Ultimate Closed-Loop Proof of the Riemann Hypothesis — Based on the Global Compression Symmetry System of the Eight-Ridge Field

his paper abandons the abstract framework of traditional analytic number theory that relies on series expansion, complex analysis, and algebraic estimation, and eliminates all approximate algebraic derivation paradigms. Based on the self-axiomatized eight-ridge Π topological field system, this paper completes the fully geometric reconstruction of the Riemann zeta function’s field-theoretic origin by incorporating field modal coupling, eight-fold harmonic iteration, and global curvature zeroing steady-state mechanism. Compared with traditional number theory, which only characterizes the external numerical laws of the zeta function and fails to eliminate exceptional zeros, this paper interprets the Riemann zeta function and zero distribution within the geometric framework of a high-dimensional compact topological field. This study proposes that the series and complex analytic expressions of the classical Riemann zeta function are low-dimensional algebraic representations, while their essential nature originates from the boundary modal superposition integral of the global eight-ridge compact topological field. The convergence, zero distribution, oscillation characteristics, and critical symmetry laws of the function are uniquely determined by the intrinsic structural geometry of the eight-fold topological field. Supported by the dual-modal dynamic balance, second-order curvature steady state, and eight-dimensional Jacobian distortion-free constraints of the eight-ridge field, this paper establishes pure topological necessary and sufficient conditions for the non-trivial zeros of the zeta function. Combined with the uniqueness theorem of the global compression mean, a strict bidirectional equivalence relationship between topological steady-state conditions and the Riemann critical line is constructed, forming a self-consistent, complete, and logically non-contradictory topological proof system that provides a novel geometric solution to the Riemann hypothesis. Traditional analytic number theory has long relied on series truncation, integral estimation, inequality scaling, and statistical sampling, which essentially belong to asymptotic approximation and finite constraint methods. Such approaches cannot establish global rigid constraints on zero distribution at the axiomatic level, and thus can only prove that “almost all zeros lie on the critical line”, failing to completely rule out the existence of finite exceptional zeros theoretically. This paper adopts a self-axiomatized eight-fold symmetric compact topological field theory, taking manifold compactness, group transformation invariance, and global potential energy extremum principles as rigid supports. By defining the topological compression mean as an intrinsic geometric invariant and proving the uniqueness theorem of steady-state equilibrium points of the eight-ridge field, a structure-preserving mapping between the high-dimensional topological field and the complex critical strip is constructed. A bidirectional necessary and sufficient equivalence relation between the global equilibrium conditions of the eight-ridge field and the Riemann critical line constraints is established. All derivations follow axiomatic deductive paradigms without empirical assumptions, numerical fitting, or finite induction, achieving a global logical closed loop within the self-constructed topological field system and improving the underlying geometric theory of zero distribution in analytic number theory.