TEBAC Hilbert--Pólya Program for the Riemann Hypothesis: Finite Trace-Lift Version

This v6 preprint presents a strengthened, claim-safe version of the TEBAC Hilbert--Pólya program for the Riemann Hypothesis. The manuscript develops a non-circular spectral-determinant framework aimed at the identity\[Dcomp(s)=ξ(s),\]where \(Dcomp\) is the completed TEBAC Hilbert--Pólya determinant and \(ξ\) is the completed Riemann \(ξ\)-function. Compared with earlier versions, the present manuscript hardens the terminal analytic layer by isolating the remaining obstruction in a shifted prime-trace form rather than in ordinary determinant or unshifted phase form. In particular, the unresolved local analytic burden is organized around the shifted obstruction package and its trace-level realization through finite truncation, shell/tail control, and compatibility of the trace-lift mechanism. The official terminal front is therefore no longer an ordinary relative-determinant closure statement, but a shifted prime-trace closure problem expressed through the finite-level and limiting behavior of the shifted prime-trace terms. The manuscript also records the finite shifted prime-trace structure in explicit theorem-facing form, including the truncated shifted prime-power contribution\[∑p^k≤ N Cp,kˢʰ(s),\]together with the associated shell, tail, and compatibility reductions that make the remaining analytic gates referee-checkable rather than schematic. The final spectral-square Hilbert--Pólya implication is retained in a claim-safe form: if the upstream determinant identity is established non-circularly, then the nontrivial zeros of \(ξ\) are transferred to the spectral-square condition\[(s-12)^2∈ R≤0,\]and hence lie on the critical line. This version should therefore be read as a near-terminal proof manuscript and analytic hardening stage, not as a final completed unconditional proof of the Riemann Hypothesis. The remaining global verification is concentrated in the determinant-identification and trace-lift compatibility gates explicitly recorded in the manuscript.