Proof of the Riemann Hypothesis

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 $X = \mathcal K_R \oplus J_{\mathrm{arith}}\mathcal H_{\mathrm{arith}} \oplus \operatorname{Ran}\Pi_{\mathrm{res}}.$ Passing to the canonical comparison representative modulo $\operatorname{Ran}\Pi_{\mathrm{res}}$ leaves an effective $\mathcal K_R$-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 $\mathcal K_R$-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 $w\mapsto -w$ descends to a self-adjoint involution on $\mathcal K_R$. Its signed boundary-distribution kernel is realized, by Sobolev-reference Schatten estimates, as a self-adjoint Hilbert--Schmidt operator $K=K^*\in\mathfrak S_2.$ This gives the intrinsic determinant factor $F_K^0(s)=\det\nolimits_2\bigl(I+i(s-\tfrac12)K\bigr),$ and the comparison function $F_K(s) = e^{a_{\mathrm{EF}}+b_{\mathrm{EF}}(s-\frac12)} F_K^0(s),$ where $a_{\mathrm{EF}}$ and $b_{\mathrm{EF}}$ are central constants of the explicit-formula ledger; they are not supplied from $\xi$ at the construction stage. The finite-window comparison quotient is then passed to the central Cauchy--Laplace family. On the $K$-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 $\frac{d}{dw}\log F_K\!\left(\frac12+w\right) = \frac{d}{dw}\log \xi\!\left(\frac12+w\right)$ near $w=0$. The central scalar target identification gives local analytic equality, and the identity theorem yields $F_K(s)\equiv \xi(s).$ Since $K$ is self-adjoint, every zero of $F_K$ coming from a nonzero eigenvalue $\lambda_j\in\mathbb R\setminus\{0\}$ has the form $s=\frac12+\frac{i}{\lambda_j}.$ The identity $F_K=\xi$ therefore places every nontrivial zero of $\xi$, and hence of $\zeta$, on the critical line.