Telling coordination from coincidence: a no-interaction twin for collective systems
Eric J. Couture
physics.soc-ph
Nov 7, 2026 · v1
TL;DR
Proves in Lean 4 that, for any pairwise force obeying Newton's third law over any commutative ring, the centre of mass follows the no-interaction law exactly.
Abstract
Measures of emergence based on information theory or causal closure ask whether a group variable has a predictable law of its own, but not whether that law was produced by the members' interactions: the centre of mass of non-interacting random walkers passes the standard criterion $Ψ$. We introduce the interaction share S, the fraction of the forecast error of a system's no-interaction twin (the same members with every interaction removed) that the real group law eliminates, and a projected estimator with expectation zero without interaction. For Gaussian forecasts, $-\frac{1}{2}\ln(1-S)$ is the expected Kullback-Leibler divergence of the real group law from the twin's. S is exactly zero for the centre of mass of linearly coupled walkers, and a machine-checked proof extends this to any pairwise force obeying Newton's third law. Across magnets, oscillators and flocking models it rises from 0 to about 1 where collective order appears. With time-shifted virtual groups and positive controls, it separates real fish schools (0.54-0.91, 17 recordings, two species) from groups that never interacted (-0.10 to 0.07), and it falls to zero when zebrafish are placed in the dark. In 221 mouse brain areas and in US stock markets the shares are small (about 1-3%) and were confirmed in pre-registered tests on unseen data. For linear forecasts the share is a memory gap, computable from two autocorrelation curves without fitting a twin; it matches measured shares (Spearman 0.97) and predicted fresh markets within 0.004. All code, data pipelines and hashed pre-registrations are public.
Problem
Information-theoretic emergence criteria test whether a group variable has a predictable law of its own, but not whether that law comes from interactions among members. For example, the centre of mass of non-interacting random walkers passes the standard criterion Psi.
Approach
The authors define the interaction share S: the fraction of a no-interaction twin's forecast error that the real group law eliminates. They pair it with a projected estimator whose expectation is zero when there is no interaction. For Gaussian forecasts, S is related to a KL divergence. A Lean 4 proof shows that, for any pairwise force obeying Newton's third law, the sum of positions follows the no-interaction law exactly, and that the twin's law is then the only exact law.
Results
S rises from 0 to about 1 where collective order appears in magnets, oscillators and flocking models. It separates real fish schools (0.54-0.91) from groups that never interacted (-0.10 to 0.07), and falls to zero for zebrafish in the dark. Mouse brain areas and US stock markets show small shares (1-3%), confirmed by pre-registered tests; a linear memory-gap formula matches measured shares (Spearman 0.97).
