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 D_n, forcing all zeros onto the critical line.