Positive Curvature of the Spectral Trace at the Critical Line

This paper is the sixth in a series developing an operator-theoretic framework for studying the distribution of the non-trivial zeros of the Riemann zeta function. Building on the exact trace formula established in Paper 5 of the series, it analyses the second-order curvature of the spectral trace at the critical line via a Guinand–Weil decomposition of a smoothed prime–zero sum. No hypothetical input is used: the Riemann Hypothesis, the GUE conjecture, the Montgomery pair correlation conjecture, and the Hilbert–Pólya postulate are explicitly avoided throughout. What is proved. For every prime p and every ε > 0, the leading term of the Guinand–Weil decomposition is strictly negative. The other-prime error is non-positive, and the zero-sum truncation error at N = 100 is bounded by 3×10⁻⁶¹ relative to the main term. An algebraic curvature–bias identity O″ (½) = −2B, connecting the second derivative of the oscillatory trace component to the direct curvature sum, is derived unconditionally from the trace formula of Paper 5. What is numerical. At the reference parameters (κ, ε, N) = (53, 0. 05, 100), the gamma contribution is numerically consistent with linear scaling in ε (log–log slope 1. 001), the constant-term ratio appears to converge to a finite limit rₚ ∈ 0. 75, 0. 80 for every prime p ≤ 53, and the sign-crossover of the pointwise bias is localised on a tested grid to the interval (0. 020, 0. 025). The integrated bias is negative, Bᵢnt (0. 05) = −42. 21 0 by the curvature–bias identity. What is conditional. Under the stationarity hypothesis O′ (½) = 0 at the reference parameters — stated explicitly and left as an open problem — σ = ½ is a strict local minimum of the oscillatory component O, and the spectral trace attains a strict local maximum at the critical line. What remains open. Five questions separate the present work from a full curvature proof: (1) an analytic stationarity bound, (2) an analytic proof of asymptotic pointwise bias and rₚ < 1, (3) uniformity of positive curvature across the parameter space, (4) the integrated-to-direct transfer, and (5) the bridge to Weil positivity in the Lagarias formulation. All five are named precisely in the paper; none is used as a hypothesis for the unconditional claims.