Finite Observation

We study finite observation in irreducible finite-state continuous-time Markov nonequilibrium steady states through the local Donsker-Varadhan Hessian at the steady-state minimiser. In reduced Fisher coordinates, the nonequilibrium correction factors through a weighted signal operator and the detailed-balance backbone. For fixed-rank orthogonal observation in this reduced Euclidean Fisher geometry, the visible correction is controlled by a Ky Fan envelope, together with an associated retention hierarchy, hidden tail, and weighted stable rank. This hierarchy determines the spectral profile of the visible nonequilibrium correction and, equivalently, the corresponding singular-energy profile of the weighted signal operator, and therefore its nonzero singular-value magnitudes.

We also identify the optimal orthogonal observer at fixed rank and show that a Haar-random rank-constrained observer retains, on average, exactly the corresponding dimensional fraction of the available signal. A separate statistical question arises after comparison with the observed detailed-balance backbone: in the compressed-Hessian Gaussian model, distinguishability is governed by a positive shadow operator whose spectrum can differ from the raw visibility spectrum except in special commuting or isotropic-backbone regimes. The same weighted geometry extends to abstract finite-dimensional metric settings, and hence to the linearised BKM-symmetrised operator setting.