The study of formation mechanisms of heavy-quarkonium states, the so-called “hydrogen atoms” of QCD, stands as a valuable tool for unveiling core aspects of the strong force. Quarkonium physics bridges precision studies of perturbative QCD and explorations of the proton structure. In this seminar, we present progress on the Heavy-Flavor Non-Relativistic evolution (HF-NRevo) framework, a novel approach
for addressing quarkonium production within the collinear fragmentation approximation. We draw on next-to-leading order Non-Relativistic-QCD (NRQCD) calculations for parton fragmentation channels into both vector and pseudoscalar quarkonia, which we use as proxies for initial-scale inputs. This results in the construction of a complete set of variable-flavour number-scheme fragmentation functions,
named NRFF1.0, which are derived through standard DGLAP evolution. Statistical errors are assessed using a Monte Carlo, replica-like approach that also accounts for Missing Higher-Order Uncertainties (MHOUs). Looking ahead, we will also discuss the potential of using HF-NRevo to address quarkonium in-jet fragmentation.