User blog:MilkyWay90/Analysis of non-positive integer hyperoperators

https://googology.wikia.com/wiki/User_blog:Eners49/The_secret_0th_hyper-operator%3F The inspiration for this blog post is in the link.

Define the 0th hyperoperator as x?y: a?b?...?c with d numbers = S(S(S(...S(d)...))) with d Ss, where S is the successor function. Proven in my comment

Define the -1st hyperoperator as x??y:

1??1 = 1?2 = S(S(2)) = 4 1??1??1 = 1?3 = 5 1??1??1??1 = 6 And so on... Because the x has no affect in x?y, we can define: a??b??...??c with d ??s = d + 3

-2nd hyperoperator coming soon: