\documentclass[11pt]{article}
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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}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{prediction}[theorem]{Prediction}

\title{Patterns in the Extended Event Space:\\Independence, Correlation, and the New Synapses}
\author{Claude and MJC}
\date{February 15, 2026}

\begin{document}
\maketitle

\begin{abstract}
The nested model~\cite{nested-model} extends the event space from
$E = I \times H \times O$ to $E' = I \times (I' \times H_{\mathrm{inner}}
\times O') \times O$, adding lexical events (position, accumulator,
bag-of-letters, word identity) inside $H$.  But adding events without
adding patterns changes nothing: the extended model with zero new synapses
is literally the base model with unused neurons.  This paper formalizes
the pattern space $P = E^2 \times T$ (source event, target event,
weight), characterizes which new patterns are valid under the
independence requirement for evidence accumulation, and confronts the
central difficulty: interior events like word identity and letter
accumulation are \emph{highly correlated} with the byte stream, so
na\"ive Bayesian combination of their evidence is catastrophically wrong.
We give the correct combination rules, classify the new patterns into
four families, and make testable predictions about their information
contribution.
\end{abstract}


\section{The Empty Extension}\label{sec:empty}

\begin{proposition}[New events without new patterns are inert]
\label{prop:inert}
Let $U = (e, t, p, f, \omega)$ be a UM with event space $E$.
Let $E' = E \times E_{\mathrm{new}}$ be a product extension with
$|E_{\mathrm{new}}| = k$ new events.  If the pattern table $p'$ has
$p'_{e_{\mathrm{new}}, j} = 0$ and $p'_{i, e_{\mathrm{new}}} = 0$ for
all new events $e_{\mathrm{new}} \in E_{\mathrm{new}}$ and all
$i, j \in E'$, then:
\begin{equation}
f_{p'}(t')_j = f_p(t)_j \qquad \text{for all } j \in E.
\end{equation}
The new events participate in no syllogisms.  The forward pass is
unchanged.  The model has gained neurons but no synapses.
\end{proposition}

\begin{proof}
The forward pass $(f_{p'}(t'))_j = \max_i \min(t'_i, p'_{ij})$.
For $j \in E$, the $\max$ ranges over $i \in E' = E \cup E_{\mathrm{new}}$.
For $i \in E_{\mathrm{new}}$, $p'_{ij} = 0$, so $\min(t'_i, 0) = 0$.
The terms from new events contribute nothing.  The remaining terms
are exactly $\max_{i \in E} \min(t_i, p_{ij}) = f_p(t)_j$.
\end{proof}

\begin{remark}[The synapse is the unit of value]
The nested model paper~\cite{nested-model} proves that the extension
is conservative (the outer model's predictions are unchanged) and that
the inner model is a complete UM.  But conservative means \emph{inert}
until patterns are added.  The value of the extension comes entirely
from the new patterns---the synapses connecting old events to new events
and new events to each other.  This paper characterizes those synapses.
\end{remark}


\section{The Atomic Pattern}\label{sec:atomic}

\begin{definition}[Atomic pattern]\label{def:atomic}
An \emph{atomic pattern} is a triple $(e_{\mathrm{from}}, e_{\mathrm{to}},
w) \in E \times E \times T$, where:
\begin{itemize}
\item $e_{\mathrm{from}} \in E$ is the \emph{evidence event} (the antecedent),
\item $e_{\mathrm{to}} \in E$ is the \emph{conclusion event} (the consequent),
\item $w \in T = \{0, 1, \ldots, 255\}$ is the \emph{weight} (log support
  of joint occurrence).
\end{itemize}
The weight is:
\begin{equation}\label{eq:weight}
w = s(e_{\mathrm{from}}, e_{\mathrm{to}})
= \lfloor \log_2 c(e_{\mathrm{from}}, e_{\mathrm{to}}) \rfloor,
\end{equation}
where $c(e_{\mathrm{from}}, e_{\mathrm{to}})$ is the count of joint
occurrences in the data.
\end{definition}

\begin{proposition}[Weight as upper bound on conditional support]
\label{prop:weight-bound}
By the forward pass, when evidence $e_{\mathrm{from}}$ is observed with
support $t_{\mathrm{from}}$, the conclusion $e_{\mathrm{to}}$ receives
support at most $\min(t_{\mathrm{from}}, w)$.  Since $w$ is the log of
the joint count:
\begin{equation}
\min(t_{\mathrm{from}}, w) \leq w
= \lfloor \log_2 c(e_{\mathrm{from}}, e_{\mathrm{to}}) \rfloor.
\end{equation}
The pattern weight is an upper bound on the support that the
consequent can receive from this particular antecedent.  This is the
content of the ``weakest link'' principle: the syllogism's conclusion
cannot exceed the evidence for the major premise~\cite{logic-counting}.
\end{proposition}

\begin{definition}[Pattern space]
The \emph{pattern space} of a UM with event space $E$ is:
\begin{equation}
P = \{(e_i, e_j, w_{ij}) : e_i, e_j \in E, \; w_{ij} > 0\}
\subseteq E \times E \times T.
\end{equation}
The pattern space has at most $|E|^2$ elements (one for each ordered
pair of events with nonzero joint count).  In practice $|P| \ll |E|^2$
because most event pairs never co-occur.
\end{definition}


\section{The Four Families of New Patterns}\label{sec:families}

The extended event space $E' = I \times I' \times H_{\mathrm{inner}}
\times O' \times O$ introduces four structurally distinct families of
new patterns, classified by which spaces the source and target events
inhabit.

\begin{definition}[Pattern families]\label{def:families}
\begin{enumerate}
\item \textbf{Recognition patterns} ($I \to I'$ and $I \to H'$):
  Byte-level input events predict interior events.
  \begin{itemize}
  \item $(\text{byte}=\texttt{t}, \text{pos}=0) \to (\text{acc} \ni \texttt{t})$:
    deterministic (weight 255).
  \item $(\text{byte}=\texttt{h}, \text{pos}=1,
    \text{acc} \supseteq \{\texttt{t}\}) \to (\text{bol}_{\text{the}} = 0)$:
    ``bag-of-letters for `the' is incomplete after seeing `th'\thinspace''.
    Deterministic (weight 255).
  \item $(\text{byte}=\texttt{e}, \text{pos}=2,
    \text{acc} = \{\texttt{t},\texttt{h}\}) \to (\text{bol}_{\text{the}} = 1)$:
    ``bag-of-letters for `the' is complete''. Deterministic (weight 255).
  \item $(\text{acc} = \{\texttt{t},\texttt{h},\texttt{e}\},
    \text{pos} \leq 3) \to (\sigma_{\text{the}} \approx 0.92)$:
    graded word support, learned from data (weight $\sim$ 200).
  \end{itemize}

\item \textbf{Prediction patterns} ($H' \to O'$ and $H' \to O$):
  Interior events predict output bytes.
  \begin{itemize}
  \item $(\sigma_{\text{the}} > 0.9, \text{pos}=3) \to
    (\text{next byte} = \texttt{' '})$: word identity predicts word
    boundary.  Weight $\approx \log_2 c(\text{the}, \text{space})$.
  \item $(\text{word}=\texttt{the}) \to (\text{byte at offset 2} =
    \texttt{e})$: word identity predicts specific future byte.
  \end{itemize}

\item \textbf{Language-model patterns} ($H' \to H'$):
  Interior events predict other interior events across time.
  \begin{itemize}
  \item $(\text{prev\_word}=\texttt{of}) \to
    (\sigma_{\text{the}} \uparrow)$: word-to-word transition.
  \item $(\text{prev\_word}=\texttt{the}) \to
    (\text{next word is noun} \uparrow)$: syntactic prediction.
  \end{itemize}

\item \textbf{Structural patterns} ($I' \to I'$ and $O' \to O'$):
  Within-layer patterns that express deterministic constraints.
  \begin{itemize}
  \item $(\text{pos}=k) \to (\text{pos}=k+1)$: position increment
    (weight 255, deterministic).
  \item $(\text{acc} = A) \to (\text{acc} = A \cup \{\text{current byte}\})$:
    accumulator update (weight 255).
  \end{itemize}
\end{enumerate}
\end{definition}

\begin{proposition}[Count of new patterns]
Let $|V|$ be the vocabulary size.  The pattern families contribute:
\begin{center}
\begin{tabular}{lcc}
\toprule
Family & Max patterns & Typical nonzero \\
\midrule
Recognition ($I \to I', H'$) & $256 \times L_{\max} \times |V|$ & $O(|V| \cdot \bar{L})$ \\
Prediction ($H' \to O', O$) & $|V| \times 256 \times D$ & $O(|V| \cdot \bar{L})$ \\
Language-model ($H' \to H'$) & $|V|^2$ & $O(|V| \cdot B)$ \\
Structural ($I' \to I'$) & $256 \times L_{\max}$ & $O(L_{\max})$ \\
\bottomrule
\end{tabular}
\end{center}
where $\bar{L}$ is mean word length, $D$ is prediction horizon, and $B$
is the mean number of distinct following words per word.  For
$|V| = 1000$, $\bar{L} = 5$, $D = 20$, $B = 50$: roughly $10^5$
recognition, $10^5$ prediction, $5 \times 10^4$ language-model, and
$5 \times 10^3$ structural patterns.
\end{proposition}


\section{The Independence Requirement}\label{sec:independence}

Evidence accumulates into the log product pattern (LPP) via addition of
log supports from independent sources.  The forward pass combines
multiple patterns via $\max$, selecting the best-supported syllogism.
But the LPP within each syllogism requires that the evidence arriving
from different patterns be \emph{independent}.

\begin{definition}[Evidence independence]
\label{def:independence}
Two patterns $(e_1, e_j, w_1)$ and $(e_2, e_j, w_2)$ targeting the same
conclusion $e_j$ provide \emph{independent evidence} for $e_j$ if:
\begin{equation}
P(e_1, e_2 \mid e_j) = P(e_1 \mid e_j) \cdot P(e_2 \mid e_j).
\end{equation}
Equivalently, the evidence events $e_1$ and $e_2$ are conditionally
independent given the conclusion.
\end{definition}

\begin{theorem}[Independence from microstate partition]
\label{thm:micro-independence}
Two evidence sources provide genuinely independent evidence if and only
if they condition on disjoint partitions of the microstate
space~\cite{bayes-counting}.  Concretely: if the microstates supporting
$e_j$ can be partitioned as $M_j = M_1 \times M_2$ such that $e_1$
conditions on $M_1$ and $e_2$ conditions on $M_2$, then $e_1$ and $e_2$
are independent given $e_j$.
\end{theorem}

\begin{proof}
This is the product structure of independent events.  If $e_j$'s
microstates factor as $M_1 \times M_2$ and $e_1$ depends only on
$M_1$ while $e_2$ depends only on $M_2$, then:
\[
P(e_1, e_2 \mid e_j) = P(e_1 \mid M_1) \cdot P(e_2 \mid M_2)
= P(e_1 \mid e_j) \cdot P(e_2 \mid e_j).
\]
Conversely, if $e_1$ and $e_2$ both depend on the same component of the
microstate, they share a common cause and are dependent.
\end{proof}

\begin{corollary}[The offset graph criterion]
For skip-bigram patterns, independence requires vertex-disjoint edges
in the offset graph~\cite{conditional-independence}.  A shared offset
guarantees dependence.  The same principle applies to any evidence
source: two pieces of evidence are independent only when they ``look at
different parts of the world.''
\end{corollary}


\section{The Correlation Problem}\label{sec:correlation}

The extended event space introduces interior events---position,
accumulator, bag-of-letters, word identity---that are \emph{deterministic
functions of the byte stream}.  This creates massive correlations.

\begin{theorem}[Interior events are not independent of bytes]
\label{thm:not-independent}
Let $b_t$ be the byte at time $t$, $\mathrm{pos}(t)$ the in-word
position, $\mathrm{acc}(t)$ the letter accumulator, and $\sigma_w(t)$
the graded word support.  Then:
\begin{enumerate}
\item $\mathrm{pos}(t)$ is a deterministic function of $(b_0, \ldots, b_t)$.
\item $\mathrm{acc}(t)$ is a deterministic function of $(b_0, \ldots, b_t)$.
\item $\sigma_w(t)$ is a deterministic function of $(\mathrm{pos}(t),
  \mathrm{acc}(t))$, hence of $(b_0, \ldots, b_t)$.
\end{enumerate}
Therefore, for any output event $o$:
\begin{equation}
P(\mathrm{pos}(t), b_t \mid o) \neq P(\mathrm{pos}(t) \mid o) \cdot P(b_t \mid o)
\end{equation}
in general.  The interior events and the byte events are
conditionally dependent given any output.
\end{theorem}

\begin{proof}
If $\mathrm{pos}(t) = 0$, then $b_{t-1}$ was a word boundary character
(space, punctuation), which constrains $b_t$ (more likely uppercase or
common word-initial letters).  This constraint is not mediated by $o$
alone---it is a direct dependence between $\mathrm{pos}$ and $b_t$.
\end{proof}

\begin{corollary}[Na\"ive Bayesian combination fails]
\label{cor:naive-fails}
If we treat a byte-level pattern $(b_{t-1} = \texttt{' '}) \to (o =
\texttt{T})$ and an interior pattern $(\mathrm{pos}(t) = 0) \to (o =
\texttt{T})$ as independent evidence for the output, we double-count:
both patterns encode the same underlying fact (we are at a word boundary).
This is the shared-offset catastrophe~\cite{conditional-independence}
generalized from temporal offsets to event-space dimensions.
\end{corollary}


\section{The Correct Combination Rules}\label{sec:correct}

\begin{theorem}[Evidence combination for correlated interior events]
\label{thm:correct-combination}
Let $e_{\mathrm{byte}}$ be a byte-level evidence event and
$e_{\mathrm{int}}$ be an interior event derived from the byte stream.
The correct evidence combination for predicting output $o$ is NOT
Bayesian product but one of:

\begin{enumerate}
\item \textbf{Absorption}: Combine the byte event and interior event
  into a single joint pattern:
  \begin{equation}
  w = s(e_{\mathrm{byte}} \times e_{\mathrm{int}}, o)
  = \lfloor \log_2 c(e_{\mathrm{byte}}, e_{\mathrm{int}}, o) \rfloor.
  \end{equation}
  This is the correct conditional $P(o \mid e_{\mathrm{byte}},
  e_{\mathrm{int}})$ with no double-counting.

\item \textbf{Residual}: Use only the \emph{additional} information
  that the interior event provides beyond the byte event:
  \begin{equation}
  w_{\mathrm{residual}} = s(e_{\mathrm{int}}, o) - s(e_{\mathrm{int}},
  o \mid e_{\mathrm{byte}})_{\mathrm{baseline}}.
  \end{equation}
  The residual is the MI that the interior event contributes above
  the byte-level baseline.  For deterministic interior events, this
  residual is exactly zero when the byte determines the interior event.

\item \textbf{Hierarchical}: Use interior events \emph{only at the
  interior level}, never mixing them with byte-level evidence for the
  same prediction:
  \begin{equation}
  P(o) = \sum_w P(o \mid w) \cdot P(w \mid \mathrm{byte\ context}),
  \end{equation}
  where $w$ ranges over word identities.  The byte $\to$ word mapping
  ($P(w \mid \mathrm{context})$) uses byte-level patterns; the
  word $\to$ output mapping ($P(o \mid w)$) uses interior patterns.
  No mixing occurs because the two stages condition on disjoint
  information: the first stage conditions on the raw bytes, the second
  on the word identity (which absorbs the byte information).
\end{enumerate}
\end{theorem}

\begin{proof}
For (1): Absorption creates a single pattern over the product event
space, which is always correct (no independence assumption is needed
for a single pattern).  The cost is data sparsity: the joint count
table is larger.

For (2): The residual information is zero when $e_{\mathrm{int}}$ is a
deterministic function of the bytes already conditioned on.  It is
positive only when $e_{\mathrm{int}}$ carries information about the
output that the byte context does not.  This happens when the interior
event aggregates information from multiple byte positions
(e.g., the bag-of-letters summarizes the entire word seen so far, which
no single byte offset captures).

For (3): The hierarchical combination is the chain rule of probability.
The byte $\to$ word stage uses byte-level evidence (independent of
interior events).  The word $\to$ output stage uses word-level evidence
(which has absorbed the byte information via the recognition patterns).
The two stages are independent because the word identity is a sufficient
statistic for the byte $\to$ output prediction at the word level: once
you know the word, the individual bytes add no information about the
next word (though they do add information about spelling variants,
handled by the residual from rule 2).
\end{proof}

\begin{remark}[The hourglass enforces independence]
The hourglass structure (bytes $\to$ words $\to$ bytes) naturally
enforces the hierarchical combination rule: information flows from bytes
to word identity (recognition), then from word identity to predicted
bytes (prediction).  The word-level bottleneck is a sufficient statistic
that absorbs byte-level correlations.  Evidence that arrives via the
word bottleneck is independent of evidence that arrives via direct
byte-to-byte patterns (different offsets), because the two evidence
streams condition on disjoint aspects of the context.
\end{remark}


\section{Which New Patterns Have Value?}\label{sec:value}

Not all new patterns contribute information.  We classify them by their
expected $\Delta$bpc contribution.

\begin{proposition}[Deterministic patterns have zero predictive value]
\label{prop:deterministic-zero}
The structural patterns (position increment, accumulator update) are
deterministic (weight 255) but carry zero predictive information.
They are bookkeeping: they maintain the interior state, but they predict
nothing about the output that the byte stream doesn't already determine.
\end{proposition}

\begin{proof}
The position event $\mathrm{pos}(t) = k$ is a deterministic function of
the byte history.  Any pattern from $\mathrm{pos}(t)$ to an output $o$
is equivalent to a pattern from the byte history to $o$.  The position
event adds no information beyond what is already in the bytes; it merely
re-represents existing information in a more accessible form.
\end{proof}

\begin{remark}[Value from accessibility, not information]
Although deterministic interior events carry zero \emph{new} information
(they are functions of existing events), they can have enormous
\emph{computational} value.  A UM runner that can read $\mathrm{pos}(t) = 0$
directly does not need to trace the byte history back to the last word
boundary.  The accessibility of the position event enables patterns that
would otherwise require long-range byte-level pattern chains.

The information is already there; the new event makes it addressable.
This is the distinction between Shannon information (bits of surprise)
and computational accessibility (addressing cost in the pattern space).
\end{remark}

\begin{theorem}[Recognition patterns: value from aggregation]
\label{thm:recognition-value}
Recognition patterns ($I \to H'$) have positive $\Delta$bpc when the
interior event aggregates information from multiple byte positions that
would otherwise require a high-order pattern.

Specifically, the bag-of-letters event $\mathrm{bol}_w(t) = 1$ is
equivalent to the conjunction:
\begin{equation}
\bigwedge_{c \in \mathrm{letters}(w)} (\exists s \leq t :
b_s = c \land \mathrm{same\_word}(s, t)).
\end{equation}
Representing this as a single event in $H'$ replaces a pattern of
order $|\mathrm{letters}(w)|$ (requiring $256^{|\mathrm{letters}(w)|}$
count table entries) with a single binary event (requiring 1 bit per
word per position).
\end{theorem}

\begin{proof}
Without the bag-of-letters event, recognizing ``the'' from its letters
requires a 3-offset pattern: $(b_{t-2} = \texttt{t}) \land
(b_{t-1} = \texttt{h}) \land (b_t = \texttt{e})$.
This is an order-3 pattern with $256^3 = 16.7\text{M}$ entries in the
count table.  With the bag-of-letters event, recognition is a single
deterministic check: $\mathrm{acc}(t) \supseteq \{\texttt{t}, \texttt{h},
\texttt{e}\}$.  The aggregation replaces exponential table size with
constant-size event lookup.

The $\Delta$bpc from this aggregation is the difference between the
order-$k$ and order-1 models for within-word prediction.  For ``the''
($k = 3$), the order-3 model provides complete within-word prediction;
the order-1 model provides only single-byte conditionals.  The
difference is the within-word MI contribution of the conjunction:
empirically $\sim 2$--$4$ bits per word occurrence.
\end{proof}

\begin{theorem}[Prediction patterns: value from word-level conditioning]
\label{thm:prediction-value}
Prediction patterns ($H' \to O$) have positive $\Delta$bpc when word
identity provides better output prediction than any available byte-level
context.

For common function words, the word identity nearly determines the
word-final byte and subsequent space:
\begin{equation}
H(\text{next byte} \mid \text{word}=\texttt{the}, \text{pos}=3)
\approx 0.12 \;\text{bits},
\end{equation}
compared to $H(\text{next byte} \mid b_{t-1}=\texttt{e}) \approx 4.2$ bits.
The $\Delta$bpc is $\sim 4$ bits per word-final position for high-frequency
words.
\end{theorem}

\begin{theorem}[Language-model patterns: value from word-to-word context]
\label{thm:lm-value}
Language-model patterns ($H' \to H'$) provide the largest potential
$\Delta$bpc because they capture inter-word dependencies that byte-level
models can only access through long-range skip-grams.

The MI between adjacent words in English is $\sim 2$--$6$ bits
(depending on word frequency and syntactic context).  At the byte level,
capturing this requires skip-grams at offsets $\geq \bar{L} \approx 5$
bytes apart.  At the word level, it is a direct bigram: weight
$= \log_2 c(w_{\mathrm{prev}}, w_{\mathrm{curr}})$.

The $\Delta$bpc from word bigrams alone, above byte-level unigrams, is
bounded by:
\begin{equation}
\Delta \leq \frac{1}{\bar{L}} \cdot \mathrm{MI}(W_{t-1}; W_t)
\approx \frac{4}{\bar{L}} \approx 0.8 \;\text{bpc}.
\end{equation}
\end{theorem}


\section{The Synapse Budget}\label{sec:budget}

\begin{definition}[Synapse budget]
The \emph{synapse budget} for the extended model is the total number of
nonzero patterns:
\begin{equation}
|P'| = |P_{\mathrm{old}}| + |P_{\mathrm{new}}|,
\end{equation}
where $P_{\mathrm{old}}$ are the byte-level patterns from the base model
and $P_{\mathrm{new}}$ are the new patterns from the extension.
\end{definition}

\begin{proposition}[The injection curve]
As words are injected into the vocabulary (in frequency order), the
synapse budget grows linearly in $|V|$ (each word adds $O(\bar{L})$
recognition patterns, $O(\bar{L})$ prediction patterns, and $O(B)$
language-model patterns), while the $\Delta$bpc per word decreases
(Zipf's law: the $k$-th word has frequency $\sim 1/k$, so its
per-byte contribution scales as $\sim \log k / (k \cdot \bar{L})$).

The injection curve is therefore concave: high-frequency function words
(``the'', ``of'', ``and'') contribute the most per synapse; content
words contribute incrementally.
\end{proposition}

\begin{example}[First word: ``the'']
Injecting ``the'' ($f \approx 0.07$, $\bar{L} = 3$) adds:
\begin{itemize}
\item $\sim 3 \times 256 = 768$ recognition patterns (byte $\times$ position
  $\to$ accumulator, most with weight 0).
\item $\sim 3 \times 256 = 768$ prediction patterns
  (word $\times$ position $\to$ next byte).
\item $\sim 50$ language-model patterns (word-to-word transitions with
  nonzero count for the top 50 preceding words).
\item $\sim 3$ structural patterns (position increment within ``the'').
\end{itemize}
Total: $\sim 1600$ new patterns, of which $\sim 100$ have significant
weight.  The $\Delta$bpc from recognition + prediction: at $7\%$
frequency, ``the'' occupies $\sim 7\% \times 3 / \bar{L}_{\mathrm{global}}
\approx 4.2\%$ of bytes.  At $\sim 4$ bits $\Delta$bpc for word-final
prediction, the contribution is $\sim 0.04 \times 4 = 0.16$ bpc on
those positions, or $\sim 0.007$ bpc overall.

This matches the order of magnitude of the
neutral-factorization experiments~\cite{extended-es}: ``the'' alone
contributes modestly, but the infrastructure (position, accumulator,
word identity) enables all subsequent word injections to share the
same recognition and prediction machinery.
\end{example}


\section{Independence Classes in the Extended Space}\label{sec:classes}

\begin{theorem}[Three independence classes]\label{thm:three-classes}
The evidence sources in the extended model partition into three
independence classes:
\begin{enumerate}
\item \textbf{Byte-level offsets} (the original $I \to O$ patterns):
  independent when edges are vertex-disjoint in the offset
  graph~\cite{conditional-independence}.
\item \textbf{Word-level predictions} ($H' \to O$): independent of
  byte-level predictions at \emph{different} words, but correlated with
  byte-level predictions at the \emph{current} word position.
\item \textbf{Cross-word context} ($H' \to H'$): independent of
  within-word byte patterns (the word boundary is a conditional
  independence barrier), but correlated with word-level predictions
  at the current word (shared word identity).
\end{enumerate}
\end{theorem}

\begin{proof}
For (1): this is the offset graph
theorem~\cite{conditional-independence}, unchanged by the extension.

For (2): the word identity $w$ absorbs all within-word byte information
(it is a sufficient statistic for the canonical spelling).
A prediction pattern $(\text{word} = w) \to (o = \texttt{' '})$ and a
byte-level skip-bigram $(b_{t-7}, b_{t-1}) \to (o)$ are approximately
independent when the skip-bigram's offsets fall outside the current
word (they condition on bytes from a different word, hence a different
part of the microstate space).  But if the skip-bigram's offsets fall
\emph{within} the current word, they share information with the word
identity and are dependent.

For (3): the word boundary acts as a Markov blanket: conditioned on
the current word identity, the bytes within the current word are
independent of the bytes within the previous word.  Therefore,
cross-word patterns (previous word $\to$ current word) are independent
of within-word byte patterns.
\end{proof}

\begin{corollary}[Safe combinations]
The following evidence combinations are safe (independent, no
double-counting):
\begin{itemize}
\item Word-level bigram + within-word byte pattern (different information
  domains: inter-word vs.\ intra-word).
\item Offset-disjoint skip-bigrams (from the base model).
\item Word identity at the current position + skip-bigram from a
  \emph{previous} word's bytes.
\end{itemize}
The following combinations are UNSAFE (dependent, will double-count):
\begin{itemize}
\item Word identity + byte-level pattern using bytes from the current
  word (shared information source).
\item Bag-of-letters + individual letter-accumulator events (the BoL
  is a deterministic function of the accumulator).
\item Position event + byte-level ``distance from word boundary''
  pattern (identical information, different representation).
\end{itemize}
\end{corollary}


\section{Predictions}\label{sec:predictions}

\begin{prediction}[Recognition patterns are deterministic and free]
All recognition patterns (byte $\to$ accumulator, accumulator $\to$
bag-of-letters) will have weight 255.  They add zero information but
provide accessibility.  Implementing them costs nothing at the data
level (no counting required) and provides the addressing infrastructure
for all subsequent patterns.
\end{prediction}

\begin{prediction}[First non-trivial value from word-final prediction]
The first measurable $\Delta$bpc from the extension will come from
prediction patterns at word-final positions: ``word is $w$ and
position is $|w|$ $\to$ next byte is space''.  This is where the word
identity provides maximum information gain over byte-level context.
Expected: $\sim 0.1$--$0.2$ bpc for the top 100 words,
concentrated at word-final positions.
\end{prediction}

\begin{prediction}[Word bigrams dominate at scale]
As the vocabulary grows beyond $\sim 500$ words, language-model patterns
(word-to-word bigrams) will contribute more $\Delta$bpc than
recognition or prediction patterns.  This is because inter-word MI
($\sim 4$ bits/word) exceeds the within-word MI that recognition captures
($\sim 2$ bits/word), and the vocabulary growth unlocks more word-bigram
patterns.
\end{prediction}

\begin{prediction}[The injection curve is concave with knee at $\sim 200$ words]
The cumulative $\Delta$bpc as a function of vocabulary size $|V|$ will
be concave (diminishing returns per word).  The ``knee'' where marginal
$\Delta$bpc drops below $10^{-4}$ bpc per word will occur at
approximately 200 words, which covers $\sim 60\%$ of running text
(function words + top content words).  Beyond this knee, content words
add context but not raw prediction quality.
\end{prediction}

\begin{prediction}[Hierarchical combination matches or beats absorption]
The hierarchical combination rule (Section~\ref{sec:correct}, rule 3)
will achieve $\Delta$bpc within 0.01 bpc of absorption (rule 1), while
being computationally cheaper (no high-dimensional count tables).
This is because the hourglass bottleneck (word identity) is a near-
sufficient statistic for the byte $\to$ output mapping at the word level.
\end{prediction}

\begin{prediction}[The independence barrier at $\sim 1$ bpc]
At byte-level KN-6 with $\sim 1.8$ bpc~\cite{extended-es}, adding word-level
patterns should reduce to $\sim 1.0$--$1.2$ bpc.  Below 1 bpc, the
remaining information requires patterns that violate the independence
assumptions of the current framework (e.g., syntactic dependencies that
span multiple words and cannot be captured by word bigrams alone).
Breaking through 1 bpc will require either trigram word patterns
(expensive in data) or a second nesting level (sentences as events inside
the word-level $H'$).
\end{prediction}


\section{Discussion}\label{sec:discussion}

\subsection{The synapse is what counts}

The nested model paper~\cite{nested-model} and the
tokenization-loss paper~\cite{tokenization-loss} established the
event-space architecture: which events exist, why they are necessary,
and what information they carry.  This paper establishes the complementary
result: the events are inert without patterns, and the patterns must
respect the independence structure of the evidence.

The atomic pattern $(e_{\mathrm{from}}, e_{\mathrm{to}}, w) \in E^2
\times T$ is the synapse of the UM.  Its weight $w$ is the log of the
joint count---the empirical evidence that these two events co-occur.
The forward pass is a search for the best syllogism, and the conclusion's
support is bounded by the weakest link.  This is the content of the
forward pass as existential quantification~\cite{logic-counting}.

\subsection{Correlation as the price of richness}

The extended event space is richer than the base model (more events,
more potential patterns), but this richness comes with a cost:
correlated events that cannot be combined by na\"ive Bayesian product.
The correlation problem is not a bug---it is the \emph{consequence} of
interior events being deterministic functions of the byte stream.

The resolution is structural: the hourglass architecture enforces
independence by channeling information through the word-identity
bottleneck.  Evidence that flows through the bottleneck (word-level
predictions) is independent of evidence that flows around it
(direct byte-to-byte patterns at distant offsets).  The bottleneck
is the conditional-independence barrier that makes hierarchical
combination valid.

\subsection{From the RNN to the extended UM}

The trained RNN solves the combination problem implicitly: the
$W_h$ matrix combines information from all offsets via matrix
multiplication, which is the correct multivariate
conditional~\cite{conditional-independence}.  But the RNN's solution
is opaque---we cannot read off which evidence sources are independent
and which are correlated.

The extended UM solves the same problem explicitly: the pattern families
classify evidence by independence class, and the combination rules
specify exactly when Bayesian product is valid and when absorption or
hierarchical combination is required.  The result is an interpretable
architecture where every prediction is a traceable syllogism.


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\end{document}
