We build a model to predict from first principles the properties of major mergers. We predict these from the coalescence of peaks and saddle points in the vicinity of a given larger peak, as one increases the smoothing scale in the initial linear density field as a proxy for cosmic time. To refine our results, we also ensure, using a suite of ∼400 power-law Gaussian random fields smoothed at ∼30 different scales, that the relevant peaks and saddles are topologically connected: They should belong to a persistent pair before coalescence. Our model allows us to (a) compute the probability distribution function of the satellite-merger separation in Lagrangian space: They peak at three times the smoothing scale; (b) predict the distribution of the number of mergers as a function of peak rarity: haloes typically undergo two major mergers (>1:10) per decade of mass growth; (c) recover that the typical spin brought by mergers: it is of the order of a few tens of per cent.