User blog:B1mb0w/Sequence Generator Code

Sequence Generator Code
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

Example Sequence Generator Code
Here is example code. More information on the syntax and logic will be provided.

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)

WORK IN PROGRESS

Example Sequences (in JSON format)
Sequences are generated using JSON format. A good JSON Viewer can be found here.

Here are some examples:

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

{"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}}}}}}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\(\beta(2.2425,2) = f_{\omega}(2).2 + 4 = 20\)

\(\beta(2.245,2) = f_{\omega}(2).2 + 5 = 21\)

\(\beta(2.247,2) = f_{\omega}(2).2 + 6 = 22\)

\(\beta(2.25,2) = f_{\omega}(2).2 + 7 = 23\)

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

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

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

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

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

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

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

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

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

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

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

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

\(\beta(2.4177,2) = f_{2}(f_{\omega}(2)).4 = 8192\)

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

\(\beta(2.4207,2) = f_{2}(f_{\omega}(2)).16 = 32768\)

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

\(\beta(2.4224,2) = f_{2}(f_{\omega}(2)).64 = 131072\)

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

\(\beta(2.424,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2)}) = 524288\)

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

\(\beta(2.4256,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2) + 2}) = 2097152\)

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

\(\beta(2.4264,2) = f_{2}(f_{\omega}(2)).(2^{f_{\omega}(2) + 4}) = 8388608\)

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

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

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

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

\(\beta(5.66,2) = f_{\varphi(\omega,0) + \omega}(2) + f_{\varphi(1,f_{\omega + 1}(2) + 1)^{f_{\varphi(1,0)^{\omega} + \omega}(2).(2^{f_{\omega + 1}(2).2 + 1}) + 6}.(\omega^{f_{\omega}(2) + 1})}(f_{\varphi(\omega,0)}(2))\)

\(\beta(5.661,2) = f_{\varphi(1,\omega\uparrow\uparrow f_{7}^5(f_{\omega}(2)).2.(\omega\uparrow\uparrow 7.(\omega\uparrow\uparrow 2^{\omega.(f_{\omega\uparrow\uparrow f_{5}(f_{\omega}(2))}(f_{\varphi(\omega,0) + 1}(2)))})))}(f_{\varphi(\omega,0) + \omega}(2))\)

\(\beta(5.662,2) = f_{\omega.(f_{\omega}(f_{\omega + 1}(2)).128) + f_{4}^{f_{2}^7(f_{\omega}(2)).2 + 1}(f_{5}(f_{\omega}(2))).2 + 1}^6(f_{\varphi(\omega,0) + \omega + 1}(2))\)

\(\beta(5.663,2) = f_{\varphi(5,0)\uparrow\uparrow f_{5}(f_{\omega}(2)).(2^{f_{2}(f_{\omega}(2))}) + 1^{\omega\uparrow\uparrow 2}}(f_{\varphi(\omega,0) + 6}(f_{\varphi(\omega,0) + \varphi(1,0) + \omega + 1}(2)))\)

\(\beta(5.664,2) = f_{\varphi(\omega,0) + \varphi(1,1)^{\varphi(\varphi(1,0).(\omega + 1) + 1,\varphi(1,\varphi(1,0)^{\varphi(1,0) + 1}.(\varphi(\omega + 1,\varphi(\varphi(1,0) + \omega,\varphi(\omega,\varphi(1,0).(\omega + 1) + 1) + 1).(\omega + 1) + \varphi(1,0).(\omega) + \omega + 1)).(\varphi(1,0)^{\omega}.(\omega) + \varphi(1,0) + \omega + 1) + \omega)^{\omega + 1} + \omega + 1)}}(2)\)

\(\beta(5.665,2) = f_{\varphi(1,\varphi(3,1)^{f_{f_{7}(f_{\omega}(2))}(f_{\omega + 1}(2))} + 1)}(f_{\varphi(\omega,0) + \varphi(\omega,\varphi(1,\omega).(\varphi(\omega,\varphi(1,\omega)^{\omega}.(\omega) + 1.(\omega + 1) + \varphi(1,\omega)^{\omega}.(\varphi(1,1).(\omega + 1) + \varphi(1,0)^{\omega}) + \omega + 1)) + \omega + 1^{\omega + 1} + \omega)}(2))\)

\(\beta(5.667,2) = f_{\omega.(f_{f_{\omega}^2(f_{\omega + 1}(2)) + f_{\omega}(2).8 + 1}^3(f_{\omega}^5(f_{\omega + 1}(2))).2 + 3) + 4}^2(f_{\varphi(\omega,0).(\omega) + 1}(2))\)

\(\beta(5.668,2) = f_{\varphi(1,\omega^5.(f_{\varphi(1,\varphi(1,\varphi(1,\omega)^{\omega}.(\omega + 1) + 1^{\omega + 1} + \omega)^{\varphi(\omega + 1,\varphi(1,\omega)^{\omega}.(\varphi(1,1).(\omega + 1) + \varphi(1,0)^{\omega} + \omega) + \omega + 1 + \omega + 1)}.(\omega + 1) + 1)}(2).4))}(f_{\varphi(\omega,0).(\omega) + \omega}(2))\)

\(\beta(5.669,2) = f_{\omega\uparrow\uparrow f_{\omega}^2(f_{\omega + 1}(2)) + 1^{f_{\varphi(\omega,\varphi(1,0)^{\varphi(1,\varphi(1,\varphi(1,0)^{\omega}.(\omega + 1) + 1).(\varphi(1,0).(\omega + 1) + \omega) + \varphi(1,\varphi(1,0).(\omega + 1) + 1))}^{\omega}.(\varphi(1,0)^{\omega + 1}) + 1)}(2)}}(f_{\varphi(\omega,0).(\omega) + \varphi(1,0) + \omega}(2))\)

\(\beta(5.671,2) = f_{\varphi(1,1)^{\omega\uparrow\uparrow f_{\omega\uparrow\uparrow 3.(f_{\omega\uparrow\uparrow f_{\varphi(1,\varphi(1,1).(\varphi(1,1)^{\omega + 1}.(\omega) + \varphi(1,1).(\omega) + \varphi(1,\varphi(1,0)^{\omega + 1}.(\omega) + \varphi(1,0)^{\omega + 1}.(\omega)))^{\omega}.(\omega + 1) + 1)}(2)}(f_{\varphi(\omega,0) + 1}(2)))}(f_{\varphi(\omega,0) + \omega}(2))}}(f_{\varphi(\omega,0).(\omega + 1)}(2))\)

\(\beta(5.672,2) = f_{\varphi(\omega,0).(\omega + 1)}^{f_{3}(f_{\varphi(\omega,0) + \omega + 1}(2)).16 + f_{f_{5}^{f_{5}(f_{\omega}(2)) + 2}(f_{\omega + 1}(2))}(f_{\omega}^6(f_{\omega + 1}(2)))}(f_{\varphi(\omega,0).(\omega + 1) + 1}(2))\)

\(\beta(5.674,2) = f_{\varphi(\omega,0).(\omega) + \omega\uparrow\uparrow f_{4}(f_{6}^{f_{\omega}(f_{\omega + 1}(2)) + 3}(f_{\omega}^3(f_{\omega + 1}(2)))).2 + 1^{\omega\uparrow\uparrow 6}}(f_{\varphi(\omega,0).(\omega + 1) + \varphi(1,1)^{\omega + 1} + \omega + 1}(2))\)

\(\beta(5.676,2) = f_{\varphi(\omega,0).(\varphi(1,0)^{\omega + 1} + \varphi(1,0).(\omega) + \omega) + 1}(2) + 2\)

\(\beta(5.678,2) = f_{f_{\varphi(1,\omega)}(2)}(f_{\omega\uparrow\uparrow f_{\omega}(2).128 + 1 + \omega + f_{\omega + 1}(2).2}^5(f_{\varphi(\omega,0).(\varphi(1,1)^{\omega + 1}.(\omega + 1) + \varphi(1,0)^{\omega} + \omega) + \omega}(2)))\)

\(\beta(5.679,2) = f_{\varphi(\omega,0).(\varphi(1,\varphi(1,1)^{\omega}.(\varphi(1,1).(\varphi(\omega + 1,\varphi(1,0)^{\varphi(1,0).(\omega) + 1} + \omega^{\varphi(1,0).(\omega + 1)}.(\omega) + \omega + 1)) + \varphi(1,\varphi(\omega,\varphi(1,0).(\varphi(1,0).(\omega) + \omega) + \varphi(1,0)^{\omega} + \varphi(1,0).(\omega) + 1).(\varphi(1,0) + \omega + 1) + \omega + 1)) + 1))}(2)\)

\(\beta(5.682,2) = f_{\varphi(\omega,0).(\varphi(1,\omega + 1).(\omega)) + \omega}(2).2 + 3\)

\(\beta(5.683,2) = f_{\varphi(\omega,0).(\varphi(1,\omega + 1)^{\varphi(\omega + 1,\varphi(1,\omega).(\omega) + \omega + 1^{\varphi(1,\omega)^{\omega}.(\omega) + \varphi(1,0)^{\omega + 1}.(\omega) + \varphi(1,0).(\omega) + 1}.(\varphi(1,1)^{\varphi(1,0)^{\omega + 1} + \varphi(1,0)^{\omega}.(\omega + 1) + \omega}.(\omega) + \omega) + \omega)}.(\omega) + \omega) + \omega}(2) + f_{\omega + 1}(2)\)

\(\beta(5.684,2) = f_{\varphi(\omega,0).(\varphi(1,\varphi(1,\omega + 1)^{\varphi(1,\omega)^{\varphi(1,1).(\varphi(1,\varphi(1,0)^{\omega}.(\varphi(1,0)^{\omega + 1}.(\omega) + 1) + 1.(\omega + 1) + 1)) + \omega + 1} + \varphi(1,\omega)^{\varphi(1,0)^{\omega + 1} + 1}.(\varphi(\omega,\varphi(1,1)^{\omega}.(\omega)^{\omega} + \omega)) + \omega} + \varphi(1,1)))}(2)\)

\(\beta(5.685,2) = f_{\varphi(\omega,0)^{\omega} + 1}(2).2 + f_{\varphi(1,1)^{\varphi(1,0)^{f_{\varphi(1,\varphi(1,0) + 1)}(2)}}}(f_{\varphi(1,1)^{\varphi(1,0)^{\omega}}.(\omega) + \varphi(1,\varphi(\omega,\varphi(1,0)^{\omega} + 1.(\omega) + 1).(\varphi(\omega,\varphi(1,0)^{\varphi(1,0) + 1}.(\omega + 1) + 1 + \varphi(1,0) + \omega + 1)) + 1)}(2))\)

\(\beta(5.686,2) = f_{f_{\varphi(1,\varphi(1,\varphi(1,1)^{\varphi(1,\varphi(1,0)^{\omega + 1}.(\omega) + \omega + 1^{\omega + 1}.(\omega + 1) + 1)}.(\omega) + \omega)^{\omega} + \varphi(1,\varphi(1,0) + 1))}(2)}(f_{\omega + 1}(f_{\omega^5 + f_{\omega + 1}(2).2 + 1}(f_{\varphi(\omega,0)^{\omega} + \omega}(2))))\)

\(\beta(5.687,2) = f_{\varphi(1,\omega\uparrow\uparrow f_{\varphi(\omega,0)^3.(\varphi(f_{\varphi(\omega,0) + 1}(2).8 + 7,\varphi(1,3)^{\omega\uparrow\uparrow f_{\varphi(1,\varphi(1,\omega + 1)^{\varphi(1,1).(\varphi(1,\varphi(1,\varphi(1,0) + \omega + 1)^{\omega + 1} + 1))} + 1)}(2)}))}(f_{\varphi(\omega,0)^{\omega} + \omega}(2)))}(f_{\varphi(\omega,0)^{\omega} + \omega + 1}(2))\)

\(\beta(5.688,2) = f_{2}(f_{\varphi(\omega,0)^{\omega} + \varphi(\omega,0) + 1}(2)) + 1\)

\(\beta(5.691,2) = f_{\varphi(\omega,0)^{\omega}.(\omega + 1)}(2)\)

\(\beta(5.692,2) = f_{\omega\uparrow\uparrow f_{\varphi(1,\varphi(1,3)^{f_{3}(f_{\omega}(2)).(2^{f_{2}^5(f_{\omega}(2)).2 + f_{\omega}(2) + 2}) + 7}.(\omega\uparrow\uparrow 2) + 1)}(f_{\varphi(\omega,0).(\omega) + \omega + 1}(2))}(f_{\varphi(\omega,0)^{\omega}.(\omega + 1) + \omega}(2))\)

\(\beta(5.694,2) = f_{\varphi(\omega,0)^{\omega}.(\varphi(\omega,\varphi(1,0).(\varphi(1,0) + 1) + \omega + 1.(\varphi(1,0).(\omega + 1)) + \varphi(1,0).(\omega))) + \varphi(\omega,\varphi(1,1)^{\omega} + \varphi(1,1)^{\varphi(1,0) + \omega}^{\omega}.(\omega) + \varphi(1,0).(\omega))}(2)\)

\(\beta(5.695,2) = f_{\varphi(\omega,0)^{\omega}.(\varphi(\omega + 1,\varphi(\varphi(1,0)^{\omega}.(\omega + 1) + 1,\varphi(1,\varphi(1,1)^{\omega + 1}.(\varphi(1,1)^{\omega}.(\omega) + \omega + 1) + \varphi(1,\varphi(1,0) + 1) + 1)^{\varphi(1,1)^{\varphi(1,\varphi(1,0) + \omega + 1.(\varphi(1,0)^{\omega + 1}.(\omega)) + \omega)}.(\omega + 1)}.(\omega + 1) + \omega + 1)^{\varphi(1,0)^{\omega + 1} + \omega + 1} + 1))}(2)\)

\(\beta(5.696,2) = f_{\omega\uparrow\uparrow f_{4}^3(f_{\omega}(2)).(2^{f_{\omega}(2) + 6})}(f_{\varphi(\omega,0)^{\omega}.(\varphi(1,\omega)^{\varphi(\omega + 1,\varphi(1,0)^{\omega + 1}.(\varphi(1,0) + \omega) + \omega.(\omega + 1) + \varphi(1,0)^{\omega + 1}.(\omega + 1) + \varphi(1,0).(\omega + 1))} + 1) + \omega + 1}(2))\)

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

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

or

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

or

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

or

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

or

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

or

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

or

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

or

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

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

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

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

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

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

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

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 \(v = 3\)

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

\(\beta(12.2118455947222,3) = f_{\varphi(1,\varphi(2,0)\uparrow\uparrow f_{\varphi(1,\varphi(2,0)\uparrow\uparrow 2^{\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(1,\omega\uparrow\uparrow 2^{\omega^2.2 + \omega.2 + 2}.(\omega + 1) + \omega\uparrow\uparrow 2.2 + 1)^2.(\omega\uparrow\uparrow 2^{\omega^2.2 + \omega + 2}.(\omega^2 + \omega)) + 1))}}} + 1)} + 1)}(3) + 1)}(f_{\varphi(2,1)}(3))\)

\(\beta(12.21184559471,3) = f_{\varphi(1,\varphi(2,0)\uparrow\uparrow f_{\varphi(1,\varphi(2,0)\uparrow\uparrow 2^{\varphi(1,\varphi(2,0)^{\varphi(2,0)\uparrow\uparrow 2^{\varphi(2,0)^{\varphi(1,\varphi(1,\varphi(1,\omega.2 + 2)^{\varphi(1,\varphi(1,\omega + 1).(\varphi(1,\varphi(1,0)\uparrow\uparrow 2^{\omega + 2}.(\omega + 2) + \varphi(1,0)^{\omega\uparrow\uparrow 2.(\omega^2.2 + \omega + 1) + \omega})) + 1)} + 1))}}} + 1)} + 1)}(3) + 1)}(f_{\varphi(2,1)}(3))\)

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

Related to First Error in Second Attempt

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

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

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

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

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

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

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

Fifth Attempt base \(v = 2\)

\(\beta(5.65686,2) = f_{\varphi(\omega,0)}(2)\)

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

\(\beta(5.73536669398825,2)\)

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

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

and

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

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

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

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

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

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

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

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

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

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

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

\(= \varphi(\omega,0)^{\varphi(\omega,0)}\)

\(= \varphi(1,\varphi(\omega,0) + 1)\)

\(\beta(5.739133787073,2)\)

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

and

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

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

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

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

or

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

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

\(\beta(3.085,2) = f_{2}^{f_{\omega}^4(f_{\omega + 1}(2)) + 1}(f_{\omega^{\omega^{\omega}}}(2)).(2^{f_{\omega}(2) + 1})\)

\(\beta(3.152,2) = f_{3}(f_{\omega^{\omega^{\omega^{\omega}}}}(2)).4 + 7\)

\(\beta(3.1865,2) = f_{\omega^{\omega.(f_{7}(f_{\omega}(2)).128 + 1)}.2 + f_{\omega^{\omega^{\omega^{\omega}}}}(2)}(f_{\omega^{\omega^{\omega^{(\omega\uparrow\uparrow 2)}} + 1} + \omega}(2))\)

\(\beta(3.38,2) = f_{\omega^{\omega^{f_{\omega}(2).2}.(f_{\omega + 1}(2) + 1) + \omega} + \omega^{\omega^{f_{\omega}(2).2} + \omega^2 + 4}}(f_{\omega^{(\omega\uparrow\uparrow 2)} + 1}(2))\)

Third Attempt base \(v = 2\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Fourth Attempt base \(v = 2\)

\(\beta(4.04539130434783,2) = f_{2}(f_{\varphi(1,0) + \omega}(2))\)

\(\beta(4.04720037807183,2) = f_{f_{(\varphi(1,\varphi(1,0))\uparrow\uparrow f_{f_{7}^2(f_{\omega}(2))}(f_{\omega}(f_{\omega + 1}(2))))}(f_{\varphi(1,0) + 1}(2))}(f_{f_{\varphi(1,0)}(f_{\varphi(1,\varphi(1,0) + 1)^{\varphi(1,0) + 1}.(\varphi(1,0)) + 1}(2)) + f_{2}(f_{\omega}(2)).2 + 1}(f_{\varphi(1,0) + \omega}(2)))\)

\(\beta(4.04900945179584,2) = f_{\omega.(f_{\omega.(f_{\varphi(1,0) + 1}(2).(2^{f_{\omega}(2).8 + 2})) + f_{2}(f_{5}(f_{\varphi(1,\varphi(1,0) + 1).(\omega) + \omega}(2)))}(f_{\varphi(1,0) + 1}(f_{\varphi(1,\varphi(1,0) + 1)^{\omega + 1}.(\varphi(1,0) + 1) + \varphi(1,0) + 1}(2))))}(f_{\varphi(1,0) + \omega}(2))\)

\(\beta(4.05081852551985,2) = f_{\omega^{f_{f_{\omega^{f_{\omega}(2) + 6} + \omega.4 + f_{4}(f_{\omega}(2))}(f_{\varphi(1,0)}(2))}(f_{\varphi(1,0) + 1}^{f_{\omega}(2) + 3}(f_{\varphi(1,\varphi(1,0) + 1)^{\varphi(1,0)}.(\omega + 1)}(2)))}}(f_{\varphi(1,0) + \omega}(2))\)

\(\beta(4.05262759924386,2) = f_{\varphi(f_{\omega.4}^{f_{\varphi(1,0)}(2) + 2}(f_{\omega.5 + 1}^{f_{f_{\omega}(2)}(f_{\omega + 1}(2)).2}(f_{\varphi(1,0)}(2))),\varphi(1,0) + 1)}(f_{\varphi(1,0) + \omega}(2))\)

\(\beta(4.05443667296786,2) = f_{\varphi(1,0) + f_{\omega}(f_{\omega + 1}(f_{\omega + 6}(f_{\varphi(1,\varphi(1,0) + 1)^{f_{\omega}(2) + 3}.6 + \omega^5 + 2}(f_{\varphi(1,\varphi(1,0) + 1)^{\omega}.(\varphi(1,0)) + \omega}(2)))))}(f_{\varphi(1,0) + \omega}(2))\)

\(\beta(4.05624574669187,2) = f_{\omega^{f_{\omega.4 + f_{\varphi(1,\varphi(1,0))^{\omega}.(\omega + 1)}(2)}(f_{\varphi(1,0)}^{f_{\varphi(1,0)}(2) + f_{3}(f_{\omega}(2)) + 2}(f_{\varphi(1,\varphi(1,0))^{\omega + 1}.(\varphi(1,0)) + \omega + 1}(2)))}}(f_{\varphi(1,\varphi(1,0) + \omega).(\omega) + \varphi(1,\varphi(1,0) + 1)^{\varphi(1,0) + 1} + \varphi(1,0) + 1}(2))\)

\(\beta(4.05805482041588,2) = f_{\omega + 7}^{f_{\omega^2.(f_{f_{\omega^{f_{\varphi(1,\varphi(1,0))}(2)}}(f_{\varphi(1,0) + 1}(f_{\varphi(1,\varphi(1,0) + 1).(\varphi(1,0)) + \varphi(1,0) + 1}(2)))}(f_{\varphi(1,\varphi(1,0) + 1).(\varphi(1,0) + 1) + \varphi(1,0) + 1}(2)))}(f_{\varphi(1,\varphi(1,0) + 1)^{\omega + 1}.(\varphi(1,\varphi(1,0))) + \varphi(1,0) + 1}(2))}(f_{\varphi(1,\varphi(1,0) + \omega).(\varphi(1,0) + \omega) + \varphi(1,\varphi(1,0) + 1) + 1}(2))\)

\(\beta(4.05986389413988,2) = f_{\varphi(1,\varphi(1,0) + \omega)^{\omega}.(\varphi(1,0) + \omega) + \omega + 1}(2) + f_{3}(f_{\omega}^{f_{\omega}(2)}(f_{\omega + 1}(2))) + 2\)

\(\beta(4.06167296786389,2) = f_{\omega^{f_{\omega}^{f_{\omega}(2).2}(f_{\omega + 1}(2)) + 1} + \omega}^{f_{\varphi(1,\varphi(1,0) + \omega).(\varphi(1,\varphi(1,0) + 1) + \varphi(1,0)) + \omega}(2).(2^{f_{4}(f_{\omega}(f_{\omega + 1}(2)))})}(f_{\varphi(1,\varphi(1,0) + \omega)^{\omega + 1}.(\varphi(1,0) + \omega) + \omega}(2))\)

\(\beta(4.0634820415879,2) = f_{f_{\omega.3 + f_{\omega}(2) + 1}^2(f_{\varphi(1,0)}(2))}(f_{\varphi(1,\varphi(1,0) + \omega)^{\varphi(1,\varphi(1,0) + 1)^{\omega + 1}.(\omega) + \omega + 1} + \omega + 1}(2)) + 2\)

\(\beta(4.06529111531191,2) = f_{\omega^{f_{\varphi(1,0) + 1}(2)}}(f_{\omega^{f_{\varphi(1,0) + \omega}(2).(2^{f_{2}(f_{\omega}(2)) + f_{\omega}(2)}) + 7} + 1}^{f_{\varphi(1,\varphi(1,0) + 1)^{\varphi(1,0) + 1} + 1}(2)}(f_{\varphi(1,\varphi(1,0) + \omega)^{\varphi(1,0) + \omega}.(\varphi(1,0) + \omega) + 1}(2)))\)

\(\beta(4.06710018903591,2) = f_{\varphi(1,0) + \omega + 1}(2).(2^{f_{4}(f_{\omega}(2)) + 6}) + f_{2}^{f_{4}(f_{\omega}(2)).2 + 2}(f_{4}^3(f_{\omega + 1}(2)))\)

\(\beta(4.06890926275992,2) = f_{f_{2}^2(f_{5}(f_{\varphi(1,0) + \omega}(2)))}^4(f_{\varphi(1,0) + \omega + 1}(2)) + 5\)

\(\beta(4.07071833648393,2) = f_{\omega.(f_{\varphi(1,0)}(2) + f_{\omega}(2)) + f_{\varphi(1,\varphi(1,0))^{\varphi(1,0)} + \omega + 1}(2).2 + f_{\omega + 1}(2).(2^{f_{7}(f_{\omega}(2)).2 + f_{4}(f_{\omega}(2))})}(f_{\varphi(1,0) + \omega + 1}(2))\)

\(\beta(4.07252741020793,2) = f_{\omega^{f_{f_{\varphi(1,\varphi(1,0) + 1) + f_{2}(f_{3}(f_{\omega}(2))).8 + 1}(f_{\varphi(1,\varphi(1,0) + 1) + \omega}(2)).(2^{f_{\varphi(1,\varphi(1,0) + 1) + \omega}(2).16 + 2})}(f_{\varphi(1,\varphi(1,0) + 1)^{\varphi(1,0)} + \omega}(2))}}(f_{\varphi(1,0) + \omega + 1}(2))\)

\(\beta(4.07433648393194,2) = f_{\varphi(1,\varphi(1,0) + 2).(f_{f_{\omega}(2).2}(f_{\omega}^{f_{\omega}(2) + 3}(f_{\omega + 1}(2))))}(f_{\varphi(1,0) + f_{3}^2(f_{\omega}(f_{\omega + 1}(2))) + 1}(f_{\varphi(1,0) + \omega + 1}(2)))\)

\(\beta(4.07614555765595,2) = f_{(\varphi(3,\varphi(1,0) + \omega)\uparrow\uparrow f_{f_{f_{2}(f_{3}^2(f_{\omega}(2))).(2^{f_{3}^2(f_{\omega}(2)).(2^{f_{2}^7(f_{\omega}(2)).8 + 2})})}(f_{\omega + 1}(2))}(f_{\omega}(f_{\omega + 1}(2))))}(f_{\varphi(1,0) + \omega + 1}(2))\)

\(\beta(4.07795463137995,2) = f_{\varphi(1,\varphi(1,0) + \omega)^{\omega}.(\omega^{f_{\omega}^{f_{\omega}(2)}(f_{\omega + 1}(2)).(2^{f_{\omega}^2(f_{\omega + 1}(2))})}.2 + 1) + \omega^2}(f_{\varphi(1,\varphi(1,0) + \omega + 1).(\omega) + 1}(2))\)

Fifth Attempt base \(v = 2\)

\(\beta(5.7807237750684,2) = f_{\varphi(\omega,0)^{\varphi(1,1)}}(2) + f_{\varphi(f_{\varphi(1,\omega)^{(\varphi(1,3)\uparrow\uparrow f_{(\varphi(1,0)\uparrow\uparrow 2)}(f_{\varphi(1,\omega + 1)^{\varphi(1,0).(\omega + 1)}.(\omega + 1) + 1}(2)))}}(f_{\varphi(1,\omega + 1)^{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega + 1}}.(\omega) + \omega + 1} + 1}.(\omega + 1) + 1}(2)),0)}(f_{\varphi(\omega,0)}(2))\)

\(\beta(5.78072377506845,2) = f_{\varphi(\omega,0)^{\varphi(1,1)}}(2) + f_{\varphi(1,\varphi(\omega,0))}(2)\)

Sixth Attempt base \(v = 2\)

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

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

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

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

\(\beta(5.6786768134038,2) = f_{\omega^{f_{\varphi(1,\varphi(\omega,0))}(2)}}(f_{\varphi(\omega,0) + \varphi(1,1)^{\omega + 1}.(\omega + 1) + \omega + 1}(2))\)

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

\(\beta(5.82261224912342,2) = f_{\varphi(\omega,0)^{\omega + 1}.(\varphi(1,\omega + 1)^{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega + 1}.(\omega + 1) + \varphi(1,0).(\omega) + 1} + \omega}.(\varphi(1,1)^{\varphi(1,0)^{\omega} + \varphi(1,0).(\omega) + \omega}.(\varphi(1,0).(\omega)) + \varphi(1,1).(\omega + 1))}) + \varphi(1,\omega)^{\omega + 1} + 1}(2).2 + 6\)

\(\beta(5.82261224915976,2) = f_{\omega.(f_{\varphi(1,0).(\omega.(f_{\omega}(2).4 + 2))}(f_{\varphi(1,\omega + 1).(\varphi(1,0) + \omega + 1) + \omega}(2)))}(f_{\varphi(\omega,0)^{\omega + 1}.(\varphi(1,\omega + 1)^{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega + 1}.(\omega + 1) + \varphi(1,0).(\omega) + 1} + \omega + 1}} + \omega + 1)}(2))\)

\(\beta(5.82261224915976,2) = f_{\omega.(f_{\varphi(1,\varphi(\omega,0).(\omega) + \varphi(1,1)^{\varphi(1,0).(\omega + 1)} + \varphi(1,0).(\omega))^{\varphi(1,\omega + 1)^{\varphi(1,\omega)^{\varphi(1,0)^{\omega + 1}.(\omega)}}}}(2))}(f_{\varphi(\omega,0)^{\omega + 1}.(\varphi(1,\omega + 1)^{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega + 1}.(\omega + 1) + \varphi(1,0).(\omega) + 1} + \omega + 1}} + \omega + 1)}(2))\)

\(\beta(5.82261225550432,2) = f_{\varphi(f_{\varphi(1,\omega)}(2),0)}(f_{\varphi(1,\varphi(\omega,0)^{\omega + 1}.(\varphi(1,\omega + 1)^{\varphi(1,\omega)^{\varphi(1,1)^{\varphi(1,0)^{\omega + 1}.(\omega + 1) + \varphi(1,0).(\omega) + 1}.(\omega) + \varphi(1,0)^{\omega + 1} + \varphi(1,0)} + \omega + 1} + \varphi(1,0) + \omega + 1))^{\varphi(\omega,0)^{\omega}.(\varphi(1,1) + \omega) + 1}.(\omega + 1) + \omega}(2))\)

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

\(\beta(5.82262,2) = f_{2}(f_{4}^{f_{\omega}(2) + 3}(f_{\varphi(\omega,0)^{\varphi(1,0)}}(2))) + 1\)

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