The Riemann Hypothesis: Proof via Herglotz Analyticity and Explicit Construction of the Hilbert-Pólya Operator

We prove the Riemann Hypothesis (RH) unconditionally via the classical theory of orthogonal polynomials and self-adjoint Jacobi operators. Using the polylogarithmic deformation family F (s, w) at w=1, we construct a representation of the Riemann zeta function in which no primes appear explicitly. The proof proceeds via the three-term decomposition of the Bose-Einstein kernel into completely monotone kernels, establishing angular monotonicity of the completed zeta function, which is equivalent to the Herglotz property of the generating function G (z). The Herglotz representation then yields strict positivity of all Hankel determinants Dₙ, forcing all zeros onto the critical line.