We prove a classification theorem for objective-collapse theories: any model that is (i) sharp insingle runs, (ii) signaling-safe in ensembles (linear CPTP evolution), and (iii) defined by a record-determined stopping time, is mathematically equivalent to conditioning on a quantum instrument.In this sense, signaling-safe “physical collapse” is isomorphic to information updating within aconsistent instrument framework. When collapse is triggered by a threshold in a record-adaptedsharpness statistic, collapse-time laws are restricted to first-passage families; in the canonicaldrift–diffusion limit, this yields the Inverse–Gaussian (Wald) distribution. We provide falsifiabletests and an operational validity checklist that distinguish instrument-consistent stopping-timemodels from nonlinear update rules that necessarily violate convex-linearity or require hiddennonlocal dependence.Scope note. The classification result is strongest for sharp-event models where a run becomesε-sharp at a stopping time with respect to an accessible classical record. Continuous or record-freecollapse dynamics may evade the theorem by denying the existence of such a stopping time orby relegating the driving record to an inaccessible (“hidden”) ontology; we make these escapehatches explicit in §6.