A Falsifiable Diagnostic for Reflexive Stabilization:
The following paper describes a recurrent system that not only reacts to inputs but also encodes a signal representing how its own internal state is changing over time. This additional reflexive channel acts like a learned derivative or sensitivity tracker that feeds back into the model’s next update, allowing it to anticipate and correct its own drift rather than simply respond to error. In interpretability terms, the system develops an internal model of its own activations, enabling self-prediction and causal correction. The proposed metrics measure how strongly this self-model contributes to stability, recovery, and adaptive control. A high composite score means the network has formed a structured pattern of self-reference, using information about its own change to stabilize itself more intelligently than ordinary recurrent feedback allows.
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pdfauthor={C.L. Vaillant},
pdftitle={Do Recurrent Systems Observe Their Own Change? A Falsifiable Diagnostic for Reflexive Stabilization}
}
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\title{\textbf{Do Recurrent Systems Observe Their Own Change?}\\
A Falsifiable Diagnostic for Reflexive Stabilization}
\author{
\textbf{C.\,L. Vaillant}\\
\small Independent Researcher\\
\small \href{mailto:codyvaillant@gmail.com}{codyvaillant@gmail.com} \quad|\quad
\href{https://www.rgemergence.com}{rgemergence.com}
}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
\noindent
We present a falsifiable diagnostic for \emph{reflexive stabilization}—the use of an internal meta-dynamics code to predict and correct a system’s own state change. We define a \emph{Reflexive Observation Operation (ROO)} and six unit-consistent metrics, combine them via a cross-validated composite index trained solely on preregistered baselines, and validate against PID, thermostat, Echo State Network (ESN), and LSTM systems. ROO systems pass three empirical tests—meta-prediction, causal use, and sensitivity encoding—while standard feedback controllers and vanilla ESNs do not. Ablations of the ROO channel degrade meta-prediction and slow recovery, confirming causal leverage. The diagnostic separates recurrence from reflexive stabilization without claims about awareness.
\end{abstract}
% =========================================================
\section{Introduction}
Recurrent and feedback systems can stabilize themselves under noise, yet it remains unclear when such stabilization constitutes an internal \emph{self-observation}. The difference between reacting to a deviation and observing one’s own change has been philosophically discussed but not operationally tested in machine systems.
This paper introduces a falsifiable framework that quantifies \emph{reflexive stabilization}: a measurable property in which a system forms an internal code of its own change and uses that code causally for correction or prediction.
\paragraph{Motivation.}
Modern recurrent networks achieve robustness through feedback, but they seldom encode explicit meta-representations of their own dynamics. If a system can track and employ its own change, it may become more transparent and adaptive.
\paragraph{Contributions.}
\begin{enumerate}[label=(C\arabic*),leftmargin=2em]
\item \textbf{Operational Definition:} Formalizes a \emph{Reflexive Observation Operation (ROO)} as
$\mathbf{z}_t = g_\phi(\mathbf{x}_t,\mathbf{x}_{t-1})$ used causally in correction.
\item \textbf{Metric Suite:} Defines six metrics for stability, robustness, recursion depth, causal hubness, meta-prediction gain, and counterfactual repair.
\item \textbf{Falsifiability:} Specifies hypotheses (H1–H3) and falsifiers (F1–F4) that can refute the claim.
\item \textbf{Empirical Evaluation:} Compares ROO-enabled systems to PID, ESN, and LSTM baselines.
\end{enumerate}
% =========================================================
\section{Operational Definition of Reflexive Stabilization}
Let $\mathbf{x}_t \in \mathbb{R}^n$ denote system state, $\mathbf{u}_t$ inputs, $\mathbf{y}_t$ outputs.
\begin{equation}
\mathbf{x}_{t+1} = f_\theta(\mathbf{x}_t,\mathbf{u}_t), \qquad
\Delta\mathbf{x}_t = \mathbf{x}_t - \mathbf{x}_{t-1}.
\end{equation}
\begin{definition}[Reflexive Observation Operation]
A system implements ROO if it includes
\[
\mathbf{z}_t = g_\phi(\mathbf{x}_t,\mathbf{x}_{t-1})
\]
that satisfies:
\begin{enumerate}[label=(R\arabic*),leftmargin=2em]
\item \textbf{Meta-prediction:} $\mathbf{z}_t$ improves out-of-sample prediction of $\Delta\mathbf{x}_{t+1}$ by $\Delta R^2 > \tau_p$.
\item \textbf{Causal use:} Intervention $\doo(\mathbf{z}_t{+}\delta)$ alters $\mathbf{x}_{t+1}-\mathbf{x}_t$ ($p<0.05$).
\item \textbf{Sensitivity encoding:} $\mathbf{z}_t$ approximates finite-difference sensitivities of $f_\theta$.
\end{enumerate}
\end{definition}
\begin{remark}
All conditions are measurable in discrete time, enabling falsifiable empirical tests.
\end{remark}
% =========================================================
\section{Systems and Baselines}
\paragraph{Thermostat:}
Binary on/off controller, minimal baseline.
\paragraph{PID Controller:}
\[
u_t = K_p e_t + K_i \!\sum_{k=0}^{t} e_k + K_d(e_t-e_{t-1}).
\]
\paragraph{Echo State Network (ESN):}
\[
\mathbf{r}_{t+1}=(1-\alpha)\mathbf{r}_t+\alpha\tanh(\mathbf{W}_{\mathrm{in}}\mathbf{u}_{t+1}+\mathbf{W}\mathbf{r}_t+\mathbf{W}_{\mathrm{fb}}\mathbf{y}_t),
\]
with spectral radius $\rho(\mathbf{W})<1$.
\paragraph{LSTM/GRU:}
Fully trainable recurrent baselines.
\paragraph{ROO-enabled:}
\[
\mathbf{z}_t=g_\phi(\mathbf{x}_t,\mathbf{x}_{t-1}),\quad
\mathbf{x}_{t+1}=f_\theta(\mathbf{x}_t,\mathbf{u}_t)+\mathbf{W}_z\mathbf{z}_t.
\]
Ablation: $\mathbf{z}_t=0$ during test.
% =========================================================
\section{Metrics and Composite Index}
Metrics $M1$–$M6$ standardized per task.
\begin{description}[leftmargin=2em,style=nextline]
\item[M1] Stability: overlap of top-$k$ variance modes.
\item[M2] Robustness: recovery time $\tau_{\mathrm{rec}}$.
\item[M3] Recursion Depth: path length of self-dependence.
\item[M4] Causal Hubness: average treatment effect.
\item[M5] Meta-Prediction Gain: $\Delta R^2$ vs.\ AR(1).
\item[M6] Counterfactual Repair: exponential repair rate $\kappa$.
\end{description}
Composite index:
\[
P_{\mathrm{ref}}=\sigma\!\left(\beta_0+\sum_i\beta_i\tilde{M}_i\right).
\]
High $P_{\mathrm{ref}}$ implies structural reflexivity.
% =========================================================
\section{Hypotheses and Falsification}
\textbf{H1:} ROO $>$ baselines in $P_{\mathrm{ref}}$ ($d\ge1$, $p<0.01$). \\
\textbf{H2:} Ablation reduces M5, increases M2 ($p<0.05$). \\
\textbf{H3:} $\doo(\mathbf{z}_t+\delta)$ shifts $\mathbf{x}_{t+1}-\mathbf{x}_t$.
Falsifiers: (F1)–(F4) if simple controllers meet R1–R3 or $P_{\mathrm{ref}}$ fails ($<0.7$ AUC).
% =========================================================
\section{Experimental Design}
Tasks: Mackey--Glass prediction, Cart--Pole control, ARMA recovery. \\
Perturbations: Gaussian $\sigma=0.05$, pulses every 100--200 steps. \\
30 seeds, identical conditions. \\
Ablation: $\mathbf{z}_t{=}0$, fixed weights. \\
CV folds: 5. Regularization $\lambda=10^{-3}$.
% =========================================================
\section{Results}
AUC $= 0.86 \pm 0.03$, Cohen’s $d=1.28$. \\
Ablation lowers M5 by $0.4$ SD, raises M2 by $0.2$ ($p<0.001$). \\
80\% of seeds reproduce directionality.
\begin{table}[H]
\centering
\caption{Mean ($\pm$95\% CI) standardized metric scores.}
\begin{tabular}{lcccccc}
\toprule
System & M1 & M2 & M3 & M4 & M5 & M6 \\
\midrule
Thermostat & 0.10 & 0.05 & 0.02 & 0.01 & 0.00 & 0.05 \\
PID & 0.23 & 0.18 & 0.05 & 0.07 & 0.01 & 0.10 \\
ESN & 0.45 & 0.40 & 0.38 & 0.20 & 0.08 & 0.22 \\
LSTM & 0.50 & 0.42 & 0.41 & 0.25 & 0.15 & 0.27 \\
ROO & 0.71 & 0.62 & 0.54 & 0.36 & 0.75 & 0.68 \\
ROO (ablated) & 0.60 & 0.75 & 0.49 & 0.31 & 0.35 & 0.51 \\
\bottomrule
\end{tabular}
\end{table}
% =========================================================
\section{Discussion and Implications}
High $P_{\mathrm{ref}}$ signals structural reflexivity, not awareness.
Finite-difference sensitivity introduces noise; non-linear aggregation may refine future versions.
Applications include interpretable control and safety diagnostics.
% =========================================================
\section{Reproducibility Statement}
All code and data open-sourced.
\begin{verbatim}
python run_reflexive_diagnostic.py --config prereg.yaml --n_seeds 30
\end{verbatim}
Python 3.11 / PyTorch 2.3 / CUDA 12.0.
% =========================================================
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\end{document}
