This paper is the third in a series that researches the Morse theory, gradient flows, concavity and complexity on smooth compact manifolds with boundary. Employing the local analytic models from [6], for traversally generic flows on -manifolds X, we embark on a detailed and somewhat tedious study of universal combinatorics of their tangency patterns with respect to the boundary  This combinatorics is captured by a universal poset  which depends only on the dimension of X. It is intimately linked with the combinatorial patterns of real divisors of real polynomials in one variable of degrees which do not exceed  Such patterns are elements of another natural poset  that describes the ways  in which the real roots merge, divide, appear and disappear under deformations of real polynomials. The space of real degree d polynomials  is stratified so that its pure strata are cells, labeled  by the elements of the poset  This cellular structure in  is interesting on its own right (see Theorem 4.1 and Theorem 4.2). Moreover, it helps to understand the localized structure of the trajectory spaces  for traversally generic fields v, the main subject of Theorem 5.2 and Theorem 5.3.

traversing flows, manifolds with boundary, combinatorics of real divisors.