How quickly does a disappearing population actually disappear? In compartment models, that question matters because a quantity can approach zero while remaining significant for far longer than an exponential prediction suggests. This candidate research note gives a four-compartment counterexample to a proposed exponential-extinction rule. The model satisfies the stated restrictions on its transition and dependency graphs, yet the disappearing compartment decays as an inverse square in time. Its total accumulated presence remains finite. Four compartments are also shown to be the minimum needed for this phenomenon under the paper’s assumptions. A family of larger models produces still slower power-law decay, approaching the boundary beyond which accumulated presence would cease to be finite. The paper then explains what separates exponential and slower extinction. A matrix obtained by linearising the disappearing compartments at the limiting state provides the criterion. A strictly negative dominant spectral value gives exponential decay; a zero value allows slower behaviour. The revised argument also identifies the exact long-time logarithmic rate of the total disappearing mass. Exact arithmetic checks cover twenty-four finite examples and three parameter regimes, with deliberate errors tested to ensure the checker rejects them. These checks support the algebra but do not replace the general proofs. This is a candidate research release, not a claim of formal verification or external human peer review. Published on 20 October 2026, the paper and evidence are linked on the algebraic-extinction compartments release at Evidence Press. This briefing uses an AI-generated voice and is not additional scientific evidence.