130 lines
5.0 KiB
TeX
130 lines
5.0 KiB
TeX
%% SCRAP: experiments/02-experiments/james-law/protocol
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%% SOURCE: docs/working/experiments/02-experiments/james-law/protocol.md
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%% STATUS: WORKING
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%% FITS: experiments/ch-james-law, vol3-research/ch-james-law
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%% EDITORIAL: lifted — prose rewritten to press voice
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%% PATENT: Section on gravitational collapse threshold and W_collapse
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%% may constitute patentable subject matter. Do NOT expand claim
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%% language here. Flag for patent counsel review.
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\section{James Law — Window Scaling Experiment Protocol}
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\label{sec:james-law-protocol}
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\subsection{Objective}
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The window scaling experiment tests whether the conservation invariant
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$\Lambda \times (\mathrm{DoF} + 1) = W$ holds across the full range of
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achievable window sizes, and whether a critical window capacity $W^*$ exists
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at which the Steady-State Machine undergoes a phase transition.
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\subsection{Conservation Invariant}
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Empirical validation at $W = 4096$ yielded:
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\begin{equation}
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\Lambda(\mathrm{DoF}) \times (\mathrm{DoF} + 1) = 4096.0 \quad
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(\text{CV} = 0.00\%)
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\label{eq:james-law}
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\end{equation}
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The hypothesis to be tested is that this relationship generalises:
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\begin{equation}
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\Lambda(\mathrm{DoF}) = \frac{W}{\mathrm{DoF} + 1}
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\end{equation}
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for all valid window sizes $W$, and that a critical threshold $W^*$ exists
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beyond which the invariant breaks down and the SSM collapses to the ground
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state (all loops off, configuration \texttt{0000000}).
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%% PATENT: The existence and characterisation of W* — a predictable collapse
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%% threshold for adaptive feedback systems — is patent-adjacent.
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%% Do not draft claim language here. Flag for counsel.
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\subsection{Experimental Design}
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\paragraph{Phase 1 — Coarse sweep.}
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Eight window sizes: 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536~bytes.
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Five configurations: ground state (\texttt{0000000}), best static from
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$2^7$ DoE (\texttt{0100011}), L8 adaptive choice (\texttt{0110111}), worst
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static (\texttt{1111100}), and \texttt{L8\_ADAPTIVE}. 100~replicates per
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(window, configuration) pair; $8 \times 5 \times 100 = 4{,}000$ total runs.
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\paragraph{Phase 2 — Critical zone refinement.}
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A fine sweep of seven window sizes within the interval identified as the
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critical zone by Phase~1, at 200~replicates per pair. Objective: locate
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$W^*$ within $\pm 1{,}024$~bytes.
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\paragraph{Phase 3 — Workload independence validation.}
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Three window sizes centred on $W^*$, four workload shapes (baseline, damped
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sine, square wave, triangle), five configurations, 100~replicates. Tests
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whether $W^*$ is shape-invariant.
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\begin{center}
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\begin{tabular}{lrrr}
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\toprule
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Phase & Window sizes & Replicates & Total runs \\
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\midrule
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1 — Coarse sweep & 8 & 100 & 4{,}000 \\
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2 — Fine refinement & 7 & 200 & 7{,}000 \\
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3 — Workload validation & 3 & 100 & 6{,}000 \\
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\bottomrule
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\end{tabular}
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\end{center}
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\subsection{Key Metrics}
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\begin{itemize}
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\item \textbf{Stability score} $= 1/\overline{\mathrm{CV}}$ for each
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window size.
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\item \textbf{$\Lambda$ deviation} $= |\Lambda_{\text{measured}} -
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W/(\mathrm{DoF}+1)|$.
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\item \textbf{Collapse probability} $= P(\text{config} = 0 \mid
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\texttt{L8\_ADAPTIVE})$.
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\item \textbf{Heat density} $= \text{total\_heat} / W$.
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\item \textbf{Variance inflation} $= \mathrm{CV}(W) / \mathrm{CV}(4096)$.
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\end{itemize}
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\subsection{Collapse Threshold Detection}
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Three independent methods identify $W^*$:
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\begin{enumerate}
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\item \textbf{Mode selection transition.} Plot $P(\text{config}=0 \mid
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\texttt{L8\_ADAPTIVE})$ versus $W$. The threshold is the window size
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at which this probability crosses 50\%.
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\item \textbf{Variance explosion.} Plot CV versus $W$. The threshold is
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where $\mathrm{d CV}/\mathrm{d W} \to \infty$ (divergence).
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\item \textbf{Conservation breakdown.} Plot $\Lambda \times (\mathrm{DoF}+1)$
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versus $W$. The threshold is where the product departs from the
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conservation value.
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\end{enumerate}
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Agreement across all three methods constitutes strong evidence for a
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genuine phase transition.
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\subsection{Success Criteria}
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\begin{enumerate}
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\item A reproducible $W^*$ is identified with $\delta W / W^* < 0.2$.
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\item $\Lambda \times (\mathrm{DoF}+1) = W$ holds for all $W < W^*$.
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\item $W^*$ is independent of workload shape (Phase~3 ANOVA $p > 0.05$).
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\item $W^* = k W_0$ for a small integer $k$, where $W_0 = 4096$.
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\end{enumerate}
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\subsection{Planned Extensions}
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\begin{itemize}
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\item \textbf{Two-dimensional phase diagram.} Vary both $W$ and $\mathrm{DoF}$
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simultaneously; plot the stability boundary in ($W$, DoF) space.
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\item \textbf{Hysteresis testing.} Start at $W < W^*$, increase past $W^*$,
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decrease back, and test whether the system recovers --- analogous to
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magnetic hysteresis.
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\item \textbf{Adaptive window sizing.} Allow the VM to adjust $W$ dynamically
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and test whether it self-organises toward $W \approx W_0$.
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\end{itemize}
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%% TODO(bob): confirm whether Phase 1 was executed and results are in the
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%% archive; cross-reference campaigns/window_scaling/README.md
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