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, 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 Guinand–Weil term 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. The 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 scales linearly in ε (log–log slope 1. 001) ; the constant-term proxy ratio rₚ^∞ is consistent with decrease toward ≤ ½; the sign-crossover of the pointwise bias is localised on a sampled grid between ε = 0. 020 and ε = 0. 025. The integrated bias Bᵢnt^+ (0. 05, 100) = −42. 21 0 by the curvature–bias identity. At the reference parameters, O′ (½) = +2. 475 ≠ 0. What is conditional. If, at any parameter point or in a specified limiting regime, the stationarity condition O′ (½) = 0 holds together with O″ (½) > 0, then σ = ½ is a strict local minimum of O. The stationarity condition is left as an open problem; it does not hold at the finite reference parameters. What remains open. Five problems separate the present work from a full curvature proof: (1) an analytic first-derivative cancellation bound, (2) an analytic proof of asymptotic pointwise bias and rₚ^∞ = ½, (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; none is used as a hypothesis.
Ulrich Tehrani (Mon,) studied this question.