User blog comment:Wythagoras/A far better bound for TREE(3)/@comment-5529393-20140622122929/@comment-5529393-20140623032709

The thing is, you can't have [][][][]...[][][][] or [][][]...[][][] - we are dealing with trees, not forests.

I guess you could go from (([])) to ([][]...[][]), and then to ([][]...[](((...()...)))) or probably better ([][]...[]...). You don't want to use the Kirby-Paris hydra procedure here - as I'm sure you know, the Kirby-Paris hydra ordering goes up to ε_0. In the SVO ordering that I'm aware of, first you go to (((......))) for the finite ordinals, and (...) represents the really big ordinals.

Okay, from ([][][]...[]) you can reduce one-color trees n times, so it's at level SVO+1. (([])) is at level SVO+2. (([]())) is at level SVO+3. (([])) is at level SVO+ω. (([])) is at level SVO+ε_0. (([][])) is at level SVO*2. (([][][])) is at level SVO*3. ((([]))) is at level SVO*ω. ((([]))) is at level SVO*ω+1. ((([])[])) is at level SVO*(ω+1). ((([])([]))) is at level SVO*ω2. ((([]))) is at level SVO*ω^2. ((([][]))) is at level SVO*SVO. ((([][][]))) is at level SVO^3. (((([])))) is at level SVO^ω. ((((([]))))) is at level ω^ω^ω^(SVO+1). [] is at level ε_(SVO+1).

So it looks like it works out. Good job finding this improvement!