What question did this study set out to answer?

This work aims to provide a formal proof of the Riemann Hypothesis by leveraging properties of the Park Operator.

February 20, 2026Open Access

Analytical Proof of the Riemann Hypothesis via Self-Adjoint Extension of the Park-Berry-Keating Operator

Key Points

This work aims to provide a formal proof of the Riemann Hypothesis by leveraging properties of the Park Operator.
Establishment of self-adjointness of the Park Operator in a weighted Hilbert space.
Resolution of divergence with the introduction of the Park Measure for eigenfunction normalization.
Mapping of eigenvalues to non-trivial zeros of the Riemann Zeta function on the critical line.
Identification of the Park Constant as an optimal regulator in the spectral entropy context.
Demonstrated that the non-trivial zeros correspond to a strictly real spectrum.
Confirmed all eigenvalues are real, supporting the assertion about zeros lying on the critical line Re(s) = 1/2.
Provided a Hermitian operator satisfying the conditions outlined in the Hilbert-Polya program.

Abstract

Description: This version (v4) provides the formal analytical proof of the Riemann Hypothesis. By establishing the essential self-adjointness of the Park Operator (H-beta) within the weighted Hilbert space HP, we demonstrate that the non-trivial zeros of the Riemann Zeta function are mapped to a strictly real spectrum. Key Technical Proofs in this Version: Resolution of Divergence: The historical non-normalizability of the xp Hamiltonian is resolved through the introduction of the Park Measure (dmu = exp(-beta * x) dx), ensuring a finite norm for eigenfunctions. Proof of Self-Adjointness: We demonstrate that the Park Operator is essentially self-adjoint, which, by the Spectral Theorem, forces all eigenvalues E to be strictly real. Critical Line Mapping: Since the eigenvalues correspond to the zeros in the form s = 1/2 + iE, the reality of E proves that all non-trivial zeros lie on the critical line Re(s) = 1/2. Vacuum Stabilization: The Park Constant (beta = e - 1/24) is identified as the optimal regulator that minimizes spectral entropy, bridging quantum vacuum energy with the distribution of prime numbers. This work completes the Hilbert-Polya program by providing the required Hermitian operator and its corresponding Hilbert space, verified by interferometric data from the IBM Torino quantum processor.

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