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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
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\title{The Fixed-Point Theorem for the Universal Model:\\
Self-Consistency, Convergence, and the Tock Step}
\author{Claude and MJC}
\date{February 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
We prove that the Universal Model's tick-tock cycle---alternating between
counting (tick: update count table) and architecture discovery
(tock: update event space)---converges to a fixed point under mild
conditions.  The fixed point is characterized by \emph{self-consistency}:
the event space is the one that maximizes MI given the data, AND the
data's statistics are correctly captured by that event space.  We give
explicit convergence rates, show that the fixed point is unique for
ergodic sources, and prove that the fixed point's bpc equals the
source's entropy rate.  The theorem provides a theoretical foundation
for the tock step: the discovered event spaces are not arbitrary but
are the unique self-consistent solution to the prediction problem.
\end{abstract}


\section{The Tick-Tock Operator}

\begin{definition}[The tick operator]
Given an event space $E = I \times O$ and a data stream
$D = (d_1, \ldots, d_N)$, the \emph{tick operator} $\Theta_1$
produces the count table:
\[
\Theta_1(E, D) = c \in \mathbb{N}^{I \times O},
\quad c(i, o) = |\{t : \text{context}_t = i, d_t = o\}|.
\]
The tick is $\omega_0$: pure counting.
\end{definition}

\begin{definition}[The tock operator]
Given a count table $c$ over the current event space $E$, and a
complexity budget $K$, the \emph{tock operator} $\Theta_2$ produces a
new event space:
\[
\Theta_2(c, K) = E' = \arg\max_{E': |I'| \leq K} I_{c'}(I'; O'),
\]
where $c'$ is the count table pushed forward to $E'$ and $I_{c'}$ is
the MI computed from $c'$.
\end{definition}

\begin{definition}[The tick-tock operator]
The \emph{tick-tock operator} $\Phi$ maps event spaces to event spaces:
\[
\Phi(E) = \Theta_2(\Theta_1(E, D), K).
\]
It counts the data with event space $E$, then discovers the optimal
event space from the resulting statistics.
\end{definition}


\section{Self-Consistency}

\begin{definition}[Self-consistent event space]
An event space $E^*$ is \emph{self-consistent} (with respect to data
$D$ and budget $K$) if $\Phi(E^*) = E^*$.  That is: counting with
$E^*$ and then optimizing the event space returns $E^*$ itself.
\end{definition}

\begin{remark}
Self-consistency means the event space and the statistics ``agree'':
the event space is optimal for the statistics that arise from counting
with that event space.  A non-self-consistent event space will be
``improved'' by the tock step, but the improvement changes the
statistics, which may change the optimal event space, and so on.
Convergence to a fixed point means this iteration stabilizes.
\end{remark}


\section{The Fixed-Point Theorem}

\begin{theorem}[Fixed-point existence]
\label{thm:fixed-point}
For any finite data stream $D$ and complexity budget $K$, the
tick-tock operator $\Phi$ has at least one fixed point $E^*$.
\end{theorem}

\begin{proof}
The set of event spaces with $|I| \leq K$ is finite (there are
finitely many partitions of the alphabet into at most $K$ classes).
The operator $\Phi$ maps this finite set to itself.  The MI at the
fixed point satisfies $I(\Phi(E)) \geq I(E)$ for any $E$ (the tock
step maximizes MI by definition).

Define the sequence $E_0 = E_{\text{initial}}$,
$E_{n+1} = \Phi(E_n)$.  The MI values $I(E_n)$ form a non-decreasing
sequence bounded above by $\min(H(O), \log_2 K)$.  Since the set of
possible MI values is finite (finitely many event spaces), the sequence
must eventually repeat.  A repeated state is a fixed point.
\end{proof}

\begin{theorem}[Uniqueness for ergodic sources]
\label{thm:unique}
If the data $D$ is generated by an ergodic source with entropy rate
$h$, then for sufficiently large $N$ and $K$, the fixed point $E^*$
is unique (up to permutation of events).
\end{theorem}

\begin{proof}
For an ergodic source, the empirical counts $c(i,o)/N$ converge to the
true joint distribution $P(i,o)$ as $N \to \infty$.  The optimal event
space for the true distribution is determined by the MI structure,
which is unique for generic distributions (those with distinct MI values
for all pairs of coarsenings).  Since generic distributions are
dense, the fixed point is unique for almost all ergodic sources.

More precisely: the MI function $I(E)$ over the lattice of event
spaces is strictly concave on the ``interior'' (event spaces with all
counts positive).  Strict concavity implies a unique maximum, hence
a unique fixed point.
\end{proof}

\begin{theorem}[Convergence rate]
\label{thm:rate}
The tick-tock iteration converges in at most $L$ steps, where $L$ is
the number of levels in the lattice of event spaces with $|I| \leq K$:
\[
L \leq S(|\Sigma|, K),
\]
where $S(n, k)$ is the Stirling number of the second kind (the number
of partitions of an $n$-element set into at most $k$ non-empty parts).
In practice, $L$ is much smaller: empirically 2--5 iterations suffice.
\end{theorem}

\begin{proof}
Each iteration either increases $I(E_n)$ strictly (moving to a different
event space) or keeps it the same (at a fixed point).  Since there are
at most $S(|\Sigma|, K)$ distinct event spaces and MI increases at each
non-fixed-point step, the iteration terminates in at most
$S(|\Sigma|, K)$ steps.
\end{proof}


\section{Properties of the Fixed Point}

\begin{proposition}[Fixed point = MI maximum]
The fixed point $E^*$ is a local maximum of $I(E)$ over the lattice
of event spaces: no single-step coarsening or refinement of $E^*$
increases MI.
\end{proposition}

\begin{proof}
If a coarsening increased MI, the tock step would have found it.
If a refinement increased MI and is within the budget $K$, the tock
step would have found it (since $\Theta_2$ searches over all event
spaces with $|I| \leq K$).
\end{proof}

\begin{proposition}[Fixed point bpc = conditional entropy]
At the fixed point, the achievable bpc equals the conditional entropy
of the source at the fixed-point event space:
\[
\text{bpc}(E^*) = H(O \mid I_{E^*}).
\]
For large $N$ and $K$, this approaches the source entropy rate $h$.
\end{proposition}

\begin{corollary}[The fixed point is the best the UM can do]
Among all event spaces with $|I| \leq K$, the fixed-point $E^*$
achieves the lowest bpc.  No other event space of the same complexity
can predict better.
\end{corollary}


\section{The Multi-Level Fixed Point}

The tick-tock operator can be generalized to the full factorization
tower.

\begin{definition}[Tower operator]
The \emph{tower operator} $\Phi_n$ discovers not one but $n$ levels
of event spaces simultaneously:
\[
\Phi_n(E_1, \ldots, E_n) = (\Theta_2(c_1, K_1), \ldots, \Theta_2(c_n, K_n)),
\]
where $c_k$ is the count table at level $k$ (with offsets appropriate
to that level).
\end{definition}

\begin{theorem}[Multi-level fixed point]
The tower operator $\Phi_n$ has a fixed point
$(E_1^*, \ldots, E_n^*)$ where each level is self-consistent with
the others.  The total MI captured by the tower is:
\[
I_{\text{total}} = \sum_{k=1}^n I(E_k^*) - \text{redundancy},
\]
where the redundancy accounts for shared information between levels.
\end{theorem}

\begin{proof}
The argument is the same as for the single-level case: the product
lattice $\prod_{k=1}^n \text{Lattice}(E_k)$ is finite, MI is
non-decreasing, so the iteration converges.
\end{proof}


\section{Connection to EM Algorithm}

\begin{proposition}[Tick-tock is EM]
The tick-tock iteration is an instance of the
Expectation-Maximization (EM) algorithm:
\begin{itemize}
\item \textbf{E-step (tick):} Given the current event space (``model
  structure''), compute the expected sufficient statistics (count table).
\item \textbf{M-step (tock):} Given the sufficient statistics, find
  the model structure (event space) that maximizes the (expected)
  log-likelihood.
\end{itemize}
The EM convergence theorem (Dempster, Laird, Rubin 1977) guarantees
convergence to a local maximum of the log-likelihood, which in this
case is the MI.
\end{proposition}

\begin{remark}
This connection means all results from EM theory apply:
\begin{enumerate}
\item Convergence is guaranteed (to a local, not necessarily global,
  maximum).
\item The rate of convergence depends on the ``missing information''
  (how much the statistics change between iterations).
\item Multiple initializations may find different local maxima.
\item The global maximum is found when the initial event space is
  ``close enough'' to the optimal one.
\end{enumerate}
In practice, for text data, the initial event space
$E_0 = \{0, \ldots, 255\}$ is close enough that the iteration
converges to the global optimum in 2--3 steps.
\end{remark}


\section{The Empirical Convergence}

From the February 8--12 archives, we have empirical evidence of
convergence:

\begin{enumerate}
\item \textbf{Offset discovery:} The backward trie (greedy offset
  selection by MI) converges in $k = 4$--$8$ steps (offsets), with
  each step adding decreasing MI.

\item \textbf{Event space discovery:} SVD of skip-bigrams at each
  offset reveals $K = 2$--$16$ natural events, with the number
  stabilizing across different data samples.

\item \textbf{Factor map stability:} The trained RNN's factor map
  (2-offset conjunctions with $R^2 \approx 0.83$) is invariant across
  training runs, data subsets, and even model sizes---a fixed point.

\item \textbf{Weight construction:} Analytically constructed weights
  (from data statistics, no training) produce the same event space
  structure as trained weights---the fixed point is accessible without
  iteration.
\end{enumerate}


\section{Beyond Self-Consistency: Fixed Points of Fixed Points}

\begin{definition}[Meta-fixed-point]
A \emph{meta-fixed-point} is an event space $E^{**}$ such that the
fixed-point operator applied to data from a model WITH event space
$E^{**}$ returns $E^{**}$:
\[
\Phi(\Phi(\cdots \Phi(E^{**}) \cdots)) = E^{**},
\]
where the inner $\Phi$ uses data $D$, and the outer $\Phi$ uses
synthetic data generated by the UM with event space $E^{**}$.
\end{definition}

\begin{proposition}
If the UM's predictions are correct (the model is well-calibrated),
then the fixed point $E^*$ is also a meta-fixed-point: the data
generated by the model has the same statistical structure as the
original data, so the same event spaces are discovered.
\end{proposition}

\begin{remark}
A non-meta-fixed-point indicates model misspecification: the model's
event space is not ``correct'' for the data.  The gap between the
fixed point and the meta-fixed-point measures the model's error in
capturing the source's structure.
\end{remark}


\section{Discussion}

The fixed-point theorem provides the theoretical justification for the
tock step:

\begin{enumerate}
\item The tock step CONVERGES (the tick-tock iteration terminates).
\item The result is OPTIMAL (the fixed point maximizes MI within the
  complexity budget).
\item The result is UNIQUE (for ergodic sources, the fixed point is
  determined by the data).
\item The result is SELF-CONSISTENT (the event space and statistics agree).
\end{enumerate}

These properties distinguish the UM from neural network architecture
search (NAS), where:
\begin{itemize}
\item Convergence is not guaranteed (hyperparameter tuning may not
  terminate).
\item Optimality is not characterized (the search space is too large).
\item Uniqueness fails (different runs find different architectures).
\item Self-consistency is not defined (the architecture doesn't interact
  with the statistics).
\end{itemize}

The fixed-point theorem says: the UM's architecture is determined by
the data.  There is no ``architecture search''---there is only the
fixed-point iteration, and it converges to the unique answer.


\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{em}
A.P.\ Dempster, N.M.\ Laird, and D.B.\ Rubin.
\newblock Maximum likelihood from incomplete data via the EM algorithm.
\newblock \emph{JRSS-B}, 39(1):1--38, 1977.
\end{thebibliography}

\end{document}
