1 Introduction↩︎

Phase transitions in networks of interacting units underpin neural computation [2], social opinion phenomena and collective behavior [3], [4], and, by structural analogy, multi-agent AI architectures with emergent behavior [1]. In every canonical model, be it Ising, Hopfield, voter or Potts, interactions are pairwise. The energy is a sum over dyadic terms.

Biological neural systems and a growing class of artificial architectures, by contrast, exhibit higher-order dependencies that resist reduction to dyadic terms [5]. The formation of functional neural assemblies and the synchronization of task-specific computational modules all involve coordinated activity among three or more units simultaneously [6]. Capturing such dependencies in artificial systems clearly exceeds a dyadic structure. Even the pairwise-weight Boltzmann machine requires the simultaneous statistics of hidden-unit assemblies to represent joint distributions that cannot be encoded by pairwise connections alone [7].
The structural claim of this paper is general. In any AI architecture whose dominant collective observable is a \(k\)-body correlator \(O_k\!\equiv\!\langle\phi^k\rangle\) over a \(\mathbb{Z}_2\)-symmetric mean-field order parameter \(m\), composite-operator scaling theory [8], [9] implies effective exponents \[\beta_k = \frac{k}{2}, \quad \gamma_k = 2-k, \quad \text{for} \quad k \geq 1. \label{eq:general}\tag{1}\] These follow from the scaling dimension \(\Delta_{O_k} = k\Delta_\phi\) at \(d_c=4\), where \(\Delta_\phi=1\) is the mean-field scaling dimension of the fluctuating field operator. The standard pairwise case (\(k\!=\!1\), observable \(= m\)) recovers \((\beta,\gamma) = (1/2,\,1)\). For \(k\!=\!2\), the susceptibility is finite at \(T_c\) rather than diverging, but for \(k\!\geq\!3\) it vanishes. The non-diverging, vanishing susceptibility profile is a qualitative signature of triadic (or higher) observable structure, absent from all scalar-order-parameter networks, with \(k\!=\!3\) being the first instance of this vanishing-susceptibility regime and the relevant one for triadic AI architectures.

We derive the \(k\!=\!3\) case in full, motivated by the COGENT\(^{\boldsymbol{3}}\) framework [1] in which artificial cognition is formalized via continuous dynamical triadic hypergraphs, building on the work by [10]. In Sec. 3 we show that the formation transition of COGENT\(^{\boldsymbol{3}}\) reduces, undercontrolled universality and mean-field arguments, to the Boolean triadic Ising model analyzed here. Our goals for this paper are simple: (i) solve the triad partition function exactly on \(\{-1,+1\}^3\); (ii) derive the thermodynamic-limit transition by mean-field theory; (iii) determine the critical scaling of the formation order parameter \(\Psi_{\mathrm{form}}\equiv\langle\phi_i\phi_j\phi_k\rangle\); and (iv) compute the Mori–Zwanzig memory correction to the dynamical critical exponent.

2 The Triadic Ising Model↩︎

Hamiltonian↩︎

Define \(\tau=\{i,j,k\}\) as a triad with binary formation fields \(\phi_a\in\{-1,+1\}\) and Hamiltonian given by \[H_\tau = -J\!\sum_{\langle ab\rangle\subset\tau}\!\phi_a\phi_b - h\sum_{a\in\tau}\phi_a + \frac{\gamma}{2}\sum_{\langle ab\rangle\subset\tau} w_{ab}(\phi_a-\phi_b)^{2} \label{eq:ham}\tag{2}\] where \(J>0\) is the pairwise alignment coupling, \(h\) an external symmetry-breaking field, \(\gamma>0\) the gradient coefficient, and \(w_{ab}=w\) denote uniform edge weights.

Exact reduction↩︎

On \(\{-1,+1\}\) the following identity \[(\phi_a-\phi_b)^2 = 2(1-\phi_a\phi_b) \label{eq:identity}\tag{3}\] holds exactly. Applying 3 to all three edges of \(\tau\), and inserting into 2 , yields \[H_\tau = -J_{\mathrm{eff}}\!\sum_{\langle ab\rangle}\!\phi_a\phi_b - h\sum_a\phi_a + E_0 \label{eq:hamred}\tag{4}\] with the renormalized coupling and offset \[J_{\mathrm{eff}}= J+\gamma w \quad \text{with} \quad E_0 = 3\gamma w. \label{eq:jeff}\tag{5}\] The gradient term does not frustrate the system but cooperates with the alignment interaction, renormalizing \(J\) upward by the amount \(\gamma w\). The energy is thus exactly pairwise, with the triadic physics entering through the composite observable \(\Psi_{\mathrm{form}}\) rather than the Hamiltonian structure. Equation 5 is exact on the binary hypercube: the minimal triad maps exactly onto a renormalized Ising model, enabling analytical solution of the baseline mechanics before evaluation of the composite higher-order observables. No approximation is involved.

Exact partition function↩︎

At \(h=0\) the eight configurations of \(\{-1,+1\}^3\) cluster into two symmetry classes under \(\phi\to-\phi\), as shown in Table 1.

Crossover temperature↩︎

Class A dominates Class B when \(2e^{3\beta J}>6e^{-\beta(J+4\gamma w)}\), or equivalently \(e^{4\beta(J+\gamma w)}>3\). Taking logarithms yields the crossover scale directly.

Note also that \(T^*/T_c = 4/\ln3 \approx 3.64\), hence the crossover of the isolated triad occurs at roughly \(3.6\times T_c\), which is well within the disordered phase of the network.

3 Reduction to the Triadic Ising Model↩︎

We derive that the critical behavior of the full COGENT\(^{\boldsymbol{3}}\) system near its formation transition \(T_c^{\mathrm{form}}\) is governed exactly by the triadic Ising model defined in Sec. 2, whose mean-field analysis and composite-operator scaling are developed in Secs. 4–6. All COGENT\(^{\boldsymbol{3}}\) notation and definitions follow Ref. [1] (specific section references are given inline). The argument requires three steps: (i) symmetry reduction of the formation field; (ii) collapse of the dynamic topology onto a static all-to-all hypergraph; and (iii) demonstration that the remaining Hamiltonian sectors contribute only irrelevant operators at \(T_c^{\mathrm{form}}\).

Symmetry reduction of the formation field↩︎

The formation field \(\phi_i(t)\in\mathbb{R}\) in COGENT\(^{\boldsymbol{3}}\) carries a global \(\mathbb{Z}_2\) symmetry \(\phi_i\!\to\!-\phi_i\), inherited from the symmetry of the formation correlator \(\Psi_{\mathrm{form}}\) under interchange of formation roles (Sec. 5 therein). Near \(T_c^{\mathrm{form}}\), the effective free energy for \(\phi_i\) takes the standard \(\mathbb{Z}_2\)-symmetric Landau form \[\mathcal{F}[\phi] = \sum_{i}\bigl[a(T)\,\phi_i^{2} + b\,\phi_i^{4}\bigr] + \mathcal{F}_{\mathrm{int}}[\phi] \label{eq:landau}\tag{8}\] where \(a(T)=a_0(T-T_c^{\mathrm{form}})\) changes sign at \(T_c^{\mathrm{form}}\), \(b>0\) ensures stability, and \(\mathcal{F}_{\mathrm{int}}\) collects the interaction terms from \(H_{\mathrm{group}}\).

The correct justification for replacing the continuous field \(\phi_i\) with binary spins is universality. Both the continuous \(\phi^4\) Landau–Ginzburg–Wilson theory and the discrete Ising model flow to the same Wilson-Fisher fixed point under renormalization, and therefore share identical critical exponents. The binary discretization \(\phi_a\in\{-1,+1\}\) preserves the \(\mathbb{Z}_2\) symmetry that controls the universality class, and no critical information is lost. Away from \(T_c^{\mathrm{form}}\), amplitude fluctuations \(\delta|\phi_i|\) are short-ranged with mass \(\propto|a(T)|>0\) and decouple from the long-wavelength critical mode. At \(T_c^{\mathrm{form}}\) itself the reduction is justified by the universality of the fixed point, not by a separation of modes.

Dynamic topology reduces to all-to-all in the thermodynamic limit↩︎

In COGENT\(^{\boldsymbol{3}}\) the active triad set \(E^{(3)}(t)\) evolves dynamically. Let \(\rho_N\) denote the mean fraction of agent triples forming active triads. The minimal condition for a non-trivial thermodynamic formation transition is \(\rho_N\to\rho>0\) as \(N\to\infty\), so that each agent participates in \(\sim\rho N^2\) triads.

Under this condition, the effective field on agent \(i\) from \(H_{\mathrm{group}}\) in the mean-field approximation is \[h_i^{\mathrm{eff}} = \frac{J_{\mathrm{eff}}}{N^2} \sum_{\tau\ni i}\,\sum_{\substack{j\in\tau \\ j\neq i}} \phi_j \label{eq:heff}\tag{9}\] a sum of \(n = 2K_i\) spin values, where \(K_i = |\{\tau \ni i : \tau \in E^{(3)}\}| \approx \rho N^2/2\) is the number of active triads containing agent \(i\). Writing \(\phi_j = m + \delta_j\) with \(\mathbb{E}[\delta_j]=0\) and \(\delta_j \in [-(1+m),\, 1-m] \subset [-2,2]\), the mean of \(h_i^{\mathrm{eff}}\) is \(\rho J_{\mathrm{eff}}m\) and its fluctuation is \(\Delta_i \equiv h_i^{\mathrm{eff}} - \rho J_{\mathrm{eff}}m = (J_{\mathrm{eff}}/N^2)\sum_\ell\delta_{j_\ell}\).

Although the formal sum contains \(n = 2K_i \approx \rho N^2\) terms, there are only \(N-1\) distinct agents. Each \(\phi_j\) appears in approximately \(c_{ij}\approx\rho N\) triads containing \(i\). Grouping by distinct agent, define \(Y_j = c_{ij}\,\delta_j\) with each \(Y_j\) bounded in \([-2\rho N,\, 2\rho N]\). Applying Hoeffding’s inequality [11] to the \(N-1\) grouped terms with threshold \(t = \varepsilon N^2/J_{\mathrm{eff}}\) gives \[P\!\left(|\Delta_i| \geq \varepsilon\right) \leq 2\exp\!\left(-\frac{\varepsilon^2\, N}{8\,\rho^2\,J_{\mathrm{eff}}^2}\right) \quad \forall\,\varepsilon > 0. \label{eq:hoeffding}\tag{10}\] For any fixed \(\varepsilon > 0\) the right-hand side decays exponentially in \(N\). The effective field concentrates on \(\rho J_{\mathrm{eff}}m\) as \(N\to\infty\), and the dynamic topology is asymptotically equivalent to the complete static hypergraph (\(\rho = 1\)). Every agent participates in \(\binom{N-1}{2} \sim N^2/2\) triads, giving \(h_i^{\mathrm{eff}} = (J_{\mathrm{eff}}/N^2)\cdot\rho N^2\cdot m = J_{\mathrm{eff}}m\) in agreement with 12 .

Consequently, the dynamic topology enters only through a renormalization of \(J_{\mathrm{eff}}\). The universality class and all critical exponents are independent of \(\rho>0\).

Remaining Hamiltonian components are irrelevant at \(T_c^{\mathrm{form}}\)↩︎

There are three residual components of the COGENT\(^{\boldsymbol{3}}\) Hamiltonian that must be accounted for.¹
Formation component \(H_{\mathrm{form}}\) – The GAN parameter variables \((G_\tau^{(m)},D_\tau)\) carry no \(\mathbb{Z}_2\) charge under \(\phi_i\!\to\!-\phi_i\), as they reside in the parameter space of the formation networks and are invariant under sign reversal of the formation field (Sec. 4.6.1 therein). Therefore, they cannot couple to the \(\mathbb{Z}_2\) order parameter at odd order. The leading coupling is \(\phi_{\mathrm{align},\tau}^2\,\phi_i^2\), a \(\phi^2\) mass renormalization that shifts \(T_c^{\mathrm{form}}\) but leaves the universality class unchanged.
Memory component \(H_{\mathrm{mem}}\) – The memory field \(M_i\) undergoes its own transition at \(T_c^{\mathrm{mem}}\neq T_c^{\mathrm{form}}\) (Sec. 8.1 therein). At \(T_c^{\mathrm{form}}\) \(M_i\) is non-critical, with finite correlation length \(\xi_{\mathrm{mem}}<\infty\). Integrating it out generates local operators \(\delta a\,\phi_i^2+\delta b\,\phi_i^4+\cdots\) that renormalize the Landau coefficients in 8 , without introducing new operators at the Ising fixed point.
Coupling component \(H_{\mathrm{coupling}}\) – The term \(H_{\mathrm{coupling}}=-\sum_\tau\gamma_\tau\phi_{\mathrm{align},\tau}\,m_\tau\) ties the non-critical (gapped) formation intensity \(\phi_{\mathrm{align},\tau}\) to local opinion coherence \(m_\tau\). Integrating out \(\phi_{\mathrm{align},\tau}\) at \(T_c^{\mathrm{form}}\) generates an effective pairwise term \(\delta J_{\mathrm{eff}}\sum_{\langle ij\rangle}\phi_i\phi_j\), which is absorbed into the renormalized coupling constant \(J_{\mathrm{eff}}\) (Eq. 5 ).
Taken together, the three reduction steps above show that the effective theory governing COGENT\(^{\boldsymbol{3}}\) near \(T_c^{\mathrm{form}}\) is precisely the triadic Ising model with coupling \(J_{\mathrm{eff}}\).

4 Mean-Field Theory↩︎

Consider now \(N\) agents in a complete \(3\)-uniform hypergraph, with Hamiltonian \[\mathcal{H} = -\frac{J_{\mathrm{eff}}}{N^2}\sum_{\tau\in E^{(3)}} \sum_{\langle ab\rangle\subset\tau}\phi_a\phi_b - h\sum_{i=1}^{N}\phi_i \label{eq:Hnet}\tag{11}\] where the normalization by \(N^{-2}\) ensures extensivity. In a mean-field approximation, \(\phi_j\to m\equiv\langle\phi_j\rangle\) for all \(j\ne i\). As previously noted, agent \(i\) participates in \(\binom{N-1}{2}\sim N^2/2\) triads. Each such triad contributes two terms of the form \(J_{\mathrm{eff}}\phi_i\phi_j/N^2\) to \(\mathcal{H}\). Replacing \(\phi_j\to m\) and summing over all \(\sim N^2/2\) triads gives \(2J_{\mathrm{eff}}m\cdot(N^2/2)/N^2=J_{\mathrm{eff}}m\) in the limit \(N\to\infty\). The effective single-site Hamiltonian from collecting all contributions is \[H_i^{\rm MF} = -(J_{\mathrm{eff}}m + h)\phi_i + \mathcal{O}(N^{-1}) \label{eq:HiMF}\tag{12}\] from which the self-consistency equation \[m = \tanh(\beta J_{\mathrm{eff}}m + \beta h) \label{eq:mf}\tag{13}\] follows immediately. Linearizing in \(m\) and \(h\) at \(h=0\) reveals the critical condition \(\beta_cJ_{\mathrm{eff}}=1\), giving:

5 Triadic Formation Order Parameter↩︎

The single-spin magnetization \(m\) is not the physically natural observable in triadic networks. The emergence of a coherent computational structure within a triad is measured by the three-spin correlator \[\Psi_{\mathrm{form}}= \Bigl\langle\frac{1}{|E^{(3)}|} \sum_{\tau=\{i,j,k\}}\phi_i\phi_j\phi_k\Bigr\rangle_T \label{eq:Pform}\tag{16}\] which is zero in the disordered phase (by \(\mathbb{Z}_2\) symmetry) and nonzero whenever a triad achieves coherent configuration. It corresponds to the formation order parameter \(\Psi_\mathrm{form}\) introduced in [1] (Eq. (95) therein) but derived here from first principles in the Boolean limit of the group-interaction sector.

Mean-field factorization↩︎

In the \(N \to \infty\) limit on a fully connected hypergraph, the standard cumulant expansion for the three-spin correlator is \[\langle\phi_i\phi_j\phi_k\rangle = m^3 + 3m\underbrace{\langle\phi_i\phi_j\rangle_c}_{\sim\,\mathcal{O}(N^{-1})} + \underbrace{\langle\phi_i\phi_j\phi_k\rangle_c}_{\sim\,\mathcal{O}(N^{-2})}. \label{eq:factorize}\tag{17}\]

Critical exponent↩︎

Substituting \(m\sim(T_c-T)^{1/2}\) from 15 into 18 yields:

This exponent does not coincide with the primary order-parameter exponent of mean-field Ising (\(\beta_\mathrm{Ising}^\mathrm{MF} = \tfrac{1}{2}\)), mean-field XY (\(\beta_\mathrm{XY}^\mathrm{MF} = \tfrac{1}{2}\)), or mean-field Heisenberg (\(\beta_\mathrm{Heis}^\mathrm{MF} = \tfrac{1}{2}\)) universality classes, all of which share \(\beta^\mathrm{MF} = 1/2\) regardless of the spin symmetry group (since mean-field theory is insensitive to the number of order-parameter components). However, it is precisely the composite-operator exponent \(n\beta_\mathrm{Ising}^\mathrm{MF}\) evaluated at \(n = 3\) which follows from \(\Delta_{\Psi_\mathrm{form}} = 3\Delta_\phi\) [8], [9].

Equation 20 is an exact consequence of the three-body structure. Coherent formation requires three spins to align simultaneously, imposing a cubic relation between \(\Psi_\mathrm{form}\) and the underlying magnetization. The exponent \(\beta_\mathrm{TF} = 3/2\) is an algebraic consequence of the composite-operator structure, and any order parameter constructed as the cube of a mean-field Ising mode inherits this effective exponent via \(\Delta_{\Psi_\mathrm{form}} = 3\Delta_\phi\).

Absence of susceptibility divergence↩︎

Let \(h_3\) couple extensively to the formation order parameter as \(\mathcal{H}\to\mathcal{H} - h_3 N\Psi_{\mathrm{form}}\). The thermodynamically consistent, intensive linear response is \[\chi_{\mathrm{TF}}\equiv \frac{\partial\Psi_{\mathrm{form}}}{\partial h_3}\bigg|_{h_3=0} = \beta\,N\,\mathrm{Var}(\Psi_{\mathrm{form}}). \label{eq:chiTF}\tag{21}\] It parallels the standard Ising definition \(\chi_\mathrm{Ising} = \beta N\,\mathrm{Var}(m)\), with \(\Psi_{\mathrm{form}}\) playing the role of order parameter. Both are \(\mathcal{O}(1)\) in \(N\): since \(\mathrm{Var}(m) = \chi_\mathrm{Ising}/(\beta N) = \mathcal{O}(N^{-1})\), the factor of \(N\) in 21 cancels exactly, giving \(\chi_{\mathrm{TF}}= \beta N\cdot(3m^2)^2\mathrm{Var}(m) = 9m^4\chi_\mathrm{Ising}\), which is genuinely intensive.

Using \(\Psi_{\mathrm{form}}=m^3\) and the delta-method approximation \(\mathrm{Var}(m^3)\approx(3m^2)^2\mathrm{Var}(m)\), together with \(m^2\sim(T_c-T)\) and \(\chi_\mathrm{Ising}\sim(T_c-T)^{-1}\) near \(T_c\):

The delta-method captures the leading power law. Near \(T_c\) fluctuations in \(m\) modify the prefactor but not the exponent \(\gamma_{\mathrm{TF}}=-1\).

As shown in the next section, the scaling exponent \(\gamma_{\mathrm{TF}}= -1\) is confirmed independently and without recourse to the variance argument by the field-theoretic two-point function. This is in sharp contrast to Ising criticality (\(\chi_{\rm Ising}\sim |T-T_c|^{-1}\) diverges on both sides). In principle, the non-diverging, monotone vanishing of \(\chi_{\mathrm{TF}}\) at \(T_c\) (which approaches \(T_c\) with a kink rather than a divergence) constitutes a signature of the TF scaling regime (see Sec. 7).

Scaling consistency↩︎

Since all TF exponents are defined via \(\Psi_\mathrm{form} = m^3\) and the relation \(\Delta_{\Psi_\mathrm{form}} = 3\Delta_\phi\), the Rushbrooke equality \[\alpha_\mathrm{TF} + 2\beta_\mathrm{TF} + \gamma_\mathrm{TF} = 0 + 3 + (-1) = 2 \label{eq:rushbrooke}\tag{23}\] is satisfied by algebraic construction and constitutes a self-consistency check on the exponent definitions, not an independent thermodynamic test.

An independent non-trivial check is provided by the field-theoretic two-point function at criticality. At the upper critical dimension \(d_c = 4\), the scalar field \(\phi\) has mean-field scaling dimension \(\Delta_\phi = 1\). The composite operator \(\Psi_\mathrm{form} = \phi^3\) then has dimension \(\Delta_{\Psi_\mathrm{form}} = 3\). Its momentum-space susceptibility scales as \[\chi_\mathrm{TF}(q) \sim |q|^{2\Delta_{\Psi_\mathrm{form}} - d} = |q|^{6-4} = |q|^2 \label{eq:chiTF95q}\tag{24}\] so the uniform susceptibility \(\chi_\mathrm{TF}(q{=}0) \sim \xi^{d - 2\Delta_{\Psi_\mathrm{form}}} = \xi^{-2} \sim t^{+1}\), giving \(\gamma_\mathrm{TF} = -1\) independently of Result 6. This confirms the exponent by an entirely distinct route [12].

6 Memory-Modified Dynamical Exponent↩︎

The architecture of COGENT\(^{\boldsymbol{3}}\) assigns each agent a memory field. To capture the history-dependent nature of cognitive processing without resolving the full microscopic fast-variable dynamics, we describe this field using a phenomenological Mori–Zwanzig memory ansatz \(K(t)\) [13]–[15]. Following [1] (Sec. 8.3.2), in the vicinity of a critical point this memory kernel modifies the dynamical exponent \(z\) (relating relaxation time to correlation length, \(\tau_{\rm relax}\sim\xi^z\)) to \[z_\alpha = z_\alpha^{(0)} + \gamma_K\int_0^\infty K(t)\,t^{\theta_\alpha-1}\,\mathrm{d}t \label{eq:zdyn}\tag{25}\] with \(\theta_\alpha\) the autocorrelation decay exponent of the observable \(\alpha\). For \(\Psi_\mathrm{form}=m^3\), the exponent \(\theta_\mathrm{TF}\) is obtained by analyzing the six-point correlation function in the disordered phase \(T>T_c\), where Gaussian fluctuations dominate.

The calculation is straightforward. In the disordered phase \(T > T_c\), the free energy is dominated by the quadratic term \(F \sim \tfrac{1}{2}N a(T) m^2\) with \(a(T) = a_0(T - T_c) > 0\). Hence, the \(m\)-fluctuations are Gaussian to leading order in \(N^{-1}\), with \(\langle m^2\rangle = [Na(T)]^{-1}\), and Isserlis’ theorem applies exactly. Denoting \(C(t) \equiv \langle m(t)m(0)\rangle_c\), the six-point function factorizes as \[\langle m(t)^3 m(0)^3 \rangle = 6\,C(t)^3 + 9\,\langle m^2\rangle^2\,C(t). \label{eq:wick}\tag{26}\]

Both terms scale identically in \(N\) because \(\langle m^2\rangle^2 \sim N^{-2}\) and \(C(t) \sim \langle m^2\rangle g(t) \sim N^{-1}g(t)\), giving \(6C(t)^3 \sim N^{-3} g(t)^3\) and \(9\langle m^2\rangle^2 C(t) \sim N^{-3} g(t)\). The linear term in \(C(t)\) is not suppressed by any additional power of \(N\), as it is of the same order as the cubic term. For a single-exponential correlation \(C(t)=\langle m^2\rangle e^{-t/\tau_0}\), the autocorrelation of \(\Psi_\mathrm{form}\) becomes \[\langle \Psi_\mathrm{form}(t)\Psi_\mathrm{form}(0)\rangle \propto \langle m^2\rangle^3\bigl[6e^{-3t/\tau_0} + 9e^{-t/\tau_0}\bigr]. \label{eq:corr}\tag{27}\]

At long times, \(e^{-t/\tau_0}\) dominates \(e^{-3t/\tau_0}\) so the \(\Psi_{\mathrm{form}}\) autocorrelation is controlled by the same exponential scale as \(m\). This off-critical argument (valid for all \(T>T_c\)) shows that \(\Psi_{\mathrm{form}}\) and \(m\) share the same long-time decay mode. By continuity as \(T\to T_c^+\), where the exponential relaxation time \(\tau_0\sim\xi^z\) diverges and the decay crosses over to a critical power law \(C(t)\sim t^{-\theta_0}\), the composite correlator inherits the same power-law exponent. As a result, \(\theta_\mathrm{TF} = \theta_0\), the critical autocorrelation exponent of the single-spin magnetization \(m\).

For the canonical single-exponential kernel \(K(t) = \tau_K^{-1}e^{-t/\tau_K}\), the integral in 25 evaluates to \(\tau_K^{\theta_0-1}\Gamma(\theta_0)\) via the Mellin transform.

For \(\theta_0>1\), in the limit \(\tau_K\to0\) (memoryless), \(\Delta z\to0\), and the standard exponent is recovered. When \(\tau_K\to\infty\) (persistent memory) we have \(\Delta z\to\infty\), and the system exhibits complete dynamical arrest, characteristic of glassy behavior. For \(0<\theta_0<1\) the limits are reversed (long memory suppresses the dynamical correction and short memory amplifies it).

7 Discussion↩︎

The seven results presented above characterize the TF composite scaling regime. Table 2 summarizes and compares the exponents.

Three remarks are now in order.
Exactness of \(\beta_{\mathrm{TF}}=3/2\) – The result is not perturbative but follows from the exact factorization \(\Psi_{\mathrm{form}}=m^3\) in the \(N\to\infty\) mean-field limit, where connected three-body correlations are suppressed by \(\mathcal{O}(N^{-2})\) (Result 4). Corrections at finite \(N\) modify the exponent by terms \(\sim N^{-1}\) arising from the connected three-body correlator.
Negative effective \(\gamma_{\mathrm{TF}}\) – The susceptibility \(\chi_{\mathrm{TF}}\to0\) at \(T_c\) is an exact consequence of the cubic relation 18 and is physically interpretable. Three agents must be coherently perturbed by the field \(h_3\) to produce a response, and the probability of joint triadic alignment vanishes as the system approaches the disordered phase from below. This feature is in principle separable from Ising criticality: whereas \(\chi_\mathrm{Ising}\) diverges symmetrically on both sides of \(T_c\), a direct measurement of \(\chi_\mathrm{TF}\) requires applying a field \(h_3\) conjugate to the three-body product \(\phi_i\phi_j\phi_k\) and recording the response of \(\Psi_\mathrm{form}\). In a multi-agent AI system, \(h_3\) corresponds to a coordinated triadic bias. Its experimental identification and the resulting non-diverging susceptibility profile provide a concrete protocol for distinguishing TF from scalar-Ising criticality, deferred to future work.
Tunable dynamical scaling – Result 7 shows that the TF regime exhibits a continuously tunable dynamical exponent parametrized by \(\tau_K\), qualitatively analogous (although mechanistically distinct) to the Kosterlitz–Thouless line in two-dimensional systems.
The validity of mean-field theory and upper critical dimension – The mean-field results (Results 3–7) are exact in the \(N\to\infty\) limit of the fully connected hypergraph. For a spatially embedded system in dimension \(d\), mean-field theory is valid above the upper critical dimension \(d_c\) determined by the Ginzburg criterion. Near \(T_c\), the Landau free energy expanded in \(m\) takes the standard \(\phi^4\) form (the cubic term vanishes by \(\mathbb{Z}_2\) symmetry) so the gradient-term coefficient and the quartic coupling are unchanged from those of the scalar Ising model. The Ginzburg criterion therefore gives \(d_c = 4\), identical to the Ising case: the three-body structure of \(\Psi_\mathrm{form}\) modifies the observable but not the free energy topology.² For \(d < 4\) the exponents \(\beta_\mathrm{TF}\), \(\gamma_\mathrm{TF}\), and \(z_\mathrm{TF}\) will receive renormalization-group corrections. Determining whether these corrections preserve the composite-operator relation \(\beta_\mathrm{TF}^\mathrm{RG} = 3\beta_\mathrm{Ising}^\mathrm{RG}\) at fixed points below \(d_c\) is a natural extension.
The present work characterizes the TF scaling regime at the mean-field level. Natural extensions include (i) fluctuation corrections in spatial dimension \(d<d_c=4\) via explicit \(\epsilon\)-expansion renormalization-group methods, verifying that the Ginzburg criterion argument of Sec. 7 extends to the composite operator \(\phi^3\) at the Wilson-Fisher fixed point, and determining whether the composite-operator relation \(\beta_\mathrm{TF}^\mathrm{RG}=3\beta_\mathrm{Ising}^\mathrm{RG}\) is preserved below \(d_c\); (ii) extension to the COGENT\(^{\boldsymbol{3}}\) role-transition sector, which introduces a \(\mathbb{Z}_3\) (Potts-like) component coupled to the Ising \(\Psi_{\mathrm{form}}\), potentially giving rise to a multicritical point; and (iii) identification of measurable experimental signatures in multi-agent AI systems, where the non-diverging susceptibility and the \(\beta_{\rm TF}=3/2\) scaling provide direct tests of the TF scaling regime.

8 Conclusions↩︎

In this paper we have derived two unconditionally exact results (Results 1–2, valid on \(\{-1,+1\}^3\)) and five results exact within mean-field theory (Results 3–7, valid in the \(N\to\infty\) limit of the fully connected 3-uniform hypergraph) for the triadic Ising model and its network generalization. The key finding is that the triadic formation correlator \(\Psi_\mathrm{form} = \langle\phi_i\phi_j\phi_k\rangle\) scales as \((T_c-T)^{3/2}\) near criticality, with effective exponent \(\beta_\mathrm{TF} = 3/2\) that is the \(k=3\) instance of the composite-operator exponent in 1 , phenomenologically identifiable from any scalar order-parameter exponent. The associated susceptibility vanishes (rather than diverges) at \(T_c\), which is independently confirmed by the field-theoretic two-point function (Sec. 5). Memory kernels introduce a tunable dynamical exponent, completing the characterization of the composite-operator critical observables in the TF sector.

These results provide the statistical-mechanical foundation for phase transitions in AI systems with a higher-order interaction structure. Crucially, we demonstrate that when an AI architecture computes via triadic units, it accesses a composite scaling regime that is fundamentally smoother and more intrinsically robust to critical noise than standard pairwise networks.

References↩︎

---

1. A full renormalization-group analysis of these components is deferred to future work.↩︎

2. Below \(d_c=4\) the composite operator \(\phi^3\) may acquire an anomalous dimension, so \(\beta_{\mathrm{TF}}^{\mathrm{RG}} = 3\beta_{\mathrm{Ising}}^{\mathrm{RG}}\) is not guaranteed a priori and requires explicit RG computation.↩︎