The Relativity of AGI: Distributional Axioms, Fragility, and Undecidability

We study whether Artificial General Intelligence (AGI) admits a coherent theoretical definition that supports absolute claims of existence, robustness, or self-verification. We formalize AGI axiomatically as a distributional, resource-bounded semantic predicate, indexed by a task family, a task distribution, a performance functional, and explicit resource budgets. Under this framework, we derive four classes of results. First, we show that generality is inherently relational: there is no distribution-independent notion of AGI. Second, we prove non-invariance results demonstrating that arbitrarily small perturbations of the task distribution can invalidate AGI properties via cliff sets, precluding universal robustness. Third, we establish bounded transfer guarantees, ruling out unbounded generalization across task families under finite resources. Fourth, invoking Rice-style and G"odel--Tarski arguments, we prove that AGI is a nontrivial semantic property and therefore cannot be soundly and completely certified by any computable procedure, including procedures implemented by the agent itself. Consequently, recursive self-improvement schemes that rely on internal self-certification of AGI are ill-posed. Taken together, our results show that strong, distribution-independent claims of AGI are not false but undefined without explicit formal indexing, and that empirical progress in AI does not imply the attainability of self-certifying general intelligence.