obtained global
existence and uniqueness for small periodic perturbations of a Maxwellian
in weighted spatial Sobolev spaces~\cite[Theorem~1.2]{Ukai1974}.
Duan, Huang, Wang, and Yang later allowed large but bounded amplitudes
on the torus or in the whole space, assuming small relative entropy and
a small spatially integrated velocity supremum of the
perturbation~\cite[Theorem~1.1]{DuanHuangWangYang2017}.
Our datum instead has unbounded amplitudes at shrinking spatial scales.
The small relative entropy used in its construction therefore does not
place it in these uniqueness classes.

For the spatially homogeneous hard-sphere equation, finite mass and
energy suffice for uniqueness within the conservative class
\cite[Theorem~1.1]{MischlerWennberg1999}. Wennberg's homogeneous
nonuniqueness examples for hard potentials allow increasing
energy~\cite{Wennberg1999}. The mechanism studied here uses spatial
concentration and preserves energy locally. Thus the distinction between
homogeneous and inhomogeneous evolution, as well as the precise
conservation requirements, matters for comparison.

Recent instability results address a different aspect of the Cauchy
problem. Chen and Holmer prove failure of uniform continuity in weighted
Sobolev spaces for the constant collision kernel on
\(\R^3_x\times\R^3_v\)~\cite[Theorem~1.3]{ChenHolmer2024}.
Chen, Shen, and Zhang extended this analysis to a range of cutoff soft
potentials~\cite{ChenShenZhang2024}. For hard spheres on the whole space,
Chen, Guo, Shen, and Zhang prove failure of uniform continuity from
Gaussian-weighted initial data to the unweighted \(L^2_vH^s_x\) norm
for \(0\le s<1\)
\cite[Theorem~1.1]{ChenGuoShenZhang2026}. These results compare nearby
initial data, whereas the two solutions constructed here have exactly
the same datum. For the Krieger--Strain (isotropic Landau) model,
Gismondi, Golding, and Novack construct rough stationary solutions whose
mollified collision operators tend to zero, and conjecture nonuniqueness
of renormalized solutions to the inhomogeneous Landau and Boltzmann
Cauchy problems
\cite[Section~1.2]{GismondiGoldingNovack2026}.

\subsection{The equation and the solution class}

Let \(\T^3=\R^3/\mathbb Z^3\) have volume one and put
\(\Omega=\T^3_x\times\R^3_v\). Write
\(\D=\partial_t+v\cdot\nabla_x\). For \(v,v_*\in\R^3\) and
\(\omega\in S^2\), define the elastic collision by
\[
 a=(v-v_*)\cdot\omega,\qquad
 v'=v-a\omega,\qquad v_*'=v_*+a\omega.
\]
The sphere carries ordinary surface measure. The hard-sphere equation is
\begin{equation}\label{eq:collision}
 \D F=Q(F,F)=Q^+(F,F)-Q^-(F,F),\qquad F\ge0,
\end{equation}
where
\[
\begin{split}
 Q^+(F,F)(v)&=\int_{\R^3\times S^2}|a|F(v')F(v_*')\dd v_*\dd\omega,\\
 Q^-(F,F)(v)&=\int_{\R^3\times S^2}|a|F(v)F(v_*)\dd v_*\dd\omega.
\end{split}
\]
Every factor has the same time and position. We use primes and stars with
the same convention in the entropy dissipation below.

Set \(M(v)=(2\pi)^{-3/2}e^{-|v|^2/2}\) and \(G=F/M\). Define
\begin{equation}\label{eq:entropy}
\begin{split}
 \mathcal H_M(F)&=\int_\Omega
       \bigl(F\log(F/M)-F+M\bigr)\dd x\dd v,\\
 \mathcal D(F)&=\frac14\int_{\T^3\times\R^3\times\R^3\times S^2}
 |a|(F'F_*'-FF_*)\log\frac{F'F_*'}{FF_*}
       \dd x\dd v\dd v_*\dd\omega.
\end{split}
\end{equation}
The dissipation integrand takes its nonnegative lower-semicontinuous
extended value at zero. We set \(0|\log0|=0\).

\begin{definition}[Entropy solution with local conservation]
\label{def:admissible}
Let \(F_0\ge0\) have finite mass, energy, and absolute entropy.
A global entropy solution with local conservation is a nonnegative
function \(F\) satisfying the following conditions.
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(F\in C_w([0,\infty);L^1(\Omega))\), \(F(0)=F_0\), with
continuity against every bounded measurable test. For every finite \(T\),
\[
 \sup_{0\le t\le T}\int_\Omega
 F(t)(1+|v|^2+|\log F(t)|)\dd x\dd v<\infty.
\]
\item The two functions \(Q^\pm(F,F)/(1+G)\) belong to
\(L^1([0,T]\times\Omega)\) for every finite \(T\).
For every \(\beta\in C^1([0,\infty))\) with \(\beta(0)=0\) and
\(|\beta'(z)|\le C_\beta/(1+z)\), and every smooth \(\varphi\) periodic
in \(x\) and compactly supported in \([0,\infty)\times\R^3_v\),
\begin{equation}\label{eq:renormalized}
\begin{split}
 &\int_0^\infty\!\!\int_\Omega
 \left[M\beta(G)\D\varphi+\beta'(G)Q(F,F)\varphi\right]
                  \dd x\dd v\dd t\\
 &\hspace{15mm}
 +\int_\Omega M\beta(F_0/M)\varphi(0,x,v)\dd x\dd v=0.
\end{split}
\end{equation}
\item Total momentum is conserved. At every \(t\ge0\),
\[
 \int_\Omega |v|^2F(t)\dd x\dd v\le\int_\Omega |v|^2F_0\dd x\dd v,
 \qquad
 \mathcal H_M(F(t))+\int_0^t\mathcal D(F(s))\dd s
      \le\mathcal H_M(F_0).
\]
\item For every finite \(T\),
\[
 \int_0^T\!\!\int_\Omega (1+|v|^3)F\dd x\dd v\dd t<\infty.
\]
For every \(\psi\in\{1,v_1,v_2,v_3,|v|^2/2\}\) and
\(\phi\in C_c^\infty([0,\infty)\times\T^3)\),
\begin{equation}\label{eq:local-balances}
 \int_0^\infty\!\!\int_\Omega F\psi\D\phi\dd x\dd v\dd t
   +\int_\Omega F_0\psi\phi(0,x)\dd x\dd v=0.
\end{equation}
\end{enumerate}
\end{definition}

The last condition is local conservation of mass, momentum, and kinetic
energy, including the initial traces. In particular it is stronger than
requiring only the corresponding global inequalities.

\begin{theorem}\label{thm:main}
There exist \(R<\infty\) and a nonnegative datum \(F_0\) with
\[
 \supp_vF_0\subset B_R(0),\qquad
 \int_\Omega F_0(1+|v|^2+|\log F_0|)\dd x\dd v<\infty,
\]
for which \eqref{eq:collision} has two global entropy solutions with local
conservation in the sense of Definition~\ref{def:admissible}, differing on
a set of positive \((t,x,v)\)-measure. Both solutions belong to
\(C([0,\infty);L^1(\Omega))\). Moreover, for every finite \(T\) and
every \(k\ge0\), they satisfy
\begin{equation}\label{eq:strong-collision-integrability}
 \int_0^T\!\!\int_\Omega
 \langle v\rangle^k\bigl[Q^+(F,F)+Q^-(F,F)\bigr]
                       \dd x\dd v\dd t<\infty,
 \qquad \langle v\rangle=(1+|v|^2)^{1/2}.
\end{equation}
\end{theorem}

\subsection{The construction and its new global estimate}

The starting point is the concentrating-jet mechanism of
\cite{OpenAIPeriodic}. That construction produces nonuniqueness in an
entropy class. Its continuation theorem does not supply the exact local
momentum and energy identities required here. We retain the jet mechanism
and construct the global continuation directly. The kernel estimates,
the detection of a nonzero rescaled solution, and its branching growth
are reproduced below with attribution; the new estimates concern the
interaction of those jets with a surrounding gas over arbitrarily long
times.

A cold jet at scale \(s\) occupies a thin transverse annulus and has a
small velocity width. Its velocity-integrated density can be of order \(1/s\), but its total
mass is of order \(s^2\). Geometrically decreasing scales therefore allow
singular collision frequencies near time zero while retaining finite
mass, energy, and entropy. The opposite axial streams are separated into
fine spatial stripes. Under dilation those stripes average to a
two-stream medium; a small additional population then follows a linear
equation in that medium.

There are two obstacles to extending this mechanism while preserving all
local conservation laws. First, thin jets eventually meet their periodic
copies. Second, an entropy bound alone does not give the separate
integrability of gain and loss needed to pass every local balance to a
limit. We address both by giving each scale a finite lifetime as a
separate component. The jet axis is chosen in a quadratic-irrational
direction, which prevents intersections before that lifetime ends. A
nearly Maxwellian bath supplies a uniform positive collision frequency
after a fixed short time. The jets then decay exponentially, so that each
can be added to a remainder with a uniform weighted bound at the end of
its lifetime.

The central continuation estimate combines three facts. On a fixed
weighted order interval, small velocity mass makes the normalized gain
small. Small relative entropy controls the spatial integral of the
excess over the Maxwellian. Finally, integration over velocities along
free flights turns this spatial smallness into a pointwise mass bound.
These facts close a global weighted supremum estimate even though the
initial cold components have no common finite supremum. The same
construction makes the contributions of the small scales to the
weighted collision integrals summable. This yields
\eqref{eq:strong-collision-integrability}, which is the decisive input
for exact local conservation and strong time traces.

For nonuniqueness, the approximation with no additional initial
population will be called dormant. In a second family, a vanishing seed
is varied until its evolution attains a prescribed time-dependent bound.
Dilation at an attaining time produces a nonzero solution of the linear
equation in the two-stream medium; nonconcentration prevents the attained
size from disappearing. Branching growth forces the rescaled limit to
have a finite time horizon. The dormant solution, meanwhile, tends to
zero away from the velocity axis under the normalized dilation by the
original horizon. Finiteness makes the two spatial and temporal dilation
scales comparable. If the two global limits agreed at arbitrarily small
horizons, comparing their dilations would force the nonzero limit to
vanish.

\paragraph{Organization.}
Section~\ref{sec:framework} fixes the gain normalization and the velocity
weight. Section~\ref{sec:kernel} proves the kernel estimates.
Sections~\ref{sec:construction}--\ref{sec:early} construct the data,
control the cold jets, and select the short-time approximations.
Section~\ref{sec:global} proves the global continuation estimate, and
Section~\ref{sec:compactness} proves admissibility and local conservation
of the limits. Sections~\ref{sec:dilation}--\ref{sec:branching} establish
the nonzero rescaled limit and its finite horizon.
Section~\ref{sec:separation} completes the proof of
Theorem~\ref{thm:main}.
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//01-introduction.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//02-framework.tex ====
\section{Coordinates and collision estimates}\label{sec:framework}

We first fix the normalization used in the estimates and explain the
singular velocity weight. Throughout the construction we rotate space
and velocity by the same orthogonal map, taking the original direction
\((1,\sqrt2,0)/\sqrt3\) to the third coordinate axis. The period lattice
becomes a rotation \(\Lambda\) of \(\mathbb Z^3\), still of covolume one.
The equation and all the properties in Definition~\ref{def:admissible}
are invariant under this change of coordinates. We work on
\(\R^3/\Lambda\), or in its periodic lift, until the last step of the
proof. Write \(x=(y,z)\in\R^2\times\R\) and
\(v=(\bar v,v_3)\). Dilations in space are always taken in the lift.

Let \(P\) denote the bilinear gain
\[
 P(f,g)(v)=\int_{\R^3\times S^2}
 |(v-v_*)\cdot\omega|f(v')g(v_*')\dd v_*\dd\omega.
\]
For the hard-sphere kernel it is symmetric in \(f,g\), as verified in
Lemma~\ref{lem:carleman}; in particular \(P(F,F)=Q^+(F,F)\).
The loss frequency is
\begin{equation}\label{eq:frequency}
 \nu_f(v)=2\pi\int_{\R^3}|v-p|f(p)\dd p,
 \qquad Q^-(F,F)=F\nu_F.
\end{equation}
A superscript \(b\), where \(b\ge1\), means that the collision kernel is
multiplied by
\begin{equation}\label{eq:cutoff}
 \min\{1,b/|v-v_*|\}.
\end{equation}
Its value at zero is immaterial. The same factor is used in gain and
loss; thus \(\nu_f^b(v)=2\pi\int\min\{b,|v-p|\}f(p)\dd p\).
It preserves particle interchange and the elastic change of variables,
and is exactly one on any fixed bounded velocity region for all
sufficiently large \(b\).

Fix \(1<q<2\) and \(m=100\). Our velocity weight is
\begin{equation}\label{eq:weight}
 H(v)=e^{-|v|^2}\langle v\rangle^{-m}W(v),
 \qquad W(v)=1+|\bar v|^{-q},\qquad
 \|f\|_H=\esssup_{x,v}\frac{|f(x,v)|}{H(v)}.
\end{equation}
The value assigned on the velocity axis is irrelevant. All supremum
estimates are essential estimates away from that axis. Since its
transverse dimension is two and \(q<2\), \(H\) is integrable with every
polynomial velocity weight; it also belongs to \(L^{1+\varepsilon}\)
for small enough \(\varepsilon>0\). The singularity permits gain
estimates uniform as a cold velocity distribution collapses onto the
axis. The inequality \(q>1\) is used in the corresponding plane
integrals.

Constants denoted \(C\) may depend on fixed construction parameters,
but not on the approximation index or the shrinking horizon unless
explicitly stated. The choices of the four parameters governing
branching growth will be made in Section~\ref{sec:branching}; all earlier
arguments hold for any fixed parameters in the indicated ranges. The
remaining small parameters are then chosen in the order specified by
the continuation proof.

\paragraph{Transport representatives.}
An integrable distributional equation \(\D f=R\) is read in free
coordinates, \(f^\sharp(t,x,v)=f(t,x+tv,v)\). Its time sections are
absolutely continuous for almost every \((x,v)\), with derivative
\(R^\sharp\). We use these representatives in path inequalities.
Fubini's theorem makes such inequalities valid for almost every
endpoint at any prescribed time. At the finite approximations the
gain kernels have absolutely continuous hot-input marginals; together
with free translations this permits iteration on common full-measure
endpoint sets. The later limiting path formulas will be established
from their integrable sources. No assertion depends on values assigned
to the velocity axis or other null sets.
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//02-framework.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//03-kernel.tex ====
\section{Collision kernels and a velocity majorant}
\label{ker:section}\label{sec:kernel}

We establish gain bounds for narrow velocity distributions and for
the weight $H$. They control the finite evolutions and the global
continuation. The sheet and quadratic estimates are adapted from
\cite[Section~3]{OpenAIPeriodic}; their proofs are included below,
followed by the small-mass estimate needed for continuation.

Write $v=(\bar v,v_3)\in\R^2\times\R$ and set
\[
 W(v)=1+|\bar v|^{-q},\qquad
 H(v)=e^{-|v|^2}\langle v\rangle^{-m}W(v),
 \qquad 1<q<2,\quad m=100,
\]
as in \eqref{eq:weight}. Values on the velocity axis are irrelevant to
all the almost-everywhere estimates below. The function $H$ has every
polynomial moment and belongs to $L^p(\R^3)$ whenever $1\leq p<2/q$.
Indeed, its only local singularity is $|\bar v|^{-q}$ in two dimensions,
and its decay at infinity is Gaussian. Every estimate below is proved
for an arbitrary fixed $q\in(1,2)$; it may therefore also be applied
with a different fixed exponent in that interval.

A model for the cold velocities, in parameters
$w=(\bar w,w_3)\in\R^3$, is
\[
 \bar p=d\bar w,\qquad p_3=\upsilon(z)+dw_3,
\]
where $d>0$ is small and $\upsilon$ is bounded, nondecreasing, and
$C^1$. At a fixed position after free transport, the axial parameter
relation and its Jacobian become
\[
 p_3=\upsilon(z-tp_3)+dw_3,\qquad
 \frac{\partial p_3}{\partial w_3}
   =\frac{d}{1+t\upsilon'(z-tp_3)}.
\]
A fixed parameter law can therefore produce strong axial
concentration. We will estimate gains using transverse Gaussian
majorants while imposing no density bound on the axial law. For any
nonnegative velocity density $c$ and $Z\geq0$, positivity gives
\[
 0\leq h\leq ZH\quad\Longrightarrow\quad
 \frac{P(c,h)}{H}\leq Z\frac{P(c,H)}{H}.
\]
Thus $P(c,H)/H$ controls the gain in the weighted supremum. The
collision formula below realizes this coefficient as the mass of a
normalized measure on the two input velocities.

\subsection{The exact Carleman formula}

The hyperplane representation used here is a Carleman formula;
see also \cite[Appendix~C]{GambaPanferovVillani2009}.
We derive its coefficient for the normalization of this paper.

\begin{lemma}[Collision geometry]\label{lem:carleman}
For the normalization in \eqref{eq:collision}, the symmetric bilinear
gain is
\begin{equation}\label{ker:carleman}
 P(c,h)(v)
 =2\int_{\R^3}c(p)\,\dd p
       \int_{\R^3}
       \delta\bigl((v-p)\cdot(\xi-v)\bigr)h(\xi)\,\dd\xi.
\end{equation}
For fixed $p\ne v$, the inner delta denotes surface measure on the
plane $(v-p)\cdot(\xi-v)=0$, divided by $|v-p|$.
For incoming velocities $p,\xi$, the outgoing measure has total mass
$2\pi|p-\xi|$ and is uniform on the sphere with diameter $[p,\xi]$.
Formula~\eqref{ker:carleman} also defines the gain when $c$ is a
nonnegative velocity measure and $h$ is a nonnegative density, whenever
the displayed integral is finite almost everywhere.
\end{lemma}

\begin{proof}
Put $g=p-\xi$. Reflection sends $g$ to
$g-2(g\cdot\omega)\omega$. On either hemisphere of the normal sphere,
the change from the polar angle of $\omega$ to the reflected direction
shows that $|g\cdot\omega|\,\dd\omega$ pushes forward to
$(|g|/4)\,\dd\sigma$. The two hemispheres therefore give
$(|g|/2)\,\dd\sigma$, where $\sigma\in S^2$ is the outgoing relative
direction. Equivalently, for every nonnegative test function $\psi$,
the one-output measure is
\begin{equation}\label{ker:sphere-law}
 \frac{|p-\xi|}{2}\int_{S^2}
 \psi\left(\frac{p+\xi}{2}
                 +\frac{|p-\xi|}{2}\sigma\right)\dd\sigma.
\end{equation}
Its mass is $2\pi|p-\xi|$. The affine map in this integral multiplies
surface area by $|p-\xi|^2/4$, so the density on its image sphere is
$2/|p-\xi|$.

The same sphere is the zero set of
$f(v)=(v-p)\cdot(\xi-v)$. On that sphere,
$|\nabla_v f(v)|=|p-\xi|$. The coarea formula consequently identifies
the measure in \eqref{ker:sphere-law} with $2\delta(f(v))\,\dd v$.
Integration against the two incoming densities and Tonelli's theorem
give \eqref{ker:carleman}. The usual pre/post-collisional change of
variables identifies this incoming-pair expression with the gain in
\eqref{eq:collision}. Coincident incoming velocities have zero
collision rate. The same identities can first be integrated against a
velocity measure $c(\dd p)$, proving the asserted extension.
\end{proof}

In the cutoff gain, the integrand in \eqref{ker:carleman} is multiplied
by $\min(1,b/|p-\xi|)$. The relative input speed equals the relative
output speed, so this factor preserves the collision symmetries.
It is at most one. All upper estimates below consequently hold for
$P^b$ as well, uniformly in $b\geq1$.

\subsection{A geometric slice estimate}

For $d>0$, let $\mathcal G_d$ be the centered transverse Gaussian
probability measure with density
\[
 \frac{1}{\pi d^2}e^{-|\bar p|^2/d^2},
\]
and let $\mathcal G_0=\delta_0$. We shall also use the uniform
probability measure $\mathcal U_d$ on $\{\bar p:|\bar p|<d\}$.

\begin{lemma}[Transverse averaging and axial slices]
\label{ker:geometric-slice}
Let $\lambda$ be a probability mixture of the measures
$\mathcal U_d$, $d>0$, and $\delta_0$, and let $\rho$ be any axial
probability measure. No bound on its density or its support is
required. Put $c=\lambda\otimes\rho$. For $R\geq1$, $\bar v\ne0$,
and an interval $I\subset\R$ of length $0<\ell\leq2R$, one has
\begin{align}
 &\frac{1}{W(v)}\int c(\dd p)
   \int_{|\xi-v|\leq R}
    \delta\bigl((v-p)\cdot(\xi-v)\bigr)W(\xi)\,\dd\xi
     \leq C_q R^2,\label{ker:geometric-total}\\
 &\frac{1}{W(v)}\int c(\dd p)
   \int_{\substack{|\xi-v|\leq R\\ \xi_3\in I}}
    \delta\bigl((v-p)\cdot(\xi-v)\bigr)W(\xi)\,\dd\xi
     \leq C_q\bigl(R\ell+R^{q-1}\ell^{2-q}\bigr).
     \label{ker:geometric-interval}
\end{align}
The constants are independent of all transverse widths and of $\rho$.
In particular, these statements hold with $\lambda=\mathcal G_d$ for
every $d\geq0$.
\end{lemma}

\begin{proof}
Set $r=|\bar v|$, $n=\bar v-\bar p$, and $e_n=n/|n|$. Except on a
set of transverse measure zero, $n\ne0$. For fixed $p_3,\xi_3$, the
delta in the transverse variables is line measure divided by $|n|$,
on a line whose signed distance from zero is
\[
 A-B(\xi_3-v_3),\qquad
 A=e_n\cdot\bar v,\quad B=\frac{v_3-p_3}{|n|}.
\]
The regular part of its integral over $|\xi-v|\leq R$ is at most
$2R/|n|$. The singular part is bounded by its integral over the full
line, namely
\begin{equation}\label{ker:line-integral}
 \frac{1}{|n|}\int_{\R}
      \bigl((A-B(\xi_3-v_3))^2+s^2\bigr)^{-q/2}\,\dd s
 =\frac{a_q}{|n|}|A-B(\xi_3-v_3)|^{1-q},
\end{equation}
where $a_q=\int_{\R}(1+s^2)^{-q/2}\,\dd s<\infty$ because $q>1$.
Infinite values on a zero-distance slice are harmless in the
nonnegative iterated integral.

For every interval $J\subset[-R,R]$ of length at most $\ell$,
\begin{equation}\label{ker:scalar-interval}
 \int_J |A-Bx|^{1-q}\,\dd x
 \leq C_q R^{q-1}\ell^{2-q}|A|^{1-q}.
\end{equation}
If $2R|B|\leq|A|$, the integrand is at most
$C_q|A|^{1-q}$, and the assertion follows from $\ell\leq2R$.
Otherwise, integration about the possible root gives an upper bound
$C_q|B|^{1-q}\ell^{2-q}$; since $1-q<0$ and
$|B|>|A|/(2R)$, this proves \eqref{ker:scalar-interval}. When $A=0$
the displayed upper bound is understood in the extended sense.

We next prove
\begin{equation}\label{ker:transverse-expectation}
 \int\frac{1+|e_n\cdot e_{\bar v}|^{1-q}}{|n|}\,
          \lambda(\dd\bar p)\leq\frac{C_q}{r}.
\end{equation}
It suffices to consider a disk of radius $d$. If $d\leq r/2$, then
$|n|\geq r/2$ and $e_n\cdot e_{\bar v}\geq1/3$, which proves the
bound. If $d>r/2$, use polar coordinates for $n$. Its range lies in
$|n|<r+d<3d$. The factor $|n|^{-1}$ cancels the radial Jacobian, while
\[
 \int_0^{2\pi}\bigl(1+|\cos\theta|^{1-q}\bigr)\,\dd\theta<\infty
\]
because $q<2$. Division by the disk area gives $C_q/d\leq2C_q/r$.
The point mass at zero satisfies the same estimate directly.
Finally, the Gaussian is a probability mixture of these disks: for
$d>0$ its mixture density in the disk radius $a$ is
\[
 \frac{2a^3}{d^4}e^{-a^2/d^2}\,\dd a,
\]
whose integral is one. This proves the Gaussian assertion as well.

Integrating \eqref{ker:line-integral} in $\xi_3$, applying
\eqref{ker:scalar-interval}, and then
\eqref{ker:transverse-expectation}, bounds the expression in
\eqref{ker:geometric-interval} by
\[
 C_q R\ell\frac{r^{-1}}{1+r^{-q}}
 +C_qR^{q-1}\ell^{2-q}\frac{r^{-q}}{1+r^{-q}}.
\]
Both fractions are bounded for $q>1$. These bounds are independent of
$p_3$, so integration against the arbitrary probability $\rho$
preserves them. Taking the entire axial range of length $2R$ also
gives \eqref{ker:geometric-total}.
\end{proof}

\subsection{Sheets and normalized kernel measures}

\begin{definition}[Uniform sheet class]\label{ker:sheet-class}
Fix $L\geq1$. An elementary sheet in $\mathcal S_L$ is a product
probability measure
\begin{equation}\label{ker:sheet-definition}
 c(\dd p)=\mathcal G_d(\dd\bar p)\rho(\dd p_3),\qquad
 0\leq d\leq\tfrac18,\qquad
 \int_{\R}e^{8a^2}\rho(\dd a)\leq L.
\end{equation}
We also allow probability mixtures of elementary sheets. An upper
bound by a sheet with amplitude $A_0$ means domination as measures by
$A_0$ times such a mixture.
\end{definition}

Every axial probability supported in $[-A,A]$ satisfies this
definition with $L=e^{8A^2}$. In particular, axial atoms and limits of
arbitrarily narrow axial densities are permitted. The exponential
moment is used only to remove large axial velocities, and imposes no
axial regularity.

For such a sheet and $\bar v\ne0$, define its normalized kernel by
\begin{equation}\label{ker:normalized-kernel}
 \mathcal K_c(v;\dd p\,\dd\xi)
 =2\,c(\dd p)\,
       \delta\bigl((v-p)\cdot(\xi-v)\bigr)
       \frac{H(\xi)}{H(v)}\,\dd\xi.
\end{equation}
The total mass is $P(c,H)(v)/H(v)$. Kernel restrictions always refer
to this complete measure, including the ratio $H(\xi)/H(v)$.

\begin{lemma}[Uniform sheet gain bound]\label{lem:sheet-bound}
For every $L<\infty$ there is $C_{q,m,L}<\infty$ such that
\begin{equation}\label{ker:sheet-gain}
 P(c,H)(v)\leq C_{q,m,L}H(v)
 \quad\hbox{for almost every }v,
 \qquad c\in\mathcal S_L.
\end{equation}
More quantitatively, for $R\geq1$,
\begin{equation}\label{ker:shell-tail}
 \mathcal K_c\bigl(v;
      R<|p|+|\xi-v|\leq2R\bigr)
 \leq C_{q,m,L}R^2e^{-cR^2},
\end{equation}
where $c>0$ is an absolute constant. The same conclusions, multiplied
by $A_0$, hold under sheet domination with amplitude $A_0$.
\end{lemma}

\begin{proof}
By positivity it suffices to treat an elementary sheet. Put
$k=\xi-v$ and $w=p+k$. On the collision plane,
\begin{equation}\label{ker:energy-factor}
 |p|^2+|\xi|^2=|v|^2+|w|^2,
 \qquad
 \frac{H(\xi)}{H(v)}
 =e^{|p|^2-|w|^2}
    \frac{\langle v\rangle^m}{\langle\xi\rangle^m}
    \frac{W(\xi)}{W(v)}.
\end{equation}
The same identity implies
$\langle v\rangle\leq\langle p\rangle\langle\xi\rangle$.
Since $\langle p\rangle^m\leq C_m e^{|p|^2}$, the kernel is bounded
by
\begin{equation}\label{ker:tilted-kernel}
 C_m e^{2|p|^2}c(\dd p)\,
 e^{-|w|^2}\delta\bigl((v-p)\cdot k\bigr)
       \frac{W(v+k)}{W(v)}\,\dd k.
\end{equation}

The tilted cold measure retains a product majorant. For $0<d\leq1/8$,
\begin{equation}\label{ker:transverse-tilt}
 e^{2|\bar p|^2}\mathcal G_d(\dd\bar p)
 \leq4e^{-46|\bar p|^2}\mathcal G_{2d}(\dd\bar p).
\end{equation}
This follows by dividing the two Gaussian densities: their ratio is
$4\exp[-(3/(4d^2)-2)|\bar p|^2]$, and
$3/(4d^2)-2\geq46$. The assertion at $d=0$ has the evident
interpretation. The axial tilted measure
$\widetilde\rho(\dd a)=e^{2a^2}\rho(\dd a)$ has mass at most $L$,
and satisfies
\begin{equation}\label{ker:axial-tilt-tail}
 \widetilde\rho\{|a|>u\}\leq Le^{-6u^2},\qquad u\geq0.
\end{equation}
Consequently $e^{2|p|^2}c(\dd p)$ is bounded by $4L$ times a product
of a centered transverse Gaussian and an axial probability. Moreover,
its restriction to $|p|>u$ is bounded by
$CLe^{-cu^2}$ times a sum of at most two such product probabilities.
To see the last assertion, split into $|\bar p|>u/\sqrt2$ and
$|p_3|>u/\sqrt2$, and apply
\eqref{ker:transverse-tilt} and \eqref{ker:axial-tilt-tail},
respectively. The axial probabilities obtained by normalization may
have large support; Lemma~\ref{ker:geometric-slice} permits this.

On the shell in \eqref{ker:shell-tail}, either $|p|>R/4$ or
$|w|>R/2$, because $|p|+|k|\leq2|p|+|w|$. In the first case the
preceding tilted-measure tail contributes $CLe^{-cR^2}$. In the
second case $e^{-|w|^2}\leq e^{-R^2/4}$. In either case the remaining
integral has $|k|\leq2R$ and is bounded by $C_qR^2$ using
Lemma~\ref{ker:geometric-slice}. This proves
\eqref{ker:shell-tail}. The base region $|p|+|k|\leq2$ is bounded
by the same geometric lemma and the total tilted mass. Summing the
dyadic shells proves \eqref{ker:sheet-gain} and also the finiteness
of \eqref{ker:normalized-kernel}.
\end{proof}

The sheet gain bound controls the total normalized mass. We also
need uniform control of the input tails and decay for large output
speed, so that later path estimates can be restricted to bounded
velocity regions.

\begin{lemma}[Uniform tightness and large-output decay]
\label{lem:kernel-tightness}
Fix $1\leq L<\infty$. Uniformly for $c\in\mathcal S_L$ and outgoing
velocities off the axis, the tails in $|p|+|\xi-v|$ are negligible:
\begin{equation}\label{ker:uniform-tail}
 \lim_{R\to\infty}\sup_{c\in\mathcal S_L}\esssup_v
 \mathcal K_c(v;|p|+|\xi-v|>R)=0.
\end{equation}
Moreover,
\begin{equation}\label{ker:large-output}
 \lim_{M\to\infty}\sup_{c\in\mathcal S_L}
 \esssup_{|v|>M}\mathcal K_c(v;\R^3\times\R^3)=0.
\end{equation}
Both conclusions hold under uniformly bounded sheet amplitudes.
\end{lemma}

\begin{proof}
The dyadic estimate \eqref{ker:shell-tail} proves
\eqref{ker:uniform-tail}. Write $k=\xi-v$. We may consequently work on
$|p|+|k|\leq R$ and remove that restriction at the end of each
argument. On this region the exponential and polynomial factors in
\eqref{ker:energy-factor} are bounded by a constant depending on $R$.

For \eqref{ker:large-output}, if $|\bar v|>2R+1$, then
$|\bar v-\bar p|\geq|\bar v|/2$ and
$|\bar\xi|\geq|\bar v|-R$. The transverse line estimate, followed
by the axial integration over a length at most $2R$, gives
$C_R/|\bar v|$. If $|v_3|>2R+1$, integrate the delta in $\xi_3$
instead. Its divisor is $|v_3-p_3|\geq|v_3|/2$, and
\begin{equation}\label{ker:disk-singularity}
 \sup_{a\in\R^2}\int_{|\zeta-a|\leq R}
       (1+|\zeta|^{-q})\,\dd\zeta
 \leq C_q(R^2+R^{2-q}).
\end{equation}
For completeness, if $|a|\leq2R$ the disk is contained in the disk
of radius $3R$ about zero; if $|a|>2R$, then $|\zeta|>|a|/2>R$ on
the disk, giving the same bound. We obtain $C_R/|v_3|$. At least one
of the two outgoing components becomes large with $|v|$, proving
\eqref{ker:large-output}.
\end{proof}

\subsection{Gaussian parameter laws and compressed axial maps}

The next formulation records the uniformity needed when a cold
profile is parameterized before its axial change of variables.

\begin{lemma}[Gaussian parameter tails]
\label{ker:parameter-sheets}
Let a measure in parameters $w=(\bar w,w_3)\in\R^2\times\R$ have
density $j$ satisfying
\[
 0\leq j(w)\leq C_0\pi^{-3/2}e^{-|w|^2}.
\]
Let its velocity image be given by
\begin{equation}\label{ker:parameter-map}
 \bar p=d\bar w,\qquad p_3=\Phi(w_3),\qquad
 0\leq d\leq\tfrac1{16},\qquad
 |\Phi(a)|\leq A+\delta|a|,\quad0\leq\delta\leq\tfrac1{16},
\end{equation}
where $\Phi$ is merely measurable. The image measure is dominated
by $C_0$ times a sheet in $\mathcal S_{L_A}$, with $L_A$ depending
only on $A$. Its gain satisfies Lemma~\ref{lem:sheet-bound} with a
constant depending only on $q,m,A,C_0$.

If $c_{>B}$ is the image of the restricted parameter measure
$\ind_{|w|>B}j(w)\,\dd w$, then
\begin{equation}\label{ker:parameter-tail}
 \esssup_v\frac{P(c_{>B},H)(v)}{H(v)}
 \leq C_{q,m,A,C_0}e^{-B^2/2},\qquad B\geq0.
\end{equation}
Thus parameter truncations are valid in the normalized collision
measure, even when $\Phi$ compresses intervals or has atoms in its
image law. Probability mixtures of measurable families with common
$q,m,A,C_0$ and the width bounds in \eqref{ker:parameter-map} preserve
these assertions. More generally, a positive mixing measure of total
mass $B_0<\infty$ multiplies the domination amplitude and quantitative
upper bounds by $B_0$; the sheet parameter $L_A$ is unchanged.
For \eqref{ker:parameter-tail}, impose the restriction $|w|>B$ in
each component before mixing. Additional measurable masks taking
values in $[0,1]$ preserve the assertions.
\end{lemma}

\begin{proof}
The Gaussian majorant is a product measure. Its transverse image is
$\mathcal G_d$, and its axial image is the probability
$\rho=\Phi_*(\pi^{-1/2}e^{-a^2}\,\dd a)$. There is no need to
differentiate or invert $\Phi$. From \eqref{ker:parameter-map},
\[
 e^{8|\Phi(a)|^2}\leq e^{16A^2}e^{16\delta^2a^2}.
\]
Even under the wider axial probability
$(2\pi)^{-1/2}e^{-a^2/2}\,\dd a$, the integral of the right side
is
\[
 e^{16A^2}(1-32\delta^2)^{-1/2}\leq2e^{16A^2}.
\]
We may therefore take $L_A=2e^{16A^2}$ for both the original and
the wider parameter laws. The original image has the required
sheet domination.

On the parameter tail,
\[
 \ind_{|w|>B}e^{-|w|^2}
 \leq e^{-B^2/2}e^{-|w|^2/2}.
\]
After normalization the right side is $2^{3/2}e^{-B^2/2}$ times a
Gaussian probability with parameter widths enlarged by $\sqrt2$.
Its transverse velocity width is at most $\sqrt2/16<1/8$, and
its axial image has the exponential moment just proved.
Lemma~\ref{lem:sheet-bound} applied to this dominating image gives
\eqref{ker:parameter-tail}.

For a probability mixture, mix the dominating sheets just constructed;
this is again a member of $\mathcal S_{L_A}$. If the mixing measure
has mass $B_0>0$, normalize it by $B_0$ and retain that factor in
the domination amplitude. If $B_0=0$, the mixed image is zero.
The same argument applied to the wider Gaussian majorants of the
component tails proves the asserted factor in
\eqref{ker:parameter-tail}. Thus the estimates are uniform over
families of mixtures whose masses $B_0$ have a common finite bound
independent of the tail cutoff. Masks taking values in $[0,1]$ only
decrease the measures, whether imposed before or after the pushforward.
\end{proof}

Return to the profile at the start of the section:
$p_3=\upsilon(z-tp_3)+d w_3$, with $\upsilon$ nondecreasing, and
let $u$ solve $u=\upsilon(z-tu)$. The function
$p\mapsto p-\upsilon(z-tp)$ is increasing with inverse Lipschitz
constant at most one. Hence
\begin{equation}\label{ker:compressed-displacement}
 |p_3-u|\leq d|w_3|.
\end{equation}
If $|u|\leq A$, Lemma~\ref{ker:parameter-sheets} applies. A large
Jacobian compression of the axial law does not affect its constants.
Gaussian majorants with any fixed positive exponent in place of one
are covered by a fixed rescaling of the parameters and a corresponding
decrease of the upper bounds on the physical widths.

\subsection{The hot--hot gain}

\begin{lemma}[Quadratic gain bound]\label{lem:hot-bound}
There is a constant $C_{q,m}$ such that
\begin{equation}\label{ker:hot-gain}
 P(H,H)\leq C_{q,m}H
 \quad\hbox{almost everywhere}.
\end{equation}
In fact the proof below works for every $m>5$.
\end{lemma}

\begin{proof}
The integrand is symmetric in $p,\xi$. On the collision plane,
$|v|^2\leq|p|^2+|\xi|^2$, so at least one incoming speed is at
least $|v|/2$. It suffices, at a factor of two, to retain
$|\xi|\geq|v|/2$. Formula~\eqref{ker:energy-factor} now gives
\begin{align}
 \frac{P(H,H)(v)}{H(v)}
 &\leq C_m\int_{|\xi|\geq|v|/2}
 \delta\bigl((v-p)\cdot(\xi-v)\bigr)
 e^{-|p+\xi-v|^2}\langle p\rangle^{-m}
 \frac{W(p)W(\xi)}{W(v)}\,\dd p\,\dd\xi,
 \label{ker:hot-reduced}
\end{align}
because $\langle v\rangle^m/\langle\xi\rangle^m\leq2^m$.

Again write $k=\xi-v$ and $w=p+k$. On
$R<|p|+|k|\leq2R$, $R\geq1$, either $|p|>R/4$ or $|w|>R/2$.
Consequently
\begin{equation}\label{ker:hot-shell-weight}
 \langle p\rangle^{-m}e^{-|w|^2}
 \leq C_mR^{-m}e^{-|w|^2/2}.
\end{equation}
In the second alternative this uses the finite supremum of
$s^m e^{-s^2/2}$; in the first it follows directly from the
polynomial factor.

The measure $W(p)\ind_{|p|\leq2R}\,\dd p$ is dominated by
\[
 \bigl(1+|\bar p|^{-q}\bigr)
 \ind_{|\bar p|\leq2R}\,\dd\bar p
 \ \otimes\ \ind_{|p_3|\leq2R}\,\dd p_3.
\]
Its transverse factor is a positive mixture of centered disks.
Indeed, for $D>0$,
\[
 |\bar p|^{-q}\ind_{|\bar p|\leq D}
 =D^{-q}\ind_{|\bar p|\leq D}
   +q\int_0^D a^{-q-1}\ind_{|\bar p|\leq a}\,\dd a
\]
almost everywhere. Upon normalizing the disks, the mixture has
total mass
$\pi D^2+2\pi D^{2-q}/(2-q)$. The axial factor has mass $4R$.
For $D=2R$ the resulting product mixture therefore has total mass
at most $C_qR^3$.

Discard $e^{-|w|^2/2}$ and the incoming-speed restriction in
\eqref{ker:hot-reduced}. Apply
Lemma~\ref{ker:geometric-slice} to each normalized product in this
mixture, with $|k|\leq2R$. In conjunction with
\eqref{ker:hot-shell-weight}, this yields the shell estimate
\begin{equation}\label{ker:hot-shell}
 \frac{P(H,H)(v)}{H(v)}\bigg|_{R<|p|+|\xi-v|\leq2R}
 \leq C_{q,m}R^{5-m}.
\end{equation}
The notation on the left denotes the restricted nonnegative
integral from \eqref{ker:hot-reduced}, including its fixed symmetry
factor. The base region is bounded by the same disk-mixture
argument with $R=1$. Summation over dyadic $R$ proves the result
for $m>5$.
\end{proof}

\subsection{A small-mass gain estimate}

The quadratic estimate controls a gain by a weighted supremum. For
global continuation we need a stronger conclusion when one input has
small mass while staying below a fixed multiple of $H$. The relevant
compactness is in the velocity integrals themselves: large speeds,
small transverse speeds, and the remaining collision diagonal can
all be removed with uniformly small normalized cost.

\begin{lemma}[Gain estimate from small mass]
\label{lem:gain-small-mass}
For each $0\leq B<\infty$ and $\varepsilon>0$ there is a finite
constant $C_{B,\varepsilon}$ such that every measurable density $j$
with $0\leq j\leq BH$ satisfies
\begin{equation}\label{ker:gain-small-mass}
 P(H,j)(v)\leq H(v)
 \left(\varepsilon+C_{B,\varepsilon}
                         \int_{\R^3}j(p)\,\dd p\right)
 \qquad\hbox{for almost every }v.
\end{equation}
The same constant works with $P^b$ in place of $P$.
\end{lemma}

\begin{proof}
The case $B=0$ is immediate. Assume $B>0$. We first establish
\begin{equation}\label{ker:hot-large-output}
 \lim_{M\to\infty}\esssup_{|v|>M}
                  \frac{P(H,H)(v)}{H(v)}=0.
\end{equation}
As in the proof of Lemma~\ref{lem:hot-bound}, symmetry reduces the
integral to $|\xi|\geq|v|/2$, up to a factor of two.
The shell estimate \eqref{ker:hot-shell} then removes
$|p|+|\xi-v|>R$ with an error tending to zero uniformly in $v$ as
$R\to\infty$. On the retained region, the right side of
\eqref{ker:hot-reduced} is bounded by a constant times
\[
 \int_{|p|\leq R}W(p)
 \left[\int_{|\xi-v|\leq R}
   \delta\bigl((v-p)\cdot(\xi-v)\bigr)
               \frac{W(\xi)}{W(v)}\,\dd\xi\right]\dd p.
\]
If $|\bar v|>2R+1$, the transverse divisor is at least
$|\bar v|/2$, and $W(\xi)$ is bounded on the retained region. The
transverse line integral and the axial interval of length $2R$ give
an upper bound $C_R/|\bar v|$ after integration in $p$. If
$|v_3|>2R+1$, integrating the delta in $\xi_3$ instead and using
\eqref{ker:disk-singularity} gives $C_R/|v_3|$. Here
$\int_{|p|\leq R}W(p)\,\dd p<\infty$, since $q<2$.
At least one output component tends to infinity with $|v|$.
First letting $|v|\to\infty$ and then $R\to\infty$ proves
\eqref{ker:hot-large-output}.

Choose $M$ so large that $BP(H,H)\leq\varepsilon H$ on $|v|>M$.
It remains to prove the desired estimate for $|v|\leq M$. On this
output ball, the part of the normalized quadratic gain with
$|p|+|\xi|>L$ tends uniformly to zero as $L\to\infty$. Indeed, this
deletion is symmetric in the two inputs. After the same reduction to
$|\xi|\geq|v|/2$, it implies
\[
 |p|+|\xi-v|>L-M,
\]
so \eqref{ker:hot-shell} bounds its cost by a summable tail.
The condition $j\leq BH$ transfers this estimate to $P(H,j)$.

Next retain both inputs in $|p|,|\xi|\leq L$. The Gaussian and
polynomial factors in the normalized quadratic integrand are then
bounded by a constant depending on $M,L$, leaving
\[
 \delta\bigl((v-p)\cdot(\xi-v)\bigr)
                       \frac{W(p)W(\xi)}{W(v)}.
\]
For $0<\delta<1$, the restriction $|\bar p|<\delta$ is dominated in
its $p$ variable by the product measure
\[
 (1+|\bar p|^{-q})\ind_{|\bar p|<\delta}\,\dd\bar p
       \otimes\ind_{|p_3|\leq L}\,\dd p_3.
\]
Its transverse factor is a positive mixture of centered disks, by
the layer-cake formula used in Lemma~\ref{lem:hot-bound}, and its
total mass is
\[
 2L\left(\pi\delta^2+
                  \frac{2\pi}{2-q}\delta^{2-q}\right).
\]
After normalization, \eqref{ker:geometric-total}, with radius at
least $L+M$, therefore bounds the cost of this restriction by
$C_{M,L}(\delta^2+\delta^{2-q})$. Symmetry gives the same bound
for $|\bar\xi|<\delta$. These estimates are uniform even when the
output velocity approaches its axis. Again $j\leq BH$ transfers
them to the mixed gain.

We have proved that, for a suitable compact set
\[
 K=\{p:|p|\leq L,\ |\bar p|\geq\delta\},
\]
the truncations $H_K=H\ind_K$ and $j_K=j\ind_K$ satisfy
\[
 0\leq P(H,j)-P(H_K,j_K)\leq\tfrac12\varepsilon H
       \qquad (|v|\leq M),
\]
uniformly over $0\leq j\leq BH$.
The function $H_K$ is bounded and supported in the ball of radius
$L$. By symmetry of $P$, put $j_K$ outside the plane integral in
\eqref{ker:carleman}. The area of a plane section of that ball is
at most $\pi L^2$, so
\[
 P(H_K,j_K)(v)\leq C_{L,\delta}
              \int_K\frac{j_K(p)}{|v-p|}\,\dd p.
\]
For $0<\rho<1$, the part with $|v-p|<\rho$ is at most
$C_{B,L,\delta}\rho^2$, because $j_K$ is bounded. The complement
is at most
$C_{L,\delta}\rho^{-1}\int j$.
Finally, $H$ has a positive lower bound on $|v|\leq M$ off its
null axis. Choose $\rho$ small enough that the near part is at most
$\varepsilon H/2$ there. Absorbing the other fixed factors into
$C_{B,\varepsilon}$ proves \eqref{ker:gain-small-mass}.
The cutoff assertion follows from $P^b\leq P$ on nonnegative inputs.
\end{proof}
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//03-kernel.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//04-construction.tex ====
\section{The datum and the finite colored evolutions}
\label{sec:construction}

The datum combines narrow jets with a bounded background close to a
Maxwellian.  The background supplies collisions at later times, whereas
an initial spatial hole keeps it apart from the jets during the short
interval on which we select the two families of solutions.  We first
construct these densities and their finite approximations.  The estimates
in this section and the next two sections hold for every fixed choice
\begin{equation}
 K\ge1,\qquad 0<\eta\le\log2,\qquad
 0<\alpha\le\tfrac14,\qquad 0<\gamma\le1.
 \label{con:parameters}
\end{equation}
The growth argument will fix these four parameters.  All remaining
small parameters are chosen after them.  The annular jets and the
colored evolution adapt the construction of
\cite[Sections 4--5]{OpenAIPeriodic}; the spatial orientation, the
background, and the later transfer of colors will allow uniform bounds
for arbitrarily large physical times.

\subsection{An irrational axis and annular jets}

Use the orthonormal frame
\[
 \frac{(-\sqrt2,1,0)}{\sqrt3},\qquad (0,0,1),\qquad
 \frac{(1,\sqrt2,0)}{\sqrt3}.
\]
In these coordinates write \(x=(y,z)\), \(v=(\bar v,v_3)\), and let
\(\Lambda\) be the image of \(\mathbb Z^3\) under the change of
coordinates.  The spatial torus is now \(\mathbb R^3/\Lambda\), with
covolume one.  Rotation preserves transport and the hard-sphere
collision operator, so this change does not alter the equation.

\begin{lemma}[Separation from the axis]\label{con:lattice}
There is a numerical constant \(c>0\) such that
\begin{equation}
 |\bar n|\ge\frac{c}{1+|n|}\qquad
             (n\in\Lambda\setminus\{0\}).
 \label{con:diophantine}
\end{equation}
\end{lemma}
\begin{proof}
An original integer vector \((k,m,\ell)\) has transverse coordinates
\(((m-\sqrt2 k)/\sqrt3,\ell)\).  If \(\ell\ne0\), its transverse
length is at least one.  Otherwise \((k,m)\ne(0,0)\), and
\[
 |m-\sqrt2 k|
 =\frac{|m^2-2k^2|}{|m+\sqrt2 k|}
 \ge\frac{1}{C(1+|k|+|m|)}.
\]
The numerator is a nonzero integer.  Since rotation preserves length,
this proves the claim.
\end{proof}

Fix a nonnegative \(J\in C_c^\infty(\mathbb R^3)\), supported in the
unit ball and of integral one.  With \(0<s_0\le1/8\), define
\begin{equation}
 s_j=s_0e^{-j\eta},\quad p_j=s_j^2,\quad
 c_\eta=\frac{e^\eta-1}{16},\quad
 d_s=d_0s^{10},\quad S(r)=\operatorname{sech}^2r,
 \qquad j\ge1,
 \label{con:scales}
\end{equation}
where \(0<d_0\le1\) is fixed sufficiently small.  It is enough to
make all Gaussian widths below, including their fixed widenings, fit
within the sheet estimates of Section~\ref{ker:section}.  Further
conditions involving positive powers of \(s_0\) will be obtained by
reducing \(s_0\).  Set
\[
 a_s^\sigma(z)=\sigma+\alpha\tanh(\gamma z/s),\qquad
 a=\tfrac18,\qquad b_{\mathrm{sp}}=\tfrac1{16},
 \qquad \sigma\in\{-1,+1\}.
\]
For each scale let \(E_j^\sigma\subset\mathbb R^2\) be the annulus
\[
 s_j+c_\eta p_j<|y|<e^\eta s_j-c_\eta p_j
\]
intersected with the stripes
\[
 y_1/p_j\pmod1\in
 \begin{cases}
 (0,1/4),&\sigma=+1,\\
 (1/2,3/4),&\sigma=-1.
 \end{cases}
\]
Both sets are nonempty for small \(s_0\): the radial interval is
nonempty, and a circle at an interior radius crosses both stripe
patterns because their period is \(s_j^2\).  Each pattern has spatial
fraction \(1/4\); no exact fraction is asserted after intersection
with an annulus.

A jet label is \(i=(j,\sigma,n)\), with \(n=(\bar n,n_3)\in\Lambda\).
For \(s=s_j\) and \((y_c,z_c)=x-n\), define
\begin{equation}
 c_i^0(x,v)=
 \ind_{E_j^\sigma}(y_c)\ind_{(-a,a)}(z_c)
 \frac K{s}S(\gamma z_c/s)d_s^{-3}
 J\!\left(\frac{v-a_s^\sigma(z_c)e_3}{d_s}\right).
 \label{con:jet-datum}
\end{equation}
All sums over labels include the lattice copies.  Translating \(x\)
by an element of \(\Lambda\) permutes these labels.

\subsection{The background and the initial entropy}

For any positive Maxwellian \(\rho\), write
\[
 \mathcal H_\rho(F)=\int
       \bigl(F\log(F/\rho)-F+\rho\bigr)\dd x\dd v.
\]
This agrees with the entropy already defined when \(\rho=M\).
Fix a sufficiently large numerical \(\beta_0\), and write
\(\mu(v)=e^{-\beta_0|v|^2}\).  Let \(0<\delta<1/32\), and let
\(D_{\mathrm{hole}}\) be the open \(2\delta\)-neighborhood in the
spatial torus of the axial segment of half-length \(1/3\) centered at
zero.  For a velocity cutoff \(R_\mu<\infty\), put
\begin{equation}
 u^0(x,v)=\ind_{D_{\mathrm{hole}}^c}(x)
             \mu(v)\ind_{\{|v|\le R_\mu\}},\qquad
 F_0=u^0+\sum_i c_i^0.
 \label{con:datum-equation}
\end{equation}
The hole has volume tending to zero as \(\delta\downarrow0\), by the
usual volume bound for a cylinder and its two endpoint neighborhoods.
For \(s_0\) sufficiently small compared with \(\delta\), every jet
lies in this hole.  The different jet supports are disjoint: annuli
and opposite stripe patterns are disjoint within a single copy, and
the unshifted supports lie in a ball of radius less than \(1/2\), so
distinct lattice copies cannot overlap.

Figure~\ref{fig:jet-geometry} summarizes the initial geometry.
\input{figures/jet-geometry}

Choose also a nonzero nonnegative
\(\chi\in C_c^\infty(\mathbb R^3_x\times\mathbb R^3_v)\), and a
constant \(L_0\) with \(L_0\|\chi\|_H>1\).  For the approximation
retaining \(j\le N\), the additional initial density is either zero
or
\begin{equation}
 h^0_{N,T,\lambda}(x,v)
 =\lambda m_Ts_N\sum_{n\in\Lambda}
        \chi\bigl((x-n)/s_N,v\bigr),\qquad
 m_T=T^p,\qquad 0\le\lambda\le L_0.
 \label{con:seed}
\end{equation}
Here \(0<T\le t_1\le1/32\); Section~\ref{sec:early} fixes
\(t_1>0\) and then \(p>0\).  We reduce \(s_0\) so that the seed's
lattice copies have disjoint spatial supports.  Its weighted norm is
then \(\lambda m_Ts_N\|\chi\|_H\), with the unperiodized norm on the
right.

\begin{lemma}[Datum bounds and approximation]\label{con:datum}
The density \(F_0\) is nonnegative, has bounded velocity support, and
satisfies
\[
 \int F_0(1+|v|^2+|\log F_0|)\dd x\dd v<\infty.
\]
Its truncations with \(j\le N\), with or without a seed
\eqref{con:seed}, converge to \(F_0\) in weighted \(L^1\) for every
fixed polynomial velocity weight and in relative entropy with respect
to either \(M\) or \(\mu\).  The convergence is uniform for
\(0<T\le t_1\) and \(0\le\lambda\le L_0\).
Moreover their initial \(\mu\)-relative entropy can be made arbitrarily
small, uniformly in \(N,T,\lambda\), by first decreasing \(\delta\)
and increasing \(R_\mu\), then decreasing \(s_0\).
\end{lemma}
\begin{proof}
Integration over one period of the lattice sum equals integration of
each unshifted prototype over \(\mathbb R^3_x\).  The transverse area
is \(O(s^2)\), while
\[
 \int_{-a}^a S(\gamma z/s)\dd z\le 2s/\gamma.
\]
The velocity factor has integral one and support in \(\{|v|<2\}\).
Thus every fixed velocity moment of a sign prototype is \(O(s^2)\).
On its positive set the logarithm of the density splits into
\[
 \log((K/s)d_s^{-3}),\qquad \log S(\gamma z/s),\qquad \log J(w).
\]
The first term is \(O(1+|\log s|)\), and
\(\int S|\log S|<\infty\), \(\int J|\log J|<\infty\).
Its absolute entropy is therefore
\(O(s^2(1+|\log s|))\).  The corresponding series over the geometric
scales converges.  Disjointness proves the asserted integrability and
convergence for the cold sum.  The background is bounded by \(\mu\),
and its velocities satisfy \(|v|\le R_\mu\), giving the datum claims.

For a seed \(b=h^0_{N,T,\lambda}\), scaling \(x\) gives
\[
 \int b\langle v\rangle^k\dd x\dd v\le C_k s_N^4,\qquad
 \int b|\log b|\dd x\dd v
       \le C s_N^4(1+|\log s_N|).
\]
These constants are uniform because \(\lambda m_T\) stays in a fixed
bounded interval and \(r|\log r|\) is bounded on such an interval.
The preexisting truncated density \(f=u^0+\sum_{j\le N}c_i^0\)
satisfies \(f\le C s_N^{-31}\).  For \(f,b\ge0\), integration of
\(1+\log(f+r)\) over \(0<r<b\) gives
\[
 |(f+b)\log(f+b)-f\log f|
 \le b\{2+\log^+(f+b)\}+b|\log b|.
\]
Indeed, the positive logarithm is bounded by its terminal value, while
\(\int_0^b\log^-(f+r)\dd r\le b(1+|\log b|)\).
The entropy change caused by the seed is consequently
\(O(s_N^4(1+|\log s_N|))\).  Its reference terms are controlled by
its mass and energy.  This proves all convergence assertions.

Removing the hole and the velocity tail from \(\mu\) costs precisely
the removed mass in \(\mathcal H_\mu\).  The inserted jets have
bounded velocities, so their contribution is bounded by the preceding
absolute entropy, mass, and energy sums, which tend to zero with
\(s_0\).  The same is true uniformly for the seeds.  The stated order
of choices therefore makes the initial relative entropy arbitrarily
small.
\end{proof}

\subsection{Colored equations and transfers at large times}

Use the collision cutoff obtained by multiplying the hard-sphere
kernel by
\[
 \vartheta_b(|v-v_*|)=\min\{1,b/|v-v_*|\},\qquad b\ge1.
\]
Its value at zero is immaterial.  Write \(P^b\) for the symmetric
cutoff gain and
\(\nu_f^b(v)=2\pi\int\min\{b,|v-p|\}f(p)\dd p\).
The cutoff is invariant under particle exchange and the elastic
pre/post-collisional substitution.  It can equivalently be inserted
in the incoming relative speed in the Carleman and sphere formulas.
Fix \(b=b_N\to\infty\), and put
\(\mathcal I_N=\{(j,\sigma,n):1\le j\le N,\ \sigma=\pm1,
 n\in\Lambda\}\).

Until \(t_1\), write \(c=\sum_{i\in\mathcal I_N}c_i\),
\(F=c+u+h\), and solve
\begin{equation}
 \begin{aligned}
 \D c_i+\nu_F^b c_i&=P^b(c_i,c_i),&
 \D u+\nu_F^b u&=P^b(u,u),\\
 \D h+\nu_F^b h&=P^b(h,h)+2P^b(c+u,h)+\mathcal S,&
 \mathcal S&=2P^b(c,u)+\sum_{i\ne i'}P^b(c_i,c_{i'}).
 \end{aligned}
 \label{con:colors}
\end{equation}
The last sum is ordered.  Thus \(\D F=Q^b(F,F)\).

At \(t_1\), replace \(u+h\) by a single component \(w\).  A jet at
scale \(s\) remains separate until its retirement time
\begin{equation}
 L_s=s^{-1/2}>1.
 \label{con:deadline}
\end{equation}
At \(L_s\), add all colors at that scale to \(w\) and remove their
labels.  For the active labels, retain their equations in
\eqref{con:colors}; between retirements set \(F=c+w\) and solve
\begin{equation}
 \D w+\nu_F^b w
   =P^b(w,w)+2P^b(c,w)+\sum_{i\ne i'}P^b(c_i,c_{i'}).
 \label{con:late-colors}
\end{equation}
The total density has no jump at a retirement.  It satisfies the same
cutoff Boltzmann equation across the entire evolution.

\begin{proposition}[Finite evolution and its identities]
\label{con:finite-evolution}
For fixed \(N,b,T,\lambda\), the colored equations have a unique
nonnegative local mild solution with bounded total \(H\)-norm.  It
continues while that norm remains bounded, with the prescribed
transfers at retirement times, and depends continuously in this norm
on its initial data on every common bounded interval of existence.
The norm of each component is continuous between transfers.

On every such interval, total mass, momentum, and energy are conserved.
The exact local collision-invariant balances hold, with all polynomial
velocity fluxes integrable.  For \(\rho=M\) or \(\rho=\mu\), one has
at every time of existence
\begin{equation}
 \mathcal H_\rho(F(t))+
       \int_0^t\mathcal D^b(F(s))\dd s
       \le\mathcal H_\rho(F(0)),
 \label{con:finite-entropy}
\end{equation}
where \(\mathcal D^b\) is the entropy dissipation with the same cutoff.
\end{proposition}
\begin{proof}
The gain estimate \(P(H,H)\le CH\) and
\[
 \|\nu_f^b\|_\infty\le
  2\pi b\|f\|_H\int H(p)\dd p
\]
make the gain and loss locally Lipschitz in the weighted norm.
For the colored system use the norm of the pointwise sum of the
absolute values of the components, and the sum of absolute differences
for two solutions.  All quadratic estimates close in this norm.
The initially disjoint lattice copies have finite total norm, and
periodic covariance is preserved by uniqueness.  Picard iteration in
free coordinates gives the local solution; the loss integrating factor
and positive gain iteration give nonnegativity.  Repetition on short
subintervals gives continuation and the difference estimate.  These
arguments use the integral equation along flights and require no norm
continuity of spatial translation for measurable data.  On an interval
with bounded norm, the increments along flights are Lipschitz in that
norm.  Since free transport is isometric, each component's norm is
continuous.  Transfers only regroup finitely many scales and preserve
the total density.

The weight \(H\) has every polynomial moment and belongs to
\(L^{1+\epsilon}(\mathbb R^3)\) for some \(\epsilon>0\).  Hence the
cutoff equation has integrable gain and loss with all polynomial
weights on every bounded interval under consideration.  Integrating
its path formula and using particle interchange and the elastic
substitution yields conservation of the collision invariants.  The
same calculation with a smooth spatial test gives each local balance,
including its initial trace.

Here are details of the entropy calculation at a fixed approximation.
Put \(g(v)=e^{-2|v|^2}\) and \(F_\epsilon=F+\epsilon g\).  Differentiate
\(F_\epsilon\log(F_\epsilon/\rho)-F_\epsilon+\rho\) along flights;
its derivative is \(Q^b(F,F)\log(F_\epsilon/\rho)\).
Replacing \(Q^b(F,F)\) by \(Q^b(F_\epsilon,F_\epsilon)\) changes its
integral by a quantity tending to zero.  Indeed, the operator difference
is \(O(\epsilon)H\) on the fixed interval, while
\[
 |\log(F_\epsilon/\rho)|
 \le C\bigl(1+|\log\epsilon|+|v|^2+
                           \log^+(1/|\bar v|)\bigr).
\]
The right-hand side is integrable against \(H\), and the resulting
error is \(O(\epsilon(1+|\log\epsilon|))\).
For the shifted density the usual symmetric calculation is now
justified by this integrability.  The logarithm of either Maxwellian
reference is a collision invariant.  Symmetrization therefore gives
minus the cutoff dissipation.  The regularized entropies converge by
dominated convergence using the same bounds, and Fatou's lemma applies
to the nonnegative dissipation.  This proves
\eqref{con:finite-entropy}.  The integrated path formula includes every
time section, so the conservation and entropy statements hold at every
time, also when colors are transferred.
\end{proof}

We record the consequence of the small initial entropy that will drive
the later estimates.  Define
\begin{equation}
 D_F(t,x)=\int\langle v\rangle|F(t,x,v)-\mu(v)|\dd v,\qquad
 D_-(t,x)=\int\langle v\rangle(\mu(v)-F(t,x,v))_+\dd v.
 \label{con:entropy-defect}
\end{equation}
For any prescribed sufficiently small \(\varepsilon_E>0\), the
parameters in Lemma~\ref{con:datum} can be chosen so that, throughout
every interval of existence of every finite approximation,
\begin{equation}
 \|D_F(t)\|_{L^1_x}\le\varepsilon_E,\qquad
 0\le D_-(t,x)\le\int\langle v\rangle\mu(v)\dd v.
 \label{con:entropy-distance}
\end{equation}
To see the first assertion, use
\((\sqrt z-1)^2\le z\log z-z+1\) and Cauchy--Schwarz to obtain
\[
 \|D_F(t)\|_{L^1_x}
 \le\mathcal H_\mu(F(t))^{1/2}
       \left(2\int\langle v\rangle^2(F(t)+\mu)\dd x\dd v\right)^{1/2}.
\]
The second factor has a uniform fixed bound by mass and energy
conservation and the initial estimates.  The first factor is uniformly
small by \eqref{con:finite-entropy}.  No pointwise smallness of \(D_F\)
is asserted here; recovering the necessary pointwise control after
transport is the purpose of the global continuation argument.
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//04-construction.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//05-cold.tex ====
\section{Cold estimates up to the retirement times}
\label{sec:cold}

A jet remains a separate color until \(L_s=s^{-1/2}\), which tends to
infinity as its scale tends to zero.  We now control it throughout that
interval.  The essential facts are that the free jet remains in a
narrow spatial tube, its self-collisions create only a small Gaussian
remainder, and the active tubes stay disjoint on the torus.  These are
long-time versions of the cold estimates in
\cite[Section 4]{OpenAIPeriodic}.  We give the details because both the
long deadlines and the change of lattice matter for global continuation.
All constants may depend on the four fixed parameters
\eqref{con:parameters}, on \(J\), and on the fixed choice of \(d_0\),
but not on \(N,b,T\) or the size of the hot component.

\subsection{Disjoint spatial enlargements}

For \(i=(j,\sigma,n)\), put
\begin{equation}
 \widehat E_j^\sigma
   =\{y:\dist(y,E_j^\sigma)\le(c_\eta/4)s_j^2\},\qquad
 \mathcal O_i(t)=n+\widehat E_j^\sigma\times
     [-a-2t-b_{\mathrm{sp}},a+2t+b_{\mathrm{sp}}].
 \label{cold:enlargements}
\end{equation}
These sets enlarge the spatial supports for estimates; the original
stripe masks in \(c_i^0\) are always retained when taking a limit.

\begin{lemma}[Disjointness before retirement]\label{cold:geometry}
For sufficiently small \(s_0\), the sets \(\mathcal O_i(t)\) for
active labels are pairwise disjoint in the periodic lift.  More
precisely, \(\mathcal O_i(t)\cap\mathcal O_{i'}(t)=\varnothing\)
for distinct labels whenever
\(t\le\min\{L_{s_i},L_{s_{i'}}\}\).  Also
\[
 \widehat E_j^\sigma\subset\{s_j<|y|<e^\eta s_j\}.
\]
\end{lemma}
\begin{proof}
Within a fixed copy, the radial margins are larger than the
corresponding enlargement radii.  Adjacent radial cores have gap
\(c_\eta(s_j^2+s_{j+1}^2)\), whereas the sum of their enlargement
radii is one quarter of that gap.  Opposite stripe patterns at a
single scale have gap \(s_j^2/4\), larger than the sum
\(c_\eta s_j^2/2\) of their enlargement radii.  This proves the
assertions for one copy.

For distinct copies let \(S_* =\max\{s_i,s_{i'}\}\).  An intersection
would force
\[
 |\bar n-\bar n'|\le4S_*,\qquad
 |n-n'|\le C(1+t)\le C(1+S_*^{-1/2}).
\]
By \eqref{con:diophantine}, the first quantity is at least
\(c/(1+|n-n'|)\ge c' S_*^{1/2}\).  For sufficiently small \(s_0\)
this contradicts \(4S_*\).  The choice is uniform over all pairs of
scales and all lattice copies.
\end{proof}

At a point \((t,x)\), the \emph{main label} is the unique active
label whose enlargement contains \(x\), if one exists.  Every other
active label is called nonmain at that point.  Write \(\mathcal O(t)\)
for the union, in the torus, of the active enlargements.  Including
additional scales absent from the finite approximation is harmless,
provided their deadlines have not passed.  At a retirement endpoint,
these definitions may use the colors just before they are transferred.

\subsection{A supersolution for one color}

For all \(s>0\), signs \(\sigma\), and \(t\ge0\), define the
axial profile \(U_{s,\sigma}(t,z)\) by
\begin{equation}
 U_{s,\sigma}(t,z)
       =a_s^\sigma\bigl(z-tU_{s,\sigma}(t,z)\bigr).
 \label{cold:profile}
\end{equation}
This is uniquely and smoothly defined: the map
\(\tau\mapsto\tau+t a_s^\sigma(\tau)\) is onto and has derivative
at least one.  In particular \(|U_{s,\sigma}|\le1+\alpha\).
For a fixed label, abbreviate \(s=s_j\), \(L=L_s\),
\(U=U_{s,\sigma}(t,z_c)\), and set
\[
 \tau=z_c-tU,\qquad \xi=v-Ue_3,\qquad
 D_y=\dist(y_c,E_j^\sigma),\qquad
 D_z=\dist(\tau,[-a,a]).
\]
The free transport and its majorant are
\begin{equation}
 f_i(t,x,v)=c_i^0(x-tv,v),
 \label{cold:free-core}
\end{equation}
\begin{equation}
 m_i(t,x,v)=C_m\frac K{s}d_s^{-3}e^{t/L}
 \exp\left[-\frac{(2-t/L)|\xi|^2}{d_s^2}
           -\frac{D_y^2+D_z^2}{8d_s^2Lt}\right],\qquad 0<t\le L.
 \label{cold:barrier-definition}
\end{equation}
Its value at time zero is the trace along free characteristics.

\begin{lemma}[Barrier estimates]\label{cold:barrier}
A fixed sufficiently large \(C_m\) gives, for \(0\le t\le L_s\),
\begin{align}
 f_i&\le m_i,\qquad
 \D m_i\ge \frac1{L_s}
       \left(1+\frac78\frac{|\xi|^2}{d_s^2}\right)m_i,
 \label{cold:barrier-derivative}\\
 P(m_i,m_i)&=m_i\nu_{m_i}
       \le C\frac K{s}(d_s+|\xi|)m_i.
 \label{cold:barrier-gain}
\end{align}
The differential inequality holds almost everywhere along flights.
For sufficiently small \(s_0\),
\(f_i(t,x,v)>0\) implies \(x\in\mathcal O_i(t)\) throughout the
same time interval.
\end{lemma}
\begin{proof}
Put \(A=(a_s^\sigma)'(\tau)\ge0\).  Differentiation of
\eqref{cold:profile} gives
\[
 U_z=\frac A{1+tA}\ge0,\qquad U_t=-UU_z,\qquad
 \D U=(v_3-U)U_z,\qquad \D\tau=\frac{v_3-U}{1+tA}.
\]
Thus \(\D|\xi|^2=-2U_z\xi_3^2\).  The distance functions are locally
Lipschitz along every flight, and
\[
 |\D D_y|\le|\bar\xi|,\qquad |\D D_z|\le|\xi_3|.
\]
With \(r=(D_y,D_z)\), differentiation of the logarithm of the barrier
therefore yields
\[
 \D\log m_i
 =\frac1L+\frac{|\xi|^2}{Ld_s^2}
   +\frac{2(2-t/L)U_z\xi_3^2}{d_s^2}
   +\frac{|r|^2}{8d_s^2Lt^2}
   -\frac{r\cdot\D r}{4d_s^2Lt}.
\]
The last two terms are at least
\(-|\D r|^2/(8d_s^2L)\), by completing a square.  Since
\(|\D r|\le|\xi|\) and \(2-t/L\ge1\), this proves the derivative
estimate.  The large axial derivative \(U_z\) has a favorable sign.

For a flight starting in the core at \(x^0\), write
\(v=a_s^\sigma(z_c^0)e_3+d_sw\), \(|w|\le1\).
At \(x=x^0+tv\), the increasing function
\(p\mapsto p-a_s^\sigma(z_c-tp)\) has derivative at least one,
value \(d_sw_3\) at \(p=v_3\), and value zero at \(p=U\).
Consequently
\[
 |\xi|\le d_s,\qquad
 D_y\le td_s|\bar w|,\qquad D_z\le td_s|w_3|.
\]
The distance exponent in \eqref{cold:barrier-definition} is at most
\(t/(8L)\), and its velocity exponent is at most two.  A fixed
\(C_m\), depending only on \(\|J\|_\infty\), thus gives \(f_i\le m_i\).
Since \(L_sd_s=d_0s^{19/2}\ll s^2\), the transverse displacement is
less than the enlargement radius.  The axial speed is below two, so
the flight also stays inside the axial interval of
\eqref{cold:enlargements}.

At fixed \((t,x)\), the majorant is a scalar multiple of a Maxwellian
centered at \(Ue_3\).  Detailed balance gives
\(P(m_i,m_i)=m_i\nu_{m_i}\).  Its mass is at most \(CK/s\), and its
first moment about \(Ue_3\) is at most \(CKd_s/s\), uniformly for
\(t\le L_s\).  The inequality
\(|v-p|\le|\xi|+|p-Ue_3|\) gives \eqref{cold:barrier-gain}.

Finally, along a flight starting on the closed initial spatial
support, both distance functions vanish initially and are locally
Lipschitz, so their squared-distance-over-time terms tend to zero.
Away from that support the barrier tends to zero.  These observations
give its path trace and allow the differential inequalities to be
integrated from zero.  Boundary conventions for the masks change only
null sets; no spatial indicator is differentiated.
\end{proof}

\begin{proposition}[Control of every active color]\label{cold:control}
There is a fixed constant \(C_*\) such that, with
\[
 \epsilon_s=C_*L_sd_s/s,\qquad
 \mathcal U_i=f_i+\epsilon_s m_i,
\]
and sufficiently small \(s_0\), every finite colored evolution obeys
\begin{equation}
 0\le c_i(t,x,v)\le\mathcal U_i(t,x,v)\qquad (0\le t\le L_{s_i})
 \label{cold:upper}
\end{equation}
as long as that color is active and the solution exists.
If, on an interval beginning at \(t_1\), one also has
\(\nu_F^b\ge\kappa>0\), then
\begin{equation}
 c_i(t,x,v)\le e^{-\kappa(t-t_1)}\mathcal U_i(t,x,v)\qquad
                   (t_1\le t\le L_{s_i})
 \label{cold:decay}
\end{equation}
throughout that interval.  None of the constants in
\eqref{cold:upper} uses a bound on the bath or hot density.
\end{proposition}
\begin{proof}
Reduce \(s_0\) so that \(\epsilon_s\le1\) at every scale.  Then
\(\mathcal U_i\le2m_i\), and Lemma~\ref{cold:barrier} gives
\[
 P^b(\mathcal U_i,\mathcal U_i)
 \le4C\frac K{s}(d_s+|\xi|)m_i,\qquad
 \D\mathcal U_i\ge\frac{\epsilon_s}{L_s}
          \left(1+\frac78\frac{|\xi|^2}{d_s^2}\right)m_i.
\]
Since \(1+z^2\ge c(1+z)\), a fixed large \(C_*\) makes
\(\D\mathcal U_i\ge P^b(\mathcal U_i,\mathcal U_i)\).
Dropping the nonnegative loss in the equation for \(c_i\) permits
positive gain comparison from the initial trace.

For completeness, this comparison is valid in the measurable solution
class of Proposition~\ref{con:finite-evolution}.  At a fixed
approximation \(m_i\le C_sH\) throughout the interval, and
\(z_i=(c_i-\mathcal U_i)_+\) has a pathwise upper integral comparison
from zero with source
\(2P^b(\mathcal U_i,z_i)+P^b(z_i,z_i)\).
The bilinear \(H\)-norm estimate and Gronwall's inequality give
\(\|z_i\|_H=0\).  One may first integrate from a positive time and
then use the trace established in Lemma~\ref{cold:barrier}.

For the second assertion,
\(e^{-\kappa(t-t_1)}\mathcal U_i\) is a supersolution for
\(\D+\kappa\): its linear derivative has factor
\(e^{-\kappa(t-t_1)}\), whereas its quadratic gain has the smaller
factor \(e^{-2\kappa(t-t_1)}\).  Its value at \(t_1\) dominates
\(c_i(t_1)\).  The same comparison proves \eqref{cold:decay}.
\end{proof}

\subsection{Velocity measures and sums over colors}

The preceding comparison bounds a narrow free core and a small
remainder separately.  The next lemma identifies the mass of that
core at a fixed spatial point.  Axial expansion produces the factor
\((t+s)^{-1}\), even though the initial velocity-integrated density is of order \(s^{-1}\).

\begin{lemma}[Core formula and sheet amplitudes]\label{cold:core-amplitude}
At a fixed \((t,x)\), the velocity measure \(f_i(t,x,p)\dd p\) is the
pushforward of
\begin{equation}
 \frac{K S(\gamma(z_c-tp_3)/s)}
          {s+t\alpha\gamma S(\gamma(z_c-tp_3)/s)}\,J(w)\dd w,\qquad
 \bar p=d_s\bar w,\quad
 p_3=a_s^\sigma(z_c-tp_3)+d_sw_3,
 \label{cold:core-formula}
\end{equation}
with the initial spatial masks evaluated at \(x-tp\).
For \(t\le L_s\), it is dominated by probability sheets with total
amplitude \(C/(t+s)\).  The remainder
\(\epsilon_s m_i\dd p\) has sheet amplitude at most
\begin{equation}
 C L_sd_s/s^2\le Cs^{15/2}\le C/(t+s).
 \label{cold:remainder-amplitude}
\end{equation}
Thus the full upper bound \(\mathcal U_i\dd p\) has sheet amplitude \(C/(t+s)\).
In particular, for every fixed \(k\ge0\), a main color satisfies
\begin{equation}
 \int\langle p\rangle^k c_i(t,x,p)\dd p\le\frac{C_k}{t+s},\qquad
 \int_{|p|>R}\langle p\rangle^k c_i(t,x,p)\dd p
       \le\frac{C_k e^{-c_kR^2}}{t+s}\quad(R\ge3).
 \label{cold:moments}
\end{equation}
\end{lemma}
\begin{proof}
The transverse Jacobian is \(d_s^2\), while differentiation of the
implicit equation for \(p_3\) gives
\[
 \frac{\partial p_3}{\partial w_3}
 =\frac{d_s}{1+t(a_s^\sigma)'(z_c-tp_3)}.
\]
Multiplying by the density in \eqref{con:jet-datum} proves
\eqref{cold:core-formula}.  Its coefficient is at most
\(C/(s+t)\), since \(0\le S\le1\) and \(\alpha\gamma>0\).
A fixed product Gaussian dominates \(J\).  Moreover, the monotonicity
argument in Lemma~\ref{cold:barrier} gives
\(|p_3-U|\le d_s|w_3|\).  The Gaussian parameter estimate
(Lemma~\ref{ker:parameter-sheets}) therefore supplies the asserted
sheet domination, with constants independent of the axial compression.

The remainder is a Gaussian centered at \(Ue_3\), of width comparable
to \(d_s\), whose mass is at most
\(C\epsilon_s K/s=C L_sd_s/s^2\).  Its transverse Gaussian width is
within the sheet class and its axial exponential moment is uniformly
bounded because \(|U|\le1+\alpha\).  Since
\(L_sd_s/s^2=d_0s^{15/2}\) and \(t\le s^{-1/2}\), this also gives the
last inequality of \eqref{cold:remainder-amplitude}.  Uniform Gaussian
moments and tails, together with \eqref{cold:upper}, prove
\eqref{cold:moments}.
\end{proof}

\begin{lemma}[Nonmain tails]\label{cold:nonmain}
There is a function \(\theta_0(s_0)\to0\) as \(s_0\downarrow0\)
such that, uniformly at all times of existence,
\begin{equation}
 \sum_{\substack{i\ \mathrm{active}\\x\notin\mathcal O_i(t)}}
       \mathcal U_i(t,x,v)
       \le\theta_0(s_0)e^{-4|v|^2}.
 \label{cold:nonmain-bound}
\end{equation}
For \(0<t\le1\), and any fixed \(A,B>0\), the coefficient can instead
be taken to be \(C_{A,B}t^A s_0^B\).  Both assertions remain valid
for an arbitrary subset of the active labels.
\end{lemma}
\begin{proof}
Free cores vanish off their enlargements.  For a nonmain label either
\(D_y\ge(c_\eta/4)s^2\), or
\(|z_c|>a+2t+b_{\mathrm{sp}}\), in which case
\(D_z\ge b_{\mathrm{sp}}\) because \(|U|<2\).
Completing the velocity square in the remainder gives
\[
 \mathcal U_i(t,x,v)
 \le C s^{-31}e^{-4|v|^2}
       \exp\left(-\frac{D_y^2+D_z^2}{8d_s^2L_st}\right).
\]
Here the Gaussian rate is at least eight once the widths are fixed
small, and its centers stay bounded.  Half the distance exponential
is at most
\(\exp[-c/(s^{16}L_st)]\).  Since \(t\le L_s\), this is bounded by
\(e^{-c/s^{15}}\), which absorbs every polynomial in \(1/s\).
If also \(t\le1\), it absorbs arbitrary powers of \(t\) and \(s\).

It remains to sum the other half over lattice copies.  Uniformly,
\[
 D_y\ge(|y_c|-C)_+,\qquad
 D_z\ge(|z_c|-C(1+t))_+.
\]
The Gaussian distance scale \(d_s\sqrt{L_st}\) is bounded, and the
central spatial region has diameter \(O(1+L_s)\).  The lattice has
unit separation, hence at most \(C(R+1)^3\) points in a ball of
radius \(R\).  Summing in successive shells gives at worst a fixed
polynomial loss in \(1/s\).  The extracted exponential absorbs this
loss and the factor \(s^{-31}\).  Finally sum over the geometric
scales.  This proves both claims, including their stated uniformity.
\end{proof}

\begin{lemma}[Size at retirement and volume of active tubes]
\label{cold:retirement-size}
For some fixed constants \(C,C'>0\),
\begin{equation}
 \left\|\sum_{i:\,s_i=s}\mathcal U_i(L_s)\right\|_H
       \le C s^{-C'}.
 \label{cold:retirement-bound}
\end{equation}
Moreover, uniformly over the active label sets and all times,
\begin{equation}
 |\mathcal O(t)|\le\sum_{j\ge1}C s_j^2(1+L_{s_j})
                    \le C s_0^{3/2}.
 \label{cold:tube-volume}
\end{equation}
\end{lemma}
\begin{proof}
For \eqref{cold:retirement-bound}, use \(f_i\le m_i\) and the same
Gaussian lattice sum as in Lemma~\ref{cold:nonmain}, without extracting
a small distance factor.  The prefactor and the number of central
copies are polynomial in \(1/s\).  Completing the velocity square
bounds the remaining velocity factor by \(Ce^{-4|v|^2}\le CH(v)\).
For \eqref{cold:tube-volume}, the transverse area at scale \(s_j\)
is \(O(s_j^2)\), and its axial length while active is
\(O(1+L_{s_j})\).  Projection onto the torus does not increase volume.
Summation gives \(C\sum_j(s_j^2+s_j^{3/2})\le Cs_0^{3/2}\).
\end{proof}

All pointwise assertions above mean essential bounds.  They also hold
at the time sections defined by the finite path equations: free
coordinates preserve measure, and Fubini's theorem supplies a
full-measure set of paths on which the inequalities can be integrated.
For each fixed terminal time this gives a full-measure endpoint set.
Countably many approximation indices, lattice copies, and interval
choices can be handled simultaneously.  This convention will also be
used for the path estimates in the next section.
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//05-cold.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//06-early.tex ====
\section{Selection and continuation on the initial time interval}
\label{sec:early}

The cold estimates hold without an a priori bound on the hot part.
We use them first to select a hot seed whose evolution touches a prescribed
cap, and then to continue every selected evolution to a fixed time $t_1$.
The second step must be uniform as the cap horizon $T$ tends to zero.
The bath remains weak inside the hole at very small times; away from time
zero, the thinness of the jet tubes gives the additional smallness we need.
All constants in this section may depend on the fixed jet parameters.
They are independent of $N$, the collision cutoff, and the seed parameter,
unless explicitly stated otherwise.

\subsection{The bath and the initial source}

\begin{lemma}[Bath barrier]\label{early:bath}
There are $t_1\in(0,1/32]$, $\beta'>4$, and constants $C,c>0$,
independent of $\delta$, $R_\mu$, and $s_0$, such that every finite evolution
of \eqref{con:colors} satisfies, on its interval of existence,
\begin{equation}\label{early:bath-bound}
 0\le u(t,x,v)\le C e^{-\beta'|v|^2}
       \exp\!\left(-\frac{cD_0(x)^2}{t}\right),
 \qquad 0<t\le t_1,
 \qquad D_0(x)=\dist(x,D_{\rm hole}^c).
\end{equation}
For sufficiently small $s_0$ depending on $\delta$, one also has
\begin{equation}\label{early:hole-distance}
 D_0(x)\ge\delta
 \qquad (0\le t\le t_1,\ x\in\mathcal O(t)).
\end{equation}
\end{lemma}

\begin{proof}
The distance $D_0$ is $1$-Lipschitz. Fix $4<\beta'<\beta_0$ and a
positive absolute constant $c$, and consider
\[
 B(t,x,v)=\exp(C_1t)\exp\!\left(
       -(\beta_0-C_1t)|v|^2-\frac{cD_0(x)^2}{t}\right).
\]
At almost every point of a free path,
\begin{align*}
 \D\log B
 &=C_1+C_1|v|^2+\frac{cD_0^2}{t^2}
                  -\frac{2cD_0}{t}\,v\cdot\nabla D_0\\
 &\ge C_1+(C_1-c)|v|^2.
\end{align*}
The last inequality is completion of the square in $D_0/t$.
At fixed $(t,x)$, $B$ is a Maxwellian in velocity. Energy conservation in a
collision therefore gives $P(B,B)=B\nu_B$, and, as long as
$\beta_0-C_1t\ge\beta'>4$ and $e^{C_1t}\le2$,
\[
 P^b(B,B)\le P(B,B)\le C(1+|v|)B.
\]
Choose $C_1$ large enough that the preceding lower bound for
$\D\log B$ dominates $C(1+|v|)$, and then choose $t_1\le1/32$ small
enough to maintain both restrictions. These choices use only $\beta_0$
and the collision normalization.

The trace of $B$ along a path starting outside the hole dominates $u^0$.
Indeed, if $D_0(x)=0$, then $D_0(x+tv)\le t|v|$, so the distance factor
tends to one. If $D_0(x)>0$, its trace is zero, as is $u^0(x,v)$.
The differential inequality can be integrated from a positive time and
then passed to this trace. The positive-gain comparison used for
Proposition~\ref{cold:control}, with the nonnegative loss omitted, yields $u\le B$.
This uses only absolute continuity along paths at positive times and
boundedness in the $H$ norm; it does not differentiate the initial spatial
mask. Since $e^{C_1t}$ is bounded, \eqref{early:bath-bound} follows.

For $t\le1/32$, the axial interval defining $\mathcal O_i(t)$ has
half-length
\[
 a+2t+b_{\rm sp}\le\frac18+\frac1{16}+\frac1{16}=\frac14<\frac13.
\]
Its transverse distance from the corresponding lifted axis is at most
$s_0$. Thus every point of $\mathcal O(t)$ lies within distance $s_0$ of
the segment defining the hole. Since the hole is its open $2\delta$
neighborhood, the triangle inequality gives
$D_0\ge2\delta-s_0\ge\delta$ when $s_0\le\delta$.
\end{proof}

We fix $t_1$ from now on. Recall that
\[
 \mathcal S=2P^b(c,u)+\sum_{i\ne i'}P^b(c_i,c_{i'})
\]
is the source in the hot equation: it consists of collisions between
different cold colors and collisions between a cold color and the bath.

\begin{lemma}[Source estimates]\label{early:source-estimates}
For sufficiently small $s_0$ depending on $\delta$, every finite evolution
satisfies
\begin{equation}\label{early:source}
 P(c+u,H)\le\frac{C}{t+s_N}H,
 \qquad
 0\le\mathcal S\le C_{A,\delta}t^{A-1}H,
 \qquad 0<t\le t_1,
\end{equation}
on its interval of existence, for every fixed $A>0$.
Here $C$ is independent of $\delta$ and $R_\mu$, and the constants are
uniform for all sufficiently small $s_0$. No bound on $h$ is assumed.
\end{lemma}

\begin{proof}
At each $(t,x)$ there is at most one main color. Denote it by $a$, taking
$a=0$ if no main color is present, and put $r=c-a$.
The main-color measure bound in Lemma~\ref{cold:core-amplitude} and the sheet
gain estimates, including Lemma~\ref{ker:parameter-sheets}, give
\[
 P(a,H)\le\frac{C}{t+s_N}H.
\]
By \eqref{cold:nonmain-bound}, for each $M>0$,
\[
 0\le r\le\rho(t)H,\qquad \rho(t)\le C_Mt^M,
\]
uniformly for sufficiently small $s_0$. The bath bound gives $u\le CH$.
The quadratic gain bound of Lemma~\ref{lem:hot-bound} proves the first
estimate in \eqref{early:source}, absorbing the bounded terms since
$t+s_N\le2$.

If $a\ne0$, \eqref{early:hole-distance} improves the bath estimate to
$u\le C e^{-c\delta^2/t}H$ at this position. The ordered cross-color sum
is bounded by $2P(a,r)+P(r,r)$, so positivity yields
\[
 \mathcal S\le
 C\left(\frac{e^{-c\delta^2/t}+\rho(t)}{t+s_N}
                 +\rho(t)+\rho(t)^2\right)H.
\]
The exponential is bounded by any prescribed power of $t$, with a
constant depending on that power and on $\delta$. Taking $M=A$ proves
the second estimate. When $a=0$, only the last two types of terms remain,
and the same conclusion follows.
\end{proof}

Set $Z(t)=\|h(t)\|_H$. Dropping the loss in the hot path equation and
using \eqref{early:source}, we obtain, on every interval where $Z\le1$,
\begin{equation}\label{early:hot-comparison}
 Z(t)\le Z(r)+\int_r^t\left(
       \frac{C_2}{s+s_N}Z(s)+C_{A,\delta}s^{A-1}\right)\dd s,
 \qquad 0\le r\le t\le t_1.
\end{equation}
The constant $C_2$ includes the quadratic hot gain, because $Z^2\le Z$
and $s+s_N\le2$. It is independent of $\delta$, $R_\mu$, $s_0$, and $N$.
We will use integral inequalities such as \eqref{early:hot-comparison},
without assuming differentiability of the norm $Z$.

\subsection{Selecting a solution that touches the cap}

Fix the exponent in the seed \eqref{con:seed} so that
\begin{equation}\label{early:exponents}
 p>C_2+1,
 \qquad A>p+C_2+3,
 \qquad m_T=T^p.
\end{equation}
We call the solution with zero hot seed the \emph{dormant solution}.
The following selection adapts the cap argument of
\cite[Section~5]{OpenAIPeriodic}; we give the continuity argument here for
the present cutoff and color system.

\begin{proposition}[Cap selection]\label{early:cap-selection}
For fixed $\delta>0$, there are $\bar s_0,T_0>0$, uniform in $N$ and
the collision cutoff, with the following property.
For each $0<s_0\le\bar s_0$, each $0<T\le T_0$, and each finite
approximation, a seed parameter
$\lambda\in(0,L_0)$ gives a solution on $[0,T]$ satisfying
\begin{equation}\label{early:cap}
 Z(t)\le m_T(t+s_N)\quad(0\le t\le T),
 \qquad
 \max_{0\le t\le T}\frac{Z(t)}{m_T(t+s_N)}=1.
\end{equation}
The dormant solution exists on a fixed initial interval and satisfies
\begin{equation}\label{early:dormant-bound}
 Z(t)\le C'_{A,\delta}t^A.
\end{equation}
The interval and constant in \eqref{early:dormant-bound} are independent
of $N$, the collision cutoff, and sufficiently small $s_0$.
\end{proposition}

\begin{proof}
For the dormant solution, the function $Y(t)=C_dt^A$ is a supersolution
of \eqref{early:hot-comparison} if
\[
 C_d(A-C_2)\ge C_{A,\delta},
\]
because $(t+s_N)^{-1}\le t^{-1}$. At each fixed $N$, comparison begins
at the regular initial section $t=0$, where both functions vanish.
Choosing an initial interval on which $C_dt^A<1/2$ closes the condition
$Z\le1$. The cold and bath bounds are finite in the $H$ norm at each
fixed approximation. Thus the continuation criterion of
Proposition~\ref{con:finite-evolution} also proves existence on that
interval and gives \eqref{early:dormant-bound}.

Reduce $T_0$ until this interval contains $[0,T_0]$ and
$C_dT_0^{A-p-1}<1$. Then, for $0<t\le T\le T_0$,
\[
 \frac{Z(t)}{T^p(t+s_N)}
 \le\frac{C_dt^{A-1}}{T^p}
 \le C_dT^{A-p-1}<1.
\]
At $t=0$ the ratio is zero. Hence the dormant solution stays strictly
below the cap. At the other endpoint, $\lambda=L_0$, the ratio at time
zero equals $L_0\|\chi\|_H>1$.

Fix $T,N$, and the cutoff. Let $I$ be the set of
$\lambda\in[0,L_0]$ whose solutions exist on $[0,T]$ and remain strictly
below the cap there. Continuous dependence in the local theory and
continuation at the terminal section make $I$ relatively open.
It contains $0$ and does not contain $L_0$. Let $\lambda_*$ be the
right endpoint of the connected component of $I$ containing zero.
Along a sequence $\lambda_k\uparrow\lambda_*$ within that component,
all hot parts are bounded by the cap. The cold and bath estimates bound
the remaining colors uniformly in $k$ in the fixed-approximation norm.
The difference estimate in Proposition~\ref{con:finite-evolution}
therefore makes these solutions Cauchy, uniformly on $[0,T]$ in free
coordinates. Their limit is the solution with parameter $\lambda_*$
and satisfies the non-strict cap inequality.

The norm $Z$ is continuous in time, and the cap denominator is positive,
so the displayed ratio has a maximum. If that maximum were less than
one, continuous dependence and continuation would place a neighborhood
of $\lambda_*$ in $I$, contrary to its endpoint property. Its maximum
is therefore one. Since $0$ has a neighborhood in $I$, while
$L_0\|\chi\|_H>1$ also excludes a neighborhood of $L_0$, one has
$0<\lambda_*<L_0$. A final reduction of $T_0$ makes the cap at most one
on $[0,T]$ for every $T\le T_0$.
\end{proof}

\subsection{Smallness through the fixed bath time}

The pointwise estimate in \eqref{early:source} is very strong near zero,
but its constant depends on the hole width. To reach the fixed time
$t_1$ with an arbitrarily small hot part, we also use the shrinking
spatial support of the main colors.

\begin{lemma}[Small source along free paths]\label{early:path-source}
Fix $0<t_b<t_1$. There is a function $E_{t_b}(s_0)$ tending to zero as
$s_0\to0$ such that, for every finite evolution and every terminal time
$t\in[t_b,t_1]$ in its interval of existence,
\begin{equation}\label{early:source-integral}
 \left\|\int_{t_b}^t
    \mathcal S(s,x-(t-s)v,v)\dd s\right\|_H
 \le E_{t_b}(s_0).
\end{equation}
This estimate is uniform in the hot seed, the cap horizon, the
approximation index, and the collision cutoff.
\end{lemma}

\begin{proof}
On $[t_b,t_1]$, the nonmain bound \eqref{cold:nonmain-bound} gives
$r\le\varepsilon(s_0)H$, with $\varepsilon(s_0)\to0$.
The main-color sheet amplitude is bounded by $C/t_b$.
Consequently all terms of $\mathcal S$ containing a nonmain factor
are bounded by $C_{t_b}\varepsilon(s_0)H$, after enlarging
$\varepsilon$ if needed.

For the remaining main--bath term, define a deterministic majorant
\[
 \mathcal U_{\mathrm{main}}(s,x,v)
   =\sum_i\ind_{\mathcal O_i(s)}(x)\mathcal U_i(s,x,v),
 \qquad U_{\mathrm{bath}}(v)=C e^{-\beta'|v|^2},
\]
\[
 \mathcal S_{\mathrm{mb}}(s,x,v)
   =2P(\mathcal U_{\mathrm{main}}(s,x,\cdot),U_{\mathrm{bath}})(v).
\]
Here the sum includes all scales, so it bounds every finite selection;
at each position it has at most one nonzero summand.
By Proposition~\ref{cold:control} and \eqref{early:bath-bound},
\begin{equation}\label{early:common-majorant}
 0\le\mathcal S(s,x,v)
 \le C_{t_b}\varepsilon(s_0)H(v)+\mathcal S_{\mathrm{mb}}(s,x,v),
 \qquad t_b\le s\le t_1.
\end{equation}
The sheet gain estimates give $\mathcal S_{\mathrm{mb}}\le C_{t_b}H$.
The uniform large-output estimate \eqref{ker:large-output} gives
\begin{equation}\label{early:output-tail}
 \sup_{s,x,\,|v|>R}\frac{\mathcal S_{\mathrm{mb}}(s,x,v)}{H(v)}
 \le\omega_{t_b}(R),\qquad \omega_{t_b}(R)\longrightarrow0
 \quad(R\to\infty),
\end{equation}
with essential suprema in the velocity variable.
To handle velocities near the axis, fix $q'\in(1,q)$ and write
\[
 H_{q'}(v)=e^{-|v|^2}\langle v\rangle^{-100}
                 (1+|\bar v|^{-q'}).
\]
Since $U_{\mathrm{bath}}\le C_{q'}H_{q'}$, the same sheet lemma with exponent $q'$
implies $\mathcal S_{\mathrm{mb}}\le C_{t_b,q'}H_{q'}$. Hence, for $0<r\le1$,
\begin{equation}\label{early:output-axis}
 \sup_{s,x,\,0<|\bar v|<r}\frac{\mathcal S_{\mathrm{mb}}(s,x,v)}{H(v)}
 \le C_{t_b,q'}r^{q-q'}.
\end{equation}
Both estimates include the Gaussian-parameter measures in
Lemma~\ref{cold:core-amplitude}, by Lemma~\ref{ker:parameter-sheets}.

It remains to consider the compact velocity region
$|v|\le R$, $|\bar v|\ge r$. The spatial support of $\mathcal S_{\mathrm{mb}}$ is contained,
for every $s\le t_1$, in the projection of the fixed union
\[
 \bigcup_{n\in\Lambda}
     \bigl(n+B^2_{s_0}\times[-1/4,1/4]\bigr),
\]
where $B^2_{s_0}$ is the transverse disk of radius $s_0$.
A lifted free path with $|v|\le R$ and duration at most $t_1$ can meet
only $C_R$ of these cylinders, uniformly in its endpoint: their centers
must lie in a fixed bounded enlargement of a segment of length at most
$Rt_1$, and the lattice has a fixed separation.
For any one cylinder, its transverse projection is crossed in time at
most $2s_0/|\bar v|\le2s_0/r$.
Thus the normalized integral of $\mathcal S_{\mathrm{mb}}$ along such a path is bounded by
$C_{t_b,R}s_0/r$. Combining the three velocity regions gives
\[
 \left\|\int_{t_b}^t
     \mathcal S(s,x-(t-s)v,v)\dd s\right\|_H
 \le C_{t_b}\varepsilon(s_0)
      +t_1\omega_{t_b}(R)
      +C_{t_b,q'}t_1r^{q-q'}
      +C_{t_b,R}\frac{s_0}{r}.
\]
First take $R$ large, then $r$ small, and finally $s_0$ small.
This proves \eqref{early:source-integral}.

All estimates above concern the common pointwise majorant
\eqref{early:common-majorant}, rather than values chosen separately on
individual paths. To pass an almost-everywhere inequality to path
integrals, use free coordinates $(s,x_0,v)\mapsto(s,x_0+sv,v)$.
This map preserves measure, so Fubini's theorem gives the inequality at
almost every time on almost every free path. Integration on those paths
then defines representatives at every terminal section. Since free
transport preserves $\|\cdot\|_H$, the resulting bound is precisely the
essential norm in \eqref{early:source-integral} at each fixed terminal
time. The construction applies on any existing solution interval, so
this lemma does not presuppose continuation to $t_1$.
\end{proof}

\begin{proposition}[Uniform early continuation]\label{early:continuation}
Fix $\delta>0$ and $\zeta\in(0,1)$.
There are $\bar s_0>0$ and $T_0\in(0,t_1)$, depending on $\delta$ and
$\zeta$ and on the fixed jet parameters, such that for every
$0<s_0\le\bar s_0$, every finite approximation, and every
$0<T\le T_0$, the dormant solution and every solution selected in
Proposition~\ref{early:cap-selection} continue to $t_1$ and satisfy
\begin{equation}\label{early:small-hot}
 \sup_{0\le t\le t_1}\|h(t)\|_H\le\zeta.
\end{equation}
The choices are uniform in $N$, the collision cutoff, $R_\mu$, and $T$.
For dormant solutions, \eqref{early:dormant-bound} also holds on an
initial interval independent of these indices, with the fixed exponent
$A>p+1$ chosen in \eqref{early:exponents}.
\end{proposition}

\begin{proof}
We choose an intermediate time $t_b\in(0,t_1)$ and then require
$T_0<t_b$. At the cap horizon, both the selected and dormant solutions
satisfy
\[
 Z(T)\le T^p(T+s_N)\le2T^p,
\]
using only $T,s_N\le1$; no relation between $T$ and $s_N$ is needed.
For $T\le t\le t_b$, \eqref{early:hot-comparison} and
$(s+s_N)^{-1}\le s^{-1}$ give, while $Z\le1$,
\begin{equation}\label{early:intermediate-bound}
 Z(t)\le2T^{p-C_2}t^{C_2}
                  +\widetilde C_{A,\delta}t^A.
\end{equation}
Indeed multiplication of the scalar comparison by $t^{-C_2}$ integrates
the source to at most
$C_{A,\delta}t^{A-C_2}/(A-C_2)$.

From $t_b$ onward, retain the source inside the path integral rather
than using its pointwise bound. The same hot gain estimates and
Lemma~\ref{early:path-source} give
\[
 Z(t)\le Z(t_b)+E_{t_b}(s_0)
                 +\int_{t_b}^t\frac{C_2}{s}Z(s)\dd s.
\]
Consequently, as long as $Z\le1$,
\begin{equation}\label{early:final-comparison}
 \sup_{t_b\le t\le t_1}Z(t)
 \le\left(\frac{t_1}{t_b}\right)^{C_2}
       \left(2T_0^{p-C_2}t_b^{C_2}
              +\widetilde C_{A,\delta}t_b^A+E_{t_b}(s_0)\right),
\end{equation}
where the supremum is restricted to the current interval of existence
until continuation has been established.

The order of choices now matters. Since $A>C_2$, choose $t_b$ so that
\[
 \widetilde C_{A,\delta}t_1^{C_2}t_b^{A-C_2}<\frac\zeta6.
\]
Since $p>C_2$, reduce $T_0<t_b$, within the range of
Proposition~\ref{early:cap-selection}, so that
\[
 2T_0^{p-C_2}t_1^{C_2}<\frac\zeta6,
 \qquad 2T_0^p<\frac\zeta2.
\]
Finally take $\bar s_0$ small enough for all preceding cold and bath
estimates and for
\[
 \left(\frac{t_1}{t_b}\right)^{C_2}E_{t_b}(s_0)<\frac\zeta6
 \qquad(0<s_0\le\bar s_0).
\]
The last choice is possible by Lemma~\ref{early:path-source}.
Equation~\eqref{early:final-comparison} is then strictly below
$\zeta/2$. The same first two choices make
\eqref{early:intermediate-bound} strictly below $\zeta/2$ on
$[T,t_b]$, and the cap gives the same bound on $[0,T]$.

These estimates close the condition $Z\le1$ with a strict margin.
At each fixed approximation the cold sum and bath have bounded
$H$ norm through $t_1$, so the finite-evolution continuation criterion
excludes a finite endpoint before $t_1$.
This proves existence and \eqref{early:small-hot} uniformly in all the
stated indices. The dormant estimate was obtained independently on its
initial interval and is preserved under this continuation.
\end{proof}
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//06-early.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//07-global.tex ====
\section{A uniform global bound for the finite evolutions}
\label{sec:global}

The estimates before $t_1$ give both families of finite evolutions a
common starting point for continuation.  We now prove that they exist
for all time, with a uniform bound on the part of the density that
contains the bath, the hot component, and the retired jets.  The
characteristic comparisons retain the color decomposition used in
\cite[Sections~5--6]{OpenAIPeriodic}.  The additional control comes from
the bath: the resulting small entropy distance of $F$ from $\mu$ supplies a
positive collision frequency and makes the nonlinear gain uniformly
bounded after averaging along free flights.
Entropy smallness and characteristic estimates also underlie the
bounded-amplitude theory of
\cite[Section~3]{DuanHuangWangYang2017}. Here the cold components remain
separate until retirement, while Lemma~\ref{lem:gain-small-mass}
controls the gain using a majorant singular on the velocity axis.

Recall from \eqref{con:late-colors} that, for $t\ge t_1$,
\begin{equation}\label{global:decomposition}
 F=c+w,\qquad c=\sum_{i\text{ active}}c_i,
 \qquad w(t_1)=u(t_1)+h(t_1).
\end{equation}
Between retirement times the equation for $w$ is
\begin{equation}\label{global:w-equation}
 (\D+\nu_F^b)w=P^b(w,w)+2P^b(c,w)+\mathcal S_c,
 \qquad
 \mathcal S_c=\sum_{i\ne i',\ i,i'\text{ active}}P^b(c_i,c_{i'}).
\end{equation}
The sum is ordered.  At $L_s=s^{-1/2}$ we add the jets of scale $s$ to
$w$ and remove them from $c$, so that $F$ has no jump.  All bounds below
hold for the characteristic representatives, at each time section and
almost everywhere in $(x,v)$, as in the finite evolution construction.

The bath bound in \cref{early:bath} and
\cref{early:continuation}, with $\zeta<1$, give a constant $B_0$ such that
\begin{equation}\label{global:initial-bound}
 \|u(t)+h(t)\|_H\le B_0\quad(0\le t\le t_1).
\end{equation}
Here $B_0$ is independent of the hole, the bath cutoff, the smallest
scale, and the approximation.  Fix provisionally $B>B_0+2$ and consider
an interval of existence beginning at $t_1$ on which
\begin{equation}\label{global:bootstrap}
 0\le w(t,x,v)\le B H(v).
\end{equation}
We will improve this bound by a constant independent of $B$, after
making the entropy error and the remaining small parameters sufficiently
small depending on $B$.

\subsection{Averaging and a positive collision frequency}

The spatial entropy estimate is an $L^1_x$ bound, whereas our
continuation argument needs estimates at almost every spatial point.
Velocity integration along flights makes this passage possible.  We
first record the elementary averaging estimate in a form that also
applies over arbitrarily long time intervals.  Write
$M_H=\int H(v)\dd v<\infty$.

\begin{lemma}[Averaging along free flights]\label{global:flight-average}
Let $A:[0,\infty)\times(\R^3/\Lambda)\to[0,\infty)$ be measurable,
write $A_s=A(s,\cdot)$, and suppose
\[
 \sup_s\|A_s\|_\infty\le A_*,\qquad
 \sup_s\|A_s\|_1\le a_*.
\]
For a fixed bounded velocity set $V$ and $\rho>0$,
\begin{equation}\label{global:ball-average}
 \int_V A_s(x-\tau v)\dd v
 \le C_{V,\rho}a_*
 \qquad(\tau\ge\rho),
\end{equation}
uniformly in $x,s,\tau$.  If $A_*$ is fixed, the following quantities
tend to zero as $a_*\to0$:
\begin{align}
 &\sup_{t\ge t_1,x}
 \int_{t_1}^t e^{-\kappa(t-s)}
       \int H(v)A_s(x-(t-s)v)\dd v\dd s,
       \label{global:damped-average}\\
 &\sup_{t\ge t_1,x}
 \int_0^{t_1}\int H(v)A_s(x-(t-s)v)\dd v\dd s,
       \label{global:early-average}
\end{align}
where $\kappa>0$ and $t_1>0$ are fixed.  The supremum in $x$ is essential.
\end{lemma}

\begin{proof}
Contain $V$ in a ball of radius $R$.  The change of variables
$z=x-\tau v$ gives a factor $\tau^{-3}$.  The image ball meets at most
$C(1+\tau R)^3$ translates of a fixed bounded fundamental cell of
$\Lambda$.  Periodicity therefore gives
\[
 \int_V A_s(x-\tau v)\dd v
 \le C\tau^{-3}(1+\tau R)^3\|A_s\|_1,
\]
which proves \eqref{global:ball-average}.  In particular, its constant
does not grow when $\tau$ tends to infinity.

For $L\ge1$ let
\[
 H_L(v)=\min\{H(v),L\}\ind_{\{|v|\le L\}},\qquad
 \delta_L=\int(H-H_L)\dd v.
\]
Then $\delta_L\to0$, and \eqref{global:ball-average} implies, for
$\tau\ge\rho$,
\begin{equation}\label{global:weighted-average}
 \int H(v)A_s(x-\tau v)\dd v
 \le A_*\delta_L+C_{L,\rho}a_*.
\end{equation}
For $0\le\tau<\rho$, the same integral is at most $A_*M_H$.
Consequently the expression in \eqref{global:damped-average} is at most
\[
 \rho A_*M_H+\kappa^{-1}
       (A_*\delta_L+C_{L,\rho}a_*),
\]
and that in \eqref{global:early-average} is at most
\[
 \rho A_*M_H+t_1(A_*\delta_L+C_{L,\rho}a_*).
\]
Choose first $\rho$ small, then $L$ large, and finally $a_*$ small.
Both upper bounds tend to zero in this order.  The same proof works
when the time integrals are restricted to an interval of existence.
\end{proof}

Recall the entropy quantities from \eqref{con:entropy-distance}:
\begin{equation}\label{global:entropy-distance}
 \begin{aligned}
 D_F(t,x)&=\int\langle v\rangle|F(t,x,v)-\mu(v)|\dd v,
 &\|D_F(t)\|_1&\le\varepsilon_E,\\
 D_-(t,x)&=\int\langle v\rangle(\mu(v)-F(t,x,v))_+\dd v.
 \end{aligned}
\end{equation}
The second quantity is bounded pointwise by
$\int\langle v\rangle\mu(v)\dd v$, independently of the bootstrap.
The first has a pointwise bound
\begin{equation}\label{global:moment-bound}
 D_F(t,x)\le C(B),\qquad t\ge t_1/2,
\end{equation}
as long as \eqref{global:bootstrap} holds after $t_1$.  Indeed, before
$t_1$ use \eqref{global:initial-bound}; after $t_1$ use $w\le BH$.
At every point there is at most one active main jet, whose mass and
first moment are bounded by $C/(t+s)\le2C/t_1$ by
\cref{cold:core-amplitude}.  The other jets are controlled by
\cref{cold:nonmain}.  These bounds prove \eqref{global:moment-bound}
uniformly in the number of retained scales.

\begin{lemma}[A collision frequency independent of the bootstrap size]
\label{global:frequency}
There is a constant $\kappa>0$, depending only on the fixed Maxwellian
$\mu$ and $t_1$, with the following property.  For each fixed $B>B_0+2$,
if $\varepsilon_E$ is sufficiently small depending on $B$, then every
finite evolution satisfying \eqref{global:bootstrap} obeys
\begin{equation}\label{global:frequency-bound}
 \nu_F^b(t,x,v)\ge\kappa\qquad(t\ge t_1).
\end{equation}
The constant is uniform for $b\ge1$ and for all approximation indices.
\end{lemma}

\begin{proof}
Fix two closed velocity balls $V_1,V_2$ of positive radius with
$\dist(V_1,V_2)>3$, and put $V=V_1\cup V_2$.  All constants until the
choice of the entropy tolerance may depend on these fixed balls.
Since $b\ge1$ and $\mu$ is a Maxwellian,
\[
 P^b(\mu,\mu)(v)=\mu(v)\nu_\mu^b(v)
 \ge \mu(v)\nu_\mu^1(v)\ge m_0>0
 \qquad(v\in V).
\]
At a fixed $(t,x)$ write $d=(\mu-F)_+$.  Positivity of the gain and
$F\ge\mu-d$ imply
\[
 P^b(F,F)\ge P^b(\mu-d,\mu-d)
 \ge P^b(\mu,\mu)-2P(\mu,d).
\]
By the plane formula \eqref{ker:carleman}, the Gaussian integral over
each plane is bounded, so
\[
 P(\mu,d)(v)\le C\int\frac{d(p)}{|p-v|}\dd p.
\]
For $|p-v|<r$, use $d\le\mu\le1$ to bound this integral by $Cr^2$;
for $|p-v|\ge r$, it is at most $Cr^{-1}D_F(t,x)$.  Choose $r>0$
once so that the first contribution to $2P(\mu,d)$ is at most $m_0/2$.
With $m_*=m_0/2$ and a fixed constant $C$, we obtain
\begin{equation}\label{global:gain-lower}
 P^b(F,F)(t,x,v)\ge m_*-CD_F(t,x),\qquad
 \nu_F^b(t,x,v)\le C+CD_F(t,x),\qquad v\in V.
\end{equation}
The second inequality follows directly from
$\min\{b,|v-p|\}\le |v-p|\le C\langle p\rangle$ on $V$.

Let $\Delta=t_1/2$, fix $t\ge t_1$, and write
$X_\tau=x-(t-\tau)v$.  The Duhamel formula for the total density,
whose value is continuous through retirements, gives
\[
 F(t,x,v)\ge
 \int_{t-\Delta}^{t-\Delta/2}
 e^{-\int_s^t\nu_F^b(\tau,X_\tau,v)\dd\tau}
 P^b(F,F)(s,X_s,v)\dd s.
\]
Using \eqref{global:gain-lower}, $e^{-a}\ge1-a$ for $a\ge0$, and the
upper bound $1$ for a survival factor, the integrand is bounded below by
\begin{equation}\label{global:survival-lower}
 m_*e^{-C\Delta}
 \left(1-C\int_s^t D_F(\tau,X_\tau)\dd\tau\right)
 -CD_F(s,X_s).
\end{equation}
This inequality remains valid when its right side is negative.

Integrate \eqref{global:survival-lower} over $s$ and over either ball
$V_k$.  The positive constant contributes
\[
 a_k:=|V_k|(\Delta/2)m_*e^{-C\Delta}>0.
\]
Choose $0<\rho<\Delta/2$.  For $t-\tau\ge\rho$,
\eqref{global:ball-average} and \eqref{global:entropy-distance} give
\begin{equation}\label{global:entropy-flight}
 \int_{V_k}D_F(\tau,x-(t-\tau)v)\dd v
 \le C_{V_k,\rho}\varepsilon_E.
\end{equation}
The term $D_F(s,X_s)$ always has $t-s\ge\Delta/2$.  Its integrated
cost is therefore $C_\rho\varepsilon_E$.  For the double time integral
in \eqref{global:survival-lower}, the portion with $t-\tau\ge\rho$ has
the same type of bound, whereas the remaining portion costs at most
$C(B)\rho$ by \eqref{global:moment-bound}.  Thus
\[
 \int_{V_k}F(t,x,v)\dd v
 \ge a_k-C_\rho\varepsilon_E-C(B)\rho.
\]
After $B$ has been fixed, choose $\rho$ to make the last term at most
$\min(a_1,a_2)/4$, and then choose $\varepsilon_E$ to make the middle
term at most the same quantity.  It follows that
\[
 \int_{V_k}F(t,x,v)\dd v\ge a_k/2\qquad(k=1,2).
\]
For any velocity $v$, one of the balls has distance at least $1$ from
$v$: otherwise their mutual distance would be less than $2$.
On that ball $\min\{b,|v-p|\}\ge1$.  Consequently
\[
 \nu_F^b(t,x,v)
 =2\pi\int\min\{b,|v-p|\}F(t,x,p)\dd p
 \ge\pi\min(a_1,a_2)=:\kappa.
\]
Only the entropy tolerance, and not this value of $\kappa$, depended
on $B$.
\end{proof}

The positive frequency gives exponential decay to every active jet.
Specifically, \eqref{cold:decay} yields
\begin{equation}\label{global:cold-decay}
 c_i(t,x,v)\le e^{-\kappa(t-t_1)}\mathcal U_i(t,x,v),
 \qquad t_1\le t\le L_{s_i}.
\end{equation}
At a point in the active tube union $\mathcal O(t)$ there is at most
one main color.  Its sheet amplitude is at most $C/t_1$; outside the
union all colors are nonmain.  Applying \cref{lem:sheet-bound} and
\cref{cold:nonmain}, and enlarging its small coefficient if necessary,
gives constants $C_3,C_4$ independent of $B$ such that
\begin{align}
 P(c,H)&\le C_3e^{-\kappa(t-t_1)}H
            \bigl(\ind_{\mathcal O(t)}+\theta_0\bigr),
              \label{global:cold-gain}\\
 \mathcal S_c&\le C_4\theta_0H,
              \label{global:cold-source}
\end{align}
where $\theta_0=\theta_0(s_0)\to0$ as $s_0\to0$.
For the second estimate, if $c_*$ is the main color and $r$ the sum of
the remaining colors, the ordered cross sum is at most
$2P(c_*,r)+P(r,r)$.  Here $r\le\theta_0e^{-4|v|^2}$ and the sheet bound
controls $P(c_*,H)$; if there is no main color use $c\le
\theta_0e^{-4|v|^2}$ and \cref{lem:hot-bound}.

At a retirement time, \cref{cold:retirement-size} and
\eqref{global:cold-decay} bound the increment of $w$ by $J_sH$, where
\begin{equation}\label{global:retirement}
 J_s=C s^{-C}e^{-\kappa(L_s-t_1)},\qquad
 \sum_sJ_s\le J_0(s_0):=
 \sum_{j\ge1}C s_j^{-C}e^{-\kappa(s_j^{-1/2}-t_1)}\longrightarrow0.
\end{equation}
This follows for the one-sided trace just before retirement, using
only the bootstrap up to that time.  To check the last limit explicitly,
the exponential dominates every power of $s$: for sufficiently small
$s$, $Cs^{-C}e^{-\kappa(s^{-1/2}-t_1)}\le s^2$.  The remaining geometric
sum is at most $s_0^2/(e^{2\eta}-1)$.  In particular the sum of all
retirement increments can be made small uniformly in $N$.

\subsection{Small excess mass and a bounded nonlinear gain}

The remaining obstruction is the quadratic term $P^b(w,w)$: estimating
it directly from $w\le BH$ would produce a constant of order $B^2$.
Instead we compare $w$ with the fixed Maxwellian.  Entropy bounds the
spatial integral of its positive excess, and the preceding flight
estimate converts that bound into small excess mass at each position.
The small-mass gain estimate then controls the quadratic term by a
constant chosen before $B$.

\begin{lemma}[Uniform nonlinear gain]\label{global:gain}
Let $B>B_0+2$ be fixed and suppose \eqref{global:bootstrap} holds on an
interval beginning at $t_1$.  After taking
$\varepsilon_E$, $\zeta$, and $s_0$ sufficiently small depending on $B$,
one has on that interval
\begin{equation}\label{global:gain-bound}
 P^b(w,w)\le C_5H,
 \qquad C_5:=\|P(\mu,\mu)\|_H+1.
\end{equation}
In particular, $C_5$ is independent of $B$, the approximation, and the
cap horizon.  The estimate holds also at the appropriate one-sided
sections of retirement times.
\end{lemma}

\begin{proof}
Choose the entropy tolerance small enough for
\cref{global:frequency}, so that the damping constant $\kappa$ and
\eqref{global:cold-gain}--\eqref{global:retirement} are available.
Define
\begin{equation}\label{global:excess}
 j=(w-\mu)_+,\qquad m_j(t,x)=\int j(t,x,v)\dd v.
\end{equation}
Since $w\le F$ and $j\le BH$,
\begin{equation}\label{global:excess-mass}
 0\le m_j\le BM_H,
 \qquad\|m_j(t)\|_1\le\varepsilon_E.
\end{equation}
Write $C_\mu=\|\mu\|_H<\infty$ and $A_B=2C_\mu+B$.
The inequality $w\le\mu+j$, bilinearity, and positivity give
\begin{equation}\label{global:gain-difference}
 P^b(w,w)-P^b(\mu,\mu)
 \le2P(\mu,j)+P(j,j)\le A_BP(H,j).
\end{equation}
The frequency deficit has the bound
\begin{equation}\label{global:loss-difference}
 (\nu_\mu^b-\nu_F^b)\mu
 \le2\pi\mu(v)\int|v-p|(\mu(p)-F(p))_+\dd p
 \le CH(v)D_-.
\end{equation}
Here $\mu(v)\langle v\rangle\le CH(v)$.

Between retirements, subtracting the stationary Maxwellian equation
from \eqref{global:w-equation} gives
\[
 (\D+\nu_F^b)(w-\mu)
 =P^b(w,w)-P^b(\mu,\mu)
   +(\nu_\mu^b-\nu_F^b)\mu+2P^b(c,w)+\mathcal S_c.
\]
The chain rule for the positive part on almost every free flight
therefore gives an upper comparison for $j$ with damping $\kappa$.
Using \cref{lem:gain-small-mass} in \eqref{global:gain-difference},
and then \eqref{global:loss-difference},
\eqref{global:cold-gain}, and \eqref{global:cold-source}, shows that
for each $\epsilon>0$ its positive source is at most $H(v)R(s,x)$,
where
\begin{equation}\label{global:excess-source}
 R(s,x)=\epsilon+C_{B,\epsilon}m_j(s,x)+CD_-(s,x)
           +C(B)\bigl(\ind_{\mathcal O(s)}(x)+\theta_0\bigr).
\end{equation}
Constants have been enlarged to absorb $A_B$.  At a retirement time,
the increase of $j$ is at most the increase of $w$, because
$(a+d)_+-a_+\le d$ for $d\ge0$.  Thus its increments are bounded by
the same $J_sH$ as in \eqref{global:retirement}.

We also need to estimate the excess already present in the bath at
$t_1$.  Put $j_u=(u-\mu)_+$ and $m_{j_u}=\int j_u\dd v$ for
$0\le s\le t_1$.  The bath starts below $\mu$, so $j_u(0)=0$;
\cref{early:bath} gives the fixed bound $j_u\le B_0H$.
Since $u\le F$,
\begin{equation}\label{global:bath-excess-mass}
 \|m_{j_u}(s)\|_\infty\le B_0M_H,
 \qquad\|m_{j_u}(s)\|_1\le\varepsilon_E.
\end{equation}
The bath equation is $(\D+\nu_F^b)u=P^b(u,u)$.  Subtracting the
Maxwellian equation and applying the same two difference estimates as
above gives, now discarding the nonnegative damping, a source bounded
by $H(v)R_u(s,x)$, where
\begin{equation}\label{global:bath-excess-source}
 R_u(s,x)=\epsilon+C_{B_0,\epsilon}m_{j_u}(s,x)+CD_-(s,x).
\end{equation}
The cold density appears only through the frequency deficit, which is
controlled by $D_-$.  No positive source involving a cold gain occurs
in this bath equation.

At $t_1$, the hot bound gives $j(t_1)\le j_u(t_1)+\zeta H$.
Continue the same backward free flight through $t_1$, and write
$X_s=x-(t-s)v$.  The two comparisons, together with all intervening
retirements, imply
\begin{equation}\label{global:excess-path}
 \frac{j(t,x,v)}{H(v)}
 \le\zeta+J_0
   +\int_0^{t_1}R_u(s,X_s)\dd s
   +\int_{t_1}^t e^{-\kappa(t-s)}R(s,X_s)\dd s.
\end{equation}
We have dropped damping from the initial and retirement contributions;
doing so also allows the bath comparison to run along that one flight
all the way back to zero.

We claim that the velocity integral of the right side times $H$ can
be made smaller than any prescribed number $\tau>0$, uniformly in
$(t,x)$ on the bootstrap interval.  The constant terms contribute at
most
\begin{equation}\label{global:constant-errors}
 M_H\left(\zeta+J_0+\epsilon(t_1+\kappa^{-1})
                      +\kappa^{-1}C(B)\theta_0\right).
\end{equation}
For the remaining terms, all the spatial functions in question have
a fixed pointwise upper bound once $B$ is fixed:
$m_j\le BM_H$, $m_{j_u}\le B_0M_H$,
$D_-\le\int\langle v\rangle\mu\dd v$, and
$\ind_{\mathcal O(s)}\le1$.  Their spatial integrals are respectively
bounded by $\varepsilon_E$, $\varepsilon_E$, $\varepsilon_E$, and
$Cs_0^{3/2}$, the last by \eqref{cold:tube-volume}.
Apply \cref{global:flight-average} to each term, with its coefficient
in \eqref{global:excess-source} or \eqref{global:bath-excess-source}.
For clarity, the choices are made in the following order.  First choose
$\epsilon$ so that its contribution to \eqref{global:constant-errors}
is less than $\tau/4$; all the coefficients $C_{B,\epsilon}$ and
$C_{B_0,\epsilon}$ are then fixed.  For each of the finitely many
averaged terms, choose the time deletion length $\rho$ and the
truncation $H_L$ in the proof of that lemma so that the endpoint and
velocity-tail errors, including its coefficient, have total less than
$\tau/4$.  Next make $\varepsilon_E$ and $s_0$ sufficiently small for
the remaining averaged errors to total less than $\tau/4$.  Finally
take $\zeta$ small and, if necessary, reduce $s_0$ further to make the
other terms in \eqref{global:constant-errors} less than $\tau/4$.
All these choices are uniform in the endpoint time.  Thus
\begin{equation}\label{global:small-excess}
 m_j(t,x)\le\tau.
\end{equation}
The same calculation on a pre-retirement section includes only the
jumps that have already occurred, so it does not require a post-jump
bootstrap hypothesis.

It remains to select the particular smallness threshold that controls
the gain.  Apply \cref{lem:gain-small-mass} once more, now with
$\epsilon'=1/(2A_B)$.  Write its constant as $C_{B,\epsilon'}$ and set
\[
 \tau=\frac{1}{2A_B\max\{1,C_{B,\epsilon'}\}}.
\]
Choose the parameters in the preceding argument to give
\eqref{global:small-excess} for this $\tau$.  Then
\[
 A_BP(H,j)\le
 A_BH\bigl(\epsilon'+C_{B,\epsilon'}m_j\bigr)\le H.
\]
Equation \eqref{global:gain-difference} and $P^b(\mu,\mu)\le
P(\mu,\mu)$ prove \eqref{global:gain-bound}.  The latter has finite
$H$ norm because $P(\mu,\mu)=\mu\nu_\mu$ is a Gaussian times a function
of at most linear growth.
\end{proof}

\subsection{Parameter choice and continuation}

We can now choose the bootstrap size and then realize all the
smallness requirements imposed above.  This order is essential: both
the damping rate and the ultimate gain bound have already been fixed
independently of the provisional value $B$.

\begin{proposition}[Uniform global continuation]\label{prop:global-control}
The construction parameters can be fixed so that every dormant finite
evolution and every cap solution with $0<T\le T_0$ continues globally.
There are constants $B<\infty$ and $\kappa>0$, independent of $N$,
$b_N\ge1$, and $T$, such that
\begin{equation}\label{global:uniform-bound}
 \|w(t)\|_H\le B,\qquad \nu_F^{b_N}(t,x,v)\ge\kappa,
 \qquad t\ge t_1.
\end{equation}
Every active jet satisfies \eqref{global:cold-decay} through its
retirement, the total retirement increment is bounded by
\eqref{global:retirement}, and the hot component satisfies the small
bound of \cref{early:continuation} on $[0,t_1]$.
\end{proposition}

\begin{proof}
On a bootstrap interval let $Z_w(t)=\|w(t)\|_H$.  The Duhamel formula
for \eqref{global:w-equation}, with all retirement increments included,
and \eqref{global:frequency-bound}, \eqref{global:cold-gain},
\eqref{global:cold-source}, and \eqref{global:gain-bound}, give
\begin{align*}
 Z_w(t)\le {}&e^{-\kappa(t-t_1)}B_0
  +\frac{C_5+C_4\theta_0}{\kappa}+J_0\\
 &+2C_3(1+\theta_0)
     \int_{t_1}^t e^{-\kappa(t-s)}e^{-\kappa(s-t_1)}Z_w(s)\dd s.
\end{align*}
Take $s_0$ small enough that $\theta_0,J_0\le1$ and discard the first
exponential in the last integrand.  We obtain
\begin{equation}\label{global:scalar-bootstrap}
 Z_w(t)\le B_0+\frac{C_4+C_5}{\kappa}+1
       +4C_3\int_{t_1}^t e^{-\kappa(s-t_1)}Z_w(s)\dd s.
\end{equation}
The integral version of Gronwall's inequality therefore gives
\begin{equation}\label{global:strict-improvement}
 Z_w(t)\le
 \left(B_0+\frac{C_4+C_5}{\kappa}+1\right)
       \exp\left(\frac{4C_3}{\kappa}\right)=:B_*.
\end{equation}
The constants defining $B_*$ were fixed before $B$.

Here is the complete order of parameter choices.  Fix the kernel and
jet parameters, the narrow width factor, the Maxwellian $\mu$, and the
short time $t_1$ as in the preceding sections; also fix the early
growth exponent $p$ there.  The bath estimate fixes $B_0$.
The proof of \cref{global:frequency} fixes $\kappa$ from $\mu$, $t_1$,
and the two chosen balls, without imposing its entropy tolerance yet.
The cold estimates then fix $C_3,C_4$ uniformly for sufficiently small
$s_0$, while $C_5=\|P(\mu,\mu)\|_H+1$ is already fixed.  Choose
\[
 B>\max\{B_0+2,B_*+3\}.
\]
For this $B$, prescribe the entropy tolerance required by
\cref{global:frequency}, and the possibly smaller entropy tolerance,
hot tolerance $\zeta$, and upper bounds on $s_0$ required by
\cref{global:gain}.  Include $\theta_0,J_0\le1$ among these requirements.

To realize the entropy tolerance, take the hole radius $\delta$ small
and the bath velocity cutoff $R_\mu$ large.  Their contributions to
the initial relative entropy can then be made arbitrarily small.
With these parameters fixed, the jet and seed contributions become
arbitrarily small as $s_0\to0$, as proved in
\eqref{con:entropy-distance} and its derivation.  To realize the
prescribed $\zeta$, use \cref{early:continuation}: choose its auxiliary
time $t_b$ and horizon upper bound $T_0$ after $\delta,\zeta$, and then
take $s_0$ sufficiently small for its source estimate.  Decrease $s_0$
further to meet the entropy, separation, cold-gain, and retirement
requirements already listed.  Reducing $T_0$ when required by the
early estimates preserves all later estimates.  These are fixed
construction parameters; none depends on $N$, on the chosen cap solution,
or on its horizon $T\in(0,T_0]$.

For completeness, the strict estimate closes the bootstrap even at
retirement times.  At $t_1$ it starts by
\eqref{global:initial-bound}.  On an interval with no retirement,
the characteristic equations make the weighted norm continuous, and
\eqref{global:strict-improvement} bounds it by $B_*<B-3$ as long as
it is at most $B$.  Thus it cannot first reach $B$ there.  At a
retirement time $L_s$, apply the argument only up to the left-hand
section.  The cold comparison on that interval and
\eqref{global:retirement} give
\[
 Z_w(L_s+)\le Z_w(L_s-)+J_s\le B_*+1<B-2.
\]
This starts the bootstrap on the next interval without having assumed
its conclusion across the jump.  The total density and its entropy
bound are unchanged by this relabeling.

Finally, at each fixed approximation only finitely many scales occur.
The cold barriers bound the sum of the active colors in $H$ norm on
every finite interval, with constants allowed to depend on that
approximation.  The uniform bound on $w$ therefore bounds the total
$H$ norm on every such interval.  The cutoff local existence and
continuation argument from the construction excludes a finite maximal
existence time.  There are only finitely many retirements at that
approximation, after which $F=w$.  This proves global existence and
\eqref{global:uniform-bound}, with all the stated auxiliary estimates.
\end{proof}
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//07-global.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//08-compactness.tex ====
\section{Global limits and local conservation}\label{sec:compactness}

The uniform bounds of Proposition~\ref{prop:global-control} allow us to
remove both the collision cutoff and the truncation of the initial jets.
The relevant compactness is stronger than a bound on mass and entropy:
the same-position collision products are uniformly integrable, with
polynomial velocity weights. This will preserve the local energy flux
and rule out defects in all five collision-invariant balances.
The argument extends the fixed-horizon compactness proof of
\cite[Section~6]{OpenAIPeriodic} to the global approximations constructed
here; we give the proof, including the averaging step.

\begin{proposition}\label{prop:global-limits}
Fix the construction parameters supplied by
Proposition~\ref{prop:global-control}, and let $b_N\to\infty$.
Either the dormant family, or a family of cap solutions at one fixed horizon
$T\in(0,T_0]$, has a subsequence of total densities $F_N$ converging
weakly in $L^1([0,\tau]\times\Omega)$ for every $\tau<\infty$ to a
nonnegative density $F$. This subsequence can be chosen so that
$F_N(t)\rightharpoonup F(t)$ in $L^1(\Omega)$ at every $t\ge0$,
with uniformly small polynomially weighted velocity tails on each
bounded time interval.

The limit has initial datum $F_0$ and is a global entropy solution
obeying local conservation in the sense of
Definition~\ref{def:admissible}. More precisely,
\begin{equation}\label{comp:strong-properties}
 F\in C([0,\infty);L^1(\Omega)),\qquad
 \int_0^\tau\!\int_\Omega
 \langle v\rangle^k\bigl(F+Q^+(F,F)+Q^-(F,F)\bigr)
 \dd x\dd v\dd t<\infty
 \quad(k\ge0,\ \tau<\infty).
\end{equation}
Total mass, momentum, and energy are conserved at every time.
The common early-time pointwise majorant of the dormant approximations
also bounds any dormant limit.
\end{proposition}

Throughout the proof, $T$ denotes the fixed cap horizon when the cap
family is used, whereas $\tau$ denotes an arbitrary finite observation
time. Constants may depend on $\tau$, but not on $N$ or $b_N$.
We write
\[
 Q_N^\pm=Q^{\pm,b_N}(F_N,F_N),\qquad
 R_{k,N}(t,x)=\int_{\R^3}\langle v\rangle^kF_N(t,x,v)\dd v.
\]
The initial convergence is supplied by Lemma~\ref{con:datum};
the finite-approximation entropy inequalities and local conservation
laws are those of Proposition~\ref{con:finite-evolution}.

\subsection{Small scales and uniform integrability}

Recall that a color of scale $s=s_j$ remains active until
$L_s=s^{-1/2}$. Let $\mathcal A_s(t)$ be the union, in one periodic
cell, of its two signs' enlarged sets $\mathcal O_i(t)$, including
all lattice copies, when $t\le L_s$; set
$\mathcal A_s(t)=\varnothing$ for $t>L_s$.
We use these sets for every scale, even if that scale is absent from a
particular finite approximation. The active enlarged sets are disjoint.
Their axial lengths on $[0,\tau]$ are bounded by $C_\tau$, so
\[
 |\mathcal A_s(t)|\le C_\tau s^2
 \qquad(0\le t\le\tau).
\]
The free-core amplitude estimate, the cold remainder bound, and the
bounds on the bath and the hot or regular part imply
\begin{equation}\label{comp:moment-envelope}
 R_{k,N}(t,x)\le C_{k,\tau}
 \left(1+\sum_{j\ge1}
       \frac{\ind_{\mathcal A_{s_j}(t)}(x)}{t+s_j}\right).
\end{equation}
Here the bounded term includes all nonmain colors and all retired
colors: after retirement they belong to $w\le BH$.
For active main colors the estimate follows from
Lemma~\ref{cold:core-amplitude} and Proposition~\ref{cold:control};
the nonmain contribution is controlled by Lemma~\ref{cold:nonmain}.

The square of this envelope controls the collision products.
Disjointness ensures that no product of two distinct singular terms
appears when it is squared, and one scale contributes at most
\begin{equation}\label{comp:annular-square}
 \int_0^\tau\frac{C_\tau s^2}{(t+s)^2}\dd t
 =C_\tau s^2\left(\frac1s-\frac1{\tau+s}\right)
 \le C_\tau s.
\end{equation}
For $\varepsilon>0$, put
$\mathcal A_{<\varepsilon}(t)=
\bigcup_{s_j<\varepsilon}\mathcal A_{s_j}(t)$.
Since the scales form a geometric sequence,
\begin{equation}\label{comp:small-annuli}
 \sup_N\int_0^\tau\!\int_{\mathcal A_{<\varepsilon}(t)}
          R_{k,N}(t,x)^2\dd x\dd t
 \le C_{k,\tau}\sum_{s_j<\varepsilon}(s_j+s_j^2)
 \longrightarrow0
 \qquad(\varepsilon\downarrow0).
\end{equation}
The first power of the envelope gives, uniformly at individual times,
\begin{equation}\label{comp:fixed-time-small}
 \sup_{N,\,0\le t\le\tau}
 \int_{\mathcal A_{<\varepsilon}(t)}R_{k,N}(t,x)\dd x
 \le C_{k,\tau}\sum_{s_j<\varepsilon}(s_j+s_j^2)
 \longrightarrow0.
\end{equation}
These estimates include retirement times, because a color may be
included in its last active enlargement at that section and is covered
by the bound on $w$ thereafter.

Outside $\mathcal A_{<\varepsilon}(t)$, a main color can have only
one of the finitely many scales $s_j\ge\varepsilon$.
Their thermal widths are positive and fixed. The Gaussian velocity
bounds and the summation over lattice copies therefore give
\begin{equation}\label{comp:fixed-majorant}
 F_N(t,x,v)\le \Psi_{\varepsilon,\tau}(v)
 \quad\text{if }x\notin\mathcal A_{<\varepsilon}(t),
 \qquad
 0\le\Psi_{\varepsilon,\tau}\le C_{\varepsilon,\tau}H.
\end{equation}
The same majorant includes the bounded hot, bath, and retired parts,
and the Gaussian nonmain tails. In particular it has every polynomial
velocity moment; its singularity on the velocity axis is integrable.

Equations~\eqref{comp:small-annuli}--\eqref{comp:fixed-majorant}
prove uniform integrability in the full spacetime and two-velocity
variables of
\begin{equation}\label{comp:weighted-pairs}
 \langle v\rangle^k\langle v_*\rangle^k
 F_N(t,x,v)F_N(t,x,v_*),\qquad k\ge0.
\end{equation}
Indeed their integrals on the removed spatial sets are bounded by
\eqref{comp:small-annuli}; on the complement they are dominated by the
fixed integrable function
$\langle v\rangle^k\Psi_{\varepsilon,\tau}(v)
 \langle v_*\rangle^k\Psi_{\varepsilon,\tau}(v_*)$.
First make the removed contribution small, and then use absolute
continuity of this fixed integral on arbitrary small measurable sets.
The same decomposition gives uniformly small weighted velocity tails.
Using the first power instead of the square proves the corresponding
one-particle assertions, both in spacetime and, by
\eqref{comp:fixed-time-small}, uniformly at all individual times.

The separate collision terms satisfy the same spacetime conclusions.
Collision symmetry and conservation of the sum of squared speeds give
\begin{equation}\label{comp:collision-moments}
 \int_{\R^3}\langle v\rangle^kQ_N^\pm(t,x,v)\dd v
 \le C_k R_{k+1,N}(t,x)^2.
\end{equation}
For the gain, this follows by placing the velocity weight on an
outgoing particle and using
$\langle v'\rangle^k|v-v_*|
 \le C_k\langle v\rangle^{k+1}\langle v_*\rangle^{k+1}$;
the loss is estimated directly. The cutoff factor is at most one.
Thus \eqref{comp:small-annuli} controls their weighted integrals on the
removed sets. On the complement, positivity gives
\[
 Q_N^+\le P(\Psi_{\varepsilon,\tau},\Psi_{\varepsilon,\tau}),
 \qquad
 Q_N^-\le\Psi_{\varepsilon,\tau}\nu_{\Psi_{\varepsilon,\tau}}.
\]
Both majorants have every polynomial moment, either by
\eqref{comp:collision-moments} for the fixed density
$\Psi_{\varepsilon,\tau}$ or by the bounds on $H$.
Consequently $\{Q_N^\pm\}$ is uniformly integrable with every fixed
polynomial velocity weight. Its integrals over time intervals whose
length tends to zero vanish uniformly in $N$: the small-scale
contribution can first be discarded, and the remaining fixed
majorants have integrals proportional to the length of the interval.

\subsection{Velocity averaging with uniformly integrable sources}

Weak compactness alone does not identify a product evaluated at the
same spatial point. We obtain the needed compactness of velocity
averages from the transport equation. The following elementary form of
the averaging principle of Golse, Lions, Perthame, and Sentis
\cite{GolseLionsPerthameSentis1988} is also proved in
\cite[Section~6]{OpenAIPeriodic}.

\begin{lemma}\label{comp:averaging}
Let $f_n,g_n$ be locally uniformly integrable in $L^1$ on an open
region of time, space, and velocity, and suppose that
$(\partial_t+v\cdot\nabla_x)f_n=g_n$ there in distributions.
For every smooth compactly supported velocity test $\psi$, the averages
$\int\psi(v)f_n(t,x,v)\dd v$ are relatively compact in local
$L^1_{t,x}$ wherever the hypotheses hold for all velocities in a
neighborhood of $\supp\psi$.
The spatial domain may be a flat torus or an open subset of $\R^3$.
\end{lemma}

\begin{proof}
Localize to a compact time-space set and a compact velocity
neighborhood of $\supp\psi$, using smooth cutoffs. Denote the localized
functions by $q_n$, extend them by zero, and set
$r_n=(1+\partial_t+v\cdot\nabla_x)q_n$.
The equation shows that $r_n$ is uniformly integrable in $L^1$ and has
support in a fixed bounded set of time and velocity, and also of space
in the Euclidean case. Spatial cutoff derivatives cause no difficulty
because velocity has been localized. Integration from a time preceding
the support gives
\[
 q_n(t,x,v)=\int_0^\infty e^{-s}r_n(t-s,x-sv,v)\dd s=:Rr_n(t,x,v).
\]
The damped inverse $R$ is an $L^1$ contraction, since free transport
preserves Lebesgue measure.

Truncate $r_n$ at height $L$, preserving its sign, to obtain
$r_n^{(L)}$. Uniform integrability implies
\[
 \sup_n\|r_n-r_n^{(L)}\|_1\longrightarrow0
 \quad(L\to\infty),\qquad
 \sup_n\|r_n^{(L)}\|_2<\infty
 \quad\text{for fixed }L.
\]
The velocity averages of $R(r_n-r_n^{(L)})$ are therefore uniformly
small in $L^1$. It remains to prove compactness for a bounded $L^2$
family of right-hand sides.

Take Fourier transform in $(t,x)$, with Fourier series in the torus
case. For $u_n=Rr_n^{(L)}$, Cauchy--Schwarz yields
\[
 \left|\int\psi(v)\widehat u_n(\sigma,k,v)\dd v\right|^2
 \le I_\psi(\sigma,k)
       \int|\widehat r_n^{(L)}(\sigma,k,v)|^2\dd v,
 \qquad
 I_\psi(\sigma,k)=\int
 \frac{|\psi(v)|^2}{1+(\sigma+k\cdot v)^2}\dd v.
\]
If $\supp\psi\subset\{|v|\le R_0\}$, integration in the direction
of $k$ gives $I_\psi\le C_\psi/|k|$ for $k\ne0$.
When $|\sigma|>2R_0|k|$, the denominator instead gives
$I_\psi\le C_\psi/(1+|\sigma|^2)$.
Thus $I_\psi\to0$ uniformly as $|\sigma|+|k|\to\infty$.
Plancherel's theorem shows that the averages have uniformly small
high-frequency $L^2$ tails. Convolution with a fixed smooth approximate
identity consequently approximates them uniformly in $L^2$.
On every compact time-space set these convolutions are bounded and
equicontinuous, by Cauchy--Schwarz applied to translates of the
convolution kernel and its derivatives. They are relatively compact
there. This proves local $L^2$, and hence local $L^1$, compactness at
fixed $L$; the uniform $L^1$ error as $L\to\infty$ completes the proof.
The argument applies unchanged to the dual lattice of a rotated torus.
\end{proof}

\subsection{Identification of the collision terms}

The uniform integrability and velocity tightness give a subsequence
with $F_N\rightharpoonup F$ in
$L^1([0,\tau]\times\Omega)$.
Since $\D F_N=Q_N^+-Q_N^-$, Lemma~\ref{comp:averaging} applies.
After extraction for a countable dense family of compact velocity
tests, all smooth compact velocity averages converge strongly in local
$L^1_{t,x}$ to the corresponding averages of $F$.
The extension from the countable family follows from the uniform
integrated mass bound.

Set $W_N(t,x,v,v_*)=F_N(t,x,v)F_N(t,x,v_*)$.
By \eqref{comp:weighted-pairs}, a further subsequence has a weak $L^1$
limit $W$. For smooth compactly supported $\psi,\chi$,
\[
 \int\psi(v)\chi(v_*)W_N(t,x,v,v_*)\dd v\dd v_*
 =\left(\int\psi F_N\dd v\right)
  \left(\int\chi F_N\dd v_*\right).
\]
The right-hand side converges in measure locally in $(t,x)$.
It is uniformly integrable there: its absolute value is bounded by the
integral of $|\psi(v)\chi(v_*)|W_N$, and the full pair-product bounds
control this integral on small time-space sets. Vitali's theorem
therefore gives convergence in local $L^1$.
It follows that $W$ and $F(t,x,v)F(t,x,v_*)$ have identical integrals
against separated velocity tests, for almost every $(t,x)$.
A countable determining family gives a common full-measure set;
finite sums of separated tests are dense among continuous functions on
compact two-velocity sets. Hence
\begin{equation}\label{comp:pair-limit}
 W_N\rightharpoonup F(t,x,v)F(t,x,v_*)
 \quad\text{in }L^1([0,\tau]\times\Omega\times\R^3).
\end{equation}
In particular the product on the right is integrable.
The uniform integrability removes time strips near $0$ and $\tau$,
so interior averaging suffices. Weighted tightness extends the
identification to each fixed polynomial velocity weight. One may
extract simultaneously for integer moment orders; domination by a
higher integer order then supplies every order $k\ge0$.

Adjoining the finite angular measure preserves weak convergence.
For each $\omega$, the elastic substitution on $(v,v_*)$ is
measure preserving and involutive. Consequently
\eqref{comp:pair-limit} also gives weak convergence of the primed pair
products on the collision space. On bounded input velocities the
collision factor is bounded and the cutoff equals one for all large
$N$. The estimates with higher velocity moments remove the complement.
For the gain this uses the preserved sum of squared speeds when an
output test is pulled back to the inputs. These observations identify
any weak limit of the separately uniformly integrable gains and losses:
\begin{equation}\label{comp:collision-limit}
 Q_N^\pm\rightharpoonup Q^\pm(F,F)
 \quad\text{in }L^1([0,\tau]\times\Omega),
\end{equation}
also after multiplication by any fixed polynomial velocity weight.
The argument permits bounded measurable tests: after the angular
integration and elastic substitution, such a test remains bounded on
each truncated collision domain.

We can now pass to the unrenormalized equation, including the initial
boundary term, since the initial densities converge in $L^1$.
Thus $\D F=Q(F,F)$ in distributions with datum $F_0$, and both
separate collision terms have the weighted integrability in
\eqref{comp:strong-properties}. The remaining work concerns the time
representative, entropy, and the local invariant balances.

\subsection{A single representative at every time}

For a smooth periodic $\varphi(x,v)$ compactly supported in velocity,
the cutoff path equations give
\[
 \big|\langle F_N(t)-F_N(s),\varphi\rangle\big|
 \le\int_s^t\!\int_\Omega
 \left(F_N|v\cdot\nabla_x\varphi|
       +(Q_N^++Q_N^-)|\varphi|\right)\dd x\dd v\dd u.
\]
The first term is bounded by $C_\varphi|t-s|$.
The second has a modulus tending to zero with $|t-s|$, uniformly in
$N$, by the short-time-integral conclusion above.
For a countable family dense in the uniform norm on each compact
velocity set, Arzel\`a--Ascoli therefore supplies uniform convergence
of the pairings on $[0,\tau]$, after one extraction.

At any individual time, \eqref{comp:fixed-time-small} and
\eqref{comp:fixed-majorant} make $\{F_N(t)\}$ relatively weakly
compact in $L^1(\Omega)$. Every cluster point has the smooth pairings
just obtained, and hence is the same density. Thus this one subsequence
converges weakly in $L^1$ at every time, defining a representative
$F(t)$ whose smooth pairings are continuous and which agrees with the
spacetime limit. Weighted fixed-time tightness gives the same conclusion
for moments, so the conserved cutoff mass, momentum, and energy pass
at every time. In particular the initial trace of this representative
is $F_0$.

Let $\Phi(A,B)=(A-B)\log(A/B)$ with its nonnegative
lower-semicontinuous extension. It is jointly convex: it is the sum of
the two convex perspectives $A\log(A/B)$ and $B\log(B/A)$.
On any bounded collision domain, \eqref{comp:pair-limit} and its primed
version give joint weak convergence of the two arguments of $\Phi$.
Restriction to $[0,t]$ preserves this convergence for every fixed
$t\in[0,\tau]$. Weak lower semicontinuity therefore applies to the
integral of $|a|\Phi$; the weight can be absorbed into the arguments
by $\Phi(cA,cB)=c\Phi(A,B)$ for $c\ge0$.
On $|v|,|v_*|\le R$ the cutoff is absent once $b_N\ge2R$.
Exhausting the velocity space yields
\[
 \int_0^t\mathcal D(F(s))\dd s
 \le\liminf_N\int_0^t\mathcal D^{b_N}(F_N(s))\dd s.
\]
The same convex lower-semicontinuity argument applies to
$\mathcal H_M(F_N(t))$ under the weak $L^1$ convergence at that time.
Combining the two nonnegative terms with the finite-approximation
entropy inequality and initial entropy convergence gives
\begin{equation}\label{comp:limit-entropy}
 \mathcal H_M(F(t))+\int_0^t\mathcal D(F(s))\dd s
 \le\mathcal H_M(F_0),\qquad 0\le t\le\tau.
\end{equation}
The extraction is independent of $t$; no exceptional set of time
endpoints is introduced by the lower-semicontinuity argument.

For completeness, the negative part of $F\log F$ is bounded by mass,
energy, and a fixed Gaussian integral. On
$e^{-1-|v|^2}\le F<1$, use
$-F\log F\le(1+|v|^2)F$; below that threshold, monotonicity of
$-r\log r$ on $(0,e^{-1})$ gives the integrable bound
$(1+|v|^2)e^{-1-|v|^2}$.
The positive part is controlled by the relative entropy integrand and
a multiple of $F$. Hence the absolute logarithmic integral is bounded
uniformly on $[0,\tau]$.

\subsection{Strong traces, renormalization, and the invariant balances}

The integrability of the unrenormalized collision operator now yields
strong continuity. In free coordinates the distributional equation,
including its initial trace, reads
\[
 \partial_t\bigl[F(t,x+tv,v)\bigr]=Q(F,F)(t,x+tv,v).
\]
The right-hand side belongs to spacetime $L^1$.
Subtracting its Bochner integral leaves a distribution with zero time
derivative and initial value $F_0$. Therefore
\begin{equation}\label{comp:path-formula}
 F(t,x+tv,v)=F_0(x,v)
       +\int_0^t Q(F,F)(s,x+sv,v)\dd s
\end{equation}
holds as an $L^1_{x,v}$ identity, first for almost every time.
Its right-hand side is absolutely continuous in $L^1$.
Free transport is a strongly continuous group on $L^1(\Omega)$,
as follows by approximation by continuous functions with compact
velocity support. Thus \eqref{comp:path-formula} defines a strongly
continuous representative. Its smooth pairings agree almost
everywhere in time with those of the representative already chosen;
continuity makes them agree at every time.
In particular $F\in C([0,\tau];L^1)$, $F(0)=F_0$ strongly, and
pairings with every bounded measurable test are continuous.
All these observations also hold with any fixed polynomial velocity
weight, since both the datum and the source have that integrability.

Equation~\eqref{comp:path-formula} holds on almost every individual
free path. On such a path $M(v)>0$ is constant. For a renormalization
$\beta$ from Definition~\ref{def:admissible}, the ordinary absolutely
continuous chain rule gives
\[
 \D\bigl[M\beta(F/M)\bigr]=\beta'(F/M)Q(F,F).
\]
The source is integrable because $|\beta'|\le C_\beta$.
Moreover, for nonnegative $f,g$,
\[
 |M\beta(f/M)-M\beta(g/M)|\le C_\beta|f-g|,
 \qquad |M\beta(f/M)|\le C_\beta f.
\]
The strong initial and terminal traces consequently pass through the
renormalization. Integrating the path identity against a smooth test
proves exactly \eqref{eq:renormalized}, including its initial term.
The separate integrability requirement follows from
$0\le Q^\pm/(1+F/M)\le Q^\pm$.

Finally, let
$\psi\in\{1,v_1,v_2,v_3,|v|^2/2\}$ and let
$\phi\in C_c^\infty([0,\infty)\times(\R^3/\Lambda))$.
At the cutoff level the local identity is
\[
 \int_0^\infty\!\int_\Omega
 F_N\psi(v)(\partial_t\phi+v\cdot\nabla_x\phi)
 \dd x\dd v\dd t
 +\int_\Omega F_N(0,x,v)\psi(v)\phi(0,x)\dd x\dd v=0.
\]
It follows from the path equation and collision symmetry, all integrals
being absolutely convergent. Weighted weak convergence of $F_N$ and
weighted convergence of the initial data pass this identity to $F$.
For the energy identity the largest required weight is $|v|^3$,
which is covered by \eqref{comp:strong-properties}.
This proves all the exact local balances \eqref{eq:local-balances}
with their initial traces. Equivalently, their collision terms vanish
in the limit by absolute weighted collision integrability; no
moment defect is present.

Apply the preceding extractions successively on $[0,m]$, $m\in\N$,
and take a diagonal subsequence, retaining the countable moment and
velocity-test families. The representatives on overlapping intervals
agree by their smooth pairings, so they define one global strongly
continuous density. All integrability, entropy, and conservation
statements hold on every finite interval. For a dormant sequence, sum
the cold majorants over all scales and add the bath and dormant-hot
majorants on their common early interval. This is a locally integrable
pointwise upper bound independent of $N$. Testing against nonnegative
compactly supported functions and passing to the weak limit preserves
it. Its velocity integral is also integrable in space at each fixed
time, by the first-power estimate used in
\eqref{comp:fixed-time-small}. The every-time weak convergence therefore
preserves the bound at each such time section as well. This completes
the proof of Proposition~\ref{prop:global-limits}.
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//08-compactness.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//09-dilation.tex ====
\section{The cap dilation and its linear limit}\label{sec:dilation}

The cap selects finite solutions whose hot component has a prescribed
weighted size at some time before the horizon. We now magnify that event.
The bath and the quadratic hot interactions disappear under this
magnification; the cold particles converge to two axial velocity atoms
whose densities retain the annular geometry. This is the dilation
argument of \cite[Section~7]{OpenAIPeriodic}, with the additional seed
factor and bath included explicitly below.

Choose horizons \(T_N\downarrow0\), terminal scales
\(s_N/T_N\to0\), and cutoffs \(b_N\to\infty\), and take the cap
solutions furnished by Proposition~\ref{early:cap-selection}. Indices may be
relabeled after passage to a subsequence. Write
\(m_N=m_{T_N}=T_N^p\), and choose a time
\(t_{*,N}\in[0,T_N]\) at which the cap is attained. On the periodic
lift, define
\begin{equation}\label{eq:rescaling}
\begin{gathered}
r_N=t_{*,N}+s_N,\qquad
h_{r_N}(t,x,v)=\frac{h_N(r_Nt,r_Nx,v)}{m_Nr_N},\qquad
C_{r_N}(t,x,v)=r_Nc_N(r_Nt,r_Nx,v),\\
l_N=\frac{s_N}{r_N}\longrightarrow l\in[0,1],\qquad
U_N=\frac{T_N}{r_N}\longrightarrow U\in[1,\infty].
\end{gathered}
\end{equation}
The limits are obtained by subselection, allowing the second ratio to
tend to infinity. We have \(r_N\to0\), \(r_N\le2T_N\) eventually,
and
\(U_N\ge(1+s_N/T_N)^{-1}\), which proves \(U\ge1\).
The rescaled cap and its attained value are
\begin{equation}\label{dil:cap-bound}
0\le h_{r_N}(t,x,v)\le(t+l_N)H(v)\quad(0\le t\le U_N),
\qquad \|h_{r_N}(1-l_N)\|_H=1.
\end{equation}
When no confusion is possible, we suppress the index on \(r_N\).
\input{figures/scales}

\subsection{The limiting mesh and background}

The dilated annular scales are \(\varrho_{N,j}=s_j/r_N\), \(j\leq N\).
Their ratio is \(e^\eta\), their smallest member is \(l_N\), and
their largest member tends to infinity. Subselect their logarithmic
phase modulo \(\eta\). The grids then converge on compact
subintervals of \((0,\infty)\) to a geometric mesh \(\mathcal M\)
unbounded above. If \(l>0\), its members are
\(l,e^\eta l,e^{2\eta}l,\ldots\). If \(l=0\), it is a two-sided
geometric mesh with no smallest member. To verify this description,
choose a grid point in \((e^{-\eta},1]\), pass to its limit, and
multiply it by the fixed factors \(e^{j\eta}\). When the smallest
point stays positive its integer distance in the grid from that chosen
point is bounded, so can also be fixed by subselection. Coincident phase
endpoints give the same mesh.

For each \(\varrho>0\) and \(\sigma\in\{-1,1\}\), let
\(u_{\varrho,\sigma}(t,z)\) be the unique solution of
\[
u=\sigma+\alpha\tanh\frac{\gamma(z-tu)}\varrho.
\]
Thus \(u_{\varrho,\sigma}=U_{\varrho,\sigma}\) in the notation of the cold profiles. The left side minus the right side has derivative in \(u\) equal to
\(1+t\alpha\gamma S(\gamma(z-tu)/\varrho)/\varrho>0\); its values have
opposite signs for sufficiently large positive and negative \(u\).
This proves existence and uniqueness. The implicit function theorem
gives smoothness, and
\[
|u_{\varrho,\sigma}-\sigma|\leq\alpha,
\qquad
\partial_z u_{\varrho,\sigma}
=\frac{\alpha\gamma S(\gamma(z-tu)/\varrho)}
       {\varrho+t\alpha\gamma S(\gamma(z-tu)/\varrho)}\geq0.
\]
Put \(\theta=1/4\), the fraction of each transverse stripe occupied
by one sign. On an open mesh annulus \(\varrho<|y|<e^\eta\varrho\), define the
nonnegative velocity measure
\begin{equation}\label{eq:background}
\begin{split}
B(t,x,\dd p)
&=\sum_{\sigma=\pm1}b_{\varrho,\sigma}(t,z)
                         \delta_{u_{\varrho,\sigma}(t,z)e_3}(\dd p),\\
b_{\varrho,\sigma}(t,z)
&=\frac{\theta K S(\gamma(z-tu_{\varrho,\sigma})/\varrho)}
 {\varrho+t\alpha\gamma S(\gamma(z-tu_{\varrho,\sigma})/\varrho)}.
\end{split}
\end{equation}
Set \(B=0\) in the hole \(|y|<l\) if \(l>0\), and assign arbitrary
values obeying the same bounds on annular interfaces and the transverse
axis. These exceptional positions have zero Lebesgue measure. In
particular, for every \(k\geq0\),
\begin{equation}\label{dil:background-bound}
\int\langle p\rangle^kB(t,x,\dd p)
\leq\frac{C_k}{t+\varrho}\leq\frac{C_k}{t}
\quad(t>0)
\end{equation}
on an active annulus. If \(l>0\), the bound is instead uniformly
\(C_k/(t+l)\) for all \(t\geq0\).

\begin{proposition}\label{prop:blowup-limit}
Every sequence \eqref{eq:rescaling} has a subsequence for which
\(h_{r_N}\rightharpoonup g\) in local spacetime \(L^1\) on
\((0,U)\times\R^3_x\times\R^3_v\), where
\begin{equation}\label{eq:limit-equation}
0\leq g(t,x,v)\leq(t+l)H(v),\qquad
\D g=2P(B,g)-g\nu_B.
\end{equation}
Here \(B\) is \eqref{eq:background}; its gain and loss are defined by
integration against the velocity measure. Both terms on the right are
locally integrable, with a velocity majorant uniform in position on
each compact positive-time interval. Compactly supported velocity
averages of \(h_{r_N}\) converge strongly in local \(L^1_{t,x}\).
The limit has a locally strongly continuous \(L^1\) representative
at positive times and obeys the integrating-factor formula
\eqref{dil:mild} below.

If \(l>0\), pass also to a limit \(\lambda_N\to\lambda_0\) of the
seed parameters. The conclusion and its local continuity extend to
time zero, with
\begin{equation}\label{dil:initial-trace}
g(0,x,v)=\lambda_0 l\,\chi(x/l,v).
\end{equation}
\end{proposition}

\subsection{Cold convergence: survival, Jacobian, and stripe averaging}

We first identify the cold measure independently of the hot limit.
Fix a compact time-space region
\[
a\leq t\leq A<U,\qquad |x|\leq R,
\]
where \(a>0\). Since \(U_N\to U\) and \(A<U\), eventually
\(r_NA<T_N\). The cap therefore holds on the entire initial
physical interval \([0,r_NA]\), as required for the survival
estimate below.
By Proposition~\ref{cold:control}, the dilated
cold velocity moments are bounded there, and indeed uniformly in
position, by
\begin{equation}\label{dil:cold-moment-bound}
\int\langle p\rangle^k C_{r_N}(t,x,p)\dd p
\leq C_k\left(\frac1{t+l_N}+1\right).
\end{equation}
The nonmain tails have the still smaller bound
\(C_M r_N^{M+1}t^Me^{-4|p|^2}\) for any fixed \(M>0\).
Uniformly decreasing velocity tails hold under these moment bounds.

First restrict to annuli with \(\varrho_{N,j}\) in a fixed compact
interval \([\varepsilon,R_1]\subset(0,\infty)\). Their number is
bounded independently of \(N\). A free-core path at scale
\(s=r_N\varrho_{N,j}\) has bounded velocity and remains in its own
spatial enlargement. At its position, all free velocities in that
color are within \(C d_s\) of the same center, so the contribution
to its loss frequency is at most \(C d_s/s\).
The remainder in Proposition~\ref{cold:control} has velocity moment
amplitude \(O(s^{15/2})\), by Lemma~\ref{cold:core-amplitude}.
The nonmain colors have the bound of Lemma~\ref{cold:nonmain}; the
bath bound and the cap bound give bounded frequencies at these
velocities. Hence, over physical times \(0\le t_{\rm old}\le r_NA\),
the optical depth on every free-core path is at most
\[
 C_A r_N\bigl(d_s/s+s^{15/2}+1\bigr)=o(1)
\]
uniformly on this compact scale range. The cutoff only decreases
this frequency. Keeping the freely transported term in the positive
path formula of \eqref{con:colors} gives
\[
 e^{-\epsilon_N}f_i\le c_i\le f_i+\epsilon_s m_i,
 \qquad \epsilon_N\longrightarrow0,
\]
where \(f_i\) is the free core of \eqref{cold:free-core}; the symbols
\(\epsilon_N\) and \(\epsilon_s\) denote different quantities.
After multiplication by \(r_N\), the remainder has velocity mass
\begin{equation}\label{dil:remainder-power}
 r_N O(s^{15/2})=O(r_N^{17/2}\varrho_{N,j}^{15/2}).
\end{equation}
The same estimates hold with every fixed polynomial velocity weight.
Consequently the dilated actual color and its free core differ by
\(o(1)\) in velocity total variation, uniformly on the compact
regions under consideration.

On bounded dilated spatial sets, only the copy centered at the origin
can be main for sufficiently large \(N\). Indeed, an intersecting
copy would have a lattice center \(n\) with
\(|\bar n|\le s_0+o(1)\) and
\(|n_3|\le 1/8+b_{\rm sp}+o(1)\). Since \(1/8+b_{\rm sp}=3/16\)
and \(s_0\) is small, this contradicts \(|n|\ge1\) unless
\(n=0\). This argument uses the Euclidean spacing of the rotated
lattice and identifies the free cores directly on its lift.

We compute a centered free core exactly. Fix one sign and one such annulus,
write \(\varrho_N=s/r_N\), and use the velocity parameters
\[
\bar p=d_s\bar w,\qquad
p_3=\sigma+\alpha\tanh\frac{\gamma(z-tp_3)}{\varrho_N}+d_sw_3.
\]
The axial map depends only on \(w_3\). Its derivative is
\[
\frac{\partial p_3}{\partial w_3}
=\frac{d_s}
 {1+t\alpha\gamma S(\gamma(z-tp_3)/\varrho_N)/\varrho_N}.
\]
Thus Equation~\eqref{cold:core-formula}, after dilation, says that the
velocity integral of the free core against a test \(\psi\) is
\begin{equation}\label{dil:free-core-integral}
\begin{split}
\int r_N f_i(r_Nt,r_Nx,p)\psi(p)\dd p
={}&\int J(w)
 \frac{K S(\gamma(z-tp_3)/\varrho_N)}
      {\varrho_N+t\alpha\gamma S(\gamma(z-tp_3)/\varrho_N)}
 \psi(d_s\bar w,p_3)\\
&\quad\cdot\ind_{E_j^\sigma}(r_Ny-r_Ntd_s\bar w)
 \ind_{\{|r_N(z-tp_3)|<1/8\}}\dd w.
\end{split}
\end{equation}
On the compact regions under consideration the last axial indicator
equals one for all sufficiently large \(N\). Since the implicit
velocity map has derivative at least one in \(p_3\),
\[
|p_3-u_{\varrho_N,\sigma}(t,z)|\leq d_s|w_3|.
\]
After the mesh subselection, \(\varrho_N\to\varrho>0\), so the velocity
test and the smooth amplitude in \eqref{dil:free-core-integral}
converge uniformly for \(w\in\supp J\) to
\[
\psi(u_{\varrho,\sigma}e_3)
\frac{K S(\gamma(z-tu_{\varrho,\sigma})/\varrho)}
 {\varrho+t\alpha\gamma S(\gamma(z-tu_{\varrho,\sigma})/\varrho)}.
\]

It remains to average the transverse indicator. In dilated coordinates
its period is
\[
\frac{s^2}{r_N}=r_N\varrho_N^2\longrightarrow0.
\]
The transverse shift of its argument is \(td_s\bar w\), whose ratio
to that period is at most
\[
\frac{Ad_s}{r_N\varrho_N^2}=A d_0 r_N^9\varrho_N^8\longrightarrow0.
\]
For the indicator of the fixed periodic stripe pattern, a translation
by this vanishing relative amount changes its integral on a bounded
set by \(o(1)\), uniformly in its phase and in bounded \(t,w\).
One can verify this by summing the lengths of the small intervals
around stripe endpoints. The annular margins have dilated width
\(O(s^2/r_N)=O(r_N\varrho_N^2)\), so their area also tends to zero on
the annulus. The outer annular radii converge; their boundaries can
likewise be removed in sets of arbitrarily small area.

Finally the periodic stripe indicators converge weak-* to their mean
\(\theta\). Explicitly, if \(q\) is the one-period indicator and
\(q-\theta\) has bounded periodic primitive \(Q\), then for a smooth
compactly supported one-dimensional test \(f\), period \(p\), and
arbitrary phase \(\zeta\),
\[
\left|\int f(y_1)(q(y_1/p+\zeta)-\theta)\dd y_1\right|
\leq p\|Q\|_\infty\|f'\|_1.
\]
Fubini's theorem gives the same assertion with the other coordinates
and parameters present. Approximation in \(L^1\) extends it to the
annular restrictions. Integrating \(J\), whose mass is one, in
\eqref{dil:free-core-integral} produces exactly the coefficient
\(b_{\varrho,\sigma}\) in \eqref{eq:background}.

These arguments identify the cold limit on all compact positive
scale annuli. To remove the lower scale restriction, use
\eqref{dil:cold-moment-bound}: on the slab \(t\geq a\), the integral
over \(|y|<C\varepsilon\), with bounded \(z\), is at most
\(C_{a,A,R,k}\varepsilon^2\). The same estimate holds for the
candidate \(B\). Scales larger than a fixed multiple of \(R\) have
no main color in the chosen spatial compact set, and their escaped
tails vanish. We have proved that for every continuous compactly
supported velocity test \(\psi\),
\begin{equation}\label{dil:cold-weakstar}
\int C_{r_N}(t,x,p)\psi(p)\dd p
\stackrel{*}{\rightharpoonup}
\int\psi(p)B(t,x,\dd p)
\quad\hbox{in local }L^\infty_{t,x}
\end{equation}
on \((0,U)\times\R^3\). Indeed the computation first gives weak
convergence against smooth compact time-space tests, and the common
\(L^\infty\) bound extends it to all \(L^1\) tests. Polynomial
velocity tails can also be removed uniformly. If \(l>0\), choose
\(\varepsilon<l/2\); all the bounds used above hold down to time
zero, so \eqref{dil:cold-weakstar} does too.

\subsection{The hot equation and the size of every error}

The transport derivative of \(h_N(r_Nt,r_Nx,v)/(m_Nr_N)\)
is \(m_N^{-1}(\D h_N)(r_Nt,r_Nx,v)\). Thus the equation for
\(h_N\) gives exactly
\begin{equation}\label{dil:prelimit-equation}
\D h_{r_N}
=2P^{b_N}(C_{r_N},h_{r_N})
 -h_{r_N}\nu^{b_N}_{C_{r_N}}+E_N,
\end{equation}
where, writing \(u_N^r(t,x,v)=u_N(r_Nt,r_Nx,v)\),
\[
\begin{split}
E_N={}&r_N\bigl[2P^{b_N}(u_N^r,h_{r_N})
                   -h_{r_N}\nu^{b_N}_{u_N^r}\bigr]\\
 &+m_Nr_N^2\bigl[P^{b_N}(h_{r_N},h_{r_N})
                   -h_{r_N}\nu^{b_N}_{h_{r_N}}\bigr]
   +m_N^{-1}\mathcal S_N(r_Nt,r_Nx,v).
\end{split}
\]
Here \(\mathcal S_N\) is the nonnegative source in
\eqref{con:colors}, including the cold--bath and distinct-color gains.
The bath estimate \eqref{early:bath-bound} gives \(u_N^r\le C H\). For every fixed
\(A<\infty\), the cap and the gain estimates therefore imply, on
\(0\le t\le\min(A,U_N)\),
\begin{equation}\label{dil:error-bounds}
 |E_N(t,x,v)|
 \le C_A\bigl(r_N+m_Nr_N^2+T_N^{M-1-p}\bigr)
             \langle v\rangle H(v),
 \qquad M>p+1.
\end{equation}
For the last term we used the arbitrary-power source bound of
\eqref{early:source}:
\[
 \frac{\mathcal S_N(r_Nt,r_Nx,v)}{m_N}
 \le C_M\frac{r_N^{M-1}}{T_N^p}t^{M-1}H(v)
 \le C_{A,M}T_N^{M-1-p}H(v).
\]
Thus every error tends to zero locally in spacetime \(L^1\), including
down to time zero on bounded dilated intervals. The use of
\(r_N\le2T_N\) here requires no positive lower bound for
\(r_N/T_N\).

The main cold sheets and Lemma~\ref{lem:sheet-bound} also give
\[
P^{b_N}(C_{r_N},H)\leq
 C\left(\frac1{t+l_N}+1\right)H,
\qquad
\nu^{b_N}_{C_{r_N}}(v)\leq
 C\left(\frac1{t+l_N}+1\right)\langle v\rangle.
\]
The nonmain cold component is bounded by a vanishing multiple of
\(e^{-4|v|^2}\leq CH\), so its gain is covered by
Lemma~\ref{lem:hot-bound}. Equations \eqref{dil:cap-bound} and
\eqref{dil:prelimit-equation} now imply, on \(a\leq t\leq A<U\),
\begin{equation}\label{dil:transport-majorant}
|\D h_{r_N}(t,x,v)|\leq C_{a,A}\langle v\rangle H(v).
\end{equation}
This is a genuine integrable velocity majorant, uniform in position.
If \(l>0\), the same conclusion holds on \(0\leq t\leq A\), since
\(l_N\geq l/2\) eventually.

The cap gives weak compactness of \(h_{r_N}\) in local \(L^1\),
and \eqref{dil:transport-majorant} permits application of
Lemma~\ref{comp:averaging}. Subselect on an exhaustion of
\((0,U)\times\R^3\). The resulting limit \(g\) satisfies the
bound in \eqref{eq:limit-equation}, and each compactly supported
velocity average converges strongly in local \(L^1_{t,x}\).

\subsection{Passing the cold--hot products}

The dilated cold densities concentrate in velocity, so the
two-density compactness argument of Section~\ref{sec:compactness}
does not apply to \(C_{r_N}\). The needed product passage uses
\eqref{dil:cold-weakstar} and hot averaging instead.

For compact velocity tests \(\psi,\chi\), put
\[
a_N(t,x)=\int\psi(p)C_{r_N}(t,x,p)\dd p,\qquad
\zeta_N(t,x)=\int\chi(v)h_{r_N}(t,x,v)\dd v.
\]
On every compact positive-time region, \(a_N\) is bounded in
\(L^\infty\) and converges weak-* to
\(a=\int\psi\dd B\), while
\(\zeta_N\to\zeta=\int\chi g\dd v\)
strongly in \(L^1\). For a bounded time-space test \(\varphi\),
\[
\int\varphi(a_N\zeta_N-a\zeta)
=\int\varphi a_N(\zeta_N-\zeta)
 +\int\varphi(a_N-a)\zeta\longrightarrow0.
\]
Finite sums of separated velocity tests approximate continuous tests
uniformly on compact velocity products. The total product mass is
uniformly bounded on compact time-space regions by
\eqref{dil:cold-moment-bound} and the cap. We conclude
\begin{equation}\label{dil:mixed-product}
C_{r_N}(t,x,p)h_{r_N}(t,x,v)\dd p\dd v\dd x\dd t
\rightharpoonup B(t,x,\dd p)g(t,x,v)\dd v\dd x\dd t
\end{equation}
against continuous compact velocity tests, with arbitrary smooth
compact time-space dependence. Uniform moment bounds in both velocities
allow removal of velocity cutoffs in tests with any fixed polynomial
growth for which they are used below.

Here is an explicit weak collision test suitable for this convergence.
For a continuous output test \(\phi\), let
\[
\mathcal K_\phi(p,v)
=\frac{|p-v|}{2}\int_{S^2}
 \phi\left(\frac{p+v}{2}+\frac{|p-v|}{2}\omega\right)\dd\omega.
\]
Lemma~\ref{lem:carleman} gives
\(\int\phi P(c,h)=\iint c(p)h(v)\mathcal K_\phi(p,v)\dd p\dd v\).
The test \(\mathcal K_\phi\) is continuous in both incoming
velocities, including \(p=v\), and is bounded in absolute value by
\(2\pi|p-v|\|\phi\|_\infty\). The loss test is
\(2\pi|p-v|\phi(v)\), with the same properties. Inserting smooth
time-space dependence in \(\phi\) does not change these facts.
On bounded velocity products the factor
\(\min(1,b_N/|p-v|)\) is eventually one; its complement is
uniformly negligible by the weighted product tails. Applying
\eqref{dil:mixed-product} proves the required gain and loss limits.

The limiting gain is an integrable function. To check the possible
exception at the velocity axis, use the sphere formula: every
nondegenerate diameter sphere meets that axis in at most two points,
which have zero sphere-area measure. A degenerate sphere has zero
collision rate. Tonelli's theorem therefore gives no output mass on
the axis. Off the axis, the collision-plane formula of
Lemma~\ref{lem:carleman} gives a density for each axial partner.
Each summand of \(B\) is an allowed zero-width axial sheet in
Lemma~\ref{lem:sheet-bound}; hence on
\(a\leq t\leq A\),
\[
0\leq P(B,g)\leq C_{a,A}H,
\qquad 0\leq g\nu_B\leq C_{a,A}\langle v\rangle H.
\]
These bounds dominate the entire weak gain measure, including across
the velocity axis. Passing \eqref{dil:prelimit-equation}, using
\eqref{dil:error-bounds}, proves the equation in
\eqref{eq:limit-equation}.

\subsection{A positive lower mesh endpoint and the local mild formula}

When \(l>0\), \eqref{dil:transport-majorant} holds uniformly down to
zero. The rescaled initial hot density, on every fixed spatial compact
set for large \(N\), is exactly
\[
h_{r_N}(0,x,v)=\lambda_N l_N\chi(x/l_N,v).
\]
The other periodic copies have moved outside that compact set.
Subselect \(\lambda_N\to\lambda_0\). Smoothness and compact support
of \(\chi\), and \(l_N\to l>0\), give strong local \(L^1\)
convergence to \eqref{dil:initial-trace}. Passing the weak transport
identity with tests touching time zero is justified by the uniform
source majorant and yields that initial trace for \(g\). The cold
product argument also extends to zero in this case, as already noted.
For \(l=0\) the coefficient can be of order \(1/t\), and the
proposition only asserts the equation on the open positive-time
interval.

On any \([a,A]\subset(0,U)\), the limiting transport source is
locally integrable and has the preceding velocity majorant uniformly
in position. Changing to free coordinates, first on bounded velocities and enlarged
spatial compact sets, gives a locally strongly continuous \(L^1\)
representative. The uniform integrable velocity tails then remove the
velocity truncation. The cap bound passes to every trace of this
representative. Almost every free path obeys the scalar absolutely
continuous equation
\(\D g+\nu_Bg=2P(B,g)\). Its loss coefficient is locally integrable
along the path, since
\(\nu_B(t,x,v)\leq C_a\langle v\rangle\) for \(t\geq a\).
The integrating factor therefore gives, for \(0<a<t<U\),
\begin{equation}\label{dil:mild}
\begin{split}
g(t,x,v)={}&g(a,x-(t-a)v,v)
 \exp\left[-\int_a^t\nu_B(\tau,x-(t-\tau)v,v)\dd\tau\right]\\
&+2\int_a^t P(B,g)(s,x-(t-s)v,v)
 \exp\left[-\int_s^t\nu_B(\tau,x-(t-\tau)v,v)\dd\tau\right]\dd s.
\end{split}
\end{equation}
This is an almost-everywhere identity, or equivalently an identity in
local \(L^1\) at the displayed times. It holds with \(a=0\) when
\(l>0\). Altering the medium on the null annular interfaces does not
change the identity: free transport preserves phase-space measure, so
Fubini's theorem discards the affected set of paths. These statements
complete the proof of Proposition~\ref{prop:blowup-limit}.

\subsection{The free-core comparison used for nonconcentration}

Weak convergence alone need not preserve the attained weighted norm in
\eqref{dil:cap-bound}. The next section will exclude its concentration
near the velocity axis or in a small set of collision histories. Its
input is an upper comparison involving only the explicit free cores.

\begin{lemma}\label{dil:free-core-comparison}
For all sufficiently large \(N\), define the velocity measure
\[
 A_N(t,x,\dd p)
   =r_N\sum_{i:\,j\le N}f_i(r_Nt,r_Nx,p)\dd p,
       \qquad 0\le t\le1.
\]
This definition uses free transport of the initial colors and is
meaningful even if \(t>U_N\). At each position, at most one summand
is nonzero. Its velocity moments and normalized gain obey
\[
 \int\langle p\rangle^k A_N(t,x,\dd p)
      \le\frac{C_k}{t+l_N},\qquad
 P(A_N,H)\le\frac{C}{t+l_N}H.
\]
There are numbers \(\varepsilon_N\downarrow0\) such that, on
\(0\le t\le\min(1,U_N)\), the hot component obeys the pathwise
upper comparison
\begin{equation}\label{dil:positive-comparison}
 \D h_{r_N}\le2P(A_N,h_{r_N})+\varepsilon_NH.
\end{equation}
The bounds and comparison hold on full-measure endpoint sets at each
time section, with representatives given by their path formulas.
\end{lemma}

\begin{proof}
The supports of the free cores lie in their disjoint spatial
enlargements through physical time \(r_N\); no retirement occurs
on this interval for large \(N\). The exact Jacobian formula
\eqref{dil:free-core-integral}, or its version before centering a
periodic copy, gives the moment estimate because
\(r_N/(r_Nt+s_j)\le1/(t+l_N)\). The sheet gain estimate gives
the second bound.

In the upper cold estimate, the main-color remainder contributes at
most \(C r_Ns_j^{15/2}H\) to its gain against \(H\).
The sum of the nonmain contributions has a vanishing Gaussian
majorant after multiplication by \(r_N\). Uniformly over all
positions, therefore,
\[
 P(C_{r_N},H)\le P(A_N,H)+o(1)H.
\]
More precisely this inequality follows by applying the positive gain
to the nonnegative upper error in \(C_{r_N}\le A_N+R_N\), where
\(P(R_N,H)\le o(1)H\); it does not assert a signed total-variation
comparison at all scales. On the present time range
\(h_{r_N}\le2H\). Drop every nonnegative loss in
\eqref{dil:prelimit-equation}, use \(P^{b_N}\le P\), and bound the
positive bath, hot, and source terms exactly as in
\eqref{dil:error-bounds}. This proves
\eqref{dil:positive-comparison} with an error independent of
\(t,x,v\).

All finite approximations obey their equations in free coordinates.
Fubini's theorem gives the comparisons along almost every free path
and on full-measure endpoint sets at every specified terminal time.
The countably many approximation indices and lattice translations
can be treated simultaneously. This is the form of the comparison
used in the next section.
\end{proof}
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//09-dilation.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//10-nonconcentration.tex ====
\section{A cap event produces a nonzero limit}
\label{sec:nonconcentration}
\label{nc:section}

The weak limit in Proposition~\ref{prop:blowup-limit} is obtained on an
open time interval.  A supremum attained by an approximation could in
principle disappear through concentration, or through escape from that
interval.  The purpose of this section is to exclude both possibilities.
The proof develops the nonconcentration mechanism of
\cite[Section~8]{OpenAIPeriodic}: two successive gains exclude an
axial limiting velocity, whereas five gains regularize the full
spacetime--velocity variables away from the axis. We give both
arguments here, including the localization needed for the rotated
period lattice. All constants may depend on the fixed jet parameters,
but none depends on the approximation index.

\begin{proposition}[Nonvanishing of every cap limit]
\label{prop:nonzero-limit}
Consider any sequence of cap solutions with horizons \(T_n\to0\),
approximation indices \(N_n\to\infty\), and
\(s_{N_n}/T_n\to0\).  Make the rescaling
\eqref{eq:rescaling}, and pass to any subsequence supplied by
Proposition~\ref{prop:blowup-limit}.  Its limit satisfies
\[
 g\not\equiv0\qquad\text{on }(0,U)\times\R^3_x\times\R^3_v.
\]
This conclusion includes \(l=1\), \(U=1\), and \(U=\infty\).
\end{proposition}

We first describe the positive kernels used in the proof.  Subsequently
we establish the two different mechanisms needed near and away from the
velocity axis.

\subsection{A positive majorizing equation and its kernels}

For this section relabel the sequence by \(n\), set \(U_n=T_n/r_n\),
and write
\[
 r_n=t_{*,n}+s_{N_n},\quad
 \ell_n=\frac{s_{N_n}}{r_n},\quad
 \tau_n=\frac{t_{*,n}}{r_n}=1-\ell_n,\quad
 h_n=h_{r_n},\quad C_n=C_{r_n},\quad f_n=\frac{h_n}{H}.
\]
Then \(\ell_n\to l\). In particular \(0\le\ell_n\le1\),
\(0\le\tau_n\le1\). The cap gives, on \(0\le t\le U_n\),
\begin{equation}
\label{nc:cap-normalized}
 0\le f_n(t,x,v)\le t+\ell_n,\qquad
 \left\|f_n(\tau_n)\right\|_\infty=1.
\end{equation}
Essential supremum norms and pointwise formulas involving \(H\) always
ignore the velocity axis, a null set.  The mild representatives of the
finite approximations will be used at the cap times.
Choose Borel representatives when composing densities with the maps
below, enforcing their asserted bounds on exceptional null sets as
well.  Characteristic identities and endpoint choices are still made
only on their common full-measure sets.

\begin{lemma}[Core kernels on the dilated scale]
\label{nc:core-kernels}
There are nonnegative velocity measures \(A_n(t,x,\dd p)\) and numbers
\(\varepsilon_n\to0\) with the following properties.
After discarding finitely many terms,
the measures \(A_n\) are defined from free transport on the whole interval
\(0\le t\le1\), independently of the cap interval \([0,U_n]\).
The bounds on \(A_n\) below hold on that whole interval; comparison with
\(h_n\) is used only while the cap bound holds.

\begin{enumerate}[label=\textup{(\roman*)}]
\item On \(0<t<\min(1,U_n)\), along almost every free characteristic,
\begin{equation}
\label{nc:positive-equation}
 \D h_n\le 2P(A_n,h_n)+\varepsilon_n H .
\end{equation}
At each point \(A_n\) contains at most one main free core.  Its dilated
scale \(\varrho=s/r_n\) satisfies \(|y|\asymp\varrho\), with coordinates measured
from its periodic copy, and
\[
 \frac{2P(A_n,H)}{H}\le\frac{C}{t+\ell_n}.
\]

\item Before spatial masks are imposed, a core of scale
\(\varrho=s/r_n\) and sign \(\sigma\) is the pushforward of
\begin{equation}
\label{nc:core-parameters}
 \frac{K S\bigl(\gamma(z-tp_3)/\varrho\bigr)}
      {\varrho+t\alpha\gamma S\bigl(\gamma(z-tp_3)/\varrho\bigr)}
 J(w)\dd w ,
 \qquad
 \bar p=d_s\bar w,\quad
 p_3=\sigma+\alpha\tanh\bigl(\gamma(z-tp_3)/\varrho\bigr)+d_sw_3 .
\end{equation}
The actual masks are measurable factors between zero and one.
The density in \eqref{nc:core-parameters} is at most
\(C(t+\varrho)^{-1}J(w)\).

\item For \(t\ge a>0\), the normalized gain measure
\begin{equation}
\label{nc:normalized-kernel}
 \mathcal K_n(t,x,v;\dd p\,\dd\xi)
 =
 4A_n(t,x,\dd p)\,
 \delta\bigl((v-p)\cdot(\xi-v)\bigr)
 \frac{H(\xi)}{H(v)}\dd\xi
\end{equation}
has total mass at most \(C/a\).  Uniformly in its base point, the
contributions from large \(|p|+|\xi-v|\), or from large retained
Gaussian parameters in a product-sheet majorant, tend to zero.
Its total mass tends uniformly to zero as \(|v|\to\infty\).
\end{enumerate}
\end{lemma}

\begin{proof}
Use the free-core measure from Lemma~\ref{dil:free-core-comparison}.
More explicitly, if the main retained label at \((r_nt,r_nx)\) is
\(i\), set
\[
 A_n(t,x,\dd p)=r_n f_i(r_nt,r_nx,p)\dd p,
 \qquad f_i(t,x,p)=c_i^0(x-tp,p),
\]
and otherwise set \(A_n=0\). The disjointness in
Lemma~\ref{cold:geometry} holds on the whole physical interval
\([0,r_n]\) for all sufficiently large \(n\), and no color is
retired there. Free transport therefore defines \(A_n\) on
\([0,1]\), regardless of \(U_n\). Equation~\eqref{dil:positive-comparison}
gives \eqref{nc:positive-equation}. Its error already includes
the bath, the source divided by \(m_{T_n}=T_n^p\), the rescaled
hot self-gain, and the difference between actual and free cold cores.

The enlargement of a main transverse annulus remains between its
original radial boundaries. Thus, relative to its period point,
\(\varrho<|y|<e^\eta\varrho\), where \(\varrho=s/r_n\). The change of velocity
variables in \eqref{cold:core-formula}, multiplied by \(r_n\), is
exactly \eqref{nc:core-parameters}; equivalently, this is the
parameter formula in \eqref{dil:free-core-integral} before velocity
testing. For \(0\le S\le1\),
\[
 \frac{KS}{\varrho+t\alpha\gamma S}
 \le \frac{C}{t+\varrho}.
\]
Indeed, multiplying by \(t+\varrho\) bounds the \(\varrho\)-term by \(K\)
and the \(t\)-term by \(K/(\alpha\gamma)\). The smallest scale is
\(\ell_n\), so the amplitude is at most \(C/(t+\ell_n)\).

The implicit axial map depends only on \(w_3\), and its monotonicity
gives \(|p_3|\le1+\alpha+d_s|w_3|\). The smooth compactly supported
\(J\) has a fixed product Gaussian majorant. Lemma~\ref{ker:parameter-sheets}
therefore dominates each core by a sheet, with the preceding amplitude
and a fixed axial moment bound. The gain bound in
Lemma~\ref{lem:sheet-bound} proves (i). The factor \(4\) in
\eqref{nc:normalized-kernel} is the product of the linear gain
coefficient \(2\) and the coefficient \(2\) in the collision-plane
formula of Lemma~\ref{lem:carleman}. Lemma~\ref{lem:kernel-tightness}
now gives (iii); Lemma~\ref{ker:parameter-sheets} also controls
Gaussian parameter tails before the compressed axial change of
variables. For the exact free cores the parameters already lie in
\(\supp J\). All these estimates allow measurable masks bounded by
one, which can only decrease the nonnegative integrals.
\end{proof}

For a fixed \(a>0\), define the backward Volterra operator
\begin{equation}
\label{nc:volterra}
 (\mathcal V_n F)(t,x,v)
 =
 \int_a^t\int
 F\bigl(s,x-(t-s)v,\xi\bigr)
 \mathcal K_n\bigl(s,x-(t-s)v,v;\dd p\,\dd\xi\bigr)\dd s .
\end{equation}
\emph{Representatives in the finite-temperature iteration.}
For each finite \(n\), the core measure has a velocity density,
\(A_n(s,x,\dd p)=a_n(s,x,p)\dd p\).
Integrating first in \(p\) in \eqref{nc:normalized-kernel} shows that,
for \(\xi\ne v\), its input marginal is
\[
\mathcal K_n(s,x,v;\R^3,\dd\xi)
=\frac{4H(\xi)}{|\xi-v|H(v)}
  \left[\int_{(v-p)\cdot(\xi-v)=0}
       a_n(s,x,p)\dd A_p\right]\dd\xi.
\]
There is no atom at \(\xi=v\): each plane-area measure gives that
point zero mass, and \(p=v\) is null for the cold density.
For a null set in \((s,X,\xi)\), integrate also in the endpoint
variables and use the measure-preserving change
\(X=x-(t-s)v\). Tonelli's theorem then shows that altering the input
on that set changes \(\mathcal V_nF(t)\) only on a null set of
endpoints, at each fixed terminal time. Finite repetition preserves
this property. A countable sequence of lower times \(a\downarrow0\)
suffices below; intersecting the good endpoint sets for these times,
the approximation indices, and the finitely many iterations gives
the common full-measure sets used at the cap events.

Positivity and Lemma~\ref{nc:core-kernels} give, for a bounded
nonnegative \(F\),
\begin{equation}
\label{nc:iteration-norm}
 \|\mathcal V_n^jF(t)\|_\infty
 \le \|F\|_\infty\frac{\bigl(C\log(t/a)\bigr)^j}{j!},
 \qquad a\le t\le1.
\end{equation}
Indeed, the ordered \(j\)-time integral of
\(\prod_{i=1}^j C/t_i\) equals
\((C\log(t/a))^j/j!\).  In particular every fixed number of further
gains has bounded mass on this slab.

\subsection{Reduction to a finite gain integral}

We begin the proof of Proposition~\ref{prop:nonzero-limit}. The case
\(l>0\) is settled directly. For \(l=0\), the cap will force a
positive contribution from five successive gains; the remaining
subsections show that this contribution vanishes if the weak limit is zero.


Pass to a subsequence on which the seed parameters converge,
\(\lambda_n\to\lambda_\infty\).  If \(l>0\) and
\(\lambda_\infty>0\), Proposition~\ref{prop:blowup-limit}
gives the initial hot trace
\[
 g(0,x,v)=\lambda_\infty l\,\chi(x/l,v).
\]
This trace is nonzero.  More explicitly, the limiting loss
frequency obeys
\[
 \nu_B(t,x,v)\le
       \frac{C(1+|v|)}{t+l}.
\]
On any bounded velocity set and any finite interval beginning
at zero its integral along a flight is finite.  The positive
mild formula for \eqref{eq:limit-equation} therefore retains
the initial trace multiplied by a strictly positive survival
factor.  A bounded positive portion of the smooth nonzero bump
gives positive interior mass.

If \(l>0\) and \(\lambda_\infty=0\), use
Lemma~\ref{nc:core-kernels} from time zero.  The initial ratio
has norm at most
\(\lambda_n\ell_n\|\chi/H\|_\infty\), independently of the
spatial scaling of the bump.  On \(0\le t\le\tau_n\le1\),
eventually \(t+\ell_n\ge l/2\).  The positive integral
inequality and the elementary scalar integrating-factor estimate
give
\[
 \sup_{0\le t\le\tau_n}\|f_n(t)\|_\infty
 \le
 \bigl(\lambda_n\ell_n\|\chi/H\|_\infty+\varepsilon_n\bigr)
 e^{2C/l}\longrightarrow0.
\]
This contradicts \eqref{nc:cap-normalized}.  Both alternatives
also apply when \(l=1\), even if \(\tau_n\to0\) or some hit
occurs at time zero.

It remains to consider \(l=0\), so \(\tau_n\to1\).
Suppose, to obtain a contradiction, that \(g=0\) throughout
the open interval \((0,U)\).  Nonnegativity upgrades this zero
weak limit to local \(L^1\) convergence: if \(K\) is a compact
subset of the open spacetime--velocity region, take a
nonnegative smooth compact test at least one on \(K\).  Its
pairing with \(h_n\) tends to zero, hence
\(\int_K h_n\to0\).  Away from the velocity axis \(H\) has a
positive lower bound on compact velocity sets.  Consequently
\begin{equation}
\label{nc:zero-in-measure}
 f_n\longrightarrow0
 \quad\text{in measure on compact subsets of }
 (0,U)\times\R^3_x\times\{\bar v\ne0\}.
\end{equation}

Fix \(0<a<1/4\), so that \(a<\tau_n\) for large \(n\).
The positive mild inequality \eqref{nc:positive-equation}
reads
\[
 f_n\le d_n+\mathcal V_nf_n+e_n,
\]
where \(d_n(t,x,v)=
 f_n(a,x-(t-a)v,v)\), \(\|d_n\|_\infty\le a+\ell_n\),
and \(0\le e_n\le\varepsilon_n\) on this unit-length interval.
Iterate the gain term five times.
Using positivity and \eqref{nc:iteration-norm} gives
\begin{equation}
\label{nc:five-residual}
 f_n(t,x,v)
 \le
 (a+\ell_n+\varepsilon_n)
 \sum_{j=0}^4\frac{(C\log(1/a))^j}{j!}
 +(\mathcal V_n^5 f_n)(t,x,v),
 \qquad a\le t\le\tau_n .
\end{equation}
Every formula is used on a common full-measure set of
characteristics for the finite approximation.


The term outside \(\mathcal V_n^5f_n\) becomes arbitrarily small when
first \(n\to\infty\) and then \(a\downarrow0\). We therefore need to
show that the five-fold term tends to zero at endpoints carrying a fixed
positive fraction of the cap. We first control those endpoints, then
separate axial and nonaxial limiting velocities.

\subsection{Compactness of the relevant endpoints}

\begin{lemma}[Endpoint and path truncations]
\label{nc:endpoint-tightness}
Fix \(a>0\), and let \(t_n\in[a,1]\).  In a five-fold integral
\((\mathcal V_n^5F_n)(t_n,x_n,v_n)\), assume \(0\le F_n\le2\).
For every prescribed error, one may restrict all
\(|p_i|+|v_i-v_{i-1}|\) and sheet parameters to fixed bounded sets at
that error, uniformly in \(n,t_n,x_n,v_n\).  Moreover:
\begin{enumerate}[label=\textup{(\roman*)}]
\item if \(|v_n|\to\infty\), the whole integral tends to zero;
\item after retaining bounded endpoint velocities, an endpoint
escaping every bounded neighborhood of every dilated periodic origin
also has a vanishing integral.
\end{enumerate}
Consequently any sequence along which the five-fold integral has a
positive lower bound has, after periodic translations and
subselection, convergent endpoint positions and velocities.
\end{lemma}

\begin{proof}
At any one gain, a removed normalized tail has uniformly small mass
by Lemma~\ref{nc:core-kernels}. Bound the other four gains and the
input by \eqref{nc:iteration-norm}, and sum over the five possible
locations of a removed tail. This proves the first assertion.
The same argument at the first gain, using its uniform decay for
large outgoing velocities, proves (i).

Retain bounded endpoint velocities and bounded increments. Then all
velocities are bounded by a fixed \(V\), and every collision position
is within distance \(V\) of the endpoint, since the total free-flight
time is at most one. Suppose
\[
 d_n:=\dist(x_n,r_n^{-1}\Lambda)\longrightarrow\infty.
\]
We show that the first kernel becomes uniformly small on these paths.
A main core of dilated scale \(\varrho\ge M\) has normalized mass at
most \(C/M\), by \eqref{nc:core-parameters} and the sheet bound.
Here \(M\) can be chosen arbitrarily large, independently of \(n\).
For a main core with \(\varrho\le M\), let \(q/r_n\), \(q\in\Lambda\),
be its period point. Write \((y,z)\) for the first collision position
relative to this point. Its transverse coordinate obeys
\(|y|\le e^\eta M\), while its distance to this point is at least
\(d_n-V\). Consequently \(|z|\to\infty\), uniformly over all
encountered cores of these scales. For bounded \(p_3,t\), their
parameter amplitudes satisfy
\begin{equation}
\label{nc:axial-escape}
 \frac{K S(\gamma(z-tp_3)/\varrho)}
      {\varrho+t\alpha\gamma S(\gamma(z-tp_3)/\varrho)}
 \le \frac K\varrho S(\gamma(z-tp_3)/\varrho)\longrightarrow0
\end{equation}
uniformly for \(0<\varrho\le M\). In fact, when
\(|z-tp_3|\ge Z\), the right side is at most
\(4K\varrho^{-1}e^{-2\gamma Z/\varrho}\), whose supremum tends to zero
as \(Z\to\infty\). Applying the sheet bound with this vanishing
amplitude proves the assertion for \(\varrho\le M\). The remaining
gains have bounded mass. First let \(n\to\infty\), then
\(M\to\infty\), and finally remove the initial tail error.
This proves (ii).

It follows that a sequence with a positive lower bound for its
five-fold integral has bounded velocities and bounded distance from
\(r_n^{-1}\Lambda\), after subselection. Choose period points
\(q_n/r_n\) realizing a bounded distance, and translate the endpoints
by these periods. The resulting positions have a convergent
subsequence. Such translations preserve both the kernels and the
periodic hot inputs used below.

We record also the geometry along the translated bounded paths.
Every physical main enlargement for times \(r_nt\le r_n\) lies
in a fixed ball of radius less than \(1/2\) about its period point,
for sufficiently small \(s_0,r_n\). Distinct points of \(\Lambda\)
are at least distance one apart. A path in a bounded dilated set
lies at physical distance \(O(r_n)\) from the chosen origin, so
for all large \(n\) it can encounter only that copy. Its encountered
scales satisfy \(s/r_n\le C\), by the transverse annular restriction.
Thus the largest physical width along these paths is
\(O(r_n^{10})\), which tends to zero.
\end{proof}

\subsection{Kernel estimates near the velocity axis}

The preceding localization reduces the problem to bounded endpoint
velocities. To distinguish axial and nonaxial limits, we now refine
the normalized sheet bounds of Section~\ref{sec:kernel}. The estimates
below, from \cite[Section~3]{OpenAIPeriodic}, allow transverse
concentration of the hot input but prevent concentration in its axial
coordinate. We recall that $\mathcal S_L$ is the class of probability
mixtures of sheets $\mathcal G_d\otimes\rho$, where
$0\leq d\leq1/8$ and $\int e^{8a^2}\rho(\dd a)\leq L$, and that
$\mathcal K_c$ denotes the normalized measure in
\eqref{ker:normalized-kernel}.

\begin{lemma}[Kernel estimates near the axis]
\label{lem:axis-kernel-estimates}
Fix $1\leq L<\infty$. The following conclusions hold uniformly for
$c\in\mathcal S_L$ and for outgoing velocities off the axis.
\begin{enumerate}[label=\textup{(\roman*)}]
\item For every fixed $M,\delta>0$,
\begin{align}
 &\lim_{\varepsilon\downarrow0}
 \sup_{c\in\mathcal S_L}
 \esssup_{\substack{|v|\leq M\\0<|\bar v|\leq\varepsilon}}
 \mathcal K_c(v;|\bar\xi|\geq\delta)=0,
 \label{ker:axis-input}\\
 &\lim_{\varepsilon\downarrow0}
 \sup_{c\in\mathcal S_L}
 \esssup_{\substack{|v|\leq M\\0<|\bar v|\leq\varepsilon}}
 \mathcal K_c(v;|v_3-p_3|\geq\delta)=0.
 \label{ker:axis-mismatch}
\end{align}
\item The axial input marginals are uniformly non-atomic. More
precisely, for each $R\geq1$ and each interval $I$ of length
$0<\ell\leq1$,
\begin{equation}\label{ker:axial-interval}
 \mathcal K_c(v;|p|+|\xi-v|\leq R,\ \xi_3\in I)
 \leq C_{q,m,L,R}\bigl(\ell+\ell^{2-q}\bigr).
\end{equation}
In combination with \eqref{ker:uniform-tail}, this gives
\begin{equation}\label{ker:axial-modulus}
 \lim_{\ell\downarrow0}\sup_{c\in\mathcal S_L}\esssup_v
 \sup_{|I|\leq\ell}\mathcal K_c(v;\xi_3\in I)=0.
\end{equation}
\item Let $\mathcal S_L(\varepsilon)$ denote the subclass whose
elementary transverse widths are at most $\varepsilon$. For every
$M<\infty$ and $r_0>0$,
\begin{equation}\label{ker:off-axis-small-input}
 \lim_{\delta\downarrow0}\limsup_{\varepsilon\downarrow0}
 \sup_{c\in\mathcal S_L(\varepsilon)}
 \esssup_{\substack{|v|\leq M\\|\bar v|\geq r_0}}
 \mathcal K_c(v;|\bar\xi|<\delta)=0.
\end{equation}
\end{enumerate}
All these conclusions hold under uniformly bounded sheet amplitudes.
\end{lemma}

\begin{proof}
By \eqref{ker:uniform-tail}, we may work on
$|p|+|\xi-v|\leq R$ and remove that restriction at the end of each
argument. On this region the exponential and polynomial factors in
\eqref{ker:energy-factor} are bounded by a constant depending on $R$.

For \eqref{ker:axis-input}, on $|\bar\xi|\geq\delta$ the factor
$W(\xi)$ is bounded by $1+\delta^{-q}$. The regular line estimate
and \eqref{ker:transverse-expectation} therefore give
\[
 C_{R,\delta}\frac{|\bar v|^{-1}}{1+|\bar v|^{-q}}
 \leq C_{R,\delta}|\bar v|^{q-1}
 \qquad(0<|\bar v|\leq1).
\]
For \eqref{ker:axis-mismatch}, integrate in $\xi_3$ first, using
$|v_3-p_3|^{-1}\leq\delta^{-1}$ and
\eqref{ker:disk-singularity}. The bound is
$C_{R,\delta}/W(v)\leq C_{R,\delta}|\bar v|^q$.
These are uniform in the axial law. For
\eqref{ker:axial-interval}, discard the bounded exponential and
polynomial factors and apply
\eqref{ker:geometric-interval}. This also proves
\eqref{ker:axial-modulus} after the tail removal.

It remains to prove \eqref{ker:off-axis-small-input}. Fix the
truncation radius $R$. The part $|\bar p|>r_0/4$ is uniformly
negligible as the transverse widths tend to zero. Indeed,
\[
 \ind_{|\bar p|>r_0/4}\mathcal G_d(\dd\bar p)
 \leq2e^{-r_0^2/(32d^2)}\mathcal G_{\sqrt2d}(\dd\bar p),
\]
and Lemma~\ref{lem:sheet-bound} applies to the wider Gaussian for
all sufficiently small $d$. Now retain $|\bar p|\leq r_0/4$.
Write $n=\bar v-\bar p$, $A=(n/|n|)\cdot\bar v$, and
$B=(v_3-p_3)/|n|$, as in Lemma~\ref{ker:geometric-slice}. Then
\[
 |n|\geq\tfrac34r_0,\qquad A\geq\tfrac35r_0.
\]
If $|v_3-p_3|\leq r_0^2/(8R)$, then for
$|\xi_3-v_3|\leq R$,
\[
 |B(\xi_3-v_3)|\leq\tfrac16r_0.
\]
The transverse collision line stays at distance greater than
$r_0/3$ from zero, and cannot meet $|\bar\xi|<\delta$ when
$\delta<r_0/3$. In the complementary case,
$|v_3-p_3|^{-1}\leq8R/r_0^2$. Integrating first in $\xi_3$ bounds
the small-input contribution by
\[
 C_{R,r_0}\int_{|\bar\xi|<\delta}
               (1+|\bar\xi|^{-q})\,\dd\bar\xi
 \leq C_{R,r_0}(\delta^2+\delta^{2-q}).
\]
First let the widths tend to zero, then $\delta$ tend to zero, and
finally remove the truncation. This proves the last assertion.
\end{proof}

These conclusions apply to the core kernels
\eqref{nc:normalized-kernel}: for $t\geq a>0$ the proof of
Lemma~\ref{nc:core-kernels} gives sheet domination with a fixed axial
moment bound and amplitude $C/a$. On the localized paths, the
transverse widths tend uniformly to zero, as shown at the end of
Lemma~\ref{nc:endpoint-tightness}. This is the additional hypothesis
needed for \eqref{ker:off-axis-small-input}.

\begin{remark}[A sharp axial endpoint]\label{ker:matching-axis}
A single normalized gain need not become small as the outgoing
velocity approaches the axis. Recall
$a_q=\int_{\R}(1+s^2)^{-q/2}\dd s$ from \eqref{ker:line-integral}. Take
$c=\delta_{(0,0,u)}$ and
$v_\varepsilon=(\varepsilon,0,u)$. The collision plane is
$\xi_1=\varepsilon$, and
\[
 \frac{P(c,H)(v_\varepsilon)}{H(v_\varepsilon)}
 =\frac{2}{\varepsilon H(v_\varepsilon)}
      \int_{\R^2}H(\varepsilon,y,z)\,\dd y\,\dd z.
\]
The regular part of $W(\varepsilon,y,z)$ contributes
$O(\varepsilon^{q-1})$. In its singular part substitute
$y=\varepsilon s$. Dominated convergence, with dominating function
$(1+s^2)^{-q/2}e^{-z^2}\langle z\rangle^{-m}$, gives
\[
 \lim_{\varepsilon\downarrow0}
 \frac{P(c,H)(v_\varepsilon)}{H(v_\varepsilon)}
 =2a_q e^{u^2}\langle u\rangle^m
        \int_{\R}e^{-z^2}\langle z\rangle^{-m}\,\dd z>0.
\]
The corresponding axial input marginal converges to the absolutely
continuous measure
\[
2a_q e^{u^2}\langle u\rangle^m
 e^{-z^2}\langle z\rangle^{-m}\,\dd z.
\]
Thus transverse concentration is compatible with the uniform axial
interval estimate. This is the reason for using two successive gains
in the axial argument that follows.
\end{remark}

\subsection{Two gains at an axial endpoint}

For \(\varrho>0\) and \(\sigma\in\{-1,1\}\), denote by
\(u_{\varrho,\sigma}(t,z)\) the solution of
\[
 u=\sigma+\alpha\tanh\bigl(\gamma(z-tu)/\varrho\bigr).
\]
This notation extends the profile independently of every annular
or stripe mask.

\begin{lemma}[Characteristic matching and small scales]
\label{nc:characteristic-matching}
On every compact set with \(t\ge a>0\), the following statements
hold uniformly.
\begin{enumerate}[label=\textup{(\roman*)}]
\item If \(a\le t_2\le t_1\), \(\Delta=t_1-t_2\), and
\(z_2=z_1-\Delta w\), then
\begin{equation}
\label{nc:matching-inequality}
 |w-u_{\varrho,\sigma}(t_1,z_1)|
 \le |w-u_{\varrho,\sigma}(t_2,z_2)|.
\end{equation}
\item As \(\varrho\downarrow0\), the distance of
\(u_{\varrho,\sigma}(t,z)\) to
\[
 \{z/t,\ \sigma-\alpha,\ \sigma+\alpha\}
\]
tends to zero uniformly.  For the perturbed axial velocity in
\eqref{nc:core-parameters},
\begin{equation}
\label{nc:thermal-speed-error}
 |p_3-u_{\varrho,\sigma}(t,z)|\le d_s|w_3|.
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
The profile satisfies
\[
 \partial_z u_{\varrho,\sigma}
 =\frac{(\alpha\gamma/\varrho)S}
        {1+(t\alpha\gamma/\varrho)S}\ge0,
 \qquad
 \partial_tu_{\varrho,\sigma}
       +u_{\varrho,\sigma}\partial_zu_{\varrho,\sigma}=0.
\]
For fixed \(t_1,t_2,z_1\), the function
\(F(w)=w-u_{\varrho,\sigma}(t_2,z_1-\Delta w)\) has derivative
\(F'(w)\ge1\).  Its root is
\(u_{\varrho,\sigma}(t_1,z_1)\): the defining implicit equation shows
that this velocity is unchanged along the straight segment between
the two points.  Integrating \(F'\ge1\) proves
\eqref{nc:matching-inequality}.

Fix \(0<\delta<\alpha\).  If \(u_{\varrho,\sigma}\) is within
\(\delta\) of an endpoint speed, there is nothing to prove.
Otherwise
\[
 \left|\frac{u_{\varrho,\sigma}-\sigma}{\alpha}\right|
 \le1-\frac{\delta}{\alpha},
 \qquad
 |z-tu_{\varrho,\sigma}|
 \le\frac{\varrho}{\gamma}
       \operatorname{arctanh}(1-\delta/\alpha).
\]
It follows that
\[
 |u_{\varrho,\sigma}-z/t|
 \le\frac{\varrho}{\gamma a}
       \operatorname{arctanh}(1-\delta/\alpha).
\]
First choose \(\delta\), then \(\varrho\) small.  This proves the
uniform assertion in (ii).
Finally, the function
\[
 p\longmapsto p-\sigma-\alpha\tanh\bigl(\gamma(z-tp)/\varrho\bigr)
\]
has derivative at least one.  Its values at \(u_{\varrho,\sigma}\)
and at the perturbed \(p_3\) are \(0\) and \(d_sw_3\).
This proves \eqref{nc:thermal-speed-error}.
\end{proof}

\begin{lemma}[Vanishing of two axial gains]
\label{nc:two-axial-gains}
Fix \(a>0\).  Let \(t_n\in[a,1]\), and suppose the endpoint
positions and velocities are bounded with \(\bar v_n\to0\).
Then
\[
 (\mathcal V_n^2 1)(t_n,x_n,v_n)\longrightarrow0.
\]
The conclusion continues to hold if all subsequent factors in a
longer finite gain integral are bounded uniformly.
\end{lemma}

\begin{proof}
Remove normalized parameter and increment tails at an arbitrarily
small cost, using Lemma~\ref{nc:core-kernels}.  On the retained
two-gain paths, all velocities and positions lie in fixed compact
sets, the times lie in \([a,1]\), and every active scale satisfies
\(0<\varrho\le M\).  In particular the largest possible physical width
is \(d_0(r_nM)^{10}\to0\).

Write \(t_1,x_1=(y_1,z_1)\) for the first collision point, and
\(\xi\) for its input velocity.  The next point has axial coordinate
\(z_2=z_1-(t_1-t_2)\xi_3\).  All estimates below are uniform when
\(t_1,x_1\) vary in their retained compact sets.  At the first
gain, Lemma~\ref{lem:axis-kernel-estimates} supplies a common modulus
\(\omega(\rho)\downarrow0\) for the mass of any axial interval of
length \(\rho\); the interval's center is arbitrary.  It also says
that for each fixed \(b>0\), the first-kernel mass on
\(\{|\bar\xi|>b\}\) tends to zero as \(\bar v_n\to0\).

Here is an order of choices that handles all scales at the second
point.  Fix an error \(\epsilon>0\), including the fixed factors
from the two time integrations and the bound \(C/a\).  Choose
\(\delta>0\) so that the sum of the first-kernel masses of the
\(\delta\)-neighborhoods of the five numbers
\begin{equation}
\label{nc:collapsed-speeds}
 z_1/t_1,\quad 1-\alpha,\quad1+\alpha,\quad
 -1-\alpha,\quad-1+\alpha
\end{equation}
is at most \(\epsilon\).  This is possible uniformly because
there are five intervals and their lengths are \(2\delta\).
Lemma~\ref{nc:characteristic-matching} then gives
\(\varrho_0>0\), chosen uniformly on the retained compact sets, so that
every profile with \(\varrho<\varrho_0\) is within \(\delta/4\) of its
corresponding three collapsed speeds.

The geometric mesh has a uniformly bounded number \(J_0\) of
scales in \([\varrho_0,M]\).  Choose \(b_0>0\) so small that the
first-kernel mass of the union of the \(b_0\)-neighborhoods of
\[
 \{u_{\varrho,\sigma}(t_1,z_1):
       \varrho\in[\varrho_0,M]\text{ in the current mesh},\
       \sigma=\pm1\}
\]
is at most \(\epsilon\).  For example it suffices that
\(2J_0\omega(2b_0)\le\epsilon\), after incorporating the fixed
amplitude bound into \(\omega\).  This list may depend on \(n\);
its cardinality bound does not.  Its centers may depend on
\(t_1,z_1\), which are held fixed when the first kernel is
integrated.  Thus neither dependence weakens the interval bound.

Let \(E_n(t_1,z_1)\) be the union of the neighborhoods of the five
collapsed speeds in \eqref{nc:collapsed-speeds} and the neighborhoods
of the finite-scale profile speeds just chosen.
Outside \(E_n\), every retained cold speed at the second point has
an axial mismatch bounded below by a fixed positive number for
all large \(n\).  For \(\varrho\ge\varrho_0\), this follows from
\eqref{nc:matching-inequality} and
\eqref{nc:thermal-speed-error}, with lower bound \(b_0/2\).
For a constant collapsed speed it follows immediately, with lower
bound \(\delta/2\).  For the remaining branch use the exact identity
\[
 \xi_3-\frac{z_2}{t_2}
 =\frac{t_1}{t_2}
       \left(\xi_3-\frac{z_1}{t_1}\right);
\]
the factor \(t_1/t_2\ge1\) preserves the lower bound.
The errors \(\delta/4\) in small-scale approximation and the
vanishing thermal errors can be absorbed.  We have therefore
obtained, on the complement of \(E_n\),
\[
 |\xi_3-p_3^{(2)}|
 \ge c_*:=\tfrac14\min(\delta,b_0)>0
\]
for every retained second-core parameter.

The second-kernel axial mismatch estimate in
Lemma~\ref{lem:axis-kernel-estimates} now tends uniformly to zero as
\(|\bar\xi|\to0\).  Choose \(b>0\) so small that this second
kernel has mass at most \(\epsilon\) whenever
\(|\bar\xi|\le b\) and the displayed mismatch holds.
For this fixed \(b\), the first-kernel mass on
\(|\bar\xi|>b\) tends to zero.  Its mass on \(E_n\) is at most
\(2\epsilon\), while the remaining first-kernel mass is bounded.
Combining the three regions, then removing the initial tail error,
gives a bound tending to zero as \(\epsilon\downarrow0\).
All choices preceded the final passage \(n\to\infty\), and were
uniform in the two times.  This proves the lemma.

In a longer integral, all factors following these two gains are
bounded by \eqref{nc:iteration-norm}; the same proof applies.
\end{proof}

\subsection{Finite-dimensional regularization away from the axis}

Two gains have now ruled out endpoints approaching the velocity axis.
At a nonaxial endpoint the kernel mass need not vanish. Instead, we
will show that five gains cannot concentrate their input on a set of
small spacetime--velocity volume. The next three lemmas make this
statement precise: they parameterize the collision planes, give an
abstract estimate for composing measurable functions with smooth maps,
and verify the required rank for five successive collisions.

\begin{lemma}[Redundant plane parameters]
\label{nc:plane-parameters}
Let \(n\ne0\), \(e=n/|n|\), and let \(\zeta\) be a bounded
nonnegative function supported in \([-1,1]\) with integral one.
For every nonnegative measurable \(F\),
\begin{equation}
\label{nc:plane-identity}
 \int_{n^\perp} F(k)\frac{\dd A(k)}{|n|}
 =
 \int_{\R^3}F(n\times L)|n|\zeta(L\cdot e)\dd L .
\end{equation}
When \(\delta\le|n|\le R\) and the plane integral is restricted
to \(|k|\le R_1\), the corresponding parameter measure
\(\ind_{\{|n\times L|\le R_1\}}|n|\zeta(L\cdot e)\dd L\)
is supported in
\[
 |L|\le\sqrt{(R_1/\delta)^2+1}
\]
and has density at most \(R\|\zeta\|_\infty\). This bound concerns
the parameter measure before multiplication by \(F(n\times L)\).
The identity can be applied successively when each normal is a
measurable function of parameters chosen at earlier gains.
\end{lemma}

\begin{proof}
Write \(L=L^\perp+\lambda e\).  The map
\(L^\perp\mapsto n\times L^\perp\) multiplies two-dimensional
area by \(|n|^2\).  Integration in \(\lambda\) contributes
\(\int\zeta=1\), proving \eqref{nc:plane-identity} and the support
and density bounds.  For successive gains, fix all previous
parameters before applying the identity.  The current normal is
then fixed; Fubini's theorem for nonnegative functions justifies
iteration.  No derivatives of the variable normals enter this
conditional change of variables.
\end{proof}

\begin{lemma}[Uniform pullback estimate]
\label{nc:uniform-pullback}
Let \(\mathcal O\subset\R^k\) be open, \(k\ge d\), and suppose
\(\Phi_n,\Phi:\mathcal O\to\R^d\) satisfy
\(\Phi_n\to\Phi\) in \(C^1_{\mathrm{loc}}\) and
\(\operatorname{rank}D\Phi=d\) almost everywhere.
Let \(K\Subset\mathcal O\) and \(B\subset\R^d\) be compact.
Let \(a_n\) be measurable, supported in \(K\), with
\(|a_n|\le A\).  Let \(F_n\) be measurable, supported in \(B\),
with \(|F_n|\le M\), and converging to zero in measure.  Then
\begin{equation}
\label{nc:pullback-conclusion}
 \int_{\mathcal O}|F_n(\Phi_n(q))a_n(q)|\dd q\longrightarrow0 .
\end{equation}
No convergence or continuity of the densities \(a_n\) is required.
\end{lemma}

\begin{proof}
Boundedness and convergence in measure on the finite-volume set
\(B\) imply \(\|F_n\|_{L^1(\R^d)}\to0\): for every
\(\eta>0\), bound the integral by
\(\eta|B|+M|\{|F_n|>\eta\}|\).

Fix \(\epsilon>0\).  The critical set in \(K\) has measure zero,
so choose an open neighborhood \(E\) of it with
\(|E\cap K|<\epsilon\).  The compact set \(K\setminus E\)
consists of regular points.  At such a point \(q_0\), choose \(d\)
input coordinates giving a nonzero minor of \(D\Phi(q_0)\).
Let \(\pi\) be projection onto the complementary \(k-d\)
coordinates, and define
\[
 G(q)=(\Phi(q),\pi q),\qquad
 G_n(q)=(\Phi_n(q),\pi q),\qquad A_0=DG(q_0).
\]
The matrix \(A_0\) is invertible.  Choose a convex ball
\(V\Subset\mathcal O\) about \(q_0\) so small that, for all
sufficiently large \(n\),
\[
 \sup_{q\in V}\|A_0^{-1}DG_n(q)-I\|<\tfrac12.
\]
Integration on the line segment between \(q,q'\in V\) gives
\[
 |A_0^{-1}(G_n(q)-G_n(q'))-(q-q')|
 \le\tfrac12|q-q'|.
\]
Thus \(G_n\) is injective on \(V\), and its Jacobian has a
positive lower bound \(c_V\) independent of \(n\).  The inverse
function theorem consequently makes it a diffeomorphism onto
its image.

Cover \(K\setminus E\) by finitely many such balls \(V_j\).
The change-of-variables formula gives
\[
 \begin{split}
 \int_{V_j\cap K}|F_n(\Phi_n(q))|\dd q
 &\le c_{V_j}^{-1}
       \int_{G_{n,j}(V_j\cap K)}|F_n(z)|\dd z\,\dd w\\
 &\le c_{V_j}^{-1}|\pi_j(V_j)|
       \|F_n\|_{L^1(\R^d)}.
 \end{split}
\]
When \(k=d\), the zero-dimensional factor has value one.
The integral on \(E\cap K\), including its density, is at most
\(AM\epsilon\). Multiplying the regular-chart estimates by \(A\)
and summing over the finite cover gives
\[
 \limsup_{n\to\infty}
 \int_{\mathcal O}|F_n(\Phi_n(q))a_n(q)|\dd q
 \le AM\epsilon.
\]
Let \(\epsilon\downarrow0\).
This proves \eqref{nc:pullback-conclusion}. The argument used the
densities only through \(|a_n|\le A\), so arbitrary measurable masks
and restrictions may be included in them without any convergence
assumption.
\end{proof}

\begin{lemma}[Five-gain rank and robust pullback]
\label{lem:five-gain-rank}
Fix a nonaxial endpoint velocity \(v_0\), an endpoint
\((t_0,x_0)\) with \(t_0>0\), five positive scales
\(\varrho_i\), and five signs \(\sigma_i\).
Given positive times \(t_i\) and \(L_i\in\R^3\), define recursively
\begin{equation}
\label{nc:rank-map}
 \begin{split}
 x_i&=x_{i-1}-(t_{i-1}-t_i)v_{i-1},\\
 p_i&=u_{\varrho_i,\sigma_i}(t_i,z_i)e_3,\qquad
 n_i=v_{i-1}-p_i,\\
 k_i&=n_i\times L_i,\qquad v_i=v_{i-1}+k_i ,
 \qquad i=1,\ldots,5 .
 \end{split}
\end{equation}
The map
\[
 \Phi:(t_1,\ldots,t_5,L_1,\ldots,L_5)
       \longmapsto(t_5,x_5,v_5)
\]
is real analytic on the connected open set
\((0,\infty)^5\times\R^{15}\), and has rank seven almost everywhere.
In particular it is a submersion almost everywhere on the ordered
region \(0<t_5<\cdots<t_1<t_0\).

For convergent endpoint sequences with a nonaxial limiting velocity,
convergent positive scale sequences, and the finite-temperature
core formulas \eqref{nc:core-parameters}, the analogous maps converge
in \(C^1\) on compact parameter sets to \(\Phi\).
Bounded measurable parameter densities supported in a common
compact set therefore satisfy
Lemma~\ref{nc:uniform-pullback}, including after any measurable
spatial, velocity, or time restrictions.
\end{lemma}

\begin{proof}
For \(t>0\), the derivative with respect to \(u\) in the defining
implicit equation is
\[
 1+\frac{t\alpha\gamma}{\varrho}
 S\bigl(\gamma(z-tu)/\varrho\bigr)>0.
\]
The implicit analytic function theorem gives an analytic profile
for every \(z\in\R\).  Uniqueness, which also follows from this
strict monotonicity, identifies the local profiles on overlaps.
The recursive formulas \eqref{nc:rank-map} are consequently
analytic.  They make sense even without ordering the positive
times: a negative formal free-flight duration merely changes
the point at which the next analytic profile is evaluated.
This supplies the connected open domain stated in the lemma.

We exhibit one rank-seven point.  Fix \(t_5=\tau\) with
\(0<\tau<t_0\), and set all five collision times equal to
\(\tau\).  After a rotation about the velocity axis, write
\(v_0=(R,0,z)\), \(R>0\).  All collision positions now coincide
at \(x_0-(t_0-\tau)v_0\), independently of the angular parameters.
Thus write the five axial velocities simply as \(p_i=a_i e_3\);
their values \(a_i\) may be arbitrary.

Choose
\[
 k_1=(0,1,0),\qquad k_2=(1,-R,0).
\]
They satisfy \(k_1\perp n_1\), \(k_2\perp n_2\), and span the
transverse plane.  Put
\[
 w=\bar v_2=(R+1,1-R)\ne0,\qquad b_3=v_{2,3}-a_3,
\]
and choose a sufficiently small nonzero number \(\epsilon\) and
\[
 k_3=\epsilon\left(-\frac{b_3w}{|w|^2},1\right).
\]
Then \(k_3\perp n_3\), its axial component is nonzero, and
\(\bar v_3\ne0\).  Hence \(k_1,k_2,k_3\) are linearly independent.
Writing \(\bar v_3=(X,Y)\), take
\[
 k_4=(-Y,X,0),\qquad k_5=0.
\]
The fourth increment is perpendicular to \(n_4\), and
\[
 \det(\bar n_4,\bar n_5)=X^2+Y^2>0.
\]
Thus \(n_4,n_5\) are nonparallel.  All the normals in this
construction are nonzero, and every chosen increment can be
represented by \(L_i=-n_i\times k_i/|n_i|^2\).  Choose \(L_5=0\).

At the equal-time configuration,
\[
 x_5=x_0-v_0(t_0-t_1)
           -\sum_{i=1}^4v_i(t_i-t_{i+1}).
\]
All derivatives of \(x_5\) in angular directions vanish.
Keeping \(t_5\) fixed, its first three time columns are
\[
 \partial_{t_1}x_5=-k_1,\qquad
 \partial_{t_2}x_5=-k_2,\qquad
 \partial_{t_3}x_5=-k_3.
\]
Derivatives of the \(v_i\), including derivatives of the
position-dependent \(p_i\), are multiplied by zero free-flight
durations in this formula.  In the final velocity block,
\[
 \delta_{L_4}v_5=n_4\times\delta L_4,\qquad
 \delta_{L_5}v_5=n_5\times\delta L_5,
\]
because \(L_5=0\) and the \(p_i\) have zero angular position
derivatives at equal times.  The sum of the planes
\(n_4^\perp+n_5^\perp\) is \(\R^3\).  Choose three angular
columns spanning this sum.  Together with the three time columns
above, they give an invertible \(6\times6\) block for
\((x_5,v_5)\): its upper-right block is zero and both diagonal
blocks are invertible.  Appending the \(t_5\) column and target
coordinate gives a nonzero \(7\times7\) minor.

This minor is a nontrivial analytic function on the connected
domain.  Its zero set has Lebesgue measure zero.  For completeness,
at any zero of a nontrivial analytic function some derivative of
finite order is nonzero; otherwise its Taylor series vanishes on
a neighborhood, and analytic continuation through overlapping
balls in the connected domain makes the function identically
zero.  Choosing a derivative of minimal nonzero order puts the
point on a regular zero set of a derivative of one lower order.
There are countably many such derivatives, and each regular zero
set is covered by countably many smooth hypersurface charts.
Their union is null.  Rank failure is contained in the zero set
of our particular minor, so it is null as well.  Continuity also
shows directly that the witness persists at nearby strictly
ordered times.  It need not obey any spatial mask.

For the prelimit maps replace \(p_i\) by
\eqref{nc:core-parameters} and include \(w_i\in\R^3\) among
the parameters.  On compact sets the scales have positive lower
bounds, \(d_{r_n\varrho_i}\to0\), and all implicit derivatives have
the same positive lower bound one.  Differentiating the implicit
equations, or applying their implicit function formulas, gives
\(C^1\) convergence in the times, positions, and all bounded
\(L_i,w_i\).  The limit is \(\Phi\), independent of the fifteen
spectator \(w_i\)-coordinates.  Its rank is still seven almost
everywhere on the enlarged parameter domain, by Fubini.
Lemma~\ref{nc:uniform-pullback} now proves the final assertion.
\end{proof}

\begin{lemma}[Vanishing of five nonaxial gains]
\label{nc:five-nonaxial-gains}
Suppose \(\ell_n\to0\), the hit endpoints converge after periodic
translation, and their limiting velocity is nonaxial.  Suppose
also that \(0\le f_n\le2\) on \(a\le t\le\tau_n\) and
\(f_n\to0\) in measure on every compact subset of
\[
 (0,U)\times\R^3_x
       \times\{v\in\R^3:\bar v\ne0\}.
\]
Then
\[
 (\mathcal V_n^5f_n)(\tau_n,x_n,v_n)\longrightarrow0.
\]
\end{lemma}

\begin{proof}
Fix an arbitrarily small error tolerance.  Remove the tails of
Lemma~\ref{nc:endpoint-tightness}.  All path velocities and
positions are now in fixed bounded sets.  At every encountered
main core, \(s\le Cr_n\), so its transverse width tends uniformly
to zero.

Starting from a positive lower bound for the endpoint transverse
speed, apply the nonaxial small-input estimate in
Lemma~\ref{lem:axis-kernel-estimates} successively at gains
\(1,\ldots,5\).  Choose positive numbers \(b_1,\ldots,b_5\)
so that deleting \(|\bar v_i|<b_i\) costs less than the prescribed
error at step \(i\).  At each step the previous lower bound has
already been fixed; the later gain masses are bounded by
\eqref{nc:iteration-norm}.  Thus this is a finite successive
choice, followed by \(n\to\infty\).  The sum of the five deleted
contributions is arbitrarily small.  Write \(b>0\) for a lower
bound on all retained transverse speeds, including \(v_0\).
For large \(n\), every retained normal \(v_{i-1}-p_i\) has length
at least \(b/2\).

Next delete collision points with \(|y_i|<\rho\).
Conditional on all earlier parameters, \(y_i\) is an affine
function of \(t_i\) with derivative \(\bar v_{i-1}\).
Consequently
\[
 \left|\{t_i:|y_i|<\rho\}\right|\le\frac{2\rho}{b}.
\]
Integrate the current kernel using its mass bound \(C/a\),
and bound the later kernels by \eqref{nc:iteration-norm}.
Summing over the five steps bounds this deletion by \(C_a\rho/b\).
Choose \(\rho>0\) after the speed thresholds.  The retained
scales now belong to a compact subinterval
\([\varrho_-,\varrho_+]\subset(0,\infty)\).
Mesh convergence gives a fixed finite list of scales on this
interval, with positive limits, after an immaterial enlargement
of its endpoints.  There are consequently only finitely many
five-tuples of scales and signs to consider.

The tested time \(t_5\) must remain inside the weak-limit
interval.  If \(U>1\), it already lies in the compact interval
\([a,1]\Subset(0,U)\).  If \(U=1\), delete \(t_5>1-\delta\).
Ordering forces all five times into \((1-\delta,\tau_n)\),
whose length is at most \(\delta\).  This costs at most
\begin{equation}
\label{nc:upper-endpoint-cost}
 2(C/a)^5\frac{\delta^5}{5!}.
\end{equation}
Choose \(\delta>0\) to make this as small as desired.  No
convergence at the event time is being asserted.

We also remove a thin neighborhood of the boundary of the ordered
time region. Put \(t_0=\tau_n\), and discard paths on which
\(t_5-a<\kappa\) or \(t_{i-1}-t_i<\kappa\) for some
\(1\le i\le5\). Integrating velocities successively with the
bound \(C/a\), their total contribution is at most
\(C_a\kappa\): each of the six time strips has five-dimensional
volume at most \(\kappa\) in the unit cube. Choose \(\kappa>0\)
after the preceding tolerances. On the retained set all consecutive
time gaps are at least \(\kappa\), and the lower time exceeds
\(a+\kappa\). Since \(\tau_n\to1\), the retained times lie in a
fixed compact subset of \(0<t_5<\cdots<t_1<1\) for all large \(n\).

Fix one retained scale/sign tuple. Drop all spatial masks in
constructing its maps, and retain the masks in its nonnegative
integrand.  Each plane increment is bounded, all normals are
bounded above and below, and each cold parameter is bounded.
Apply Lemma~\ref{nc:plane-parameters} successively.  The integral
becomes an integral over five times, fifteen \(L\)-coordinates,
and fifteen cold parameter coordinates.  Its support lies in
a fixed compact subset of
\((0,\infty)^5\times\R^{30}\), and its density is uniformly
bounded: the core amplitudes are at most \(C/a\), the factors
\(|n_i|\zeta(L_i\cdot n_i/|n_i|)\) are bounded, and all retained
ratios \(H(v_i)/H(v_{i-1})\) are bounded.  The masks, time
ordering, and cut conditions are arbitrary measurable factors
between zero and one in this density.

The target \((t_5,x_5,v_5)\) on the actual support lies in a
fixed compact set \(B\) inside the open weak-limit region, away
from the velocity axis.  Set
\(F_n=f_n\ind_B\ind_{\{a\le t\le\tau_n\}}\) and extend it by zero;
it is bounded by two and tends to zero in measure.  The maps converge
in \(C^1\) to the map in Lemma~\ref{lem:five-gain-rank}.
That lemma and Lemma~\ref{nc:uniform-pullback} show that the
integral for this tuple tends to zero.  Notice that the entire
parameter compact need not map into \(B\); only the support of
the measurable density must do so.

There are finitely many tuples, so their sum tends to zero.
The total limsup of the original integral is bounded by the
arbitrarily small errors from the preceding deletions.
Letting those tolerances tend to zero proves the lemma.
\end{proof}

\subsection{Completion of the nonvanishing argument}

\begin{proof}[Completion of the proof of Proposition~\ref{prop:nonzero-limit}]
Assume \(l=0\) and \(g=0\), as in the reduction above.
Choose endpoints \((x_n,v_n)\) at time \(\tau_n\) in the common
full-measure set fixed above such that \(f_n(\tau_n,x_n,v_n)\ge1/2\).
For each fixed \(a\), the residual in
\eqref{nc:five-residual} tends to zero along any subsequence
of these endpoints.  Indeed, Lemma~\ref{nc:endpoint-tightness}
first excludes large velocities and escaping positions.
Otherwise choose convergent endpoints after a periodic
translation.  If the limiting velocity is axial, the first
two gains vanish by Lemma~\ref{nc:two-axial-gains}, and the
last three have bounded mass by \eqref{nc:iteration-norm}.
If it is nonaxial, use \eqref{nc:zero-in-measure} in
Lemma~\ref{nc:five-nonaxial-gains}.  This covers every
subsequence; hence the residual tends to zero for the
original endpoint sequence as well.

Taking the limsup in \eqref{nc:five-residual} at these
endpoints gives
\[
 \frac12
 \le a\sum_{j=0}^4\frac{(C\log(1/a))^j}{j!}.
\]
Finally let \(a\downarrow0\).  The right side tends to zero,
a contradiction.  Throughout this argument \(U\) was allowed
to be infinite, and the case \(U=1\) was handled by
\eqref{nc:upper-endpoint-cost}.  This proves the proposition.
\end{proof}
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//10-nonconcentration.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//11-branching.tex ====
\section{Branching growth and the dilated horizon}\label{sec:branching}

Proposition~\ref{prop:nonzero-limit} gives a nonzero solution of the
linear equation \eqref{eq:limit-equation}. We prove that this solution
cannot persist for all positive dilated times under the cap
\(g\le(t+l)H\). The argument is the branching construction of
\cite[Section~9]{OpenAIPeriodic}, whose proof we give in full. Its inputs
are the limiting medium \eqref{eq:background}, the cap, and the local
transport formula established in Proposition~\ref{prop:blowup-limit}.

\begin{proposition}\label{prop:finite-dilated-horizon}
There are fixed choices
\[
K\ge1,\qquad 0<\eta\le\log2,\qquad
0<\alpha\le\tfrac14,\qquad 0<\gamma\le1
\]
with the following property. For every allowed geometric mesh with
lower endpoint \(l\in[0,1]\), the only nonnegative density solving
\eqref{eq:limit-equation} on \(0<t<\infty\), with locally integrable
gain and loss, the positive-time mild formula \eqref{dil:mild}, and the
bound \(g(t,x,v)\le(t+l)H(v)\), is zero.
Consequently every nonzero limit in
Propositions~\ref{prop:blowup-limit} and~\ref{prop:nonzero-limit} has
\(U<\infty\). The four parameters can be fixed before the thermal,
seed, bath, and final horizon parameters of the construction. Their
choice is independent of the mesh phase and of \(l\).
\end{proposition}

The proof compares growth with spatial spreading. In a constant medium,
we construct positive collision histories that multiply mass while
keeping all speeds bounded and prescribing a small mean transverse
velocity. Finitely many compact families of these histories remain
effective when the medium changes measurably within a small tolerance. Far enough from
the initial time, the actual background has precisely this form on
short intervals whose lengths are proportional to their starting times.
Iteration then makes mass grow faster than the cap allows in spatial
balls of radius proportional to time. All histories begin at a strictly
positive time, where the collision rates are bounded on bounded velocity
sets.

The decisive comparison is between a multiplication factor and the
volume of these balls. Their volume is of order \(t^3\), and the cap
contributes one further power of \(t\). We will obtain a factor greater
than \(3\) between times \(t\) and \((1+a_{\rm br})t\), where the fixed number
\(a_{\rm br}>0\) is small enough that \((1+a_{\rm br})^4<3\).

\subsection{Positive histories and their normalization}

For \(v,w\in\R^3\) and \(n\in S^2\), set
\begin{equation}\label{br:sphere-map}
\Phi(v,w,n)=\frac{v+w}{2}+\frac{|v-w|}{2}n,
\qquad \dd\varsigma(n)=\frac{\dd n}{4\pi}.
\end{equation}
Thus \(\varsigma\) is a probability measure. By
Lemma~\ref{lem:carleman}, a collision with incident velocities \(v,w\)
has rate \(2\pi|v-w|\), and the marginal law of either child is the
pushforward of \(\varsigma\) under \(\Phi(v,w,\cdot)\). The two marginal
laws agree because \(n\mapsto-n\) preserves \(\varsigma\). Consequently,
for a nonnegative velocity measure \(B\) and a density \(f\),
\begin{equation}\label{br:gain-mass}
\int_{\R^3}P(B,f)(v)\dd v=\int_{\R^3}\nu_B(v)f(v)\dd v.
\end{equation}
The coefficient \(2\) in the gain of \eqref{eq:limit-equation} therefore
counts both children, while the loss removes their parent once.

More generally, fix Borel representatives of the coefficients of a
two-beam medium
\[
\widetilde B(t,x)=\sum_{\sigma=\pm1}b_\sigma(t,x)
                  \delta_{w_\sigma(t,x)},\qquad
q_\sigma(t,x,v)=2\pi b_\sigma(t,x)|v-w_\sigma(t,x)|,
\qquad \nu=\sum_\sigma q_\sigma.
\]
All pointwise path conditions below refer to these representatives.
On \([a,b]\), a history with ordered collision times
\(a<\tau_1<\cdots<\tau_j<b\), signs \(\sigma_i\), and directions \(n_i\)
follows free flights between collisions and uses
\[
V_{\tau_i}=\Phi(V_{\tau_i-},w_{\sigma_i}(\tau_i,X_{\tau_i}),n_i).
\]
Its weight relative to \(\prod_i\dd\tau_i\dd\varsigma(n_i)\) is
\begin{equation}\label{br:history-weight}
\exp\left(-\int_a^b\nu(s,X_s,V_s)\dd s\right)
\prod_{i=1}^j2q_{\sigma_i}(\tau_i,X_{\tau_i},V_{\tau_i-}).
\end{equation}
For \(j=0\) the empty product is one.

\begin{lemma}\label{br:positive-histories}
Let \(0<a<b<\infty\). Suppose \(f\ge0\) solves
\(\D f=2P(\widetilde B,f)-\nu f\) in the local transport sense,
both collision terms are locally integrable, and the chosen rate
representatives \(q_\sigma\) are bounded on compact positive-time
phase-space sets. Start with any nonnegative bounded, integrable
portion of \(f(a)\). Integration of any measurable collection of bounded
finite histories with weights \eqref{br:history-weight} gives a measure
dominated by \(f(b)\).
\end{lemma}

\begin{proof}
Set \(G=2P(\widetilde B,f)\). In free coordinates
\(f^\sharp(t,x,v)=f(t,x+tv,v)\), the equation becomes
\[
\partial_t f^\sharp=(G-\nu f)^\sharp.
\]
On bounded tubes its right side is locally integrable, so \(f^\sharp\)
has absolutely continuous time sections for almost every \((x,v)\)
and local \(L^1\) time traces.
The rate bounds make \(\nu^\sharp\) integrable along every bounded
characteristic segment in a compact positive-time tube. The one-dimensional integrating-factor
formula therefore consists of the freely
transported initial density multiplied by its survival factor, plus the
time integral of the nonnegative gain with the same survival factor.
Repeated substitution into the gain gives \eqref{br:history-weight},
by \eqref{br:sphere-map} and \eqref{br:gain-mass}. Every remainder is
nonnegative. Restricting the histories, or the initial density, preserves
the lower bound. Tonelli's theorem justifies the substitutions. On bounded
portions with total gain rate at most \(C\), the \(j\)-collision term is
bounded by \(C^j(b-a)^j/j!\) times the initial mass.

These formulas apply to point-velocity partners. Indeed
Lemma~\ref{lem:carleman} gives a density for the gain against each such
partner, with mass \eqref{br:gain-mass}. Thus free flight and each gain
preserve absolute continuity when the starting measure has a density.
Null spatial sets in representatives of the coefficients do not affect
the integrated formula.
\end{proof}

For the background \eqref{eq:background}, \(|u_\sigma|\le1+\alpha\) and
\begin{equation}\label{br:global-rate}
0\le b_\sigma(t,x)\le\frac{\theta K}{t\alpha\gamma},
\qquad
\nu_B(t,x,v)\le\frac{C(1+|v|)}{t}\quad(t>0).
\end{equation}
At each mesh interface choose the value from the outer adjacent
annulus, and on the transverse axis set the medium to zero.
These are Borel representatives obeying the same bounds.
All bounded finite flights starting at
positive time have strictly positive survival. The cap and
Lemma~\ref{lem:sheet-bound} give locally integrable collision terms.
The preceding transport argument therefore supplies local \(L^1\)
time traces of \(g\) on bounded regions. For the extracted limits,
these agree with the representatives in
Proposition~\ref{prop:blowup-limit}. Testing
nonnegative compact functions and taking a time limit shows that
\(g(t)\le(t+l)H\) also holds at these traces. We use this representative.

\subsection{A reference flight with a positive potential}

We convert the branching history weights into expectations along a
single Markov flight with a positive potential. This is a jump-process
Feynman--Kac representation, related to the many-to-one formulas for
branching systems; see \cite[Section~4.1]{HarrisRoberts2017} for general
probabilistic context. The finite-history calculation below proves the
precise representation for the present collision law.

Fix
\[
D=5,\qquad E=\{v:|v|\le D\},\qquad
B_0=\delta_{e_3}+\delta_{-e_3},\qquad
\nu_0(v)=2\pi\bigl(|v-e_3|+|v+e_3|\bigr).
\]
Let \(J(v,\dd v')\) be the single-child law obtained by choosing a sign
with probabilities proportional to \(|v-\sigma e_3|\) and then using
\eqref{br:sphere-map}. Put \(p(v)=J(v,E)\).

\begin{lemma}\label{br:retention}
For every \(v\in E\), \(p(v)\ge25/26\).
The reference history sum restricted to velocities in \(E\) is represented
by a Markov flight with velocity jump rate
\(\lambda(v)=2p(v)\nu_0(v)\), jump law
\(J(v,\dd v')\ind_E(v')/p(v)\), and multiplicative weight
\[
\exp\left(\int_0^A \mathcal W(V_s)\dd s\right),
\qquad \mathcal W(v)=(2p(v)-1)\nu_0(v).
\]
One may take
\begin{equation}\label{br:potential-bounds}
0<c_0=\frac{48\pi}{13}\le \mathcal W(v),\qquad
\lambda(v)\le C_0=48\pi\quad(v\in E).
\end{equation}
\end{lemma}

\begin{proof}
Set \(s=|v|^2\), \(z=v_3\), and
\(R=\sqrt{(s+1)^2-4z^2}\). For either partner sign the squared child
speed has the law
\[
|v'|^2=\frac{s+1+RZ}{2},
\qquad Z\text{ uniform on }[-1,1].
\]
This follows by expanding the square in \eqref{br:sphere-map}: a
coordinate of a uniform point on \(S^2\) is uniform on \([-1,1]\).
The maximum is at most \(s+1\), so there is no exit for \(s\le24\).
For \(24<s\le25\), the exit probability, if positive, is
\[
\frac12+\frac{s-49}{2R}
\le\frac12+\frac{s-49}{2(s+1)}
=\frac{s-24}{s+1}\le\frac1{26}.
\]
The inequality uses \(R\le s+1\) and \(s-49<0\). If \(R=0\), the
squared speed is deterministic and at most 25. Mixing the signs
preserves the bound.

A Markov history of rate \(\lambda\) has the survival factor
\(e^{-\int\lambda}\), and one rate times jump-law factor per collision.
Multiplication by \(e^{\int(\lambda-\nu_0)}\) changes this exactly to
\eqref{br:history-weight}, restricted to \(E\). Both history sums
converge absolutely on finite intervals because their rates are bounded.
Finally \(4\pi\le\nu_0(v)\le24\pi\) on \(E\), and
\(2p(v)-1\ge12/13\), giving \eqref{br:potential-bounds}.
\end{proof}

\subsection{Three collisions give a volume minorization}

The positive potential gives growth along every retained reference
path. To control where that growth occurs, we also need cancellation in
the integrated velocity. Write \(P_t(v,\cdot)\) for the transition law
of the preceding velocity process. We prove that these laws approach a
symmetric stationary distribution by establishing a Doeblin-type
minorization: every starting velocity has a common positive portion of
its transition law \cite{Doeblin1940}. Three collisions supply this
common portion.

\begin{lemma}\label{br:minorization}
There are \(r_0>0\) and \(\varepsilon_0>0\) such that
\begin{equation}\label{br:doeblin}
P_1(v,A)\ge\varepsilon_0|A\cap B(0,r_0)|\quad(v\in E)
\end{equation}
for every Borel set \(A\subset E\). For each \(\zeta>0\), there is also
\(\varepsilon_\zeta>0\) such that
\[
P_1(v,B(e_1,\zeta)\cap\{|v'|<D\})\ge\varepsilon_\zeta\quad(v\in E).
\]
\end{lemma}

\begin{proof}
Given \(v\in E\), choose a sign with \(|v-\sigma e_3|\ge1\).
Use this partner for the first collision, and select an output within
a small fixed distance \(\delta\) of \(\sigma e_3\). Its probability
has a uniform positive lower bound: the incident distance lies in
\([1,6]\), and a cap of fixed Euclidean radius \(\delta\) at that
endpoint has area probability bounded below uniformly on the
corresponding diameter spheres. Such first outputs lie strictly in \(E\).

Use partner \(-\sigma e_3\) next. If the first output is exactly
\(\sigma e_3\), the second sphere is the unit sphere; restrict its
output \(u\) near \(e_1\). For the third collision use partner \(e_3\).
The last output is
\[
T(u,n)=\frac{u+e_3}{2}+\frac{|u-e_3|}{2}n.
\]
At \(u=e_1\), \(n_0=-(e_1+e_3)/\sqrt2\), this output is zero.
The two angular derivatives in \(n\) span \((e_1+e_3)^\perp\).
The vector \(e_3\) is tangent to the preceding unit sphere at \(e_1\),
and direct differentiation gives
\begin{equation}\label{br:rank-derivative}
\partial_{e_3}T(e_1,n_0)=\frac{e_1+3e_3}{4},\qquad
(e_1+e_3)\cdot\partial_{e_3}T(e_1,n_0)=1.
\end{equation}
Thus the map of the four angular coordinates in the last two collisions
to the last velocity has rank three.

Take the two coordinates of \(n\) and the coordinate of \(u\) with
derivative \(e_3\) as distinguished variables, and hold the fourth
coordinate in a short interval. The inverse function theorem, with
this fourth coordinate and the first output as parameters, gives
neighborhoods on which the distinguished-variable map is one-to-one,
its Jacobian and inverse Jacobian are bounded, and its image contains
a fixed ball \(B(0,r_0)\). Shrink these neighborhoods and \(\delta\)
so this holds for every first output within \(\delta\) of
\(\sigma e_3\), for both signs. Sphere area in these coordinates has
a positive lower density. Change of variables in the three distinguished
coordinates, and integration in the fourth, give a fixed positive
multiple of volume on \(B(0,r_0)\).

Force the three times into three fixed, disjoint intervals inside
\((0,1)\), and prohibit other jumps. The first selected relative speed
is at least 1, the second is close to 2, and the third close to
\(\sqrt2\). In the reference Markov process the selected transition
intensity is
\[
4\pi|v-\sigma e_3|\,
\ind_E(\Phi(v,\sigma e_3,n))\dd\varsigma(n).
\]
It has positive uniform lower density on the patches just specified.
The no-additional-jump factor is at least \(e^{-C_0}\).
Integrating the times proves \eqref{br:doeblin}. Keeping only the first
two collisions, selecting a small patch near \(e_1\) in the second,
and prohibiting later jumps proves the last assertion.
\end{proof}

\begin{lemma}\label{br:mixing}
The reference process has a stationary probability law \(\pi\) with
\(\int v\dd\pi(v)=0\). There are \(C,c>0\) such that
\[
\sup_{v\in E}\|P_t(v,\cdot)-\pi\|_{\mathrm{TV}}\le Ce^{-ct},
\qquad
\sup_{v\in E}\mathbb E_v\left|
A^{-1}\int_0^A V_s\dd s\right|^2\le\frac C A\quad(A\ge1).
\]
\end{lemma}

\begin{proof}
The normalized measure on the right side of \eqref{br:doeblin} is a
common part of positive mass \(\epsilon\) in all rows of \(P_1\).
Removing this common part shows that the map \(\mathfrak m\mapsto\mathfrak m P_1\)
contracts total variation distance by \(1-\epsilon\). Probability
measures form a complete space in that metric, so iteration yields a
unique fixed point \(\pi\) and uniform geometric convergence.
Since \(P_t\) commutes with \(P_1\), \(\pi P_t\) is another fixed point
of \(P_1\), hence equals \(\pi\). Contractivity of Markov kernels gives
the estimate between integer times.

Negation of velocity interchanges the two signs and preserves \(E\),
so uniqueness of \(\pi\) makes it symmetric and its mean zero.
Consequently \(\sup_{v\in E}|\mathbb E_vV_t|\le Ce^{-ct}\).
The Markov property and bounded velocities imply
\[
|\mathbb E(V_s\cdot V_t)|\le Ce^{-c(t-s)}\quad(0\le s\le t)
\]
uniformly over starting states. Integrating this estimate on
\([0,A]^2\) bounds the second moment of \(\int_0^A V_s\dd s\) by \(CA\).
\end{proof}

\subsection{A displacement block and measurable perturbations}

Fix \(d=1/10\), and choose once and for all
\begin{equation}\label{br:kappa-choice}
0<\varkappa<\min\left\{\frac14,\frac{c_0}{2C_0},\frac{r_0}{4}\right\}.
\end{equation}
In particular a small ball about \(\varkappa e_1\) lies in the
minorization ball.

To repeat growth in the cold background, we will transport mass through
boxes whose transverse distance from the axis grows proportionally to time:
\[
\mathcal B_t=\{(x,v):|x-\varkappa te_1|\le d\varkappa t,\ |v|\le D\},
\qquad t>0.
\]
For a duration \(\Delta>0\), a displacement within
\(d\varkappa\Delta\) of \(\varkappa\Delta e_1\)
carries every starting point of \(\mathcal B_t\) into
\(\mathcal B_{t+\Delta}\), provided the final velocity remains in
\(E\). Mixing makes the integrated velocity small over most of a
reference block. We reserve a final fraction \(\varkappa\) of the block
for a flight with velocity close to \(e_1\), producing the displacement
\(\varkappa A e_1\). The positive potential will compensate for the cost
of prohibiting collisions during that final flight.

\begin{lemma}\label{br:reference-block}
There is a finite \(L\) such that, for all
\[
A\in I_L:=\left[\frac L{1+d},\frac L{1-d}\right],\qquad v\in E,
\]
reference branching histories with velocities in \(E\), final velocity
strictly inside \(E\), and displacement \(Z_A\) satisfying
\begin{equation}\label{br:strict-drift}
|Z_A-\varkappa Ae_1|<\frac34d\varkappa A
\end{equation}
have total weight greater than 8.
\end{lemma}

\begin{proof}
Put \(A_0=(1-\varkappa)A-1\). On the first interval of length \(A_0\),
require the reference Markov flight to have integrated velocity of
norm at most \(Ad\varkappa/4\). Lemma~\ref{br:mixing} and Chebyshev's
inequality give probability at least \(1/2\), uniformly in the start
when \(A\) is sufficiently large. Over the next unit interval require
the last velocity to lie in \(B(e_1,d/4)\). The conditional probability
is at least a fixed \(\varepsilon>0\), by
Lemma~\ref{br:minorization}. Require no jump during the final interval
of length \(\varkappa A\), whose conditional probability is at least
\(e^{-C_0\varkappa A}\).

The errors in displacement from \(\varkappa Ae_1\) on the three intervals
are at most \(Ad\varkappa/4\), \(D\), and \(Ad\varkappa/4\).
For \(A>4D/(d\varkappa)\) their sum is less than \(3Ad\varkappa/4\).
The final velocity lies strictly in \(E\).
Lemma~\ref{br:retention} bounds the branching weight below by
\[
\frac{\varepsilon}{2}\exp((c_0-C_0\varkappa)A).
\]
Its exponent is positive by \eqref{br:kappa-choice}. Choose \(L\)
large enough that this exceeds 8, and all the preceding lower bounds
on \(A\) hold throughout \(I_L\).
\end{proof}

The background coefficients jump across annular interfaces. To use the
reference growth there, we first restrict it to compact families with
bounded collision counts and strict velocity and displacement margins.
These margins allow comparison with a merely measurable medium: its
values along each actual path need only remain close to the constant
reference values.

\begin{lemma}\label{br:robust-block}
For these fixed \(\varkappa,L\), there are \(J_{\mathrm{br}}\in\N\) and
\(\varepsilon_*\in(0,1/4)\) with the following property. Suppose a
measurable medium obeys, throughout every speed-\(D\) excursion from
a specified starting point over a duration \(A\in I_L\),
\begin{equation}\label{br:medium-closeness}
|b_\sigma-1|\le\varepsilon_*,
\qquad |w_\sigma-\sigma e_3|\le\varepsilon_*
\quad(\sigma=\pm1).
\end{equation}
From each starting velocity in \(E\), histories with between \(1\) and
\(J_{\mathrm{br}}\) collisions, speed at most \(D\) on every free flight, every
postcollision velocity and the final velocity strictly inside \(E\), and
\begin{equation}\label{br:robust-drift}
|Z_A-\varkappa Ae_1|<d\varkappa A
\end{equation}
have total weight greater than 3. The initial free flight may have
speed \(D\). No continuity of the medium is needed.
\end{lemma}

\begin{proof}
Fix a reference start and duration \((v,A)\in E\times I_L\).
The zero-collision term has weight \(e^{-\nu_0(v)A}\le1\).
Discarding it from the family in Lemma~\ref{br:reference-block}
leaves weight greater than 7. Monotone convergence in the positive
history count permits restricting to finitely many counts
\(j\ge1\) while retaining weight greater than 7. Every postcollision
velocity lies strictly in \(E\) almost surely: the intersection of a
nondegenerate diameter sphere with \(\partial E\) has sphere area zero,
and a degenerate collision has zero rate. This includes the final
velocity. No inward margin is required for the initial velocity.
The displacement inequality
is strict. Ordered times may be kept apart and away from the endpoints.
For each retained collision count and sign sequence, parameterize the
ordered times as fractions of \(A\) and partition the angular parameter
space into finitely many measurable pieces lying in sphere coordinate
charts. Inner regularity of the
resulting finite parameter measures allows a further restriction to
compact sets with total weight greater than 6. These compact sets have
positive distances from the forbidden time coincidences and from every
postcollision speed boundary. They also have a positive margin in
\(|Z_A-\varkappa Ae_1|<d\varkappa A\), since the reference histories obey the
stronger inequality \eqref{br:strict-drift}. We call these finitely many
compact parameter sets a test for \((v,A)\).

The reference paths and weights are continuous in these finite
parameters and in \((v,A)\). The fixed compact tests therefore retain
weight greater than 5 and at least half their original positive
margins in a sufficiently small relative neighborhood of that start
and duration in \(E\times I_L\).
These neighborhoods cover the compact set
\(E\times I_L\). A finite subcover gives finitely many tests, a maximum
count \(J_{\mathrm{br}}\), and uniform positive margins.

Use the same initial velocity, signs, directions, and fractional times
for each perturbed history and its reference history. Their first
flights have speed at most \(D\), including boundary starts.
The sphere map satisfies
\[
|\Phi(v,w,n)-\Phi(\widetilde v,\widetilde w,n)|
\le |v-\widetilde v|+|w-\widetilde w|.
\]
Induction bounds the velocity error after \(j\) collisions by
\(j\varepsilon_*\) and the displacement error by \(AJ_{\mathrm{br}}\varepsilon_*\).
This comparison does not compare the medium at two nearby positions:
its value at the actual perturbed position is close to the same
constant reference partner. A sufficiently small \(\varepsilon_*\)
preserves all the strict velocity and displacement margins. The
actual paths thus have speed at most \(D\), so they stay within the
region on which closeness was assumed. Formally this follows by
induction up to the first possible failure; the preserved margins
exclude such a failure.

The relative-speed errors and full loss-rate errors on these histories
are bounded by \(C_{J_{\mathrm{br}}}\varepsilon_*\). The density error gives the same
bound for the gain-rate factors. The interval has bounded length,
so the error in the integrated loss and its exponential is bounded
by \(C_{J_{\mathrm{br}},L}\varepsilon_*\). Finite products of the bounded gain factors
then show that \eqref{br:history-weight} changes by at most
\(C_{J_{\mathrm{br}},L}\varepsilon_*\) uniformly on each test. The tests have finite
parameter measure. Decreasing \(\varepsilon_*\) makes their integrals
greater than 3. All coefficient estimates were pointwise; measurability
is sufficient.
\end{proof}

\subsection{Uniform choices of the construction parameters}

Lemma~\ref{br:robust-block} reduces the growth argument to a comparison
of the background with two constant beams. We now choose the
construction parameters to make that comparison uniform on the moving
boxes. Before doing so, we reserve a velocity tolerance that will also
allow positive mass to enter such a box. For this purpose we will steer
its velocity close to \(\varkappa e_1\), then let it fly freely until it
enters the moving box. There is
\(\alpha_{\mathrm{acc}}>0\), depending only on \(\varkappa\) and
Lemma~\ref{br:minorization}, with this property: after a first collision
puts a velocity sufficiently close to \(\sigma e_3\), two more
collisions, with partners within \(\alpha_{\mathrm{acc}}\) of
\(-\sigma e_3,e_3\), have positive angular probability of ending in
\begin{equation}\label{br:access-velocity}
B(\varkappa e_1,d\varkappa/4).
\end{equation}
Indeed the inverse-coordinate construction in that lemma, with first
output exactly \(\sigma e_3\), maps a positive-measure angular set into
\(B(\varkappa e_1,d\varkappa/8)\), which is inside \(B(0,r_0)\).
Keep a compact subset of positive measure in this set. Its outputs
have a positive distance from the complement of
\eqref{br:access-velocity}; the intermediate velocities and relative
speeds remain in the neighborhoods of that construction. Uniform
continuity of the two sphere maps preserves this for sufficiently
small perturbations of the first output and partners, including
measurable partners at the actual path positions. Take the smaller
tolerance for the two possible signs. We also require
\(\alpha_{\mathrm{acc}}\) to be less than half the permitted radius
of the first-output neighborhood.

The constants \(\varkappa,L,J_{\mathrm{br}},\varepsilon_*,\alpha_{\mathrm{acc}}\) are now
fixed and do not involve any construction parameter. Fix preliminary
bounds \(\eta\le1\), \(\alpha\le1\), \(K\ge1\). For \(t>0\), put
\begin{equation}\label{br:moving-box}
\Delta=\frac{L\varkappa t}{\theta K},
\qquad a_{\rm br}=\frac{L\varkappa}{\theta K}.
\end{equation}
At a starting point \(x_0=(y_0,z_0)\) in \(\mathcal B_t\),
\[
(1-d)\varkappa t\le|y_0|\le(1+d)\varkappa t.
\]
Every speed-\(D\) excursion over \([t,t+\Delta]\) satisfies
\[
\big||y|-|y_0|\big|\le D\Delta,\qquad
|z|\le d\varkappa t+D\Delta.
\]
Set \(\delta_K=DL/(\theta K(1-d))\). By increasing \(K\), impose
\(\delta_K\le1/4\) and \(a_{\rm br}\le1/4\). Then
\[
|y|\ge(1-\delta_K)(1-d)\varkappa t.
\]
For all sufficiently large \(t\), every such excursion is outside the
lower mesh hole, since \(l\le1\). On its active annulus
\(e^{-\eta}|y|\le\varrho\le|y|\), for every mesh phase. Thus, with a fixed
constant \(C_1\) depending only on \(d,D\),
\begin{equation}\label{br:comparison-quantities}
\begin{split}
\left|\frac{|y_0|}{\varrho}-1\right|&\le C_1(\eta+\delta_K),\\
\frac{\tau}{\varrho}&\le\frac{C_1}{\varkappa},\\
\frac{|z-\tau u_\sigma|}{\varrho}&\le\frac{C_1}{\varkappa}
\qquad(t\le\tau\le t+\Delta).
\end{split}
\end{equation}
The last estimate uses \(|u_\sigma|\le2\).

For a fixed starting point \(x_0\), set
\(\rho_0=\theta K/|y_0|\) and introduce the coordinates
\[
 s=\rho_0(\tau-t),\qquad \xi=\rho_0(x-x_0).
\]
Velocities are unchanged in these coordinates. The duration
\(\Delta\) is the same for every start in \(\mathcal B_t\), whereas
\(\rho_0\) varies with \(x_0\). This is the reason the reference block
was proved for an interval of durations: here its duration is exactly
\begin{equation}\label{br:duration-scaling}
A=\rho_0\Delta=\frac{L\varkappa t}{|y_0|}\in I_L.
\end{equation}
With \(S_\sigma=S(\gamma(z-\tau u_\sigma)/\varrho)\), the density ratio
in these units is
\begin{equation}\label{br:density-ratio}
\frac{b_\sigma(\tau,x)}{\rho_0}
=\frac{|y_0|}{\varrho}\,
\frac{S_\sigma}{1+(\tau\alpha\gamma/\varrho)S_\sigma}.
\end{equation}
Because \(1-\operatorname{sech}^2 h=\tanh^2h\le h^2\),
\eqref{br:comparison-quantities} gives, when the right-hand small
quantities are at most one,
\begin{equation}\label{br:uniform-medium-error}
\left|\frac{b_\sigma}{\rho_0}-1\right|
\le C_2\left(\eta+\delta_K+
              (\gamma/\varkappa)^2+\alpha\gamma/\varkappa\right),
\qquad |u_\sigma-\sigma|\le\alpha.
\end{equation}
Here \(C_2\) is independent of \(t,l\), and mesh phase.

Choose \(0<\eta\le\log2\) small, then
\[
0<\alpha<\min\{1/4,\varepsilon_*,\alpha_{\mathrm{acc}}\},
\]
then \(0<\gamma\le1\) small, so that the first, third, and fourth terms
on the right of \eqref{br:uniform-medium-error}, including their factor
\(C_2\), sum to less than \(\varepsilon_*/2\).
Finally increase \(K\) so the \(\delta_K\) term is less than
\(\varepsilon_*/2\), the preliminary restrictions hold, and
\begin{equation}\label{br:growth-exponent}
(1+a_{\rm br})^4<3.
\end{equation}
This establishes \eqref{br:medium-closeness} for all the scaled blocks,
with no choice depending on a limit, mesh, or solution.

These choices precede the remaining choices in the approximation.
Making \(\alpha\gamma\) small increases only fixed constants in
\eqref{br:global-rate} and the earlier comparison estimates.
Proposition~\ref{cold:control} is valid for every fixed
\(K,\eta,\alpha,\gamma\), after the thermal width is made sufficiently
small. Proposition~\ref{early:cap-selection} then permits the seed
exponent and the final short horizon to be chosen after those constants.
The later choices of bath, geometric scale, and horizon therefore
preserve the block estimates just proved.

\begin{lemma}[Growth in the limiting medium]\label{br:block-growth}
For the parameters just fixed, let \(a_{\rm br}=L\varkappa/(\theta K)\).
There is a time \(t_{\mathrm{br}}>0\), independent of the mesh and
\(l\in[0,1]\), such that every density satisfying the equation, cap,
and mild formula of Proposition~\ref{prop:finite-dilated-horizon} on
an interval containing \([t,(1+a_{\rm br})t]\), with \(t\ge t_{\mathrm{br}}\),
obeys, at its positive-time traces,
\begin{equation}\label{br:mass-growth}
 M((1+a_{\rm br})t)\ge3M(t),\qquad
 M(s):=\int_{\mathcal B_s}g(s,x,v)\dd x\dd v.
\end{equation}
\end{lemma}

\begin{proof}
Choose \(t_{\mathrm{br}}\) so that
\((1-\delta_K)(1-d)\varkappa t_{\mathrm{br}}>1\).
Then every speed-\(D\) excursion in
\eqref{br:moving-box} stays outside the hole for every
\(l\in[0,1]\) and every \(t\ge t_{\mathrm{br}}\).
Put \(\Delta=a_{\rm br}t\).
At each starting point of \(\mathcal B_t\), apply
Lemma~\ref{br:robust-block} using the simultaneous space--time scaling
\eqref{br:duration-scaling}. Translating back its displacement
restriction gives
\[
|x_{\mathrm{end}}-x_{\mathrm{start}}-\varkappa\Delta e_1|
<d\varkappa\Delta.
\]
The starting radius \(d\varkappa t\), followed by the triangle inequality,
places the endpoint in \(\mathcal B_{t+\Delta}\).
The velocity belongs to \(E\): the selected positive-count histories
have a strictly interior final velocity. Their first flight need
only have speed at most \(D\), exactly as in the medium comparison.
History weights are unchanged by this scaling: each gain rate is
divided by \(\rho_0\), each time differential is multiplied by
\(\rho_0\), and the integrated loss is invariant. The selected
histories therefore have total weight greater than 3 uniformly over
starts in \(\mathcal B_t\).

Use the kernel obtained by integrating all histories with
\(1\le j\le J_{\mathrm{br}}\) that satisfy the block's velocity and displacement
restrictions. The finite compact tests above certify its weight
greater than 3; no choice of a test for each starting point is needed.
For each integer \(m\ge1\), integrate this positive kernel against
\(\min\{g(t),m\}\ind_{\mathcal B_t}\). This is a bounded, integrable
portion of the trace, so Lemma~\ref{br:positive-histories} applies.
Letting \(m\to\infty\) and using monotone convergence gives
\eqref{br:mass-growth}.
\end{proof}

\subsection{Access to a late box and the growth contradiction}

We have obtained a uniform multiplication factor for mass already in a
moving box. The remaining point is to put some positive mass there.
The following argument needs no lower bound for the size of that mass:
any positive amount suffices for the eventual growth contradiction.

\begin{lemma}\label{br:accessibility}
For the fixed parameters above, let \(g\not\equiv0\) solve
\eqref{eq:limit-equation} on \(0<t<\infty\) with
\(0\le g\le(t+l)H\), for any allowed mesh and \(l\in[0,1]\).
Then \(\int_{\mathcal B_t}g(t,x,v)\dd x\dd v>0\) at arbitrarily large
finite times.
\end{lemma}

\begin{proof}
At some positive time \(a_0\), a bounded, integrable portion of
\(g(a_0)\) has positive mass. Such a portion is obtained by restricting
to bounded phase-space sets and truncating the density. The velocity axis has Lebesgue measure zero, so a
further portion of positive mass satisfies
\[
|x|\le R,\qquad |v|\le R,\qquad |\bar v|\ge\delta>0.
\]
For a sufficiently long but finite free flight of length \(s\),
\[
|y+s\bar v|\ge s\delta-R>l+2
\]
on this portion. Its survival is bounded below by a positive constant
by \eqref{br:global-rate}. Lemma~\ref{br:positive-histories} gives
positive mass in a bounded region strictly outside the hole at
time \(a_1=a_0+s\). This freely transported portion remains bounded
and integrable.

Choose \(h>0\) small so that paths whose initial speed is at most
\(R\) and whose later speeds are at most 3 remain in a bounded region
with \(|y|>l+1\) during \([a_1,a_1+h]\). Both sign densities have
strictly positive lower bounds there. To see this directly, the time,
axial coordinate, and transverse radius are bounded, the transverse
radius has a positive lower bound, and
\(e^{-\eta}|y|\le\varrho\le|y|\). The argument of \(S\) is therefore
bounded, its numerator is positive, and the denominator in
\eqref{eq:background} is bounded above. The rates on the selected
bounded-speed paths also have finite upper bounds.

Confine three collisions to this short interval. Choose a reference
sign with \(|v-\sigma e_3|\ge1\) for the first collision. The actual
relative speed is at least \(1-\alpha\ge3/4\).
Choose its output in a sufficiently small cap about the actual
partner. The cap probability has a positive lower bound for inputs
bounded by \(R\). Since the actual partner is within \(\alpha\) of
\(\sigma e_3\), this first output lies in the reserved first-output
neighborhood. Next use the signs \(-\sigma\) and \(+1\), and the
compact angular tests defining \(\alpha_{\mathrm{acc}}\). Their final
velocity lies in \eqref{br:access-velocity}. The second output stays
near \(e_1\), so the second and third relative speeds have positive
lower bounds; all postcollision speeds are at most 3.
Put the three times in disjoint subintervals and prohibit other
collisions. Positive lower bounds for the selected gain factors,
angular probabilities, time-interval lengths, and survival give
strictly positive total weight. The sphere-map comparison used here
requires only uniform partner closeness, and therefore remains valid
for measurable coefficients.

There is consequently a positive portion at time \(a_2=a_1+h\), with
bounded spatial support and \(|v-\varkappa e_1|<d\varkappa/4\).
Truncating its density gives a bounded, integrable subportion of
positive mass.
Let \(R_2\) bound \(|x-\varkappa a_2e_1|\) on that portion.
After free flight to any sufficiently large finite time \(t\),
\[
|x+(t-a_2)v-\varkappa te_1|
\le R_2+\frac{d\varkappa}{4}(t-a_2)<d\varkappa t.
\]
Its speed is strictly less than \(D\). For each finite \(t\), survival
remains strictly positive by \eqref{br:global-rate}. The zero-collision
term of Lemma~\ref{br:positive-histories} proves the assertion.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:finite-dilated-horizon}]
Suppose that a nonzero density \(g\) solves
\eqref{eq:limit-equation} for all positive times and obeys the cap.
Lemma~\ref{br:accessibility} gives a time \(t_0\ge t_{\mathrm{br}}\)
with positive mass \(m_0\) in \(\mathcal B_{t_0}\).

Apply Lemma~\ref{br:block-growth} successively.
At \(t_n=(1+a_{\rm br})^nt_0\), induction gives
\[
\int_{\mathcal B_{t_n}}g(t_n,x,v)\dd x\dd v\ge3^n m_0.
\]
The cap and \(l\le1\), on the other hand, give
\[
\int_{\mathcal B_{t_n}}g(t_n,x,v)\dd x\dd v
\le (t_n+1)\frac{4\pi}{3}(d\varkappa t_n)^3\int_E H(v)\dd v
\le C(1+t_n)^4.
\]
This contradicts \eqref{br:growth-exponent} as \(n\to\infty\).
The positive mass \(m_0\) and starting time may depend on the particular
density; their sizes are immaterial to this comparison. The parameters
governing the blocks were fixed before the mesh and \(l\) were chosen.
This rules out every nonzero global density in the proposition.
Applying this conclusion to the nonzero limits of
Proposition~\ref{prop:nonzero-limit} gives \(U<\infty\).
\end{proof}
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//11-branching.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//12-separation.tex ====
\section{Separation of two global solutions}\label{sec:separation}

The preceding sections give two different ways to take limits. At a
fixed physical horizon, the finite solutions have global entropy limits
with local conservation. At a shrinking horizon, magnification at a
cap event produces a nonzero linear solution, and the branching argument
bounds its time interval. We now show that these facts force a cap limit
to differ from a dormant limit. The separation argument follows
\cite[Section~10]{OpenAIPeriodic}; the normalization here also removes
the factor \(m_T=T^p\).

All construction parameters are henceforth fixed, with the choices in
Proposition~\ref{prop:finite-dilated-horizon} and the subsequent smallness
requirements of the global estimates. Choose one dormant global limit
\(F^{\mathrm d}\) supplied by Proposition~\ref{prop:global-limits}.

\begin{lemma}\label{sep:dormant-vanishing}
On the periodic lift centered at the origin,
\[
 \frac{F^{\mathrm d}(Tt,Tx,v)}{T m_T}\longrightarrow0
 \qquad(T\downarrow0)
\]
in local \(L^1\) on
\[
 \mathcal V=(0,1)\times\R^3_x\times
                 \{v\in\R^3:\bar v\ne0\}.
\]
\end{lemma}

\begin{proof}
The early upper bounds for the dormant approximations are independent
of the approximation index. Their sum is locally integrable, so testing
against nonnegative functions passes these bounds to \(F^{\mathrm d}\).
We estimate them on a compact set on which
\[
 0<\tau_0\le t\le A<1,\qquad |x|+|v|\le R,
                  \qquad |\bar v|\ge\varepsilon>0.
\]
By \eqref{early:dormant-bound}, the hot upper bound is
\(C(Tt)^M H(v)\), for a fixed \(M>p+1\). Division by
\(T m_T=T^{p+1}\) makes it tend to zero uniformly relative to
\(H\) on this compact set.

The bath bound \eqref{early:bath-bound} is exponentially small here.
Indeed \(Tx\) lies inside the fixed initial hole with
\(D_0(Tx)\ge\delta\) for all sufficiently small \(T\). Hence its
normalized contribution is bounded by
\[
 C T^{-p-1}e^{-\beta'|v|^2}
       \exp\left(-\frac{c\delta^2}{TA}\right),
\]
which tends to zero.

For the cold upper bound, a main spatial enlargement at transverse
position \(Ty\), with \(y\) bounded, has scale \(s\le C_R T\).
For small \(T\), only the copy centered at the origin can be main
on this compact set: any other such copy would have a lattice center
with transverse distance at most \(s_0+o(1)\) from the origin and
axial distance at most \(a+b_{\rm sp}+o(1)<1/2\), contradicting
the unit minimum length of a nonzero lattice vector. The other copies
are covered by the nonmain bound.

The transverse Gaussian in the main-color bound of
Proposition~\ref{cold:control}, together with \(|\bar v|\ge\varepsilon\),
gives
\[
 c_{\rm main}(Tt,Tx,v)
 \le C s^{-31}
       \exp\left(-\frac{c\varepsilon^2}{d_0^2s^{20}}\right).
\]
The supremum of the right side over \(0<s\le C_R T\) tends to
zero faster than every power of \(T\). At each point there is at
most one main color. Lemma~\ref{cold:nonmain} bounds the remaining
sum by \(C_M(Tt)^M e^{-4|v|^2}\) for every fixed \(M>0\).
Taking \(M>p+1\) handles the same normalization for this sum.
All three components therefore vanish in local \(L^1\), as claimed.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Take the datum \(F_0\) defined in \eqref{con:datum-equation}.
Lemma~\ref{con:datum} gives its finite mass, energy, and absolute
entropy integral, as well as bounded velocity support. For every
\(0<T\le T_0\), Proposition~\ref{early:cap-selection} gives cap
approximations, and Proposition~\ref{prop:global-limits} gives a global
entropy limit satisfying all the required local conservation laws and
having datum \(F_0\). The dormant limit \(F^{\mathrm d}\) has these
same properties.

Suppose there are horizons \(T_k\downarrow0\) at which a cap limit
equals \(F^{\mathrm d}\) almost everywhere on \((0,T_k)\).
For each fixed \(T_k\), choose approximations along the subsequence
converging to that particular common limit. Let \(K_j\) be a compact
exhaustion of \(\mathcal V\), and choose nonnegative smooth compact
functions \(\zeta_j\) on \(\mathcal V\) with
\(\zeta_j\ge1\) on \(K_j\). A diagonal choice of indices \(N_k\)
can ensure both \(s_{N_k}/T_k\to0\) and
\begin{equation}\label{sep:diagonal}
 f_k(t,x,v):=\frac{F_k(T_kt,T_kx,v)}{T_km_{T_k}}
       \longrightarrow0
       \quad\hbox{in local }L^1(\mathcal V),
\end{equation}
where \(F_k\) is the chosen finite solution.

Here is the diagonal choice explicitly. At stage \(k\), require
\(s_{N_k}/T_k<1/k\) and make the differences between the integrals
of the normalized approximation and normalized common limit against
\(\zeta_1,\ldots,\zeta_k\) smaller than \(1/k\). Each test, when
written in physical coordinates at the fixed horizon \(T_k\), is
a fixed smooth test of finite norm; periodizing its spatial factor
gives a test on the torus. Thus fixed-horizon weak convergence permits
these finitely many requirements. Lemma~\ref{sep:dormant-vanishing}
makes the integrals of the common limit tend to zero. Since the
approximations are nonnegative and \(\zeta_j\ge1\) on \(K_j\),
their integrals over \(K_j\) tend to zero. This proves
\eqref{sep:diagonal} without a convergence rate uniform in \(T_k\).

Choose a cap time \(t_{*,k}\) of each selected solution and set
\(r_k=t_{*,k}+s_{N_k}\). Proposition~\ref{prop:blowup-limit} gives,
after subselection,
\[
 \frac{s_{N_k}}{r_k}\to l,\qquad
 \frac{T_k}{r_k}\to U\in[1,\infty],\qquad
 \frac{h_k(r_kt,r_kx,v)}{m_{T_k}r_k}\rightharpoonup g
\]
locally in spacetime on \(0<t<U\). By
Proposition~\ref{prop:nonzero-limit}, \(g\not\equiv0\); by
Proposition~\ref{prop:finite-dilated-horizon}, \(U<\infty\).
Consequently
\[
 \rho_k=\frac{r_k}{T_k}\longrightarrow\rho=\frac1U>0.
\]
This positive lower bound permits comparison of the two dilations.
Since \(F_k=c_k+u_k+h_k\ge h_k\) through its cap horizon,
\begin{equation}\label{sep:scale-comparison}
 0\le\frac{h_k(r_kt,r_kx,v)}{m_{T_k}r_k}
       \le\rho_k^{-1}f_k(\rho_kt,\rho_kx,v).
\end{equation}

Fix a compact subset \(K\) of
\((0,U)\times\R^3_x\times\{\bar v\ne0\}\).
For all sufficiently large \(k\), the images of \(K\) under
\((t,x,v)\mapsto(\rho_kt,\rho_kx,v)\) lie in one fixed compact
subset \(K'\) of \(\mathcal V\). The four-dimensional
space--time Jacobian is \(\rho_k^4\), so
\eqref{sep:scale-comparison} yields
\[
 \int_K\frac{h_k(r_kt,r_kx,v)}{m_{T_k}r_k}\dd t\dd x\dd v
 \le\rho_k^{-5}\int_{K'}f_k(t,x,v)\dd t\dd x\dd v
 \longrightarrow0.
\]
Thus \(g=0\) off the velocity axis. The cap
\(0\le g\le(t+l)H\) makes \(g\) a density, and the axis has
Lebesgue measure zero. Hence \(g=0\) almost everywhere, contradicting
Proposition~\ref{prop:nonzero-limit}.

It follows that for all sufficiently small horizons no cap limit can
equal the fixed dormant limit on the corresponding early interval.
Choose one such cap limit. Both it and \(F^{\mathrm d}\) are global
entropy solutions with the same datum and all local conservation laws,
and they differ on a set of positive spacetime--velocity measure.
Rotating back from the lattice \(\Lambda\) to \(\mathbb Z^3\)
preserves these properties and gives the assertion on \(\mathbb T^3\).
\end{proof}
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//12-separation.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//jet-geometry.tex ====
\begin{figure}[tbp]
\centering
\resizebox{0.98\linewidth}{!}{%
\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\small,
  jetplus/.style={fill=blue!55!black,draw=none},
  jetminus/.style={fill=orange!75!black,draw=none},
  guide/.style={draw=black!40,line width=.35pt}]
  \path[use as bounding box] (0,0) rectangle (13.7,9.45);

  \node[anchor=west,font=\small\bfseries] at (0.15,9.15)
    {(a) Transverse annuli and stripes};
  \begin{scope}[shift={(3.1,6.65)}]
    \foreach \inner/\outer/\period in
      {1.025/1.475/.36,.678/0.989/.16,.449/0.661/.0711}{
      \begin{scope}[even odd rule]
        \clip (0,0) circle[radius=\outer]
          (0,0) circle[radius=\inner];
        \foreach \k in {-23,...,23}{
          \path[jetplus] ({\k*\period},-1.6)
            rectangle ({(\k+.25)*\period},1.6);
          \path[jetminus] ({(\k+.5)*\period},-1.6)
            rectangle ({(\k+.75)*\period},1.6);
        }
      \end{scope}
    }
    \foreach \radius in {1.5,1,.6667,.4444}
      \draw[guide] (0,0) circle[radius=\radius];
    \draw[->,black!65] (-1.85,0) -- (1.98,0) node[right] {$y_1$};
    \draw[->,black!65] (0,-1.8) -- (0,1.82) node[above] {$y_2$};
    \fill (0,0) circle[radius=.025];
    \node[anchor=east,fill=white,inner sep=1pt] at (-.06,-.15) {$0$};
    \draw[->,thin] (1.58,1.36) -- (1.13,.985);
    \node[anchor=west] at (1.53,1.48) {$e^\eta s_j$};
    \draw[->,thin] (1.78,.78) -- (.84,.54);
    \node[anchor=west] at (1.79,.83) {$s_j$};
  \end{scope}
  \node[align=center] at (3.1,4.63)
    {Geometric annuli; stripe period $p_j=s_j^2$.};

  \node[anchor=west,font=\small\bfseries] at (.15,3.76)
    {(b) A stripe period, magnified};
  \node[anchor=west] at (.25,3.28) {$y_1/p_j\pmod 1$};
  \path[jetplus] (.65,2.35) rectangle (1.9,2.92);
  \path[jetminus] (3.15,2.35) rectangle (4.4,2.92);
  \draw[black!60] (.65,2.35) rectangle (5.65,2.92);
  \foreach \xx in {1.9,3.15,4.4}
    \draw[black!45] (\xx,2.35)--(\xx,2.92);
  \node[text=white] at (1.275,2.635) {$+$};
  \node[text=white] at (3.775,2.635) {$-$};
  \node at (2.525,2.635) {empty};
  \node at (5.025,2.635) {empty};
  \foreach \xx/\lab in {.65/0,1.9/{\tfrac14},3.15/{\tfrac12},4.4/{\tfrac34},5.65/1}
    \node[below] at (\xx,2.30) {$\lab$};
  \node[align=center] at (3.15,1.12)
    {Each sign occupies one quarter\\
     of every stripe period.};

  \node[anchor=west,font=\small\bfseries] at (7.05,9.15)
    {(c) Jets inside the bath hole};
  \path[fill=black!9] (7.1,5.05) rectangle (13.55,8.15);
  \path[fill=white,draw=black!55]
    (8.8,5.90) -- (11.8,5.90)
    arc[start angle=-90,end angle=90,radius=.70]
    -- (8.8,7.30)
    arc[start angle=90,end angle=270,radius=.70] -- cycle;
  \draw[->,black!50,dashed] (7.4,6.60) -- (13.25,6.60)
    node[above] {$z$};
  \draw[black!45,line width=1pt] (8.8,6.60)--(11.8,6.60);
  \foreach \xx/\lab in {8.8/{-\tfrac13},10.3/0,11.8/{\tfrac13}}
    {\draw[black!60] (\xx,6.51)--(\xx,6.69);
     \node[below,fill=white,inner sep=1pt] at (\xx,6.49) {$\lab$};}
  \path[jetplus] (9.7375,6.85) rectangle (10.8625,6.94);
  \path[jetminus] (9.7375,6.98) rectangle (10.8625,7.07);
  \draw[<->,thin] (9.7375,7.20)--(10.8625,7.20)
    node[midway,above,inner sep=1pt] {$|z|<a$};
  \draw[->,thin] (12.65,7.58)--(12.31,7.02);
  \node[anchor=east] at (13.40,7.81) {$D_{\rm hole}$};
  \node[anchor=west,align=left] at (7.3,5.45)
    {Initial bath on $D_{\rm hole}^{\,c}$};
  \node[align=center] at (10.3,4.63)
    {Schematic axial view; $a=1/8$.};

  \node[anchor=west,font=\small\bfseries] at (7.05,3.76)
    {(d) Axial velocity centers};
  \draw[->,black!55] (7.7,1.87)--(13.30,1.87) node[right] {$z$};
  \draw[->,black!55] (10.3,.30)--(10.3,3.23) node[above] {$v_3$};
  \draw[dotted,black!45] (7.9,2.55)--(12.7,2.55);
  \draw[dotted,black!45] (7.9,1.19)--(12.7,1.19);
  \node[anchor=east,fill=white,inner sep=1pt] at (10.22,2.55) {$1$};
  \node[anchor=east,fill=white,inner sep=1pt] at (10.22,1.19) {$-1$};
  \draw[blue!55!black,line width=1.1pt,domain=-2.2:2.2,samples=70]
    plot ({10.3+\x},{2.55+.23*tanh(1.35*\x)});
  \draw[orange!75!black,line width=1.1pt,domain=-2.2:2.2,samples=70]
    plot ({10.3+\x},{1.19+.23*tanh(1.35*\x)});
  \node[anchor=west,text=blue!55!black,fill=white,inner sep=1pt]
    at (12.55,2.78) {$a_s^+$};
  \node[anchor=west,text=orange!75!black,fill=white,inner sep=1pt]
    at (12.55,1.40) {$a_s^-$};
  \node[align=center] at (10.3,.04)
    {$a_s^\sigma(z)=\sigma+\alpha\tanh(\gamma z/s)$};
\end{tikzpicture}%
}
\caption{The initial jets and bath near one lattice point.
The annular picture is schematic: stripe widths and radial gaps are enlarged
for visibility, and only three annuli are drawn.  Panel (b) records the exact
stripe fractions, which later give a coefficient $1/4$ for each sign after
averaging.  Panel (c) places the jets schematically inside the bath hole,
the tube of radius $2\delta$ about the axial segment $[-1/3,1/3]$;
the vertical placement of the two signs is only illustrative.
Panel (d) shows the centers of the compact velocity profiles of width $d_s$.
All spatial panels are local views in the rotated lift; they do not depict
a closed axis on the torus.}
\label{fig:jet-geometry}
\end{figure}
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//jet-geometry.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Nonuniqueness with local conservation\\
for the hard-sphere Boltzmann equation}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove nonuniqueness for the three-dimensional periodic hard-sphere
Boltzmann equation among global entropy solutions satisfying exact local
conservation of mass, momentum, and kinetic energy. We construct one
nonnegative initial density with bounded velocity support and finite mass,
energy, and absolute entropy that gives rise to two distinct such solutions.
Both are strongly continuous in \(L^1\), and their collision gain and loss
terms are integrable with every polynomial velocity weight on every bounded
time interval.
\end{abstract}
\tableofcontents
\input{paper/01-introduction}
\input{paper/02-framework}
\input{paper/03-kernel}
\input{paper/04-construction}
\input{paper/05-cold}
\input{paper/06-early}
\input{paper/07-global}
\input{paper/08-compactness}
\input{paper/09-dilation}
\input{paper/10-nonconcentration}
\input{paper/11-branching}
\input{paper/12-separation}
\bibliographystyle{plain}
\bibliography{paper/references}
\end{document}
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//main.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//preamble.tex ====
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=1.1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{microtype}
\usepackage{enumitem,booktabs}
\usepackage{xcolor,graphicx,flafter}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning,calc,patterns,decorations.pathreplacing}
\PassOptionsToPackage{hyphens}{url}
\usepackage{hyperref}
\usepackage[capitalise,nameinlink,noabbrev]{cleveref}
\hypersetup{colorlinks=true,linkcolor=blue!45!black,
 citecolor=green!35!black,urlcolor=blue!45!black,
 pdftitle={Nonuniqueness with local conservation for the hard-sphere Boltzmann equation},
 pdfauthor={OpenAI}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
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\setlist[itemize]{itemsep=3pt,topsep=5pt}
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  \originalthebibliography{#1}%
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==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//preamble.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//scales.tex ====
\begin{figure}[!htbp]
\centering
\resizebox{0.98\linewidth}{!}{%
\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\small,
  timeaxis/.style={->,black!65,line width=.55pt},
  hittick/.style={blue!55!black,line width=1.1pt},
  horizontick/.style={orange!75!black,line width=1.1pt}]
  \path[use as bounding box] (0,-.32) rectangle (13.7,5.97);
  \node[anchor=west,align=left] at (0,5.32)
    {Physical time\\$t$};
  \node[anchor=west,align=left] at (0,3.33)
    {Hit dilation\\$t/r_N$};
  \node[anchor=west,align=left] at (0,1.33)
    {Horizon dilation\\$t/T_N$};

  \draw[black!20,dashed] (2.45,.99)--(2.45,5.37);
  \draw[blue!35!black,dashed] (6.70,.99)--(6.70,5.37);
  \draw[orange!55!black,dashed] (12.05,.99)--(12.05,5.37);
  \foreach \yy in {5.30,3.30,1.30}{
    \draw[timeaxis] (2.20,\yy)--(13.15,\yy);
    \draw[black!60] (2.45,\yy-.10)--(2.45,\yy+.10);
    \draw[hittick] (6.70,\yy-.13)--(6.70,\yy+.13);
    \draw[horizontick] (12.05,\yy-.13)--(12.05,\yy+.13);
    \node[below,fill=white,inner sep=2pt] at (2.45,\yy-.14) {$0$};
  }
  \node[above,text=blue!55!black] at (6.70,5.51) {cap hit};
  \node[above,text=orange!75!black] at (12.05,5.51) {horizon};
  \node[below,fill=white,inner sep=2pt] at (6.70,5.12)
    {$t_{*,N}=r_N-s_N$};
  \node[below,fill=white,inner sep=2pt] at (12.05,5.12) {$T_N$};
  \node[below,fill=white,inner sep=2pt] at (6.70,3.12) {$1-l_N$};
  \node[below,fill=white,inner sep=2pt] at (12.05,3.12) {$U_N=T_N/r_N$};
  \node[below,fill=white,inner sep=2pt] at (6.70,1.12)
    {$\dfrac{1-l_N}{U_N}$};
  \node[below,fill=white,inner sep=2pt] at (12.05,1.12) {$1$};

  \node[align=center,fill=white,inner sep=3pt] at (7.25,4.25)
    {$r_N=t_{*,N}+s_N,\qquad l_N=s_N/r_N$};
  \node[align=center,fill=white,inner sep=3pt] at (7.25,2.24)
    {$t/T_N=(t/r_N)/U_N,\qquad x/T_N=(x/r_N)/U_N$};
  \node[align=center] at (7.05,-.12)
    {$U_N\longrightarrow U<\infty
      \quad\Longrightarrow\quad r_N/T_N\longrightarrow 1/U>0$};
\end{tikzpicture}%
}
\caption{The same hit and horizon in the two dilations.
The rows align physical events and are schematic; the hit may occur at either
endpoint, and no ordering of $r_N$ and $T_N$ is assumed.
Sections~\ref{sec:dilation}--\ref{sec:branching} will show that the hit
dilation has a nonzero weak limit on $0<t<U$ and that $U<\infty$.
The latter conclusion makes the change to the horizon dilation nondegenerate:
compact subsets of $0<t<U$ map inside $0<t<1$.
Thus vanishing in the horizon dilation would force vanishing in the hit
dilation, which is the contradiction used to separate the solutions.}
\label{fig:scales}
\end{figure}
==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//scales.tex ====

==== BEGIN Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//references.bib ====

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==== END Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation//references.bib ====