A Mathematical Framework for Stable Concepts and Cognition through Representational Dynamics:
Concepts, cognition, and even physical organization can be described as stable attractor states that persist when the curvature of their representational manifold ($\lambda_{\min}$) exceeds the noise variance ($\sigma^2$) in the substrate that realizes them. Physical substrates give rise to stable informational states; these stabilize into conceptual attractors, which recursively interact to generate higher-order attractors such as cognition, culture, and values. Each layer imposes feedback and constraint on the one below, forming a continuous hierarchy through which matter self-organizes into mind. In this view, information with sufficient feedback becomes self-maintaining—a physically measurable property that unites life, thought, and structure under a single mathematical principle.
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\title{From Concept to Cognition:\\
A Recursive Generative Emergence Framework for Representational Stability}
\author{C.\,L. Vaillant}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
We present a mathematical and empirical framework for understanding how concepts arise and persist in both biological and artificial systems. Drawing on attractor network theory, representation alignment, and the meta-framework of \emph{Recursive Generative Emergence} (RGE), we formalize concepts as stable attractors in a shared latent space linking physical substrates and semantic structure.
We derive a stability condition relating the curvature of conceptual attractors to noise in the underlying substrate, predicting the breakdown of coherent representation under specific dynamical constraints. Empirical pathways—embedding alignment, recurrent simulations, and neural manifold analysis—are outlined to test these predictions.
The framework provides a tractable, falsifiable bridge between cognitive neuroscience and AI alignment, reframing conceptual stability as an emergent property of recursive generative processes.
\end{abstract}
\section{Introduction}
Concepts enable systems—biological or artificial—to compress experience into reusable representations. Yet the mathematical conditions under which a concept can form, persist, and generalize across substrates remain unclear.
We address this by treating concepts as \emph{attractor states} within a latent space whose geometry reflects both internal dynamics and environmental noise.
This formulation connects theories of attractor networks \cite{KhonaFiete2022}, representation alignment \cite{Goldstein2023}, and variational free-energy minimization \cite{Fountas2020}.
Situated within RGE, we interpret cognition as a recursive generative process: information patterns iteratively refine themselves through feedback between stability and novelty, producing structured emergence across scales.
\section{Mathematical Framework}
\subsection{Latent Concept Space}
Let $\mathcal{C}=\mathbb{R}^d$ denote a latent concept space.
Each stimulus $T$ is mapped to $\phi(T)\in \mathcal{C}$ via encoder $M:S\to\mathcal{C}$ from substrate $S$ (neural or computational).
A concept $I$ is represented by prototype $G_I=\mathbb{E}_{T\sim I}[\phi(T)]$, normalized as $\hat{G}_I=G_I/\|G_I\|$.
\subsection{Concept Similarity and Potential}
Similarity of a token to a concept is
\begin{equation}
P_I(T)=\frac{\langle \phi(T),\hat{G}_I\rangle}{\|\phi(T)\|}.
\end{equation}
We define a quadratic potential governing convergence toward $G_I$:
\begin{equation}
\Psi_I(\phi)=\tfrac12\|\phi-\hat{G}_I\|^2+\sum_{J\neq I}\beta_{I,J}\langle \phi,\hat{G}_J\rangle,
\end{equation}
where cross-terms $\beta_{I,J}$ encode interference between attractors.
\subsection{Dynamics and Stability}
Assuming stochastic gradient descent on $\Psi_I$ with noise $\xi_t$:
\begin{equation}
\dot{\phi}_t=-\nabla_\phi\Psi_I(\phi_t)+\xi_t,\qquad \mathbb{E}[\xi_t\xi_t^\top]=\sigma^2 I.
\end{equation}
Linearizing near equilibrium $\phi^\ast=\hat{G}_I$ yields
\begin{equation}
\dot{\delta\phi}_t=-H_I\,\delta\phi_t+\xi_t,
\end{equation}
where $H_I=\nabla^2_\phi\Psi_I(\hat{G}_I)$ is the local Hessian.
\textbf{Stability condition:}
\begin{equation}
\lambda_{\min}(H_I)>\sigma^2 C,
\end{equation}
ensuring convergence rather than diffusion.
Here $C\propto \|DM\|_2^2$ reflects the Lipschitz constant of the mapping $M$ from substrate to concept space.\footnote{Derived by linearizing the stochastic differential equation and applying Lyapunov analysis; stability requires all real parts of eigenvalues of $-H_I$ to remain below noise-scaled curvature.}
\section{Empirical Directions}
\subsection{Text Embedding Validation}
Compute $G_I$ from clustered exemplars using Sentence-BERT or late-layer embeddings from large language models.
Perturb tokens with controlled corruption to estimate $\sigma^2$; measure the critical point where classification coherence collapses to validate Eq.\,(5).
\subsection{Toy Attractor Simulation}
Train a recurrent network or neural ODE to reproduce attractor dynamics of Eq.\,(3).
Vary $\sigma$ to map transitions from stable to chaotic regimes, visualizing basin curvature and cross-interference $\beta_{I,J}$.
\subsection{Neural Simulation}
Implement a continuous-time RNN with weight matrix $W$ trained via FORCE learning to embed fixed points at $M^{-1}(G_I)$.
Additive Gaussian state noise models biological variability; observe breakdown of attractors as $\sigma$ increases, estimating $\lambda_{\min}$ and $r$ experimentally from manifold curvature and noise amplitude.
\section{Discussion}
This framework unifies three perspectives:
\begin{enumerate}
\item \textbf{Neuroscience:} concepts correspond to attractors on neural manifolds; stability reflects curvature-noise tradeoffs.
\item \textbf{Machine learning:} embedding collapse, mode averaging, and drift are special cases of attractor instability.
\item \textbf{Philosophy of mind:} RGE formalizes how ideas emerge recursively within physical substrates without invoking metaphysical dualism.
\end{enumerate}
Thus, cognition is viewed as a recursive generative process governed by stability constraints on representational dynamics.
\section{Limitations and Future Work}
Cross-substrate equivalence remains an open problem: aligning conceptual spaces between brains and models requires robust nonlinear mappings.
Experimental validation demands methods for estimating manifold curvature and noise directly from population activity.
Future extensions could generalize $\Psi_I$ to non-quadratic convex forms or incorporate temporally recursive attractors to model learning and creativity.
\section{Conclusion}
We propose a unified mathematical framework situating conceptual stability within Recursive Generative Emergence.
By connecting attractor geometry, noise dynamics, and empirical measurability, the work provides a concrete path to testable theories of cognition across natural and artificial systems.
The stability condition $\lambda_{\min}(H_I)>\sigma^2C$ offers a compact criterion for when meaning holds or dissolves—a quantitative bridge between concept and consciousness.
\section*{References}
\begin{thebibliography}{9}
\bibitem{KhonaFiete2022} Khona, M. \& Fiete, I. ``Attractor and Continuous-Manifold Models of Neural Dynamics.'' \emph{Nature Neuroscience}, 2022.
\bibitem{Goldstein2023} Goldstein, P. et al. ``Representational Alignment between Artificial and Biological Neural Networks.'' \emph{PNAS}, 2023.
\bibitem{Fountas2020} Fountas, Z. et al. ``A Neural Network Model of the Free Energy Principle.'' \emph{Frontiers in Computational Neuroscience}, 2020.
\end{thebibliography}
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\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{figure1_attractor.png}
\caption{Attractor landscape of conceptual representations: stable basins correspond to coherent concepts; noise flattens curvature, leading to diffusion.}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{figure2_stability.png}
\caption{Stability geometry linking curvature $\lambda_{\min}$ and noise $\sigma$: boundary marks transition between coherent and chaotic regimes.}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{figure3_RGE_hierarchy.png}
\caption{Recursive Generative Emergence hierarchy: from information to pattern to cognition; each layer generates and stabilizes the next.}
\end{figure}
\end{document}
