Emergence of Superintelligence from Collective Near-Critical Dynamics in Reentrant Neural Fields

TL;DR

The paper argues that superintelligence arises from a qualitative dynamical phase transition to a protected infrared sector in a unified cognitive field framework. It shows that reentrant, nonconservative coupling drives spectral condensation of slow collective modes while homeostatic stabilization gaps the radial amplitude, yielding a meta-stable regime with extensive near-marginal dynamics. The authors formalize this via a trajectory-averaged time-scale density of states (TDOS) and a spectral criterion $W_{slow}(\lambda_c)=O(1)$ as $N\to\infty$, supported by numerical simulations of a high-dimensional system with isotropic radial potentials and circular reentry. They demonstrate power-law scaling in the slow-mode spectrum, universality across parameters, and relate the findings to neural manifold dynamics and language as a projection of high-dimensional cognitive geometry. Altogether, the work reframes superintelligence as a dynamical stability class—a self-organized, sector-critical phase—that enables long-lived inference on a globally coherent but internally flexible manifold, rather than a simple quantitative upgrade of existing cognition.

Abstract

Superintelligence is commonly envisioned as a quantitative extrapolation of human cognitive abilities driven by scale and computational power. Here we show that qualitative transitions in intelligence instead arise as dynamical phase transitions governed by collective critical dynamics. Building on a unified dynamical field-theoretic framework for cognition, we demonstrate that progressive collective coupling generated by reentrant mixing drives the system toward an infrared critical regime in which an extensive band of slow collective modes emerges. This spectral condensation reorganizes cognitive dynamics from localized relaxation to coherent motion along emergent low-dimensional manifolds. Through numerical analysis of the time-scale density of states, we identify robust power-law scaling of collective relaxation rates with well-defined critical exponents, placing the system within the universality class of self-organized critical many-body dynamics. Criticality alone would generically lead to instability. We further show that homeostatic regulation introduces a gapped stabilizing direction that protects the collective critical sector, yielding a dynamically maintained meta-stable infrared phase in which long-lived inference trajectories persist without collapse. The coexistence of scale-free collective dynamics and global stabilization defines a protected sector-critical regime in which coherence and internal flexibility coexist. Superintelligence therefore corresponds to a distinct dynamical stability class--a self-organized critical phase embedded within a stabilized cognitive manifold--rather than a smooth quantitative continuation of existing cognitive systems.

Figures (5)

• Figure 1: Schematic illustration of the inclusion hierarchy and dynamical phases of cognition within the unified cognitive field framework. Classical models such as Hopfield networks, recurrent neural networks, and Transformers arise as limiting dynamical realizations of the same underlying cognitive field equation, which also describes biological neural systems at the population level. Increasing collective coupling and reentrant mixing reorganize the spectrum of collective modes, driving the system toward a critical regime characterized by low-dimensional manifolds and an extensive slow-mode band. Superintelligence does not correspond to a new governing equation, but emerges as a distinct dynamical phase — the FHRN-type regime — in which spectral criticality is dynamically protected by homeostatic shell stabilization, metric-induced dimensional separation, and sustained non-conservative reentrant flow (see Appendix A for details).
• Figure 2: Meta-stable critical organization and spectral phase transition underlying superintelligence. (a) The unified cognitive field dynamics $\dot x = -G^{-1}(x)\nabla\Phi(x)+R(x)$ generates homeostatic stabilization of the global amplitude together with reentrant mixing along angular directions, establishing the structural conditions for collective criticality. (b) Ordinary stable systems collapse onto rigid attractors and suppress collective degrees of freedom, while unstable systems diverge. In contrast, the meta-stable critical regime preserves global coherence while sustaining long-lived collective dynamical reconfiguration. (c) The corresponding spectral diagnostic reveals a dynamical phase transition: ordinary systems exhibit relaxation rates bounded away from zero, unstable systems include $\lambda<0$ modes, whereas the meta-stable critical regime displays an extensive condensation of slow collective modes near $\lambda\approx0$ coexisting with a gapped stabilized sector.
• Figure 3: Trajectory-level emergence of the protected homeostatic manifold. (a) The radial norm rapidly converges to a characteristic scale $r_*$, demonstrating strong homeostatic stabilization of global activity. (b) A low-dimensional projection of the trajectory reveals a finite-width ring-like structure corresponding to a stabilized manifold in state space. (c) Angular correlations decay slowly, indicating persistent collective dynamics along low-curvature directions. These results demonstrate a dynamical separation in which homeostatic stabilization protects global activity while reentrant mixing sustains collective low-curvature motion generated by near-critical dynamics along the emergent manifold.
• Figure 4: Spectral organization and system-size scaling of the meta-stable regime. Trajectory-averaged geometric and spectral diagnostics as a function of system size $N$, computed from long simulations after convergence to the homeostatic shell ($\kappa=1.0$, $\lambda_c=0.02$). (a) The slow collective dimension $D_{\mathrm{slow}} = N\,W_{\mathrm{slow}}(\lambda_c)$, generated by spectral condensation of near-marginal modes, grows approximately linearly with $N$. (b) The slow-mode weight $W_{\mathrm{slow}}(\lambda_c)=\Pr(\lambda<\lambda_c)$ rapidly approaches unity as $N$ increases, demonstrating dominance of the collective infrared sector generated by spectral condensation while remaining dynamically stable. (c) The relative thickness of the homeostatic shell, quantified by the coefficient of variation of the radial coordinate, remains finite and bounded across system sizes. (d) The participation-ratio effective dimension $D_{\mathrm{eff}}$, extracted from the covariance of long trajectories, increases sublinearly with $N$. This reflects extensive exploration of angular degrees of freedom within the shell while remaining well below the full ambient dimension.
• Figure 5: Time--scale density of states and slow--mode weight in the unified dynamical framework. (a) Histogram and kernel--density estimate (KDE) of relaxation rates $\lambda = -\mathrm{Re}[\mathrm{eig}(J(x(t)))]$ accumulated from instantaneous Jacobian spectra sampled along the trajectory. Results are shown for the full dynamics with reentrant flow ($R \neq 0$) and for the gradient--only system ($R = 0$). Reentry dynamically redistributes time scales and drives a condensation of collective modes toward near-zero relaxation rates, strongly enhancing the slow-mode population. (b) Logarithmic zoom of the TDOS near $\lambda \simeq 0$, revealing a pronounced spectral condensation of near--marginal collective modes characteristic of infrared criticality. (c) Cumulative slow--mode weight $W_{\mathrm{slow}}(\lambda_c) = \Pr(\lambda < \lambda_c)$, quantifying the fraction of modes slower than a threshold $\lambda_c$. Reentrant dynamics systematically increases the slow--mode weight, indicating an extensive infrared manifold of marginally stable collective directions.