Harmonically oriented neo-Schenkerian analysis and generative theories have in common the concept that a single chord can be "prolonged" over a given span of music, even if not every (or even any) note in the chord is playing at every moment during that span. When is it valid to say "chord x is prolonged throughout passage y?" As an answer, I could imagine either a general definition/theory, or a list of common prolongational patterns; either would be welcomed.
Unfortunately, I don't believe there is any concise answer to your question. Schenkerian theory takes years of dedicated study to master, in part because there are so many different methods of prolongation, so many ways in which these methods can manifest themselves, and so much repertoire to study to see these concepts "in the flesh" and how they vary from composer to composer.
In short, the answer to your question—"when is it valid to say 'chord x is prolonged throughout passage y'" in Schenkerian and other generative theories—is when these theories say that they do.
If there were such concise answers to Schenkerian theory, or to Lerdahl and Jackendoff's generative theory, there wouldn't be articles and dissertations and books about them, nor would graduate musicians need entire semesters (sometimes two!) to learn the fundamentals of the approaches, nor would sixty-year-old professors who have spent their careers studying Schenker still open up Der freie Satz every day to learn something new.