I really like the conversion to a representation that's more familiar but I suspect it's also confusing me. Here are the questions that I have to start:

Does 'there-is' evaluate to a boolean (is boolean a thing? is evaluation a thing?)

I was surprised that the axioms don't use there-is but I sense I shouldn't be?

what does '(when 0 (or 0 1))' mean, it's described as 'when 0, then there is either 0 or 1' but I'm not sure what that means? is '(when 0 0)' true and '(when 1 0)' false? Why not use '(when A B)' as equals?

Not a question but in the axioms I assume pairs should be pears and applies should be apples.

How goes 'implies' get introduced by '(when (when q r) (when (or p q) (or p r))'? Also, this introduces a category in the example i.e. fruits but not sure how that happened.

Why do we need to have second '(not (factor? x 3^22 ...))', isn't it sufficient to say that (factor? x 3^21 ...) to establish that the 2nd term is 'there-is'?