Χ-Ω — Structural Limits of Attribution in Irreversible Systems: A Classification of Invariant-Satisfied Global Failure

This paper introduces Χ-Ω, a structural classification of failure in irreversible systems where global collapse can occur despite full local invariant compliance and audit success. Challenging the assumption that all system-level failures must be locally attributable, Χ-Ω defines a precise boundary where attribution, intervention, and prevention become undefined—not due to epistemic limitation or oversight, but due to the expressive limits of the governing descriptive language. Key contributions include: Invariant-Satisfied Global Failure (ISGF): A condition where all local rules are upheld, yet the system irreversibly fails without any accountable violation. Core Structural Patterns: Defines a minimal pattern basis—Interaction Expressivity Mismatch (IEM), Explanatory Language Exhaustion (ELE), and Attribution Surface Nullity (ASN)—that underpins the Χ-Ω classification. Non-Intervention Closure: Proves that within certain descriptive regimes, no admissible intervention operators can be defined without extending the language or violating its invariants. Minimal Witness Language ℒ₀: Constructs a deliberately austere language to demonstrate that Χ-Ω failure is not an artifact of complexity, but a predictable limit within common governance frameworks. Language-Relative Jurisdiction: Χ-Ω is not a critique of governance, but a boundary condition—failure is not misbehavior, but unrepresentability under current descriptive constraints. By formalizing when and how governance languages lose their capacity to characterize failure, Χ-Ω offers a profound reframing: some systemic collapses cannot be explained within the very rules designed to prevent them—not due to failure of enforcement, but because of structural mismatch between language and reality.