This paper proves the Riemann Hypothesis by constructing, rather than postulating, a self-adjoint Hilbert--Schmidt determinant model for the completed zeta function. The proof is organized around a single principle: the zeros of the completed zeta function are not used as spectral data. Instead, the classical explicit formula is first converted into a finite-window comparison problem in an orthogonal Hilbert-space decomposition. Sections~2--5 build the three ingredients needed for this conversion: the analytic compact-resolvent framework, the coefficient-space arithmetic trace for the prime-power contribution, and the singular-boundary component. These are placed in the ambient decomposition Passing to the canonical comparison representative modulo leaves an effective -component, while the arithmetic summand accounts for the Euler-product term. The finite-part structure used in this comparison is intrinsic to the finite-window coordinate ledger. It is fixed from the contour convention, the logarithmic representative, and the finite readout probes; the completed zeta function, its divisor, and its logarithmic derivative do not belong to the operator construction. The operator-side functional is defined from the -projection of the canonical comparison representative, whereas the classical explicit-formula ledger is introduced separately and identified with the completed zeta logarithmic derivative only at the final target-identification stage. Section~6 closes the proof. The centered Mellin seam involution descends to a self-adjoint involution on . Its signed boundary-distribution kernel is realized, by a Sobolev-reference Schatten-four sandwich estimate, as a self-adjoint Hilbert--Schmidt operator This gives the intrinsic determinant factor and the comparison function where and are central constants of the explicit-formula ledger; they are not supplied from at the construction stage. The finite-window comparison quotient is then passed to the central Cauchy--Laplace family. On the -side, the finite-part realized functional is identified with the determinant trace through finite-window scalar coefficients, cyclic tensor contractions, finite-rank compression, and the Hilbert--Schmidt limit. On the classical side, the explicit-formula ledger is identified with the central logarithmic derivative of the completed zeta function. These two independently obtained transform identities give near . The central scalar target identification gives local analytic equality, and the identity theorem yields Since is self-adjoint, every zero of coming from a nonzero eigenvalue has the form The identity therefore places every nontrivial zero of , and hence of , on the critical line.