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)) = D_SEL − O(σ) by a two-line algebraic argument, valid for all σ without analytic assumptions. The decomposition B = Σ_p (log p)² Re(Z_p⁽²⁾), where Z_p⁽²⁾ is the γ²-weighted phasor sum, proved from first principles. The Guinand–Weil object Z̃_p^GW and the ordinate proxy Z_p^∞ are defined and distinguished; their relation is conditional on the GW bridge (open). What is numerical. The renormalized energy asymmetry η_ren(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 — Z_p^∞(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 B_int < 0; the full γ²-weighted curvature sign B < 0 requires additionally a Weighted Bias Bridge from B_int to B. No use of the Riemann Hypothesis, the GUE conjecture, or the Hilbert–Pólya postulate is made anywhere.