A Spectral Construction Toward the Riemann Hypothesis

This short note presents a constructive spectral framework related to the Hilbert–Pólya program.A self-adjoint operator is defined whose spectrum reproduces the imaginary parts of the non-trivial zeros of the Riemann zeta function. The construction combines inverse spectral theory with the classical explicit formula, yielding: a self-adjoint operator with prescribed spectrum, a trace relation consistent with prime number distribution, and a positivity condition aligned with Weil’s criterion. Together, these elements form a coherent operator-theoretic structure consistent with the Riemann Hypothesis.