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This paper presents an operator-theoretic proof of the Riemann Hypothesis. The proof is organized so as not to identify the zeros of the completed zeta function with eigenvalues at the outset. Instead, three independent pieces of data are built: an analytic operator setting on a weighted Hilbert space, a coefficient-space arithmetic trace that evaluates the Euler-product prime-power contribution, and singular-boundary data constructed inside the analytic Hilbert-space framework. These data are then placed in a common Hilbert space $X = \mathcal{K}_R \oplus J_{\mathrm{arith}}\mathcal{H}_{\mathrm{arith}} \oplus \operatorname{Ran}\Pi_{\mathrm{res}}$, where the prime-power term is evaluated exactly on the arithmetic summand and the residual part is removed by passing to the canonical representative modulo $\operatorname{Ran}\Pi_{\mathrm{res}}$. The remaining effective $\mathcal{K}_R$-projected component is thus represented as the $\Pi_R$-projection onto the singular-boundary subspace $\mathcal{K}_R$. From this residual-free $\mathcal{K}_R$-component, a boundary-distribution comparison map is constructed. The functional equation for $\xi$ induces a boundary reflection $\Theta_R$, which descends to a bounded self-adjoint involution $\mathcal{S}_R$ on $\mathcal{K}_R$. The resulting signed boundary-distribution comparison kernel is realized, by Schatten-class smoothing estimates, as a self-adjoint Hilbert--Schmidt operator $K=K^* \in \mathfrak{S}_2$. This construction uses the functional equation, the boundary-distribution framework, and the orthogonal projection structure; it does not assume the location of the zeros of $\xi$, nor any positivity, Herglotz, or spectral localization statement equivalent to the Riemann Hypothesis. The operator $K$ defines the regularized Fredholm determinant $F_K(s) = e^{a_K+b_K(s-1/2)} \det_2(I+i(s-1/2)K)$, where the constants $a_K, b_K$ fix only the value and first logarithmic derivative at $s=1/2$. The comparison with the completed zeta function is carried out through a central Cauchy--Laplace regularization. The central comparison topology is fixed independently of the pairings $\mu_L$ and $\mu_\xi$. The finite-window counterterm is defined algebraically from central and endpoint finite jets before either pairing is evaluated, so the regularization does not encode the desired equality. Finite-window central cutoffs converge to the central kernel in this topology, and the two central pairings extend continuously to it. The finite-window residual-free equality therefore passes to the central limit and gives equality of the central logarithmic derivatives of $F_K$ and $\xi$. Together with the central normalization, this yields a local analytic equality, and the identity theorem gives $F_K(s) \equiv \xi(s)$ on the whole complex plane. Finally, since $K$ is self-adjoint, every zero of $F_K$ arises from a nonzero eigenvalue $\lambda_j$ of $K$ and is therefore of the form $s=1/2+i/\lambda_j$ for $\lambda_j \in \mathbb{R}\setminus\{0\}$. The global identity $F_K \equiv \xi$ therefore places every nontrivial zero of $\xi$, and hence of $\zeta$, on the critical line.