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This is the first paper of a series of two devoted to develop a practicalmethod to describe the growth history of bound virialized objects in thegravitational instability scenario without resorting to $N$-body simulations.Here we present the basic tool of this method, ``the confluent systemformalism'', which allows us to follow the filtering evolution of peaks in arandom Gaussian field of density fluctuations. This is applied to derive thetheoretical mass function of objects within the peak model framework. Along theprocess followed for the derivation of this function, we prove that theGaussian window is the only one consistent with the peak model ansatz. We alsogive a well justified derivation of the density of peaks with density contrastupcrossing a given threshold in infinitesimal ranges of scale and correct thisscale function for the cloud-in-cloud effect. Finally, we characterize the formof the mass vs. scale and the critical overdensity vs. collapse time relationswhich are physically consistent with the peak model in an Einstein-de Sitteruniverse with density field endowed with different power spectra. The result isa fully justified semianalytical mass function which is close to the Press \&Schechter (1974) one giving good fits to $N$-body simulations. But the interestof the confluent system formalism is not merely formal. It allows us todistinguish between accretion and merger events, which is essential for thedetailed modelling of the clustering process experienced by objects.Comment: 24 pages, uuencoded compressed postscript file including 4 figures  (164 kb). To be published in ApJ, 1st November issu

The Confluent System Formalism. I. The Mass Function of Objects in the Peak Model