User blog comment:Bubby3/Proof of termination of pair sequence system/@comment-35470197-20181119225559

> This is exactly the same as Bucholoz hydra, without the adding another node with label k-1 if the label is less than k-1.

Unfortunately, this paragraph states nothing unless you specify an explicit well-defined way to convert pair sequences which you are mentioning to Buchholz's hydras.