User blog:B1mb0w/Program Code Version 3

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

Program Code Version 3
This VBA visual basic code replaces Version 2 and 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.

WORK IN PROGRESS

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 version 3.

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