If the LHC is able to produce dark matter particles, they would appear at the end of cascade decay chains, manifesting themselves as missing transverse energy. However, such "dark matter candidates" may decay invisibly later on. We propose to test for this possibility by studying the effect of particle widths on the observable invariant mass distributions of the visible particles seen in the detector. We consider the simplest non-trivial case of a two-step two-body cascade decay and derive analytically the shapes of the invariant mass distributions, for generic values of the widths of the new particles. We demonstrate that the resulting distortion in the shape of the invariant mass distribution can be significant enough to measure the width of the dark matter "candidate", ruling it out as the source of the cosmological dark matter.PACS numbers: 95.35.+d, 13.85.Qk Introduction. / E T events at the Large Hadron Collider (LHC) at CERN are motivated by the dark matter problem -the dark matter particles are stable and weakly interacting, and, once produced in the LHC collisions, will escape without leaving a trace inside the detector. This will lead to an imbalance in the transverse momentum of the event, known as "missing transverse momentum" / P T .[29] However, the reverse statement is not so obvious -if we observe an excess of / E T events at the LHC, how can one be sure that what we are seeing is indeed the cosmological dark matter?The question of proving that a / E T signal observed at the LHC is indeed due to dark matter, has attracted a lot of attention in the past [1-8]. The basic idea was to test whether the newly discovered weakly interacting massive particle (WIMP) was consistent with being a thermal relic or not. The general approach was to assume a specific model, most often some version of low-energy supersymmetry, and then attempt to measure all relevant model parameters affecting the thermal relic density calculation. Unfortunately, such an approach is modeldependent; applies only to thermal relics (for alternative non-thermal scenarios, see [9,10]); requires full understanding of the early cosmology; and typically demands a large number of additional measurements, possibly at future (or futuristic) facilities.Given that proving the discovery of dark matter at the LHC is such a difficult task, perhaps one should focus on the opposite question -how to disprove that the newly found invisible particle is the cosmological dark matter. One possibility is to perform a precise measurement of its mass, and if the mass is consistent with zero, it may just be one of the Standard Model (SM) neutrinos instead of a brand new particle [11]. However, this logic is not ironclad either -there exist examples where the dark matter particles are very light [12,13] and cannot be ruled out just on the basis of their small mass.A much more direct approach would be to test whether the particle which is the source of the / E T is indeed stable -after all, we only know that it did not decay inside the detector. If its lifet...