User blog:B1mb0w/Alpha Function Code Version 8

Alpha Function Code Version 8
This version of The Alpha Function has been changed cosmetically from the last version to 're-calibrate' the function and make the range more interesting.

The function code is still based on the The S Function (Version 2). The growth rate of the function is still well beyond \(f_{LVO}(n)\).

Changes in Version 8
Version 8 has been cosmetically changed from Version 7 to 're-calibrate' the function and make the range more interesting.

The first change is to reduce the range of the input real number parameter to between 0 and 10,000. The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. The real number is manipulated by Sequence Generating Code (see below) to create a finite sequence of finite integers that represents a unique combination of S and T functions which can be translated into unique finite integers (up to and beyond \(f_{LVO}(n)\) for any n).

The Alpha Function will still translate unique real numbers into any and every finite integer (up to and beyond \(f_{LVO}(n)\) for any \(n\)).

The second change is to set the value of \(n\) in all the resulting \(S\) function output to \(n = 2\). The previous version had set this to \(n = 3\) instead.

Example of Changes in Version 8
Version 8 will generate string combinations of S and T functions. Each combination uniquely belongs to an ascending order of all sequences. Therefore each sequence can be assigned a finite ordinal values. The growth rate of these combinations is well beyond \(f_{LVO}(n)\) for any \(n\)).

\(\alpha(0) = 0\)

\(\alpha(0.5) = 1\)

\(\alpha(1) = 2\)

\(\alpha(1.4) = 3\)

\(\alpha(1.45) = S(3,0,1) = 4\)

\(\alpha(1.5) = S(3,0,2) = 5\)

The growth rate can be seen to accelerate when we start introducing T functions:

\(\alpha(1.78) = S(3,2,1) = 24\)

\(\alpha(2) = S(3,T(0),1) = f_{\omega}(3)\)

Growth Rate of the Alpha Function
The Alpha Function is now 're-calibrated' to accept real number inputs up to \(10,000\) at which point the Alpha Function will generate an S Function approaching:

\(\alpha(10000) = S(2,T^{\omega}(0),1)\)

Version 8 Code
The code for the Alpha Function uses Sequence Generator Code Syntax to generate sequences of finite integers.

Refer to my previous Version 7 blog for the actual code. The only change has been to set a variable in the first line of code \(V_v = 2\) instead of (V_v = 3\).

a_x = (V_v=3,V_t=2,n_x,n_x)

n_x = (C_d<2,C_d>(0:(i_x,i_x),(V_n=1,s_0,s_0)))

s_x = (x>(0:V_v,s_x-1),

t_x(0:(n_c(0:,[S([],[t_x],[n_c])])),

(n_C(0:(i_x,i_x),

(t_a,[T([t_a])],

t_b<(1,t_a),

t_b>(0:(t_c<(1,t_a),t_c>(0:,[S([],0,[t_c])])),

(t_C<(1,t_a),[S([],[t_b],[t_C])]))))

i_x = (C_x<V_v-U_x,C_x]+[U_x)

Test Bed for Version 7
Below is the test bed and various results using version 7.

\(\alpha(0) = 0\)

\(\alpha(0.5) = 1\)

\(\alpha(1) = 2\)

\(\alpha(1.5) = 2 + 1\)

\(\alpha(1.75) = f_{1}(2)\)

\(\alpha(2) = f_{1}(2) + 1\)

\(\alpha(2.25) = f_{1}(2) + 2\)

Fifth Attempt on 8 Jun 2016

\(\alpha(0) = 0\)

\(\alpha(0.5) = 1\)

\(\alpha(1) = 2\)

\(\alpha(1.5) = 3\)

\(\alpha(1.75) = f_{1}(2)\)

\(\alpha(2) = f_{1}(2) + 1\)

\(\alpha(2.25) = f_{1}(2) + 2\)

\(\alpha(2.5) = f_{1}(2) + 3\)

\(\alpha(2.85) = f_{\omega}(2)\)

\(\alpha(2.86) = f_{\omega}(2) + 1\)

Next Attempt on 14 Jun 2016 using S Function Version 2

\(\alpha(0) = 0\)

\(\alpha(0.5) = 1\)

\(\alpha(1) = 2\)

\(\alpha(1.4) = 3\)

\(\alpha(1.45) = S(3,0,1)\)

\(\alpha(1.5) = S(3,0,2)\)

\(\alpha(1.56) = S(3,1,1)\)

\(\alpha(1.58) = S(S(3,1,1),0,1)\)

\(\alpha(1.6) = S(S(3,1,1),0,2)\)

\(\alpha(1.62) = S(S(3,1,1),0,3)\)

\(\alpha(1.64) = S(S(3,1,1),0,S(3,0,1))\)

\(\alpha(1.66) = S(S(3,1,1),0,S(3,0,2))\)

\(\alpha(1.68) = S(3,1,2)\)

\(\alpha(1.7) = S(S(3,1,2),0,1)\)

\(\alpha(1.72) = S(S(3,1,2),0,2)\)

\(\alpha(1.73) = S(S(3,1,2),0,3)\)

\(\alpha(1.74) = S(S(3,1,2),0,S(3,0,1))\)

\(\alpha(1.745) = S(S(3,1,2),0,S(3,0,2))\)

\(\alpha(1.75) = S(S(3,1,2),0,S(3,1,1))\)

\(\alpha(1.755) = S(S(3,1,2),0,S(S(3,1,1),0,1))\)

\(\alpha(1.76) = S(S(3,1,2),0,S(S(3,1,1),0,2))\)

\(\alpha(1.765) = S(S(3,1,2),0,S(S(3,1,1),0,3))\)

\(\alpha(1.77) = S(S(3,1,2),0,S(S(3,1,1),0,S(3,0,1)))\)

\(\alpha(1.775) = S(S(3,1,2),0,S(S(3,1,1),0,S(3,0,2)))\)

\(\alpha(1.78) = S(3,2,1)\)

\(\alpha(1.89) = S(3,2,2)\)

\(\alpha(2) = S(3,T(0),1)\)

\(\alpha(2.02) = S(S(3,T(0),1),0,1)\)

\(\alpha(2.05) = S(S(3,T(0),1),0,S(S(3,1,2),0,1))\)

\(\alpha(2.1) = S(S(S(3,T(0),1),1,3),0,2)\)

\(\alpha(2.2) = S(S(S(S(3,T(0),1),2,S(S(3,2,2),1,2)),1,3),0,S(S(S(3,T(0),1),2,S(S(3,2,2),1,1)),0,1))\)

\(\alpha(2.5) = S(S(S(3,S(T(0),0,1),1),1,S(S(3,T(0),2),0,S(S(3,T(0),1),0,1))),0,S(S(S(3,T(0),1),2,S(3,0,1)),0,2))\)

\(\alpha(3) = S(S(3,S(T(0),0,2),1),1,2)\)

\(\alpha(4) = S(S(S(S(S(3,S(T(0),1,1),1),S(T(0),0,1),S(S(3,T(0),2),0,S(S(3,1,1),0,2))),2,2),1,1),0,S(S(S(S(3,S(T(0),0,1),1),2,2),1,1),0,S(S(S(3,T(0),1),2,2),1,S(S(3,T(0),1),0,S(3,2,2)))))\)

\(\alpha(6) = S(S(S(S(3,S(T(0),1,2),2),S(T(0),0,1),2),1,1),0,1)\)

\(\alpha(8) = S(S(S(S(S(3,S(T(0),2,1),2),S(T(0),0,1),1),T(0),S(S(3,1,1),0,2)),1,1),0,S(S(S(S(S(3,S(T(0),0,1),2),T(0),1),2,2),1,1),0,S(S(S(S(3,T(0),1),2,2),1,1),0,S(3,T(0),1))))\)

\(\alpha(10) = S(S(3,S(T(0),2,2),2),0,S(S(3,S(T(0),1,1),2),S(T(0),0,2),1))\)

\(\alpha(20) = S(S(S(3,S(T(1),1,2),1),1,1),0,S(S(S(S(3,T(0),1),2,2),1,S(S(3,1,2),0,2)),0,S(S(S(S(3,T(0),1),2,2),1,2),0,S(S(3,1,1),0,3))))\)

\(\alpha(40) = S(S(S(3,S(T(1),1,T(0)),1),1,3),0,S(S(3,1,1),0,S(3,0,1)))\)

\(\alpha(75) = S(S(S(3,S(T(1),1,S(T(0),1,1)),2),1,2),0,2)\)

\(\alpha(100) = S(S(S(S(3,S(T(1),1,S(T(0),1,2)),2),S(T(0),0,1),2),2,1),0,S(3,1,2))\)

\(\alpha(200) = S(S(S(S(3,S(T(1),2,1),1),2,1),1,S(S(S(3,S(T(0),0,2),1),1,S(S(3,2,1),0,1)),0,S(3,0,1))),0,S(S(S(3,S(T(0),1,1),2),S(T(0),0,1),S(S(S(3,S(T(0),1,1),1),2,2),0,1)),0,2))\)

\(\alpha(300) = S(S(S(3,S(T(1),2,1),2),2,S(S(S(3,2,2),1,S(S(3,1,1),0,2)),0,2)),0,S(S(S(3,2,1),1,1),0,2))\)

\(\alpha(400) = S(S(S(3,S(T(1),2,2),1),1,2),0,2)\)

\(\alpha(500) = S(S(S(3,S(T(1),2,2),1),S(T(1),0,S(T(0),2,2)),1),0,S(S(3,2,2),0,S(S(3,1,2),0,S(3,1,1))))\)

\(\alpha(600) = S(S(3,S(T(1),2,2),2),1,S(S(S(S(3,S(T(1),0,2),1),S(T(1),0,1),1),1,3),0,S(S(S(S(S(3,S(T(0),1,2),2),S(T(0),1,1),1),S(T(0),0,2),2),T(0),S(S(S(3,S(T(0),1,2),1),1,S(3,0,1)),0,1)),2,S(3,2,1))))\)

\(\alpha(700) = S(S(S(S(S(S(S(3,S(T(1),2,2),2),S(T(0),2,2),2),S(T(0),2,1),S(S(3,1,2),0,1)),S(T(0),1,2),1),S(T(0),1,1),S(S(3,1,2),0,1)),S(T(0),0,2),2),T(0),S(S(3,S(T(1),0,S(T(0),1,2)),1),S(T(0),1,2),S(3,T(0),1)))\)

\(\alpha(800) = S(S(S(3,S(T(1),2,T(0)),1),2,2),0,S(S(3,2,1),0,S(3,0,1)))\)

\(\alpha(900) = S(S(S(S(S(S(3,S(T(1),2,T(0)),2),S(T(1),0,S(T(0),1,1)),S(3,1,2)),S(T(0),2,1),1),S(T(0),1,1),2),1,S(S(3,S(T(0),0,1),1),2,S(S(S(3,T(0),2),1,1),0,S(S(3,1,2),0,1)))),0,2)\)

\(\alpha(1000) = S(S(3,S(T(1),2,S(T(0),0,1)),2),0,2)\)

\(\alpha(1500) = S(S(3,S(T(1),2,S(T(0),1,1)),2),1,S(S(3,S(T(1),0,1),2),T(0),S(S(S(S(3,S(T(1),0,1),1),T(1),S(S(3,T(1),2),0,S(S(3,1,1),0,S(3,0,1)))),S(T(0),2,1),S(3,T(0),1)),2,2)))\)

\(\alpha(2000) = S(S(S(3,S(T(1),2,S(T(0),1,2)),2),S(T(0),1,2),1),1,S(S(S(3,2,2),1,1),0,1))\)

\(\alpha(2500) = S(S(S(3,S(T(1),2,S(T(0),2,1)),2),2,S(3,0,1)),0,3)\)

\(\alpha(5000) = S(S(S(S(3,S(T(1),T(0),2),1),S(T(1),T(0),1),S(S(3,S(T(1),T(0),1),2),0,S(S(3,2,2),0,S(S(S(3,2,1),1,2),0,3)))),S(T(0),1,1),S(S(3,1,2),0,S(3,0,1))),S(T(0),0,2),S(3,S(T(0),1,2),1))\)

\(\alpha(7500) = S(S(S(S(3,S(T(1),T(0),S(T(0),1,2)),1),S(T(1),0,T(0)),2),2,1),1,S(S(S(3,S(T(1),2,2),1),T(1),S(3,1,2)),0,2))\)

\(\alpha(10000) = S(S(S(S(S(3,S(T(1),S(T(0),0,1),1),1),S(T(0),0,2),S(3,0,1)),T(0),2),1,2),0,1)\)

\(\alpha(20000) = S(S(S(3,S(T(1),S(T(0),0,1),S(T(0),1,2)),1),2,S(S(3,1,1),0,3)),0,3)\)

\(\alpha(50000) = S(S(S(S(S(S(3,S(T(1),S(T(0),0,2),S(T(0),1,1)),1),S(T(0),0,1),2),T(0),3),2,S(S(3,2,2),0,1)),1,1),0,S(3,0,2))\)

\(\alpha(100000) = S(S(3,S(T(1),S(T(0),1,1),2),1),0,2)\)

\(\alpha(1000000) = S(S(3,S(T(1),S(T(0),1,2),S(T(0),1,2)),2),1,2)\)

\(\alpha(10000000) = S(S(S(3,S(T(1),S(T(0),2,2),2),1),S(T(0),2,2),S(3,S(T(1),2,S(T(0),2,2)),1)),2,S(S(3,1,2),0,2))\)

\(\alpha(100000000) = S(S(S(3,S(T(2),0,2),2),1,S(S(S(S(3,S(T(1),S(T(0),2,2),T(0)),1),2,2),1,2),0,2)),0,2)\)

\(\alpha(1000000000) = S(S(3,S(T(2),1,1),2),S(T(1),1,S(T(0),2,1)),2)\)

\(\alpha(10000000000) = S(S(S(3,S(T(2),1,S(T(1),S(T(0),2,2),2)),1),S(T(1),2,S(T(0),1,1)),1),0,S(S(3,2,1),0,S(3,0,1)))\)

\(\alpha(100000000000) = S(S(S(3,S(T(2),2,S(T(1),0,1)),1),1,S(3,0,1)),0,2)\)

\(\alpha(1000000000000) = S(S(S(S(3,S(T(2),S(T(0),1,1),1),1),S(T(2),T(0),S(T(1),1,S(T(0),2,2))),S(S(3,T(0),2),0,1)),S(T(0),2,1),1),0,S(S(3,1,2),0,S(3,0,2)))\)

\(\alpha(10000000000000) = S(S(S(S(3,S(T(2),S(T(0),2,2),1),2),S(T(0),2,1),S(S(3,1,2),0,1)),S(T(0),1,1),2),S(T(0),0,1),1)\)

\(\alpha(100000000000000) = S(S(S(3,S(T(2),S(T(1),2,2),1),2),1,S(3,1,2)),0,2)\)

\(\alpha(1000000000000000) = S(S(3,S(T(2),S(T(1),S(T(0),2,1),S(T(0),1,2)),1),1),0,S(S(S(S(S(3,S(T(2),S(T(1),T(0),S(T(0),0,2)),2),1),S(T(2),0,2),S(S(3,2,2),0,1)),2,S(S(S(3,2,2),1,S(3,0,2)),0,1)),1,1),0,S(3,0,2)))\)

\(\alpha(10000000000000000) = S(S(3,S(T(S(T(0),0,1)),S(T(T(0)),2,2),2),1),0,S(3,1,2))\)

\(\alpha(100000000000000000) = S(S(S(3,S(T(S(T(0),1,1)),S(T(T(0)),1,S(T(1),2,2)),2),2),T(0),1),0,S(S(3,1,1),0,2))\)

\(\alpha(1000000000000000000) = S(S(S(S(3,S(T(S(T(0),2,1)),1,S(T(S(T(0),1,1)),2,S(T(1),S(T(0),2,2),2))),2),2,2),1,1),0,S(S(S(3,S(T(T(0)),1,2),1),1,2),0,1))\)

\(\alpha(10000000000000000000) = S(S(S(S(3,S(T(S(T(0),2,2)),S(T(S(T(0),1,1)),S(T(1),S(T(0),1,1),S(T(0),1,2)),S(T(2),0,S(T(0),1,1))),1),1),S(T(2),2,S(T(0),2,1)),1),S(T(1),S(T(0),2,1),1),3),S(T(0),2,2),S(3,S(T(S(T(0),2,2)),T(T(0)),1),1))\)

\(\alpha(1E+20) = S(S(S(3,S(T(S(T(1),1,2)),S(T(1),2,1),S(T(0),1,2)),1),S(T(0),1,2),2),S(T(0),1,1),S(S(3,S(T(1),S(T(0),0,2),S(T(0),1,2)),1),S(T(0),2,2),S(S(3,S(T(1),S(T(0),0,2),S(T(0),1,1)),2),T(0),S(3,S(T(0),1,1),1))))\)

\(\alpha(1E+21) = S(S(3,S(T(S(T(1),2,S(T(0),2,2))),2,1),2),0,2)\)

\(\alpha(1E+22) = S(S(3,S(T(S(T(1),S(T(0),1,1),S(T(0),2,2))),S(T(2),S(T(1),S(T(0),1,2),1),S(T(0),2,2)),S(T(S(T(1),S(T(0),1,1),1)),S(T(2),T(1),S(T(1),S(T(0),2,1),S(T(0),1,1))),S(T(0),2,2))),1),0,S(3,T(S(T(0),2,2)),1))\)

\(\alpha(1E+23) = S(S(S(3,S(T(T(2)),1,1),2),2,1),0,3)\)

\(\alpha(1E+24) = S(S(3,S(T(S(T(2),1,S(T(0),0,2))),S(T(2),S(T(0),1,1),S(T(1),2,S(T(0),2,2))),1),1),0,S(S(3,1,1),0,S(3,0,1)))\)

\(\alpha(1E+25) = S(S(S(S(3,S(T(S(T(2),T(0),S(T(0),2,2))),0,2),2),2,1),1,2),0,S(3,0,1))\)

\(\alpha(1E+26) = S(3,S(T(S(T(2),S(T(1),0,S(T(0),0,2)),S(T(0),1,2))),1,S(T(S(T(1),1,2)),S(T(0),2,2),S(T(S(T(1),1,1)),S(T(S(T(0),2,1)),S(T(T(0)),0,S(T(2),S(T(1),1,S(T(0),2,1)),S(T(0),2,1))),S(T(S(T(0),0,1)),S(T(1),T(0),1),1)),1))),1)\)

\(\alpha(1E+27) = S(S(3,S(T(S(T(T(0)),S(T(0),2,1),1)),2,S(T(1),0,1)),1),0,S(S(3,S(T(S(T(1),S(T(0),0,1),1)),S(T(S(T(1),T(0),2)),1,2),S(T(1),S(T(0),2,2),1)),2),0,S(S(3,2,1),1,1)))\)

\(\alpha(1E+28) = S(S(S(S(3,S(T(S(T(S(T(0),1,2)),2,S(T(S(T(0),1,1)),S(T(T(0)),2,S(T(0),1,1)),1))),0,1),2),S(T(2),1,2),2),1,2),0,S(S(S(3,T(S(T(1),1,2)),2),S(T(1),S(T(0),2,1),2),S(S(3,1,2),0,3)),T(0),1))\)

\(\alpha(1E+29) = S(S(3,S(T(S(T(S(T(1),0,S(T(0),0,1))),2,S(T(S(T(0),1,2)),1,S(T(S(T(0),1,1)),S(T(0),2,2),2)))),0,2),1),1,2)\)

\(\alpha(1E+30) = S(S(3,S(T(T(S(T(1),S(T(0),0,2),1))),2,S(T(S(T(S(T(0),1,2)),0,S(T(2),S(T(0),2,1),2))),S(T(S(T(S(T(0),0,2)),1,2)),1,S(T(S(T(2),S(T(1),S(T(0),1,2),S(T(0),0,2)),2)),2,2)),1)),1),S(T(0),2,1),3)\)

\(\alpha(1E+31) = S(S(3,T(S(T(S(T(2),0,T(1))),S(T(0),2,1),S(T(1),2,1))),2),1,S(S(S(S(S(S(3,S(T(1),S(T(0),2,1),S(T(0),2,1)),2),S(T(0),1,1),1),S(T(0),0,1),S(3,0,2)),T(0),2),2,1),1,S(3,T(0),1)))\)

\(\alpha(1E+32) = S(S(3,S(T(S(T(S(T(2),S(T(0),1,2),2)),2,S(T(1),S(T(0),1,1),S(T(0),1,2)))),2,S(T(S(T(2),S(T(0),2,1),S(T(1),S(T(0),2,2),2))),1,S(T(1),1,S(T(0),1,2)))),1),S(T(0),2,2),S(3,T(S(T(1),T(0),S(T(0),2,2))),1))\)

\(\alpha(1E+33) = S(S(3,S(T(S(T(S(T(S(T(0),0,2)),S(T(T(0)),1,2),2)),S(T(S(T(T(0)),S(T(1),2,1),2)),S(T(2),1,S(T(1),S(T(0),2,1),2)),S(T(2),2,2)),1)),S(T(1),S(T(0),1,2),S(T(0),0,2)),2),1),S(T(2),2,S(T(0),1,1)),1)\)

\(\alpha(1E+34) = S(3,S(T(S(T(S(T(S(T(1),2,1)),0,S(T(1),0,T(0)))),1,S(T(S(T(S(T(1),1,2)),2,S(T(S(T(1),1,1)),S(T(S(T(1),0,S(T(0),1,2))),2,S(T(S(T(1),0,2)),2,2)),2))),T(0),1))),T(T(S(T(S(T(0),1,1)),1,T(0)))),1),1)\)

\(>> S(3,S(T(S(T(S(T(S(T(1),2,1)),0,T(1))),1,S(T(S(T(S(T(1),1,2)),2,S(T(S(T(1),1,1)),S(T(S(T(1),0,S(T(0),1,2))),2,S(T(T(1)),2,2)),2))),T(0),1))),T(T(S(T(S(T(0),1,1)),1,T(0)))),1),1)\)

\(>> S(3,S(T(S(T(T(S(T(1),2,1))),1,S(T(S(T(S(T(1),1,2)),2,S(T(S(T(1),1,1)),S(T(S(T(1),0,S(T(0),1,2))),2,S(T(T(1)),2,2)),2))),T(0),1))),T(T(S(T(S(T(0),1,1)),1,T(0)))),1),1)\)

\(>> S(3,S(T(S(T(T(S(T(1),2,1))),1,S(T(S(T(S(T(1),1,2)),2,S(T(S(T(1),1,1)),S(T(S(T(1),0,S(T(0),1,2))),2,S(T(T(1)),2,2)),2))),T(0),1))),T(T(T(T(0)))),1),1)\)

\(>> S(3,S(T(S(T(T(S(T(1),2,1))),1,S(T(S(T(S(T(1),1,2)),2,S(T(S(T(1),1,1)),S(T^2(1),2,S(T^2(1),2,2)),2))),T(0),1))),T^4(0),1),1)\)

\(>> S(3,S(T(S(T(T(S(T(1),2,1))),1,S(T(S(T(S(T(1),1,2)),2,S(T^2(1),S(T^2(1),2,S(T^2(1),2,2)),2))),T(0),1))),T^4(0),1),1)\)

\(>> S(3,S(T(S(T(T(T(1))),1,S(T(S(T(S(T(1),1,2)),2,S(T^2(1),S(T^2(1),2,S(T^2(1),2,2)),2))),T(0),1))),T^4(0),1),1)\)

\(>> S(3,S(T(S(T(T(T(1))),1,S(T(S(T^2(1),2,S(T^2(1),S(T^2(1),2,S(T^2(1),2,2)),2))),T(0),1))),T^4(0),1),1)\)

\(>> S(3,S(T(S(T(T(T(1))),1,S(T(S(T^2(1),2,S(T^2(1),S(T^2(1),2,T^2(1)),2))),T(0),1))),T^4(0),1),1)\)

\(>> S(3,S(T(S(T(T(T(1))),1,S(T(S(T^2(1),2,S(T^2(1),T^2(1),2))),T(0),1))),T^4(0),1),1)\)

\(>> S(3,S(T(S(T(T(T(1))),1,S(T(S(T^2(1),2,T^2(1))),T(0),1))),T^4(0),1),1)\)

\(>> S(3,S(T(S(T(T(T(1))),1,S(T(T^2(1)),T(0),1))),T^4(0),1),1)\)

\(>> S(3,S(T(S(T(T(T(1))),1,T(T^2(1)))),T^4(0),1),1)\)

\(>> S(3,S(T(T(T(T(1)))),T^4(0),1),1)\)

\(>> S(3,S(T^4(1),T^4(0),1),1)\)

\(>> S(3,T^4(1),1) >> S(3,T^3(0),1) >> f_{\psi(\Omega\uparrow\uparrow\omega)}(3)\) i.e. using the Bachmann-Howard ordinal To be confirmed

\(\alpha(1E+35) = S(S(3,S(T(S(T(S(T(S(T(2),2,1)),S(T(S(T(0),1,1)),S(T(2),2,1),S(T(0),2,2)),1)),S(T(T(1)),1,S(T(1),S(T(0),2,2),1)),2)),0,S(T(2),0,S(T(0),0,2))),1),1,2)\)

\(\alpha(1E+36) = S(3,T(S(T(S(T(S(T(S(T(0),2,1)),S(T(2),S(T(1),2,S(T(0),0,2)),1),1)),S(T(0),1,1),S(T(T(S(T(0),2,1))),2,1))),2,T(S(T(S(T(S(T(0),1,1)),1,S(T(2),1,2))),1,T(S(T(T(0)),S(T(0),1,1),2)))))),1)\)

\(\alpha(1E+37) = S(3,T(T(S(T(S(T(S(T(2),S(T(1),2,2),2)),1,S(T(S(T(0),0,2)),S(T(S(T(0),0,1)),T(T(0)),1),S(T(2),1,T(0))))),S(T(S(T(1),0,1)),T(0),S(T(2),2,S(T(0),0,1))),S(T(1),S(T(0),1,2),T(0))))),1)\)

\(\alpha(1E+38) = S(3,T(S(T(S(T(S(T(T(S(T(S(T(1),S(T(0),0,2),1)),T(0),S(T(S(T(1),T(0),S(T(0),1,1))),0,1)))),2,2)),2,S(T(2),T(0),2))),T(T(T(T(S(T(S(T(0),2,2)),S(T(2),S(T(0),1,2),1),1))))),1)),1)\)

\(>> S(3,T(S(T(S(T(S(T(T(S(T(S(T(1),S(T(0),0,2),1)),T(0),S(T(S(T(1),T(0),T(0))),0,1)))),2,2)),2,S(T(2),T(0),2))),T(T(T(T(S(T(T(0)),S(T(2),T(0),1),1))))),1)),1)\)

\(>> S(3,T(S(T(S(T(S(T(T(S(T(S(T(1),T(0),1)),T(0),S(T(T(1)),0,1)))),2,2)),2,S(T(2),T(0),2))),T(T(T(T(S(T(T(0)),T(2),1))))),1)),1)\)

\(>> S(3,T(S(T(S(T(S(T(T(S(T(T(1)),T(0),T(T(1))))),2,2)),2,T(2))),T(T(T(T(S(T(T(0)),T(2),1))))),1)),1)\)

\(>> S(3,T(S(T(S(T(S(T(T(T(T(1)))),2,2)),2,T(2))),T(T(T(T(T(T(0)))))),1)),1)\)

\(= S(3,T(S(T(S(T(S(T^4(1),2,2)),2,T(2))),T^6(0),1)),1)\)

\(>> S(3,T(S(T(T(T^4(1))),T^6(0),1)),1)\)

\(= S(3,T(S(T^6(1),T^6(0),1)),1)\)

\(>> S(3,T(T^6(1)),1)\)

\(= S(3,T^7(1),1)\)