This paper presents a novel structural framework exploring a potential proof of the Riemann Hypothesis by constructing a spectral representation over the adelic class space \(C_{\mathbb{A}}^{1}\). We define a self-adjoint Hamiltonian operator \(\mathcal{H}\) on the pure adelic Hilbert space \(L_{0}^{2}(C_{\mathbb{A}}^{1})\), aiming to establish a correspondence between the non-trivial zeros of the Riemann Zeta function and the real eigenvalues of \(\mathcal{H}\). We further investigate cohomological properties of the adelic configuration that may impose structural constraints on the distribution of zeros. This work is presented as an exploratory study proposing a novel approach and does not constitute a completed proof of the Riemann Hypothesis.