Before revision:  The first result about a persistence-perfect gradient vector-field is that the number of its critical cells yield inequalities bounding the Betti tables of persistence modules , as a sort of Morse inequalities for multi-parameter persistence. The second result is that, at least in low dimensions, persistence-perfect gradient vector-fields not only exist but can be constructed by an algorithm based on local homotopy expansions .
After revision:  The first result about a persistence-perfect gradient vector-field is that the number of its critical cells yield inequalities bounding the Betti tables of persistence modules from above and below, via the number of critical cells. Our last result is the proof that existing algorithms based on local homotopy expansions .