Rank-Wise Approximations of Hyper-Operations

[Catullus]

1. \(\forall x\:\tau(x,1)=1\)
   
2. \(\forall y\in\Bbb{C-N}\:\tau(x,-y)=x\uparrow\tau(x,-y-1)\)
   
3. For a given k, using \(\tau\) to do \(\text{sexp(slog(}x)+k)\) produces the same iteration of exponentials, (Or principled logarithms, if k is negative, and if possible,) for any branch of slog, if slog happens to branch.
   
4. If \(\tau(x,y)\) approaches any of the fixed points or n-cycles of the function \(x\uparrow k\), it will approach continuously iterating \(m*y+b\).
   
5. \(\tau\) is mostly holomorphic.
   

Does \(\tau\) exist?
If so, is it unique?