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 separated by type. The contour-coordinate ledger and the seam/Gram/LCI transport are fixed from the finite-window contour convention, the logarithmic representative, and finite readout coordinates. The scalar probe used for the classical comparison is fixed separately by the geometric Guinand--Weil contour normal form, which uses the classical boundary normal form of the completed zeta logarithmic derivative, namely the gamma-factor, Euler-product, and functional-equation normal forms, but takes no zero divisor as input; it is not chosen from prescribed zero-side values, from the determinant side, from a \(\mathcal K_R\)-side comparison equality, or from RH. 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 a Sobolev-reference Schatten-four sandwich estimate, as a self-adjoint Hilbert--Schmidt operator \(K=K^*\in\mathfrak S_2\). This gives the intrinsic determinant factor \(F_K^0(s)=\det\nolimits_2(I+i(s-\tfrac12)K)\) 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 a bookkeeping total, so that after the Archimedean and arithmetic contributions are removed its zero-side residual is identified, by a finite-window Guinand--Weil residue theorem, with the central logarithmic derivative of the completed zeta function; the zeros of \(\xi\) enter only as the residue side of Cauchy's theorem. These two independently obtained transform identities give \(\frac{d}{dw}\log F_K(\frac12+w)=\frac{d}{dw}\log \xi(\frac12+w)\) 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.