Designing collective behaviour within a fixed coarse-grained description

Abstract: A fixed coarse-grained description need not uniquely determine the underlying microscopic rules. Variations in these rules can alter collective behaviour while leaving the retained observables unchanged. Here we develop a method to identify and use such variations for collective-state design. We construct families of microscopic models with exactly matched coarse observables and use the response derivative within each family to identify independent first-order output changes. In a nonreciprocal active mixture, the largest first-order change in interfacial growth rate under a pointwise kernel budget also determines the leading minimax error of predictions based only on the matched data. Near a Hopf bifurcation, this response map gives an initial turning-kernel design, which is refined by jointly solving for the periodic kinetic state and the kernel. Fixed-kernel integrations realize prescribed oscillation powers and frequencies while preserving the selected angular relaxation rates. In a reaction-diffusion family, cubic amplitude equations guide the design of stripe and hexagon amplitudes while the full linear operator remains fixed. Additional constraints reduce the available response even when its rank is unchanged, whereas the matched angular moments retain their exact relaxation rates. These results provide a constructive route to collective-state design within a prescribed coarse-grained description.