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.

翻译：部分信息分解（PID）通过反链格表示多元互信息，旨在指定哪些源组可以恢复目标的哪些信息成分。对于三个或更多源，广泛期望的PID公理变得互不相容。这通常被视为公理调优问题。本文论证，这种障碍源于表示层面，其根源在于反链索引本身，因此仅靠反链格结构内的公理调整无法在一般情况下解决该问题。我们首先针对无目标的三变量特例引入系统信息分解（SID），通过操作性的冗余定义获得自洽的熵分解。更进一步，我们随后证明，对于一般多元PID，不存在通用规则能从反链索引的信息原子中恢复分解后的互信息。特别地，两个系统可能共享完全相同的原子（无论任何公理如何），却具有不同的互信息。这些结果揭示了反链格的局限性，并推动了基于关系的多元信息测度基础研究。