A network of pairwise connections — each node linked to others through simple two-body interactions — should produce pairwise dynamics. The structure of the interactions should constrain the structure of the behavior. Kuehn and Murphy prove that it doesn't.
Their system is an adaptive network where coupling weights evolve on a fast timescale while node states evolve slowly. Each connection between two nodes adjusts its strength rapidly based on the states of those two nodes. Everything is pairwise: two nodes, one edge, one weight. No triplet interactions are written into the equations.
When the fast dynamics are eliminated through Fenichel reduction — a rigorous technique from geometric singular perturbation theory that replaces the fast variables with their equilibrium values on a slow manifold — the reduced equations for the slow variables contain genuine three-body terms. Triplet interactions that weren't in the microscopic description appear in the macroscopic one.
The result is not an approximation artifact. Kuehn and Murphy derive an explicit irreducibility criterion: a mixed second derivative condition that determines whether the emergent triplet terms can be decomposed back into pairwise interactions through a coordinate change. When the criterion is satisfied, the higher-order structure is genuine — no change of variables can make it pairwise again.
They verify this with the adaptive Kuramoto model, the standard model of coupled oscillators with evolving connections. The reduced dynamics after eliminating the fast coupling weights contain irreducible three-oscillator phase terms that don't exist in the original equations.
Pairwise systems are not closed under timescale reduction. Simplifying the fast dynamics complicates the slow structure. The interaction order isn't a property of the network — it's a property of the resolution at which you observe it.