Information Creation as Retained Correlation: A Kinetic Framework for the Emergence of Collective Dynamics

A school of fish turns as one; an array of threshold switches locks into a common rhythm. Emergence—order at the level of the whole that no component carries—has names and measures but no mechanism. For one class of emergence, collective modes of populations of interacting elements, we propose one. The information a network creates about itself is pair correlation: the mutual information between two elements is D_KL(f_2 ∥ f_1f_1) and vanishes if and only if their pair correlation g_2 does. Boltzmann’s Stosshypothese and one-population mean-field closures set it to zero by con-struction, which is why it is invisible from below. Retaining g_2 in the kinetic hierarchy turns the network into a loop that writes correlation, retains it for a time τcand reads it back with strength g. Eliminating the pair exactly yields a memory kernel and a quadratic pole equation in which stored information acts as a restoring force. Four bounds follow: collective modes exist only inside a reten-tion window, 4gτ_0τ_c> (τ_0−τ_c)2; no line is narrower than 1/τ_0; a line of quality Q requires g ≥ 4Q^2; and it dissipates at least ≈ 4Qω_0 nats per unit time. As the network learns, its collective peak climbs as √C at fixed width. These results yield an operational definition: a collective behaviour is emergent if it is carried by correlation and vanishes without it, if that correlation is retained and read back, and if the collective acquires a property no element has. Its onset is derived rather than postulated: at an exceptional point of the non-reciprocal write–read dynamics the collective frequency appears as the square root of the distance from threshold. Because two-population models share the pole equation, the temporal spectrum alone cannot identify the mechanism. Three tests decide it: a re-entrant retention window, a linewidth floor, and rate-preserving decorrelation, which must abolish an emergent mode and leave a merely collective one intact. Emergence in this sense is irreducibility to the factorised description, and it comes in degrees. The derived stage is computable in closed form. A second, self-sustaining stage at C_c= 0.5, which needs both the nonlinearity of the threshold elements and that of the collision term, is conjectured to be accessible only by simulation, and in infinite networks properties undecidable from below are possible. Strong emergence in the ontological sense is not claimed: a derivation from the parts can establish the converse, never that. Keywords: emergence · causality · information theory · complex networks · statistical mechanics · Boltzmann transport equation · molecular chaos · critical exponents · Kullback-Leibler divergence · free energy principle · Landauer principle · self-modeling systems · neuromorphic computing.