Why Axioms Fail Across Domains: Structural Constraints on Consensus and Cognition

Are axioms truly universal? This paper argues they are not. Through examples from geometry, arithmetic, and language, it shows that what appears necessary holds only under specific Recognition-Realms and structural conditions, and fails once transferred across domains with different structural bases. The paper further demonstrates that meaning, computation, and value judgment cannot be universally shared, due to differences in Causal-Base and Coloring. These differences are traced to the generative constraints of Recognition-Realms, where differentiation, compression, and convergence constitute the minimal conditions for structured cognition. On this basis, consensus is derived as a local structural alignment under four conditions: homogeneity of field, isomorphism of base, shared dimensionality of sensing, and proximity of Coloring. Axioms are thus defined as irreducible consensus—the starting points of structural assertion, rather than domain-independent truths. This work directly challenges the assumption of universal axioms in cognition, neuroscience, and artificial intelligence, and has direct implications for generalization and representation in AI systems.