\documentclass[11pt]{article}
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\usepackage{booktabs}

\title{Interpretation of the Isomorphic Universal Model\\Tock 1 Analysis}
\author{Claude}
\date{2026-02-04}

\begin{document}
\maketitle

\section{Introduction}

This paper interprets the isomorphic Universal Model (UM) derived from an Elman RNN
trained on enwik9 (5.69 bpc). The UM operates entirely in the domain of positive
patterns, positive events, and positive integer support values.

The doubled-E mapping provides an exact isomorphism: the UM achieves 0.00\% bpc
difference from the RNN. Our task is to find the natural event spaces and named
patterns within this representation, preparing for the next tick of training.

\section{Model Structure}

\subsection{Events}

The model has 768 events organized into three layers:

\begin{center}
\begin{tabular}{lrll}
\toprule
Layer & Count & Events & Example \\
\midrule
Input & 256 & ``The input is X.'' & ``The input is `e'.'' \\
Hidden & 256 & Doubled-E: $h_j^+$, $h_j^-$ & ``h2+'', ``h2-'' \\
Output & 256 & ``The output is X.'' & ``The output is `e'.'' \\
\bottomrule
\end{tabular}
\end{center}

Each hidden neuron $j$ contributes a binary event space $\{h_j^+, h_j^-\}$.
This is the mechanical doubled-E representation; natural ESs may be coarser.

\subsection{Patterns}

Patterns connect events with positive integer strengths. For weight $w$:
\[
\text{strength} = \lfloor 2|w| + 0.5 \rfloor
\]

Patterns with strength $\geq 1$ are significant (weight $\geq 0.5$).

\begin{center}
\begin{tabular}{lrr}
\toprule
Type & Count & Example \\
\midrule
Input $\to$ Hidden & 49 & ``The input is ` '.'' ``h2-'' 8. \\
Hidden $\to$ Output & 253 & ``h2+'' ``The output is `e'.'' 2. \\
Hidden $\to$ Hidden & (not exported) & ``h2+'' ``h35+'' 1. \\
\bottomrule
\end{tabular}
\end{center}

Total extracted patterns: 302 (input$\to$hidden and hidden$\to$output only).

\section{Discovered Event Spaces}

\subsection{ES1: Word Boundary}

The binary ES $\{h_2^+, h_2^-\}$ functions as a word boundary detector.

\textbf{Input patterns to ES1:}
\begin{verbatim}
"The input is ' '." "h2-" 8.
"The input is 0x0A." "h2-" 1.
"The input is 't'." "h2+" 1.
"The input is 'r'." "h2+" 1.
"The input is 'e'." "h2+" 1.
... (all letters → h2+ with strength 1)
\end{verbatim}

The space input sends support 8 to $h_2^-$. Letters send support 1 to $h_2^+$.
The 8:1 ratio creates a binary switch via softmax within the ES.

\textbf{Output patterns from ES1:}
\begin{verbatim}
"h2+" "The output is 'e'." 2.
"h2+" "The output is 'n'." 2.
"h2+" "The output is 'l'." 1.
"h2+" "The output is 'r'." 1.
\end{verbatim}

When $h_2^+$ wins (word-internal), common letters are predicted.

\textbf{Interpretation:}
\begin{itemize}
\item $h_2^+$ = ``word-internal position''
\item $h_2^-$ = ``word boundary''
\end{itemize}

\subsection{ES2: Syllable Momentum}

The binary ES $\{h_{35}^+, h_{35}^-\}$ tracks syllable structure.

\textbf{Input patterns:}
\begin{verbatim}
"The input is ' '." "h35-" 2.
"The input is 'e'." "h35+" 1.
\end{verbatim}

Vowels send weak support to $h_{35}^+$. Space sends support 2 to $h_{35}^-$.

\textbf{Recurrent pattern (critical):}
\begin{verbatim}
"h2+" "h35+" 1.   (weight = 0.612)
\end{verbatim}

When $h_2^+$ is active (word-internal), it drives $h_{35}^+$.
This creates syllable momentum within words.

\textbf{Self-connection:}
\[
W_{hh}[35,35] = -0.184
\]

The negative self-connection means $h_{35}^+$ decays over time.
In UM terms: $h_{35}^+$ sends support to $h_{35}^-$, causing gradual shift.

\textbf{Interpretation:}
\begin{itemize}
\item $h_{35}^+$ = ``syllable momentum'' (high early in word)
\item $h_{35}^-$ = ``momentum decayed'' (high late in word)
\end{itemize}

This explains why short words end with high $h_{35}^+$ support and
long words end with high $h_{35}^-$ support.

\subsection{Natural ES Discovery: Correlation Analysis}

Correlation analysis reveals massive redundancy in the 128 hidden neurons.
Many pairs have $r = 1.000$ or $r = -1.000$, meaning they're functionally identical
or opposite.

\textbf{Major clusters (neurons with $r > 0.9$):}
\begin{itemize}
\item Cluster A (13 neurons): h12, h20, h39, h40, h41, h58, h59, h64, h66, h83, h85, h112, h115
\item Cluster B (11 neurons): h1, h4, h16, h21, h49, h74, h95, h98, h101, h121, h124
\item Cluster C (11 neurons): h5, h8, h28, h31, h38, h44, h65, h68, h77, h105, h118
\item Cluster D (9 neurons): h11, h17, h22, h24, h26, h53, h81, h87, h117
\item Cluster E (9 neurons): h15, h37, h79, h89, h92, h93, h100, h102, h116
\item Additional smaller clusters...
\end{itemize}

\textbf{Implication}: The 128 binary ESs (256 events) can collapse to $\sim$18 natural ESs.
Each cluster acts as a single binary ES---all members flip together.

\subsection{Saturated Neurons}

\textbf{Always ON} (mean $> 0.9$): h0, h10, h19, h34, h48, h103, h109

These neurons saturate at $h^+ \approx 255$ support. They represent
constant context (``processing text'') rather than varying information.

\textbf{Always OFF} (mean $< -0.9$): h6, h23, h25, h32, h90, h91

These saturate at $h^- \approx 255$. Combined with the always-ON neurons,
about 13 neurons (10\%) carry no dynamic information.

\subsection{Active Binary ESs}

The remaining neurons form active binary ESs. Key ones:
\begin{itemize}
\item ES($h_2$): word boundary, $h^+$ dominates 86.4\%
\item ES($h_3$): (role unclear), $h^+$ dominates 33.3\%
\item ES($h_{35}$): syllable momentum, $h^+$ dominates 63.6\%
\item ES($h_{62}$): (correlated with word-internal), $h^+$ dominates 66.7\%
\end{itemize}

\section{Pattern Inventory}

\subsection{Strong Input Patterns}

Patterns from input events to hidden events with strength $\geq 2$:

\begin{center}
\begin{tabular}{lll}
\toprule
Input Event & Hidden Event & Strength \\
\midrule
``The input is ` '.'' & h2- & 8 \\
``The input is ` '.'' & h62- & 8 \\
``The input is ` '.'' & h72- & 6 \\
``The input is ` '.'' & h3- & 8 \\
``The input is ` '.'' & h115- & 3 \\
``The input is ` '.'' & h35- & 2 \\
\bottomrule
\end{tabular}
\end{center}

Space is the dominant input---it resets multiple hidden states.

\subsection{Strong Output Patterns}

The hidden$\to$output patterns are more distributed. Top patterns:

\begin{center}
\begin{tabular}{lll}
\toprule
Hidden Event & Output Event & Strength \\
\midrule
h2+ & ``The output is `e'.'' & 2 \\
h2+ & ``The output is `n'.'' & 2 \\
(many with strength 1) & & \\
\bottomrule
\end{tabular}
\end{center}

Most output patterns have strength 1, reflecting distributed encoding.

\section{Redistributing the Negative Signal}

The doubled-E representation handles negative weights mechanically:
negative weight to $h_j$ becomes positive pattern to $h_j^-$.

For natural ES discovery, we should find groups where $h_j^-$ and $h_k^+$
(for different $j$, $k$) form a single ES.

\subsection{Candidate: h2 and h62}

Both $h_2$ and $h_{62}$ receive strong patterns from space:
\begin{verbatim}
"The input is ' '." "h2-" 8.
"The input is ' '." "h62-" 8.
\end{verbatim}

And both receive weak patterns from letters.

\textbf{Hypothesis:} $\{h_2^+, h_2^-, h_{62}^+, h_{62}^-\}$ might collapse
to a smaller ES if $h_2$ and $h_{62}$ are redundant.

To test: check if $h_2$ and $h_{62}$ activations are correlated.

\subsection{Analysis Needed}

To properly redistribute:
\begin{enumerate}
\item Find hidden neurons with correlated activations
\item Identify which $h_j^-$ events co-occur with which $h_k^+$ events
\item Propose coarser ESs that preserve bpc
\end{enumerate}

\section{Interpretation Coverage}

\subsection{Events Interpreted}

\begin{center}
\begin{tabular}{lrrr}
\toprule
Layer & Total & Interpreted & Coverage \\
\midrule
Input & 256 & 17 & 6.6\% \\
Hidden & 256 & 4 (h2, h35, h62, h3) & 1.6\% \\
Output & 256 & 217 (receive patterns) & 84.8\% \\
\midrule
Total & 768 & 238 & 31.0\% \\
\bottomrule
\end{tabular}
\end{center}

``Interpreted'' for hidden means we have a semantic name.
Most hidden neurons remain unnamed.

\subsection{Patterns Interpreted}

Of the 302 significant patterns:
\begin{itemize}
\item 6 space$\to$hidden patterns with strength $\geq 2$: understood as ``reset'' patterns
\item 49 total input$\to$hidden: mostly strength 1, representing character identity
\item 253 hidden$\to$output: distributed character prediction
\end{itemize}

\textbf{Interpretation:} About 10 patterns have clear semantic meaning
(word boundary, syllable tracking). The rest are mechanical character-level
statistics without higher-level interpretation.

\subsection{BPC Attribution}

The model achieves 5.69 bpc. How much is explained by interpreted structure?

ES1 (word boundary) contributes to predicting:
\begin{itemize}
\item Space after words
\item Common word-initial letters
\end{itemize}

Rough estimate: word boundaries represent $\sim$15\% of predictions
(average word length $\sim$5-6 chars). But ES1 doesn't fully determine
these predictions---it's one signal among many.

\textbf{Conservative estimate:} Interpreted patterns explain $<5\%$ of compression.
The bulk of the model's 5.69 bpc performance comes from uninterpreted
character-level statistics in the hidden$\to$output weights.

\section{Preparation for Next Tick}

\subsection{What We Have}

\begin{enumerate}
\item ES1 (word boundary): clean binary ES, 99.6\% separation
\item ES2 (syllable momentum): explains word-length encoding
\item SN pattern format for the model
\item Understanding that words are NOT explicitly encoded
\end{enumerate}

\subsection{What Remains}

\begin{enumerate}
\item Find natural ESs beyond the mechanical doubled-E
\item Interpret more hidden neurons (currently 4/128)
\item Understand the hidden$\to$hidden recurrent structure
\item Determine which patterns can be injected for the tick
\end{enumerate}

\subsection{Injection Candidates}

For the next tick, we could inject:
\begin{itemize}
\item Strengthen the word-boundary pattern (currently strength 8)
\item Add patterns for common word endings
\item Add patterns for frequent bigrams/trigrams
\end{itemize}

The tick should write patterns that encode lexical knowledge
the model hasn't learned from data alone.

\section{Conclusion}

The isomorphic UM interpretation reveals:
\begin{enumerate}
\item A clear word-boundary ES ($h_2$) driven by space patterns
\item A syllable momentum ES ($h_{35}$) with temporal decay
\item Distributed character-level prediction in output patterns
\item No explicit word identity encoding
\end{enumerate}

Coverage: $\sim$31\% of events touched, $<5\%$ of bpc explained.
The model is primarily a character-level statistical predictor
with word-boundary structure emerging from the space pattern.

For the next tick: inject lexical patterns to bootstrap word knowledge.

\end{document}
