User blog:B1mb0w/Beta Function Code Version 4

Beta Function - Sequence Generating Code
The Beta Function has been defined using program code shown below.

A separate blog is being written to explain how Sequence Generator Code is compiled and executed using a normal programming language ... Work in Progress.

Sequence Generating Code Version 4
Version 4 should now correctly access all Veblen ordinals to any level of tetration. This corrects the errors discussed in my Version 3 and Version 2 blogs.

The Sequence Generating Ruleset has been modified in Version 4 to correct the logic for accessing complex tetration examples of nested ordinals. The Beta Function is equivalent to a sequence of finite integers of the form:

\(\beta(r,v) == (v,h_0)\)

using this Sequence Generating RuleSet:
 * \(h_x = (C_d<2,C_d=(0:C_x<V_v-U_x,(E_h=1,f_0)))\)
 * \(f_x = (g_x,g_x=(0:h_u,(h_U,E_h=C_d..h_U,f_x+1<g_x)))\)
 * \(g_x = (C_g<V_v+1,C_g=(0:h_x,(C_g=(1:V_n=0,V_n=C_g+0),n_0<C_g)))\)
 * \(n_x = (V_n=(0:,(g_{[V_n]},V_n=(1:V_y=M_{g_Y},V_y=V_n+M_{g_{v_n}}-1)),x=(0:C_n<V_y,C_n<V_y+1),h_T,V_g=(C_g..V_y,0),g_E<E_h_T,g_C<C_g..h_T,C_n=(0:g_a<C_g..g_E,(g_A<C_g..g_E,n_{x+1}<g_Y..g_{C_n})))\)
 * \(E_{z_x} = (V_g,M_{z_x}=(1:z_E<z_x,g_E<E_{z_E}))\)
 * Min Function \(M_{z_z} = 0\) if sequence \(z_z\) only contains zeros, else \(M_{z_z} = 1\)

The syntax used for this ruleset is explained in my blog on Sequence Generator Code, please refer to that blog for the best description of the syntax.

The sequence generating ruleset is a complete and sufficient description for the Beta Function. It can be relied upon to generate any Veblen ordinal or FGH function required to access every individual finite integer up to \(f_{SVO}(v)\) for any base \(v\).

Tetration Example using Version 4
These examples will illustrate the fine detail that can be accessed by the Beta Function. The missing values will be added to this blog when I can find them (!). I have various search programs that are trying to find the values but this will take time.

\(\beta(17.5818575378532,6) = f_{(\omega\uparrow\uparrow 3)^{10}}(f_{(\omega\uparrow\uparrow 4)}(6))\)

\(\beta(17.5818611839716,6) = f_{(\omega\uparrow\uparrow 3)^{100}}(f_{(\omega\uparrow\uparrow 4)}(6))\)

\(\beta(17.5818623993446,6) = f_{(\omega\uparrow\uparrow 3)^{f_{2}(6)}}(f_{(\omega\uparrow\uparrow 4)}(6)) + 1\)

\(\beta(17.5818770032733,6) = f_{(\omega\uparrow\uparrow 3)^{f_{\omega}^3(6)} + 4}^2(f_{(\omega\uparrow\uparrow 4)}(6)).16 + 1\)

\(\beta(\) TBA \(,6) = f_{(\omega\uparrow\uparrow 3)^{\omega}}(f_{(\omega\uparrow\uparrow 4)}(6))\)

\(\beta(\) TBA \(,6) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)}}(f_{(\omega\uparrow\uparrow 4)}(6))\)

\(\beta(\) TBA \(,6) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega - 1} - 2}}(f_{(\omega\uparrow\uparrow 4)}(6))\)

\(\beta(17.58199,6) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega - 1} - 1}}(f_{(\omega\uparrow\uparrow 4)}(6)) = f_{(\omega\uparrow\uparrow 4)}^2(6)\)

Granularity Examples \(\beta(8.9,3)\) to \(\beta(9.0,3)\)
When we use base \(v = 3\) the transitions up to \(\varphi(1,0)\) are correct:

\(\beta(8.9,3) = f_{f_{f_{f_{2}^{f_{4}^3(f_{\omega}^2(3))}(f_{5}^3(f_{9}^2(f_{12}^2(f_{\omega}^2(3)))))}(f_{\omega + 1}^2(3))}(f_{\omega.2 + 2}^2(3))}(f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2) + 1}^2(3))\)

\(\beta(8.99,3) = f_{\omega^{f_{19}^{f_{8}^{f_{4}^3(f_{\omega + 1}^2(3))}(f_{9}^2(f_{\omega + 1}^2(3)))}(f_{\omega.2 + 2}(3))}}(f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2) + \omega^2 + \omega + 2}(3))\)

\(\beta(8.999,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2).2}^2(3) + 2\)

\(\beta(8.9999,3) = f_{2}^2(f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.2}(3)).(2^{f_{(\omega\uparrow\uparrow 2)^{f_{\omega^{f_{\omega}(3)}}(f_{(\omega\uparrow\uparrow 2)^{\omega}.2 + \omega.2}^2(3))}}(f_{(\omega\uparrow\uparrow 2)^{\omega}.(\omega) + 2}(3))})\)

\(\beta(8.99999,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2.2 + 2) + 2}^2(3) + 7\)

\(\beta(8.999999,3) = f_{f_{\omega.(f_{\omega + 1}^{f_{f_{\omega}(f_{\omega + 1}(3))}(f_{\omega + 1}^2(3))}(f_{\omega + 2}^2(3)))}(f_{\omega^2}(f_{\omega^2 + 1}(3)))}(f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2.2 + \omega.2 + 2) + 1}^2(3))\)

\(\beta(8.9999999,3) = f_{2}^2(f_{6}(f_{f_{2}^2(3).4 + 2}^2(f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega + 2}.(\omega + 1) + \omega + 1}^2(3))))\)

\(\beta(8.99999999,3) = f_{2}^{f_{2}(3).4 + 4}(f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2}.(\omega^2.2) + \omega}(3)) + f_{2}(3).2 + 2\)

\(\beta(8.999999999,3) = f_{(\omega\uparrow\uparrow 2)^{\omega + 2}.(\omega.2)}(f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2}.(\omega^2.2 + \omega.2 + 1) + (\omega\uparrow\uparrow 2)^{\omega}.(\omega + 2) + \omega^2 + \omega + 2}(3))\)

\(\beta(8.9999999999,3) = f_{f_{2}(3)}^{f_{(\omega\uparrow\uparrow 2)^2 + \omega^2}(3)}(f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega}.2 + 2}^2(3))\)

\(\beta(8.99999999999,3) = f_{f_{\omega}^3(f_{\omega + 2}^2(3))}(f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega + 2}.(\omega.2 + 2) + \omega^2 + \omega + 1}^2(3))\)

\(\beta(8.999999999999,3) = f_{\omega^2}(f_{\omega^2 + 2}^2(f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega + 2}.(\omega^2.2 + \omega.2) + 2}^2(3)))\)

\(\beta(8.9999999999999,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2) + 2}(3) + 18\)

\(\beta(9,3) = f_{\varphi(1,0)}(3)\)

\(\beta(9.000000000001,3) = f_{\varphi(1,0)}(3)\)

\(\beta(9.000000001,3) = f_{\varphi(1,0)}(3)\)

\(\beta(9.000001,3) = f_{\varphi(1,0)}(3)\)

\(\beta(9.00001,3) = f_{\varphi(1,0)}(3) + 2\)

\(\beta(9.0001,3) = f_{3}(f_{5}(f_{\varphi(1,0)}(3))).16\)

\(\beta(9.001,3) = f_{\omega^2.(f_{2}^{f_{f_{3}(f_{\omega + 2}^{f_{\omega}(f_{\omega + 2}(3)) + 5}(f_{\omega.2}(3)))}(f_{\omega.2}^2(3))}(f_{\omega.2 + 2}(3)))}(f_{\varphi(1,0)}^2(f_{\varphi(1,0) + 1}(3)))\)

\(\beta(9.01,3) = f_{(\omega\uparrow\uparrow f_{\varphi(1,0).2 + \omega^2.2 + 2}^2(3) + 1)^{f_{\omega^2.2 + 1}^2(3) + 13}.2}(f_{\varphi(1,0).(\omega.2) + (\omega\uparrow\uparrow 2)^{\omega.2}.2 + (\omega\uparrow\uparrow 2)^2.2}(3))\)

Granularity Examples \(\beta(10.4,4)\) to \(\beta(10.0794,4)\)
When we use base \(v = 4\) the transitions up to \(\omega\uparrow\uparrow 3\) are correct:

\(\beta(10,4) = f_{3}^{f_{2}^3(4) + f_{2}(4) + 2}(f_{\omega + 3}^3(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.2 + 3} + 3}^2(4))).4\)

\(\beta(10.07,4) = f_{(\omega\uparrow\uparrow 2)^3.(f_{(\omega\uparrow\uparrow 2)^2 + f_{\omega}(f_{\omega^2.2 + \omega + 2}^2(f_{\omega^2.3 + \omega.3}^3(4)))}(f_{(\omega\uparrow\uparrow 2)^{\omega.3 + 2}.(\omega^2.3)}^3(4)))}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.3 + \omega + 3}(4))\)

\(\beta(10.079,4) = f_{\omega^6.(f_{\omega^2.(f_{\omega^{f_{\omega}(f_{(\omega\uparrow\uparrow 2) + \omega.2}^2(4))}}(f_{(\omega\uparrow\uparrow 2).2 + 3}^3(4)))}(f_{(\omega\uparrow\uparrow 2).(\omega^2.3 + \omega.3) + 2}(4)))}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega.2) + 2}^2(4))\)

\(\beta(10.0793,4) = f_{3}^{f_{(\omega\uparrow\uparrow 2).3 + \omega.2}(4).4 + f_{\omega^3.3 + \omega^2 + 1}^3(4).4 + f_{3}(4)}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3) + \omega^3 + \omega^2.2 + \omega.2 + 2}^3(4))\)

\(\beta(10.07936,4) = f_{f_{4}(f_{5}^{f_{2}(4) + 6}(f_{17}^{21}(f_{\omega.2 + 1}^2(4))))}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 1) + 2}^2(4))\)

\(\beta(10.079368,4) = f_{f_{f_{2}^2(4) + 7}^{f_{2}(f_{\omega}^3(4))}(f_{\omega.2 + 1}(4))}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3) + (\omega\uparrow\uparrow 2)^3.(\omega^3.2 + \omega.3 + 2) + (\omega\uparrow\uparrow 2).(\omega^3 + 1) + \omega^2.3 + \omega.2}^3(4))\)

\(\beta(10.0793683,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3) + (\omega\uparrow\uparrow 2)^{\omega^2.2 + 3}.(\omega^2.3 + 2) + 3}^2(4) + 2\)

\(\beta(10.07936839,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3) + (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3}.3 + (\omega\uparrow\uparrow 2).(\omega^2.2 + 1) + 1}^3(4) + 3\)

\(\beta(10.079368399,4) = f_{\omega^3.(f_{\omega.2 + 3}^2(4).(2^{f_{23}(f_{\omega}(f_{\omega + 1}^2(4)))}))}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3) + (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 2}.(\omega^3.3 + 1) + \omega^2.3 + \omega.2 + 1}^3(4))\)

\(\beta(10.0793683991,4) = f_{7}^{f_{f_{(\omega\uparrow\uparrow 2).(\omega^2.2 + \omega)}(4)}(f_{(\omega\uparrow\uparrow 2)^3.3 + 1}^2(4))}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3) + (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 2}.(\omega^3.3 + \omega.3 + 1) + (\omega\uparrow\uparrow 2)^{\omega + 2}}^2(4))\)

\(\beta(10.07936839915,4) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega.(f_{\omega^2.2 + \omega.3 + f_{\omega.2 + 2}(4)}(f_{\omega^2.3 + \omega.3 + 3}^3(4))))}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3) + (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 2}.(\omega^3.3 + \omega^2.3 + 3) + \omega^3.2 + 1}(4))\)

\(\beta(10.079368399158,4) = f_{3}^2(f_{4}^2(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3) + (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 2}.(\omega^3.3 + \omega^2.3 + \omega.3 + 2) + \omega^2 + 1}^3(4))).4 + 47\)

\(\beta(10.0793683991589,4) = f_{\omega.(f_{2}^2(4).2 + f_{2}(4) + 36) + f_{3}(4)}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3) + (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 2}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3) + (\omega\uparrow\uparrow 2)^3 + 3}^3(4))\)

\(\beta(10.079368399159,4) = f_{(\omega\uparrow\uparrow 3)}(4)\)

\(\beta(10.0793683992,4) = f_{(\omega\uparrow\uparrow 3)}(4)\)

\(\beta(10.0793684,4) = f_{(\omega\uparrow\uparrow 3)}(4)\)

\(\beta(10.0794,4) = f_{(\omega\uparrow\uparrow 3)}(4)\)

\(\beta(10.08,4) = f_{(\omega\uparrow\uparrow 3)}(4).(2^{f_{\omega.2 + 3}^{f_{\omega^2.15 + \omega.(f_{2}(f_{3}^2(f_{\omega^2.2 + 3}(4))) + f_{\omega}(4))}(f_{\omega^3 + \omega + 2}^2(4))}(f_{\omega^3.3 + \omega^2.3 + \omega + 2}^2(4))})\)

\(\beta(10.08,4) = f_{(\omega\uparrow\uparrow 3)}(4).(2^{f_{\omega.2 + 3}^{f_{\omega^2.15 + \omega.(f_{2}(f_{3}^2(f_{\omega^2.2 + 3}(4))) + f_{\omega}(4))}(f_{\omega^3 + \omega + 2}^2(4))}(f_{\omega^3.3 + \omega^2.3 + \omega + 2}^2(4))})\)

Granularity Examples near \(\beta(16,4)\)
The undesired values in Version 3 of the code have now been fixed.

These examples correctly transition up to \(\omega\uparrow\uparrow 4\) in base \(v = 4\):

\(\beta(15.9896716635699,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 1}.(\omega^2.3 + 2) + 1} + (\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2).(\omega^3.2 + \omega^2.2 + 1)) + 1}(4) + f_{(\omega\uparrow\uparrow 2)^{\omega^2}}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.2}.(\omega + 2)}(4))\)

\(\beta(15.9917925394842,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 2} + 1}.((\omega\uparrow\uparrow 2)^{\omega^2 + \omega + 3}.3 + \omega^3.2 + \omega^2.3 + 2) + \omega^2.3 + \omega.2}^2(4).(2^{f_{3}^2(4).4 + 4}) + 1\)

\(\beta(16,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 2}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3) + (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 1}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3) + (\omega\uparrow\uparrow 2)^{\omega^2.3 + 1} + (\omega\uparrow\uparrow 2)^{\omega^2 + 3}.(\omega^2.3 + \omega.2 + 2) + 1}.(\omega^3.3 + 3) + \omega}(4)\)

\(\beta(16.0000000000001,4) = f_{\varphi(1,0)}(4)\)

\(\beta(16.0000000001,4) = f_{\varphi(1,0)}(4)\)

\(\beta(16.0000001,4) = f_{\varphi(1,0)}(4)\)

\(\beta(16.000001,4) = f_{\varphi(1,0)}(4) + 2\)

\(\beta(16.00001,4) = f_{2}^{f_{3}(4) + 13}(f_{\varphi(1,0)}(4)).(2^{f_{20}(f_{(\omega\uparrow\uparrow 3)^{\omega.2 + 2}.(\omega + 1) + (\omega\uparrow\uparrow 2)^{\omega.3}.(\omega.3 + 3) + \omega^3 + 2}^3(4))})\)

\(\beta(16.0001,4) = f_{(\omega\uparrow\uparrow 3)^3.(f_{3}(4).(1073741824) + 43) + f_{f_{2}^8(f_{\omega}^3(f_{\omega + 1}^2(4)))}(f_{\omega.2}(4))}(f_{\varphi(1,0)}^2(4))\)

\(\beta(16.001,4) = f_{(\omega\uparrow\uparrow f_{\varphi(1,0) + \omega^2.3 + 1}^3(4).2 + 1)^3.((\omega\uparrow\uparrow 48)^2.(\omega^3.2 + \omega^2.(f_{\omega^2 + 3}(4))))}(f_{\varphi(1,0) + (\omega\uparrow\uparrow 2)^2.2 + 3}^2(4))\)

\(\beta(16.01,4) = f_{2}^6(f_{6}^2(f_{\omega.2 + 1}(f_{\varphi(1,0)^2.2}(4)))) + f_{f_{3}^2(4).16}(f_{(\omega\uparrow\uparrow 3).2 + (\omega\uparrow\uparrow 2).(\omega^3.2 + \omega + 3) + \omega.2 + 3}^2(4))\)

Granularity Examples \(\beta(16.7,5)\) to \(\beta(16.7185077,5)\)
The undesired values in Version 3 of the code have now been fixed.

These examples correctly transition up to \(\omega\uparrow\uparrow 4\) in base \(v = 5\):

\(\beta(16.7,5) = f_{(\omega\uparrow\uparrow 2)^{f_{2}^4(5) + 17}.(\omega.9 + f_{3}^4(5))}(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega.4 + 3}.(\omega^3.4 + 4) + \omega^3.2 + 2}.4 + 3}^4(5))\)

\(\beta(16.718507,5) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 3}.(\omega^4.4 + \omega^3 + 4) + (\omega\uparrow\uparrow 2)^{\omega^3 + 1}.(\omega^3.4) + (\omega\uparrow\uparrow 2)^{\omega^3}.(\omega^2 + \omega + 2) + \omega}}(5)\)

\(\beta(16.718507624,5) = f_{(\omega\uparrow\uparrow 2)^3.(\omega^5)}(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 3}.(\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.4 + 3) + \omega.2 + 3}.2 + 3}(5))\)

\(\beta(16.7185076244,5) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 3}.(\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.4 + 4) + (\omega\uparrow\uparrow 2)^{\omega^3 + 3}.(\omega.3 + 2) + (\omega\uparrow\uparrow 2)^4.3}}(5)\)

\(\beta(16.71850762441,5) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 3}.(\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.4 + 4) + (\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2 + \omega.4 + 1} + \omega^2.3}}(5)\)

\(\beta(16.7185076244105,5) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 3}.(\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.4 + 4) + (\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.2 + 1}.(\omega^3.3 + 1) + 2}.(\omega)}(5)\)

\(\beta(16.7185076244106,5) = f_{(\omega\uparrow\uparrow 4)}(5)\)

\(\beta(16.7185076245,5) = f_{(\omega\uparrow\uparrow 4)}(5)\)

\(\beta(16.7185077,5) = f_{(\omega\uparrow\uparrow 4)}(5)\)

\(\beta(16.7186,5) = f_{(\omega\uparrow\uparrow 4)}(5) + 3\)

\(\beta(16.719,5) = f_{(\omega\uparrow\uparrow 4)}(5).(2^{f_{f_{(\omega\uparrow\uparrow 2)^4.4 + 4}^4(5) + 2}(f_{(\omega\uparrow\uparrow 2)^4.(\omega^4 + 2) + 4}^4(5)) + 1}) + 20\)

\(\beta(16.72,5) = f_{3}^{f_{(\omega\uparrow\uparrow 2)^4.4}(5)}(f_{4}^3(f_{8}^{f_{4}^3(5).4 + 1}(f_{24}^3(f_{(\omega\uparrow\uparrow 4)}(5)))))\)

\(\beta(16.8,5) = f_{f_{(\omega\uparrow\uparrow 4)}(5).8 + f_{4}(5).8 + 4}^{f_{(\omega\uparrow\uparrow 2)}(f_{(\omega\uparrow\uparrow 4) + 1}^3(5))}(f_{(\omega\uparrow\uparrow 4) + 3}^4(5))\)

Granularity Examples near \(\beta(25,5)\)
The undesired values in Version 3 of the code have now been fixed.

These examples correctly transition up to \(\omega\uparrow\uparrow 5\) in base \(v = 5\):

\(\beta(24.9,5) = f_{f_{(\omega\uparrow\uparrow 2)^{\omega.4}}(5)}(f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + 2}.(\omega.4 + 4) + \omega^3.3 + \omega^2.3 + \omega.3}.4 + \omega^3.3 + 3}.2 + 2}^4(5))\)

\(\beta(24.999,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4}.(\omega^3.3 + \omega^2 + \omega.4 + 4) + 3}.((\omega\uparrow\uparrow 2)^2.(\omega.2 + 4) + 3)}.((\omega\uparrow\uparrow 3)^{\omega^4.3 + 3}.3 + 2) + \omega}(5)\)

\(\beta(24.99999,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 3}.(\omega^2.3 + \omega + 4) + \omega^4.3 + 4}.((\omega\uparrow\uparrow 2).4 + 1) + 2}.3 + (\omega\uparrow\uparrow 3)^{\omega^3}}(5)\)

\(\beta(24.9999999,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 3}.(\omega^4.4 + \omega^3.2 + \omega.4 + 4) + (\omega\uparrow\uparrow 2)^4.(\omega^3.2 + \omega) + 4}.3 + 3}.((\omega\uparrow\uparrow 2)^{\omega})}(5)\)

\(\beta(24.999999999,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 3}.(\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.2) + \omega^2.2 + \omega.2 + 3}.3 + (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3}}}(5)\)

\(\beta(24.99999999999,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 3}.(\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.4 + 4) + (\omega\uparrow\uparrow 2)^4.4 + 1}.(\omega.2) + (\omega\uparrow\uparrow 3)^{\omega^3}}}(5)\)

\(\beta(25,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 3}.(\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.4 + 4) + (\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega + 4}.2 + 4}.4}}(5)\)

\(\beta(25.0000000000001,5) = f_{\varphi(1,0)}(5)\)

\(\beta(25.0000001,5) = f_{\varphi(1,0)}(5) + 1\)

\(\beta(25.001,5) = f_{\varphi(1,0).2 + \omega.4 + 2}^2(5) + f_{2}(f_{\omega^3.4 + \omega + 1}^3(5)) + 1\)

\(\beta(25.1,5) = f_{(\varphi(1,0)\uparrow\uparrow 3)^{\omega^3.2 + 1}.((\varphi(1,0)\uparrow\uparrow 2).((\omega\uparrow\uparrow 3)^3.3 + 2) + (\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2) + \omega^4.4 + 2}.2 + (\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2) + 2}.(\omega))}(5)\)

Granularity Examples \(\beta(17.5,6)\) to \(\beta(17.581,6)\)
The undesired values in Version 3 of the code have now been fixed.

These examples correctly transition up to \(\omega\uparrow\uparrow 4\) in base \(v = 6\):

\(\beta(17.5,6) = f_{6}^5(f_{f_{2}(6).64 + f_{2}(6) + 3}^5(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.3 + \omega^3.4 + 2}.4 + 2}.(\omega^2.2 + 4) + (\omega\uparrow\uparrow 3)^{\omega^4.2 + \omega^2.2 + 2}.2 + (\omega\uparrow\uparrow 2)^3.(\omega^4.4 + 5) + 2}^4(6))).16 + 1\)

\(\beta(17.58,6) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{f_{5}^5(6).8 + f_{4}(6).4 + 5}.(f_{3}(f_{(\omega\uparrow\uparrow 2)^{\omega^3 + \omega.5 + 5}.4 + \omega^2 + 5}^4(6)))}}(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + 3}.5 + 4}.(\omega^5.4 + 4) + 5}^5(6))\)

\(\beta(17.5809,6) = f_{2}^{f_{\omega^{f_{(\omega\uparrow\uparrow 2).(\omega^2.3 + \omega)}(f_{(\omega\uparrow\uparrow 2)^3.(\omega.3 + 4) + 1}^2(6))}}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.4 + \omega + 3}.4 + 2}^4(6))}(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + 5}.4 + \omega^3 + \omega.5 + 1}.5 + 5}^3(6))\)

\(\beta(17.580936,6) = f_{3}^3(f_{4}^2(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^3.4 + 2) + \omega.2 + 4}.5 + 5}^2(6))).16 + 5\)

\(\beta(17.5809363,6) = f_{2}^{98}(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + \omega^2 + \omega + 1) + 4}.2 + (\omega\uparrow\uparrow 2)^{\omega.4 + 2}.3 + 1}^4(6)).4 + 4\)

\(\beta(17.5809363095,6) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.4 + \omega.5 + 5) + 2}.(\omega^4.3 + \omega.4 + 3) + \omega^3.2 + 4}^2(6) + 19\)

\(\beta(17.5809363095,6) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.4 + \omega.5 + 5) + 2}.(\omega^4.3 + \omega.4 + 3) + \omega^3.2 + 4}^2(6) + 19\)

\(\beta(17.5809363095011,6) = f_{\omega^{f_{4}^6(f_{(\omega\uparrow\uparrow 3)^2.5 + 5}^3(6))}}(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4) + 1}.((\omega\uparrow\uparrow 2)^{\omega.4 + 5}.4) + (\omega\uparrow\uparrow 3)^5.(\omega^4.5 + \omega^3.2 + 1)}^4(6))\)

\(\beta(17.5809363095012,6) = f_{(\omega\uparrow\uparrow 4)}(6)\)

\(\beta(17.58094,6) = f_{(\omega\uparrow\uparrow 4)}(6) + 1\)

\(\beta(17.581,6) = f_{(\omega\uparrow\uparrow 4)}(6).32 + 42\)

\(\beta(17.59,6) = f_{(\omega\uparrow\uparrow 2).(\omega^5.(f_{f_{(\omega\uparrow\uparrow 2)^{\omega + 2}.4 + (\omega\uparrow\uparrow 2)^{\omega}.(\omega^5.4 + 5) + 3}(6)}(f_{(\omega\uparrow\uparrow 2)^{\omega.4}.(\omega^3.2 + 5) + (\omega\uparrow\uparrow 2)^2.(\omega^2.2 + 4)}^3(6))))}(f_{(\omega\uparrow\uparrow 4) + 1}^4(6))\)

\(\beta(17.6,6) = f_{2}^{f_{\omega + 3}^{f_{4}^{f_{f_{\omega}(6)}(f_{\omega + 1}^5(6))}(f_{\omega^2.4 + \omega.4 + 2}^4(6))}(f_{\omega + f_{2}^5(6) + 3}(f_{\omega.2 + 4}^5(f_{(\omega\uparrow\uparrow 2)^3.4 + 3}^5(6))))}(f_{3}^5(f_{(\omega\uparrow\uparrow 4) + 3}^4(6)))\)

Granularity Examples \(\beta(25.15,6)\) to \(\beta(25.16,6)\)
The undesired values in Version 3 of the code have now been fixed.

These examples correctly transition up to \(\omega\uparrow\uparrow 5\) in base \(v = 6\):

\(\beta(25.15,6) = f_{2}^2(f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + 2} + \omega^3.2 + \omega^2 + 3}.4 + \omega.2}.5 + (\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^4.3 + 5} + 5}^4(6)).(2^{f_{2}(6).32}) + 1\)

\(\beta(25.15777,6) = f_{f_{2}^3(6) + 4}^6(f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega + 5}.(\omega^5.3 + 1) + 2}.5 + (\omega\uparrow\uparrow 2)^{\omega.5 + 3}.2 + (\omega\uparrow\uparrow 2)^{\omega.4 + 3}.4 + 5}.(\omega.4 + 2) + (\omega\uparrow\uparrow 2)^{\omega^2.5 + \omega.4 + 2}.3 + 2}^4(6))\)

\(\beta(25.1577762,6) = f_{4}^{f_{2}^4(6) + f_{2}^3(6) + 2}(f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^3.3 + \omega^2.3 + \omega.5 + 1) + \omega^2.5 + \omega.3 + 5} + 4} + (\omega\uparrow\uparrow 4)^2.(\omega.4 + 3) + 5}^2(6)) + f_{2}^6(f_{4}(6))\)

\(\beta(25.157776275,6) = f_{3}^2(f_{4}(f_{8}^{57}(f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + \omega^4) + 5}.3 + \omega^2.3 + \omega.5 + 3}.((\omega\uparrow\uparrow 2)^2.(\omega^5.3 + 5) + 1) + 1}^3(6)))) + f_{\omega}(6)\)

\(\beta(25.15777627577,6) = f_{(\omega\uparrow\uparrow 3)^3.3 + \omega^5.(f_{2}(f_{4}^3(6)))}(f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + \omega^4.5 + \omega^3.4 + \omega.4 + 3) + (\omega\uparrow\uparrow 2)^{\omega.3}.3 + 5}.(\omega^2.3 + 2) + 5}.(\omega^5.5 + 2) + 3}^2(6))\)

\(\beta(25.1577762757768,6) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.2 + 5) + \omega^2.4 + 3}.3 + (\omega\uparrow\uparrow 2)^{\omega^2.4 + \omega.5 + 5}.(\omega.3 + 4) + (\omega\uparrow\uparrow 2)^5 + 4} + (\omega\uparrow\uparrow 4)^{\omega^2 + \omega.4 + 3}.(\omega^2.5 + 4) + \omega}(6)\)

\(\beta(25.1577762757769,6) = f_{(\omega\uparrow\uparrow 5)}(6)\)

\(\beta(25.1578,6) = f_{(\omega\uparrow\uparrow 5)}(6) + 4\)

\(\beta(25.16,6) = f_{f_{4}^{f_{3}^{f_{2}^2(f_{3}(6)).(2^{f_{2}(f_{3}(6)).(2^{f_{3}(6).4 + 15}) + 6}) + 7}(f_{4}^4(6)) + 2}(f_{(\omega\uparrow\uparrow 5)}(6)) + 3}^{54}(f_{(\omega\uparrow\uparrow 5)}^2(6))\)

Granularity Examples near \(\beta(36,6)\)
The undesired values in Version 3 of the code have now been fixed.

These examples correctly transition up to \(\omega\uparrow\uparrow 4\) in base \(v = 6\):

\(\beta(35.9,6) = f_{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{\omega^3.3 + \omega.5 + 4}.((\omega\uparrow\uparrow 2)^{\omega.5 + 2}.2 + \omega^3 + \omega.3 + 5) + 3}.(\omega^2.2 + \omega.3 + 1) + (\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{\omega^2.5 + \omega.3 + 1}.3 + 1}.(\omega.3 + 5) + 5}.(\omega^5.4 + \omega^3.4 + 1) + \omega}(6)\)

\(\beta(35.99,6) = f_{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.5}.(\omega.3 + 3) + \omega^2.3 + 5}.((\omega\uparrow\uparrow 2).4 + \omega.5) + (\omega\uparrow\uparrow 3)^5.((\omega\uparrow\uparrow 2)^{\omega.5 + 4}.2 + 5) + 3}.((\omega\uparrow\uparrow 2)^{\omega.5 + 1}.(\omega^4.2 + \omega^3.5 + \omega^2 + 3) + (\omega\uparrow\uparrow 2))}}(6)\)

\(\beta(35.99999,6) = f_{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.2 + 5}.2 + 4}.5 + \omega.3 + 3}.((\omega\uparrow\uparrow 2)^{\omega^2.2 + 3}.(\omega^5.3 + \omega^2.2 + \omega.2 + 1) + 4) + (\omega\uparrow\uparrow 2)^{\omega^4 + 5}.(\omega^4) + 5}.((\omega\uparrow\uparrow 2))}(6)\)

\(\beta(35.9999999,6) = f_{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 3}.3 + 1}.(\omega^4.4 + \omega.2 + 2) + \omega^2.4 + 1} + 4}.4 + (\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + 3}.(\omega^2.5 + 3) + 3}.((\omega\uparrow\uparrow 2)^{\omega^3})}}(6)\)

\(\beta(35.999999999,6) = f_{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + 1) + 3}.5}.((\omega\uparrow\uparrow 2)^5.(\omega^3.4) + 1) + (\omega\uparrow\uparrow 2).(\omega^5 + \omega.5) + \omega^2.4 + \omega + 4}.((\omega\uparrow\uparrow 4)^2.5) + (\omega\uparrow\uparrow 3)^{\omega}}(6)\)

\(\beta(35.99999999999,6) = f_{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + \omega^4.5 + \omega.5 + 1) + (\omega\uparrow\uparrow 2)^4.4 + \omega^4.5 + \omega.3}.((\omega\uparrow\uparrow 2)^{\omega^2.4 + 1} + (\omega\uparrow\uparrow 2)^{\omega^2.2 + 1}.(\omega^5.3 + 3) + 2) + 3}}}(6)\)

\(\beta(35.9999999999999,6) = f_{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.4 + \omega.3 + 4) + 3}.((\omega\uparrow\uparrow 2)^{\omega^4 + 4}.(\omega^2 + 4) + \omega^4 + \omega^2.4 + \omega.5 + 1) + 4}.4 + (\omega\uparrow\uparrow 2)^3.2 + (\omega\uparrow\uparrow 2)^2}}(6)\)

\(\beta(36,6) = f_{\varphi(1,0)}(6)\)

\(\beta(36.0000001,6) = f_{\varphi(1,0)}(6) + 4\)

\(\beta(36.00001,6) = f_{(\omega\uparrow\uparrow 4)^{f_{(\omega\uparrow\uparrow f_{2}(f_{4}^2(6)) + 3)^{f_{\omega^{10}.2 + \omega^4.3}^{f_{\omega}(6)}(f_{(\omega\uparrow\uparrow 2)^5.(\omega^2.2 + 1) + (\omega\uparrow\uparrow 2)^2.3 + 2}^2(6))}}(f_{\varphi(1,0)}(6))}}(f_{\varphi(1,0)}^2(6))\)

\(\beta(36.01,6) = f_{(\omega\uparrow\uparrow 6)}(f_{\varphi(1,0)^3.((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega.5 + 1}.(\omega.5 + 3)} + (\omega\uparrow\uparrow 3)^{\omega^3.2 + \omega^2.4 + 2}.((\omega\uparrow\uparrow 2)^{\omega^3.5 + 4}.(\omega.4 + 3) + (\omega\uparrow\uparrow 2)^{\omega^2.5 + 1}.(\omega^3.2 + 3) + \omega^2.3 + 1) + \omega^2.3) + \omega^4 + \omega.4 + 4}^3(6))\)

Valid Sequence Counts
WORK IN PROGRESS

Test Bed for Version 4
Below is the test bed and various results using version 4.

\(\beta(2.43,5) = f_{2}^4(5)\)

\(\beta(2.63,5) = f_{3}(5)\)

\(\beta(2.85,5) = f_{3}^2(5)\)

\(\beta(3.09,5) = f_{3}^3(5)\)

\(\beta(3.35,5) = f_{3}^4(5)\)

\(\beta(3.63,5) = f_{4}(5)\)

\(\beta(3.93,5) = f_{4}^2(5)\)

\(\beta(4.26,5) = f_{4}^3(5)\)

\(\beta(3.94,5) = f_{4}^2(5) + 1\)

\(\beta(4.62,5) = f_{4}^4(5)\)

\(\beta(4.95,5) = f_{3}^5(f_{4}^4(5))\)

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

\(\beta(5.07,5) = f_{\omega + 2}^4(5).4 + 1\)

\(\beta(5.16,5) = f_{\omega.2 + 1}^2(5).2 + 1\)

\(\beta(5.44,5) = f_{\omega.4 + 1}^4(5).512 + 5\)

\(\beta(5.54,5) = f_{2}^22(f_{\omega^2}^4(5)).64\)

\(\beta(5.6,5) = f_{\omega^2 + \omega}^2(5).2\)

\(\beta(5.67,5) = f_{\omega^2.2}(5) + 4\)

\(\beta(5.84,5) = f_{\omega^2.3 + 1}^4(5) + 2\)

\(\beta(5.94,5) = f_{\omega^2.3 + \omega.3 + 4}(5).16 + 2\)

\(\beta(5.97,5) = f_{\omega^2.4}^3(5) + f_{2}^2(5) + 7\)

\(\beta(6,5) = f_{\omega^2.4 + 2}^3(5)\)

\(\beta(6.16,5) = f_{\omega^3}^4(f_{\omega^3 + 2}^4(5))\)

\(\beta(6.18,5) = f_{\omega^3 + 4}^2(5) + 4\)

\(\beta(6.27,5) = f_{\omega^3.2}(5) + 3\)

\(\beta(6.33,5) = f_{2}(f_{3}^3(f_{\omega^3.2 + 3}^4(5))) + 4\)

\(\beta(6.43,5) = f_{\omega^3.3}(5).(2^{f_{\omega^3.2 + 3}(5) + 1})\)

\(\beta(6.61,5) = f_{\omega^3.4 + 1}(5) + 2\)

\(\beta(6.66,5) = f_{\omega^3.4 + 4}(5)\)

\(\beta(6.73,5) = f_{2}^2(f_{\omega^3.4 + \omega^2.2}^3(5))\)

\(\beta(6.77,5) = f_{\omega^4}^3(5).4 + 2\)

\(\beta(6.89,5) = f_{2}^3(f_{\omega^4 + \omega^2.3}(5)) + 2\)

\(\beta(7.03,5) = f_{\omega^4.2 + \omega.2 + 1}(5) + 2\)

\(\beta(7.31,5) = f_{2}^5(f_{\omega^4.4 + 1}(5)) + 1\)

\(\beta(7.32,5) = f_{5}(f_{120}^2(f_{\omega^4.4 + 1}^3(5))) + 2\)

\(\beta(8.7,5) = f_{(\omega\uparrow\uparrow 2)^4 + 1}(5).512 + 8\)

\(\beta(8.73,5) = f_{40}^3(f_{(\omega\uparrow\uparrow 2)^4 + \omega^3.2 + \omega.2}^2(5))\)

\(\beta(8.77,5) = f_{(\omega\uparrow\uparrow 2)^4.2 + 3}^4(5).8 + 3\)

\(\beta(8.81,5) = f_{(\omega\uparrow\uparrow 2)^4.3 + 1}(5) + 3\)

\(\beta(8.82,5) = f_{4}^{f_{4}^2(5) + f_{2}^4(5).512 + 3}(f_{(\omega\uparrow\uparrow 2)^4.3 + 2}^4(5))\)

\(\beta(8.86,5) = f_{(\omega\uparrow\uparrow 2)^4.4}(5) + 5\)

\(\beta(8.89,5) = f_{(\omega\uparrow\uparrow 2)^4.4 + \omega.3 + 2}^2(5) + f_{2}^2(5) + 48\)

\(\beta(9.04,5) = f_{(\omega\uparrow\uparrow 2)^4.(\omega^3 + \omega^2 + \omega.3 + 1) + 3}^3(5) + 4\)

\(\beta(9.06,5) = f_{(\omega\uparrow\uparrow 2)^4.(\omega^3.3 + 2)}^3(5) + 4\)

\(\beta(10,5) = f_{2}^3(f_{(\omega\uparrow\uparrow 2)^{\omega^2.4 + 1}.3 + 1}^2(5)).(2^{f_{5}(f_{\omega^2.3 + 2}(f_{\omega^3 + \omega^2.3 + \omega.4 + 2}^3(5)))})\)

\(\beta(11,5) = f_{\omega^3.27 + \omega^2 + \omega.4}^{f_{(\omega\uparrow\uparrow 2)^{\omega^4.2 + 3}.2}(5)}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.3 + \omega^2 + \omega.4 + 3}.4 + 4}^3(5))\)

\(\beta(11.1,5) = f_{\omega^4.(f_{4}(5).2 + 3) + \omega^2.(f_{(\omega\uparrow\uparrow 2)^2.(\omega^{f_{\omega}(5)})}(f_{(\omega\uparrow\uparrow 2)^3.4 + 3}^2(5)))}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.4 + 4}.3 + 1}(5))\)

\(\beta(11.15,5) = f_{\omega^2.2 + \omega.(f_{\omega^2}(5))}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega^2.3 + \omega.3 + 2}.(\omega + 4) + (\omega\uparrow\uparrow 2).2 + \omega^3 + \omega.2 + 2}^3(5))\)

\(\beta(11.16,5) = f_{(\omega\uparrow\uparrow 2).(\omega^{f_{8}^{f_{\omega^2.4 + \omega}(5)}(f_{\omega^3 + \omega.4 + 1}^3(5))})}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega^3 + \omega + 1}.(\omega^4.3 + 3) + 3}^4(5))\)

\(\beta(11.17,5) = f_{\omega^{f_{2}^8(f_{3}(f_{4}(5))) + 2}.(f_{2}^2(f_{4}^4(5)))}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega^3.3 + 2}.3 + \omega^4.2 + \omega^2 + \omega.4 + 2}(5))\)

\(\beta(11.18,5) = f_{f_{4}^4(5).32}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega^3.4 + \omega^2.4}.(\omega^2.3 + \omega.4) + (\omega\uparrow\uparrow 2)^{\omega^4 + 2}.2 + \omega + 1}^2(5))\)

\(\beta(11.1801,5) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.(f_{\omega^3}(f_{\omega^4.2 + \omega^2.4 + \omega}^4(5)))}}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega^3.4 + \omega^2.4 + 3}.4 + \omega^2 + \omega.4 + 4}^2(5))\)

\(\beta(11.1802,5) = f_{(\omega\uparrow\uparrow 2)^{f_{3}(5).256 + f_{3}(5).8 + 2}.(\omega)}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega + 4}.(\omega^2.3 + \omega.4) + \omega^4.4}(5))\)

\(\beta(11.1803,5) = f_{(\omega\uparrow\uparrow 2)^{\omega^3 + \omega + f_{\omega^4.3 + 2}^3(5)}}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.4}.(\omega^2 + \omega + 1) + 4}^3(5))\)

\(\beta(11.18031,5) = f_{\omega^4.(f_{\omega.(f_{\omega}(f_{\omega^3.2}(5)))}(f_{\omega^3.2 + 1}^3(5)))}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.4 + 1}.(\omega^2.4 + \omega)}^2(5))\)

\(\beta(11.18032,5) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.15 + f_{2}^3(5)}}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.4 + 2}.(\omega^3.3 + 4) + \omega^3.2 + \omega.4}^4(5))\)

\(\beta(11.18033,5) = f_{\omega^{f_{\omega.(f_{\omega}^{f_{2}^3(5)}(f_{\omega + 4}^2(5)))}(f_{\omega^3.3 + 1}^2(5))}}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.4 + 3}.(\omega^4.2 + 4) + 2}^4(5))\)

\(\beta(11.18034,5) = f_{(\omega\uparrow\uparrow 3)}(5)\)

\(\beta(11.1902,5) = f_{\omega^{f_{3}(5) + 5}.3}^4(f_{(\omega\uparrow\uparrow 3) + 1}^2(5)) + 6\)

\(\beta(11.1903,5) = f_{\omega^{f_{4}^{f_{\omega + f_{2}^{f_{2}^4(5).(131072) + 2}(f_{3}(5)) + 6}^3(f_{\omega.2}(5))}(f_{(\omega\uparrow\uparrow 3)}(f_{(\omega\uparrow\uparrow 3) + 1}(5)))}}(f_{(\omega\uparrow\uparrow 3) + 1}^2(5))\)

\(\beta(11.1904,5) = f_{(\omega\uparrow\uparrow 2)^{f_{3}^2(5).32 + 6}.2 + \omega^5 + \omega^4 + \omega^2.4 + f_{(\omega\uparrow\uparrow 2)^{\omega}}(5)}(f_{(\omega\uparrow\uparrow 3) + 1}^2(5))\)

\(\beta(11.1905,5) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.4 + 4}.(\omega^4 + f_{f_{2}^{f_{2}(5) + 2}(f_{3}^2(5)) + 6}^3(f_{(\omega\uparrow\uparrow 3)}^2(5)))}(f_{(\omega\uparrow\uparrow 3) + 1}^2(5))\)

\(\beta(11.1906,5) = f_{2}^{f_{3}^2(5).8 + 3}(f_{(\omega\uparrow\uparrow 3)}(f_{(\omega\uparrow\uparrow 3) + 1}^2(5))).8 + f_{f_{2}^3(f_{3}^2(5))}(f_{(\omega\uparrow\uparrow 2).3 + \omega.3 + 4}^4(5))\)

\(\beta(11.1907,5) = f_{(\omega\uparrow\uparrow 2)^{f_{\omega.2}(f_{(\omega\uparrow\uparrow 2)^{\omega^3.2 + 4}.3 + \omega^4.3 + \omega^2.2 + 4}(5))}}(f_{(\omega\uparrow\uparrow 3)}^3(f_{(\omega\uparrow\uparrow 3) + 1}^2(5)))\)

\(\beta(11.19071,5) = f_{2}^{f_{2}^2(f_{4}(5)).(2^{f_{3}(5).(2^{f_{2}^2(5).256 + 6}) + 15}) + f_{2}^2(5) + 7}(f_{(\omega\uparrow\uparrow 3)}^4(f_{(\omega\uparrow\uparrow 3) + 1}^2(5))).(2^{f_{2}^3(5)})\)

\(\beta(11.19072,5) = f_{f_{f_{f_{\omega^3}(5)}(f_{\omega^4 + \omega.2 + 2}^3(f_{\omega^4 + \omega.3 + 2}^3(5)))}(f_{\omega^4 + \omega^2.3 + 4}^3(5))}(f_{(\omega\uparrow\uparrow 3)}^4(f_{(\omega\uparrow\uparrow 3) + 1}^2(5)))\)

\(\beta(11.19073,5) = f_{\omega^{f_{2}^42(f_{3}^{f_{3}^4(5) + 16}(f_{4}^2(5))).16}.(f_{2}^4(5) + 3) + f_{(\omega\uparrow\uparrow 2)^{\omega}}(5)}(f_{(\omega\uparrow\uparrow 3)}^4(f_{(\omega\uparrow\uparrow 3) + 1}^2(5)))\)

\(\beta(11.19074,5) = f_{(\omega\uparrow\uparrow 2)^{\omega^3.7 + \omega^2.(f_{4}^2(f_{(\omega\uparrow\uparrow 3) + 1}(5)) + 1) + f_{f_{\omega.2 + 2}^3(5)}(f_{(\omega\uparrow\uparrow 3) + 1}^2(5))}}(f_{(\omega\uparrow\uparrow 3)}^4(f_{(\omega\uparrow\uparrow 3) + 1}^2(5)))\)

\(\beta(11.19075,5) = f_{(\omega\uparrow\uparrow 2)^{f_{2}(f_{3}^4(5)).64 + f_{2}^3(f_{3}^2(5)).512 + f_{3}(5).(2^{f_{2}^2(5).4})}}(f_{(\omega\uparrow\uparrow 3)}^7(f_{(\omega\uparrow\uparrow 3) + 1}^2(5)))\)

\(\beta(11.19076,5) = f_{(\omega\uparrow\uparrow 2)^{\omega^3 + \omega}}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.24 + 3}.4 + 1}^27(f_{(\omega\uparrow\uparrow 3)}^19(f_{(\omega\uparrow\uparrow 3) + 1}^2(5))))\)

\(\beta(11.19077,5) = f_{\omega^2.(f_{2}^7(f_{3}^2(f_{4}(5))) + 7) + f_{2}^3(f_{\omega}(5)).32}(f_{(\omega\uparrow\uparrow 3)}^89(f_{(\omega\uparrow\uparrow 3) + 1}^2(5)))\)

\(\beta(11.19078,5) = f_{f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 3}.2 + (\omega\uparrow\uparrow 2)^3.(\omega.3)}(5)}(f_{(\omega\uparrow\uparrow 3)}^{f_{2}^3(5) + f_{2}(5).(562949953421312) + 81}(f_{(\omega\uparrow\uparrow 3) + 1}^2(5)))\)

\(\beta(11.19079,5) = f_{27}^{f_{(\omega\uparrow\uparrow 2)^{\omega^2 + \omega.2}.(\omega.4 + 1) + (\omega\uparrow\uparrow 2)^{\omega^2 + \omega + 4}.(\omega.2)}(5)}(f_{(\omega\uparrow\uparrow 3)}^{f_{3}^2(5) + 7}(f_{(\omega\uparrow\uparrow 3) + 1}^2(5)))\)

\(\beta(11.1908,5) = f_{\omega.2 + 42}^5(f_{(\omega\uparrow\uparrow 3)}^{f_{2}^97(f_{3}^4(5)).8 + 3}(f_{(\omega\uparrow\uparrow 3) + 1}^2(5)))\)

\(\beta(11.1909,5) = f_{(\omega\uparrow\uparrow 3) + 1}^3(5) + 1\)

\(\beta(11.2,5) = f_{3}^22(f_{(\omega\uparrow\uparrow 3) + 2}^4(5)).(2^{f_{3}(5).2 + 3}) + 7\)

\(\beta(11.85,5) = f_{2}^2(f_{(\omega\uparrow\uparrow 3)^2.2 + 2}^3(5)).(4.83570327845852E+24) + 2\)

\(\beta(12.13,5) = f_{2}^3(f_{(\omega\uparrow\uparrow 3)^2.(\omega^2.4 + \omega.2 + 4) + \omega^3.2 + 1}^3(5)) + 2\)

\(\beta(12.41,5) = f_{(\omega\uparrow\uparrow 3)^3 + \omega^3.2 + 3}^3(5).16\)

\(\beta(12.53,5) = f_{2}^4(f_{(\omega\uparrow\uparrow 3)^3.3 + 1}^2(5)).(2199023255552) + 41\)

\(\beta(12.8,5) = f_{(\omega\uparrow\uparrow 3)^3.(\omega^4 + 1) + \omega.4 + 2}(5).64 + 2\)

\(\beta(12.81,5) = f_{(\omega\uparrow\uparrow 3)^2.(\omega.4 + 1) + f_{2}^{f_{2}^{f_{4}(5).32}(f_{4}^2(5))}(f_{4}^2(f_{\omega + 3}(5)))}(f_{(\omega\uparrow\uparrow 3)^3.(\omega^4.2 + 1) + 4}^3(5))\)

\(\beta(12.82,5) = f_{2}^2(f_{(\omega\uparrow\uparrow 3)^3.(\omega^4.3 + 1) + 4}(5))\)

\(\beta(12.83,5) = f_{(\omega\uparrow\uparrow 3)^3.(\omega^2.(f_{(\omega\uparrow\uparrow 2)^{\omega^4.2 + \omega^3 + \omega + 4}.3 + 4}(5).(2^{f_{4}^3(5).4})))}(f_{(\omega\uparrow\uparrow 3)^3.(\omega^4.4 + 1) + 3}(5))\)

\(\beta(12.85,5) = f_{2}^{f_{4}(5).(1024)}(f_{f_{3}^4(5) + f_{3}^3(5) + 4}(f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2).(\omega.2 + 4) + 1) + \omega^4 + \omega^2.4 + \omega.2 + 2}^3(5)))\)

\(\beta(12.86,5) = f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^2 + 4) + 1}^4(5).(281474976710656) + 1\)

\(\beta(12.87,5) = f_{f_{f_{4}^3(5) + 5}(f_{(\omega\uparrow\uparrow 3)^2.3 + 4}^2(5))}(f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^2.(\omega.2 + 2) + 4) + (\omega\uparrow\uparrow 2)^{\omega^4.4 + 3}.(\omega^2.4 + 2)}^4(5))\)

\(\beta(12.88,5) = f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^3 + 3) + 2}^4(5) + 2\)

\(\beta(12.89,5) = f_{2}^{f_{4}(5).16}(f_{f_{2}(5) + 3}^4(f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^3.(\omega.2) + \omega.4 + 3) + 1}^4(5))) + f_{(\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2))}(5)\)

\(\beta(12.9,5) = f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^4 + 2) + 2}^3(5) + f_{3}^4(f_{4}(5)) + 1\)

\(\beta(12.92,5) = f_{23}(f_{81}^3(f_{f_{2}(5).16 + 9}(f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{\omega}.3 + 2)}^4(5)))) + f_{2}^3(5)\)

\(\beta(12.93,5) = f_{3}^81(f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{\omega.3} + (\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega.4) + \omega^4.4 + 2) + 4}^4(5)).2 + 1\)

\(\beta(12.94,5) = f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{\omega.4 + 4}.(\omega^4.3 + \omega^2 + 3) + (\omega\uparrow\uparrow 2)^{\omega.4}.(\omega^3.2 + \omega^2 + \omega.3 + 1) + \omega + 4) + 3}^4(5)\)

\(\beta(12.96,5) = f_{3}^2(f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{\omega^2.4 + \omega.4 + 1}.(\omega^4 + 4) + (\omega\uparrow\uparrow 2)^4.3 + \omega.4) + 4}^2(5)) + 3\)

\(\beta(12.97,5) = f_{9}(f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{\omega^3.2 + \omega^2.3 + 1}.(\omega^2 + 3) + 3) + (\omega\uparrow\uparrow 2)^{\omega.4}.(\omega^4.2 + 2) + 2}^2(5)) + 3\)

\(\beta(12.98,5) = f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.2 + \omega + 1}.(\omega^3.4 + \omega^2.4 + 2) + 2) + 4}^2(5).2 + 2\)

\(\beta(13.01,5) = f_{f_{2}(5).64 + 87}^{f_{85}(f_{\omega^2 + 3}(f_{\omega^3 + \omega}^3(5)))}(f_{(\omega\uparrow\uparrow 3)^4 + 1}(5))\)

\(\beta(13.02,5) = f_{3}^{f_{f_{2}^4(5).(2^{f_{2}(5).32})}(f_{f_{3}^4(5) + f_{3}(5).(16777216) + 3}(f_{(\omega\uparrow\uparrow 2)^4.(\omega.4)}^4(5)))}(f_{(\omega\uparrow\uparrow 3)^4 + 2}^2(5))\)

\(\beta(13.03,5) = f_{(\omega\uparrow\uparrow 3)^4 + 3}^3(5).(8388608) + f_{2}^4(5) + 4\)

\(\beta(13.04,5) = f_{\omega^4.(f_{(\omega\uparrow\uparrow 3) + 2}^3(5).(2^{f_{5}(f_{(\omega\uparrow\uparrow 3) + 1}^2(5))}))}(f_{(\omega\uparrow\uparrow 3) + 2}^3(f_{(\omega\uparrow\uparrow 3)^4}(f_{(\omega\uparrow\uparrow 3)^4 + 4}^3(5))))\)

\(\beta(13.05,5) = f_{2}^2(f_{f_{3}(5).4 + 1}^{f_{2}^3(5) + 1}(f_{(\omega\uparrow\uparrow 3)^4 + \omega^3.2 + 2}(5))).(2^{f_{4}^4(f_{\omega + 4}^4(5))})\)

\(\beta(13.06,5) = f_{\omega + f_{2}^{f_{3}(5)}(f_{4}(5))}(f_{\omega.3 + 4}^48(f_{(\omega\uparrow\uparrow 3)^4 + (\omega\uparrow\uparrow 2)^3.4 + \omega^2.4 + \omega.2}(5)))\)

\(\beta(13.08,5) = f_{9}^2(f_{(\omega\uparrow\uparrow 3)^4 + (\omega\uparrow\uparrow 3)^3.3 + 4}(5)).(2^{f_{5}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega.2 + 3} + (\omega\uparrow\uparrow 2)^4.(\omega^2.2 + 3) + 2}^4(5))})\)

\(\beta(13.09,5) = f_{5}^2(f_{6}(f_{7}^{126}(f_{8}^24(f_{f_{2}^2(5) + 81}^2(f_{(\omega\uparrow\uparrow 3)^4.2}^4(5))))))\)

\(\beta(13.1,5) = f_{2}^3(f_{3}^3(f_{(\omega\uparrow\uparrow 3)^4.2 + 2}(5))).(2^{f_{(\omega\uparrow\uparrow 3)^2.(f_{(\omega\uparrow\uparrow 3).2}(5))}(f_{(\omega\uparrow\uparrow 3)^2.(\omega^2.4 + 1) + \omega^4 + \omega^2.4 + 2}(5))})\)

\(\beta(13.11,5) = f_{(\omega\uparrow\uparrow 3)^4.2 + 3}^2(5) + 4\)

\(\beta(13.13,5) = f_{(\omega\uparrow\uparrow 3)^4.2 + f_{(\omega\uparrow\uparrow 2)^{\omega^3.3 + \omega.4 + 2}.(\omega^3.4 + 3) + (\omega\uparrow\uparrow 2)^3.(\omega^4.4)}(5)}(f_{(\omega\uparrow\uparrow 3)^4.2 + \omega^2.3 + \omega + 1}(5))\)

\(\beta(13.14,5) = f_{(\omega\uparrow\uparrow 3)^4.2 + (\omega\uparrow\uparrow 2)^2 + \omega^3}^4(5).(2^{f_{\omega^{f_{(\omega\uparrow\uparrow 2) + \omega^2.3}(5)}}(f_{(\omega\uparrow\uparrow 2)^4}^3(5))})\)

\(\beta(13.15,5) = f_{(\omega\uparrow\uparrow 3)^4.2 + (\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega + 3}.(\omega^2.3 + 1) + (\omega\uparrow\uparrow 2).(\omega^4.2 + \omega.3 + 3) + \omega^3.3 + \omega.4 + 3}^4(5)\)

\(\beta(13.17,5) = f_{(\omega\uparrow\uparrow 3)^4.3}^3(5).16 + f_{\omega^4.4 + \omega^2.(f_{2}^4(5) + 5)}(f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega.3 + 2}.2 + 4) + 3}(5))\)

\(\beta(13.18,5) = f_{(\omega\uparrow\uparrow 3)^2.(\omega^{f_{\omega.2 + f_{f_{3}^{f_{\omega}(5)}(f_{\omega.3}^3(5))}(f_{\omega.4 + 2}^2(5))}(f_{\omega^2.4 + \omega + 4}^2(5))})}(f_{(\omega\uparrow\uparrow 3)^4.3 + 1}^3(5))\)

\(\beta(13.19,5) = f_{(\omega\uparrow\uparrow 2)^{f_{2}^3(f_{\omega + 2}(f_{\omega + 4}^4(5))).(2^{f_{f_{3}^2(5).128 + 4}^{f_{2}(5).32}(f_{\omega}^2(5))})}}(f_{(\omega\uparrow\uparrow 3)^4.3 + 2}^4(5))\)

\(\beta(13.2,5) = f_{127}^5(f_{\omega^3.4 + \omega^2 + \omega + 3}^{f_{2}^4(5) + f_{2}^3(5).8 + 2}(f_{(\omega\uparrow\uparrow 3)^4.3 + 4}(5)))\)

\(\beta(13.22,5) = f_{\omega^11.(f_{2}^{f_{\omega^3}(5)}(f_{41}(f_{(\omega\uparrow\uparrow 2)^2.3 + 2}^4(5))))}(f_{(\omega\uparrow\uparrow 3)^4.3 + \omega^4.3 + \omega^3 + 2}(5))\)

\(\beta(13.23,5) = f_{4}^{f_{f_{\omega^4}(5)}(f_{\omega^4.4 + \omega^2.3 + 3}^4(5))}(f_{(\omega\uparrow\uparrow 3)^4.3 + (\omega\uparrow\uparrow 2)^{\omega^2 + \omega.3 + 1} + \omega^3.3 + \omega.4 + 2}(5))\)

\(\beta(13.24,5) = f_{(\omega\uparrow\uparrow 2)^{f_{f_{\omega.4 + 3}^3(5).8}(f_{\omega^2 + \omega.3 + 4}^2(5))}}(f_{(\omega\uparrow\uparrow 3)^4.3 + (\omega\uparrow\uparrow 3)^2 + (\omega\uparrow\uparrow 3).3 + \omega^3.4 + 3}^3(5))\)

\(\beta(13.25,5) = f_{\omega^{f_{2}^3(5).(2^{f_{2}^2(5).2 + 5}) + f_{2}(5)}.(f_{28}^{f_{\omega^4.2 + \omega^2.3}(5)}(f_{\omega^4.4}^3(5)))}(f_{(\omega\uparrow\uparrow 3)^4.4}(5))\)

\(\beta(13.26,5) = f_{f_{3}^{f_{\omega.2}^{f_{\omega + 5}^2(f_{\omega.2 + 1}(5)).16 + f_{\omega.2}(5)}(f_{\omega.2 + 1}^2(5))}(f_{\omega^4}(5))}(f_{(\omega\uparrow\uparrow 3)^4.4 + 1}^2(5))\)

\(\beta(13.27,5) = f_{4}(f_{(\omega\uparrow\uparrow 3)^4.4 + 2}^3(5)).(2^{f_{4}^3(5) + 2}) + 7\)

\(\beta(13.28,5) = f_{(\omega\uparrow\uparrow 3)^4.4 + 1}^{f_{2}^{f_{3}^2(5) + 1}(f_{4}(5)) + 4}(f_{(\omega\uparrow\uparrow 3)^4.4 + 3}^3(5)).16 + 2\)

\(\beta(13.29,5) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.(f_{3}^2(5).2 + f_{2}^3(5) + 2) + \omega + 1}.4 + \omega^3.2 + f_{\omega}(5)}(f_{(\omega\uparrow\uparrow 3)^4.4 + 4}^4(5))\)

\(\beta(13.3,5) = f_{2}^{f_{\omega^3.3 + \omega^2.3 + \omega.3 + 3}^4(5)}(f_{3}^{f_{3}(5) + f_{2}^3(5) + 42}(f_{(\omega\uparrow\uparrow 3)^4.4 + \omega^3.3 + \omega^2.4 + \omega}(5)))\)

\(\beta(13.31,5) = f_{2}^36(f_{9}^3(f_{(\omega\uparrow\uparrow 3)^4.4 + (\omega\uparrow\uparrow 2)^4.2 + (\omega\uparrow\uparrow 2).4 + 1}^2(5))) + f_{(\omega\uparrow\uparrow 2).(\omega.3 + 2) + \omega}(5)\)

\(\beta(13.32,5) = f_{2}^5(f_{f_{(\omega\uparrow\uparrow 3)^2.2}(5).16 + 3}^3(f_{(\omega\uparrow\uparrow 3)^4.4 + (\omega\uparrow\uparrow 3).3 + (\omega\uparrow\uparrow 2)^4.(\omega^3.4 + \omega.4 + 1) + 4}(5)))\)

\(\beta(13.33,5) = f_{f_{2}^11(f_{3}^4(5)) + 82}^{f_{f_{2}^3(5)}(f_{(\omega\uparrow\uparrow 2)^4.2 + 1}^2(5))}(f_{(\omega\uparrow\uparrow 3)^4.4 + (\omega\uparrow\uparrow 3)^3.(\omega + 2) + 4}(5))\)

\(\beta(13.34,5) = f_{7}^{f_{(\omega\uparrow\uparrow 2)^{\omega^3.3 + \omega + 4}.(\omega^2.3)}(5)}(f_{9}^6(f_{(\omega\uparrow\uparrow 3)^4.(\omega + 3) + (\omega\uparrow\uparrow 2).4 + \omega^4.2 + 4}(5)))\)

\(\beta(13.35,5) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.9 + 3}.(\omega^{f_{2}^{f_{2}(5)}(f_{\omega^3.3 + 2}^2(5)).(2^{f_{\omega^2.3}(f_{\omega^2.4 + 1}^3(5))})})}(f_{(\omega\uparrow\uparrow 3)^4.(\omega.2 + 3) + 4}^2(5))\)

\(\beta(13.36,5) = f_{2}^{f_{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{\omega^4})}(5)}(f_{f_{3}^4(5).8 + f_{2}(5).(4.05648192073033E+31) + 6}^3(f_{(\omega\uparrow\uparrow 3)^4.(\omega.3 + 3) + 2}(5)))\)

\(\beta(13.37,5) = f_{f_{\omega}(f_{(\omega\uparrow\uparrow 2)^{\omega^3.3 + \omega.3 + 4} + \omega^2.3 + \omega.3 + 1}^4(5))}(f_{(\omega\uparrow\uparrow 3)^4.(\omega.4 + 2) + (\omega\uparrow\uparrow 3)^3.4 + \omega^2.3 + 3}^3(5))\)

\(\beta(13.39,5) = f_{4}(f_{f_{3}^3(5) + 4}(f_{(\omega\uparrow\uparrow 3)^4.(\omega^2.2 + 4) + (\omega\uparrow\uparrow 3)^3.3 + 1}^2(5))) + 1\)

\(\beta(13.4,5) = f_{f_{3}^2(f_{4}^2(5)) + 2}^20(f_{(\omega\uparrow\uparrow 3)^4.(\omega^2.3 + 4) + 4}^3(5)).4 + 9\)

\(\beta(13.42,5) = f_{f_{23}^{f_{\omega}(f_{(\omega\uparrow\uparrow 3).(\omega^4.2 + 3) + 4}^2(5))}(f_{(\omega\uparrow\uparrow 3)^4.3 + 3}^3(5))}(f_{(\omega\uparrow\uparrow 3)^4.(\omega^3 + 3) + 4}^2(5))\)

\(\beta(13.44,5) = f_{f_{(\omega\uparrow\uparrow 3)^4.4 + 1}^2(5) + f_{f_{\omega^4}(5)}(f_{(\omega\uparrow\uparrow 3)^2.(\omega^3.4 + \omega^2.2 + 1)}^2(5))}(f_{(\omega\uparrow\uparrow 3)^4.(\omega^3.3 + 2) + 3}^4(5))\)

\(\beta(13.45,5) = f_{(\omega\uparrow\uparrow 2)^{\omega^3.44 + 3}.(\omega^2.(f_{\omega^2.2 + \omega.4 + 1}^2(5)))}(f_{(\omega\uparrow\uparrow 3)^4.(\omega^3.4 + 1) + (\omega\uparrow\uparrow 3).3 + \omega^3.3}(5))\)

\(\beta(13.46,5) = f_{f_{2}(f_{3}^2(f_{4}^3(5))) + f_{2}(5)}^4(f_{(\omega\uparrow\uparrow 3)^4.(\omega^4 + 1) + 3}(5)) + 10\)

\(\beta(13.47,5) = f_{(\omega\uparrow\uparrow 3)^2.(f_{2}(f_{4}(5))) + \omega^3.3}(f_{(\omega\uparrow\uparrow 3)^4.(\omega^4.2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 4}.2 + \omega}(5))\)

\(\beta(13.48,5) = f_{2}^3(f_{(\omega\uparrow\uparrow 3)^4.(\omega^4.3) + 2}(5)).(2^{f_{2}^3(5) + 3}) + f_{\omega^2.2 + \omega.2 + f_{\omega}(f_{\omega.2 + 1}^4(5))}(f_{\omega^2.2 + \omega.4}^3(5))\)

\(\beta(13.49,5) = f_{(\omega\uparrow\uparrow 2)^{f_{f_{2}^{f_{4}^2(5).8 + 4}(f_{3}^5(f_{4}^2(5))) + f_{2}^3(5)}(f_{\omega^3.3 + \omega.2 + 2}(5))}}(f_{(\omega\uparrow\uparrow 3)^4.(\omega^4.3 + \omega^3.4 + 1) + 3}^2(5))\)

\(\beta(13.5,5) = f_{(\omega\uparrow\uparrow 3)^3.3 + f_{f_{2}(5) + 82}^5(f_{\omega^2.3 + 1}(5))}(f_{(\omega\uparrow\uparrow 3)^4.(\omega^4.4 + \omega^3.2 + \omega^2 + 2) + 3}^3(5))\)

\(\beta(13.51,5) = f_{(\omega\uparrow\uparrow 3).2 + f_{2}^4(5) + f_{2}^2(5).4 + f_{2}^2(5).2 + 85}(f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2).4 + 3) + (\omega\uparrow\uparrow 3)^3.3 + (\omega\uparrow\uparrow 3).3 + 2}^4(5))\)

\(\beta(13.52,5) = f_{f_{(\omega\uparrow\uparrow 2)^{\omega^2.4 + \omega.4}}(5)}(f_{\omega^3.4 + \omega + 1}^4(f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2).(\omega^4 + \omega.4 + 3)) + 4}^3(5)))\)

\(\beta(13.53,5) = f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^2.3 + (\omega\uparrow\uparrow 2).(\omega^2.3 + \omega + 1)) + 1}^4(5) + 40\)

\(\beta(13.54,5) = f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^2.(\omega^3.3 + \omega.4 + 4) + \omega^4.2 + 2) + 1}^4(5) + 2\)

\(\beta(13.55,5) = f_{(\omega\uparrow\uparrow 2)^{\omega.(f_{2}^5(f_{3}(5)).(67108864) + 4) + f_{3}^{f_{2}^4(f_{3}(5)).2}(f_{4}^3(5))}}(f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^3.3 + 4) + 4}^2(5))\)

\(\beta(13.57,5) = f_{2}(f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^4.2 + (\omega\uparrow\uparrow 2)^3.4 + 3) + (\omega\uparrow\uparrow 2).(\omega^2)}^3(5)).8 + 2\)

\(\beta(13.58,5) = f_{f_{\omega + f_{\omega + 8}^{f_{\omega}(5)}(f_{\omega + f_{2}^2(5) + 7}^2(f_{\omega.2 + 4}^3(5)))}(f_{\omega.3 + 4}^3(5))}(f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^4.(\omega^2.3 + \omega.3 + 2) + 2)}^2(5))\)

\(\beta(13.59,5) = f_{f_{4}^2(5) + f_{2}(5).2 + f_{2}(5) + 17}^{f_{\omega^4 + \omega^3.3 + \omega}(5)}(f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^{\omega + 3}.(\omega.4 + 4) + 2) + 1}^2(5))\)

\(\beta(13.62,5) = f_{\omega^4.2 + f_{2}^3(f_{3}^2(5))}(f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^{\omega^2.3 + 3}.3 + (\omega\uparrow\uparrow 2)^4.(\omega^4 + \omega^3.2 + \omega.4) + 1) + 3}^2(5))\)

\(\beta(13.63,5) = f_{(\omega\uparrow\uparrow 3)^4.(\omega^{f_{2}^4(5).8}.3 + 4) + (\omega\uparrow\uparrow 2)^31.5}(f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^{\omega^3 + 2} + 3)}(5))\)

\(\beta(13.64,5) = f_{2}^5(f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^{\omega^3.3}.(\omega^2.3 + 4) + (\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega.2 + 3} + \omega.4 + 4) + (\omega\uparrow\uparrow 2)^4.4 + 4}^3(5))\)

\(\beta(13.65,5) = f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.3 + \omega.4 + 3}.(\omega^2.2 + \omega.2 + 3) + \omega^2.2 + 1) + 1}(5) + 2\)

\(\beta(13.67,5) = f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2) + \omega^3.2 + \omega)}(f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega.4 + 4}.4 + 1) + \omega^3.3 + \omega.3}^3(5))\)

\(\beta(13.68,5) = f_{(\omega\uparrow\uparrow 3)^{\omega}.4 + (\omega\uparrow\uparrow 2)^{\omega^3.3 + \omega^2.3 + 1}.(\omega^2 + \omega.2 + 1) + \omega^3.4}^4(5) + 1\)

\(\beta(13.69,5) = f_{2}^6(f_{4}^{f_{4}^4(5) + 3}(f_{(\omega\uparrow\uparrow 3)^{\omega + 1} + 4}^3(5))).(2^{f_{(\omega\uparrow\uparrow 2)^2.(\omega) + \omega^4.3 + \omega.2 + 2}^3(5)})\)

\(\beta(13.7,5) = f_{(\omega\uparrow\uparrow 3)^{\omega + 1}.(\omega^2 + f_{2}^20(f_{\omega^4.2 + 1}^2(5)).(1048576))}(f_{(\omega\uparrow\uparrow 3)^{\omega + 1}.(\omega^3 + \omega^2.2) + 2}^2(5))\)

\(\beta(13.71,5) = f_{f_{4}(5) + f_{2}^6(f_{3}(5)).2 + f_{2}^4(5)}(f_{(\omega\uparrow\uparrow 3)^{\omega + 2}.2 + (\omega\uparrow\uparrow 2)^4.(\omega^2.4) + (\omega\uparrow\uparrow 2)^3.3 + 3}^4(5))\)

\(\beta(13.73,5) = f_{(\omega\uparrow\uparrow 2)^{\omega.3 + f_{\omega^{f_{2}^2(f_{4}^3(5)) + 2}.5}(f_{(\omega\uparrow\uparrow 3)^2.3 + \omega.2 + 1}^2(5))}}(f_{(\omega\uparrow\uparrow 3)^{\omega + 3}.4}(5))\)

\(\beta(13.74,5) = f_{(\omega\uparrow\uparrow 3)^{\omega + 3}.((\omega\uparrow\uparrow 2)^{\omega^3.3 + 4}.2 + \omega^2.3)}(5) + f_{f_{f_{3}(f_{4}^3(5))}(f_{\omega + 2}^4(5))}(f_{\omega.3 + 1}^4(5))\)

\(\beta(13.75,5) = f_{35}^{f_{3}^{f_{(\omega\uparrow\uparrow 2)^2}(5)}(f_{(\omega\uparrow\uparrow 2)^4.2 + (\omega\uparrow\uparrow 2).(\omega^2.4 + 1) + 2}^3(5))}(f_{(\omega\uparrow\uparrow 3)^{\omega + 4}.(\omega.3 + 1) + 2}(5))\)

\(\beta(13.76,5) = f_{(\omega\uparrow\uparrow 2)^3.(\omega^{f_{2}(f_{3}^2(5))}.2 + 5)}(f_{(\omega\uparrow\uparrow 3)^{\omega.2} + (\omega\uparrow\uparrow 3)^4.4 + (\omega\uparrow\uparrow 2)^{\omega^3.4} + 1}(5))\)

\(\beta(13.77,5) = f_{2}^5(f_{f_{(\omega\uparrow\uparrow 3).(\omega^2.2 + 4) + \omega^4.4 + 2}^4(5) + 3}^{f_{2}^2(5) + 1}(f_{(\omega\uparrow\uparrow 3)^{\omega.2}.(\omega^4 + 2) + 2}^2(5)))\)

\(\beta(13.81,5) = f_{(\omega\uparrow\uparrow 3)^{\omega.2 + 2}.((\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega^2.2 + \omega + 3}.2 + (\omega\uparrow\uparrow 2)^{\omega^4 + \omega + 3}.(\omega.3 + 4) + 3) + \omega^4.3 + \omega.4}(5)\)

\(\beta(13.82,5) = f_{3}^9(f_{(\omega\uparrow\uparrow 3)^{\omega.2 + 3}.(\omega^2 + \omega.2 + 3) + 4}(5)).2 + 1\)

\(\beta(13.83,5) = f_{\omega^{f_{4}^{f_{4}^2(5) + 9}(f_{\omega^4.2 + 2}^2(5)).(3.09485009821345E+26) + f_{2}^{f_{2}(5).2}(f_{3}^3(5))}}(f_{(\omega\uparrow\uparrow 3)^{\omega.2 + 4}.2 + 1}^4(5))\)

\(\beta(13.84,5) = f_{2}^{f_{(\omega\uparrow\uparrow 3)^4.(\omega^4.4 + f_{2}^3(5))}(f_{(\omega\uparrow\uparrow 3)^{\omega + 3}.(\omega^3.3 + 4) + 1}^4(5))}(f_{(\omega\uparrow\uparrow 3)^{\omega.2 + 4}.(\omega^4.3 + 3) + 1}^4(5))\)

\(\beta(13.85,5) = f_{f_{\omega^3.(f_{\omega^2.4 + f_{5}(f_{\omega^2}^2(5))}(f_{\omega^3 + 4}^4(5)))}(f_{\omega^4.2 + 1}^4(5))}(f_{(\omega\uparrow\uparrow 3)^{\omega.3}.3 + 3}^4(5))\)

\(\beta(13.87,5) = f_{\omega^3.3 + \omega.(f_{f_{3}(f_{4}^3(5))}(f_{(\omega\uparrow\uparrow 2)^{\omega^4.2 + 4}.(\omega.4)}^2(5)))}(f_{(\omega\uparrow\uparrow 3)^{\omega.3 + 1}.4 + \omega^2.3 + 3}^3(5))\)

\(\beta(13.88,5) = f_{(\omega\uparrow\uparrow 3)^{\omega.3 + 2} + 1}^3(5).16 + f_{\omega^3.(f_{2}^2(5) + 2) + \omega^2.(f_{(\omega\uparrow\uparrow 2)^2.4 + 2}^3(5))}(f_{(\omega\uparrow\uparrow 2)^3.3 + 4}^2(5))\)

\(\beta(13.89,5) = f_{(\omega\uparrow\uparrow 3)^{\omega.3 + 2}.(\omega^2.2 + \omega.3 + 4) + (\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^{\omega^4.4 + 1}.4 + 3)}^4(5) + 6\)

\(\beta(13.9,5) = f_{\omega^2 + \omega.(f_{f_{2}^2(5).(2^{f_{2}(5).8 + 38}) + 7}^{f_{3}(5).32}(f_{\omega}(5)))}(f_{(\omega\uparrow\uparrow 3)^{\omega.3 + 3}.2 + 3}(5))\)

\(\beta(13.91,5) = f_{(\omega\uparrow\uparrow 3)^{\omega.3 + 3}.13 + 2}^3(f_{(\omega\uparrow\uparrow 3)^{\omega.3 + 3}.(\omega^4.4 + 1)}^4(5)).(2^{f_{2}^4(5).4 + 42}) + f_{\omega^4}(5)\)

\(\beta(13.92,5) = f_{f_{(\omega\uparrow\uparrow 2)^{\omega + 5}}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega.4 + 2}.4 + (\omega\uparrow\uparrow 2).(\omega^2.3 + 3) + 1}^2(5))}(f_{(\omega\uparrow\uparrow 3)^{\omega.3 + 4}.3 + 4}^3(5))\)

\(\beta(13.93,5) = f_{(\omega\uparrow\uparrow 2)^{\omega + f_{\omega^3.4 + \omega}^4(5) + 2}.(\omega.(f_{2}^3(5)))}(f_{(\omega\uparrow\uparrow 3)^{\omega.3 + 4}.((\omega\uparrow\uparrow 2)^{\omega + 2}.4 + 3) + 4}^3(5))\)

\(\beta(13.94,5) = f_{(\omega\uparrow\uparrow 2)^{\omega^4.(f_{\omega^3.(f_{5}(f_{\omega^2 + 1}^3(5)))}(f_{\omega^4.2 + \omega^2.3 + \omega.4 + 3}(5)))}}(f_{(\omega\uparrow\uparrow 3)^{\omega.4}.4 + \omega^3.2 + 2}(5))\)

\(\beta(13.95,5) = f_{f_{3}^4(5)}^3(f_{(\omega\uparrow\uparrow 3)^{\omega.4 + 1} + 1}^3(5)) + f_{4}(5)\)

\(\beta(13.96,5) = f_{5}^{f_{(\omega\uparrow\uparrow 3)^3}(5)}(f_{24}^4(f_{(\omega\uparrow\uparrow 3)^{\omega.4 + 1}.(\omega^2.2 + \omega.2 + 4) + (\omega\uparrow\uparrow 2)^3.(\omega) + \omega^2.4 + 4}(5)))\)

\(\beta(13.97,5) = f_{(\omega\uparrow\uparrow 3)^{\omega.2 + f_{\omega^{f_{2}(5).2 + 2}.5}(f_{(\omega\uparrow\uparrow 2)^3 + \omega^3}^2(5))}}(f_{(\omega\uparrow\uparrow 3)^{\omega.4 + 2}.2 + 2}^3(5))\)

\(\beta(13.99,5) = f_{(\omega\uparrow\uparrow 3)^{\omega.4 + 3}.3 + 3}^4(5) + f_{16}^{f_{(\omega\uparrow\uparrow 2)}(5)}(f_{(\omega\uparrow\uparrow 2)^{\omega^3.4 + 1}.4 + \omega^4.3 + \omega^2 + \omega.4 + 2}^4(5))\)

\(\beta(14,5) = f_{(\omega\uparrow\uparrow 3)^{\omega.4 + 3}.((\omega\uparrow\uparrow 2)^4.(\omega^2.3) + \omega^3 + \omega^2.4 + \omega) + (\omega\uparrow\uparrow 2)^{\omega^4.3 + 3}.3 + (\omega\uparrow\uparrow 2)^{\omega^3 + \omega^2}}(5)\)

\(\beta(14.01,5) = f_{\omega^{f_{3}^4(5) + 6}.(f_{\omega^4 + 4}^4(5).4 + f_{f_{2}^2(5) + 40}(f_{\omega}^3(5)))}(f_{(\omega\uparrow\uparrow 3)^{\omega.4 + 4}.4 + 4}^3(5))\)

\(\beta(14.02,5) = f_{f_{2}^2(f_{f_{2}^4(f_{4}^4(5)) + 23}(f_{\omega^4.2 + \omega.4 + 1}^3(5))).2}(f_{(\omega\uparrow\uparrow 3)^{\omega^2}}(5))\)

\(\beta(14.03,5) = f_{2}^84(f_{6}^8(f_{f_{2}^2(5).4 + 4}^3(f_{f_{3}^2(5) + 4}^3(f_{(\omega\uparrow\uparrow 3)^{\omega^2 + 1}.2}^4(5))))) + 4\)

\(\beta(14.05,5) = f_{f_{\omega^2 + f_{\omega^3.2 + 4}^3(5).16 + f_{3}^{f_{\omega}(5)}(f_{\omega^3.2 + 4}(5))}(f_{\omega^3.2 + \omega.2 + 4}^3(5))}(f_{(\omega\uparrow\uparrow 3)^{\omega^2 + 3}.4 + 2}(5))\)

\(\beta(14.07,5) = f_{(\omega\uparrow\uparrow 2)^4.(\omega^20)}(f_{(\omega\uparrow\uparrow 3)^{\omega^2 + \omega + 2}.(\omega^3.2 + \omega^2.4 + 1) + (\omega\uparrow\uparrow 3)^3.2 + (\omega\uparrow\uparrow 3).2 + 4}^3(5))\)

\(\beta(14.08,5) = f_{8}^{f_{3}^55(f_{\omega^2.4}^3(5)) + f_{\omega^2.3}(5)}(f_{11}^3(f_{(\omega\uparrow\uparrow 3)^{\omega^2 + \omega.2 + 2}.2 + 4}(5)))\)

\(\beta(14.09,5) = f_{f_{2}^2(5).16 + 2}^{f_{f_{\omega.3}^3(5)}(f_{\omega.3 + 1}^2(5))}(f_{(\omega\uparrow\uparrow 3)^{\omega^2 + \omega.3 + 1}.(\omega^4 + 3) + \omega^2.2 + 4}^3(5))\)

\(\beta(14.1,5) = f_{\omega^2.2 + \omega.23 + f_{(\omega\uparrow\uparrow 2)^4.3 + (\omega\uparrow\uparrow 2)^3.(\omega^4.3 + \omega)}(5)}(f_{(\omega\uparrow\uparrow 3)^{\omega^2 + \omega.4 + 1}.2 + \omega^4.4}(5))\)

\(\beta(14.11,5) = f_{f_{\omega.(f_{f_{\omega.2 + 2}^3(5)}(f_{(\omega\uparrow\uparrow 2)^2 + 2}^3(5)))}(f_{(\omega\uparrow\uparrow 2)^3}^2(5))}(f_{(\omega\uparrow\uparrow 3)^{\omega^2.2}.2 + 4}^2(5))\)

\(\beta(14.12,5) = f_{(\omega\uparrow\uparrow 3)^{\omega^2.2 + 1}.3 + 4}^3(5) + f_{3}^4(5).32 + 2\)

\(\beta(14.13,5) = f_{f_{\omega + f_{\omega.(f_{\omega}(5))}(f_{\omega^2 + \omega}^4(f_{\omega^2 + \omega + 1}(5)))}(f_{\omega^2 + \omega + 3}^4(5))}(f_{(\omega\uparrow\uparrow 3)^{\omega^2.2 + 2}.4 + 4}^3(5))\)

\(\beta(14.14,5) = f_{f_{2}^7(f_{4}^4(5)) + f_{2}^2(5).4 + 40}^{f_{\omega^3.5}(f_{\omega^4 + 4}^2(5))}(f_{(\omega\uparrow\uparrow 3)^{\omega^2.2 + 3}.(\omega.4 + 3) + 4}^4(5))\)

\(\beta(14.15,5) = f_{4}^{f_{3}^4(5)}(f_{f_{2}^{f_{2}^3(5) + 8}(f_{3}^2(5)).(2^{f_{3}(5) + 4}) + 1}^3(f_{(\omega\uparrow\uparrow 3)^{\omega^2.2 + 4}.(\omega^3.4 + \omega.3 + 3) + 2}^2(5)))\)

\(\beta(14.16,5) = f_{f_{(\omega\uparrow\uparrow 3)^4.(\omega^3.2)}(5)}(f_{(\omega\uparrow\uparrow 3)^{\omega^2.2 + \omega + 3}.2 + (\omega\uparrow\uparrow 2)^{\omega^2.4 + \omega.4 + 4}.2 + \omega^4.2 + \omega^3.4 + 1}^2(5))\)

\(\beta(14.17,5) = f_{\omega^{f_{5}(f_{f_{\omega.3}^3(5).4 + 2}^2(f_{\omega.3 + 3}^3(5)))}}(f_{(\omega\uparrow\uparrow 3)^{\omega^2.2 + \omega.2 + 2}.(\omega^4.2 + \omega^2.4 + 2)}^3(5))\)

\(\beta(14.18,5) = f_{60}(f_{(\omega\uparrow\uparrow 3)^{\omega.3 + 4} + 1}^3(f_{(\omega\uparrow\uparrow 3)^{\omega^2.2 + \omega.3 + 2}.2 + \omega^4 + \omega^2.3 + \omega.3 + 1}^2(5)))\)

\(\beta(14.19,5) = f_{(\omega\uparrow\uparrow 3)^{\omega^2.2 + \omega.4 + 1}.(\omega^4 + 1) + 2}^4(5).16 + 4\)

\(\beta(14.2,5) = f_{(\omega\uparrow\uparrow 2)^{f_{\omega^2.(f_{2}^2(f_{4}^4(5)).8 + 67) + \omega}(f_{\omega^4.2 + 1}^2(5))}}(f_{(\omega\uparrow\uparrow 3)^{\omega^2.3}.3 + 3}^2(5))\)

\(\beta(14.21,5) = f_{(\omega\uparrow\uparrow 3)^{\omega^2.3 + 1}.4 + 3}(5).4 + 2\)

Next Attempt Base \(v = 5\) on 21 May 2016

\(\beta(2,5) = f_{2}(5).4 + 86\)

\(\beta(2.1,5) = f_{2}^2(5) + 4\)

\(\beta(2.11,5) = f_{2}^2(5) + 6\)

\(\beta(2.111,5) = f_{2}^2(5) + 7\)

\(\beta(2.1111,5) = f_{2}^2(5) + 7\)

\(\beta(2.112,5) = f_{2}^2(5) + 7\)

\(\beta(2.12,5) = f_{2}^2(5) + 12\)

\(\beta(2.121,5) = f_{2}^2(5) + 14\)

\(\beta(2.63,5) = f_{3}(5)\)

\(\beta(3.63,5) = f_{4}(5)\)

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

\(\beta(5.07,5) = f_{\omega + 2}^4(5).4 + 1\)

\(\beta(5.16,5) = f_{\omega.2 + 1}^2(5).2 + 1\)

\(\beta(5.44,5) = f_{\omega.4 + 1}^4(5).512 + 5\)

\(\beta(5.6,5) = f_{\omega^2 + \omega}^2(5).2\)

\(\beta(5.67,5) = f_{\omega^2.2}(5) + 4\)

\(\beta(6.16,5) = f_{\omega^3}^4(f_{\omega^3 + 2}^4(5))\)

\(\beta(6.18,5) = f_{\omega^3 + 4}^2(5) + 4\)

\(\beta(6.27,5) = f_{\omega^3.2}(5) + 3\)

\(\beta(6.33,5) = f_{2}(f_{3}^3(f_{\omega^3.2 + 3}^4(5))) + 4\)

\(\beta(6.43,5) = f_{\omega^3.3}(5).(2^{f_{\omega^3.2 + 3}(5) + 1})\)

\(\beta(6.61,5) = f_{\omega^3.4 + 1}(5) + 2\)

\(\beta(6.66,5) = f_{\omega^3.4 + 4}(5)\)

\(\beta(6.73,5) = f_{2}^2(f_{\omega^3.4 + \omega^2.2}^3(5))\)

Next Attempt Base \(v = 6\) on 22 May 2016

\(\beta(17.5880909390219,6)\)

\(= f_{(\omega\uparrow\uparrow 3)^{f_{(\omega\uparrow\uparrow 2)^{f_{f_{2}^3(f_{3}^4(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^4 + \omega^3.5 + 3}.3 + 4}^5(6))).2}(f_{(\omega\uparrow\uparrow 4) + 1}(6))}}(f_{(\omega\uparrow\uparrow 4)}^5(f_{(\omega\uparrow\uparrow 4) + 1}(6)))}}(f_{(\omega\uparrow\uparrow 4) + 1}^2(6))\)

\(<< \beta(17.5881568554761,6) = f_{(\omega\uparrow\uparrow 4)}(f_{(\omega\uparrow\uparrow 4) + 1}^2(6)) + 2\)

and

\((\omega\uparrow\uparrow 3)^{f_{(\omega\uparrow\uparrow 2)^{f_{f_{2}^3(f_{3}^4(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^4 + \omega^3.5 + 3}.3 + 4}^5(6))).2}(f_{(\omega\uparrow\uparrow 4) + 1}(6))}}(f_{(\omega\uparrow\uparrow 4)}^5(f_{(\omega\uparrow\uparrow 4) + 1}(6)))}\)

\(<< \omega\uparrow\uparrow 4\)

and

\({f_{(\omega\uparrow\uparrow 2)^{f_{f_{2}^3(f_{3}^4(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^4 + \omega^3.5 + 3}.3 + 4}^5(6))).2}(f_{(\omega\uparrow\uparrow 4) + 1}(6))}}(f_{(\omega\uparrow\uparrow 4)}^5(f_{(\omega\uparrow\uparrow 4) + 1}(6)))}\)

\(<< (\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega - 1} - 1}\)

\(\beta(36,6) = f_{\varphi(1,0)}(6)\)

\(\beta(36.02,6) = f_{\varphi(1,0)^{\omega^3.5 + 5}.((\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2).2 + \omega^3.3 + \omega^2.3 + \omega.4 + 5} + (\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{\omega^4.3 + \omega^2 + 3}.(\omega.2 + 5) + \omega^5.2 + \omega^4.2 + \omega^2.2 + 5) + (\omega\uparrow\uparrow 3)^2.((\omega\uparrow\uparrow 2).(\omega^4.3 + \omega^3.5) + \omega^3.3 + \omega))}(6)\)

\(\beta(36.021,6) = f_{\varphi(1,0)^{\omega^2.3 + \omega.3 + 4}.((\omega\uparrow\uparrow 2)^{f_{2}^5(6) + 1}.(\omega))}(f_{\varphi(1,0)^{\omega^5.2 + 5}.(\omega^2.4 + \omega.5 + 4) + (\omega\uparrow\uparrow 5)^5.((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2).2 + 4}.((\omega\uparrow\uparrow 2)^{\omega^3.4 + 4}.5 + \omega^3.2 + 5) + 1) + 1}(6))\)

\(\beta(36.021507,6) = f_{\varphi(1,0)^{f_{2}^5(f_{3}^5(f_{5}^2(6))) + f_{3}^3(6) + 46}.(f_{\varphi(1,0)^5.4 + \varphi(1,0)^2}(6))}(f_{\varphi(1,0)^{\omega^5.5 + \omega^4.5 + \omega^3 + \omega + 2}.2 + (\omega\uparrow\uparrow 2)^{\omega^2 + \omega.2 + 1} + (\omega\uparrow\uparrow 2)^2.(\omega^4.2 + \omega) + (\omega\uparrow\uparrow 2).2 + 5}^2(6))\)

\(\beta(36.02150753,6) = f_{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^2.(\omega^{f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)}}(6)})}(f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^{\omega^3 + \omega.3}.3 + 5}.2 + \omega^5.4 + \omega^2.4 + 5}^5(6)))}(f_{\varphi(1,0)^{\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.5}.(\omega^2.2 + \omega.5 + 5)}^5(6))\)

\(\beta(36.021507535,6) = f_{\varphi(1,0)^{\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.4 + 4}.((\omega\uparrow\uparrow 4)^4.((\omega\uparrow\uparrow 3)^{\omega^3.5} + \omega^3 + 3) + (\omega\uparrow\uparrow 2)^{\omega^4.2 + 4}.(\omega^2.4) + (\omega\uparrow\uparrow 2)^{\omega^2} + \omega^2.4 + \omega + 5) + (\omega\uparrow\uparrow 5)^4.((\omega\uparrow\uparrow 3))}(6)\)

\(\beta(36.0215075357,6) = f_{\varphi(1,0)^{\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 5}.((\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2).(\omega.4) + 4}.(\omega^2 + 2) + 3) + (\omega\uparrow\uparrow 5).((\omega\uparrow\uparrow 2)^{\omega^3.4 + 4}.4 + 2) + \omega^4 + \omega.4 + 1}^2(6) + f_{(\omega\uparrow\uparrow 3)^2.5 + (\omega\uparrow\uparrow 3) + \omega^3.3 + \omega}(6)\)

\(\beta(36.02150753571,6) = f_{\varphi(1,0)^{\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 5}.((\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 2)^5.(\omega^2.3 + \omega.4 + 3) + \omega^4.3 + \omega^2.2 + 5}.((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^4.5 + \omega.5 + 5}.4 + 4) + (\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 2)^2.2 + \omega^4.2 + \omega^3.5 + \omega^2 + 3}.((\omega\uparrow\uparrow 3)))}(6)\)

\(\beta(36.021507535713,6) = f_{\varphi(1,0)^{\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 5}.((\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^{\omega^2.5 + \omega.4}.(\omega^5.5 + \omega^4 + 5) + \omega^4.3 + \omega.2 + 3}.4 + (\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^3.4 + 3}.2 + (\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^3.2}})}(6)\)

\(\beta(36.0215075357131,6) = f_{\varphi(1,0)^{\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 5}.((\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{\omega^4.2 + \omega^2 + \omega.4 + 1}.3 + 1) + (\omega\uparrow\uparrow 2).(\omega^3.5 + 3) + 5}.4 + 4}.((\omega\uparrow\uparrow 4)^4.(\omega^4.5 + \omega^3.3 + \omega.4) + (\omega\uparrow\uparrow 3)))}(6)\)

\(\beta(36.0215075357132,6) = f_{\varphi(1,0)^{(\omega\uparrow\uparrow 2)}}(6)\)

\(\beta(36.021507535714,6) = f_{\varphi(1,0)^{(\omega\uparrow\uparrow 2)}}(6)\)

\(\beta(36.02150753572,6) = f_{\varphi(1,0)^{(\omega\uparrow\uparrow 2)}}(6)\)

\(\beta(36.0215075358,6) = f_{\varphi(1,0)^{(\omega\uparrow\uparrow 2)}}(6) + f_{\varphi(1,0)^{f_{(\omega\uparrow\uparrow 4)^{\omega^3.4 + \omega^2.3 + \omega.4 + 1}.((\omega\uparrow\uparrow 3)^{\omega^4.3}.((\omega\uparrow\uparrow 2)^5.(\omega^3.5 + \omega.4 + 4) + 2) + 5) + (\omega\uparrow\uparrow 3)^{\omega}}(6)}}(f_{\varphi(1,0)^{\omega^2.3 + 4}.((\omega\uparrow\uparrow 5).2 + 2)}^4(6))\)

\(\beta(36.021507536,6) = f_{3}^2(f_{\varphi(1,0)^{(\omega\uparrow\uparrow 2)}}(6)) + 3\)

\(\beta(36.02150754,6) = f_{f_{f_{f_{\omega^5.5 + \omega^2.4 + 4}^{f_{\omega^4 + \omega^2.(f_{\omega^4.2}(6))}(f_{\omega^4.2 + \omega.3 + 2}^5(6))}(f_{\omega^5.5 + \omega^2.4 + \omega + 2}(6))}(f_{\varphi(1,0)^2.3 + \varphi(1,0).((\omega\uparrow\uparrow 3)^5.3 + 3) + 5}^2(6))}(f_{\varphi(1,0)^3.5 + 2}^5(6))}(f_{\varphi(1,0)^{(\omega\uparrow\uparrow 2)}}^2(6))\)

\(\beta(36.0215076,6) = f_{\omega^{f_{\varphi(1,0)^2.(f_{(\omega\uparrow\uparrow 2)^{\omega^4.2 + \omega^3.3 + \omega^2.3 + 2}.(\omega^3 + \omega^2.4 + \omega.3 + 3)}(6))}(f_{\varphi(1,0)^4.((\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^2.4 + 3}.5 + (\omega\uparrow\uparrow 2)^{\omega^3 + 3}.(\omega) + \omega.2 + 5) + 5}(6))}}(f_{\varphi(1,0)^{(\omega\uparrow\uparrow 2)} + 3}^5(6))\)

\(\beta(36.021508,6) = f_{f_{f_{(\omega\uparrow\uparrow 2).(f_{\omega^3.3 + \omega}(6))}(f_{(\omega\uparrow\uparrow 2).(\omega^5 + 4) + \omega^2.5 + 1}^3(6))}(f_{(\omega\uparrow\uparrow 2).(\omega^5 + 4) + \omega^5.2 + 3}^4(6))}(f_{\varphi(1,0)^{(\omega\uparrow\uparrow 2)}.2 + (\omega\uparrow\uparrow 4)^4 + (\omega\uparrow\uparrow 2)^{\omega^2.5 + \omega.3 + 4} + 4}^2(6))\)

\(\beta(36.02151,6) = f_{4}^6(f_{\varphi(1,0)^{(\omega\uparrow\uparrow 2)}.((\omega\uparrow\uparrow 4)^3.5 + (\omega\uparrow\uparrow 2)^{\omega^2.4 + \omega}.2 + (\omega\uparrow\uparrow 2)^5.(\omega^4 + 3) + 5) + \varphi(1,0)^{\omega.2 + 5}.((\omega\uparrow\uparrow 2)^3.4 + 1) + (\omega\uparrow\uparrow 2)^{\omega^2.2 + 2}.(\omega^2 + 2) + \omega^4.5 + \omega + 5}^4(6))\)

\(\beta(36.0216,6) = f_{\varphi(1,0)^{(\omega\uparrow\uparrow 2).3 + \omega.4 + 5}.((\omega\uparrow\uparrow 2)^2.(\omega^2.2 + \omega.2 + 5) + \omega^5.2 + 5) + \varphi(1,0)^4.((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.3 + \omega.5 + 4} + \omega^5.3 + \omega^3.5 + \omega.4 + 2}.(\omega^3.3 + \omega^2 + 5) + (\omega\uparrow\uparrow 3)^{\omega + 4}.(\omega^4.4 + \omega^2.3 + \omega))}(6)\)

\(\beta(36.022,6) = f_{\omega^3.(f_{\varphi(1,0)^2.(f_{f_{f_{\omega^{f_{(\omega\uparrow\uparrow 2)}(6)}}(f_{(\omega\uparrow\uparrow 3)}(6))}(f_{(\omega\uparrow\uparrow 3)}^2(6))}(f_{(\omega\uparrow\uparrow 3)}^4(6)))}(f_{\varphi(1,0)^3.4 + \varphi(1,0) + 4}(6)))}(f_{\varphi(1,0)^2.4}^4(f_{\varphi(1,0)^2.4 + 3}^5(f_{\varphi(1,0)^{(\omega\uparrow\uparrow 2)^2.4 + \omega^5.4 + 5} + 2}^3(6))))\)

\(\beta(36.03,6) = f_{\varphi(1,0)^{f_{2}^2(f_{5}^2(6)) + f_{3}^4(6).64 + f_{2}^3(f_{3}(6)).32 + 54}}(f_{\varphi(1,0)^{(\omega\uparrow\uparrow 4)^4.((\omega\uparrow\uparrow 2).(\omega^5 + 2) + 3) + (\omega\uparrow\uparrow 4)^3.(\omega.3 + 5) + 5}.(\omega^5.2 + \omega^3.3 + 2) + 4}^3(6))\)