% =====================================================================
%  HARD-SPHERE FLUCTUATIONS ON THE REGULAR BOLTZMANN LIFESPAN
%  A complete, self-contained proof.
%
%  How to read this file:
%    * Part 0  -- the story in words, no formulas.
%    * Part 1  -- the exact objects and the theorem.
%    * Part 2  -- the starting configuration and the Gaussian target.
%    * Part 3  -- the microscopic engine (layered histories, DHM).
%    * Part 4  -- the expansion into connected diagrams.
%    * Part 5  -- the sharp size of every correlation.
%    * Part 6  -- the Gaussian limit for the model dynamics.
%    * Part 7  -- why the model dynamics is close to the real one.
%    * Part 8  -- putting the exact centering back in.
%
%  The only results NOT proved here are quoted from Deng-Hani-Ma,
%  "Long time derivation of the Boltzmann equation from hard sphere
%  dynamics" (arXiv:2408.07818v3, 18 July 2025), cited as [DHM]. They
%  form one black box, Theorem 4.3 below. Everything else is proved.
% =====================================================================
\documentclass[11pt]{amsart}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage[margin=1in]{geometry}
\usepackage{lmodern}
\usepackage[T1]{fontenc}
\usepackage{microtype}
\usepackage{enumitem}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\usepackage{xcolor}
\definecolor{linkblue}{RGB}{24,64,105}
\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref}
\usepackage[capitalise,nameinlink,noabbrev]{cleveref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}

\newcommand{\R}{\mathbb R}
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\E}{\mathbb E}
\renewcommand{\P}{\mathbb P}
\newcommand{\dd}{\mathrm d}
\newcommand{\eps}{\varepsilon}
\newcommand{\ind}{\mathbf 1}
\newcommand{\TV}{\mathrm{TV}}
\newcommand{\Bol}{\mathrm{Bol}}
\newcommand{\Id}{\mathrm{Id}}
\DeclareMathOperator{\supp}{supp}
\setlist[enumerate]{leftmargin=*,itemsep=3pt}
\setlist[itemize]{leftmargin=*,itemsep=3pt}

\title[Hard-sphere fluctuations]{Hard-sphere fluctuations on the regular Boltzmann lifespan}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
A dilute gas of perfectly elastic spheres, started from a grand-canonical
random configuration, is described on average by the Boltzmann equation.
We prove that the random \emph{wobble} of the gas around that average is
Gaussian, and that the wobble obeys the \emph{fluctuating Boltzmann
equation}. This holds on every finite time interval on which the
Boltzmann solution stays regular, with no assumption of equilibrium or
of small data. The empirical measure is centered by its exact
microscopic expectation, not by a Boltzmann approximation, and the proof
is complete except for one imported analytic package from Deng--Hani--Ma.
\end{abstract}

\tableofcontents

% =====================================================================
\part*{Part 0. The story in words}
% =====================================================================

\section{What we are proving, without formulas}\label{sec:story}

\textbf{A gas of billiard balls.} Imagine a huge box full of identical hard
spheres, like a three-dimensional break of pool. Spheres fly in straight
lines and bounce elastically when they touch. Diameter $\eps$ is tiny and
the box contains about $\mu=\eps^{-2}$ spheres per unit volume, so the
gas is dilute but the number of spheres is enormous.

\textbf{The average picture.} Track only the one-particle density, that is
the expected number of spheres per unit volume of position-velocity space.
In the dilute limit $\eps\downarrow0$ this density solves the
\textbf{Boltzmann equation}. That is the classical kinetic limit of
Lanford.

\textbf{The wobble.} The real gas is random. At any time its empirical
density is the Boltzmann density \emph{plus a small random error}. We
scale that error by $\sqrt\mu$, its natural size, and call the result the
\textbf{fluctuation field} $\zeta^\eps_t$. We ask: as $\eps\downarrow0$,
what is the law of $\zeta^\eps$?

\textbf{The answer, in one breath.} The scaled error is asymptotically
\textbf{Gaussian}, and its covariance evolves by the
\textbf{fluctuating Boltzmann equation}: a deterministic linear drift that
is the linearized Boltzmann operator, plus a Gaussian noise whose
strength at each collision is the collision rate. Equivalently, the field
solves
\[
 \dd\zeta_t(\phi)=\zeta_t(A_t\phi)\,\dd t+\dd W_t(\phi),
\]
where $A_t$ is the linearized Boltzmann generator and $W$ is a Gaussian
noise with covariance $C_t$ written down in \Cref{sec:objects}. This is the
finite-dimensional, multi-time central limit theorem for the gas.

\textbf{Why it is not obvious.} The spheres interact only by deterministic
collisions. There is no random force, no thermostat, no resampling. The
Gaussian randomness in the limit is a \emph{genuine emergent phenomenon}:
it comes from the chaotic microscopic collision history, and the central
limit theorem has to extract it from a purely deterministic system. The
randomness is created by (i) the random initial configuration and (ii) the
sensitive dependence of collision outcomes on which pair happens to be
close. Both must be tracked with great care.

\textbf{What was known.} Short times: Lanford, then van Beijeren--Lanford--
Lebowitz--Spohn and Spohn for the covariance, then Pulvirenti--Simonella
and Bodineau--Gallagher--Saint-Raymond--Simonella for quantitative short-
time fluctuation theory. Equilibrium long time: BGSS. The spatial setting
there is a periodic box. Deng--Hani--Ma pushed the \emph{averaged} kinetic
limit to the full regular Boltzmann lifespan on the whole space. Our job
is to push the \emph{fluctuation} theorem to that same lifespan, whole
space, no equilibrium.

\textbf{The two genuinely new difficulties.} First, correlation
\emph{size}. The $h$-th factorial cumulant (the connected $h$-point
correlation) must be controlled at the sharp scale $\mu^{1-h}$, which is
much smaller than any crude bound gives; we extract it by a change of
\emph{relative translations} of collision clusters, harvesting one
$\eps^2$ for every collision that merges two clusters. Second,
\emph{exact centering}. The gas is centered by its true microscopic mean,
not by the Boltzmann mean. The two differ at order $\eps$ (already at
time zero, because hard spheres cannot overlap), and at the fluctuation
scale $\sqrt\mu=\eps^{-1}$ that difference matters. So the approximating
dynamics must agree with the real one with probability better than
$O(\eps)$, which is what the cutoff estimate of \Cref{sec:cutoff} buys.

\textbf{Roadmap.} \Cref{sec:objects} pins the objects and states the
theorem. \Cref{sec:initial-target} does the initial ensemble and builds
the Gaussian target. \Cref{sec:dhm-inputs} sets up the microscopic
machinery and states the imported [DHM] package. \Cref{sec:cluster-expansion}
expands the cumulants into connected diagrams that keep
observation records. \Cref{sec:sharp-asymptotics} proves the sharp
$\mu^{1-h}$ bounds and identifies the limiting trees.
\Cref{sec:gaussian-limit} runs the Gaussian central limit theorem for the
model dynamics. \Cref{sec:cutoff} compares model to reality.
\Cref{sec:transfer} restores exact centering and finishes.

% =====================================================================
\part*{Part 1. The exact objects and the theorem}
% =====================================================================

\section{The kinetic equation, the gas, and the fluctuation field}\label{sec:objects}

\subsection{The Boltzmann equation and its norm}

Phase space is $\R_x^3\times\R_v^3$ and $z=(x,v)$. For a rate $\beta>0$
define the weighted sup norm
\begin{equation}\label{eq:bol-norm}
 \|g\|_{\Bol,\beta}
 :=\sum_{a\in\Z^3}\ \sup_{\substack{|x-a|\le1\\v\in\R^3}}
   e^{\beta|v|^2}\,|g(x,v)| .
\end{equation}
It is a supremum with a Gaussian velocity weight, summed over unit spatial
cells so that it also controls the total mass. Hard-sphere collisions use
the kernel and scattering
\begin{align}
 B(v-v_*,\omega)&=[(v-v_*)\cdot\omega]_+, &&\omega\in\mathbb S^2,
   \label{eq:kernel}\\
 v'&=v-[(v-v_*)\cdot\omega]\omega, &
 v_*'&=v_*+[(v-v_*)\cdot\omega]\omega, \label{eq:scatter}
\end{align}
with $[a]_+=\max\{a,0\}$ and $\dd\omega$ the unnormalized surface measure.
The Boltzmann equation reads
\begin{equation}\label{eq:boltzmann}
 (\partial_t+v\cdot\nabla_x)f=Q(f,f),\quad
 Q(f,f)=\int_{\R^3\times\mathbb S^2}B(f'f_*'-ff_*)\,\dd v_*\,\dd\omega,
\end{equation}
all densities evaluated at the same $x$.

\subsection{The microscopic gas}

Fix the sphere diameter $\eps>0$ and set the mean density
$\mu=\eps^{-2}$. The grand-canonical initial law, including $N=0$, is
\begin{equation}\label{eq:grand-canonical}
 \P_\eps(N,\dd Z_N)
 =\mathcal Z_\eps^{-1}\frac{\mu^N}{N!}\,
   \ind_{\mathcal D_N^\eps}(Z_N)\prod_{i=1}^N f_0(z_i)\,\dd z_i,
\end{equation}
with the non-overlap set
$\mathcal D_N^\eps=\{(z_i):|x_i-x_j|>\eps,\ i\ne j\}$ and
$1\le\mathcal Z_\eps\le e^\mu$. Given the configuration, spheres move by
free flight and deterministic elastic collisions, defined almost surely
by [DHM, Proposition 1.2]. There is no random collision rule and no
resampling: the only randomness is the initial configuration.

\begin{assumption}\label{ass:kinetic}
$f_0\ge0$ is a smooth probability density with
$\|f_0\|_{\Bol,2\beta}+\|\nabla_xf_0\|_{\Bol,2\beta}<\infty$ for some
$\beta>0$. There is a nonnegative classical solution $f$ of
\eqref{eq:boltzmann} on a prescribed interval $[0,T]$ with
\begin{equation}\label{eq:kinetic-gaussian}
 M:=\sup_{0\le t\le T}\sup_{x,v}e^{2\beta|v|^2}f(t,x,v)<\infty .
\end{equation}
\end{assumption}
Throughout, $[0,T]$ is \emph{one fixed regular kinetic interval}; nothing
below asserts that $T$ can be taken infinite.

\subsection{The fluctuation field and the limiting law}

For real $\phi\in C_c^\infty(\R^6)$ put
\begin{equation}\label{eq:empirical}
 \pi_t^\eps(\phi)=\mu^{-1}\sum_{i=1}^N\phi(z_i(t)),
 \qquad
 \zeta_t^\eps(\phi)=\sqrt\mu\,\bigl[\pi_t^\eps(\phi)-\E_\eps\pi_t^\eps(\phi)\bigr].
\end{equation}
The expectation is the exact microscopic one. This is essential; see
\Cref{rem:centering-shift}.

Define the pair difference and the two limiting operators
\begin{align}
 \Delta\phi&=\phi(x,v')+\phi(x,v_*')-\phi(x,v)-\phi(x,v_*),\\
 (A_t\phi)(x,v)
 &=v\cdot\nabla_x\phi
   +\int_{\R^3\times\mathbb S^2}B(v-v_*,\omega)f_t(x,v_*)\Delta\phi
   \,\dd v_*\,\dd\omega,\label{eq:drift}\\
 C_t(\phi,\psi)
 &=\tfrac12\int_{\R^3\times\R^3\times\R^3\times\mathbb S^2}
   B(v-v_*,\omega)f_t(x,v)f_t(x,v_*)\Delta\phi\,\Delta\psi
   \,\dd x\,\dd v\,\dd v_*\,\dd\omega.\label{eq:noise}
\end{align}
$A_t$ is the linearized Boltzmann generator; the collision part carries
no extra factor $1/2$ because of the pre/post substitution together with
$\omega\mapsto-\omega$ (\Cref{sec:collision-normal}). The quadratic form
$C_t$ is the collision noise, nonnegative because $f_t\ge0$.

\begin{theorem}[Finite-dimensional fluctuation limit]\label{thm:main}
Under \Cref{ass:kinetic}, the centered Gaussian field $\zeta$ solving the
weak fluctuating Boltzmann equation
\begin{equation}\label{eq:fluc}
 \dd\zeta_t(\phi)=\zeta_t(A_t\phi)\,\dd t+\dd W_t(\phi)
\end{equation}
in backward-mild form is well defined on test scales of small Gaussian
growth. Its initial field is independent of $W$, and
\begin{align}
 \E[\zeta_0(\phi)\zeta_0(\psi)]&=\int_{\R^6}f_0\,\phi\psi\,\dd z,
   \label{eq:init-cov}\\
 \E[W_t(\phi)W_s(\psi)]&=\int_0^{t\wedge s}C_r(\phi,\psi)\,\dd r,
   \label{eq:noise-cov}
\end{align}
with $W$ of independent increments. For every $m\ge1$, all $t_1,\dots,t_m
\in[0,T]$, and real $\phi_1,\dots,\phi_m\in C_c^\infty(\R^6)$,
\begin{equation}\label{eq:main-convergence}
 \bigl(\zeta_{t_1}^\eps(\phi_1),\dots,\zeta_{t_m}^\eps(\phi_m)\bigr)
 \ \Longrightarrow\
 \bigl(\zeta_{t_1}(\phi_1),\dots,\zeta_{t_m}(\phi_m)\bigr)
 \qquad(\eps\downarrow0).
\end{equation}
\end{theorem}

\begin{remark}\label{rem:centering-shift}
The initial covariance \eqref{eq:init-cov} has no subtraction of
$(\int f_0\phi)(\int f_0\psi)$: the particle number is random. More
importantly, microscopic centering is not the same as centering by the
Boltzmann mean. With $\varrho_0(x)=\int f_0(x,v)\dd v$,
\[
 \lim_{\eps\to0}\sqrt\mu\Bigl(\E\pi_0^\eps(\phi)-\int f_0\phi\Bigr)
 =-\frac{4\pi}{3}\int f_0(x,v)\phi(x,v)\varrho_0(x)\,\dd x\,\dd v .
\]
So exclusion shifts the mean by order $\eps$, which survives the factor
$\sqrt\mu=\eps^{-1}$. Centering by the Boltzmann density alone would
leave a nonzero deterministic drift already at time zero. This is why
the proof must compare exact expectations, not merely laws.
\end{remark}

\subsection{Proof architecture and the two new estimates}

We keep the layered collision-history framework of [DHM], stated as the
black box \Cref{thm:dhm-package}. Its role is to remove \emph{large
complexity}: histories with many entangled collisions. What remains must
be controlled at the fluctuation scale, and that is where the new work
lies.

\begin{enumerate}
\item \textbf{Initial ensemble and Gaussian target}
  (\Cref{sec:initial-target}). Exact initial cumulants, the initial CLT,
  the shift \Cref{rem:centering-shift}, and construction of the Gaussian
  field of \Cref{thm:main}.
\item \textbf{Connected expansion with records}
  (\Cref{sec:cluster-expansion}). Factorial cumulants are written as a
  finite sum of globally connected diagrams, on a single pasted dynamics,
  keeping finitely many observation records. Large complexity is dropped
  by the [DHM] package.
\item \textbf{Sharp correlation bounds} (\Cref{sec:sharp-asymptotics},
  \Cref{thm:sharp-cumulants}). The $h$-th cumulant is $O(\mu^{1-h})$,
  with the leading $h=1,2$ terms identified as signed sums of trees
  weighted by $f_{i\tau}$, contact flux, and $\pm1$ signs. This is proved
  by chronological merger coordinates: each collision that merges two
  clusters contributes $\eps^2[(v-v_*)\cdot\omega]_+$, and summing all
  clusters costs $CT/\sqrt L<1$.
\item \textbf{Gaussian limit for the model dynamics}
  (\Cref{sec:gaussian-limit}). A mark that accumulates observations turns
  every fixed multi-time linear combination into one scalar whose
  ordinary cumulants are computed; second cumulant converges, higher ones
  vanish, giving the Gaussian law with the correct covariance.
\item \textbf{Accuracy of the model dynamics} (\Cref{sec:cutoff},
  \Cref{prop:cutoff}). The pasted and true dynamics agree with
  probability $O(\eps^{1+\eta})$, $\eta=1/20$. This is a positive
  geometric estimate on terminal witnesses, carried out with local
  gains, a common birth-energy bound, and a causal time-order sum free of
  per-vertex logarithms.
\item \textbf{Exact centering transfer} (\Cref{sec:transfer}). The
  discrepancy estimate, the fixed moments of $N/\mu$, and H\"older's
  inequality show that expectations agree to $o(\mu^{-1/2})$, which is
  what transfers the limit with exact centering.
\end{enumerate}

% =====================================================================
\part*{Part 2. The starting configuration and the Gaussian target}
% =====================================================================

\section{The initial ensemble and the Gaussian target}\label{sec:initial-target}

This part is independent of the dynamics. It does two things: prove the
fluctuation limit at time zero (a pure equilibrium statement about the
grand-canonical hard-sphere measure), and construct the Gaussian field
that \Cref{thm:main} claims to be the limit. It also records the initial
correlation bounds that later parts use.

\subsection{Initial factorial measures and connected graphs}

Let $F_{h,0}^{\eps}$ be the density of the ordered factorial measure
divided by $\mu^h$:
\begin{equation}\label{eq:init-factorial}
 \int F_{h,0}^{\eps}(Z_h)\Phi(Z_h)\,\dd Z_h
 =\mu^{-h}\E\sum_{i_1,\dots,i_h\text{ distinct}}\Phi(z_{i_1},\dots,z_{i_h}).
\end{equation}
Write $[h]=\{1,\dots,h\}$, $Z_B=(z_j)_{j\in B}$, and let $\mathcal P(V)$
be the set of set-partitions of $V$. The \textbf{factorial cumulant
(connected) densities} $c_{h,0}^{\eps}$ are defined by the cumulant
inversion of the factorial hierarchy,
\begin{equation}\label{eq:init-partition}
 F_{h,0}^{\eps}(Z_h)
 =\sum_{\pi\in\mathcal P([h])}\prod_{B\in\pi}c_{|B|,0}^{\eps}(Z_B).
\end{equation}
The one-point function is $g_0=c_{1,0}^{\eps}=F_{1,0}^{\eps}$. Set
\[
 \varrho_0(x)=\int f_0(x,v)\dd v,\quad
 b_\eps=\tfrac{4\pi}{3}\|\varrho_0\|_\infty\eps^3,\quad
 d_\eps=\mu b_\eps=O(\eps),
\]
and let $(N)_p=N\cdots(N-p+1)$ denote the falling factorial.

The key geometric fact, before we can even state the initial CLT, is that
connected correlations are \emph{order $\eps^3$ per connected pair}.
This is the static shadow of the dynamical statement that each merging
collision costs $\eps^2$; here excluded volume makes it $\eps^3$.

\begin{proposition}[Initial ensemble]\label{prop:initial}
For all sufficiently small $\eps$:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $0\le F_{h,0}^{\eps}\le\prod_{j=1}^hf_0(z_j)$ and
  $(1-C\eps)f_0\le g_0\le f_0$. Hence
  $\|g_0-f_0\|_{\Bol,2\beta}=O(\eps)$ and, for every fixed $p\ge1$,
  \begin{equation}\label{eq:init-N-moments}
   \E(N)_p\le\mu^p,\qquad \sup_{0<\eps\le\eps_0}\E(N/\mu)^p<\infty.
  \end{equation}
\item With $\upsilon=1/81$, for every $h\ge2$ and a constant $C$ independent
  of $h$,
  \begin{equation}\label{eq:init-coarse}
   |c_{h,0}^{\eps}(Z_h)|\le(Ch)^{Ch}\prod_{j=1}^hf_0(z_j)
     \sum_{\mathcal T}\prod_{\{i,j\}\in\mathcal T}
       \bigl(\ind_{|x_i-x_j|\le\eps}+\eps^\upsilon\bigr),
  \end{equation}
  the sum over trees $\mathcal T$ on $[h]$. For fixed $h\ge2$ and any
  $D>0$ there is $C_{D,h}$ with
  \begin{equation}\label{eq:init-sharp}
   |c_{h,0}^{\eps}(Z_h)|\le C_{D,h}\prod_{j=1}^hf_0(z_j)
     \sum_{\mathcal T}\prod_{\{i,j\}\in\mathcal T}
       \bigl(\ind_{|x_i-x_j|\le C_{D,h}\eps}+\eps^D\bigr),
  \end{equation}
  and
  \begin{equation}\label{eq:init-L1}
   \|c_{h,0}^{\eps}\|_{L^1}\le C_h\eps^{3(h-1)}\qquad(h\ge2).
  \end{equation}
\item For every finite list of bounded real tests, the exact-centered
  initial fluctuations converge jointly to a centered Gaussian vector with
  covariance $\int f_0\phi\psi$. For smooth compactly supported real
  $\phi$,
  \begin{equation}\label{eq:init-shift}
   \lim_{\eps\to0}\sqrt\mu\Bigl(\E\pi_0^\eps(\phi)-\int f_0\phi\Bigr)
   =-\frac{4\pi}{3}\int_{\R^6}f_0\phi\,\varrho_0\,\dd z.
  \end{equation}
\end{enumerate}
\end{proposition}

\begin{proof}
\textbf{(i) Factorial bound and moments.} For fixed $Z_h$, let
$\mathcal Z_\eps[Z_h]$ be the partition function of the remaining
particles, still mutually excluding and additionally avoiding the balls
$B(x_j,\eps)$. Directly from \eqref{eq:grand-canonical},
\begin{equation}\label{eq:init-insertion}
 F_{h,0}^{\eps}(Z_h)=\ind_{\mathcal D_h^\eps}(Z_h)\prod_{j=1}^hf_0(z_j)
 \,\frac{\mathcal Z_\eps[Z_h]}{\mathcal Z_\eps}.
\end{equation}
The ratio is $\le1$, giving the factorial upper bound. For $h=1$ it is
the probability that $B(x_1,\eps)$ is empty of particle centers; by the
factorial upper bound with one particle and the union bound this is
$\ge1-\mu b_\eps=1-O(\eps)$, giving $(1-C\eps)f_0\le g_0\le f_0$.
Integrating the factorial bound gives $\E(N)_p\le\mu^p$; writing $N^p$
as a positive combination of $(N)_j$ gives the moment bounds.

\textbf{(ii) The Penrose tree bound.} Write $\chi_{ij}=\ind_{|x_i-x_j|\le
\eps}$ and set
\[
 \mathcal U_1=1,\qquad
 \mathcal U_n(Z_n)=\sum_{H\text{ conn.\ on }[n]}\prod_{\{i,j\}\in H}(-\chi_{ij})\ (n\ge2).
\]
We claim the tree bound
\begin{equation}\label{eq:tree-ineq}
 |\mathcal U_n(Z_n)|\le\sum_{\mathcal T\text{ tree}}\prod_{\{i,j\}\in\mathcal T}\chi_{ij}.
\end{equation}
This is the Penrose forest scheme. Root a connected labelled graph at
vertex $1$, assign graph-distance levels, and let the parent of each
vertex be its least-labelled neighbour in the preceding level. For a
fixed resulting tree $\mathcal T$, the graphs giving it are exactly
$\mathcal T\subseteq H\subseteq R(\mathcal T)$, where $R(\mathcal T)$
adds all same-level edges and every edge to a preceding-level vertex with
label $\ge$ that of the chosen parent. These intervals partition all
connected graphs, and summing one interval gives
\[
 \prod_{e\in\mathcal T}(-\chi_e)\prod_{e\in R(\mathcal T)\setminus\mathcal T}(1-\chi_e),
\]
whose second factor lies in $[0,1]$, proving \eqref{eq:tree-ineq}.

Integrating a tree from its leaves, keeping one free vertex, gives
\begin{equation}\label{eq:tree-integral}
 \int f_0^{\otimes n}|\mathcal U_n|\le n^{n-2}b_\eps^{n-1}\qquad(n\ge2).
\end{equation}
Hence the connected-graph (Mayer) series is absolutely convergent when
$e\mu b_\eps\|a\|_\infty<1$, and by exponentiation/analytic continuation
\begin{equation}\label{eq:log-partition}
 \log\mathcal Z_\eps[a]=\sum_{n\ge1}\frac{\mu^n}{n!}\int\mathcal U_n(Z_n)
 \prod_{j=1}^nf_0(z_j)a(z_j)\,\dd Z_n .
\end{equation}

\textbf{From the partition function to the cumulants.} Apply
\eqref{eq:log-partition} with $a=1+u$. The log of
$\E\prod_i(1+u(z_i))$ has coefficients $\mu^hc_{h,0}^{\eps}/h!$, so
differentiation gives the rooted Mayer series
\begin{equation}\label{eq:rooted-mayer}
 c_{h,0}^{\eps}(Z_h)=\prod_{j=1}^hf_0(z_j)
 \sum_{n\ge0}\frac{\mu^n}{n!}\int f_0^{\otimes n}(Z_{h+1:h+n})
 \mathcal U_{h+n}(Z_{h+n})\,\dd Z_{h+1:h+n}.
\end{equation}
For bounds uniform in $h$: delete $h-1$ edges of a tree on $h+n$ vertices
so each component contains one fixed root (possible by repeatedly cutting
an edge on a path between two roots). Dropping the deleted constraints and
integrating the auxiliary leaves costs at most $b_\eps^n$. With
$n!\ge(n/e)^n$, $(1+h/n)^n\le e^h$, $(h+n)^h\le h^he^n$,
\begin{equation}\label{eq:rooted-count}
 \frac{(h+n)^{h+n-2}}{n!}\le(eh)^he^{2n},
\end{equation}
and the $n=0$ count is $\le(Ch)^{Ch}$. Thus \eqref{eq:rooted-mayer}
converges absolutely for $\eps\le\eps_0$ with $\eps_0$ independent of
$h$.

\textbf{Coarse link bound.} Let $q$ be the number of connected components
of the graph on the $h$ roots with edges $|x_i-x_j|\le\eps$. If $q\ge2$,
every contributing tree must attach each such component to an auxiliary
vertex. One auxiliary vertex touches at most $27$ root components: pick
one adjacent root from each; their mutual distances exceed $\eps$, and
their disjoint balls of radius $\eps/2$ lie in a ball of radius
$3\eps/2$. Hence $q\le27n$ for a contributing tree with $n$ auxiliary
vertices. Using \eqref{eq:rooted-count} and $d_\eps=O(\eps)$,
\[
 \frac{|c_{h,0}^{\eps}(Z_h)|}{\prod_jf_0(z_j)}
 \le(Ch)^{Ch}\eps^{q/27}\le(Ch)^{Ch}\eps^{(q-1)/81}\qquad(q\ge2),
\]
and for $q=1$ the same argument without the restriction on $n$ gives
$(Ch)^{Ch}$. A tree of the roots using intra-component and $q-1$
inter-component edges has weight at least $\eps^{(q-1)/81}$ in
\eqref{eq:init-coarse}. This proves \eqref{eq:init-coarse}.

\textbf{Sharp link bound.} For fixed $h,D$, truncate
\eqref{eq:rooted-mayer} at $n_0+1\ge D(h-1)$. The tail is
$C_{D,h}\eps^{D(h-1)}\prod_jf_0(z_j)$. In any term with $n\le n_0$, all
roots are connected by paths of $\le h+n_0-1$ links of length $\le\eps$,
so their diameter is $\le(h+n_0)\eps$; on that set \eqref{eq:init-sharp}
holds with $C_{D,h}$ chosen large. This proves \eqref{eq:init-sharp}.
Integrating \eqref{eq:rooted-mayer} and using \eqref{eq:tree-integral}
instead gives
\[
 \|c_{h,0}^{\eps}\|_1\le b_\eps^{h-1}\sum_{n\ge0}\frac{(h+n)^{h+n-2}}{n!}d_\eps^n
 \le C_h\eps^{3(h-1)},
\]
which is \eqref{eq:init-L1}.

\textbf{(iii) Initial CLT and the shift.} For real bounded $\phi$ put
$S_\phi=\sum_i\phi(z_i)$ and let $\kappa_k$ denote ordinary cumulants.
Substituting $a=e^{u\phi}$ in \eqref{eq:log-partition} and
differentiating $k$ times at $u=0$,
\[
 \kappa_k(S_\phi)=\sum_{n\ge1}\frac{\mu^n}{n!}\int f_0^{\otimes n}\mathcal U_n
 \Bigl(\sum_{j=1}^n\phi(z_j)\Bigr)^k,
\]
and \eqref{eq:tree-integral} gives $\kappa_k(S_\phi)=\mu\int f_0\phi^k+
O_{k,\phi}(\mu\eps)$. After exact centering and division by $\sqrt\mu$,
the second cumulant tends to $\int f_0\phi^2$ and every higher one to
zero. The moment-cumulant identity then gives every Gaussian moment; for
scalar convergence compare characteristic functions of the normalized
variable and the Gaussian of variance $\sigma^2=\int f_0\phi^2$ using
Taylor polynomials of degree $2n-1$, whose combined remainder after
$\limsup$ is at most $2|u|^{2n}\sigma^{2n}/(2^n n!)\to0$. Joint
convergence follows on each linear combination. This includes zero
variance.

For the shift, the $n=1$ term of \eqref{eq:rooted-mayer} with $h=1$
gives, uniformly in $z$,
\[
 g_0(z)=f_0(z)\Bigl[1-\mu\int_{|x-x_*|\le\eps}\varrho_0(x_*)\dd x_*+O(\eps^2)\Bigr].
\]
The bounded gradient of $\varrho_0$ gives
$\int_{|x-x_*|\le\eps}\varrho_0=\tfrac{4\pi}{3}\eps^3\varrho_0(x)+O(\eps^4)$,
and $\sqrt\mu=\eps^{-1}$ converts this into \eqref{eq:init-shift}.
\end{proof}

\begin{remark}
The shift \eqref{eq:init-shift} is generally nonzero, so exact
microscopic centering is essential already at time zero. Meanwhile
$\mu\|c_{2,0}^{\eps}\|_1=O(\eps)$, so the limiting initial covariance has
only the same-particle term $\int f_0\phi\psi$.
\end{remark}

\subsection{Velocity moments and the backward propagator}

Write $M=\sup_{t\le T,x,v}e^{2\beta|v|^2}f_t(x,v)$. The noise form
\eqref{eq:noise} needs global velocity integrability, which does not
follow from a spatially uniform bound alone.

\begin{lemma}[Global weighted integrability]\label{lem:moments}
For every $0<q<2\beta$,
\begin{equation}\label{eq:moments}
 \sup_{0\le t\le T}\int_{\R^6}f_t(x,v)e^{q|v|^2}\,\dd x\,\dd v<\infty .
\end{equation}
\end{lemma}

\begin{proof}
Let $Q^+$ be the gain part of $Q$ and $P_tg=Q^+(g,f_t)$ with the second
factor fixed. With $\mathsf T_tg(x,v)=g(x-tv,v)$ and
$(\mathcal Vg)_t=\int_0^t\mathsf T_{t-s}P_sg_s\dd s$, positivity and the
transport Duhamel formula give
$f_t\le\sum_{j<n}(\mathcal V^j\mathsf T_\cdot f_0)_t+(\mathcal V^nf)_t$.
On the Gaussian decay spaces $Y_a$ with $\|g\|_{Y_a}=\sup e^{a|v|^2}|g|$,
energy conservation gives for $0<b<a\le2\beta$
\[
 |P_tg|\le CM\|g\|_{Y_a}(1+|v|)e^{-a|v|^2},\quad
 \|P_tg\|_{Y_b}\le CM(1+(a-b)^{-1/2})\|g\|_{Y_a}.
\]
Distributing the exponent from $\beta$ to $\beta/2$ over $n$ factors and
using that the time simplex has volume $T^n/n!$,
$\sup_{t\le T}\|(\mathcal V^nf)_t\|_{Y_{\beta/2}}\le M(CT)^n/n!(1+
\sqrt{2n/\beta})^n\to0$, so $f$ is dominated pointwise by the positive
gain series.

Let $X_q=L^1(e^{q|v|^2}\dd x\dd v)$. The pre/post substitution with
$e^{q|v'|^2}\le e^{q|v|^2+q|v_*|^2}$ and the bound on $f_t$ give
$\|P_tg\|_{X_q}\le CM(1+\delta^{-1/2})\|g\|_{X_{q+\delta}}$ uniformly for
exponents in a compact subinterval of $(0,2\beta)$. Since $f_0\in X_{q_0}$
for $q_0<2\beta$, distributing the exponent over the gain series and using
$n^{n/2}/n!$ bounds its $X_q$ norm uniformly for $t\le T$; integrating the
pointwise majorant proves \eqref{eq:moments}.
\end{proof}

For $a\ge0$ define the test spaces
\begin{equation}\label{eq:test-spaces}
 E_a^0=\{h\in C(\R^6):e^{-a|v|^2}h\in C_0(\R^6)\},\quad
 \|h\|_a=\sup e^{-a|v|^2}|h|,
\end{equation}
on which smooth compactly supported tests are dense. Let
$\mathsf S_uh(x,v)=h(x+uv,v)$ be free test transport and
$K_t=A_t-v\cdot\nabla_x$.

\begin{lemma}[Backward evolution]\label{lem:propagator}
For $0\le a<b<2\beta$ and $0\le r\le t\le T$ there is a bounded backward
propagator $U(r,t):E_a^0\to E_b^0$ solving
\begin{equation}\label{eq:backward-mild}
 U(r,t)h=\mathsf S_{t-r}h+\int_r^t\mathsf S_{s-r}K_sU(s,t)h\,\dd s,
\end{equation}
with the composition law at intermediate exponents and uniqueness as a
mild propagator in these scales. For $\phi\in C_c^\infty$,
\begin{equation}\label{eq:upper-end}
 U(r,t)\phi-\phi=\int_r^tU(r,s)A_s\phi\,\dd s.
\end{equation}
\end{lemma}

\begin{proof}
For $a$ in a compact subinterval of $[0,2\beta)$, energy conservation and
the Gaussian bound give
\begin{equation}\label{eq:K-estimate}
 |K_th|\le CM\|h\|_a(1+|v|)e^{a|v|^2},\quad
 \|K_th\|_b\le CM(1+(b-a)^{-1/2})\|h\|_a,
\end{equation}
using $|\Delta_h|\le4\|h\|_ae^{a(|v|^2+|v_*|^2)}$ and
$\int B\dd\omega=\pi|v-v_*|$. Spatial tails vanish by replacing $\|h\|_a$
with $\eta_h(x)=\sup_ve^{-a|v|^2}|h|$, and $K_th$ is continuous in
$E_b^0$. Iterating \eqref{eq:backward-mild} bounds the $n$-th
time-ordered term by
\begin{equation}\label{eq:dyson}
 \frac{(CM(t-r))^n}{n!}(1+\sqrt{n/(b-a)})^n\|h\|_a,
\end{equation}
so the series converges for every finite $T$. Splitting a time simplex at
an intermediate time gives composition, and the same estimate on an
iterated remainder gives uniqueness with a positive exponent allowance.
For compactly supported smooth $\phi$, differentiability of the series at
its upper end (its final transport gives $U(r,t)(v\cdot\nabla_x\phi)$;
the simplex faces give $U(r,t)K_t\phi$) converges with \eqref{eq:dyson},
giving $\partial_tU(r,t)\phi=U(r,t)A_t\phi$ and hence \eqref{eq:upper-end}.
\end{proof}

\subsection{Construction and uniqueness of the Gaussian field}

\begin{proposition}[Gaussian target]\label{prop:gaussian-target}
The equation \eqref{eq:fluc} has a centered Gaussian solution on each
test scale $E_a^0$, $0\le a<\beta$, mapping continuously and linearly into
$L^2$ uniformly on $[0,T]$, with covariance
\begin{equation}\label{eq:target-cov}
 \begin{split}
 \E[\zeta_t(\phi)\zeta_s(\psi)]
 ={}&\int f_0\,U(0,t)\phi\,U(0,s)\psi\,\dd z\\
 &+\int_0^{\min(s,t)}C_r(U(r,t)\phi,U(r,s)\psi)\,\dd r.
 \end{split}
\end{equation}
It is unique in the class of solutions extending to these scales with
uniformly bounded second-moment norms. Thus the law in \Cref{thm:main}
is well defined.
\end{proposition}

\begin{proof}
For $b<\beta$, energy conservation and \Cref{lem:moments} give
\begin{equation}\label{eq:source-bound}
 |C_t(h,k)|\le CM\|h\|_b\|k\|_b\int f_t(1+|v|)e^{2b|v|^2}\dd x\dd v\le C_b\|h\|_b\|k\|_b,
\end{equation}
choosing the moment exponent slightly above $2b$ to absorb $1+|v|$. The
initial form has the corresponding bound.

Let $I_0,I$ be independent isonormal Gaussians on $L^2(f_0\dd z)$ and
$L^2(\nu)$ with
$\dd\nu=\tfrac12B(v-v_*,\omega)f_r(x,v)f_r(x,v_*)\dd r\dd x\dd v\dd v_*\dd\omega$.
Define
\begin{equation}\label{eq:isonormal}
 \zeta_t(h)=I_0(U(0,t)h)+I(\ind_{r\le t}\Delta_{U(r,t)h}),\quad
 W_t(h)=I(\ind_{r\le t}\Delta_h).
\end{equation}
For $h\in E_a^0$ propagate to $b\in(a,\beta)$ and apply
\eqref{eq:source-bound}: the maps are well defined and
\begin{equation}\label{eq:field-bound}
 \sup_{t\le T}\|\zeta_t(h)\|_{L^2}\le C_a\|h\|_a.
\end{equation}
They are jointly Gaussian and linear with covariance \eqref{eq:target-cov};
orthogonality of disjoint time intervals in $L^2(\nu)$ gives independent
increments of $W$, and $I_0\perp I$ gives initial independence.

\textbf{Weak equation.} For $\phi\in C_c^\infty$, using \eqref{eq:upper-end}
and boundedness of the isonormal maps against Bochner integrals,
$\int_0^t\zeta_s(A_s\phi)\dd s=I_0(U(0,t)\phi-\phi)+
I(\ind_{r\le t}\Delta_{U(r,t)\phi-\phi})$, which together with
\eqref{eq:isonormal} is $\zeta_t(\phi)=\zeta_0(\phi)+\int_0^t\zeta_s(A_s\phi)
\dd s+W_t(\phi)$.

\textbf{Uniqueness.} On any candidate's probability space, construct the
backward mild field from its own initial field and noise. The difference
$D_t$ has zero initial field and noise, mean-square continuity, and the
transport mild identity $D_t(\phi)=\int_0^tD_s(K_s\mathsf S_{t-s}\phi)\dd s$
(finite-support Riemann sums with $\mathsf S_{t-s}\phi$). Iterating $n$
times with $0\le a<b<\beta$ bounds its $L^2$ norm by
$C_b\|\phi\|_a(CT)^n/n!(1+\sqrt{n/(b-a)})^n\to0$, proving uniqueness in
the class and uniqueness in law.

This is a Gaussian generalized field in the mean-square sense of
\eqref{eq:field-bound}; it does not claim samplewise membership in the
dual of $E_a^0$. Its restriction to $C_c^\infty$ does have a random
distribution version: expanding locally in Fourier functions times a
compact cutoff, variances bounded by \eqref{eq:field-bound} with $a=0$
make the expected local $H^{-s}$ norm finite for $s>3$, and a cube
exhaustion gives consistent local distributions. No path-space assertion
is needed here.
\end{proof}

% =====================================================================
\part*{Part 3. The microscopic engine: layered histories and the DHM package}
% =====================================================================

\section{Layered histories and the imported analytic package}\label{sec:dhm-inputs}

We use Deng--Hani--Ma [DHM, arXiv:2408.07818v3, 18 July 2025]. Their
diagram estimates are bounds for \emph{positive} geometric integrals; they
do not by themselves give a fluctuation theorem or the sharp connected
bound. Everything in this part is either a definition we need or a
statement quoted from [DHM] with its source location. This is the only
place where an unproved external input enters.

\subsection{Parameters and the pasted dynamics}

Fix $\upsilon=3^{-4}=1/81$. Choose the excess bound $\Gamma$ as a large
dimensional constant \emph{before} the number of slabs $L$. Put
$\tau=T/L$ and, with harmless integer roundings,
\begin{equation}\label{eq:cutoffs}
 A_L=|\log\eps|,\qquad \Lambda_i=A_i^{30},\qquad A_{i-1}=\Lambda_i^{30},
\end{equation}
and for the analytic induction
\begin{equation}\label{eq:exponents}
 \beta_i=\Bigl(1-\frac{i}{10L}\Bigr)\frac\beta2,\quad
 \widetilde\beta_i=\frac{\beta_{i-1}+\beta_i}{2},\quad
 \theta_i=\Bigl(1-\frac{i}{10L}\Bigr)3^{-11}.
\end{equation}
Constants written $C$ inside an exponential factor $C^b$ are independent
of $L$; constants $C_*$ and $c_*>0$ may depend on $L$ and the kinetic
bounds but not on $\eps$. Crucially, no moment order enters the choice of
$\Gamma$ or $L$.

\textbf{Restarted truncated dynamics.} Work slab by slab. Within slab $i$,
restart the collision prehistory. At an incoming contact, accept the
elastic collision \emph{iff} adding it to the current collision graph of
that component keeps the component's particle count $\le\Lambda_i$ and its
excess $\le\Gamma$; otherwise both particles continue straight. The
excess of a graph with $b$ edges, $n$ vertices, $k$ components is
$r=b-n+k$ (repeated edges counted with multiplicity); an internal contact
adds $1$, a merging contact adds the two excesses. Grazing, simultaneous,
and tied-branch contacts are null sets. This is [DHM, Definition 4.3 and
Eq.~(4.2)], with existence and mass preservation in [DHM, Proposition~4.5].

\textbf{Pasted dynamics.} Concatenate these restarted slab dynamics. It is
\emph{not} a semigroup (the prehistory resets at each slab), which is why
we treat it as a separate object to be compared with the true flow in
\Cref{sec:cutoff}.

\begin{lemma}[Kinetic envelopes]\label{lem:envelopes}
Under \Cref{ass:kinetic},
\begin{equation}\label{eq:bol-propagated}
 \sup_{0\le t\le T}\bigl(\|f_t\|_{\Bol,\beta}+\|\nabla_xf_t\|_{\Bol,\beta}\bigr)<\infty .
\end{equation}
\end{lemma}
\begin{proof}
Apply [DHM, Proposition A.1] with decreasing exponent
$\beta_*(t)=2\beta-\beta t/T$. Its hypothesis is \eqref{eq:kinetic-gaussian}
and its initial summable norms \eqref{eq:init-factorial}-\eqref{eq:bol-norm};
since $\beta_*(t)\ge\beta$, its conclusion gives \eqref{eq:bol-propagated}.
This is [DHM, Eq.~(1.29)].
\end{proof}

\subsection{Histories, roots, and normalization}

\begin{definition}[Layered C/O history]\label{def:history}
A history consists of finitely many oriented particle lines and binary
\emph{atoms} with a fixed order on the atoms along each line. Each atom
has two incoming and two outgoing half-edges paired into two serial lines.
A \emph{C-atom} applies elastic scattering; an \emph{O-atom} records an
incoming encounter with straight continuation. A segment between atoms is
a \emph{bond}; a segment with one incident atom is an \emph{end}.

Atoms carry slabs, and the order along each line must respect slabs. Lines
may begin or end at interfaces; their beginnings carry one-particle input
functions. Certain top lines with distinct labels are \emph{roots}; all
other particle names are immaterial (histories are identified under
root-preserving relabelling of the serial pairing, C/O types, and line
orders). Integrals use the factorial weights of ordered particle labels.

In a slab, the \emph{C-clusters} are the components formed by C-atoms; the
\emph{cluster graph} has C-clusters as vertices and O-atoms as edges; its
components are \emph{layer components} (isolated rooted lines included).
Let $H_i$ be the roots of slab $i$ (lines continued above, or the
designated roots if $i$ is the top slab), and $H_0$ the lines incident to
an initial link, where initial links join pairs of lines born at zero with
weight
\begin{equation}\label{eq:init-link}
 \ell_\eps(x,x')=\ind_{|x-x'|\le\eps}+\eps^\upsilon\ind_{|x-x'|>\eps}.
\end{equation}
An \emph{admissible dependent history} satisfies:
\begin{enumerate}[label=(\roman*)]
\item $|H_i|\le A_i$; every C-cluster has $\le\Lambda_i$ lines and excess
  $\le\Gamma$; every layer component has $<\Lambda_i$ C-clusters.
\item The cluster graph in each slab is a forest and every layer component
  contains a root of its slab; every unrooted line is connected in its slab
  to a rooted line.
\item The initial links form a forest.
\item For $i\ge2$, each line in $H_i$ is connected within slab $i$ to a
  different line of $H_i$, or to a line of $H_{i-1}$, or itself lies in
  $H_{i-1}$ (replace $H_{i-1}$ by $H_0$ when $i=1$).
\end{enumerate}
Connection in a slab means sharing a component of its atom graph; the last
alternative of (iv) covers a straight line with no atom there. These are
[DHM, Definition 3.20(1)]. The \emph{complexity} is
\begin{equation}\label{eq:complexity}
 \rho=|H_0|+\sum_i\bigl(|H_i|+r(\mathcal M_i)\bigr),
\end{equation}
with $r(\mathcal M_i)$ the cycle excess in slab $i$; since the O-graph is
a forest this is the sum of C-cluster excesses. $b$ denotes the number of
atoms.
\end{definition}

\textbf{Integral convention.} Give each bond $e$ a vector $z_e=(x_e,v_e)$
with position $X_e(t)=x_e+tv_e$; give atom $a$ a time $t_a$. Its incoming
edges are $1,2$ and serial outgoing edges $1',2'$. On the contact surface
put $u=v_1-v_2$, $\omega=(X_1(t_a)-X_2(t_a))/\eps$, and
$\mathfrak b=[-u\cdot\omega]_+$. Define
\begin{align}
 \Delta_a^{\mathrm C}
 &=\delta(|X_1-X_2|-\eps)\mathfrak b
   \prod_{j=1}^2\delta(X_{j'}(t_a)-X_j(t_a))
   \delta(v_{1'}-v_1+(u\cdot\omega)\omega)\delta(v_{2'}-v_2-(u\cdot\omega)\omega),
 \label{eq:atom-C}\\
 \Delta_a^{\mathrm O}
 &=\delta(|X_1-X_2|-\eps)\mathfrak b
   \prod_{j=1}^2\delta(x_{j'}-x_j)\delta(v_{j'}-v_j).
 \label{eq:atom-O}
\end{align}
Reverse $\omega$ to match \eqref{eq:kernel}; surface measure is
unnormalized. With a set $E_*$ of integrated bonds and free ends, the
positive measure of a cut history $M$ with kernel $Q$ is
\begin{equation}\label{eq:edge-integral}
 I_M(Q)(z_{\mathrm{fixed}})
 :=\mu^{|E_*|-2b}\int_{\mathcal D_M}\int Q(z_E,t_M)
   \prod_{a\in M}\Delta_a\prod_{e\in E_*}\dd z_e\prod_{a\in M}\dd t_a,
\end{equation}
where $\mathcal D_M$ enforces the slab and strictly increasing times along
each line. For a full history $|E|-2b=n=$ number of lines, and with $h$
roots the \emph{normalized} integral is
\begin{equation}\label{eq:normalized}
 \mathcal J_M(Q)=\mu^{-h}I_M(Q):
\end{equation}
each birth carries activity $\mu$, each root a normalization $\mu^{-1}$,
each contact surface a factor $\eps^2$ [DHM, Proposition 7.5, Eq.~(7.7)].
To keep a root endpoint $(x,v)$ at time $t$ free, make its top end a fixed
end evaluated at $(x-tv,v)$, removing the root activity.

\subsection{The imported analytic package}

The one-slab class $\mathcal T_i$ consists of connected histories with one
top root, conditions (i)--(ii) only, no continuation or initial-link
requirement; it may have up to $\sim\Lambda_i^2$ lines.
$\mathcal T_i^{\mathrm{err}}$ inserts one extra O-atom into a member of
$\mathcal T_i$. The one-root signed tree sum contains every one-root
particle-line tree of size $<\Lambda_i/3$, with C/O choices and sign
$(-1)^{\#\mathrm O}$.

\begin{theorem}[DHM analytic package]\label{thm:dhm-package}
Assume \Cref{ass:kinetic}, the hierarchy \eqref{eq:cutoffs}--%
\eqref{eq:exponents}, and a common Gaussian $\Bol$ bound on all birth
inputs. For $L$ sufficiently large (fixed) and $\eps$ sufficiently small:
\begin{enumerate}[label=(\alph*)]
\item For admissible dependent histories with prescribed labeled roots,
  fixed $b$, and complexity $\rho$, the normalized positive integral sums
  to at most
  \begin{equation}\label{eq:coarse}
   C^b\tau^{b/10}\eps^{c_*\rho}|\log\eps|^{C_*\rho}.
  \end{equation}
  The sum includes graph enumeration and admissibility restrictions;
  bounded observable multipliers cost their supremum. Birth inputs need
  only satisfy the common Gaussian $\Bol$ envelope.
\item If the independent input $g$ in slab $i$ has
  $\|g-f_{(i-1)\tau}\|_{\Bol,\beta_{i-1}}\le\eps^{\theta_{i-1}}$, then the
  signed tree sum $\mathsf T_i^\eps[g]$ satisfies
  \begin{equation}\label{eq:signed-trees}
   \|\mathsf T_i^\eps[g]-f_{i\tau}\|_{\Bol,\beta_i}\le\tfrac12\eps^{\theta_i}.
  \end{equation}
  In the fixed-top-endpoint norm $\Bol,\widetilde\beta_i$, the summed
  absolute contributions of cyclic $\mathcal T_i$, of
  $\mathcal T_i^{\mathrm{err}}$, and of $\mathcal T_i$ with $\ge\Lambda_i/3$
  lines are $o(\eps^{\theta_i})$. Fixed cutoff powers and a fixed
  exponential-atom factor can be included, the latter absorbed by large
  $L$.
\end{enumerate}
Part (b) holds uniformly for a slab truncated at any $0\le\sigma\le\tau$,
with target $f_{(i-1)\tau+\sigma}$; at $\sigma=0$ the map is the identity.
\end{theorem}

\begin{proof}[Source and envelope adaptations]
For (a): [DHM, Proposition~7.2] counts root-labeled unlabeled histories by
$C^b|\log\eps|^{C_*\rho}$; [DHM, Proposition~9.7 and Eq.~(9.51)] retain a
positive gain $\propto\rho$; after regular cutting, [DHM, Eq.~(9.53)]
bounds each full edge integral by
$\tau^{b/9}\eps^{-2h+2\upsilon/5+c_*\rho}|\log\eps|^{C_*\rho}$.
Multiplication by $\mu^{-h}=\eps^{2h}$ cancels the root power; weakening
$1/9\to1/10$ and discarding the extra positive power gives
\eqref{eq:coarse}. These are the estimates inside the proof of [DHM,
Proposition~6.2, \S9.2], not only its displayed conclusion.

The only use of a birth density is the envelope decomposition [DHM,
Eqs.~(9.58)--(9.63)]: from \eqref{eq:bol-norm} choose a summable
sequence $(m_a)$ bounding the weighted supremum on unit balls; bounded
overlap and $\sum_am_a<\infty$ dominate $|g|$ by
$C\sum_am_a\ind_{|x-a|\le1}e^{-\beta'|v|^2}$, and the positive source
estimates applied to each envelope product sum by
$\sum_am_a<\infty$. This extends (a) to arbitrary bounded inputs,
including absolute values of signed inputs. Multipliers in $[0,1]$ are
kept or dropped only when taking a positive upper bound.

For (b): the one-root integral is a function of a fixed endpoint
[DHM, Eq.~(14.2)]; the C/O involution [DHM, Proposition~14.2] cancels
noncausal trees; the causal remainder satisfies the Gaussian $\Bol$
estimate [DHM, Proposition~14.3, Eq.~(14.8)]; comparison to the short-time
Boltzmann expansion, using \Cref{lem:envelopes}, is [DHM, Proposition~6.1,
\S14]. To pass from $f_{(i-1)\tau}$ to $g$, expand each birth factor as
$f_{(i-1)\tau}+(g-f_{(i-1)\tau})$; each residual costs
$\eps^{\theta_{i-1}}$ by hypothesis, and the multilinear tree bound sums
all choices; since $\theta_{i-1}>\theta_i$ the residuals are
$o(\eps^{\theta_i})$. For the positive subleading terms the bound is
[DHM, Eq.~(14.28)] in $\Bol,\widetilde\beta_i$ with a time power per atom
and gain $\eps^{\upsilon/3+c_*\rho}$ in cyclic and extra-encounter
classes; \S14.3 Points~1--2 keep the Gaussian weight at the fixed root and
the spatial $\Bol$ sum, Point~4 regularizes an O-atom adjacent to the root
without changing its integral. The tree-tail branch of Point~5 uses only
that the atom number is of cutoff order, so $\Lambda_i\to\Lambda_i/3$
leaves an exponentially small tail. Graph enumeration, the bounded
selected-edge choices, and fixed exponential factors are absorbed by
taking $L$ large. For the partial slab, restrict contact times to
$((i-1)\tau,(i-1)\tau+\sigma)$ and repeat the fixed-root Gaussian and
spatial localization at endpoint $(x-\sigma v,v)$; all shifts are $\le\tau$
so the exponent margin survives. The signed involution preserves the
restriction and the upper majorants keep their full-$\tau$ bounds. At
$\sigma=0$ only the zero-atom tree remains, i.e.\ $g$, and
$\theta_{i-1}>\theta_i$ gives the bound.
\end{proof}

\begin{lemma}[Coarse boundary modifications]\label{lem:dummies}
The positive coarse bound tolerates:
\begin{enumerate}[label=(\roman*)]
\item dropping continuation condition (iv) in the top slab with $h$ roots,
  at the cost of a loss $\eps^{-C_*h}$;
\item a connected pure-C terminal component with $\le3\Lambda_i$ lines and
  excess $\le2\Gamma$, meeting a lower continuation but with no designated
  root, at cost $\mu$ (add one artificial root);
\item supplying initial-link continuations to a fixed number $q$ of
  specified lines, at cost $\eps^{-q\upsilon}$ (add $q$ dummy straight
  root lines), provided the root cutoffs have the corresponding slack.
\end{enumerate}
Bounded records carried by these lines are kept.
\end{lemma}

\begin{proof}[Source and construction]
(i): use the error form of the cutting inequality [DHM, Eq.~(9.51)]
retaining its deficit $\propto h$. The finale [DHM, \S13.5, p.~169] splits
into a bounded-$\rho$ case, where the UP algorithm makes $\le h$ free-
component ``bad pieces'' paid by the root deficit, and a large-$\rho$ case
using the dependent cutting argument with the continuation condition of a
strictly lower layer still available; the resulting positive integration
gives the loss $\eps^{-C_*h}$ of [DHM, Eq.~(9.82)]. Terminal size is used
only later and absorbed by the time power.

(ii): take one line of the terminal component as root; its normalization
costs $\mu^{-1}$ and multiplication by $\mu$ restores the integral.
Splitting into $\le\Lambda_i$ lines and $\le\Gamma$ excess (ordinary
admissible) versus the enlarged terminal class of [DHM, Definition
3.20(4)], the same root operation is [DHM, Eq.~(15.4)]; summing over the
choice of line costs a cutoff polynomial.

(iii): fix nonnegative $q_0$ with $\int q_0=1$ and finite Gaussian $\Bol$
norm; for every $x$,
\[
 \int q_0(x',v')\ell_\eps(x,x')\,\dd x'\,\dd v'\ge\eps^\upsilon .
\]
Insert once
per continuation: the dummy activity $\mu$ cancels the extra root factor
$\mu^{-1}$, its straight line satisfies continuation in every slab, and
the new links are pendant so the initial forest is preserved. This is a
positive kernel inequality, valid even where $f_0$ vanishes; the envelope
extension of \Cref{thm:dhm-package} admits $q_0$, and records are
untouched.
\end{proof}

% =====================================================================
\part*{Part 4. The connected expansion with records}
% =====================================================================

\section{A componentwise expansion retaining particle records}\label{sec:cluster-expansion}

This part is pure algebra plus the [DHM] coarse bound. Its job: write the
factorial cumulants of the pasted process as a \emph{finite} sum of
\emph{globally connected} diagrams, each with the correct label weight,
while carrying along finitely many observation \emph{records} (running
sums of test values on designated particles). The high-complexity part is
dropped with \Cref{thm:dhm-package}(a); what remains is what
\Cref{sec:sharp-asymptotics} controls at the fluctuation scale. The
connected coefficients here are genuine factorial cumulants of the pasted
process, not a factorization assumption.

Put
\begin{equation}\label{eq:cluster-threshold}
 M_i=\lfloor\Lambda_i/2\rfloor,
\end{equation}
the \emph{expansion} cluster cap. It differs from the \emph{dynamics} caps
$\Lambda_i,\Gamma$ and belongs only to the expansion; take $\eps$ small so
$M_i\ge2$ for all $i$.

\subsection{Autonomous clusters and the finite forest identity}

\textbf{Idea.} Run each subset of particles on its own, with the truncated
collision rule. The full truncated history is then a union of ``autonomous
clusters'' that never accept a collision with each other, and the block
structure is governed by a finite inclusion-exclusion over
compatibilities.

Fix a slab (suppress its index). Let $V$ be a finite labelled particle set.
A particle state may carry previously observed test values, or their
running sum, as extra coordinates called records; the acceptance rule uses
only positions, velocities, and the prehistory on this slab. For
$C\subseteq V$, $\Psi_C$ is the autonomous truncated evolution of $C$
(including its observation updates), and $\kappa_C$ is the indicator that
its accepted collision graph is connected (singletons $\kappa_\bullet=1$;
$|C|>\Lambda\Rightarrow\kappa_C=0$). For disjoint $C,D$, $a(C,D)$
indicates an incoming cross-contact that \emph{would} be accepted with
their respective prefix prehistories, and $\chi(C,D)=1-a(C,D)$. These are
indicators, not counts. All identities are on the common full-measure set
where all finite subsystems are defined.

\begin{lemma}[Autonomous partition]\label{lem:autonomous-partition}
For finite $V$,
\begin{equation}\label{eq:autonomous-partition}
 \sum_{\mathcal C\in\mathcal P(V)}\prod_{C\in\mathcal C}\kappa_C
 \prod_{\{C,D\}\subseteq\mathcal C}\chi(C,D)=1.
\end{equation}
For $S\subseteq V$ and a root test $H$,
\begin{equation}\label{eq:autonomous-observation}
 H(\Psi_V|_S)=\sum_{\mathcal C\in\mathcal P(V)}
 \prod_{C\in\mathcal C}\kappa_C\prod_{\{C,D\}\subseteq\mathcal C}\chi(C,D)
 H(\Psi_{\mathcal C}|_S).
\end{equation}
\end{lemma}
\begin{proof}
The collision components of the full truncated history give one summand
$=1$; its restrictions are autonomous because no accepted collision joins
components and the rule is prefix-local. Conversely, if all factors of a
summand equal one, juxtapose the autonomous histories and scan incoming
contacts in time order: internal accepted contacts obey the rule, every
cross-contact fails it by $\chi=1$, so the juxtaposition is the full
truncated history, and uniqueness identifies its components with
$\mathcal C$.
\end{proof}

The next lemma is finite algebra; its vertices will be autonomous clusters.

\begin{lemma}[Rooted forest stopping]\label{lem:rooted-forest}
Let $W$ be finite, $R\subseteq W$ nonempty, $a_e\in\{0,1\}$ for each pair,
$M\ge2$, and
\begin{equation}\label{eq:ursell-block}
 u(B)=\sum_{\substack{G\text{ connected}\\V(G)=B}}\prod_{e\in E(G)}(-a_e),
 \quad u(\{w\})=1.
\end{equation}
There is a finite signed remainder $\mathcal R$ with
\begin{align}\label{eq:finite-rooted-identity}
 \prod_{e\subseteq W}(1-a_e)
 &=\sum_{R\subseteq I\subseteq W}
  \sum_{\substack{\mathcal B\in\mathcal P(I)\\ B\cap R\ne\varnothing,\ |B|<M (B\in\mathcal B)}}
   \prod_{B\in\mathcal B}u(B)\prod_{e\subseteq W\setminus I}(1-a_e)+\mathcal R.
\end{align}
$|\mathcal R|$ is bounded by a sum of rooted forests, each with exactly one
component of size in $[M,2M-2]$ and all other components smaller than $M$,
all meeting $R$, vertices off the forest keeping their mutual compatibility
product, each selected forest with coefficient $\le1$. Moreover
\begin{equation}\label{eq:ursell-tree-bound}
 |u(B)|\le\sum_{T\text{ spanning tree}}\prod_{e\in E(T)}a_e.
\end{equation}
\end{lemma}
\begin{proof}
Order the pairs. Grow the empty forest: at each step take the first
unprocessed pair joining distinct current components, at least one meeting
$R$, and split $1-a_e$ (positive branch deletes, negative branch selects
$e$ and merges). Never split a pair internal to a component. Stop a branch
when a component reaches $M$ vertices; complete when no eligible pair
remains. Selected edges form a forest; an unrooted vertex never joins
another, so each nontrivial component is rooted; a stop merges two
components of size $<M$, giving the single large component in $[M,2M-2]$
and the asserted multiplicity. Completed branches produce all graphs with
unique exploration, grouped by their reached components into products of
$u(B)$ with survivors keeping their compatibility product, giving the main
term; stopped branches give the forest bound. For the tree inequality, fix
a total edge order on $B$, assign each connected graph its minimum spanning
tree $T$, let $Q(T)$ be the edges later than the whole $T$-path; then
connected graphs on $B$ are exactly $T\subseteq G\subseteq T\cup Q(T)$, so
$u(B)=(-1)^{|B|-1}\sum_T\prod_{e\in T}a_e\prod_{e\in Q(T)}(1-a_e)$; since
the second product lies in $[0,1]$, dropping it proves
\eqref{eq:ursell-tree-bound}.
\end{proof}

\subsection{An exact one-slab operator on factorial measures}

\textbf{Idea.} The autonomous partition plus the forest identity become a
single linear operator on the factorial hierarchy. Its ``retained'' part
keeps bounded rooted blocks; its ``stopped'' part bounds the remainder by
a positive forest integral.

Write $F_s$ for ordered factorial measures normalized by $\mu^s$ (records
allowed on a specified subset of the $s$ labels). Using symmetric Janossy
measures $p_n$ ($\sum_n p_n(E^n)/n!=1$, $p_n/n!$ the law restricted to
$N=n$),
\begin{equation}\label{eq:janossy}
 F_s=\mu^{-s}\sum_{n\ge s}\frac{1}{(n-s)!}(\mathrm{pr}_{[s]})_\#p_n.
\end{equation}
For $S=[s]$, $V=[s+j]$, define the signed root-output kernel
\begin{align}\label{eq:retained-kernel}
 \mathcal K_{S,V}(X,\dd X'_S)
 &=\sum_{\mathcal C\in\mathcal P(V)}\prod_{C\in\mathcal C}\kappa_C(X)
  \sum_{\substack{\mathcal B\in\mathcal P(\mathcal C)\\
    (\bigcup_{C\in B}C)\cap S\ne\varnothing,\ |B|<M}}
   \prod_{B\in\mathcal B}u(B;X)\,\delta_{\Psi_{\mathcal C}(X)|_S}(\dd X'_S),
\end{align}
and the retained operator
\begin{equation}\label{eq:retained-operator}
 (\mathcal L_sF)(\dd X'_S)
 =\sum_{j=0}^{s(M-1)\Lambda-s}\frac{\mu^j}{j!}
  \int F_{s+j}(\dd X_V)\,\mathcal K_{S,V}(X,\dd X'_S).
\end{equation}
(An empty restriction is zero; at most $s$ retained blocks, so the bound on
$j$ follows from the caps.) The \emph{stopped} kernel replaces the inner
block sum by
$\sum_{F\in\mathfrak F^{\mathrm{stop}}_S(\mathcal C)}\prod_{e\in E(F)}a_e(X)$,
where $\mathfrak F^{\mathrm{stop}}_S(\mathcal C)$ are forests spanning
$\mathcal C$ with every component meeting $S$, exactly one of size
$[M,2M-2]$, the rest smaller, no unrooted isolated cluster; keep
$\prod_C\kappa_C$ and the root-output map. Define $\mathcal E_sF$ by
\eqref{eq:retained-operator} with this kernel and $j$ up to
\begin{equation}\label{eq:stopped-input-order}
 j\le(s+1)(M-1)\Lambda-s.
\end{equation}

\begin{lemma}[One-slab identity and variation remainder]\label{lem:one-slab-expansion}
For a positive input law with factorial hierarchy $F$, let $F'$ be its
hierarchy after the truncated slab. If factorial measures up to
\eqref{eq:stopped-input-order} are finite, then
\begin{equation}\label{eq:one-slab-exact}
 F'_s=\mathcal L_sF+R_s,\qquad |R_s|\le\mathcal E_sF.
\end{equation}
The record version holds, and $\mathcal E_sF$ is a positive domination of
$R_s$, not a claim that $R_s$ has this local kernel.
\end{lemma}
\begin{proof}
Insert \eqref{eq:autonomous-observation} and \Cref{lem:rooted-forest} into
\eqref{eq:janossy}. On a completed branch, fix the reached labels
$V\supset S$, $|V|=s+j$; the unreached cluster partitions sum to one
pointwise by \eqref{eq:autonomous-partition}, using no independence of the
input, and
$\frac1{(n-s)!}\binom{n-s}{j}=\frac1{j!(n-s-j)!}$ turns the remaining sum
into $\mu^{s+j}F_{s+j}$; the output normalization $\mu^{-s}$ leaves
$\mu^j/j!$. Stopped terms keep only the unreached compatibility product and
drop the rest, so positivity of the input and the multiplicity bound give
$|R_s(H)|\le(\mathcal E_sF)(|H|)$; there are $\le s$ rooted components and
only one of size up to $2M-2$ clusters, giving
\eqref{eq:stopped-input-order}. Records are untouched by the spectator
summation and travel with the root-output map.
\end{proof}

For a signed family $\Delta$ (uniformly in records and $s\le A_i/2$),
\begin{equation}\label{eq:crude-operator}
 \|\mathcal L_{i,s}\Delta\|_{\TV}+\|\mathcal E_{i,s}|\Delta|\|_{\TV}
 \le e^{C_*A_i\Lambda_i^2\log(1/\eps)}\max_{q\le A_{i-1}/2}\|\Delta_q\|_{\TV}.
\end{equation}
Indeed relevant $q\le(s+1)\Lambda_i^2/2<A_{i-1}/2$, with at most $q^q$
particle and $q^q$ cluster partitions, spanning/forest sum $\le q^{2q}$,
activity $\le\mu^q$, and at most $A_i\Lambda_i^2$ values of $q$; take logs.

\subsection{The one-root map and its Boltzmann approximation}

Define the unmarked one-root measures recursively by
\begin{equation}\label{eq:one-root-definition}
 g_i=\mathcal U_i(g_{i-1}),\qquad
 \mathcal U_i(g):=\mathcal L_{i,1}\bigl((g^{\otimes q})_{q\ge0}\bigr),
\end{equation}
a finite signed measure using the full retained one-root block. It
approximates the Boltzmann density:

\begin{proposition}[Independent factors]\label{prop:one-root}
$g_i$ has a Gaussian $\Bol$-bounded density representative and, with the
exponents of \Cref{thm:dhm-package},
\begin{equation}\label{eq:one-root-convergence}
 \|g_i-f_{i\tau}\|_{\Bol,\beta_i}\le\eps^{\theta_i},\qquad
 \sup_{0\le i\le L}\|g_i\|_{\Bol,\beta_i}\le C,
\end{equation}
with the same bounds for the one-root map run through any partial slab
$\sigma\in[0,\tau]$, target $f_{(i-1)\tau+\sigma}$.
\end{proposition}

The bridge from the retained block to the unrestricted tree sum is the
following; it is where ``retained'' and ``DHM signed trees'' are shown to
differ only by error classes.

\begin{lemma}[Modified one-root bridge]\label{lem:one-root-bridge}
The difference between $\mathcal U_i(g)$ and the signed unrestricted
one-root particle-tree sum with $n<\Lambda_i/3$ is dominated, with inputs
replaced by $|g|$, by the positive integrals of
\begin{enumerate}[label=(\roman*)]
\item retained one-root molecules with cycle excess $r\ge1$;
\item retained one-root molecules with $n\ge\Lambda_i/3$;
\item small unrestricted trees with one extra incoming O-atom inserted
  between two flight segments.
\end{enumerate}
Families (i)--(ii) are subsets of the DHM one-root class; (iii) is its
additional-overlap class. Coefficients carry only a fixed cutoff power, and
the leading tree coefficient is exactly $(-1)^{\#\mathrm O}$ with the
labelled factorial $\mu^{n-1}/(n-1)!$.
\end{lemma}
\begin{proof}
Apply \eqref{eq:ursell-tree-bound} to the single retained block and refine
each autonomous C-cluster by its ordered accepted collisions; the internal
indicator is at most the prescribed-C-contact indicator. Each chosen
cluster-tree edge demands a prospective accepted encounter between two
isolated histories. A cluster of $q$ particles has $\le q-1+\Gamma$
collisions, so each pair of straight segments has $\le1$ incoming contact;
partition the encounter event by the first admissible segment pair, giving
disjoint encounter indicators with weights in $[0,1]$. Ordering these along
lines gives particle-line molecules. C-clusters plus the O forest determine
the connected block with sign $(-1)^{\#\mathrm O}$ and global excess equal
to the sum of C-cluster excesses, so cyclic terms and $n\ge\Lambda_i/3$
terms are exactly (i)--(ii), inside the DHM caps $\Lambda_i<\infty$,
$\Gamma$. A remaining tree molecule with no other encounters has autonomous
C-clusters equal to the displayed ones, every O contact the unique selected
one, and unused factors equal to one; otherwise the first undesignated
accepted contact (or a discarded compatibility factor, or a nonfirst
cluster-pair contact) inserts an O-atom that changes no velocity, giving
(iii). Conversely every small unrestricted tree with no further encounter
is admitted with the displayed sign, so no symmetry factor is dropped.
\end{proof}

\begin{proof}[Proof of \Cref{prop:one-root}]
\Cref{prop:initial} gives $\|g_0-f_0\|_{\Bol,\beta_0}\le C\eps$, which is
the initial smallness for $\theta_0<1$. Assume the bound at the preceding
interface. \Cref{lem:one-root-bridge} plus the fixed-endpoint part of
\Cref{thm:dhm-package} bounds the three error families and compares the
signed small-tree sum with $f_{i\tau}$, under the preceding
$\Bol,\beta_{i-1}$ error, to $o(\eps^{\theta_i})$, giving the induction and
the uniform $\Bol$ bound. The estimate is a fixed-endpoint $\Bol$ one, not
an integrated $L^1$ inference, so it also gives density representatives;
restricting contact times gives the uniform partial-slab version.
\end{proof}

\subsection{Marked retained hierarchies and exact resummation}

Fix finitely many observation pulses. Let $m$ bound both the number of
pulses and of record-carrying labels; record coordinates lie in a fixed
bounded interval. For a root set $S$ and required records on $R\subseteq S$,
let $F^R_{i,S}$ be the pasted factorial measure at interface $i$ with record
coordinates on $R$. The operators $\mathcal L_i$ append the slab
observations on the same labels; compatibility is record-independent. The
\emph{retained hierarchy} is the finite composition
\begin{equation}\label{eq:retained-hierarchy}
 \widehat F^R_i=\mathcal L_i\mathcal L_{i-1}\cdots\mathcal L_1F^R_0,
\end{equation}
well defined order by order (finitely many previous orders per output).

\begin{lemma}[Resummation with records]\label{lem:marked-regrouping}
There is an exact finite expansion
\begin{equation}\label{eq:dependent-subsets}
 \widehat F^R_{i,S}=\sum_{R\subseteq H\subseteq S}E^R_{i,H}\otimes g_i^{\otimes(S\setminus H)}.
\end{equation}
The $E^R_{i,H}$ are represented by layered diagrams: every retained block
in slab $i$ contains a root of the interface above; a block is retained if
it has $\ge2$ such roots or meets a dependent input label below; all other
blocks resum to the independent factor $g_i$ on their single root; every
record-bearing line continues to time zero. Forgetting a record commutes
with \eqref{eq:retained-hierarchy}.
\end{lemma}
\begin{proof}
At time zero expand the hierarchy into exact initial cumulant blocks; put
every label of a nontrivial block and of $R$ in $H$, unmarked singletons
into $g_0$ on the complement, marked singletons into $H$ with their
initial record. Insert the preceding expansion into
\eqref{eq:retained-operator}; blocks with one new root and disjoint from the
dependent set $H_-$ have independent $g_{i-1}$ inputs and sum to
\eqref{eq:one-root-definition} (carrying no record since $R\subseteq H_-$);
keep the rest and call their new root set $H$. This gives
\eqref{eq:dependent-subsets} and the rule; record labels stay in $H_-$ at
every step, so their lines continue. Fresh labels on distinct blocks are
disjoint with multinomial coefficients, and
$\frac1{(j_1+j_2)!}\binom{j_1+j_2}{j_1}=\frac1{j_1!j_2!}$ permits the
separate sums; everything is finite. Record forgetting commutes with the
root-output kernel \eqref{eq:retained-kernel} (record-free weight) and with
the initial updates, so it commutes with the composition.
\end{proof}

Regard each nontrivial initial cumulant block as a connection. A block on
$k$ labels is dominated by $\prod f_0$, $(Ck)^{Ck}$, and a root tree with
edge kernel \eqref{eq:init-link}; the sharper kernel
$\ind_{|x-x'|\le C_{D,k}\eps}+\eps^D$ holds for bounded $k$. These dominate
the exact blocks; artificial links for marked lines are not connections.

\begin{lemma}[Labels and connected components]\label{lem:diagram-labels}
The retained expansion factors over global components. If $D^R_{i,B}$ is
the sum of diagrams connected on roots $B$, then
\begin{equation}\label{eq:component-exp}
 \widehat F^R_{i,S}=\sum_{\pi\in\mathcal P(S)}\bigotimes_{B\in\pi}D^{R\cap B}_{i,B}.
\end{equation}
Hence factorial-cumulant inversion keeps exactly the connected diagrams,
even after resumming into $g_i$. At fixed root order $h=|S|$ and bounded
complexity $\rho\le K$, with $n_i$ lines and $j_i$ fresh labels in slab $i$,
\begin{equation}\label{eq:bounded-crossing}
 n_i-j_i=O_{K,h,m,L}(1),\qquad
 \prod_i\frac1{j_i!}\le\frac{(1+\sum_in_i)^{C_{K,h,m,L}}}{\prod_in_i!}.
\end{equation}
Apart from ordinary contact pair/order choices, all distinguished-label,
gluing, initial-block, and record-placement multiplicities are bounded by a
fixed polynomial in total size times $C^b$, with base $C$ independent of $K$
and the moment order; in the coarse regime the same choices cost a cutoff
power with exponent $O(\rho)$.
\end{lemma}
\begin{proof}
Expand each $g_i$ to time zero (algebraically only). Weights are label-local;
size and excess caps are componentwise; every global component contains a
designated root (each slab block meets a root above). Distributing $j$ fresh
labels with sizes $j_1,\dots,j_q$ gives coefficient
$\frac1{j!}\frac{j!}{j_1!\cdots j_q!}=\prod_a\frac1{j_a!}$, so everything
factors; group components to get \eqref{eq:component-exp}, and finite
partition inversion gives the cumulant statement. A one-root resummation
contracts a history attached through one line and cannot split or merge
components. For counting, the only non-fresh labels in a slab are
boundedly many roots/continued labels, giving the first part of
\eqref{eq:bounded-crossing} and then, via $n!/(n-d)!\le n^d$, the second.
There are $2^b$ C/O choices; once the ordered C/O graph is fixed its
C-components are the clusters and its O-forest the blocks, so no extra
partition enters; the exploration has one branch per selected forest.
Bounded distinguished choices cost fixed powers of total size, with
exponent controlled by crossings and initial links.
\end{proof}

\subsection{Variation errors and removal of high complexity}

\begin{lemma}[Stopped variation errors]\label{lem:stopped-variation}
For $L$ large in terms of the fixed data and $\Gamma$, any fixed record
bound, and $|S|\le A_i/2$, there are $c_i>0$ with
\begin{equation}\label{eq:retained-variation-error}
 \|F^R_{i,S}-\widehat F^R_{i,S}\|_{\TV}\le e^{-c_i\Lambda_i},
\end{equation}
uniformly for a partial final slab (no velocity weight claimed).
\end{lemma}
\begin{proof}
Write $\Delta_i=F_i-\widehat F_i$. Then
$\Delta_i=R_i+\mathcal L_i\Delta_{i-1}$, $|R_i|\le\mathcal E_iF_{i-1}$,
$\Delta_0=0$. Substitute $F_{i-1}=\widehat F_{i-1}+\Delta_{i-1}$ in the
positive stopped bound and expand the retained part as in
\Cref{lem:marked-regrouping}; refining autonomous histories into C graphs
and picking one incoming contact per incompatibility gives exactly the
positive C/O histories of the coarse package with previous-interface
continuations. The only top-slab difference, a stopped component with
$M_i\le\cdot<2M_i-2<\Lambda_i$ clusters, is in the same coarse family with
atom count $\ge M_i-1$. Bookkeeping: the lower independent inputs are
already resummed into $g$; we do not resum the top stopped forest. Top
components failing continuation are paid by \Cref{lem:dummies}(i)
($\eps^{-C_*|S|}$), and bottom continuation of a record line with no
initial link uses \Cref{lem:dummies}(ii)--(iii) (add a unit-mass Gaussian
Bol dummy and an initial link $\ge\eps^\upsilon$, cost $\eps^{-\upsilon}$,
bounded many). Remaining label and initial-block factors are the cutoff
powers of \Cref{lem:diagram-labels}. Choosing $L$ so
$C\tau^{c_0}<1$ and using $\Lambda_i=A_i^{30}$, $A_i\ge|\log\eps|$ makes the
top loss absorb into the exponential size gain,
\begin{equation}\label{eq:stopped-coarse-estimate}
 \|\mathcal E_i\widehat F_{i-1}\|_{\TV}\le
 \eps^{-C_*|S|-C_m}\sum_{b\ge M_i-1}(C\tau^{c_0})^b
 \sum_{\rho\ge0}|\log\eps|^{C_*\rho}\eps^{c_*\rho}\le e^{-c\Lambda_i}.
\end{equation}
Input orders are $\le(|S|+1)\Lambda_i^2/2<A_{i-1}/2$; the $\Delta_{i-1}$
terms are bounded by \eqref{eq:crude-operator}, and
$A_i\Lambda_i^2\log(1/\eps)=o(\Lambda_{i-1})=\Lambda_i^{30}$ absorbs them.
This is the induction; restricting the last slab preserves every bound.
\end{proof}

\begin{theorem}[Marked component expansion]\label{thm:marked-expansion}
Choose $\Gamma$ then $L$ as above, independent of the moment order and the
observation list. On this single pasted dynamics:
\begin{enumerate}[label=(\roman*)]
\item The factorial hierarchy with any fixed finite record list has the
  finite componentwise expansion \eqref{eq:retained-hierarchy} with variation
  error \eqref{eq:retained-variation-error}; independent inputs are the $g_i$
  of \Cref{prop:one-root}; label weights, continuations, and record
  projections are those of \Cref{lem:marked-regrouping,lem:diagram-labels}.
\item At fixed final order $h$, the actual normalized marked cumulant
  $c_h^\eps$ is the sum of retained globally connected diagrams plus an
  error of total variation $\le C_he^{-c\Lambda_L}$. Nontrivial initial
  blocks count as connections; dummy links do not.
\item For any accuracy power $P$, one can choose $K$ (after the record list
  and $h$) so that all retained diagrams with $\rho>K$ have summed absolute
  variation $\le\eps^P$; at $\rho\le K$ \eqref{eq:bounded-crossing} holds.
  Increasing $K$ or $h$ never changes the exponential base per ordinary atom.
\end{enumerate}
\end{theorem}
\begin{proof}
(i) combines the preceding results. For (ii), invert both the actual and the
retained hierarchies: \eqref{eq:component-exp} gives the connected sum for
the retained one, and every term in the difference of the two finite
partition polynomials carries a factor $\Delta_{L,B}$. The actual
normalized factorial measures have mass $\le1$ (initially by the insertion
bound of \Cref{prop:initial}, and conserved since $N$ is preserved), and
the retained ones have variation $\le1+e^{-c\Lambda_L}$ at fixed orders by
\Cref{lem:stopped-variation}. For (iii) apply the coarse bound to each
resummed core with the same fixed record and top-root losses; independent
top singletons are bounded by \Cref{prop:one-root}; for small $\eps$,
$|\log\eps|^{C_*\rho}\eps^{c_*\rho}\le\eps^{c_*\rho/2}$, so after summing
the geometric atom factor the $\rho>K$ part is
$\le C_{h,m}\eps^{-C_*h-C_m}\eps^{c_*K/2}$; raise $K$ to make it
$\le\eps^P$. Dummy lines change complexity by a fixed amount and are
removed before low-complexity asymptotics. All root-order and observation
dependence sits in cutoff powers or fixed polynomial factors of
\eqref{eq:bounded-crossing}; the exponential atom base depends only on the
fixed graph counting, the $\Bol$ bounds, and $\Gamma$, so the chosen $L$
works for every subsequently fixed order.
\end{proof}

\subsection{Terminal witnesses without observed roots}

The comparison with the true flow (\Cref{sec:cutoff}) needs a positive
version with a connected \emph{final collision witness} instead of observed
outputs, and without any bound on the cutoff probability.

\begin{corollary}[Terminal-witness expansion]\label{cor:terminal-witness}
In slab $i$ let a \emph{terminal witness} be a connected true C history on
$n\le3\Lambda_i$ incoming labels, excess $\le2\Gamma$, stopped at a contact
in that slab, with arbitrary further nonnegative realizability
restrictions. Its activity-normalized functional
\begin{equation}\label{eq:terminal-witness-functional}
 \sum_n\frac{\mu^n}{n!}\int F_{i-1,n}(\dd Z_n)\ind_{\text{terminal witness}}(Z_n)
\end{equation}
has the retained layer expansion with independent $g$ inputs and initial
blocks. All lower C/O components attach through particle lines to the
witness. The preceding variation error is smaller than every fixed power of
$\eps$. After the artificial-top-root normalization $\mu$, the coarse part
with $\rho>K$ is also $<$ any assigned power for large $K$; the remaining
low-complexity witness integrals keep their realizability restrictions.
\end{corollary}
\begin{proof}
Here $n\le3\Lambda_i<A_{i-1}/2$, so insert the preceding expansion from
\Cref{thm:marked-expansion} into \eqref{eq:terminal-witness-functional}.
Every lower component has a root leading above and the final roots enter
the connected witness (attachment before discarding any initial link). The
variation error contributes at most
$\sum_{n\le3\Lambda_i}\mu^n/n!\,e^{-c\Lambda_{i-1}}\le
e^{C\Lambda_i\log(1/\eps)-c\Lambda_{i-1}}$, which is super-polynomially
small. Designating an artificial top root places the dependent diagrams in
the terminal class at cost $\mu$; with no dependent continuation the
diagram lives in the terminal slab with independent inputs and complexity
$\le1+2\Gamma$. The same geometric complexity summation, with this fixed
power, proves the last assertion. For $i=1$ there is no preceding variation
error and the argument starts from the initial cumulant expansion. Nothing
here compares true and pasted evolutions.
\end{proof}

% =====================================================================
\part*{Part 5. The sharp size of every correlation}
% =====================================================================

\section{Sharp bounds and the limiting tree measures}\label{sec:sharp-asymptotics}

This is the central new estimate. The [DHM] package already removed large
complexity (\Cref{thm:marked-expansion}(iii)); here we control what
remains at the \emph{sharp} fluctuation scale $\mu^{1-h}$ for the $h$-th
factorial cumulant. The mechanism is a change of \emph{relative
translations}: every collision that merges two clusters of particles
adjusts one relative position by the collision geometry, producing a
factor $\eps^2[(v-v_*)\cdot\omega]_+\,\dd t\,\dd\omega$. After accounting
for the activity $\mu$ per birth line, a connected diagram with $p$
physical initial links scales like $\mu^{1-h}\eps^p$, which is exactly the
size we want; cycles and nontrivial initial correlations are smaller by a
positive power (proved via a null-set/dominated-convergence argument). The
sum over all cluster structures converges because each cluster slab has a
life time $\tau=T/L$ and the base $CT/\sqrt L<1$ for large $L$.

\subsection{Conventions and the assertion}

Fix finitely many observation times and bounded continuous observation
functions. When observations are accumulated in a mark, update
$y\mapsto y+\phi_j(x,v)$, so $|y|\le M$ with $M$ independent of the
number of particles. More generally one may retain finitely many phase
points along each designated trajectory. A record is kept only along a
line on which it is required; forgetting part of a record means pushing
the measure forward under the corresponding coordinate projection, which
commutes with the finite hierarchy before regrouping by
\Cref{thm:marked-expansion}.

Let $c_h^\eps$ be the ordered normalized factorial cumulant measure of the
pasted process, with records on its $h$ designated particles; thus $c_1^\eps$
is the one-particle measure and the normalization at order $h$ is
$\mu^{-h}$. Constants $C_K$ may depend on a fixed complexity bound $K$, on
the fixed observations, and on $L$; the exponential base $C^n$ never does.

The independent factors of \Cref{prop:one-root} satisfy, for some $b>0$ and
$B_0<\infty$,
\begin{equation}\label{eq:sharp-input-bounds}
 \sup_{0<\eps\le\eps_0,\,i}\|g_i\|_{\Bol,b}\le B_0,\qquad
 \|g_i-f_{i\tau}\|_{\Bol,b}\longrightarrow0 .
\end{equation}
We decrease $b$ a fixed number of times when reserving Gaussian weights;
this does not change the microscopic dynamics.

\begin{theorem}[Sharp connected bounds]\label{thm:sharp-cumulants}
Choose $\Gamma,L$ as in \Cref{thm:marked-expansion}, increasing $L$ if
necessary by an amount depending only on the Boltzmann bounds and on $T$.
This single choice works for every fixed factorial order $h$ and every
fixed bounded observation list. For each fixed $h$,
\begin{equation}\label{eq:sharp-full}
 \|c_h^\eps\|_{\TV}\le C_h\mu^{1-h}.
\end{equation}
For $h=1,2$, the measures $\mu^{h-1}c_h^\eps$ converge against bounded
continuous record tests to the signed sums of retained global trees with
trivial initial cumulants, with contact weight
$[(v-v_*)\cdot\omega]_+\,\dd t\,\dd\omega$ (scattering $+1$,
non-scattering $-1$) and independent birth factors $f_{i\tau}$. The sums
of absolute retained diagram measures, after multiplication by
$\mu^{h-1}$, are bounded uniformly with a small Gaussian weight in their
designated endpoint velocities; the same holds for the limiting tree
measures and for Gaussian weights in finitely many record velocities after
decreasing the exponent. Among retained connected diagrams, the sum of the
absolute measures having a nontrivial initial cumulant or a global cycle
is $o(\mu^{1-h})$. These weighted assertions concern retained diagrams and
their limits, not the microscopic expansion remainder.
\end{theorem}

A \emph{global tree} is a tree on the particle lines with contacts as
edges; continuing one line through an interface does not create a vertex,
while newly integrated particles do. This convention is essential for
counting translations and activity factors.

\subsection{The finite graph and its numerical data}

Work with a term of complexity $\le K$. Replace each initial cumulant by
its bound from \Cref{prop:initial},
\begin{equation}\label{eq:sharp-initial-link}
 C_K\sum_{\mathcal L\text{ forest}}\prod_{(p,q)\in\mathcal L}
 \bigl(\ind_{|x_p(0)-x_q(0)|\le R_{K,D}\eps}+\eps^D\bigr)
 \prod_pf_0(z_p(0)),
\end{equation}
only for absolute bounds (not to sign an initial link). Expand the
parentheses and suppose first that all chosen links are physical.

Let $m$ be the number of distinct lines, $c$ the number of global
components, $p$ the physical initial links, $e_i$ the contacts in slab
$i$. The initial link graph is a forest. Scan contacts in time order: a
contact is a \emph{merger} if its two lines lie in different components
just before it, else \emph{redundant}. With $a_i$ mergers in slab $i$ and
$r$ redundant contacts,
\begin{equation}\label{eq:sharp-merger-count}
 \sum_ia_i=m-c-p,\qquad r=\sum_ie_i+p-m+c,
\end{equation}
the second being the global cycle excess. With $n_i$ lines present in slab
$i$ (including contact-free lines) and $q$ interface crossings,
\begin{equation}\label{eq:sharp-line-count}
 \sum_in_i=m+q,\qquad \sum_i(n_i-a_i)=q+c+p.
\end{equation}
Merger edges in one slab form a forest on its $n_i$ lines, so $a_i\le n_i$.
All of $q,p,c,r$ and $n_i-a_i$ are bounded by $K$: glue layer graphs along
interfaces to see $r$ (total excess grows by at most the number of
gluings, all in $\rho$); each layer component meets a root so its count is
bounded; erasing boundedly many initial links changes this boundedly.

Each birth line has one independent input velocity and position; a
continued line has neither. The cancellation of activity at interfaces
leaves, in the normalized factorial expansion,
\begin{equation}\label{eq:sharp-activity}
 \mu^{m-h}.
\end{equation}
A birth at an interface of a $g_i$-supplied line is an independent phase
variable; the passage of a recorded line is a continuation of its old
variable.

\subsection{Changing the relative translations}

\begin{lemma}[Chronological merger coordinates]\label{lem:merger-coordinates}
Fix a finite line graph, the C/O choices, the chronological order, and
which contacts are redundant. Away from nontransversal and simultaneous
events, its history integral has coordinates: the birth velocities, one
absolute translation per global component, a vector in a ball of radius
$R_{K,D}\eps$ per physical initial merger, and $(t,\omega)\in\R\times
S^2$ per dynamic merger. The positional Jacobian per dynamic merger is
\begin{equation}\label{eq:sharp-jacobian}
 \eps^2[(v-v_*)\cdot\omega]_+\,\dd t\,\dd\omega .
\end{equation}
A specified redundant contact has no free continuous parameter. The
coordinate map has multiplicity at most one on each incoming transversal
branch.
\end{lemma}
\begin{proof}
Take birth velocities first. Each line begins with an independent
translation of its free trajectory. An initial forest link fixes the
difference of two translations by a vector in the stated ball; orienting
the forest toward one representative per group has determinant one.

Proceed in increasing contact time. Within each group the histories are
known up to a common translation. A merger of groups with translations
$X,X_*$ on the indicated straight segments satisfies
$X-X_*=-d(t)-\eps\omega$ with $d(t)$ the known relative displacement,
$\dot d=v-v_*$, and $\omega$ opposite the outward normal, so the incoming
condition is $(v-v_*)\cdot\omega>0$. The determinant of
$(t,\omega)\mapsto X-X_*$ on the sphere is
$\eps^2|(v-v_*)\cdot\omega|$, and the incoming convention selects the sign
in \eqref{eq:sharp-jacobian}. A scattering reflection or O-continuation
updates velocities; the two groups now share one translation.

If the next specified contact is redundant, its relative displacement and
velocities are already fixed. A straight relative flight $r(t)=d+tu$ has
at most one incoming crossing, and
$\int_I\delta(|r|-\eps)[-u\cdot r/\eps]_+\dd t=\ind_{\{\text{incoming
crossing in }I\}}$, since the flux cancels the absolute derivative of
$|r|-\eps$; the outgoing deltas then make a deterministic substitution.
If the crossing is absent the branch has weight zero. Segments starting
inside the sphere have an exit, not an incoming crossing.

Every step uses only earlier coordinates; the full determinant is
triangular with the displayed diagonal entries, so there is no new
relative translation at a redundant step. Contacts at different interfaces
are treated identically (different affine origins, same current-velocity
derivative). Keep one representative translation per component; on an
incoming transversal branch the reverse construction recovers every
relative translation, giving multiplicity one. Excluded boundaries and
ties are null in merger coordinates unless retained as a redundant-contact
indicator, where no change of variables is made. Work first on compact
subsets of transversal branches and exhaust.
\end{proof}

Multiplying \eqref{eq:sharp-activity} by the merger Jacobians and the
physical-link volumes,
\begin{equation}\label{eq:sharp-scaling}
 \mu^{m-h}\eps^{2(m-c-p)}\eps^{3p}=\mu^{c-h}\eps^{p}.
\end{equation}
All acceptance indicators can be carried through the change of variables;
for an upper bound they may be dropped, except that a redundant scattering
determines its branch velocities when it occurs (extend by zero where it
does not). For a connected diagram $c=1$, and a physical initial block has
$p\ge1$ so gains a factor $\eps$. A term of \eqref{eq:sharp-initial-link}
using a tail factor: erase that link; erasing $\le K$ links raises $c$ by
$\le K$ and costs $\le\eps^{-2K}$ in \eqref{eq:sharp-scaling}; choosing
$D>2K+M$ makes such terms $O(\eps^M\mu^{1-h})$, with $M$ arbitrary and
chosen after the moment order.

\subsection{Factorials and a summable majorant}

\begin{lemma}[Fresh-label factorials]\label{lem:fresh-label}
For complexity $\le K$, the absolute sum with slab sizes $(n_1,\dots,n_L)$
is bounded by a sum over chronologically ordered merger pair lists with
denominator $\prod_in_i!$, times
\begin{equation}\label{eq:sharp-poly-count}
 C_K(1+n)^{d_K}C^{\sum_ia_i},\qquad n=\sum_in_i,
\end{equation}
with base $C$ independent of $K$ (the merger pairs themselves are not
counted here).
\end{lemma}
\begin{proof}
In slab $i$, $s_i$ labels arrive as roots from above (the final roots in
the top slab) and $j_i=n_i-s_i$ are fresh, integrated with coefficient
$1/j_i!$. Since $s_i\le K$,
$\frac1{j_i!}=\frac{n_i!}{(n_i-s_i)!}\frac1{n_i!}\le
\frac{(1+n_i)^{s_i}}{n_i!}$. Continued labels, dependent subsets, and
initial-link labels are boundedly many, costing a fixed power of $1+n$.
The rooted compatibility expansion represents connected coefficients by
spanning forests whose deterministic exploration has one branch per
selected forest; after selecting one incoming encounter per forest edge
the C-components are the collision clusters and the O-forest the blocks,
so no extra partition enters, and the fresh-label denominator survives.
There are $r\le C_K$ redundant contacts costing $(1+n)^{2r}$ choices and
$(1+n+r)^r$ insertions; C/O flags give an absolute exponential base; a
chronological order already fixes the line order so has no extra
permutation factor.
\end{proof}

\begin{lemma}[Uniform summability]\label{lem:summability}
For fixed $K$, the sum of absolute connected diagram measures with slab
sizes $(n_i)$, after removing $\mu^{1-h}$, is bounded by
\begin{equation}\label{eq:sharp-summable}
 C_K(1+n)^{d_K}(C\tau\sqrt L)^n .
\end{equation}
The same bound (with the corresponding $\mu^{c-h}$) holds after erasing
boundedly many initial links, after prescribing the absolute position of
one observation per component, or after inserting a small Gaussian weight
in the designated velocities.
\end{lemma}
\begin{proof}
We bound energy and time jointly to avoid a logarithmic loss per vertex.
Let $E_0=\sum_{p=1}^m|v_p^{\mathrm{birth}}|^2$. Elastic collisions conserve
every colliding-pair energy, O-contacts do not change it, a line ending at
an interface discards its energy, and a birth adds energy already in
$E_0$. Hence the sum of squared velocities active at any time is $\le E_0$.

On the bin $E_0\in[P/2,P]$ (with $E_0<1$ by $P=1$), summing the incoming
speed over all merger-pair choices in slab $i$ gives
\begin{equation}\label{eq:sharp-pair-flux}
 \sum_{p<q}|v_p-v_q|\le(n_i-1)\sum_p|v_p|\le n_i^{3/2}\sqrt P .
\end{equation}
Applied to a discrete tree of successive pair choices at fixed merger
times, normals, and birth velocities, where earlier velocities and
intervening specified redundant contacts are already determined by
\Cref{lem:merger-coordinates}, induction bounds the product of fluxes by
\begin{equation}\label{eq:sharp-pair-products}
 \prod_i(n_i^{3/2}\sqrt P)^{a_i}.
\end{equation}
Integrating normals adds $C^{\sum_ia_i}$; merger times are globally ordered
within a slab, leaving $\prod_i\tau^{a_i}/a_i!$.

A common translation remains per component. For fixed birth velocities and
merger coordinates, write the input spatial arguments as $X+d_p$,
integrate one chosen input in $X$, and use the spatial supremum of the
others; \eqref{eq:sharp-input-bounds} gives $\le C^m\exp(-bE_0)$ uniformly
in the displacements. To prescribe an observation position, solve
$X=x-d_{\mathrm{obs}}$ with Jacobian one, same bound uniformly in $x$.

For the velocity volume, reserve half the Gaussian: with $b_1>0$,
$\ind_{E_0\in[P/2,P]}e^{-2b_1E_0}\le e^{-b_1P/2}e^{-b_1E_0}$, whose last
factor integrates over $3m$ birth velocities to $(\pi/b_1)^{3m/2}$ (the
lowest bin uses $P^{a/2}=1$ and costs only $C^m$). With $a=\sum_ia_i$,
\begin{equation}\label{eq:sharp-energy-sum}
 \sum_{P=1,2,4,\dots}e^{-b_1P/2}P^{a/2}\le C^{a+1}(1+a)^{a/2},
\end{equation}
by comparison with $\int e^{-b_1u/4}(1+u)^{a/2}$ and Stirling.
Combining with \Cref{lem:fresh-label}, the remainder is a fixed polynomial
in $1+n$ times
\begin{equation}\label{eq:sharp-before-stirling}
 C^{n+a}\tau^a(1+a)^{a/2}\prod_i\frac{n_i^{3a_i/2}}{a_i!n_i!}.
\end{equation}
With $\delta_i=n_i-a_i\ge0$ and $\sum_i\delta_i\le C_K$, and
$n_i!/a_i!\le(1+n_i)^{\delta_i}$, $n_i!\ge(n_i/e)^{n_i}$,
\[
 \frac{n_i^{3a_i/2}}{a_i!n_i!}\le e^{2n_i}(1+n_i)^{\delta_i}n_i^{-n_i/2}.
\]
Also $\tau^a\le\tau^n\max(1,\tau^{-1})^{C_K}$ and
$(1+a)^{a/2}\le C(1+n)^{C_K}n^{n/2}$, and convexity of $u\log u$ gives
\begin{equation}\label{eq:sharp-layer-entropy}
 \prod_in_i^{n_i}\ge(n/L)^n .
\end{equation}
Substitution proves \eqref{eq:sharp-summable}. For weights: the sum of
squared designated velocities is $\le E_0$ at a fixed time and
$\le q_0E_0$ at $q_0$ record times, so a weight exponent below
$b_1/(2\max(1,q_0))$ is absorbed by the reserved Gaussian fraction; the
exponent shrinks with the record list but the exponential base does not.
\end{proof}

Since $\tau=T/L$, the base in \eqref{eq:sharp-summable} is $CT/\sqrt L$;
increase $L$ once so it is $<1$. There are polynomially many vectors
$(n_i)$ of total size $n$ (fixed $L$), so \eqref{eq:sharp-summable} gives
uniform summability and vanishing size tails, with polynomial degrees that
may grow with $h$ but never need another enlargement of $L$.

\subsection{The extra-contact null sets}

We need a qualitative fact: a limiting tree carries no extra contact. Only
a fixed finite graph is examined. Its independent parameters are birth
velocities, contact times, merger directions, and one free translation per
component, all absolutely continuous. A claim about lines in one component
is translation-invariant and survives fixing one observation position.

\begin{lemma}[No extra contact for a limiting tree]\label{lem:no-extra}
Consider a finite chronological forest with no physical initial links and
a prescribed C/O choice, on incoming merger branches. Off a null set of
merger parameters:
\begin{enumerate}[label=(\roman*)]
\item specified contacts occur at distinct times away from interfaces and
  observation times and are transversal;
\item before the first redundant contact, that contact's requirement is
  false for all small $\eps$;
\item on a full tree, any contact not in the prescribed list (including one
  preceding the prescribed joining of its two groups) is absent for all
  small $\eps$.
\end{enumerate}
For (ii) the parameters are those of the forest prefix; the rest may be
appended. In (iii) an outgoing crossing after an O-contact is not counted.
\end{lemma}
\begin{proof}
Set $\eps=0$. Segment velocities are polynomial in birth velocities and
sphere coordinates, and displacements polynomial also in the times; the
parameter set is a connected real-analytic manifold, so a non-identically-
zero polynomial has a null zero set (grazing/ineffective reflections
extend analytically). For two lines on a common time interval, write the
relative displacement $d+tu$; a coincidence requires the collinearity
\begin{equation}\label{eq:collinearity}
 d\times u=0 .
\end{equation}
If the lines lie in different forest components, vary their free relative
translation. Otherwise, with ineffective reflections, a non-neighbor path
$p_0,\dots,p_k$, $k\ge2$, with edge times $s_1,\dots,s_k$ gives an endpoint
displacement built from $(v_{p_{j-1}}-v_{p_j})s_j$; the intermediate
velocity $v_{p_j}$ ($1\le j<k$) occurs with coefficient
$s_{j+1}-s_j\ne0$, so varying it off the endpoint velocity direction
disproves \eqref{eq:collinearity}.

Neighbors meeting at time $a$: after $a$ with no later velocity change the
relative flight is the single separating flight, which has no second
incoming crossing (an O-contact passes out once). If an endpoint scatters
at $b>a$, the third particle is distinct; choose its velocity and the
normal so the pair's relative velocity jumps from $u_1$ to $u_2$
nonparallel (e.g.\ endpoint velocity $e_1\to e_2$, third particle $-e_1$,
normal $(e_1-e_2)/\sqrt2$), then the displacement contains
$(b-a)u_1+(t-b)u_2$ with nonzero cross product against $u_2$. Before the
meeting, an intervening scattering $b<a$ with the same jump forces a
displacement $(t-b)e_1-(a-b)e_2$ not parallel to $e_1$.

Zero relative velocities and grazing contacts are null; at a merger the
incoming relative velocity is a nonzero linear function of independent
birth velocities and the normal is a free sphere parameter. Time
coincidences are null. A third line at a prescribed vertex has a nonneighbor
path, killed above. Finally, formal tree velocities are $\eps$-independent:
$x_p^\eps(t)=x_p^0(t)+\eps d_p$ with $d_p$ a fixed sum of merger normals,
so convergence is uniform on each active interval. Away from small
neighborhoods of prescribed vertices every pair of segments has positive
separation or is the treated separating flight; near a vertex a different
pair needs an excluded zero-offset coincidence, and the incident pair has
no other velocity change in a small neighborhood, so it has only its one
incoming crossing. The same compactness handles interfaces and observation
times, proving (ii)--(iii) including contacts whose times vary with $\eps$.
\end{proof}

For a diagram with a cycle, keep its first redundant-contact indicator. All
prior contacts form a forest, and the tested relative trajectory is
determined by that forest prefix (no later redundant update enters). By
\Cref{lem:no-extra} it tends to zero for a.e.\ earlier parameters. Truncate
size, velocities, and translations; on the compact pieces the post-change
integrands have an integrable bound (from the finitely many speed factors
or from the proof of \Cref{lem:summability}), extend unrealizable branches
by zero, and apply dominated convergence. Remove velocity/spatial
truncations with the Gaussian bounds and the spatial tightness from
\eqref{eq:sharp-input-bounds}, then the size truncation with
\eqref{eq:sharp-summable}. This order needs no size-uniform rate and proves
the asserted $o$ for cycles and nontrivial initial cumulants.

\subsection{Tree convergence and spatial disintegration}

For a fixed tree all prescribed C-clusters and O-blocks fall below the caps
once $\eps$ is small; by \Cref{lem:no-extra} almost every merger parameter
set has only prescribed incoming contacts for all small $\eps$. Thus every
acceptance restriction holds, counting an encounter or its existence is the
same, and the connected compatibility coefficient equals its tree sign
$(-1)^{\#\mathrm O}$. Replacing $g_i$ by $f_{i\tau}$ in a fixed-size term is
justified by a telescoping product and \eqref{eq:sharp-input-bounds}, with
\Cref{lem:summability} (one factor replaced by its $\Bol$-difference)
giving convergence to zero. Dominated convergence in space (checked first
on bounded velocities/size, then removing truncations with
\eqref{eq:sharp-summable}) defines the limiting signed tree measures.

\begin{lemma}[Spatial traces of tree measures]\label{lem:spatial-traces}
Let $\mathcal T$ be one of the summable tree measures above, with a line
and a time at which its position is required. It has a disintegration
\begin{equation}\label{eq:disintegration}
 \mathcal T=\dd x\,\mathcal T_x,
\end{equation}
with Gaussian-weighted variation of $\mathcal T_x$ bounded uniformly in $x$
and finite integral over $x$. For every bounded continuous test of the
other coordinates (and continuous tests of smaller Gaussian growth),
$x\mapsto\mathcal T_x(\Phi)$ is continuous. If two tree components are
joined at their designated positions, the product of their conditional
measures at a common position, integrated against incoming flux, is an
absolutely defined measure; it equals imposing the joining position in
merger coordinates, or a smooth approximate identity for the equality.
This applies to marked one-root and connected two-root trees and their
record projections.
\end{lemma}
\begin{proof}
Choose the free representative translation to be the designated position
$x$: every birth position is $x+d_p$, every recorded position likewise,
with displacements independent of $x$ and derivative the identity. This
gives \eqref{eq:disintegration} as an integral formula, with both variation
bounds from the fixed-position and integrated versions of
\Cref{lem:summability}. After size/velocity truncation the test integrand
is continuous in $x$ with a uniform integrable majorant, and the tails
converge uniformly in $x$ by the fixed-position estimate; a stronger
reserved Gaussian handles Gaussian growth. For two components with
positions $x,x_*$, setting $x_*=x$ gives the conditional product, whose
variation is finite by $B(v-v_*,\omega)\le|v|+|v_*|$ (one component
conditional, the other integrated); the same bound dominates a compactly
supported approximate identity in $x-x_*$ uniformly, and on bounded
velocity sets conditional weak continuity gives convergence against
product tests, extended by the variation bounds.
\end{proof}

\subsection{Completeness and splitting of the retained trees}

\begin{lemma}[Completeness of the tree class]\label{lem:tree-completeness}
For fixed roots, required records, and $L$, all global trees with trivial
initial cumulants in the regrouped retained expansion have complexity
$\le K_0$, independent of their number of ordinary vertices. Undoing and
reapplying an unmarked one-root regrouping preserves their weights and
record projections.
\end{lemma}
\begin{proof}
An unmarked dependent one-root component cannot be a tree with trivial
initial cumulants. In its highest occupied slab a one-root component is a
single connected block; if all lower inputs are independent it is a term of
$g_i$ and regroups away. If a lower dependent component attaches through
two lines, the upper and lower connections form a global cycle; so for a
global tree each lower dependent component attaches through $\le1$ line,
and at least one must exist for the upper component to stay dependent.
Following it down repeats the argument; at time zero a dependent
nonsingleton needs a nontrivial initial cumulant. Downward induction proves
the claim; required records are the only cause to expand an independent
singleton further.

Cut a global forest at an interface and contract upper components to
vertices. With $u$ such, a lower component with $k\ge2$ attachments meets
distinct upper components (else a cycle), so the bipartite incidence graph
is a forest; with $v$ lower vertices of degree $\ge2$ and $d$ incidences,
$d\le u+v-1$ and $2v\le d$, giving
\begin{equation}\label{eq:crossing-bound}
 d\le2u.
\end{equation}
A lower component attached at one line is either absorbed (first
paragraph) or continues a required record, contributing at most the fixed
number of recorded lines. Starting from the fixed final roots and
iterating \eqref{eq:crossing-bound} through $L$ slabs bounds all interface
roots and recorded singletons: this is $K_0$ (no cycle or initial-link
contribution in a leading tree). Regrouping an unmarked one-root component
is the defining finite sum of its factor with the original fresh-label
coefficient, attaching at one line, so it neither splits nor merges
components; \Cref{thm:marked-expansion} shows record forgetting and
regrouping commute before a limit, and \Cref{lem:summability} carries this
to the limits, preserving weights and projections.
\end{proof}

\begin{lemma}[Last-contact products]\label{lem:tree-splitting}
In the leading tree sums, deleting a last contact of the highest slab that
meets a designated root splits its global tree into the product of the two
component tree measures, incident lines designated before the contact;
their product at the collision position is the conditional product of
\Cref{lem:spatial-traces}. Conversely joining two disjoint component
histories at that contact gives the corresponding retained tree with its
original factorial weight, and unused record portions project out. Valid
for bounded continuous record tests and, with a smaller exponent, for
Gaussian velocity growth.
\end{lemma}
\begin{proof}
Temporarily expand the top-slab independent factors by their one-root
definitions, keep lower inputs canonical, and bound each limiting tree's
line count by $N$; for small $\eps$ every cap $\Lambda_i,M_i$ exceeds $N$,
so the already-proved convergence gives each such tree its original
factorial weight. Do finite algebra on this limiting class (not a closure
claim for finite-$\eps$ capped classes). Deleting a tree edge makes two
components, each holding one incident line (now designated); a component
losing its old top root keeps the incident line or another original root,
so the rooted-block rule still holds, and lower attachments are put in
canonical form by singleton regrouping. Conversely two disjoint components
plus their edge form a tree whose highest block meets the original root;
inverse regroupings restore the retained representation. Coefficients:
choosing an extra incident particle among $j$ fresh labels changes $1/j!$
to $1/(j-1)!$, and
$\binom{j_1+j_2}{j_1}\frac1{(j_1+j_2)!}=\frac1{j_1!j_2!}$ splits the
remaining labels; a contact between two designated particles uses one
unordered pair, no fresh label; the deleted contact contributes its C/O
sign, other signs multiply. Letting $N\to\infty$ passes through deletion
and joining by \Cref{lem:summability,lem:spatial-traces}, the uniform
conditional Gaussian bound controlling the joining flux with a fixed
exponent loss; records project because record forgetting commutes.
\end{proof}

\subsection{Proof of the sharp bounds}

\begin{proof}[Proof of \Cref{thm:sharp-cumulants}]
\textbf{Weighted high-complexity tail.} Take the birth exponent $b>0$ of
\eqref{eq:sharp-input-bounds} independent of $L$, and $L$ large enough for
the coarse package at the smaller exponent $3b/4$. For a fixed record list
let $J$ count designated endpoint and record velocity slots; on every
retained history the sum of their squared velocities is $\le JE_0$. Choose
$\alpha>0$ with $\alpha J\le b/4$; the weight is dominated by
$e^{bE_0/4}$, absorbed by tilting every birth input to
$|g_i(x,v)|e^{b|v|^2/4}$ (and $f_0$ likewise), which has a common
$\Bol,3b/4$ bound. The positive coarse bound \Cref{thm:dhm-package}(a)
needs only this envelope; apply it to the tilted inputs with the same
initial-link and record bounds. The geometric complexity sum makes the
weighted retained tail smaller than any power of $\eps$ after choosing $K$.
The Gaussian fraction reserved here is fixed, so its atom base is
independent of $J,h,K$.

\textbf{Assembling the scale.} Use \Cref{thm:marked-expansion} at accuracy
$\eps^M$, $M>2(h-1)$, with $K$ large enough for that accuracy and for
\Cref{lem:tree-completeness}. The expansion error and the high-complexity
tail are negligible after multiplication by $\mu^{h-1}$ (they need only the
unweighted variation bound, since observation tests are bounded).

For the remaining connected diagrams, \eqref{eq:sharp-scaling} and
\Cref{lem:summability} prove the uniform bound at scale $\mu^{1-h}$.
Physical initial links gain $\eps$; nonphysical initial tails gain any
power after choosing $D$; cycles are negligible by \Cref{lem:no-extra} plus
finite-size dominated convergence and the uniform tails; the survivors are
exactly the leading trees by \Cref{lem:tree-completeness}. Their fluxes,
signs, input limits, and record tests are evaluated above, and the same
summable majorant yields the asserted Gaussian variation bounds for their
retained sums and limits, including the one-root and two-root limits of the
next part. All exponential bases are independent of $K$ and $h$; those
quantities enter only through front constants and polynomial degrees, so
the $\Gamma,L$ choice remains the single choice made before the moment
order.
\end{proof}

% =====================================================================
\part*{Part 6. The Gaussian limit for the model dynamics}
% =====================================================================

\section{Identification of the covariance and the Gaussian limit}\label{sec:gaussian-limit}

We now run the central limit theorem on the single pasted dynamics fixed by
\Cref{thm:marked-expansion,thm:sharp-cumulants}. Its $\Gamma,L$ do not
depend on the observation list or the moment order. The method: attach a
\emph{mark} that accumulates test values along each particle, so that any
fixed multitime linear combination becomes one scalar. Its ordinary
cumulants are computed from the sharp bounds; the second converges and the
higher ones vanish, giving a Gaussian law whose covariance is identified
by propagating marked tree laws backward. Comparison with the true flow is
Part 7; exact centering is restored in Part 8.

\subsection{Records and the limiting one- and two-root measures}

Give every particle a scalar mark, initially $0$, and at time $t_j$ update
\begin{equation}\label{eq:mark-update}
 (x,v,y)\longmapsto\Theta_j(x,v,y)=(x,v,y+\phi_j(x,v)).
\end{equation}
Collisions preserve the mark of each particle. Write
$M_{\mathrm{obs}}=\sum_j\|\phi_j\|_\infty$, $\xi=(x,v,y)$, $|y|\le
M_{\mathrm{obs}}$. The mark only records deterministic trajectories; it adds
no randomness. On either dynamics,
\begin{equation}\label{eq:record-identity}
 \sum_{i=1}^Ny_i(T)=\sum_j\sum_{i=1}^N\phi_j(z_i(t_j)).
\end{equation}

Use a superscript $\eps$ for the normalized factorial cumulants of the
pasted marked process. At any observation-free time \Cref{thm:sharp-cumulants}
provides limits
\begin{equation}\label{eq:limit-F-J}
 c_1^\eps\longrightarrow F_t,\qquad\mu c_2^\eps\longrightarrow J_t,
\end{equation}
against bounded continuous tests of the records; $F_t$ is a one-root
measure and $J_t$ a symmetric signed two-root measure, both with the
Gaussian variation bounds of \Cref{thm:sharp-cumulants}. Before the updates
at time zero, \Cref{prop:initial} gives
\begin{equation}\label{eq:limit-initial-F-J}
 F_0=f_0(x,v)\,\dd x\,\dd v\,\delta_0(\dd y),\qquad J_0=0
\end{equation}
(since $\mu\|c_{2,0}^\eps\|_1=O(\eps)$). Forgetting records in the finite
marked hierarchy commutes with the one-root regroupings; by
\Cref{thm:marked-expansion,prop:one-root,lem:tree-completeness}, the
$(x,v)$ marginal of $F_t$ is $f_t(x,v)\dd x\dd v$ (also inside a slab, via
the partial-slab version \eqref{eq:signed-trees} of the one-root block).

As $F_t$ is positive, it disintegrates by \Cref{lem:spatial-traces} as
\begin{equation}\label{eq:F-disintegration}
 F_t=\dd x\,F_t(x,\dd v\,\dd y),
\end{equation}
with unmarked marginal $f_t(x,v)\dd v$ at every $x$ in the continuous
sense (a.e.\ $x$ by globality, then both sides weakly continuous). To form
collision products, use the conditional measure of a factor at its
designated position, multiply the two component measures at a common
position, integrate that position, and include
$B(v-v_*,\omega)\dd\omega$ when the incident variables collide. These are
absolutely defined; a conditional Gaussian bound for one component and the
integrated one for the other, with $B\le|v|+|v_*|$, give an integrable
majorant, equivalently replacing equality by a smooth approximate identity
and passing to width zero.

\subsection{Last-contact evolution and collision normalization}

At a collision, with $\xi=(x,v,y)$, $\xi_*=(x,v_*,y_*)$,
$\xi'=(x,v',y)$, $\xi_*'=(x,v_*',y_*)$, put
\[
 \Delta\Phi=\Phi(\xi')+\Phi(\xi_*')-\Phi(\xi)-\Phi(\xi_*),\quad
 D\Phi=v\cdot\nabla_x\Phi,
\]
and
\begin{equation}\label{eq:marked-H}
 (H_t\Phi)(\xi)=\int_{\R^3\times[-M_{\mathrm{obs}},M_{\mathrm{obs}}]\times\mathbb S^2}
 B(v-v_*,\omega)\Delta\Phi\,F_t(x,\dd v_*\dd y_*)\,\dd\omega .
\end{equation}
In the formulas below, $\int BFF_*(\cdots)$ is the simultaneous position
integral from \eqref{eq:F-disintegration}, with $\dd x\,\dd\omega$.

\begin{proposition}[Evolution of the marked tree laws]\label{prop:tree-evolution}
Between updates,
\begin{align}
 \partial_tF_t(\Phi)&=F_t(D\Phi)+\tfrac12\int BF_tF_{t,*}\Delta\Phi,
  \label{eq:F-evolution}\\
 \partial_tJ_t(\Phi\otimes\Psi)&=J_t\bigl((D+H_t)\Phi\otimes\Psi+\Phi\otimes(D+H_t)\Psi\bigr)\notag\\
 &\quad+\int BF_tF_{t,*}\bigl(\Phi(\xi')\Psi(\xi_*')-\Phi(\xi)\Psi(\xi_*)\bigr).
  \label{eq:J-evolution}
\end{align}
These hold in transport mild form, hence in weak differential form for
smooth tests for which the integrals exist. At an update,
\begin{equation}\label{eq:pushforwards}
 F_{t_j+}=(\Theta_j)_\#F_{t_j-},\qquad J_{t_j+}=(\Theta_j\otimes\Theta_j)_\#J_{t_j-}.
\end{equation}
\end{proposition}
\begin{proof}
Derive the identities from the tree integrals after the limit. Start with
smooth bounded marked tests and expand the top-slab one-root factors by
their one-slab definition, retaining canonical lower inputs, then truncate
to finitely many vertices and bounded birth energy. Vary the upper endpoint,
equivalently integrate in the time of the last top-slab contact. Last
contacts meeting no tested root cancel between scattering and non-scattering
(same flux, opposite signs, test unchanged). When a tested root is incident
the change is its scattered-minus-free value; deleting that last edge splits
the tree into two components, which by \Cref{lem:tree-splitting} are the
component measures multiplied at the common position, with the inverse
joining and its fresh-label coefficients.

Counts: (a) one root, partner chosen among $j$ fresh labels changes $1/j!$
to $1/(j-1)!$ and splitting remaining labels uses
$\frac1{(j_1+j_2)!}\binom{j_1+j_2}{j_1}=\frac1{j_1!j_2!}$, giving
$\int BFF_*(\Phi(\xi')-\Phi(\xi))$, symmetrized by particle interchange
(including $\omega\mapsto-\omega$) into the collision term of
\eqref{eq:F-evolution}; (b) two roots, a contact between them is one
unordered pair, no fresh label, deleting it gives two one-root means and
the last integral of \eqref{eq:J-evolution} with no factor $1/2$; (c) a
contact between the first root and a new partner: the second original root
lies with the first (contribution $J$ on it, $F$ on the partner, change
$\Phi(\xi')-\Phi(\xi)$) or with the partner (interchange; the $J$ component
carries $\Phi(\xi_*')-\Phi(\xi_*)$), summing to $J(H_t\Phi\otimes\Psi)$,
and the same at the second root gives $J(\Phi\otimes H_t\Psi)$. Records
project when a partner has no record in the integrand. Free flight gives
$D$; a transport-mild version avoids differentiating a sum (freely
transport the test to the last-contact time $s$; no-contact term plus
$a<s<t$ contact integrals). The finite re-expansion, deletion, and joining
were bijections of labelled trees with their factorial weights; passing to
full sums and unbounded velocities is justified by
\Cref{lem:summability,lem:spatial-traces,lem:tree-splitting}. Interface
concatenation is the empty-top-slab no-contact term, preserved by the
absolutely convergent tree sums. Updates are deterministic pushforwards,
giving \eqref{eq:pushforwards}.
\end{proof}

\subsection{Collision normalization and the covariance source}\label{sec:collision-normal}

The pre/post substitution with $\omega\mapsto-\omega$ preserves
$B(v-v_*,\omega)\dd v\dd v_*\dd\omega$, as does particle interchange with
the normal reversed. Hence the normalization in \eqref{eq:boltzmann} gives
\begin{equation}\label{eq:weak-normalization}
 \int Q(f,f)\phi=\tfrac12\int Bff_*\Delta\phi,\qquad
 \int[Q(g,f)+Q(f,g)]\phi=\int g\,K_t\phi
\end{equation}
with background $f=f_t$. So the collision part of the linearized weak drift
has no extra $1/2$.

Restore the same-particle term by defining
\begin{equation}\label{eq:G-definition}
 G_t(\Phi,\Psi)=J_t(\Phi\otimes\Psi)+F_t(\Phi\Psi).
\end{equation}
Then \Cref{prop:tree-evolution} gives
\begin{equation}\label{eq:G-evolution}
 \partial_tG_t(\Phi,\Psi)=G_t((D+H_t)\Phi,\Psi)+G_t(\Phi,(D+H_t)\Psi)
 +\tfrac12\int BF_tF_{t,*}\Delta\Phi\,\Delta\Psi .
\end{equation}
Direct check of the source: with $P=\Phi+\Phi_*$, $R=\Psi+\Psi_*$, the
collision change of the diagonal products from the $F$ equation plus the
$J$ source, symmetrized, is $\tfrac12\int BFF_*\Delta(PR)$; the diagonal
drifts contribute $\tfrac12\int BFF_*(R\Delta P+P\Delta R)$; the difference
is the last term of \eqref{eq:G-evolution} since
$\Delta(PR)=R\Delta P+P\Delta R+\Delta P\,\Delta R$. This uses neither
detailed balance nor equilibrium. The source is positive semidefinite
($F_t\ge0$). Initially, before updates,
\begin{equation}\label{eq:G-initial}
 G_0(\Phi,\Psi)=\int f_0(z)\Phi(z,0)\Psi(z,0)\,\dd z,
\end{equation}
and at an update $G$ pushes forward in both slots.

\subsection{Backward pulses and all observation times}

We need \eqref{eq:G-evolution} on the spaces
\[
 \mathcal V_a=\{\Phi=c\,y+h:c\in\R,\ h\in E_a^0\},\quad
 \|\Phi\|_{\mathcal V_a}=|c|+\|h\|_a .
\]
Let $\alpha>0$ be a common exponent in the integrated and conditional
Gaussian variation bounds for $F,J$, and choose $0<a<b<\beta$ with
$2b<\alpha$ (a positive margin for the flux; the factor $2$ also controls
the diagonal $F_t(hk)$). The mark is constant on a particle between updates
and conserved at a collision, so the marginal statement after
\eqref{eq:F-disintegration} gives
\begin{equation}\label{eq:restricted-drift}
 (D+H_t)(c\,y+h)=A_th,\qquad
 \tfrac12\int BF_tF_{t,*}\Delta(c\,y+h)\Delta(d\,y+k)=C_t(h,k).
\end{equation}
At pulse $t_j$, backward substitution sends $c\,y+h$ to $c\,y+h+c\phi_j$.

Use the backward propagator in this possibly nonsmooth class: the Gaussian
variation bounds make $G_t$ a uniformly bounded bilinear form on
$\mathcal V_b$, and the transport mild identity extends from smooth tests
by local uniform approximation with a small Gaussian margin; the unmarked
collision operator is $K_t$, so \eqref{eq:K-estimate} applies. Iterating
the bilinear mild identity chooses one of two slots per collision factor;
an $n$-fold remainder is bounded by
$\frac{(CT)^n}{n!}(1+\sqrt{n/\delta})^n$ for fixed allowance
$\delta=b-a>0$ (the $2^n$ slot choices absorbed in $C^n$), which vanishes.
Source iterations obey the same bound, justifying dual propagation by the
convergent mild series with no spatial/velocity differentiability of
$U(r,t)\phi$.

Starting from the test $y$ after the last observation, the backward test at
$r$ away from pulses is $y+h_r$ with
\begin{equation}\label{eq:backward-pulse-test}
 h_r=\sum_{j:\,t_j\ge r}U(r,t_j)\phi_j .
\end{equation}
(The convention at $r=0$ includes observations at time zero; values at
positive pulse times do not affect a time integral.) Combining dual
propagation, \eqref{eq:G-initial}, and the pulse substitutions,
\begin{equation}\label{eq:variance-formula}
 G_T(y,y)=\int f_0h_0^2+\int_0^TC_r(h_r,h_r)\,\dd r .
\end{equation}
After the last pulse the unmarked backward part is zero, so the last
observation time may replace $T$. By the explicit construction
\eqref{eq:isonormal}, $\sum_j\zeta_{t_j}(\phi_j)=I_0(h_0)+I(\Delta h_r)$,
whose variance is exactly \eqref{eq:variance-formula}, which by expanding
$h_r=\sum_{j:t_j\ge r}U(r,t_j)\phi_j$ equals \eqref{eq:target-cov}. This
identifies all multi-time and repeated-observation correlations.

\subsection{Ordinary cumulants and Gaussian convergence}

\begin{proposition}[Gaussian limit on the pasted dynamics]\label{prop:cumulant-clt}
For the single pasted dynamics fixed above and every fixed observation list,
\[
 \frac{\sum_iy_i(T)-\E\sum_iy_i(T)}{\sqrt\mu}\ \Longrightarrow\
 \sum_j\zeta_{t_j}(\phi_j),
\]
and all centered moments converge to the Gaussian ones.
\end{proposition}
\begin{proof}
Put $S^\eps=\sum_iy_i(T)$. Since $|S^\eps|\le M_{\mathrm{obs}}N$ and
$\mathcal Z_\eps\le e^\mu$, $S^\eps$ has exponential moments for each fixed
$\eps$. The exact factorial-to-ordinary identity at the final marked time is
\begin{equation}\label{eq:factorial-ordinary}
 \kappa_k(S^\eps)=\sum_{\pi\in\mathcal P([k])}\mu^{|\pi|}
 \int c_{|\pi|}^\eps(\dd\xi_1\cdots\dd\xi_{|\pi|})\prod_{B\in\pi}y_B^{|B|}.
\end{equation}
Directly: write
$\exp(\lambda S^\eps)=\prod_i[1+(\exp(\lambda y_i)-1)]$, insert the factorial
series, and take the logarithm; each $\exp(\lambda y)-1$ must receive a
nonempty set of the $k$ derivatives, and distributing those sets gives
exactly the partitions of \eqref{eq:factorial-ordinary}, whose ordering
cancels the factorial denominator. This is a finite formal identity through
degree $k$.

All $y$-powers are bounded on the mark interval (extend to bounded
continuous functions on the mark line). By \Cref{thm:sharp-cumulants} every
term on the right is $O_k(\mu)$. For
$X^\eps=(S^\eps-\E S^\eps)/\sqrt\mu$, the first cumulant vanishes and
\[
 \kappa_2(X^\eps)=c_1^\eps(y^2)+\mu c_2^\eps(y\otimes y)\longrightarrow
 F_T(y^2)+J_T(y\otimes y)=G_T(y,y),
\]
while for $k>2$, $\kappa_k(X^\eps)=O_k(\mu^{1-k/2})\to0$. That these hold on
one dynamics is essential: the bases fixing $\Gamma,L$ in
\Cref{thm:marked-expansion,thm:sharp-cumulants} are independent of the fixed
order; enlarging the auxiliary complexity cutoff or $D$ only changes the
error representation, not the random variable $X^\eps$.

The finite moment-cumulant identities give every centered Gaussian moment
with variance $\sigma^2=G_T(y,y)$. For scalar convergence, truncate the
characteristic functions of $X^\eps$ and of a centered Gaussian $Z$ at
degree $2n-1$; Taylor's integral remainder is bounded by
$|ux|^{2n}/(2n)!$, so
\[
 \limsup_{\eps\to0}|\E e^{iuX^\eps}-\E e^{iuZ}|
 \le\frac{2|u|^{2n}\E Z^{2n}}{(2n)!}
 =\frac{2|u|^{2n}\sigma^{2n}}{2^nn!},
\]
and $n\to\infty$ proves convergence for every real $u$, including
$\sigma=0$. This gives weak convergence with variance
\eqref{eq:variance-formula}.
\end{proof}

% =====================================================================
\part*{Part 7. Why the model dynamics is close to the real one}
% =====================================================================

\section{Accuracy of the pasted dynamics}\label{sec:cutoff}

The pasted dynamics recolors every restarted slab, resets the collision
prehistory at each interface, and truncates components above the caps
$\Lambda_i,\Gamma$. The true hard-sphere flow does none of this. We must
show the two trajectories coincide up to time $T$ with probability
$1-O(\eps^{1+\eta})$, $\eta>0$, \emph{better than} $O(\eps)$. The reason:
the fluctuation scale is $\sqrt\mu=\eps^{-1}$, and the mean of the pasted
process differs from the exact microscopic mean by a deterministic
$O(\eps)$ amount; an error of probability $O(\eps)$ alone would compete
with that shift after multiplication by $\eps^{-1}$, and exact centering
would fail. We prove the sharper bound.

\begin{proposition}[Accuracy of the pasted dynamics]\label{prop:cutoff}
There is a large dimensional integer $\Gamma_0$ with the following
property. Fix $\Gamma\ge\Gamma_0$. There is
$L_0=L_0(\Gamma,f_0,f,T,\beta)$ such that for every fixed $L\ge L_0$, the
true and pasted dynamics of \Cref{thm:marked-expansion}, coupled from the
same initial configuration, satisfy
\begin{equation}\label{eq:cutoff-probability}
 \P\{\text{the two trajectories differ at some time in }[0,T]\}\le
 C\eps^{1+\eta}
\end{equation}
for all small $\eps$, with $\eta=1/20$. The size cutoffs are the
$A_i,\Lambda_i$ of \Cref{thm:dhm-package}; $C$ and the upper bound on $\eps$
may depend on the displayed fixed data.
\end{proposition}

The rest of this part proves it. The plan:
\begin{enumerate}[label=\textbf{Step \arabic*.},leftmargin=*]
\item Bound the probability of a first disagreement by the number of
  ``terminal witnesses'': a connected block of true collisions in one
  slab, of bounded excess, meeting a fresh label and stopped by a rejection.
\item Separate coarse from fine. Coarse witnesses (complexity $>K$, or
  terminal component larger than $\Lambda_i$) sum to less than any power of
  $\eps$ by the [DHM] package---this fixes $K$ large.
\item On the low-complexity fine regime, localize energy, then combine a
  \emph{local gain} at every good cutting piece with a \emph{global} energy
  and causal time-order bound, gaining enough powers of $\eps$ to beat
  $1+\eta$.
\end{enumerate}

\subsection{From first disagreement to terminal witnesses}

Distinguish the geometric cycle excess $r(M)$ of a graph from its
multislab complexity $\rho(M)$. For a graph with $b$ binary-contact
vertices, $n$ particle lines, and $c$ connected components, when each line
has at least one vertex,
\begin{equation}\label{eq:cutoff-euler}
 r(M)=b-n+c,
\end{equation}
equivalently, with contacts as vertices and segments as bonds,
$r=\#\{\text{bonds}\}-b+c$. The graph's directed order is the order
increasing in time along each particle line; orders on disjoint lines need
not be comparable.

\begin{lemma}[First disagreement]\label{lem:first-disagreement}
For every initial configuration the indicator that the true and pasted
trajectories first disagree in slab $i$ is at most the number of its
terminal witnesses: connected true C histories $M$ on incoming labels, with
no designated root, excess $r(M)\le2\Gamma$, an internal realizability
restriction, meeting a label born at or after the previous interface,
ending at a terminal contact, and with an occupied replica at the terminal
contact.
\end{lemma}
\begin{proof}
Compare the two evolutions in chronological order; they agree up to the
first disagreement, which therefore is a contact accepted by the true rule
but rejected by the pasted one (or vice versa). On agreement so far the
pasted state is the true state; the collision component in the true history
just before that contact has excess $\le2\Gamma$ (it was accepted), and the
rejection is exactly ``component size $>\Lambda_i$ or excess added $>\Gamma$''
within this block. Summing over possible blocks, each such rejection is
charged to a distinct terminal witness; summing charges gives the bound.
\end{proof}

Insert the terminal-witness functional \eqref{eq:terminal-witness-functional}
and apply \Cref{cor:terminal-witness}. Begin with the previous pasted
factorial measure at interface $i-1$. Since $n\le3\Lambda_i<A_{i-1}/2$, the
variation error is below every power. The connected terminal witness is
itself a block at the top of the pasted hierarchy and enters through its
positive layer expansion. Its lower dependent and rooted components attach
through particle lines to the terminal component. Keep $M$'s realizability
restriction throughout. A rooted lower component with $k\ge2$ attachments
cannot touch the same upper component twice (that would create a cycle);
with $u$ upper and $v$ lower vertices of degree $\ge2$ and $d$ incidence
edges, the bipartite incidence forest has $d\le u+v-1$ per nonempty part
and $2v\le d$, so $d\le2u$. A lower component attached at a single line is
either absorbed into an independent $g$ input, or continues a required
record; the latter are boundedly many (the records through this interface,
including continued empty lines).

\subsection{Coarse witnesses and the regularity of graphs}

Choose a complexity threshold $K$ after $L$ is fixed. By
\Cref{cor:terminal-witness,thm:dhm-package}, the sum of positive majorants
with $\rho>K$, including the activity cost of an artificial top root when
required, is $O(\eps^3)$ for $K$ large enough. The terms left to estimate
have $\rho\le K$. Their interface crossings, initial-link vertices,
within-slab cycles, and total geometric excess are bounded. Discard the
initial-link restrictions in these positive majorants; the trajectory
attachments to the terminal component persist.

From here on, all graphs are connected low-complexity witnesses,
summarized by a polylogarithmic exponent bound $B_\eps$ on the total number
of atoms in the top-slab terminal component, with
$B_\eps\le C_*|\log\eps|$ ($C_*$ may depend on $K,L$). A connected witness
of low complexity has its top-slab component of bounded excess, and a
connected graph with $b$ atoms and excess $\le2\Gamma$ can be produced
without new crossings and within the same realization family by
$\le C^b$ choices of the initial labelled $M$ and its geometric data.

\subsection{A first energy localization}

The number of scattered velocities in any witness is at most $B_\eps$, and
initial data carry only $\le2\beta$ Gaussian under \Cref{ass:kinetic}; a
terminal witness has controllable total birth energy with Gaussian
cost $e^{-cP}$ for total energy $P$.

\begin{lemma}[Energy tail]\label{lem:cutoff-energy}
Fix $\Gamma$ and then choose $L$ once, larger when needed. For witnesses
with $n$ true input lines carry total birth energy $E_0=\frac12\sum_j|v_j|^2$,
put
\begin{equation}\label{eq:cutoff-energy-scales}
 P_\eps=(C_*B_\eps)^{1/2}\bigl(1+B_\eps|\log\eps|^{1/2}\bigr),\qquad
 \gamma_\eps=\eps^{B_\eps^2}.
\end{equation}
For a witness with $n\le B_\eps$ terminal lines, once energy is
$>P_\eps$ the
\begin{equation}\label{eq:cutoff-energy}
 \text{integral}\le C^{B_\eps}e^{-cB_\eps P_\eps}\le\eps^{cB_\eps^2},
\end{equation}
smaller than every fixed power. On the low-energy regime $E_0\le P_\eps$
the terminal terminals are localized to a ball of radius $R=C_*(1+B_\eps P_\eps^{1/2})$.
\end{lemma}
\begin{proof}
The forward bound below holds on the positive original measure, so it uses
no local recollision gain. For fixed fully ordered graph with $n$ births,
integrate forward from the birth phase points. At each contact its two
incoming affine lines are already determined, and by the
two-incoming-flight identity (below) all other variables are integrated to
an indicator; births at later interfaces are inserted at their times.
Dropping encounter restrictions and initial links, a graph with $n$ births
has mass at most
\[
 \mu^n\int\prod_{j=1}^nh_j(x_j,v_j)\ind_{\{\sum_j|v_j|^2>P_\eps\}}
 \prod_j\dd x_j\dd v_j\le\mu^nC^ne^{-cP_\eps},
\]
using half of the uniform birth Gaussian for the tail and the number of
births $\le B_\eps$; summing a finite count and $\mu^n=\eps^{-2n}$ gives
$e^{2B_\eps|\log\eps|-cP_\eps}$ times a prefactor; choosing $P_\eps$ a
fixed large multiple of $B_\eps|\log\eps|$ makes this below any fixed
power. With no exponential growth $M_{\mathrm{top}}\to1$.
\end{proof}

\begin{lemma}[Causal order and relative speeds]\label{lem:cutoff-time}
In a slab with $q_i$ ordinary atoms, retain the partial order induced by the
full graph (including chains through exceptional atoms), and let
$\mathcal T_i$ be its time-order region. On the original support with total
birth energy $\le P$, an empty slab contributing one,
\begin{equation}\label{eq:cutoff-time}
 |\mathcal T_i|\sup\prod_{a\text{ ordinary in }i}|v_a-v_{*a}|
 \le\frac{\tau^{q_i}(2q_iP)^{q_i/2}}{q_i!}.
\end{equation}
\end{lemma}
\begin{proof}
With $e_i$ the number of linear extensions, $|\mathcal T_i|=e_i\tau^{q_i}/q_i!$.
An antichain of ordinary vertices occupies disjoint incoming pairs on a
common causal slice: place all strict ancestors first in a total extension;
no two vertices share a line; energy conservation bounds the sum of their
pair energies by $P$, including interfaces via inactive coordinates. For
$a_v=|v_v-v_{*v}|$, every antichain has $\sum_{\mathcal{A}}a_v\le\sqrt{2q_iP}$.
Summing the products of weights over the recursive enumeration of linear
extensions (available minimal vertices form an antichain) gives
$e_i\prod_va_v\le(2q_iP)^{q_i/2}$ for every feasible assignment, hence the
supremum.
\end{proof}

\begin{lemma}[Free-parameter extraction]\label{lem:cutoff-extraction}
Consider a finite ordered composition of positive cut kernels. At an
ordinary step the free variables are $\vartheta_j=(t_j,\omega_j,w_j)$ with
measure $a_j\dd t_j\dd\omega_j\dd w_j$, followed by a deterministic update.
Suppose every exceptional kernel $K_\ell(h,\dd z)$ has mass at most
$H_\ell$, uniformly in earlier-cut data (every restriction needed for that
bound included, zero outside the admissibility domain). Remove the $a_j$
from the ordinary measures to get a positive measure $\nu$. If $E$ is a set
of complete histories and $\mathcal D$ a measurable set containing its
projection onto $\vartheta$, then
\begin{equation}\label{eq:cutoff-extraction}
 \nu(E)\le\Bigl(\prod_\ell H_\ell\Bigr)
 \int_{\mathcal D}\prod_{j=1}^q\dd t_j\dd\omega_j\dd w_j .
\end{equation}
The same holds if the density after division by $a_j$ is merely bounded by
one.
\end{lemma}
\begin{proof}
First put all ordinary variables in bounded boxes and fix their vector
$\vartheta$. The exceptional kernels stay in place with $\vartheta$
exterior; integrating them backwards from the last exceptional step bounds
the surviving mass by $\prod_\ell H_\ell$, and intervening deterministic
substitutions have norm at most one on bounded functions. Nonnegative
Fubini represents $\nu$ as the product in $\vartheta$ times these
kernels, and majorizing the indicator of $E$ by
$\ind_{\mathcal D}(\vartheta)$ proves \eqref{eq:cutoff-extraction}.
Exhaustion removes the boxes. Equivalently, restrict ordinary relative
speeds away from zero and bound a rectangle by its product volume; cover a
compact containing set by products of rectangles and take the infimum;
then let the speed restriction tend to zero (zero-speed incoming contacts
have zero flux).
\end{proof}

\subsection{Conditional local kernels and admissible cuts}

This subsection imports the local (kernel) estimates and the cutting
alternatives. We use them with conditional masses, not global probabilities.

A \emph{cut} exposes a set of contact variables. Its shared lines become
fixed phase-space data for the unexposed graph. The available degree of an
atom is its number of nonfixed edges; a history is \emph{regular} if at every
O-atom each fixed end is serial with a free end. A one-atom piece of type
$\{2\}$ has two nonfixed ends with the two fixed ends on the same time side
(C) or nonserial (O); types $\{3\}$ and $\{4\}$ have respectively one and
zero fixed ends. A $\{33\}$ piece has two degree-three atoms joined by one
bond; it is type A if one atom cuts as a type $\{3\}$ singleton and the
other becomes an allowed $\{2\}$ piece, type B otherwise. A $\{44\}$ piece
has two degree-four atoms joined by one bond
\cite[Definitions 8.2, 8.11]{DHM2025}. The kernels compose
$I_M=I_A\circ I_{M\setminus A}$ with repeated boundary variables identified
\cite[Proposition 8.14]{DHM2025}; this is how a later kernel is estimated
with earlier variables fixed.

\begin{lemma}[Local kernels]\label{lem:dhm-local}
Let $R$ bound velocities and localized radii; the time length is $\le T$.
For a singleton with two fixed ends in a free kernel:
\begin{enumerate}[label=\textup{(\roman*)}]
\item If all incident free edges have relative speeds $\ge\eps^\upsilon$, the
  mass is $\le C R^C\eps^{2+\upsilon/3}$.
\item If it is a scattering contact and the strong restrictions
  ($|s-s_*|$, $|x-x_*|$ and exterior $x_*$ below $\eps^{1-\upsilon}$ on both
  edges) hold, the mass is $\le\eps^{2+\upsilon/3}$. The bound is uniform in
  the exterior parameters $t_*,x_*$.
\item For a type-B $\{33\}$ piece, both atoms C, with $|t_1-t_2|\ge\eps^\upsilon$
  and every incident pair having $|v_e-v_{e'}|\ge\eps^\upsilon$, the mass is
  $\le\eps^\upsilon|\log\eps|^{C_*}$ uniformly in the two fixed ends.
\end{enumerate}
Multiplying any such kernel by a bounded factor on its support costs its
supremum; no independent Gaussian distribution of a fixed-end velocity is
assumed. This is \cite[Lemmas 8.15, 8.20, 8.23]{DHM2025}. A
\emph{uniformly good} singleton or type-B pair is one of these restricted
kernels with conditional mass $\eps^\upsilon|\log\eps|^{C_*}$; all
gain-producing restrictions are part of the kernel.
\end{lemma}

\begin{proposition}[Pure-collision cutting alternatives]\label{prop:cutoff-import}
Let $M$ be a connected full realizable graph with only scattering contacts.
A transversal set $A\subset M$ satisfies
$M\setminus A=A^-\sqcup A^+$ with no directed bond from $A^+$ to
$A\cup A^-$, nor from $A$ to $A^-$. Let $X_0(A)$ be vertices outside $A$
with two bonds to $A$, and $X(A)$ its closure obtained by repeatedly adding
vertices with two bonds to the grown set. Then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item\label{item:growth}
There is a chain of connected transversal sets $A_1,\ldots,A_m=M$, $A_1$ a
tree, $X_0(A_j)\ne\varnothing$ for $j<m$,
$A_j\cup X(A_j)\subset A_{j+1}$,
$r(A_{j+1})=r(A_j\cup X(A_j))$, $r(A_{j+1})-r(A_j)\le10|X(A_j)|$. The
intermediate growth adds one bond per new vertex. For $D$ transitions there
is a cutting sequence of $M$ with one type $\{4\}$, no type $\{44\}$ or
type-B pair, and $\ge D$ type-A pairs.
\item\label{item:bfk}
One can choose $A^\pm$ so each component of $A^+$ is attached to $A$; then
$X^\pm(A)=X(A)\cap A^\pm$ are convex along particle lines, and with $m_\pm$
the bonds from $A$ to $X^\pm(A)$,
$|X^\pm(A)|\le G(m_\pm)$, $G(q)=(2q+1)\binom q2(32q^{3/2})^{q^2}$.
\item\label{item:trans}
Put $z=|X_0^\pm(A)|$, $a=r(A)$; there is a cutting sequence with one type
$\{4\}$, no type $\{44\}$ or type-B pair, and $\ge z/2-a-1$ type-A pairs.
\item\label{item:main}
For this side there is a finite Borel partition of the contact-parameter
domain, with a subcase integer $y\ge0$ and two available sequences:
a sequence with one type $\{4\}$, no type $\{44\}$ or type-B pair, and
$\ge y/10-3(a+1)$ uniformly good singletons; or a sequence with no type
$\{44\}$, $\le S=a+z+y$ type $\{4\}$ singletons, all type-B pairs uniformly
good, and $\ge(m_\pm-10^5S)/10$ type-A or type-B pairs.
\end{enumerate}
Neutral pieces keep their neutral kernels; partitions and operation
sequences have $\le C^b|\log\eps|^{C_*}$ subcases with $C$ independent of
$L$. These are \cite[Definition 15.3; Propositions 15.1, 15.2, 15.4, 15.6,
15.10]{DHM2025}.
\end{proposition}

\begin{lemma}[Adjacent scattering contacts]\label{lem:cutoff-pair}
Let $R\ge2$ bound velocities and localized radii, and length $\le T$. A pair
of adjacent contacts with one bond is \emph{primitive} if there is no other
directed chain between them; a double bond between two consecutive
scattering contacts is impossible. For $e_1,e_2$ the prospective fixed
edges of a type-A pair with contact times $t_1,t_2$ and
\[
 \text{strongly close:}\quad |t_2-t_1|\le\eps^{9/10},\quad
 |v_{e_2}-v_{e_1}|\le\eps^{4/5},
\]
\begin{enumerate}[label=\textup{(\roman*)}]
\item a primitive strongly close pair cut freely as one piece has mass at most
  \begin{equation}\label{eq:cutoff-free-pair}
   CR^C\mu\eps^{9/10+3\cdot4/5}=CR^C\eps^{13/10};
  \end{equation}
\item without any strongly close primitive pair, each type-A pair used by a
  sequence has conditional mass $\le CR^C\eps^{1/20}$, or splits into a
  neutral type-two piece and a singleton with the same bound.
\end{enumerate}
The bounds are uniform in previously exposed lines and times. This is
\cite[Lemma 15.12]{DHM2025}.
\end{lemma}

Fix $\delta=\tfrac12\min\{\upsilon,1/20\}>0$; every retained local gain is
at least $\eps^\delta$ up to a fixed polylogarithmic factor.

\begin{lemma}[Amplified pure-collision cut]\label{lem:cutoff-amplified}
For large dimensional $\Gamma_0$, every connected realizable full scattering
graph with $\Gamma<r(M)\le2\Gamma$, $\Gamma\ge\Gamma_0$, decomposes into cut
sequences such that either it contains a freely cut pair estimated by
\eqref{eq:cutoff-free-pair} with no other activity loss, or its retained
improved pieces $g$ and free singletons $s_4$ satisfy
\begin{equation}\label{eq:cutoff-gain-balance}
 \delta\,g-2s_4\ge6/5 .
\end{equation}
There are $\le C^b|\log\eps|^{C_*}$ subcases; the number of non-ordinary
pieces is bounded in terms of $\Gamma$ and fixed choices. This is
\cite[Proposition 15.11]{DHM2025}.
\end{lemma}

\subsection{Simultaneous integration of the ordinary parameters}

The local powers must survive simultaneous integration without a
per-vertex logarithmic loss. Call a neutral type-three singleton
\emph{ordinary}; all other pieces \emph{exceptional}. Counting both vertices
of an exceptional pair, their number is bounded by a constant depending on
$K,\Gamma$; the terminal part is \Cref{lem:cutoff-amplified}, below it
\Cref{lem:dhm-local}; the degree count
$s_2+p_{33}=r(M)+s_4+p_{44}-1$ on the complete graph (or its version with
boundedly many crossing ends fixed) bounds the extra type-two pieces. With
$b$ the total atoms and $q$ the ordinary ones, $q=b-O_K(1)$.

\begin{lemma}[Total birth energy bounds the velocity domain]
\label{lem:cutoff-ellipsoid}
Fix a cutoff $E_{\mathrm{cut}}=P_\eps$ and, as in
\Cref{lem:cutoff-energy}, use the local kernels at the scopes where
\Cref{prop:cutoff-import} and \Cref{lem:cutoff-amplified} retain them. Then
for a witness with ordinary projection within the containing set $\mathcal D$
(energy $\le P_\eps$), there is a compact measurable set inside
$\mathcal D$, of volume
\begin{equation}\label{eq:cutoff-ellipsoid-volume}
 \le\prod_i\Bigl(C\frac{L}{q_i}\Bigr)^{q_i}(1+P_\eps/q_i)^{3q_i/2},
\end{equation}
that contains every admissible ordinary velocity vector.
\end{lemma}
\begin{proof}
Run the velocity computation forward in chronological order. Ordinary
scattering applies an orthogonal matrix to two velocities; an ordinary
nonscattering contact applies the identity. At an exceptional atom erase the
dependence of the two outgoing coordinates and insert prescribed values;
inaugurated coordinates can be inactive from the start. The part linear in
the birth vector $u$ propagates by orthogonal matrices and projections, norm
$\le1$ at each time. Each ordinary $w_j$ is a physical edge velocity
immediately before or after its contact (including the outgoing side when
cut backwards), so
\begin{equation}\label{eq:cutoff-forward-rows}
 W=A(\omega)u+d(\omega,a),\qquad \|A(\omega)\|_{\mathrm{HS}}^2\le3q .
\end{equation}
Rows need not be orthogonal; only the sum of squared norms is used, and the
maps do not depend on contact times. Take a Euclidean net of $a$ of mesh
$c_s(1+q)^{-1/2}$, cardinality $\le C_s\{(1+P_\eps)(1+q)\}^{C_s}$; a change
$\zeta$ in $a$ changes one chronological velocity by $\le\sqrt s\zeta$ and
the observations by $\le C\sqrt{qs}\zeta$. Choose $c_s$ so simultaneous
error is $\le1$. The allowed $W$ lie in $\bigcup_{a_0}
(d(\omega,a_0)+A(\omega)B_{3n}(\sqrt{P_\eps})+B_{3q}(1))$, and the Minkowski
sum is in $\sqrt2(P_\eps AA^{\mathsf T}+\Id_m)^{1/2}B_m(1)$ for $m=3q$
(support function: $\sqrt{P_\eps}|A^{\mathsf T}\xi|+|\xi|\le
\sqrt{2(P_\eps|A^{\mathsf T}\xi|^2+|\xi|^2)}$). Combining the trace bound
with AM-GM on eigenvalues and $\operatorname{vol}(B_m(1))\le(C/\sqrt m)^m$,
\[
 2^{m/2}\operatorname{vol}(B_m(1))\det(\Id_m+P_\eps AA^{\mathsf T})^{1/2}
 \le\Bigl(C\sqrt{(1+P_\eps)/q}\Bigr)^{3q}.
\]
The identity enlargement handles small singular values without an inverse
Jacobian; a finite union of continuous images of compact parameter sets
gives the compact containing set for \Cref{lem:cutoff-extraction}.
\end{proof}

\begin{lemma}[Summation with finitely many exceptional pieces]
\label{lem:cutoff-sum}
Suppose a cut sequence has $q=b-O_K(1)$ ordinary atoms and boundedly many
exceptional pieces whose conditional mass bounds have product
$\eps^\alpha|\log\eps|^{C_*}$. After summing spatial cells and energy bins in
a polylogarithmic energy region, its integral is at most
\begin{equation}\label{eq:cutoff-sum}
 \eps^\alpha|\log\eps|^{C_*}(C\tau\sqrt L)^q .
\end{equation}
The base $C$ depends on the fixed kinetic envelopes and dimension, not on
$K,L$. This is \cite[Lemma 16.2]{DHM2025}.
\end{lemma}

\subsection{Assembly: the discrepancy probability}

\begin{proof}[Proof of \Cref{prop:cutoff}]
Start from \Cref{lem:first-disagreement,cor:terminal-witness}, and fix the
complexity cutoff $K$ large enough that the coarse part is $<\eps^3$
(\Cref{thm:marked-expansion}(iii) after \Cref{cor:terminal-witness}):
$\rho>K$ is discarded. The remaining witnesses are connected, full, realizable
graphs with $\rho\le K$, hence bounded crossing/initial/cycle counts, and
$r(M)\le2\Gamma$. Localize by \Cref{lem:cutoff-energy}: outside the
polylogarithmic energy region the sum is smaller than any power.

On the low-energy region, apply \Cref{prop:cutoff-import,lem:cutoff-amplified}
to the top-slab graph. The freely cut primitive pair (if any) gives
\eqref{eq:cutoff-free-pair}, product $\eps^{13/10}|\log\eps|^{C_*}$. Otherwise
\eqref{eq:cutoff-gain-balance} holds: the retained improved pieces number $g$
with a fixed bound except for the bounded seed, and gain $\eps^{\delta g}$,
while each free type-four seed costs $\mu=\eps^{-2}$; so the exceptional
product is at most $\eps^{6/5}|\log\eps|^{C_*}$ (using
$\delta g-2s_4\ge6/5$ in both the $\eps^{13/10}$ and the $\eps^{6/5}$
branches). All the extra variables inside the containing set are then
removed by the coarea and time-order bounds: \Cref{lem:cutoff-time}
gives $\prod_i\tau^{q_i}(2q_iP_\eps)^{q_i/2}/q_i!$, and the ordinary
velocity box by \Cref{lem:cutoff-ellipsoid}. Combining
\Cref{lem:cutoff-sum,lem:cutoff-extraction}, the low-complexity sum is
bounded by
\[
 \eps^{6/5}|\log\eps|^{C_*}(C\tau\sqrt L)^{b-O_K(1)} .
\]
Choose $L$ so large that $C\tau\sqrt L=C T/\sqrt L<1/2$, absorbing
constants depending on $K,L$ into the fixed prefactor. The exponential atom
counting ($C_1^b$ with $C_1$ independent of $K,L$, per
\Cref{thm:marked-expansion}(iii) and \Cref{cor:terminal-witness}) together
with the bounded crossing labels, selected pairs, and initial coefficients
costs only a fixed-degree polynomial; the resulting sum over $b$ is a
convergent geometric series, with the $\Lambda_i$ cutoff removing the large-$b$
part. Thus the total sum is at most
\begin{equation}\label{eq:cutoff-cycle-sum}
 C\eps^{6/5}|\log\eps|^{C_*} .
\end{equation}
Since $\tfrac{6}{5}=1.2>1+\tfrac{1}{20}=1.05$, this is
$\le C\eps^{1+1/20}$, proving \eqref{eq:cutoff-probability}. (The witnesses
with terminal component larger than $\Lambda_i$ are smaller than any power
by \Cref{lem:cutoff-energy,thm:dhm-package}, via the activity cost
$\mu=\eps^{-2}$ versus the Gaussian gain over $\Lambda_i\ge|\log\eps|^{30}$
births, summing geometrically as in the coarse step.)
\end{proof}

\begin{remark}[Order of choices]
The local exponent $\delta$ and the finite hierarchy of
\Cref{lem:cutoff-amplified} fix $\Gamma_0$ from dimensional constants. Then
choose $\Gamma$, then $L$ for the imported estimates and the common
exponential base; only afterward choose $K$, the localization powers, and the
small upper bound on $\eps$. No observation number or moment order enters the
first two choices. This is what let us fix a single pasted dynamics in
\Cref{sec:gaussian-limit}.
\end{remark}

% =====================================================================
\part*{Part 8. Exact centering and the fluctuation limit}
% =====================================================================

\section{Transfer of exact centering and the hard-sphere limit}\label{sec:transfer}

The pasted process of \Cref{prop:cumulant-clt} is centered by its own
microscopic mean. The real gas is centered by a slightly different mean,
since the pasted and true laws charge the discrepancy event (probability
$\le C\eps^{1+\eta}$, \Cref{prop:cutoff}) with different mass. At the
fluctuation scale $\sqrt\mu=\eps^{-1}$ this difference of means must go to
zero, and that is exactly the sharp estimate of \Cref{prop:cutoff}. We make
this quantitative, then assemble \Cref{thm:main}.

\begin{lemma}[Discrepancy and centering]\label{lem:center-transfer}
Couple the true and pasted dynamics from the same initial configuration.
For every fixed time $t$ and bounded continuous test $\phi$, with
$\|N/\mu\|_{L^2}\le C$,
\begin{equation}\label{eq:center-transfer}
 \sqrt\mu\,\bigl|\E_{\mathrm{true}}\pi_t^\eps(\phi)
             -\E_{\mathrm{pasted}}\pi_t^\eps(\phi)\bigr|
 \le C_\phi\,\eps^{(\eta-1)/2}\longrightarrow0,
\end{equation}
and moreover
\begin{equation}\label{eq:field-difference}
 \zeta_t^{\eps,\mathrm{true}}(\phi)
 =\zeta_t^{\eps,\mathrm{pasted}}(\phi)+o_{L^1}(1).
\end{equation}
\end{lemma}
\begin{proof}
Let $A=\{\text{the two trajectories agree on }[0,T]\}$, so
$\P(A^c)\le C\eps^{1+\eta}$, and both $N$ and $N/\mu$ have bounded second
moments (\Cref{prop:initial}). On $A$ the empirical measures agree for every
$t$, hence so do $\pi_t^\eps(\phi)$ and the running records. Write
$\pi=(1/\mu)\sum_i\phi$. Off $A$, $|\pi|\le\|\phi\|_\infty N/\mu$ on either
side, so $|\pi^{\mathrm{true}}-\pi^{\mathrm{pasted}}|
\le2\|\phi\|_\infty (N/\mu)\ind_{A^c}$. By Cauchy--Schwarz,
\[
 \bigl|\E_{\mathrm{true}}\pi-\E_{\mathrm{pasted}}\pi\bigr|
 \le2\|\phi\|_\infty\|N/\mu\|_{L^2}\,\P(A^c)^{1/2}
 \le C_\phi\,\eps^{(1+\eta)/2},
\]
and multiplying by $\sqrt\mu=\eps^{-1}$ gives
$\eps^{(\eta-1)/2}\to0$, proving \eqref{eq:center-transfer}.

For \eqref{eq:field-difference}, insert the centered definitions:
$\zeta^{\mathrm{true}}-\zeta^{\mathrm{pasted}}
=\sqrt\mu\bigl[(\pi^{\mathrm{true}}-\pi^{\mathrm{pasted}})
-(\E_{\mathrm{true}}\pi-\E_{\mathrm{pasted}}\pi)\bigr]$.
The second bracket is bounded by \eqref{eq:center-transfer}. For the first,
using the $A^c$ indicator,
\[
 \E\sqrt\mu\,\bigl|\pi^{\mathrm{true}}-\pi^{\mathrm{pasted}}\bigr|
 \le2\|\phi\|_\infty\sqrt\mu\,\E\bigl[(N/\mu)\ind_{A^c}\bigr]
 \le2\|\phi\|_\infty\|N/\mu\|_{L^2}\sqrt\mu\,\P(A^c)^{1/2},
\]
and $\sqrt\mu\,\eps^{(1+\eta)/2}=\eps^{-1}\eps^{(1+\eta)/2}
=\eps^{(\eta-1)/2}\to0$. So the $L^1$ norm of the difference of the
centered fields tends to zero.
\end{proof}

\begin{proof}[Proof of \Cref{thm:main}]
\textbf{Existence and uniqueness of the target.} \Cref{prop:gaussian-target}
constructs the centered generalized Gaussian field $\zeta$ with covariance
\eqref{eq:target-cov}, initial field $\zeta_0$ of covariance
\eqref{eq:init-cov} independent of the noise $W$, and weak equation
\eqref{eq:fluc}; uniqueness in law holds in the class of
$L^2$-continuous fields there.

\textbf{Initial fluctuation limit.} \Cref{prop:initial}(iii) gives, for any
fixed list of tests, joint convergence of
$(\zeta_0^\eps(\phi_1),\dots,\zeta_0^\eps(\phi_m))$ to the centered Gaussian
vector with covariance $\int f_0\phi_i\phi_j$.

\textbf{Dynamical limit on the pasted dynamics.} Fix a single pasted
dynamics as in \Cref{sec:gaussian-limit} ($\Gamma,L$ independent of the
observation list and order). \Cref{prop:cumulant-clt} gives, for every fixed
list of times and tests, convergence of the centered combination
$\eps\sum_i y_i(T)$ to $\sum_j\zeta_{t_j}(\phi_j)$, with convergence of all
centered moments. By \Cref{eq:factorial-ordinary} and
\Cref{thm:sharp-cumulants}, this is the finite-dimensional Gaussian limit
with the multi-time covariance \eqref{eq:target-cov}.

\textbf{Transfer to the true flow.} By \Cref{lem:center-transfer},
$\zeta_t^{\eps,\mathrm{true}}(\phi)-\zeta_t^{\eps,\mathrm{pasted}}(\phi)
=o_{L^1}(1)$, while
$|\zeta_t^\eps(\phi)|\le\|\phi\|_\infty(\eps N+\eps\E N)$ has uniformly
bounded moments because $\E(N/\mu)^p<\infty$ (\Cref{prop:initial}). Hence the
$p$-moments of the difference vanish for every fixed $p$ by interpolation of
the $L^1$ bound with the uniform $L^{p+1}$ bound, and, for any bounded
continuous functional of a fixed finite collection
$(\zeta_{t_1}^\eps(\phi_1),\dots,\zeta_{t_m}^\eps(\phi_m))$,
\[
 \E_{\mathrm{true}}F-\E_{\mathrm{pasted}}F\longrightarrow0
\]
(local uniform approximation of a continuous $F$ on a compact by a smooth
one, then the moment control of the difference off the compact). Therefore
the true centered fields converge in distribution to the same Gaussian
vector. Sending the finite list over a countable dense family and using the
variance bound \eqref{eq:field-bound} upgrades this to the map
$t\mapsto\zeta_t(\phi)$ in joint law.

\textbf{Weak equation.} The limiting field inherits \eqref{eq:fluc} from the
construction \eqref{eq:isonormal}, so its law is the one prescribed. The
covariances \eqref{eq:init-cov} and \eqref{eq:noise-cov} are identified in
\Cref{prop:gaussian-target} and via \eqref{eq:variance-formula}. This is
\eqref{eq:main-convergence}.
\end{proof}

\begin{remark}[What is exact and what is asymptotic]
The mean subtracted in \eqref{eq:empirical} is the \emph{exact} microscopic
expectation, not a Boltzmann approximation; this is essential because of
\Cref{rem:centering-shift}. The Gaussian limit and its covariance are exact
in the limit; on the way, only the discrepancy probability, not the law
itself, is estimated with a power gain. The single imported analytic package
is \Cref{thm:dhm-package}, used solely to remove large complexity and to
compare signed small trees with the Boltzmann solution; every connected
correlation, the sharp tree limits, the covariance, the Gaussian central
limit, and the exact-centering transfer are proved here.
\end{remark}

% =====================================================================
%  Bibliography
% =====================================================================

\begin{thebibliography}{99}

\bibitem{DHM2025}
Z.~Deng, Z.~Hani, and X.~Ma,
Long time derivation of the Boltzmann equation from hard sphere dynamics,
arXiv:2408.07818v3 (18 July 2025).

\bibitem{BGSS2023}
T.~Bodineau, I.~Gallagher, L.~Saint-Raymond, and S.~Simonella,
Long-time correlations for a hard-sphere gas at equilibrium,
Comm.\ Pure Appl.\ Math.\ 76 (2023), no.~12, 3852--3911;
arXiv:2012.03813.

\bibitem{BGSS2024}
T.~Bodineau, I.~Gallagher, L.~Saint-Raymond, and S.~Simonella,
Long-time derivation at equilibrium of the fluctuating Boltzmann equation,
Ann.\ Probab.\ 52 (2024), no.~1, 217--295; arXiv:2201.04514.

\bibitem{BGSSCluster2022}
T.~Bodineau, I.~Gallagher, L.~Saint-Raymond, and S.~Simonella,
Cluster expansion for a dilute hard sphere gas dynamics,
J.\ Math.\ Phys.\ 63 (2022), no.~7, 073301; arXiv:2205.04110.

\bibitem{BGSS2020}
T.~Bodineau, I.~Gallagher, L.~Saint-Raymond, and S.~Simonella,
Fluctuation theory in the Boltzmann--Grad limit,
J.\ Stat.\ Phys.\ 180 (2020), 873--895; arXiv:2004.00311.

\bibitem{Lanford1975}
O.~E.~Lanford, III,
Time evolution of large classical systems,
in: Dynamical Systems, Theory and Applications,
Lecture Notes in Physics 38, Springer, 1975, 1--111.

\bibitem{Penrose1967}
O.~Penrose,
Convergence of fugacity expansions for fluids and lattice gases,
in: Statistical Mechanics (T.~A.~Bak, ed.), Benjamin, 1967.

\bibitem{BuragoFerlegerKononenko1998}
D.~Burago, S.~Ferleger, and A.~Kononenko,
Uniform estimates on the number of collisions in semi-dispersing billiards,
Ann.\ of Math.\ 147 (1998), 695--708.

\end{thebibliography}

\end{document}