The construction counts an arbitrary length list of numbers, each which can count an integer > 0. The case between a number < 1 and a number >=1 is also addressed.
If that holds true without missing any possibilities, then this maps to the continued fractions. I've gone through several such mappings that missed at least one before, so looking to see if this one has any such as well.
There's a much better introduction to write here, so I'll jump right into avoiding one.
I realize this isn't an expected possibility, but I've reached a point where I am not able to uncover where it might have gone wrong.
I've tried to keep things direct and in point form, to (attempt to) make it easier to spot any assumptions made.
This might not make it as readable as it could be, and specifically avoids the formalization required for a proof, but I have more to add if it holds up.
Thank you in advance for any feedback provided or questions to clarify on.
Fantastic show for sure.