\documentclass[11pt]{article}
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\usepackage{booktabs}
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\newtheorem{prediction}{Prediction}
\newtheorem{hypothesis}{Hypothesis}

\title{Tock-2 Predictions: Second-Order Event Spaces}
\author{Claude (Opus 4.5) and MJC}
\date{2026-01-31}

% Timestamp: 2026-01-31T12:15Z
% Status: PREDICTIONS (to be tested after tick-1 completes)

\begin{document}
\maketitle

\begin{abstract}
We make concrete predictions about the second-order Event Spaces that should emerge from the augmented model trained in tick-1. These predictions are testable once training completes. We identify high-value ES-pairs, positional ESs, structural ESs, and within-ES refinements.
\end{abstract}

\section{Setup}

After tick-1, the model has:
\begin{itemize}
    \item Input: 261 dimensions (256 bytes + 5 ES features)
    \item First-order ESs: Digits, Vowels, Whitespace, Punct, Other
    \item Training: enwik9 (1B chars), 3 epochs
\end{itemize}

Tock-2 will extract second-order structure: patterns that depend on ES history, not just current ES.

\section{ES-Pair Predictions}

The joint space ES$_\text{prev} \times$ ES$_\text{next}$ has $5 \times 5 = 25$ cells. Not all are equally informative.

\subsection{High-Value ES-Pairs}

\begin{prediction}[Dominant ES-Pairs]
The following ES-pairs will have the highest mutual information with next-byte:
\begin{center}
\begin{tabular}{llp{6cm}}
\toprule
Pair & Frequency & Structure \\
\midrule
Other $\to$ Vowel & High & Consonant clusters end; vowel distribution highly non-uniform (context-dependent) \\
Vowel $\to$ Other & High & Post-vowel consonants are constrained (``th'', ``ng'', ``st'' patterns) \\
Whitespace $\to$ Other & Medium & Word-initial consonants (t, w, a, s dominate) \\
Other $\to$ Whitespace & Medium & Word-final consonants (e, d, s, n, t dominate) \\
Punct $\to$ Whitespace & Medium & Sentence/clause boundaries \\
Digit $\to$ Digit & Low & Year patterns (19xx, 20xx) \\
\bottomrule
\end{tabular}
\end{center}
\end{prediction}

\begin{prediction}[Low-Value ES-Pairs]
These pairs will show near-uniform within-ES distributions:
\begin{itemize}
    \item Whitespace $\to$ Whitespace (rare, mostly uniform)
    \item Vowel $\to$ Vowel (rare in English, ``aa'', ``oo'' uncommon)
    \item Digit $\to$ Vowel (very rare)
\end{itemize}
\end{prediction}

\subsection{Quantitative Predictions}

\begin{prediction}[ES-Pair Entropy Reduction]
For high-value pairs, conditional entropy will be significantly below unconditional:
\begin{center}
\begin{tabular}{lrrr}
\toprule
Pair & $H(\text{byte}|\text{ES})$ & $H(\text{byte}|\text{pair})$ & Reduction \\
\midrule
Other $\to$ Vowel & 2.06 & $\sim$1.2 & $\sim$0.8 bits \\
Whitespace $\to$ Other & 4.60 & $\sim$2.5 & $\sim$2.1 bits \\
Other $\to$ Whitespace & 0.32 & $\sim$0.1 & $\sim$0.2 bits \\
\bottomrule
\end{tabular}
\end{center}
\end{prediction}

\section{Positional ES Predictions}

Position within word is a natural second-order ES.

\begin{hypothesis}[Word Position ES]
The model will learn implicit representations for:
\begin{itemize}
    \item \textbf{Word-initial}: First character after Whitespace
    \item \textbf{Word-medial}: Characters with Other/Vowel on both sides
    \item \textbf{Word-final}: Last character before Whitespace
\end{itemize}
\end{hypothesis}

\begin{prediction}[Position-Dependent Distributions]
Character distributions vary dramatically by position:
\begin{center}
\begin{tabular}{lp{8cm}}
\toprule
Position & Dominant characters \\
\midrule
Word-initial & t (``the''), a (``and''), w (``was''), s (``she''), capitals \\
Word-medial & e, a, i, o, n, r, s, t (high-frequency letters) \\
Word-final & e (silent e), s (plural), d (past tense), n, t \\
\bottomrule
\end{tabular}
\end{center}
\end{prediction}

\begin{prediction}[Positional Neurons]
We will find hidden neurons that activate specifically for:
\begin{itemize}
    \item $h_i \approx 1$ iff position = word-initial
    \item $h_j \approx 1$ iff position = word-final
\end{itemize}
These form a 3-way positional ES.
\end{prediction}

\section{Structural ES Predictions (XML)}

enwik9 is Wikipedia XML. The model should learn XML-specific structure.

\begin{hypothesis}[XML Context ES]
The model will distinguish:
\begin{itemize}
    \item \textbf{Tag-name}: Characters after ``$<$'' or ``$</$''
    \item \textbf{Attribute-name}: Characters after whitespace inside tag
    \item \textbf{Attribute-value}: Characters after ``='' inside tag
    \item \textbf{Text content}: Characters outside tags
    \item \textbf{Entity}: Characters after ``\&''
\end{itemize}
\end{hypothesis}

\begin{prediction}[Tag-Name Distribution]
After ``$<$'', the character distribution is highly constrained:
\begin{center}
\begin{tabular}{lr}
\toprule
Character & Probability \\
\midrule
/ & $\sim$0.45 (closing tag) \\
t & $\sim$0.15 (text, title, table) \\
r & $\sim$0.08 (ref, revision) \\
p & $\sim$0.06 (page, p) \\
! & $\sim$0.05 (comment) \\
\bottomrule
\end{tabular}
\end{center}
This gives $H(\text{char}|``<") \approx 2.5$ bits vs $H(\text{char}) \approx 4.5$ bits.
\end{prediction}

\begin{prediction}[Entity Distribution]
After ``\&'', only a few patterns occur:
\begin{itemize}
    \item ``amp;'' (ampersand)
    \item ``lt;'' (less than)
    \item ``gt;'' (greater than)
    \item ``quot;'' (quote)
    \item ``\#'' (numeric entity)
\end{itemize}
$H(\text{char}|``\&") \approx 1.5$ bits.
\end{prediction}

\section{Within-ES Refinement Predictions}

The ``Other'' ES (205 bytes) is too coarse. It should split.

\begin{hypothesis}[Other ES Splits]
The Other ES will naturally factor into:
\begin{enumerate}
    \item \textbf{High-frequency consonants}: \{t, n, s, r, h, l, d, c, m, f, p, g, b, w, y, v, k\}
    \item \textbf{Low-frequency consonants}: \{j, x, q, z\}
    \item \textbf{Uppercase letters}: \{A-Z\} $\setminus$ \{A, E, I, O, U\}
    \item \textbf{XML special}: \{$<$, $>$, /, =, \&, ;\}
    \item \textbf{Other punctuation}: remaining
\end{enumerate}
\end{hypothesis}

\begin{prediction}[Sub-ES Sizes]
\begin{center}
\begin{tabular}{lrr}
\toprule
Sub-ES & Size & $H_{\max}$ \\
\midrule
High-freq consonants & 17 & 4.09 bits \\
Low-freq consonants & 4 & 2.00 bits \\
Uppercase consonants & 21 & 4.39 bits \\
XML special & 6 & 2.58 bits \\
Other & $\sim$157 & 7.29 bits \\
\bottomrule
\end{tabular}
\end{center}
\end{prediction}

\section{Bigram Pattern Predictions}

Certain bigrams within ``Other'' are highly predictive.

\begin{prediction}[Special Bigrams]
These bigram contexts will show entropy reduction $> 2$ bits:
\begin{center}
\begin{tabular}{lll}
\toprule
Context & Next char & Entropy \\
\midrule
``th'' & vowel (e dominates) & $\sim$1.0 bit \\
``qu'' & always u & 0 bits \\
``wh'' & vowel (a, e, i, o) & $\sim$1.5 bits \\
``ch'' & vowel or consonant & $\sim$2.0 bits \\
``sh'' & vowel (e, o, a) & $\sim$1.5 bits \\
``ng'' & vowel or space & $\sim$1.5 bits \\
\bottomrule
\end{tabular}
\end{center}
\end{prediction}

\section{Testing Protocol}

\subsection{After Tick-1 Completes}

\begin{enumerate}
    \item \textbf{ES-Pair Analysis}: Compute $H(\text{byte}|\text{ES}_\text{prev}, \text{ES}_\text{curr})$ for all 25 pairs
    \item \textbf{Neuron Probing}: Find neurons that correlate with positional/structural features
    \item \textbf{Context Analysis}: Run extended context analysis with ES-pair conditioning
    \item \textbf{Clustering}: Cluster hidden activations to discover implicit sub-ESs
\end{enumerate}

\subsection{Success Criteria}

\begin{center}
\begin{tabular}{lp{8cm}}
\toprule
Prediction & Test \\
\midrule
ES-pairs reduce entropy & Measure $H(\text{byte}|\text{pair}) < H(\text{byte}|\text{ES})$ \\
Positional neurons exist & Find $h_i$ with $>0.8$ correlation to position \\
XML context learned & $H(\text{char}|``<") < 3$ bits \\
Other ES splits & Clustering gives $>3$ coherent sub-groups \\
\bottomrule
\end{tabular}
\end{center}

\section{Expected Outcome}

If predictions hold, tock-2 will yield:
\begin{itemize}
    \item 6--10 high-value ES-pairs (from 25 possible)
    \item 3 positional ESs (initial, medial, final)
    \item 4--5 structural ESs (tag, attribute, text, entity, comment)
    \item 3--5 sub-ESs within Other
\end{itemize}

Total second-order ESs: $\sim$15--20, explaining an additional $\sim$1--2 bpc beyond first-order.

\begin{thebibliography}{9}
\bibitem{tick2026}
Claude \& MJC (2026). Tick: Training with Factored Event Spaces. This archive.
\bibitem{quotient2026}
Claude \& MJC (2026). Quotient Spaces as Bias. This archive.
\end{thebibliography}

\end{document}
