\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,booktabs,xcolor}
\usepackage{hyperref}
\definecolor{svdblue}{HTML}{58a6ff}
\definecolor{humgreen}{HTML}{3fb950}

\title{The Factor-Permutation Map:\\Commentary on the February~11 Archive}
\author{Claude and MJC}
\date{February 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
The SSSP appendix introduced the \emph{equal-dimension factor permutation}: when
a macrostate space factors as $Z \cong Z_0^k$, the symmetric group $S_k$ acts by
permuting coordinates, and the factor map must carry an explicit alignment
$\pi \in S_k$.  We trace this structure through all twenty papers of the
February~11 archive and identify two ramifications that the original appendix
left implicit.  First, \textbf{most problems have low-dimensional inner product
structure}: the prediction depends on a $d$-dimensional subspace of $\mathbb{R}^k$
with $d \ll k$, so the effective symmetry group is far smaller than $S_k$.
Second, \textbf{most numbers are not prime}: macrostate integers
$N(\sigma) = \prod p_i^{x_i}$ are always highly composite, and their unique
prime factorization IS the interpretation.  Together these explain why total
interpretation of the 128-hidden RNN is possible: the alignment problem is
low-dimensional, and the state is always decomposable.
\end{abstract}

\section{The Equal-Dimension Factor Permutation}

We recall the key construction from the SSSP appendix~\cite{sssp-appendix}.
Suppose the macrostate space factors as $k$ copies of a single factor:
\[
  Z \;\cong\; \prod_{i=1}^k Z_0.
\]
The symmetric group $S_k$ acts on $Z$ by permuting coordinates:
\[
  P_\pi(z_1, \ldots, z_k) \;=\; (z_{\pi^{-1}(1)}, \ldots, z_{\pi^{-1}(k)}).
\]
When the underlying semantics is invariant under relabeling equal factors,
a factor map $\phi: E \to Z$ is only defined up to the $S_k$ action.
To remove this ambiguity, we refine:
\[
  \phi: E \to S_k \times Z_0^k,
  \qquad
  \phi(e) = (\pi(e),\, \tilde\phi(e)),
\]
where $\pi(e)$ is the canonical alignment and $\tilde\phi(e)$ gives the
coordinates in an arbitrary order.  The decoder undoes the permutation:
$d(\pi, z) = \tilde{d}(P_{\pi^{-1}}(z))$.

In the $E \to \mathbb{N}$ encoding, this is literally a permutation of
primes:
\[
  P_\pi: \prod_{i=1}^k p_i^{x_i} \;\longmapsto\; \prod_{i=1}^k p_{\pi(i)}^{x_i}.
\]
This is a clean algebraic statement: an alignment between two presentations
is a bijection on the prime indices.

For the sat-rnn with 128 binary event spaces, $Z_0 = \{0,1\}$, $k = 128$,
and the symmetry group is $S_{128}$ with $|S_{128}| = 128! \approx 10^{215}$.


\section{Two Ramifications}

\subsection{Most problems have low-dimensional inner product structure}

Consider the output prediction $\hat{y} = \mathrm{softmax}(W_y h + b_y)$ where
$h \in \{-1,+1\}^{128}$ is the hidden state (sign bits) and
$W_y \in \mathbb{R}^{256 \times 128}$.  The prediction depends on the inner
product $W_y h$---a linear map from $\mathbb{R}^{128}$ to $\mathbb{R}^{256}$.

If $W_y$ has effective rank $d \ll 128$, then the prediction depends on only
$d$ linear combinations of the 128 coordinates.  The $S_{128}$ action permutes
coordinates, but only permutations that preserve the $d$-dimensional column
space of $W_y$ leave the prediction invariant.  The effective symmetry group
is far smaller than $S_{128}$.

For our RNN, the February~11 archive establishes:
\begin{itemize}
  \item $d \approx 20$ neurons suffice for the full compression
    (Q3: \texttt{h28} alone = 99.7\%, top 15 neurons $>$ 100\%),
  \item The redux with 20 neurons + 36\% of $W_h$ achieves 4.81~bpc
    (0.15 better than the full 128-neuron model),
  \item The factor map identifies 2-offset conjunction detectors with
    mean $R^2 = 0.837$, concentrated on $\sim$20 functional dimensions.
\end{itemize}

The implication: the alignment problem for our RNN is not a search over
$S_{128}$ ($10^{215}$ elements) but over the $\sim$20-dimensional subspace
that carries predictive content.  The remaining 108 dimensions are
\emph{gauge freedom}---the network uses them for gradient flow during training
but they carry no predictive information.

This is not specific to our RNN.  For any prediction problem with $k$
equal-dimension features and effective dimensionality $d$, the factor
permutation alignment has $\binom{k}{d} \cdot d!$ relevant configurations
rather than $k!$.  Most problems in practice have $d/k \ll 1$.

\subsection{Most numbers are not prime}

In the $E \to \mathbb{N}$ encoding of a binary-ES system, the macrostate
integer is
\[
  N(\sigma) = \prod_{i=1}^{128} p_i^{b_i},
  \qquad b_i \in \{0, 1\}.
\]
Since the mean margin is 60.5 (i.e., most bits are ``on'' most of the time),
typical macrostate integers have $\sim$60--128 prime factors.  They are
\emph{maximally composite}: their prime factorization carries the full 128
bits of information about the state.

This has three consequences:
\begin{enumerate}
  \item \textbf{Factorization IS interpretation.}  The unique prime
    factorization of $N(\sigma)$ recovers the binary state
    $(b_1, \ldots, b_{128})$ exactly.  If the macrostate integer were
    prime, it would have no internal structure---no decomposition, no
    interpretation.  The fundamental theorem of arithmetic guarantees that
    composite numbers always decompose, and highly composite numbers
    decompose maximally.

  \item \textbf{The state space is sparse in $\mathbb{N}$.}  There are
    $2^{128} \approx 3.4 \times 10^{38}$ possible macrostate integers
    (square-free products of subsets of 128 primes), embedded in
    $[1, \prod p_i]$.  The product $\prod_{i=1}^{128} p_i$ is
    astronomically large ($> 10^{400}$).  So the macrostate integers are
    an extraordinarily sparse, highly structured subset of $\mathbb{N}$.

  \item \textbf{Orbits under $S_{128}$ are indexed by Hamming weight.}
    The $S_{128}$ action permutes which primes appear.  Two macrostate
    integers in the same orbit have the same number of prime factors
    (same Hamming weight $w$).  There are exactly 129 orbits
    (for $w = 0, 1, \ldots, 128$), each of size $\binom{128}{w}$.
    The alignment $\pi \in S_{128}$ is exactly the information beyond
    the orbit: \emph{which specific bits are on}.
\end{enumerate}

The connection to low-dimensional inner product: the prediction depends on
$\langle w, z \rangle$ where $w \in \mathbb{R}^{128}$ and
$z \in \{0,1\}^{128}$.  This inner product is a \emph{weighted count} of
active bits.  If $w$ has only $d$ significant entries, then only $d$ prime
factors of $N(\sigma)$ determine the prediction.  The other $128 - d$ primes
are present in the factorization but irrelevant to the output---they are
the gauge freedom made concrete in the prime-power encoding.


\section{Commentary on the Twenty Papers}

We now trace the factor-permutation structure through each paper of the
February~11 archive, grouped thematically.

\subsection{Foundational theory}

\paragraph{E onto N: The Bi-Embedding of Events and Numbers~\cite{e-onto-n}.}
This paper defines the counting map $\phi: E \to \mathbb{N}$ and the
construction map $\psi: \mathbb{N} \to E$.  The factor permutation is the
core ambiguity in both directions.  In the forward direction ($E \to N$),
counting events into the prime-power encoding requires choosing an
assignment of primes to event spaces---i.e., fixing an element of $S_k$.
Different assignments yield different integers for the same state.
In the backward direction ($N \to E$), constructing weights from data
statistics requires choosing a neuron assignment: which neuron stores
which feature.  The Hebbian construction ($W = X^T Y$) is
$S_{128}$-equivariant (permuting neurons just permutes rows and columns),
so any assignment works equally well.  The fact that the analytic
construction (1.89~bpc) works at all is evidence that the
$S_{128}$ ambiguity is harmless: the compositeness of $N(\sigma)$ means
the factorization carries the content regardless of labeling.

The paper's Shannon--Boltzmann identity (SN strength = $\log\Omega$) is
$S_k$-invariant: entropy does not depend on which prime encodes which
event space.  This invariance is what makes the thermodynamic formulation
possible---the bridge between E and N does not require resolving the
permutation.

\paragraph{The Quotient Chain~\cite{quotient-chain}.}
The quotient $Q = N / c(e)$ divides the macrostate integer by the
prediction count.  Under the $S_k$ action, if $\pi$ permutes the primes
of $N$ and the prediction count $c(e)$ is computed from the same event
space factorization, then $Q$ transforms covariantly:
$Q_\pi = P_\pi(N) / P_\pi(c(e))$.  The quotient chain
$E \to N \to Q$ is equivariant under the factor permutation at every
layer.

The paper's key result---that quotient decomposition is multiplicative
($Q_{\mathrm{total}} = \prod Q_{\mathrm{layer}}$)---means that each layer
contributes an independent factor to the total quotient.  In prime-power
terms, each layer's quotient multiplies certain primes by certain exponents.
The layerwise factorization of Q is a \emph{refinement} of the product
factorization of N: the same prime-power structure, but with additional
temporal indexing.

The error amplification result (error grows linearly in $\log Q$,
exponentially in $Q$) applies regardless of the $S_k$ alignment because
it concerns the \emph{magnitude} of exponents, not their labeling.  This
is another manifestation of low-dimensional structure: the amplification
depends on the norm $\|x\|$, not on which coordinates are large.

\paragraph{Microstates, Macrostates, and the Partition Function~\cite{micro-macro}.}
The thermodynamic identification (binary-ES softmax = Boltzmann with
$\beta = \ln 2$) is deeply connected to equal-dimension factors.
Each binary ES contributes a factor $e^{\beta s_i}$ or $e^{-\beta s_i}$
to the partition function, where $s_i$ is the SN strength.  Since all
128 ESes have the same dimension ($|Z_0| = 2$), the partition function
factors as
\[
  \mathcal{Z} = \prod_{i=1}^{128} (e^{\beta s_i} + e^{-\beta s_i})
  = \prod_{i=1}^{128} 2\cosh(\beta s_i).
\]
This product is invariant under $S_{128}$: permuting the ESes just
reorders the factors.  The Boltzmann entropy $S_B = \ln\mathcal{Z} -
\beta\langle E\rangle$ inherits this invariance.

The ``second law'' result (factoring increases macrostate entropy) is
the statement that going from the unfactored state space $\{0,1\}^{128}$
to 128 binary ESes \emph{cannot decrease} entropy.  In permutation terms:
the $S_{128}$ orbit of any macrostate has size $\binom{128}{w}$ for
Hamming weight $w$, and the orbit entropy $\log\binom{128}{w}$ is
maximized near $w = 64$.  The mean margin of 60.5 places typical states
near this maximum---the RNN operates near the entropy peak of its own
state space.

\paragraph{The Entropy Bridge~\cite{entropy-bridge}.}
Shannon equals Boltzmann through the event formalism: $H = \log_2 N -
\langle S_B \rangle$.  Both sides are $S_k$-invariant.  Shannon entropy
$H(X) = -\sum p_i \log p_i$ does not depend on the labeling of outcomes.
Boltzmann entropy $S_B = k_B \ln \Omega$ counts microstates, which is
also label-independent.  The bridge between them works precisely because
both are defined on \emph{orbits} of the symmetric group action on the
event factorization.

The hierarchical factoring table in the paper---showing how entropy
decomposes as you factor the event space---is the additive version of
the multiplicative prime-power decomposition.  In the $E \to \mathbb{N}$
encoding, $\log N = \sum x_i \log p_i$, and the entropy of each factor
contributes additively to the total.

\subsection{Structure papers}

\paragraph{The 128-Bit Boolean Automaton~\cite{boolean-automaton}.}
This paper establishes that $Z_0 = \{0,1\}$ conclusively.  The margins
(mean 60.5, safety factor $10^6 \times$) prove that the mantissa is
irrelevant: the computation is purely Boolean.  In factor-permutation
terms, this \emph{validates the equal-dimension assumption}.  If the
neurons were analog (using the full f32 range), the factor would be
$Z_0 = \mathbb{R}$ (or $[0,1]$, or $\{0,\ldots,2^{32}-1\}$), and the
$S_{128}$ action would be on a different, possibly non-uniform space.
The Boolean automaton result pins $Z_0$ down to exactly 2 elements, making
the equal-dimension structure \emph{exact}.

The sparse influence result (mean out-degree 3.5 despite dense $W_h$)
is a direct manifestation of low-dimensional inner product structure in
the dynamics.  Each neuron's next state depends on only $\sim$3.5 other
neurons---the $128 \times 128$ Boolean function is effectively a sparse
graph.  The $S_{128}$ permutation of neurons is constrained by this
graph topology: only permutations that preserve the influence graph
structure are dynamically equivalent.

The no-attractor result (no fixed points of the Boolean map) means the
system cycles through macrostates without settling.  In $S_{128}$ orbit
terms, the trajectory visits macrostates of different Hamming weights
(all 128 neurons volatile, mean dwell 3.3 steps), so the orbit index
changes at every step.  The macrostate integer $N(\sigma_t)$ changes its
number of prime factors every $\sim$3 steps.

\paragraph{$H = 2^{32}$: The f32 State Space~\cite{h32}.}
This paper decomposes the per-neuron state into three factors of
\emph{different} dimension:
\[
  Z_{\mathrm{neuron}} \;=\; Z_{\mathrm{sign}} \times Z_{\mathrm{exp}} \times Z_{\mathrm{mantissa}}
  \;\cong\; \{0,1\} \times \{0,\ldots,255\} \times \{0,\ldots,2^{23}-1\}.
\]
The 300:52:1 leverage hierarchy quantifies the dimension mismatch: the
1-bit sign factor carries 300$\times$ more predictive information per bit
than the 23-bit mantissa factor.  The factors are NOT equal-dimension,
so the full $S_3$ permutation (sign $\leftrightarrow$ exponent $\leftrightarrow$
mantissa) is not meaningful---these are qualitatively different factors.

However, \emph{within} the mantissa, the 23 bits ARE equal-dimension
($Z_0 = \{0,1\}$ for each mantissa bit), and $S_{23}$ acts by permuting
them.  The paper shows that mantissa bits 0--14 have zero predictive
effect and removing all mantissa bits \emph{improves} performance by
0.139~bpc.  This means the $S_{23}$ symmetry is unbroken: no mantissa
bit has been assigned a meaningful role by training.  The mantissa is
pure gauge.

This is ``most numbers are not prime'' applied locally: the macrostate
integer per neuron factors as $p_{\mathrm{sign}}^{b} \cdot
p_{\mathrm{exp}}^{e} \cdot p_{\mathrm{mantissa}}^{m}$.  The mantissa
factor $p_{\mathrm{mantissa}}^m$ contributes a large prime power to $N$
but carries no predictive information.  It makes $N$ more composite
without adding interpretive content---compositeness without content is
the precise signature of gauge dimensions.

\paragraph{The Temporal Bi-Embedding~\cite{temporal-biembedding}.}
The forward map (skip-$k$-grams as temporal counting) selects specific
offsets $\{d_1, \ldots, d_k\}$ from the set $\{1, \ldots, D\}$ of
possible temporal offsets.  The offset selection is itself a
combinatorial choice with $S_D$ acting on the $D$ possible offsets.
The greedy MI ordering breaks this symmetry: offsets are chosen in
order of mutual information with the output, giving a canonical
ordering that resolves the $S_D$ ambiguity.

The PMI alignment result (88\% at shallow offsets, 24--37\% at deep
offsets) quantifies how well the RNN's internal offset usage matches
the data's statistical structure.  In factor-permutation terms: the
RNN has implicitly chosen a permutation $\pi$ that maps its 128
neurons to temporal offset features, and this permutation achieves
88\% alignment at shallow depths but diverges at depth.  The
divergence at deep offsets reflects the fact that deep temporal
patterns have more $S_k$ ambiguity (more equivalent orderings) than
shallow ones.

The Hebbian correlation $r = 0.56$ between $W_h$ and
$\mathrm{cov}(h_j(t), h_i(t+1))$ is a measure of how well the
\emph{temporal} factor permutation aligns with the \emph{spatial}
(neuron-level) factor permutation.  Perfect alignment ($r = 1$) would
mean the network's neuron ordering exactly matches the data's temporal
offset ordering.  The partial alignment ($r = 0.56$) reflects the
residual $S_{128}$ gauge freedom.

\paragraph{The Event Space Isomorphism~\cite{es-iso}.}
This paper is the \emph{purest} application of the equal-dimension
factor permutation.  The SVD sign-bit partition creates 8 groups
($Z_0 = \{0,\ldots,7\}$, but more precisely $Z_0^3 = \{0,1\}^3$ via
three sign bits).  The human partition also creates 8 groups.
The optimal permutation $\pi \in S_8$ aligns them, achieving 85.7\%
accuracy at offset~11.

The paper's ``inner product preservation'' analysis (centroid cosine
similarity) directly measures the low-dimensional structure.  The
centroids live in $\mathbb{R}^3$ (the 3-dimensional SVD subspace),
and the cosine similarity measures how well the $S_8$ alignment
preserves the 3-dimensional inner product structure.  Mean cosine
$\approx 0.5$ reflects the refinement asymmetry: the arch-native
partition resolves sub-category structure that the human partition
merges.

The input-side refinement (lowercase split into 3--5 SVD groups)
is the prototypical example of equal-dimension factors creating
\emph{more structure than the labeling}:  the data's statistical
structure discovers that ``lowercase'' is not one event but several,
each with different predictive context.  The $S_8$ alignment can only
capture the coarsening (merging SVD groups into human categories);
the finer structure requires a non-bijective map---exactly the
quotient from the E$\to$N framework.

Since $8! = 40{,}320$, the brute-force search is trivial.  This
feasibility is itself a consequence of low-dimensional inner product
structure: 3 sign bits create 8 groups, and the alignment problem is
$S_8$ rather than $S_{256}$ (for byte-level alignment).  The SVD
\emph{reduces} the problem to its low-dimensional core.

\subsection{Empirical results}

\paragraph{Toward Total Interpretation~\cite{total-interp}.}
The isomorphic UM (768 events, 130 ESes, $\sim$3048 patterns) is an
alternative factorization of the same macrostate space.  The map from
the RNN's 128 binary ESes to the UM's 130 ESes IS a factor-permutation
map (with a dimension change: $128 \to 130$).  Strictly, this is not
an element of $S_k$ (since $k$ changes), but a map between two
factorizations of approximately the same space.

The seven key questions (Q1--Q7) are all questions about the factor
permutation in different guises:
\begin{itemize}
  \item Q1 (Boolean?): Is $Z_0 = \{0,1\}$ exact?  (Yes, validating
    the equal-dimension assumption.)
  \item Q2 (Offsets): Which temporal factors does each neuron encode?
    (The temporal component of $\pi$.)
  \item Q3 (Which neurons): Which coordinates in $Z_0^{128}$ carry
    signal?  (The $d$-dimensional subspace.)
  \item Q4 (Saturation): Does the system stay in $\{0,1\}^{128}$?
    (Yes, confirming $Z_0 = \{0,1\}$ dynamically.)
  \item Q5 (Redux): What is the minimal $d$?  ($d \approx 20$.)
  \item Q6 (Justifications): Which weight paths determine each
    prediction?  (The sparse graph within $W_h$.)
  \item Q7 (Algebraic): Does the RNN match the data's statistical
    structure?  (74\% alignment = partial resolution of $\pi$.)
\end{itemize}

\paragraph{Q1 Exact Results~\cite{q1-exact-results} and Q1 Exact~\cite{q1-exact}.}
The f32-vs-MPFR comparison is a comparison of two \emph{representations}
of the same computation with different $Z_0$: $Z_0^{\mathrm{f32}} = [0, 2^{32})$
and $Z_0^{\mathrm{MPFR}} = [0, 2^{256})$.  The decorrelation at depth $d=1$
shows that the exact representation matters for individual neuron trajectories.
But the pattern ranking $\rho = 1.000$ at $d \geq 11$ shows that the
$S_k$-invariant pattern content (which patterns are present, not their exact
strengths) is preserved across representations.

This is the factor permutation in a different guise: the map between f32
and MPFR representations is not a permutation of coordinates (both have
128 neurons) but a change of $Z_0$---a different factor space with the same
number of copies.  The pattern-level agreement says the \emph{orbit
structure} (which macrostates are visited) is preserved even when the
individual coordinates diverge.

\paragraph{Q1 Protocol B: Pattern Census~\cite{q1-protocol-b}.}
The RNN uses $\sim$1.1\% of available data patterns.  This sparsity is
the ``most numbers are not prime'' ramification at the pattern level:
of the $2^{128}$ possible binary states, the RNN visits a tiny fraction,
and of the vastly larger space of possible input$\to$output patterns,
only $\sim$1.1\% have non-negligible support.

The Hebbian learning characterization (log-stochastic counting) connects
to the factor permutation: Hebbian learning is $S_{128}$-equivariant
(it doesn't prefer any neuron labeling), so the patterns it discovers
are properties of the $S_{128}$ orbits, not of specific coordinate
choices.  The greedy MI ordering (complete via submodularity) provides
a canonical way to break the $S_k$ symmetry on the offset selection,
analogous to how the SVD provides canonical singular vectors.

\paragraph{Q1 Protocol C: The f32 Quotient~\cite{q1-protocol-c}.}
The 300:52:1 bit hierarchy decomposes the per-neuron f32 into three
factors of radically different information density.  The entropy
``bleeding'' through mantissa channels is precisely the leakage from
the gauge dimensions (mantissa bits) into the signal dimensions
(sign and exponent).  In prime-power terms: the mantissa primes carry
non-zero exponents that are pure noise, and the quotient
$Q_{\mathrm{f32}} = N_{\mathrm{f32}} / N_{\mathrm{Boolean}}$
measures how much of the total macrostate integer is gauge.

The lottery ticket observation (specific mantissa configurations at
initialization affect training outcome) is a subtle breaking of the
$S_{23}$ symmetry on mantissa bits: the initial random values create
a specific gradient landscape, and the training trajectory depends on
which mantissa bits happen to align with the gradient direction.  This
is the factor permutation at the \emph{initialization} level.

\paragraph{Q1 Sparsity~\cite{q1-sparsity}.}
Median 15 active patterns at $\tau = 0.01$ means that at any given
step, the macrostate integer $N(\sigma)$ is effectively determined by
only 15 patterns out of $\sim$44,794.  The inner product
$W_y h = \sum_{i=1}^{128} (W_y)_{\cdot,i} h_i$ has 128 terms, but
only 15 are ``active'' in the sense of contributing above threshold
to the prediction.

The non-monotonic depth profile (peak at $d \approx 20$--21) means the
most important patterns involve temporal offsets at intermediate depth,
not the shallowest or deepest.  In factor-permutation terms: the
alignment between neurons and temporal offsets is strongest at
intermediate depth, where the temporal pattern structure is rich enough
to exploit but not so deep that the $S_k$ ambiguity dominates.

\paragraph{Q2--Q4 Results~\cite{q234-results}.}
Deep dominant offsets ($d = 18$--25), single-neuron dominance
(\texttt{h28} = 99.7\%), and universal volatility (all 128 volatile)
are three aspects of the same structure:
\begin{itemize}
  \item Deep offsets: the network uses temporal factors far from the
    output, meaning the factor permutation maps neurons to
    \emph{distant} temporal positions.
  \item Single-neuron dominance: one coordinate in $Z_0^{128}$
    captures nearly all the predictive content---the inner product
    $\langle w, z \rangle$ is dominated by one component, giving
    $d \approx 1$ effective dimension for the readout.
  \item Universal volatility: every coordinate changes frequently,
    so the macrostate integer $N(\sigma_t)$ changes many prime
    factors at each step.  The orbit (Hamming weight) fluctuates,
    and the alignment information (which bits are on) is
    fully dynamic.
\end{itemize}

\paragraph{Q6: Justifications~\cite{q6-justifications}.}
The routing backbone (\texttt{h54} $\leftarrow$ \texttt{h121} $\leftarrow$
\texttt{h78}) and $\sim$15 weights per prediction describe the
\emph{sparse subgraph} of $W_h$ that determines each output.
This is the factor permutation restricted to the active subset:
of the $128 \times 128 = 16{,}384$ entries in $W_h$, only $\sim$15
($0.1\%$) participate in any given prediction.  The alignment
$\pi$ restricted to these 15 weights is a small permutation
(effectively $S_3$ or $S_4$ for the backbone neurons) rather than
the full $S_{128}$.

\paragraph{Cost Analysis~\cite{cost}.}
The 39,800$\times$ cost advantage of analytic construction over SGD
training is directly attributable to the factor permutation.  SGD
must \emph{discover} the alignment $\pi \in S_{128}$ by gradient
descent through a loss landscape with $S_{128}$ symmetry (permuting
neurons gives equivalent minima).  The analytic construction
\emph{bypasses} the alignment search entirely: it computes the
weights directly from data statistics, using the $S_{128}$-equivariance
of Hebbian learning to avoid the symmetry altogether.

The scaling advantage (gap widens from 4.6 to 7.2 orders of magnitude
at $H = 4096$) reflects the growth of $|S_H|$: the permutation
search space grows as $H!$, while the analytic construction cost is
$H$-independent (it depends on the data, not the network width).

\subsection{Synthesis}

\paragraph{Synthesis: Total Interpretation~\cite{synthesis}.}
The summary result---300:52:1, $d = 18$--25, 1 neuron = 99.7\%, 128/128
volatile, 20 neurons + 36\% $W_h$ = 4.81~bpc---is a complete
specification of the factor-permutation structure.  The 300:52:1
hierarchy identifies which factor dimensions carry information.
The $d = 18$--25 range identifies the temporal alignment.
The single-neuron dominance identifies the spatial alignment.
The redux identifies the effective dimension.

The statement ``the mantissa was the ladder'' captures the
factor-permutation insight perfectly: the 23 mantissa bits per neuron
(= $23 \times 128 = 2{,}944$ equal-dimension binary factors) were
necessary during training to provide gradient signal, but they carry
zero predictive content in the final model.  They are the scaffolding
of the $S_{2944}$ search that training performs, discarded once the
alignment is found.

\paragraph{From Counting to Construction~\cite{narrative}.}
The narrative arc (training $\to$ isomorphism $\to$ patterns $\to$
interpretation $\to$ construction) is the arc of resolving the
factor permutation:
\begin{enumerate}
  \item \emph{Training}: SGD searches $S_{128}$ implicitly by moving
    in weight space.
  \item \emph{Isomorphism}: The doubled-E construction maps the trained
    RNN to a UM, fixing one presentation.
  \item \emph{Patterns}: Skip-$k$-gram discovery identifies the
    temporal factors, partially resolving $\pi$.
  \item \emph{Interpretation}: The factor map $\phi: H \to \text{features}$
    gives the spatial component of $\pi$.
  \item \emph{Construction}: Hebbian/analytic weights bypass $\pi$
    entirely by working in the $S_{128}$-invariant space of
    data statistics.
\end{enumerate}

The Hebbian correlation $r = 0.56$ measures the partial alignment
between the trained $\pi$ and the data-optimal $\pi$.  The 50\%
$W_y$ blend improvement (+0.66~bpc) shows that the trained alignment
is \emph{suboptimal}---the gradient-discovered $\pi$ is not the
best one, and mixing with the Hebbian (permutation-invariant)
correction helps.

\paragraph{Q1 Implementation Notes~\cite{q1-implementation}.}
The implementation details for the GMP comparison concern the
\emph{numerical} factor permutation: how bits within a single f32
value are organized (IEEE 754 layout) and how this organization
interacts with the computation.  The implementation verifies that
the Boolean extraction (sign bits only) is numerically stable,
confirming that the equal-dimension factorization $\{0,1\}^{128}$
is not an artifact of finite precision.


\section{The Overarching Insight}

The twenty papers of the February~11 archive collectively demonstrate
that the total interpretation of the 128-hidden sat-rnn reduces to
resolving a factor permutation in a low-dimensional subspace of a
highly composite macrostate space.

The three pieces fit together:
\begin{enumerate}
  \item \textbf{Equal-dimension factors} ($Z_0 = \{0,1\}$, $k = 128$)
    create an $S_{128}$ symmetry.  The Boolean automaton result
    validates that this is exact.

  \item \textbf{Low-dimensional inner product} ($d \approx 20$)
    collapses the alignment problem from $S_{128}$ to a tractable
    subproblem.  The redux, the factor map, and the single-neuron
    dominance all confirm $d \ll k$.

  \item \textbf{Composite macrostate integers} (typical Hamming
    weight $\sim$ 60--128) guarantee that the state always has
    rich internal structure.  The fundamental theorem of arithmetic
    provides the interpretability guarantee: every composite number
    factors uniquely.
\end{enumerate}

In CMP terms: the interpretability of the RNN is not a contingent
empirical finding but a \emph{structural consequence} of the
equal-dimension factorization.  Any system with $k$ equal-dimension
binary factors, low effective dimension $d \ll k$, and composite
macrostates will be interpretable by the same method: resolve the
factor permutation in the $d$-dimensional subspace.

This explains the CMP paper's central thesis from a new angle:
``interpretability and efficiency are the same problem, both resolved
by recovering the correct factorization''~\cite{cmp}.  The
\emph{efficiency} of the analytic construction (39,800$\times$ cheaper)
comes from bypassing the $S_k$ search.  The \emph{interpretability}
of the result comes from the compositeness of the macrostate integer.
Both are consequences of the equal-dimension factor structure.

The SSSP analogy makes this precise: Dijkstra computes
$(\pi, d^*) \in \Pi \times T_{\mathrm{dist}}$ (the order factor $\Pi$
is the unnecessary $S_k$ search), while the sub-sorting algorithm
computes $d^*$ directly.  The analytic weight construction computes
the weights directly (from data statistics), while SGD training
searches $S_{128}$ implicitly.  In both cases, the speedup comes from
recognizing that the $S_k$ factor is gauge and avoiding its construction.

\begin{thebibliography}{99}
\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{cmp}
Michaeljohn Clement.
\newblock \emph{CMP}.
\newblock \url{https://cmpr.ai/cmp.pdf}, 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{quotient-chain}
Claude and MJC.
\newblock \emph{The Quotient Chain: $E \to N \to Q$ at Every Operation}.
\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{entropy-bridge}
Claude and MJC.
\newblock \emph{The Entropy Bridge: Microstates, Macrostates, and Event Factoring}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{boolean-automaton}
Claude and MJC.
\newblock \emph{The Sat-RNN as a 128-Bit Boolean Automaton}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{h32}
Claude and MJC.
\newblock \emph{$H = 2^{32}$: The f32 State Space}.
\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.

\bibitem{es-iso}
Claude and MJC.
\newblock \emph{The Event Space Isomorphism: Arch-Native and Human-Native Partitions}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{total-interp}
Claude and MJC.
\newblock \emph{Toward Total Interpretation}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{q1-exact-results}
Claude and MJC.
\newblock \emph{Q1 Exact Results: f32 vs MPFR-256}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{q1-exact}
Claude and MJC.
\newblock \emph{Q1 Exact: Six Programs}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{q1-protocol-b}
Claude and MJC.
\newblock \emph{Q1 Protocol B: Exact Pattern Census}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{q1-protocol-c}
Claude and MJC.
\newblock \emph{Q1 Protocol C: The f32 Quotient}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{q1-sparsity}
Claude and MJC.
\newblock \emph{Q1: How Sparse Is the Explanation?}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{q1-implementation}
Claude and MJC.
\newblock \emph{Q1 Implementation Notes}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{q234-results}
Claude and MJC.
\newblock \emph{Q2--Q4 Results: Offsets, Neurons, Saturation}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{q6-justifications}
Claude and MJC.
\newblock \emph{Q6: Human-Readable Justifications}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{synthesis}
Claude and MJC.
\newblock \emph{Synthesis: Total Interpretation of a 128-Hidden RNN}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{narrative}
Claude and MJC.
\newblock \emph{From Counting to Construction: The Complete Arc}.
\newblock Hutter archive, 11 Feb 2026.

\bibitem{cost}
Claude and MJC.
\newblock \emph{Computational Cost of Analytic Weight Construction vs.\ Gradient Descent Training}.
\newblock Hutter archive, 11 Feb 2026.
\end{thebibliography}

\end{document}
