Folding a Perfect Pentagon from a Square

At the IOIO - 2025 (International Origami Internet Olympiad X), there was quite a bit of debate regarding the method of folding a perfect pentagon from a square. While the margin of error in the reference points for the standard method is only about 0.3%, many participants found this unacceptable for the ″Task 3. Penrose Punch (by Alphonsus Rouis)″ challenge (which translates to roughly 3 mm per 1 meter of paper).

So, my question is: what are the known, purely mathematical methods for obtaining a perfect pentagon with zero theoretical error?

@Andrey Ermakov :

about 0.3%

And not only... The diagram′s method is quite challenging to execute precisely—especially step 6, where the crease intersects the diagonal at a very sharp angle. On a large sheet of paper, it′s very easy to make an error in that step. Not to mention the accumulated error you′re talking about.

@LEV :

sharp angle

At the time the diagram was drawn, no better solution had been found. Good thing it didn′t discourage most competitors!