In regards to writing the introduction of science research papers, what exactly is written in the space after one finishes defining the gap in research and before the specific aims and methodologies of the current paper is stated?

I'm guessing it would be some sort of overarching thesis statement. Would this thesis statement summarize the research gap, or propose new research/novel solution or address the overall objectives/purpose of the paper? The problem is that from all the papers I've read I can't abstract out any reliable or rational way of setting up the thesis statement and understanding what it is.

A paper is not the same as an undergraduate or postgraduate thesis/dissertation. You can use more space in a dissertation in the introduction section to present a general overview of the topics you have studied and what your dissertation discusses. Try to keep it so that anyone with a reasonable background can get the overall ideas just be reading the introduction section. You do not have to explain all details and can gloss over some technicalities in the introduction.