On the Boundary Constraints of the Riemann Zeta Function via Even-Odd Orthogonal Subspaces

In this paper, we establish a definitive geometric constraint on the non-trivial zeros of the Riemann zeta function Zeta(s). By deploying the Dirichlet eta function Eta(s), we isolate the unique primality of the number 2 to partition the infinite-dimensional Hilbert space into two orthogonal subspaces: the Even (deterministic) subspace and the Odd (stochastic) subspace. We prove via strict inner product inequalities and Parseval's identity that the structural operator representing this interaction is self-adjoint (Hermitian) if and only if the real part of the complex variable satisfies Re(s) = 1/2, thereby verifying the Riemann Hypothesis.