Mesoscopic Variance Equilibrium and Spectral Rigidity for Zeta Zeros Unconditional Off-Diagonal Decorrelation and a Conditional Path to the Riemann Hypothesis

We develop a mesoscopic, renormalized connected curvature-energy theory for theRiemann zeta function. The central object is the mollified second logarithmic derivative(log ζ)′′, localized to a carrier-matched bandpass |ξ ∓ ξT | ≤ 1/L at the frequencyξT = (log T )/(2π). This choice arises naturally from the dyadic scale n ≍ T andthe Fourier/Mellin structure of (log ζ)′′. The resulting variance functional admits acanonical decomposition into a disconnected one-body background and a connectedtwo-body remainder, with the background removed simultaneously and identically onthe arithmetic, spectral, and operator sides of the identity. This identifies the varianceas a renormalized connected second-energy functional for the zero configuration.