User blog:Rgetar/Classification of special cases of generalized Veblen function

This is generalized Veblen function:

φ(X) is (1 + leo(X))-th common fixed point of all functions α = φ(Y), Y ∈ X0{·}α

where

\(X\{·\}a = \left\{\begin{array}{lcr} \{lbest(X; β)\} \quad \text{if} \; X' = 0\\ \left.\begin{array}{lcr} \{lbest(X; β), \langle Y \rangle a\} \quad \text{if} \; X'^\prime = 0\\ \{lbest(X; β), \langle Y \rangle 1\} \quad \text{if} \; X'^\prime ≠ 0\\ \end{array}\right\} \; \text{if} \; X' ≠ 0\\ \end{array}\right. \quad ,β < lbeo(X), \; Y ∈ X'\{·\}a \)

and other designation used in this blog, such as leo, lbeo, lbest, X' etc. see here.

X--1
Let's introduce new designation X--1: it is array X with its last base element decreased by 1 (if lbeo(X) is successor ordinal).

(It is like X-1, but with decreasing of last base element instead of last element).

So,

X--1 = lbest(X; lbeo(X)-1)

I already added X--1 to my list of designations.

Fundamental sequence system
This is fundamental sequence system for generalized Veblen function:

\(δ = \left\{\begin{array}{lcr} 0 \qquad \qquad \qquad \, \text{if} \; leo(X) = 0\\ φ(X-1) + 1 \quad \text{if} \; leo(X) ≠ 0\\ \end{array}\right. \)

"l." means "limit ordinal", and "s." means "successor ordinal".

\(X[n]_α = \left\{\begin{array}{lcr} lbest(X; lbeo(X)[n]) \quad \text{if} \; lbeo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} X--1, \langle X'[n]_α\rangle 1 \quad \text{if} \; leo(X') \; \text{- not s.}\\ X--1, \langle X'-1\rangle α \quad \text{if} \; leo(X') \; \text{- s.}\\ \end{array}\right\} \; \text{if} \; lbeo(X) \; \text{- s.}\\ \end{array}\right. \)

To get fundamental sequence of Cantor normal form, replace its last term with fundamental sequence of the last term.

Fundamental sequence for Cantor normal form term:

\(φ(X)[n] = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(X[n]_0) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(lest(X^0[n]_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.} \; or \; n=0\\ φ(X^0[n]_{φ(X)[n-1]}) \quad \text{if} \; ilbeo(X_0) ∌ \; \text{l.} \; and \; n>0\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \)

or

\(φ(X)[n] = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(lest(X; leo(X)[n])) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(lest(X^0[n]_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.} \; or \; n=0\\ φ(X^0[n]_{φ(X)[n-1]}) \quad \text{if} \; ilbeo(X_0) ∌ \; \text{l.} \; and \; n>0\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \)

Other definitions
There are other possible definitions of the generalized Veblen function. For example, we can replace X{·}α with this subset of X{·}α

\(\left\{\begin{array}{lcr} \{lbest(X; lbeo(X)[n])\} \quad \text{if} \; X' = 0\\ \left.\begin{array}{lcr} \{lbest(X; lbeo(X)[n]), \langle Y \rangle a\} \quad \text{if} \; X'^\prime = 0\\ \{lbest(X; lbeo(X)[n]), \langle Y \rangle 1\} \quad \text{if} \; X'^\prime ≠ 0\\ \end{array}\right\} \; \text{if} \; X' ≠ 0\\ \end{array}\right. \quad Y ∈ \; \text{this set for} \; X' \)

for some fundamental sequence system (if lbeo(X) is successor ordinal then lbeo(X)[n] is its predecessor),

or with this superset of X{·}α:

X{··}α = X{·}α ∪ Y{··}α, Y ∈ X{·}α

or with this subset of X{··}α:

\(X\{·\}a = \left\{\begin{array}{lcr} \{lbest(X; lbeo(X)[n])\} \quad \text{if} \; X' = 0 \; or \; lbeo(X) - limit ordinal\\ \left.\begin{array}{lcr} \{lbest(X; lbeo(X)[n]), \langle Y \rangle a\} \quad \text{if} \; X'^\prime = 0\\ \{lbest(X; lbeo(X)[n]), \langle Y \rangle 1\} \quad \text{if} \; X'^\prime ≠ 0\\ \end{array}\right\} \; \text{if} \; X' ≠ 0\\ \end{array}\right. \quad Y ∈ \; \text{this set for} \; X' \)

for some fundamental sequence system (if lbeo(X) is successor ordinal then lbeo(X)[n] is its predecessor).

Further in this blog I will use this last set with forementioned fundamental sequence system. (I will use this set because it simplifies the definition for many special cases).

Note: this is the same generalized Veblen function, just different definitions.

Types of special cases
There are four types of all special cases of generalized Veblen function φ(X): in type 1 φ(X) is defined through exponentiation, in types 2, 3 φ(X) is defined through fundamental sequence, that is φ(X) = sup(φ(X)[n]), and in type 4 φ(X) is defined through one fixed point.

This is list of definitions of generalized Veblen function for each case.

Type 1
X = α, where α is successor ordinal or 0.

φ(X) = φ(α) = ωα

Type 2
leo(X) is limit ordinal.

φ(X[n] = φ(lest(X; leo(X)[n]))

Type 3
X0 ≠ 0, leo(X) is not limit ordinal, ilbeo(X0) ∋ limit ordinal.

There are subtypes, and each subtype has two subsubtypes - for leo(X) = 0 and for leo(X) - successor ordinal:

3.1. lbeo(X0) - limit ordinal.

3.1.1. leo(X) = 0

φ(X)[n] = φ(lbest(X0; lbeo(X0)[n]), 0)

3.1.2. leo(X) - successor ordinal

φ(X)[n] = φ(lbest(X0; lbeo(X0)[n]), φ(X-1))

3.2. lbeo(X0) - successor ordinal, lbeo(X0') - limit ordinal.

3.2.1. leo(X) = 0

φ(X)[n] = φ(X0--1, ⟨lbest(X0'; lbeo(X0')[n])⟩1, 0)

3.2.2. leo(X) - successor ordinal

φ(X)[n] = φ(X0--1, ⟨lbest(X0'; lbeo(X0')[n])⟩1, φ(X-1))

3.3. lbeo(X0), lbeo(X0') - successor ordinals, lbeo(X0") - limit ordinal.

3.3.1. leo(X) = 0

φ(X)[n] = φ(X0--1, ⟨X0'--1, ⟨lbest(X0"; lbeo(X0")[n])⟩1⟩1, 0)

3.3.2. leo(X) - successor ordinal

φ(X)[n] = φ(X0--1, ⟨X0'--1, ⟨lbest(X0"; lbeo(X0")[n])⟩1⟩1, φ(X-1))

3.4. lbeo(X0), lbeo(X0'), lbeo(X0") - successor ordinals, lbeo(X0"') - limit ordinal.

3.4.1. leo(X) = 0

φ(X)[n] = φ(X0--1, ⟨X0'--1, ⟨X0"--1, ⟨lbest(X0"'; lbeo(X0"')[n])⟩1⟩1⟩1, 0)

3.4.2. leo(X) - successor ordinal

φ(X)[n] = φ(X0--1, ⟨X0'--1, ⟨X0"--1, ⟨lbest(X0"'; lbeo(X0"')[n])⟩1⟩1⟩1, φ(X-1))

3.5. lbeo(X0), lbeo(X0'), lbeo(X0"), lbeo(X0"') - successor ordinals, lbeo(X0"") - limit ordinal.

3.5.1. leo(X) = 0

φ(X)[n] = φ(X0--1, ⟨X0'--1, ⟨X0"--1, ⟨X0"'--1, ⟨lbest(X0"'; lbeo(X0"')[n])⟩1⟩1⟩1⟩1, 0)

3.5.2. leo(X) - successor ordinal

φ(X)[n] = φ(X0--1, ⟨X0'--1, ⟨X0"--1, ⟨X0"'--1, ⟨lbest(X0"'; lbeo(X0"')[n])⟩1⟩1⟩1⟩1, φ(X-1))

etc.

Type 4
X0 ≠ 0, leo(X) is not limit ordinal, ilbeo(X0) ∌ limit ordinal.

There are subtypes, and each subtype has two subsubtypes - for leo(X) = 0 and for leo(X) - successor ordinal. For leo(X) = 0 φ(X) is least fixed point, and for leo(X) - successor ordinal φ(X) is (1 + leo(X))-th fixed point, that is least fixed point larger than φ(X-1):

4.1. lbeo(X0), lbeo(X0') - successor ordinals, lbeo(X0") = 0.

4.1.1. leo(X) = 0

φ(X) is least ordinal such as

α = φ(X0--1, ⟨X0′−1⟩α)

4.1.2. leo(X) - successor ordinal

φ(X) is least ordinal such as

α = φ(X0--1, ⟨X0′−1⟩α) and α > φ(X-1)

4.2. lbeo(X0), lbeo(X0'), lbeo(X0") - successor ordinals, lbeo(X0"') = 0.

4.2.1. leo(X) = 0

φ(X) is least ordinal such as

α = φ(X0--1, ⟨X0'--1, ⟨X0"−1⟩α⟩1)

4.2.2. leo(X) - successor ordinal

φ(X) is least ordinal such as

α = φ(X0--1, ⟨X0'--1, ⟨X0"−1⟩α⟩1) and α > φ(X-1)

4.3. lbeo(X0), lbeo(X0'), lbeo(X0"), lbeo(X0"') - successor ordinals, lbeo(X0"") = 0.

4.3.1. leo(X) = 0

φ(X) is least ordinal such as

α = φ(X0--1, ⟨X0'--1, ⟨X0"--1, ⟨X0"'−1⟩α⟩1⟩1)

4.3.2. leo(X) - successor ordinal

φ(X) is least ordinal such as

α = φ(X0--1, ⟨X0'--1, ⟨X0"--1, ⟨X0"'−1⟩α⟩1⟩1) and α > φ(X-1)

4.4. lbeo(X0), lbeo(X0'), lbeo(X0"), lbeo(X0"'), lbeo(X0"") - successor ordinals, lbeo(X0""') = 0.

4.4.1. leo(X) = 0

φ(X) is least ordinal such as

α = φ(X0--1, ⟨X0'--1, ⟨X0"--1, ⟨X0"'--1, ⟨X0""−1⟩α⟩1⟩1⟩1)

4.4.2. leo(X) - successor ordinal

φ(X) is least ordinal such as

α = φ(X0--1, ⟨X0'--1, ⟨X0"--1, ⟨X0"'--1, ⟨X0""−1⟩α⟩1⟩1⟩1) and α > φ(X-1)

4.5. lbeo(X0), lbeo(X0'), lbeo(X0"), lbeo(X0"'), lbeo(X0""), lbeo(X0""') - successor ordinals, lbeo(X0""") = 0.

4.5.1. leo(X) = 0

φ(X) is least ordinal such as

α = φ(X0--1, ⟨X0'--1, ⟨X0"--1, ⟨X0"'--1, ⟨X0""--1, ⟨X0""'−1⟩α⟩1⟩1⟩1⟩1)

4.5.2. leo(X) - successor ordinal

φ(X) is least ordinal such as

α = φ(X0--1, ⟨X0'--1, ⟨X0"--1, ⟨X0"'--1, ⟨X0""--1, ⟨X0""'−1⟩α⟩1⟩1⟩1⟩1) and α > φ(X-1)

etc.

Examples
Here are examples for each type, subtype and subsubtype of generalized Veblen function with fundamental sequences for the forementioned fundamental sequence system.

1
φ(15) = ω15

ω15[n] = ω14n

that is

ω15[0] = 0

ω15[1] = ω14

ω15[2] = ω142

ω15[3] = ω143

ω15[4] = ω144

ω15[5] = ω145

...

2
φ(5, 6, ω3)[n] = φ(5, 6, ω3[n]) = φ(5, 6, ω2 + n)

that is

φ(5, 6, ω3)[0] = φ(5, 6, ω2)

φ(5, 6, ω3)[1] = φ(5, 6, ω2 + 1)

φ(5, 6, ω3)[2] = φ(5, 6, ω2 + 2)

φ(5, 6, ω3)[3] = φ(5, 6, ω2 + 3)

φ(5, 6, ω3)[4] = φ(5, 6, ω2 + 4)

φ(5, 6, ω3)[5] = φ(5, 6, ω2 + 5)

...

3.1.1
φ(ω, 0, 0)[n] = φ(ω[n], 0, 0) = φ(n, 0, 0)

that is

φ(ω, 0, 0)[0] = φ(0, 0, 0) = φ(0) = 1

φ(ω, 0, 0)[1] = φ(1, 0, 0) = Γ0

φ(ω, 0, 0)[2] = φ(2, 0, 0)

φ(ω, 0, 0)[3] = φ(3, 0, 0)

φ(ω, 0, 0)[4] = φ(4, 0, 0)

φ(ω, 0, 0)[5] = φ(5, 0, 0)

...

Another example:

φ(⟨ω2⟩ω)[n] = φ(⟨ω2⟩ω[n]) = φ(⟨ω2⟩n)

that is

φ(⟨ω2⟩ω)[0] = φ(⟨ω2⟩0) = φ(0) = 1

φ(⟨ω2⟩ω)[1] = φ(⟨ω2⟩1)

φ(⟨ω2⟩ω)[2] = φ(⟨ω2⟩2)

φ(⟨ω2⟩ω)[3] = φ(⟨ω2⟩3)

φ(⟨ω2⟩ω)[4] = φ(⟨ω2⟩4)

φ(⟨ω2⟩ω)[5] = φ(⟨ω2⟩5)

...

3.1.2
φ(ω, 0, 6)[n] = φ(ω[n], 0, φ(ω, 0, 5) + 1) = φ(n, 0, φ(ω, 0, 5) + 1)

that is

φ(ω, 0, 6)[0] = φ(0, 0, φ(ω, 0, 5) + 1) = ωφ(ω, 0, 5) + 1

φ(ω, 0, 6)[1] = φ(1, 0, ωφ(ω, 0, 5) + 1) = Γωφ(ω, 0, 5) + 1

φ(ω, 0, 6)[2] = φ(2, 0, ωφ(ω, 0, 5) + 1)

φ(ω, 0, 6)[3] = φ(3, 0, ωφ(ω, 0, 5) + 1)

φ(ω, 0, 6)[4] = φ(4, 0, ωφ(ω, 0, 5) + 1)

φ(ω, 0, 6)[5] = φ(5, 0, ωφ(ω, 0, 5) + 1)

...

Another example:

φ(⟨ω2⟩ω, 6)[n] = φ(⟨ω2⟩ω[n], φ(⟨ω2⟩ω, 5) + 1)[n] = φ(⟨ω2⟩n, φ(⟨ω2⟩ω, 5) + 1)

that is

φ(⟨ω2⟩ω, 6)[0] = φ(⟨ω2⟩0, φ(⟨ω2⟩ω, 5) + 1)

φ(⟨ω2⟩ω, 6)[1] = φ(⟨ω2⟩1, φ(⟨ω2⟩ω, 5) + 1)

φ(⟨ω2⟩ω, 6)[2] = φ(⟨ω2⟩2, φ(⟨ω2⟩ω, 5) + 1)

φ(⟨ω2⟩ω, 6)[3] = φ(⟨ω2⟩3, φ(⟨ω2⟩ω, 5) + 1)

φ(⟨ω2⟩ω, 6)[4] = φ(⟨ω2⟩4, φ(⟨ω2⟩ω, 5) + 1)

φ(⟨ω2⟩ω, 6)[5] = φ(⟨ω2⟩5, φ(⟨ω2⟩ω, 5) + 1)

...

3.2.1
φ(⟨ω2⟩8)[n] = φ(⟨ω2⟩7, ⟨ω2[n]⟩1) = φ(⟨ω2⟩7, ⟨ωn⟩1)

that is

φ(⟨ω2⟩8)[0] = φ(⟨ω2⟩7, 1)

φ(⟨ω2⟩8)[1] = φ(⟨ω2⟩7, ⟨ω⟩1)

φ(⟨ω2⟩8)[2] = φ(⟨ω2⟩7, ⟨ω2⟩1)

φ(⟨ω2⟩8)[3] = φ(⟨ω2⟩7, ⟨ω3⟩1)

φ(⟨ω2⟩8)[4] = φ(⟨ω2⟩7, ⟨ω4⟩1)

φ(⟨ω2⟩8)[5] = φ(⟨ω2⟩7, ⟨ω5⟩1)

...

3.2.2
φ(⟨ω2⟩8, 6)[0] = φ(⟨ω2⟩7, φ(⟨ω2⟩8, 5) + 1)

φ(⟨ω2⟩8, 6)[n] = φ(⟨ω2⟩7, ⟨ω2[n]⟩1, φ(⟨ω2⟩8, 5) + 1) = φ(⟨ω2⟩7, ⟨ωn⟩1, φ(⟨ω2⟩8, 5) + 1) for n > 0

that is

φ(⟨ω2⟩8, 6)[0] = φ(⟨ω2⟩7, φ(⟨ω2⟩8, 5) + 1)

φ(⟨ω2⟩8, 6)[1] = φ(⟨ω2⟩7, ⟨ω⟩1, φ(⟨ω2⟩8, 5) + 1)

φ(⟨ω2⟩8, 6)[2] = φ(⟨ω2⟩7, ⟨ω2⟩1, φ(⟨ω2⟩8, 5) + 1)

φ(⟨ω2⟩8, 6)[3] = φ(⟨ω2⟩7, ⟨ω3⟩1, φ(⟨ω2⟩8, 5) + 1)

φ(⟨ω2⟩8, 6)[4] = φ(⟨ω2⟩7, ⟨ω4⟩1, φ(⟨ω2⟩8, 5) + 1)

φ(⟨ω2⟩8, 6)[5] = φ(⟨ω2⟩7, ⟨ω5⟩1, φ(⟨ω2⟩8, 5) + 1)

...

3.3.1
φ(⟨⟨ω2⟩4⟩8)[n] =

φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω2[n]⟩1⟩1) =

φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω + n⟩1⟩1)

that is

φ(⟨⟨ω2⟩4⟩8)[0] = φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω⟩1⟩1)

φ(⟨⟨ω2⟩4⟩8)[1] = φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω + 1⟩1⟩1)

φ(⟨⟨ω2⟩4⟩8)[2] = φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω + 2⟩1⟩1)

φ(⟨⟨ω2⟩4⟩8)[3] = φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω + 3⟩1⟩1)

φ(⟨⟨ω2⟩4⟩8)[4] = φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω + 4⟩1⟩1)

φ(⟨⟨ω2⟩4⟩8)[5] = φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω + 5⟩1⟩1)

...

3.3.2
φ(⟨⟨ω2⟩4⟩8, 6)[n] =

φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω2[n]⟩1⟩1, φ(⟨ω2⟩8, 5) + 1) =

φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω + n⟩1⟩1, φ(⟨ω2⟩8, 5) + 1)

that is

φ(⟨⟨ω2⟩4⟩8, 6)[0] = φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω⟩1⟩1, φ(⟨ω2⟩8, 5) + 1)

φ(⟨⟨ω2⟩4⟩8, 6)[1] = φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω + 1⟩1⟩1, φ(⟨ω2⟩8, 5) + 1)

φ(⟨⟨ω2⟩4⟩8, 6)[2] = φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω + 2⟩1⟩1, φ(⟨ω2⟩8, 5) + 1)

φ(⟨⟨ω2⟩4⟩8, 6)[3] = φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω + 3⟩1⟩1, φ(⟨ω2⟩8, 5) + 1)

φ(⟨⟨ω2⟩4⟩8, 6)[4] = φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω + 4⟩1⟩1, φ(⟨ω2⟩8, 5) + 1)

φ(⟨⟨ω2⟩4⟩8, 6)[5] = φ(⟨⟨ω2⟩4⟩7, ⟨⟨ω2⟩3, ⟨ω + 5⟩1⟩1, φ(⟨ω2⟩8, 5) + 1)

...

3.4.1
φ(⟨⟨⟨ωω⟩11⟩4⟩8)[n] =

φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ωω[n]⟩1⟩1⟩1) =

φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ωn⟩1⟩1⟩1)

that is

φ(⟨⟨⟨ωω⟩11⟩4⟩8)[0] = φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨1⟩1⟩1⟩1)

φ(⟨⟨⟨ωω⟩11⟩4⟩8)[1] = φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ω⟩1⟩1⟩1)

φ(⟨⟨⟨ωω⟩11⟩4⟩8)[2] = φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ω2⟩1⟩1⟩1)

φ(⟨⟨⟨ωω⟩11⟩4⟩8)[3] = φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ω3⟩1⟩1⟩1)

φ(⟨⟨⟨ωω⟩11⟩4⟩8)[4] = φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ω4⟩1⟩1⟩1)

φ(⟨⟨⟨ωω⟩11⟩4⟩8)[5] = φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ω5⟩1⟩1⟩1)

...

3.4.2
φ(⟨⟨⟨ωω⟩11⟩4⟩8, 6)[n] =

φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ωω[n]⟩1⟩1⟩1, φ(⟨⟨⟨ωω⟩11⟩4⟩8, 5) + 1) =

φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ωn⟩1⟩1⟩1, φ(⟨⟨⟨ωω⟩11⟩4⟩8, 5) + 1)

that is

φ(⟨⟨⟨ωω⟩11⟩4⟩8, 6)[0] = φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨1⟩1⟩1⟩1, φ(⟨⟨⟨ωω⟩11⟩4⟩8, 5) + 1)

φ(⟨⟨⟨ωω⟩11⟩4⟩8, 6)[1] = φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ω⟩1⟩1⟩1, φ(⟨⟨⟨ωω⟩11⟩4⟩8, 5) + 1)

φ(⟨⟨⟨ωω⟩11⟩4⟩8, 6)[2] = φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ω2⟩1⟩1⟩1, φ(⟨⟨⟨ωω⟩11⟩4⟩8, 5) + 1)

φ(⟨⟨⟨ωω⟩11⟩4⟩8, 6)[3] = φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ω3⟩1⟩1⟩1, φ(⟨⟨⟨ωω⟩11⟩4⟩8, 5) + 1)

φ(⟨⟨⟨ωω⟩11⟩4⟩8, 6)[4] = φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ω4⟩1⟩1⟩1, φ(⟨⟨⟨ωω⟩11⟩4⟩8, 5) + 1)

φ(⟨⟨⟨ωω⟩11⟩4⟩8, 6)[5] = φ(⟨⟨⟨ωω⟩11⟩4⟩7, ⟨⟨⟨ωω⟩11⟩3, ⟨⟨ωω⟩10, ⟨ω5⟩1⟩1⟩1, φ(⟨⟨⟨ωω⟩11⟩4⟩8, 5) + 1)

...

3.5.1
φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8)[n] =

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨ω[n]⟩1⟩1⟩1⟩1) =

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨n⟩1⟩1⟩1⟩1)

that is

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8)[0] = φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, 1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8)[1] = φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨1⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8)[2] = φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨2⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8)[3] = φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨3⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8)[4] = φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨4⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8)[5] = φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨5⟩1⟩1⟩1⟩1)

...

3.5.2
φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 6)[n] =

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨ω[n]⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 5) + 1) =

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨n⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 5) + 1)

that is

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 6)[0] = φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, 1⟩1⟩1⟩1, φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 5) + 1)

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 6)[1] = φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨1⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 5) + 1)

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 6)[2] = φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨2⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 5) + 1)

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 6)[3] = φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨3⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 5) + 1)

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 6)[4] = φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨4⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 5) + 1)

φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 6)[5] = φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩7, ⟨⟨⟨⟨ω⟩5⟩11⟩3, ⟨⟨⟨ω⟩5⟩10, ⟨⟨ω⟩4, ⟨5⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨ω⟩5⟩11⟩4⟩8, 5) + 1)

...

...

4.1.1
φ(⟨2⟩8) = φ(8, 0, 0) is least α such as

α = φ(⟨2⟩7, ⟨1⟩α) = φ(7, α, 0)

φ(8, 0, 0)[0] = φ(7, 0, 0)

φ(8, 0, 0)[1] = φ(7, φ(7, 0, 0), 0)

φ(8, 0, 0)[2] = φ(7, φ(7, φ(7, 0, 0), 0), 0)

φ(8, 0, 0)[3] = φ(7, φ(7, φ(7, φ(7, 0, 0), 0), 0), 0)

φ(8, 0, 0)[4] = φ(7, φ(7, φ(7, φ(7, φ(7, 0, 0), 0), 0), 0), 0)

φ(8, 0, 0)[5] = φ(7, φ(7, φ(7, φ(7, φ(7, φ(7, 0, 0), 0), 0), 0), 0), 0)

...

4.1.2
φ(⟨2⟩8, 6) = φ(8, 0, 6) is least α such as

α = φ(⟨2⟩7, ⟨1⟩α) = φ(7, α, 0) and α > φ(8, 0, 5)

φ(8, 0, 6)[0] = φ(7, 0, φ(8, 0, 5) + 1)

φ(8, 0, 6)[1] = φ(7, φ(7, 0, φ(8, 0, 5) + 1), 0)

φ(8, 0, 6)[2] = φ(7, φ(7, φ(7, 0, φ(8, 0, 5) + 1), 0), 0)

φ(8, 0, 6)[3] = φ(7, φ(7, φ(7, φ(7, 0, φ(8, 0, 5) + 1), 0), 0), 0)

φ(8, 0, 6)[4] = φ(7, φ(7, φ(7, φ(7, φ(7, 0, φ(8, 0, 5) + 1), 0), 0), 0), 0)

φ(8, 0, 6)[5] = φ(7, φ(7, φ(7, φ(7, φ(7, φ(7, 0, φ(8, 0, 5) + 1), 0), 0), 0), 0), 0)

...

4.2.1
φ(⟨⟨3⟩4⟩8) is least α such as

α = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩α⟩1)

φ(⟨⟨3⟩4⟩8)[0] = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3⟩1)

φ(⟨⟨3⟩4⟩8)[1] = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3⟩1)⟩1)

φ(⟨⟨3⟩4⟩8)[2] = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3⟩1)⟩1)⟩1)

φ(⟨⟨3⟩4⟩8)[3] = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3⟩1)⟩1)⟩1)⟩1)

φ(⟨⟨3⟩4⟩8)[4] = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3⟩1)⟩1)⟩1)⟩1)⟩1)

φ(⟨⟨3⟩4⟩8)[5] = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3⟩1)⟩1)⟩1)⟩1)⟩1)⟩1)

...

4.2.2
φ(⟨⟨3⟩4⟩8, 6) is least α such as

α = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩α⟩1) and α > φ(⟨⟨3⟩4⟩8, 5)

φ(⟨⟨3⟩4⟩8, 6)[0] = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3⟩1, φ(⟨⟨3⟩4⟩8, 5) + 1)

φ(⟨⟨3⟩4⟩8, 6)[1] = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3⟩1, φ(⟨⟨3⟩4⟩8, 5) + 1)⟩1)

φ(⟨⟨3⟩4⟩8, 6)[2] = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3⟩1, φ(⟨⟨3⟩4⟩8, 5) + 1)⟩1)⟩1)

φ(⟨⟨3⟩4⟩8, 6)[3] = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3⟩1, φ(⟨⟨3⟩4⟩8, 5) + 1)⟩1)⟩1)⟩1)

φ(⟨⟨3⟩4⟩8, 6)[4] = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3⟩1, φ(⟨⟨3⟩4⟩8, 5) + 1)⟩1)⟩1)⟩1)⟩1)

φ(⟨⟨3⟩4⟩8, 6)[5] = φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3, ⟨2⟩φ(⟨⟨3⟩4⟩7, ⟨⟨3⟩3⟩1, φ(⟨⟨3⟩4⟩8, 5) + 1)⟩1)⟩1)⟩1)⟩1)⟩1)

...

4.3.1
φ(⟨⟨⟨19⟩12⟩4⟩8) is least α such as

α = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩α⟩1⟩1)

φ(⟨⟨⟨19⟩12⟩4⟩8)[0] = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11⟩1⟩1)

φ(⟨⟨⟨19⟩12⟩4⟩8)[1] = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11⟩1⟩1)⟩1⟩1)

φ(⟨⟨⟨19⟩12⟩4⟩8)[2] = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11⟩1⟩1)⟩1⟩1)⟩1⟩1)

φ(⟨⟨⟨19⟩12⟩4⟩8)[3] = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11⟩1⟩1)⟩1⟩1)⟩1⟩1)⟩1⟩1)

φ(⟨⟨⟨19⟩12⟩4⟩8)[4] = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11⟩1⟩1)⟩1⟩1)⟩1⟩1)⟩1⟩1)⟩1⟩1)

φ(⟨⟨⟨19⟩12⟩4⟩8)[5] = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11⟩1⟩1)⟩1⟩1)⟩1⟩1)⟩1⟩1)⟩1⟩1)⟩1⟩1)

...

4.3.2
φ(⟨⟨⟨19⟩12⟩4⟩8, 6) is least α such as

α = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩α⟩1⟩1) and α > φ(⟨⟨⟨19⟩12⟩4⟩8, 5)

φ(⟨⟨⟨19⟩12⟩4⟩8, 6)[0] = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11⟩1⟩1, φ(⟨⟨⟨19⟩12⟩4⟩8, 5) + 1)

φ(⟨⟨⟨19⟩12⟩4⟩8, 6)[1] = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11⟩1⟩1, φ(⟨⟨⟨19⟩12⟩4⟩8, 5) + 1)⟩1⟩1)

φ(⟨⟨⟨19⟩12⟩4⟩8, 6)[2] = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11⟩1⟩1, φ(⟨⟨⟨19⟩12⟩4⟩8, 5) + 1)⟩1⟩1)⟩1⟩1)

φ(⟨⟨⟨19⟩12⟩4⟩8, 6)[3] = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11⟩1⟩1, φ(⟨⟨⟨19⟩12⟩4⟩8, 5) + 1)⟩1⟩1)⟩1⟩1)⟩1⟩1)

φ(⟨⟨⟨19⟩12⟩4⟩8, 6)[4] = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11⟩1⟩1, φ(⟨⟨⟨19⟩12⟩4⟩8, 5) + 1)⟩1⟩1)⟩1⟩1)⟩1⟩1)⟩1⟩1)

φ(⟨⟨⟨19⟩12⟩4⟩8, 6)[5] = φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11, ⟨18⟩φ(⟨⟨⟨19⟩12⟩4⟩7, ⟨⟨⟨19⟩12⟩3, ⟨⟨19⟩11⟩1⟩1, φ(⟨⟨⟨19⟩12⟩4⟩8, 5) + 1)⟩1⟩1)⟩1⟩1)⟩1⟩1)⟩1⟩1)⟩1⟩1)

...

4.4.1
φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8) is least α such as

α = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩α⟩1⟩1⟩1)

φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8)[0] = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18⟩1⟩1⟩1)

φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8)[1] = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18⟩1⟩1⟩1)⟩1⟩1⟩1)

φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8)[2] = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)

φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8)[3] = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)

φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8)[4] = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)

φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8)[5] = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)

...

4.4.2
φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 6) is least α such as

α = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩α⟩1⟩1⟩1) and α > φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 5)

φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 6)[0] = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18⟩1⟩1⟩1, φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 5) + 1)

φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 6)[1] = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18⟩1⟩1⟩1, φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 5) + 1)⟩1⟩1⟩1)

φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 6)[2] = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18⟩1⟩1⟩1, φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 5) + 1)⟩1⟩1⟩1)⟩1⟩1⟩1)

φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 6)[3] = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18⟩1⟩1⟩1, φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 5) + 1)⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)

φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 6)[4] = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18⟩1⟩1⟩1, φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 5) + 1)⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)

φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 6)[5] = φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18, ⟨4⟩φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨5⟩19⟩12⟩3, ⟨⟨⟨5⟩19⟩11, ⟨⟨5⟩18⟩1⟩1⟩1, φ(⟨⟨⟨⟨5⟩19⟩12⟩4⟩8, 5) + 1)⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)⟩1⟩1⟩1)

...

4.5.1
φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8) is least α such as

α = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩α⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8)[0] = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8)[1] = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8)[2] = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8)[3] = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8)[4] = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8)[5] = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)

...

4.5.2
φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 6) is least α such as

α = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩α⟩1⟩1⟩1⟩1) and α > φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 5)

φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 6)[0] = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 5) + 1)

φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 6)[1] = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 5) + 1)⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 6)[2] = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 5) + 1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 6)[3] = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 5) + 1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 6)[4] = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 5) + 1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)

φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 6)[5] = φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4, ⟨ω⟩φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩7, ⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩3, ⟨⟨⟨⟨ω + 1⟩5⟩19⟩11, ⟨⟨⟨ω + 1⟩5⟩18, ⟨⟨ω + 1⟩4⟩1⟩1⟩1⟩1, φ(⟨⟨⟨⟨⟨ω + 1⟩5⟩19⟩12⟩4⟩8, 5) + 1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)⟩1⟩1⟩1⟩1)

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