Möbius Decomposition of Multipartite Correlations: Typed Atoms, a Faithfulness Calculus for Order-k Claims, and Higher-Order Synergy in NISQ and Many-Body Systems

This version updates the DAGI Möbius methods paper to v2.2, with typed atoms, a dual reliability-and-faithfulness admissibility frontier, and certified-budget refinements for order-k claims. The paper presents a subset-poset Möbius decomposition for multipartite correlations. For cumulative information observables on variable sets, Möbius inversion yields irreducible atoms assigned to sub-configurations. The manuscript specializes the construction to total correlation, introduces a positive-tail statistic for active irreducible constraints, summarizes computation and reliability frontiers, and connects the method to NISQ and many-body settings. Version 2.2 supersedes v2.1, v2.0, v1.1, and v1.0.