An Explicit Formula for the Trace of the Riemann Zero Log-Gas Laplacian

Let LN denote the weighted graph Laplacian of the first N Riemann zeros, as defined in Wright (2026b). We prove two theorems connecting LN to classical analytic number theory. Theorem 1 (the Stiffness Identity): the local stiffness cj = Σk≠j 2/(γj −γk)2 equals twice the principal value of the second logarithmic derivative of the completed Riemann xi function evaluated at the zero ρj = 1/2 + iγj: cj = 2·PVd2/ds2 log ξ(s)s=ρj . The proof is a one-paragraph consequence of the Hadamard product. Theorem 2 (the Prime Sum Formula): the trace of LN satisfies Tr(LN) = 2NBN + 2 Σ Σ m≥1 (log2 p)/pm/2 · ReΣj=1 p N p−imγj, expressing the Laplacian trace as a prime sum modulated by oscillatory exponentials over the zeros. Together these theorems close the triangle: primes ↔ (explicit formula) ↔ zeros ↔ (Stiffness Identity) ↔ Laplacian. We state a precise open problem (Open Problem 5.1) connecting Tr(LN) to the Rodgers–Tao local-equilibrium framework, and identify the remaining analytic obstacle as a new entrance to the core difficulty of RH.