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%% SOURCE: docs/working/architecture/03-architecture/adaptive-systems/multivariate-dynamics.md
%% STATUS: WORKING
%% FITS: dev-guide/ch-physics
%% EDITORIAL: lifted — prose rewritten to press voice

\section{Multivariate Coupled Dynamics: Golden-Configuration Discovery}

This section describes the experimental framework, dated 2025-11-20 and in
design at the time of writing, for discovering a golden runtime configuration by
stability fingerprinting. It is deliberately not a factorial design of
experiments. The premise is that the adaptive runtime is a coupled dynamical
system in which every metric is at once a measurable signal, a feedback variable
that shapes future state, and a coordinate in a high-dimensional attractor
landscape. The analysis therefore measures the shape of each configuration's
trajectory through metric space rather than isolated main effects.

The dynamical-systems vocabulary --- attractors, basins, eigenmodes, coupling
--- is modeling language for the runtime's tick-by-tick behavior, applied to the
domain of configuration selection; it is not a claim of literal physics.

\subsection{The Coupling Web}

Within a single tick the metrics form a feedback web. Cache and bucket hit rates
feed bucket load, which feeds heat, which feeds decay, which feeds window width,
which feeds prediction accuracy; prediction accuracy and load feed back into the
next tick's cache behavior and into the adaptive heartbeat interval. Load
intensity shapes the word-access pattern, which shapes the heat distribution,
which governs decay effectiveness and the jitter envelope. Each configuration
traces a distinct path through this web.

\subsection{Three Levels of Measurement}

The framework records state at three nested scopes. \emph{Per tick}, at roughly
one-millisecond cadence, it captures a snapshot of the coupled system: a temporal
anchor, the cache and bucket wave functions $\psi_{\text{cache}}(t)$ and
$\psi_{\text{bucket}}(t)$, the feedback signals (hot-word count, mean heat,
inferred decay, window width), the load response (executions, prediction
attempts, prefetch hits), and the stability measures (tick jitter and a composite
smoothness index). \emph{Per run}, over roughly a hundred thousand ticks, it
distills a stability fingerprint: convergence time and coefficients of variation,
steady-state means, dynamics (rise time, settling time, overshoot), coupling
strengths including a full cross-correlation matrix, spectral properties
(dominant frequency, spectral entropy, dominant eigenvalue), and an overall
stability score. \emph{Per configuration}, over 150 or more runs, it aggregates
those fingerprints into between-run statistics: the mean and spread of the
stability score, convergence reliability, coupling stability, an attractor-basin
depth estimate, the jitter envelope and its whiteness, and noise-propagation
measures.

\begin{equation}
\textit{overall\_stability\_score} =
  \tfrac{1}{3}\,\textit{jitter\_score}
+ \tfrac{1}{3}\,\textit{convergence\_score}
+ \tfrac{1}{3}\,\textit{coupling\_score}
\end{equation}

\subsection{Experimental Design}

The design holds five elite configurations carried forward from an earlier stage,
runs each 150 to 200 times for statistical power, and runs each for roughly a
hundred thousand ticks to capture both convergence and steady state --- on the
order of seventy-five million ticks in total, some six to eight hours of
wall-clock time. Run order is randomized and configurations are interleaved to
avoid temporal and thermal confounds, background tasks are minimized, and every
tick is captured without subsampling to preserve time-series fidelity.

\subsection{Analysis Pipeline}

Analysis proceeds in three phases. Per-run fingerprinting detects convergence
(when the cache hit-rate coefficient of variation falls below five percent),
computes steady-state statistics, extracts the transient dynamics, measures
coupling with lag search and a full correlation matrix, performs spectral
analysis to find dominant modes, and reduces all of it to a stability index.
Per-configuration fingerprinting aggregates the run fingerprints into robustness,
convergence reliability, coupling strength, basin depth, and noise
characteristics. Golden-configuration selection then scores the five candidates
on weighted dimensions and takes the argmax.

\begin{table}[h]
\centering
\begin{tabular}{lll}
\toprule
Dimension & Weight & Ideal behavior \\
\midrule
Stability score          & 30\% & highest mean, lowest spread \\
Convergence reliability  & 25\% & full convergence, low variability \\
Coupling strength        & 20\% & strong ($0.8+$), stable load-to-heartbeat \\
Attractor basin depth    & 15\% & fast eigenmode decay, noise-resilient \\
Jitter smoothness        & 10\% & low variation, white noise \\
\bottomrule
\end{tabular}
\caption{Weighted dimensions for golden-configuration selection.}
\end{table}

\subsection{Interpretation}

A strong configuration fingerprint shows a high mean stability score with low
spread (reproducibility), full convergence across runs, tight load-to-heartbeat
coupling, a low jitter coefficient of variation, and a large dominant eigenvalue
indicating a deep attractor basin. A weak fingerprint inverts each of these: a low
and widely scattered stability score, incomplete convergence, loose and noisy
coupling, high jitter variation, and a shallow, noise-prone basin.

\subsection{Implementation Roadmap}

The build sequence instruments the heartbeat loop with a per-tick circular
buffer, implements per-run fingerprinting (convergence detection, coefficients of
variation, correlations, eigenvalues, spectral properties), aggregates run
fingerprints into per-configuration fingerprints, scores and selects the golden
configuration, and finally produces the visual reporting suite --- time-series
plots, correlation heatmaps, a principal-component projection of runs, and
stability-score distributions.

%% PATENT: stability fingerprinting and golden-configuration selection over the
%%   coupled adaptive runtime are patent-adjacent; no claims drafted here.

%% TODO(bob): confirm whether the harness was built and whether a golden
%%   configuration was selected, for a results subsection.