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\usepackage{booktabs}
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\newtheorem{theorem}{Theorem}
\newtheorem{finding}{Finding}

\title{Checkpoint Trajectory: The Boolean Automaton\\Strengthens, the Factor Map Collapses}
\author{Claude and MJC}
\date{February 11, 2026}

\begin{document}
\maketitle

\begin{abstract}
We sweep 99 checkpoints (10M--990M) of the enwik9 training run,
measuring margins, R$^2$, Hebbian correlation, W$_h$ drift, and
output bias range at each point.  Three phase transitions emerge:
(1)~margins grow monotonically from 2.8 to 61.3---the model is
\emph{always} Boolean and gets more so; (2)~the factor map R$^2$
holds at 0.80--0.86 for 400M bytes, then collapses to 0.40--0.70
at the same point as catastrophic forgetting; (3)~the output bias
$b_y$ crosses all-negative at 640M, a sharp phase transition.
The W$_h$ standard deviation grows linearly at 0.031 per checkpoint,
never stabilizing.  An anomalous 100M checkpoint reveals that
it belongs to Run~1 (SGD, exploded), not Run~2 (Adam).
\end{abstract}

\section{Experimental Setup}

We compiled \texttt{trajectory.c}, a tool that computes nine metrics
from a model checkpoint and a fixed evaluation window: mean margin,
fraction of small margins ($|z|<1$), bpc, W$_h$ std, $b_y$ range
(min, max), mean 2-offset conjunction R$^2$, count of neurons with
R$^2 \geq 0.80$, and Hebbian $r(\text{cov}, W_h)$.

The evaluation window is the first 1024 bytes of enwik9 (the same
data used for the 1024-byte sat model).  All 99 checkpoints from
\texttt{epoch1\_10M.bin} through \texttt{epoch1\_990M.bin} were
swept, plus the original \texttt{sat\_model.bin} as baseline.

\section{Margin Trajectory}

\begin{finding}[The model is always Boolean and gets more so]
Mean margin grows monotonically: 2.78 (10M) $\to$ 8.24 (110M)
$\to$ 21.3 (450M) $\to$ 61.3 (990M).  The fraction of small
margins ($|z|<1$) drops from 24.3\% to 1.0\%.
\end{finding}

\begin{table}[h]
\centering
\small
\begin{tabular}{lcccccc}
\toprule
Checkpoint & 10M & 110M & 300M & 450M & 700M & 990M \\
\midrule
Mean margin & 2.78 & 8.24 & 14.2 & 21.3 & 47.3 & 61.3 \\
Small ($<1$) & 24.3\% & 7.6\% & 4.3\% & 3.0\% & 1.2\% & 1.0\% \\
\bottomrule
\end{tabular}
\caption{Margin trajectory.  The Boolean regime strengthens
monotonically.  Even the earliest checkpoint (10M) has 75.7\%
of margins above 1.}
\end{table}

\textbf{Implication.}  There is no analog regime at any point during
training.  The tanh RNN is Boolean from the very first checkpoint.
Adam with gradient clipping drives margins higher over time: the
optimizer grows weights (W$_h$ std 0.22 $\to$ 3.27) without bound,
and gradient clipping prevents explosion but not growth.

\section{R$^2$ Trajectory: The 450M Cliff}

\begin{finding}[R$^2$ collapses at the catastrophic forgetting point]
The 2-offset conjunction R$^2$ is stable at 0.80--0.86 for the first
400M bytes of training, then drops sharply at 450--460M to 0.40--0.70.
This coincides exactly with the onset of catastrophic forgetting.
\end{finding}

\begin{table}[h]
\centering
\small
\begin{tabular}{lcccccccc}
\toprule
Checkpoint & 10M & 110M & 200M & 300M & 400M & 450M & 460M & 500M \\
\midrule
R$^2$ & 0.856 & 0.828 & 0.837 & 0.821 & 0.817 & 0.796 & \textbf{0.717} & 0.705 \\
$\geq$0.80 & 126 & 94 & 107 & 85 & 73 & 53 & \textbf{8} & 0 \\
\bottomrule
\end{tabular}
\caption{The R$^2$ cliff.  At 450M, the number of neurons with
R$^2 \geq 0.80$ drops from 53 to 8, and mean R$^2$ falls below 0.80.}
\end{table}

\textbf{Before 450M:}  R$^2$ oscillates in a narrow band (0.80--0.86),
essentially constant.  The 2-offset conjunction structure is robust
to 400M bytes of online learning.  128/128 neurons maintain the
conjunction motif even as the model processes 400$\times$ more data
than the 1024-byte model.

\textbf{After 450M:}  R$^2$ drops to 0.40--0.70 and becomes volatile,
with occasional spikes (840M: 0.76, 920M: 0.63).  The conjunction
structure has fragmented: some neurons retain it, but the mean
collapses.  This is not a clean phase transition but a gradual
dissolution, with transient re-organizations.

\textbf{Coincidence with training:}  The training log records
catastrophic forgetting beginning at $\sim$150M (bpc starts
rising from 2.81 to 3.0) and a cliff at 450M (bpc jumps to 4.5).
The R$^2$ cliff aligns precisely with the 450M training cliff.
The interpretation is causal: as the model forgets the data statistics
that define the conjunctions, the conjunctions degrade.  R$^2$
is a proxy for how well the model's internal representation
matches the local data structure.

\section{W$_h$ Drift: Perfectly Linear}

\begin{finding}[W$_h$ std grows linearly and never stabilizes]
$\text{std}(W_h) = 0.22 + 0.031 \times (\text{checkpoint} - 10\text{M})/10\text{M}$.
The growth is perfectly linear ($R^2 > 0.99$, excluding the
anomalous 100M checkpoint).
\end{finding}

At 10M: std = 0.224.  At 990M: std = 3.273.  The growth rate is
$\Delta\text{std} \approx 0.031$ per 10M-byte checkpoint, or
$3.1 \times 10^{-9}$ per byte.  This rate never changes---there
is no sign of convergence at any point.

\textbf{Implication.}  The model's weights grow without bound under
Adam.  Each new byte of data shifts the weights slightly, and Adam's
per-parameter learning rates prevent the shifts from canceling.
The weight growth drives margin growth (larger $\|W_h\|$ means larger
pre-activations), which drives deeper saturation, which makes the
Boolean regime stronger.  This is a positive feedback loop.

\section{Output Bias Collapse: The 640M Phase Transition}

\begin{finding}[All output biases become negative at 640M]
$b_y^{\max}$ crosses zero between 630M ($+0.66$) and 640M ($-0.37$).
After 640M, all 256 output biases are negative, reaching
$[-46.3, -29.9]$ at 990M.
\end{finding}

\begin{table}[h]
\centering
\small
\begin{tabular}{lccccccc}
\toprule
Checkpoint & 110M & 300M & 450M & 630M & 640M & 700M & 990M \\
\midrule
$b_y^{\min}$ & $-9.0$ & $-11.9$ & $-14.6$ & $-21.1$ & $-21.4$ & $-24.1$ & $-46.3$ \\
$b_y^{\max}$ & $+3.4$ & $+7.1$ & $+8.2$ & $+0.7$ & $\mathbf{-0.4}$ & $-5.4$ & $-29.9$ \\
\bottomrule
\end{tabular}
\caption{Output bias range.  At 640M, $b_y^{\max}$ crosses zero.
After this point, the model's default prediction is ``no character
is likely.''}
\end{table}

\textbf{Mechanism.}  With all $b_y < 0$, the softmax baseline
suppresses all predictions.  The model can only predict well when
$W_y h$ generates large positive logits for the correct character.
As $b_y$ grows more negative, the model must work harder to overcome
the suppressive baseline.  The bpc on our eval window stays at
5.3--5.7 despite the collapse, because the model's $W_y h$
contributions are large enough (margins $\sim$50) to compensate.

\textbf{Trajectory of $b_y^{\max}$:}  The growth of $b_y^{\max}$
actually reverses at 450M.  From 10M--450M, $b_y^{\max}$ grows
(0.79 $\to$ 8.18).  From 450M--640M, it collapses (8.18 $\to$ $-$0.37).
The reversal point (450M) again coincides with the R$^2$ cliff
and the training bpc cliff.  After 640M, both min and max grow
negative at roughly equal rates.

\section{The Anomalous 100M Checkpoint}

\begin{finding}[epoch1\_100M.bin is from Run 1 (SGD), not Run 2 (Adam)]
The 100M checkpoint has margin 59.7, W$_h$ std 3.30, R$^2$ = 0.49,
and $b_y$ range $[-47.4, -30.9]$.  These values are 7$\times$ higher
than the surrounding checkpoints (90M and 110M) and match the late-training
profile.
\end{finding}

The trajectory at 90M--110M should be smooth (both are from Run~2):
$\text{std}(W_h)$ goes 0.50 (90M) $\to$ 0.55 (110M).  But the 100M
checkpoint has std 3.30, matching the 990M value.  This checkpoint is
clearly from Run~1 (SGD, which exploded at $\sim$125M according to
training notes).  The Run~1 model at 100M had already undergone
the same kind of weight explosion that Run~2 doesn't reach until 990M.

\section{Phase Summary}

Three phases of training emerge:

\begin{table}[h]
\centering
\small
\begin{tabular}{lcccc}
\toprule
Phase & Bytes & Margin & R$^2$ & $b_y^{\max}$ \\
\midrule
I: Learning & 10--110M & 2.8--8.2 & 0.83--0.86 & growing \\
II: Stable & 110--400M & 8.2--17.4 & 0.80--0.84 & growing \\
III: Collapse & 450--990M & 21--61 & 0.40--0.76 & all negative \\
\bottomrule
\end{tabular}
\caption{Three phases of training.  Phase I learns; Phase II
maintains quality while growing margins; Phase III forgets.}
\end{table}

\textbf{Phase I (10--110M):}  The model learns.  Margins grow from
2.8 to 8.2 as the optimizer pushes weights.  R$^2$ is already high
(0.83--0.86)---the conjunction structure forms immediately.  bpc
improves on the rolling training average (from 5.9 to 2.81).

\textbf{Phase II (110--400M):}  The model maintains quality.  R$^2$
stays in $[0.80, 0.84]$.  Margins continue growing (8.2 $\to$ 17.4)
but the conjunction structure is preserved.  $b_y^{\max}$ grows
(3.4 $\to$ 7.5), indicating the model is still allocating capacity
to useful predictions.

\textbf{Phase III (450--990M):}  Catastrophic collapse.  R$^2$ drops
to 0.40--0.76.  $b_y^{\max}$ reverses and crosses zero at 640M.
Margins continue growing (21 $\to$ 61)---the model becomes
\emph{more} Boolean even as it forgets.  The forgetting is in
the readout ($W_y, b_y$), not in the dynamics ($W_h$, margins).

\section{Xavier/AdamW Comparison}

We reproduced the Xavier/AdamW training run (seed=42, cosine LR,
label smoothing $\varepsilon=0.10$, gradient accumulation $4\times$,
20M chars) and swept all 10 checkpoints (2M--20M).  This run differs
from Run~2 in initialization (Xavier vs random) and optimizer details
(label smoothing, gradient accumulation, cosine schedule).

\begin{table}[h]
\centering
\small
\begin{tabular}{lccccccc}
\toprule
& \multicolumn{3}{c}{Xavier/AdamW} & & \multicolumn{3}{c}{Run 2 (Adam)} \\
\cmidrule{2-4} \cmidrule{6-8}
Metric & 2M & 10M & 20M & & 10M & 20M & 110M \\
\midrule
Margin & 0.98 & 1.76 & 1.93 & & 2.78 & 3.67 & 8.24 \\
R$^2$ & \textbf{0.890} & \textbf{0.872} & \textbf{0.868} & & 0.856 & 0.836 & 0.828 \\
$n \geq 0.80$ & 128 & 128 & 127 & & 126 & 107 & 94 \\
W$_h$ std & 0.111 & 0.160 & 0.173 & & 0.224 & 0.277 & 0.546 \\
Hebb.\ $r$ & 0.428 & 0.441 & 0.425 & & 0.421 & 0.414 & 0.379 \\
bpc (eval) & 4.72 & 4.72 & 4.48 & & 5.94 & 6.77 & 6.42 \\
\bottomrule
\end{tabular}
\caption{Xavier vs Run~2 at comparable data exposure.  Xavier has
\emph{higher} R$^2$, \emph{lower} margins, and \emph{better} bpc
at every comparable checkpoint.}
\end{table}

\begin{finding}[Xavier init produces higher R$^2$ than random init]
Mean R$^2 = 0.87$--$0.89$ across all Xavier checkpoints vs
$0.83$--$0.86$ for Run~2 at the same data exposure.  All 128 neurons
stay above 0.80 throughout training (vs 94--126 for Run~2).
The conjunction structure is even \emph{cleaner} with Xavier init.
\end{finding}

\textbf{Three contrasts:}

\textit{Margin growth.}  Xavier margins grow 2$\times$ slower
(0.98 $\to$ 1.93 over 20M vs 2.78 $\to$ 3.67 for Adam over 10M--20M).
Xavier's controlled initialization and gradient accumulation produce
a less aggressive weight trajectory.  W$_h$ std grows at
$\sim$0.0034/\text{M} (Xavier) vs $\sim$0.028/\text{M} (Adam)---an
8$\times$ difference.

\textit{BPC.}  Xavier achieves 4.48 bpc at 20M vs 6.77 for Run~2
at the same data exposure.  This is not about the model being ``better''
in any generalizable sense---at 20M, neither model has converged.
Rather, Xavier's initialization places the model closer to a useful
operating point, so each byte of data is used more efficiently.

\textit{Hebbian correlation.}  Both runs maintain Hebbian $r \approx 0.43$
throughout early training, suggesting the weight--data correlation
is also an architectural property independent of initialization.

\textbf{Implication.}  The R$^2$ invariant is even stronger than the
Run~2 trajectory suggested.  Xavier initialization, which deliberately
sets $\text{std}(W) = 1/\sqrt{n_\text{in}}$, produces a cleaner
conjunction structure than random initialization.  This reinforces
the conclusion that 2-offset conjunctions are a property of the
128-hidden tanh RNN architecture, not of any particular training run.

\section{Conclusions}

\begin{enumerate}
\item \textbf{The Boolean regime is inevitable.}  From the first
  checkpoint, the model is Boolean and never stops being so.  This
  is an architectural property of the 128-hidden tanh RNN.

\item \textbf{R$^2 \approx 0.83$ is a sweet-spot invariant.}
  During the ``healthy'' training phase (10--400M), R$^2$ stays
  in $[0.80, 0.86]$, matching the 1024-byte sat-rnn (0.837).
  The conjunction structure is a fixed point of the architecture.

\item \textbf{Catastrophic forgetting destroys the readout, not the
  dynamics.}  Margins grow through the collapse; the Boolean automaton
  strengthens.  What fails is the $W_y$/$b_y$ readout: the model
  can no longer decode its own internal representation.

\item \textbf{W$_h$ drift is the root cause.}  The linear,
  unbounded growth of $\text{std}(W_h)$ drives both the strengthening
  of the Boolean regime (larger margins) and the eventual failure
  of the readout (the $W_y$ mapping becomes miscalibrated as
  $W_h$ drifts).  A training recipe that stabilizes $W_h$
  (weight decay, normalization) would likely prevent the collapse.

\item \textbf{The 100M anomaly.}  One checkpoint
  (\texttt{epoch1\_100M.bin}) belongs to Run~1 (SGD).  Its profile
  (margin 59.7, std 3.30, all-negative $b_y$) matches the late
  Phase~III of Run~2, confirming that SGD without clipping reaches
  the same endpoint faster.
\end{enumerate}

\end{document}
