A POLYA–HILBERT OPERATOR FROM THE RIGOROUS SPECTRAL PARAMETRISATION OF RIEMANN ZEROS VIA THE LAMBERT W FUNCTION

Abstract. Building on the unconditional spectral parametrisationF11(n) established in our companion paper [1], we construct anexplicit, self-adjoint operator S on a separable Hilbert space whosespectrum is exactly {γn : n ≥ 1}, the imaginary parts of thenontrivial zeros of the Riemann zeta function. The constructionrequires no assumption on the Riemann Hypothesis.The operator is S = D + K, where D is a diagonal opera-tor with eigenvalues F11(n) and K is a compact, self-adjoint per-turbation whose spectral shift encodes the oscillatory correctionTosc(n) = −S(Tbase(n))/ρ(Tbase(n)). Self-adjointness follows fromthe Kato–Rellich theorem; spectral identification relies on a classi-cal separation-of-measures argument applied to the trace formulaof [1].We further show that the existence of S does not imply the Rie-mann Hypothesis: self-adjointness guarantees only that eigenvaluesare real, which is trivially satisfied for the real numbers γn regard-less of the location of the corresponding zeros ρn = βn + iγn. Thepaper thus delineates precisely where the P´olya–Hilbert programmesucceeds (unconditional const