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