Hilbert polya operator for Riemann Zeta function

We develop an intrinsic geometric framework in which logarithmic scaling, global holonomy, and inversion symmetry together give rise to a self-adjoint operator whose spectral parameters lie on the line ℜ(s) = 1 2 . Starting from multiplicativity and the logarithmic coordinate u = log r, we introduce an independent phase variable and study uniform transport along curves. Global consistency of three local complex representations leads to a holonomy condition producing a discrete set of geometric modes. A modified Frenet-Serret analysis yields a skew-adjoint generator and an associated self-adjoint operator. Imposing multiplicative inversion symmetry fixes the real part of the spectral parameter uniquely. Arithmetic enters only at the final stage, through Mellin analysis and logarithmic sampling on the integers, where the completed zeta function emerges as canonical arithmetic detector compatible with the symmetries of the construction. Discreteness arises geometrically, while arithmetic serves to detect, rather than determine, the spectrum. Numerical comparisons are included as consistency checks illustrating detectability. No properties of the Riemann zeta function are assumed in advance.