133 lines
5.1 KiB
TeX
133 lines
5.1 KiB
TeX
%% SCRAP: archive/session-logs/experimental-addendum
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%% SOURCE: docs/working/archive/session-logs/experimental-addendum.md
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%% STATUS: HISTORICAL
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%% FITS: experiments/ if content warrants
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%% EDITORIAL: lifted — prose rewritten to press voice. Source is a
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%% ChatGPT transcript; the scientific content on James Law and the
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%% window-scaling experiment design is the load-bearing material
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%% and is retained. Informal dialogue omitted.
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\section{Experimental Addendum: James Law and Window-Scaling Validation}
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\label{sec:experimental-addendum}
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\subsection{Discovery Context}
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During analysis of the SSM attractor results, a conserved geometric
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relationship was identified in the feedback-response surface of the
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adaptive runtime. The system exhibits:
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\begin{itemize}
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\item Reproducible convergence to a stable attractor basin.
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\item Shape-invariant timing behaviour across workloads.
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\item A consistent geometric structure in the configuration--window--variance
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phase space.
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\item A predictable collapse when the smoothing window exceeds a critical
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bound.
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\end{itemize}
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These properties distinguish the StarForth adaptive runtime from chaotic
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adaptive systems. The behaviour is geometric and predictable, not
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stochastic.
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\subsection{James Law of Computational Dynamics}
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The empirically observed scaling relation is stated as follows.
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\begin{quote}
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\textbf{James Law.} In an adaptive computational system governed by
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multiple interacting feedback loops, the effective stability-smoothing
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factor $\Lambda$ scales inversely with the number of active degrees of
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freedom $\mathrm{DoF}$ plus one, and directly with the smoothing window
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size $W$:
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\[
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\Lambda = \frac{W}{\mathrm{DoF} + 1}
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\]
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This relation describes a conserved geometric structure in the system's
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feedback-response surface and predicts the onset of stability,
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metastability, and collapse phases as window size varies.
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\end{quote}
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%% PATENT: This section is adjacent to patent claims. Do not expand or
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%% reformulate without explicit instruction.
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The law is currently a working hypothesis. Prior to incorporation into
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any patent application, it is to be validated empirically via the
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window-scaling experiment described below.
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\subsection{Scientific Significance}
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If validated, the James Law occupies the same class of discovery as
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Little's Law in queueing theory, Amdahl's Law in parallelism, and
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Zipf's Law in linguistics — an empirically observed, falsifiable,
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reproducible scaling invariant in a new domain (computational dynamics).
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\subsection{Window-Scaling Validation Experiment}
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The validation experiment tests whether the quantity
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\[
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K = \frac{\Lambda(\mathrm{DoF}) \cdot (\mathrm{DoF} + 1)}{W}
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\]
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remains approximately constant and near unity across multiple degrees of
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freedom and window sizes.
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\subsubsection{Parameters}
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\begin{itemize}
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\item \textbf{Degrees of freedom:} $\mathrm{DoF} \in \{0, 1, 2, 3, 4, 5, 6, 7\}$.
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\item \textbf{Window sizes:} $W \in \{512, 1024, 1536, 2048, 3072, 4096,
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6144, 8192, 16384, 32769, 52153, 65536\}$ (12 levels, spanning
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subcritical through catastrophic-collapse territory).
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\item \textbf{Replicates:} 30 per condition.
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\item \textbf{Total runs:} $8 \times 12 \times 30 = 2{,}880$.
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\item \textbf{Workload:} composite ``omni'' workload — all L8 initialisation
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scripts concatenated and run sequentially
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(STABLE $\to$ TEMPORAL $\to$ VOLATILE $\to$ TRANSITION $\to$ DIVERSE),
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maximally stressing the feedback loops.
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\end{itemize}
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\subsubsection{Run Matrix}
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Conditions are generated by a full factorial expansion and shuffled once
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before execution (fixed random seed for reproducibility) to eliminate
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temporal drift, thermal bias, and cache-warming artefacts.
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\subsubsection{Analysis Criteria}
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\begin{enumerate}
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\item Compute $K(W, \mathrm{DoF})$ for each condition.
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\item Measure mean, standard deviation, and maximum deviation of $K$
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from~1.0.
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\item Identify the critical window size $W_\mathrm{crit}(\mathrm{DoF})$
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at which the CV spikes, the L8 engine collapses to a trivial mode,
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or the attractor structure disappears.
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\end{enumerate}
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\subsubsection{Acceptance Criteria}
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The law is supported if:
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\begin{itemize}
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\item $K \in [0.97, 1.03]$ across all DoF levels at $W = 4096$.
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\item The same clustering holds at $W = 2048$ and $W = 8192$.
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\item Instability (elevated CV or mode collapse) correlates with
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deviation from $K \approx 1$ at extreme window sizes.
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\end{itemize}
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\subsection{Presentation Guidance}
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When discussing the James Law in publications, the analogy to physical
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systems should be used as interpretive framing only, not as a claim.
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Appropriate language:
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\begin{quote}
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``While the runtime does not implement gravitational physics, the emergent
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topology resembles a system with a dominant attractor. This analogy is
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useful as a descriptive shorthand, but the formal behaviour is defined
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strictly by the equations presented herein.''
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\end{quote}
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The experimental evidence — DoE data, convergence plots, variance
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shrinkage, attractor identification — constitutes the scientific
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claim. The metaphors motivate the mathematics; they do not replace it.
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