Mathematics Has Already Chosen Federalism: The Logic of Domain Separation

Mathematics, as actually practiced, operates as a federated system: practitioners work within autonomous domain-specific axiomatizations (geometry, algebra, analysis) and construct explicit bridges only when cross-domain reasoning is required. This organization is not accidental; it is a structural adaptation that safeguards local decidability and algorithmic efficiency. Yet the dominant foundational narrative still operates on the Compiler Myth—the belief that all mathematics must theoretically compile down into ZFC set theory to achieve rigor. We argue that this monolithic reductionism confuses representational universality with logical priority. Embedding a decidable (tame) domain into an undecidable (wild) one does not clarify foundations; it imposes a crippling epistemic overhead. It buries efficient, domain-specific decision procedures under general proof search and destroys the native structural immunities of the object. We introduce the Decidability Threshold — a litmus test based on Negation, Representability, and Discrete Unboundedness — to explain why mathematicians instinctively isolate tame domains from wild ones. Finally, we distinguish the Mathematician (builder of formal systems) from the Scientist (consumer modeling empirical reality). We argue that federalism is not a failure of unification, but the primary safeguard preventing the scientist from inadvertently importing uncomputable, undecidable paradoxes into physical theories. We show that for empirical applications, syntactic safety is insufficient; valid scientific modeling must be strictly confined to the constructively computable sub-fragments of these domains.