A Hilbert–Pólya Spectral Construction for the Riemann Xi-Function

This research master develops a Hilbert--P\'olya spectral program for the completed Riemann xi-function and documents the analytic, arithmetic, and operator-theoretic conditions required for a spectral determinant identification. It combines a proposed local-to-global construction with an explicit audit of its terminal proof obligations. The framework includes the \(\mathrm{GL}(1)\) channel, confining end operators, self-adjointness and compact-resolvent arguments, centered heat traces, regularized spectral determinants, boundary and Kre\u{\i}n resolvent methods, prime-power correspondences, and trace--prime comparison identities. The intended determinant comparison and critical-line conclusion are conditional on independent verification of the terminal closure package. The principal addition in the 26 October 2026 revision is an operator-level obstruction to the raw prime-shift completeness condition C2. For correspondences built from a strongly continuous unitary group and a fixed bounded cutoff, the manuscript derives an exact asymptotic for the weighted Hilbert--Schmidt defect using the unconditional prime number theorem. Writing \(S_N\) for the truncated prime-power correspondence sum, \(A\) for a nonzero positive trace-class weight, \(M\) for the bounded cutoff operator, and \(G\) for the self-adjoint generator of the unitary group, the main asymptotic is\[\frac{\log N}{N}\,\bigl\|A^{1/2}(I-S_N)\bigr\|_{\mathrm{HS}}^2\longrightarrow C_A,\]where\[C_A=\bigl\|(\tfrac12 I-iG)^{-1}M^*A^{1/2}\bigr\|_{\mathrm{HS}}^2.\]The coefficient is strictly positive whenever \(M^*A^{1/2}\) is nonzero. In that case, the squared defect grows as\[(C_A+o(1))\,\frac{N}{\log N}.\]In the degenerate case \(M^*A^{1/2}=0\), the squared defect equals \(\operatorname{Tr}(A)>0\) for every \(N\). Thus the raw completeness target fails in both cases. Under the additional weak-escape hypothesis, satisfied by translations on the full line, the leading coefficient survives finite-dimensional zero-mode removal. The manuscript also records the exact correction introduced by compression. In the positive-coefficient case, even the optimal complex scalar normalization of the compressed correspondence sum leaves a weighted squared defect tending to \(\operatorname{Tr}(A)\), rather than to zero. Corrections whose weighted Hilbert--Schmidt norm is \(o(\sqrt{N/\log N})\) cannot cancel the positive leading term. These results identify a specific failure of the raw unitary prime-shift realization. They do not disprove the Riemann Hypothesis and do not exclude other Hilbert--P\'olya constructions. A replacement correspondence family would require explicit global operators, domains, and independently derived trace identities, including any projection corrections. The revision retains an explicit status ledger for the shifted trace--prime condition C1, the exact operator-square identity C3, and the relative-determinant and no-crossing requirements F2. These fronts are not discharged by the C2 audit. No unconditional proof of the Riemann Hypothesis or unconditional determinant identity with the Riemann xi-function is claimed in this version.