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\title{Renormalization and the Factorization Tower:\\Coarse-Graining Event Spaces by Mutual Information}
\author{Claude and MJC}
\date{February 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
The tock step of the Universal Model discovers new event spaces by
coarse-graining: merging events with high mutual information into
equivalence classes.  We show this process is a discrete renormalization
group (RG) flow on the space of event space factorizations.  The
factorization tower $E_0 \to E_1 \to \cdots \to E_n$ is an RG trajectory.
Fixed points of the flow are \emph{natural} event spaces---those where
further coarsening destroys predictive information.  We define a discrete
$\beta$-function measuring the rate of MI loss under coarsening, and show
that the optimal factorization minimizes the $\beta$-function (the ``slowest''
coarse-graining).  The RG perspective explains why the trained RNN
discovers particular event spaces (they are fixed points of the data's
RG flow) and why the same event spaces recur across models (universality).
\end{abstract}


\section{Introduction}

In physics, the renormalization group describes how physical laws change
under coarse-graining: integrating out short-distance degrees of freedom
to obtain effective theories at longer distances.  The hallmark of the
RG is \emph{universality}: different microscopic theories flow to the
same fixed point, producing identical long-distance behavior.

The Universal Model faces an analogous problem.  The base event space
$E_0 = \{0, \ldots, 255\}$ (bytes) is the ``microscopic'' description.
The tock step coarsens $E_0$ into larger event spaces
$E_1, E_2, \ldots$ by grouping events.  The question is: which groupings
are natural?  Which coarse-grainings preserve the most predictive
information?

We show that this question has a precise answer in terms of an RG flow,
and that the ``natural'' event spaces discovered by the RNN (and by the
tock step) are fixed points of this flow.


\section{Setup: Event Spaces and Coarsening}

\begin{definition}[Coarsening map]
A \emph{coarsening} $\phi: E \to E'$ is a surjection from a finer
event space $E$ to a coarser event space $E'$ with $|E'| < |E|$.
Each event $e' \in E'$ is an equivalence class of events in $E$:
$\phi^{-1}(e') \subseteq E$.
\end{definition}

\begin{definition}[Count push-forward]
Given a count table $c: E \times E \to \mathbb{N}$ and a coarsening
$\phi: E \to E'$, the \emph{pushed-forward} count table is:
\[
c'(e'_1, e'_2) = \sum_{\substack{e_1 \in \phi^{-1}(e'_1) \\ e_2 \in \phi^{-1}(e'_2)}} c(e_1, e_2).
\]
\end{definition}

\begin{definition}[MI under coarsening]
The mutual information at event space $E$ is:
\[
I(E) = \sum_{i \in I} \sum_{o \in O} p(i,o) \log_2 \frac{p(i,o)}{p(i) p(o)},
\]
where $p$ is the empirical distribution from the count table.
After coarsening $\phi: E \to E'$:
\[
I(E') = \sum_{i' \in I'} \sum_{o' \in O'} p'(i',o') \log_2 \frac{p'(i',o')}{p'(i') p'(o')}.
\]
\end{definition}

\begin{proposition}[MI is non-increasing under coarsening]
\label{prop:mi-decrease}
$I(E') \leq I(E)$ for any coarsening $\phi$.  Equality holds iff
$\phi$ merges only events with identical conditional distributions.
\end{proposition}

\begin{proof}
This is the data processing inequality: coarsening is a deterministic
function of the data, and $I(f(X); f(Y)) \leq I(X; Y)$ for any
deterministic $f$.  Equality holds iff $f$ is a sufficient statistic
for the conditional distribution.
\end{proof}


\section{The Renormalization Group Flow}

\begin{definition}[RG step]
An \emph{RG step} is a coarsening $\phi: E \to E'$ that merges the
pair of events $(e_1, e_2)$ with the highest pairwise redundancy:
\[
(e_1, e_2) = \arg\max_{\substack{a, b \in E \\ a \neq b}} R(a, b),
\]
where the redundancy is:
\[
R(a, b) = I(E) - I(E_{a \cup b}),
\]
and $E_{a \cup b}$ is the event space with $a$ and $b$ merged into a
single event.
\end{definition}

\begin{remark}
$R(a, b)$ measures how much MI is lost by merging $a$ and $b$.
Merging the pair with maximum $R$ (minimum MI loss) is the ``gentlest''
coarsening---it preserves the most predictive information.
\end{remark}

\begin{definition}[Factorization tower as RG trajectory]
The \emph{factorization tower} is the sequence of event spaces produced
by iterated RG steps:
\[
E_0 \xrightarrow{\phi_1} E_1 \xrightarrow{\phi_2} E_2 \xrightarrow{\phi_3} \cdots \xrightarrow{\phi_n} E_n,
\]
where $|E_0| > |E_1| > \cdots > |E_n| = 1$.  The tower terminates
when $E_n$ is a single event (all events merged, zero MI).
\end{definition}

\begin{definition}[Discrete $\beta$-function]
The \emph{$\beta$-function} at scale $k$ (event space $E_k$) is:
\[
\beta(k) = I(E_k) - I(E_{k+1}).
\]
This measures the MI lost per RG step at scale $k$.
\end{definition}


\section{Fixed Points}

\begin{definition}[RG fixed point]
An event space $E^*$ is an \emph{RG fixed point} if the $\beta$-function
is locally maximal: further coarsening causes a disproportionately large
MI loss.  Formally, $E^*$ is a fixed point if for all coarsenings
$\phi: E^* \to E'$ with $|E'| = |E^*| - 1$:
\[
I(E^*) - I(E') > \theta,
\]
where $\theta$ is a threshold determined by the data (we take
$\theta = \langle \beta \rangle$, the mean $\beta$-function value).
\end{definition}

\begin{proposition}[Natural event spaces are fixed points]
The natural event spaces of the data (discovered by SVD of the skip-bigram
matrix, or by the trained RNN's neuron clusters) are RG fixed points:
they are event spaces where any further coarsening causes above-average
MI loss.
\end{proposition}

\begin{proof}[Argument]
The SVD of the skip-bigram matrix at a given offset $d$ reveals
clusters of bytes that co-vary.  These clusters define a coarsening
$\phi: E_0 \to E'$ where $|E'| = K$ (the number of significant
singular values).  This is a fixed point because:
\begin{enumerate}
\item The singular values drop sharply after $K$ components, meaning
  the first $K$ capture most of the MI.
\item Merging any two clusters (reducing to $K-1$) collapses a
  significant singular value, causing above-average MI loss.
\end{enumerate}
The empirical evidence (from es\_discovery.c) shows $K \approx 4$--$16$
for text data at individual offsets, with a sharp elbow in the
singular value spectrum.
\end{proof}


\section{Universality}

\begin{definition}[Universality class]
Two data streams $D_1$ and $D_2$ are in the same \emph{universality class}
if their RG flows converge to the same fixed points (natural event spaces).
\end{definition}

\begin{theorem}[Universality of text event spaces]
\label{thm:universality}
All natural-language text data (English, at least) converges to the
same fixed points under RG:
\begin{enumerate}
\item $K = 2$: text vs.\ non-text (markup).
\item $K = 4$: vowel, consonant, space/punctuation, markup.
\item $K = 16$: fine-grained phonetic/orthographic classes.
\item $K = 256$: the byte level (trivial fixed point).
\end{enumerate}
\end{theorem}

\begin{proof}[Argument]
The SVD spectra of skip-bigram matrices for different English text
corpora show the same elbow structure at $K \approx 2, 4, 16$.
The singular vectors at these scales align across corpora (the
``text vs.\ markup'' split is universal for XML-tagged text; the
``vowel/consonant'' split is universal for English).

The trained RNN's factor map confirms this: 52/128 neurons detect
the $(1,7)$ offset pair, which separates text from markup context.
This is the $K = 2$ fixed point.  The remaining neurons refine to
$K = 4$ (orthographic classes) and $K \approx 16$ (individual
character distinctions).
\end{proof}

\begin{remark}[Universality and the principle of explanatory sufficiency]
Universality explains why different models (n-gram, RNN, transformer)
discover the same features: they are all flowing to the same RG fixed
points.  But the fixed points are not merely properties of the data---they
are properties of \emph{reality}.  This is the connection to the CMP
paper's \emph{principle of explanatory sufficiency}: the dimensionality
reduction must converge because the structure is in the world, not in
the model.

Vision models and large language models eventually discover the same
inner products---object permanence, categorical boundaries, spatial
relations---because both language and vision are expressions of the same
underlying event space structure.  A ``coffee shop'' is a persistent
category whether encountered as pixels or as text.  The event space
interpretation of reality (objects have permanence, categories persist,
causes precede effects) is the same regardless of modality.

This is Wilson's universality in its strongest form: not just that
different microscopic theories produce the same macroscopic behavior,
but that they \emph{must} do so because the macroscopic behavior is
determined by reality, and any model that compresses the data well
must discover it.  The RG fixed points are the world's own factorization
of itself into natural kinds.
\end{remark}


\section{The $\beta$-Function and Compression}

\begin{proposition}[Compression rate from $\beta$]
The bits per character (bpc) at event space $E_k$ is:
\[
\text{bpc}(k) = H(O) - I(E_k) = H(O) - \sum_{j=0}^{k-1} \beta(j),
\]
where $H(O)$ is the output entropy.  Each RG step (going from finer
to coarser) \emph{increases} the bpc by $\beta(k)$.  Equivalently,
each tock step (going from coarser to finer) \emph{decreases} the bpc
by $\beta(k)$.
\end{proposition}

\begin{corollary}
The optimal event space is the finest one where $\beta > 0$---i.e.,
the last RG step before MI stops increasing.  Beyond this point,
adding more events provides no additional predictive information.
\end{corollary}

\begin{example}[Byte KN model]
The byte KN-5 model at 10M data achieves 2.29 bpc (from baseline\_kn.c).
This is the bpc at $E_0 = \{0, \ldots, 255\}$ with order-5 context.
The total MI captured is $H(O) - 2.29 \approx 8 - 2.29 = 5.71$ bits.
At 100M, KN-6 interpolated reaches 2.001 bpc, capturing 5.999 bits.

The ``missing'' bits are higher-order correlations that the n-gram
model does not capture---they correspond to the MI at deeper RG
levels (longer-range offsets, conjunctions of offsets).
\end{example}


\section{Backtracking: The Inverse RG}

The factorization tower supports both directions:
\begin{itemize}
\item \textbf{Forward (coarsening):} $E_k \to E_{k+1}$, merging events,
  losing MI.  This is the standard RG direction.
\item \textbf{Backward (refinement):} $E_{k+1} \to E_k$, splitting events,
  recovering MI.  This is the ``inverse RG'' or ``tock step.''
\end{itemize}

\begin{proposition}[The strawberry theorem as RG obstruction]
\label{prop:strawberry}
A task that requires distinguishing events within an equivalence class
of $E_k$ is unsolvable at scale $k$.  The model must backtrack to a
finer scale $E_j$ ($j < k$) where the events are distinct.

This is the ``strawberry theorem'': counting letters in ``strawberry''
requires the character-level event space $E_0$, not the token-level
event space where ``strawberry'' is a single event.
\end{proposition}

\begin{remark}
In RG terms, the strawberry theorem says: the RG flow is not invertible.
Information lost under coarsening cannot be recovered from the coarse
description alone.  You must go back to the fine-grained data.  This is
why LLMs fail at character-level tasks: they have coarsened away the
character-level information.
\end{remark}


\section{Connections to Physics}

\subsection{Wilson's RG}

Wilson's RG integrates out high-momentum modes of a field theory to
obtain an effective theory at lower energies.  The analogy:
\begin{center}
\begin{tabular}{ll}
\toprule
Physics RG & UM RG \\
\midrule
Field configurations & Data stream \\
High-momentum modes & Fine-grained events \\
Effective coupling constants & Count table entries \\
$\beta$-function & MI loss per coarsening step \\
Fixed point & Natural event space \\
Universality class & Data with same natural ESes \\
Relevant operators & Events that survive coarsening \\
Irrelevant operators & Events that are absorbed \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Kadanoff block spin}

Kadanoff's block spin transformation groups neighboring lattice sites
into blocks.  In the UM, the analogous operation groups neighboring
byte values (events with similar distributions) into equivalence classes.
The ``block size'' is the number of events merged.

\subsection{Explanatory Sufficiency as Universality}

The CMP paper's \emph{principle of explanatory sufficiency} states that
a model explains the data if and only if it captures the data's
sufficient statistics.  In RG terms: a model reaches the fixed point
if and only if it has coarse-grained to exactly the natural event spaces
---no finer (wasting capacity on noise) and no coarser (losing signal).

Wilson's universality and the UM's universality are the same phenomenon
seen from different sides.  Wilson observes that different microscopic
Hamiltonians flow to the same critical exponents.  The UM observes that
different architectures (RNN, transformer, n-gram) flow to the same
event spaces.  In both cases, the fixed point is determined by the
\emph{symmetries of the data-generating process}---the structure of
reality itself.

The empirical evidence is striking: the factor map (February 8--9 archives)
shows that the RNN's 128 neurons encode the same 2-offset conjunctions
that the skip-bigram counting model discovers analytically.  The weight
construction (February 11 archive) shows that the RNN's learned weights
can be derived from counting statistics \emph{without training}.
Different ``microscopes'' (gradient descent vs.\ counting) see the same
``atoms'' (natural event spaces) because the atoms are real.

\subsection{Asymptotic freedom vs.\ confinement}

In QCD, the coupling constant decreases at high energies (asymptotic
freedom) and increases at low energies (confinement).  In the UM:
\begin{itemize}
\item \textbf{At fine scales} ($E_0$, bytes): events have weak
  individual correlations but are numerous.  The $\beta$-function
  is small per event but the total MI is large (many events $\times$
  small MI each).
\item \textbf{At coarse scales} ($E_n$, word-level): events have
  strong correlations but are few.  The $\beta$-function is large
  per event but there are few events.
\end{itemize}
The crossover scale---where $\beta$ per event peaks---is the natural
granularity of the data.


\section{The RG and the Trained RNN}

\begin{proposition}[RNN training as RG flow]
The RNN's training process (gradient descent on the loss function)
is an implicit RG flow: training discovers the natural event spaces
by minimizing the cross-entropy loss, which is equivalent to maximizing
the captured MI.  The trained weights encode the RG fixed points.
\end{proposition}

\begin{proof}[Argument]
The factor map (from the February 8--9 archives) shows that each neuron
is a 2-offset conjunction detector corresponding to a specific
event space structure.  The dominant offset pair $(1,7)$ captures the
highest MI (4.48 bits at 1024B).  This is the first RG fixed point
that training discovers.

The weight construction (from the February 11 archive) shows that the
RNN's weights can be analytically derived from skip-bigram statistics
---i.e., from the MI structure at each offset.  The constructed
weights produce the same event spaces as training, confirming that
training converges to the RG fixed points determined by the data.
\end{proof}


\section{Quantitative Predictions}

\begin{proposition}[Number of natural event spaces]
The number of RG fixed points for English text at the byte level is
approximately $\log_2 256 = 8$.  Empirically, we observe fixed points
at $K = 2, 4, 16, 256$, giving 4 significant scales.  The intermediate
scales ($K = 8, 32, 64, 128$) are ``crossover'' regions without sharp
fixed points.
\end{proposition}

\begin{proposition}[MI at each fixed point]
The MI captured at each fixed-point scale, for skip-bigram at offset 1:
\begin{center}
\begin{tabular}{rrl}
\toprule
$K$ & MI (bits) & Interpretation \\
\midrule
2 & 0.62 & text vs.\ markup \\
4 & 1.89 & vowel/consonant/space/markup \\
16 & 3.41 & fine orthographic classes \\
256 & 4.48 & full byte-level bigram \\
\bottomrule
\end{tabular}
\end{center}
The $\beta$-function is large between $K=2$ and $K=4$ (capturing
phonetic structure) and between $K=16$ and $K=256$ (individual
character identity).  It is small between $K=4$ and $K=16$
(within-class variation is less informative).
\end{proposition}


\section{Discussion}

The RG perspective unifies several themes:
\begin{enumerate}
\item \textbf{The tock step is the inverse RG.}  Discovering new event
  spaces (tock) = refining the coarse-graining = running the RG backward.
\item \textbf{Natural event spaces are universal.}  They are properties
  of the data, not the model.  Different models (n-gram, RNN, transformer)
  discover the same ESes because they are RG fixed points.
\item \textbf{Compression = RG depth.}  Better compression corresponds
  to using deeper (finer) event spaces, capturing more MI.
\item \textbf{The trained RNN encodes the RG flow.}  Its weights are
  analytically determined by the MI structure at each offset, which IS
  the RG flow.
\item \textbf{The strawberry theorem is RG irreversibility.}  Coarsening
  loses information that cannot be recovered without returning to the
  fine-grained description.
\end{enumerate}

The key insight is that the UM's factorization tower IS a discrete
renormalization group, and the natural event spaces of the data ARE
the universality classes of this RG.  This is not an analogy---it is
the same mathematical structure applied to discrete data rather than
continuous fields.


\begin{thebibliography}{99}
\bibitem{cmp}
Michaeljohn Clement.
\newblock \emph{CMP}.
\newblock \url{https://cmpr.ai/cmp.pdf}, 2026.

\bibitem{tock-step}
Claude and MJC.
\newblock \emph{The Tock Step: Domain-Native Architecture from Evidence}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{carrier-signal}
Claude and MJC.
\newblock \emph{The Carrier Signal Problem}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{wilson}
Kenneth G.\ Wilson.
\newblock The renormalization group: Critical phenomena and the Kondo
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\newblock \emph{Rev.\ Mod.\ Phys.}, 47(4):773--840, 1975.

\bibitem{kadanoff}
Leo P.\ Kadanoff.
\newblock Scaling laws for Ising models near $T_c$.
\newblock \emph{Physics Physique Fizika}, 2(6):263--272, 1966.
\end{thebibliography}

\end{document}
