claude suggested my this paper

http://webdoc.sub.gwdg.de/ebook/serien/e/IMPA_A/721.pdf about partitioning the complex plane to generate Julia sets with certainty of their shape (thanks again). I am currently trying to implement their algorithm, but there is one major point I do not understand:

Where does the first black square come from?

I understand their partitioning concept of white squares (all points lie on orbits that escape, hence in the attractor basin A(inf) of infinity), black squares (all points lie on orbits that are bounded) and gray ones: unknown so far.

The rule for turning gray cells into white is clear to me: if the image of a gray square A, named f(A) only intersects with white cells (all orbits escaping), then it also must be white.

The rule for black cells states: "Color black each gray leaf cell such that no path from it reaches a white cell". That part is not clear to me.

If I knew the exact attractor basin of infinity, surely, if I could say this gray cell does not intersect with A(inf), then it must be black. But the only thing I know, are the so far computed white cells, and the union of those is contained in the true A(inf). The union surely can't be bigger (then the algorithm would have classified points falsely) but since the refinement process is discrete that computed white cells union always is strictly a subset of A(inf), otherwise one could at some point color all remaining gray cells black and would now have straight lines as boundaries between white and black - which is not the case for a fractal (or am I mistaken hiere?). So what I'm trying to say is, bluntly spoken, "there's always more white to come", so some future refinement step will eventually color a gray cell to white.

Since at the beginning of the algorithm the black region is empty, the phrase "such that no path ever reaches a white cell" means that all paths only intersect with gray cells (again, am I mistaken here?). But since more white is to come, how can I be sure that those intersected gray cells do not eventually turn into white (or some subsquare in further refinement steps)?

I would unterstand the rule "if all paths eventually lead to black cells", but since the black region is empty at the beginning, the rule would never apply.

If someone could shed light on what I am missing here, that would really help me a lot. Thanks in advance.