Axions are a leading candidate to explain both dark matter and to solve the CP problem of the strong interactions, which makes it one of the most well-motivated extensions of the Standard Model. Recent ideas about topology and generalized symmetries suggest that non-perturbative defects (e.g., monopoles) might reshape axion phenomenology, opening signal space we have not yet probed. This program unites theory, phenomenology, and observation to identify testable consequences for axions and related axion-like particles. On the experimental side, these ideas translate into targets and strategies for searches (ABRACADABRA/DMRadio, ADMX, CASPEr, MADMAX, high-Q cavities, and SRF resonators) and on the astronomical side, we will pinpoint the most sensitive probes across stellar cooling, magnetars and pulsars, black-hole superradiance, and cosmological datasets. We will also test these ideas with targeted questions tied to recent work on compact objects: Which stellar mass–radius windows and compositions best probe sourced axion effects, and how should revised finite-density axion–nucleon couplings update supernova and neutron-star constraints? The timing is right: flagship searches are moving from prototypes to science runs, while design choices and downselection remain fluid. POutcomes will include prioritized parameter regions, suggestions for modest design tweaks with large impact, and a post-discovery plan to extract the underlying theory and its cosmological implications.