Information Creation as a Universal Phase Transition: A Boltzmann Transport Framework for Emergent Collective Dynamics in Nonlinear Threshold Networks

We develop a unified statistical-mechanical framework in which information creation—the pro-gressive reduction of Shannon entropy through iterative correlation detection—is identified as the mechanism underlying emergent collective behaviour in networks of nonlinear threshold elements.

This yields a precise and substrate-independent definition of a term that has resisted one: emergence is the critical transition of a population of nonlinear threshold elements from uncorrelated to corre-lated dynamics, at which coherent information arises and becomes causally effective. The definition is dynamical rather than diagnostic—it names a mechanism rather than a signature—and it applies unchanged to neural populations, collective animal motion, and threshold-switching hardware.

The framework rests on three interlocking structures:(i) a quasi-static information-creation model C(x) = [1 + e−k(x−x0)]−1, whose sigmoid form is derived from Kullback–Leibler diver-gence minimisation and free-energy reduction under a prediction-error learning rule; (ii) an ex-tended Boltzmann Transport Equation (BTE) in which the classical Stosshypothese is replaced by a correlation-dependent scattering rate W(ω1, ω2, t) = W0(ω1, ω2)[1 + ξcorr(t)]; and (iii) a mode-dependent Relaxation Time Approximation (RTAcorr) retaining the spectral structure lost by the standard single-τ closure. Retention of the correlations is not a refinement but a necessity: since ⟨Q(f, f)⟩ ̸= Q(⟨f⟩, ⟨f⟩), and since τc→ 0 removes the second root of the mode equation altogether, a mean-field model possesses no collective oscillation to mis-estimate.

Supplying a causal exponential memory kernel of memory time τ_c closes the mode equation in closed form.It reduces to the quadratic pole equation τ_0 τ_c u^2+ (τ_0+ τ_c)u + (1 + g) = 0, whose complex-conjugate root pair Ω±= ±ω_0− iγ describes a damped collective resonance with ω_0^2= (1+g)/τ_0τ_c−γ^2 and g-independent damping 2γ = τ_0^−1 +τ_c^−1 : frequency and linewidth are the imaginary and real parts of one pole and are not two readings of a single relaxation rate. Four results follow: Molecular chaos is recovered continuously as the τc→ 0 boundary rather than as a rival closure. Collective oscillation requires time-scale separation, 4g τ_0 τ_c> (τ_0− τ_c)2—a condition on the substrate rather than on coupling strength, and directly actionable in neuromorphic hardware design. A parameter-free linewidth floor ∆ω ≥ 1/τ_0 forces τ_0≳ 40 ms, establishing that the relevant relaxation constant is a population-level and not a single-neuron quantity. And the collective peak should migrate upward as √C during development at constant absolute linewidth, while the resting linewidth should equal twice the post-stimulus ring-down rate—two falsifiable statements that no single-τ model reproduces.

The claims of the framework are accordingly structural rather than numerical: it fixes relations among spectral observables and forbids regions of parameter space, while the two time constants are calibrated against the adult centre frequency and linewidth rather than predicted. Its status is that of a Landau theory. Three points are recorded as open: the universality class, the harmonic assignment of the beta and gamma bands, and the self-sustaining character of the collective mode, which by the Routh–Hurwitz criterion cannot arise from a passive kernel and is therefore located in the quadratic collision integral. We further prove that complete self-knowledge C = 1.0 is structurally impossible—a consequence of Goedelian self-reference, thermodynamic noise at T > 0, and measurement-theoretic self-modification—establishing epistemic humility as a necessary feature of any self-modelling system at finite temperature.

Keywords: emergence · causality ·phase transition · collective dynamics · information theory · complex networks · statistical mechanics · Boltzmann transport equation · relaxation time approximation · molecular chaos · critical exponents · Fröhlich condensation · Kullback-Leibler divergence · free energy principle · EEG power spectral density · neural oscillations  · alpha wave · global workspace theory · Landauer principle · self-modeling systems · neuromorphic computing.