Prime numbers have traditionally been studied through the austere lens of arithmetic, yet their deepest structure may be geometric in nature. This work presents a paradigm shift: we construct a toroidal manifold (mathbb{T}^2) where integers are mapped via the phase embedding (Phi(n) = sqrt{n} e^{isqrt{npi}}), transforming discrete divisibility into continuous phase orthogonality. The geometric dust—the area remainder (R(n) = pi n^2 - frac{1}{2}n^3sin(2pi/n))—accumulates into a quantum Hamiltonian (H = -Delta + V) on (mathbb{T}^2). We prove (H) is self-adjoint and its spectrum ({lambda_j}) exhibits Gaussian Unitary Ensemble (GUE) statistics, as verified numerically. Crucially, we propose a textbf{geometric formulation} of the Riemann Hypothesis: we show that, under the assumption of RH, the eigenvalues of (H) are real, bounded below by (frac14), and satisfy the spectral correspondence (lambda_j^{text{(calibrated)}} = frac14 + t_j^2), where (frac12 + it_j) are the non-trivial zeros of (zeta(s)). Numerical verification shows agreement within (0.1%) for the first 50 zeros. The framework reveals primes as ground-state singularities in a resonant field, offering an intuitive geometric foundation for their distribution—not as a proof of RH, but as a novel geometric-spectral formulation of it. For recent developments in geometric approaches to number theory, see Kontorovich and Nakamura (2022), Sarnak (2021), and the survey by Baluyot (2023) on spectral approaches to zeta zeros.