Composite-Operator Scaling on Triadic Hypergraphs: Formation Transitions in Multi-Agent Architectures with Three-Body Coupling
We study phase transitions on dynamic triadic hypergraphs, in which a continuous formation field evolves under stochastic Ginzburg--Landau dynamics with a cubic three-body coupling $g_τϕ_iϕ_jϕ_k$, while a discrete opinion variable $s_i\in\{-1,+1\}$ undergoes Kawasaki exchange under a Hamiltonian with pairwise alignment and an irreducible three-body energy $-λ_τ\prod_{a\inτ}s_a$. Near the formation critical point the cubic coupling is subleading and the transition remains continuous, controlled at leading order by a pairwise Ising baseline with renormalized coupling $J_{\rm eff}=J+γw$. The dominant observable is the triadic formation correlator $Ψ_{\rm form}\equiv\langleϕ_iϕ_jϕ_k\rangle$, a $k=3$ composite operator built over the underlying $\mathbb{Z}_2$-symmetric order parameter. Composite-operator scaling yields the effective exponents $β_{\rm TF}=3/2$ and $γ_{\rm TF}=-1$. The susceptibility conjugate to $Ψ_{\rm form}$ vanishes at the critical temperature $T_c$ rather than diverging, in contrast to the divergence characterizing scalar (pairwise) order parameters. The exact partition function of the minimal triad on $\{-1,+1\}^3$ identifies a crossover scale $T^*=4J_{\rm eff}/\ln 3$. A field-theoretic two-point function argument reproduces the same vanishing susceptibility. Restoring the three-body coupling ($λ\neq0$) makes the transition first-order, with a critical endpoint at $λ=0$. The exponent relations $β_{\rm TF}=3β_{\rm Ising}$ and $γ_{\rm TF}=γ_{\rm Ising}-4β_{\rm Ising}$ hold exactly on dense hypergraphs via cluster decomposition, and the vanishing-susceptibility signature persists for $d\geq3$ but fails in $d=2$. A Mori--Zwanzig memory kernel yields a continuously tunable dynamical exponent $z_{\rm TF}$, completing the composite-operator scaling regime.