Zeration

[bo198214]

The problem, as you also correctly said, is to define an operation (ONE hyperop) that would be the unique operation, exactly fitting in the hyperops hierarchy, at rank 0.

Only to summarize and clarify the current situation:
If we agree on the law a[n+1](b+1)=a[n](a[n+1]b) for all hyperoperations [n] for integer n and agree that a[1]b=a+b then it stringently follows (without assumptions about initial values) that a[0]b=b+1 and it also stringently follows for all hypo operations that a[-n]b=b+1. (This was shown in this thread by Andrew and me.)
I would call this "exactly fitting in the hyper operations hierarchy".

If we otherwise dont accept the above law, then there are of course multiple possibilities to define an operation [0] and if we impose the conditions
a[0]a=a+2
a[0](a+n)=a+n+1 for n>1
the operation [0] is still is not uniquely defined not even on the integer numbers a.
One example of an operation obeying these rules is Konstantin's zeration and another example was given by me in this thread. Till now there was no set of equations presented for which Konstantin's zeration is the only solution.