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## 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. Please refer to that blog for more detailed information.

## 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,[[C_x]+[U_x]]),

(V_h=1,f_0,[[f_0]])))

f_x = (g_x,

[f_{[g_x]}],

g_x>(0:(h_u<V_h,[[]^[h_u]]),

(h_U<V_h,

V_h=C_d..h_U,

f_x+1<g_x,

x>(0:[[]([V_v])]),[[f_x+1]([])])))

g_x = (C_g<V_v+1,

C_g>(0:(h_x<V_h,[[h_x]]),

1:(V_t=1,t_g,[\omega[t_g]]),

(n_0<C_g,[[n_0]]))))

n_x = (x>(0:(g_[C_g],V_n=C_g-1,[\varphi([])]),

(g_[C_n],V_n=C_n),[\varphi([],[n_x-1])]),

V_n>(1:C_n<1+M_g_Y,C_n<V_n+M_g_V_n),

V_t=(C_g..V_n,0),

C_n>(0:(t_a,[[][t_a]]),

(t_A,[[][t_A]],n_x+1<g_Y..g_C_n)))

t_x = (h_T<V_h,

g_E<u_T,

g_C<C_g..h_T,

U_x+M_h_T+M_g_E+M_g_C>(0:(g_a<C_g..g_E,[[g_a]]),(g_A<C_g..g_E,[[g_A]])),

[^^[h_T]^[g_E]*[g_C]+[]])

u_x = (V_t,

M_h_x>(1:(h_E<h_x,g_E<U_E)))


and these predefined functions:

• Min Function $$M_{z_z} = 0$$ if sequence $$z_z$$ only contains zeros, else $$M_{z_z} = 1$$
• Uppercase Function $$U_x = 0$$ if $$x$$ is a digit or lowercase character, else $$U_x = 1$$ if $$x$$ is an uppercase character

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

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

Here is another example around $$\varphi(1,0) = \epsilon_0$$. 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.

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

$$\beta(36.0000031521921,6) = f_{10}(f_{\varphi(1,0)}(6))$$

$$\beta(36.000003401027,6) = f_{100}(f_{\varphi(1,0)}(6))$$

$$\beta(36.0000034831992,6) = f_{300}(f_{\varphi(1,0)}(6))$$

$$\beta(36.000003483978,6) = f_{f_{2}(6)}(f_{\varphi(1,0)}(6))$$

$$\beta(36.0000044793989,6) = f_{f_{4}^{f_{3}^{f_{5}^2(6)}(f_{4}^{f_{2}(f_{3}^{f_{2}^{f_{2}^4(6).(2048) + 8}(f_{3}^5(6)) + 4}(f_{4}^{f_{4}^3(6) + 3}(f_{5}^3(6)))) + f_{2}^3(6)}(f_{5}^4(6)))}(f_{5}^5(6))}(f_{\varphi(1,0)}(6))$$

$$\beta(36.000004479399,6) = f_{3}^{f_{2}^{f_{2}^{45}(f_{3}(6)).2 + 5}(f_{5}^3(6)) + f_{3}^3(f_{5}(6)) + 4}(f_{f_{\omega}(6)}(f_{\varphi(1,0)}(6))).(2^{f_{f_{(\omega\uparrow\uparrow 2)^3.(\omega)}(6)}(f_{(\omega\uparrow\uparrow 2)^5.(\omega^3.4 + 2) + \omega^3.5 + 3}(6))})$$

$$\beta(36.000005673906,6) = f_{f_{\omega^3.(f_{2}^{f_{f_{f_{\omega}(6)}(f_{\omega^2 + 4}^5(6))}(f_{\omega^2.2}^2(6))}(f_{\omega^3.4 + 1}^5(6)))}(f_{\omega^5.3 + \omega^3 + \omega.4 + 5}^5(6))}(f_{f_{(\omega\uparrow\uparrow 5)}(6) + 3}^4(f_{\varphi(1,0)}(6)))$$

$$\beta(36.0000056739061,6) = f_{5}^2(f_{f_{(\omega\uparrow\uparrow 5)}(6) + 4}(f_{\varphi(1,0)}(6))) + 5$$

$$\beta(36.000005972532,6) = f_{f_{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.2 + 4}.2 + 5}.5 + (\omega\uparrow\uparrow 2)^2.(\omega) + 5}.5 + (\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega.3}.(\omega^4.2 + \omega.4) + 3}.4 + (\omega\uparrow\uparrow 3)^{\omega}}(6)}(f_{\varphi(1,0)}(6))$$

$$\beta($$ TBA $$,6) = f_{(\omega\uparrow\uparrow 10)}(f_{\varphi(1,0)}(6))$$

$$\beta($$ TBA $$,6) = f_{(\omega\uparrow\uparrow 100)}(f_{\varphi(1,0)}(6))$$

$$\beta($$ TBA $$,6) = f_{(\omega\uparrow\uparrow 300)}(f_{\varphi(1,0)}(6))$$

$$\beta($$ TBA $$,6) = f_{(\omega\uparrow\uparrow f_{\omega}(6))}(f_{\varphi(1,0)}(6))$$

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

## Veblen Ordinal Examples using Version 4

These examples will illustrate the fine detail that can be accessed by the Beta Function.

First Set of Examples

Lets look at these two results:

$$\beta(30.1,3) = f_{\varphi(1,\varphi(2,\omega^2.2 + \omega.2 + 2)^{\omega.2} + \omega + 1,\varphi(1,(\omega\uparrow\uparrow 2)^{\omega^2 + 2} + \omega + 2,1)^{\varphi(1,1).((\omega\uparrow\uparrow 2)^{\omega.2 + 1}.2 + 1) + (\omega\uparrow\uparrow 2)}) + 1}(3)$$

$$\beta(30.4,3) = f_{(\varphi(1,(\varphi(\omega.2 + 1,\omega + 2)\uparrow\uparrow 2)^2.2 + 2,(\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega^2.2 + \omega + 2) + \omega^2.2 + 2)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega + 1}.(\omega^2 + 2) + (\omega\uparrow\uparrow 2)^2.2 + \omega}}(3)$$

And strip away the functions to focus on the root ordinals only:

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

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

And lets compare the parameters in the Veblen ordinal functions

No. $$\beta(30.1,3)$$ $$\beta(30.4,3)$$
1 $$1$$ $$1$$
2 $$\varphi(2,\omega^2.2 + \omega.2 + 2)^{\omega.2} + \omega + 1$$ $$(\varphi(\omega.2 + 1,\omega + 2)\uparrow\uparrow 2)^2.2 + 2$$
3 $$\varphi(1,(\omega\uparrow\uparrow 2)^{\omega^2 + 2} + \omega + 2,1)^{\varphi(1,1).((\omega\uparrow\uparrow 2)^{\omega.2 + 1}.2 + 1) + (\omega\uparrow\uparrow 2)}$$ $$(\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega^2.2 + \omega + 2) + \omega^2.2 + 2)\uparrow\uparrow 2$$

The second parameter is clearly larger for the second example when we focus on the root ordinal and compare the parameters of those Veblen ordinal functions:

No. $$\beta(30.1,3)$$ $$\beta(30.4,3)$$
2 $$\varphi(2,\omega^2.2 + \omega.2 + 2)^{\omega.2} + \omega + 1$$ $$(\varphi(\omega.2 + 1,\omega + 2)\uparrow\uparrow 2)^2.2 + 2$$
Root $$\varphi(2,\omega^2.2 + \omega.2 + 2)$$ $$\varphi(\omega.2 + 1,\omega + 2)$$
1 $$2$$ $$\omega.2 + 1$$
2 $$\omega^2.2 + \omega.2 + 2$$ $$\omega + 2$$

Second Set of Examples

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.

$$\beta(36.0000059725321,6) = f_{\omega}(f_{\varphi(1,0)}(6))$$

$$\beta(36.0000597253652,6) = f_{\varphi(1,0) + 1}(6)$$

$$\beta(36.0003583536775,6) = f_{\varphi(1,0) + \omega}(6)$$

$$\beta(36.0035836973001,6) = f_{\varphi(1,0).(\omega)}(6)$$

$$\beta(36.007167769,6) = f_{\varphi(1,0)^2}(6)$$

$$\beta(36.0358693359926,6) = f_{\varphi(1,0)^{f_{5}^{f_{3}^2(f_{5}(6)).(2^{f_{3}^5(6) + 10}) + f_{3}^2(f_{5}(6)).2 + 54}(f_{f_{2}^4(f_{4}^2(6))}^3(f_{\varphi(1,0)^4.(\omega^3 + 1) + \varphi(1,0).2 + 2}(6)))}}(f_{\varphi(1,0)^{\omega}.4 + (\omega\uparrow\uparrow 2)^2.4 + \omega^4.5}^5(6))$$

$$\beta(36.0358693359927,6) = f_{\varphi(1,0)^{\omega}.4 + (\omega\uparrow\uparrow 2)^2.4 + \omega^4.5 + 1}(6).4 + 2$$

$$\beta($$ TBA $$,6) = f_{(\varphi(1,0)\uparrow\uparrow 2)}(f_{\varphi(1,0)}(6))$$

$$\beta($$ TBA $$,6) = f_{(\varphi(1,0)\uparrow\uparrow 5)}(f_{\varphi(1,0)}(6))$$

$$\beta($$ TBA $$,6) = f_{\varphi(1,0)}^2(6)$$

$$\beta($$ TBA $$,6) = f_{\varphi(1,0)}^5(6)$$

$$\beta($$ TBA $$,6) = f_{\varphi(1,1)}(6)$$

$$\beta(36.4,6) = f_{(\omega\uparrow\uparrow 10)^{40}.(\omega^{f_{(\omega\uparrow\uparrow 2).(\omega^5.2 + \omega^4)}(6)})}(f_{\varphi(1,1)^{(\omega\uparrow\uparrow 4)^3.(\omega^2.3 + 4) + \omega^2.2 + 5}.(\varphi(1,0)^2 + (\omega\uparrow\uparrow 4)^{\omega^4.5 + 1} + (\omega\uparrow\uparrow 4)^{\omega^4 + \omega.4 + 5}.5 + \omega^2.5 + 3) + 2}^5(6))$$

$$\beta($$ TBA $$,6) = f_{\varphi(1,2)}(6)$$

$$\beta(37.5111,6) = f_{2}^{f_{(\omega\uparrow\uparrow 3)^{f_{2}^{12}(f_{3}^2(6))}}(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2).(\omega.3) + 3}.((\omega\uparrow\uparrow 2)^{\omega.3 + 3}.5 + (\omega\uparrow\uparrow 2)^5.4 + 2) + 5}^4(6))}(f_{\varphi(1,4)^{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3)^5.3 + 4}.2 + 4}.4}^4(6))$$

$$\beta($$ TBA $$,6) = f_{\varphi(1,5)}(6)$$

$$\beta($$ TBA $$,6) = f_{\varphi(1,\omega)}(6)$$

$$\beta(38.41,6) = f_{2}^{f_{2}^2(f_{\omega^5.4 + \omega^3.3 + \omega.2 + 1}^5(6)) + f_{\omega^4.4 + \omega^2.5 + \omega.4 + 2}(6) + 4}(f_{(\varphi(1,\omega^3 + \omega.3 + 2)\uparrow\uparrow 2)^3.5 + (\omega\uparrow\uparrow 2)^2.3 + (\omega\uparrow\uparrow 2).4 + \omega.4}^2(6)).32$$

$$\beta(38.512,6) = f_{\varphi(1,f_{f_{(\omega\uparrow\uparrow 3)^2.(\omega^{f_{3}^4(6) + 10}.(f_{(\omega\uparrow\uparrow 2)^3}(f_{(\omega\uparrow\uparrow 2)^4.3}^2(6))))}(f_{(\omega\uparrow\uparrow 4).(\omega^3) + 3}^2(6))}(f_{(\omega\uparrow\uparrow 4)^2}^3(6)))}(f_{\varphi(1,\omega^4.2 + 2)^3.3 + (\omega\uparrow\uparrow 4)^{\omega^3 + 4} + 3}^5(6))$$

$$\beta(38.521,6) = f_{(\varphi(1,\omega.(f_{5}^5(6) + 5) + 4)\uparrow\uparrow f_{2}^{124}(f_{3}^5(6)).(2^{f_{2}(6)}))}(f_{(\varphi(1,\omega^4.2 + \omega.5 + 4)\uparrow\uparrow 3)^5.((\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3)^{\omega^2.5}.((\omega\uparrow\uparrow 2)^5.(\omega^4.5 + \omega^2.2 + 4)) + 3}.(\omega^2.2 + \omega.4 + 1) + 3)}^4(6))$$

$$\beta(39.5131,6) = f_{5}^5(f_{f_{2}(6).16 + 1}^3(f_{\varphi(1,(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^2.(\omega^4 + 3) + 1}.(\omega.4 + 5) + 3)^3.(\omega.3) + \omega.4 + 1}^2(6))) + 3$$

$$\beta(40.11114,6) = f_{(\varphi(1,(\omega\uparrow\uparrow 5).(\omega^2 + 4) + (\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3).(\omega^2.4 + \omega.2 + 3)}.5 + \omega^2.5)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3).2 + (\omega\uparrow\uparrow 2)^5.(\omega^2.2 + 4) + \omega^5.3 + \omega.4}.5 + \omega^3.5}.(\varphi(1,3))}(6)$$

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

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

$$\beta($$ TBA $$,6) = f_{\varphi(2,0)}(6)$$

$$\beta($$ TBA $$,6) = f_{\varphi(1,\varphi(2,0) + 1)}(6)$$

$$\beta($$ TBA $$,6) = f_{\varphi(1,\varphi(2,0) + \omega)}(6)$$

## 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.

Base Input Value Version 3 Output
$$v = 4$$ From 15.98716909 $$f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 2}}}(4)$$
To 16.0000001 $$f_{\varphi(1,0)}(4)$$

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.

Base Input Value Version 3 Output
$$v = 5$$ From 16.71839813828 $$f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 4}}}(5)$$
To 16.71852 $$f_{(\omega\uparrow\uparrow 4)}(5)$$

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.

Base Input Value Version 3 Output
$$v = 5$$ 24.9999727132664 $$f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 3}.(\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.4 + 4) + (\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega}.(\omega^2.2 + 2) + \omega}}}(5)$$
From 24.9999727132665 $$f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.4 + 4}}}}(5) + 1$$
To 25.0000001 $$f_{\varphi(1,0)}(5)$$

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.

Base Input Value Version 3 Output
$$v = 6$$ 17.5809352157230 $$f_{4}^{f_{(\omega\uparrow\uparrow 2)^2.3 + \omega}(6)}(f_{5}^{3}(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5) + 2}.(\omega^5.3 + 2) + (\omega\uparrow\uparrow 3)^3.2 + (\omega\uparrow\uparrow 3)^2.3 + (\omega\uparrow\uparrow 2)^2.(\omega.4 + 4) + (\omega\uparrow\uparrow 2).2 + \omega^4 + \omega^3.5 + 2}^{2}(6)))$$
From 17.5809352157231 $$f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 5}}}(6)$$
To 17.58093631 $$f_{(\omega\uparrow\uparrow 4)}(6)$$

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.

Base Input Value Version 3 Output
$$v = 6$$ 25.1577760149164 $$f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 3) + 4}.3 + (\omega\uparrow\uparrow 3)^{\omega^3.2 + \omega.5}.((\omega\uparrow\uparrow 2).2 + 1) + (\omega\uparrow\uparrow 2)^{\omega^4.4 + 1}.(\omega^3.3 + 5) + 4}.((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^3.(\omega)})}(6)$$
From 25.1577760149165 $$f_{(\omega\uparrow\uparrow 4)^{f_{\omega^{f_{3}^{5}(6) + f_{1}^{2}(6) + 3}}(f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{\omega^2 + 5}}.3 + 2}^{5}(6))}}(f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 5}}}}^{2}(6))$$
To 25.15778 $$f_{(\omega\uparrow\uparrow 5)}(6)$$

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.

Base Input Value Version 3 Output
$$v = 6$$ 35.9999999533395 $$f_{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 4}.(\omega^5.5 + \omega^4.5 + \omega^3.5 + \omega^2.4) + 3}.2 + (\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^2.3 + \omega^2.4 + 1)}.(\omega^2.3 + \omega.2 + 1)}.4 + (\omega\uparrow\uparrow 3)^{\omega}}(6)$$
From 35.9999999533396 $$f_{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.5 + \omega^3.5 + \omega^2.5 + \omega.5 + 5}}}}.(\omega^2.5 + 3) + (\omega\uparrow\uparrow 3)^{\omega^3.2 + 1}.(\omega^4.4 + \omega^3.5 + \omega.3 + 4) + (\omega\uparrow\uparrow 3)^{\omega.4}.3 + (\omega\uparrow\uparrow 2)^{\omega^4}}(6)$$
To 36 $$f_{\varphi(1,0)}(6)$$

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

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

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

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

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

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

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

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

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

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

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

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

WORK IN PROGRESS

## Test Bed for Version 4

Below is the test bed and various results using version 4.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Next Attempt Base $$v = 3$$ on 26 May 2016

$$\beta(0,3) = 0$$

$$\beta(1,3) = 3$$

$$\beta(2,3) = 18$$

$$\beta(3,3) = f_{\omega}(3)$$

$$\beta(4,3) = f_{\omega^2}^2(3).(2097152) + 2$$

$$\beta(5,3) = f_{2}^{14}(f_{4}^{f_{2}^{12}(f_{\omega + 2}(3)).(2^{f_{2}(3).2 + 2}) + 1}(f_{\omega^2.2 + \omega + 2}^2(3))).2 + f_{\omega}^{f_{\omega}^2(3) + 2}(f_{\omega + 1}(3)).8$$

$$\beta(6,3) = f_{4}^{f_{2}^{f_{2}^2(3).32 + 2}(f_{3}^2(f_{4}(f_{(\omega\uparrow\uparrow 2)^2}(3)))).(2^{f_{\omega^{f_{2}(3) + 11}.2}(f_{(\omega\uparrow\uparrow 2) + 1}(3))})}(f_{(\omega\uparrow\uparrow 2)^2 + 1}(3))$$

$$\beta(7,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}.(\omega.3 + 18) + (\omega\uparrow\uparrow 2)^{f_{2}^2(f_{(\omega\uparrow\uparrow 2)^2 + (\omega\uparrow\uparrow 2).(\omega + 2) + \omega + 1}^2(3)) + f_{\omega^2.2}(3)}}(f_{(\omega\uparrow\uparrow 2)^{\omega + 1}}(3))$$

$$\beta(8,3) = f_{12}(f_{f_{2}(f_{(\omega\uparrow\uparrow 2)^{\omega.2}.(\omega^2) + 2}^2(3)).(2^{f_{2}(3)}) + 1}^2(f_{(\omega\uparrow\uparrow 2)^{\omega^2 + 1}.(\omega.2 + 1) + (\omega\uparrow\uparrow 2)^{\omega^2}.(\omega^2 + 1) + \omega.2 + 1}^2(3)))$$

$$\beta(10,3) = f_{(\varphi(1,\omega + 1)\uparrow\uparrow 14).(f_{\varphi(1,2)^{(\omega\uparrow\uparrow 2)^{\omega^2.2 + 2} + \omega^2 + \omega}.2 + (\omega\uparrow\uparrow 2)^2.2 + (\omega\uparrow\uparrow 2) + 1}^2(3))}(f_{(\varphi(1,\omega.2)\uparrow\uparrow 2)^2.(\omega^2 + \omega + 1) + 1}^2(3))$$

$$\beta(11,3) = f_{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)}^{f_{3}(f_{(\omega\uparrow\uparrow 2) + 2}(3))}(f_{(\omega\uparrow\uparrow 2) + 2}^2(3)))}(f_{(\varphi(1,(\varphi(1,\omega.2)\uparrow\uparrow 2)^2.((\omega\uparrow\uparrow 2)^{\omega + 2}.(\omega^2.2)) + \omega^2.2 + 2)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).2 + \omega^2 + \omega.2} + 2}(3))$$

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

$$\beta(15,3) = f_{\varphi(2,(\varphi(4,\omega^2 + 2)\uparrow\uparrow 2).(\omega^2 + 2) + \varphi(3,(\omega\uparrow\uparrow 2)^2 + (\omega\uparrow\uparrow 2) + \omega^2.2 + 2)^2.2 + \varphi(1,(\omega\uparrow\uparrow 2)^{\omega + 2} + 1)^2.(\varphi(1,\omega^2 + \omega.2) + 1)) + 1}(3)$$

$$\beta(17,3) = f_{\varphi(\omega.2,(\varphi(1,0)\uparrow\uparrow 2)^{\omega^2 + \omega + 2}.(\varphi(1,0)^{\omega^2 + 2} + 2) + (\varphi(1,0)\uparrow\uparrow 2)^{\omega.2 + 2}.(\varphi(1,0).(\omega^2 + \omega.2)))^2.((\varphi(1,(\omega\uparrow\uparrow 2).2)\uparrow\uparrow 2)^{(\varphi(1,1)\uparrow\uparrow 2)^2.(\varphi(1,1)^{(\omega\uparrow\uparrow 2)^2})})}(3)$$

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

$$\beta(20,3) = f_{\varphi(f_{2}^2(3).512 + f_{2}(3) + 3,\omega^3) + 1}(f_{(\varphi(\omega^2.2 + \omega + 1,(\omega\uparrow\uparrow 2)^2 + (\omega\uparrow\uparrow 2).(\omega^2.2 + \omega) + \omega^2)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).(\omega.2) + 1}.2 + 2}(3))$$

$$\beta(21,3) = f_{\varphi((\omega\uparrow\uparrow 2).2 + 2,1)^{(\omega\uparrow\uparrow 2)^2.2 + \omega + 2}.((\omega\uparrow\uparrow 2)^{\omega^2 + 1}.(\omega^2.2 + \omega + 1) + (\omega\uparrow\uparrow 2)^2.2 + \omega^2 + 1) + (\omega\uparrow\uparrow 2)^{\omega + 2} + \omega + 1}^2(3)$$

$$\beta(22,3) = f_{\varphi((\omega\uparrow\uparrow 2)^2,\omega + f_{(\varphi((\omega\uparrow\uparrow 2).2 + \omega^2.2 + 1,2)\uparrow\uparrow 2).(\varphi(1,(\omega\uparrow\uparrow 2)^2.(\omega^2) + (\omega\uparrow\uparrow 2).(\omega^2)) + 1)}(3)) + 1}(f_{\varphi((\omega\uparrow\uparrow 2)^2,\omega^2)^2.(\omega) + 2}(3))$$

$$\beta(28,3) = f_{\varphi(1,2,(\omega\uparrow\uparrow 2).2)^{\varphi((\omega\uparrow\uparrow 2)^2.(\omega^2 + \omega.2 + 1),\varphi(1,(\omega\uparrow\uparrow 2)^{\omega^2.2 + 1} + 1)^{\omega^2 + \omega + 2}.2 + \varphi(1,(\omega\uparrow\uparrow 2)^{\omega^2.2}.(\omega)) + 1) + 1}}(3)$$

$$\beta(33,3) = f_{(\varphi(2,1,1)\uparrow\uparrow 2)^{\varphi(\omega + 1,(\omega\uparrow\uparrow 2)^{\omega^2.2 + 2} + 1)^{\omega + 2}.2 + \omega^2 + 2}.(\varphi(1,0,\omega^2 + \omega.2 + 1)^{(\varphi(2,\omega^2.2 + 1)\uparrow\uparrow 2)})}(3)$$

$$\beta(39,3) = f_{\varphi(\omega,1,2)^{(\varphi(1,\omega^2 + 1)\uparrow\uparrow 2)^{\varphi(1,1).(\omega^2 + \omega + 2) + \omega.2 + 2}.(\varphi(1,\omega^2 + 1)^{(\omega\uparrow\uparrow 2)^2.(\omega^2.2 + \omega + 1) + 2}.((\omega\uparrow\uparrow 2)^2.2 + 2) + \omega) + (\varphi(1,1)\uparrow\uparrow 2)}}(3)$$

$$\beta(53,3) = f_{\varphi((\omega\uparrow\uparrow 2)^{\omega.2 + 1}.2 + \omega.2 + 2,\varphi(1,(\omega\uparrow\uparrow 2)^2 + 2)^{\omega^2.2 + 1}.(\omega^2 + \omega + 2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2 + 1) + 1,\omega^2.2 + \omega + 2)^{(\omega\uparrow\uparrow 2)}}(3)$$

$$\beta(55.7,3) = f_{\varphi((\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega + 1}.(\omega.2 + 1),0,1)^{\omega^2 + 1} + (\varphi(\omega.2,(\varphi(1,0)\uparrow\uparrow 2)^2.2 + (\omega\uparrow\uparrow 2)^2.2)\uparrow\uparrow 2)^{\omega + 2}.((\omega\uparrow\uparrow 2)^{\omega.2 + 1}.2 + 1) + (\omega\uparrow\uparrow 2)}(3)$$

$$\beta(56,3) = f_{(\varphi((\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega.2 + 1}.2 + (\omega\uparrow\uparrow 2)^{\omega.2 + 2}.2 + (\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega.2) + 1,2,2)\uparrow\uparrow 2) + (\omega\uparrow\uparrow 2)^{\omega^2 + \omega}.2 + \omega}(3)$$

$$\beta(56.1,3) = f_{(\varphi((\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega.2 + 2}.2 + \omega^2 + \omega.2,\omega^2.2 + 1,2)\uparrow\uparrow 2)^{\varphi(1,2,(\varphi(\omega.2 + 1,(\omega\uparrow\uparrow 2).(\omega.2 + 2) + 2)\uparrow\uparrow 2)^2.(\varphi(2,\varphi(2,(\omega\uparrow\uparrow 2)^2.(\omega)) + 1) + 1)) + 1}}(3)$$

$$\beta(57.5,3) = f_{\varphi(\varphi(1,2)^{\omega^2}.2 + (\varphi(1,1)\uparrow\uparrow 2)^{\omega + 1}.(\omega^2.2 + \omega.2 + 2) + \omega,\varphi(1,1)^2.(\omega^2.2 + \omega + 2) + \omega.2 + 2,(\varphi(1,1,1)\uparrow\uparrow 2)^{\varphi(1,2,\varphi(2,(\omega\uparrow\uparrow 2)^{\omega^2}) + 1) + 1}) + 1}(3)$$

$$\beta(58.9,3) = f_{\varphi((\varphi(1,(\omega\uparrow\uparrow 2).2 + \omega^2.2 + 1)\uparrow\uparrow 2)^2.((\omega\uparrow\uparrow 2)^2.(\omega^2 + \omega.2 + 2) + \omega) + \omega^2 + \omega + 1,(\omega\uparrow\uparrow 2)^{\omega + 2}.(\omega + 2) + \omega^2.2,(\omega\uparrow\uparrow 2)^{\omega^2}) + 1}(3)$$

$$\beta(59.4,3) = f_{(\varphi(\varphi(1,(\omega\uparrow\uparrow 2)^{\omega.2 + 1} + \omega^2 + 1)^2.((\omega\uparrow\uparrow 2)^2.(\omega.2 + 1) + \omega.2) + \omega^2 + 2,0,1)\uparrow\uparrow 2)^{\varphi(1,(\omega\uparrow\uparrow 2)^2.(\omega^2 + 2) + \omega^2) + 1}}(3)$$

$$\beta(63.1,3) = f_{\varphi(\varphi(2,2)^{(\omega\uparrow\uparrow 2)^{\omega + 2}.2 + (\omega\uparrow\uparrow 2).(\omega + 1) + \omega^2 + \omega + 1}.((\varphi(1,1)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^2 + \omega^2.2 + 1}.((\omega\uparrow\uparrow 2).(\omega + 2) + 1) + 1),2,1)^{(\omega\uparrow\uparrow 2)^{\omega^2}}}(3)$$

$$\beta(65.5,3) = f_{(\varphi((\varphi(2,\varphi(1,1)^{(\omega\uparrow\uparrow 2)^{\omega}.2 + 2} + \omega + 2)\uparrow\uparrow 2)^2 + \omega.2 + 1,1,(\omega\uparrow\uparrow 2)^{\omega.2}.2 + (\omega\uparrow\uparrow 2) + \omega + 1)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)}}(3)$$

$$\beta(67.6,3) = f_{\varphi((\varphi(\omega,2)\uparrow\uparrow 2)^{\omega} + \omega + 2,2,\varphi((\omega\uparrow\uparrow 2)^2.(\omega) + (\omega\uparrow\uparrow 2).2 + \omega^2.2 + \omega.2,\varphi(1,(\omega\uparrow\uparrow 2) + 1)^{\omega}.((\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega})) + 1) + 1}(3)$$

Second Attempt Base $$v = 3$$

$$\beta(10,3) = f_{(\varphi(1,\omega + 1)\uparrow\uparrow 14).(f_{\varphi(1,2)^{(\omega\uparrow\uparrow 2)^{\omega^2.2 + 2} + \omega^2 + \omega}.2 + (\omega\uparrow\uparrow 2)^2.2 + (\omega\uparrow\uparrow 2) + 1}^2(3))}(f_{(\varphi(1,\omega.2)\uparrow\uparrow 2)^2.(\omega^2 + \omega + 1) + 1}^2(3))$$

$$\beta(10.1,3) = f_{\varphi(1,\omega^2)^{(\omega\uparrow\uparrow 2)^{\omega.2}.(\omega^2 + 1) + (\omega\uparrow\uparrow 2) + \omega^2 + 2}.(\varphi(1,\omega.2)^2.(\omega^2 + \omega) + \omega^2.2) + (\varphi(1,\omega)\uparrow\uparrow 2).(\omega^2.2 + 1) + (\omega\uparrow\uparrow 2)}(3)$$

$$\beta(11.8,3) = f_{\varphi(1,\varphi(2,(\varphi(12,1)\uparrow\uparrow 2).((\omega\uparrow\uparrow 2)^{\omega.2} + \omega + 1) + \varphi(2,\omega.2)^2.(\varphi(2,\omega)^{\omega + 1}.((\omega\uparrow\uparrow 2)^{\omega^2 + 2}.2 + \omega.2))) + 1) + 1}(3)$$

$$\beta(13.11,3) = f_{\omega^7.(f_{(\omega\uparrow\uparrow 2)^{\omega} + 1}^{12}(f_{(\omega\uparrow\uparrow 2)^{\omega} + 2}^2(3)))}(f_{(\varphi(2,\omega + 2)\uparrow\uparrow 2)^2.2 + \varphi(1,\omega^2 + \omega)^{\omega.2 + 1} + (\varphi(1,\omega.2 + 2)\uparrow\uparrow 2).(\omega) + 1}(3))$$

$$\beta(13.3,3) = f_{(\varphi(2,\omega^2)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).2 + \omega.2 + 2}.(\omega + 2) + 1}^2(3).16 + f_{\varphi(2,1)^3 + 1}(f_{\varphi(2,1)^{(\omega\uparrow\uparrow 2)^{\omega}.(\omega.2 + 2)}.(\omega + 2) + (\omega\uparrow\uparrow 2)^{\omega^2.2 + 2}.2 + 1}^2(3))$$

$$\beta(13.4,3) = f_{2}^{f_{2}(f_{4}^2(f_{(\omega\uparrow\uparrow 2)^{\omega^2 + 1}.(\omega + 1) + \omega.2}^2(3))) + 3}(f_{\varphi(2,\omega^2 + \omega.2)^{\omega^2.2 + \omega.2 + 2} + (\omega\uparrow\uparrow 2)^2.2 + \omega.2 + 2}^2(3))$$

$$\beta(13.8,3) = f_{(\omega\uparrow\uparrow f_{f_{f_{5}^2(f_{\omega}(f_{\omega + 1}(3)))}(f_{\omega + 1}^2(3))}(f_{\omega.2}^2(f_{\omega.2 + 2}^2(3))))}(f_{\varphi(2,(\omega\uparrow\uparrow 2)^2.2 + 2)^{(\omega\uparrow\uparrow 2)^2.(\omega^2 + \omega) + \omega + 2}}^2(3))$$

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

$$\beta(19.5,3) = f_{\varphi(\omega^2.2 + 1,(\omega\uparrow\uparrow 2)^2.(\omega) + \omega^2 + 2)^2.(\varphi(\omega.2 + 1,(\omega\uparrow\uparrow 2)^2.(\omega^2.2))^{(\omega\uparrow\uparrow 2).(\omega) + 1}.(\varphi(\omega.2 + 1,2)^2 + \omega.2))}(3)$$

$$\beta(19.7,3) = f_{f_{\omega^5.12}(f_{\omega^{15} + 20}(f_{(\omega\uparrow\uparrow 2)^2.2}^2(3)))}(f_{\varphi(\omega^2.2 + 2,\omega^2 + \omega.2 + 2)^2}(3))$$

$$\beta(19.811,3) = f_{\varphi(\omega^2.2 + 2,(\varphi(1,f_{\omega + f_{5}^2(6).32}(f_{\omega.8}(f_{\omega^2 + 2}^3(6))).256 + 1)\uparrow\uparrow 2)) + 1}(3)$$

$$\beta(19.812,3) = f_{\varphi(\omega^2.2 + 2,\varphi(1,f_{\omega^3.9 + \omega.4 + f_{3}^4(f_{4}^2(6)).8}(f_{\omega^5.2 + \omega.3}^2(f_{\omega^5.2 + \omega.3 + 2}(6)))) + 1) + 1}(3)$$

$$\beta(19.82,3) = f_{\varphi(\omega^2.2 + 2,\varphi(1,f_{(\varphi(1,2)\uparrow\uparrow 5)^{\varphi(1,\varphi(2,2)^5.((\omega\uparrow\uparrow 2)^{\omega^2 + 5} + 4) + \varphi(2,2)^4.5 + (\omega\uparrow\uparrow 5).3 + \omega^4.2 + \omega^2) + 1}}(6)) + 1) + 1}(3)$$

$$\beta(20,3) = f_{\varphi(f_{2}^2(3).512 + f_{2}(3) + 3,\omega^3) + 1}(f_{(\varphi(\omega^2.2 + \omega + 1,(\omega\uparrow\uparrow 2)^2 + (\omega\uparrow\uparrow 2).(\omega^2.2 + \omega) + \omega^2)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).(\omega.2) + 1}.2 + 2}(3))$$

$$\beta(20.2,3) = f_{f_{\omega.2 + f_{(\omega\uparrow\uparrow 2)^{\omega + 1}.2 + (\omega\uparrow\uparrow 2)^{\omega}.2}(3)}(f_{(\omega\uparrow\uparrow 2)^{\omega^2 + \omega.2 + 1} + \omega^2}(3))}(f_{(\varphi(\omega^2.2 + \omega.2,2)\uparrow\uparrow 2)^2.(\omega^2) + \omega.2 + 1}(3))$$

$$\beta(20.3,3) = f_{13}^{f_{2}(f_{f_{\omega.2 + 2}^2(3).2 + 7}^{f_{2}(3) + 1}(f_{\omega^2.2 + 1}^2(3))).4}(f_{\varphi(\omega^2.2 + \omega.2 + 1,1)^{(\omega\uparrow\uparrow 2).2 + 2}.2 + 1}^2(3))$$

$$\beta(20.5,3) = f_{\varphi(\omega^2.2 + \omega.2 + 2,\varphi(1,4)^2.(\varphi(1,2)^{(\omega\uparrow\uparrow 2)^{\omega}.2 + (\omega\uparrow\uparrow 2) + \omega.2 + 2}.2 + (\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega.2 + 2}.(\omega + 2) + 2) + \varphi(1,4).((\varphi(1,3)\uparrow\uparrow 2).(\varphi(1,3)^{\omega^2}))) + 1}(3)$$

$$\beta(22.2,3) = f_{\varphi((\omega\uparrow\uparrow 2)^2 + \omega.2 + 1,(\omega\uparrow\uparrow 2)^{\omega^2.2}.(\omega.2 + 1) + (\omega\uparrow\uparrow 2)^2 + 1)^{(\omega\uparrow\uparrow 2)^{\omega + 1} + 2} + \omega.2}(3).4 + f_{3}(f_{\omega.2 + 1}(3))$$

$$\beta(23.6,3) = f_{\varphi((\omega\uparrow\uparrow 2)^{\omega} + (\omega\uparrow\uparrow 2)^2.(\omega.2) + \omega^2 + 1,\omega)^{\omega}.2 + \varphi(\omega.2 + 1,\varphi(1,0)^{\omega.2 + 1}.((\omega\uparrow\uparrow 2).(\omega.2 + 2) + \omega.2)) + 1}(3)$$

$$\beta(24.51,3) = f_{\varphi((\omega\uparrow\uparrow 2)^{\omega.2}.(\omega + 1),(\omega\uparrow\uparrow 2) + \omega^2.2)^{\omega^2.2 + \omega + 2}.(\omega^2.2) + \varphi(1,\varphi(2,2)^{(\omega\uparrow\uparrow 2).(\omega^2 + 2) + 2} + \omega.2) + 1}(3)$$

$$\beta(24.6,3) = f_{(\varphi((\omega\uparrow\uparrow 2)^{\omega.2}.(\omega^2 + \omega.2 + 2),(\omega\uparrow\uparrow 2).2 + \omega.2 + 2)\uparrow\uparrow 2)^2.(\omega^2) + 1}(3).2 + 2$$

$$\beta(27.6,3) = f_{(\omega\uparrow\uparrow 5)^{\omega^{f_{\varphi(1,0,\varphi(2,\omega^2 + \omega + 2)^{\omega} + 1) + 1}(3)}}}(f_{(\varphi(1,1,(\omega\uparrow\uparrow 2)^{\omega}.(\omega + 1) + \omega + 1)\uparrow\uparrow 2).(\omega)}(3))$$

$$\beta(28.111,3) = f_{\varphi(1,2,\varphi(\omega.2,\omega.2 + 2).(\varphi(1,\omega.2)^{(\omega\uparrow\uparrow 2).(\omega^2.2) + 2}.(\omega.2 + 2) + 1) + \varphi(\omega,\varphi(1,0) + \omega^2 + \omega.2 + 1)^{(\omega\uparrow\uparrow 2)^{\omega.2}}) + 1}(3)$$

$$\beta(28.8,3) = f_{\varphi(1,\omega^2.2 + 1,\varphi(\omega^2.2 + 1,1)^2 + (\omega\uparrow\uparrow 2)^{\omega^2.2}.2 + (\omega\uparrow\uparrow 2)^2 + \omega.2 + 2)^{(\omega\uparrow\uparrow 2)^2.(\omega.2 + 2)}.2 + \omega^2}(3)$$

$$\beta(29.1,3) = f_{(\varphi(1,(\omega\uparrow\uparrow 2)^2 + 2,1)\uparrow\uparrow 2)^{\varphi(1,2,(\omega\uparrow\uparrow 2)^{\omega}.(\omega.2) + (\omega\uparrow\uparrow 2).(\omega + 2))^{\varphi(\omega^2 + \omega,(\omega\uparrow\uparrow 2)^{\omega.2}.2 + \omega^2.2 + \omega.2) + 1}}}(3)$$

$$\beta(29.2,3) = f_{(\varphi(1,(\omega\uparrow\uparrow 2)^2.(\omega.2 + 1) + \omega + 2,\varphi(2,2)^2.(\omega^2.2 + 1) + (\omega\uparrow\uparrow 2)^{\omega^2 + \omega + 1}.(\omega.2))\uparrow\uparrow 2)^{\varphi(2,\varphi(f_{2}(3) + 2,2)^{(\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega)}) + 1}}(3)$$

$$\beta(29.21,3) = f_{\varphi(1,(\omega\uparrow\uparrow 2)^2.(\omega.2 + 2) + (\omega\uparrow\uparrow 2).(\omega^2.2 + \omega + 2),\varphi((\omega\uparrow\uparrow 2).(\omega^2.2) + \omega.2,(\omega\uparrow\uparrow 2)^2.2 + 2)^{\omega}.(\varphi(\omega^2 + 1,1)^2)) + 1}(3)$$

$$\beta(29.3,3) = f_{(\varphi(1,(\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega.2 + 2) + 2,1)\uparrow\uparrow 2)^{(\varphi(1,2,\varphi(\omega.2,\omega.2 + 1)^{\omega^2 + 1}.((\omega\uparrow\uparrow 2).2 + \omega^2) + (\omega\uparrow\uparrow 2).(\omega^2 + 2) + 1)\uparrow\uparrow 2)}}(3)$$

$$\beta(29.41,3) = f_{\varphi(1,(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega.2) + \omega^2 + \omega + 1,\varphi(\omega^2.2,2)^{(\omega\uparrow\uparrow 2)^{\omega + 1}.2 + (\omega\uparrow\uparrow 2).2 + 1}.2 + \varphi(1,(\omega\uparrow\uparrow 2)^2 + \omega^2.2) + 1) + 1}(3)$$

$$\beta(30.1,3) = f_{\varphi(1,\varphi(2,\omega^2.2 + \omega.2 + 2)^{\omega.2} + \omega + 1,\varphi(1,(\omega\uparrow\uparrow 2)^{\omega^2 + 2} + \omega + 2,1)^{\varphi(1,1).((\omega\uparrow\uparrow 2)^{\omega.2 + 1}.2 + 1) + (\omega\uparrow\uparrow 2)}) + 1}(3)$$

$$\beta(30.4,3) = f_{(\varphi(1,(\varphi(\omega.2 + 1,\omega + 2)\uparrow\uparrow 2)^2.2 + 2,(\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega^2.2 + \omega + 2) + \omega^2.2 + 2)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega + 1}.(\omega^2 + 2) + (\omega\uparrow\uparrow 2)^2.2 + \omega}}(3)$$

$$\beta(33,3) = f_{(\varphi(2,1,1)\uparrow\uparrow 2)^{\varphi(\omega + 1,(\omega\uparrow\uparrow 2)^{\omega^2.2 + 2} + 1)^{\omega + 2}.2 + \omega^2 + 2}.(\varphi(1,0,\omega^2 + \omega.2 + 1)^{(\varphi(2,\omega^2.2 + 1)\uparrow\uparrow 2)})}(3)$$

$$\beta(33.6,3) = f_{(\varphi(2,2,\omega^2 + \omega + 1)\uparrow\uparrow 2)^2.(\varphi(1,\varphi(1,(\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2.2 + 2) + \omega^2.2 + 2)^2.(\omega.2) + 1)^{(\omega\uparrow\uparrow 2).(\omega^2.2 + 2) + \omega.2})}(3)$$

$$\beta(34.2,3) = f_{\varphi(2,\omega.2,\varphi(f_{\omega}(3) + 9,0,1)^{\varphi(\omega.2,(\varphi(1,0)\uparrow\uparrow 2).2 + \varphi(1,0)^{\omega^2 + \omega}.((\omega\uparrow\uparrow 2)^2 + (\omega\uparrow\uparrow 2).2 + \omega.2)) + 1}) + 1}(3)$$

$$\beta(34.3,3) = f_{\varphi(2,\omega.2 + 2,(\omega\uparrow\uparrow 2)^{\omega^2} + 1)^{\varphi((\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega + 1}.(\omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega}.(\omega.2) + 2,\varphi(5,(\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega) + \omega.2) + 1) + 1}}(3)$$

$$\beta(34.8,3) = f_{(\varphi(2,(\omega\uparrow\uparrow 2).2 + 2,(\omega\uparrow\uparrow 2)^{\omega + 2}.(\omega^2.2) + \omega.2 + 1)\uparrow\uparrow 2)^{\omega.2 + 1} + \varphi(\omega^2.2 + \omega + 1,(\omega\uparrow\uparrow 2)^{\omega + 2}.(\omega^2) + (\omega\uparrow\uparrow 2)) + 1}(3)$$

$$\beta(35.4,3) = f_{\varphi(2,(\omega\uparrow\uparrow 2)^{\omega^2 + \omega + 2} + \omega.2 + 2,(\omega\uparrow\uparrow 2).(\omega^2 + \omega) + \omega^2 + 2)^{(\omega\uparrow\uparrow 2)^{\omega^2.2} + (\omega\uparrow\uparrow 2)^{\omega^2 + 2}.2}}(3)$$

$$\beta(35.511,3) = f_{\varphi(2,(\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega.2}.2,\varphi(2,2,(\varphi(\omega.2 + 1,\omega.2 + 1)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega^2.2 + 2}.2 + (\omega\uparrow\uparrow 2)^{\omega^2.2}.(\omega^2.2 + 1) + \omega}) + 1) + 1}(3)$$

$$\beta(35.512,3) = f_{(\varphi(2,(\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega.2}.2 + \omega.2 + 2,\omega.2)\uparrow\uparrow 2)^{\omega^2}.(\omega.2) + (\varphi(2,\omega.2 + 2,(\omega\uparrow\uparrow 2)^2 + 2)\uparrow\uparrow 2)^{\omega}}(3)$$

$$\beta(39.3,3) = f_{(\varphi(\omega,\varphi(2,(\omega\uparrow\uparrow 2)^{\omega^2 + 1}.2 + \omega.2)^{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega + 1)}.((\omega\uparrow\uparrow 2).2),(\omega\uparrow\uparrow 2)^{\omega^2.2 + 1}.(\omega.2))\uparrow\uparrow 2)^2}(3)$$

$$\beta(40.5,3) = f_{\varphi(\omega + 2,(\varphi(2,1)\uparrow\uparrow 2).2 + \omega^2.2,\varphi(\omega,1).((\omega\uparrow\uparrow 2)^2.(\omega^2.2 + 1) + (\omega\uparrow\uparrow 2) + 1))^{\omega + 2}.(\omega + 1) + 1}^2(3)$$

$$\beta(41.3,3) = f_{(\varphi(\omega.2,(\varphi(1,0,2)\uparrow\uparrow 2).(\omega^2 + 1) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2.2 + 2),\varphi(1,1)^2.2 + 1)\uparrow\uparrow 2)^{\varphi(1,2,\varphi(1,3,\varphi(2,\varphi(2,\omega.2) + 1) + 1) + 1) + 1}}(3)$$

$$\beta(42.2,3) = f_{\varphi(\omega.2 + 2,\omega + 1,(\varphi((\omega\uparrow\uparrow 2)^2.(\omega + 1) + \omega^2 + \omega,\omega^2.2 + \omega.2 + 2)\uparrow\uparrow 2)^2 + 2)^{(\omega\uparrow\uparrow 2).2 + \omega^2}.2 + \omega^2}(3)$$

$$\beta(43.1,3) = f_{\varphi(\omega^2 + 1,\omega.2 + 1,\varphi(1,3,(\omega\uparrow\uparrow 2)^{\omega.2}.(\omega.2 + 2) + \omega^2 + \omega + 1)^{\varphi(2,(\omega\uparrow\uparrow 2).(\omega + 1) + \omega^2.2)^2 + \varphi(1,(\omega\uparrow\uparrow 2)) + 1}) + 1}(3)$$

$$\beta(46.5,3) = f_{\varphi(\omega^2.2 + \omega.2 + 1,\varphi(1,1)^2.(\omega^2 + 2) + (\omega\uparrow\uparrow 2)^{\omega + 2}.2 + 2,(\varphi(\omega^2.2 + \omega + 2,\omega^2 + 1)\uparrow\uparrow 2) + 1)^{(\omega\uparrow\uparrow 2)^{\omega.2}.(\omega^2.2) + (\omega\uparrow\uparrow 2)}}(3)$$

$$\beta(46.9,3) = f_{(\varphi((\omega\uparrow\uparrow 2) + 1,\varphi(1,1).2 + (\omega\uparrow\uparrow 2)^{\omega^2 + 2} + 1,\varphi(1,2)^2)\uparrow\uparrow 2)^{\varphi(1,0,\varphi(1,\omega.2)^{\omega}.(\omega^2.2 + 1) + (\omega\uparrow\uparrow 2)) + 1}}(3)$$

$$\beta(48.1,3) = f_{\varphi((\omega\uparrow\uparrow 2).(\omega + 2) + \omega^2,1,\varphi(4,\omega^2 + 2,1)^{\varphi(\omega.2 + 1,(\omega\uparrow\uparrow 2).(\omega^2 + \omega + 2) + \omega.2 + 1)^2.(\varphi(\omega.2 + 1,\omega^2) + 1)}) + 1}(3)$$

$$\beta(50.8,3) = f_{\varphi((\omega\uparrow\uparrow 2)^2.(\omega^2 + 2) + (\omega\uparrow\uparrow 2) + 2,(\omega\uparrow\uparrow 2) + \omega.2,(\omega\uparrow\uparrow 2).2 + \omega)^{(\omega\uparrow\uparrow 2)}.2 + \omega^2 + \omega.2}(3)$$

$$\beta(51.4,3) = f_{\varphi((\omega\uparrow\uparrow 2)^{\omega}.2 + \omega + 2,1,1)^{\varphi(2,(\omega\uparrow\uparrow 2)^2.2 + (\omega\uparrow\uparrow 2).(\omega^2 + \omega + 1) + \omega^2 + \omega)^{(\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega + 2)}.2 + \omega.2 + 2}.(\omega)}(3)$$

Next Attempt Base $$v = 3$$ and Sequence Depth $$d = 50$$ on 28 May 2016

$$\beta(0,3) = 0$$

$$\beta(1,3) = 3$$

$$\beta(2,3) = 18$$

$$\beta(3,3) = f_{\omega}(3)$$

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

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

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

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

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

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

$$\beta(10,3) = f_{\varphi(1,\omega^{\varphi(2,(\omega\uparrow\uparrow f_{\varphi((\varphi(1,2)\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).(\varphi(\varphi((\omega\uparrow\uparrow 6),0) + 1,0,0) + 1)}}(3))}(3)),0) + 1},0,0) + 1}(3))) + 1}) + 1}(3)$$

$$\beta(11,3) = f_{\varphi(1,\varphi(1,\omega^{\varphi(2,(\varphi(1,2)\uparrow\uparrow 2)^{\varphi((\varphi(1,0)\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{\varphi(\omega^{\omega^{\varphi(\varphi(1,0) + 1,0,0) + 1}},0,0) + 1}(3)),0,0) + 1},0,0) + 1}) + 1}) + 1) + 1}(3)$$

$$\beta(12,3) = f_{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^2.((\varphi(1,1)\uparrow\uparrow 19)^{\varphi((\omega\uparrow\uparrow 3),0,0) + 1})}(3)),0) + 1}}(3)$$

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

$$\beta(14,3) = f_{\varphi(2,(\omega\uparrow\uparrow f_{\varphi(\omega^{\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{(\varphi(2,\omega^2.2 + 1)\uparrow\uparrow 2)^{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow f_{\varphi(2,(\omega\uparrow\uparrow f_{\omega}(3))) + 1}(3)),0) + 1}(3)),0) + 1,0,0) + 1}}(3)),0) + 1},0,0) + 1},0) + 1}(3))) + 1}(3)$$

$$\beta(15,3) = f_{\varphi(2,\varphi(4,(\omega\uparrow\uparrow 2)^2.(\varphi((\omega\uparrow\uparrow 4)^{\varphi(\varphi((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 4).(\varphi(\varphi(\omega,0) + 1,0,0) + 1)}(3))},0) + 1,0,0) + 1},0) + 1)) + 1) + 1}(3)$$

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

$$\beta(17,3) = f_{\varphi(\omega^{\varphi(\omega^2.(\varphi(2,(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow f_{\varphi(\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi(1,1,\varphi(1,(\varphi(2,0)\uparrow\uparrow f_{\omega}(3)) + 1) + 1) + 1}(3))}(3)),0) + 1,0,0) + 1}(3))}) + 1),0) + 1},0) + 1}(3)$$

$$\beta(18,3) = f_{\varphi((\omega\uparrow\uparrow 2).((\omega\uparrow\uparrow 4)^{\omega^{\varphi(1,1,(\varphi(2,2)\uparrow\uparrow 2)^{\varphi((\varphi(2,0)\uparrow\uparrow f_{\varphi(\varphi(1,0) + 1,0,0) + 1}(3)) + 1,0,0) + 1}) + 1}}),0) + 1}(3)$$

$$\beta(19,3) = f_{\varphi((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow f_{\varphi(1,(\omega\uparrow\uparrow 2)^{\varphi(2,\varphi(1,\varphi(1,1)^{\omega^{\varphi((\omega\uparrow\uparrow 18),0) + 1}}) + 1) + 1}) + 1}(3))},0) + 1}(3)$$

$$\beta(20,3) = f_{\varphi((\omega\uparrow\uparrow 2)^{\varphi(2,\varphi(1,\varphi(1,\varphi(2,(\varphi(12,\omega.2 + 2)\uparrow\uparrow 6) + 1) + 1) + 1) + 1,0) + 1},0) + 1}(3)$$

$$\beta(21,3) = f_{\varphi((\omega\uparrow\uparrow 6)^2.(\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 9)^{\varphi(1,\varphi(2,\varphi(2,2,0) + 1,0) + 1,0) + 1}}(3)),0,0) + 1),0) + 1}(3)$$

$$\beta(22,3) = f_{\varphi((\omega\uparrow\uparrow f_{\omega.(\varphi(1,\varphi(2,(\omega\uparrow\uparrow 2)^{\varphi(1,\varphi(1,(\omega\uparrow\uparrow 2).(\varphi(2,\varphi(f_{2}^2(3),0) + 1) + 1),0) + 1,0) + 1}) + 1) + 1)}(3)),0) + 1}(3)$$

$$\beta(23,3) = f_{\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^{\omega^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\varphi(2,1)^{\varphi((\varphi(2,0)\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{2}(3))}(3)) + 1,0,0) + 1}}}}}}(3))}(3)),0) + 1}(3)$$

$$\beta(24,3) = f_{\varphi((\omega\uparrow\uparrow f_{\varphi(2,1)^{\varphi(\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^{\varphi(2,(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi(2,\varphi(3,(\omega\uparrow\uparrow 2)^2.((\omega\uparrow\uparrow f_{\varphi(1,0,0) + 1}(3))),0) + 1,0) + 1}(3))}(3))) + 1}}(3)),0) + 1,0,0) + 1}}(3)),0) + 1}(3)$$

$$\beta(25,3) = f_{\varphi((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow f_{(\varphi(1,2)\uparrow\uparrow 2)^{(\varphi(2,0)\uparrow\uparrow f_{\varphi(1,1)^{(\varphi(1,0)\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi(\varphi(1,0) + 1,0,0) + 1}(3))}(3)) + 1}}(3)) + 1}}(3)),0) + 1}(3)),0) + 1}(3)$$

$$\beta(26,3) = f_{\varphi((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow f_{2}^2(3).2 + f_{2}^2(3) + 5)^2.(\varphi((\omega\uparrow\uparrow f_{\varphi(1,\omega) + 1}(3)),0) + 1)},0,0) + 1}(3)),0) + 1}(3)$$

$$\beta(27,3) = f_{\varphi(1,0,0) + 1}(3)$$

$$\beta(28,3) = f_{\varphi(1,2,(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^{\varphi(\varphi((\omega\uparrow\uparrow 2)^{\varphi(1,1)^{(\omega\uparrow\uparrow 2)^{\varphi(1,(\omega\uparrow\uparrow 3)) + 1}}},0) + 1,0,0) + 1}}(3)),0) + 1}(3))}(3))}(3))}) + 1}(3)$$

$$\beta(29,3) = f_{\varphi(1,(\omega\uparrow\uparrow 11)^{\varphi(2,(\omega\uparrow\uparrow 2)^2.2 + \varphi(\omega^{\varphi(\omega^{\varphi(2,\varphi(1,\omega) + 1) + 1},0) + 1},0) + 1) + 1},0) + 1}(3)$$

$$\beta(30,3) = f_{\varphi(1,(\varphi(2,1)\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow f_{\varphi(1,(\omega\uparrow\uparrow f_{2}^2(3).4 + 5)^{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi(\omega^{\varphi(1,0) + 1},0) + 1}(3)),0) + 1,0,0) + 1}) + 1}(3)),0) + 1}(3)) + 1,0) + 1}(3)$$

$$\beta(31,3) = f_{\varphi(1,\varphi(1,0,(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi(\omega^{(\varphi(2,0)\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi(1,\varphi(1,0,\varphi((\omega\uparrow\uparrow f_{\omega^{\varphi((\omega\uparrow\uparrow 3),0) + 1}}(3)),0) + 1) + 1,0) + 1}(3))}(3)) + 1},0) + 1}(3))}(3))) + 1,0) + 1}(3)$$

$$\beta(32,3) = f_{\varphi(1,\varphi(2,(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 5)^{\varphi((\omega\uparrow\uparrow f_{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi(\omega^{\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{2}(3) + 4)^{\varphi(1,0) + 1}}(3)),0) + 1},0,0) + 1}(3)),0) + 1,0,0) + 1}(3)),0,0) + 1}}(3)),0) + 1,0) + 1}(3)$$

$$\beta(33,3) = f_{(\varphi(2,1,1)\uparrow\uparrow f_{\varphi(\omega^{\omega^{(\omega\uparrow\uparrow f_{\omega^2.(\varphi((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^{\varphi(\omega^{\varphi(1,(\omega\uparrow\uparrow 3)) + 1},0) + 1}}(3))},0) + 1)}(3))}},0) + 1}(3)) + 1}(3)$$

$$\beta(34,3) = f_{\varphi(2,\omega^2.2 + 2,(\omega\uparrow\uparrow f_{\varphi(1,2,\varphi(2,0).(\varphi(1,(\omega\uparrow\uparrow 2)^{\varphi(\omega^{\omega},0) + 1}) + 1)) + 1}(3))) + 1}(3)$$

$$\beta(35,3) = f_{\varphi(2,(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2).(\varphi(2,\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{\varphi(\varphi((\omega\uparrow\uparrow f_{\omega^{(\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3)},0) + 1}(3))}}(3)),0) + 1,0,0) + 1}(3)),0) + 1},0) + 1,0) + 1)}(3))}}(3)),0) + 1}(3)$$

$$\beta(36,3) = f_{\varphi(2,\varphi(2,1)^{\varphi(\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^{\varphi(2,(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi(2,\varphi(3,(\omega\uparrow\uparrow 2)^2.((\omega\uparrow\uparrow 2)),0) + 1,0) + 1}(3))}(3))) + 1}}(3)),0) + 1,0,0) + 1},0) + 1}(3)$$

$$\beta(37,3) = f_{\varphi(2,\varphi((\omega\uparrow\uparrow f_{\varphi(1,(\omega\uparrow\uparrow 14)^{\omega^{(\omega\uparrow\uparrow 2)^2.((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^2.((\omega\uparrow\uparrow 2)^{\omega^{\varphi((\omega\uparrow\uparrow 3),0,0) + 1}}),0,0) + 1}(3)))}}) + 1}(3)),0) + 1,0) + 1}(3)$$

$$\beta(38,3) = f_{\varphi(2,\varphi(2,\varphi(2,\varphi(1,\varphi(2,\varphi(1,\varphi(2,\varphi(3,(\omega\uparrow\uparrow f_{\omega^{\omega.(\varphi(1,0) + 1)}}(3))) + 1) + 1) + 1,0) + 1,0) + 1,0) + 1,0) + 1,0) + 1}(3)$$

$$\beta(39,3) = f_{\varphi(\omega.2,(\varphi(1,0,0)\uparrow\uparrow 2)^{\omega^{(\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^2.(\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2).(\varphi((\omega\uparrow\uparrow 3),0,0) + 1)}(3)),0) + 1),0) + 1}(3))}},0) + 1}(3)$$

$$\beta(40,3) = f_{\varphi(\omega^{\omega^{\varphi((\omega\uparrow\uparrow 4)^{\varphi((\omega\uparrow\uparrow f_{2}^2(3) + 13)^{(\omega\uparrow\uparrow f_{\varphi(\omega^{(\omega\uparrow\uparrow 2).(\varphi(1,0,0) + 1)},0,0) + 1}(3))},0) + 1},0) + 1}},0,0) + 1}(3)$$

$$\beta(41,3) = f_{\varphi(\omega^{\varphi(2,0)^2.(\varphi((\omega\uparrow\uparrow 8)^{\varphi((\omega\uparrow\uparrow 2)^{\omega^{(\varphi(1,0)\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow f_{\varphi(1,0,0) + 1}(3)),0) + 1}(3)) + 1}},0) + 1},0) + 1)},0,0) + 1}(3)$$

$$\beta(42,3) = f_{\varphi(\omega^{\varphi(2,\varphi(2,2,\varphi(2,\omega^{(\varphi(1,1)\uparrow\uparrow f_{\varphi(\omega^{\varphi(1,\varphi((\omega\uparrow\uparrow 5)^{\varphi(1,0) + 1},0) + 1,0) + 1},0,0) + 1}(3)) + 1}) + 1) + 1,0) + 1},0,0) + 1}(3)$$

$$\beta(43,3) = f_{\varphi((\omega\uparrow\uparrow 2).(\varphi((\omega\uparrow\uparrow 2)^{\omega^{(\omega\uparrow\uparrow 9)^{(\omega\uparrow\uparrow 2)^{\omega^{(\omega\uparrow\uparrow 2)^{\varphi(1,0)^{\varphi(1,0,0) + 1} + 1}}}}}},0) + 1),0,0) + 1}(3)$$

$$\beta(44,3) = f_{\varphi((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\varphi(1,\varphi(2,(\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 19)^{\varphi(1,\omega) + 1},0,0) + 1}(3))) + 1) + 1}}},0,0) + 1}(3)$$

$$\beta(45,3) = f_{\varphi((\omega\uparrow\uparrow 2)^{(\varphi(2,2)\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{\varphi(1,(\omega\uparrow\uparrow 2)^{\omega^2.(\varphi(1,0)^{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi(1,\varphi(1,0,0) + 1,0) + 1}(3)),0) + 1,0,0) + 1})}) + 1}(3)),0,0) + 1}},0,0) + 1}(3)$$

$$\beta(46,3) = f_{\varphi((\omega\uparrow\uparrow 2)^{\varphi(2,\varphi((\omega\uparrow\uparrow 2)^{(\varphi(2,1)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 5).((\varphi(1,1)\uparrow\uparrow f_{\varphi(2,1,\varphi(1,0) + 1) + 1}(3)) + 1)}},0) + 1,0) + 1},0,0) + 1}(3)$$

$$\beta(47,3) = f_{\varphi((\omega\uparrow\uparrow 3)^{\varphi(\varphi((\omega\uparrow\uparrow f_{2}^2(3).2 + 1).(\varphi((\omega\uparrow\uparrow f_{(\varphi(1,0)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{\varphi(1,0) + 1}(3)),0,0) + 1}}}(3)),0,0) + 1),0) + 1,0,0) + 1},0,0) + 1}(3)$$

$$\beta(48,3) = f_{\varphi((\omega\uparrow\uparrow 14)^{(\omega\uparrow\uparrow f_{\varphi(\varphi(2,\omega^{\varphi(2,1,\omega^{\varphi((\omega\uparrow\uparrow f_{\varphi(2,\varphi(12,0,0) + 1,0) + 1}(3)),0,0) + 1}) + 1}) + 1,0,0) + 1}(3))},0,0) + 1}(3)$$

$$\beta(49,3) = f_{\varphi((\omega\uparrow\uparrow f_{\omega.(\varphi(\omega^2.((\omega\uparrow\uparrow 5).(\omega^{\varphi(\omega^{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^2.(\omega^{\varphi(1,0,0) + 1})}(3))},0) + 1})),0) + 1)}(3)),0,0) + 1}(3)$$

$$\beta(50,3) = f_{\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^{\varphi(\omega^{(\varphi(1,2)\uparrow\uparrow f_{\varphi(\omega^2.(\varphi(\omega^{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi(1,\varphi(1,\varphi(1,1) + 1) + 1) + 1}(3)),0) + 1,0,0) + 1},0,0) + 1),0,0) + 1}(3)) + 1},0,0) + 1}}(3)),0,0) + 1}(3)$$

$$\beta(51,3) = f_{\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi(1,\varphi(2,1,\varphi(5,2,\varphi(1,\varphi(\omega.(\varphi((\omega\uparrow\uparrow f_{\varphi(\varphi(2,\omega^{\omega}) + 1,0,0) + 1}(3)),0) + 1),0) + 1,0) + 1) + 1) + 1,0) + 1}(3))}(3)),0,0) + 1}(3)$$

$$\beta(52,3) = f_{\varphi((\omega\uparrow\uparrow f_{(\varphi(2,2)\uparrow\uparrow 2)^{\varphi(1,(\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 8)^{(\omega\uparrow\uparrow 16)^{\varphi(1,0) + 1}},0,0) + 1}(3)),0) + 1}}(3)),0,0) + 1}(3)$$

$$\beta(53,3) = f_{\varphi((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow 2)^{\varphi(\omega^{\varphi(1,\varphi(1,\omega^{\varphi((\omega\uparrow\uparrow f_{2}^2(3).16 + 1)^{\varphi(\omega,0,0) + 1},0) + 1}) + 1) + 1},0,0) + 1},0,0) + 1},0) + 1}(3)),0,0) + 1}(3)$$

$$\beta(54,3) = f_{\varphi((\omega\uparrow\uparrow f_{\varphi(1,\varphi(1,\varphi(1,\varphi(\omega^{\varphi(1,1,(\varphi(2,2)\uparrow\uparrow 2)^{\varphi((\varphi(2,0)\uparrow\uparrow f_{\varphi(\varphi(1,0) + 1,0,0) + 1}(3)) + 1,0,0) + 1}) + 1},0) + 1,0) + 1,0) + 1,0) + 1}(3)),0,0) + 1}(3)$$

$$\beta(55,3) = f_{\varphi((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow f_{\omega + (\omega\uparrow\uparrow 2)^{\varphi(f_{\omega}(3),0) + 1}}(f_{\omega^2.2}(f_{\omega^2.2 + 2}(3)))),0,0) + 1}(3)),0,0) + 1}(3)$$

$$\beta(56,3) = f_{\varphi((\omega\uparrow\uparrow f_{\varphi(\varphi((\omega\uparrow\uparrow f_{2}^2(3).16 + 5)^{\varphi((\omega\uparrow\uparrow f_{\varphi(\varphi(2,(\omega\uparrow\uparrow 2)^2.(\varphi((\omega\uparrow\uparrow f_{\varphi(1,0,0) + 1}(3)),0,0) + 1)) + 1,0,0) + 1}(3)),0,0) + 1},0) + 1,0,0) + 1}(3)),0,0) + 1}(3)$$

$$\beta(57,3) = f_{\varphi((\varphi(1,1)\uparrow\uparrow 2)^{\varphi(2,\varphi((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow f_{\omega^{(\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{\varphi(2,(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow 3),0,0) + 1}}(3))}}}(3))) + 1}(3)),0) + 1}}}(3)),0) + 1}(3)),0) + 1,0) + 1},0,0) + 1}(3)$$

$$\beta(58,3) = f_{\varphi(\varphi(1,\omega^2.(\varphi(1,\omega^{\varphi(\omega^{\varphi(2,(\omega\uparrow\uparrow 2)^{\varphi(2,(\omega\uparrow\uparrow 2) + \omega^{\varphi((\omega\uparrow\uparrow f_{\varphi(\omega,0) + 1}(3)),0) + 1}) + 1},0) + 1},0,0) + 1},0) + 1)) + 1,0,0) + 1}(3)$$

$$\beta(59,3) = f_{\varphi(\varphi(1,(\omega\uparrow\uparrow f_{2}^2(3).2 + 1)^{(\varphi(1,2)\uparrow\uparrow f_{\varphi(1,(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^2.((\omega\uparrow\uparrow f_{\omega}(3)))},0) + 1}(3)) + 1}) + 1,0,0) + 1}(3)$$

$$\beta(60,3) = f_{\varphi(\varphi(1,(\varphi(1,2)\uparrow\uparrow f_{\varphi(\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow 2).((\omega\uparrow\uparrow f_{(\varphi(2,1)\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{2}^2(3)),0) + 1}}(3))),0) + 1},0) + 1,0,0) + 1}(3)) + 1) + 1,0,0) + 1}(3)$$

$$\beta(61,3) = f_{\varphi(\varphi(1,\varphi(2,\omega^{\varphi(\varphi((\omega\uparrow\uparrow f_{2}(3).2 + 9)^{\varphi(1,0,0) + 1},0) + 1,0,0) + 1}) + 1) + 1,0,0) + 1}(3)$$

$$\beta(62,3) = f_{\varphi((\varphi(2,0)\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi(1,(\varphi(1,0)\uparrow\uparrow f_{2}^2(3))^{\varphi(1,\varphi(1,\varphi(1,2,0) + 1,0) + 1,0) + 1}) + 1}(3))}(3)) + 1,0,0) + 1}(3)$$

$$\beta(63,3) = f_{\varphi((\varphi(2,2)\uparrow\uparrow 2)^2.((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^{\varphi(1,(\varphi(2,2)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow f_{\varphi(\varphi(1,1) + 1,0,0) + 1}(3)),0,0) + 1}(3))}) + 1}}(3))),0,0) + 1}(3)$$

$$\beta(64,3) = f_{\varphi(\varphi(2,(\omega\uparrow\uparrow 2).(\varphi(\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^{\varphi(2,(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi(2,\varphi(3,(\omega\uparrow\uparrow 2)^2.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow f_{\omega}(3))}),0) + 1,0) + 1}(3))}(3))) + 1}}(3)),0) + 1,0,0) + 1)) + 1,0,0) + 1}(3)$$

$$\beta(65,3) = f_{\varphi(\varphi(2,(\omega\uparrow\uparrow f_{\varphi(2,2)^{\varphi(1,\varphi(2,0,\varphi(2,\omega^{\varphi(2,(\varphi(1,2)\uparrow\uparrow f_{\omega}(3)) + 1) + 1},0) + 1) + 1,0) + 1}}(3))) + 1,0,0) + 1}(3)$$

$$\beta(66,3) = f_{\varphi(\varphi(2,(\varphi(2,1)\uparrow\uparrow 2)^{\varphi(\omega^{\varphi(1,0)^{\varphi(2,\varphi(8,\varphi(f_{\varphi(1,0,0) + 1}(3),0,0) + 1,0) + 1,0) + 1}},0) + 1}) + 1,0,0) + 1}(3)$$

$$\beta(67,3) = f_{\varphi(\varphi(2,(\varphi(14,2)\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{2}(3) + 1)^{(\omega\uparrow\uparrow 3)^{\varphi(1,\varphi(2,0) + 1,0) + 1}}}(3)) + 1) + 1,0,0) + 1}(3)$$

$$\beta(68,3) = f_{\varphi(\varphi(\omega^2.((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\omega^{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 7).(\omega^{\varphi(2,\varphi(2,(\omega\uparrow\uparrow 5)) + 1) + 1})}(3))}}(3))}(3))),0) + 1,0,0) + 1}(3)$$

$$\beta(69,3) = f_{\varphi(\varphi(\omega^{(\omega\uparrow\uparrow f_{\varphi(\varphi((\omega\uparrow\uparrow 2)^{\varphi(\omega^{\varphi(\omega^2.((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\omega^{\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{\omega^2}(3)),0,0) + 1},0) + 1}}(3))}(3))),0) + 1},0,0) + 1},0) + 1,0,0) + 1}(3))},0) + 1,0,0) + 1}(3)$$

$$\beta(70,3) = f_{\varphi(\varphi(\omega^{\varphi(2,(\omega\uparrow\uparrow f_{\varphi(2,\varphi(1,\omega^{\varphi(1,(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2).(\varphi(2,\varphi(1,\omega^{\varphi(\varphi(\omega,0) + 1,0,0) + 1}) + 1) + 1)}(3))) + 1}) + 1,0) + 1}(3)),0) + 1},0) + 1,0,0) + 1}(3)$$

$$\beta(71,3) = f_{\varphi(\varphi((\omega\uparrow\uparrow 2).(\varphi(\varphi((\omega\uparrow\uparrow 2)^2.(\varphi((\omega\uparrow\uparrow f_{(\varphi(1,2)\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{\varphi(1,\varphi(2,1,\varphi(2,\varphi(1,0) + 1,0) + 1) + 1,0) + 1}(3)),0) + 1}}(3)),0,0) + 1),0) + 1,0,0) + 1),0) + 1,0,0) + 1}(3)$$

$$\beta(72,3) = f_{\varphi(\varphi((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 2)^2.((\varphi(1,1)\uparrow\uparrow 19)^{\varphi((\omega\uparrow\uparrow 6),0,0) + 1})}(3))}(3))},0) + 1,0,0) + 1}(3)$$

$$\beta(73,3) = f_{\varphi(\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{\varphi(\omega^{\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{2}(3).4 + 2)^{\varphi(\omega^{\varphi((\omega\uparrow\uparrow f_{\varphi(2,0) + 1}(3)),0,0) + 1},0) + 1}}(3)),0) + 1}(3)),0,0) + 1},0,0) + 1},0) + 1}(3)),0) + 1},0) + 1,0,0) + 1}(3)$$

$$\beta(74,3) = f_{\varphi(\varphi((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow f_{\varphi(1,\varphi((\omega\uparrow\uparrow f_{(\varphi(1,0)\uparrow\uparrow f_{2}^2(3).4 + f_{2}(3)) + 1}(3)),0) + 1,0) + 1}(3))},0) + 1}(3))},0) + 1,0,0) + 1}(3)$$

$$\beta(75,3) = f_{\varphi(\varphi((\omega\uparrow\uparrow 19)^{\varphi(2,0,\varphi((\omega\uparrow\uparrow 2).((\omega\uparrow\uparrow 3).(\varphi(\varphi(\omega,0) + 1,0,0) + 1)),0) + 1) + 1},0) + 1,0,0) + 1}(3)$$

$$\beta(76,3) = f_{\varphi(\varphi((\omega\uparrow\uparrow f_{\omega^{\varphi((\omega\uparrow\uparrow 2)^{(\varphi(1,0)\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 3).(\varphi(2,\omega^{\varphi((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 3)}(3)),0,0) + 1}(3)),0) + 1},0) + 1),0,0) + 1}(3)),0) + 1}},0,0) + 1}}(3)),0) + 1,0,0) + 1}(3)$$

$$\beta(77,3) = f_{\varphi(\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{(\varphi(2,1)\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 9)^2.((\omega\uparrow\uparrow 2)^{\varphi(\omega^{(\omega\uparrow\uparrow 2)^2},0) + 1}),0,0) + 1}(3))}(3)) + 1}(3))}(3)),0) + 1,0,0) + 1}(3)$$

$$\beta(78,3) = f_{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi(2,\omega^{\varphi(\omega^{\varphi((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi(2,\varphi(1,\omega^2.((\omega\uparrow\uparrow 12)^{\varphi(1,0) + 1})) + 1,0) + 1}(3))}(3))},0,0) + 1},0,0) + 1}) + 1}(3)),0) + 1,0,0) + 1}(3)$$

$$\beta(79,3) = f_{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow f_{\varphi(\omega^2.((\omega\uparrow\uparrow f_{(\varphi(1,2)\uparrow\uparrow f_{2}^2(3).(2^{f_{2}(3) + 3})) + 1}(3))),0) + 1}(3)),0) + 1}(3)),0) + 1,0,0) + 1}(3)$$

$$\beta(80,3) = f_{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 6)^{\varphi(2,\varphi(2,\omega^{\omega^{(\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2),0) + 1}(3))}}) + 1) + 1},0,0) + 1}(3))}(3)),0,0) + 1}(3))},0,0) + 1}(3)),0) + 1,0,0) + 1}(3)$$

$$\beta(80.9999999,3) = f_{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi(\varphi((\omega\uparrow\uparrow f_{\varphi(\varphi(1,(\omega\uparrow\uparrow f_{(\omega\uparrow\uparrow 5)^{\varphi(1,\omega^{\varphi(1,\omega^{\varphi(\omega,0,0) + 1}) + 1}) + 1}}(3))) + 1,0,0) + 1}(3)),0) + 1,0,0) + 1}(3)),0) + 1,0,0) + 1}(3)),0) + 1,0,0) + 1}(3)),0) + 1,0,0) + 1}(3)),0) + 1,0,0) + 1}(3)$$

Second Attempt Base $$v = 3$$

These examples use increasing Sequence Depths from 50 to 64, and the Sequence Generating Code progressively generates higher numbers of finite integers for the Sequence and therefore more detail in the output number:

Sequence Depths from 50 to 54

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

$$\beta(13,3) = f_{\varphi(2,\omega.(\varphi(1,(\omega\uparrow\uparrow 3)^{\varphi(\varphi(\omega.((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{\varphi(1,\omega^{\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow 6),0) + 1},0,0) + 1}) + 1},0,0) + 1}(3))),0) + 1,0,0) + 1}) + 1)) + 1}(3)$$

$$\beta(13,3) = f_{\varphi(2,\omega.(\varphi(1,(\omega\uparrow\uparrow 3)^{\varphi(\varphi(\omega.((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{\varphi(1,\omega^{\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow 12),0) + 1},0,0) + 1}) + 1},0,0) + 1}(3))),0) + 1,0,0) + 1}) + 1)) + 1}(3)$$

Sequence Depths from 56 to 60

$$\beta(13,3) = f_{\varphi(2,\omega.(\varphi(1,(\omega\uparrow\uparrow 3)^{\varphi(\varphi(\omega.((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{\varphi(1,\omega^{\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow 12),0) + 1},0,0) + 1}) + 1},0,0) + 1}(3))),0) + 1,0,0) + 1}) + 1)) + 1}(3)$$

$$\beta(13,3) = f_{\varphi(2,\omega.(\varphi(1,(\omega\uparrow\uparrow 3)^{\varphi(\varphi(\omega.((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{\varphi(1,\omega^{\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow 15),0) + 1},0,0) + 1}) + 1},0,0) + 1}(3))),0) + 1,0,0) + 1}) + 1)) + 1}(3)$$

$$\beta(13,3) = f_{\varphi(2,\omega.(\varphi(1,(\omega\uparrow\uparrow 3)^{\varphi(\varphi(\omega.((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{\varphi(1,\omega^{\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow 15),0) + 1},0,0) + 1}) + 1},0,0) + 1}(3))),0) + 1,0,0) + 1}) + 1)) + 1}(3)$$

Sequence Depths from 62 to 64

$$\beta(13,3) = f_{\varphi(2,\omega.(\varphi(1,(\omega\uparrow\uparrow 3)^{\varphi(\varphi(\omega.((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{\varphi(1,\omega^{\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow 15),0) + 1},0,0) + 1}) + 1},0,0) + 1}(3))),0) + 1,0,0) + 1}) + 1)) + 1}(3)$$

$$\beta(13,3) = f_{\varphi(2,\omega.(\varphi(1,(\omega\uparrow\uparrow 3)^{\varphi(\varphi(\omega.((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{\varphi(1,\omega^{\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow 15)^{\varphi(1,0,0) + 1},0) + 1},0,0) + 1}) + 1},0,0) + 1}(3))),0) + 1,0,0) + 1}) + 1)) + 1}(3)$$

Third Attempt Base $$v = 3$$

Unfortunately there is an introduced error in the code since the last calculation on 26 May.

26 May

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

28 May

$$\beta(13,3) = f_{\varphi(2,\omega.(\varphi(1,(\omega\uparrow\uparrow 3)^{\varphi(\varphi(\omega.((\omega\uparrow\uparrow f_{\varphi((\omega\uparrow\uparrow 2)^{\varphi(1,\omega^{\varphi((\omega\uparrow\uparrow 2)^{\varphi((\omega\uparrow\uparrow 15)^{\varphi(1,0,0) + 1},0) + 1},0,0) + 1}) + 1},0,0) + 1}(3))),0) + 1,0,0) + 1}) + 1)) + 1}(3)$$