Dual Extreme Value Topological Isomorphic Field Theory: A Global Unified System of π Geometric Continuous Extremum and Prime Discrete Extremum (Final Complete Version)

Since the separation of number theory and geometry in mathematics, discrete prime systems and continuous transcendental systems have long been regarded as mutually independent branches without underlying homologous structures. The distribution of primes has always been described as a statistical asymptotic phenomenon lacking rigorous geometric essential interpretation, and the circular constant π has never been endowed with fundamental number-theoretic significance. This paper firstly establishesthe dual extreme value topological isomorphic field theory, and proves that the discrete extreme value system of primes and the continuous geometric extreme value system of π belong to two projective forms of the same universal mathematical field. This research constructs the global coupled topological invariant Ω and establishes the third-order residual zeroing correction system, realizing one-to-one zero-residual global matching between finite π precision from 1 to 100 digits and prime extreme values. By deriving 60 original closed-loop formulas, this work achieves unification of finiteness and infinity, real and complex domains, number theory and geometry, as well as statistical laws and field mechanisms. The study fundamentally explains the essence of prime distribution, uniqueness of the Riemann critical line, formation mechanism of prime interval fluctuation, and dynamic convergence of extreme values, thoroughly eliminating the fundamental separation between discreteness and continuity in modern mathematics. The system is globally self-consistent, fitting-free, exception-free, and reversibly derivable, constituting a paradigm-level fundamental revolution in modern mathematics.