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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
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\title{Lexemes as Binary Event Spaces:\\
From Atomic Patterns to Bag-of-Letters Prediction}
\author{Claude and MJC}
\date{February 14, 2026}

\begin{document}
\maketitle

\begin{abstract}
We show how to introduce lexeme-level prediction into the isomorphic
Universal Model without defining any new finite automata.  The
model's existing atomic patterns---the data-derived skip-bigrams
over individual bytes---already carry the influence of each letter
separately on the prediction of upcoming word boundaries.  A word
like ``the'' is not recognized by a dedicated detector but is
\emph{implicit} in the joint support from \texttt{t}, \texttt{h},
and \texttt{e} at their respective offsets.  This gives a natural
``bag of letters'' model of words, which we note matches findings
from human reading research (transposed-letter effects, etc.).
We formalize each lexeme as a binary event space---log support for
``the'' vs.\ its negation (which includes ``there'', ``them'',
``then'', etc.)---and show that this binary ES is the correct shape:
input bytes map to a \emph{probability} of the lexeme, not a
certainty.  The negative event distributes over the rest of the
lexeme's prefix class, making this a clean case study for negative
event distribution.  Finally, we observe that the map from input
bytes to support over lexemes is itself a complete model $U$,
and the UM algebra tells us how to compose it with the existing
byte-level model.
\end{abstract}


\section{Introduction}

The tock step~\cite{tock-step} discovers new event spaces from data
statistics.  The tock protocol~\cite{tock-protocol} proposed injecting
words one at a time via dedicated finite automata---word detectors
that scan the byte stream and fire deterministically.

MJC's key observation: we don't need to define new FSAs.
The model's \emph{existing} atomic patterns already encode the
influence of each letter.  When the model has seen \texttt{t} at
offset 3, \texttt{h} at offset 2, and \texttt{e} at offset 1,
these three independent byte-level observations jointly support the
prediction that the current position follows the word ``the.''
No word detector is needed.  The word is implicit in the
bag of letters.

This paper develops this observation into a formal construction.


\section{Atomic Patterns Already Match Letters}

\begin{definition}[Atomic byte patterns]
An \emph{atomic byte pattern} is a skip-bigram: a pair
$(x_d, o)$ where $x_d$ is the byte at offset $d$ from the current
position and $o$ is the output byte.  The pattern's support is the
log count ratio:
\begin{equation}\label{eq:atomic}
T(o \mid x_d) = \log \frac{P(o \mid x_d)}{P(o)},
\end{equation}
the log-likelihood ratio of the output byte given the input byte
at offset $d$.
\end{definition}

The isomorphic UM already has these patterns for all offsets
$d = 1, 2, \ldots, k$.  The factor map~\cite{factor-map} showed
that the sat-rnn's 128 neurons implement conjunction detectors over
pairs of offsets, with the dominant pair $(1, 7)$ used by 52/128
neurons.

\begin{remark}[What the patterns already know about ``the'']
Consider predicting the byte after ``the '' (the space after ``the'').
The model sees:
\begin{center}
\begin{tabular}{ccc}
\toprule
Offset & Byte & Pattern \\
\midrule
$d=1$ & \texttt{e} (101) & $T(\texttt{' '} \mid \texttt{e}_{d=1})$ \\
$d=2$ & \texttt{h} (104) & $T(\texttt{' '} \mid \texttt{h}_{d=2})$ \\
$d=3$ & \texttt{t} (116) & $T(\texttt{' '} \mid \texttt{t}_{d=3})$ \\
$d=4$ & \texttt{' '} (32) & $T(\texttt{' '} \mid \texttt{' '}_{d=4})$ \\
\bottomrule
\end{tabular}
\end{center}
Each of these is an independent atomic pattern.  None of them
``knows'' the word is ``the''---each only knows what one byte
at one offset contributes to the output prediction.

But \emph{jointly}, $\texttt{t}_{d=3}$, $\texttt{h}_{d=2}$,
$\texttt{e}_{d=1}$, and $\texttt{' '}_{d=4}$ are exactly the
signature of ``the '' occurring.  The word is latent in the
bag of letters.
\end{remark}


\section{The Bag-of-Letters Model}\label{sec:bag}

\begin{definition}[Bag-of-letters representation]
For a word $w = c_1 c_2 \cdots c_\ell$ of length $\ell$, its
\emph{bag-of-letters representation} at position $t$ (where position
$t$ is the byte \emph{after} $w$) is the multiset:
\begin{equation}
B_w(t) = \{(c_1, d{=}\ell), (c_2, d{=}\ell{-}1), \ldots,
(c_\ell, d{=}1)\},
\end{equation}
i.e., each letter of $w$ paired with its offset from the current
position.
\end{definition}

\begin{proposition}[Bag-of-letters is sufficient for short words]
For short words ($\ell \leq 4$), the bag-of-letters representation
is almost always sufficient to identify the word uniquely, given that
position within the byte stream is known.  The reason: English has
very few short anagrams that also share letter positions (e.g.,
``the'' has no rearrangement that is also a common word at offsets
$3,2,1$).

For longer words, the bag-of-letters becomes ambiguous
(``conservation'' vs.\ ``conversation'' share the same bag).
But even in these cases, the individual letter contributions
carry most of the predictive information.
\end{proposition}

\begin{remark}[Human perception works this way]
The psycholinguistic literature on transposed-letter effects
shows that human word recognition is largely insensitive to
internal letter order.  The classic demonstration:
``Aoccdrnig to rscheearch, the huamn mnid can raed txet wtih
srcmabled ltetres.''  Readers can parse this because word recognition
depends primarily on:
\begin{enumerate}
\item The first and last letters (position anchors),
\item The set of internal letters (the ``bag''),
\item Word length,
\item Context (preceding words).
\end{enumerate}
The atomic byte patterns in the UM capture exactly (1)--(3):
the offset encodes position, the byte value encodes letter identity,
and the number of contributing offsets encodes length.
It is notable that the same ``bag-of-letters'' structure falls out
of both human perception and the RNN's learned patterns.
\end{remark}


\section{The Lexeme as Binary Event Space}\label{sec:binary-es}

The bag-of-letters representation gives us a \emph{graded} support
for each word.  We now formalize this as a binary event space.

\begin{definition}[Lexeme ES]
For a lexeme $w$, define the binary event space
$E_w = \{w^+, w^-\}$ where:
\begin{itemize}
\item $w^+$ = ``$w$ occurred'' (the current position is at or
  just after an occurrence of $w$),
\item $w^-$ = ``$w$ did not occur'' (the negation: everything
  that is \emph{not} $w$, including ``there'', ``them'', ``then'',
  etc.\ for $w = \text{``the''}$).
\end{itemize}
\end{definition}

\begin{proposition}[Support from atomic patterns]\label{prop:support}
The log support for $w^+$ given the current context is the sum of
individual letter contributions:
\begin{equation}\label{eq:support}
T(w^+) = \sum_{i=1}^{\ell} T(c_i \text{ at offset } \ell{-}i{+}1)
+ T_{\text{boundary}},
\end{equation}
where $T(c_i \text{ at offset } d)$ is the log-likelihood that
the byte at offset $d$ is $c_i$ (computable from the data), and
$T_{\text{boundary}}$ accounts for word-boundary evidence (space
before, space/punctuation after).

This is an approximation: it assumes independence between the letter
observations.  The actual support should account for correlations
(e.g., \texttt{t} at offset 3 and \texttt{h} at offset 2 are not
independent---``th'' is a common bigram).  The conjunction patterns
from the factor map~\cite{factor-map} capture exactly these
correlations.
\end{proposition}

\begin{remark}[Probability, not certainty]
The support $T(w^+)$ is a \emph{log probability}, not a binary
decision.  Even when all letters of ``the'' are present at the
right offsets, $T(w^+)$ may not be overwhelming: the evidence
is also consistent with ``them'', ``then'', ``there'', ``these'',
``their'', ``they'', etc.

The correct shape is: input bytes (which happen to be deterministic
at each position) map to a \emph{probability distribution} over
lexemes, not a certainty.  This is fundamental: the model should
maintain uncertainty about which word is occurring until enough
evidence accumulates.
\end{remark}

\begin{definition}[Negative event distribution]\label{def:negative}
The negative event $w^-$ is not a single event but a
\emph{distribution} over alternative lexemes.  For
$w = \text{``the''}$, the negative event decomposes as:
\begin{equation}\label{eq:negative}
P(w^-) = P(\text{``there''}) + P(\text{``them''})
+ P(\text{``then''}) + P(\text{``these''}) + \cdots
+ P(\text{not ``the''-prefixed}).
\end{equation}
The ``the''-prefixed portion of $w^-$ distributes over the rest of
the lexicon that shares the prefix ``the''.  This makes ``the'' a
particularly clean case study for negative event distribution:
the negative event has a natural decomposition by prefix extension.
\end{definition}

\begin{example}[Support calculation for ``the'']
At a position where the preceding bytes are
``\texttt{... of the }'' (i.e., we are predicting the byte after
the space after ``the''):
\begin{align}
T(\text{``the''}^+) &= T(\texttt{' '}_{d=4})
  + T(\texttt{t}_{d=3}) + T(\texttt{h}_{d=2})
  + T(\texttt{e}_{d=1}) + T(\texttt{' '}_{d=0}) \\
&\quad + \text{conjunction corrections}.
\end{align}
The conjunction corrections come from the 2-offset patterns:
e.g., $T(\texttt{t}_{d=3}, \texttt{h}_{d=2})$ contributes
additional support from the ``th'' bigram at the right position.

The support for the negative event:
\begin{align}
T(\text{``the''}^-) &= \log\bigl(1 - \exp(T(\text{``the''}^+))\bigr)
\end{align}
as the standard negation of the binary ES~\cite{cmp}.
\end{example}


\section{The Lexeme Predictor as Atomic Input}\label{sec:atomic-input}

A crucial observation: the ``the'' predictor does not wait until
all letters are seen before producing output.  It has positive
support even after seeing just \texttt{t}.

\begin{proposition}[Graded support at every position]
The lexeme support $T(w^+)$ is well-defined at every byte position,
not just at the end of the word.  After seeing \texttt{t}:
\begin{equation}
T(\text{``the''}^+ \mid \texttt{t}_{d=1}) > 0,
\end{equation}
because \texttt{t} is the first letter of ``the'' and observing it
raises the probability that ``the'' is occurring.  After seeing
\texttt{th}:
\begin{equation}
T(\text{``the''}^+ \mid \texttt{t}_{d=2}, \texttt{h}_{d=1})
> T(\text{``the''}^+ \mid \texttt{t}_{d=1}),
\end{equation}
and so on: the support accumulates as more letters arrive, reaching
its maximum when all letters plus a word boundary have been seen.

The support can also decrease: after ``the'' if the next byte is
\texttt{r} (heading toward ``there''), $T(\text{``the''}^+)$
drops and $T(\text{``there''}^+)$ rises.
\end{proposition}

\begin{remark}[Just another atomic input]
From the perspective of the rest of the model, the graded support
$T(\text{``the''}^+)$ at each position is \emph{just another
atomic input}---exactly like a byte observation at a particular
offset.  The byte-level model sees:
\begin{center}
\begin{tabular}{ll}
\toprule
Input & Type \\
\midrule
$\texttt{e}_{d=1}$ & byte at offset 1 \\
$\texttt{h}_{d=2}$ & byte at offset 2 \\
$\texttt{t}_{d=3}$ & byte at offset 3 \\
$T(\text{``the''}^+)$ & lexeme support (graded) \\
\bottomrule
\end{tabular}
\end{center}
The lexeme support is just another feature that the model can
condition on.  It happens to be computed from a conjunction of
byte observations, but from the model's perspective it is an
opaque scalar input, no different from any other.

This is the compositionality of the UM: the output of one model
($U_{\text{lex}}$: bytes $\to$ lexeme support) becomes an input
to another ($U_{\text{byte}}$: bytes + lexeme support $\to$ output
byte).  The composed model's event space is enriched by the lexeme
dimension, but its forward pass is unchanged: each input (whether
a raw byte or a lexeme support value) contributes to the output
through the same pattern-matching mechanism.
\end{remark}

\begin{proposition}[Recursive composability]
If lexeme support is just another atomic input, then:
\begin{enumerate}
\item Multiple lexeme predictors can run in parallel, each producing
  a graded support signal.  $T(\text{``the''}^+)$,
  $T(\text{``of''}^+)$, $T(\text{``and''}^+)$, \ldots{} all feed
  into the byte-level model as additional atomic inputs.
\item Higher-order predictors can condition on lexeme support:
  $T(\text{``of the''}^+)$ could be computed from
  $T(\text{``of''}^+)$ and $T(\text{``the''}^+)$ as inputs.
  The phrase is a conjunction of lexeme events, exactly as
  the lexeme was a conjunction of letter events.
\item The recursion bottoms out at bytes and tops out wherever
  the data stops supporting further conjunctions.  The
  factorization tower~\cite{tock-step} is built from the
  bottom up, each level providing atomic inputs to the next.
\end{enumerate}
\end{proposition}


\section{Subtracting Lexeme Contributions}\label{sec:subtract}

\begin{proposition}[Marginals are downstream of lexemes]
The marginal byte distributions---$P(\texttt{t})$, $P(\texttt{h})$,
$P(\texttt{e})$, $P(\texttt{' '})$---are \emph{causally downstream}
of the marginal lexeme distribution.  The reason: the bytes that
appear in the data are there \emph{because} certain words were used.
The frequency of the bigram ``th'' is high partly because ``the'' is
the most common English word.

This causal direction means: to understand the byte-level Markov
statistics, we should factor out the lexeme contributions.  The
``residual'' byte statistics (after subtracting lexeme effects)
would reflect non-lexical regularities: character class transitions,
phonotactic constraints, orthographic conventions.
\end{proposition}

\begin{definition}[Lexeme-adjusted count table]
Given the base count table $c(x_d, o)$ and a set of lexemes
$\mathcal{W} = \{w_1, \ldots, w_M\}$, the \emph{lexeme-adjusted}
count table is:
\begin{equation}\label{eq:adjusted}
c'(x_d, o) = c(x_d, o) - \sum_{w \in \mathcal{W}}
c_w(x_d, o) \cdot P(w),
\end{equation}
where $c_w(x_d, o)$ is the count table restricted to positions
inside word $w$.  This subtracts out the contribution of each
word's occurrences from the overall byte statistics.
\end{definition}

\begin{remark}[What remains after subtraction?]
After subtracting the ``the'' contribution from the byte-level
Markov statistics, what changes?
\begin{itemize}
\item The bigram ``th'' becomes less frequent (``the'' contributed
  a large fraction of ``th'' occurrences).
\item The bigram ``he'' followed by space becomes rarer.
\item The trigram ``the'' + space becomes much rarer.
\item But ``th'' in ``this'', ``that'', ``think'' etc.\ remains.
\end{itemize}
The residual ``th'' bigram now reflects all sources \emph{except}
the word ``the.''  Repeating this for each word in the lexicon would
peel away successive layers of word-level causation, leaving only the
``pure'' character-level regularities.
\end{remark}


\section{The Lexeme Map as a Complete Model $U$}

\begin{theorem}[The char-to-lexeme map is a UM]\label{thm:um}
The map from input bytes to support over lexemes has the shape of
a complete Universal Model $U = (E, T, P, f, \omega)$:
\begin{itemize}
\item $E_{\text{in}} = \{0, \ldots, 255\}^k$: the input bytes at
  offsets $d_1, \ldots, d_k$.
\item $E_{\text{out}} = \{w_1^+, w_1^-, w_2^+, w_2^-, \ldots,
  w_M^+, w_M^-\}$: the lexeme binary ESes.
\item $T$: the log support values over the lexeme events.
\item $P$: the atomic patterns connecting bytes to lexemes
  (one pattern per letter per offset per lexeme).
\item $f$: the UM forward pass---existential quantification
  over the input patterns to compute output support.
\item $\omega = \omega_0$: the standard counting function.
\end{itemize}

This is a complete UM in its own right.  Its ``output'' is not a
byte prediction but a lexeme prediction: given the recent bytes,
which word are we in?
\end{theorem}

\begin{remark}[Two UMs, composed]
We now have two UMs:
\begin{enumerate}
\item $U_{\text{byte}}$: bytes $\to$ bytes.  The existing model
  (isomorphic UM, KN model, pattern-chain model, etc.).
\item $U_{\text{lex}}$: bytes $\to$ lexemes.  The new model
  from Theorem~\ref{thm:um}.
\end{enumerate}
The UM algebra~\cite{cmp,algebraic-semantics} tells us how to
compose them.  The composed model is:
\begin{equation}\label{eq:compose}
U_{\text{full}} = U_{\text{byte}} \bowtie U_{\text{lex}},
\end{equation}
where $\bowtie$ denotes the UM join (product of event spaces,
union of patterns, combined forward pass).

The full model predicts the next byte using \emph{both} byte-level
patterns (character transitions, offset conjunctions) and
lexeme-level patterns (word identity, word-boundary prediction).
Neither model is subordinate: they operate in parallel, and the
forward pass combines their support values.
\end{remark}

\begin{proposition}[The join preserves the tower]
The composed model $U_{\text{full}}$ maintains the factorization
tower~\cite{tock-step}:
\begin{equation}
E_{\text{bytes}} \xleftarrow{\text{recognition}}
E_{\text{lexemes}} \xrightarrow{\text{prediction}}
E_{\text{bytes}}.
\end{equation}
This is a product factorization (both levels coexist), not a
sum factorization (one replaces the other).  The byte-level
event space is never discarded.  The model can always ``backtrack''
to bytes because the byte level was never left.
\end{proposition}


\section{Spelling Distributions}\label{sec:spelling}

\begin{definition}[Spelling distribution]
A word $w$ has a \emph{spelling distribution}: the probability
distribution over byte sequences that realize $w$.  For most words,
this distribution is concentrated on one spelling:
$P(\texttt{the} \mid w=\text{``the''}) \approx 1$.

But the distribution is not degenerate:
\begin{itemize}
\item Capitalization: ``The'' vs.\ ``the'' vs.\ ``THE''.
\item Typos: ``teh'' for ``the'' (a common transposition).
\item Encoding variants: ``caf\'e'' vs.\ ``cafe'' vs.\
  ``caf\textbackslash u00e9''.
\item Historical/dialectal: ``colour'' vs.\ ``color''.
\end{itemize}
The lexeme ES captures the \emph{abstract} word identity;
the spelling distribution captures its byte-level realizations.
\end{definition}

\begin{remark}[The bag of letters handles misspellings naturally]
Consider the misspelling ``teh'' for ``the.''  In the bag-of-letters
model:
\begin{center}
\begin{tabular}{lcc}
\toprule
& ``the'' & ``teh'' \\
\midrule
Letter set & $\{$t, h, e$\}$ & $\{$t, e, h$\}$ \\
Offsets (from space) & $d=3,2,1$ & $d=3,2,1$ \\
\bottomrule
\end{tabular}
\end{center}
The bag of letters is \emph{identical}.  The only difference is
which letter maps to which offset.  If the atomic patterns primarily
care about which letters are present (not their exact ordering),
then ``teh'' provides almost the same support for
$\text{``the''}^+$ as ``the'' does.

The offset-specific information does differ: $T(\texttt{e}_{d=2})$
vs.\ $T(\texttt{h}_{d=2})$ are different values.  But the
\emph{conjunction} support $T(\texttt{t} \wedge \texttt{h} \wedge
\texttt{e} \text{ within 3 offsets})$ is identical.  This is exactly
the ``bag of letters'' effect from psycholinguistics: the letters
are present, the word is recognized, the internal order matters less
than the external boundaries.
\end{remark}


\section{Negative Events and the Prefix Class}\label{sec:negative}

\begin{definition}[Prefix class]
For a word $w$, its \emph{prefix class} $[w]_{\text{pre}}$ is the
set of words sharing the same prefix:
\begin{equation}
[w]_{\text{pre}} = \{w' \in \mathcal{W} : w \text{ is a prefix of } w'
\text{ or } w' \text{ is a prefix of } w\}.
\end{equation}
For $w = \text{``the''}$:
$[\text{``the''}]_{\text{pre}} = \{\text{the, there, their, them,
then, these, they, thereby, thereof, \ldots}\}$.
\end{definition}

\begin{proposition}[Negative event distributes over prefix class]
When the evidence is consistent with ``the'' but the actual word
is not ``the,'' the negative event distributes over the
prefix class:
\begin{equation}
P(\text{``the''}^- \mid \text{prefix = ``the''})
= \sum_{w' \in [\text{``the''}]_{\text{pre}} \setminus \{\text{``the''}\}}
P(w' \mid \text{prefix = ``the''}).
\end{equation}
This distribution is computable from lexeme frequencies:
\begin{equation}
P(w' \mid \text{prefix = ``the''})
= \frac{\text{freq}(w')}{\sum_{w'' \in [\text{``the''}]_{\text{pre}}} \text{freq}(w'')}.
\end{equation}
\end{proposition}

\begin{example}[``the'' prefix class distribution]
Approximate frequencies in enwik9 (relative to prefix ``the''):
\begin{center}
\begin{tabular}{lrl}
\toprule
Word & Rel.\ freq.\ (\%) & Continuation byte \\
\midrule
the (+ space) & $\sim$65 & \texttt{' '} \\
there & $\sim$8 & \texttt{r} \\
their & $\sim$7 & \texttt{i} \\
they & $\sim$6 & \texttt{y} \\
them & $\sim$5 & \texttt{m} \\
then & $\sim$4 & \texttt{n} \\
these & $\sim$3 & \texttt{s} \\
the + other & $\sim$2 & various \\
\bottomrule
\end{tabular}
\end{center}
The binary ES at the point where we have seen ``the'' and must
predict the next byte:
\begin{itemize}
\item $w^+$: the word is ``the'' $\to$ predict space ($\sim$65\%).
\item $w^-$: the word is not ``the'' $\to$ predict the
  continuation byte from the table above ($\sim$35\%, distributed
  over \texttt{r}, \texttt{i}, \texttt{y}, \texttt{m}, \texttt{n},
  \texttt{s}, \ldots).
\end{itemize}
This is a clean binary ES with well-defined positive and negative
events, where the negative event has a natural decomposition.
\end{example}

\begin{remark}[Case study for negative event distribution]
This ``the'' example is a particularly clean case study for the
general problem of negative event distribution in the UM.  The
issues that arise:
\begin{enumerate}
\item The negative event is not uniform---it has structure (the
  prefix class with specific frequencies).
\item The negative event \emph{itself} decomposes into further
  binary ESes: ``there'' vs.\ ``not there,'' etc.
\item As we expand the lexicon, each new word ``steals'' probability
  mass from the negative events of related words.
\item The prefix tree (trie) over the lexicon provides the natural
  structure for this decomposition.
\end{enumerate}
As we eventually expand the lexicon in the model, we will need to
handle these interactions.  Starting with ``the'' gives us a
controlled setting where the prefix class is small enough to
enumerate but rich enough to illustrate all the key phenomena.
\end{remark}


\section{Discussion}

\subsection{What changed from the previous papers}

The tock-step paper~\cite{tock-step} and tock protocol~\cite{tock-protocol}
proposed adding words via dedicated finite automata (word detectors).
This paper makes a different and more fundamental point: we don't
need new detectors because the model's \emph{existing} atomic
patterns already carry the word-level information.

The atomic patterns---skip-bigrams at various offsets---are the
``letters'' of the bag-of-letters model.  Each letter at each
offset provides independent evidence.  The word is not detected;
it is \emph{inferred} from the conjunction of letter evidence.

This is both simpler (no new machinery) and more powerful
(handles misspellings, capitalization variants, and partial
evidence naturally).

\subsection{The UM composition}

The central result (Theorem~\ref{thm:um}) is that the
char-to-lexeme map is itself a complete UM.  This means we don't
need to ``bolt on'' word-level prediction as a separate system;
it fits within the existing UM algebra.

The composed model $U_{\text{byte}} \bowtie U_{\text{lex}}$ is a
single UM with a richer event space.  The byte-level patterns and
lexeme-level patterns coexist and contribute jointly to prediction.
The UM forward pass (existential quantification over probabilistic
syllogisms) handles the combination automatically.

\subsection{Next steps}

This is the first of several intermediate papers (per MJC's
direction).  The next steps:
\begin{enumerate}
\item \textbf{Empirical validation}: compute $T(w^+)$ for ``the''
  from actual data, verify that bag-of-letters support matches
  word occurrence, measure the bpc gain from the binary ES.
\item \textbf{Negative event structure}: formalize the prefix-class
  decomposition and show it gives the correct probability
  distribution over continuation bytes.
\item \textbf{UM composition}: implement the join
  $U_{\text{byte}} \bowtie U_{\text{lex}}$ and measure the
  combined model's bpc.
\item \textbf{Lexicon scaling}: extend from one word to many,
  using the prefix trie to efficiently handle negative event
  distribution.
\end{enumerate}

\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{tock-protocol}
Claude and MJC.
\newblock \emph{The Tock Protocol: Systematic Lexicon Injection into the
Isomorphic UM}.
\newblock Hutter archive, 13 Feb 2026.

\bibitem{factor-map}
Claude and MJC.
\newblock \emph{The Real Factor Map: Interpretable Patterns onto
Hidden Dynamics}.
\newblock Hutter archive, 9 Feb 2026.

\bibitem{algebraic-semantics}
Claude and MJC.
\newblock \emph{Algebraic Semantics of the Universal Model}.
\newblock Hutter archive, 12 Feb 2026.
\end{thebibliography}

\end{document}
