Structural Impossibility of Antichain-Lattice Partial Information Decomposition

2026-04-04

AI Summary

This work addresses a fundamental inconsistency in multivariate partial information decomposition (PID) when applied to systems with three or more sources: the antichain lattice–based representation fails to simultaneously satisfy widely accepted axioms, leading to theoretical incoherence. The authors propose a target-free information decomposition (SID) framework for trivariate systems and demonstrate that the root of this axiom conflict lies in an inherent structural limitation of the antichain lattice—its information atoms cannot uniquely reconstruct mutual information. By integrating information theory, lattice theory, and system decomposition principles, they prove that identical information atoms may correspond to distinct mutual information values. This finding establishes that the antichain lattice is ill-suited to support a consistent and complete multivariate information decomposition, advocating instead for a relational foundation in the development of next-generation multivariate information measures.

Abstract

Partial Information Decomposition (PID) represents multivariate mutual information via antichain-lattice that aims to specify which source groups can recover which informational components of a target. For three or more sources, widely desired PID axioms become mutually incompatible. This is often treated as an axiomatic tuning issue. This paper argues that the obstruction is representational, rooted in the antichain indexing itself, so that purely axiomatic adjustments within an antichain-lattice structure cannot resolve it in general. We first introduce System Information Decomposition (SID) for the special target-free three-variable setting, obtaining a self-consistent entropy decomposition with an operational redundancy definition. More fundamentally, we then show that for general multivariate PID, there is no universal rule that recovers the decomposed mutual information from the antichain-indexed information atoms. In particular, two systems can share identical atoms regardless of any axioms while having different mutual information. These results reveal the limits of antichain-lattice and motivate relation-based foundations for multivariate information measures.

Problem

• Partial Information Decomposition
• antichain-lattice
• multivariate mutual information
• information atoms
• representational limitation

Innovation

• Partial Information Decomposition
• antichain-lattice
• System Information Decomposition
• multivariate mutual information
• information atoms