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

The Hilbert-Pólya conjecture proposes that the Riemann Hypothesis could be proved by constructing a self-adjoint operator whose eigenvalues are the imaginary parts of the non-trivial zeros of the Riemann zeta function. This paper presents a candidate for such an operator, derived from the four dynamic primitives (Order, Amplitude, Acceleration, Polarity) of the canvas model of fundamental physics. What this paper provides: · An explicit self-adjoint operator on \ell^2(\mathbb{N}): (\hat{H}a)_n = \frac{a_{n+1} - 2a_n + a_{n-1}}{(\ln n)^2} - V_0 (\ln n) a_n with Dirichlet boundary condition a_0 = 0 and V_0 = \sqrt{2} + 1/2. The constant V_0 emerges from the canvas model's primitive ratio c_{\text{eff}}/d_{\text{eff}} = \pi/2.· Rigorous proofs of essential self-adjointness via the Carleman criterion, establishing that the spectrum of \hat{H} is real — a necessary condition for a Hilbert-Pólya operator.· Unitary equivalence to the hyperbolic Laplacian on the modular surface \operatorname{SL}(2,\mathbb{Z})\backslash\mathbb{H} in the continuum limit, via the Kontorovich-Lebedev transform and Liouville transformation. The Bessel functions K_{i\tau}(x) are exact eigenfunctions of the continuum operator, linking \hat{H} to the spectral theory of automorphic forms.· A formal trace formula matching the Riemann-Weil explicit formula, providing structural evidence that the spectral determinant of \hat{H} is related to the Riemann zeta function.· A conjectured spectral determinant identity: \det\nolimits_{\text{reg}}(\hat{H} - \lambda) = C(\lambda) \cdot \zeta\left(\frac{1}{2} + \sqrt{\lambda}\right)^{V_0/2} where C(\lambda) is an entire function without zeros. If this conjecture holds, the eigenvalues \lambda_k of \hat{H} are in bijection with the non-trivial zeros \rho_k of \zeta(s) via \rho_k = 1/2 + \sqrt{\lambda_k}, and self-adjointness of \hat{H} forces \operatorname{Re}(\rho_k) = 1/2 — the Riemann Hypothesis.· Identification of critical open problems required to prove the conjecture: rigorous spectral determinant regularization, analytic continuation of the resolvent, trace computation with Mellin transform, and generalization to all automorphic L-functions.· A note on numerical impossibility: Direct numerical verification would require matrix size N \sim e^{t_k^2/V_0}, which for the tenth zero (t_{10} \approx 50) is \sim 10^{565} — astronomically beyond feasible computation. The conjecture must be settled by analytic methods. What this paper does not claim: This paper does not claim to have proved the Riemann Hypothesis within ZFC. It presents a candidate operator, provides rigorous supporting results (self-adjointness, unitary equivalence to the modular Laplacian, Bessel eigenfunction property), and formulates a spectral determinant conjecture. The resolution of the conjecture is left as an open problem for future work. The paper is a theoretical framework, not a finished proof. Why this matters: The canvas model's four primitives — Order (\mathbb{N}), Amplitude (\ln p_n), Acceleration (\Delta^2), and Polarity (\mu(n)) — provide a unified foundation for physics and number theory. The Prime Wave Operator is the bridge between them. Whether that bridge can be crossed depends on the resolution of the spectral determinant conjecture. This paper maps the crossing. Keywords: Riemann Hypothesis, Hilbert-Pólya operator, Prime Wave Operator, canvas model, self-adjoint operator, spectral determinant, Kontorovich-Lebedev transform, Bessel functions, modular surface, automorphic forms, explicit formula, Grand Riemann Hypothesis, number theory