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%% SOURCE: docs/working/experiments/02-experiments/james-law/protocol.md
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%% PATENT: Section on gravitational collapse threshold and W_collapse
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\section{James Law — Window Scaling Experiment Protocol}
\label{sec:james-law-protocol}

\subsection{Objective}

The window scaling experiment tests whether the conservation invariant
$\Lambda \times (\mathrm{DoF} + 1) = W$ holds across the full range of
achievable window sizes, and whether a critical window capacity $W^*$ exists
at which the Steady-State Machine undergoes a phase transition.

\subsection{Conservation Invariant}

Empirical validation at $W = 4096$ yielded:

\begin{equation}
  \Lambda(\mathrm{DoF}) \times (\mathrm{DoF} + 1) = 4096.0 \quad
  (\text{CV} = 0.00\%)
  \label{eq:james-law}
\end{equation}

The hypothesis to be tested is that this relationship generalises:

\begin{equation}
  \Lambda(\mathrm{DoF}) = \frac{W}{\mathrm{DoF} + 1}
\end{equation}

for all valid window sizes $W$, and that a critical threshold $W^*$ exists
beyond which the invariant breaks down and the SSM collapses to the ground
state (all loops off, configuration \texttt{0000000}).

%% PATENT: The existence and characterisation of W* — a predictable collapse
%%   threshold for adaptive feedback systems — is patent-adjacent.
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\subsection{Experimental Design}

\paragraph{Phase 1 — Coarse sweep.}
Eight window sizes: 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536~bytes.
Five configurations: ground state (\texttt{0000000}), best static from
$2^7$ DoE (\texttt{0100011}), L8 adaptive choice (\texttt{0110111}), worst
static (\texttt{1111100}), and \texttt{L8\_ADAPTIVE}. 100~replicates per
(window, configuration) pair; $8 \times 5 \times 100 = 4{,}000$ total runs.

\paragraph{Phase 2 — Critical zone refinement.}
A fine sweep of seven window sizes within the interval identified as the
critical zone by Phase~1, at 200~replicates per pair. Objective: locate
$W^*$ within $\pm 1{,}024$~bytes.

\paragraph{Phase 3 — Workload independence validation.}
Three window sizes centred on $W^*$, four workload shapes (baseline, damped
sine, square wave, triangle), five configurations, 100~replicates. Tests
whether $W^*$ is shape-invariant.

\begin{center}
\begin{tabular}{lrrr}
\toprule
Phase & Window sizes & Replicates & Total runs \\
\midrule
1 — Coarse sweep & 8 & 100 & 4{,}000 \\
2 — Fine refinement & 7 & 200 & 7{,}000 \\
3 — Workload validation & 3 & 100 & 6{,}000 \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Key Metrics}

\begin{itemize}
  \item \textbf{Stability score} $= 1/\overline{\mathrm{CV}}$ for each
        window size.
  \item \textbf{$\Lambda$ deviation} $= |\Lambda_{\text{measured}} -
        W/(\mathrm{DoF}+1)|$.
  \item \textbf{Collapse probability} $= P(\text{config} = 0 \mid
        \texttt{L8\_ADAPTIVE})$.
  \item \textbf{Heat density} $= \text{total\_heat} / W$.
  \item \textbf{Variance inflation} $= \mathrm{CV}(W) / \mathrm{CV}(4096)$.
\end{itemize}

\subsection{Collapse Threshold Detection}

Three independent methods identify $W^*$:

\begin{enumerate}
  \item \textbf{Mode selection transition.} Plot $P(\text{config}=0 \mid
        \texttt{L8\_ADAPTIVE})$ versus $W$. The threshold is the window size
        at which this probability crosses 50\%.
  \item \textbf{Variance explosion.} Plot CV versus $W$. The threshold is
        where $\mathrm{d CV}/\mathrm{d W} \to \infty$ (divergence).
  \item \textbf{Conservation breakdown.} Plot $\Lambda \times (\mathrm{DoF}+1)$
        versus $W$. The threshold is where the product departs from the
        conservation value.
\end{enumerate}

Agreement across all three methods constitutes strong evidence for a
genuine phase transition.

\subsection{Success Criteria}

\begin{enumerate}
  \item A reproducible $W^*$ is identified with $\delta W / W^* < 0.2$.
  \item $\Lambda \times (\mathrm{DoF}+1) = W$ holds for all $W < W^*$.
  \item $W^*$ is independent of workload shape (Phase~3 ANOVA $p > 0.05$).
  \item $W^* = k W_0$ for a small integer $k$, where $W_0 = 4096$.
\end{enumerate}

\subsection{Planned Extensions}

\begin{itemize}
  \item \textbf{Two-dimensional phase diagram.} Vary both $W$ and $\mathrm{DoF}$
        simultaneously; plot the stability boundary in ($W$, DoF) space.
  \item \textbf{Hysteresis testing.} Start at $W < W^*$, increase past $W^*$,
        decrease back, and test whether the system recovers --- analogous to
        magnetic hysteresis.
  \item \textbf{Adaptive window sizing.} Allow the VM to adjust $W$ dynamically
        and test whether it self-organises toward $W \approx W_0$.
\end{itemize}

%% TODO(bob): confirm whether Phase 1 was executed and results are in the
%%   archive; cross-reference campaigns/window_scaling/README.md