Triadic Phase Transitions in AI Networks: Composite-Operator Scaling in Cognitive Architectures

TL;DR

This paper explores triadic phase transitions in multi-agent AI networks, revealing unique critical behaviors and scaling laws that differ from traditional pairwise models, with exact solutions and dynamic scaling insights.

Contribution

It introduces a novel triadic Ising model for AI networks, deriving exact critical points and scaling laws for composite operators, expanding understanding of collective AI behavior.

Findings

Exact partition function for the triadic Ising model

Crossover temperature and critical point derived analytically

Scaling of formation correlator as (T_c - T)^{3/2}

Abstract

Multi-agent AI architectures whose dominant collective observable is a $k$-body spin correlator $O_k\equiv\langle\phi^k\rangle$ over a $\mathbb{Z}_2$-symmetric order parameter exhibit composite-operator criticality with effective exponents $\beta_k = k/2$ and $\gamma_k = 2-k$, thereby producing a finite susceptibility for $k\geq2$ and a vanishing susceptibility for $k\geq3$. This is a qualitative departure from all pairwise-network universality classes. We derive these results for the first non-trivial case $k=3$ as presented in COGENT$^3$ (Salazar, 2026). The formation transition of COGENT$^3$ and comparable models, under controlled universality and mean-field arguments, reduces to an exactly solvable triadic Ising model. The minimal triad Hamiltonian admits an exact partition function on $\{-1,+1\}^3$, with crossover temperature $T^*=4(J+\gamma w)/\ln 3$ and mean-field critical point $T_c=J+\gamma w$ (gradient coupling cooperatively enhancing order). The formation correlator $\Psi_{\rm form}\equiv\langle\phi_i\phi_j\phi_k\rangle$ scales as $(T_c-T)^{3/2}$. Its conjugate susceptibility vanishes at $T_c$, confirmed by an independent field-theoretic two-point function argument. A Mori-Zwanzig memory ansatz yields a continuously tunable dynamical exponent, completing the composite-operator scaling regime.