Gravitational-Wave Image Multiplicity as a Topological Probe of Dark-Matter Core Collapse

Dark-matter self-interactions can drive subhalos through gravothermal core collapse, but conventional lensing observables vary continuously with uncertain inner profiles. We show that gravitational-wave image multiplicity provides a discrete alternative. Near a macro-critical minimum, increasing subhalo central convergence opens two caustics in sequence, converting one detectable waveform into three and then five stationary images. Using weighted SASHIMI subhalos and an ET+CE selection model, we forecast almost 2 systems with at least three copies detectable in five years if dense remnants survive, compared with the prediction of 0.36 in 5yrs for standard CDM. The much more diagnostic second case is rare: the corresponding 5yrs yield with at least four detectable copies is 0.149, versus 1.55 times10-6 for CDM. According to the Poisson statistics, this gives only a 13.8% probability of at least one event detected in 5 years. Thus this is a rare-event, exposure-limited test rather than a guaranteed detection: a resolved second pair would be difficult to reconcile with the fiducial single-halo CDM population, whereas a five-year null result would not by itself constrain remnant survival strongly.