A Proof of the Riemann Hypothesis: Exclusion of Off-Critical-Line Zeros via Series Structure

Starting from the Dirichlet series definition of the Riemann zeta function, anodd-even splitting is introduced to establish an equivalence condition for the existence of non-trivial zeros. It is then demonstrated that off the critical line (σ = 1/2),the asymmetric structure of term moduli between odd and even parts prevents theodd-even components from simultaneously vanishing. On the critical line (σ = 1/2),however, the uniform decay of term moduli allows complete phase cancellation atspecific t-values, giving rise to zeros. The functional equation guarantees pairedsymmetry of zeros. Consequently, all non-trivial zeros are confined to the criticalline Re(s) = 1/2, thereby proving the Riemann Hypothesis. On this basis, thedetermination of zero positions provides a rigorous foundation for the completeoscillatory behavior of the prime-counting function