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. We establish an exact trace formula Tr (T̃ (σ) ) = DSEL − O (σ) by a two-line algebraic argument, valid for all σ without analytic assumptions. We prove the decomposition B = Σₚ (log p) ² Re (Zₚ^ (2) ), where Zₚ^ (2) is the γ²-weighted phasor sum, and identify a formally isolated dominant term for the Guinand–Weil representation of Z̃ₚ (ε), with an explicit normalization convention. What is numerical. The asymptotic energy ηₒrig (κ) converges to η_∞ ≈ 0. 81 as κ → ∞. Three spectral signatures of T̃ (σ) at σ = 1/2 are observed numerically. What is open. The Bias Conjecture Re (Z̃ₚ (ε) ) < 0 is the central open problem. Its proof would close the unweighted bias Bᵢnt < 0; the full γ²-weighted curvature B < 0 is confirmed numerically. We do not prove the Bias Conjecture in this paper. No use of the Riemann Hypothesis, the GUE conjecture, or the Hilbert–Pólya postulate is made anywhere.