The √2 is inherent in any square as π is inherent in any circle knowledge of neither is required for the completion of either task.
That we employ terms like π, φ, √2, √3 or √5, using decimals and angles is perfectly understandable, because that is way we were trained. It is not the way Ancient Egyptians were trained. They were trained in unit fractions, binary multiplication and division by the Eye of Horace, seked and sadly therein lies the problem. In the Ancient Egyptians world of mathematics based on unit fractions our irrational values simply did not exist.
In this real world, it is not for you to adapt Ancient Egyptians methods to conform to your hypothesis, since their methods are history and kind of set in stone. To resolve this issue it will be up to you to adapt your hypothesis to their methods and see if it still works.