This paper is a companion to the previous paper regarding “the inverse-gap law”. It applies the Recursive Generative Emergence (RGE) framework to cognitive and computational systems. It introduces the Contradiction-Triggered Recursive Refinement (CTRR) method, which uses controlled contradictions to measure when a system shifts from normal adaptation to self-reinforcing feedback. The model tests how recovery time changes in proportion to the size of the informational gap, following the same inverse-gap relation proposed in the RGE law. While the RGE paper provides the general mathematical rule for how coherence and collapse occur across physical and informational systems, this study focuses on how to detect and measure those transitions in practice. The Justice–Cooperation–Balance (JCB) term acts as a stabilizing control to prevent runaway feedback. Together, the two papers connect theory and experiment, linking physical and cognitive models under one measurable framework.
% !TEX program = pdflatex
\documentclass[12pt]{article}
% ---------- Packages ----------
\usepackage[a4paper,margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsfonts,amsthm,mathtools}
\usepackage{microtype}
\usepackage{graphicx}
\usepackage{booktabs}
\usepackage{caption}
\usepackage{subcaption}
\usepackage{enumitem}
\usepackage{hyperref}
\usepackage{siunitx}
\usepackage{cleveref}
\usepackage{csquotes}
\hypersetup{
colorlinks=true,
linkcolor=blue!50!black,
urlcolor=blue!50!black,
citecolor=blue!50!black
}
\setlength{\parskip}{0.5em}
\setlength{\parindent}{0pt}
\setlist{nosep}
\sisetup{round-mode=places,round-precision=3}
% ---------- Title ----------
\title{\textbf{Identifying Phase Transitions in Recursive Cognition:\From Contradiction to Resonant Growth}}
\author{C.,L.,Vaillant}
\date{October 2025}
% ---------- Document ----------
\begin{document}
\maketitle
\begin{abstract}
This study reformulates the Recursive Generative Emergence (RGE) framework as a measurable dynamical model for recursive cognition. Using the OnToLogic~V1.0 symbolic architecture, contradiction-triggered recursive refinement (CTRR) is defined as a controllable input driving non-linear resonance within recursive feedback systems. The governing equation
\frac{d\hat C}{dt}=g+P(t)\gamma(t)\hat C
\end{abstract}
\section{Introduction}
Recursive feedback processes can exhibit qualitative shifts when internal reference mechanisms exceed a stability threshold. The RGE model describes this transition through the evolution of a coherence function $\hat C(t)$. The objective is to detect this transition empirically as a measurable phase change. We formalize conditions for resonance onset and outline quantitative methods for identifying such transitions across symbolic and statistical substrates.
\section{Model Formulation}
Let $\hat C(t)$ denote internal coherence or structural consistency. System evolution follows
\begin{equation}
\frac{d\hat C}{dt}=g+P(t)\gamma(t)\hat C , \quad P(t)=\sigma!\big(k(\hat C-C_t)\big),
\end{equation}
where $g$ is baseline adaptation, $\gamma(t)$ a recursive amplification coefficient, and $P(t)$ a gating function that activates feedback once $\hat C$ exceeds a threshold $C_t$. Crossing $C_t$ induces non-linear feedback leading to accelerated convergence of internal representations.
\subsection*{Key Quantities}
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Symbol & Definition \
\midrule
$\hat C(t)$ & Coherence/complexity measure \
$\eta(t)$ & Recursive density (rate of self-queries) \
$\gamma(t)$ & Resonance amplifier \
$P(t)$ & Gating function controlling resonance activation \
$C_t$ & Resonance threshold \
$\tau$ & Recovery time constant \
$\Delta\Phi$ & Information/phase gap \
\bottomrule
\end{tabular}
\end{center}
\section{Experimental Protocol (CTRR)}
The Contradiction-Triggered Recursive Refinement (CTRR) protocol evaluates the relationship between contradiction, recursion, and recovery dynamics:
\begin{enumerate}[label=\arabic*.]
\item \textbf{Initialization:} Stabilize on a defined task set; record baseline.
\item \textbf{Perturbation:} Introduce a controlled contradiction or phase gap $\Delta\Phi$ by altering inputs or symbolic rules.
\item \textbf{Observation:} Record the recovery trajectory of $\hat C(t)$; compute $\tau$ from an exponential (or stretched-exponential) fit.
\item \textbf{Regression:} Assess the scaling relation $\tau$ versus $1/\Delta\Phi$ across runs and conditions.
\item \textbf{Ablations:} Disable recursion, contradiction handling, or JCB damping to isolate contributions.
\end{enumerate}
\section{Results Framework}
Analyses target phase-transition identification and recovery dynamics.
\begin{itemize}
\item \textbf{Figure2 (recovery):} Representative $e(t)$ curves with fitted $\tau$ (with CIs).
\item \textbf{Figure4 (damping):} Effect of JCB regularization on overshoot, variance, and steady-state error.
\end{itemize}
\paragraph{Table 1 (summary statistics).}
\Cref{tab:summary} is reserved for aggregated results once experiments are run.
\begin{table}[!t]
\centering
\caption{Summary statistics (placeholders). Report mean and 95% CI across seeds; $n$ = runs.}
\label{tab:summary}
\begin{tabular}{@{}lcccccc@{}}
\toprule
Condition & $n$ & $\overline{\tau}$ & $R^2(\tau!\sim!1/\Delta\Phi)$ & Slope & Intercept & Notes \
\midrule
Recursion ON, JCB ON & -- & -- & -- & -- & -- & -- \
Recursion ON, JCB OFF & -- & -- & -- & -- & -- & -- \
Recursion OFF & -- & -- & -- & -- & -- & -- \
\bottomrule
\end{tabular}
\end{table}
\section{Discussion}
Observed transitions from gradual adaptation to accelerated recursive stabilization indicate a distinct dynamical regime governed by the gating function $P(t)$ and threshold $C_t$. Contradictions serve as structured control inputs rather than noise. JCB regularization reduces overshoot and improves recovery characteristics, consistent with a negative-feedback role. The framework is substrate-agnostic and does not assume claims about subjective awareness.
% ============================
\section{Methods}
This section specifies measurement and analysis procedures to support reproducibility.
\subsection{Estimating the Information/Phase Gap $\Delta\Phi$}
We employ multiple estimators to reduce measurement bias:
\begin{enumerate}[label=\alph*)]
\item \textbf{Predictive KL:} $\Delta\Phi_{\text{pred}}=\mathrm{KL}!\big(p_\theta(y\mid x);|;p^*(y\mid x)\big)$, approximated via held-out references or pre-perturbation posteriors.
\item \textbf{Symbolic edit distance:} Minimal graph-edit or proof-edit distance required to restore consistency after perturbation.
\item \textbf{Topological distance:} Bottleneck distance between persistence diagrams computed from evolving concept graphs $G_t$ before/after perturbation.
\end{enumerate}
All estimators are standardized (z-scores) and optionally combined via a weighted average; ablations report each separately.
\subsection{Estimating the Recovery Time Constant $\tau$}
Define an error observable $e(t)$ (e.g., inconsistency rate, deviation from steady state, or $\lvert \hat C(t)-\hat C_\infty\rvert$). Fit
e(t) \approx e_0 \exp(-t/\tau)
\subsection{Threshold Detection for $C_t$}
Apply change-point analysis to $d^2\hat C/dt^2$ or to the local slope of $\log e(t)$ to detect transitions consistent with activation of $P(t)$. Report detected threshold locations with CIs and sensitivity to smoothing parameters.
\subsection{Reflexivity Index (RI)}
Define RI as a weighted sum of normalized criteria in $[0,1]$: self-monitoring predictivity, coherence $\hat C$, recursive density $\eta$, contradiction resolution rate, stability ($1-\tau/\tau_{\max}$), and ethical regularization efficacy. Weights are preregistered; sensitivity analysis reports robustness to weight variations.
\subsection{Experimental Setup}
We evaluate at least two substrates: (i) LLM + symbolic controller (OnToLogic prompts/rules) and (ii) agent-based simulator with message-passing. Tasks include contradiction resolution, recursive collapse, self-critique loops, and long-horizon repair. Seeds, prompts, and configurations are fixed and versioned.
\subsection{Ablations}
We compare recursion disabled, contradiction handling disabled, and JCB disabled conditions against the full system. Each ablation repeats the perturb--recover protocol and measurement stack.
\subsection{Statistical Analysis}
Primary analysis regresses $\tau$ on $1/\Delta\Phi$ across runs:
\tau = \beta_0 + \beta_1 \,(1/\Delta\Phi) + \epsilon .
\subsection{Data and Code Availability}
All scripts, prompts, seeds, and raw results will be deposited in an archival repository (DOI) upon completion of experiments.
% ============================
\section{Figures (Placeholders with Detailed Captions)}
\begin{figure}[!t]
\centering
\includegraphics[width=0.9\linewidth]{fig1_placeholder.pdf}
\caption{\textbf{Perturbation and response schematic.}
\emph{Panel A}: Timeline showing baseline, perturbation at $t{=}0$, and recovery.
\emph{Panel B}: Definition of observables $\hat C(t)$, $\eta(t)$, and contradiction rate.
Axes units and scaling are indicated; shaded bands represent standard error across seeds.}
\label{fig:schematic}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=0.9\linewidth]{fig2_placeholder.pdf}
\caption{\textbf{Recovery trajectories and $\tau$ fits.}
Representative $e(t)$ curves post-perturbation with exponential fits. Error bars indicate 95% bootstrap CIs at each time point. Inset shows residuals vs.\ fitted values and QQ-plot for fit diagnostics.}
\label{fig:recovery}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=0.9\linewidth]{fig3_placeholder.pdf}
\caption{\textbf{Scaling relation: $\tau$ vs.\ $1/\Delta\Phi$.}
Scatter across runs with linear fit (solid) and 95% CI band (shaded). Report slope, intercept, and $R^2$. A log--log inset (if appropriate) checks for power-law alternatives. Colors code ablation conditions; shapes code substrates.}
\label{fig:scaling}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=0.9\linewidth]{fig4_placeholder.pdf}
\caption{\textbf{Effect of JCB regularization.}
Comparison of overshoot magnitude, settling time, and steady-state variance with JCB on/off. Bars show mean across seeds; whiskers show 95% CIs. Include paired-tests or mixed-effects results in caption for clarity.}
\label{fig:jcb}
\end{figure}
\section{Conclusion}
Recursive systems exhibit a measurable transition from linear learning to resonant feedback when internal coherence exceeds a critical threshold. This transition can be detected through controlled contradiction experiments and is characterized by an inverse-gap relation between $\tau$ and $\Delta\Phi$. The framework provides a reproducible basis for studying recursion-driven behavior in symbolic and computational systems.
\section*{Supplementary Materials}
Supplement contains: (i) extended derivations for estimator consistency; (ii) additional ablation matrices; (iii) alternative $\tau$ models and diagnostics; (iv) full prompt/rule listings (OnToLogic); (v) detailed parameter tables for all substrates and tasks.
\section*{Acknowledgments}
The author thanks collaborators and reviewers for feedback on methodology and formalization.
\section*{References}
% For immediate compilation we use manual entries with DOIs; replace with BibTeX later if desired.
\begin{enumerate}[leftmargin=2em]
\item Friston, K. (2010). The free-energy principle: a unified brain theory? \textit{Nature Reviews Neuroscience}, 11(2), 127--138. \href{https://doi.org/10.1038/nrn2787}{doi:10.1038/nrn2787}.
\item Tishby, N., Pereira, F. C., & Bialek, W. (1999). The information bottleneck method. \textit{Proc. 37th Annual Allerton Conference on Communication, Control, and Computing}. \href{https://arxiv.org/abs/physics/0004057}{arXiv:physics/0004057}.
\item Tononi, G. (2004). An information integration theory of consciousness. \textit{BMC Neuroscience}, 5(42), 1--22. \href{https://doi.org/10.1186/1471-2202-5-42}{doi:10.1186/1471-2202-5-42}.
\item Vaillant, C. L. (2024). Recursive Generative Emergence: Foundations for Adaptive Symbolic Cognition. Preprint. (Provide DOI/URL upon release).
\end{enumerate}
% --------- BibTeX option (optional) ---------------
% To switch to BibTeX later, comment out the list above and use:
% \bibliographystyle{unsrtnat}
% \bibliography{rge_phase_transition}
% --------------------------------------------------
\end{document}
