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 the
Riemann 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 frequency
ξT = (log T )/(2π). This choice arises naturally from the dyadic scale n ≍ T and
the Fourier/Mellin structure of (log ζ)′′. The resulting variance functional admits a
canonical decomposition into a disconnected one-body background and a connected
two-body remainder, with the background removed simultaneously and identically on
the arithmetic, spectral, and operator sides of the identity. This identifies the variance
as a renormalized connected second-energy functional for the zero configuration.