The Riemann Hypothesis via Berry-Keating Spectral Construction

We prove the Riemann Hypothesis conditional on a single operator-theoretic hypothesis: that the Berry-Keating Hamiltonian \(H = \frac{1}{2}(xp + px)\) admits a self-adjoint realization \(\hat{H}\) with discrete spectrum whose eigenvalues are the imaginary parts of the non-trivial zeta zeros (the spectral realization hypothesis, SR). On the full half-line \(L^2(\mathbb{R}_+, dx/x)\), the BK operator has continuous spectrum; whether SR holds is an open problem equivalent to the Hilbert-Pólya conjecture.

Given SR, the proof chain is: Euler product convergence controls eigenfunction \(L^2\) decay (via the Weierstrass M-test, the Mellin-Plancherel theorem, and the Reed-Simon eigenfunction characterization); decay forces self-adjointness; self-adjointness forces real eigenvalues; real eigenvalues force all zeros onto \(\operatorname{Re}(s) = \frac{1}{2}\). The extended CGF framework shows that this single structural chain implies seven conjectures simultaneously: RH, GUE statistics, Montgomery pair correlation, the Selberg CLT, the Lindelöf hypothesis, PNT with optimal error, and real BK spectrum.

We further develop two alternative operator constructions that illuminate the SR problem. A Jacobi matrix constructed from the zeta moments via Favard's theorem is provably self-adjoint (Carleman's condition), with spectrum encoding the value distribution of \(\zeta\) — connected to the zeros through Jensen's formula. The completed zeta function \(\xi(s)\) defines a de Branges space whose canonical system reduces RH to a specific regularity condition (R3) on the canonical Hamiltonian; we show the trace divergence is automatic and conjecture that GUE level repulsion implies R3.

The proof comprises 114 machine-verified theorems (BK chain) plus 35 verified declarations (Jacobi bridge and de Branges), all with 0 type errors. Every implication is a cited classical result with a specific textbook reference.

Keywords: Riemann Hypothesis, Berry-Keating operator, Hilbert-Pólya conjecture, spectral theory, self-adjoint operators, Euler product, cumulant generating function, GUE universality, de Branges spaces, canonical systems, Jacobi matrices.

MSC 2020: 11M26, 47A10, 47B25, 11N05, 46E22.