Quantitative Spectral Bounds for the Riemann Zero Log-Gas Laplacian, an Unconditional Prime Sum Identity, and a Program Summary

We present two new unconditional results for the weighted graph Laplacian LN of the first N Riemann zeros, introduced in Wright (2026a–e), and use them to close the research program with a precise account of what has been established and what remains open. First (Theorem 2.2), we prove the unconditional identity TrHad(N) = Pfull(N) + 2N¯ BN , where Pfull is a prime sum over the full complex zero positions ρj , without assuming the Riemann Hypothesis (RH). Second (Corollary 3.6), we prove matching bounds on the spectral gap: the lower bound λ2(LN ) ≥ π2/(N2 δ2 j ) follows from Selberg’s zero-gap theorem and a path-graph comparison, and the upper bound λ2(LN ) ≤ 12 j<k (j−k)2/(γj−γk )2/N(N2 −1) is new and follows from the linear test vector vj= j−(N+1)/2. Both bounds are exact in terms of actual zero spacings; using Ingham’s unconditional max-gap bound and the Selberg gap respectively, they give explicit forms C1(log TN )2/N41/12 ≤λ2(LN ) ≤C2(log TN )2/N unconditionally. The upper bound is numerically tight to within a factor of 1.7–2.0 for N ≤50. We close with a complete program summary: the structural reason the framework cannot by itself prove RH, the connection to the Baluyot–Goldston–Suriajaya–Turnage-Butterbaugh unconditional pair correlation program, and a precise statement of what new external mathematics would be required to close the gap.