%% SCRAP: experiments/campaigns/window_scaling/README
%% SOURCE: docs/working/experiments/campaigns/window_scaling/README.md
%% STATUS: CURRENT
%% FITS: experiments/ch-window-scaling
%% EDITORIAL: lifted — prose rewritten to press voice
\section{Window Scaling Experiment: James Law Validation}
\textbf{Objective:} Empirically validate James Law:
\[
\Lambda = \frac{W}{\mathrm{DoF} + 1}
\]
where $\Lambda$ is the effective smoothing capacity per degree of freedom,
$W$ is the rolling window size in bytes, and $\mathrm{DoF}$ is the number
of active feedback loops (0--7).
%% PATENT: James Law ($\Lambda = W / (\text{DoF}+1)$) and the window-scaling
%% invariant are patent-adjacent. Do not draft claims here.
\subsection{Hypothesis}
The quantity $K = \Lambda \times (\mathrm{DoF}+1) / W$ should remain
approximately constant across multiple window sizes, multiple degrees of
freedom, and diverse workload patterns. $K \approx 1.0$ across all
conditions validates James Law as a fundamental scaling relationship in
adaptive computational systems.
\subsection{Experimental Design}
\begin{center}
\begin{tabular}{lll}
\toprule
Variable & Levels & Values \\
\midrule
DoF & 8 & 0--7 \\
Window size & 12 & 512, 1{,}024, 1{,}536, 2{,}048, 3{,}072, 4{,}096, \\
& & 6{,}144, 8{,}192, 16{,}384, 32{,}769, 52{,}153, 65{,}536 \\
Replicate & 30 & 1--30 \\
\midrule
\textbf{Total runs} & & $8 \times 12 \times 30 = 2{,}880$ \\
\bottomrule
\end{tabular}
\end{center}
Fixed workload: \texttt{init-l8-omni.4th} (the mega-workload combining all
six L8 workload patterns). Run order is shuffled to eliminate temporal bias.
Primary metrics: \texttt{ns\_per\_word}, CV. The $K$ statistic is derived
as $K = \texttt{win\_final\_bytes} / (\mathrm{DoF}+1) / W$.
\subsection{Validation Criteria}
\begin{itemize}
\item $\mathrm{Mean}(K) \in [0.95,\,1.05]$
\item $\mathrm{Std}(K) < 0.1$
\item $\max|K - 1.0| < 0.1$
\end{itemize}
\subsection{Expected Outcomes}
\begin{description}
\item[Scenario A — Law holds.] $K \approx 1.0$ across all conditions.
James Law validated as a universal scaling relationship. Implication:
system behavior is predictable and governed by geometric invariants.
\item[Scenario B — Critical threshold exists.] Law holds for
$W < W_\text{critical}$, then degrades. A phase transition is
present (``gravitational collapse''). The prior hypothesis places
$W_\text{critical} \approx 16{,}384$ bytes ($4 \times W_0$).
\item[Scenario C — DoF-dependent scaling.] $K$ varies systematically with
DoF but not randomly. A more complex relationship (logarithmic or
power-law) governs the system.
\end{description}
\subsection{Running the Experiment}
\begin{lstlisting}[language=bash]
# Step 1: Generate run matrix
cd scripts/
./generate_run_matrix.R
# Step 2: Pre-build all 96 configurations (~1.5 hours)
./prebuild_all_configs.sh
# Step 3: Execute 2,880 runs (~4-5 hours)
./run_window_sweep.sh
# Step 4: Analyze
./analyze_results.R
\end{lstlisting}
\subsection{Analysis Plan}
The R analysis script computes:
\begin{enumerate}
\item $K$ for all runs and its distribution by condition (DoF $\times$ window)
\item ANOVA to assess significance of DoF and window size effects on $K$
\item The critical window $W_\text{critical}$ where CV exceeds a threshold
(elbow detection)
\item Five plots: $K$ vs.\ DoF (faceted by $W$), $K$ vs.\ $W$ (faceted
by DoF), CV vs.\ $W$ (phase transition), 3D stability surface,
and $K$-deviation heatmap
\end{enumerate}
\subsection{Baseline Validation}
From prior DoE experiments at the reference window $W_0 = 4{,}096$:
\[
\Lambda(\mathrm{DoF}) \times (\mathrm{DoF}+1) = 4{,}096.0 \pm 0.0
\quad \text{(CV = 0.00\%)}
\]
This experiment extends this result to arbitrary window sizes.
