Axiomatic Structures: A ZFC–Internal Framework for Projection, Measure, and Curvature

We introduce a minimal ZFC–internal axiom system for axiomatic structures: tuples(X, A, μ, R, I, ΠR, G, E0, η) coupling a finitely additive measure, an idempotent retraction,and an idempotent symmetric relation via a single coupling law (Axiom III). In the identity–retraction regime ΠR = id, the axioms force G to be an equivalence relation and reduce thecoupling law to a blockwise normalization constraint μ(Ck) = 11−η on G–components. Weprove satisfiability in ZFC by explicit finite and countable models, including finite familieswith η̸ = 0. We also record basic structural consequences (measure–preserving projection onR, equivalence–like behavior of G on components, and non–generation of closed componentsby projection) and give several finite examples. We also note that in scaling limits where thetotal observational weight is not probability-normalized, the global constraint shifts to localinstances of the coupling law, a viewpoint compatible with normalization-based continuumprocedures. This provides a compact rigidity framework in which projection, measure, andclosure interact, with potential applications in analysis, geometry, and mathematical physics.