We develop a geometric framework for observation systems by treating the forward map from parameters to observations, together with the observation covariance, as a single structured object. This object — the resolution geometry — packages the reference metric on observation space, the resolved sector spanned by the forward response, the null sector orthogonal to it, and the full covariance decomposed into resolved, null, and cross‑sector blocks. The central organizing principle is that the resolved–null covariance coupling is not an incidental term but the primary invariant of the geometry. It is the quantity that determines rigidity, estimator performance, classification, and dynamical behavior across all layers of the theory. In the static layer, the response (Fisher) metric emerges as a derived quantity: it is exactly the inverse Schur complement of the covariance. This identity shows that the Fisher metric is blind to the null sector whenever the coupling vanishes, and that nonzero coupling rigidifies the automorphism group of the geometry. The static layer therefore identifies coupling as the structural source of both distinguishability and symmetry breaking. In the inferential layer, the coupling plays a quantitative role. A coupling‑aware estimator strictly dominates a resolved‑only estimator, and the improvement is governed by the canonical correlations between the resolved and null sectors. The gain in log‑volume of the estimator covariance is exactly equal to the resolved–null mutual information. This identity links statistical efficiency directly to information flow and shows that coupling is the operational quantity controlling how much additional information the null sector contributes to inference. In the classificatory layer, resolution geometries of fixed dimensions (r,q) form a moduli space of dimension rq+r+q. A complete invariant consists of the spectra of the resolved and null blocks together with the coupling expressed in their block eigenbases, modulo a discrete gauge. The coupling occupies rq of the moduli‑space dimensions and is therefore an irreducible coordinate. Canonical correlations separate geometries but are incomplete once both sectors have dimension at least two. This layer provides a full structural classification of resolution geometries and identifies coupling as the essential coordinate of the moduli space. In the dynamical layer, a linear stochastic evolution generates coupling at first order through the cross‑sector drift and propagates it exactly as a Mori–Zwanzig memory kernel. The circulating component of this kernel is the microscopic source of entropy production, linking coupling directly to irreversible behavior. This establishes coupling as the dynamical quantity that governs memory, transport, and the thermodynamic arrow of time. The constituent mathematical tools — Schur complements, canonical correlations, Lyapunov equations, Mori–Zwanzig projection, and entropy‑production identities — are classical. The contribution of the framework is their organization around the coupling as the central invariant, together with the classification and identity results, which appear to be new. Scope and non‑claims are stated precisely: the theory is first‑order and local, and several results apply specifically to the Gaussian and linear‑stochastic settings.
Matěj Rada (Fri,) studied this question.