FANDOM


Alpha Function

This version contains errors and has been replaced by Version 3.


Program Code Version 2

The Alpha Function has been defined using program code shown below.

This code replaces Version 1 and was written during Late Feb 2016 to align the code to changes in my blog on Fundamental Sequences.

This VBA visual basic code, will run as a macro in Microsoft Excel. This function creates a string literal of a \(J_8\) Function equal to the Alpha function with any Real number input. The program does not attempt to evaluate the function and the run time is therefore very fast.

Option Explicit
Dim vReal As Double, vMax As Integer, vCount As Integer
Const vSize = 100
Dim vArray(0 To vSize, 0 To 7) As Integer

Function Alpha(r As Double, s As Boolean) As String
Dim i As Integer, j As Integer

vMax = 0
vCount = 0

vReal = Log(r) / Log(10)
vMax = Int(vReal)

vReal = vReal - vMax

i = vMax
vGamma i, 0, vInit(0, 0), False

j = vMax
i = IIf(j = 0, 0, 2) + vGet6(0)
If j + 1 > i Then j = j + 1 Else j = i

If s Then
    Alpha = "f_{" & vVeblen(0) & "}^{" & i & "}("
Else
    Alpha = "J_8(" & vRead(0) & "," & i & ","
    End If

Alpha = "\(" & Alpha & vGet6(0) + IIf(i > 0, j, 0) & ")\)"
End Function

Function vGetI(i As Integer, j As Integer) As Integer

vReal = vReal * (j - i + 1)
vGetI = Int(vReal)

vReal = vReal - vGetI
End Function

Function vGet6(j As Integer) As Integer
Dim k As Integer

If j = 0 Then
    vGet6 = 0
    Do: k = vGetI(0, 5)
        vGet6 = vGet6 + IIf(k = 5, 4, k)
        Loop Until k < 5
Else
    vGet6 = vGetI(0, j)
    End If

If vGet6 > vMax Then vMax = vGet6
End Function

Function vInit(vgR As Integer, vgC As Integer) As Integer

vInit = vCount
vCount = vCount + 1

vArray(vInit, 0) = 0
vArray(vInit, 1) = 0
vArray(vInit, 2) = 0
vArray(vInit, 3) = 0
vArray(vInit, 4) = 0
vArray(vInit, 5) = 0

vArray(vInit, 6) = vgR
vArray(vInit, 7) = vgC
End Function

Function vGamma(q As Integer, vgA As Integer, vgR As Integer, vShadow As Boolean) As Integer
Dim k As Integer, m As Integer, v As Integer

vGamma = vgR
vArray(vgR, 0) = q

v = vgA

If q = 0 Then
    vArray(vgR, 5) = vGet6(0) + v
Else
    If q > 1 Then
        vArray(vgR, 1) = -vInit(vgR, 1)
        
        For k = 2 To q
            vInit vgR, 1
            Next k
                
        For k = 1 To q
            vGamma vGet6(q), IIf(k = 1, 1, 0), -vArray(vgR, 1) - 1 + k, True
            Next k
        End If
        
    If Not vShadow Then
        vArray(vgR, 2) = vGet6(0) + 1
    Else
        k = vArray(vArray(vgR, 6), 2)
        If k = 1 Then
            vArray(vgR, 2) = 1
        Else
            vArray(vgR, 2) = vGet6(k - 1) + 1
            vShadow = (vArray(vgR, 2) = k)
            End If
        End If
    
    If q > 1 Or (q = 1 And vArray(vgR, 2) > 1) Then k = vGet6(q) Else k = 0
    If Not vShadow Then
        If k = 0 Then k = vGet6(0) + 1 Else k = -vGamma(k, 1, vInit(vgR, 3), True)
    Else
        If k = 0 Then
            m = vArray(vArray(vgR, 6), 3) - 2
            If m = 0 Then
                k = 1
            Else
                k = vGet6(m) + 1
                End If
            vShadow = (k = m + 1)
        Else
            k = -vGamma(k, 1, vInit(vgR, 3), True)
            End If
        End If
    vArray(vgR, 3) = k
        
    If k <> 1 Then v = 0
        
    If vTest(vgR, 2) Then k = vGet6(q) Else k = 0
    If k = 0 Then k = vGet6(0) + 1 Else k = -vGamma(k, 1, vInit(vgR, 4), True)
    vArray(vgR, 4) = k
    
    If k <> 1 Then v = 0
    
    If vTest(vgR, 3) Then k = vGet6(q) Else k = 0
    If k = 0 Then k = vGet6(0) + v Else k = -vGamma(k, v, vInit(vgR, 5), True)
    vArray(vgR, 5) = k
    End If
End Function

Function vTest(vtR As Integer, vtC As Integer) As Boolean

vTest = True
If vtC = 3 And vTestA(vtR, 3) Then If vTestA(vtR, 2) Then vTest = Not vTestA(vtR, 0)

If vtC = 2 And vTestA(vtR, 2) Then vTest = Not vTestA(vtR, 0)
End Function
Function vTestA(vtR As Integer, vtC As Integer) As Boolean

vTestA = (vArray(vtR, vtC) = 1)
End Function

Function vRead(vrRow As Integer) As String
Dim i As Integer

i = vArray(vrRow, 0)
Select Case i
Case 0
    vRead = "<" & vArray(vrRow, 0) & "," & vArray(vrRow, 5) & ">"
Case 1
    vRead = "<" & vArray(vrRow, 0) & "," & vArray(vrRow, 2) & "," & vCells(vrRow) & ">"
Case Else
    vRead = "<" & vArray(vrRow, 0) & "," & vReadL(-vArray(vrRow, 1), i) & "," & vArray(vrRow, 2) & "," & vCells(vrRow) & ">"
    End Select
End Function

Function vCells(vcRow As Integer) As String

vCells = vCell(vArray(vcRow, 3)) & "," & vCell(vArray(vcRow, 4)) & "," & vCell(vArray(vcRow, 5))
End Function
Function vCell(vcI As Integer) As String

If vcI >= 0 Then vCell = "<0," & vcI & ">" Else vCell = vRead(-vcI)
End Function

Function vReadL(vrRow As Integer, vrRows As Integer) As String
Dim i As Integer

vReadL = "<"
For i = vrRow To vrRow + vrRows - 1
    vReadL = vReadL & vRead(i) & IIf(i = vrRow + vrRows - 1, ">", ",")
    Next i
End Function

Function vVeblen(vvRow As Integer) As String
Dim i As Integer, j As Integer, s As String

i = vArray(vvRow, 0)
Select Case i
Case 0
    vVeblen = vArray(vvRow, 5)
    Exit Function
Case 1
    vVeblen = "\omega"
    
Case Else
    vVeblen = vLambda(-vArray(vvRow, 1), i)
    End Select

If vArray(vvRow, 2) > 1 Then vVeblen = "(" & vVeblen & "\uparrow\uparrow " & vArray(vvRow, 2) & ")"


For j = 3 To 5
    i = vArray(vvRow, j)
    If i >= 0 Then s = i Else s = vVeblen(-i)
    
    Select Case j
    Case 3
        If i <> 1 Then vVeblen = vVeblen & "^{" & s & "}"
    Case 4
        If i <> 1 Then
            If i < 0 Then s = "(" & s & ")"
            vVeblen = vVeblen & "." & s
            End If
    Case 5
        If i <> 0 Then vVeblen = vVeblen & " + " & s
        End Select
    Next j
End Function

Function vLambda(vlRow As Integer, vlRows As Integer) As String
Dim i As Integer

vLambda = "\varphi("
For i = vlRow To vlRow + vlRows - 1
    vLambda = vLambda & vVeblen(i) & IIf(i = vlRow + vlRows - 1, ")", ",")
    Next i
End Function


How the Function Works

A description of how the code works will be provided here ... Work in Progress.

  • VBA Constants
  • VBA Data Structures
  • VBA Functions
    • Alpha Function
    • Work In Progress


Test Bed for Version 3

Version 2 contains quite a few errors which will be fixed. Below is the test bed for the next version. It uses a test version (2.5) so the examples below will not necessarily be the same results as the version 2 code above.

\(\alpha(100) = f_{\varphi(1,0)}^{2}(2)\)

  • Comments: The Alpha function uses Log10 to set the initial value q. In this case \(\alpha(100) == q=2\) and because the remainder is zero, then the rest of the function is derived by definition.

\(\alpha(100.1) = f_{(\varphi(1,0)\uparrow\uparrow 2)^{5} + 5}^{7}(25)\)

  • Comments: This is a complex result but also a rare example of a legible FGH function. It is equivalent to:
    • \(= f_{\epsilon_0^{\epsilon_0.5} + 5}^{7}(25)\)

\(\alpha(100.2) = f_{(\varphi(1,0)\uparrow\uparrow 3)^{\omega^{4}}.((\omega\uparrow\uparrow 2)^{3}.(\omega^{10}.6 + \omega^{6} + \omega^{2}.4 + 2) + 2) + (\varphi(\omega^{0} + 4,11)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).(\omega^{5}.4 + 1) + 3}.(\omega^{4}.9 + 5) + \omega^{6}.6 + \omega^{4}.3 + 5}.5 + \varphi^2(12_*,1)^{\omega^{0}.3 + \omega^{1}.4 + 3}.4 + \omega^{1}.3 + 5}^{6}(18)\)

  • Comments: This is a typical result of an unwieldy FGH function. It is also incorrect. The Alpha function program code will rely on generating a 'root' Veblen function which will appear first in the FGH ordinal sequence. For all Alpha function values from 100 to 100.5891227, the root Veblen function will be \(\varphi(1,0) = \epsilon_0\) and all other occurrences of ordinals will be strictly less than the root or in special cases the same value but never greater.
  • Therefore the following ordinals in this result are wrong and due to 1st Error in the program code:
    • \(\varphi(\omega^{0} + 4,11)\uparrow\uparrow 2\) and
    • \(\varphi^2(12_*,1)\)
  • The 2nd Error in the program code generates a 0 value exponent for \(\omega\) in the first of these ordinals that will also have to be fixed.

\(\alpha(100.3) = f_{(\varphi(1,0)\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 3).2 + 3}.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).4 + \omega.3}.(\omega.12 + 5) + 9}.18 + (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).4 + \omega.3}.(\omega.12 + 5) + 9}.(\omega^{4}.2 + 1) + (\omega\uparrow\uparrow 2)^{0}.5) + (\omega\uparrow\uparrow 2)^{4}.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{1}.17 + 4}.12 + 9) + (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2) + 1}}}^{2}(19)\)

  • Comments: This result illustrates how difficult to read these FGH functions can become. However, the few errors in the result seem to be the 0 value exponent (2nd Error) identified in the last comment and 1 other. The ordinals follow a descending sequence defined in my Extended Normal Form blog. The descending ordinals begin with a root ordinal:
    • \((\varphi(1,0)\uparrow\uparrow 4)^{\gamma_e}\) where
    • \(\gamma_e = (\omega\uparrow\uparrow 3)^{\gamma_{e_2}}.\gamma_c + \gamma_a\) where
      • \(\gamma_{e_2} = (\omega\uparrow\uparrow 3).2 + 3\)
      • \(\gamma_c = (\omega\uparrow\uparrow 2)^{\gamma_{e_3}}.\gamma_{c_2} + \gamma_{a_2}\) where
        • \(\gamma_{e_3} = (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).4 + \omega.3}.(\omega.12 + 5) + 9\)
        • \(\gamma_{c_2} = 18\)
        • \(\gamma_{a_2} = (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).4 + \omega.3}.(\omega.12 + 5) + 9}.(\omega^{4}.2 + 1) + (\omega\uparrow\uparrow 2)^{0}.5\)
      • \(\gamma_a = (\omega\uparrow\uparrow 2)^{4}.((\omega\uparrow\uparrow 2)^{\omega^{\omega}.17 + 4}.12 + 9) + (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2) + 1}\)
  • The 3rd Error occurs in the value of \(\gamma_{e_2}\) which should not equal or exceed the value of \(\omega^{\omega}\), for reasons explained in my Extended Normal Form blog. The same error recurs throughout the rest of the result.

\(\alpha(100.5) = f_{(\varphi(1,0)\uparrow\uparrow 6)^{(\varphi(\omega^{0}.3 + 5,3)\uparrow\uparrow 3)^{4}.(\omega^{3}.10 + 5)}.((\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{0}.5 + (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega^{4}.6 + \omega.9 + 5}.9 + (\omega\uparrow\uparrow 2) + (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).(\omega^{6}.2 + 1) + (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{1}.(\omega^{5}.3 + \omega.6 + 1) + (\omega\uparrow\uparrow 2)}}}}})}^{2}(11)\)

\(\alpha(100.51) = f_{(\varphi(1,0)\uparrow\uparrow 7)^{(\omega\uparrow\uparrow 3)^{3}.2 + 5}.((\varphi(1,\omega^{0}.10 + 3)\uparrow\uparrow 7)^{(\omega\uparrow\uparrow 7) + 4}.(\omega.7 + 1) + 5) + \varphi(1,(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega.6 + 1}.((\omega\uparrow\uparrow 2)^{0}.(\omega^{4}.3) + (\omega\uparrow\uparrow 2)^{1}.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega^{6}.2 + \omega^{4}.3 + \omega.2 + 2}.2 + 3}.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega + 1}}})))})}^{2}(11)\)

\(\alpha(100.52) = f_{(\varphi(1,0)\uparrow\uparrow 8)^{6}.((\omega\uparrow\uparrow 6)^{(\omega\uparrow\uparrow 5)^{\omega^{4}.11 + 2}.((\omega\uparrow\uparrow 5)^{2}.3) + 4} + 19) + (\varphi(\omega^{0} + \omega^{0}.6 + \omega^{1}.17 + 1,5)\uparrow\uparrow 7)^{7}.((\varphi^6(4_*,(\omega\uparrow\uparrow 4)^{\omega^{5}.3 + \omega^{3}.6 + 4}.9 + (\omega\uparrow\uparrow 3)^{5}.6 + 3)\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3) + 1})}^{2}(20)\)

\(\alpha(100.53) = f_{(\varphi(1,0)\uparrow\uparrow 9)^{3} + 2}^{3}(10)\)

\(\alpha(100.54) = f_{(\varphi(1,0)\uparrow\uparrow 9)^{(\varphi^5(\omega^{0}.4 + 1_*,\omega^{0} + 3)\uparrow\uparrow 5)^{(\varphi((\omega\uparrow\uparrow 2)^{4}.6 + 1,3)\uparrow\uparrow 5)^{(\varphi^2(\omega^{0}.2 + \omega^{1}.5 + \omega.6 + 2_*,\omega^{1}.3 + 1)\uparrow\uparrow 4)^{6}.((\varphi(3,2)\uparrow\uparrow 2)^{5}.6 + \varphi((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{0}.5 + (\omega\uparrow\uparrow 2)^{1}.(\omega^{4}.2 + \omega^{2}.7 + \omega.6 + 1) + (\omega\uparrow\uparrow 2)},0))}}}}^{2}(10)\)

\(\alpha(100.55) = f_{(\varphi(1,0)\uparrow\uparrow 10)^{(\varphi^2(4_*,\omega^{0}.3 + 11)\uparrow\uparrow 7)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^{\omega^{5}.2}.((\omega\uparrow\uparrow 2)^{\omega^{6}.4 + 3}.((\omega\uparrow\uparrow 2)^{4}.4 + \omega^{5}.11 + \omega.3 + 12) + (\omega\uparrow\uparrow 2)^{\omega^{11}.2 + 3}.(\omega^{5}.2 + \omega^{2}.6 + 2) + (\omega\uparrow\uparrow 2)^{9}.5 + (\omega\uparrow\uparrow 2)^{\omega^{3}.2 + 4}.6 + 3) + \omega}}}}^{2}(13)\)

\(\alpha(100.56) = f_{(\varphi(1,0)\uparrow\uparrow 11)^{(\omega\uparrow\uparrow 9)^{5} + 7}.12 + (\omega\uparrow\uparrow 9)^{(\omega\uparrow\uparrow 4)^{5} + 3}.(\omega^{3}.12 + \omega.2 + 8)}^{6}(15)\)

\(\alpha(100.57) = f_{(\varphi(1,0)\uparrow\uparrow 12)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 3)^{0}.3 + 2}.(\omega^{6}.5 + 2) + 4}.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{0}.((\omega\uparrow\uparrow 2)^{\omega^{2}.6 + \omega.4 + 2}.((\omega\uparrow\uparrow 2).6 + 4) + 1) + \omega.2 + 4}.(\omega^{2}.5 + \omega + 4) + (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega^{6}.3 + \omega^{4}.7}.3 + \omega^{6} + 3} + (\omega\uparrow\uparrow 2)^{\omega + 1})}^{2}(13)\)

\(\alpha(100.58) = f_{(\varphi(1,0)\uparrow\uparrow 14)^{(\varphi^2(2_*,3)\uparrow\uparrow 8)^{\varphi((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2).2 + (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{3}.(\omega^{5}.3 + 5) + 5}.((\omega\uparrow\uparrow 2)^{\omega^{5} + 3}.4 + 10) + 3}.((\omega\uparrow\uparrow 3)^{\omega^{5}.5}.(\omega^{3}.3) + \omega^{4}.6 + \omega^{2}.3 + \omega.3) + (\omega\uparrow\uparrow 3)^{\omega^{4}.2 + \omega^{2}.7 + \omega.6 + 1}.((\omega\uparrow\uparrow 2)^{\omega + 1}),0) + 1}}}^{2}(15)\)

\(\alpha(100.581) = f_{(\varphi(1,0)\uparrow\uparrow 15)^{6}.((\varphi(\omega^{0}.2 + 1,\omega^{0} + \omega^{0}.4 + 3)\uparrow\uparrow 12).(\omega^{10}.6 + \omega^{6}.3 + 5) + (\omega\uparrow\uparrow 9)^{(\omega\uparrow\uparrow 9)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 4)^{\omega^{2}.7 + 4}.9 + \omega^{11}.3 + \omega^{3}.5 + 4}.2 + \omega^{3}.5 + \omega.2 + 4}.3 + (\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3)^{2}.(\omega^{2}.3 + 2) + \omega^{6}.2 + \omega^{3}}})}^{2}(16)\)

\(\alpha(100.582) = f_{(\varphi(1,0)\uparrow\uparrow 15)^{(\varphi^2(\omega^{0}.6 + \omega^{0}.3 + 3_*,\omega^{0}.4 + \omega^{0}.20 + \omega^{1}.6 + 2)\uparrow\uparrow 7)^{(\omega\uparrow\uparrow 4)^{\omega^{13}.2}.4 + \omega + 16}.((\varphi((\omega\uparrow\uparrow 4)^{5}.2 + \omega^{11} + \omega^{4}.4 + \omega^{2}.2 + 2,3)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2).3 + 4}.(\varphi(4,5)^{8}.(\varphi^2(\omega^{1}.7 + 13_*,\omega^{0}.6 + 3)^{\omega^{1}.4 + \omega})))}}^{2}(21)\)

\(\alpha(100.583) = f_{(\varphi(1,0)\uparrow\uparrow 16)^{(\omega\uparrow\uparrow 6)^{6}.6 + (\omega\uparrow\uparrow 6)^{4}.6 + (\omega\uparrow\uparrow 5)^{2}.2 + 3}.2}^{3}(20)\)

\(\alpha(100.584) = f_{(\varphi(1,0)\uparrow\uparrow 17).((\omega\uparrow\uparrow 11)^{(\omega\uparrow\uparrow 11)^{0}.2 + 3}.((\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3)^{\omega^{2}.6 + 2}.(\omega^{5}.5 + \omega^{2}.4 + 2) + (\omega\uparrow\uparrow 3)^{\omega + 2}.((\omega\uparrow\uparrow 2).((\omega\uparrow\uparrow 2)^{\omega^{6}.5 + 3}.5) + \omega^{2}.4 + 2) + 1}.5 + 5}.((\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{0}.(\omega^{6}.10 + \omega^{4}.3 + 2) + 3}.6 + 9}.((\omega\uparrow\uparrow 5)^{\omega^{6}.5 + 3}.4 + \omega))))}^{2}(18)\)

\(\alpha(100.585) = f_{(\varphi(1,0)\uparrow\uparrow 17)^{(\omega\uparrow\uparrow 14)^{(\omega\uparrow\uparrow 13)^{5}.6 + \omega^{4}.3 + 5}.2 + (\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{6}.3 + \omega^{2}.2 + 6}.((\omega\uparrow\uparrow 4)^{5}.((\omega\uparrow\uparrow 2)^{\omega^{4}.5 + \omega^{2}}.((\omega\uparrow\uparrow 2).((\omega\uparrow\uparrow 2)^{0}.6 + 3) + 10)) + (\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega^{4}.4 + \omega}}})}}^{2}(18)\)

\(\alpha(100.586) = f_{(\varphi(1,0)\uparrow\uparrow 18)^{4}.((\varphi^2(\omega^{0}.3 + \omega^{1}.4_*,\omega^{0}.3 + \omega^{1}.2 + 4)\uparrow\uparrow 18)^{\varphi^2(5_*,(\omega\uparrow\uparrow 12).((\omega\uparrow\uparrow 6)^{(\omega\uparrow\uparrow 3)^{7}.8 + 12}.2 + (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{0}.((\omega\uparrow\uparrow 2)^{\omega^{2}.12 + 9}.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2) + 1}))})) + 1})}^{2}(19)\)

\(\alpha(100.587) = f_{(\varphi(1,0)\uparrow\uparrow 18)^{(\varphi(5,11)\uparrow\uparrow 17)^{\varphi((\omega\uparrow\uparrow 11)^{(\omega\uparrow\uparrow 6)^{5}.((\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{\omega^{8}.5 + 5}.((\omega\uparrow\uparrow 2)^{4}.((\omega\uparrow\uparrow 2)^{2}.((\omega\uparrow\uparrow 2)^{\omega^{6} + 2}.((\omega\uparrow\uparrow 2)^{\omega^{4}.12 + \omega^{2}.5 + \omega.2 + 1}.(\omega^{12}.4 + \omega^{8}.6 + 5) + 2) + (\omega\uparrow\uparrow 2)^{2} + 6) + (\omega\uparrow\uparrow 2)^{\omega + 1}))})},0) + 1}}}^{2}(19)\)

\(\alpha(100.588) = f_{(\varphi(1,0)\uparrow\uparrow 21)^{(\omega\uparrow\uparrow 7)^{(\omega\uparrow\uparrow 2)^{2}.(\omega^{3}.4 + 5) + 1}.3 + \omega.6 + 1}.((\varphi(4,\omega^{0}.9 + \omega^{1}.2 + \omega.6 + 2)\uparrow\uparrow 11)^{(\omega\uparrow\uparrow 2)^{4}.(\omega^{3} + 12) + 2}.((\varphi^2(1_*,\omega^{2} + 4)\uparrow\uparrow 8)^{(\varphi(\omega^{5} + \omega^{3}.5 + \omega.15 + 4,4)\uparrow\uparrow 6)^{4}.((\omega\uparrow\uparrow 4)^{\omega^{7}.2 + 1}.6 + (\omega\uparrow\uparrow 2)^{\omega^{7}.7 + \omega^{4}.6 + 3}.(\omega + 1))}))}^{2}(22)\)

\(\alpha(100.589) = \)

\(\alpha(100.5891) = f_{(\varphi(1,0)\uparrow\uparrow 33)^{(\omega\uparrow\uparrow 15)^{(\omega\uparrow\uparrow 2)^{\omega^{2}.4 + 9}.6 + 2}.2 + 4}.2 + (\varphi^2(\omega^{0}.6 + 2_*,5)\uparrow\uparrow 13)^{9}.6 + 4}^{5}(34)\)

\(\alpha(100.58911) = f_{(\varphi(1,0)\uparrow\uparrow 35)^{(\omega\uparrow\uparrow 15)^{\omega^{2}.3 + 2}.5 + (\omega\uparrow\uparrow 6)^{4}.7}.((\omega\uparrow\uparrow 15)^{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{6}.((\omega\uparrow\uparrow 2)^{2}.2 + 3) + 2}.2 + (\omega\uparrow\uparrow 4)^{5}.3 + 4}.((\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^{2}.3 + 1}.3 + 1) + 3) + (\varphi(\omega^{0}.11 + \omega^{1}.3 + 3,\omega^{0}.4 + 5)\uparrow\uparrow 20)^{\varphi^2(4_*,(\omega\uparrow\uparrow 15)^{\omega^{10}.10 + \omega^{8}}) + 1}}^{2}(36)\)

\(\alpha(100.58912) = f_{(\varphi(1,0)\uparrow\uparrow 40)^{6} + 4}^{5}(45)\)

\(\alpha(100.589121) = f_{(\varphi(1,0)\uparrow\uparrow 41)^{(\omega\uparrow\uparrow 37)^{(\omega\uparrow\uparrow 22)^{2}.7 + (\omega\uparrow\uparrow 18)^{3}.((\omega\uparrow\uparrow 12)^{6}.3 + 5) + (\omega\uparrow\uparrow 2)^{2}.5} + (\omega\uparrow\uparrow 17)^{6} + 1}.((\omega\uparrow\uparrow 9)^{24}.((\omega\uparrow\uparrow 8)^{6}.(\omega^{2}.12 + \omega) + 5) + (\omega\uparrow\uparrow 2)^{\omega^{6}.6 + \omega^{3}.3 + 1}.6 + 4) + (\omega\uparrow\uparrow 39)^{(\omega\uparrow\uparrow 21)^{2}.((\omega\uparrow\uparrow 2)^{5}.6 + 3) + \omega}}^{2}(42)\)

\(\alpha(100.589122) = f_{(\varphi(1,0)\uparrow\uparrow 43)^{4}.((\varphi^2(\omega^{0}.7 + \omega^{1} + 5_*,\omega^{0}.3 + \omega^{1}.4 + 13)\uparrow\uparrow 7)^{5} + 5) + (\varphi^2(6_*,(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 2)^{\omega^{2}}.(\omega^{3} + 10) + (\omega\uparrow\uparrow 2).(\omega^{16}.3 + 1) + 1}.2)\uparrow\uparrow 39)^{(\omega\uparrow\uparrow 23).((\omega\uparrow\uparrow 7)^{(\omega\uparrow\uparrow 5).4 + (\omega\uparrow\uparrow 3).5 + (\omega\uparrow\uparrow 3)^{\omega^{4} + 3}.8 + \omega^{6}.9 + 5}.((\omega\uparrow\uparrow 3)^{\omega^{6}.2 + \omega^{3}}))}}^{2}(44)\)

\(\alpha(100.5891221) = f_{(\varphi(1,0)\uparrow\uparrow 44)^{4}.((\omega\uparrow\uparrow 2)^{4}.(\omega^{6} + \omega^{3}.3 + \omega.7 + 3) + 1) + (\omega\uparrow\uparrow 33)^{3}.4 + (\omega\uparrow\uparrow 29)^{6}.7 + 8}^{3}(46)\)

\(\alpha(100.5891222) = f_{(\varphi(1,0)\uparrow\uparrow 45)^{3}.((\varphi(17,10)\uparrow\uparrow 38)^{(\varphi^3((\omega\uparrow\uparrow 3)^{\omega^{2}.6 + \omega + 9}.3_*,(\omega\uparrow\uparrow 21)^{\omega^{6} + 4}.3 + 11)\uparrow\uparrow 3)^{3}.((\varphi^2(\omega^{0}.6 + 5_*,11)\uparrow\uparrow 3)^{(\varphi^10(5_*,\omega.4 + 5)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{0}.2 + 1}.5 + 5}.6 + 5}.6 + 5) + 5}.6 + 5) + 5}^{3}(51)\)

\(\alpha(100.5891223) = f_{(\varphi(1,0)\uparrow\uparrow 46)^{3}.12 + 4}^{3}(51)\)

\(\alpha(100.5891224) = f_{(\varphi(1,0)\uparrow\uparrow 47)^{2}.((\varphi(7,5)\uparrow\uparrow 32)^{\varphi((\omega\uparrow\uparrow 10)^{2}.((\omega\uparrow\uparrow 7)^{3}.2 + 1) + (\omega\uparrow\uparrow 4)^{3}.((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^{6}.8 + \omega^{4}.4 + \omega.5 + 6}.((\omega\uparrow\uparrow 2)^{3}.(\omega^{4}.6 + \omega.6 + 4) + 6) + 1}.(\omega^{2}.3 + 2) + \omega^{6}.2 + \omega^{3}),0) + 1})}^{2}(48)\)

\(\alpha(100.5891225) = f_{(\varphi(1,0)\uparrow\uparrow 48)^{2}.9}^{2}(49)\)

\(\alpha(100.5891226) = f_{(\varphi(1,0)\uparrow\uparrow 49)^{6}.((\omega\uparrow\uparrow 28)^{4}.((\omega\uparrow\uparrow 6)^{4}.((\omega\uparrow\uparrow 3)^{11}.((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega^{5}.4 + \omega^{3}.6 + 4}.6 + \omega^{4}.3 + 2}.5 + \omega^{10}.3 + 1}.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{2} + (\omega\uparrow\uparrow 2).((\omega\uparrow\uparrow 2)^{0}.(\omega^{6} + \omega.6 + 4) + (\omega\uparrow\uparrow 2)^{0})})))))}^{2}(50)\)

\(\alpha(100.5891227) = f_{\varphi(1,1)}^{2}(2)\)

Work in Progress

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