User blog:Hyp cos/TON, stable ordinals, and my array notation

Many people don't understand Taranovsky's ordinal notation (TON). Some of them understand normal ordinal collapsing functions (OCFs), and want to know comparisons between other notations and OCFs. However, OCFs beyond \(\Pi_3\)-reflection suddenly become complicated, such as the ones for \(\Pi_4\)-reflection, 1-stable, \(\beta\)-stable for any constant \(\beta\), and further, yet TON remains simple. Here I show some details of TON with my understanding.

Definition of TON
To understand the definition of this kind of notation is not easy, although it might look simpler than normal OCF. We need 5 steps to understand it.

Terms
Terms are the things the system work on. In the n-th system of TON, terms are constructed using these following 2 rules.
 * 1) \(0\) and \(\Omega_n\) are terms
 * 2) If \(a\) and \(b\) are terms, then \(C(a,b)\) is a term

Comparisons
Terms can be compared and connected with ">", "<" or "=".

To compare terms \(a\) and \(b\), first write them in postfix form, i.e. delete all the "(", ")" and "," and then reverse the string. In postfix form, "terms" are constructed only with 3 symbols - "0", "\(\Omega_n\)" and "C".

Then the ordering of "terms" are done in lexicographical order. i.e.
 * 1) \(a=b\) if \(a\) is the same string as \(b\)
 * 2) Empty string is less than any other string
 * 3) If both \(a\) and \(b\) are not empty, then let \(a_1\) (\(b_1\) respectively) be the first symbol of \(a\) (\(b\) respectively), and \(a^-\) (\(b^-\) respectively) be the string without the first symbol.
 * 4) If \(a_1< b_1\) (\(a_1>b_1\) respectively), then \(a< b\) (\(a>b\) respectively)
 * 5) If \(a_1=b_1\), then the order of \(a\) and \(b\) is the same as the order of \(a^-\) and \(b^-\)

A binary relation
Now it's the most important part of TON - the binary relation called "is m-built from below from", denoted \(B_m\) here, where m is a non-negative integer. For terms \(a\) and \(b\), To avoid "is a subterm of", this binary relation can also be described using a ternary relation \(T_m\). For terms \(a,\ b,\ c\), \(T_m(a,b,c)\) if at least one of the 5 is true. Then \(B_m(a,b)\) iff \(T_m(a,b,0)\).
 * \(B_0(a,b)\) if \(a< b\).
 * \(B_{m+1}(a,b)\) if for every proper subterm \(c\) of \(a\), "\(c<a\)" or "\(c\) is a subterm of \(d\), where \(d\) is a subterm of \(a\) and \(B_m(d,b)\)".
 * 1) \(a< b\)
 * 2) \(a=\Omega_n\) and \(m\ge1\)
 * 3) \(a=\Omega_n<c\)
 * 4) \(a=C(d,e)\), where \(m\ge1\), \(T_{m-1}(d,b,a)\) and \(T_{m-1}(e,b,a)\)
 * 5) \(a=C(d,e)<c\), where \(T_m(d,b,c)\) and \(T_m(e,b,c)\)

Standard terms
Some of the terms are standard, but some are not. In the n-th system of TON,
 * \(0\) and \(\Omega_n\) are standard.
 * \(C(a,b)\) is standard if all the following 3 are true.
 * Both \(a\) and \(b\) are standard
 * If \(b=C(c,d)\), then \(a\le c\)
 * \(B_n(a,C(a,b))\)

Ordinals
In TON, one standard term means one ordinal, and different standard terms mean different ordinals. The ordering of ordinals is defined to be exactly the ordering of standard terms.

So the least standard term, 0, corresponds to the least ordinal, 0. The standard term \(a\) larger than \(b_1,\ b_2,\cdots\) corresponds to an ordinal larger than what \(b_1,\ b_2,\cdots\) correspond to.

But there is a problem: is the ordering of standard terms well-founded? It's unknown yet, and Taranovsky is working on it.

Comparisons of TON
TON has some basic properties: And, using \(\Omega_n=C(\Omega_{n+1},0)\) we can combine all the n-th system into one notation.
 * \(C(a,b)>b\)
 * \(C(a,b)=b+\omega^a\) iff \(C(a,b)\ge a\)
 * \(C(a,b)\) is monotonic in both \(a\) and \(b\), and continuous in \(a\).

Comparing terms, checking \(T_n\), \(B_n\) and standard are computable, so all ordinals from TON are computable. But why there are some corresponence above \(\omega^\text{CK}_1\)? Because Taranovsky set gaps below some ordinals, e.g. \(\Omega_1=\omega^{CK}_1\) instead of BHO, which is the case in 1st system if there is no "gap".

To determine "how large the gaps is" is hard. For set theoretical propose, the gaps should fit large ordinal axioms in KP set theory; for googological propose, we want a diagonalizer large enough for collapsing in any possible further extensions.

Up to \(\Pi_3\)-reflection
The least example is \(\Omega_1\). It's larger than not only \(C(\Omega_1^{\Omega_1^{\Omega_1^\cdots}},0)\), but also \(C(C(\Omega_2^{\Omega_2^{\Omega_2^\cdots}},0),0)\) in 2nd system, \(C(C(C(\Omega_3^{\Omega_3^{\Omega_3^\cdots}},0),0),0)\) in 3rd system, and more systems and possible extensions beyond the limit of \(\underbrace{C(C(\cdots C(}_n\Omega_n2,0)\cdots,0),0)\). The notation may extend up to (but not including) \(\omega^\text{CK}_1\), so we can set \(\Omega_1=\omega^\text{CK}_1\).

From \(\Omega_1=\omega^\text{CK}_1\) we also have \(C(0,\Omega_1)=\omega^\text{CK}_1+1\), \(C(\Omega_1,\Omega_1)=\omega^\text{CK}_12\), \(C(C(\Omega_2,C(\Omega_2,0)),\Omega_1)=\varepsilon_{\omega^\text{CK}_1+1}\), \(C(C(\Omega_2+1,0),\Omega_1)=\theta(\Omega_\omega,\omega^\text{CK}_1+1)\), and so on. These computable application over \(\omega^\text{CK}_1\) would be very strong, but still computable. To make \(C(\Omega_2,C(\Omega_2,0))\) large enough for collapsing, we set \(C(\Omega_2,C(\Omega_2,0))=\omega^\text{CK}_2\) - the 2nd admissible ordinal.

And so on. Making \(C(\Omega_2,\beta)\) from \(\beta\) results a next admissible. Thus \(C(\Omega_2+1,0)=\omega^\text{CK}_\omega\), and \(C(\Omega_2+C(\Omega_22,0),0)\) is the first fixed point of \(\alpha\mapsto\omega^\text{CK}_\alpha\). Higher level fixed points can be \(C(\Omega_2+C(\Omega_22,0)+1,0)\), \(C(\Omega_2+C(\Omega_22,0)2,0)\), \(C(\Omega_2+C(\Omega_22,0)^{C(\Omega_22,0)},0)\), \(C(\Omega_2+C(C(\Omega_2,C(\Omega_22,0)),C(\Omega_22,0)),0)=C(\Omega_2+\varepsilon_{C(\Omega_22,0)+1},0)\), \(C(\Omega_2+C(C(\Omega_22+1,0),C(\Omega_22,0)),0)\), \(C(\Omega_2+C(C(\Omega_2^{\Omega_2^\cdots},0),C(\Omega_22,0)),0)\), \(C(\Omega_2+C(C(C(\Omega_3^{\Omega_3^\cdots},0),0),C(\Omega_22,0)),0)\), etc.. All these terms have a supremum \(C(\Omega_2+C(\Omega_2,C(\Omega_22,0)),0)\). It should corresponds to an ordinal large enough to express all these levels of admissibles, limits or fixed points. Set theories provide a candidate for this - the first recursively inaccessible ordinal, which is admissible and limit of admissibles. Then the gap below \(C(\Omega_2+C(\Omega_2,C(\Omega_22,0)),0)\) makes \(C(\Omega_2+C(\Omega_2,C(\Omega_22,0)),0)\) the first recursively inaccessible.

Let \(d=C(\Omega_2,C(\Omega_22,0))\), then
 * \(C(\Omega_2+d,0)\) is first recursively inaccessible
 * \(C(\Omega_2,C(\Omega_2+d,0))\) is first admissible after first recursively inaccessible
 * \(C(\Omega_2+d,C(\Omega_2+d,0))\) is 2nd recursively inaccessible
 * \(C(\Omega_2+d+1,0)\) is first limit of recursively inaccessibles
 * \(C(\Omega_2+d,C(\Omega_2+d+1,0))\) is \(\omega\)-th recursively inaccessible
 * \(C(\Omega_2+d+C(\Omega_22,0),0)\) is first recursively inaccessible-fixed-point
 * \(C(\Omega_2+d2,0)\) is first recursively 2-inaccessible (a.k.a. level-2 recursively inaccessible)
 * \(C(\Omega_2+d\omega,0)\) is limit of recursively n-inaccessible
 * \(C(\Omega_2+d,C(\Omega_2+d\omega,0))\) is recursively inaccessible limit of recursively n-inaccessible
 * \(C(\Omega_2+d\omega+d,0)\) is first recursively \(\omega\)-inaccessible
 * \(C(\Omega_2+d C(\Omega_22,0),0)\) is the fixed point of \(\alpha\mapsto\)recursively \(\alpha\)-inaccessible
 * \(C(\Omega_2+d C(\Omega_22,0)+d,0)\) is first recursively (1,0)-inaccessible (a.k.a. level-(1,0) recursively inaccessible)
 * \(C(\Omega_2+d^2,0)\) is first recursively Mahlo
 * \(C(\Omega_2+d^2,C(\Omega_2+d^2,0))\) is 2nd recursively Mahlo
 * \(C(\Omega_2+d^2+1,0)\) is first limit of recursively Mahlos
 * \(C(\Omega_2+d^2,C(\Omega_2+d^2+1,0))\) is \(\omega\)-th recursively Mahlo
 * \(C(\Omega_2+d^2+d,0)\) is first recursively inaccessible limit of recursively Mahlos
 * \(C(\Omega_2+d^22,0)\) is first recursively Mahlo limit of recursively Mahlos
 * \(C(\Omega_2+d^3,0)\) is first recursively 2-Mahlo (a.k.a. level-2 recursively Mahlo)
 * \(C(\Omega_2+d^\omega,0)\) is limit of recursively n-Mahlo
 * \(C(\Omega_2+d^\omega+d^2,0)\) is recursively Mahlo limit of recursively n-Mahlo
 * \(C(\Omega_2+d^{\omega+1},0)\) is first recursively \(\omega\)-Mahlo (limit points of recursively n-Mahlo are stationary in this ordinal)
 * \(C(\Omega_2+d^{C(\Omega_22,0)+1},0)\) is first recursively (1,0)-Mahlo
 * \(C(\Omega_2+d^d,0)\) is first \(\Pi_3\)-reflecting
 * \(C(\Omega_2+d^d,C(\Omega_2+d^d,0))\) is 2nd \(\Pi_3\)-reflecting
 * \(C(\Omega_2+d^d+d,0)\) is first recursively inaccessible limit of \(\Pi_3\)-reflectings
 * \(C(\Omega_2+d^d+d^2,0)\) is first recursively Mahlo limit of \(\Pi_3\)-reflectings
 * \(C(\Omega_2+d^d2,0)\) is first \(\Pi_3\)-reflecting limit of \(\Pi_3\)-reflectings

Up to \(a^{++}\)-stable
However, Taranovsky's corresponding beyond the \(\Pi_3\)-reflecting ordinal might have problems. OCF for \(\Pi_4\)-reflection is more complicated than we originally think, even more complicated than the OCF for \(\Pi_n\)-reflection in Taranovsky's corresponding. I have not compared it but conjecture that the \(\Pi_4\)-reflecting ordinal should be \(C(\Omega_2+d^{d^d},0)\).

Taranovsky's corresponding
Let \(d=C(\Omega_2,C(\Omega_22,0))\), then \(\alpha\) is \(\beta\)-stable if \(L_\alpha\prec_{\Sigma_1}L_{\alpha+\beta}\). This property can even exceed "\(\alpha\) is \(\alpha\)-stable", such as "\(\alpha\) is \(\varepsilon_{\alpha+1}\)-stable", "\(\alpha\) is \(\alpha^+\)-stable" (\(\alpha^+\) is the next admissible ordinal after \(\alpha\)), "\(\alpha\) is \(\beta\)-stable where \(\beta\) is next \(\Pi_3\)-reflecting after \(\alpha\)", "doubly 1-stable" (i.e. \(L_\alpha\prec_{\Sigma_1}L_\beta\prec_{\Sigma_1}L_{\beta+1}\)), "triply 1-stable" (i.e. \(L_\alpha\prec_{\Sigma_1}L_\beta\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+1}\)), and "\(\omega\)-ply-stable" whose existence with KP yields \(\Pi^1_2\text{-CA}_0\). Now we use the notation of lambda abstraction for stable levels. Ordinal \(\alpha\) is \(f(\alpha)\)-stable can be written as an \(f\)-stable ordinal, where \(f\) is an abstraction or ordinal function. Beyond that, we can have \(C(\Omega_2+d^{d^{C(C(\Omega_2^{\Omega_2^{\Omega_2^\cdots}},0),C(\Omega_22,0))}+1},0)\), \(C(\Omega_2+d^{d^{C(C(C(\Omega_3^{\Omega_3^{\Omega_3^\cdots}},0),0),C(\Omega_22,0))}+1},0)\), and more to come. The "term supremum" of them should correspond to first \(\lambda a.a^+\)-stable. Beyond that, we can have \(C(\Omega_2+C(C(\Omega_2^{\Omega_2^{\Omega_2^\cdots}},0),d),0)\), \(C(\Omega_2+C(C(C(\Omega_3^{\Omega_3^{\Omega_3^\cdots}},0),0),d),0)\), and more to come. The "term supremum" of them, \(C(\Omega_2+C(\Omega_2,C(\Omega_2,C(\Omega_22,0))),0)\), should correspond to first \(\lambda a.a^{++}\)-stable.
 * \(C(\Omega_2+d^{d+1},0)\) is first \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings (i.e. \(\Pi_3\)-reflectings are stationary in this ordinal)
 * \(C(\Omega_2+d^{d+2},0)\) is first \(\Pi_2\)-reflecting onto \(\Pi_2\)-reflectings onto \(\Pi_3\)-reflectings
 * \(C(\Omega_2+d^{d+C(\Omega_22,0)},0)\) is fixed point of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings
 * \(C(\Omega_2+d^{d+C(\Omega_22,0)}2,0)\) is fixed point of "\(\alpha\mapsto\)level-\(\alpha\) \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings, which is limit of fixed points of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_2\)-reflectings onto \(\Pi_3\)-reflectings"
 * \(C(\Omega_2+d^{d+C(\Omega_22,0)+1},0)\) is first level-(1,0) \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings
 * \(C(\Omega_2+d^{d2},0)\) is first \(\Pi_3\)-reflecting onto \(\Pi_3\)-reflectings
 * \(C(\Omega_2+d^{d C(\Omega_22,0)+1},0)\) is first level-(1,0) \(\Pi_3\)-reflecting
 * \(C(\Omega_2+d^{d^2},0)\) is first \(\Pi_4\)-reflecting
 * \(C(\Omega_2+d^{d^2}2,0)\) is first \(\Pi_4\)-reflecting limit of \(\Pi_4\)-reflectings
 * \(C(\Omega_2+d^{d^2+1},0)\) is first \(\Pi_2\)-reflecting onto \(\Pi_4\)-reflectings
 * \(C(\Omega_2+d^{d^2+d},0)\) is first \(\Pi_3\)-reflecting onto \(\Pi_4\)-reflectings
 * \(C(\Omega_2+d^{d^22},0)\) is first \(\Pi_4\)-reflecting onto \(\Pi_4\)-reflectings
 * \(C(\Omega_2+d^{d^3},0)\) is first \(\Pi_5\)-reflecting
 * \(C(\Omega_2+d^{d^4},0)\) is first \(\Pi_6\)-reflecting
 * \(C(\Omega_2+d^{d^\omega},0)\) is first limit of \(\Pi_n\)-reflecting
 * \(C(\Omega_2+d^{d^\omega}+d^d,0)\) is first \(\Pi_3\)-reflecting limit of \(\Pi_n\)-reflecting
 * \(C(\Omega_2+d^{d^\omega}+d^{d^2},0)\) is first \(\Pi_4\)-reflecting limit of \(\Pi_n\)-reflecting
 * \(C(\Omega_2+d^{d^\omega}2,0)\) is first level-2 limit of \(\Pi_n\)-reflecting
 * \(C(\Omega_2+d^{d^\omega+1},0)\) is first \(\Pi_\omega\)-reflecting, or first 1-stable
 * \(C(\Omega_2+d^{d^\omega+1}2,0)\) is first 1-stable limit of 1-stables
 * \(C(\Omega_2+d^{d^\omega+2},0)\) is first \(\Pi_2\)-reflecting onto 1-stables
 * \(C(\Omega_2+d^{d^\omega+d},0)\) is first \(\Pi_3\)-reflecting onto 1-stables
 * \(C(\Omega_2+d^{d^\omega+d^2},0)\) is first \(\Pi_4\)-reflecting onto 1-stables
 * \(C(\Omega_2+d^{d^\omega2},0)\) is first limit of \(\Pi_n\)-reflecting onto 1-stables
 * \(C(\Omega_2+d^{d^\omega2+1},0)\) is first level-2 1-stable
 * \(C(\Omega_2+d^{d^\omega C(\Omega_22,0)+1},0)\) is first level-(1,0) 1-stable
 * \(C(\Omega_2+d^{d^{\omega+1}},0)\) is first 1-\(\Pi_1\)-reflecting
 * \(C(\Omega_2+d^{d^{\omega+2}},0)\) is first 1-\(\Pi_2\)-reflecting
 * \(C(\Omega_2+d^{d^{\omega2}},0)\) is first limit of 1-\(\Pi_n\)-reflecting
 * \(C(\Omega_2+d^{d^{\omega2}+1},0)\) is first 2-stable
 * \(C(\Omega_2+d^{d^{C(\Omega_2,0)}+1},0)\) is first \(\omega^\text{CK}_1\)-stable
 * \(C(\Omega_2+d^{d^{C(\Omega_2+d^{d^\omega+1},0)}+1},0)\) is first "first 1-stable ordinal"-stable
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)}},0)\) is fixed point of \(\alpha\mapsto\alpha\)-stable
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)}}2,0)\) is "fixed point of \(\alpha\mapsto\alpha\)-stable" limit of "fixed points of \(\alpha\mapsto\alpha\)-stable"
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)}+1},0)\) is the ordinal \(\alpha\) that is \(\alpha\)-stable
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)}+1},0)\) is first \(\lambda a.a\)-stable
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)+\omega}+1},0)\) is first \(\lambda a.a+1\)-stable
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)2}+1},0)\) is first \(\lambda a.a2\)-stable
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)^2}+1},0)\) is first \(\lambda a.a^2\)-stable
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)^{C(\Omega_22,0)}}+1},0)\) is first \(\lambda a.a^a\)-stable
 * \(C(\Omega_2+d^{d^{\varepsilon_{C(\Omega_22,0)+1}}+1},0)\) is first \(\lambda a.\varepsilon_{a+1}\)-stable
 * \(C(\Omega_2+d^{d^d},0)\) is first \(\lambda a.a^+\)-stable
 * \(C(\Omega_2+d^{d^{d+\omega}+1},0)\) is first \(\lambda a.a^++1\)-stable
 * \(C(\Omega_2+d^{d^{d+C(\Omega_22,0)}+1},0)\) is first \(\lambda a.a^++a\)-stable
 * \(C(\Omega_2+d^{d^{d+C(\Omega_22,0)^2}+1},0)\) is first \(\lambda a.a^++a^2\)-stable
 * \(C(\Omega_2+d^{d^{d2}},0)\) is first \(\lambda a.a^+2\)-stable
 * \(C(\Omega_2+d^{d^{d\omega}},0)\) is first limit of \(\lambda a.a^+n\)-stable
 * \(C(\Omega_2+d^{d^{d\omega}+1},0)\) is first \(\lambda a.a^+\omega\)-stable
 * \(C(\Omega_2+d^{d^{d C(\Omega_22,0)}+1},0)\) is first \(\lambda a.a^+a\)-stable
 * \(C(\Omega_2+d^{d^{d^2}},0)\) is first \(\lambda a.(a^+)^2\)-stable
 * \(C(\Omega_2+d^{d^{d^\omega}+1},0)\) is first \(\lambda a.(a^+)^\omega\)-stable
 * \(C(\Omega_2+d^{d^{d^d}},0)\) is first \(\lambda a.(a^+)^{a^+}\)-stable
 * \(C(\Omega_2+\varepsilon_{d+1},0)\) is first limit of \(\lambda a.a^+\uparrow\uparrow n\)-stable
 * \(C(\Omega_2+\varepsilon_{d+1}d,0)\) is first \(\lambda a.\varepsilon_{a^++1}\)-stable
 * \(C(\Omega_2+\Gamma_{d+1}d,0)\) is first \(\lambda a.\Gamma_{a^++1}\)-stable

Stronger than Taranovsky's corresponding
Let \(d=C(\Omega_2,C(\Omega_22,0))\), then Beyond that, we can have \(C(\Omega_2+\varphi(C(C(\Omega_2^{\Omega_2^{\Omega_2^\cdots}},0),C(\Omega_22,0)),d2),0)\), \(C(\Omega_2+\varphi(C(C(C(\Omega_3^{\Omega_3^{\Omega_3^\cdots}},0),0),C(\Omega_22,0)),d2),0)\), and more to come. The "term supremum" of them should be - Beyond that, we can have \(C(\Omega_2+C(C(\Omega_2^{\Omega_2^{\Omega_2^\cdots}},0),d),0)\), \(C(\Omega_2+C(C(C(\Omega_3^{\Omega_3^{\Omega_3^\cdots}},0),0),d),0)\), etc. very similar to the case of Taranovsky's corresponding. So \(C(\Omega_2+C(\Omega_2,C(\Omega_2,C(\Omega_22,0))),0)\) is still first \(\lambda a.a^{++}\)-stable.
 * \(C(\Omega_2+d^{d+1},0)\) is first \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings (i.e. \(\Pi_3\)-reflectings are stationary in this ordinal)
 * \(C(\Omega_2+d^{d+2},0)\) is first \(\Pi_2\)-reflecting onto \(\Pi_2\)-reflectings onto \(\Pi_3\)-reflectings
 * \(C(\Omega_2+d^{d+C(\Omega_22,0)},0)\) is fixed point of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings
 * \(C(\Omega_2+d^{d+C(\Omega_22,0)}2,0)\) is fixed point of "\(\alpha\mapsto\)level-\(\alpha\) \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings, which is limit of fixed points of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_2\)-reflectings onto \(\Pi_3\)-reflectings"
 * \(C(\Omega_2+d^{d+C(\Omega_22,0)+1},0)\) is first level-(1,0) \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings
 * \(C(\Omega_2+d^{d2},0)\) is first \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings
 * \(C(\Omega_2+d^{d2+1},0)\) is first \(\Pi_2\)-reflecting onto "\(\Pi_3\)-reflectings that are \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings"
 * \(C(\Omega_2+d^{d3},0)\) is first \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto "\(\Pi_3\)-reflectings that are \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings"
 * \(C(\Omega_2+d^{d^2},0)\) is first \(\Pi_3\)-reflecting onto \(\Pi_3\)-reflectings
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)}},0)\) is fixed point of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)}}+d^d,0)\) is \(\Pi_3\)-reflecting fixed point of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)}+1},0)\) is \(\Pi_2\)-reflecting onto fixed points of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)}+d},0)\) is \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto fixed points of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)}+d^2},0)\) is level-2 \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto fixed points of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)}2},0)\) is fixed point of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting onto fixed points of \(\alpha\mapsto\)level-\(\alpha\) \(\Pi_3\)-reflecting
 * \(C(\Omega_2+d^{d^{C(\Omega_22,0)+1}},0)\) is first level-(1,0) \(\Pi_3\)-reflecting
 * \(C(\Omega_2+d^{d^d},0)\) is first \(\Pi_4\)-reflecting
 * \(C(\Omega_2+d^{d^d}2,0)\) is first \(\Pi_4\)-reflecting limit of \(\Pi_4\)-reflectings
 * \(C(\Omega_2+d^{d^d+1},0)\) is first \(\Pi_2\)-reflecting onto \(\Pi_4\)-reflectings
 * \(C(\Omega_2+d^{d^d+d},0)\) is first \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto \(\Pi_4\)-reflectings
 * \(C(\Omega_2+d^{d^d2},0)\) is first \(\Pi_4\)-reflecting that is \(\Pi_2\)-reflecting onto \(\Pi_4\)-reflectings
 * \(C(\Omega_2+d^{d^{d+1}},0)\) is first \(\Pi_3\)-reflecting onto \(\Pi_4\)-reflectings
 * \(C(\Omega_2+d^{d^{d2}},0)\) is first \(\Pi_4\)-reflecting that is \(\Pi_3\)-reflecting onto \(\Pi_4\)-reflectings
 * \(C(\Omega_2+d^{d^{d^2}},0)\) is first \(\Pi_4\)-reflecting onto \(\Pi_4\)-reflectings
 * \(C(\Omega_2+d^{d^{d^d}},0)\) is first \(\Pi_5\)-reflecting
 * \(C(\Omega_2+\varepsilon_{d+1},0)\) is limit of \(\Pi_n\)-reflecting
 * \(C(\Omega_2+\varepsilon_{d+1}+d^d,0)\) is \(\Pi_3\)-reflecting limit of \(\Pi_n\)-reflecting
 * \(C(\Omega_2+\varepsilon_{d+1}2,0)\) is limit of "\(\Pi_m\)-reflecting limit of \(\Pi_n\)-reflecting"
 * \(C(\Omega_2+\varepsilon_{d+1}d,0)\) is \(\Pi_2\)-reflecting onto limits of \(\Pi_n\)-reflecting
 * \(C(\Omega_2+\varepsilon_{d+1}d^d,0)\) is \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto limits of \(\Pi_n\)-reflecting
 * \(C(\Omega_2+\varepsilon_{d+1}^2,0)\) is limit of "\(\Pi_m\)-reflecting that is \(\Pi_2\)-reflecting onto limits of \(\Pi_n\)-reflecting"
 * \(C(\Omega_2+\varepsilon_{d+1}^d,0)\) is \(\Pi_3\)-reflecting onto limits of \(\Pi_n\)-reflecting
 * \(C(\Omega_2+\varepsilon_{d+1}^{\varepsilon_{d+1}},0)\) is limit of "\(\Pi_m\)-reflecting that is \(\Pi_3\)-reflecting onto limits of \(\Pi_n\)-reflecting"
 * \(C(\Omega_2+\varepsilon_{d+1}^{\varepsilon_{d+1}^d},0)\) is \(\Pi_4\)-reflecting onto limits of \(\Pi_n\)-reflecting
 * \(C(\Omega_2+\varepsilon_{d+2},0)\) is limit of "\(\Pi_m\)-reflecting onto limits of \(\Pi_n\)-reflecting" (i.e. level-2 limit of \(\Pi_n\)-reflecting)
 * \(C(\Omega_2+\varepsilon_{d+C(\Omega_22,0)},0)\) is level-(1,0) limit of \(\Pi_n\)-reflecting
 * \(C(\Omega_2+\varepsilon_{d2},0)\) is first 1-stable
 * \(C(\Omega_2+\varepsilon_{d2+1},0)\) is limit of \(\Pi_n\)-reflecting onto 1-stable
 * \(C(\Omega_2+\varepsilon_{d2+2},0)\) is level-2 limit of \(\Pi_n\)-reflecting onto 1-stable
 * \(C(\Omega_2+\varepsilon_{d3},0)\) is first level-2 1-stable
 * \(C(\Omega_2+\varepsilon_{d^2},0)\) is first 1-\(\Pi_1\)-reflecting
 * \(C(\Omega_2+\varepsilon_{d^d},0)\) is first 1-\(\Pi_2\)-reflecting
 * \(C(\Omega_2+\varepsilon_{\varepsilon_{d2}},0)\) is first 2-stable
 * \(C(\Omega_2+\zeta_{d+1},0)\) is limit of n-stable
 * \(C(\Omega_2+\zeta_{d+2},0)\) is level-2 limit of n-stable
 * \(C(\Omega_2+\zeta_{d2},0)\) is first \(\omega\)-stable
 * \(C(\Omega_2+\zeta_{\varepsilon_{d2}},0)\) is first \(\omega+1\)-stable
 * \(C(\Omega_2+\zeta_{\zeta_{d2}},0)\) is first \(\omega2\)-stable
 * \(C(\Omega_2+\varphi(3,d2),0)\) is first \(\omega^2\)-stable
 * \(C(\Omega_2+\varphi(3,\varepsilon_{d2}),0)\) is first \(\omega^2+1\)-stable
 * \(C(\Omega_2+\varphi(4,d2),0)\) is first \(\omega^3\)-stable
 * \(C(\Omega_2+\varphi(\omega,d2),0)\) is first \(\omega^\omega\)-stable
 * \(C(\Omega_2+\varphi(C(\Omega_22,0),d+1),0)\) is fixed point of \(\alpha\mapsto\alpha\)-stable
 * \(C(\Omega_2+\varphi(\omega,\varphi(C(\Omega_22,0),d+1)+d),0)\) is \(\omega^\omega\)-stable over fixed points of \(\alpha\mapsto\alpha\)-stable
 * \(C(\Omega_2+\varphi(C(\Omega_22,0),d+2),0)\) is level-2 fixed point of \(\alpha\mapsto\alpha\)-stable
 * \(C(\Omega_2+\varphi(C(\Omega_22,0),d2),0)\) is first \(\lambda a.a\)-stable
 * \(C(\Omega_2+\varphi(C(\Omega_22,0),d2+1),0)\) is fixed point of "\(\alpha\mapsto\alpha\)-stable over \(\lambda a.a\)-stables"
 * \(C(\Omega_2+\varphi(C(\Omega_22,0),d3),0)\) is first level-2 \(\lambda a.a\)-stable
 * \(C(\Omega_2+\varphi(C(\Omega_22,0),\varepsilon_{d2}),0)\) is first \(\lambda a.a+1\)-stable
 * \(C(\Omega_2+\varphi(C(\Omega_22,0),C(\Omega_22,0),d+1)),0)\) is fixed point of \(\alpha\mapsto\lambda a.a+\alpha\)-stable
 * \(C(\Omega_2+\varphi(C(\Omega_22,0),C(\Omega_22,0),d2)),0)\) is first \(\lambda a.a2\)-stable
 * \(C(\Omega_2+\varphi(C(\Omega_22,0)+1,d2),0)\) is first \(\lambda a.a\omega\)-stable
 * \(C(\Omega_2+\varphi(C(\Omega_22,0)2,d2),0)\) is first \(\lambda a.a^2\)-stable
 * \(C(\Omega_2+\varphi(C(\Omega_22,0)^2,d2),0)\) is first \(\lambda a.a^a\)-stable
 * \(C(\Omega_2+\varphi(\varepsilon_{C(\Omega_22,0)+1},d2),0)\) is first \(\lambda a.\varepsilon_{a+1}\)-stable
 * \(C(\Omega_2+\varphi(d,1),0)\) is first \(\lambda a.a^+\)-stable
 * \(C(\Omega_2+\varphi(C(\Omega_22,0),\varphi(d,1)+1),0)\) is fixed point of \(\alpha\mapsto\alpha\)-stable over \(\lambda a.a^+\)-stables
 * \(C(\Omega_2+\varphi(C(\Omega_22,0),\varphi(d,1)+d),0)\) is first \(\lambda a.a\)-stable over \(\lambda a.a^+\)-stables
 * \(C(\Omega_2+\varphi(d,2),0)\) is first level-2 \(\lambda a.a^+\)-stable
 * \(C(\Omega_2+\varphi(d,\varepsilon_{d2}),0)\) is first \(\lambda a.a^++1\)-stable
 * \(C(\Omega_2+\varphi(d,\varphi(C(\Omega_22,0),d2)),0)\) is first \(\lambda a.a^++a\)-stable
 * \(C(\Omega_2+\varphi(d,\varphi(d,1)),0)\) is first \(\lambda a.a^+2\)-stable
 * \(C(\Omega_2+\varphi(d+1,0),0)\) is limit of \(\lambda a.a^+n\)-stable
 * \(C(\Omega_2+\varphi(d+1,d),0)\) is first \(\lambda a.a^+\omega\)-stable
 * \(C(\Omega_2+\varphi(d+C(\Omega_22,0),0),0)\) is fixed point of \(\alpha\mapsto\lambda a.a^+\alpha\)-stable
 * \(C(\Omega_2+\varphi(d+C(\Omega_22,0),d),0)\) is first \(\lambda a.a^+a\)-stable
 * \(C(\Omega_2+\varphi(d2,0),0)\) is first \(\lambda a.(a^+)^2\)-stable
 * \(C(\Omega_2+\varphi(d^2,0),0)\) is first \(\lambda a.(a^+)^{a^+}\)-stable
 * \(C(\Omega_2+\varphi(\varepsilon_{d+1},d),0)\) is first \(\lambda a.\varepsilon_{a^++1}\)-stable
 * \(C(\Omega_2+\varphi(\varphi(d,1),d),0)\) is first \(\lambda a.\varphi(a^+,1)\)-stable
 * \(C(\Omega_2+\varphi(\Gamma_{d+1},d),0)\) is first \(\lambda a.\Gamma_{a^++1}\)-stable

The two interpretations almost "catch up" at \(\lambda a.\Gamma_{a^++1}\)-stable, and "catch up" at \(\lambda a.a^{++}\)-stable. Beyond that, the difference will be small.

Up to "next \(\Pi_3\)-reflecting"-stable
Next I'll use Taranovsky's corresponding to compare. Let \(d_0=C(\Omega_22,0)\) and \(d_{n+1}=C(\Omega_2,d_n)\).
 * \(C(\Omega_2+d_2,0)\) is first \(\lambda a.a^{++}\)-stable
 * \(C(\Omega_2+d_1^{d_1^{d_2+\omega}+1},0)\) is first \(\lambda a.a^{++}+1\)-stable
 * \(C(\Omega_2+d_1^{d_1^{d_2+d_0}+1},0)\) is first \(\lambda a.a^{++}+a\)-stable
 * \(C(\Omega_2+d_1^{d_1^{d_2+d_1}},0)\) is first \(\lambda a.a^{++}+a^+\)-stable
 * \(C(\Omega_2+d_1^{d_1^{d_22}},0)\) is first \(\lambda a.a^{++}2\)-stable
 * \(C(\Omega_2+d_1^{d_1^{d_2\omega}},0)\) is limit of \(\lambda a.a^{++}n\)-stable
 * \(C(\Omega_2+d_1^{d_1^{d_2\omega}+1},0)\) is first \(\lambda a.a^{++}\omega\)-stable
 * \(C(\Omega_2+d_1^{d_1^{d_2^2}},0)\) is first \(\lambda a.(a^{++})^2\)-stable
 * \(C(\Omega_2+\varepsilon_{d_2+1}d_1,0)\) is first \(\lambda a.\varepsilon_{a^{++}+1}\)-stable
 * \(C(\Omega_2+d_3,0)\) is first \(\lambda a.a^{+++}\)-stable (i.e. "next 3rd admissible"-stable)
 * \(C(\Omega_2+d_4,0)\) is first \(\lambda a.a^{++++}\)-stable (i.e. "next 4th admissible"-stable)
 * \(C(\Omega_2+C(\Omega_2+1,d_0),0)\) is limit of "next n-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+1,d_0)d_1,0)\) is first "next \(\omega\)-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2,C(\Omega_2+1,d_0)),0)\) is first "next \(\omega+1\)-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+2,d_0)d_1,0)\) is first "next \(\omega^2\)-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+\varepsilon_0,d_0)d_1,0)\) is first "next \(\varepsilon_0\)-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2,0),d_0)d_1,0)\) is first "next \(\omega^\text{CK}_1\)-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2+d_1^{d_1^\omega+1},0),d_0)d_1,0)\) is first "next \(C(\Omega_2+d_1^{d_1^\omega+1},0)\)-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_0,d_0),0)\) is fixed point of \(\alpha\mapsto\)"next \(\alpha\)-th admissible"-stable (this ordinal is not 1-stable)
 * \(C(\Omega_2+C(\Omega_2+d_0,d_0)d_1,0)\) is such \(\alpha\) that is "next \(\alpha\)-th admissible"-stable ("next (1,0)-th admissible"-stable)
 * \(C(\Omega_2+C(\Omega_2,C(\Omega_2+d_0,d_0)),0)\) is first "next (1,0)+1-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_0,C(\Omega_2+d_0,d_0))d_1,0)\) is first "next (1,1)-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_0+1,d_0)d_1,0)\) is first "next \((1,\omega)\)-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_02,d_0)d_1,0)\) is first "next (2,0)-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_0^2,d_0)d_1,0)\) is first "next (1,0,0)-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1,d_0),0)\) is first "next recursively inaccessible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1,C(\Omega_2+d_1,d_0)),0)\) is first "next 2nd recursively inaccessible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1+d_0,d_0)d_1,0)\) is first "next (1,0)-th recursively inaccessible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_12,d_0),0)\) is first "next level-2 recursively inaccessible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1d_0,d_0),0)\) is fixed point of \(\alpha\mapsto\)"next level-\(\alpha\) recursively inaccessible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1d_0,d_0)d_1,0)\) is first "next level-(1,0) recursively inaccessible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^2,d_0),0)\) is first "next recursively Mahlo"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^2+d_1,d_0),0)\) is first "next recursively inaccessible limit of recursively Mahlos"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^22,d_0),0)\) is first "next recursively Mahlo limit of recursively Mahlos"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^3,d_0),0)\) is first "next level-2 recursively Mahlos"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^{d_0},d_0)d_1,0)\) is first "next level-(1,0) recursively Mahlos"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^{d_1},d_0),0)\) is first "next \(\Pi_3\)-reflecting"-stable

Up to \(\omega\)-ply-stable
Next I'll use "stronger than Taranovsky's corresponding" to compare. Let \(\sigma=d_0=C(\Omega_22,0)\) and \(d_{n+1}=C(\Omega_2,d_n)\). For an \(\omega\)-ply-stable ordinal \(\alpha\), \(L_\alpha\prec_{\Sigma_1}L_\beta\), where \(\beta\) is still \(\omega\)-ply-stable. So \(\omega\)-ply-stable ordinals make a chain \(\{\alpha_n\}_{n<\omega}\) such that \(L_{\alpha_n}\prec_{\Sigma_1}L_{\alpha_{n+1}}\), with a supremum \(\alpha_\omega=\sup\{\alpha_n|n<\omega\}\), which is not admissible.
 * \(C(\Omega_2+C(\Omega_2+d_1^{d_1},\sigma),0)\) is first "next \(\Pi_3\)-reflecting"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^{d_1}+d_1,\sigma),0)\) is first "next recursively inaccessible limit of \(\Pi_3\)-reflectings"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^{d_1}2,\sigma),0)\) is first "next \(\Pi_3\)-reflecting limit of \(\Pi_3\)-reflectings"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^{d_1+1},\sigma),0)\) is first "next \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^{d_12},\sigma),0)\) is first "next \(\Pi_3\)-reflecting that is \(\Pi_2\)-reflecting onto \(\Pi_3\)-reflectings"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^{d_1^2},\sigma),0)\) is first "next \(\Pi_3\)-reflecting onto \(\Pi_3\)-reflectings"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^{d_1^{d_1}},\sigma),0)\) is first "next \(\Pi_4\)-reflecting"-stable
 * \(C(\Omega_2+C(\Omega_2+d_1^{d_1^{d_1^{d_1}}},\sigma),0)\) is first "next \(\Pi_5\)-reflecting"-stable
 * \(C(\Omega_2+C(\Omega_2+\varepsilon_{d_1+1},\sigma),0)\) is limit of "next \(\Pi_n\)-reflecting"-stable
 * \(C(\Omega_2+\varphi(C(\Omega_2+\varepsilon_{d_1+1},\sigma),d_1),0)\) is first "next limit of \(\Pi_n\)-reflecting"-stable
 * \(C(\Omega_2+C(\Omega_2+\varepsilon_{d_12},\sigma),0)\) is first "next 1-stable"-stable (i.e. doubly 1-stable)
 * \(C(\Omega_2+C(\Omega_2+\varepsilon_{\varepsilon_{d_12}},\sigma),0)\) is first doubly 2-stable
 * \(C(\Omega_2+C(\Omega_2+\Gamma_{d_12},\sigma),0)\) is first doubly \(\lambda x.\Gamma_{x^++1}\)-stable
 * \(C(\Omega_2+C(\Omega_2+d_2,\sigma),0)\) is first doubly "next 2nd admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+d_3,\sigma),0)\) is first doubly "next 3rd admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2+1,\sigma),\sigma),0)\) is limit of doubly "next n-th admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2+\sigma,\sigma),\sigma),0)\) is fixed point of \(\alpha\mapsto\)doubly "next \(\alpha\)-th admissible"-stable
 * \(C(\Omega_2+\varphi(C(\Omega_2+C(\Omega_2+\sigma,\sigma),\sigma),d_1),0)\) is first "next fixed point of \(\alpha\mapsto\)'next \(\alpha\)-th admissible'-stable"-stable
 * \(C(\Omega_2+C(\Omega_2+\varphi(C(\Omega_2+\sigma,\sigma),d_1),\sigma),0)\) is first "next \(\alpha\) that is 'next \(\alpha\)-th admissible'-stable"-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2+d_1,\sigma),\sigma),0)\) is first doubly "next recursively inaccessible"-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2+d_1^2,\sigma),\sigma),0)\) is first doubly "next recursively Mahlo"-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2+d_1^{d_1},\sigma),\sigma),0)\) is first doubly "next \(\Pi_3\)-reflecting"-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2+d_1^{d_1^{d_1}},\sigma),\sigma),0)\) is first doubly "next \(\Pi_4\)-reflecting"-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2+\varepsilon_{d_12},\sigma),\sigma),0)\) is first triply 1-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2+d_2,\sigma),\sigma),0)\) is first triply "next 2nd admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2+C(\Omega_2+d_1,\sigma),\sigma),\sigma),0)\) is first triply "next recursively inaccessible"-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2+C(\Omega_2+d_2,\sigma),\sigma),\sigma),0)\) is first quadruply "next 2nd admissible"-stable
 * \(C(\Omega_2+C(\Omega_2+C(\Omega_2+C(\Omega_2+C(\Omega_2+d_1,\sigma),\sigma),\sigma),\sigma),0)\) is first quadruply "next recursively inaccessible"-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma),0)\) is limit of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)+d_1,0)\) is recursively inaccessible limit of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)+d_2,0)\) is "next 2nd admissible"-stable limit of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)+C(\Omega_2+d_1,\sigma),0)\) is "next recursively inaccessible"-stable limit of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)+C(\Omega_2+d_2,\sigma),0)\) is doubly "next 2nd admissible"-stable limit of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)+C(\Omega_2+C(\Omega_2+d_2,\sigma),\sigma),0)\) is triply "next 2nd admissible"-stable limit of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)+C(\Omega_2+C(\Omega_22,\sigma),\sigma),0)\) is limit of n-ply-stable limits of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)+C(\Omega_2+C(\Omega_22,\sigma),\sigma)2,0)\) is limit of n-ply-stable limits of n-ply-stable limits of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)+C(\Omega_2+C(\Omega_22,\sigma),\sigma)d_1,0)\) is \(\Pi_2\)-reflecting over limits of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)+\varphi(C(\Omega_2+C(\Omega_22,\sigma),\sigma),1),0)\) is limit of n-ply-stable over limits of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)+\varphi(C(\Omega_2+C(\Omega_22,\sigma),\sigma),d_1),0)\) is first \(\beta\)-stable where \(\beta\) is limit of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)+C(\Omega_2+C(\Omega_22,\sigma)+\varphi(C(\Omega_2+C(\Omega_22,\sigma),\sigma),d_1),\sigma),0)\) is first doubly \(\beta\)-stable where \(\beta\) is limit of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)2,0)\) is limit of n-ply \(\beta\)-stable where \(\beta\) is limit of n-ply-stable
 * \(C(\Omega_2+C(\Omega_22,\sigma)d_1,0)\) is first \(\omega\)-ply-stable

A related concept about \(\omega\)-ply-stable ordinal is nonprojectable ordinal. \(\beta\) is nonprojectable if \(\sup\{\alpha<\beta|L_\alpha\prec_{\Sigma_1}L_\beta\}=\beta\), meaning that there are infinite amount of \(\alpha\)'s which are stable to the same target \(\beta\). Because higher stability implies lower stability:
 * Let \(\alpha<\beta<\gamma\). If \(L_\alpha\prec_{\Sigma_1}L_\gamma\), then \(L_\alpha\prec_{\Sigma_1}L_\beta\).

nonprojectability thus implies the existence of \(\omega\)-ply-stability:
 * If \(\alpha\in A=\{\alpha<\beta|L_\alpha\prec_{\Sigma_1}L_\beta\}\) and \(\beta=\sup A\), then \(\alpha\) is \(\omega\)-ply-stable.

Proof: There are infinite amount of \(\alpha_n\)'s above the \(\alpha=\alpha_0\). Let \(\alpha_i<\alpha_{i+1}\), then \(L_{\alpha_i}\prec_{\Sigma_1}L_\beta\), so \(L_{\alpha_i}\prec_{\Sigma_1}L_{\alpha_{i+1}}\).

The structure of stable ordinals beyond that is unclear for me. So I stop here, with some guesses.

Guess 1 is more -ply-stables. \(\alpha=\alpha_0\) is \(\beta\)-ply-stable if there exists sequence \(\{\alpha_\gamma\}_{\gamma<\beta}\) such that \(L_{\alpha_\gamma}\prec_{\Sigma_1}L_{\alpha_{\gamma+1}}\) for \(\gamma<\beta\) and \(\alpha_\gamma=\sup\{\alpha_\delta|\delta<\gamma\}\) for limit \(\gamma<\beta\).

Guess 2 is more stable ordinals targeting the same ordinal. \(\alpha=\alpha_0\) is \(\beta\)-target if there exists ordinal \(\delta\), sequence \(\{\alpha_\gamma\}_{\gamma<\beta}\) such that \(L_{\alpha_\gamma}\prec_{\Sigma_1}L_\delta\) for all \(\gamma<\beta\), and \(\gamma<\eta\rightarrow\alpha_\gamma<\alpha_\eta\). Then an \(\omega\)-target ordinal is \(\omega\)-ply-stable. A nonprojectable \(\delta\) adds the requirement \(\delta=\sup\{\alpha_\gamma|\gamma<\beta\}\), so it's stronger.

Guess 3 combines the above two. \(\alpha=\alpha_0\) is \(\beta\)-chained if there exists sequence \(\{\alpha_\gamma\}_{\gamma<\beta}\) such that \(L_{\alpha_\gamma}\prec_{\Sigma_1}L_{\alpha_\delta}\) for all \(\gamma<\delta<\beta\), and \(\gamma<\delta\rightarrow\alpha_\gamma<\alpha_\delta\). So \(\beta+1\)-chained ordinals are \(\beta\)-target.

Details under incomputable stability
From the above comparisons, it seems that some expressions in Taranovsky's ordinal notation with a \(C(\Omega_2+C(\Omega_22,C(\Omega_22,0)),0)\) reaches \(\Pi^1_2\text{-CA}_0\). The least computable ordinal with occurence of a \(C(\Omega_2+C(\Omega_22,C(\Omega_22,0)),0)\) is \(C(C(\Omega_22+C(\Omega_2+C(\Omega_22,C(\Omega_22,0)),0),0),0)\), so a question naturally comes: does \(C(C(\Omega_22+C(\Omega_2+C(\Omega_22,C(\Omega_22,0)),0),0),0)\) reach \(\Pi^1_2\text{-CA}_0\)?

To get it, let's focus on the first incomputable stability - "next admissible"-stable ordinals. Inside an ordinal collapsing function, it's "large enough", and works as a "diagonalizer" over the expression \(f(\alpha)\) in "\(\alpha\) is \(f(\alpha)\)-stable". For example, the least collapsing of "next admissible"-stable should be the limit of \(\theta(\)1-stable\(,0)\), \(\theta(\lambda a.\theta(1\)-stable\(,a)\)-stable\(,0)\), \(\theta(\lambda a.\theta(\lambda b.\theta(1\)-stable\(,b)\)-stable\(,a)\)-stable\(,0)\), etc.

When we talk about \(\lambda a.\Gamma_{a+1}\)-stable ordinals, the "\(a\)" is so large that we need larger ordinal (\(a^+\)) for collapsing such as \(\lambda a.\theta(a^+,a)\)-stable or \(\lambda a.\psi_{a^+}((a^+)^{a^+})\)-stable. The notation \(\lambda a.\theta(\Omega,a)\)-stable just leads to \(\lambda a.\varphi(\Omega,a+1)\)-stable (where \(\Omega\) means admissible, \(\Omega_{a+n}=a^{\overbrace{++\cdots+}^n}\)). In this sense, the "1-stable" in \(\lambda a.\theta(1\)-stable\(,a)\)-stable actually means the next 1-stable ordinal after \(a\).

Let \(d_0=C(\Omega_22,0)\) and \(d_{n+1}=C(\Omega_2,d_n)\). \(C(\Omega_2+d_1^{d_1^{d_1}},0)\) thus works as the bound varible in abstraction-stable, \(C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))\) works as the inner bound varible in \(\lambda a.\psi_{\Omega_{a+1}}(\text{abstraction-stable})\)-stable, and so on.
 * \(C(\Omega_2+d_1^{d_1^{C(d_1,d_0)}+1},0)\) is first \(\lambda a.\varepsilon_{a+1}\)-stable, or first \(\lambda a.\theta(1,a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(d_1d_0,d_0)}+1},0)\) is first \(\lambda a.\varphi(a,1)\)-stable, or first \(\lambda a.\theta(a,a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(d_1^2,d_0)}+1},0)\) is first \(\lambda a.\Gamma_{a+1}\)-stable, or first \(\lambda a.\theta(a^+,a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(d_2,d_0)}+1},0)\) is first \(\lambda a.\theta(a^{++},a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_2+1,d_0),d_0)}+1},0)\) is first \(\lambda a.\theta(\Omega_{a+\omega},a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_2+d_0,d_0),d_0)}+1},0)\) is first \(\lambda a.\theta(\Omega_{a2},a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_2+d_1,d_0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22,d_0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+1,0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\omega})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_0,0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\psi_I(0)})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_0+1,0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\psi_I(\omega)})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1,0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}I})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1+1,0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}I_\omega})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1}+1,0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}K_\omega})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^\omega+1},0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\text{first 1-stable}})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^\omega+1},0),0),d_0)}+1},0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\text{first }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}\text{first 1-stable}})\text{-stable}})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}a})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},0)2,0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}a^2})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(C(\Omega_2,C(\Omega_2+d_1^{d_1^{d_1}},0)),C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}^{\Omega_{a+1}^{\Omega_{a+1}}}\varepsilon_{a+1}})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(d_0,C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\psi_{I_{a+1}}(0))\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1,C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(I_{a+1})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(K_{a+1})\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^\omega+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }1\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(\Omega_2+d_1^{d_1^{d_1}},0)}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.a\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_0}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.b\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+1,0),d_0)}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}\omega})\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},0),0),d_0)}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}a})\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^\omega+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}\text{next }1\text{-stable after }a})\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+1,0),d_0)}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}\text{next }\lambda c.\psi_{\Omega_{c+1}}(\Omega_{\Omega_{c+1}^{\Omega_{c+1}^{\Omega_{c+1}}}\omega})\text{-stable after }a})\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}b})\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))+C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)+\omega}+1},C(\Omega_2+d_1^{d_1^{d_1}},0)),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}b})+1\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))2,0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\Omega_{\Omega_{b+1}^{\Omega_{b+1}^{\Omega_{b+1}}}b^2})\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(d_0,C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\psi_{I_{b+1}}(0))\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1,C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(I_{b+1})\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^\omega+1},C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\text{next }1\text{-stable after }b)\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},C(\Omega_2+d_1^{d_1^{d_1}},0))),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\text{next }\lambda c.\psi_{\Omega_{c+1}}(\Omega_{\Omega_{c+1}^{\Omega_{c+1}^{\Omega_{c+1}}}c})\text{-stable after }b)\text{-stable after }a)\)-stable
 * \(C(\Omega_2+d_1^{d_1^{C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}}+1,0),0),d_0)}+1},0)\) is first \(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\)-stable
 * \(C(C(\Omega_22+C(\Omega_2+d_1^{d_1^{d_1}}+1,0),0),0)=\psi(\lambda a.\Omega_{a+1}\text{-stable})\)

Due to the inconvenience applying the diagonalizer behavior in Taranovsky's ordinal notation, I next compare stable ordinals with my array notation. Let separator ■ = {1{1,,2,,}2,,}
 * {1{1,,2{2,,}2,,}2} corresponds with 1-stable
 * {1{1,,2{1{1■2}2,,}2,,}2} corresponds with \(\lambda a.a\)-stable
 * {1{1,,2{2{1■2}2,,}2,,}2} corresponds with \(\lambda a.a\omega\)-stable
 * {1{1,,2{1{1■2}3,,}2,,}2} corresponds with \(\lambda a.a^2\)-stable
 * {1{1,,2{1{1■2}1{1■2}2,,}2,,}2} corresponds with \(\lambda a.a^a\)-stable
 * {1{1,,2{1{2■2}2,,}2,,}2} corresponds with \(\lambda a.a^{a^\omega}\)-stable
 * {1{1,,2{1{1{1■2}2■2}2,,}2,,}2} corresponds with \(\lambda a.a^{a^a}\)-stable
 * {1{1,,2{1{1,,2■2}2,,}2,,}2} corresponds with \(\lambda a.\varepsilon_{a+1}\)-stable
 * {1{1,,2{1{1,,3■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{a+2})\)-stable
 * {1{1,,2{1{1,,1{1■2}2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{a2})\)-stable
 * {1{1,,2{1{1,,1{1,,2■2}2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a+1}})\)-stable
 * {1{1,,2{1{1,,1{1,,1{1■2}2■2}2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\Omega_{\Omega_{a2}})\)-stable
 * {1{1,,2{1{1,,1,,2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\psi_{I_{a+1}}(0))\)-stable
 * {1{1,,2{1{1{1,,2{2,,}2,,}2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }1\text{-stable after }a)\)-stable
 * {1{1,,2{1{1{1,,2{1{1■2}2,,}2,,}2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.a\text{-stable after }a)\)-stable
 * {1{1,,2{1{1■3}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.b\text{-stable after }a)\)-stable
 * {1{1,,2{1{1■4}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\text{next }\lambda c.c\text{-stable after }b)\text{-stable after }a)\)-stable
 * s(n,n{1■1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1{1{1,,2{1{1■1,2}2,,}2,,}2}2{1,,2{1{1■1,2}2,,}2,,}2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable}+\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\text{-stable})\)
 * {1{1,,2{1{1■1,2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\)-stable
 * {1{1,,2{1{1{1{1,,2{1{1■1,2}2,,}2,,}2■2}2{1,,2{1{1■1,2}2,,}2,,}2■2}2,,}2,,}2} corresponds with \(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable}+\lambda b.\psi_{\Omega_{b+1}}(\lambda c.\Omega_{c+1}\text{-stable})\text{-stable})\)-stable
 * s(n,n{1{1■1,2}2■1,2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable}\cdot2)\)
 * s(n,n{1,,2■1,2}2) has growth rate \(\psi(\text{next admissible after }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1,,1{1,,1,,2■1,2}1{1,,1,,2■1,2}2■1,2}2) has growth rate \(\psi(\text{next recursively inaccessible after }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1{1,,2,,}2■1,2}2) has growth rate \(\psi(\text{next recursively Mahlo after }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1{1,,1,,2,,}2■1,2}2) has growth rate \(\psi(\text{next }\Pi_3\text{-reflecting after }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1{1{2,,}2,,}2■1,2}2) has growth rate \(\psi(\text{next }1\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■2,2}2) has growth rate \(\psi(\text{next }\lambda a.a\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■3,2}2) has growth rate \(\psi(\text{next }\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.b\text{-stable after }a)\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■4,2}2) has growth rate \(\psi(\text{next }\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\text{next }\lambda c.c\text{-stable after }b)\text{-stable after }a)\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■1,3}2) has growth rate \(\psi(\text{next }\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■2,3}2) has growth rate \(\psi(\text{next }\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.b\text{-stable after }\lambda b.\Omega_{b+1}\text{-stable})\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■1,4}2) has growth rate \(\psi(\text{next }\lambda a.\psi_{\Omega_{a+1}}(\text{next }\lambda b.\psi_{\Omega_{b+1}}(\lambda c.\Omega_{c+1}\text{-stable})\text{-stable after }\lambda b.\Omega_{b+1}\text{-stable})\text{-stable after }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■1,1,2}2) has growth rate \(\psi(2\text{nd }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■1,1,1,2}2) has growth rate \(\psi(3\text{rd }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■1`2}2) has growth rate \(\psi(\Omega\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■1{1,,1,,2}2}2) has growth rate \(\psi(I\text{th }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■1{1{1,,2{2,,}2,,}2}2}2) has growth rate \(\psi(1\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■1{1{1,,2{1{1■2}2,,}2,,}2}2}2) has growth rate \(\psi(\lambda a.a\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■1{1{1,,2{1{1■1,2}2,,}2,,}2}2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\Omega_{b+1}\text{-stable})\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■1{1{1,,2{1{1■1{1{1,,2{1{1■1,2}2,,}2,,}2}2}2,,}2,,}2}2}2) has growth rate \(\psi(\lambda a.\psi_{\Omega_{a+1}}(\lambda b.\psi_{\Omega_{b+1}}(\lambda c.\Omega_{c+1}\text{-stable})\text{-stable th }\lambda b.\Omega_{b+1}\text{-stable})\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable})\)
 * s(n,n{1■1{1■2}2}2) has growth rate \(\psi(\lambda a.\Omega_{a+1}\text{-stable th }\lambda a.\Omega_{a+1}\text{-stable})\)