User blog:QuasarBooster/Second-order worms ordinal analysis dump

I recently analysed normal worms so I could get a better intuition of their ordinals and why the limit ordinal is epsilon-0. I proceeded to do the same for order-2 worms, though obviously it took a lot more work (for me at least). Here's a bunch of ordinals I reached on the way to finding the limit ordinal for second-order worms.

Some disclaimers: I had to space out a few brackets so that they would show up. The reason this analysis starts at [ [1] ] is that if an order-2 worm's entries contain just zeroes then it behaves just like a corresponding first-order worm. So the limit of [ [0,0,0,...] ] = [ [1] ] is epsilon-0. Obviously I might have made mistakes so I'd love to know if you spot any. Finally, I'm aware that some of the fundamental sequences I used aren't standard. Please bear with me on that (unless even then there's a mistake, in which case, gosh dang it). Order-2 limit: φ(ε_0,0) = φ(φ(1,0),0)
 * [[1] ] = ε_0
 * 1],[0],[1 = ε_0^2
 * 1],[0],[1],[0],[1 = ε_0^3
 * 1],[0,0 = ε_0^ω
 * 1],[0,0],[0],[1],[0,0 = ε_0^(ω2)
 * 1],[0,0],[0,0 = ε_0^(ω^2)
 * 1],[0,0],[0,0,0 = ε_0^(ω^ω)
 * 1],[0,0],[0,0,0],[0,0 = ε_0^(ω^(ω+1))
 * 1],[0,0],[0,0,0],[0,0,0 = ε_0^(ω^(ω^2))
 * 1],[0,0],[0,0,0,0 = ε_0^(ω^(ω^ω))
 * 1],[0,0],[0,0,0,0],[0,0,0 = ε_0^(ω^(ω^(ω+1)))
 * 1],[0,0],[0,0,0,0],[0,0,0,0 = ε_0^(ω^(ω^(ω^2)))
 * 1],[0,0],[1 = ε_0^ε_0
 * 1],[0,0],[1],[0,0],[1 = ε_0^(ε_0^2)
 * 1],[0,0],[1],[0,0],[1],[0,0],[1 = ε_0^(ε_0^3)
 * 1],[0,0,0 = ε_0^(ε_0^ω)
 * 1],[0,0,0],[0,0,0 = ε_0^(ε_0^(ω^2))
 * 1],[0,0,0],[0,0,0,0 = ε_0^(ε_0^(ω^ω))
 * 1],[0,0,0],[0,0,0,0],[0,0,0 = ε_0^(ε_0^(ω^(ω+1)))
 * 1],[0,0,0],[0,0,0,0],[0,0,0,0 = ε_0^(ε_0^(ω^(ω^2)))
 * 1],[0,0,0],[1 = ε_0^(ε_0^ε_0)
 * 1],[0,0,0],[1],[0,0,0],[1 = ε_0^(ε_0^(ε_0^2))
 * 1],[0,0,0,0],[1 = ε_0^(ε_0^(ε_0^ε_0))
 * 1],[1 = ε_1
 * 1],[1],[0],[1],[0,0],[0,0,0 = ε_1*ε_0^(ω^ω)
 * 1],[1],[0],[1],[0,0],[0,0,0,0 = ε_1*ε_0^(ω^(ω^ω))
 * 1],[1],[0],[1],[0,0],[1 = ε_1*ε_0^ε_0
 * 1],[1],[0],[1],[1 = ε_1^2
 * 1],[1],[0],[1],[1],[0],[1],[1 = ε_1^3
 * 1],[1],[0,0 = ε_1^ω
 * 1],[1],[0,0],[0,0,0 = ε_1^(ω^ω)
 * (...same as with ε_0...)
 * 1],[1],[1 = ε_2
 * 1],[1],[1],[1 = ε_3
 * [[1,0] ] = ε_ω
 * 1,0],[1 = ε_(ω+1)
 * 1,0],[1],[1,0 = ε_(ω2)
 * 1,0],[1,0 = ε_(ω^2)
 * 1,0],[1,0],[1,0 = ε_(ω^3)
 * [[1,0,0] ] = ε_(ω^ω)
 * 1,0,0],[1],[1,0,0 = ε_((ω^ω)2)
 * 1,0,0],[1,0 = ε_(ω^(ω+1))
 * 1,0,0],[1,0],[1,0,0 = ε_(ω^(ω2))
 * 1,0,0],[1,0,0 = ε_(ω^(ω^2))
 * 1,0,0],[1,0,0],[1,0,0 = ε_(ω^(ω^3))
 * [[1,0,0,0] ] = ε_(ω^(ω^ω))
 * [[1,0,1] ] = ε_(ε_0)
 * 1,0,1],[1,0],[1,0,0 = ε_(ε_0*ω^ω)
 * 1,0,1],[1,0],[1,0,0,0 = ε_(ε_0*ω^(ω^ω))
 * 1,0,1],[1,0],[1,0,1 = ε_(ε_0^2)
 * (...similar to ε_0^2...)
 * 1,0,1],[1,0,0 = ε_(ε_0^ω)
 * 1,0,1],[1,0,0],[1,0,1 = ε_(ε_0^ε_0)
 * 1,0,1],[1,0,0,0],[1,0,1 = ε_(ε_0^(ε_0^ε_0))
 * 1,0,1],[1,0,1 = ε_(ε_1)
 * (...should be similar to ε_1 - ε_2...)
 * 1,0,1],[1,0,1],[1,0,1 = ε_(ε_2)
 * [[1,0,1,0] ] = ε_(ε_ω)
 * (...similar to ε_ω - ε_(ε_0)...)
 * [[1,0,1,0,1,0] ] = ε_(ε_(ε_ω))
 * [[1,1] ] = ζ_0 = φ(2,0)
 * 1,1],[0,0],[1 = ζ_0^ε_0
 * 1,1],[0,0],[1],[0,0,0],[1 = ζ_0^(ε_0^ε_0)
 * 1,1],[0,0],[1,1 = ζ_0^ζ_0
 * 1,1],[0,0,0],[1,1 = ζ_0^(ζ_0^ζ_0)
 * 1,1],[1 = ζ_1 = φ(2,1)
 * (still works like ε... I think)
 * [[1,1,0] ] = ζ_ω
 * [[1,1,0,0] ] = ζ_(ω^ω)
 * [[1,1,0,1] ] = ζ_(ε_0)
 * [[1,1,0,1,1] ] = ζ_(ζ_0) = φ(2,φ(2,0))
 * [[1,1,0,1,1,0,1,1] ] = ζ_(ζ_(ζ_0))
 * [[1,1,1] ] = φ(3,0)
 * [[1,1,1,1] ] = φ(4,0)
 * [[2] ] = φ(ω,0)
 * [[2,1] ] = φ(ω+1,0)
 * [[2,2] ] = φ(ω^2,0)
 * [[2,2,2] ] = φ(ω^3,0)
 * [[3] ] = φ(ω^ω,0)
 * [[4] ] = φ(ω^(ω^ω),0)