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I love these various theoretical explanations, but what would really land the plane is some suitably small unit test date doing the forward and reverse transforms.

This is the general grief with math: in enshrining the ideas in a single letter based formula, the mathematician has produced a kind of sculpture that makes sense to the initiate, but baffles the lay reader.

This is not a complaint; such sculpture filters out the riff-raff. One has to want it. And maybe filling in that unit test data is just part of the journey to understanding.