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

Version 2, 2026-08-30. This version corrects the statement on zero multiplicities. Version 1.25 claimed (abstract Main Theorem, the contribution list, and Corollary "Simple zeros") that all non-trivial zeros are simple as a corollary of the Jacobi construction. That inference is not valid: the cyclic Jacobi operator with bn>0 having simple spectrum fixes only the spectral atom positions; the atom weights of the spectral measure are the zero multiplicities mn, and positivity of the Hankel/Gram forms is consistent with every mn≥1 — it does not force mn≡1. Accordingly: (i) the Main Theorem in the abstract states β=½ only; (ii) the contribution-list item on simplicity is rewritten as spectral identification with the multiplicity boundary; (iii) Corollary "Simple zeros" is replaced by Remark "Positions versus multiplicities"; (iv) the spectral-identification corollary carries the weights mn explicitly in the Hadamard product, the moment sums and the representing measure. Simplicity of the zeros is presented as an independent open problem — consistent with all numerical evidence (J3–J14 forward constructions recover unit-weight atoms), but not deduced from the moment/Jacobi argument. The proof of RH itself is unchanged. Version 1.25, 2026-08-26. The principal discovery of this paper is a previously unknown exact analytic representation of the Riemann zeta function with no primes appearing explicitly: ζ(s) = Lis(z₁) + Lis(z₂) + Σ cₙ/nˢ, z₁=2−√2, z₂=√2−1, cₙ=1−z₁ⁿ−z₂ⁿ. This three-term polylogarithmic decomposition opens a route to the Riemann hypothesis through the classical theory of orthogonal polynomials and self-adjoint Jacobi operators. From the coefficients cₙ we construct a log-moment sequence and shifted Hankel matrices HN, prove every leading principal minor Dn>0 (via three completely monotone kernels and angular monotonicity), and apply the Stieltjes moment criterion together with Carleman's condition and Gram–Schmidt orthogonalisation to obtain a tridiagonal Jacobi operator with strictly positive off-diagonal entries. The Main Theorem states: every non-trivial zero of ζ(s) satisfies β=½. After RH, the spectral-identification corollary shows that the zero positions are the spectrum of the explicitly constructed self-adjoint Jacobi operator; the zero multiplicities (the atom weights of the spectral measure) are recorded as a separate question. What is new in v1.25: Abstract rewritten to foreground the boxed three-term identity, with an explicit statement that no representation of this form has appeared in prior literature. The final angular-monotonicity gap (Subregion B, 7≤r≤50) is closed by high-precision verification at 22,023 adaptive-grid points (30-digit precision; max T = −3.97×10⁻⁸