Explicit Self-Adjoint Operator for the Riemann Zeros: Jacobi Matrix Construction, Cardano Formula for J3, and High-Precision Numerical Localisation

We construct an explicit self-adjoint Jacobi operator whose spectrum is given by the imaginary parts of the nontrivial Riemann zeros, conditionally assuming the Riemann Hypothesis. The construction proceeds via a Stieltjes moment problem with a spectral measure supported on the squared ordinates of the zeros. The main contributions are: A systematic derivation of the Jacobi matrix coefficients from the Taylor expansion of the Riemann xi function at s=1/2; An explicit Cardano formula for the 3x3 Jacobi submatrix J3, yielding closed-form eigenvalues that approximate the first three Riemann zeros; A complete sequence of finite Jacobi matrices J3 through J14, computed at 120-digit precision using mpmath; Numerical localisation of the Riemann zeros to high precision via eigenvalues of the truncated Jacobi matrices, demonstrating convergence to the known zero ordinates as matrix size increases. All numerical results are reproducible using the accompanying Python script (paper_jacobi_reproducible.py), which uses mpmath at 120-digit working precision. This paper explicitly assumes the Riemann Hypothesis and does not claim a proof thereof; the goal is to elucidate the spectral structure underlying the Hilbert-Polya conjecture.