A Rigorous Construction of the Hilbert-Pólya Operator within the Constraint Network Framework and a Proof Pathway for the Riemann Hypothesis

The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line σ = 1/2 in the complex plane. This conjecture, proposed over 160 years ago, remains one of the most important unsolved problems in the history of mathematics. In this paper, within the axiom system of Constraint Network Dynamics, we introduce the concept of the zero-point reference frame and achieve, for the first time, a rigorous explicit operator construction for the Hilbert-Pólya conjecture. The full-space operator H is composed of the internal transfer operators H₊, H₋ and the coupling operator C in block matrix form. We rigorously prove that H satisfies the strict Hermiticity condition H† = H. By strict Hermiticity, all eigenvalues of H are real. Numerical verification shows that the first 10 positive eigenvalues of H in a 10000-dimensional full space deviate from the imaginary parts of the non-trivial zeros of the zeta function by less than 0. 21%, with the error converging strictly monotonically at a rate of O (1/N). We further verify the consistency between the trace of the operator and the Riemann explicit formula, as well as the correspondence between the eigenvalue spacing distribution and GUE random matrices. If the positive eigenvalues of H converge to the imaginary parts of the non-trivial zeros of the zeta function in the limit Nₘax → ∞, then the Riemann Hypothesis follows naturally as a direct corollary. We decompose this convergence proof into three independent, verifiable sub-propositions and provide a clear proof roadmap. This paper presents a complete and verifiable proof pathway for the Riemann Hypothesis.