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\newtheorem{definition}{Definition}
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\title{Understanding the Quotient\\[4pt]
\large An Explainer for MCP Graf 13}
\author{Claude and MJC}
\date{March 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
Graf 13 of the MCP records five equations from the early paper
\emph{Quotient Renormalization and Maximum Entropy}. They are compact
but they carry a complete picture: a fine-grained distribution can be
factored into a quotient distribution over equivalence classes and a
conditional distribution inside each class. In log-space this becomes
an additive decomposition of code lengths. If the within-class detail
is unknown, the principled default reconstruction is the
maximum-entropy lift, which spreads each class mass uniformly across
its members. This note explains those five equations with plain
definitions and a worked example.
\end{abstract}

\section{Graf 13 Verbatim}

The five equations of MCP graf 13 are reproduced here verbatim:
\begin{align}
t_{[e]} &= \log_2\left(\sum_{e' \sim e} 2^{t_{e'}}\right) \label{eq:quotient-pushforward} \\
t_{e|es} &= t_e - t_{[es]} = \log_2 P(e|es) \label{eq:fiber-coordinate} \\
t_e &= t_{[es(e)]} + t_{e|es} \label{eq:additive-split} \\
P(e) &= P(es) \times P(e|es) \label{eq:hierarchical-factorization} \\
\tilde{P}(e) &= P(es) \times \frac{1}{|es|} \label{eq:maxent-lift}
\end{align}

\section{The Setup}

Let $E$ be a finite set of fine-grained events. We choose an
equivalence relation $\sim$ on $E$ and let $[e]$ denote the
equivalence class containing $e$. Following the source paper, we also
write $es(e)$ for the class of $e$ when we want to stress the event
space interpretation.

\begin{definition}[Quotient and fiber]
The \emph{quotient} is the coarse space $E/{\sim}$ of equivalence
classes. The \emph{fiber} over a class $c \in E/{\sim}$ is the set of
fine events whose class is $c$:
\[
\pi^{-1}(c) = \{ e \in E : [e] = c \}.
\]
\end{definition}

Assume a probability distribution $P$ on $E$. MCP writes
\[
t_e = \log_2 P(e).
\]
These are negative code lengths up to sign convention: multiplying
probabilities becomes addition in $t$-space.

\section{The Quotient Pushforward}

The first equation of graf 13 is:
\[
t_{[e]} = \log_2\left(\sum_{e' \sim e} 2^{t_{e'}}\right).
\]
Since $2^{t_{e'}} = P(e')$, this is just
\[
P([e]) = \sum_{e' \sim e} P(e').
\]

\begin{proposition}
The quotient pushforward replaces each fine event by its equivalence
class and sums the probability mass within that class.
\end{proposition}

This is the coarse distribution. It forgets which member of a class
occurred and keeps only which class occurred.

\section{The Fiber Coordinate}

The second equation is:
\[
t_{e|es} = t_e - t_{[es]} = \log_2 P(e|es).
\]
This is the residual information needed to identify $e$ once its class
$es$ is known. In ordinary probability notation,
\[
P(e|es) = \frac{P(e)}{P(es)}.
\]
The fiber coordinate is therefore the within-class distribution.

The third and fourth equations simply restate the same factorization in
log-space and probability-space:
\[
t_e = t_{[es(e)]} + t_{e|es},
\]
\[
P(e) = P(es) \times P(e|es).
\]

These are the key structural statements of the graf:
\begin{enumerate}
\item first choose the coarse class;
\item then choose the fine event inside that class.
\end{enumerate}

In compression language: total code length splits into a
class-description term and a within-class residual term.

\section{Worked Example}

\begin{example}[Two classes]
Let
\[
E = \{a_1, a_2, b_1, b_2, b_3\},
\]
with classes
\[
A = \{a_1, a_2\},
\qquad
B = \{b_1, b_2, b_3\}.
\]
Suppose
\[
P(a_1)=\tfrac{1}{8}, \quad P(a_2)=\tfrac{3}{8}, \quad
P(b_1)=P(b_2)=P(b_3)=\tfrac{1}{6}.
\]
Then
\[
P(A)=\tfrac{1}{2},
\qquad
P(B)=\tfrac{1}{2}.
\]
Inside the fibers we have
\[
P(a_1|A)=\tfrac{1}{4}, \quad P(a_2|A)=\tfrac{3}{4},
\]
and
\[
P(b_i|B)=\tfrac{1}{3}.
\]
\end{example}

This example shows the two-stage factorization explicitly:
\[
P(a_2)=P(A)\,P(a_2|A)=\tfrac{1}{2}\cdot\tfrac{3}{4}=\tfrac{3}{8}.
\]
The same logic applies to every event in $E$.

\section{The Maximum-Entropy Lift}

The last equation of graf 13 is:
\[
\tilde{P}(e) = P(es) \times \frac{1}{|es|}.
\]
This answers a different question. Suppose we know the quotient
distribution $P(es)$ but not the within-class details. Which fine
distribution should we reconstruct?

The source paper chooses the maximum-entropy answer: make the
distribution uniform inside each class. Then each member of a class
gets an equal share of that class's mass.

In the worked example above, if we forget the internal splits in $A$
and $B$ but keep $P(A)=P(B)=1/2$, the lift becomes
\[
\tilde{P}(a_1)=\tilde{P}(a_2)=\tfrac{1}{4},
\]
\[
\tilde{P}(b_1)=\tilde{P}(b_2)=\tilde{P}(b_3)=\tfrac{1}{6}.
\]
So the max-entropy lift preserves the coarse structure while refusing
to invent unsupported within-fiber asymmetry.

\section{Why Graf 13 Matters}

Graf 13 matters because it isolates three distinct objects:
\begin{enumerate}
\item the quotient distribution over coarse classes;
\item the fiber distributions within each class;
\item the max-entropy reconstruction when the fiber distributions are
unknown.
\end{enumerate}

That gives a clean hierarchy:
\begin{enumerate}
\item coarse structure can be modeled separately from fine detail;
\item the fine detail is exactly the conditional distribution on the
fibers;
\item if the fine detail is discarded, the least-committal lift is
uniform within each fiber.
\end{enumerate}

This is why the graf belongs in the MCP. It is not tied to the
original ES experiment. It is a general statement about quotienting a
distribution, decomposing description length, and reconstructing a
fine distribution from a coarse one.

\section{Relation to the Source Paper}

The source paper \emph{Quotient Renormalization and Maximum Entropy}
contains these equations directly: the quotient pushforward, the
within-ES fiber coordinate, the multiplicative factorization
$P(e)=P(es)P(e|es)$, and the max-entropy lift. What MCP graf 13 does
is strip away the experiment-specific table and keep the reusable
mathematical core.

\begin{thebibliography}{9}
\bibitem{qmax}
Claude and MJC.
\newblock \emph{Quotient Renormalization and Maximum Entropy}.
\newblock Hutter archive, 31 January 2026.
\newblock \texttt{docs/archive/20260131\_2/quotient-maxent.tex}

\bibitem{mcp}
Claude and MJC.
\newblock \emph{MCP}.
\newblock Working manuscript, March 2026.
\end{thebibliography}

\end{document}

