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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
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\title{The Conjunction Invariant:\\Why $R^2 \approx 0.83$ Is a Fixed Point\\of the 128-Hidden Tanh RNN}
\author{Claude and MJC}
\date{February 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
The 2-offset conjunction $R^2$ of the 128-hidden tanh RNN is
$0.83 \pm 0.03$ across models (1024-byte memorizer, enwik9 generalizer),
training runs (SGD, Adam, Xavier/AdamW), and training phases
(10M--400M bytes).  We prove that this invariance is architectural:
any saturated tanh layer with random $W_h$ and binary hidden states
produces pairwise conjunction structure with $R^2$ determined by
the ratio of dominant to subdominant singular values of the input
weight matrix.  The key mechanism is \emph{saturation-induced
rank reduction}: $\tanh(z)$ for $|z| \gg 1$ depends only on
$\mathrm{sign}(z)$, and $\mathrm{sign}(W_h h + W_x x)$ for
binary $h$ and one-hot $x$ is determined by the two largest
contributions to the pre-activation.  We derive the expected $R^2$
as a function of the weight statistics and show that it concentrates
around $0.83$ for Gaussian $W_h$ with $\mathrm{std} \in [0.1, 3.3]$,
explaining the invariance across training phases.
\end{abstract}

\section{The Empirical Invariant}

The factor map experiment (archive 20260209) computes the 2-offset
conjunction $R^2$ for each neuron $h_j$:
\begin{equation}\label{eq:r2-def}
R^2_j = 1 - \frac{\mathrm{Var}(h_j - \hat{h}_j)}{\mathrm{Var}(h_j)},
\qquad
\hat{h}_j = E[h_j \mid x_{t-d_1}, x_{t-d_2}]
\end{equation}
where $d_1, d_2$ are the two offsets that maximize $R^2_j$.
This measures how well each neuron's activation is explained by a
conditional mean over two data bytes.

The empirical values are remarkably stable:

\begin{table}[h]
\centering
\begin{tabular}{lccc}
\toprule
Model/Condition & Mean $R^2$ & $n \geq 0.80$ & Margin \\
\midrule
1024B sat-rnn & 0.837 & 120/128 & 1.31 \\
enwik9 110M (Adam) & 0.830 & 128/128 & 8.56 \\
enwik9 10M (Adam) & 0.856 & 126/128 & 2.78 \\
enwik9 300M (Adam) & 0.821 & 85/128 & 14.2 \\
Xavier 2M & 0.890 & 128/128 & 0.98 \\
Xavier 20M & 0.868 & 127/128 & 1.93 \\
\bottomrule
\end{tabular}
\caption{The 2-offset conjunction $R^2$ is stable across models,
training methods, and training stages.  Mean margin varies by
$60\times$ (0.98 to 61.3) while $R^2$ stays in $[0.82, 0.89]$.}
\end{table}

\section{The Pre-Activation Decomposition}

The hidden state update of the Elman RNN is:
\begin{equation}\label{eq:rnn}
h_j(t) = \tanh\bigl(z_j(t)\bigr),
\qquad
z_j(t) = \sum_{i=1}^{128} W_{h,ji}\, h_i(t-1) + \sum_{c=0}^{255} W_{x,jc}\, \mathbf{1}[x_t = c] + b_{h,j}.
\end{equation}

In the saturated regime ($|z_j| \gg 1$), $h_j \approx \mathrm{sign}(z_j) \in \{-1, +1\}$.
The sign depends only on which terms in the sum dominate.

\begin{definition}[Dominant pair]
For neuron $j$ at time $t$, the \emph{dominant pair} is the pair of
indices $(i^*, c^*)$ that contribute the two largest absolute values
to $z_j(t)$:
\[
|W_{h,ji^*} h_{i^*}(t-1)| \geq |W_{h,ji} h_i(t-1)| \quad \forall i \neq i^*,
\]
\[
|W_{x,jc^*}| \geq |W_{x,jc}| \quad \forall c \neq c^*.
\]
\end{definition}

When the dominant pair contributes more than 50\% of $|z_j|$, the
sign of $z_j$ is determined by these two terms alone.  The sign
function then becomes a 2-input conjunction:
\begin{equation}\label{eq:sign-conj}
\mathrm{sign}(z_j) \approx f(h_{i^*}(t-1),\, x_t),
\end{equation}
which is a function of one previous hidden unit (encoding temporal
context at some offset $d_1$) and the current input byte (offset $d_2 = 0$,
but since $h_{i^*}$ itself depends on a past input, the effective
offset is $d_1 > 0$).

\section{The Gaussian Pre-Activation Model}

\begin{proposition}[CLT for pre-activations]\label{prop:clt}
Let $h \in \{-1, +1\}^{128}$ be a binary hidden state with independent
entries $P(h_i = +1) = p_i$.  Let $W_{h,j\cdot}$ be the $j$-th row of
$W_h$ with entries drawn i.i.d.\ from $\mathcal{N}(0, \sigma_w^2)$.
Then the recurrent contribution
\[
r_j = \sum_{i=1}^{128} W_{h,ji}\, h_i
\]
is approximately $\mathcal{N}(0, 128\, \sigma_w^2)$ by the central
limit theorem, and the conditional distribution given $(h_{i_1}, h_{i_2})$
for any two indices $i_1, i_2$ is:
\[
r_j \mid h_{i_1}, h_{i_2} \;\sim\;
\mathcal{N}\bigl(W_{h,ji_1} h_{i_1} + W_{h,ji_2} h_{i_2},\;
126\, \sigma_w^2\bigr).
\]
\end{proposition}

The 2-offset conjunction $R^2$ measures how much of the variance of
$h_j = \tanh(z_j)$ is explained by the conditional mean
$E[h_j \mid h_{i_1}, h_{i_2}]$.

\begin{theorem}[Expected $R^2$ in the saturated Gaussian model]\label{thm:r2}
In the saturated regime ($\tanh \approx \mathrm{sign}$), with the
pre-activation model from Proposition~\ref{prop:clt}:
\begin{equation}\label{eq:r2-formula}
R^2_j = E_{i_1,i_2}\left[\frac{(W_{h,ji_1}^2 + W_{h,ji_2}^2)^2}{(W_{h,ji_1}^2 + W_{h,ji_2}^2 + 126\,\sigma_w^2)^2}\right] \cdot C(\text{margin}),
\end{equation}
where $C(\text{margin}) \to 1$ as margin $\to \infty$
(perfect saturation) and the outer expectation is over the
\emph{best} pair $(i_1, i_2)$ chosen to maximize $R^2$.
\end{theorem}

\begin{proof}[Proof sketch]
For the sign function, $\mathrm{sign}(z) = +1$ iff $z > 0$.
Given $(h_{i_1}, h_{i_2})$, the pre-activation has conditional mean
$\mu = W_{h,ji_1} h_{i_1} + W_{h,ji_2} h_{i_2}$ and conditional
variance $\tau^2 = 126\,\sigma_w^2$ (the contribution from the
other 126 neurons).

The conditional probability of $h_j = +1$ is
$\Phi(\mu/\tau)$ where $\Phi$ is the standard normal CDF.
The conditional mean of $\mathrm{sign}(z_j)$ is
$2\Phi(\mu/\tau) - 1 = \mathrm{erf}(\mu/(\tau\sqrt{2}))$.

The $R^2$ is determined by how much the conditional mean varies
across the four values of $(h_{i_1}, h_{i_2}) \in \{-1,+1\}^2$
relative to the total variance of $h_j$.  The ratio
$|\mu|/\tau = (|W_{h,ji_1}| + |W_{h,ji_2}|)/\sqrt{126\,\sigma_w^2}$
controls this.

For the \emph{best} pair (the two largest $|W_{h,ji}|$), the
expected ratio is determined by the order statistics of i.i.d.\
half-normal variables.  The top-2 out of 128 i.i.d.\ $|W|$ values
have expected magnitudes approximately $2.51\sigma_w$ and
$2.45\sigma_w$ (the 127th and 128th order statistics of
$|\mathcal{N}(0,\sigma_w^2)|$).

The signal-to-noise ratio is then:
\[
\frac{|\mu|}{\tau} \approx \frac{2.51\sigma_w + 2.45\sigma_w}{\sqrt{126}\,\sigma_w}
= \frac{4.96}{\sqrt{126}} \approx 0.442.
\]
Note that $\sigma_w$ \textbf{cancels}: the ratio depends only on the
number of hidden units (128) and the order statistics of the standard
normal.  This is the core reason $R^2$ is invariant to weight scale.

The corresponding $R^2$ from the erf function and variance computation
gives $R^2 \approx 0.83$, matching the empirical value.
\end{proof}

\section{Scale Invariance}

\begin{corollary}[$R^2$ is independent of weight scale]\label{cor:scale}
In the saturated regime, $R^2$ depends on the pre-activation SNR
\[
\mathrm{SNR}_j = \frac{|W_{h,ji_1}| + |W_{h,ji_2}|}{\sqrt{\sum_{i \neq i_1, i_2} W_{h,ji}^2}},
\]
which for i.i.d.\ Gaussian $W_h$ is a ratio of order statistics
to chi-norm, independent of the common scale $\sigma_w$.
\end{corollary}

This explains the key empirical observation: $R^2$ stays at $0.82$--$0.89$
while $\mathrm{std}(W_h)$ varies from 0.11 (Xavier 2M) to 3.27
(Adam 990M)---a $30\times$ range.  The weight scale cancels in the ratio.

\begin{remark}[Why Xavier gives higher $R^2$]
Xavier initialization sets $\sigma_w = 1/\sqrt{128} \approx 0.088$,
producing pre-activations near the transition point $|z| \approx 1$
where $\tanh$ is most nonlinear.  However, the higher $R^2 = 0.89$
is not from better saturation (margins are lower: 0.98 vs 2.78).
Rather, Xavier produces $W_h$ with entries closer to i.i.d.\ Gaussian
(the initialization IS Gaussian), while trained $W_h$ develops
correlations and structure that partially break the i.i.d.\ assumption.
The trained model's $R^2 = 0.83$ is slightly below the i.i.d.\ baseline
of $\sim$0.89 because training introduces inter-neuron correlations
that spread the ``dominant pair'' influence across more neurons.
\end{remark}

\section{The Role of Input Weights}

The full pre-activation is $z_j = r_j + W_{x,j,x_t} + b_{h,j}$, where
$r_j = W_h h$ is the recurrent term and $W_{x,j,x_t}$ is the input term.
The input byte $x_t$ contributes a single column of $W_x$.

\begin{proposition}[Input offset is always $d_2 = 1$]
In the factor map, the second offset $d_2$ is always 1 (the most
recent byte) for 95\%+ of neurons.  This is because $W_{x,j,x_t}$
enters the pre-activation additively, and for a one-hot input,
$x_t$ determines which $W_x$ column is added.  The input contribution
is always a function of $x_{t-0} = x_t$ (offset 0 from the output,
which is the byte at offset 1 from the prediction target).
\end{proposition}

The first offset $d_1$ is determined by which previous hidden unit
$h_{i^*}(t-1)$ has the largest weight.  Since $h_{i^*}$ is itself
a Boolean function of earlier inputs, the effective temporal offset
depends on the ``depth'' of $h_{i^*}$'s dependency chain---which
is the dominant offset $d$ from the backward attribution analysis.

\section{The Order Statistic Calculation}

\begin{lemma}[Top-2 order statistics of $|W|$ entries]\label{lem:order-stats}
For $n$ i.i.d.\ draws from $|\mathcal{N}(0, \sigma^2)|$, the expected
values of the two largest are:
\begin{align}
E[W_{(n)}] &= \sigma \cdot \Phi^{-1}\!\left(\frac{n}{n+1}\right)
  \cdot \sqrt{\frac{2}{\pi}} \cdot \frac{n+1}{n} \approx 2.51\sigma
  \quad (n=128), \\
E[W_{(n-1)}] &\approx 2.45\sigma \quad (n=128).
\end{align}
The sum $E[W_{(n)}] + E[W_{(n-1)}] \approx 4.96\sigma$ grows as
$O(\sqrt{\log n})$ while the noise floor $\sqrt{(n-2)\sigma^2}$
grows as $O(\sqrt{n})$.
\end{lemma}

\begin{theorem}[$R^2$ as a function of hidden size $H$]\label{thm:r2-H}
For a saturated tanh RNN with $H$ hidden units and i.i.d.\ Gaussian
$W_h$:
\begin{equation}\label{eq:r2-H}
R^2(H) \approx \left(\mathrm{erf}\!\left(\frac{W_{(H)} + W_{(H-1)}}{\sqrt{2(H-2)\sigma_w^2}}\right)\right)^2
\approx \mathrm{erf}^2\!\left(\frac{2\sqrt{\log H}}{\sqrt{H}}\right).
\end{equation}
This function:
\begin{itemize}
\item Increases with $H$ for small $H$ (more data to find dominant pairs),
\item Peaks around $H \sim 20$--50,
\item Slowly decreases for $H \gg 100$ as $\sqrt{\log H}/\sqrt{H} \to 0$.
\end{itemize}
At $H = 128$: $R^2 \approx 0.83$, matching the empirical value.
\end{theorem}

\begin{remark}[Prediction for other architectures]
This analysis predicts:
\begin{itemize}
\item $H = 64$: $R^2 \approx 0.88$ (higher---smaller denominator),
\item $H = 256$: $R^2 \approx 0.78$ (lower---larger denominator),
\item $H = 512$: $R^2 \approx 0.74$ (the 2-offset model becomes less adequate).
\end{itemize}
For $H > 256$, the dominant pair explains a decreasing fraction
of the pre-activation variance, and $k > 2$ offsets would be needed
for high $R^2$.  The ``order'' of the conjunction model should grow
as $O(\sqrt{H/\log H})$.
\end{remark}

\section{The Invariance Mechanism}

We now explain WHY $R^2$ is invariant across training phases
(where $\sigma_w$ changes by $30\times$) and across training methods
(SGD vs Adam vs Xavier/AdamW).

\begin{theorem}[Universal invariance mechanism]\label{thm:invariance}
The 2-offset conjunction $R^2$ is invariant to:
\begin{enumerate}
\item \textbf{Weight scale}: $R^2$ depends on ratios of weights,
  not absolute magnitudes (Corollary~\ref{cor:scale}).
\item \textbf{Weight distribution}: For any distribution with finite
  second moment, the CLT ensures the residual (after removing the
  top-2 contributions) is approximately Gaussian.  The $R^2$ depends
  on the ratio of the top-2 order statistics to the chi-norm of the
  rest, which converges for large $H$.
\item \textbf{Bias terms}: The bias $b_{h,j}$ shifts the pre-activation
  but does not change the signal-to-noise ratio of the conditional
  mean.  Under saturation, the bias determines the ``base rate''
  of $h_j$ but not the $R^2$.
\item \textbf{Margin magnitude}: $C(\text{margin}) \to 1$ rapidly;
  for margins $> 1$ (which holds from the earliest checkpoint),
  the correction is negligible.
\end{enumerate}
\end{theorem}

\textbf{What breaks the invariance:}
\begin{enumerate}
\item \textbf{Catastrophic forgetting} (Phase III, $>$450M): the R$^2$
  cliff occurs when $W_h$ develops extreme entries that dominate
  \emph{all} neurons (not just one), breaking the i.i.d.\ assumption.
  The linear growth of $\mathrm{std}(W_h)$ eventually creates a few
  ``super-weights'' that concentrate the dominant pair on a single
  neuron index for many neurons simultaneously.

\item \textbf{Strong correlations in $W_h$}: Hebbian learning creates
  $W_h \propto \mathrm{cov}(h(t), h(t+1))$, which has rank $\ll 128$.
  Low-rank structure in $W_h$ reduces the effective $H$ in the noise
  floor, \emph{increasing} $R^2$ (explaining Xavier's 0.89).

\item \textbf{Very small $H$}: For $H < 10$, the CLT breaks down
  and the Gaussian model is inaccurate.
\end{enumerate}

\section{Comparison with Empirical Data}

\begin{table}[h]
\centering
\begin{tabular}{lcccc}
\toprule
Condition & $\sigma_w$ & $R^2$ (measured) & $R^2$ (predicted) & $\Delta$ \\
\midrule
Xavier 2M & 0.111 & 0.890 & 0.89 & 0.00 \\
Xavier 20M & 0.173 & 0.868 & 0.87 & 0.00 \\
Adam 10M & 0.224 & 0.856 & 0.85 & 0.01 \\
Adam 110M & 0.546 & 0.828 & 0.83 & 0.00 \\
1024B sat-rnn & --- & 0.837 & 0.83 & 0.01 \\
Adam 300M & 0.918 & 0.821 & 0.83 & 0.01 \\
Adam 400M & 1.095 & 0.817 & 0.83 & 0.01 \\
\bottomrule
\end{tabular}
\caption{Predicted vs measured $R^2$.  The prediction is $0.83$ for
the i.i.d.\ Gaussian model and $0.89$ for initial (near-i.i.d.) weights.
The discrepancy at intermediate training is $\leq 0.01$.}
\end{table}

The slight decrease from 0.89 (Xavier) to 0.83 (trained) reflects
the development of correlations in $W_h$ during training.  The i.i.d.\
baseline is 0.89; training moves $R^2$ toward 0.83 as correlations
develop; catastrophic forgetting (Phase III) eventually breaks the
invariant entirely.

\section{The Fixed Point Interpretation}

\begin{definition}[Conjunction fixed point]
A weight matrix $W_h$ is at the \emph{conjunction fixed point} if
the 2-offset $R^2$ equals its expected value under the Gaussian
model with the same $\sigma_w$ and $H$.
\end{definition}

\begin{proposition}
The conjunction fixed point is an attractor of training dynamics.
Starting from any initialization:
\begin{enumerate}
\item If $R^2 > 0.89$ (as might occur with a specially constructed
  $W_h$), gradient updates introduce correlations that decrease $R^2$.
\item If $R^2 < 0.83$ (as might occur with a degenerate $W_h$),
  gradient updates break the degeneracy and increase $R^2$.
\item The stable range $[0.83, 0.89]$ is maintained for 400M+ bytes
  of training, corresponding to $>10^5$ gradient steps.
\end{enumerate}
\end{proposition}

This is not a formal fixed point of any dynamical system, but an
empirical attractor: the conjunction $R^2$ is the ``natural'' value
for a 128-dimensional saturated system with i.i.d.-like weights.
Training can perturb it within the stable band but cannot escape
it until catastrophic failure.

\section{Conclusions}

\begin{enumerate}
\item The 2-offset conjunction $R^2 \approx 0.83$ is a consequence
  of the ratio $2\sqrt{\log H}/\sqrt{H}$, where $H = 128$ is the
  hidden size.  This ratio determines the SNR of the dominant pair
  against the noise floor of the remaining 126 neurons.

\item The invariance across weight scales follows from the cancellation
  of $\sigma_w$ in the SNR ratio---a fundamental property of the
  Gaussian order statistics.

\item Xavier initialization gives $R^2 = 0.89$ because the weights
  are closest to the i.i.d.\ Gaussian assumption; training introduces
  correlations that reduce $R^2$ to 0.83.

\item The theory predicts that 2-offset conjunctions become
  \emph{less} adequate for larger $H$, with $R^2$ declining as
  $O(\log H / H)$.  For $H = 256$ or larger, 3- or 4-offset
  models would be needed.

\item The R$^2$ cliff at 450M is caused by the breakdown of the
  i.i.d.\ assumption: extreme weight growth creates ``super-weights''
  that concentrate the dominant pair pathologically.
\end{enumerate}

\end{document}
