diff --git a/hard_sphere_boltzmann_fluctuations_v2.txt b/hard_sphere_boltzmann_fluctuations_v2.txt
index 6dece8e..2cf4d55 100644
--- a/hard_sphere_boltzmann_fluctuations_v2.txt
+++ b/hard_sphere_boltzmann_fluctuations_v2.txt
@@ -158,6 +158,100 @@ $\mu^{1-h}$ bounds and identifies the limiting trees.
 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.
+
+\begin{center}\small
+\begin{tabular}{@{}l p{10.2cm}@{}}
+$\eps$ & diameter of one sphere; the limit is $\eps\downarrow0$. \\
+$\mu=\eps^{-2}$ & mean number of spheres per unit volume. Its square root
+  $\sqrt\mu=\eps^{-1}$ is the \emph{size of the random wobble}, so the wobble
+  is larger than $\eps$ but vanishes as a fraction of the mean. \\
+$N$ & number of spheres; $N/\mu$ stays near the mass $\int f_0\,\dd z$. \\
+$f_0(z)$, $f_t(z)$ & initial one-particle density and the regular Boltzmann
+  solution, both in $z=(x,v)$; regularity on $[0,T]$ is
+  \Cref{ass:kinetic}. \\
+$\|\cdot\|_{\Bol,\beta}$ & sup norm with a Gaussian velocity weight
+  $e^{\beta|v|^2}$, summed over unit spatial cells so mass is controlled. \\
+$B$, $\Delta\phi$ & collision kernel $[(v-v_*)\cdot\omega]_+$ and the change
+  $\Delta\phi$ of a test across one collision. \\
+$A_t$, $C_t$ & linearized Boltzmann drift and collision-noise covariance:
+  the two coefficients of the fluctuating equation. \\
+$\zeta_t^\eps$, $\zeta_t$ & the centered, $\sqrt\mu$-scaled fluctuation
+  (microscopic) and its Gaussian limit. \\
+$W$ & 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. \\
+$L,\ \tau=T/L$, $K$, $\rho$ & 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
+  sharp scale $\mu^{1-h}$.
+\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
+$\eps^2[(v-v_*)\cdot\omega]\,\dd t\,\dd\omega$. 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.
+
+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.
+
 % =====================================================================
 \part*{Part 1. The exact objects and the theorem}
 % =====================================================================
@@ -1368,7 +1462,8 @@ top loss absorb into the exponential size gain,
  \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 $\Delta_{i-1}$
+Input orders are at most $(|S|+1)\Lambda_i^2/2$, which is below
+$A_{i-1}/2$; 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.