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Alpha Function Code Version 7

This latest version of The Alpha Function has been aligned to my latest blog on The S Function (Version 2). The growth rate has been significantly increased to well beyond $$f_{LVO}(n)$$.

Changes in Version 7

Version 7 has been completely modified from Version 6 to use my latest work on The S Function.

The key change has been to generalise the string substitution procedures introduced by the S Function. The S Function is recursively defined with a sibling T Function. The T Function has been generalised to support significantly more recursive behaviour. This means the S Function can generate very long strings of S and T function combinations. Each string can be translated to a finite integer and the S Function has a growth rate well above $$f_{LVO}(n)$$.

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 can translate unique real numbers into any and every finite integer (up to and beyond $$f_{LVO}(n)$$ for any $$n$$).

Example of Changes in Version 7

Version 7 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 $$n$$).

$$\alpha(0) = 0$$

$$\alpha(0.5) = 1$$

$$\alpha(1) = 2$$

$$\alpha(1.4) = 3$$

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

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

$$\alpha(1.56) = S(3,1,1) = 6$$

$$\alpha(1.58) = S(S(3,1,1),0,1) = 7$$

$$\alpha(1.6) = S(S(3,1,1),0,2) = 8$$

$$\alpha(1.62) = S(S(3,1,1),0,3) = 9$$

$$\alpha(1.64) = S(S(3,1,1),0,S(3,0,1)) = 10$$

$$\alpha(1.66) = S(S(3,1,1),0,S(3,0,2)) = 11$$

$$\alpha(1.68) = S(3,1,2) = 12$$

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

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

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

$$\alpha(3) = S(S(3,S(T(0),0,2),1),1,2) = f_1^2(f_{\omega + 2}(3))$$

$$\alpha(10) = S(S(3,S(T(0),2,2),2),0,S(S(3,S(T(0),1,1),2),S(T(0),0,2),1)) = f_{\epsilon_0}(3)$$ approximately

$$\alpha(40) = S(S(S(3,S(T(1),1,T(0)),1),1,3),0,S(S(3,1,1),0,S(3,0,1))) >> f_{\varphi(\omega,0)}(3)$$

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

$$>> f_{\varphi(1,0,0)}(3) = f_{\Gamma_0}(3)$$

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

$$>> f_{\varphi(1,0_{[3]})}(3) = f_{SVO}(3)$$

Growth Rate of the Alpha Function

The Alpha Function is currently 'calibrated' to accept real number inputs up to $$1.10^{38}$$ which is equal to $$4^{4^3}$$. This is arbitrary and I will adjust this to provide a more interesting range.

At the present limit of $$1.10^{38}$$, the Alpha Function will approach this S Function:

$$\alpha(4^{4^3}) = S(3,T^{\omega}(0),1)$$

I estimate that:

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

Version 7 Code

The code for the Alpha Function uses Sequence Generator Code Syntax to generate sequences of finite integers. Refer to that blog for more information on how that works and is compiled.

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<V_t,V_t=t_x,

t_x>(0:(n_c<V_n,n_c>(0:,[S([],[t_x],[n_c])])),

(n_C<V_n,V_n=C_d..n_C,[S([],[t_x],[n_C])],

s_x+1<t_x,[[s_x+1]])))

t_x = (C_t<2,

C_t>(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

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

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

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

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

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

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

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

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

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

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

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

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

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