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Sequence Generator Code Version 1

Please refer to my latest blog and Version 2 of this code instead.

I am developing general purpose code to generate long strings of finite integers according to rules that can be written using the correct syntax.

This code is being used by my Beta Function code.

WORK IN PROGRESS

Sequence Generator Code needs to be be interpreted using a program in a conventional programming language. I am using VBA in Microsoft Excel to do this. The code will be added to this blog shortly. I plan to write a Javascript version in due course and make it available online on a web page.

WORK IN PROGRESS


Sequence Generator Code: Overview

Sequence Generator Code is intended to follow very general rules for generating long sequences of finite integers.

The main technique used to generate sequences is a type of cloning process where any section of the sequence can be used to build (and restrict) the values of another sequence. This technique is used in a heavily recursive process to generate arbitrary lengths of sequences. If the code syntax has been used correctly, then the sequence will be guaranteed to be finite and every integer will be finite.

Here is a very simple illustration of the cloning process used:

SequenceA = (0,1,2,0,0,0,8,1,SequenceB)

SequenceB < (2,0,0,0,8)

In this illustration, SequenceB will be generated as a clone from a section of SequenceA and then appended to SequenceA

The cloning process is straight forward. SequenceB integers may be any value from \(0\) up to the value of the relevant integer in the section being cloned.

SequenceA = (0,1,2,0,0,0,8,1,SequenceB)

SequenceB < (2,0,0,0,8)

SequenceB = (Integer1,...)

Therefore Integer1 may be either \(0, 1, 2\). The cloning process collapses and terminates if any integer in SequenceB is less than SequenceA. The process therefore would terminate if Integer1 was either \(0, 1\).

The cloning process must collapse and terminate at or before the last integer. Therefore, Integer4 for SequenceB must be either \(0, 1, 2, 3, 4, 5, 6, 7\) and never \(8\).

Using just these concepts we can see that SequenceB is restricted to a small number of valid sequences:

(0)

(1)

(2,0,0,0,0)

(2,0,0,0,1)

(2,0,0,0,2)

(2,0,0,0,3)

(2,0,0,0,4)

(2,0,0,0,5)

(2,0,0,0,6)

(2,0,0,0,7)

Refer to the section on Sequence Generator Code Syntax for more detailed explanation. A separate section will be written to explain the selection process for each finite integer ... WORK IN PROGRESS.

Sequence Generator Code needs to be be interpreted using a program in a conventional programming language. I am using VBA in Microsoft Excel to do this. The code will be added to this blog shortly. I plan to write a Javascript version in due course and make it available online on a web page.

WORK IN PROGRESS


Sequence Generator Code: Example

Here is example code from an early version of the Beta Function described on another blog. This version of the code is now superseded. Refer to my other blogs for the latest on this function.

b = (V_r='Beta Function r Parameter',V_v='Beta Function v Parameter',h_0)

h = (C_d<2,C_d=(0:C_x<V_v-U_x,(E_h=1,f_0)))

f = (g_x,g_x=(0:h_u,(h_U,E_h=C_d..h_U,f_x+1<g_x)))

g = (C_q<V_v+1,C_q=(0:h_x,1:t_a,(g_[q]<C_q..g_q-1,n_0<C_q-1,t_a)))

t = (h_T,g_E<C_q..t_x,g_C<C_q..h_T,g_x<C_q..g_E)


Simple Example

A simpler example of the code will be used to explain the syntax used for this code:

n = (V_r='real number',V_v=2,V_e=10,e_0)

e = (C_d<V_v+0,C_d=(0:C_x<V_e+0,(e_E<V_e+0,e_M<V_e+0,e_a<C_d..e_E)))

This simple example can be used to generate numbers using basic Exponentiation \(\uparrow\), Multiplication \(.\) and Addition \(+\). This is not intended to be interesting in itself, but helps to illustrate how the Sequence Generator Code does and how it can be used.

Here are the sequences generated for some real number inputs:

\(e_0=0\) = {"V_r":1,"V_v":2,"V_e":10,"e_0=0":{"C_d":0,"C_0":0}}

\(e_0=1\) = {"V_r":1.2,"V_v":2,"V_e":10,"e_0=1":{"C_d":0,"C_0":1}}

\(e_0=2\) = {"V_r":1.4,"V_v":2,"V_e":10,"e_0=2":{"C_d":0,"C_0":2}}

\(e_0=3\) = {"V_r":1.5,"V_v":2,"V_e":10,"e_0=3":{"C_d":0,"C_0":3}}

\(e_0=4\) = {"V_r":1.6,"V_v":2,"V_e":10,"e_0=4":{"C_d":0,"C_0":4}}

\(e_0=5\) = {"V_r":1.8,"V_v":2,"V_e":10,"e_0=5":{"C_d":0,"C_0":5}}

\(e_0=6\) = {"V_r":2,"V_v":2,"V_e":10,"e_0=6":{"C_d":0,"C_0":6}}

\(e_0=7\) = {"V_r":2.3,"V_v":2,"V_e":10,"e_0=7":{"C_d":0,"C_0":7}}

\(e_0=8\) = {"V_r":2.8,"V_v":2,"V_e":10,"e_0=8":{"C_d":0,"C_0":8}}

\(e_0=9\) = {"V_r":3,"V_v":2,"V_e":10,"e_0=9":{"C_d":0,"C_0":9}}

\(e_0=10\) = {"V_r":3.1623,"V_v":2,"V_e":10,"e_0=10":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=1":{"C_d":0,"C_M":0},"e_a=0":{"C_d":0,"C_a":0}}}

\(e_0=29\) = {"V_r":3.163,"V_v":2,"V_e":10,"e_0=29":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=2":{"C_d":0,"C_M":1},"e_a=9":{"C_d":0,"C_a":9}}}

\(e_0=57\) = {"V_r":3.164,"V_v":2,"V_e":10,"e_0=57":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=5":{"C_d":0,"C_M":4},"e_a=7":{"C_d":0,"C_a":7}}}

\(e_0=84\) = {"V_r":3.165,"V_v":2,"V_e":10,"e_0=84":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=8":{"C_d":0,"C_M":7},"e_a=4":{"C_d":0,"C_a":4}}}

\(e_0=87\) = {"V_r":3.1651,"V_v":2,"V_e":10,"e_0=87":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=8":{"C_d":0,"C_M":7},"e_a=7":{"C_d":0,"C_a":7}}}

\(e_0=90\) = {"V_r":3.1652,"V_v":2,"V_e":10,"e_0=90":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=9":{"C_d":0,"C_M":8},"e_a=0":{"C_d":0,"C_a":0}}}

\(e_0=92\) = {"V_r":3.1653,"V_v":2,"V_e":10,"e_0=92":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=9":{"C_d":0,"C_M":8},"e_a=2":{"C_d":0,"C_a":2}}}

\(e_0=95\) = {"V_r":3.1654,"V_v":2,"V_e":10,"e_0=95":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=9":{"C_d":0,"C_M":8},"e_a=5":{"C_d":0,"C_a":5}}}

\(e_0=98\) = {"V_r":3.1655,"V_v":2,"V_e":10,"e_0=98":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=9":{"C_d":0,"C_M":8},"e_a=8":{"C_d":0,"C_a":8}}}

\(e_0=100\) = {"V_r":3.16556,"V_v":2,"V_e":10,"e_0=100":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=10":{"C_d":0,"C_M":9},"e_a=0":{"C_d":0,"C_a":0}}}

\(e_0=101\) = {"V_r":3.1656,"V_v":2,"V_e":10,"e_0=101":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=10":{"C_d":0,"C_M":9},"e_a=1":{"C_d":0,"C_a":1}}}

\(e_0=102\) = {"V_r":3.16563,"V_v":2,"V_e":10,"e_0=102":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=10":{"C_d":0,"C_M":9},"e_a=2":{"C_d":0,"C_a":2}}}

\(e_0=103\) = {"V_r":3.16567,"V_v":2,"V_e":10,"e_0=103":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=10":{"C_d":0,"C_M":9},"e_a=3":{"C_d":0,"C_a":3}}}

\(e_0=104\) = {"V_r":3.16571,"V_v":2,"V_e":10,"e_0=104":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=10":{"C_d":0,"C_M":9},"e_a=4":{"C_d":0,"C_a":4}}}

\(e_0=105\) = {"V_r":3.16574,"V_v":2,"V_e":10,"e_0=105":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=10":{"C_d":0,"C_M":9},"e_a=5":{"C_d":0,"C_a":5}}}

\(e_0=106\) = {"V_r":3.16578,"V_v":2,"V_e":10,"e_0=106":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=10":{"C_d":0,"C_M":9},"e_a=6":{"C_d":0,"C_a":6}}}

\(e_0=107\) = {"V_r":3.16582,"V_v":2,"V_e":10,"e_0=107":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=10":{"C_d":0,"C_M":9},"e_a=7":{"C_d":0,"C_a":7}}}

\(e_0=108\) = {"V_r":3.16585,"V_v":2,"V_e":10,"e_0=108":{"C_d":1,"e_E=1":{"C_d":0,"C_E":0},"e_M=10":{"C_d":0,"C_M":9},"e_a=8":{"C_d":0,"C_a":8}}}


Sequence Generator Code: Syntax

Sequence Generator Code is intended to follow very general rules for generating long sequences of finite integers. Long sequences of final integers can be used to describe complex objects and mathematical statements, (Refer to my blog on the Beta Function for detailed code of how to generate very large numbers using the full power of the Veblen ordinals and FGH functions).

Other examples for using sequences can be to generate big numbers using simpler mathematical functions like exponentiation, multiplication and addition. This is what the Simple Example code does.

The syntax for Sequence Generator Code will be explained by following this simple example closely.

Building Different Sequence Types

n = (V_r='real number',V_v=2,V_e=10,e_0)

To build complex sequences, it is useful to define different sequence types. The syntax to do this is as follows:

SequenceTypeName = (Instruction1,Instruction2,...,InstructionN)

Every Sequence Type should be defined with instructions in one line of code. The instructions for the first sequence type of the Simple example are:

SequenceTypeName  n

Instruction1      V_r='real number'

Instruction2      V_v=2

Instruction3      V_e=10

Instruction4      e_0

There are only two instruction types used for this Sequence Type. They are:

InstructionTypeVariable  V_VariableName=Expression

InstructionTypeGenerate  SequenceTypeName_SequenceInstanceName

InstructionTypeVariable is only used to set a value to a variable. A Variable is always designated with an Uppercase V followed by a single character to designate the name of the Variable. The syntax requires an equal "=" sign and a simple mathematical expression to calculate the value of the variable. The only type of expressions supported at this time are:

  • Expression=Value
  • Expression=ValueA+ValueB
  • Expression=ValueA-ValueB

InstructionTypeGenerate is used to generate a sequence. In this example, a SequenceTypeName called \(e\) sequence will be generated. The sequence is identified using an SequenceInstanceName which in this case is \(0\).

Please note that this is a very simple example, and so far we have not actually generated any finite integers for our sequence, apart from some descriptive variables.

Generating Integers for the Sequence

e = (C_d<V_v+0,C_d=(0:C_x<V_e+0,(e_E<V_e+0,e_M<V_e+0,e_a<C_d..e_E)))

In the simple example, the finite integers for the sequence are generated by the SequenceTypeName called \(e\) using the following instructions:

SequenceTypeName  e

Instruction1      C_d<V_v+0

Instruction2      C_d=(0:C_x<V_e+0,(e_E<V_e+0,e_M<V_e+0,e_a<C_d..e_E))

Instruction2a            C_x<V_e+0

Instruction2b            (e_E<V_e+0,e_M<V_e+0,e_a<C_d..e_E)

Instruction2b1            e_E<V_e+0

Instruction2b2            e_M<V_e+0

Instruction2b3            e_a<C_d..e_E

Three new Instruction Types are used in for this Sequence Type:

InstructionTypeInteger  C_IntegerName<Expression

InstructionTypeClone    SequenceTypeName_SequenceInstance<SequenceExisting

InstructionTypeSelect   C_IntegerName=(Value1:Instruction1,Value2:Instruction2,...,InstructionN)

InstructionTypeGroup    (Instruction1,Instruction2,...,InstructionN)

WORK IN PROGRESS


Sequence Generator Code: Integer Selection

The Syntax does not explain how the value of an integer is actually selected. The rules define the range of valid values but not the process for selecting one of the valid options.

I have used real number inputs to progressively select integer values. Another possible technique would be to use a random number generator. This will be explained in more detail.

WORK IN PROGRESS


Sequence Generator Code: Sequences (in JSON format)

Sequences are generated using JSON format to make them easier to read and inspect. A good JSON Viewer can be found here.

Here are some examples of various sequences generated by early code versions. All examples are superseded. Refer to my other blogs for more recent and interesting sequences.

\(\beta(0,2) = 0\) ==

{"V_r":0,"V_v":2,"h_0":{"C_d":0,"C_0":0}}

\(\beta(0.5,2) = 1\) ==

{"V_r":0.5,"V_v":2,"h_0":{"C_d":0,"C_0":1}}

\(\beta(1,2) = 2\) ==

{"V_r":1,"V_v":2,"h_0":{"C_d":1,"E_h":1,"f_0":{"g_0":{"C_q":0,"h_0":{"C_d":0,"C_0":0}},"h_u":{"C_d":0,"C_u":0}}}}

\(\beta(1.2,2) = 3\) ==

{"V_r":1.2,"V_v":2,"h_0":{"C_d":1,"E_h":1,"f_0":{"g_0":{"C_q":0,"h_0":{"C_d":0,"C_0":0}},"h_u":{"C_d":0,"C_u":1}}}}

\(\beta(1.5,2) = 4\) ==

{"V_r":1.5,"V_v":2,"h_0":{"C_d":1,"E_h":1,"f_0":{"g_0":{"C_q":0,"h_0":{"C_d":0,"C_0":1}},"h_U":{"C_d":0,"C_U":0},"E_h":"0..5","f_1":{"g_1":{"C_q":0,"h_1":{"C_d":0,"C_1":0}},"h_u":{"C_d":0,"C_u":0}}}}}

\(\beta(1.6,2) = 5\) ==

{"V_r":1.6,"V_v":2,"h_0":{"C_d":1,"E_h":1,"f_0":{"g_0":{"C_q":0,"h_0":{"C_d":0,"C_0":1}},"h_U":{"C_d":0,"C_U":0},"E_h":"0..5","f_1":{"g_1":{"C_q":0,"h_1":{"C_d":0,"C_1":0}},"h_u":{"C_d":0,"C_u":1}}}}}

\(\beta(1.7,2) = 6\) ==

{"V_r":1.7,"V_v":2,"h_0":{"C_d":1,"E_h":1,"f_0":{"g_0":{"C_q":0,"h_0":{"C_d":0,"C_0":1}},"h_U":{"C_d":0,"C_U":0},"E_h":"0..5","f_1":{"g_1":{"C_q":0,"h_1":{"C_d":0,"C_1":0}},"h_u":{"C_d":1,"E_h":1,"f_0":{"g_0":{"C_q":0,"h_0":{"C_d":0,"C_0":0}},"h_u":{"C_d":0,"C_u":0}}}}}}}

\(\beta(1.9,2) = 7\) ==

{"V_r":1.9,"V_v":2,"h_0":{"C_d":1,"E_h":1,"f_0":{"g_0":{"C_q":0,"h_0":{"C_d":0,"C_0":1}},"h_U":{"C_d":0,"C_U":0},"E_h":"0..5","f_1":{"g_1":{"C_q":0,"h_1":{"C_d":0,"C_1":0}},"h_u":{"C_d":1,"E_h":1,"f_0":{"g_0":{"C_q":0,"h_0":{"C_d":0,"C_0":0}},"h_u":{"C_d":0,"C_u":1}}}}}}}


Sequence Generator Code: Interpreter

Sequence Generator Code needs to be be interpreted using a program in a conventional programming language. I am using VBA in Microsoft Excel to do this. The code will be added to this blog shortly. I plan to write a Javascript version in due course and make it available online on a web page.

WORK IN PROGRESS


Test Bed

Initial Attempts will be presented here until the code is stabilised:

\(\beta(0,2) = 0 = 0\)

\(\beta(0.5,2) = 1 = 1\)

\(\beta(1,2) = 2 = 2\)

\(\beta(1.2,2) = 3 = 3\)

\(\beta(1.5,2) = 4 = 4\)

\(\beta(1.6,2) = 5 = 5\)

\(\beta(1.7,2) = 6 = 6\)

\(\beta(1.9,2) = 7 = 7\)

\(\beta(2.0001,2) = f_{\omega}(2) = 8\)

\(\beta(2.05,2) = f_{\omega}(2) + 1 = 9\)

\(\beta(2.1,2) = f_{\omega}(2) + 2 = 10\)

\(\beta(2.125,2) = f_{\omega}(2) + 3 = 11\)

\(\beta(2.135,2) = f_{\omega}(2) + 4 = 12\)

\(\beta(2.15,2) = f_{\omega}(2) + 5 = 13\)

\(\beta(2.16,2) = f_{\omega}(2) + 6 = 14\)

\(\beta(2.17,2) = f_{\omega}(2) + 7 = 15\)

\(\beta(2.2,2) = f_{\omega}(2).2 = 16\)

\(\beta(2.21,2) = f_{\omega}(2).2 + 1 = 17\)

\(\beta(2.23,2) = f_{\omega}(2).2 + 2 = 18\)

\(\beta(2.24,2) = f_{\omega}(2).2 + 3 = 19\)

Next Attempt base \(v = 2\) on 10 May 2016

\(\beta(5.657,2) = f_{\varphi(\omega,0)}(2).2 + 1\)

\(\beta(5.658,2) = f_{\varphi(1,\varphi(1,\omega + f_{\omega}(2).8 + f_{\omega}(2).2 + 7)\uparrow\uparrow f_{\omega}^{f_{3}(f_{\omega}(2)) + 6}(f_{\omega + 1}(2)) + 2^{\omega\uparrow\uparrow 2} + 1)}(f_{\varphi(\omega,0)}(2))\)

\(\beta(5.659,2) = f_{\omega^2 + f_{\varphi(1,\varphi(\varphi(1,0)^{\omega + 1} + \varphi(1,0)^{\omega}.(\omega + 1) + 1,\varphi(\omega,\varphi(1,1)^{\varphi(1,0).(\omega) + 1}.(\varphi(1,0)^{\omega + 1} + \varphi(1,0)^{\omega}.(\omega + 1) + \varphi(1,0) + 1) + 1)^{\omega + 1}.(\omega) + \omega) + 1)}(2)}(f_{\varphi(\omega,0) + 1}(2))\)

Next Attempt base \(v = 2\) on 11 May 2016

Second Attempt base \(v = 3\)

\(\beta(12.2118455947222,3)\)

\(= f_{\varphi(1,\varphi(2,0)\uparrow\uparrow f_{\varphi(1,\varphi(2,0)\uparrow\uparrow 2^{\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(1,\omega\uparrow\uparrow 2^{\omega^2.2 + \omega.2 + 2}.(\omega + 1) + \omega\uparrow\uparrow 2.2 + 1)^2.(\omega\uparrow\uparrow 2^{\omega^2.2 + \omega + 2}.(\omega^2 + \omega)) + 1))}}} + 1)} + 1)}(3) + 1)}(f_{\varphi(2,1)}(3))\)

and

\(\varphi(1,\varphi(2,0)\uparrow\uparrow f_{\varphi(1,\varphi(2,0)\uparrow\uparrow 2^{\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(1,\omega\uparrow\uparrow 2^{\omega^2.2 + \omega.2 + 2}.(\omega + 1) + \omega\uparrow\uparrow 2.2 + 1)^2.(\omega\uparrow\uparrow 2^{\omega^2.2 + \omega + 2}.(\omega^2 + \omega)) + 1))}}} + 1)} + 1)}(3) + 1)\)

or

\(\varphi(1,\varphi(1,\varphi(1,\omega\uparrow\uparrow 2^{\omega^2.2 + \omega.2 + 2}.(\omega + 1) + \omega\uparrow\uparrow 2.2 + 1)^2.(\omega\uparrow\uparrow 2^{\omega^2.2 + \omega + 2}.(\omega^2 + \omega)) + 1))\)

\(< \varphi(1,\varphi(1,\varphi(1,\omega\uparrow\uparrow 2^{\omega^2.2 + \omega.2 + \omega})))\)

\(= \varphi(1,\varphi(1,\varphi(1,\omega\uparrow\uparrow 2^{\omega^2.\omega})))\)

First Error

\(\omega\uparrow\uparrow 2^{\omega^2.\omega} > \omega\uparrow\uparrow 3 > \varphi(1,0)\)

Second Error

\(\varphi(1,\varphi(1,\varphi(1,X))) > \varphi(1,X + 1)\)

Third Attempt base \(v = 3\)

\(\beta(12.2118455947223,3) = f_{\varphi(2,1)}^2(3)\)

\(\beta(12.211845594705,3) = f_{\varphi(1,\varphi(2,0)\uparrow\uparrow f_{\varphi(1,\varphi(2,0)\uparrow\uparrow 2^{\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(2,\varphi(1,2)\uparrow\uparrow 2.(\varphi(1,0)\uparrow\uparrow 2^{\omega^2 + 1}.2 + \omega + 1) + 1)^2.(\omega\uparrow\uparrow 2^2.(\omega + 1) + \omega\uparrow\uparrow 2) + \varphi(1,\varphi(1,0)\uparrow\uparrow 2 + 1)))}}} + 1)} + 1)}(3) + 1)}(f_{\varphi(2,1)}(3))\)

Related to First Error in Second Attempt

\(\varphi(1,\varphi(2,0)\uparrow\uparrow 2^{\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(2,\varphi(1,2)\uparrow\uparrow 2.(\varphi(1,0)\uparrow\uparrow 2^{\omega^2 + 1}.2 + \omega + 1) + 1)^2.(\omega\uparrow\uparrow 2^2.(\omega + 1) + \omega\uparrow\uparrow 2) + \varphi(1,\varphi(1,0)\uparrow\uparrow 2 + 1)))}}} + 1)} + 1)\)

\(\varphi(2,\varphi(1,2)\uparrow\uparrow 2.(\varphi(1,0)\uparrow\uparrow 2^{\omega^2 + 1}.2 + \omega + 1) + 1)^2.(\omega\uparrow\uparrow 2^2.(\omega + 1) + \omega\uparrow\uparrow 2) + \varphi(1,\varphi(1,0)\uparrow\uparrow 2 + 1)\)

\(\varphi(2,X)\) where \(X > 0\) is incorrect

Fifth Attempt base \(v = 2\)

\(\beta(5.73536669398825,2)\)

\(= f_{f_{\varphi(\omega,0)^{\varphi(1,0)^{\omega} + \omega} + \varphi(1,\omega)}(2)}(f_{\varphi(\omega,0)^{\varphi(1,\omega + 1)^{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega + 1} + \omega + 1}.(\omega) + \varphi(1,1)^{\omega}.(\varphi(1,0)^{\omega + 1}) + \varphi(1,1) + \varphi(1,0) + \omega + 1} + \varphi(1,\omega)^{\omega + 1} + \omega}.(\omega + 1) + \varphi(1,\omega + 1)^{\omega + 1} + \omega} + \omega}(2))\)

\(< f_{\varphi(\omega,0)^{\varphi(1,\omega + 1)^{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega + 1} + \omega + 1}.(\omega) + \varphi(1,1)^{\omega}.(\varphi(1,0)^{\omega + 1}) + \varphi(1,1) + \varphi(1,0) + \omega + 1} + \varphi(1,\omega)^{\omega + 1} + \omega}.(\omega + 1) + \varphi(1,\omega + 1)^{\omega + 1} + \omega} + \omega + 1}(2)\)

and

\(\varphi(\omega,0)^{\varphi(1,\omega + 1)^{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega + 1} + \omega + 1}.(\omega) + \varphi(1,1)^{\omega}.(\varphi(1,0)^{\omega + 1}) + \varphi(1,1) + \varphi(1,0) + \omega + 1} + \varphi(1,\omega)^{\omega + 1} + \omega}.(\omega + 1) + \varphi(1,\omega + 1)^{\omega + 1} + \omega} + \omega + 1\)

\(< \varphi(\omega,0)^{\varphi(1,\omega + 1)^{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega + 1} + \omega + 1}.(\omega) + \varphi(1,1)^{\omega}.(\varphi(1,1))}}}\)

\(< \varphi(\omega,0)^{\varphi(1,\omega + 1)^{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega + 1} + \omega + 1}.(\omega + 1)}}}\)

\(\beta(5.739133787073,2)\)

\(= f_{\varphi(\omega,0)^{\varphi(1,\varphi(1,\omega + 1)^{\varphi(1,\omega + 1)^{\varphi(1,\varphi(1,\omega)^{\varphi(1,\omega)^{\varphi(1,0) + 1}.(\omega) + \varphi(1,0)}.(\varphi(1,1).(\omega) + \omega) + \varphi(\omega,\varphi(1,\omega)^{\varphi(1,\omega)^{\omega + 1}}.(\omega) + \varphi(1,0)^{\omega + 1}.(\omega) + 1))}} + 1)}}(2)\)

and

\(\varphi(\omega,0)^{\varphi(1,\varphi(1,\omega + 1)^{\varphi(1,\omega + 1)^{\varphi(1,\varphi(1,\omega)^{\varphi(1,\omega)^{\varphi(1,0) + 1}.(\omega) + \varphi(1,0)}.(\varphi(1,1).(\omega) + \omega) + \varphi(\omega,\varphi(1,\omega)^{\varphi(1,\omega)^{\omega + 1}}.(\omega) + \varphi(1,0)^{\omega + 1}.(\omega) + 1))}} + 1)}\)

Next Attempt base \(v = 2\) on 12 May 2016

Third Attempt base \(v = 2\)

\(\beta(0,2) = 0 = 0\)

\(\beta(0.5,2) = 1 = 1\)

\(\beta(1,2) = 2 = 2\)

\(\beta(1.2,2) = 3 = 3\)

\(\beta(1.5,2) = 4 = 4\)

\(\beta(1.6,2) = 5 = 5\)

\(\beta(1.7,2) = 6 = 6\)

\(\beta(1.9,2) = 7 = 7\)

\(\beta(2.0001,2) = f_{\omega}(2) = 8\)

\(\beta(2.05,2) = f_{\omega}(2) + 1 = 9\)

\(\beta(2.1,2) = f_{\omega}(2) + 2 = 10\)

\(\beta(2.125,2) = f_{\omega}(2) + 3 = 11\)

\(\beta(2.135,2) = f_{\omega}(2) + 4 = 12\)

\(\beta(2.15,2) = f_{\omega}(2) + 5 = 13\)

\(\beta(2.16,2) = f_{\omega}(2) + 6 = 14\)

\(\beta(2.17,2) = f_{\omega}(2) + 7 = 15\)

\(\beta(2.2,2) = f_{\omega}(2).2 = 16\)

\(\beta(2.255,2) = f_{\omega}(2).2 + f_{\omega}(2) = 24\)

\(\beta(2.28,2) = f_{\omega}(2).4 = 32\)

\(\beta(2.2965,2) = f_{\omega}(2).4 + f_{\omega}(2) = 40\)

\(\beta(2.3,2) = f_{\omega}(2).4 + f_{\omega}(2).2 = 48\)

\(\beta(2.30175,2) = f_{\omega}(2).4 + f_{\omega}(2).2 + f_{\omega}(2) = 56\)

\(\beta(2.303,2) = f_{\omega}(2).8 = 64\)

\(\beta(2.33,2) = f_{\omega}(2).16 = 128\)

\(\beta(2.343,2) = f_{\omega}(2).32 = 256\)

\(\beta(2.355,2) = f_{\omega}(2).64 = 512\)

\(\beta(2.366,2) = f_{\omega}(2).128 = 1024\)

\(\beta(2.405,2) = f_{2}(f_{\omega}(2)).2 = 4096\)

\(\beta(2.4194,2) = f_{2}(f_{\omega}(2)).8 = 16384\)

\(\beta(2.4216,2) = f_{2}(f_{\omega}(2)).32 = 65536\)

\(\beta(2.4233,2) = f_{2}(f_{\omega}(2)).128 = 262144\)

\(\beta(2.4249,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2) + 1}) = 1048576\)

\(\beta(2.426,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2) + 3}) = 4194304\)

\(\beta(2.4266,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2) + 5}) = 16777216\)

\(\beta(2.42705,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2) + 7}) = 67108864\)

\(\beta(2.4277,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2).2 + 1}) = 268435456\)

\(\beta(2.42815,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2).2 + 3}) = 1073741824\)

\(\beta(2.432,2) = f_{2}^2(f_{\omega}(2))\)

\(\beta(2.4575,2) = f_{2}^4(f_{\omega}(2))\)

\(\beta(2.471,2) = f_{2}^6(f_{\omega}(2))\)

\(\beta(2.485,2) = f_{3}(f_{\omega}(2))\)

\(\beta(2.595,2) = f_{4}(f_{\omega}(2))\)

\(\beta(2.652,2) = f_{5}(f_{\omega}(2))\)

\(\beta(2.71,2) = f_{6}(f_{\omega}(2))\)

\(\beta(2.768,2) = f_{7}(f_{\omega}(2))\)

Next Attempt base \(v = 2\) on 13 May 2016

Second Attempt base \(v = 2\)

\(\beta(5.6569,2) = f_{\varphi(\omega,0) + 1}(2)\)

\(\beta(5.7391405,2) = f_{\varphi(\omega,0)^{\omega}}(2)\)

\(\beta(5.822613,2) = f_{\varphi(\omega,0)^{\varphi(1,0)}}(2)\)

\(\beta(5.6613958,2) = f_{\varphi(1,\varphi(\omega,0) + 1)}(2)\)

Third Attempt base \(v = 2\)

\(\beta(0,2) = 0 = 0\)

\(\beta(0.5,2) = 1 = 1\)

\(\beta(1,2) = 2 = 2\)

\(\beta(1.2,2) = 3 = 3\)

\(\beta(1.5,2) = 4 = 4\)

\(\beta(1.6,2) = 5 = 5\)

\(\beta(1.7,2) = 6 = 6\)

\(\beta(1.9,2) = 7 = 7\)

\(\beta(2.0001,2) = f_{\omega}(2) = 8\)

\(\beta(2.05,2) = f_{\omega}(2) + 1 = 9\)

\(\beta(2.1,2) = f_{\omega}(2) + 2 = 10\)

\(\beta(2.125,2) = f_{\omega}(2) + 3 = 11\)

\(\beta(2.135,2) = f_{\omega}(2) + 4 = 12\)

\(\beta(2.15,2) = f_{\omega}(2) + 5 = 13\)

\(\beta(2.16,2) = f_{\omega}(2) + 6 = 14\)

\(\beta(2.17,2) = f_{\omega}(2) + 7 = 15\)

\(\beta(2.2,2) = f_{\omega}(2).2 = 16\)

\(\beta(2.255,2) = f_{\omega}(2).2 + f_{\omega}(2) = 24\)

\(\beta(2.28,2) = f_{\omega}(2).4 = 32\)

\(\beta(2.2965,2) = f_{\omega}(2).4 + f_{\omega}(2) = 40\)

\(\beta(2.3,2) = f_{\omega}(2).4 + f_{\omega}(2).2 = 48\)

\(\beta(2.30175,2) = f_{\omega}(2).4 + f_{\omega}(2).2 + f_{\omega}(2) = 56\)

\(\beta(2.303,2) = f_{\omega}(2).8 = 64\)

\(\beta(2.33,2) = f_{\omega}(2).16 = 128\)

\(\beta(2.343,2) = f_{\omega}(2).32 = 256\)

\(\beta(2.355,2) = f_{\omega}(2).64 = 512\)

\(\beta(2.366,2) = f_{\omega}(2).128 = 1024\)

\(\beta(2.38,2) = f_{2}(f_{\omega}(2)) = 2048\)

\(\beta(2.405,2) = f_{2}(f_{\omega}(2)).2 = 4096\)

\(\beta(2.4194,2) = f_{2}(f_{\omega}(2)).8 = 16384\)

\(\beta(2.4216,2) = f_{2}(f_{\omega}(2)).32 = 65536\)

\(\beta(2.4233,2) = f_{2}(f_{\omega}(2)).128 = 262144\)

\(\beta(2.4249,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2) + 1}) = 1048576\)

\(\beta(2.426,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2) + 3}) = 4194304\)

\(\beta(2.4266,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2) + 5}) = 16777216\)

\(\beta(2.42705,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2) + 7}) = 67108864\)

\(\beta(2.4277,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2).2 + 1}) = 268435456\)

\(\beta(2.42815,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2).2 + 3}) = 1073741824\)

\(\beta(2.432,2) = f_{2}^2(f_{\omega}(2))\)

\(\beta(2.4575,2) = f_{2}^4(f_{\omega}(2))\)

\(\beta(2.471,2) = f_{2}^6(f_{\omega}(2))\)

\(\beta(2.485,2) = f_{3}(f_{\omega}(2))\)

\(\beta(2.595,2) = f_{4}(f_{\omega}(2))\)

\(\beta(2.652,2) = f_{5}(f_{\omega}(2))\)

\(\beta(2.71,2) = f_{6}(f_{\omega}(2))\)

\(\beta(2.768,2) = f_{7}(f_{\omega}(2))\)

\(\beta(2.85,2) = f_{\omega + 1}(2)\)

\(\beta(3.085,2) = f_{2}(f_{\omega + 1}(2))\)

\(\beta(3.152,2) = f_{4}(f_{\omega + 1}(2))\)

\(\beta(3.1865,2) = f_{6}(f_{\omega + 1}(2))\)

\(\beta(3.38,2) = f_{\omega}(f_{\omega + 1}(2))\)

\(\beta(3.749,2) = f_{\omega}^4(f_{\omega + 1}(2))\)

\(\beta(3.8306,2) = f_{\omega}^{f_{\omega}(2)}(f_{\omega + 1}(2))\)

\(\beta(3.9,2) = f_{\omega}^{f_{\omega}(2).8 + 1}(f_{\omega + 1}(2)) + f_{\omega}(2) + 7\)

\(\beta(3.95,2) = f_{3}(f_{\omega}^{f_{3}^3(f_{\omega}(2)).32 + 1}(f_{\omega + 1}(2)))\)

\(\beta(3.99,2) = f_{3}(f_{5}(f_{\omega}^{f_{7}(f_{\omega}(2)) + 3}(f_{\omega + 1}(2)))).(2^{f_{\omega}(2)}) + 1\)

\(\beta(3.995,2) = f_{7}^6(f_{\omega}^{f_{7}^2(f_{\omega}(2)).2 + 1}(f_{\omega + 1}(2))).2\)

\(\beta(3.999,2) = f_{\omega}^{f_{2}(f_{7}^6(f_{\omega}(2))) + 7}(f_{\omega + 1}(2))\)

\(\beta(3.9999,2) = f_{f_{f_{\omega + 1}(2) + 2}(f_{\omega}(f_{\omega + 1}(2)))}(f_{\omega}^{f_{5}^4(f_{7}^7(f_{\omega}(2))).2 + f_{2}(f_{\omega}(2)) + f_{\omega}(2).2 + f_{\omega}(2) + 2}(f_{\omega + 1}(2)))\)

\(\beta(3.99999,2) = f_{\omega}^{f_{3}(f_{6}^{f_{2}(f_{\omega}(2)) + f_{\omega}(2) + 7}(f_{7}^7(f_{\omega}(2)))).2 + 3}(f_{\omega + 1}(2)) + 1\)

\(\beta(3.999999,2) = f_{\omega}^{f_{6}^{f_{7}(f_{\omega}(2)).(2^{f_{2}(f_{3}(f_{4}^3(f_{\omega}(2)))).2 + f_{2}^2(f_{4}(f_{\omega}(2))).2})}(f_{7}^7(f_{\omega}(2)))}(f_{\omega + 1}(2))\)

\(\beta(3.9999999,2) = f_{\omega}^{f_{6}^{f_{7}^6(f_{\omega}(2)).8 + 1}(f_{7}^7(f_{\omega}(2))) + 1}(f_{\omega + 1}(2)).(2^{f_{\omega + 1}(2)})\)

\(\beta(3.99999999,2) = f_{\omega}^{f_{6}^{f_{6}^2(f_{7}^6(f_{\omega}(2))).2}(f_{7}^7(f_{\omega}(2))) + 3}(f_{\omega + 1}(2)).8 + 6\)

\(\beta(3.999999999,2) = f_{\omega}^{f_{6}^{f_{6}^{f_{4}^6(f_{\omega}(2)).64 + f_{2}^2(f_{4}(f_{\omega}(2)))}(f_{7}^6(f_{\omega}(2)))}(f_{7}^7(f_{\omega}(2)))}(f_{\omega + 1}(2))\)

\(\beta(3.9999999999,2) = f_{\omega}^{f_{6}^{f_{5}^{f_{\omega}(2)}(f_{6}^{f_{2}(f_{7}^3(f_{\omega}(2))) + 5}(f_{7}^6(f_{\omega}(2))))}(f_{7}^7(f_{\omega}(2)))}(f_{\omega + 1}(2))\)

\(\beta(3.99999999999,2) = f_{\omega}^{f_{2}(f_{6}^{f_{6}^{f_{4}(f_{7}^5(f_{\omega}(2))) + 1}(f_{7}^6(f_{\omega}(2))) + 1}(f_{7}^7(f_{\omega}(2))))}(f_{\omega + 1}(2))\)

\(\beta(3.999999999999,2) = f_{\omega}^{f_{6}^{f_{6}^{f_{6}^{f_{\omega}(2) + 5}(f_{7}^5(f_{\omega}(2))) + f_{2}(f_{\omega}(2))}(f_{7}^6(f_{\omega}(2)))}(f_{7}^7(f_{\omega}(2)))}(f_{\omega + 1}(2))\)

\(\beta(3.9999999999999,2) = f_{\omega}^{f_{6}^{f_{6}^{f_{6}^{f_{5}^2(f_{6}^3(f_{\omega}(2)))}(f_{7}^5(f_{\omega}(2)))}(f_{7}^6(f_{\omega}(2)))}(f_{7}^7(f_{\omega}(2)))}(f_{\omega + 1}(2))\)

\(\beta(3.99999999999999,2) = f_{\omega}^{f_{6}^{f_{6}^{f_{6}^{f_{7}^4(f_{\omega}(2)).(2^{f_{2}(f_{\omega}(2))})}(f_{7}^5(f_{\omega}(2)))}(f_{7}^6(f_{\omega}(2)))}(f_{7}^7(f_{\omega}(2)))}(f_{\omega + 1}(2))\)

\(\beta(4,2) = f_{\varphi(1,0)}(2)\)

\(\beta(4.0878,2) = f_{\varphi(1,0).(\omega)}(2)\)

\(\beta(4.17725,2) = f_{\varphi(1,0)^{\omega}}(2)\)

\(\beta(4.3621,2) = f_{\varphi(1,1)}(2)\)

\(\beta(4.4043,2) = f_{\varphi(1,1).(\omega)}(2)\)

\(\beta(4.44684,2) = f_{\varphi(1,1).(\varphi(1,0))}(2)\)

\(\beta(4.4899,2) = f_{\varphi(1,1)^{\omega}}(2)\)

\(\beta(4.62142,2) = f_{\varphi(1,1)^{\varphi(1,0)}}(2)\)

\(\beta(4.688634,2) = f_{\varphi(1,1)^{\varphi(1,0)^{\omega}}}(2)\)

\(\beta(4.7569,2) = f_{\varphi(1,\omega)}(2)\)

\(\beta(4.89625,2) = f_{\varphi(1,\omega)^{\omega}}(2)\)

\(\beta(5.039685,2) = f_{\varphi(1,\omega)^{\varphi(1,0)}}(2)\)

\(\beta(5.1875,2) = f_{\varphi(1,\omega + 1)}(2)\)

\(\beta(5.6569,2) = f_{\varphi(\omega,0)}(2)\)

\(\beta(5.718454,2) = f_{\varphi(\omega,0)^{\omega + 1}}(2)\)

\(\beta(5.7391351,2) = f_{\varphi(\omega,0)^{\varphi(1,0)}}(2)\)

\(\beta(5.78072463,2) = f_{\varphi(1,\varphi(\omega,0) + 1)}(2)\)

\(\beta(5.78536443,2) = f_{\varphi(1,\varphi(\omega,0) + \omega)}(2)\)

\(\beta(5.7946543,2) = f_{\varphi(1,\varphi(\omega,0).(\omega))}(2)\)

\(\beta(5.8648058,2) = f_{\varphi(1,\varphi(\omega,0)^{\varphi(1,0)})}(2)\)

\(\beta(5.90732,2) = f_{\varphi(\omega,1)}(2)\)

\(\beta(5.9932285,2) = f_{\varphi(\omega,1)^{\varphi(1,0)}}(2)\)

\(\beta(6.1688436,2) = f_{\varphi(\omega,\omega)}(2)\)

\(\beta(6.4730447,2) = f_{\varphi(\omega,\omega + 1).(\varphi(1,0))}(2)\)

\(\beta(6.535662,2) = f_{\varphi(\omega,\omega + 1)^{\varphi(1,0)}}(2)\)

\(\beta(6.55929912,2) = f_{\varphi(\omega,\omega + 1)^{\varphi(\omega,0)}}(2)\)

\(\beta(6.571149849,2) = f_{\varphi(\omega,\omega + 1)^{\varphi(\omega,\omega)}}(2)\)

\(\beta(6.58302796,2) = f_{\varphi(1,\varphi(\omega,\omega + 1) + 1)}(2)\)

\(\beta(6.58830582,2) = f_{\varphi(1,\varphi(\omega,\omega + 1) + \omega)}(2)\)

\(\beta(6.5896268,2) = f_{\varphi(1,\varphi(\omega,\omega + 1) + \omega)^{\omega}}(2)\)

\(\beta(6.630726,2) = f_{\varphi(1,\varphi(\omega,\omega + 1)^{\omega})}(2)\)

\(\beta(6.6787741715,2) = f_{\varphi(1,\varphi(\omega,\omega + 1)^{\varphi(1,0)})}(2)\)

\(\beta(6.70292907,2) = f_{\varphi(1,\varphi(\omega,\omega + 1)^{\varphi(\omega,0)})}(2)\)

\(\beta(6.715039255,2) = f_{\varphi(1,\varphi(\omega,\omega + 1)^{\varphi(\omega,\omega)})}(2)\)

\(\beta(6.7272,2) = f_{\varphi(\omega + 1,0)}(2)\)

\(\beta(6.9243,2) = f_{\varphi(\omega + 1,1)}(2)\)

\(\beta(7.1272,2) = f_{\varphi(\omega + 1,\omega)}(2)\)

\(\beta(7.550995,2) = f_{\varphi(\omega + 1,\varphi(1,0))}(2)\)

\(\beta(7.57830545,2) = f_{\varphi(1,\varphi(\omega + 1,\varphi(1,0)) + 1)}(2)\)

\(\beta(7.77225555,2) = f_{\varphi(\omega + 1,\varphi(1,0).(\omega))}(2)\)

\(\beta(7.786297675,2) = f_{\varphi(1,\varphi(\omega + 1,\varphi(1,0).(\omega)) + 1)}(2)\)

\(\beta(7.786558,2) = f_{\varphi(\omega,\varphi(\omega + 1,\varphi(1,0).(\omega)) + 1)}(2)\)

\(\beta(7.8998162,2) = f_{\varphi(\omega,\varphi(\omega + 1,\varphi(1,0).(\omega + 1)) + 1)}(2)\)

\(\beta(7.9,2) = f_{(\omega\uparrow\uparrow f_{\omega + 1}(2)).(\omega.(f_{\omega + 1}(2) + 6)) + \omega + f_{\varphi(\omega,\omega + 1)^{\varphi(1,\omega + 1)^{\omega + 1}}}(2)}(f_{\varphi(1,\varphi(\omega,\varphi(\omega + 1,\varphi(1,0).(\omega + 1)) + 1).(\omega + 1))}(2))\)

\(\beta(7.95,2) = f_{\varphi(\omega,\omega + f_{(\omega\uparrow\uparrow f_{2}(f_{\omega}^3(f_{\omega + 1}(2))))}(f_{\varphi(1,\varphi(\omega,\omega + 1).(\varphi(1,0).(\omega + 1) + 1) + \varphi(1,\varphi(\omega,0)^{\omega + 1} + \omega)^{\omega + 1} + 1)^{\omega^3}}(2)))}(f_{\varphi(\omega + 1,\varphi(1,0).(\omega + 1) + \omega)^{\omega + 1} + \omega}(2))\)

Fourth Attempt base \(v = 3\)

\(\beta(12.2118455947222,3) = f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow 2)^{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(1,\varphi(1,0))}})}(3)))}(f_{\varphi(2,1)}(3))\)

The following errors from the past few days should have been corrected:

\(\beta(12.2118455947222,3)\)

\(= f_{\varphi(1,\varphi(2,0)\uparrow\uparrow f_{\varphi(1,\varphi(2,0)\uparrow\uparrow 2^{\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(1,\omega\uparrow\uparrow 2^{\omega^2.2 + \omega.2 + 2}.(\omega + 1) + \omega\uparrow\uparrow 2.2 + 1)^2.(\omega\uparrow\uparrow 2^{\omega^2.2 + \omega + 2}.(\omega^2 + \omega)) + 1))}}} + 1)} + 1)}(3) + 1)}(f_{\varphi(2,1)}(3))\)

First Error

\(\omega\uparrow\uparrow 2^{\omega^2.\omega} > \omega\uparrow\uparrow 3 > \varphi(1,0)\)

Second Error

\(\varphi(1,\varphi(1,\varphi(1,X))) > \varphi(1,X + 1)\)

Lets compare the ordinals between these two results:

\(f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow 2)^{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(1,\varphi(1,0))}})}(3)\)

\(f_{\varphi(1,\varphi(2,0)\uparrow\uparrow 2^{\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(1,\omega\uparrow\uparrow 2^{\omega^2.2 + \omega.2 + 2}.(\omega + 1) + \omega\uparrow\uparrow 2.2 + 1)^2.(\omega\uparrow\uparrow 2^{\omega^2.2 + \omega + 2}.(\omega^2 + \omega)) + 1))}}} + 1)} + 1)}(3)\)

or

\(\varphi(1,(\varphi(2,0)\uparrow\uparrow 2)^{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(1,\varphi(1,0))}})\)

\(\varphi(1,\varphi(2,0)\uparrow\uparrow 2^{\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(1,\omega\uparrow\uparrow 2^{\omega^2.2 + \omega.2 + 2}.(\omega + 1) + \omega\uparrow\uparrow 2.2 + 1)^2.(\omega\uparrow\uparrow 2^{\omega^2.2 + \omega + 2}.(\omega^2 + \omega)) + 1))}}} + 1)} + 1)\)

or

\((\varphi(2,0)\uparrow\uparrow 2)^{\varphi(1,\varphi(1,0))}\) The previous errors have been corrected.

\(\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(1,\omega\uparrow\uparrow 2^{\omega^2.2 + \omega.2 + 2}.(\omega + 1) + \omega\uparrow\uparrow 2.2 + 1)^2.(\omega\uparrow\uparrow 2^{\omega^2.2 + \omega + 2}.(\omega^2 + \omega)) + 1))}}} + 1)\)

Fifth Attempt base \(v = 3\)

\(\beta(12.211845594705,3) = f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow 2)^{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(1,\varphi(1,0))}})}(3)))}(f_{\varphi(2,1)}(3))\)

The following error from the past few days should have been corrected:

\(\beta(12.211845594705,3) = f_{\varphi(1,\varphi(2,0)\uparrow\uparrow f_{\varphi(1,\varphi(2,0)\uparrow\uparrow 2^{\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(2,\varphi(1,2)\uparrow\uparrow 2.(\varphi(1,0)\uparrow\uparrow 2^{\omega^2 + 1}.2 + \omega + 1) + 1)^2.(\omega\uparrow\uparrow 2^2.(\omega + 1) + \omega\uparrow\uparrow 2) + \varphi(1,\varphi(1,0)\uparrow\uparrow 2 + 1)))}}} + 1)} + 1)}(3) + 1)}(f_{\varphi(2,1)}(3))\)

\(\varphi(2,\varphi(1,2)\uparrow\uparrow 2.(\varphi(1,0)\uparrow\uparrow 2^{\omega^2 + 1}.2 + \omega + 1) + 1)^2.(\omega\uparrow\uparrow 2^2.(\omega + 1) + \omega\uparrow\uparrow 2) + \varphi(1,\varphi(1,0)\uparrow\uparrow 2 + 1)\)

\(\varphi(2,X)\) where \(X > 0\) is incorrect

Lets compare the ordinals between these two results:

\(\varphi(1,(\varphi(2,0)\uparrow\uparrow f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow 2)^{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(1,\varphi(1,0))}})}(3)))\)

\(\varphi(1,\varphi(2,0)\uparrow\uparrow f_{\varphi(1,\varphi(2,0)\uparrow\uparrow 2^{\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(2,\varphi(1,2)\uparrow\uparrow 2.(\varphi(1,0)\uparrow\uparrow 2^{\omega^2 + 1}.2 + \omega + 1) + 1)^2.(\omega\uparrow\uparrow 2^2.(\omega + 1) + \omega\uparrow\uparrow 2) + \varphi(1,\varphi(1,0)\uparrow\uparrow 2 + 1)))}}} + 1)} + 1)}(3) + 1)\)

or

\(\varphi(1,(\varphi(2,0)\uparrow\uparrow 2)^{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(1,\varphi(1,0))}})\)

\(\varphi(1,\varphi(2,0)\uparrow\uparrow 2^{\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(2,\varphi(1,2)\uparrow\uparrow 2.(\varphi(1,0)\uparrow\uparrow 2^{\omega^2 + 1}.2 + \omega + 1) + 1)^2.(\omega\uparrow\uparrow 2^2.(\omega + 1) + \omega\uparrow\uparrow 2) + \varphi(1,\varphi(1,0)\uparrow\uparrow 2 + 1)))}}} + 1)} + 1)\)

or

\((\varphi(2,0)\uparrow\uparrow 2)^{\varphi(1,\varphi(1,0))}\) The previous error have been corrected.

\(\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(2,\varphi(1,2)\uparrow\uparrow 2.(\varphi(1,0)\uparrow\uparrow 2^{\omega^2 + 1}.2 + \omega + 1) + 1)^2.(\omega\uparrow\uparrow 2^2.(\omega + 1) + \omega\uparrow\uparrow 2) + \varphi(1,\varphi(1,0)\uparrow\uparrow 2 + 1)))}}} + 1)\)

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