39. Interaction structure and exact nonlinear quotients

Let \(X_{\mathrm f}\) be a fine state space, let \(X_{\mathrm c}\) be a coarse state space, and let \[C:X_{\mathrm f}\twoheadrightarrow X_{\mathrm c}\] be a surjective linear map. A fine trajectory satisfies \[\begin{equation} \tag{PRIQ.1.1} \dot x(t)=F_{\mathrm f}(x(t)), \qquad x(0)=x_0, \end{equation}\] and the coarse observable of interest is \[z(t)=Cx(t).\] The first question is whether \(z(t)\) obeys a closed equation of its own. Equivalently, can we define a vector field \(F_{\mathrm c}\) such that \[\begin{equation} \tag{PRIQ.1.2} C F_{\mathrm f}=F_{\mathrm c}C? \end{equation}\] When this identity holds and both initial value problems have unique solutions, aggregation commutes with evolution: \[\begin{equation} \tag{PRIQ.1.3} C\Phi_t^{\mathrm f}=\Phi_t^{\mathrm c}C. \end{equation}\] The coarse trajectory is then an exact quotient of the fine one. No fitting step is involved.