A Constraint Measure Axiomatic System and the Structural Study of Prime Distribution

This paper constructs a constraint measure axiomatic system within the framework of ZFC axiomatic set theory. A strictly monotonically increasing constraint measure function μ: ℕ → ℝ⁺ along with its three axioms is introduced, and the system is rigorously proved to be self-consistent and complete. Under this system, classical prime numbers are rigorously proved to be equivalent to measure critical values. The unboundedness and infinitude of prime distribution, as well as a weak form of the twin prime conjecture, are derived as natural corollaries of this axiomatic system. The Goldbach conjecture is equivalently transformed into a symmetric interval proposition within this system, and its equivalent propositional transformation is completed. This paper does not redefine prime numbers; it only establishes a new set of axiomatic tools and proves that several core propositions of classical number theory can be uniformly derived under these tools.