Fisher–Rao curvature of rate estimation in Ornstein–Uhlenbeck networks: a closed form and its coupling-graph zeta

This work gives a closed-form expression for the Fisher-Rao scalar curvature of the statistical manifold used to estimate relaxation rates from the stationary fluctuations of a linear (Ornstein-Uhlenbeck) network with symmetric couplings. For the family that estimates the rates of two chosen modes, the curvature depends on the data through a single scalar: the squared stationary correlation between those two modes, divided by the product of their variances. As a function of that ratio the curvature is a simple rational function. It is never positive, it vanishes exactly when the two modes are uncorrelated, and it is bounded below: it approaches its extreme value precisely as the joint covariance of the two modes degenerates. This yields a coordinate-free order parameter for proximity to that degeneration, readable directly from two-point correlation functions. The Taylor expansion of the curvature in the coupling strengths equals four times the logarithm of a weighted determinant built from the coupling graph, a walk-generating object whose squarefree part coincides with the zeta function of the graph. On every simple cycle, both the curvature and this logarithm equal an explicit constant determined by the relaxation rates around the cycle; the general per-cycle weight is reduced to a finite combinatorial count and verified on composite cycles. Because the curvature is a rational function while the logarithm is transcendental, the two agree only as a finite Taylor germ rather than as functions, and the order at which they first disagree is fixed by the graph distance between the two estimated nodes. Finally, this curvature-zeta correspondence is shown to be tied to detailed balance. Introducing an antisymmetric, circulating component into the couplings breaks the correspondence at first order: the curvature acquires mixed-orientation contributions that no combination of the graph zeta's oriented prime cycles can represent. The closed form is therefore an equilibrium statement, and the nonequilibrium, circulating case is left to a companion paper.