pDAN vs. Taranovsky’s ordinal notation

Update: under current definition of pDAN, recursion level platforms do not exist.

Here’re the comparisons between pDAN and Taranovsky’s ordinal notation. Due to its high complexity, a separate page is needed. The fundamental sequences of Taranovsky’s notation can be easily defined. Let L(α) be the amount of C’s in standard representation of α, then ${\alpha[n]=\max\{\beta|\beta<\alpha\land L(\beta)\le L(\alpha)+n\}}$ .

Just warm up

{1,,1{1,,1,,2}2} has recursion level C(C(Ω₂2,0),0)
{1,,2{1,,1,,2}2} has recursion level C(C(Ω₂,C(Ω₂2,0)),0)
{1,,3{1,,1,,2}2} has recursion level C(C(Ω₂,C(Ω₂,C(Ω₂2,0))),0)
{1,,1,2{1,,1,,2}2} has recursion level C(C(Ω₂+1,C(Ω₂2,0)),0)
{1,,1{1,,2}2{1,,1,,2}2} has recursion level C(C(Ω₂+Ω₁,C(Ω₂2,0)),0)
{1,,1{1,,3}2{1,,1,,2}2} has recursion level C(C(Ω₂+C(Ω₂,Ω₁),C(Ω₂2,0)),0)
{1,,1{1,,4}2{1,,1,,2}2} has recursion level C(C(Ω₂+C(Ω₂,C(Ω₂,Ω₁)),C(Ω₂2,0)),0)
{1,,1{1,,1,2}2{1,,1,,2}2} has recursion level C(C(Ω₂+C(Ω₂+1,0),C(Ω₂2,0)),0)
{1,,1{1,,1,3}2{1,,1,,2}2} has recursion level C(C(Ω₂+C(Ω₂+2,0),C(Ω₂2,0)),0)
{1,,1{1,,1{1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂+C(Ω₂+Ω₁,0),C(Ω₂2,0)),0)
{1,,1{1,,1{1,,1,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂+C(Ω₂+C(Ω₂+1,0),0),C(Ω₂2,0)),0)
{1,,1{1,,1{1,,1{1,,1,2}2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂+C(Ω₂+C(Ω₂+C(Ω₂+1,0),0),0),C(Ω₂2,0)),0)
{1,,1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2,C(Ω₂2,0)),0) (note that C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),0) is not a valid term)
{1,,2{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂,C(Ω₂2,C(Ω₂2,0))),0)
{1,,1{1,,1{1,,1,,2}2}3{1,,1,,2}2} has recursion level C(C(Ω₂2,C(Ω₂2,C(Ω₂2,0))),0)
{1,,1{1,,1{1,,1,,2}2}4{1,,1,,2}2} has recursion level C(C(Ω₂2,C(Ω₂2,C(Ω₂2,C(Ω₂2,0)))),0)
{1,,1{1,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω₂2+1,0),0) (note that if α is standard, and it doesn’t have subterms over Ω₂2, then C(C(Ω₂2+α,0),0) is valid)
{1,,1{1,,1{1,,1,,2}2}1{1,,2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+Ω₁,0),0)
{1,,1{1,,1{1,,1,,2}2}1{1,,1,2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+1,0),0),0)
{1,,1{1,,1{1,,1,,2}2}1{1,,1{1,,1,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂+1,0),0),0),0)
{1,,1{1,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),0),0),0)

Up to {1,,1{1,,1,,2}3}

Now let separator ■ = {1,,1,,2} and let ordinal a = C(Ω₂+C(Ω₂2,0),0).

{1,,1{1,,1■2}2{1,,1■2}2■2} has recursion level C(C(Ω₂2,C(Ω₂2+a,0)),0)
{1,,1{1,,1■2}1{1,,1■2}3■2} has recursion level C(C(Ω₂2+a,C(Ω₂2+a,0)),0)
{1,,1{1,,1■2}1{1,,1■2}1,2■2} has recursion level C(C(Ω₂2+a+1,0),0)
{1,,1{1,,1■2}1{1,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω₂2+a2,0),0)
{1,,1{2,,1■2}2■2} has recursion level C(C(Ω₂2+ω^(a+1),0),0)
{1,,1{1{1,,1■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+ω^(a2),0),0)
{1,,1{1{1,,1■2}1{1,,1■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+ω^ω^(a2),0),0)
{1,,1{1{1{1,,1■2}2,,1■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+ω^ω^ω^(a2),0),0)
{1,,1{1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+ε_a+1,0),0) = C(C(Ω₂2+C(C(Ω₂,a),a),0),0)
{1,,1{1{1,,2■2}3,,1■2}2■2} has recursion level C(C(Ω₂2+ε_a+2,0),0) = C(C(Ω₂2+C(C(Ω₂,a),C(C(Ω₂,a),a)),0),0)
{1,,1{1{1,,2■2}1,2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂,a)+1,a),0),0)
{1,,1{1{1,,2■2}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂,a)2,a),0),0)
{1,,1{1{2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(ω^(C(Ω₂,a)+1),a),0),0)
{1,,1{1{1{1,,2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(ω^(C(Ω₂,a)2),a),0),0)
{1,,1{1{1,,3■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂,a)),a),0),0)
{1,,1{1{1,,1,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+1,a),a),0),0)
{1,,1{1{1,,1{1,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+a,a),a),0),0)
{1,,1{1{1,,1{1{1,,1{1,,1■2}2■2}2,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(C(Ω₂+a,a),a),a),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),a),0),0)

So {1,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),a),0),0). (The recursion levels of {1,,1{1,,2■2}2■2}, {1{1,,1{1,,2■2}2■2}2,,1■2}, {1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2}, {1{1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2}2,,1■2}, etc. are equal. Separator {1,,1 A 2■2} (where A is any separator) reduces to an expression containing {1,,2■2}, then the {1,,2■2} acquires a layer of { ____ ,,1■2}.)

{1,,1{1,,1■2}2{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2,C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),a),0)),0)
{1,,1{1,,1■2}1{1,,1■2}2{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+a,C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),a),0)),0)
{1,,1{1{1,,2■2}2,,1■2}2{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂,a),a),C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),a),0)),0)
{1,,1{1{1,,1{1,,1■2}2■2}2,,1■2}2{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+a,a),a),C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),a),0)),0)
{1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}3■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),a),C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),a),0)),0)
{1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}1,2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),a)+1,0),0)
{1,,1{2{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+ω^(C(C(Ω₂+C(Ω₂,a),a),a)+1),0),0)
{1,,1{1{1,,2■2}2{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂,a),C(C(Ω₂+C(Ω₂,a),a),a)),0),0)
{1,,1{1{1,,1,2■2}2{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+1,a),C(C(Ω₂+C(Ω₂,a),a),a)),0),0)
{1,,1{1{1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2}2{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(C(Ω₂+C(Ω₂,a),a),a),a),C(C(Ω₂+C(Ω₂,a),a),a)),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}3,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),C(C(Ω₂+C(Ω₂,a),a),a)),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1,2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+1,a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1,,1■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+a,a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1{1,,1{1,,2■2}2■2}2,,1■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(Ω₂,a),a),a),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,a),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂,C(Ω₂,a)),C(Ω₂,a)),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂,C(Ω₂,a)),C(Ω₂,a)),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1{1,,4■2}2,,3■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂,C(Ω₂,C(Ω₂,a))),C(Ω₂,a)),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1,,4■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂,C(Ω₂,C(Ω₂,a))),C(Ω₂,a)),a),0),0) (when solving this term, by the adding rule, the {1,,4■2} will reduce to {1{1,,4■2}2,,3■2}, then {1{1{1,,4■2}2,,3■2}2,,2■2})
{1,,1{1{1,,1{1,,2■2}2■2}1{1,,1,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+1,a),C(Ω₂,a)),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+a,a),C(Ω₂,a)),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1{1,,1{1,,1■2}2■2}2,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(C(Ω₂+a,a),a),a),C(Ω₂,a)),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(C(Ω₂+C(Ω₂,a),a),a),a),C(Ω₂,a)),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a)),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}1,2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a))+1,a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a))2,a),0),0)
{1,,1{1{2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+ω^(C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a))+1),a),0),0)
{1,,1{1{1{1,,1■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+ω^(C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a))+a),a),0),0)
{1,,1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+ω^(C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a))+C(Ω₂,a)),a),0),0)

So {1{1,,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+ω^(C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a))+C(Ω₂,a)),a),0),0). (the recursion levels of {1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,1■2}, {1,,1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} and {1{1,,1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2}2,,1■2} are equal)

{1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),a),0),0)
{1,,1{1{1,,2■2}2{1{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂,a),C(C(Ω₂+C(Ω₂,a),a),a)),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}3,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),C(C(Ω₂+C(Ω₂,a),a),a)),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1,2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+1,a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,a),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1{1{1,,1{1,,1■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+a,a),C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a)),a),0),0)
{1,,1{1{2{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+ω^(C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a))+1),a),0),0)
{1,,1{1{1{1,,3■2}2{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂,C(Ω₂,a)),C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a))),a),0),0)
{1,,1{1{1{1,,1{1,,1■2}2■2}2{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+a,a),C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a))),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}3,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(Ω₂,a),a),C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a))),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1,2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(Ω₂,a),a)+1,C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,a),C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}2{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),a)),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1{1{1,,1{1,,1■2}2■2}2,,2■2}2 {1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+a,a),C(Ω₂,a)),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),a)),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1{1{1,,1{1 A 2,,1■2}2■2}2,,2■2}2 A 2,,1■2}2■2} (where A = {1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}) has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),a),a),C(Ω₂,a)),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),a)),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}2,,2■2}2 A 2,,1■2}2■2} (where A = {1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}) has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a)),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),a)),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1,2,,2■2}2{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(Ω₂,a),a)+1,C(Ω₂,a)),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),a)),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,2■2}2,,2■2}2{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,a),C(Ω₂,a)),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),a)),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}3,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),a)),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1,2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+1,a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(Ω₂,a),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1{1,,1{1,,1■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂+a,a),C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}1,2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂+C(Ω₂,a),a)+1,C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}1{1,,2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,a),C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1,2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),C(Ω₂,a))+1,a),0),0)
{1,,1{1{2{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+ω^(C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),C(Ω₂,a))+1),a),0),0)
{1,,1{1{1{1,,3■2}2{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂,C(Ω₂,a)),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),C(Ω₂,a))),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}2{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂+C(Ω₂,a),a),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),C(Ω₂,a))),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,2■2}2{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,a),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),C(Ω₂,a))),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}3,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a)),C(Ω₂,a))),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}1,2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+1,C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}1{1{1,,1{1,,2■2}2■2}1{1,,3■2}1,2,,2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))+1,C(Ω₂,a)),C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,a))2,a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{2,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+ω^(C(Ω₂,C(Ω₂,a))+1),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,4■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂,C(Ω₂,C(Ω₂,a))),C(Ω₂,C(Ω₂,a))),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,1,2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+1,a),C(Ω₂,C(Ω₂,a))),a),0),0)
{1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,C(Ω₂,a))),a),0),0)
{1,,1{1{1{2,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+ω^(C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,C(Ω₂,a)))+1),a),0),0)
{1,,1{1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+ω^(C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,C(Ω₂,a)))+C(Ω₂,a)),a),0),0)
{1,,1{1{1{1{1,,3■2}2,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+ω^(C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,C(Ω₂,a)))+C(Ω₂,C(Ω₂,a))),a),0),0)

So {1{1,,3■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+ω^(C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,C(Ω₂,a)))+C(Ω₂,C(Ω₂,a))),a),0),0). And further, the recursion level of {1{1,,k■2}2,,1{1,,2■2}2■2} is between C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,…C(Ω₂,a))),a),0),0) (with k-1 Ω₂′s in the blue part) and C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂,…C(Ω₂,a))),a),0),0) (with k Ω₂′s in the blue part), so {1{1,,1,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a),a),0),0).

{1,,1{1{1{1,,1,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}2{1{1,,1,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a),a)),0),0)
{1,,1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2{1{1,,1,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+ω^(C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a))+C(Ω₂,a)),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a),a)),0),0)
{1,,1{1{1{1,,3■2}2,,1{1,,2■2}2■2}2{1{1,,1,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+ω^(C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,C(Ω₂,a)))+C(Ω₂,C(Ω₂,a))),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a),a)),0),0)
{1,,1{1{1{1,,1,2■2}2,,1{1,,2■2}2■2}3,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a),C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a),a)),0),0)
{1,,1{1{1{1,,1,2■2}2,,1{1,,2■2}2■2}1,2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a)+1,a),0),0)
{1{1,,2■2}2{1,,1,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a)+ω^(C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a),C(Ω₂,a))+C(Ω₂,a)),a),0),0)
{1{1,,3■2}2{1,,1,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a)+ω^(C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a),C(Ω₂,C(Ω₂,a)))+C(Ω₂,C(Ω₂,a))),a),0),0)
{1{1,,1,2■2}3,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a)2,a),0),0)
{1{1,,2,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂,C(Ω₂+1,a)),a),0),0)
{1{1,,1,3■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,C(Ω₂+1,a)),a),0),0)
{1{1,,1,1,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+2,a),a),0),0)
{1{1,,1{1,,1■2}2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+a,a),a),0),0)
{1{1,,1{1{1{1,,1,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+C(C(Ω₂+C(Ω₂,a),a)+C(Ω₂+1,a),a),a),a),0),0)
{1{1,,1{1,,2■2}2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)2,a),0),0)
{1{1,,1,2■2}2{1,,1{1,,2■2}2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)2+C(Ω₂+1,a),a),0),0)
{1{1,,1{1,,2■2}2■2}3,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a)3,a),0),0)
{1{1,,1{1,,2■2}2■2}1,2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(ω^(C(Ω₂+C(Ω₂,a),a)+1),a),0),0)
{1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(ω^(C(Ω₂+C(Ω₂,a),a)2),a),0),0)
{1{2,,1{1,,2■2}2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(ω^ω^(C(Ω₂+C(Ω₂,a),a)+1),a),0),0)
{1,,2{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂+C(Ω₂,a),a)),a),0),0)
{1,,1{1,,1■2}2{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+a,C(Ω₂+C(Ω₂,a),a)),a),0),0)
{1,,1{1,,2■2}3■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),C(Ω₂+C(Ω₂,a),a)),a),0),0)
{1,,1{1,,2■2}1,2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a)+1,a),a),0),0)
{1,,1{1,,2■2}1{1,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a)2,a),a),0),0)
{1,,1{2,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+ω^(C(Ω₂,a)+1),a),a),0),0)
{1,,1{1{1,,3■2}2,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(C(Ω₂,C(Ω₂,a)),C(Ω₂,a)),a),a),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}2,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a)),a),a),0),0)
{1,,1{1{1,,1{1{1,,1{1,,2■2}2■2}2,,2■2}2■2}2,,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(C(Ω₂+C(C(Ω₂+C(Ω₂,a),a),C(Ω₂,a)),a),C(Ω₂,a)),a),a),0),0)
{1,,1{1,,3■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,C(Ω₂,a)),a),a),0),0)
{1,,1{1,,4■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,C(Ω₂,C(Ω₂,a))),a),a),0),0)
{1,,1{1,,1,2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+1,a),a),a),0),0)
{1,,1{1,,1{1,,1■2}2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+a,a),a),a),0),0)
{1,,1{1,,1{1,,2■2}2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+C(Ω₂,a),a),a),a),0),0)
{1,,1{1,,1{1,,1{1,,1■2}2■2}2■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+C(Ω₂+a,a),a),a),a),0),0)
{1,,1■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),a),0),0)

So {1,,1{1,,1,,2}3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),0)),0),0). Here’re some approximations, to make the comparisons above more simple:

{1,,1■2} (where separator ■ = {1,,1,,2}) approximately corresponds to a = C(Ω₂+C(Ω₂2,0),0)
{1{1,,2■2}2,,1■2} approximately corresponds to C(C(Ω₂,a),a)
{1,,2■2} approximately corresponds to C(Ω₂,a)
{1,,3■2} approximately corresponds to C(Ω₂,C(Ω₂,a))
{1,,1,2■2} approximately corresponds to C(Ω₂+1,a)
{1,,1{1,,1■2}2■2} approximately corresponds to C(Ω₂+a,a)
{1{1,,1{1,,1■2}2■2}2,,1■2} approximately corresponds to C(C(Ω₂+a,a),a)
{1,,1{1,,2■2}2■2} approximately corresponds to C(Ω₂+C(Ω₂,a),a)
{1{1,,1{1,,2■2}2■2}2,,1■2} approximately corresponds to C(C(Ω₂+C(Ω₂,a),a),a)
{1{1,,1{1,,2■2}2■2}3,,1■2} approximately corresponds to C(C(Ω₂+C(Ω₂,a),a),C(C(Ω₂+C(Ω₂,a),a),a))
{1{1,,1{1,,2■2}2■2}1,2,,1■2} approximately corresponds to C(C(Ω₂+C(Ω₂,a),a)+1,a)
{1,,2{1,,2■2}2■2} approximately corresponds to C(Ω₂,C(Ω₂+C(Ω₂,a),a))
{1,,1{1,,2■2}3■2} approximately corresponds to C(Ω₂+C(Ω₂,a),C(Ω₂+C(Ω₂,a),a))
{1,,1{1,,2■2}1,2■2} approximately corresponds to C(Ω₂+C(Ω₂,a)+1,a)
{1,,1{1,,2■2}1{1,,2■2}2■2} approximately corresponds to C(Ω₂+C(Ω₂,a)2,a)
{1,,1{1,,3■2}2■2} approximately corresponds to C(Ω₂+C(Ω₂,C(Ω₂,a)),a)
{1,,1{1,,1,2■2}2■2} approximately corresponds to C(Ω₂+C(Ω₂+1,a),a)
{1,,1{1,,1{1,,1■2}2■2}2■2} approximately corresponds to C(Ω₂+C(Ω₂+a,a),a)

Up to {1,,2,,2}

Still, let separator ■ = {1,,1,,2} and let ordinal a = C(Ω₂+C(Ω₂2,0),0).

{1,,1{1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),a),0),0) (the recursion levels of {1,,1■3}, {1,,1{1,,1■3}2■2}, {1{1,,1■3}2,,1■2}, {1{1,,1{1,,1■3}2■2}2,,1■2}, {1,,1{1{1,,1{1,,1■3}2■2}2,,1■2}2■2}, etc. are equal)
{1,,1{1{1,,1■3}2,,1■2}1,2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),a)+1,0),0)
{1,,1{1{1,,1■3}2,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),a)+a,0),0)
{1,,1{1{1,,1■3}2,,1■2}1{1{1,,1,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),a)+C(C(Ω₂+1,a),a),0),0)
{1,,1{1{1,,1■3}2,,1■2}1{1{1,,1{1,,1,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),a)+C(C(Ω₂+C(Ω₂+1,a),a),a),0),0)
{1,,1{1{1,,1■3}2,,1■2}1{1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),a)2,0),0)
{1,,1{2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+ω^(C(C(Ω₂2,0),a)+1),0),0)
{1,,1{1{1,,2■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂,a),C(C(Ω₂2,0),a)),0),0)
{1,,1{1{1,,3■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂,a)),C(C(Ω₂2,0),a)),0),0)
{1,,1{1{1,,1{1,,1■2}2■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+a,a),C(C(Ω₂2,0),a)),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),a),C(C(Ω₂2,0),a)),0),0)
{1,,1{1{1,,1{1,,1{1,,1■2}2■2}2■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+a,a),a),C(C(Ω₂2,0),a)),0),0)
{1,,1{1{1,,1■3}3,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(C(Ω₂2,0),a)),0),0)
{1,,1{1{1,,1■3}1,2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+1,a),0),0)
{1,,1{1{1,,1■3}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂,a),a),0),0)
{1,,1{1{1,,1■3}1{1,,1{1,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(C(Ω₂+a,a),C(Ω₂,a)),a),0),0)
{1,,1{1{1,,1■3}1{1,,1{1,,1{1,,1■2}2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(C(Ω₂+C(Ω₂+a,a),a),C(Ω₂,a)),a),0),0)
{1,,1{1{1,,1■3}1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(C(Ω₂2,0),C(Ω₂,a)),a),0),0)
{1,,1{1{2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+ω^(C(C(Ω₂2,0),C(Ω₂,a))+1),a),0),0)
{1,,1{1{1{1,,2■2}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+ω^(C(C(Ω₂2,0),C(Ω₂,a))+C(Ω₂,a)),a),0),0)

{1,,1{1{1{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),a),0),0)
{1,,1{1{1{1,,1■3}2,,2■2}1,2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+1,a),0),0)
{1,,1{1{1{1,,1■3}2,,2■2}1{1{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(C(Ω₂2,0),C(Ω₂,a)),a),0),0)
{1,,1{1{2{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+ω^(C(C(Ω₂2,0),C(Ω₂,a))+1),a),0),0)
{1,,1{1{1{1,,3■2}2{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(C(Ω₂,C(Ω₂,a)),C(C(Ω₂2,0),C(Ω₂,a))),a),0),0)
{1,,1{1{1{1,,1{1,,1■2}2■2}2{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(C(Ω₂+a,a),C(C(Ω₂2,0),C(Ω₂,a))),a),0),0)
{1,,1{1{1{1,,1■3}3,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(C(Ω₂2,0),C(C(Ω₂2,0),C(Ω₂,a))),a),0),0)
{1,,1{1{1{1,,1■3}1,2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(C(Ω₂2,0)+1,C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1■3}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂,C(Ω₂,a)),a),0),0)
{1,,1{1{1{1,,1■3}1{1,,3■2}1,2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂,C(Ω₂,a))+1,a),0),0)
{1,,1{1{1{1,,1■3}1{2,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+ω^(C(Ω₂,C(Ω₂,a))+1),a),0),0)
{1,,1{1{1{1,,1■3}1{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(C(Ω₂2,0),C(Ω₂,C(Ω₂,a))),a),0),0)
{1,,1{1{1{2,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+ω^(C(C(Ω₂2,0),C(Ω₂,C(Ω₂,a)))+1),a),0),0)
{1,,1{1{1{1{1,,3■2}2,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+ω^(C(C(Ω₂2,0),C(Ω₂,C(Ω₂,a)))+C(Ω₂,C(Ω₂,a))),a),0),0)

So {1,,1{1{1{1,,3■2}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+ω^(C(C(Ω₂2,0),C(Ω₂,C(Ω₂,a)))+C(Ω₂,C(Ω₂,a))),a),0),0). And further, the recursion level of {1,,1{1{1{1,,k■2}2,,1■3}2,,1■2}2■2} is between C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂,C(Ω₂,…C(Ω₂,a))),a),0),0) (with k-1 Ω₂′s in the blue part) and C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂,C(Ω₂,…C(Ω₂,a))),a),0),0) (with k Ω₂′s in the blue part), so {1,,1{1{1{1,,1,2■2}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂+1,a),a),0),0).

{1,,1{1{1{1,,1{1,,1■2}2■2}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂+a,a),a),0),0)
{1{1,,1{1,,2■2}2■2}2,,1■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂+C(Ω₂,a),a),a),0),0)
{1{1,,1{1,,1{1,,1■2}2■2}2■2}2,,1■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂+C(Ω₂+a,a),a),a),0),0)
{1{1,,1■3}2,,1■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)2,a),0),0)
{1,,1{1{1{1,,1,2■2}2{1,,1■3}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)2+C(Ω₂+1,a),a),0),0)
{1{1,,1■3}3,,1■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)3,a),0),0)
{1{1,,1■3}1,2,,1■3} has recursion level C(C(Ω₂2+C(ω^(C(Ω₂2,0)+1),a),0),0)
{1{1,,1■3}1{1,,1■3}2,,1■3} has recursion level C(C(Ω₂2+C(ω^(C(Ω₂2,0)2),a),0),0)
{1{2,,1■3}2,,1■3} has recursion level C(C(Ω₂2+C(ω^ω^(C(Ω₂2,0)+1),a),0),0)
{1{1,,2■3}2,,1■3} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂2,0)),a),0),0)
also, {1,,2■3} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂2,0)),a),0),0)
{1,,1{1{1{1,,1,2■2}2{1,,2■3}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂2,0))+C(Ω₂+1,a),a),0),0)
{1{1,,1■3}2{1,,2■3}2,,1■3} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂2,0))+C(Ω₂2,0),a),0),0)
{1{1{1,,2■3}2,,1■3}2{1,,2■3}2,,1■3} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂2,0))+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),a),0),0)
{1{1,,2■3}3,,1■3} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂2,0))2,a),0),0)
{1{1,,2■3}1,2,,1■3} has recursion level C(C(Ω₂2+C(ω^(C(Ω₂,C(Ω₂2,0))+1),a),0),0)
{1{2,,2■3}2,,1■3} has recursion level C(C(Ω₂2+C(ω^ω^(C(Ω₂,C(Ω₂2,0))+1),a),0),0)
{1,,3■3} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂,C(Ω₂2,0))),a),0),0)
{1,,1,2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+1,C(Ω₂2,0)),a),0),0)
{1,,1{1,,1■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+a,C(Ω₂2,0)),a),0),0)
{1,,1{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),C(Ω₂2,0)),a),0),0)
{1,,1{1,,1{1,,1■2}2■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+a,a),C(Ω₂2,0)),a),0),0)
{1,,1{1,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,C(Ω₂2,0)),a),0),0) (here again, C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),a) is not a valid term)
{1,,2{1,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂2,C(Ω₂2,0))),a),0),0)
{1,,1{1,,1■3}3■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,C(Ω₂2,C(Ω₂2,0))),a),0),0)
{1,,1{1,,1■3}1,2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2+1,0),a),0),0)
{1,,1{1,,1■3}1{1,,1■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2+a,0),a),0),0)
{1,,1{1,,1■3}1{1{1,,1{1,,1■2}2■2}2,,1■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2+C(C(Ω₂+a,a),a),0),a),0),0)
{1,,1{1,,1■3}1{1{1,,1■3}2,,1■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2+C(C(Ω₂2,0),a),0),a),0),0)
{1,,1{1,,1■3}1{1{1,,1{1,,1■3}1,2■3}2,,1■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2+C(C(Ω₂2+1,0),a),0),a),0),0)
{1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1,,1■2}2■3}2,,1■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2+C(C(Ω₂2+a,0),a),0),a),0),0)
{1,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a),0),0) (here, it hits the line of “C(C(Ω₂2+α,0),0) is valid”)
also, {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂,a),0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1,2■2} has recursion level C(C(Ω₂2+C(Ω₂,a)+1,0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂,a)+a,0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1{1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2,0),a),0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1{1{1,,1{1,,1■3}1{1,,1■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+a,0),a),0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a),0),a),0),0)
{1,,1{2{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂,a)+ω^(C(C(Ω₂2+C(Ω₂,a),0),a)+1),0),0)
{1,,1{1{1,,1■3}2{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2,0),C(C(Ω₂2+C(Ω₂,a),0),a)),0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,1■2}2■3}2{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+a,0),C(C(Ω₂2+C(Ω₂,a),0),a)),0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}3,,1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a),0),C(C(Ω₂2+C(Ω₂,a),0),a)),0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}1,2,,1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a),0)+1,a),0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a),0)+C(Ω₂,a),a),0),0)
{1,,1{1{1{1,,1,2■2}2,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a),0)+C(Ω₂+1,a),a),0),0)
{1{1,,1■3}2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a),0)+C(Ω₂2,0),a),0),0)
{1{1,,1{1,,1■3}1{1,,1■2}2■3}2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a),0)+C(Ω₂2+a,0),a),0),0)
{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a),0)2,a),0),0)
{1{1,,1{1,,1■3}1{1,,2■2}2■3}1,2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(ω^(C(Ω₂2+C(Ω₂,a),0)+1),a),0),0)
{1{2,,1{1,,1■3}1{1,,2■2}2■3}2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(ω^ω^(C(Ω₂2+C(Ω₂,a),0)+1),a),0),0)
{1,,2{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂,C(Ω₂2+C(Ω₂,a),0)),a),0),0)
{1,,1{1,,1■2}2{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂+a,C(Ω₂2+C(Ω₂,a),0)),a),0),0)
{1,,1{1,,1{1,,1■2}2■2}2{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂+C(Ω₂+a,a),C(Ω₂2+C(Ω₂,a),0)),a),0),0)
{1,,1{1,,1■3}2{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2,C(Ω₂2+C(Ω₂,a),0)),a),0),0)
{1,,1{1,,1■3}1{1,,1■2}2{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+a,C(Ω₂2+C(Ω₂,a),0)),a),0),0)
{1,,1{1,,1■3}1{1,,2■2}3■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a),C(Ω₂2+C(Ω₂,a),0)),a),0),0)
{1,,1{1,,1■3}1{1,,2■2}1,2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a)+1,0),a),0),0)
{1,,1{1,,1■3}1{1,,2■2}1{1,,1■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a)+a,0),a),0),0)
{1,,1{1,,1■3}1{1,,2■2}1{1{1,,1{1,,1■3}1{1,,2■2}1{1,,1■2}2■3}2,,1■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a)+C(C(Ω₂2+C(Ω₂,a)+a,0),a),0),a),0),0)
{1,,1{1,,1■3}1{1,,2■2}1{1,,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,a)2,0),0)
{1,,1{1,,1■3}1{2,,2■2}2■3} has recursion level C(C(Ω₂2+ω^(C(Ω₂,a)+1),0),0)
{1,,1{1,,1■3}1{1{1,,3■2}2,,2■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂,a)),C(Ω₂,a)),0),0)
{1,,1{1,,1■3}1{1{1,,1,2■2}2,,2■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+1,a),C(Ω₂,a)),0),0)
{1,,1{1,,1■3}1{1{1,,1{1,,1,2■2}2■2}2,,2■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+1,a),a),C(Ω₂,a)),0),0)
{1,,1{1,,1■3}1{1{1,,1■3}2,,2■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂,a)),0),0)
{1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1,,1■2}2■3}2,,2■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2+a,0),C(Ω₂,a)),0),0)
{1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,2■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2+C(Ω₂,a),0),C(Ω₂,a)),0),0)
{1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,2■2}2■3}2,,2■2}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2+C(C(Ω₂2+C(Ω₂,a),0),C(Ω₂,a)),0),C(Ω₂,a)),0),0)
{1,,1{1,,1■3}1{1,,3■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,C(Ω₂,a)),0),0)
{1,,1{1,,1■3}1{1,,4■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂,C(Ω₂,C(Ω₂,a))),0),0)
{1,,1{1,,1■3}1{1,,1,2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂+1,a),0),0)
{1,,1{1,,1■3}1{1,,1{1,,1■2}2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂+a,a),0),0)
{1,,1{1,,1■3}1{1,,1{1,,1{1,,1■2}2■2}2■2}2■3} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂+a,a),a),0),0)
{1,,1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),0)

It seems that {1,,1■3} approximately corresponds to C(Ω₂2,0), but it’s not so in following comparisons. Now let ordinal a = C(Ω₂+C(Ω₂2,0),0) and b = C(Ω₂+C(Ω₂2,0),a) = C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),0)).

{1,,1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω₂2+b,0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}1,2■2} has recursion level C(C(Ω₂2+b+1,0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω₂2+b+a,0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}1{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),a),0),0)
{1,,1{2{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+b+ω^(C(C(Ω₂2+b,0),a)+1),0),0)
{1,,1{1{1,,1{1,,1■3}1{1,,1■3}2■3}1,2,,1■2}2■2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+1,a),0),0)
{1{1,,1■3}2,,1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+C(Ω₂2,0),a),0),0)
{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)2,a),0),0)
{1,,2{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω₂2+b+C(C(Ω₂,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1■3}2{1,,1■3}2■3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1■3}1{1,,1■3}1,2■3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+1,0),a),0),0)
{1,,1{1,,1■3}1{1,,1■3}1{1,,1■2}2■3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+a,0),a),0),0)
{1,,1{1,,1■3}1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω₂2+b+C(Ω₂,a),0),0)
{1,,1{1,,1■3}1{1,,1■3}1{1,,1{1,,2■2}2■2}2■3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(Ω₂,a),a),0),0)
{1,,1{1,,1■3}1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω₂2+b2,0),0)
{1,,1{2,,1■3}2■3} has recursion level C(C(Ω₂2+ω^(b+1),0),0)
{1,,1{1{1,,1■2}2,,1■3}2■3} has recursion level C(C(Ω₂2+ω^(b+a),0),0)
{1,,1{1{1,,2■2}2,,1■3}2■3} has recursion level C(C(Ω₂2+ω^(b+C(Ω₂,a)),0),0)
{1,,1{1{1,,1{1,,2■2}2■2}2,,1■3}2■3} has recursion level C(C(Ω₂2+ω^(b+C(Ω₂+C(Ω₂,a),a)),0),0)
{1,,1{1{1,,1■3}2,,1■3}2■3} has recursion level C(C(Ω₂2+ω^(b2),0),0)
{1,,1{1{1,,2■3}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂,b),b),0),0)
{1,,1{1{1,,1,2■3}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+1,b),b),0),0)
{1,,1{1{1,,1{1,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+b,b),b),0),0)
{1,,1{1{1,,1{2,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+ω^(b+1),b),b),0),0)
{1,,1{1{1,,1{1{1,,1{1,,1■3}2■3}2,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+C(C(Ω₂+b,b),b),b),b),0),0)
{1,,1{1,,2■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,b),b),b),0),0)
{1,,1{1,,3■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,C(Ω₂,b)),b),b),0),0)
{1,,1{1,,1,2■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+1,b),b),b),0),0)
{1,,1{1,,1{1,,1■3}2■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+b,b),b),b),0),0)
{1,,1■4} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b),0),0)
{1,,1{1{1,,1■4}2,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+b,0),0)
{1,,1{1{1,,1■4}2,,1■3}1{1{1,,2■3}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂,b),b),0),0)
{1,,1{1{1,,1■4}2,,1■3}1{1{1,,1{1,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂+b,b),b),0),0)
{1,,1{1{1,,1■4}2,,1■3}1{1{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)2,0),0)
{1,,1{2{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω₂2+ω^(C(C(Ω₂2,0),b)+1),0),0)
{1,,1{1{1,,2■3}2{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂,b),C(C(Ω₂2,0),b)),0),0)
{1,,1{1{1,,1{1,,1■3}2■3}2{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂+b,b),C(C(Ω₂2,0),b)),0),0)
{1,,1{1{1,,1■4}3,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(C(Ω₂2,0),b)),0),0)
{1,,1{1{1,,1■4}1,2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+1,b),0),0)
{1,,1{1{1,,1■4}1{1,,2■3}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂,b),b),0),0)
{1,,1{1{1,,1■4}1{1,,1{1,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(C(Ω₂+b,b),C(Ω₂,b)),b),0),0)
{1,,1{1{1,,1■4}1{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(C(Ω₂2,0),C(Ω₂,b)),b),0),0)
{1,,1{1{2,,1■4}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+ω^(C(C(Ω₂2,0),C(Ω₂,b))+1),b),0),0)
{1,,1{1{1{1,,2■3}2,,1■4}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+ω^(C(C(Ω₂2,0),C(Ω₂,b))+C(Ω₂,b)),b),0),0)
{1,,1{1{1{1,,3■3}2,,1■4}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+ω^(C(C(Ω₂2,0),C(Ω₂,C(Ω₂,b)))+C(Ω₂,C(Ω₂,b))),b),0),0)
{1,,1{1{1{1,,1,2■3}2,,1■4}2,,1■3}2■3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂+1,b),b),0),0)
{1{1,,1■4}2,,1■4} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)2,b),0),0)
{1{1,,1■4}1,2,,1■4} has recursion level C(C(Ω₂2+C(ω^(C(Ω₂2,0)+1),b),0),0)
{1{1,,1■4}1{1,,1■4}2,,1■4} has recursion level C(C(Ω₂2+C(ω^(C(Ω₂2,0)2),b),0),0)
{1{2,,1■4}2,,1■4} has recursion level C(C(Ω₂2+C(ω^ω^(C(Ω₂2,0)+1),b),0),0)
{1,,2■4} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂2,0)),b),0),0)
{1,,1,2■4} has recursion level C(C(Ω₂2+C(C(Ω₂+1,C(Ω₂2,0)),b),0),0)
{1,,1{1,,1■2}2■4} has recursion level C(C(Ω₂2+C(C(Ω₂+a,C(Ω₂2,0)),b),0),0)
{1,,1{1,,2■2}2■4} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,a),C(Ω₂2,0)),b),0),0)
{1,,1{1,,1■3}2■4} has recursion level C(C(Ω₂2+C(C(Ω₂+b,C(Ω₂2,0)),b),0),0)
{1,,1{1,,2■3}2■4} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,b),C(Ω₂2,0)),b),0),0)
{1,,1{1,,1■4}2■4} has recursion level C(C(Ω₂2+C(C(Ω₂2,C(Ω₂2,0)),b),0),0)
{1,,1{1,,1■4}1,2■4} has recursion level C(C(Ω₂2+C(C(Ω₂2+1,0),b),0),0)
{1,,1{1,,1■4}1{1,,1■3}2■4} has recursion level C(C(Ω₂2+C(C(Ω₂2+b,0),b),0),0)
{1,,1{1,,1■4}1{1,,2■3}2■4} has recursion level C(C(Ω₂2+C(Ω₂,b),0),0)
{1,,1{1,,1■4}1{1,,3■3}2■4} has recursion level C(C(Ω₂2+C(Ω₂,C(Ω₂,b)),0),0)
{1,,1{1,,1■4}1{1,,1,2■3}2■4} has recursion level C(C(Ω₂2+C(Ω₂+1,b),0),0)
{1,,1{1,,1■4}1{1,,1{1,,1■3}2■3}2■4} has recursion level C(C(Ω₂2+C(Ω₂+b,b),0),0)
{1,,1{1,,1■4}1{1,,1{1,,1{1,,1■3}2■3}2■3}2■4} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂+b,b),b),0),0)
{1,,1{1,,1■4}1{1,,1■4}2■4} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),b),0),0)
{1,,1{1{1,,2■4}2,,1■4}2■4} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂+C(Ω₂2,0),b)),C(Ω₂+C(Ω₂2,0),b)),0),0)
{1,,1■5} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),b)),0),0)
{1,,1{1,,1■5}1{1,,2■4}2■5} has recursion level C(C(Ω₂2+C(Ω₂,C(Ω₂+C(Ω₂2,0),b)),0),0)
{1,,1{1,,1■5}1{1,,1■5}2■5} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),b)),0),0)
{1,,1■6} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),b))),0),0)

Generally, the recursion level of {1,,1■k} is between C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),…C(Ω₂+C(Ω₂2,0),0))),0),0) (with k-2 Ω₂+C(Ω₂2,0)′s in the blue part) and C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),…C(Ω₂+C(Ω₂2,0),0))),0),0) (with k-1 Ω₂+C(Ω₂2,0)′s in the blue part), so {1,,1■1,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,0),0),0). Now let ordinal a = C(Ω₂+C(Ω₂2,0),0) and b = C(Ω₂+C(Ω₂2,0)+1,0).

{1,,1{1{1,,1■1,2}2,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω₂2+b+a,0),0)
{1,,1{1{1,,1■1,2}2,,1■2}1{1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a),0),0)
{1,,1{1{1,,1■1,2}2,,1■2}1{1{1,,1■1,2}2,,1■2}2■2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),a),0),0)
{1,,1{1{1,,1■1,2}1,2,,1■2}2■2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+1,a),0),0)
{1{1,,1■1,2}2,,1■1,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)2,a),0),0)
{1{1{1,,1■1,2}2,,1■1,2}2,,1■1,2} has recursion level C(C(Ω₂2+b+C(ω^ω^(C(Ω₂2+b,0)2),a),0),0)
{1,,2■1,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1■2}2■1,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂+a,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1■3}2■1,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1■1,2}2■1,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1■1,2}1,2■1,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+1,0),a),0),0)
{1,,1{1,,1■1,2}1{1,,1■2}2■1,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+a,0),a),0),0)
{1,,1{1,,1■1,2}1{1,,2■2}2■1,2} has recursion level C(C(Ω₂2+b+C(Ω₂,a),0),0)
{1,,1{1,,1■1,2}1{1,,1■3}2■1,2} has recursion level C(C(Ω₂2+b+C(Ω₂+(Ω₂2,0),a),0),0)
{1,,1{1,,1■1,2}1{1,,1■1,2}2■1,2} has recursion level C(C(Ω₂2+b2,0),0)
{1,,1{1{1,,2■1,2}2,,1■1,2}2■1,2} has recursion level C(C(Ω₂2+C(C(Ω₂,b),b),0),0)
{1,,1{1{1,,1{1,,1■1,2}2■1,2}2,,1■1,2}2■1,2} has recursion level C(C(Ω₂2+C(C(Ω₂+b,b),b),0),0)
{1,,1{1,,2■1,2}2■1,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,b),b),b),0),0)
{1,,1{1,,1{1,,2■1,2}2■1,2}2■1,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+C(Ω₂,b),b),b),b),0),0)
{1,,1■2,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b),0),0)
{1{1,,1■2,2}2,,1■2,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)2,b),0),0)
{1,,2■2,2} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂2,0)),b),0),0)
{1,,1{1,,1■2,2}2■2,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,C(Ω₂2,0)),b),0),0)
{1,,1{1,,1■2,2}1{1,,1■1,2}2■2,2} has recursion level C(C(Ω₂2+C(C(Ω₂2+b,0),b),0),0)
{1,,1{1,,1■2,2}1{1,,2■1,2}2■2,2} has recursion level C(C(Ω₂2+C(Ω₂,b),0),0)
{1,,1■3,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),b)),0),0)
{1,,1■1,3} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,b),0),0)
{1,,1■1,1,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+2,0),0),0)
{1,,1■1{1,,1■2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+a,0),0),0)
{1,,1■1{1,,1■1,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+b,0),0),0)
{1,,1■1■2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)2,0),0),0)
{1,,1■1,2■2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,C(Ω₂+C(Ω₂2,0)2,0)),0),0)
{1,,1■1{1,,1■1■2}2■2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+C(Ω₂+C(Ω₂2,0)2,0),C(Ω₂+C(Ω₂2,0)2,0)),0),0)
{1,,1■1■3} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)2,C(Ω₂+C(Ω₂2,0)2,0)),0),0)
{1,,1■1■1,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)2+1,0),0),0)
{1,,1■1■1■2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)3,0),0),0)
{1,,1{2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(C(Ω₂2,0)+1),0),0),0)
{1,,1{1{1,,1{2,,1,,2}2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(C(Ω₂2,0)+C(Ω₂+ω^(C(Ω₂2,0)+1),0)),0),0),0)
{1,,1{1■2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(C(Ω₂2,0)2),0),0),0)
{1,,1{1■1,2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^ω^(C(Ω₂2,0)+1),0),0),0)
{1,,1{1{2,,1,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^ω^ω^(C(Ω₂2,0)+1),0),0),0)
{1,,1{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)

So {1,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0). Here’re some approximations, to make the comparisons above more simple:

{1,,1■2} (where separator ■ = {1,,1,,2}) approximately corresponds to C(Ω₂+C(Ω₂2,0),0)
{1,,2■2} approximately corresponds to C(Ω₂,C(Ω₂+C(Ω₂2,0),0))
{1,,1■3} approximately corresponds to C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),0))
{1,,2■3} approximately corresponds to C(Ω₂,C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),0)))
{1,,1■4} approximately corresponds to C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),0)))
{1,,1■1,2} approximately corresponds to C(Ω₂+C(Ω₂2,0)+1,0)
{1,,1■2,2} approximately corresponds to C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0)+1,0))
{1,,1■1,1,2} approximately corresponds to C(Ω₂+C(Ω₂2,0)+2,0)
{1,,1■1■2} approximately corresponds to C(Ω₂+C(Ω₂2,0)2,0)
{1,,1{2,,1,,2}2} approximately corresponds to C(Ω₂+ω^(C(Ω₂2,0)+1),0)
{1,,1{1■2,,1,,2}2} approximately corresponds to C(Ω₂+ω^(C(Ω₂2,0)2),0)
{1,,1{1■1■2,,1,,2}2} approximately corresponds to C(Ω₂+ω^ω^(C(Ω₂2,0)2),0)
{1,,1{1{1■2,,1,,2}2,,1,,2}2} approximately corresponds to C(Ω₂+ω^ω^ω^(C(Ω₂2,0)2),0)
{1,,1{1,,2,,2}2} approximately corresponds to C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0)

However, the ordinal C(Ω₂2,0) in Taranovsky’s ordinal notation behaves in an irregular way.

Up to {1,,1,,3}

{1,,1{1,,1,,2}1,2{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1,,2}1{1,,1{1{1,,2,,2}2,,1,,2}2}2{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0),C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1,,2}1{1,,1,,2}2{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)2,C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0)),0),0)
{1,,1{1{1,,1,,2}2,,1,,2}2{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(C(Ω₂2,0)2),C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0)),0),0)
{1,,1{1{1,,2,,2}2,,1,,2}3} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0)),0),0)
{1,,1{1{1,,2,,2}2,,1,,2}1,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0))+1,0),0),0)
{1,,1{1{1,,2,,2}2,,1,,2}1{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0))+C(Ω₂2,0),0),0),0)
{1,,1{1{1,,2,,2}2,,1,,2}1{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0))2,0),0),0)
{1{1{1,,2,,2}2,,1,,2}2{1,,2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0))2),0),0),0)
{1{1,,2,,2}3,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0))),0),0),0)
{1{1,,2,,2}1,2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0))+1,C(Ω₂2,0)),0),0),0)
{1{1,,2,,2}1{1,,2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0))2,C(Ω₂2,0)),0),0),0)
{1,,3,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)
{1,,1,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+1,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1{1,,1,,2}1,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(Ω₂+C(Ω₂2,0)+1,0),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1{1,,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0) (It’s a bit erratic here. Things like C(Ω₂+C(C(Ω₂+C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),0) are not valid)

{1,,1{1,,1,,2}1,2{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),0)),0),0)
{1,,1{1{1,,2,,2}2,,1,,2}2{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),0)),0),0)
{1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),0)),0),0)
{1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}3} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),0)),0),0)
{1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}1,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0))+1,0),0),0)
{1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0))2,0),0),0)
{1{1,,2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0))),0),0),0)
{1{1,,1{1,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0))),0),0),0)
{1{1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0))),0),0),0)
{1{1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0))),0),0),0)
{1{1,,1{1{1,,1{1,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0))),C(Ω₂2,0)),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0))),0),0),0)
{1{1,,1{1,,2,,2}2,,2}3,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0))),0),0),0)
{1{1,,1{1,,2,,2}2,,2}1,2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+1,C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}2,,2}1{1,,1,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(Ω₂2,0),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}2,,2}1{1,,2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}2,,2}1{1,,3,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(C(Ω₂,C(Ω₂,C(Ω₂2,0))),C(Ω₂,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}2,,2}1{1,,1{1,,1,,2}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}2,,2}1{1,,1{1,,2,,2}2,,2}1,2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0)))+1,C(Ω₂2,0)),0),0),0)
{1{2,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+ω^(C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0)))+1),C(Ω₂2,0)),0),0),0)
{1{1{1,,2,,2}2,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+ω^(C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0)))+C(Ω₂,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)

So {1{1,,2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+ω^(C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0)))+C(Ω₂,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0).

{1{2{1,,1{1,,2,,2}2,,2}2,,2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+ω^(C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0)))+1),C(Ω₂2,0)),0),0),0)
{1{1,,3,,2}2{1,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(C(Ω₂,C(Ω₂,C(Ω₂2,0))),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0)))),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0)))),C(Ω₂2,0)),0),0),0)
{1{1,,1{1{1,,1{1,,2,,2}2,,2}1{1,,2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0)))),C(Ω₂2,0)),0),0),0)
{1{1,,1{1{1{1,,1{1,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0)))),C(Ω₂2,0)),C(Ω₂2,0)),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0)))),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}2,,2}3,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(C(Ω₂2,C(Ω₂2,0)),C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0)))),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}2,,2}1,2,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(C(Ω₂2,C(Ω₂2,0))+1,C(Ω₂,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}2,,2}1{1,,2,,2}2,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(C(Ω₂2,C(Ω₂2,0))+C(Ω₂,C(Ω₂2,0)),C(Ω₂,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}2,,2}1{1,,3,,2}2,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(Ω₂,C(Ω₂,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}2,,2}1{1,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂,C(Ω₂2,0)))),C(Ω₂2,0)),0),0),0)
{1{2,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+ω^(C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂,C(Ω₂2,0))))+1),C(Ω₂2,0)),0),0),0)
{1{1{1,,3,,2}2,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+ω^(C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂,C(Ω₂2,0))))+C(Ω₂,C(Ω₂,C(Ω₂2,0)))),C(Ω₂2,0)),0),0),0)

So {1{1,,3,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+ω^(C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂,C(Ω₂,C(Ω₂2,0))))+C(Ω₂,C(Ω₂,C(Ω₂2,0)))),C(Ω₂2,0)),0),0),0). And further, the recursion level of {1{1,,k,,2}2,,1{1,,2,,2}2,,2} is between C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(Ω₂,C(Ω₂,…C(Ω₂,C(Ω₂2,0)))),C(Ω₂2,0)),0),0),0) (with k-1 Ω₂′s in the blue part) and C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(Ω₂,C(Ω₂,…C(Ω₂,C(Ω₂2,0)))),C(Ω₂2,0)),0),0),0) (with k Ω₂′s in the blue part), so {1{1,,1,2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(Ω₂+1,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0).

{1{1,,1{1,,1,,2}2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1{1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1{1,,1{1{1{1,,1{1,,1,,2}2,,2}2,,1{1,,2,,2}2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))+C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0))2,C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}2,,2}1{1,,1{1,,2,,2}2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(ω^(C(Ω₂2,C(Ω₂2,0))2),C(Ω₂2,0)),0),0),0)
{1{1{1,,1{1,,2,,2}2,,2}2,,1{1,,2,,2}2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(ω^ω^(C(Ω₂2,C(Ω₂2,0))2),C(Ω₂2,0)),0),0),0)
{1,,2{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)
{1,,1,2{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+1,C(Ω₂2,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1,,2}2{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)
{1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)
{1,,1{1{1,,1{1,,1,,2}2{1,,2,,2}2,,2}2,,1,,2}2{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,C(Ω₂2,0))),C(Ω₂2,0)),C(Ω₂2,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}3,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,C(Ω₂2,0))),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1,,1{1,,1,,2}2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),0),0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂,C(Ω₂2,0)),0),0),0)

Now let separator ◊ = {1,,1{1,,2,,2}1{1,,1,,2}2,,2} and let ordinal a = C(Ω₂+C(Ω₂2,0),0), and b = C(Ω₂+C(Ω₂,C(Ω₂2,0)),0).

{1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b,0),0)
{1,,1{1,,1{1,,1,,2}2}2{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2,C(Ω₂2+b,0)),0)
{1,,1{2,,1{1,,1,,2}2}2{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+ω^(a+1),C(Ω₂2+b,0)),0)
{1,,1{1{1,,1{1,,1,,2}1,2}2,,1{1,,1,,2}2}2{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,0),C(Ω₂2+b,0)),0)
{1,,1{1{1,,1{1,,2,,2}2}2,,1{1,,1,,2}2}2{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0),C(Ω₂2+b,0)),0)
{1,,1{1{1,,1{1,,1{1,,2,,2}2,,2}2}2,,1{1,,1,,2}2}2 {1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),0),C(Ω₂2+b,0)),0)
{1,,1{1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,1{1,,1,,2}2}2{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),C(Ω₂2+b,0)),0)
{1,,1{1{1,,1{1,,1{1,,2,,2}1{1,,1{1,,1,,2}2}2,,2}2}2,,1{1,,1,,2}2}2{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0),C(Ω₂2+b,0)),0)
{1,,1{1{1,,1{1,,1{1,,2,,2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,2}2}2,,1{1,,1,,2}2}2 {1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),C(Ω₂2,0)),0),C(Ω₂2+b,0)),0)
{1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}3{1,,1,,2}2} has recursion level C(C(Ω₂2+b,C(Ω₂2+b,0)),0)
{1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+1,0),0)
{1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+a,0),0)
{1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,2}3}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a),0),0)
{1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}1{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),a),0),0)
{1,,1{2{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+ω^(C(C(Ω₂2+b,0),a)+1),0),0)
{1,,1{1{1,,1◊2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+1,a),0),0)
{1,,1{1{1{1,,1,2{1,,1,,2}2}2,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+C(Ω₂+1,a),a),0),0)
{1{1,,1{1,,1,,2}3}2,,1◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+C(Ω₂2,0),a),0),0)
{1{1,,1{1,,1,,2}1,2}2,,1◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,0),0),a),0),0)
{1{1,,1◊2}2,,1◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)2,a),0),0)
{1,,2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1{1,,1,,2}2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂+a,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1{1,,1,,2}3}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1◊2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1◊2}1,2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+1,0),a),0),0)
{1,,1{1,,1◊2}1{1,,1{1,,1,,2}2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+a,0),a),0),0)
{1,,1{1,,1◊2}1{1,,2{1,,1,,2}2}2◊2} has recursion level C(C(Ω₂2+b+C(Ω₂,a),0),0)
{1,,1{1,,1◊2}1{1,,1{1,,1,,2}3}2◊2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(Ω₂2,0),a),0),0)
{1,,1{1,,1◊2}1{1,,1{1,,1,,2}1,2}2◊2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(Ω₂2,0)+1,0),0),0)
{1,,1{1,,1◊2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2◊2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1◊2}1{1,,1{1,,1{1,,2,,2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,2}2}2◊2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1◊2}1{1,,1◊2}2◊2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0),0),0)
{1,,1{2,,1◊2}2◊2} has recursion level C(C(Ω₂2+b+ω^(C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)+1),0),0)

Now let separator ◊ = {1,,1{1,,2,,2}1{1,,1,,2}2,,2} and let ordinal b = C(Ω₂+C(Ω₂,C(Ω₂2,0)),0), and a = C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0) = C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂,C(Ω₂2,0)),0),0),C(Ω₂2,0)),0).

{1,,1{1{1,,1◊2}2,,1◊2}2◊2} has recursion level C(C(Ω₂2+b+ω^(a2),0),0)
{1,,1{1{1,,2◊2}2,,1◊2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂,a),a),0),0)
{1,,1{1{1,,1,2◊2}2,,1◊2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂+1,a),a),0),0)
{1,,1{1{1,,1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2◊2}2,,1◊2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),a),a),0),0)
{1,,1{1{1,,1{1,,1◊2}2◊2}2,,1◊2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂+a,a),a),0),0)
{1,,1{1,,2◊2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂+C(Ω₂,a),a),a),0),0)
{1,,1{1,,1{1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂+C(Ω₂+C(Ω₂,a),a),a),a),0),0)
{1,,1{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a),0),0)
{1,,1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+1,0),0)
{1,,1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(Ω₂+C(Ω₂2,0),0),0),0)
{1,,1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂2+b+C(C(Ω₂2,0),a),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{2{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+ω^(C(C(Ω₂2+b+C(C(Ω₂2,0),a),0),C(Ω₂+C(Ω₂2,0),0))+1),0),0)
{1,,1{1{1,,1{1,,1,,2}2◊2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂2+b+C(C(Ω₂2,0),a),0)+1,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1,,2}3}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂2+b+C(C(Ω₂2,0),a),0)+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂2+b+C(C(Ω₂2,0),a),0)+C(Ω₂2+b,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1{1,,2◊2}2,,1◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂2+b+C(C(Ω₂2,0),a),0)+C(C(Ω₂,C(Ω₂2+b,0)),C(Ω₂2+b,0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1{1,,1{1,,1,,2}2◊2}2,,1◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂2+b+C(C(Ω₂2,0),a),0)+C(C(Ω₂2+b+C(C(Ω₂2,0),a),0),C(Ω₂2+b,0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,2◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂2+b+C(C(Ω₂2,0),a),0)+C(Ω₂,C(Ω₂2+b,0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1◊2}1,2◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂2+b+C(C(Ω₂2,0),a),0)+C(Ω₂2+b+1,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂2+b+C(C(Ω₂2,0),a),0)2,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂,C(Ω₂2+b+C(C(Ω₂2,0),a),0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂2+b+C(C(Ω₂2,0),a)+1,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(Ω₂+C(Ω₂2,0),0),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,2{1,,1,,2}2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(Ω₂,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}1,2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(Ω₂+C(Ω₂2,0)+1,0),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+a,0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂,a),a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1◊2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)+C(C(Ω₂+a,a),a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a)2,0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{2{1,,1{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+ω^(C(C(Ω₂2,0),a)+1),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1,,2}2◊2}1,2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0)+1,a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0)2,a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂,C(Ω₂2,0)),a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,C(Ω₂2,0)),a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+1,0),a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+a,0),a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂,a),0),a),0),0) (it’s different to previous comparisons on {1,,1{1,,1{1,,1,,2}k+1}1{1,,2{1,,1,,2}k}2{1,,1,,2}k+1}′s)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,2◊2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂,a),a),0),a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a),0),0)
{1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+1,0),0)
{1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2} 1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(Ω₂+C(Ω₂2,0),0),0),0)
{1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2} 1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a),0)+1,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1,,2}3}2,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a),0)+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1◊2}2,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a),0)+C(Ω₂2+b,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} 2,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a),0)2,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,2{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(C(Ω₂,C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a),0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}3{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a),C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a),0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+1,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(Ω₂+C(Ω₂2,0),0),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,2{1,,1,,2}2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(Ω₂,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}1,2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+C(Ω₂+C(Ω₂2,0)+1,0),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)+a,0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a)2,0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}1,2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0)+1,a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,2{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂,C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0)),a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a)+1,0),a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a)+a,0),a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a)+C(Ω₂,a),0),a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a)2,0),a),0),0)
{1,,1{2,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+ω^(C(Ω₂+C(Ω₂2,0),a)+1),0),a),0),0)
{1,,1{1{1,,2{1,,1,,2}2◊2}2,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(C(Ω₂,C(Ω₂+C(Ω₂2,0),a)),C(Ω₂+C(Ω₂2,0),a)),0),a),0),0)
{1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(C(Ω₂+C(Ω₂+C(Ω₂2,0),a),C(Ω₂+C(Ω₂2,0),a)),C(Ω₂+C(Ω₂2,0),a)),0),a),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(C(Ω₂+C(Ω₂,C(Ω₂+C(Ω₂2,0),a)),C(Ω₂+C(Ω₂2,0),a)),C(Ω₂+C(Ω₂2,0),a)),0),a),0),0)
{1,,1{1,,1,,2}3◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),a)),0),a),0),0)
{1,,1{1,,1{1,,1,,2}3◊2}1{1,,2{1,,1,,2}2◊2}2{1,,1,,2}3◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂,C(Ω₂+C(Ω₂2,0),a)),0),a),0),0)
{1,,1{1,,1{1,,1,,2}3◊2}1{1,,1{1,,1,,2}3◊2}2{1,,1,,2}3◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),a)),0),a),0),0)
{1,,1{1,,1,,2}1,2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,a),0),a),0),0)
{1,,1{1,,1,,2}1{1,,1◊2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+a,a),0),a),0),0)
{1,,1{1,,1,,2}1{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)2,a),0),a),0),0)
{1,,1{1{1,,2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),a),0),a),0),0)
{1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),a),0),a),0),0)
{1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),a),0),a),0),0)
{1,,1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),a),0),a),0),0)
{1,,1{1{1,,1{1,,2,,2}1{1,,1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),C(Ω₂2,0)),a),0),a),0),0)
{1,,1{1{1,,1{1,,2,,2}1{1,,1◊2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),a),0),a),0),0)
{1,,1{1{1,,1{1,,2,,2}1{1,,1{1,,1,,2}2◊2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),C(Ω₂2,0)),a),0),a),0),0)
{1,,1{1{1,,1{1,,2,,2}1{1,,1{1{1,,1{1,,2,,2}1{1,,1◊2}2,,2}2,,1,,2}2◊2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),a),0),C(Ω₂2,0)),a),0),a),0),0)
{1,,1◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),a),0),0)
{1,,1{1,,1◊3}1{1,,2{1,,1,,2}2}2◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),a)+C(Ω₂,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1◊3}1{1,,1◊2}2◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),a)+a,0),0)
{1,,1{1,,1◊3}1{1{1,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),a)2,0),0)
{1,,1{1,,1◊3}1{1{1,,1◊3}1,2,,1◊2}2◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+1,a),0),0)
{1,,1{1,,1◊3}1{1{1{1,,1◊3}2,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)2,a),0),0)
{1,,1{1,,1◊3}1{1{1,,2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1◊3}1{1{1,,1{1,,1◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂+a,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1◊3}1{1{1,,1{1,,1{1,,1,,2}2◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1,2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+1,0),a),0),0)
{1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1{1,,1◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+a,0),a),0),0)
{1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1,2◊3}2,,1◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+C(C(Ω₂2+b+1,0),a),0),a),0),0)
{1,,1{1,,1◊3}1{1,,2◊2}2◊3} has recursion level C(C(Ω₂2+b+C(Ω₂,a),0),0)
{1,,1{1,,1◊3}1{1,,1{1,,1,,2}2◊2}2◊3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(Ω₂2,0),a),0),0)
{1,,1{1,,1◊3}1{1,,1{1{1,,2,,2}2,,1,,2}2◊2}2◊3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),a),0),0)
{1,,1{1,,1◊3}1{1,,1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2◊2}2◊3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),a),0),0)
{1,,1{1,,1◊3}1{1,,1{1{1,,1{1,,2,,2}1{1,,1◊2}2,,2}2,,1,,2}2◊2}2◊3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),a),0),0)
{1,,1{1,,1◊3}1{1,,1◊3}2◊3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),a),0),0)
{1,,1◊4} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),a)),0),0)
{1,,1{1,,1◊4}1{1,,1◊4}2◊4} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),a)),0),0)
{1,,1◊5} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),a))),0),0)
{1,,1◊1,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))+1,0),0),0)
{1,,1◊1{1,,1◊1,2}2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))+1,0),0),0),0)
{1,,1◊1{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))+C(Ω₂2,0),0),0),0)
{1,,1◊1{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1◊1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1◊1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),0)
{1,,1◊1{1{1,,1{1,,2,,2}1{1,,1◊1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2}2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),C(Ω₂2,0)),0),0),0)
{1,,1◊1◊2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))2,0),0),0)

Now let ordinal b = C(Ω₂+C(Ω₂,C(Ω₂2,0)),0).

{1,,1{2,,1{1,,2,,2}1{1,,1,,2}2,,2}2} has recursion level C(C(Ω₂2+b+C(Ω₂+ω^(C(C(Ω₂2+b,0),C(Ω₂2,0))+1),0),0),0)
{1{1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1,,2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+ω^(C(C(Ω₂2+b,0),C(Ω₂2,0))2),0),0),0)
{1{1,,2,,2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(C(Ω₂2+b,0),C(Ω₂2,0))),0),0),0)
{1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}3,,1,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(C(Ω₂2+b,0),C(Ω₂2,0))),0),0),0)
{1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}1,2,,1,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0)+1,C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}1{1,,2,,2}2,,1,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0)+C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1{1,,1,2,,2}2,,1{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0)+C(Ω₂+1,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}1,2,,2}2,,1{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0)+C(Ω₂2+1,0),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0)2,C(Ω₂2,0)),0),0),0)
{1,,2{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂,C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1{1,,1,2{1,,2,,2}1{1,,1,,2}2,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂+C(C(Ω₂+1,C(Ω₂2+b,0)),C(Ω₂2,0)),C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}2{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2,C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1,2{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+1,C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1,,1{1,,1{1,,2,,2}1,2{1,,1,,2}2,,2}2}2{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,C(Ω₂2+b,0)),C(Ω₂2,0)),0),C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1,,1,,2}3,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1,,1,,2}1,2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b+1,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1,2,,2}2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b+1,0),C(Ω₂2,0)),0),0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+b2,0),0)
{1,,1{1,,2,,2}1{2,,1,,2}2,,2} has recursion level C(C(Ω₂2+ω^(b+1),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,2}2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+ω^(b+C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+ω^(b+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,2,,2}1{2,,1,,2}2,,2}2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+ω^(b+C(Ω₂+C(C(Ω₂2+ω^(b+1),0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,2,,2}1{1{1,,1,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+ω^(b2),0),0)
{1,,1{1,,2,,2}1{1{1,,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂,b),b),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,2}2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+C(Ω₂2,0),0),b),b),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0),b),b),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1{1,,2,,2}1{1{1,,2,,2}2,,1,,2}2,,2}2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+C(C(Ω₂2+C(C(Ω₂,b),b),0),C(Ω₂2,0)),0),b),b),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+b,b),b),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,b),b),b),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,b)+1,b),b),0),0)
{1,,1{1,,2,,2}1{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,b)2,b),b),0),0)
{1,,1{1,,1,2,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+1,b),b),b),0),0)
{1,,1{1,,1{1,,1,,2}2,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+b,b),b),b),0),0)
{1,,1,,3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b),0),0)

So {1,,1,,3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)),0),0). Here’re some approximations, to make the comparisons above more simple:

{1,,1{1,,1{1,,1,,2}2,,2}2} approximately corresponds to C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),0)
{1,,1{1,,1{1,,2,,2}2,,2}2} approximately corresponds to C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),0)
{1,,1{1,,1{1,,2,,2}1,2,,2}2} approximately corresponds to C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0)
{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2} approximately corresponds to C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂,C(Ω₂2,0)),0),0),C(Ω₂2,0)),0)
{1,,2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2} approximately corresponds to C(Ω₂,C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂,C(Ω₂2,0)),0),0),C(Ω₂2,0)),0))
{1,,1{1,,1,,2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2} approximately corresponds to C(Ω₂+C(Ω₂2,0),C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂,C(Ω₂2,0)),0),0),C(Ω₂2,0)),0))
{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}1,2} approximately corresponds to C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂,C(Ω₂2,0)),0),0),C(Ω₂2,0))+1,0)
{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1,2,,2}2} approximately corresponds to C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)+1,0),C(Ω₂2,0)),0)
{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2}2} approximately corresponds to C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)2,0),C(Ω₂2,0)),0)
{1,,1,,2} approximately corresponds to C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)
{1,,2,,2} approximately corresponds to C(Ω₂,C(Ω₂+C(Ω₂,C(Ω₂2,0)),0))
{1,,1{1,,1,,2}2,,2} approximately corresponds to C(Ω₂+C(Ω₂+C(Ω₂,C(Ω₂2,0)),0),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0))
{1,,1{1,,2,,2}2,,2} approximately corresponds to C(Ω₂+C(Ω₂,C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0))

However, C(Ω₂2,0) still works in an irregular way.

Up to {1,,1,,1,,2}

Now let ordinal b = C(Ω₂+C(Ω₂,C(Ω₂2,0)),0).

{1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b),0),0)
{1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,2}3}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,2}1,2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,0),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+b,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1 {1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+b2,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1{1,,1{1,,1,,3}2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+C(C(Ω₂2,0),b),0)+1,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1,,2}3}2,,1{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+C(C(Ω₂2,0),b),0)+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,1{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+C(C(Ω₂2,0),b),0)+C(Ω₂2+b,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2}2}2,,1{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+C(C(Ω₂2,0),b),0)+C(Ω₂2+b2,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1{1,,2,,2}1{1,,2,,2}2,,2}2}2,,1{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+C(C(Ω₂2,0),b),0)+C(Ω₂2+C(C(Ω₂+C(Ω₂,b)2,b),b),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1,,3}2}2,,1{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+C(C(Ω₂2,0),b),0)2,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂,C(Ω₂2+C(C(Ω₂2,0),b),0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,2}3}2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2,C(Ω₂2+C(C(Ω₂2,0),b),0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+b,C(Ω₂2+C(C(Ω₂2,0),b),0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,3}2}2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+C(C(Ω₂2,0),b),C(Ω₂2+C(C(Ω₂2,0),b),0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,3}2}1,2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+C(C(Ω₂2,0),b)+1,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(Ω₂2,0),0),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,3}2}1{1,,2{1,,1,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+b2,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1,,3}2}2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1,,2}2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂2,0),C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1,,2}1,2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(Ω₂2,0)+1,C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,2,,2}2{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1,,3}3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1,,3}1,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0))+1,0),0),0)
{1,,1{1,,1,,3}1{1,,1,,3}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0))2,0),0),0)
{1,,1{2,,1{1,,1,,3}2,,2}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+ω^(C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0))+1),0),0),0)
{1{1,,2,,2}2{1,,1{1,,1,,3}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0))),0),0),0)
{1{1,,1{1,,2,,2}1,2,,2}2{1,,1{1,,1,,3}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+1,0),C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0))),0),0),0)
{1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2{1,,1{1,,1,,3}2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+b,0),C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0))),0),0),0)
{1{1,,1{1,,1,,3}2,,2}3,,1,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(C(Ω₂2+C(C(Ω₂2,0),b),0),C(Ω₂2,0))),0),0),0)
{1{1,,1{1,,1,,3}2,,2}1,2,,1,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0)+1,C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,1,,3}2,,2}1{1,,1,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0)+C(Ω₂2,0),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0)+C(Ω₂2+b,0),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),0)2,C(Ω₂2,0)),0),0),0)
{1,,2{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂,C(Ω₂2+C(C(Ω₂2,0),b),0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}2{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2,C(Ω₂2+C(C(Ω₂2,0),b),0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1,,1,,2}2{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+b,C(Ω₂2+C(C(Ω₂2,0),b),0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}3,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b),C(Ω₂2+C(C(Ω₂2,0),b),0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}1,2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),b)+1,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+b,0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}1{1{1,,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)+C(C(Ω₂,b),b),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b)2,0),0)
{1,,1{1,,2,,2}1{1{1,,2,,2}2{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂,b),C(C(Ω₂2,0),b)),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}3,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(C(Ω₂2,0),b)),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}1,2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+1,b),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂,b),b),0),0)
{1,,1{1,,2,,2}1{1{1{1,,1,2,,2}2,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)+C(Ω₂+1,b),b),0),0)
{1{1,,1{1,,1,,3}2,,2}2,,1{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0)2,b),0),0)
{1,,2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂2,0)),b),0),0)
{1,,1,2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+1,C(Ω₂2,0)),b),0),0)
{1,,1{1,,1,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+b,C(Ω₂2,0)),b),0),0)
{1,,1{1,,2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,b),C(Ω₂2,0)),b),0),0)
{1,,1{1,,1{1,,1,,2}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+b,b),C(Ω₂2,0)),b),0),0)
{1,,1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,C(Ω₂2,0)),b),0),0)
{1,,1{1,,1{1,,1,,3}2,,2}1,2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2+1,0),b),0),0)
{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2+b,0),b),0),0)
{1,,1{1,,1{1,,1,,3}2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1,2{1,,1,,3}2,,2}2,,1,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2+C(C(Ω₂2+1,0),b),0),b),0),0)
{1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(Ω₂,b),0),0)
{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,2,,2}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂,b),b),0),0)
{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),b),0),0)
{1,,1{1{1,,2{1,,1,,3}2,,2}2,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂+C(Ω₂2,0),b)),C(Ω₂+C(Ω₂2,0),b)),0),0)
{1,,1{1,,2{1,,1,,3}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,C(Ω₂+C(Ω₂2,0),b)),C(Ω₂+C(Ω₂2,0),b)),C(Ω₂+C(Ω₂2,0),b)),0),0)
{1,,1{1,,1,,3}3,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),b)),0),0)
{1,,1{1,,1{1,,1,,3}3,,2}1{1,,1{1,,1,,3}3,,2}2{1,,1,,3}3,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),b)),0),0)
{1,,1{1,,1,,3}1,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,b),0),0)
{1,,1{1,,1,,3}1{1,,1{1,,1{1,,1,,3}1,2,,2}2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,b),0),C(Ω₂2,0)),0),b),0),0)
{1,,1{1,,1,,3}1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+b,b),0),0)
{1,,1{1,,1,,3}1{1,,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+C(Ω₂,b),b),0),0)
{1,,1{1,,1,,3}1{1,,1{1,,1,,3}2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+C(Ω₂+C(Ω₂2,0),b),b),0),0)
{1,,1{1,,1,,3}1{1,,1{1,,1,,3}1,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+C(Ω₂+C(Ω₂2,0)+1,b),b),0),0)
{1,,1{1,,1,,3}1{1,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)2,b),0),0)
{1,,1{2,,1,,3}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(C(Ω₂2,0)+1),b),0),0)
{1{1,,1,,3}2,,1,,3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(C(Ω₂2,0)2),b),0),0)
{1,,2,,3} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂2,0)),b),0),0)
{1,,1,2,,3} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+1,C(Ω₂2,0)),C(Ω₂2,0)),b),0),0)
{1,,1{1,,1,,2}2,,3} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+b,C(Ω₂2,0)),C(Ω₂2,0)),b),0),0)
{1,,1{1,,1,,3}2,,3} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),b),0),0)
{1,,1{1,,2,,3}2,,3} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),b),0),0)
{1,,1{1,,2,,3}1,2,,3} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),b),0),0)
{1,,1{1,,2,,3}1{1,,1,,2}2,,3} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),b),0),0)
{1,,1{1,,2,,3}1{1,,1{1,,1{1,,2,,3}1{1,,1,,2}2,,3}2,,2}2,,3} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),b),0),C(Ω₂2,0)),b),0),0)
{1,,1{1,,2,,3}1{1,,1,,3}2,,3} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂,C(Ω₂2,0)),b),0),0)

Now let separator ♦ = {1,,1{1,,2,,3}1{1,,1,,3}2,,3} and let ordinal a = C(Ω₂+C(Ω₂,C(Ω₂2,0)),0), and b = C(Ω₂+C(Ω₂,C(Ω₂2,0)),a) = C(Ω₂+C(Ω₂,C(Ω₂2,0)),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)).

{1,,1{1♦2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b,0),0)
{1,,1{2♦2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+ω^(C(C(Ω₂2+b,0),C(Ω₂+C(Ω₂2,0),0))+1),0),0)
{1,,1{1♦1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+1,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,1♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+C(Ω₂2+a,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1,,3}2}2,,1♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+C(Ω₂2+C(C(Ω₂2,0),a),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1{1,,1{1,,1,,3}2,,2}1,2{1,,1,,3}2,,2}2}2,,1♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+C(Ω₂2+C(C(Ω₂2+1,0),a),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2}2}2,,1♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+C(Ω₂2+C(Ω₂,a),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1{1,,1,,3}1,2,,2}2}2,,1♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,a),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1♦2}2,,1♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)2,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂,C(Ω₂2+b,0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1,,2}3}2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,C(Ω₂2+b,0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+a,C(Ω₂2+b,0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1♦2}2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,C(Ω₂2+b,0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1♦2}1,2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+1,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1♦2}1{1,,2{1,,1,,2}2}2♦2} has recursion level C(C(Ω₂2+b+C(Ω₂,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1♦2}1{1,,1{1,,1,,2}1,2}2♦2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(Ω₂2,0)+1,0),0),0)
{1,,1{1,,1♦2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2♦2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1♦2}1{1,,1{1,,1,,3}2}2♦2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),a),0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1♦2}1{1,,1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2}2}2 ♦2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+C(Ω₂,a),0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1♦2}1{1,,1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}2}2♦2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1♦2}1{1,,1♦2}2♦2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1,,2}2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1,,2}1,2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,2,,2}1,2,,2}2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+a,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1{1,,1♦2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+a,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0))+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}2,,1♦2} 2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+a,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0))2,0),0)
{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1,2,,1♦2} 2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+a,0)+1,C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1 {1{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1,2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}2,,1♦2} 2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+a+1,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1{1,,2♦2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+a+C(Ω₂,C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}3♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+a2,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1,,3}2♦2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(C(Ω₂2,0),a),0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1♦3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1♦3}1{1,,1♦2}2♦3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0))+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1♦3}1{1{1,,1♦3}2,,1♦2}2♦3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0))2,0),0)
{1,,1{1,,1♦3}1{1{1,,1♦3}1,2,,1♦2}2♦3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+1,C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1♦3}1{1{1,,2♦3}2,,1♦2}2♦3} has recursion level C(C(Ω₂2+b+C(C(Ω₂,C(Ω₂2+b,0)),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1♦3}1{1{1,,1{1,,1♦3}2♦3}2,,1♦2}2♦3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,C(Ω₂2+b,0)),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1♦3}1{1{1,,1{1,,1♦3}1,2♦3}2,,1♦2}2♦3} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+1,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1♦3}1{1,,2♦2}2♦3} has recursion level C(C(Ω₂2+b+C(Ω₂,C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1♦3}1{1,,1{1,,1,,2}2♦2}2♦3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(Ω₂2,0),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1♦3}1{1,,1♦3}2♦3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),0)),0),0)
{1,,1♦1,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))+1,0),0),0)
{1,,1♦1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))+C(C(Ω₂2+a,0),C(Ω₂2,0)),0),0),0)
{1,,1♦1♦2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))2,0),0),0)
{1,,1{2,,1♦2,,2}2} has recursion level C(C(Ω₂2+b+C(Ω₂+ω^(C(C(Ω₂2+b,0),C(Ω₂2,0))+1),0),0),0)
{1{1,,1,,2}2,,1♦2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+ω^(C(C(Ω₂2+b,0),C(Ω₂2,0))+C(Ω₂2,0)),0),0),0)
{1{1,,2,,2}2{1,,1♦2,,2}2,,1,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(C(Ω₂2+b,0),C(Ω₂2,0))),0),0),0)
{1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2{1,,1♦2,,2}2,,1,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+a,0),C(C(Ω₂2+b,0),C(Ω₂2,0))),0),0),0)
{1{1,,1♦2,,2}3,,1,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(C(Ω₂2+b,0),C(Ω₂2,0))),0),0),0)
{1{1,,1♦2,,2}1,2,,1,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0)+1,C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0)+C(Ω₂2+a,0),C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0)2,C(Ω₂2,0)),0),0),0)
{1,,2{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂,C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2,C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+a,C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+C(C(Ω₂,a),a),C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),a),C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1,2{1,,1,,3}2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2+1,0),a),C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+C(Ω₂,a),C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}2,,1,,2} 2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}1,2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,a),C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,2,,3}1,2,,3}2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),a),C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}3,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,C(Ω₂2+b,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1,2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b+1,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1,,1{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1,2,,2}2}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b+1,0),C(Ω₂2,0)),0),0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+a,0),0)
{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,0),a),0),0)
{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1,,2}2{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+a,0),a),0),0)
{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1 {1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),a),0),a),0),0)
{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1{1,,1,,3}1,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,a),0),a),0),0)
{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1{1,,1{1,,2,,3}1,2,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),a),0),a),0),0)
{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0),a),0),0)
{1,,1{1,,2,,2}1{2{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+ω^(C(C(Ω₂2+b,0),a)+1),0),0)
{1,,1{1,,2,,2}1{1{1,,1♦2,,2}1,2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+1,a),0),0)
{1,,1{1,,2,,2}1{1{1{1,,1,2,,2}2,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+C(Ω₂+1,a),a),0),0)
{1{1,,1{1,,1,,3}2,,2}2,,1♦2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)+C(Ω₂2,0),a),0),0)
{1{1,,1♦2,,2}2,,1♦2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,0)2,a),0),0)
{1,,2♦2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,2,,2}2♦2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂+C(Ω₂,a),C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1{1,,2,,2}2,,2}2♦2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂+C(Ω₂+C(Ω₂,a),a),C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1{1,,1,,3}2,,2}2♦2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1♦2,,2}2♦2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b,C(Ω₂2+b,0)),a),0),0)
{1,,1{1,,1♦2,,2}1,2♦2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+1,0),a),0),0)
{1,,1{1,,1♦2,,2}1{1{1,,1{1,,1♦2,,2}1,2♦2,,2}2,,1,,2}2♦2,,2} has recursion level C(C(Ω₂2+b+C(C(Ω₂2+b+C(C(Ω₂2+b+1,0),a),0),a),0),0)
{1,,1{1,,1♦2,,2}1{1,,2,,2}2♦2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂,a),0),0)
{1,,1{1,,1♦2,,2}1{1,,1{1,,1,,3}2,,2}2♦2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(Ω₂2,0),a),0),0)
{1,,1{1,,1♦2,,2}1{1,,1{1,,1,,3}1,2,,2}2♦2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(Ω₂2,0)+1,a),0),0)
{1,,1{1,,1♦2,,2}1{1,,1{1,,1{1,,2,,3}1,2,,3}2,,2}2♦2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),a),0),0)
{1,,1{1,,1♦2,,2}1{1,,1♦2,,2}2♦2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),a),0),0)
{1,,1{1,,1,,3}1,2♦2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(Ω₂2,0)+1,C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),a)),0),0)
{1,,1{1,,1{1,,2,,3}1,2,,3}2♦2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),a)),0),0)
{1,,1♦3,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0)),a)),0),0)
{1,,1♦1,2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0),C(Ω₂2,0))+1,a),0),0)
{1,,1{2,,1{1,,2,,3}1{1,,1,,3}2,,3}2,,2} has recursion level C(C(Ω₂2+b+C(Ω₂+ω^(C(C(Ω₂2+b,0),C(Ω₂2,0))+1),a),0),0)
{1{1,,2,,3}2♦2,,1,,3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂,C(Ω₂2,0)),C(C(Ω₂2+b,0),C(Ω₂2,0))),a),0),0)
{1♦1,2,,1,,3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0)+1,C(Ω₂2,0)),a),0),0)
{1♦2,,1{1,,2,,3}1{1,,1,,3}2,,3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,0)2,C(Ω₂2,0)),a),0),0)
{1,,2{1,,2,,3}1{1,,1,,3}2,,3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂,C(Ω₂2+b,0)),C(Ω₂2,0)),a),0),0)
{1,,1{1,,2,,3}1,2{1,,1,,3}2,,3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+1,C(Ω₂2+b,0)),C(Ω₂2,0)),a),0),0)
{1,,1{1,,2,,3}1{1,,1,,3}3,,3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b,C(Ω₂2+b,0)),C(Ω₂2,0)),a),0),0)
{1,,1{1,,2,,3}1{1,,1,,3}1,2,,3} has recursion level C(C(Ω₂2+b+C(Ω₂+C(C(Ω₂2+b+1,0),C(Ω₂2,0)),a),0),0)
{1,,1{1,,2,,3}1{1,,1,,3}1{1,,1,,3}2,,3} has recursion level C(C(Ω₂2+b2,0),0)
{1,,1{1,,2,,3}1{1{1,,2,,3}2,,1,,3}2,,3} has recursion level C(C(Ω₂2+C(C(Ω₂,b),b),0),0)
{1,,1{1,,2,,3}1{1{1,,1{1,,1,,3}2,,3}2,,1,,3}2,,3} has recursion level C(C(Ω₂2+C(C(Ω₂+b,b),b),0),0)
{1,,1{1,,2,,3}1{1{1,,1{1,,2,,3}2,,3}2,,1,,3}2,,3} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,b),b),b),0),0)
{1,,1{1,,2,,3}1{1,,2,,3}2,,3} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,b)2,b),b),0),0)
{1,,1{1,,1{1,,1,,3}2,,3}2,,3} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂+b,b),b),b),0),0)
{1,,1,,4} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),b),0),0)

And so on, the recursion level of separator {1,,1,,k} (k > 2) is C(C(Ω₂2+C(C(Ω₂2,0),α_k-2),0),0), where α₀ = 0 and α_k+1 = C(Ω₂+C(Ω₂,C(Ω₂2,0)),α_k), between C(C(Ω₂2+α_k-2,0),0) and C(C(Ω₂2+α_k-1,0),0). So {1,,1,,1,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂,C(Ω₂2,0))+1,0),0),0). Now let ordinal d = C(Ω₂,C(Ω₂2,0)).

{1,,1,,1,3} has recursion level C(C(Ω₂2+C(Ω₂+d+1,C(Ω₂+d+1,0)),0),0)
{1,,1,,1,1,2} has recursion level C(C(Ω₂2+C(Ω₂+d+2,0),0),0)
{1,,1,,1{1,,1{1,,1,,2}2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(Ω₂2,0),0),0),0),0)
{1,,1,,1{1,,1{1,,1,,2}1,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(Ω₂2,0)+1,0),0),0),0)
{1,,1,,1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),0),0)
{1,,1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(C(Ω₂2+C(Ω₂+d,0),0),C(Ω₂2,0)),0),0),0),0)
{1,,1,,1{1,,1{1,,1,,1,2}2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(C(Ω₂2+C(Ω₂+d+1,0),0),C(Ω₂2,0)),0),0),0),0)
{1,,1,,1{1,,1{1,,1,,1{1,,1{1,,1,,1,2}2}2}2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(C(Ω₂2+C(Ω₂+d+1,0),0),C(Ω₂2,0)),0),0),0),C(Ω₂2,0)),0),0),0),0)
{1,,1,,1{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+d,0),0),0),0)
{1,,1,,1,2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+1,C(Ω₂+d+C(Ω₂+d,0),0)),0),0)
{1,,1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),C(Ω₂+d+C(Ω₂+d,0),0)),0),0)
{1,,1,,1{1,,1{1,,1,,1,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(C(Ω₂2+C(Ω₂+d+1,0),0),C(Ω₂2,0)),0),C(Ω₂+d+C(Ω₂+d,0),0)),0),0)
{1,,1,,1{1,,1{1,,1,,1{1,,1,,2}2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(C(Ω₂2+C(Ω₂+d+C(Ω₂+d,0),0),0),C(Ω₂2,0)),0),C(Ω₂+d+C(Ω₂+d,0),0)),0),0)
{1,,1,,1{1,,1{1,,1,,1,2{1,,1,,2}2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(C(Ω₂2+C(Ω₂,C(Ω₂+d+C(Ω₂+d,0),0)),0),C(Ω₂2,0)),0),C(Ω₂+d+C(Ω₂+d,0),0)),0),0)
{1,,1,,1{1,,1,,2}3} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+d,0),C(Ω₂+d+C(Ω₂+d,0),0)),0),0)
{1,,1,,1{1,,1,,2}1,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+d,0)+1,0),0),0)
{1,,1,,1{1,,1,,2}1{1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+d,0)2,0),0),0)
{1,,1,,1{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂,C(Ω₂+d,0)),C(Ω₂+d,0)),0),0),0)
{1,,1,,1{1{1,,1,,3}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂2,0),C(Ω₂+d,0)),0),0),0)
{1,,1,,1{1{1,,1,,1,2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂2+C(Ω₂+d+1,0),0),C(Ω₂+d,0)),0),0),0)
{1,,1,,1{1{1,,1,,1{1{1,,1,,1,2}2,,1,,2}2}2,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂2+C(Ω₂+d+1,0),0),C(Ω₂+d,0)),0),0),C(Ω₂+d,0)),0),0),0)
{1,,1,,1{1,,2,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂,C(Ω₂+d,0)),0),0),0)
{1,,1,,1{1,,1{1,,1,,2}2,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(Ω₂+d,0),C(Ω₂+d,0)),0),0),0)
{1,,1,,1{1,,1{1,,1,,3}2,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(Ω₂2,0),C(Ω₂+d,0)),0),0),0)
{1,,1,,1{1,,1{1,,1,,1,2}2,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(C(Ω₂2+C(Ω₂+d+1,0),0),C(Ω₂2,0)),C(Ω₂+d,0)),0),0),0)
{1,,1,,1{1,,1{1,,1,,1{1,,1{1,,1,,1,2}2,,2}2}2,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(C(Ω₂2+C(Ω₂+d+C(Ω₂+C(C(Ω₂2+C(Ω₂+d+1,0),0),C(Ω₂2,0)),C(Ω₂+d,0)),0),0),C(Ω₂2,0)),C(Ω₂+d,0)),0),0),0)
{1,,1,,1{1,,1,,3}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+d,C(Ω₂+d,0)),0),0),0)
{1,,1,,1{1,,1,,1,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+d+1,0),0),0),0)
{1,,1,,1{1,,1,,1{1,,1,,2}2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+d+C(Ω₂+d,0),0),0),0),0)
{1,,1,,1{1,,1,,1{1,,1,,1,2}2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+d+C(Ω₂+d+1,0),0),0),0),0)
{1,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂2,0),0),0),0)

So {1,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂,C(Ω₂2,0))+C(Ω₂2,0),0),0),0). Here’re some approximations, to make the comparisons above more simple:

{1,,1,,2} approximately corresponds to C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)
{1,,2,,2} approximately corresponds to C(Ω₂,C(Ω₂+C(Ω₂,C(Ω₂2,0)),0))
{1,,1{1,,1,,2}2,,2} approximately corresponds to C(Ω₂+C(Ω₂+C(Ω₂,C(Ω₂2,0)),0),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0))
{1,,1{1,,1,,3}2,,2} approximately corresponds to C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0))
{1,,1{1,,1{1,,2,,3}1{1,,1,,3}2,,3}2,,2} approximately corresponds to C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂,C(Ω₂2,0)),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)),0),C(Ω₂2,0)),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0))
{1,,1,,3} approximately corresponds to C(Ω₂+C(Ω₂,C(Ω₂2,0)),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0))
{1,,2,,3} approximately corresponds to C(Ω₂,C(Ω₂+C(Ω₂,C(Ω₂2,0)),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)))
{1,,1{1,,1,,4}2,,3} approximately corresponds to C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂,C(Ω₂2,0)),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)))
{1,,1,,4} approximately corresponds to C(Ω₂+C(Ω₂,C(Ω₂2,0)),C(Ω₂+C(Ω₂,C(Ω₂2,0)),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)))
{1,,1,,1,2} approximately corresponds to C(Ω₂+C(Ω₂,C(Ω₂2,0))+1,0)
{1,,1,,1{1,,1{1,,1,,2}2}2} approximately corresponds to C(Ω₂+C(Ω₂,C(Ω₂2,0))+C(Ω₂+C(Ω₂2,0),0),0)
{1,,1,,1{1,,1,,2}2} approximately corresponds to C(Ω₂+C(Ω₂,C(Ω₂2,0))+C(Ω₂+C(Ω₂,C(Ω₂2,0)),0),0)

Beyond {1,,1,,1,,2}: Conclusion

Due to the outside layer of C(C(Ω₂2+____,0),0), the comparisons above seem erratic. But now we find the patterns on them.

The story begins from C(Ω₂+C(Ω₂2,0),0), corresponding to {1,,1{1,,1,,2}2}. Then ε_{C(Ω₂+C(Ω₂2,0),0)+1} = C(C(Ω₂,C(Ω₂+C(Ω₂2,0),0)),C(Ω₂+C(Ω₂2,0),0)) corresponds to {1{1,,2{1,,1,,2}2}2,,1{1,,1,,2}2}. When the α in C(α,C(Ω₂+C(Ω₂2,0),0)) get larger (but α < Ω₂), it’ll correspond to {1 A 2,,1{1,,1,,2}2} with higher-leveled A (here A can be very high-leveled). The least ordinal larger than all those C(α,C(Ω₂+C(Ω₂2,0),0)) is C(Ω₂,C(Ω₂+C(Ω₂2,0),0)), so C(Ω₂,C(Ω₂+C(Ω₂2,0),0)) works as the 1-separator in C(____,C(Ω₂+C(Ω₂2,0),0)), just like that {1,,2{1,,1,,2}2} is the 1-separator in { ____ ,,1{1,,1,,2}2}.

Next, C(Ω₂,C(Ω₂,C(Ω₂+C(Ω₂2,0),0))) corresponds to {1,,3{1,,1,,2}2}, and so on. C(Ω₂+C(Ω₂2,0),C(Ω₂+C(Ω₂2,0),0)) corresponds to {1,,1{1,,1,,2}3}. Generally, for β < C(Ω₂2,0), the first fixed point of x -> C(Ω₂+x,β) is C(Ω₂+C(Ω₂2,0),β), and C(Ω₂2,0) works as the 1-separator in C(Ω₂+____,β), just like that {1,,1,,2} works as the 1-separator in {1,, ____ }.

But this correspondence fails at the next key point – When the α in C(Ω₂+C(α,C(Ω₂2,0)),0) get larger (but α < Ω₂), it’ll correspond to {1,,1 A 2} with higher-leveled A (here A can be very high-leveled). Note that without the limit from outside layer of C(C(Ω₂2+____,0),0), the α can be larger than, for instance, C(Ω₂3,0). The least ordinal larger than all those C(Ω₂+C(α,C(Ω₂2,0)),0) is C(Ω₂+C(Ω₂,C(Ω₂2,0)),0), so actually C(Ω₂+C(Ω₂,C(Ω₂2,0)),0) corresponds to {1,,1,,2}, the 1-separator in {1,, ____ }. But the C(Ω₂2,0) still works as the 1-separator in α inside C(Ω₂+α,β) (α < Ω₂, β < C(Ω₂2,0)). This property is important even in further comparisons.

Next, when the α in C(α,C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)) get larger (but α < Ω₂), it’ll correspond to {1 A 2,,1,,2} with higher-leveled A (here A can be very high-leveled). The least ordinal larger than all those C(α,C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)) is C(Ω₂,C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)), so C(Ω₂,C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)) corresponds to {1,,2,,2}, the 1-separator in { ____ ,,1,,2}.

Next, C(Ω₂+C(Ω₂,C(Ω₂2,0)),C(Ω₂+C(Ω₂,C(Ω₂2,0)),0)) corresponds to {1,,1,,3}, the 1-separator in {1,, ____ ,,2}. Generally, for β < C(Ω₂2,0), when the α in C(Ω₂+C(α,C(Ω₂2,0)),β) get larger (but α < Ω₂), it can traverse very high-leveled separator A in, for instance, {1,,1…,,1,,1 A 2…}. The least ordinal larger than all those C(Ω₂+C(α,C(Ω₂2,0)),β) is C(Ω₂+C(Ω₂,C(Ω₂2,0)),β), so C(Ω₂,C(Ω₂2,0)) works as the 2-separator in C(Ω₂+____,β), just like the double comma.

But this correspondence fails at some higher point (see the later post).

Now let separator ◊ = {1,,1,,1{1,,1,,1,,2}2} and let ordinal d = C(Ω₂,C(Ω₂2,0)), and a = C(Ω₂+d+C(Ω₂2,0),0) = C(Ω₂+C(Ω₂,C(Ω₂2,0))+C(Ω₂2,0),0).

{1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+a,0),0)
{1,,1{2{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+a+ω^(C(C(Ω₂2+a,0),C(Ω₂+C(Ω₂2,0),0))+1),0),0)
{1,,1{1{1,,1◊2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,0)+1,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1,,2}1,2}2,,1◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,0)+C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,0),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,1◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,0)+C(Ω₂2+C(Ω₂+d,0),0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1{1,,1◊2}2,,1◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,0)2,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂,C(Ω₂2+a,0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1◊2}2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,C(Ω₂2+a,0)),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1◊2}1,2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a+1,0),C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1◊2}1{1,,2{1,,1,,2}2}2◊2} has recursion level C(C(Ω₂2+a+C(Ω₂,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1◊2}1{1,,1{1,,1,,2}1,2}2◊2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(Ω₂2,0)+1,0),0),0)
{1,,1{1,,1◊2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2◊2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+C(Ω₂+d,0),0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1◊2}1{1,,1◊2}2◊2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2,0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2,0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0))+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2,0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0))2,0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂,C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+1,0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+C(Ω₂,C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1,,2}1,2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,2,,2}1,2,,2}2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+C(Ω₂+d,0),0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1,,1,2}2◊2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+C(Ω₂+d+1,0),0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1◊3} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1◊3}1{1,,1◊2}2◊3} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0))+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1◊3}1{1{1,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0))2,0),0)
{1,,1{1,,1◊3}1{1{1,,1◊3}1,2,,1◊2}2◊3} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,0)+1,C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1◊3}1{1{1{1,,1{1,,1,,2}2◊2}2,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,0)+C(Ω₂2,0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1◊3}1{1{1{1,,1◊3}2,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,0)2,C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1◊3}1{1{1,,2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+a+C(C(Ω₂,C(Ω₂2+a,0)),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1,2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a+1,0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1{1,,1◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0),0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1◊3}1{1,,2◊2}2◊3} has recursion level C(C(Ω₂2+a+C(Ω₂,C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1◊3}1{1,,1{1,,1,,2}2◊2}2◊3} has recursion level C(C(Ω₂2+a+C(Ω₂+C(Ω₂2,0),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,1◊3}1{1,,1◊3}2◊3} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0)),0),0)
{1,,1◊1,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0))+1,0),0),0)
{1,,1◊1◊2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0))2,0),0),0)
{1,,1{2,,1◊2,,2}2} has recursion level C(C(Ω₂2+a+C(Ω₂+ω^(C(C(Ω₂2+a,0),C(Ω₂2,0))+1),0),0),0)
{1{1,,1,,2}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+a+C(Ω₂+ω^(C(C(Ω₂2+a,0),C(Ω₂2,0))+C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}1,2,,2}2{1,,1◊2,,2}2,,1,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+1,0),C(C(Ω₂2+a,0),C(Ω₂2,0))),0),0),0)
{1{1,,1◊2,,2}1,2,,1,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a,0)+1,C(Ω₂2,0)),0),0),0)
{1{1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a,0)2,C(Ω₂2,0)),0),0),0)
{1,,2{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂,C(Ω₂2+a,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}2{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2,C(Ω₂2+a,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1,,1,,2}2{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+C(Ω₂+d,0),C(Ω₂2+a,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+d,0)),C(Ω₂2+a,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1{1,,1,,1,2}2,,2}2,,1,,2}2{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+C(Ω₂+d+1,0),C(Ω₂2+a,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}3,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a,C(Ω₂2+a,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}1,2,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a+1,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+d,0),0),0)
{1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2,0),C(Ω₂+d,0)),0),0)
{1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}1{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,0),C(Ω₂+d,0)),0),0)
{1,,1{1,,2,,2}1{1{1,,1◊2,,2}1,2,,1,,2}2,,2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,0)+1,C(Ω₂+d,0)),0),0)
{1,,2◊2,,2} has recursion level C(C(Ω₂2+a+C(C(Ω₂,C(Ω₂2+a,0)),C(Ω₂+d,0)),0),0)
{1,,1{1,,2,,2}2◊2,,2} has recursion level C(C(Ω₂2+a+C(C(Ω₂+C(Ω₂,C(Ω₂+d,0)),C(Ω₂2+a,0)),C(Ω₂+d,0)),0),0)
{1,,1{1,,1{1,,1,,3}2,,2}2◊2,,2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2,C(Ω₂2+a,0)),C(Ω₂+d,0)),0),0)
{1,,1{1,,1◊2,,2}2◊2,,2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a,C(Ω₂2+a,0)),C(Ω₂+d,0)),0),0)
{1,,1{1,,1◊2,,2}1,2◊2,,2} has recursion level C(C(Ω₂2+a+C(C(Ω₂2+a+1,0),C(Ω₂+d,0)),0),0)
{1,,1{1,,1◊2,,2}1{1,,2,,2}2◊2,,2} has recursion level C(C(Ω₂2+a+C(Ω₂,C(Ω₂+d,0)),0),0)
{1,,1{1,,1◊2,,2}1{1,,1{1,,1,,3}2,,2}2◊2,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(Ω₂2,0),C(Ω₂+d,0)),0),0)
{1,,1{1,,1◊2,,2}1{1,,1◊2,,2}2◊2,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),C(Ω₂+d,0)),0),0)
{1,,1◊1,2,,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0))+1,C(Ω₂+d,0)),0),0)
{1{1,,1,,3}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+a+C(Ω₂+ω^(C(C(Ω₂2+a,0),C(Ω₂2,0))+C(Ω₂2,0)),C(Ω₂+d,0)),0),0)
{1{1,,1,,4}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+a+C(Ω₂+ω^(C(C(Ω₂2+a,0),C(Ω₂2,0))+C(Ω₂2,0)),C(Ω₂+d,C(Ω₂+d,0))),0),0)
{1{1,,1,,1,2}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+a+C(Ω₂+d+1,0),0),0)
{1{1,,1,,1{1,,1,,2}2}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+a+C(Ω₂+d+C(Ω₂+d,0),0),0),0)
{1◊2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+a2,0),0)
{1,,2,,1{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(C(Ω₂,a),a),0),0)
{1,,1,,2{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),a),0),0)
{1,,1{1,,1,,2{1,,1,,1,,2}2}1,2,,1{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,a),0),0)
{1,,1{1,,2,,2{1,,1,,1,,2}2}1,2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+1,0),C(Ω₂2,0)),a),0),0)
{1,,1{1,,2,,2{1,,1,,1,,2}2}1{1,,1,,2}2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+C(Ω₂+d,0),0),C(Ω₂2,0)),a),0),0)
{1,,1{1,,2,,2{1,,1,,1,,2}2}1◊2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),a),0),0)
{1,,1{1,,2,,2{1,,1,,1,,2}2}1{1,,2,,1{1,,1,,1,,2}2}2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+C(Ω₂,a),0),C(Ω₂2,0)),a),0),0)
{1,,1{1,,2,,2{1,,1,,1,,2}2}1{1,,1{1,,1,,2{1,,1,,1,,2}2}1,2,,1{1,,1,,1,,2}2}2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,a),0),C(Ω₂2,0)),a),0),0)
{1,,1{1,,2,,2{1,,1,,1,,2}2}1{1,,1,,2{1,,1,,1,,2}2}2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d,a),0),0)
{1,,1,,1,2{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+1,a),0),0)
{1,,1,,1◊2{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+a,a),0),0)
{1,,1,,1{1,,1,,1,,2}3} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂2,0),C(Ω₂+d+C(Ω₂2,0),0)),0),0)
{1,,1,,1{1,,1,,1,,2}4} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂2,0),C(Ω₂+d+C(Ω₂2,0),C(Ω₂+d+C(Ω₂2,0),0))),0),0)
{1,,1,,1{1,,1,,1,,2}1,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂2,0)+1,0),0),0)

Now let ordinal d = C(Ω₂,C(Ω₂2,0)).

{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂2,0)2,0),0),0)
{1,,1,,1{2,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+ω^(C(Ω₂2,0)+1),0),0),0)
{1,,1,,1{1{1,,1,,1,,2}2,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d+ω^(C(Ω₂2,0)2),0),0),0)
{1,,2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(d,C(Ω₂2,0)),0),0),0)
{1,,1,2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂+1,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1{1,,2,,1,,2}2,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂+C(d,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1{1,,1{1,,1,,1,,2}2,,1,,2}2,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,1,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂+d,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0) (here’s a new irregular thing)
{1,,1{1,,2,,1,,2}1,2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂+d+1,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,1,,2}1{1,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂+d+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,1,,2}1{1,,2,,1,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1,2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,1,,1{1,,1,,1,,2}2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂2+C(Ω₂+d+C(Ω₂2,0),0),0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d2,0),0),0)
{1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,1,,1,,2}1{1,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d2,0)2,0),0)
{1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,2,,1,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(C(Ω₂+C(Ω₂,C(Ω₂+d2,0))3,C(Ω₂+d2,0)),C(Ω₂+d2,0)),0),0)
{1,,1,,2,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+d2,0)),0),0)
{1,,1{1,,1{1,,1,,2,,2}2,,1,,2}1,2{1,,1,,2,,2}2,,1,,2} has recursion level C(C(Ω₂2+C(C(Ω₂2+1,0),C(Ω₂+d2,0)),0),0)
{1,,1{1,,1,,2,,2}1,2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,C(Ω₂+d2,0)),0),0)
{1,,1{1,,2,,2,,2}1{1,,1,,2,,2}2,,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+d,C(Ω₂+d2,0)),0),0)
{1,,1{1,,2,,3,,2}1{1,,1,,3,,2}2,,3,,2} has recursion level C(C(Ω₂2+C(Ω₂+d,C(Ω₂+d,C(Ω₂+d2,0))),0),0)
{1,,1,,1,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+1,C(Ω₂+d2,0)),0),0)
{1,,1,,1{1,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+d,0),C(Ω₂+d2,0)),0),0)
{1,,1,,1{1,,1,,1,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+d2,0),C(Ω₂+d2,0)),0),0)
{1,,1,,1{1,,1,,1,2,,2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂+d+1,C(Ω₂+d2,0)),C(Ω₂+d2,0)),0),0)
{1,,1,,1,,3} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂2,0),C(Ω₂+d2,0)),0),0)
{1,,1,,1{1,,1,,1,,3}1,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂2,0)+1,C(Ω₂+d2,0)),0),0)
{1,,2,,1,,3} has recursion level C(C(Ω₂2+C(Ω₂+d+C(d,C(Ω₂2,0)),C(Ω₂+d2,0)),0),0)
{1,,1{1,,2,,1,,3}2,,1,,3} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂+d,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂+d2,0)),0),0)
{1,,1{1,,2,,1,,3}1{1,,2,,1,,3}2,,1,,3} has recursion level C(C(Ω₂2+C(Ω₂+d+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂+d2,0)),0),0)
{1,,1{1,,2,,1,,3}1{1,,2,,1,,3}1{1,,1,,1,,3}2,,1,,3} has recursion level C(C(Ω₂2+C(Ω₂+d2,C(Ω₂+d2,0)),0),0)
{1,,1,,1,,4} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂2,0),C(Ω₂+d2,C(Ω₂+d2,0))),0),0)
{1,,1{1,,2,,1,,4}1{1,,2,,1,,4}1{1,,1,,1,,4}2,,1,,4} has recursion level C(C(Ω₂2+C(Ω₂+d2,C(Ω₂+d2,C(Ω₂+d2,0))),0),0)
{1,,1,,1,,1,2} has recursion level C(C(Ω₂2+C(Ω₂+d2+1,0),0),0)
{1,,1,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω₂2+C(Ω₂+d2+C(Ω₂+d2,0),0),0),0)
{1,,1,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d2+C(Ω₂2,0),0),0),0)
{1,,1,,1,,1{1,,1,,1,,1,,2}1,2} has recursion level C(C(Ω₂2+C(Ω₂+d2+C(Ω₂2,0)+1,0),0),0)
{1,,2,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d2+C(d,C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,1,,1,,2}2,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d2+C(C(Ω₂+d,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}2,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d2+C(C(Ω₂+d2,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}2,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d2+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1,2,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d2+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1{1,,1,,1,,1,,2}2,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d3,0),0),0)
{1,,1{1,,1,,2,,1,,2}1,2,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,C(Ω₂+d3,0)),0),0)
{1,,1,,1,2,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d+1,C(Ω₂+d3,0)),0),0)
{1,,1,,1,,1,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+d2+1,C(Ω₂+d3,0)),0),0)
{1,,1,,1,,1,,3} has recursion level C(C(Ω₂2+C(Ω₂+d2+C(Ω₂2,0),C(Ω₂+d3,0)),0),0)
{1,,1{1,,2,,1,,1,,3}1{1,,2,,1,,1,,3}1{1,,2,,1,,1,,3}1{1,,1,,1,,1,,3}2,,1,,1,,3} has recursion level C(C(Ω₂2+C(Ω₂+d3,C(Ω₂+d3,0)),0),0)
{1,,1,,1,,1,,1,2} has recursion level C(C(Ω₂2+C(Ω₂+d3+1,0),0),0)
{1,,1,,1,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d3+C(Ω₂2,0),0),0),0)
{1,,1,,1,,1,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+d4+C(Ω₂2,0),0),0),0)
{1{2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),0)

Something strange appears here. The recursion level of {1{1{2^,,}2}2} is only equal to that of {1{1,,2}2}, because s(n,n{1{1{2^,,}2}2}2) = s(n,n{1{1,,1,,…1,,1,,2}2}2) = (case B2, step 6) s(n,n{1{1…{1,2}…2}2}2) with n-1 1′s. Then {1{1{1{2^,,}2}2{2^,,}2}2} has equal recursion level to {1{1,,2}2,,2}. Also, {1,,1{1{1{2^,,}2}2{2^,,}2}2} has equal recursion level to {1{1,,1,,2}2,,1,,2}, {1,,1,,1{1{1{2^,,}2}2{2^,,}2}2} has equal recursion level to {1{1,,1,,1,,2}2,,1,,1,,2}, and so on – they’re all less than the recursion level of {1{2^,,}2}.

However, {1{1,,2}2{2^,,}2} has equal recursion level to {1{1,,2}2,,2}, {1{1,,1,,2}2{2^,,}2} has equal recursion level to {1{1,,1,,2}2,,1,,2}, {1{1,,1,,1,,2}2{2^,,}2} has equal recursion level to {1{1,,1,,1,,2}2,,1,,1,,2}, and so on, so {1{1{2^,,}2}2{2^,,}2} has equal recursion level to {1{2^,,}2}.

What’s more, this “recursion level platform” happens on higher level of separators, such as {1{1{2^,,}2}3{2^,,}2}, {1{1{2^,,}2}1{1{2^,,}2}2{2^,,}2}, {1{1{1{2^,,}2}2{2^,,}2}2{2^,,}2}, {1,,2{2^,,}2}, {1,,3{2^,,}2}, {1,,1,,2{2^,,}2}, {1,,1,,1,,2{2^,,}2}, and so on – until {1{2^,,}3}. Inside {1 ____ 2}, the {1{2^,,}2} actually works in the same way as {1,,2}. Further, {1,,2{2^,,}2} actually works as {1,,3}, {1,,1,,2{2^,,}2} actually works as {1,,1,,2}, and {1{2^,,}3} actually works as the real {1{2^,,}2} – {1{1{2^,,}3}2} has equal recursion level to {1{2^,,}2}.

Also, {1,,1{1{2^,,}3}2}, {1,,1,,1{1{2^,,}3}2}, and even the single {1{2^,,}3}, all have equal recursion level to {1{2^,,}2}. The main reason of the “recursion level platform” described above is at the step 5 in case B2. For separator {1,,1,,…1,,1,,2} (which is resulted from {1{2^,,}2}), the “X ,, n-1 Y” is {1,,1,,…1,,1,,1}, then lv(“X ,, n-1 Y”) = lv({1}), so we always turn into step 6 (expanding rule), with no adding between the separator {1,,1,,…1,,1,,2} and the outside {1 ____ 2}. However, for separator {1,,1,,…1,,1,,2{2^,,}2} (which is resulted from {1{2^,,}3}), the “X ,, n-1 Y” is {1,,1,,…1,,1,,1{2^,,}2}, then lv(“X ,, n-1 Y”) = lv({1{2^,,}2}), so for an outside layer lower-leveled than {1{2^,,}2} (e.g. the {1 ____ 2} layer), we can turn into step 7 (adding rule), then we leave the “recursion level platform”.

“Recursion level platform” also happens on higher {1{2^,,}1{2^,,}2}, {1{3^,,}2}, {1{1,2^,,}2}, {1{1`2^,,}2}, {1{1{1{1,2^,,}2}2^,,}2}, {1{1{1{1,,2^,,}2}2^,,}2} and even {1{1{1,,2{1,,2^,,}2}2^,,}2}. But {1{2,,2^,,}2} reduces to {1{1,,2^,,}…1{1,,2^,,}1{1,,2^,,}2}, and D({1{1,,2^,,}…1{1,,2^,,}1{1,,2^,,}2}) = {1{1,,2^,,}…1{1,,2^,,}1{1,,1^,,}2} = (rule 2) {1{1,,2^,,}…1{1,,2^,,}1,,2}, so there’s no “recursion level platform” here.

Now let ordinal d = C(Ω₂,C(Ω₂2,0)).

{1,,1{1{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),0)
{1,,1{1,,1{1{2^,,}3}2}1{1,,2{1,,1,,2}2}2{1{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂,C(Ω₂+C(Ω₂2,0),0)),0),0)
{1,,1{1,,1{1{2^,,}3}2}1{1,,1{1{2^,,}3}2}2{1{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0)),0),0),0)
{1,,1{1{2^,,}3}1,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0))+1,0),0),0)
{1,,1{2{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+ω^(C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0))+1),0),0),0)
{1{1,,1,,2}2{2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+ω^(C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0))+C(Ω₂2,0)),0),0),0)
{1,,1{1,,1{1,,1,,1{1{2^,,}3}2}2,,2}1{1,,2,,2}2{1,,1,,1{1{2^,,}3}2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂,C(Ω₂+d,0)),0),0)
{1,,1{1,,1{1,,1,,1{1{2^,,}3}2}2,,2}1{1,,1{1,,1,,1{1{2^,,}3}2}2,,2}2{1,,1,,1{1{2^,,}3}2}2,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0)),C(Ω₂+d,0)),0),0)
{1,,1{1,,1,,1{1{2^,,}3}2}1,2,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0))+1,C(Ω₂+d,0)),0),0)
{1{1,,1,,3}2,,1,,1{1{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+ω^(C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0))+C(Ω₂2,0)),C(Ω₂+d,0)),0),0)
{1{1,,1,,1,2}2,,1,,1{1{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+d+1,0),0),0)
{1{1,,1,,1{1{2^,,}2}2}2,,1,,1{1{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+d+C(Ω₂2,0),0),0),0)
{1{1,,1,,1{1{2^,,}3}2}2,,1,,1{1{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+d+C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0)),0),0),0)
{1,,1{1,,1,,2{1{2^,,}3}2}1,2,,1{1{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+C(Ω₂2,0)+1,C(Ω₂+d+C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0)),0)),0),0)
{1,,1{1,,2,,2{1{2^,,}3}2}1{1,,1,,2{1{2^,,}3}2}2,,2{1{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+d,C(Ω₂+d+C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0)),0)),0),0)
{1,,1,,1,2{1{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+d+1,C(Ω₂+d+C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0)),0)),0),0)
{1,,1,,1{1{2^,,}3}1,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+d+C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0))+1,0),0),0)
{1,,1,,1{2{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+d+ω^(C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0))+1),0),0),0)
{1{1,,1,,1,,2}2{2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+d+ω^(C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0))+C(Ω₂2,0)),0),0),0)
{1{1,,1,,1,,1,,2}2{2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)+C(Ω₂+d2+ω^(C(C(Ω₂2+C(Ω₂+ω^(d+1),0),0),C(Ω₂2,0))+C(Ω₂2,0)),0),0),0)
{1{1{2^,,}2}2{2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)2,0),0)
{1,,1{1{1{2^,,}3}2{2^,,}3}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)2,0),0)
{1{1{2^,,}3}2{2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),0)2,0),0)
{1{1{2^,,}3}1,2{2^,,}3} has recursion level C(C(Ω₂2+ω^(C(Ω₂+ω^(d+1),0)+1),0),0)
{1,,2{2^,,}3} has recursion level C(C(Ω₂2+C(C(Ω₂,C(Ω₂+ω^(d+1),0)),C(Ω₂+ω^(d+1),0)),0),0)
{1,,1,,2{2^,,}3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+ω^(d+1),0)),0),0)
{1,,1{1,,1,,2{2^,,}3}1,2,,1{2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+C(Ω₂2,0)+1,C(Ω₂+ω^(d+1),0)),0),0)
{1,,1{1,,2,,2{2^,,}3}1{1,,1,,2{2^,,}3}2,,2{2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+d,C(Ω₂+ω^(d+1),0)),0),0)
{1,,1,,1,2{2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+d+1,C(Ω₂+ω^(d+1),0)),0),0)
{1,,1,,1,,2{2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂2,0),C(Ω₂+ω^(d+1),0)),0),0)
{1,,1,,1,,1,,2{2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+d2+C(Ω₂2,0),C(Ω₂+ω^(d+1),0)),0),0)
{1{2^,,}5} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),C(Ω₂+ω^(d+1),0)),0),0)
{1{2^,,}7} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),C(Ω₂+ω^(d+1),C(Ω₂+ω^(d+1),0))),0),0)
{1{2^,,}1,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+1,0),0),0)
{1{2^,,}1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+C(Ω₂2,0),0),0),0)
{1{2^,,}1{1{2^,,}1,,2}1,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+C(Ω₂2,0)+1,0),0),0)
{1,,1{1{2^,,}1,,2}2{2^,,}1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+C(C(Ω₂+C(Ω₂2,0),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2{2^,,}1,,2}2{2^,,}1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+C(C(Ω₂+d,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2{2^,,}1,,2}1,2{2^,,}1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+C(C(Ω₂+d+1,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{1,,2{2^,,}1,,2}1{1,,2{2^,,}1,,2}2{2^,,}1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+C(C(Ω₂+d2,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{2,,2{2^,,}1,,2}2{2^,,}1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+C(C(Ω₂+ω^(d+1),C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{2,,2{2^,,}1,,2}1,2{2^,,}1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+C(C(Ω₂+ω^(d+1)+1,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{2,,2{2^,,}1,,2}1{1,,2{2^,,}1,,2}2{2^,,}1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0)),0),0),0)
{1,,1{2,,2{2^,,}1,,2}1{1,,2{2^,,}1,,2}1,2{2^,,}1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+C(C(Ω₂2+1,0),C(Ω₂2,0)),0),0),0)
{1,,1{2,,2{2^,,}1,,2}1{1,,2{2^,,}1,,2}1{1{2^,,}1,,2}2{2^,,}1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+d,0),0),0)
{1{2^,,}1,,1,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+d+1,0),0),0)
{1{2^,,}1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+d+C(Ω₂2,0),0),0),0)
{1{2^,,}1,,1,,1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)+d2+C(Ω₂2,0),0),0),0)
{1{2^,,}1{2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1)2,0),0),0)
{1{3^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+2),0),0),0)
{1{1{1{2^,,}2}2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂+ω^(d+1),0)),0),0),0)
{1{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂2,0)),0),0),0)

Now let separator ■ = {1{1,,2^,,}2} and let ordinal d = C(Ω₂,C(Ω₂2,0)), and ordinal a = C(Ω₂+ω^(d+C(Ω₂2,0)),0) = C(Ω₂+ω^(C(Ω₂,C(Ω₂2,0))+C(Ω₂2,0)),0).

{1{1■2^,,}3} has recursion level C(C(Ω₂2+a,0),0)
{1,,1{1,,1{1{1■2^,,}3}2}1{1,,1{1{1■2^,,}3}2}2{1{1■2^,,}3}2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0)),0),0),0)
{1,,1{1{1■2^,,}3}1,2} has recursion level C(C(Ω₂2+a+C(Ω₂+C(C(Ω₂2+a,0),C(Ω₂2,0))+1,0),0),0)
{1{1,,1,,2}2{1■2^,,}3} has recursion level C(C(Ω₂2+a+C(Ω₂+ω^(C(C(Ω₂2+a,0),C(Ω₂2,0))+C(Ω₂2,0)),0),0),0)
{1{1,,1,,1,,2}2{1■2^,,}3} has recursion level C(C(Ω₂2+a+C(Ω₂+d+ω^(C(C(Ω₂2+a,0),C(Ω₂2,0))+C(Ω₂2,0)),0),0),0)
{1{1,,1,,1,,1,,2}2{1■2^,,}3} has recursion level C(C(Ω₂2+a+C(Ω₂+d2+ω^(C(C(Ω₂2+a,0),C(Ω₂2,0))+C(Ω₂2,0)),0),0),0)
{1{1{2^,,}2}2{1■2^,,}3} has recursion level C(C(Ω₂2+a+C(Ω₂+ω^(d+1),0),0),0)
{1{1{1■2^,,}3}2{1■2^,,}3} has recursion level C(C(Ω₂2+a2,0),0)
{1,,2{1■2^,,}3} has recursion level C(C(Ω₂2+C(C(Ω₂,a),a),0),0)
{1,,1,,2{1■2^,,}3} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),a),0),0)
{1,,1,,1,,2{1■2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂2,0),a),0),0)
{1,,1,,1,,1,,2{1■2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+d2+C(Ω₂2,0),a),0),0)
{1{2^,,}2{1■2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),a),0),0)
{1{1{1{1■2^,,}3}2^,,}2{1■2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+a),a),0),0)
{1{1{1{1■2^,,}3}2^,,}1{2^,,}2{1■2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+a)+ω^(d+1),a),0),0)
{1{2{1{1■2^,,}3}2^,,}2{1■2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+a+1),a),0),0)
{1{1{1{1,,2{1■2^,,}3}2{1■2^,,}3}2^,,}2{1■2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(C(Ω₂,a),a)),a),0),0)
{1{1{1{1{1{1{1■2^,,}3}2^,,}2{1■2^,,}3}2{1■2^,,}3}2^,,}2{1■2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(C(Ω₂2+C(Ω₂+ω^(d+a),a),0),a)),a),0),0)
{1{1{1,,2{1■2^,,}3}2^,,}2{1■2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂,a)),a),0),0)
{1{1{1{2^,,}2{1■2^,,}3}2^,,}2{1■2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂+ω^(d+1),a)),a),0),0)
{1{1{1{1{1{1■2^,,}3}2^,,}2{1■2^,,}3}2^,,}2{1■2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂+ω^(d+a),a)),a),0),0)
{1{1■2^,,}5} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂2,0)),a),0),0)
{1{1■2^,,}1,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂2,0))+1,0),0),0)
{1{1■2^,,}1,,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂2,0))+C(Ω₂2,0),0),0),0)
{1{1■2^,,}1{2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂2,0))+ω^(d+1),0),0),0)
{1{2■2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂2,0)+1),0),0),0)
{1{1■3^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂2,0)2),0),0),0)
{1{1■1,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+ω^(C(Ω₂2,0)+1)),0),0),0)
{1{1{2{1,,2^,,}2}2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+ω^ω^(C(Ω₂2,0)+1)),0),0),0)
{1,,2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(d,C(Ω₂2,0))),0),0),0)
{1,,1{1,,2{1,,2^,,}2}2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(C(Ω₂+d,C(Ω₂2,0)),C(Ω₂2,0))),0),0),0)
{1,,1{2,,2{1,,2^,,}2}2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(C(Ω₂+ω^(d+1),C(Ω₂2,0)),C(Ω₂2,0))),0),0),0)
{1,,1{1■2,,2{1,,2^,,}2}2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(C(Ω₂+ω^(d+C(Ω₂2,0)),C(Ω₂2,0)),C(Ω₂2,0))),0),0),0)
{1,,1{1{1,,2{1,,2^,,}2}2,,2{1,,2^,,}2}2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(C(Ω₂2,C(Ω₂2,0)),C(Ω₂2,0))),0),0),0)
{1,,1{1{1,,2{1,,2^,,}2}2,,2{1,,2^,,}2}1,2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(C(Ω₂2+1,0),C(Ω₂2,0))),0),0),0)
{1,,1{1{1,,2{1,,2^,,}2}2,,2{1,,2^,,}2}1{1{1■2^,,}2}2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(C(Ω₂2+a,0),C(Ω₂2,0))),0),0),0)
{1,,1{1{1,,2{1,,2^,,}2}2,,2{1,,2^,,}2}1■2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d2),0),0),0)
{1,,1{1{1,,2{1,,2^,,}2}2,,2{1,,2^,,}2}1■1■2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d2),0)2,0),0)
{1,,1,,2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+ω^(d2),0)),0),0)
{1,,1,,1,,2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+d+C(Ω₂2,0),C(Ω₂+ω^(d2),0)),0),0)
{1,,1,,1,,1,,2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+d2+C(Ω₂2,0),C(Ω₂+ω^(d2),0)),0),0)
{1{2^,,}2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+1),C(Ω₂+ω^(d2),0)),0),0)
{1{1■2^,,}2{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂+ω^(d2),0)),C(Ω₂+ω^(d2),0)),0),0)
{1{1,,2^,,}3} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂2,0)),C(Ω₂+ω^(d2),0)),0),0)
{1{1,,2^,,}4} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d+C(Ω₂2,0)),C(Ω₂+ω^(d2),C(Ω₂+ω^(d2),0))),0),0)
{1{1,,2^,,}1,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d2)+1,0),0),0)

Now let ordinal d = C(Ω₂,C(Ω₂2,0)).

{1{1,,2^,,}1{2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d2)+ω^(d+1),0),0),0)
{1{1,,2^,,}1{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d2)+ω^(d+C(Ω₂2,0)),0),0),0)
{1{1,,2^,,}1{1,,2^,,}1{1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d2)2+ω^(d+C(Ω₂2,0)),0),0),0)
{1{2,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d2+1),0),0),0)
{1{1,,3^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d2+C(Ω₂2,0)),0),0),0)
{1,,1,,2{1,,3^,,}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+ω^(d3),0)),0),0)
{1{1,,3^,,}1,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d3)+1,0),0),0)
{1{2,,3^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d3+1),0),0),0)
{1{1,,4^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(d3+C(Ω₂2,0)),0),0),0)
{1{1,,1,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^ω^(d+1),0),0),0)
{1{1,,1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^ω^(d+C(Ω₂2,0)),0),0),0)
{1,,1,,2{1,,1,,2^,,}2} has recursion level C(C(Ω₂2+C(C(Ω₂2,0),C(Ω₂+ω^ω^(d2),0)),0),0)
{1{1,,1,,2^,,}1,2} has recursion level C(C(Ω₂2+C(Ω₂+ω^ω^(d2)+1,0),0),0)
{1{2,,1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(ω^(d2)+1),0),0),0)
{1{1,,1,,3^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^(ω^(d2)+ω^(d+C(Ω₂2,0))),0),0),0)
{1{1,,1,,1,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^ω^(d2+1),0),0),0)
{1{1,,1,,1,,2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^ω^(d2+C(Ω₂2,0)),0),0),0)
{1{1{2^,,}2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^ω^ω^(d+1),0),0),0)
{1{1{1,,2^,,}2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^ω^ω^(d+C(Ω₂2,0)),0),0),0)
{1{1{1,,1,,2^,,}2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^ω^ω^ω^(d+C(Ω₂2,0)),0),0),0)
{1{1{1{1,,2^,,}2^,,}2^,,}2} has recursion level C(C(Ω₂2+C(Ω₂+ω^ω^ω^ω^ω^(d+C(Ω₂2,0)),0),0),0)

Let f(n) = s(n,n{1{1{1…{1,2^,,}…2^,,}2^,,}2}2) where there’re n separators with a double comma. So f(n) is the limit of pDAN. It eventually outgrows all above, and has growth rate C(C(Ω₂2+C(Ω₂+C(C(Ω₂,C(Ω₂,C(Ω₂2,0))),C(Ω₂,C(Ω₂2,0))),0),0),0) = C(C(Ω₂2+C(Ω₂+ε_{C(Ω₂,C(Ω₂2,0))+1},0),0),0).

32 thoughts on “pDAN vs. Taranovsky’s ordinal notation”

Aarex Tiaokhiao says:

What is the limit of pDAN

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2016-08-27 at 19:54
- hypcos says:
  
  The notational limit of pDAN is f(n) = s(n,n{1{1{1…{1{1,,2,,}2,,}…2,,}2,,}2}2) with n pairs of {}’s.
  
  And the ordinal limit of pDAN… still not done, but I guess it’s C(C(Ω2+C(Ω+C(C(Ω,C(Ω,C(Ω2,0))),C(Ω,C(Ω2,0))),0),0),0) = C(C(Ω2+C(Ω+ε(C(Ω,C(Ω2,0))+1),0),0),0), using the n=2 system.
  
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  2016-08-28 at 08:48
  - Aarex Tiaokhiao says:
    
    My limit ordinal is around psi(psi_chi(e(M+1))(0)).
    
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    2016-08-30 at 23:07
    - Alemagno12 says:
      
      I heard that the limit of second system is compact cardinals.
      
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      2016-09-14 at 10:00
      - Aarex Tiaokhiao says:
        
        It isn’t. SOA grows rate C(W_2*w).
        
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        2016-09-14 at 19:45
  - Aarex Tiaokhiao says:
    
    I guess you are right about your guess. Congrats, now do sDAN vs TON.
    
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    2017-02-17 at 00:11
    - Username5243 says:
      
      No make numbers from EAN.
      
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      2017-02-17 at 05:09
Aarex Tiaokhiao says:

What level corresponds into {a,,b,,c,,d,,..}?

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2017-01-18 at 22:33
Username5243 says:

Can’t you use a more standard ordinal notation. I don’t understand Taranovsky’s notation, and I’m not even sure all googologists that understand ordinals this high understand it. I certainly don’t.

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2017-02-05 at 02:52
- Aarex Tiaokhiao says:
  
  Hyp cos can remember standard ordinal notations, but he is not using this anymore.
  
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  2017-02-05 at 22:19
- hypcos says:
  
  “Standard ordinal notation” can’t reach the limit of pDAN, but Taranovsky’s ordinal notation can.
  
  By the way, I only introduced Cantor’s normal form, binary φ function, binary θ function and Taranovsky’s notation in this website. If I show comparisons to higher “standard ordinal notations”, I must introduce them, but they’re too complex, such as The Pi-4 reflecting one, the Pi-n reflecting one, or even higher. Compared with them, Taranovsky states that:
  
  C(Ω+C(Ω,C(Ω2,0)),0) corresponds to the recursively inaccessible,
  C(Ω+C(Ω,C(Ω2,0))^2,0) corresponds to the recursively Mahlo,
  C(Ω+C(Ω,C(Ω2,0))^C(Ω,C(Ω2,0)),0) corresponds to the Pi-3 reflecting,
  C(Ω+C(Ω,C(Ω2,0))^C(Ω,C(Ω2,0))^(n-2),0) corresponds to the Pi-n reflecting,
  (using the 2nd system)
  
  So it’s a better choice to use Taranovsky’s ordinal notation.
  
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  2017-02-12 at 21:02
  - Aarex Tiaokhiao says:
    
    Correct. But how TON beats sDAN?
    
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    2017-02-13 at 09:48
  - SuperSpruce says:
    
    So it looks like to me that C(Ω,C(Ω2,0)) is like the stage function diagonalizer T.
    
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    2019-06-06 at 03:57
Boboris02 says:

Hypcos,can you tell me what ordinal (using standart ordinal notation) is equal to C(C(W_2*3,0),0) in Taranovsky’s notation?

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2017-03-26 at 21:26
- hypcos says:
  
  I haven’t analyse so far to C(C(W_2*3,0),0).
  Also, C(C(W_2*2+C(W_2+C(W_2,C(W_2,C(W_2*2,0))),0),0),0) already surpasses the range of “standard ordinal notations”.
  
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  2017-03-26 at 21:36
  - Boboris02 says:
    
    And how much is C(W_2*2+W_1,0)?
    
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    2017-03-26 at 22:22
    - hypcos says:
      
      C(W_2*2+W_1,0) is even larger than W_1, which is nearly the end of Taranovsky’s ordinal notation.
      
      C(C(W_2*2+W_1,0),0) = ψ(Ω_(ψ_I(0)*Ω)) in standard notation, and is the recursion level of {1,,1{1,,1{1,,1,,2}2}1{1,,2}2{1,,1,,2}2}.
      
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      2017-03-26 at 23:04
      - Boboris02 says:
        
        By C(W_2*2+W_1) I meant C(C(W_2*2+W_1),0).
        And what ordinal corresponds to {1,,1,,3} and what does the seperator drop down to?
        
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        2017-03-26 at 23:57
        
        hypcos says:
        
        Taranovsky’s notation: see the comparisons above. Standard ordinal notation: {1,,1,,3} has recursion level ψ(ψ_{I_2}(0)). {1,,1,,3} works as the 1-separator of {1,, ____ ,,2}.
        
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        2017-03-27 at 08:02
        
        Aarex Tiaokhiao says:
        
        I don’t understand how C(W_2*2,0) works.
        
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        2017-03-28 at 00:27
Username5243 says:

If I get it right then:

W = C(W_2,0) corresponds to W in standard notation
C(W_2,W) = C(W_2,C(W_2,0)) corresponds to W_2 in standard notation
C(W_2,C(W_2,C(W_2,0))) corresponds to W_3 in standard notation
C(W_2+1,0) corresponds to W_w
C(W_2+2,0) corresponds to W_(w^2)
C(W_2+W,0) corresponds to W_W
And generally W_(w^a) correspodns to C(W_2+a,0).
Then:

C(W_2*2,0) corresponds to psi_I(0)
C(W_2,C(W_2*2,0)) corresponds to W_(psi_I(0)+1)
C(W_2*2,C(W_2*2,0)) corresponds to W_(psi_I(0)*2)
C(W_2*2+1,0) corresponds to W_(psi_I(0)*w)
C(W_2*2+W,0) corresponds to W_(psi_I(0)*W)
C(W_2*2+C(W_2+C(W_2*2,0),0),0) corresponds to W_(psi_I(0)^2)

But, beyond this point, I do not understand how it works fully.

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2017-03-28 at 04:22
- Aarex Tiaokhiao says:
  
  Hmmm….
  
  C(W_2+C(W_2+C(W_2*2,0),0),0) ~ psi_I(0)
  C(W,psi_I(0)) ~ psi_I(0)+W
  C(C(C(W_2,W),W),psi_I(0)) ~ psi_I(0)+e(W+1)
  C(C(W_2,W),psi_I(0)) ~ psi_I(0)+W_2
  C(C(W_2+1,0),psi_I(0)) ~ psi_I(0)+W_w
  C(C(W_2+C(W_2+1,0),0),psi_I(0)) ~ psi_I(0)+W_W_w
  C(psi_I(0),psi_I(0)) ~ psi_I(0)2
  C(W_2,psi_I(0)) ~ W_(psi_I(0)+1)
  
  You are right.
  
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  2017-03-28 at 05:33
- hypcos says:
  
  No.
  
  C(W_2*2,0) is much stronger. You can see it as the 1-separator inside C(W_2+____,0). By Taranovsky’s words, it should be the least ordinal x such that L_x is a Σ₁ elementary substructure of L.
  
  Then C(W_2,C(W_2*2,0)) works as the 2-separator inside C(W_2+____,0), also corresponding to the double comma in pDAN.
  
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  2017-03-28 at 08:27
- Luke says:
  
  No reach no game
  
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  2023-09-01 at 19:49
Boboris02 says:

Could you tell me:
What ordinal corresponds to C(C(W_2*2+C(W_2*2,0),0),0)?
What ordinal corresponds to C(C(W_2*2+C(W_2*2,1),0),0)?
What ordinal corresponds to C(C(W_2*2+C(W,C(W_2*2,0)),0),0)?
What ordinal corresponds to a {1,,1,,1,2} seperator?

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2017-04-01 at 18:25
- hypcos says:
  
  I haven’t analysed so far to C(C(W_2*2+C(W_2*2,0),0),0).
  Also, C(W_2*2,1) is not a valid term.
  
  {1,,1,,1,2} has recursion level ψ(I_ω) in normal ordinal notations. For {1,,1,,1,2} itself, it corresponds to I_ω.
  
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  2017-04-01 at 19:50
Boboris02 says:

So C(W_2*2+C(W_2+C(W_2*2,0),0),0) is equal to psi_I(I). Is that correct?

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2017-12-14 at 01:34
- hypcos says:
  
  No. C(W_2*2+C(W_2+C(W_2*2,0),0),0) corresponds to Ω_{ψ_I(0)^2}, where ψ_I(0) is the omega fixed point.
  
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  2017-12-15 at 19:36
  - Boboris02 says:
    
    So what ordinal in TON corresponds to psi_I(I)? Is it C(W_2*2+C(W_2+C(W_2*2,0)*2,0),0)?
    
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    2017-12-16 at 06:17
Alemagno12 says:

Your analysis is wrong. C(C(W_2*2+C(C(W_2*2,0),C(W_2+C(W_2*2,0),0)),0),0) < C(C(W_2*2+C(W_2+C(W_2*2,0),C(W_2+C(W_2*2,0),0)),0),0).

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2017-12-25 at 12:23
StevenAlex wc-R1 says:

Also please support my array notation it has nested arrays and recursion

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2021-02-14 at 17:48
- Sebastian G says:
  
  Where is it?
  
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  2023-12-13 at 05:04