Finite Observation

We study finite observation in irreducible finite-state continuous-time Markov nonequilibrium steady states at the Gaussian fluctuation level set by the Donsker--Varadhan Hessian. In reduced Fisher coordinates, we prove that the nonequilibrium correction is a weighted signal operator with weighted Gram form. For rank-\(d\) orthogonal observation, this yields a closed spectral law: the visible correction is given by a Ky Fan envelope, the retention hierarchy and hidden tail are explicit, and weighted stable rank provides a natural complexity invariant. We further prove that observed Gaussian distinguishability is governed by a positive shadow operator. The compressed Hessian splits into a projected detailed-balance backbone plus visible correction, and the divergences between the observed nonequilibrium and observed detailed-balance centred Gaussian laws reduce to spectral functions of the shadow. Backbone whitening yields a second spectrum governing detectability, distinct from raw visibility. The full envelope hierarchy recovers the ordered singular-energy profile of the correction operator. This weighted geometry transfers to abstract finite-dimensional metric settings and hence to linear BKM operator geometry. Worked benchmarks and matched controls show how the visibility and detectability spectra behave across distinct nonequilibrium networks, and where low-rank concentration should not be overinterpreted.