Quantum Fluctuations from a Metric Perspective: Proof of the Quantitative Conjecture

We compute the Green–Kubo action of a collective metric phase with equilibrium overdamped dynamics. We take the cosine potential \(V_k(\varphi)=k(1-\cos\varphi)\) and assume the global observable normalization \(F_k=V_k'/\sqrt{2k}\). Solving the periodic Poisson equation gives \(\mathcal A_b=\Theta\tau[1-I_0(b)^{-2}]\), where \(b=k/\Theta\) and \(\tau=\Gamma/k\). The relative correction to \(\Theta\tau\) satisfies \(0