Impossibility Atlas Science: A Classification Science of Structural Impossibility

Impossibility Atlas Science (IA-Ω) introduces a rigorous classification framework for understanding when and why questions fail to admit invariant answers within specified formal ontologies. Departing from traditional focus on proofs, algorithms, or probabilistic modeling, IA-Ω treats impossibility as a structural outcome, not a methodological failure. Key contributions include: Ontology-Audit Framework: A formal protocol for assessing the expressibility, executability, typicality, and selection structure within a given ontology. Four Canonical Boundary Classes: IA-Ω refines Gödel-class (expressibility), Turing-class (computability), Shannon-class (typicality), and introduces the Rhys-class—capturing impossibilities due to symmetry that cannot be resolved without adding external selectors. Rhys-Ω-1 Principle: Establishes the non-selection theorem, proving that no invariant answer can exist to symmetry-respecting questions without breaking automorphism invariance. UTI Projection Tool: A minimal diagnostic projection onto universality, typicality, and external-structure axes, enabling classification via structural demands. Negative-Law Outputs: The framework yields reusable impossibility certificates, boundary tables, and structural “no-go” theorems, defining the limits of admissible reasoning across domains. IA-Ω positions itself not as a theory of failure, but as a foundational science of boundary detection—providing clarity where solution-seeking methods silently exceed their ontological scope.