  For modules over an artin algebra a linear stability condition is given by a
"central charge" and a nonlinear stability condition is given by the
wall-crossing sequence of a "green path". Finite Harder-Narasimhan
stratifications of the module category, maximal forward hom-orthogonal
sequences and maximal green sequences, defined using Fomin-Zelevinsky quiver
mutation are shown to be equivalent to finite nonlinear stability conditions
when the algebra is hereditary. This is the first of a series of three papers
whose purpose is to determine all maximal green sequences of maximal length for
quivers of affine type A and determine which are linear. See [1].
