Conditional Stationarity and Positive Curvature of the Spectral Trace at the Critical Line

This paper is the seventh 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 curvature identity O″ (½) = −2B established in Paper 6 and the exact trace formula proved in Paper 5, it addresses two of the five open problems of Paper 6: the pointwise asymptotic behaviour of the Guinand–Weil constant term and the conditional stationarity of O′ (½). Three explicit hypotheses are introduced, each formally weaker than the Riemann Hypothesis, and each clearly labelled throughout. No use of the Riemann Hypothesis, the GUE conjecture, the Montgomery pair correlation conjecture, or the Hilbert–Pólya postulate is made in the unconditional results. What is established unconditionally. An exact three-term bookkeeping decomposition of the first derivative O′ (½) into a pole contribution, a main-term contribution, and a defined residual is introduced (Definition 4. 1), with numerical values reported at the reference parameters (Observation 4. 2). The antisymmetry identity A (½ + δ) = −A (½ − δ) for the decomposition is proved (Proposition 8. 1). What is proved under Assumptions 3. 2 and 3. 3 (subleading archimedean hypothesis and proxy-transfer hypothesis). For every prime p, the asymptotic constant-term ratio rₚ^∞: = lim⏔→₀+ |Constₚ^∞ (ε) | / |Mainₚ (ε) | equals exactly ½ (Theorem 3. 1). In particular, Z₏, ∞^+ (ε) 0 (Corollary 3. 7), and for every prime cutoff κ ≥ 2 there exists εᵢnt (κ) > 0 such that the integrated bias Bᵢnt^∞ (κ, ε) 0. 200. At κ = 101, ε = 0. 05, the value O′ (½) = −137. 6 shows strong κ-dependence and motivates the Cancellation Hypothesis. The continuous stationary point lies at σ_∗ ≈ 0. 4999, not at σ = ½. What is conditional. Under the Cancellation Hypothesis (A∗∗), a constant-factor bound |O′ (½) | ≤ C · W⏔, ₍ · P (κ) is established (Proposition 7. 1). Under the additional hypotheses B < 0 and O′ (½) = 0 (exact stationarity), this yields a strict local minimum of O at σ = ½ (Corollary 7. 3). The full conditional local-selection statement thus requires three separate analytical inputs. What remains open. Six problems separate the present conditional results from a full proof: (1) the Weighted Bias Bridge from integrated to direct curvature (Open Problem 9. 1) ; (2) the Cancellation Hypothesis on prime exponential sums (Open Problem 9. 2) ; (3) the Guinand–Weil bridge with off-critical control (Open Problem 9. 3) ; (4) the scaling of εᵢnt (κ) as κ → ∞ (Open Problem 9. 4) ; (5) near-minimum behaviour across the parameter space (Open Problem 9. 5) ; and (6) the connection to Weil positivity via a finite-dimensional Weil-transfer operator (Open Problem 9. 6). All six are named precisely; none is used as a hypothesis.