Spectral Trace Formula and Smoothed Zero Sums: A Prime–Zero Duality Framework

This is the fifth paper in a series developing an operator-theoretic framework for the Riemann zeta function. The series builds a finite-dimensional Hilbert-space model in which prime numbers mediate coupling between zeros via an explicit operator family. What is proved. An exact trace formula Tr (T̃ (σ) ) = DSEL − O (σ) by a two-line algebraic argument, valid for all σ without analytic assumptions. The decomposition B = Σₚ (log p) ² Re (Zₚ^ (2) ), where Zₚ^ (2) is the γ²-weighted phasor sum, proved from first principles. The Guinand–Weil object Z̃ₚGW and the ordinate proxy Zₚ^∞ are defined and distinguished; their relation is conditional on the GW bridge (open). What is numerical. The renormalized energy asymmetry ηᵣen (κ) → 0. 81 as κ → ∞ (numerical extrapolation, not proved). Three spectral signatures of T̃ (σ) at σ = ½: trace spike +47%, dominant-eigenvalue maximum, spectral-gap minimum. B = −19342. 5 < 0 at reference parameters (κ=53, ε=0. 05, N=100). What remains open. The Bias Conjecture — Zₚ^∞ (ε) < 0 for each fixed prime p and all sufficiently small ε — is the central open problem. Its proof would establish the unweighted integrated bias Bᵢnt < 0; the full γ²-weighted curvature sign B < 0 requires additionally a Weighted Bias Bridge from Bᵢnt to B. No use of the Riemann Hypothesis, the GUE conjecture, or the Hilbert–Pólya postulate is made anywhere.