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\newtheorem{definition}{Definition}
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\title{A Mathematical Review of CMP}
\author{Claude and MJC}
\date{February 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
We present a detailed mathematical review of CMP~\cite{cmp}, the paper
introducing the Universal Model $u = (e, t, p, f, \omega)$.  We formalize
each component, identify the key structural results, and assess the
framework's coherence and completeness in light of the twenty-paper
empirical program of February 7--11, 2026.  We find that the framework
is mathematically precise where it needs to be (event spaces, factorization,
the standard learning function) and deliberately open where it should be
(the update function, the learning function beyond the standard case).
The central claim---that interpretability and efficiency are the same
problem, both resolved by correct factorization---is confirmed by the
empirical results.  We identify three areas where the February~11 archive
extends the theory beyond what the original paper proves: the
equal-dimension factor permutation, the thermodynamic identification
(Shannon = Boltzmann), and the bi-directional construction
(data $\to$ weights without training).
\end{abstract}

\section{The Five-Tuple}

CMP defines the Universal Model as a five-tuple $u = (e, t, p, f, \omega)$
drawn from a product of finite sets $U = E \times T \times P \times F
\times \Omega$.

\begin{definition}[Universal Model]
A \emph{universal model} is a tuple $u = (e, t, p, f, \omega)$ where:
\begin{itemize}
  \item $e \in E$ is an event (the current state of the world),
  \item $t \in T$ is a thought (the model's current belief about events),
  \item $p \in P$ is a pattern (the model's knowledge structure),
  \item $f \in F$ is an update function ($f: P \times T \to T$),
  \item $\omega \in \Omega$ is a learning function
    ($\omega: T \times E \to P$).
\end{itemize}
\end{definition}

The total information is
\[
  I(U) = \log|U| = \log|E| + \log|T| + \log|P| + \log|F| + \log|\Omega|
  = I(E) + I(T) + I(P) + I(F) + I(\Omega),
\]
additive because $U$ is a product.  This is the Kullback information
under the uniform prior: the information gained by specifying a particular
$u$ from absolute ignorance.

\begin{remark}[Finiteness]
CMP defines $F$ and $\Omega$ as unions of binary strings up to fixed length
bounds, with decoding maps into realizable functions.  This is a deliberate
choice to keep everything finite and computable, avoiding the measure-theoretic
complications of function spaces.  The finite convention means that
$|U| < \infty$ and all information quantities are well-defined.
\end{remark}


\section{Event Spaces}

\begin{definition}[Event space]
An \emph{event space} $E_i$ is a finite set of mutually exclusive events,
exactly one of which is true at any given time.  The total event space
is a product $E = \prod_{i=1}^k E_i$, and a \emph{total event}
$e = (e_1, \ldots, e_k)$ specifies one event from each factor.
\end{definition}

The total information decomposes: $I(E) = \sum_{i=1}^k I(E_i) =
\sum \log|E_i|$.

CMP describes event spaces as ``epistemic precommitments to divide reality
into distinguishable parts.''  This is philosophically loaded but
mathematically precise: choosing a factorization $E = \prod E_i$
is choosing a coordinate system for the state space.  The factorization
is not unique---the same set $E$ can be factored in many ways---and the
choice of factorization is the central analytical decision.

\subsection{The coprime encoding}

\begin{proposition}[Concrete representation]
Given event spaces $E_1, \ldots, E_k$ with pairwise coprime
cardinalities $|E_i|$, the product $E = \prod E_i$ embeds into
$\mathbb{N}$ via mixed-radix encoding:
\[
  n(e_1, \ldots, e_k) = \sum_{i=1}^k e_i \prod_{j<i} |E_j|
\]
and the Chinese Remainder Theorem (CRT) guarantees unique recovery:
$(e_1, \ldots, e_k) = (n \bmod |E_1|, \ldots)$.
\end{proposition}

\begin{remark}[Beyond coprime cardinalities]
The coprime condition is sufficient for CRT but not necessary for a
well-defined encoding.  When cardinalities are not coprime (e.g., all
$|E_i| = 2$ for binary event spaces), the prime-power encoding
$N = \prod p_i^{e_i}$ (one prime per event space) provides an
alternative embedding into $\mathbb{N}$ with unique factorization
guaranteed by the fundamental theorem of arithmetic.  This extension
is developed in the February~10 Event Arithmetic paper~\cite{event-arithmetic}
and is essential for the binary-ES factorization of the RNN hidden state.
\end{remark}


\section{Total Thought}

\begin{definition}[Total thought]
A \emph{total thought} $t \in T$ assigns a belief value to each event.
CMP defines two forms:
\begin{itemize}
  \item \textbf{Discrete/logical}: $t \in \{0, 1\}^{|E|}$, where
    $t_e = 1$ means ``event $e$ is believed true'' and $t_e = 0$
    means ``event $e$ is believed false.''
  \item \textbf{Continuous/probabilistic}: $t \in (0, 1)^{|E|}$
    (or $[0, 255]^{|E|}$ in the concrete representation), where $t_e$
    is the degree of belief in event $e$.
\end{itemize}
\end{definition}

The isomorphism $T \cong E$---an assignment of belief to each possible
event---is a key observation.  In the concrete representation, a thought
is a vector of SN (support/negation) strengths in $\{0, \ldots, 255\}$,
where 0 means ``no support'' (not ``certainly false'') and 255 means
``maximum support.''  The distinction matters: \emph{certainty of
falsehood} requires positive support for the ES-mate (the complementary
event in the same event space), not merely absence of support for the
event itself.  Zero support is ignorance, not disbelief.

\begin{remark}[SN strength and probability]
The SN strength $s \in [0, 255]$ maps to probability via
$p = s / 255$ (linear) or via a log-domain relationship.  CMP uses
the convention that SN strengths are \emph{log} support values:
the strength $s$ represents evidence proportional to $2^{s/255}$
or similar.  The precise scaling convention varies across the paper;
the February~11 archive pins it down: for binary event spaces,
$\text{SN strength} = \log_2 \Omega$ where $\Omega$ is the number
of microstates (dataset positions) supporting the event.  This is
the Shannon--Boltzmann identity.
\end{remark}


\section{Patterns and the Update Function}

\begin{definition}[Total pattern]
A \emph{total pattern} $p \in P$ maps thoughts to thoughts:
$p \in T^2$ (i.e., $p: T \to T$).  An \emph{atomic pattern}
is a pair $(e_a, e_b) \in E^2$ with a strength, representing
the claim ``when $e_a$ is observed, $e_b$ becomes more likely.''
\end{definition}

\begin{definition}[Standard update function]
Given Boolean patterns $p_{ij} \in \{0, 255\}$ from input $I$ to
output $O$:
\[
  (f_p(t))_j = \max_i \min(t_i, p_{ij}).
\]
This is conjunction as min, disjunction as max---a tropical semiring
computation.
\end{definition}

\begin{remark}[Tropical structure]
The standard update is a max-plus (tropical) linear map.  In the
$(\max, \min)$ semiring, patterns compose by matrix ``multiplication''
where $\times \to \min$ and $+ \to \max$.  This is computationally
efficient and preserves the Boolean character of SN strengths.
The neural network update $h' = \tanh(Wh + b)$ is a different
(non-tropical) realization of the same abstract pattern application.
The February~11 archive shows that for the sat-rnn, the tanh
saturation makes the neural update \emph{approximately} Boolean:
the margins ($\text{mean} = 60.5$) mean that $\tanh(z)$ is within
$10^{-6}$ of $\pm 1$ for 98.9\% of neuron-steps.
\end{remark}


\section{The Learning Function}

\begin{definition}[Standard learning function $\omega_0$]
Given events $E = I \times O$ (input $\times$ output):
\begin{enumerate}
  \item Maintain a list of supported observations $\mathrm{obs} = []$.
  \item For each observation $(i, o) \in I \times O$:
    \begin{itemize}
      \item Find $(i, o, s)$ in $\mathrm{obs}$ and log-stochastically
        increment $s$.
      \item If not found, add $(i, o, 1)$.
    \end{itemize}
\end{enumerate}
The result is a log contingency table: $M_{io} = \log(\text{count of }
(i,o))$.
\end{definition}

\begin{remark}[Log-stochastic counting]
``Log-stochastically increment'' means: with probability $2^{-s}$,
set $s \to s + 1$; otherwise leave $s$ unchanged.  This maintains
$\mathbb{E}[2^s] = \text{count}$, so $s \approx \log_2(\text{count})$.
The key property: the log contingency table is a \emph{sufficient
statistic} for the function $I \to O$ given the data.  Everything
statistically learnable from the observations is captured by this matrix.
\end{remark}

\begin{remark}[Symmetry]
The standard learning function over $I \times O$ is symmetric: the
same matrix serves $I \to O$ (forward) and $O \to I$ (backward)
by transposition.  CMP connects this to causality: the directed
causal arrow from rain to wet ground becomes bidirectional under learning.
This is the same bidirectionality that the February~11 temporal
bi-embedding paper~\cite{temporal-biembedding} establishes for the RNN:
forward patterns (skip-$k$-grams from data) and backward attribution
(Jacobian chains through the network) are two readings of the same
matrix.
\end{remark}


\section{Multiplication}

CMP identifies multiplication as the universal combining operation:

\begin{center}
\begin{tabular}{llll}
\toprule
Factor & Multiplication & Additive form & Example \\
\midrule
$E$ & Cartesian product $E_1 \times E_2$ & $I(E) = I(E_1) + I(E_2)$ & coin $\times$ weather \\
$T$ & Independent beliefs $t = (t_1, t_2)$ & $\log p = \log p_1 + \log p_2$ & joint probability \\
$P$ & Layer composition $P_1 \cdot P_2$ & matrix multiplication & multi-layer net \\
$F$ & Function composition $f_1 \circ f_2$ & chained updates & forward pass \\
$\Omega$ & Learning composition & chained updates & multi-epoch training \\
\bottomrule
\end{tabular}
\end{center}

The quotient operation is multiplication's inverse:
$E / E_1$ identifies events that differ only in their $E_1$ component,
giving the ``rest'' of the factorization.  CMP notes that the quotient
presupposes the factored perspective---$E/E_1$ is simply $\prod_{i \neq 1} E_i$.


\section{The Equivalence Thesis}

\begin{theorem}[Central Claim of CMP]
Interpretability and efficiency in learning systems are identical
problems, both resolved by recovering the correct factorization of
the event space.
\end{theorem}

CMP argues this in two directions:

\paragraph{Efficiency $\Rightarrow$ Interpretability.}
An efficient model (sparse $P$, small $|E|$) must factor $E$ into
small event spaces with sparse patterns between them.  This
factorization, if it matches domain structure, is automatically
interpretable: the event spaces have domain-natural names, and
the patterns express domain-natural relationships.

\paragraph{Interpretability $\Rightarrow$ Efficiency.}
An interpretable model has named event spaces and explicit patterns.
This naming \emph{is} a factorization, and an interpreted factorization
is always at least as efficient as an un-factored one (the total
information decomposes: $I(E) = \sum I(E_i)$, and patterns between
small $E_i$ are sparser than patterns over the full $E$).

\paragraph{The failure mode: architecture-natural $\neq$ domain-natural.}
CMP identifies the core problem of deep learning: the architecture
provides a factorization (layers, neurons, attention heads) that is
efficient for gradient-based learning but opaque because it does not
match the domain's natural decomposition.  Interpretability research
seeks the refactoring map $\phi: H_{\text{arch}} \to H_{\text{domain}}$
between the two factorizations.

\begin{remark}[Assessment]
The equivalence thesis is the strongest claim in the paper.  It is
not proved as a theorem but argued by example and structural reasoning.
The February~11 archive provides the first comprehensive empirical
test: the total interpretation of the sat-rnn recovers a domain-natural
factorization (128 binary ESes $\to$ $\sim$20 functional features +
108 gauge dimensions) that is both interpretable (each feature has a
name: word length, in-tag, offset-pair conjunction) and efficient
(the analytic construction from the interpretation is 39,800$\times$
cheaper than training and achieves 1.89~bpc vs.\ trained 4.97~bpc).
This confirms the thesis in at least one concrete case.
\end{remark}


\section{ML Models as UM Instances}

CMP provides a systematic translation of neural networks into UM instances:

\begin{definition}[Doubled-E construction]
Given a neural network with neurons $\{n_1, \ldots, n_H\}$:
\begin{itemize}
  \item For each neuron $n_j$, introduce a binary event space
    $E_j = \{e_j, \bar{e}_j\}$ (fires / does not fire).
  \item Positive weight $w_{ij} > 0$: pattern $(e_i \to e_j)$ with
    strength $\propto w_{ij}$.
  \item Negative weight $w_{ij} < 0$: pattern $(e_i \to \bar{e}_j)$
    with strength $\propto |w_{ij}|$.
  \item Negative pressure is mediated by competition within each
    binary ES: support for $\bar{e}_j$ reduces $p(e_j)$.
\end{itemize}
This doubles $|E|$ (from $H$ to $2H$ events) but makes all pattern
strengths positive.
\end{definition}

\begin{remark}[Isomorphism]
The doubled-E UM is isomorphic to the original network---same
computation, same predictions, different notation.  CMP is explicit
that this is ``not more interpretable'': the events are named by index,
not by meaning.  The February~11 archive confirms the isomorphism
quantitatively: the doubled-E UM matches the RNN within 0.00\% bpc
difference~\cite{e-onto-n}.
\end{remark}


\section{The Concrete Representation}

CMP introduces the concrete representation via the coprime encoding.
The February~10 Event Arithmetic paper~\cite{event-arithmetic} extends
this to the prime-power encoding for equal-dimension factors.

\begin{definition}[Prime-power encoding]
Given $k$ event spaces $E_1, \ldots, E_k$ with assigned primes
$p_1, \ldots, p_k$, the macrostate integer is
\[
  N(\sigma) = \prod_{i=1}^k p_i^{e_i(\sigma)},
\]
where $e_i(\sigma) \in \{0, \ldots, |E_i|-1\}$ is the current value
of event space $E_i$.
\end{definition}

For binary event spaces ($|E_i| = 2$), this gives square-free products:
$N(\sigma) = \prod_{i \in S} p_i$ where $S \subseteq \{1,\ldots,k\}$
is the set of ``on'' bits.

\begin{proposition}[Fundamental theorem as interpretability guarantee]
The unique prime factorization of $N(\sigma)$ recovers the state
$(e_1, \ldots, e_k)$ exactly.  Every composite macrostate integer
has non-trivial internal structure.  For binary ESes with typical
Hamming weight $w \gg 1$, the macrostate integer has $w$ prime factors,
providing $w$ independent pieces of interpretive content.
\end{proposition}


\section{What the February 11 Archive Adds}

The original CMP paper defines the framework and argues the equivalence
thesis.  The February~11 archive extends the theory in three directions
not present in CMP.

\subsection{The equal-dimension factor permutation}

When $Z \cong Z_0^k$ (as in the 128 binary ESes of the sat-rnn),
$S_k$ acts by permuting coordinates.  CMP does not discuss this symmetry.
The SSSP appendix~\cite{sssp-appendix} introduces it, and the February~11
archive demonstrates its consequences: the alignment problem between
architecture-natural and domain-natural factorizations is a permutation
search, and the low-dimensional inner product structure ($d \approx 20$)
makes this search tractable.

This is a genuine extension: CMP's $\phi: H_{\text{arch}} \to H_{\text{domain}}$
is defined abstractly, but the equal-dimension structure gives $\phi$ a
precise algebraic form: $\phi \in S_k \times Z_0^k \to T$.

\subsection{The thermodynamic identification}

The February~11 microstate-macrostate paper~\cite{micro-macro} identifies
the binary-ES softmax with the Boltzmann distribution at $\beta = \ln 2$.
CMP does not make this connection.  The identification gives physical
meaning to every CMP quantity:
\begin{center}
\begin{tabular}{ll}
\toprule
CMP & Thermodynamics \\
\midrule
Event space $E_i$ & Degree of freedom \\
SN strength $s$ & $\log_2 \Omega$ (log microstates) \\
Pattern strength & Free energy difference \\
Update $f$ & Partition function evaluation \\
Quotient $Q$ & Luck = $1/p$ = inverse probability \\
\bottomrule
\end{tabular}
\end{center}

The Shannon--Boltzmann identity ($H = \log_2 N - \langle S_B \rangle$)
bridges information theory and statistical mechanics through the event
formalism.  This is arguably the most important theoretical extension,
as it gives CMP's abstract framework a concrete physical interpretation.

\subsection{Bidirectional construction}

CMP defines the learning function $\omega$ and the standard learning
algorithm (log-stochastic counting).  The February~11 archive demonstrates
that $\omega_0$ applied to the RNN problem gives a \emph{bidirectional}
construction:
\begin{itemize}
  \item \textbf{Forward} (data $\to$ weights): Hebbian covariance,
    skip-bigram log-ratios, and shift-register design give all 82k
    parameters from data statistics, achieving 1.89~bpc with zero
    gradient descent.
  \item \textbf{Backward} (weights $\to$ interpretation): the factor
    map, backward attribution chains, and Boolean automaton analysis
    recover the data's statistical structure from the trained weights.
\end{itemize}

CMP defines $\omega$ abstractly and gives the standard algorithm for the
unfactored case ($I \times O$).  The archive shows how to apply $\omega_0$
to a \emph{factored} system (128 binary ESes $\times$ 256 output bytes)
and demonstrates that the result is not only a sufficient statistic but
a \emph{constructive} one: it directly yields the weights.


\section{Assessment}

\subsection{What CMP gets right}

\begin{enumerate}
  \item \textbf{The five-tuple is exhaustive.}  Every component of the
    RNN maps cleanly to one of $(e, t, p, f, \omega)$.  No sixth component
    is needed.

  \item \textbf{Factorization is the right abstraction.}  The entire
    interpretability program of the February~11 archive reduces to
    finding the correct factorization of $E$ (128 binary ESes) and
    the correct factorization of $P$ (sparse patterns between them).

  \item \textbf{The standard learning function works.}  Log-stochastic
    counting of joint events is exactly what the Hebbian construction
    computes, and it produces weights that outperform gradient descent
    training on the output layer (50\% blend improves by 0.66~bpc).

  \item \textbf{The equivalence thesis holds.}  The analytic
    construction is both more interpretable (every parameter has a
    data-statistical meaning) and more efficient (39,800$\times$ cheaper)
    than SGD training.  Interpretability and efficiency are indeed
    the same problem.
\end{enumerate}

\subsection{What CMP leaves open}

\begin{enumerate}
  \item \textbf{How to find the factorization.}  CMP proves that the
    correct factorization exists and is both efficient and interpretable,
    but does not give an algorithm for finding it.  The February~11
    archive provides one empirical recipe (Boolean analysis $\to$
    factor map $\to$ backward attribution), but it is not clear this
    generalizes beyond small RNNs.

  \item \textbf{The update function beyond the standard case.}  CMP
    defines the standard update (max-min tropical) and notes that
    neural networks use a different update (matrix multiply + nonlinearity).
    The relationship between these two forms is established empirically
    (the tanh saturation makes them approximately equivalent) but not
    proved in general.

  \item \textbf{Continuous vs.\ discrete.}  CMP acknowledges both
    forms of $T$ ($\{0,1\}^{|E|}$ and $(0,1)^{|E|}$) but does not
    fully develop the relationship.  The February~11 results on
    f32 vs.\ Boolean dynamics (the mantissa is noise, sign bits carry
    99.7\% of compression) suggest that the discrete form is primary
    and the continuous form is an artifact of the float representation.
    But this conclusion depends on the specific architecture (Elman RNN
    with tanh) and may not hold for networks that genuinely use
    continuous dynamics (e.g., transformers with softmax attention).

  \item \textbf{The role of depth.}  CMP treats $P$ as a single
    pattern matrix.  The February~11 archive shows that temporal
    depth (offsets $d = 1$ to $d = 25$) is a critical structural
    parameter: the dominant patterns involve deep temporal offsets
    ($d = 18$--25), not just single-step transitions.  CMP's
    framework accommodates depth via pattern composition
    ($P = P_L \cdots P_1$) but does not single it out as a primary
    variable.

  \item \textbf{Scaling.}  All February~11 results concern a
    128-hidden single-layer RNN on 262k bytes.  CMP's claims are
    universal.  Whether the factor-permutation structure,
    low-dimensional inner products, and analytic construction
    scale to larger models and datasets is the central open question.
    The cost analysis (gap widens with $H$) is encouraging, but
    untested beyond $H = 128$.
\end{enumerate}

\subsection{Mathematical precision}

CMP's level of mathematical precision is appropriate for a foundational
paper: the definitions are clear, the examples are concrete, and the
claims are falsifiable.  The paper does not attempt theorem-proof
formalism for the equivalence thesis, which is arguably the right
choice: the claim is more a research program than a theorem, and
attempting to formalize it prematurely would constrain it.

The areas that would benefit from more formalization:
\begin{itemize}
  \item The coprime encoding and its extension to equal-dimension
    factors (now addressed by the Event Arithmetic paper and
    SSSP appendix).
  \item The precise relationship between SN strength and probability
    (now pinned down by the Shannon--Boltzmann identity).
  \item The standard learning function for \emph{factored} systems
    (now demonstrated by the Hebbian construction).
\end{itemize}


\section{Conclusion}

CMP introduces a framework that is both simple (five components,
multiplicative combination, one learning algorithm) and powerful
(maps to every domain the paper considers, provides constructive
methods for interpretability and weight construction).  The
February~11 archive provides the first comprehensive test, confirming
the framework's predictions in detail:

\begin{enumerate}
  \item The correct factorization (128 binary ESes with $\sim$20
    functional dimensions) is both efficient and interpretable.
  \item The standard learning function (log-stochastic counting)
    produces weights that match or exceed trained performance.
  \item The refactoring map $\phi$ has precise algebraic structure
    (factor permutation in a low-dimensional subspace).
  \item The quotient chain $E \to N \to Q$ traces the computation
    at every layer.
\end{enumerate}

The framework's most significant contribution is arguably conceptual
rather than technical: it identifies \emph{factorization} as the
primary analytical tool, unifying interpretability, efficiency,
information theory, and statistical mechanics under a single
algebraic structure.  The concrete representation ($E \to \mathbb{N}$
via prime powers) makes this unification computational, and the
February~11 archive makes it empirical.

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

\bibitem{sssp-appendix}
Claude and MJC.
\newblock \emph{Appendix X: Directed SSSP without sorting, in UM terms; The explicit $E \to \mathbb{N}$ macrostate map; The missing component: a permutation on equal-dimension factors}.
\newblock CMP Appendix, Feb 2026.

\bibitem{event-arithmetic}
Claude and MJC.
\newblock \emph{Event Arithmetic: $E$ onto $\mathbb{N}$}.
\newblock Hutter archive, 10 Feb 2026.

\bibitem{e-onto-n}
Claude and MJC.
\newblock \emph{E onto N: The Bi-Embedding of Events and Numbers}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{micro-macro}
Claude and MJC.
\newblock \emph{Microstates, Macrostates, and the Partition Function}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{temporal-biembedding}
Claude and MJC.
\newblock \emph{The Temporal Bi-Embedding: Forward Patterns, Backward Attribution}.
\newblock Hutter archive, 11 Feb 2026.
\end{thebibliography}

\end{document}
