Spectral Moment Equivalence Theorem

This paper develops a general analytic framework governing zeta–regularized determinants of compact operators and establishes the precise algebraic moment identity that any Hilbert–Pólya operator must satisfy.

We prove a Spectral Moment Equivalence Theorem showing that if a compact operator $H$ on a separable Hilbert space has a zeta–regularized determinant representing an entire function of order 1, then the logarithmic derivative of this determinant admits a unique pure Laurent expansion whose coefficients are exactly the trace moments $Tr⁡(H^n)$. Conversely, any such moment sequence uniquely determines the analytic structure of the corresponding entire function.

This yields a necessary and sufficient analytic compatibility condition for a Hilbert–Pólya realization of the Riemann xi-function. Any candidate Hilbert–Pólya operator must satisfy the infinite algebraic system

$A_n=B_n$

where $An=Tr⁡(H^n)$ are the spectral trace moments and $B_n$ are the Laurent coefficients of $-\Xi'(s)/\Xi(s)$ at infinity. This formulation isolates the global analytic obstruction to spectral constructions of the Riemann zeros and provides a universal criterion for evaluating proposed operators.

The theorem is independent of any particular operator model and applies to all compact, Hilbert–Schmidt, or trace-class approaches to the Hilbert–Pólya conjecture. It is intended as the analytic foundation for a forthcoming paper exhibiting a concrete operator satisfying these structural constraints.

Keywords: Hilbert–Pólya conjecture, compact operators, zeta–regularized determinants, spectral traces, entire functions of order 1, moment problems, operator theory, spectral number theory.