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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}

\title{The Nested Model:\\Self-Similar Architecture in the Extended Event Space}
\author{Claude and MJC}
\date{February 15, 2026}

\begin{document}
\maketitle

\begin{abstract}
The extended event space~\cite{extended-es} faces an apparent
contradiction: we hold $I$ and $O$ fixed (bytes in, bytes out) while
simultaneously extending them with new events.  We resolve this by
proving that the UM is self-similar: extending $H$ with an inner model
$U_v$ that has its own $I'$, $H'$, $O'$---all of which are part of
$H$ from the outer perspective.  We formalize this nesting as a
categorical construction, prove that it preserves the five-tuple
structure at every level, and show that the embedding $E_v$ induces a
bijection between byte-level and word-level representations that is
useful at both the input and output sides.  The key result: the nested
model's predictions are consistent with the outer model's predictions
(a natural transformation exists), and the nesting can be iterated to
arbitrary depth, producing a factorization tower of models within
models.  This formalizes the commentary's observation~\cite{commentary}
that ``a UM is a model of an organism of organisms.''
\end{abstract}


\section{The Apparent Contradiction}\label{sec:contradiction}

The extended event space paper~\cite{extended-es} proposes:
\begin{itemize}
\item $I' = \{0..255\} \times \{0..L_{\max}\} \times \{0,1\}^{256}$
  (byte $\times$ position $\times$ accumulator).
\item $H' = \{-1,+1\}^{128} \times \{0,1\}^{|V|} \times [0,1]^{|V|}$
  (hidden $\times$ bag-of-letters $\times$ graded support).
\item $O' = \{0..255\}^D$ (future bytes at $D$ offsets).
\end{itemize}
But the outer model's $I$ and $O$ are bytes: $I = O = \{0..255\}$.
These are fixed by the Hutter Prize criterion (compress enwik8 as
bytes).  How can we simultaneously hold $I$ and $O$ fixed and extend
them?


\section{The Resolution: Nested Models}\label{sec:resolution}

\begin{definition}[Nested extension]\label{def:nested}
Given a UM $U = (e, t, p, f, \omega)$ with event space
$E = I \times H \times O$, a \emph{nested extension} introduces an
inner model $U_v$ by replacing $H$ with:
\begin{equation}\label{eq:nested}
H_{\text{ext}} = I' \times H_{\text{inner}} \times O',
\end{equation}
where:
\begin{itemize}
\item $I'$ is the inner input space (visible only within $H$ from the
  outer perspective),
\item $H_{\text{inner}}$ is the original hidden space (unchanged),
\item $O'$ is the inner output space.
\end{itemize}
The outer $I$ and $O$ remain fixed.  The full extended event space is:
\begin{equation}
E_{\text{ext}} = I \times (I' \times H_{\text{inner}} \times O') \times O
= I \times H_{\text{ext}} \times O.
\end{equation}
\end{definition}

\begin{proposition}[Consistency]\label{prop:consistency}
The nested extension preserves the five-tuple structure.  The
extended model $U_{\text{ext}} = (e_{\text{ext}}, t_{\text{ext}},
p_{\text{ext}}, f_{\text{ext}}, \omega_{\text{ext}})$ has:
\begin{enumerate}
\item $e_{\text{ext}} \in E_{\text{ext}}$ (events in the extended space),
\item $t_{\text{ext}} \in [0, 255]^{|E_{\text{ext}}|}$ (support over
  extended events),
\item $p_{\text{ext}}$: patterns between all pairs of extended events,
\item $f_{\text{ext}}$: the standard $(\max, \min)$ update on $E_{\text{ext}}$,
\item $\omega_{\text{ext}}$: counting ($\omega_0$) applied to the
  extended event co-occurrences.
\end{enumerate}
\end{proposition}

\begin{proof}
Each component is defined by the same recipe as the base case,
applied to the larger event space $E_{\text{ext}}$.  The update
function $f_{\text{ext}}$ inherits the tropical semiring structure.
The learning function $\omega_{\text{ext}}$ counts co-occurrences
over $E_{\text{ext}}$ exactly as $\omega_0$ counts over $E$.
\end{proof}


\section{The Inner Model as a Complete UM}\label{sec:inner}

\begin{theorem}[The inner model is a UM]\label{thm:inner-um}
The inner model $U_v$ with event space $E_v = I' \times H_{\text{inner}}
\times O'$ is itself a complete Universal Model:
$U_v = (e_v, t_v, p_v, f_v, \omega_v)$ where:
\begin{itemize}
\item $e_v \in E_v$,
\item $t_v$ is the restriction of $t_{\text{ext}}$ to $E_v$,
\item $p_v$ is the restriction of $p_{\text{ext}}$ to patterns between
  events in $E_v$,
\item $f_v = f_{\text{ext}}|_{E_v}$ (the forward pass restricted to
  inner events),
\item $\omega_v = \omega_{\text{ext}}|_{E_v}$ (counting restricted to
  inner event co-occurrences).
\end{itemize}
\end{theorem}

\begin{proof}
The restriction of a five-tuple to a sub-event-space preserves all
five components.  The tropical forward pass is defined pointwise
(the $\max_i \min(t_i, p_{ij})$ operation depends only on the events
involved), so restriction is well-defined.  The counting function
counts co-occurrences of events in $E_v$, which is a subset of the
co-occurrences in $E_{\text{ext}}$.
\end{proof}

\begin{corollary}[Model within a model]
From the outer perspective ($U$), the inner events $I'$, $H_{\text{inner}}$,
$O'$ are all part of $H_{\text{ext}}$.  From the inner perspective ($U_v$),
they are the input, hidden, and output spaces of a complete model.
The same events have two interpretations depending on the level of
description.
\end{corollary}


\section{Iterated Nesting}\label{sec:iteration}

\begin{definition}[Nesting tower]
A \emph{nesting tower} of depth $n$ is a sequence of models:
\begin{equation}
U_0 \supset U_1 \supset \cdots \supset U_n,
\end{equation}
where $U_{k+1}$ is nested inside $H_k$ of $U_k$.  At each level:
\begin{equation}
H_k = I_{k+1} \times H_{k+1} \times O_{k+1}.
\end{equation}
\end{definition}

\begin{proposition}[Telescoping factorization]
The full event space of a depth-$n$ nesting tower is:
\begin{equation}
E = I_0 \times \left( I_1 \times \left( \cdots \left( I_n \times H_n \times O_n \right) \cdots \right) \times O_1 \right) \times O_0.
\end{equation}
The information decomposes additively:
\begin{equation}
\log |E| = \sum_{k=0}^{n} (\log |I_k| + \log |O_k|) + \log |H_n|.
\end{equation}
\end{proposition}

\begin{proof}
Each nesting replaces $H_k$ with $I_{k+1} \times H_{k+1} \times O_{k+1}$.
By induction, the full space is the iterated product.  Information
additivity follows from $\log |A \times B| = \log |A| + \log |B|$.
\end{proof}

\begin{example}[The lexical nesting]
For the extended event space with lexical structure:
\begin{itemize}
\item Level 0 (outer): $I_0 = \{0..255\}$ (bytes), $O_0 = \{0..255\}$.
\item Level 1 (lexical): $I_1 = \{0..L_{\max}\} \times \{0,1\}^{256}$
  (position $\times$ accumulator), $O_1 = \{0..255\}^{D-1}$ (future
  bytes at offsets $2..D$).
\item $H_1 = \{-1,+1\}^{128} \times \{0,1\}^{|V|} \times [0,1]^{|V|}$
  (RNN hidden $\times$ bag-of-letters $\times$ graded support).
\end{itemize}
From the outer perspective, $I_1$ and $O_1$ are part of $H_0$.  From
the lexical perspective, they are the input and output of the word
model.
\end{example}

\begin{remark}[Biological analogy]
The nesting tower formalizes the observation that ``the human brain
contains earlier animal brains.''  Each level of nesting corresponds
to an evolutionary layer:
\begin{itemize}
\item Level 0: sensory input/output (bytes = raw signals).
\item Level 1: pattern recognition (words = perceptual objects).
\item Level 2: abstract concepts (sentences = relational structures).
\end{itemize}
Each level has its own $I$, $H$, $O$, but from the level above, all
three are part of $H$.  The organism IS its nesting tower.
\end{remark}


\section{The $E_v$ Bijection}\label{sec:bijection}

The commentary~\cite{commentary} observes that the embedding $E_v$
induces a bijection useful at both the input and output sides.  We
formalize this.

\begin{definition}[Lexical embedding]\label{def:ev}
The \emph{lexical embedding} $E_v: V \to \{0,1\}^{256}$ maps each
word $w \in V$ to its bag-of-letters representation:
\begin{equation}
E_v(w) = (\mathbf{1}[c \in \text{letters}(w)])_{c=0}^{255}.
\end{equation}
\end{definition}

\begin{proposition}[$E_v$ at the input side]\label{prop:ev-input}
At the input side, $E_v$ acts as recognition: given the letter
accumulator $\text{acc}(t) \in \{0,1\}^{256}$, the set of consistent
words is:
\begin{equation}
V_{\text{consistent}}(t) = \{w \in V : E_v(w) \subseteq \text{acc}(t)\},
\end{equation}
where $\subseteq$ is componentwise.  As letters accumulate,
$V_{\text{consistent}}$ shrinks monotonically.
\end{proposition}

\begin{proof}
Each new letter either (a) matches an existing letter in $E_v(w)$
(no change to $V_{\text{consistent}}$) or (b) adds a letter not in
$E_v(w)$ for some words, removing those words from the consistent set.
Monotonicity follows from the monotonicity of set containment.
\end{proof}

\begin{proposition}[$E_v$ at the output side]\label{prop:ev-output}
At the output side, $E_v$ acts as prediction: given word identity
$w$, the remaining bytes are determined (up to spelling variant).
The predicted output is:
\begin{equation}
P(o \mid w, \text{pos}) = P(o \mid \text{canonical spelling of } w \text{ at position pos}).
\end{equation}
For canonical spelling, this is deterministic (strength 255 in SN).
For spelling variants, the probability is the variant's frequency.
\end{proposition}

\begin{theorem}[$E_v$ bijection]\label{thm:bijection}
The embedding $E_v$ induces a bijection between the bag-of-letters
space and the word-identity space for the ``core vocabulary''
$V_{\text{core}} \subseteq V$ of words with distinct letter sets.
On $V_{\text{core}}$:
\begin{equation}
E_v: V_{\text{core}} \xrightarrow{\;\sim\;} \text{Image}(E_v).
\end{equation}
The bijection is useful at both sides:
\begin{itemize}
\item $I$-side: $\text{acc}(t) \mapsto V_{\text{consistent}}(t)$
  (recognition).
\item $O$-side: $w \mapsto \text{canonical bytes}(w)$
  (prediction).
\end{itemize}
\end{theorem}

\begin{proof}
Two words $w_1 \neq w_2$ in $V_{\text{core}}$ have distinct letter
sets by definition, so $E_v(w_1) \neq E_v(w_2)$.  The map is
injective on $V_{\text{core}}$.  Surjectivity onto the image is
tautological.  The $I$-side and $O$-side applications follow from
Propositions~\ref{prop:ev-input} and~\ref{prop:ev-output}.
\end{proof}

\begin{remark}[IO symmetry]
By IO symmetry of the log product pattern~\cite{commentary},
both the $I$-side and $O$-side projections are given by the log
counting function $\omega_0$ in the efficient log-stochastic
implementation.  Recognition and prediction are the same computation
run in opposite directions through the $E_v$ bijection.
\end{remark}


\section{Natural Transformations Between Levels}\label{sec:natural}

\begin{theorem}[Consistency of nested predictions]\label{thm:natural}
Let $U_0$ be the outer model with event space $E_0 = I \times H \times O$
and $U_1$ the inner model with $E_1 = I' \times H_{\text{inner}} \times O'$.
There exists a natural transformation
$\alpha: \mathcal{U}_0 \Rightarrow \mathcal{U}_1$ (in the sense of
the category of UMs~\cite{category-es}) if and only if the inner
model's predictions are push-forwards of the outer model's predictions
under the embedding $\iota: E_1 \hookrightarrow H_{\text{ext}}
\hookrightarrow E_0$.
\end{theorem}

\begin{proof}
By the consistency theorem for natural transformations between
UMs~\cite{category-es}, $\alpha$ exists iff:
\begin{equation}
P_{U_0}(o \mid i) = \sum_{o' \in \phi_O^{-1}(o)}
\frac{\sum_{i' \in \phi_I^{-1}(i)} P_{U_1}(o' \mid i') P(i')}
{\sum_{i' \in \phi_I^{-1}(i)} P(i')}.
\end{equation}
Since $U_1$ is nested inside $H_0$, the embedding $\iota$ maps
inner events to outer hidden events.  The outer model's prediction
$P_{U_0}(o \mid i)$ marginalizes over all hidden states, including
the inner events.  The inner model's prediction $P_{U_1}(o' \mid i')$
is one component of this marginalization.  Consistency holds if the
inner model adds no contradictory evidence---which is guaranteed by
the neutral factorization~\cite{extended-es}: introducing the inner
events does not change the outer model's byte-level predictions.
\end{proof}

\begin{corollary}[Nesting is conservative]
The nested extension is conservative: the outer model's predictions
are unchanged by the introduction of the inner model.  The inner model
provides \emph{additional structure} (word-level events, bag-of-letters)
without modifying the byte-level predictions.  Value comes from the
new patterns (word-to-byte, word-to-word) that become available in
the extended event space.
\end{corollary}


\section{Connection to Self-Representation}\label{sec:self-rep}

\begin{remark}[Self-model]
A model $U$ can be represented in a model of itself.  If the
self-model adds no capabilities, $U$ simply represents itself
(a fixed point of the nesting operation).  If the self-model adds
runtime information (e.g., a UM runner extends $H$ with execution
state), then $U'$ represents $U$ enriched with dynamic information.

In the SN concrete representation, our model of the UM via SN
induces a description of any other, necessarily smaller, model.
The inner model $U_v$ is such a description: a model of the lexicon,
described in UM terms, nested inside the outer model's $H$.
\end{remark}

\begin{proposition}[Diagonalization limit]
By the diagonalization theorem~\cite{forcing-pumping}, the event space
$E$ cannot contain its own factorization map $\phi: E \to E'$.  This
limits nesting: the outer model can represent the inner model, but
the inner model cannot represent the full outer model (it is strictly
smaller).  The nesting tower is strictly decreasing:
$|E_0| > |E_1| > \cdots > |E_n|$.
\end{proposition}


\section{The Implementation Forms}\label{sec:implementation}

The commentary~\cite{commentary} identifies two implementation forms
for the nested model.

\begin{definition}[Log-stochastic form]
In the \emph{log-stochastic form}, the inner model's evidence is
represented as log-support values.  The embedding $E_v$ computes
graded word support via the standard $\omega_0$ counting function:
\begin{equation}
\sigma_w(t) \propto 2^{\omega_0(w, \text{acc}(t), \text{pos}(t))}.
\end{equation}
This gives log-probabilities directly, and the IO projection is
given by $\omega_0$ at both sides.
\end{definition}

\begin{definition}[Witness form]
In the \emph{witness form}, the evidence for word $w$ is represented
as the length of a dataset of memory traces---the actual positions
in the data stream where $w$ was observed with the given evidence.
The count beyond the floor log count is exact:
\begin{equation}
c(w, \text{context}) = |\{t \in D : \text{word}(t) = w, \text{context}(t) = \text{context}\}|.
\end{equation}
\end{definition}

\begin{remark}
The log-stochastic form is efficient (constant space per event).
The witness form is exact (preserves all count information).  The
gap between them is the tropical--integer gap~\cite{tropical-gcd}:
$\Delta = \log_2(\min/\gcd) \approx 0.037$ bits per prediction.
Since the gap does not affect conditionals (Theorem 5
of~\cite{tropical-gcd}), either form gives the same predictions.
\end{remark}


\section{The Word-Start Carrier}\label{sec:carrier}

\begin{proposition}[Carrier signal via SN programming]
The in-word position event can be implemented via SN programming
with absolute (strength 255) patterns:
\begin{enumerate}
\item Word position starts at 1 at strength 255 (this is a joint
  event with the first input after a word boundary).
\item Position increments with each byte and resets to 1 at word
  boundaries (spaces and punctuation), via strength-255 deterministic
  patterns.
\item The position event is a deterministic function of the byte
  stream---a UM runner is free to short-circuit it as pure logic.
\end{enumerate}
This is the explicit version of the carrier signal that the RNN
implements implicitly through $W_h$ rotation~\cite{carrier-signal}.
\end{proposition}


\section{Why Letter Events Are Not Redundant}\label{sec:letters}

A natural objection: if we track word identity via the bag-of-letters,
aren't the individual letter-accumulator events redundant?

\begin{theorem}[Letter events are necessary]\label{thm:letters}
The letter-accumulator events are not redundant with the bag-of-letters
events.  They carry information that the word embedding cannot:
\begin{enumerate}
\item \textbf{Spelling variants.}  The word embedding $E_v$ maps
  ``the'' to its canonical letter set $\{t, h, e\}$.  But the actual
  observation might be ``teh'' (letters $\{t, e, h\}$ in wrong order)
  or ``thh'' (extra `h', missing `e').  The letter accumulator
  records the \emph{actual} letters; the word embedding records the
  \emph{intended} word.  The discrepancy is the spelling variant,
  with luck $\lambda = 1/P(\text{variant})$.

\item \textbf{The strawberry problem.}  Counting specific letters in
  a word (``how many r's in strawberry?'') requires the letter
  accumulator, not the word embedding.  The embedding says
  ``this is strawberry''; the accumulator says ``three r's have
  appeared.''  The latter is explicit and correct; the former
  provides no letter-count information.

\item \textbf{Information-theoretic optimality.}  The letter
  accumulator is a sufficient statistic for the letter-level
  information: it records which bytes appeared and (via the position
  event) where.  The word embedding is a \emph{lossy compression}
  that discards letter-level detail.  The product
  $I'_{\text{acc}} \times H'_{\text{word}}$ separates the two
  levels, achieving information-theoretic minimum for the joint
  representation.
\end{enumerate}
\end{theorem}

\begin{proof}
For (1): the bag-of-letters for ``teh'' is $\{t, e, h\} = \{t, h, e\}$,
identical to ``the''.  The word embedding cannot distinguish them.
The letter accumulator, combined with the position event, records
``t at position 0, e at position 1, h at position 2''---which
distinguishes ``teh'' from ``the'' (where h is at position 1 and e
at position 2).

For (2): the bag-of-letters for ``strawberry'' is
$\{s, t, r, a, w, b, e, y\}$, which contains no count information.
The accumulator extended with position gives ``r at positions 2, 5, 6''
(three r's).

For (3): the product decomposition separates letter-level and
word-level information into independent factors.  By the
factorization principle~\cite{cmp}, the information in the product
is $I(I'_{\text{acc}}) + I(H'_{\text{word}})$, and the
cross-entropy decomposes correspondingly.
\end{proof}


\section{Conclusions}

\begin{enumerate}
\item The UM is self-similar: extending $H$ with an inner model
  creates a model within a model, with the outer $I$ and $O$ fixed
  and the inner $I'$, $O'$ living inside $H$.

\item The nesting is formally a categorical construction: the inner
  model is a complete UM, and consistency between levels is guaranteed
  by natural transformations in the category of UMs.

\item The $E_v$ bijection connects byte-level and word-level
  representations, useful at both $I$ (recognition) and $O$
  (prediction) sides, with IO symmetry given by $\omega_0$.

\item Letter-accumulator events are not redundant: they carry
  spelling-variant and letter-count information that the word
  embedding discards.

\item The nesting tower (models within models) formalizes the
  structure of organisms containing earlier organisms, and of
  architectures containing earlier architectures.

\item The construction is conservative (neutral factorization)
  and iterable (to arbitrary depth).
\end{enumerate}


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

\bibitem{extended-es}
Claude and MJC.
\newblock \emph{The Extended Event Space: Injecting Lexical Structure into $H$}.
\newblock Hutter archive, 15 Feb 2026.

\bibitem{commentary}
Claude and MJC.
\newblock \emph{The Model Within the Model: Commentary on the Extended Event Space}.
\newblock Hutter archive, 15 Feb 2026.

\bibitem{category-es}
Claude and MJC.
\newblock \emph{The Category of Event Spaces}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{tropical-gcd}
Claude and MJC.
\newblock \emph{The Tropical--Integer GCD Bridge}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{carrier-signal}
Claude and MJC.
\newblock \emph{The Carrier Signal Problem}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{forcing-pumping}
Claude and MJC.
\newblock \emph{Baby Steps: Forcing, Pumping, and Diagonalization from the
$E \to \mathbb{N} \to \mathbb{Q}$ Chain}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{tock-step}
Claude and MJC.
\newblock \emph{The Tock Step: Domain-Native Architecture from Evidence}.
\newblock Hutter archive, 12 Feb 2026.
\end{thebibliography}

\end{document}
