A Hilbert--Pólya Spectral Construction for the Riemann Xi-Function via TEBAC: Shifted Trace, Supersymmetric Coboundaries, and Resonance Reduction on the GL(1) End

This preprint presents a TEBAC Hilbert--P\'olya spectral construction for the completed Riemann \(\xi\)-function. The manuscript is organized as a referee-facing chain of operator-theoretic, determinant-theoretic, heat-trace, and shifted-trace modules, with the spectral variable \(\sigma=\left(s-\frac12\right)^2\) used to encode the critical-line symmetry.

The first part of the manuscript develops the Hilbert--P\'olya spectral channel. It constructs the self-adjoint operator framework, the compact-resolvent spectral package, the centered heat trace, and the zeta-regularized determinant attached to the GL(1) channel. This part establishes the operator-theoretic environment in which the nontrivial zeros of the completed Riemann \(\xi\)-function are intended to arise from a spectral determinant.

The second part develops the determinant and normalization package. It introduces the centered determinant \(D_{\mathrm{cen}}\), the completed determinant \(D_{\mathrm{comp}}\), the reference subtraction terms, the HT2/HT3 heat-trace ledgers, and the archimedean normalization mechanism. A non-circularity ledger is used throughout to separate the spectral construction from any prior use of RH, any pre-assumed identity \(D_{\mathrm{comp}}=\xi\), or any prior information about the location of the zeros of \(\xi(s)\).

The third part develops the trace--prime comparison. It analyzes prime-power translations on the GL(1) end, shifted heat kernels, Gaussian reference traces, and the discrepancy between the free/reference prime channel and the actual confining compact-resolvent channel. The manuscript isolates the precise shifted-trace obstruction caused by the insertion of the prime translation operators \(T_{p^k}\).

The fourth part studies the complex-time rigidity and positivity mechanism. It formulates the defect kernel, the Bochner-type positivity route, and the conditions under which residual holomorphic discrepancy terms are eliminated. This part makes explicit which trace identities are formal consequences of the operator package and which identities require additional shifted-trace analysis.

The new material of this version is concentrated in the GL(1) shifted-trace and supersymmetric end analysis. The manuscript introduces a shifted supersymmetric coboundary framework for the anomaly \(\mathcal Q_a=T_a\mathcal D-\mathcal DT_a\), with \(a=k\log p\), where \(\mathcal D\) is the end Dirac operator and \(T_a\) is a prime-power translation. The shifted anomaly is analyzed through Duhamel homotopies, completed coboundary decompositions, trace-class insertion estimates, and boundary-vanishing conditions.

A harmonic-sector analysis shows that the \(P_0\)-component of the shifted anomaly is trace-neutral at the level needed for the completed shifted trace transform. In particular, the argument replaces the stronger operator-level condition \(P_0(\delta_a\mathcal V)T_aP_\perp=0\) by the weaker and sufficient traced identity \(\operatorname{Tr}\bigl(P_0\mathcal Q_a\mathcal D e^{-t\mathcal L}\bigr)=0\).

The non-harmonic sector is then treated by a Green--Sylvester primitive. Away from the opposite-eigenvalue resonance \(\lambda+\mu=0\), the shifted anomaly is written as a supercommutator and hence contributes no completed supertrace term. This reduces the remaining local obstruction to the resonant blocks.

The resonance chapter proves that the ordinary trace of the opposite-eigenvalue resonant block vanishes because the relevant blocks are off-diagonal in the \(\mathcal D\)-spectral decomposition. The graded resonance contribution is then reduced to a parity-pairing problem. The canonical pairing is constructed from the \(\mathbb Z_2\)-grading operator \(\Gamma\), which maps \(\operatorname{Ran}E_\lambda \longrightarrow \operatorname{Ran}E_{-\lambda}\).

The remaining local target is identified as the shifted-resonance skewness condition for the prime-translation resonant blocks. Thus the manuscript does not hide the last analytic burden: it localizes the final obstruction to a concrete microlocal/supersymmetric condition on the interaction between prime translations, the GL(1) end Dirac geometry, and the opposite-eigenvalue resonance structure.

This upload is intended as a status-explicit research preprint and priority record for the current TEBAC Hilbert--P\'olya proof architecture. The manuscript presents a detailed route toward an unconditional Hilbert--P\'olya proof of the Riemann Hypothesis, while explicitly identifying the remaining local analytic condition required for full closure.