diff --git a/hard_sphere_boltzmann_fluctuations_v2.txt b/hard_sphere_boltzmann_fluctuations_v2.txt
index 2cf4d55..896ad18 100644
--- a/hard_sphere_boltzmann_fluctuations_v2.txt
+++ b/hard_sphere_boltzmann_fluctuations_v2.txt
@@ -1,23 +1,4 @@
-% =====================================================================
-%  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.
-% =====================================================================
+% Hard-sphere fluctuations on the regular Boltzmann lifespan.
 \documentclass[11pt]{amsart}
 \usepackage{amsmath,amssymb,amsthm,mathtools}
 \usepackage[margin=1in]{geometry}
@@ -32,7 +13,7 @@
 \usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref}
 \usepackage[capitalise,nameinlink,noabbrev]{cleveref}
 
-\newtheorem{theorem}{Theorem}[section]
+\newtheorem{theorem}{Theorem}
 \newtheorem{proposition}[theorem]{Proposition}
 \newtheorem{lemma}[theorem]{Lemma}
 \newtheorem{corollary}[theorem]{Corollary}
@@ -41,7 +22,7 @@
 \newtheorem{assumption}[theorem]{Assumption}
 \theoremstyle{remark}
 \newtheorem{remark}[theorem]{Remark}
-\numberwithin{equation}{section}
+% equations, theorems numbered continuously (no sections)
 
 \newcommand{\R}{\mathbb R}
 \newcommand{\N}{\mathbb N}
@@ -58,110 +39,72 @@
 \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
+\textbf{A gas of billiard balls.} A large box holds identical hard spheres,
+like a three-dimensional break of pool: they fly in straight lines and
+bounce elastically on contact. The diameter $\eps$ is tiny and the density
+is $\mu=\eps^{-2}$ spheres per unit volume, so the gas is dilute yet the
+number of spheres is enormous.
+
+\textbf{The average.} The expected number of spheres per unit volume of
+position-velocity space, the one-particle density, solves the Boltzmann
+equation as $\eps\downarrow0$. That is Lanford's kinetic limit.
+
+\textbf{The wobble.} The random gas equals this deterministic density plus
+a small error. Scaling the error by $\sqrt\mu$ gives the fluctuation field
+$\zeta^\eps_t$; the question is its law as $\eps\downarrow0$.
+
+\textbf{The answer.} The scaled error is asymptotically Gaussian, with a
+covariance governed by the fluctuating Boltzmann equation: the linearized
+Boltzmann drift plus a Gaussian noise whose strength is the collision rate,
 \[
  \dd\zeta_t(\phi)=\zeta_t(A_t\phi)\,\dd t+\dd W_t(\phi),
 \]
-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.
-
-\subsection{A map of the symbols (keep this page open)}
-
-Most of the notation is fixed once, in \Cref{sec:objects}. Here is the short
-version, so a first-time reader never has to hunt for a symbol.
+where $A_t$ is the linearized generator and $W$ is a noise of covariance
+$C_t$, both fixed below. This is a finite-dimensional, multi-time central
+limit theorem for a deterministic gas.
+
+\textbf{Not obvious.} The spheres interact deterministically; there is no
+random force and no resampling. The Gaussian limit is emergent, produced by
+the random initial configuration together with the extreme sensitivity of
+which pairs collide. Both effects must be tracked, not averaged over.
+
+\textbf{What was known.} Lanford proved short-time validity of the
+Boltzmann equation. Van Beijeren--Lanford--Lebowitz--Spohn and Spohn
+computed the fluctuation covariance. Pulvirenti--Simonella and
+Bodineau--Gallagher--Saint-Raymond--Simonella built the quantitative
+short-time fluctuation theory, the latter with process-level convergence,
+the Gaussian collision noise, and exact microscopic centering, on a periodic
+box; at equilibrium the same group reached the long-time Gaussian limit.
+Deng--Hani--Ma carried the averaged kinetic limit to the full regular
+Boltzmann lifespan on all of space. The step here is to carry the
+fluctuation theorem to that same lifespan, on all of space, away from
+equilibrium.
+
+Sources, direct PDF:
+\href{https://arxiv.org/pdf/2408.07818v3}{Deng--Hani--Ma, arXiv:2408.07818v3};
+\href{https://arxiv.org/pdf/2008.10403v2}{Bodineau--Gallagher--Saint-Raymond--Simonella, arXiv:2008.10403};
+\href{https://arxiv.org/pdf/2004.00311}{fluctuation theory in the Boltzmann--Grad limit, arXiv:2004.00311};
+\href{https://arxiv.org/pdf/2205.04110}{cluster expansion for hard-sphere dynamics, arXiv:2205.04110};
+\href{https://arxiv.org/pdf/2012.03813}{equilibrium correlations, arXiv:2012.03813};
+\href{https://arxiv.org/pdf/2201.04514}{equilibrium Gaussian limit, arXiv:2201.04514};
+\href{https://doi.org/10.1007/3-540-07171-7_1}{Lanford, Time evolution of large classical systems (1975)}.
+
+\textbf{Two new difficulties.} First, size: the $h$-th factorial cumulant,
+the connected $h$-point correlation, must be pinned at the sharp scale
+$\mu^{1-h}$, far below a crude bound. A change of relative translations of
+collision clusters extracts one $\eps^2$ per merging collision and does it.
+Second, centering: the gas is centered by its true microscopic mean, which
+differs from the Boltzmann mean already at order $\eps$, since hard spheres
+cannot overlap; at the scale $\sqrt\mu=\eps^{-1}$ that difference survives,
+so the model dynamics must match the gas with probability better than
+$O(\eps)$.
+
+\textbf{A map of the symbols.} Most notation is fixed once, in the opening
+paragraphs of the main text; this table lets a first-time reader proceed
+without hunting.
 
 \begin{center}\small
 \begin{tabular}{@{}l p{10.2cm}@{}}
@@ -184,8 +127,8 @@ $\zeta_t^\eps$, $\zeta_t$ & the centered, $\sqrt\mu$-scaled fluctuation
 $W$ & the Gaussian noise driving $\zeta_t$, with independent increments. \\
 $g_i$, $f_{i\tau}$ & one-root mean at interface $i$ (the independent input)
   and the Boltzmann density $f_{i\tau}$ at the same time. \\
-$H_i,\Lambda_i,\Gamma$ & the collision bookkeeping of Part~3: root count,
-  particle-size cap, and cluster-excess cap. \\
+$H_i,\Lambda_i,\Gamma$ & collision bookkeeping of the Deng--Hani--Ma
+  machinery: root count, particle-size cap, and cluster-excess cap. \\
 $L,\ \tau=T/L$, $K$, $\rho$ & number of slabs, slab length, the complexity
   cutoff, and the complexity of a history. \\
 $c_h^\eps$ & the $h$-th factorial cumulant density; its target size is the
@@ -193,64 +136,28 @@ $c_h^\eps$ & the $h$-th factorial cumulant density; its target size is the
 \end{tabular}
 \end{center}
 
-\subsection{How to read this (even in a hurry)}
-
-Read Part~0 and \Cref{sec:objects} first; they state the problem and the
-answer in words and then in symbols. After that, the paragraph \emph{Each part in one sentence} below names the
-idea of every part, so you can always keep your bearings.
-The proof is self-contained except for a single explicit black box,
-\Cref{thm:dhm-package}, which imports the collision bookkeeping of
-Deng--Hani--Ma; everything else, including every sharp correlation bound,
-the covariance, the Gaussian limit, and the exact-centering transfer, is
-proved in these pages.
-
-\subsection{Each part in one sentence}
-
-\begin{enumerate}[label=\textbf{\arabic*.}]
-\item \textbf{Objects and theorem.} Fix the gas, the fluctuation field, and
-  the statement of the finite-dimensional limit (\Cref{thm:main}).
-\item \textbf{Start and target.} The random initial gas is already Gaussian
-  at time zero, and the limiting Gaussian field with its covariance exists.
-\item \textbf{Microscopic engine.} Slab-by-slab collision bookkeeping: the
-  one imported package, stated once and used only to discard large
-  complexity.
-\item \textbf{Connected expansion.} Rewrite correlations as a finite sum of
-  connected collision diagrams that remember the observations.
-\item \textbf{Sharp size.} A change of relative translations gives the sharp
-  size $\mu^{1-h}$ and identifies the surviving trees.
-\item \textbf{Gaussian limit.} Run the central limit theorem; the covariance
-  is the two-root part plus the diagonal.
-\item \textbf{Model versus reality.} The bookkeeping dynamics and the true
-  gas agree with probability $1-O(\eps^{1+1/20})$.
-\item \textbf{Exact centering.} Since $\eta>0$, the exact microscopic mean
-  transfers, and the theorem follows.
-\end{enumerate}
-
-\subsection{A worked picture: why one collision costs $\eps^2$}
-
-Two groups of spheres that evolved independently can join one connected
-history only if, at the contact, their relative position lands exactly on a
-sphere of radius $\eps$ around the other group. Their relative motion is a
-straight line, and a line meets that tiny sphere in a window whose cross
-section is a disk of area of order $\eps^2$, hit during a sliding time
-interval. This is the source of the factor $\eps^2$ per merging collision,
-written in Part~5 as the Jacobian
+\textbf{Why one collision costs $\eps^2$.} Two groups of spheres that
+evolved independently can join one connected history only if, at the
+contact, their relative position lands exactly on a sphere of radius $\eps$
+around the other group. Their relative motion is a straight line, and a line
+meets that tiny sphere in a window whose cross section is a disk of area of
+order $\eps^2$, hit during a sliding time interval. This is the source of the
+factor $\eps^2$ per merging collision, the merger Jacobian
 $\eps^2[(v-v_*)\cdot\omega]\,\dd t\,\dd\omega$. The gas carries activity
 $\mu=\eps^{-2}$ per particle, so a connected diagram with $h$ marked
-particles and $m$ lines weighs $\mu^{m-h}$, and each merger pays one
-$\eps^2$: the two powers of $\eps$ cancel line by line. The initial
-excluded-volume link is a \emph{ball}, not a sphere, so it costs $\eps^3$,
-and that is the only place where the static problem is even smaller than the
-dynamical one.
+particles and $m$ lines weighs $\mu^{m-h}$, and each merger pays one $\eps^2$:
+the two powers of $\eps$ cancel line by line. The initial excluded-volume link
+is a \emph{ball}, not a sphere, so it costs $\eps^3$, the only place where the
+static problem is even smaller than the dynamical one.
 
 Where does the Gaussian randomness come from? A test sees a sum of very many
 almost-independent contributions, one from each close encounter. Whether a
-given near-encounter actually happens is decided by microscopic sensitivity,
-so each contribution acts like an independent bounded random variable. A sum
-of many such variables is Gaussian by the central limit theorem, and its
-variance is the collision rate, which is exactly $C_t$. The whole paper is
-the rigorous proof that the gas is close enough to this picture for it to be
-exact, throughout the regular lifespan and on the whole space.
+given near-encounter happens is decided by microscopic sensitivity, so each
+contribution acts like an independent bounded random variable. A sum of many
+such variables is Gaussian by the central limit theorem, and its variance is
+the collision rate, exactly $C_t$. The body of the argument proves the gas is
+close enough to this picture for it to hold throughout the regular lifespan
+and on the whole space.
 
 % =====================================================================
 \part*{Part 1. The exact objects and the theorem}