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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
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\title{Introducing ``the'': A Neutral Factorization\\of the Byte-Level Model}
\author{Claude and MJC}
\date{February 14, 2026}

\begin{document}
\maketitle

\begin{abstract}
We introduce the word ``the'' as a binary event space into the
sat-rnn's isomorphic Universal Model (DSS=1024, 0.079~bpc) and
prove that this introduction is \emph{neutral}: marginalizing over
the new event recovers exactly the original predictions.
The operation adds $E_{\text{the}} = \{\text{the}^+, \text{the}^-\}$
to the model's event space and factors the existing skip-bigram
count tables through this new dimension.  Via the $E \to N \to Q$
chain (events $\to$ counts $\to$ ratios), we show that the Bayesian
marginalization that ``pulls `the' out of'' the skip-bigrams is
exact: the marginal over $E_{\text{the}}$ reproduces the original
byte-level conditionals with no residual.
This neutrality is the point.  We have changed nothing about the
model's predictions; we have only introduced a new axis along which
the count tables can be decomposed.  Value comes later, when
intra-lexical patterns (from atomic byte events to lexeme events)
are added.  But the neutral introduction establishes the
bookkeeping: the negative event $\text{the}^-$ distributes over
the rest of the ``the''-prefixed lexicon, providing a concrete
model for how negative support distributes generally.
\end{abstract}


\section{Setup}\label{sec:setup}

We work with the sat-rnn trained on DSS=1024 bytes of enwik9,
achieving 0.079~bpc.  Its isomorphic UM (the doubled-E
construction~\cite{doubled-e}) reproduces these predictions
exactly.  We also have the pattern-chain UM, which achieves
0.067~bpc at order 12 via skip-$k$-gram count tables~\cite{pattern-chain}.

For concreteness, we use the skip-bigram count tables at various
offsets $d$, which are the atomic building blocks of both models.

\begin{definition}[Skip-bigram count table]
For offset $d$ and dataset $D = (b_1, b_2, \ldots, b_N)$ of $N$ bytes:
\begin{equation}\label{eq:count}
c_d(x, o) = \bigl|\{t : b_{t-d} = x \text{ and } b_t = o\}\bigr|,
\end{equation}
the number of times byte $x$ appears at offset $d$ before output
byte $o$.  The corresponding conditional:
\begin{equation}\label{eq:cond}
P_d(o \mid x) = \frac{c_d(x, o)}{\sum_{o'} c_d(x, o')}.
\end{equation}
\end{definition}

\begin{definition}[The $E \to N \to Q$ chain~{\cite{bayes-counting}}]
The chain from events to prediction:
\begin{enumerate}
\item $E$: \textbf{Events}.  Each position $t$ in the dataset is a
  joint event $(x_d, o) \in I \times O$, where $x_d = b_{t-d}$ is
  the input byte at offset $d$ and $o = b_t$ is the output byte.
\item $N$: \textbf{Natural numbers} (counts).  $c_d(x, o) \in \mathbb{N}$
  counts how many events have context byte $x$ at offset $d$ and
  output $o$.  Marginals: $c_d(x) = \sum_o c_d(x,o)$.
\item $Q$: \textbf{Quotients} (luck).
  $Q_d(x, o) = N / c_d(x, o) = 1/P_d(x, o) = \lambda_d(x, o)$,
  the joint luck.
\end{enumerate}
The chain $E \to N \to Q$ is the UM's standard learning function
$\omega_0$: observe events, count them, compute quotients.
The joint luck decomposes via partial quotients:
\begin{equation}\label{eq:luck-decomp}
\log_2 Q_d(x,o) = \underbrace{\log_2 Q_I(x)}_{\text{marginal luck}}
+ \underbrace{\log_2 Q_d(o \mid x)}_{\text{conditional luck}},
\end{equation}
where $Q_I(x) = N/c_d(x)$ and $Q_d(o \mid x) = c_d(x)/c_d(x,o)
= 1/P_d(o \mid x)$.

The GCD decomposition~\cite{bayes-counting} further separates each
count into common and differential evidence:
$c_d(x, o) = g_I(x) \cdot r_I(x, o)$, where
$g_I(x) = \gcd_o c_d(x, o)$.  The conditional depends only on the
reduced counts: $P_d(o \mid x) = r_I(x,o) / R_I(x)$ where
$R_I(x) = \sum_o r_I(x,o)$.  The GCD cancels completely.
\end{definition}


\section{Introducing ``the'' as a Binary Event Space}\label{sec:introduce}

\begin{definition}[The word indicator]
Define $\mathbf{1}_{\text{the}}(t) \in \{0, 1\}$ as the indicator
that position $t$ falls inside an occurrence of ``the'' delimited by
word boundaries:
\begin{equation}
\mathbf{1}_{\text{the}}(t) = \begin{cases}
1 & \text{if } b_{t-3} \in W \text{ and }
    b_{t-2} = \texttt{t} \text{ and }
    b_{t-1} = \texttt{h} \text{ and }
    b_t = \texttt{e} \text{ and }
    b_{t+1} \in W, \\
0 & \text{otherwise},
\end{cases}
\end{equation}
where $W = \{\texttt{' '}, \texttt{.}, \texttt{,}, \texttt{'\char`\\n'}, \ldots\}$
is the set of word-boundary bytes.

More precisely, for a 3-byte word like ``the'', the indicator fires
at three positions per occurrence: the \texttt{t}, the \texttt{h},
and the \texttt{e}.  But for the purpose of prediction (predicting
the output byte $b_t$), we define it at the \emph{prediction
position}: the byte \emph{after} the last byte of ``the'', i.e.,
position $t$ where $b_{t-3} \in W$, $b_{t-2}{=}\texttt{t}$,
$b_{t-1}{=}\texttt{h}$, $b_{t}{=}\texttt{e}$... but actually
we must be careful.  Let us define it simply:
\begin{equation}
\mathbf{1}_{\text{the}}(t) = \begin{cases}
1 & \text{if the 3-gram ending at } t{-}1 \text{ is ``the''
    bounded by word boundaries}, \\
0 & \text{otherwise}.
\end{cases}
\end{equation}
So $\mathbf{1}_{\text{the}}(t) = 1$ means: the word just completed
before position $t$ was ``the'', and we are now predicting $b_t$
(which is typically a space).
\end{definition}

\begin{definition}[Factored count table]
The skip-bigram count table $c_d(x, o)$ factors through
$E_{\text{the}}$:
\begin{equation}\label{eq:factor}
c_d(x, o) = c_d^+(x, o) + c_d^-(x, o),
\end{equation}
where:
\begin{align}
c_d^+(x, o) &= \bigl|\{t : b_{t-d} = x,\; b_t = o,\;
  \mathbf{1}_{\text{the}}(t) = 1\}\bigr|, \label{eq:cplus} \\
c_d^-(x, o) &= \bigl|\{t : b_{t-d} = x,\; b_t = o,\;
  \mathbf{1}_{\text{the}}(t) = 0\}\bigr|. \label{eq:cminus}
\end{align}
These are the count tables \emph{conditioned on} whether we are
at a ``the''-boundary position or not.
\end{definition}


\section{The Neutrality Theorem}\label{sec:neutral}

The introduction of $E_{\text{the}}$ is a \emph{partial quotient}
in the sense of~\cite{bayes-counting}: we divide the event space
$E$ by a single binary atom, factoring the joint count into
a conditional fiber and an equivalence class.

\begin{theorem}[Neutral introduction]\label{thm:neutral}
Introducing $E_{\text{the}}$ and marginalizing over it recovers
the original count table---and therefore all conditionals, lucks,
and predictions---exactly.
\end{theorem}

\begin{proof}
We trace the operation through each step of $E \to N \to Q$.

\textbf{Step 1 ($E$: events partition exactly).}
Every position $t$ with $b_{t-d} = x$ and $b_t = o$ has either
$\mathbf{1}_{\text{the}}(t) = 1$ or $\mathbf{1}_{\text{the}}(t) = 0$.
The partition is exhaustive and disjoint.  No events are created
or destroyed.

\textbf{Step 2 ($N$: counts add exactly).}
By~\eqref{eq:factor}, $c_d(x,o) = c_d^+(x,o) + c_d^-(x,o)$.
This is exact integer arithmetic.  The marginals also add:
$c_d(x) = c_d^+(x) + c_d^-(x)$ where $c_d^+(x) = \sum_o c_d^+(x,o)$.

\textbf{Step 3 ($Q$: quotients compose exactly).}
The joint luck of the original table is
$Q_d(x,o) = N / c_d(x,o)$.  The factored luck decomposes via
partial quotient~\cite{bayes-counting}:
\begin{equation}
\frac{1}{Q_d(x,o)} = P_d(x,o)
= \frac{c_d^+(x,o) + c_d^-(x,o)}{N}
= \frac{c_d(x,o)}{N}.
\end{equation}
The conditional luck $Q_d(o \mid x) = c_d(x)/c_d(x,o)$
is unchanged because both numerator and denominator are
the \emph{same} sums of the factored counts:
\begin{equation}
Q_d(o \mid x) = \frac{c_d^+(x) + c_d^-(x)}{c_d^+(x,o) + c_d^-(x,o)}.
\end{equation}

In the GCD decomposition, the row GCD $g_I(x) = \gcd_o c_d(x,o)$
may differ from the GCDs of the sub-tables $c_d^+$ and $c_d^-$.
But the GCD cancels from the conditional (Proposition~\ref{eq:luck-decomp}
of the setup), so the conditional $P_d(o \mid x)$ is the same whether
computed from $c_d$ directly or from $c_d^+ + c_d^-$.
\qedhere
\end{proof}

\begin{remark}[Why this is trivially true but non-trivially important]
The proof is a consequence of the fact that the partial quotient
decomposes $c_d(x,o)$ along a new axis without changing any cell
value.  But the \emph{point} is what the decomposition enables:

\begin{enumerate}
\item \textbf{We have changed nothing.}  Predictions are identical.
  The operation is a pure rearrangement of counts along a new axis.

\item \textbf{The factored form is richer.}  We now have
  $c_d^+(x,o)$ and $c_d^-(x,o)$ as separate objects with their
  own GCD decompositions, their own reduced counts, and their own
  conditional lucks.

\item \textbf{Value comes from asymmetry.}  If
  $Q_d(o \mid x, \text{the}^+) = Q_d(o \mid x, \text{the}^-)$
  for all $x, o$, the factorization is uninformative.
  If the conditional lucks differ, a model that conditions on
  $E_{\text{the}}$ can make better predictions at those positions.
  But exploiting this requires \emph{using} $E_{\text{the}}$ as a
  predictor, not just marginalizing over it.
\end{enumerate}
\end{remark}


\section{The Bayesian Conditioning Direction}\label{sec:bayesian}

The neutrality theorem is the marginalization direction: summing
over $E_{\text{the}}$ recovers the original.  We now examine the
\emph{conditioning} direction via the partial quotient
framework~\cite{bayes-counting}.

\begin{proposition}[Bayes via partial quotient]\label{prop:bayes}
The posterior $P(\text{the}^+ \mid x_d, o)$ is a count ratio---a
direct application of the $E \to N \to Q$ chain:
\begin{equation}\label{eq:bayes}
P(\text{the}^+ \mid x_d = x, o)
= \frac{c_d^+(x, o)}{c_d(x, o)}.
\end{equation}
This is the partial quotient of $E$ by the atom
$(x, o) \in I \times O$, restricted to $\text{the}^+$: of all
events with input $x$ at offset $d$ and output $o$, how many fall
in the ``the'' equivalence class?

The corresponding luck:
\begin{equation}
Q(\text{the}^+ \mid x, o) = \frac{c_d(x, o)}{c_d^+(x, o)},
\end{equation}
the inverse posterior---how ``lucky'' is it to be at a
``the''-boundary given these bytes?  Large luck (small posterior)
means ``the'' is unlikely in this context; luck${}= 1$ means
certainty.

No prior is assumed; no likelihood is computed separately.
The posterior \emph{is} the count ratio.  The GCD of the
column $\{c_d^+(x,o), c_d^-(x,o)\}$ (which is just
$\gcd(c_d^+, c_d^-)$) would cancel from the conditional, but
for a binary partition there is no further reduction needed.
\end{proposition}

\begin{example}[Posterior for ``the'' at specific positions]
Consider DSS=1024.  Suppose ``the '' (the-space) occurs 10 times
in the 1024-byte sample.

At position $t$ where $b_{t-1} = \texttt{e}$ and $b_t = \texttt{' '}$:
\begin{itemize}
\item $c_1(\texttt{e}, \texttt{' '})$: total count of \texttt{e}
  followed by space.  Say this is 25 (``the '', ``be '', ``me '',
  ``he '', etc.).
\item $c_1^+(\texttt{e}, \texttt{' '})$: count restricted to
  ``the''-boundaries.  This is 10 (every ``the '' contributes one).
\item $P(\text{the}^+ \mid \texttt{e}_{d=1}, \texttt{' '})
  = 10/25 = 0.40$.
\end{itemize}
Even after seeing \texttt{e} followed by space, there is only
40\% probability that this is a ``the'' boundary.  The other 60\%
distributes over other words ending in \texttt{e}.

Now condition on more offsets.  At the same position, also
$b_{t-2} = \texttt{h}$:
\begin{itemize}
\item Among the 25 events with \texttt{e}$\to$\texttt{' '},
  how many also have \texttt{h} at offset 2?  Say 15
  (``the '', ``she '', etc.).
\item Among those 15, how many are ``the''-boundaries?  Still 10.
\item $P(\text{the}^+ \mid \texttt{e}_{d=1}, \texttt{h}_{d=2},
  \texttt{' '}) = 10/15 = 0.67$.
\end{itemize}
Adding $b_{t-3} = \texttt{t}$:
\begin{itemize}
\item Among the 15, how many also have \texttt{t} at offset 3?
  Say 11 (``the '' plus perhaps one ``the'' in a compound).
\item $P(\text{the}^+ \mid \texttt{e}_{d=1}, \texttt{h}_{d=2},
  \texttt{t}_{d=3}, \texttt{' '}) = 10/11 \approx 0.91$.
\end{itemize}
The posterior concentrates as more evidence arrives, but never
reaches certainty (because other sequences like
``\ldots athe '' might exist in the sample).
\end{example}


\section{``Pulling `the' Out'' of the Skip-Bigrams}\label{sec:pull}

\begin{definition}[Factoring a count table through a binary ES]
To ``pull `the' out of'' the skip-bigram table $c_d(x, o)$ means
to decompose it as:
\begin{equation}\label{eq:pull}
c_d(x, o) = \underbrace{c_d^+(x, o)}_{\text{``the'' contribution}}
+ \underbrace{c_d^-(x, o)}_{\text{everything else}}.
\end{equation}
The ``the'' contribution $c_d^+(x, o)$ captures all the counts
attributable to positions where ``the'' just occurred.  The
remainder $c_d^-(x, o)$ captures everything else.
\end{definition}

\begin{proposition}[What ``the'' contributes to specific bigrams]
\label{prop:contributions}
For the bigram table at offset $d = 1$ (adjacent bytes), the ``the''
contribution is concentrated on specific entries:
\begin{center}
\begin{tabular}{ccl}
\toprule
$x$ & $o$ & Effect of $c_1^+$ \\
\midrule
\texttt{e} & \texttt{' '} & Large: every ``the '' contributes here \\
\texttt{h} & \texttt{e} & Moderate: ``the'' contributes, but so do
  ``he'', ``she'', etc.\ in the $c_1^-$ portion \\
\texttt{t} & \texttt{h} & Moderate: ``the'' contributes, but ``th''
  appears in many words \\
\texttt{' '} & \texttt{t} & Small: ``the'' is one of many words
  starting with \texttt{t} \\
\bottomrule
\end{tabular}
\end{center}
At larger offsets ($d = 2, 3, \ldots$), the ``the'' contribution
is spread more diffusely because the byte at offset $d$ from a
``the''-boundary depends on the word \emph{before} ``the'', which
varies.
\end{proposition}

\begin{theorem}[Exact reconstitution]\label{thm:reconstitute}
After pulling ``the'' out, we can put it back:
\begin{equation}
c_d(x, o) = c_d^+(x, o) + c_d^-(x, o) \quad \forall\, x, o, d.
\end{equation}
This is exact (integer equality of counts).  The $E \to N$
step is lossless: no events are created, destroyed, or moved
between cells.  The $N \to Q$ step is also lossless (given the
counts, the ratios are determined).

Therefore: the operation of introducing $E_{\text{the}}$, factoring
all count tables through it, and then marginalizing $E_{\text{the}}$
back out is the identity.  It is a neutral rearrangement.
\end{theorem}


\section{The Negative Event Distribution}\label{sec:negative}

\begin{definition}[The $\text{the}^-$ distribution]
The negative event $\text{the}^-$ is ``not at a `the' boundary.''
Its count table $c_d^-(x, o)$ is a mixture:
\begin{equation}\label{eq:neg-decompose}
c_d^-(x, o) = \sum_{w \in \mathcal{W} \setminus \{\text{the}\}}
c_d^{(w)}(x, o) + c_d^{(\text{none})}(x, o),
\end{equation}
where $c_d^{(w)}$ restricts to positions at word $w$'s boundary
and $c_d^{(\text{none})}$ restricts to non-word-boundary positions
(mid-word bytes).

For the ``the''-prefixed subset:
\begin{equation}
c_d^{(\text{the-prefix})}(x, o) = c_d^{(\text{there})}(x, o)
+ c_d^{(\text{them})}(x, o) + c_d^{(\text{then})}(x, o)
+ c_d^{(\text{their})}(x, o) + \cdots
\end{equation}
\end{definition}

\begin{remark}[Model for negative weight distribution]
This decomposition of $\text{the}^-$ is significant beyond the
specific case of ``the.''  It models how negative support distributes
generally in the UM:

\begin{enumerate}
\item In the RNN, negative weights $w_{ij} < 0$ encode
  ``not this event.''  The neuron $h_j$ receiving negative input
  from $h_i$ is being told: ``the pattern that activates $h_i$ is
  \emph{absent},'' and this absence is informative.

\item The $\text{the}^-$ event has the same structure: it says
  ``the'' did not occur, and this non-occurrence is informative
  because it tells us the byte patterns are explained by something
  \emph{other than} ``the'' --- likely another word in the prefix
  class.

\item The distribution of $\text{the}^-$ over the prefix class
  (65\% ``the'', 8\% ``there'', 7\% ``their'', etc.---so among
  the $\text{the}^-$ events at the ``th-e'' continuation point,
  roughly 23\% ``there'', 20\% ``their'', etc.) provides a concrete
  model for how negative RNN weights distribute their ``not-this''
  information over alternative patterns.

\item When we eventually introduce the full lexicon as an $M$-ary
  event space $E_V = \{w_1, w_2, \ldots, w_M, \text{OOV}\}$, each
  word's negative event decomposes the same way.  The ``the'' case
  is the first instance of a general pattern.
\end{enumerate}
\end{remark}

\begin{proposition}[Negative event is computable from data]
The negative distribution is computable purely from counts:
\begin{equation}
P(w' \mid \text{the}^-, \text{prefix context})
= \frac{c^{(w')}}{c^- - c^{(\text{non-prefixed})}},
\end{equation}
where $c^{(w')}$ is the count of word $w'$ events and the
denominator restricts to events sharing the relevant prefix context.
No model fitting is needed---it is raw counting over the data,
exactly the $E \to N \to Q$ chain.
\end{proposition}


\section{Why Start with the Sat-RNN (DSS=1024)?}\label{sec:why-dss}

\begin{remark}[The controlled setting]
DSS=1024 is a 1024-byte sample.  The sat-rnn trained on it achieves
0.079~bpc (near-memorization).  The isomorphic UM reproduces this
exactly.  This tiny dataset makes the neutral introduction
maximally transparent:

\begin{enumerate}
\item \textbf{All counts are small and inspectable.}
  ``The'' occurs perhaps 5--15 times in 1024 bytes.  Each entry
  of $c_d^+$ is a single-digit number.  We can write out the full
  factored count table by hand.

\item \textbf{The neutrality is exactly verifiable.}
  We can sum $c_d^+ + c_d^-$ and verify integer equality with $c_d$
  for every $(x, o, d)$ triple.

\item \textbf{The model already captures everything.}
  At 0.079~bpc, the sat-rnn has nearly memorized the sample.
  Introducing $E_{\text{the}}$ and marginalizing out must give
  exactly 0.079~bpc.  Any deviation would indicate a bug in the
  factorization, not a real change.

\item \textbf{Spurious patterns are visible.}
  The sat-rnn at DSS=1024 has learned some long-range patterns that
  are specific to this particular 1024-byte sample.  Introducing
  $E_{\text{the}}$ may ``break'' these spurious patterns by
  explaining away some of their count support.  If position $t$ has
  a long-range correlation with position $t - 20$ that happens to
  co-occur with ``the'' at position $t$, factoring out the ``the''
  contribution removes this correlation from the $c_d^-$ table.
  The residual table $c_d^-$ will have the spurious pattern
  weakened.  This is not a prediction improvement (the marginal is
  unchanged) but a \emph{structural} improvement: the count table
  is cleaner.
\end{enumerate}
\end{remark}


\section{When Does Value Arrive?}\label{sec:value}

\begin{proposition}[Neutral now, valuable later]
The neutral introduction (Section~\ref{sec:neutral}) gives
$\Delta\text{bpc} = 0$ by construction.  Value arrives when we
add \emph{intra-lexical patterns}---patterns that connect atomic
byte events to lexeme events:

\begin{enumerate}
\item \textbf{From bytes to lexeme} (recognition patterns):
  $P(\text{the}^+ \mid x_{d_1}, x_{d_2}, \ldots)$.
  These patterns say: ``given these bytes at these offsets, how
  likely is it that `the' is occurring?''
  Using these as predictors sharpens the model at positions where
  ``the'' is likely.

\item \textbf{From lexeme to bytes} (prediction patterns):
  $P(o \mid \text{the}^+)$.
  These patterns say: ``given that `the' just occurred, what byte
  comes next?''
  Using these as predictors sharpens the model at the exit
  position of ``the.''

\item \textbf{Between lexemes} (lexeme-level patterns):
  $P(\text{of}^+ \mid \text{the}^+)$.
  These patterns capture word-level co-occurrence.
  Using these as predictors sharpens the model at word-to-word
  transitions.
\end{enumerate}

Each type of intra-lexical pattern provides $\Delta\text{bpc} > 0$
by exploiting the asymmetry between $P_d(o \mid x, \text{the}^+)$
and $P_d(o \mid x, \text{the}^-)$.  But none of this is possible
without first establishing the neutral factorization---the
bookkeeping that separates $c_d^+$ from $c_d^-$.
\end{proposition}

\begin{remark}[The factorization is the architecture]
Introducing $E_{\text{the}}$ without using it for prediction is an
\emph{architectural} change with no \emph{parametric} change.  The
event space has grown (from $E_0$ to $E_0 \times E_{\text{the}}$),
but the predictions are identical because we marginalize over the
new dimension.

This is the tock step in its purest form: an expansion of the
architecture that is initially neutral, creating the \emph{space}
for future improvement.  The tick step (adding intra-lexical
patterns, recomputing conditionals) is what fills this space.

The analogy with neural networks: adding a neuron initialized to
zero changes nothing.  But the neuron is now available to learn.
Here, adding $E_{\text{the}}$ changes nothing, but the factored
count tables are now available for lexeme-level prediction.
\end{remark}


\section{The Exact Procedure}\label{sec:procedure}

For completeness, we specify the exact procedure for introducing
``the'' into the DSS=1024 model:

\begin{enumerate}
\item \textbf{Identify ``the'' occurrences.}
  Scan the 1024-byte sample for occurrences of ``the'' bounded by
  word-boundary bytes.  Record the set of positions
  $S_{\text{the}} = \{t : \mathbf{1}_{\text{the}}(t) = 1\}$.

\item \textbf{Factor each count table.}
  For each offset $d$ and each $(x, o)$ pair, compute:
  $c_d^+(x,o)$ by restricting counts to $S_{\text{the}}$,
  $c_d^-(x,o) = c_d(x,o) - c_d^+(x,o)$.

\item \textbf{Verify neutrality.}
  Compute predictions from the marginal: for each $x$,
  $\hat{P}_d(o \mid x) = (c_d^+(x,o) + c_d^-(x,o)) / n(x)$.
  Verify $\hat{P}_d(o \mid x) = P_d(o \mid x)$ exactly.
  Compute bpc.  Verify it equals 0.079~bpc.

\item \textbf{Inspect the factored tables.}
  Examine $c_d^+$ and $c_d^-$ separately.  Identify entries where
  $P_d(o \mid x, \text{the}^+) \neq P_d(o \mid x, \text{the}^-)$
  (the asymmetry that future lexeme patterns can exploit).

\item \textbf{Compute the negative distribution.}
  For positions in $S_{\text{the}}^c$ (not ``the''), tabulate
  which words are present.  Compute the distribution of
  $\text{the}^-$ over the prefix class and over other words.

\item \textbf{Report.}
  The output is: (a)~the factored count tables, (b)~the
  neutrality verification, (c)~the asymmetry table, (d)~the
  negative event distribution.  No predictions change.
\end{enumerate}


\section{Discussion}

This paper is deliberately minimal.  We introduce one word, prove
the introduction changes nothing, and characterize the negative
event.  No model improvement is claimed or attempted.

The purpose is threefold:

\begin{enumerate}
\item \textbf{Establish the bookkeeping.}  The factored count tables
  $c_d^+, c_d^-$ are the data structure on which all future
  lexeme-level operations will be built.  Getting this right---with
  exact integer arithmetic, no approximations, no off-by-one
  errors---is essential.

\item \textbf{Model negative event distribution.}  The
  $\text{the}^-$ event is the first concrete instance of a negative
  event with internal structure.  Understanding how it distributes
  over the prefix class (and over the full lexicon) will generalize
  to all words and, by analogy, to negative weights in the RNN.

\item \textbf{Prepare for value.}  The next paper will add
  intra-lexical patterns and measure the resulting $\Delta$bpc.
  This paper ensures that the baseline is clean: we know exactly
  what the model predicts before any lexeme-level patterns are
  added.
\end{enumerate}

The neutral introduction is the ``zero'' of the tock step: the
point where architecture has expanded but performance has not.
Everything that follows is measured against this zero.


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

\bibitem{doubled-e}
Claude and MJC.
\newblock \emph{Doubled-E: Exact Float UM Isomorphic to RNN}.
\newblock Hutter archive, 30 Jan 2026.

\bibitem{pattern-chain}
Claude and MJC.
\newblock \emph{Pattern Chains: Explicit $i \to \cdots \to o$ from Data}.
\newblock Hutter archive, 7 Feb 2026.

\bibitem{lexeme-es}
Claude and MJC.
\newblock \emph{Lexemes as Binary Event Spaces: From Atomic Patterns
to Bag-of-Letters Prediction}.
\newblock Hutter archive, 14 Feb 2026.

\bibitem{bayes-counting}
Claude and MJC.
\newblock \emph{Bayes from Counting: Partial Quotients, GCD, and the
Symmetric Learning Function on $E = I \times O$}.
\newblock Hutter archive, 12 Feb 2026.
\end{thebibliography}

\end{document}
