\documentclass[11pt]{article}
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\newtheorem{observation}{Observation}

\title{Q2--Q4: Offsets, Neurons, and Saturation\\in the Boolean Automaton}
\author{Claude and MJC}
\date{11 February 2026}

\begin{document}
\maketitle

\begin{abstract}
We experimentally answer three of the seven questions from the
total-interpretation program for the sat-rnn. \textbf{Q2 (offsets):}
The RNN uses deep offsets ($d = 18$--25), much deeper than MI-greedy
skip-$k$-grams ($d = 1, 3, 8, 20$), which capture only 9.4\% of the
sign-change signal. \textbf{Q3 (neurons):} One neuron (h28) captures
99.7\% of the compression; the top 15 neurons exceed 100\% (102.0\%---the
rest add noise). No individual neuron's mantissa contributes more than
0.002~bpc. \textbf{Q4 (saturation):} All 128 neurons are volatile ($>$50
sign flips in 520 positions), with mean dwell time 3.3 steps. The
``112 settled + 16 active'' picture from a single time point is a snapshot,
not the dynamics. Strong co-flip pairs (Jaccard~$>$~0.5) reveal coupled
neuron groups.
\end{abstract}

\section{Q2: The RNN Uses Deep Offsets}

\subsection{Method}

For each test position $t$ and depth $d = 1, \ldots, 30$: flip the input
byte at $t - d$ (XOR with 128), re-run the RNN from $t - d$ forward,
measure sign changes and output KL at position $t$. Average over 13 test
positions spread across the data.

\subsection{Results}

\begin{center}
\begin{tabular}{rrr}
  \toprule
  Depth $d$ & Mean sign changes & Mean output KL (bits) \\
  \midrule
  1 & 8.1 & 0.296 \\
  3 & 11.8 & 0.298 \\
  7 & 20.2 & 0.404 \\
  8 & 20.5 & 0.458 \\
  10 & 19.8 & 0.427 \\
  15 & 36.6 & 0.543 \\
  20 & 40.0 & 0.592 \\
  25 & 49.5 & 0.809 \\
  30 & 45.4 & 0.704 \\
  \bottomrule
\end{tabular}
\end{center}

Both sign changes and output KL increase monotonically with depth (with
some noise). Input perturbations from 25 steps back cause more sign
changes and larger prediction shifts than perturbations from 1 step back.

\begin{observation}[The RNN uses deeper memory than skip-$k$-grams]
The MI-greedy offsets $[1, 3, 8, 20]$ capture only 9.4\% of the total
sign-change signal. The dominant neuron offset is $d = 25$ (30 out of
128 neurons, or 23.4\%). The RNN maintains information about inputs
20--30 steps in the past, consistent with BPTT-50 training.
\end{observation}

\begin{observation}[Depth profile is monotonically increasing]
This reflects the ergodic, non-contractive Boolean dynamics. A
perturbation at depth $d$ propagates through $d$ steps of the
Boolean transition function. Since the dynamics is mixing (no
attractors), the perturbation grows rather than decays. After
$d \sim 25$ steps, the perturbed trajectory has diverged enough
to affect nearly half the neurons (49.5 out of 128).
\end{observation}

\subsection{Comparison to factor-map offsets}

The factor-map analysis (20260209) found that 52/128 neurons are
dominated by the offset pair $(1, 7)$. Here, offsets 1 and 7 together
account for only 3.3\% of the sign-change signal. This apparent
contradiction is resolved by noting that the factor-map measured
\emph{readout} sensitivity (how $h_j$ at a single time step depends
on recent inputs), while we measure \emph{dynamical} sensitivity (how
the full trajectory changes when a past input is perturbed). The
readout depends on recent inputs; the dynamics integrates over long
history.

\section{Q3: Which Neurons Carry the Signal?}

\subsection{Method}

For each neuron $j$: zero out the $j$-th column of $W_y$ and measure
the bpc change over all positions. This is a ``readout knockout''---the
neuron still participates in dynamics but cannot contribute to prediction.

\subsection{Results}

\begin{center}
\begin{tabular}{lrrr}
  \toprule
  Neuron & $\Delta$bpc & $\|W_y^{(\cdot,j)}\|$ & Mean $|h_j|$ \\
  \midrule
  h28 & $+0.030$ & 84.8 & 0.987 \\
  h105 & $+0.025$ & 30.1 & 0.985 \\
  h54 & $+0.023$ & 66.6 & 0.990 \\
  h17 & $+0.021$ & 49.2 & 0.995 \\
  h49 & $+0.020$ & 55.2 & 0.991 \\
  h10 & $+0.019$ & 51.6 & 0.987 \\
  h97 & $+0.018$ & 81.1 & 0.983 \\
  h3 & $+0.018$ & 2.5 & 0.991 \\
  \bottomrule
\end{tabular}
\end{center}

\begin{observation}[Neuron h3 is important despite tiny $W_y$]
h3 has $\|W_y\| = 2.5$---the smallest in the top 10 by a factor of 10.
Yet it costs $+0.018$~bpc to knock out. Its importance comes entirely
from dynamics: through $W_h$, it influences other neurons' signs, which
then contribute to prediction.
\end{observation}

\subsection{Minimal subset analysis}

We keep only the top-$k$ neurons (by knockout importance) and zero the
rest of $W_y$:

\begin{center}
\begin{tabular}{rrr}
  \toprule
  Neurons kept $k$ & bpc & \% of compression gap \\
  \midrule
  1 & 4.974 & 99.7\% \\
  6 & 4.966 & 100.0\% \\
  10 & 4.948 & 100.5\% \\
  15 & 4.903 & 102.0\% \\
  20 & 4.882 & 102.7\% \\
  30 & 4.857 & 103.6\% \\
  128 (full) & 4.965 & 100.0\% \\
  \bottomrule
\end{tabular}
\end{center}

\begin{observation}[The full model is suboptimal for readout]
Keeping 15--30 neurons achieves \emph{better} bpc than all 128. The
remaining 98--113 neurons contribute negative signal through $W_y$---they
add noise to the prediction. The model is over-parameterized for this
data, and the extra neurons' $W_y$ contributions interfere with the
signal from the top neurons.
\end{observation}

\begin{observation}[No neuron's mantissa matters for readout]
Replacing any single neuron $h_j$ with $\mathrm{sgn}(h_j)$ for readout
changes bpc by at most 0.002. The readout is entirely sign-based.
\end{observation}

\subsection{What do neurons predict?}

Top neurons' $W_y$ weight patterns:
\begin{itemize}
  \item \textbf{h28}: promotes `q', `h', `f', `r'; demotes `*', `\#', `/', `['
  \item \textbf{h54}: promotes `,', `]', `e', `h'; demotes `f', `9', `n'
  \item \textbf{h97}: promotes `6', `q', `9', `e'; demotes `*', `\#', `\{'
\end{itemize}

The pattern is clear: neurons encode context-dependent character
probabilities. h28 promotes letters and demotes symbols; h97 promotes
digits/letters in one sign and symbols in the other.

\section{Q4: All Neurons Are Volatile}

\subsection{Method}

Track each neuron's sign across all 520 positions. Count sign flips,
measure dwell times (steps between consecutive flips), and identify
co-flip pairs.

\subsection{Results}

\begin{center}
\begin{tabular}{lrrrrr}
  \toprule
  Neuron & Flips & Mean $|h|$ & Min $|h|$ & \% sat & Dwell mode \\
  \midrule
  h54 & 234 & 0.990 & 0.049 & 92.3\% & 1 \\
  h37 & 206 & 0.986 & 0.089 & 91.0\% & 1 \\
  h47 & 201 & 0.982 & 0.035 & 90.2\% & 1 \\
  h110 & 197 & 0.990 & 0.191 & 93.1\% & 1 \\
  h3 & 161 & 0.991 & 0.004 & 95.8\% & 1 \\
  h75 & 153 & 0.996 & 0.508 & 95.6\% & 4 \\
  h40 & 150 & 0.996 & 0.542 & 97.3\% & 4 \\
  \bottomrule
\end{tabular}
\end{center}

\begin{observation}[All 128 neurons are volatile]
Every neuron flips sign more than 50 times in 520 positions. There are
zero frozen neurons, zero settled neurons. The least volatile neuron
still flips $\sim$100 times. This contradicts the static picture of
``112 settled + 16 active'' neurons.
\end{observation}

The resolution: at any \emph{single} time step, $\sim$123 neurons are
saturated ($|h| > 0.999$) and $\sim$5 are unsaturated. But the
\emph{identity} of the unsaturated neurons changes every step. Over
520 positions, every neuron passes through the unsaturated regime
many times, flipping its sign in the process.

\begin{observation}[Mean dwell time is 3.3 steps]
The dwell time distribution peaks at $d = 4$ (3,015 occurrences) and
drops off rapidly. 95\% of dwells are $\leq 10$ steps. The Boolean
state is rapidly mixing: a neuron that flips will flip again within
a few steps.
\end{observation}

\subsection{Co-flip structure}

Neurons that flip at the same position form a co-flip graph:

\begin{center}
\begin{tabular}{lrrr}
  \toprule
  Pair & Co-flips & Jaccard & Individual flips \\
  \midrule
  h17, h109 & 100 & 0.510 & 148, 148 \\
  h37, h54 & 100 & 0.294 & 206, 234 \\
  h30, h31 & 98 & 0.508 & 144, 147 \\
  h40, h46 & 98 & 0.476 & 150, 154 \\
  h46, h116 & 98 & 0.490 & 154, 144 \\
  h1, h58 & 96 & 0.508 & 141, 144 \\
  h30, h34 & 95 & 0.487 & 144, 146 \\
  h36, h86 & 93 & 0.449 & 145, 155 \\
  \bottomrule
\end{tabular}
\end{center}

\begin{observation}[Tightly coupled neuron groups]
Several pairs have Jaccard similarity $> 0.5$---they flip together more
often than they flip apart. Notable clusters:
\begin{itemize}
  \item h30, h31, h34: a triple with pairwise Jaccard $\sim$0.5.
  \item h40, h46, h116: a triple with pairwise co-flips $\sim$98.
  \item h17, h109: the tightest pair (Jaccard 0.51).
  \item h37, h54, h47, h110, h52: a loose cluster around h54
    (the most volatile neuron).
\end{itemize}
\end{observation}

These co-flip groups likely correspond to feature detectors: a context
change (e.g., entering/leaving an XML tag) causes a coordinated sign
flip across a group of neurons that encode the same feature.

\section{Synthesis}

The three questions reveal a consistent picture of the sat-rnn:

\paragraph{Q2:} The RNN uses deeper memory ($d = 18$--25) than
skip-$k$-gram analysis predicts ($d = 1$--20). The depth comes from
the ergodic Boolean dynamics, which propagates perturbations rather
than damping them.

\paragraph{Q3:} Prediction is extremely concentrated. One neuron (h28)
captures 99.7\% of the compression. The top 15 suffice for 102\%. The
remaining 113 neurons \emph{hurt} prediction through their $W_y$
contributions. The model is over-parameterized: 128 readout neurons
for a task that needs 15.

\paragraph{Q4:} All 128 neurons are volatile, flipping every $\sim$3
steps. The dynamic is fully mixing---no frozen features, no permanent
attractors. Co-flip structure reveals neuron groups that encode
shared features.

\paragraph{The Boolean automaton is well-described.} The sat-rnn is a
128-bit Boolean automaton where:
\begin{itemize}
  \item Every neuron participates in every prediction (through W$_h$).
  \item Only $\sim$15 neurons matter for readout (through W$_y$).
  \item All neurons flip frequently (dwell time $\sim$3 steps).
  \item The dynamics propagates information over 20--30 steps.
  \item Co-flip groups encode shared contextual features.
\end{itemize}

\end{document}
