Now, instead of making #{&_2+&_2}# an ordinal level of ψ0(ψ2(ψ2(0))), we go all the way to make (#{&_2}#){&_2}# an ordinal level of ψ0(ψ2(0) + ψ1(ψ2(0))) instead.
We now have (Bashicu matrix system and Y sequence comparison included):
#{&_2}# => ψ0(Ω_2) | ψ0(ψ2(0)) | (0,0)(1,1)(2,2) | (1,2,4,7)
(#{&_2}#)^^# => ψ0(Ω_2 + Ω) | ψ0(ψ2(0) + ψ1(0)) | (0,0)(1,1)(2,2)(1,1) | (1,2,4,7,4)
(#{&_2}#)^^## => ψ0(Ω_2 + Ω^2) | ψ0(ψ2(0) + ψ1(ψ1(0))) | (0,0)(1,1)(2,2)(1,1)(2,1) | (1,2,4,7,4,6)
(#{&_2}#)^^^# => ψ0(Ω_2 + Ω^Ω) | ψ0(ψ2(0) + ψ1(ψ1(ψ1(0)))) | (0,0)(1,1)(2,2)(1,1)(2,1)(3,1) | (1,2,4,7,4,6,8)
(#{&_2}#){&}# => ψ0(Ω_2 + Ω^Ω^2) | ψ0(ψ2(0) + ψ1(ψ1(ψ1(0) + ψ1(0)))) | (0,0)(1,1)(2,2)(1,1)(2,1)(3,1)(3,1) | (1,2,4,7,4,6,8,8)
(#{&_2}#){&^#}# => ψ0(Ω_2 + Ω^Ω^ω) | ψ0(ψ2(0) + ψ1(ψ1(ψ1(ψ0(0))))) | (0,0)(1,1)(2,2)(1,1)(2,1)(3,1)(4,0) | (1,2,4,7,4,6,8,9)
(#{&_2}#){&^&}# => ψ0(Ω_2 + Ω^Ω^Ω) | ψ0(ψ2(0) + ψ1(ψ1(ψ1(ψ1(0))))) | (0,0)(1,1)(2,2)(1,1)(2,1)(3,1)(4,1) | (1,2,4,7,4,6,8,10)
(#{&_2}#){&_2}# => ψ0(Ω_2 + ψ1(Ω_2)) | ψ0(ψ2(0) + ψ1(ψ2(0))) | (0,0)(1,1)(2,2)(1,1)(2,2) | (1,2,4,7,4,7)
We know that (#{&_2}#){&_2}# is expanded as {(#{&_2}#){&}#, (#{&_2}#){&^&}#, (#{&_2}#){&^&^&}#, ...}. We can also have:
((#{&_2}#){&_2}#){&_2}# => ψ0(Ω_2 + ψ1(Ω_2)·2) | ψ0(ψ2(0) + ψ1(ψ2(0)) + ψ1(ψ2(0))) | (0,0)(1,1)(2,2)(1,1)(2,2)(1,1)(2,2) | (1,2,4,7,4,7,4,7)
#{&_2}#># => ψ0(Ω_2 + ψ1(Ω_2)·ω) | ψ0(ψ2(0) + ψ1(ψ2(0) + ψ0(0))) | (0,0)(1,1)(2,2)(2,0) | (1,2,4,7,5)
#{&_2}#>#{&_2}# => ψ0(Ω_2 + ψ1(Ω_2 + ψ0(Ω_2))) | ψ0(ψ2(0) + ψ1(ψ2(0) + ψ0(ψ2(0)))) | (0,0)(1,1)(2,2)(2,0)(3,1)(4,2) | (1,2,4,7,5,7,10)
#{&_2}## => ψ0(Ω_2 + ψ1(Ω_2 + Ω)) | ψ0(ψ2(0) + ψ1(ψ2(0) + ψ1(0))) | (0,0)(1,1)(2,2)(2,1) | (1,2,4,7,6)
#{&_2}### => ψ0(Ω_2 + ψ1(Ω_2 + Ω·2)) | ψ0(ψ2(0) + ψ1(ψ2(0) + ψ1(0) + ψ1(0))) | (0,0)(1,1)(2,2)(2,1)(2,1) | (1,2,4,7,6,6)
#{&_2}#^# => ψ0(Ω_2 + ψ1(Ω_2 + Ω·ω)) | ψ0(ψ2(0) + ψ1(ψ2(0) + ψ1(ψ0(0)))) | (0,0)(1,1)(2,2)(2,1)(3,0) | (1,2,4,7,6,7)
#{&_2}#{&_2}# => ψ0(Ω_2 + ψ1(Ω_2 + Ω·ψ0(Ω_2))) | ψ0(ψ2(0) + ψ1(ψ2(0) + ψ1(ψ0(ψ2(0))))) | (0,0)(1,1)(2,2)(2,1)(3,0)(4,1)(5,2) | (1,2,4,7,6,7,9,12)
#{&_2+1}# => ψ0(Ω_2 + ψ1(Ω_2 + Ω^2)) | ψ0(ψ2(0) + ψ1(ψ2(0) + ψ1(ψ1(0)))) | (0,0)(1,1)(2,2)(2,1)(3,1) | (1,2,4,7,6,8)
#{&_2+&}# => ψ0(Ω_2 + ψ1(Ω_2 + Ω^Ω)) | ψ0(ψ2(0) + ψ1(ψ2(0) + ψ1(ψ1(ψ1(0))))) | (0,0)(1,1)(2,2)(2,1)(3,1)(4,1) | (1,2,4,7,6,8,10)
#{&_2+(&_2)_1}# => ψ0(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2))) | ψ0(ψ2(0) + ψ1(ψ2(0) + ψ1(ψ2(0)))) | (0,0)(1,1)(2,2)(2,1)(3,2) | (1,2,4,7,6,9)
#{&_2+(&_2+&)_1}# => ψ0(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2 + Ω))) | ψ0(ψ2(0) + ψ1(ψ2(0) + ψ1(ψ2(0) + ψ1(0)))) | (0,0)(1,1)(2,2)(2,1)(3,2)(3,1) | (1,2,4,7,6,9,8)
#{&_2+(&_2+(&_2)_1)_1}# => ψ0(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2 + ψ1(Ω_2)))) | ψ0(ψ2(0) + ψ1(ψ2(0) + ψ1(ψ2(0) + ψ1(ψ2(0))))) | (0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2) | (1,2,4,7,6,9,8,11)
And here we have #{&_2+&_2}# expanded into {#{&_2}#, #{&_2+(&_2)_1}#, #{&_2+(&_2+(&_2)_1)_1}#, ...}. Same goes for #{&_2*&_2}# => {#{&_2}#, #{&_2*(&_2)_1}#, #{&_2*(&_2*(&_2)_1)_1}#} and #{&_2^&_2}# => {#{&_2}#, #{&_2^(&_2)_1}#, #{&_2^(&_2^(&_2)_1)_1}#}. It is worth noting that #{&_2 + (&_2 + 1)_1}# is equal to #{&_2 + (&_2)_1*#}#. Here we go.
#{&_2+&_2}# => ψ0(Ω_2·2) | ψ0(ψ2(0) + ψ2(0)) | (0,0)(1,1)(2,2)(2,2) | (1,2,4,7,7)
(#{&_2+&_2}#)^^# => ψ0(Ω_2·2 + Ω) | ψ0(ψ2(0) + ψ2(0) + ψ1(0)) | (0,0)(1,1)(2,2)(2,2)(1,1) | (1,2,4,7,7,4)
(#{&_2+&_2}#){&_2+&_2}# => ψ0(Ω_2·2 + ψ1(Ω_2·2)) | ψ0(ψ2(0) + ψ2(0) + ψ1(ψ2(0) + ψ2(0))) | (0,0)(1,1)(2,2)(2,2)(1,1)(2,2)(2,2) | (1,2,4,7,7,4,7,7)
#{&_2+&_2}#># => ψ0(Ω_2·2 + ψ1(Ω_2·2)·ω) | ψ0(ψ2(0) + ψ2(0) + ψ1(ψ2(0) + ψ2(0) + ψ0(0))) | (0,0)(1,1)(2,2)(2,2)(2,0) | (1,2,4,7,7,5)
#{&_2+&_2}#>#{&_2+&_2}# => ψ0(Ω_2·2 + ψ1(Ω_2·2)·ψ0(Ω_2·2)) | ψ0(ψ2(0) + ψ2(0) + ψ1(ψ2(0) + ψ2(0) + ψ0(ψ2(0) + ψ2(0)))) | (0,0)(1,1)(2,2)(2,2)(2,0)(3,1)(4,2)(4,2) | (1,2,4,7,7,5,7,10,10)
#{&_2+&_2}## => ψ0(Ω_2·2 + ψ1(Ω_2·2 + Ω)) | ψ0(ψ2(0) + ψ2(0) + ψ1(ψ2(0) + ψ2(0) + ψ1(0))) | (0,0)(1,1)(2,2)(2,2)(2,1) | (1,2,4,7,7,6)
#{&_2+&_2+(&_2+&_2)_1}# => ψ0(Ω_2·2 + ψ1(Ω_2·2 + ψ1(Ω_2·2))) | ψ0(ψ2(0) + ψ2(0) + ψ1(ψ2(0) + ψ2(0) + ψ1(ψ2(0) + ψ2(0)))) | (0,0)(1,1)(2,2)(2,2)(2,1)(3,2)(3,2) | (1,2,4,7,7,6,9,9)
#{&_2+&_2+&_2}# => ψ0(Ω_2·3) | ψ0(ψ2(0) + ψ2(0) + ψ2(0)) | (0,0)(1,1)(2,2)(2,2)(2,2) | (1,2,4,7,7,7)
#{&_2+&_2+&_2+&_2}# => ψ0(Ω_2·4) | ψ0(ψ2(0) + ψ2(0) + ψ2(0)+ ψ2(0)) | (0,0)(1,1)(2,2)(2,2)(2,2)(2,2) | (1,2,4,7,7,7,7)
#{&_2*#}# => ψ0(Ω_2·ω) | ψ0(ψ2(ψ0(0))) | (0,0)(1,1)(2,2)(3,0) | (1,2,4,7,8)
#{&_2*#{&_2}#}# => ψ0(Ω_2·ψ0(Ω_2)) | ψ0(ψ2(ψ0(ψ2(0)))) | (0,0)(1,1)(2,2)(3,0)(4,1)(5,2) | (1,2,4,7,8,10,13)
#{&_2*&}# => ψ0(Ω_2·Ω) | ψ0(ψ2(ψ1(0))) | (0,0)(1,1)(2,2)(3,1) | (1,2,4,7,9)
#{&_2*(&_2)_1}# => ψ0(Ω_2·ψ1(Ω_2)) | ψ0(ψ2(ψ1(ψ2(0)))) | (0,0)(1,1)(2,2)(3,1)(4,2) | (1,2,4,7,9,12)
#{&_2*&_2}# => ψ0(Ω_2^2) | ψ0(ψ2(ψ2(0))) | (0,0)(1,1)(2,2)(3,2) | (1,2,4,7,10)
#{&_2*&_2*&_2}# => ψ0(Ω_2^3) | ψ0(ψ2(ψ2(0) + ψ2(0))) | (0,0)(1,1)(2,2)(3,2)(3,2) | (1,2,4,7,10,10)
#{&_2^#}# => ψ0(Ω_2^ω) | ψ0(ψ2(ψ2(ψ0(0)))) | (0,0)(1,1)(2,2)(3,2)(4,0) | (1,2,4,7,10,11)
#{&_2^&}# => ψ0(Ω_2^Ω) | ψ0(ψ2(ψ2(ψ1(0)))) | (0,0)(1,1)(2,2)(3,2)(4,1) | (1,2,4,7,10,12)
#{&_2^(&_2)_1}# => ψ0(Ω_2^ψ1(Ω_2)) | ψ0(ψ2(ψ2(ψ1(ψ2(0))))) | (0,0)(1,1)(2,2)(3,2)(4,1)(5,2) | (1,2,4,7,10,12,15)
#{&_2^(&_2*&_2)_1}# => ψ0(Ω_2^ψ1(Ω_2^2)) | ψ0(ψ2(ψ2(ψ1(ψ2(ψ2(0)))))) | (0,0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2) | (1,2,4,7,10,12,15,18)
#{&_2^&_2}# => ψ0(Ω_2^Ω_2) | ψ0(ψ2(ψ2(ψ2(0)))) | (0,0)(1,1)(2,2)(3,2)(4,2) | (1,2,4,7,10,13)
#{&_2^(&_2*&_2)}# => ψ0(Ω_2^Ω_2^2) | ψ0(ψ2(ψ2(ψ2(0) + ψ2(0)))) | (0,0)(1,1)(2,2)(3,2)(4,2)(4,2) | (1,2,4,7,10,13,13)
#{&_2^&_2^#}# => ψ0(Ω_2^Ω_2^ω) | ψ0(ψ2(ψ2(ψ2(ψ0(0))))) | (0,0)(1,1)(2,2)(3,2)(4,2)(5,0) | (1,2,4,7,10,13,14)
#{&_2^&_2^&}# => ψ0(Ω_2^Ω_2^Ω) | ψ0(ψ2(ψ2(ψ2(ψ1(0))))) | (0,0)(1,1)(2,2)(3,2)(4,2)(5,1) | (1,2,4,7,10,13,15)
#{&_2^&_2^&_2}# => ψ0(Ω_2^Ω_2^Ω_2) | ψ0(ψ2(ψ2(ψ2(ψ2(0))))) | (0,0)(1,1)(2,2)(3,2)(4,2)(5,2) | (1,2,4,7,10,13,16)
#{&_2^&_2^&_2^&_2}# => ψ0(Ω_2^Ω_2^Ω_2^Ω_2) | ψ0(ψ2(ψ2(ψ2(ψ2(ψ2(0)))))) | (0,0)(1,1)(2,2)(3,2)(4,2)(5,2)(6,2) | (1,2,4,7,10,13,16,19)
...
Now we move on #{&_3}# => {#{&_2}#, #{&_2^&_2}#, #{&_2^&_2^&_2}#}:
#{&_3}# => ψ0(Ω_3) | ψ0(ψ3(0)) | (0,0)(1,1)(2,2)(3,3) | (1,2,4,7,11)
#{&_3 + &_2}# => ψ0(Ω_3 + Ω_2) | ψ0(ψ3(0) + ψ2(0)) | (0,0)(1,1)(2,2)(3,3)(2,2) | (1,2,4,7,11,7)
#{&_3 + (&_3)_2}# => ψ0(Ω_3 + ψ2(Ω_3)) | ψ0(ψ3(0) + ψ2(ψ3(0))) | (0,0)(1,1)(2,2)(3,3)(2,2)(3,3) | (1,2,4,7,11,7,11)
#{&_3 + (&_3 + 1)_2}# => ψ0(Ω_3 + ψ2(Ω_3 + 1)) | ψ0(ψ3(0) + ψ2(ψ3(0) + ψ0(0))) | (0,0)(1,1)(2,2)(3,3)(3,0) | (1,2,4,7,11,8)
#{&_3 + (&_3 + &)_2}# => ψ0(Ω_3 + ψ2(Ω_3 + Ω)) | ψ0(ψ3(0) + ψ2(ψ3(0) + ψ1(0))) | (0,0)(1,1)(2,2)(3,3)(3,1) | (1,2,4,7,11,9)
#{&_3 + (&_3 + &_2)_2}# => ψ0(Ω_3 + ψ2(Ω_3 + Ω_2)) | ψ0(ψ3(0) + ψ2(ψ3(0) + ψ2(0))) | (0,0)(1,1)(2,2)(3,3)(3,2) | (1,2,4,7,11,10)
#{&_3 + (&_3 + (&_3)_2)_2}# => ψ0(Ω_3 + ψ2(Ω_3 + ψ2(Ω_3))) | ψ0(ψ3(0) + ψ2(ψ3(0) + ψ2(ψ3(0)))) | (0,0)(1,1)(2,2)(3,3)(3,2)(4,3) | (1,2,4,7,11,10,14)
#{&_3 + &_3}# => ψ0(Ω_3·2) | ψ0(ψ3(0) + ψ3(0)) | (0,0)(1,1)(2,2)(3,3)(3,3) | (1,2,4,7,11,11)
#{&_3*#}# => ψ0(Ω_3·ω) | ψ0(ψ3(ψ0(0))) | (0,0)(1,1)(2,2)(3,3)(4,0) | (1,2,4,7,11,12)
#{&_3*&}# => ψ0(Ω_3·Ω) | ψ0(ψ3(ψ1(0))) | (0,0)(1,1)(2,2)(3,3)(4,1) | (1,2,4,7,11,13)
#{&_3*&_2}# => ψ0(Ω_3·Ω_2) | ψ0(ψ3(ψ2(0))) | (0,0)(1,1)(2,2)(3,3)(4,2) | (1,2,4,7,11,14)
#{&_3*&_3}# => ψ0(Ω_3^2) | ψ0(ψ3(ψ3(0))) | (0,0)(1,1)(2,2)(3,3)(4,3) | (1,2,4,7,11,15)
#{&_3^&_3}# => ψ0(Ω_3^Ω_3) | ψ0(ψ3(ψ3(ψ3(0)))) | (0,0)(1,1)(2,2)(3,3)(4,3)(5,3) | (1,2,4,7,11,15,19)
#{&_4}# => ψ0(Ω_4) | ψ0(ψ4(0)) | (0,0)(1,1)(2,2)(3,3)(4,4) | (1,2,4,7,11,16)
#{&_4*&_4}# => ψ0(Ω_4^2) | ψ0(ψ4(ψ4(0))) | (0,0)(1,1)(2,2)(3,3)(4,4)(5,4) | (1,2,4,7,11,16,21)
#{&_5}# => ψ0(Ω_5) | ψ0(ψ5(0)) | (0,0)(1,1)(2,2)(3,3)(4,4)(5,5) | (1,2,4,7,11,16,22)
#{&_6}# => ψ0(Ω_6) | ψ0(ψ6(0)) | (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6) | (1,2,4,7,11,16,22,29)
#{&_7}# => ψ0(Ω_7) | ψ0(ψ7(0)) | (0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,7) | (1,2,4,7,11,16,22,29,37)
And finally, the limit is #{&_#}#, which is the Buchholz's ordinal (ψ0(Ω_ω) or ψ0(ψ{ω}(0))) It is also equal to (0,0,0)(1,1,1) in Bashicu matrix system and (1,2,4,8) in Y sequence.
Next sneak peek - #6