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Abstract Heavy elements are synthesized by ther-process in neutron star mergers and potentially in rare supernovae linked to strong magnetic fields. Expensive hydrodynamic simulations of these extreme environments are usually postprocessed to calculate the nucleosynthesis. In contrast, here we follow a site-independent approach based on three key parameters: electron fraction, entropy, and expansion timescale. Our model reproduces the results based on hydrodynamic simulations. Moreover, the 120,000 astrophysical conditions analyzed allow us to systematically and generally explore the astrophysical conditions of ther-process, also beyond those found in current simulations. Our results show that a wide range of conditions produce very similar abundance patterns explaining the observed robustness of ther-process between the second and third peak. Furthermore, we cannot find a single condition that produces the full solarr-process pattern from first to third peak. Instead, a superposition of at least two or three conditions or components is required to reproduce the typicalr-process pattern as observed in the solar system and very old stars. The different final abundances are grouped into eight nucleosynthesis clusters, which can be used to select representative conditions for comparisons to observations and investigations of the nuclear physics input.