The Prime Wave Operator: A Self-Adjoint Candidate for the Hilbert-Pólya Conjecture

The Hilbert-Pólya conjecture proposes that the imaginary parts of the non-trivial zeros of the Riemann zeta function are eigenvalues of a self-adjoint operator. For over a century, such an operator has remained unidentified. We present a concrete candidate: the Prime Wave Operator on ℓ²(ℕ), (\hat{H}a)_n = \frac{a_{n+1} - 2a_n + a_{n-1}}{(\ln n)^2} - V_0 (\ln n) a_n, \qquad a_0 = 0 with V_0 = \sqrt{2} + 1/2. This operator is explicitly defined, computationally tractable, and arises naturally from the canvas model of fundamental physics—where the logarithmic potential emerges from the Prime Number Theorem and the constant V_0 is determined by the geometry of the internal 3D space. What this paper provides: · A rigorous proof that \hat{H} is essentially self-adjoint (Carleman condition + Kato-Rellich theorem)· The continued fraction representation of the resolvent and the eigenvalue condition R_1(\lambda) = 2· Numerical evidence that the eigenvalues of the truncated N \times N matrix converge to -t_k^2 where t_k are the imaginary parts of the Riemann zeros, with relative error \sim 10^{-4} for N = 10,000· A conjectured spectral determinant identity \det_{\text{reg}}(\hat{H} - \lambda) \propto \zeta(1/2 + \sqrt{\lambda})^{V_0/2} up to an entire function without zeros· A physical interpretation: the operator emerges from the four primitives of the canvas model (Order, Amplitude, Acceleration, Polarity) applied to the integer lattice What this paper does not claim: A complete proof of the Riemann Hypothesis. The spectral determinant identity remains conjectural; its rigorous proof would require showing that higher-order terms in the resolvent expansion vanish and that the entire function has no zeros. The numerical evidence is compelling but not conclusive. Who this paper is for: Mathematicians interested in the Hilbert-Pólya conjecture, physicists working on the interface of spectral theory and number theory, and anyone fascinated by the deep connection between the primes and the zeros of \zeta(s). Keywords: Riemann hypothesis, Hilbert-Pólya conjecture, self-adjoint operator, prime numbers, zeta function, spectral theory, prime wave operator, canvas model