Synergy as the Failure of Distributivity.

The concept of emergence, or synergy in its simplest form, is widely used but lacks a rigorous definition. Our work connects information and set theory to uncover the mathematical nature of synergy as the failure of distributivity. For the trivial case of discrete random variables, we explore whether and how it is possible to get more information out of lesser parts. The approach is inspired by the role of set theory as the fundamental description of part-whole relations. If taken unaltered, synergistic behavior is forbidden by the set-theoretic axioms. However, random variables are not a perfect analogy of sets: we formalize the distinction, highlighting a single broken axiom-union/intersection distributivity. Nevertheless, it remains possible to describe information using Venn-type diagrams. The proposed multivariate theory resolves the persistent self-contradiction of partial information decomposition and reinstates it as a primary route toward a rigorous definition of emergence. Our results suggest that non-distributive variants of set theory may be used to describe emergent physical systems.