The Structural Operator

What does it mean for two dynamical systems to be structurally equivalent? This paper develops a precise mathematical answer grounded in operator geometry rather than ontological identity. We propose that dynamical systems — drawn from physics, stochastic processes, information theory, or causal modeling — can be classified by the coarse-grained geometric character of their induced operator actions on probability measure spaces, independently of their microscopic equations of motion. The framework introduces two invariants: a global contraction index κ(T) and a six-dimensional structural feature embedding Π(T), encoding the relative contributions of six canonical operator modes — compression, expansion, torsion, drift, recurrence, and directed convergence. Linear independence of these modes is established rigorously. The projection map Π(T) is interpreted throughout as a coarse-grained coordinate, not an ontological decomposition of nature. The central philosophical claim is deflationary: structural similarity across domains is real and mathematically tractable without requiring that the domains share fundamental constituents, causal mechanisms, or physical substrate. The framework belongs to the philosophy of applied mathematics and formal classification theory, and bears on debates about structural realism, inter-theoretic reduction, and the scope of mathematical explanation in science.