Beta Function - Sequence Generating Code[]
The Beta Function has been defined using program code shown below.
A separate blog is being written to explain how Sequence Generator Code is compiled and executed using a normal programming language ... Work in Progress.
Sequence Generating Code Version 3[]
The error identified in Version 2 code for the Beta Function can be corrected using this logic.
Let \(\lambda\uparrow\uparrow (T+1) = (\lambda\uparrow\uparrow T)^{\delta(T)}\)
then
\(\delta(1) = \lambda\) by definition
\(\delta(2) = \lambda^{\lambda - 1} = \lambda^{\delta(1) - 1}\)
\(\delta(3) = (\lambda\uparrow\uparrow 2)^{\delta(2) - 1}\)
and
\(\delta(T+1) = (\lambda\uparrow\uparrow T)^{\delta(T) - 1}\)
The proof for this is:
\(\lambda\uparrow\uparrow 2 = (\lambda\uparrow\uparrow 1)^{\delta(1)} = \lambda^{\lambda} = \lambda\uparrow\uparrow 2\) by definition
and
\(\lambda\uparrow\uparrow (T+1) = (\lambda\uparrow\uparrow T)^{\delta(T)} = (\lambda\uparrow\uparrow T)^{(\lambda\uparrow\uparrow (T-1))^{\delta(T-1) - 1}}\)
\(= (\lambda^{\lambda\uparrow\uparrow (T-1)})^{(\lambda\uparrow\uparrow (T-1))^{\delta(T-1) - 1}} = \lambda^{\lambda\uparrow\uparrow (T-1).(\lambda\uparrow\uparrow (T-1))^{\delta(T-1) - 1}} = \lambda^{(\lambda\uparrow\uparrow (T-1))^{\delta(T-1) - 1 + 1}}\)
\(= \lambda^{(\lambda\uparrow\uparrow (T-1))^{\delta(T-1)}} = \lambda^{\lambda\uparrow\uparrow T} = \lambda\uparrow\uparrow (T+1)\)
The Sequence Generating Ruleset has been modified to include the above logic. 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:
- \(h_x = (d<2,d(0:x<v-u(x),1:(f_0<1,P_h = 1)))\)
- \(f_x = (g_x,g_x((0,0,0):h_u,(h_U,(f_{x+1}<g_x,P_h=d-h_U)))\)
- \(g_x = (q<v+1,g_{[q]},n_0<q-1,t_x)\)
- \(n_x = (Q,t_A,g_{[Q]},n_{x+1}<Q)\)
- \(t_x = (h_T,g_E,g_C<q-h_T,g_x<q-g_E)\)
The syntax used for this ruleset will BE EXPLAINED FURTHER - in my blog on Sequence Generator Code ... Work in Progress
The constraint for \(g_E\) has been modified in Version 3 code, to closely approximate the logic described above, therefore:
\(g_E < \lambda\) when \(T = 1\)
\(g_E < \lambda^{\lambda - 1}\) when \(T = 2\)
\(g_E < (\lambda\uparrow\uparrow 2)^{\lambda^{\lambda - 1} - 1}\) when \(T = 3\)
\(g_E < (\lambda\uparrow\uparrow 3)^{(\lambda\uparrow\uparrow 2)^{\lambda^{\lambda - 1} - 1} - 1}\) when \(T = 4\)
Unfortunately Version 3 code is still an approximation because it can only apply these (almost identical) constraints:
\(g_E < \lambda\) when \(T = 1\)
\(g_E < \lambda^{\lambda - 1}\) when \(T = 2\)
\(g_E < (\lambda\uparrow\uparrow 2)^{\lambda^{\lambda - 1}}\) when \(T = 3\)
\(g_E < (\lambda\uparrow\uparrow 3)^{(\lambda\uparrow\uparrow 2)^{\lambda^{\lambda - 1}}}\) when \(T = 4\)
The number of undesired values that are still generated are substantially reduced (see below). This error will remain for now and I will try to correct it in a future version of the code.
WORK IN PROGRESS
Granularity Examples \(\beta(3.141,3)\) to \(\beta(5.1963,3)\)[]
These simple examples in base \(v = 3\) show how Version 3 code correctly transitions from \(\omega\) to \(\omega^{\omega}\) without any errors:
\(\beta(3.141,3) = f_{\omega + 1}(3)\)
\(\beta(3.4417,3) = f_{\omega.2}(3)\)
\(\beta(3.9485,3) = f_{\omega^2}(3)\)
\(\beta(4.53,3) = f_{\omega^2.2}(3)\)
\(\beta(5.1963,3) = f_{(\omega\uparrow\uparrow 2)}(3)\)
Granularity Examples \(\beta(5.1963,3)\) to \(\beta(9,3)\)[]
Version 3 code correctly transitions from \(\omega^{\omega}\) to \(\omega^{\omega^{\omega}}\) without error:
\(\beta(5.1963,3) = f_{(\omega\uparrow\uparrow 2)}(3)\)
\(\beta(5.3777,3) = f_{(\omega\uparrow\uparrow 2).2}(3)\)
\(\beta(5.5655,3) = f_{(\omega\uparrow\uparrow 2).(\omega)}(3)\)
\(\beta(5.9612,3) = f_{(\omega\uparrow\uparrow 2)^2}(3)\)
\(\beta(6.1694,3) = f_{(\omega\uparrow\uparrow 2)^2.2}(3)\)
\(\beta(6.83855,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(3)\)
\(\beta(6.917229885,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}.2}(3)\)
\(\beta(7.324573,3) = f_{(\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega)}(3)\)
\(\beta(7.84517,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(3)\)
\(\beta(8.5974,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}}(3)\)
\(\beta(8.79635,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega)}(3)\)
\(\beta(8.94865,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2)}(3)\)
\(\beta(8.99574,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2)}(3).2\)
\(\beta(8.99892723,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)}(3)\)
\(\beta(8.999463585,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega}}(3)\)
\(\beta(9,3) = f_{\varphi(1,0)}(3)\)
Granularity Examples \(\beta(6.349,4)\) to \(\beta(10.079,4)\)[]
When we use base \(v = 4\) the transitions up to \(\omega\uparrow\uparrow 3\) are correct:
\(\beta(6.34962,4) = f_{(\omega\uparrow\uparrow 2)}(4)\)
\(\beta(7.407,4) = f_{(\omega\uparrow\uparrow 2)^3}(4)\)
\(\beta(8,4) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(4)\)
\(\beta(8.314075,4) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(4)\)
\(\beta(8.979697,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2}}(4)\)
\(\beta(9.698609,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3}}(4)\)
\(\beta(9.887156,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega}}(4)\)
\(\beta(10.06323,4) = f_{3}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}}(4))\)
\(\beta(10.07937,4) = f_{(\omega\uparrow\uparrow 3)}(4)\)
Granularity Examples \(\beta(10.079,4)\) to \(\beta(16,4)\)[]
These examples transition up to \(\omega\uparrow\uparrow 4\) in base \(v = 4\):
\(\beta(10.07937,4) = f_{(\omega\uparrow\uparrow 3)}(4)\)
\(\beta(11.75788,4) = f_{(\omega\uparrow\uparrow 3)^3}(4)\)
\(\beta(12.699209,4) = f_{(\omega\uparrow\uparrow 3)^{\omega}}(4)\)
\(\beta(14.254379491,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)}}(4)\)
\(\beta(15.101989005,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega}}}(4)\)
\(\beta(15.69488142,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.2}}}(4)\)
\(\beta(15.98716909,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 2}}}(4)\)
\(\beta(15.99358326,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}}}(4)\)
\(\beta(15.9967913075,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega)}}(4)\)
\(\beta(15.9996434468,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3)}}(4)\)
\(\beta(15.9999987619545,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3)}}(4)\)
\(\beta(16.0000001,4) = f_{\varphi(1,0)}(4)\)
We can check the transition from \(\gamma\uparrow\uparrow 3\) to \(\gamma\uparrow\uparrow 4\) against these examples. The correct constraint is:
\(< (\gamma\uparrow\uparrow 2)^{\gamma^{\gamma - 1} - 1}\) or
\(<= (\gamma\uparrow\uparrow 2)^{\gamma^{\gamma - 1} - 1} - 1\)
The largest power in the examples is:
\((\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3} = (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 4 - 1} = (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.4 - 1} = (\omega\uparrow\uparrow 2)^{\omega^2.4 - 1}\)
\(= (\omega\uparrow\uparrow 2)^{\omega^3 - 1} = (\omega\uparrow\uparrow 2)^{\omega^{\omega - 1} - 1}\)
Therefore the Beta Function still generates undesired values between \(15.98716909\) to \(15.999...\) because:
\(\beta(15.98716909,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 2}}}(4) = f_{\varphi(1,0)}(4)\)
\(\beta(16.0000001,4) = f_{\varphi(1,0)}(4)\)
This error will remain for now and I will try to correct it in a future version of the code.
WORK IN PROGRESS
Undesired Values that are still generated[]
Further to the above example, here are other ranges of undesired values:
| Base | Input Value | Output | |
|---|---|---|---|
| \(v = 4\) | From | 15.98716909 | \(f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 2}}}(4) = f_{\varphi(1,0)}(4)\) |
| To | 16.0000001 | \(f_{\varphi(1,0)}(4)\) | |
| \(v = 5\) | 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 | \(f_{(\omega\uparrow\uparrow 4)}(5)\) | |
| \(v = 5\) | 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 | \(f_{\varphi(1,0)}(5)\) | |
| \(v = 6\) | 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 | \(f_{(\omega\uparrow\uparrow 4)}(6)\) | |
| \(v = 6\) | 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 | \(f_{(\omega\uparrow\uparrow 5)}(6)\) | |
| \(v = 6\) | 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 | \(f_{\varphi(1,0)}(6)\) |
This error will remain for now and I will try to correct it in a future version of the code.
WORK IN PROGRESS
Granularity Examples near \(\beta(12.2118455,3)\)[]
The Beta Function successfully generates the correct Veblen ordinals as seen in these two examples:
\(\beta(12.2118455947222,3)\)
\(= f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow (f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(2,0)^2.(\varphi(1,\varphi(1,(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2}.2 + (\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega.2 + 2) + \omega^2 + \omega + 2)^{\omega^2 + \omega.2 + 1}.((\omega\uparrow\uparrow 2))))})}(3))))}(f_{\varphi(2,1)}(3))\)
\(< f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow (f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow 3))}(3))))}(f_{\varphi(2,1)}(3))\)
\(= f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow (f_{\varphi(1,\varphi(1,\varphi(2,0) + 1))}(3))))}(f_{\varphi(2,1)}(3))\)
\(< f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow (f_{\varphi(2,1)}(3))))}(f_{\varphi(2,1)}(3))\)
\(= f_{\varphi(1,\varphi(1, \varphi(2,0) + 1)}(f_{\varphi(2,1)}(3))\)
\(< \beta(12.2118455947223,3) = f_{\varphi(2,1)}^{2}(3)\)
There is a huge gap between each of these numbers. The Beta Function requires input parameters of huge decimal precision to access the missing numbers.
Valid Sequence Counts[]
WORK IN PROGRESS
Test Bed for Version 3[]
Below is the test bed and various results using version 3.
\(\beta(3.141,3) = f_{\omega + 1}(3)\)
\(\beta(3.4417,3) = f_{\omega.2}(3)\)
\(\beta(3.9485,3) = f_{\omega^2}(3)\)
\(\beta(4.53,3) = f_{\omega^2.2}(3)\)
\(\beta(5.1963,3) = f_{(\omega\uparrow\uparrow 2)}(3)\)
\(\beta(5.3777,3) = f_{(\omega\uparrow\uparrow 2).2}(3)\)
\(\beta(5.5655,3) = f_{(\omega\uparrow\uparrow 2).(\omega)}(3)\)
\(\beta(5.9612,3) = f_{(\omega\uparrow\uparrow 2)^2}(3)\)
\(\beta(6.1694,3) = f_{(\omega\uparrow\uparrow 2)^2.2}(3)\)
\(\beta(6.34962,4) = f_{(\omega\uparrow\uparrow 2)}(4)\)
\(\beta(7.407,4) = f_{(\omega\uparrow\uparrow 2)^3}(4)\)
\(\beta(8,4) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(4)\)
\(\beta(8.314075,4) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(4)\)
\(\beta(8.979697,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2}}(4)\)
Next attempt - 2 May 2016
\(\beta(3.141,3) = f_{\omega + 1}(3)\)
\(\beta(3.4417,3) = f_{\omega.2}(3)\)
\(\beta(3.9485,3) = f_{\omega^2}(3)\)
\(\beta(4.53,3) = f_{\omega^2.2}(3)\)
\(\beta(5.1963,3) = f_{(\omega\uparrow\uparrow 2)}(3)\)
\(\beta(5.3777,3) = f_{(\omega\uparrow\uparrow 2).2}(3)\)
\(\beta(5.5655,3) = f_{(\omega\uparrow\uparrow 2).(\omega)}(3)\)
\(\beta(5.9612,3) = f_{(\omega\uparrow\uparrow 2)^2}(3)\)
\(\beta(6.1694,3) = f_{(\omega\uparrow\uparrow 2)^2.2}(3)\)
\(\beta(6.83855,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(3)\)
\(\beta(6.917229885,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}.2}(3)\)
\(\beta(6.34962,4) = f_{(\omega\uparrow\uparrow 2)}(4)\)
\(\beta(7.407,4) = f_{(\omega\uparrow\uparrow 2)^3}(4)\)
\(\beta(8,4) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(4)\)
\(\beta(8.314075,4) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(4)\)
\(\beta(8.979697,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2}}(4)\)
\(\beta(9.698609,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3}}(4)\)
\(\beta(9.887156,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega}}(4)\)
\(\beta(10.06323,4) = f_{3}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}}(4))\)
\(\beta(10.07937,4) = f_{(\omega\uparrow\uparrow 3)}(4)\)
\(\beta(11.75788,4) = f_{(\omega\uparrow\uparrow 3)^3}(4)\)
\(\beta(12.699209,4) = f_{(\omega\uparrow\uparrow 3)^{\omega}}(4)\)
\(\beta(14.254379491,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)}}(4)\)
\(\beta(12.9802463,3) = f_{\varphi(2,\omega)}(3)\)
\(\beta(13.58823285,3) = f_{\varphi(2,(\omega\uparrow\uparrow 2))}(3)\)
\(\beta(4,2) = f_{\varphi(1,0)}(2)\)
\(\beta(4.0876,2) = f_{\varphi(1,0).(\omega)}(2)\)
\(\beta(4.17725,2) = f_{\varphi(1,0)^{\omega}}(2)\)
\(\beta(4.3621,2) = f_{\varphi(1,1)}(2)\)
\(\beta(4.4043,2) = f_{\varphi(1,1).(\omega)}(2)\)
\(\beta(4.44684,2) = f_{\varphi(1,1).(\varphi(1,0))}(2)\)
\(\beta(4.4899,2) = f_{\varphi(1,1)^{\omega}}(2)\)
\(\beta(4.62142,2) = f_{\varphi(1,1)^{\varphi(1,0)}}(2)\)
\(\beta(4.688634,2) = f_{\varphi(1,1)^{\varphi(1,0)^{\omega}}}(2)\)
\(\beta(4.7569,2) = f_{\varphi(1,\omega)}(2)\)
\(\beta(4.89625,2) = f_{\varphi(1,\omega)^{\omega}}(2)\)
\(\beta(5.039685,2) = f_{\varphi(1,\omega)^{\varphi(1,0)}}(2)\)
\(\beta(5.112989,2) = f_{\varphi(1,\omega)^{\varphi(1,1)}}(2)\)
\(\beta(5.1875,2) = f_{\varphi(1,\omega + 1)}(2)\)
\(\beta(5.65686,2) = f_{\varphi(\omega,0)}(2)\)
\(\beta(5.82264,2) = f_{\varphi(\omega,1)}(2)\)
\(\beta(5.99325,2) = f_{\varphi(\omega,\omega)}(2)\)
\(\beta(6.349605,2) = f_{\varphi(\omega,\varphi(1,0))}(2)\)
\(\beta(6.4419615,2) = f_{\varphi(\omega,\varphi(1,1))}(2)\)
\(\beta(6.535662,2) = f_{\varphi(\omega,\varphi(1,\omega))}(2)\)
\(\beta(6.6307255,2) = f_{\varphi(\omega,\varphi(1,\omega + 1))}(2)\)
\(\beta(6.7272,2) = f_{\varphi(\omega + 1,0)}(2)\)
\(\beta(6.825021,2) = f_{\varphi(1,\varphi(\omega + 1,0) + 1)}(2)\)
\(\beta(6.9243,2) = f_{\varphi(\omega + 1,1)}(2)\)
\(\beta(7.1272,2) = f_{\varphi(\omega + 1,\omega)}(2)\)
\(\beta(7.550995,2) = f_{\varphi(\omega + 1,\varphi(1,0))}(2)\)
\(\beta(7.6057123,2) = f_{\varphi(\omega + 1,\varphi(1,1))}(2)\)
\(\beta(7.6608264,2) = f_{\varphi(\omega + 1,\varphi(1,\omega))}(2)\)
\(\beta(7.77225555,2) = f_{\varphi(\omega + 1,\varphi(\omega,0))}(2)\)
\(\beta(7.7909848,2) = f_{\varphi(\omega + 1,\varphi(1,\varphi(\omega,0) + 1)).(\omega) + 1}(2).2 + 4\)
Next attempt - 3 May 2016
\(\beta(11.97,3) = f_{(\varphi(2,0)\uparrow\uparrow 2)^{\omega.2}.(\omega) + 2}(3) + 1\)
\(\beta(12,3) = f_{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(1,2)^{\omega^2 + \omega}.(\omega^2 + \omega + 2) + \varphi(1,2)^{\omega^2 + 1}.(\omega.2 + 1) + 1}.(\omega^2.2 + \omega.2) + (\varphi(1,0)\uparrow\uparrow 2)^2.(\varphi(1,0)^{\omega^2.2 + \omega + 1}.2 + \varphi(1,0)^2.((\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega^2 + 2) + \omega^2))}(3)\)
\(\beta(12.2,3) = f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(2,0).(\omega^2.2 + \omega.2) + \varphi(1,1)^{\omega^2.2 + \omega + 2} + 2}.(\varphi(1,\varphi(1,\omega + 2)^{\varphi(1,0).(\omega.2 + 1) + (\omega\uparrow\uparrow 2) + \omega^2 + 2}.((\varphi(1,1)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^2.(\omega)}))))}(3)\)
\(\beta(12.21,3) = f_{\varphi(1,(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(2,0)^2.((\varphi(1,\omega)\uparrow\uparrow 2)^2.(\varphi(1,2)^{(\varphi(1,0)\uparrow\uparrow 2)^2.(\varphi(1,0)^{(\omega\uparrow\uparrow 2)^{\omega + 1}.2 + \omega.2 + 1}.((\omega\uparrow\uparrow 2).(\omega) + \omega^2.2 + \omega)) + \varphi(1,0)^{(\omega\uparrow\uparrow 2)^2.(\omega^2 + \omega + 2) + 1}.(\omega)}))})}(3)\)
\(\beta(36,6) = f_{\varphi(1,0)}(6)\)
\(\beta(36.99,6) = f_{(\varphi(1,2)\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 5)^5}.(\varphi(1,2)^{\varphi(1,1)^{(\omega\uparrow\uparrow 3)^5.(\omega^4.2 + \omega^2.5 + \omega.3 + 1) + (\omega\uparrow\uparrow 3).5 + (\omega\uparrow\uparrow 2)^5.5 + 4} + (\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^{\omega^3.2 + 4}.5 + 3)}.((\varphi(1,1)\uparrow\uparrow 3)^{(\varphi(1,1)\uparrow\uparrow (f_{(\varphi(1,0)\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^{\omega.3 + 2} + \omega^3.3 + \omega}}}(6)))}))}(6)\)
\(\beta(37,6) = f_{(\varphi(1,2)\uparrow\uparrow 4)^{(\varphi(1,0)\uparrow\uparrow 5)^5.(\varphi(1,0)^2.((\omega\uparrow\uparrow 5)^2.(\omega^3.4 + 1) + (\omega\uparrow\uparrow 3)^3.4 + 5) + 2) + 4}.((\omega\uparrow\uparrow 4)^2.5 + 1) + (\varphi(1,1)\uparrow\uparrow 3)^{(\varphi(1,1)\uparrow\uparrow (f_{1}^{4}(6) + 2))^{(\omega\uparrow\uparrow 4)^3 + 3}.(\varphi(1,1)^{(\varphi(1,0)\uparrow\uparrow 4)^{(\varphi(1,0)\uparrow\uparrow (f_{\varphi((\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^2 + 4}.(\omega^2.3 + 3),3,(\omega\uparrow\uparrow 3)^{\omega})}(6)))}})}}(6)\)
\(\beta(37.01,6) = f_{(\varphi(1,2)\uparrow\uparrow 4)^{(\varphi(1,2)\uparrow\uparrow 2)^4.(\varphi(1,2)^4.((\varphi(1,1)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{\omega^3.2 + \omega.5 + 5}.(\omega.4 + 2) + (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^2.5 + \omega.3 + 4) + 2}.(\omega + 2) + (\omega\uparrow\uparrow 3)^5.5 + 4}.((\omega\uparrow\uparrow 2)^2.3) + (\omega\uparrow\uparrow 3)^3.(\omega^4 + \omega.4 + 4) + \omega}))}}(6)\)
\(\beta(37.02,6) = f_{(\omega\uparrow\uparrow (f_{f_{1}^{f_{1}^{2}(6) + 9}(f_{4}^{4}(6)) + 4}^{6}(f_{(\omega\uparrow\uparrow 2)^{\omega^3.3 + 1}.(\omega^3 + 1) + 4}^{5}(6))))}(f_{(\varphi(1,2)\uparrow\uparrow 5).((\omega\uparrow\uparrow 3)^{\omega^3.4 + 5}.((\omega\uparrow\uparrow 2)^4.3 + 5) + (\omega\uparrow\uparrow 2)^5.(\omega^5.2 + 5) + (\omega\uparrow\uparrow 2)^3 + \omega^5.3 + 3) + 5}^{5}(6))\)
\(\beta(37.03,6) = f_{(\varphi(1,1)\uparrow\uparrow (f_{\omega}(f_{(\omega\uparrow\uparrow 3).4 + 1}^{3}(6))))}(f_{(\varphi(1,2)\uparrow\uparrow 5)^3.((\omega\uparrow\uparrow 4)^2.(\omega^3.5 + 2) + \omega^5.3 + \omega^2.3 + \omega + 3) + (\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3).3 + 1}.((\omega\uparrow\uparrow 2)^{\omega^2.3 + 5}.(\omega^4 + 5) + \omega^2.3) + (\omega\uparrow\uparrow 2)^4.2 + 4}(6))\)
\(\beta(37.04,6) = f_{(\varphi(1,2)\uparrow\uparrow 5)^5.((\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2).(\omega.3 + 3) + 4}.((\omega\uparrow\uparrow 2)^{\omega^4 + 2}.(\omega + 4) + \omega^5 + \omega^4.5 + 3) + 5) + (\varphi(1,1)\uparrow\uparrow 5)^{(\varphi(1,0)\uparrow\uparrow 4)^4.(\varphi(1,0)^{(\omega\uparrow\uparrow 4)^2.2 + (\omega\uparrow\uparrow 3)^{\omega^2.3 + \omega.3 + 5}.4}.((\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 2)^3.3 + \omega^4 + \omega.4 + 4}.(\omega)))}}(6)\)
\(\beta(37.05,6) = f_{3}^{3}(f_{(\varphi(1,2)\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 2)^{\omega.5 + 2}.(\omega^5.3 + 2) + 4}.4 + (\omega\uparrow\uparrow 2)^{\omega^4.3 + \omega^2.2 + \omega.2 + 1}.(\omega^3.3 + \omega.5 + 2) + \omega^2 + \omega.2 + 2}(6)) + f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{\omega^5 + 3}.3 + \omega^4 + 3}.5 + (\omega\uparrow\uparrow 2)^{\omega^2.5 + 4}.(\omega)}(6)\)
\(\beta(37.06,6) = f_{(\varphi(1,2)\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{\omega^5.2 + \omega^3.4 + \omega.2 + 1}.3 + \omega^5.5 + \omega + 1}.(\varphi(1,1)^{\omega^4.4 + \omega^3.3 + \omega.2 + 4}.((\omega\uparrow\uparrow 2)^{\omega^4 + \omega^3.5 + \omega^2.2 + \omega.2}.3 + 4) + (\omega\uparrow\uparrow 5)^3.((\omega\uparrow\uparrow 2)^5 + (\omega\uparrow\uparrow 2).5 + 4) + (\omega\uparrow\uparrow 5)^2.4 + (\omega\uparrow\uparrow 3)^{\omega})}(6)\)
\(\beta(37.07,6) = f_{(\varphi(1,2)\uparrow\uparrow 5)^{(\varphi(1,0)\uparrow\uparrow 3).((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega.5 + 1}}.4 + \omega^2.3 + 5) + (\varphi(1,0)\uparrow\uparrow 2)^{\varphi(1,0)^4.((\omega\uparrow\uparrow 3)^5.(\omega.3 + 5) + (\omega\uparrow\uparrow 3)^4.(\omega^4.2 + \omega^2 + 5) + (\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{\omega.3 + 1}.(\omega^2 + 5)) + 5) + (\omega\uparrow\uparrow 5)^4.((\omega\uparrow\uparrow 3))}}}(6)\)
\(\beta(37.08,6) = f_{(\varphi(1,2)\uparrow\uparrow 5)^{(\varphi(1,1)\uparrow\uparrow 4)^2.((\varphi(1,1)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^{\omega^4 + \omega^2.5 + 2}.2 + (\omega\uparrow\uparrow 2)^5.(\omega^3.2) + \omega + 4}.(\omega.5 + 3) + 4}.((\varphi(1,0)\uparrow\uparrow 5)^{(\varphi(1,0)\uparrow\uparrow (f_{2}(6) + 2))^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.4 + 2}.(\omega.2 + 1) + (\omega\uparrow\uparrow 2)^4.5 + (\omega\uparrow\uparrow 2).2 + \omega^4.4 + \omega^2.3 + \omega}}))}}(6)\)
\(\beta(37.09,6) = f_{(\varphi(1,2)\uparrow\uparrow 5)^{(\varphi(1,2)\uparrow\uparrow (f_{f_{\varphi(\varphi((\omega\uparrow\uparrow 2)^3 + \omega^5.4 + \omega^4.2 + \omega^2.3 + \omega.3 + 5,\varphi(\varphi(4,\varphi(3,(\varphi(1,(\varphi(1,(\omega\uparrow\uparrow 3).(\omega^4.2 + 5) + \omega.3 + 4)\uparrow\uparrow 2)^3 + \omega^4.4 + \omega.2 + 1)\uparrow\uparrow 2)^{(\varphi(1,(\varphi(1,(\omega\uparrow\uparrow 3).4 + 5)\uparrow\uparrow 3)^4.3 + (\omega\uparrow\uparrow 4)^2.(\omega) + 2)\uparrow\uparrow 4)^3.((\varphi(1,4)\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 4)^4.3}.(\omega^4 + \omega.4 + 4) + (\omega\uparrow\uparrow 3)^{\omega})})),0,0),0,0),0,0,0,0)}(6)}(f_{\omega^4.4}^{4}(6))))}}(6)\)
\(\beta(37.1,6) = f_{\varphi(1,3)^2.((\varphi(1,0)\uparrow\uparrow 5)^{(\varphi(1,0)\uparrow\uparrow (f_{\varphi(\varphi(1,\varphi(1,(\varphi(1,\omega^5.3 + \omega.3 + 5)\uparrow\uparrow 5)^3.4 + (\varphi(1,5)\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3)^2.(\omega^2.2 + 1) + 4} + 2}.4 + (\varphi(1,4)\uparrow\uparrow 4)^2.5 + (\varphi(1,3)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 4)^{\omega^5 + \omega^3 + 2}.(\omega^4.4 + 2) + 2}.3 + (\varphi(1,0)\uparrow\uparrow 2)^{\omega^5 + \omega^4.3 + 4}.(\varphi(1,0)^5.((\omega\uparrow\uparrow 4)^5.4 + (\omega\uparrow\uparrow 3))))),0,0,0)}(6)))})}(6)\)
\(\beta(37.2,6) = f_{(\varphi(1,3)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3).(\omega.5 + 4) + 2}.((\varphi(1,1)\uparrow\uparrow 4).((\omega\uparrow\uparrow 3)^{\omega.2 + 5}.2 + (\omega\uparrow\uparrow 3)^3.(\omega^4.5 + \omega^3 + 3) + (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.5 + 4}.(\omega.2 + 5)) + (\varphi(1,1)\uparrow\uparrow 3)^5.((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2).(\omega^4 + 4) + \omega^2.5}.2) + (\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)}})}(6)\)
\(\beta(37.3,6) = f_{(\varphi(1,3)\uparrow\uparrow 3)^{(\varphi(1,1)\uparrow\uparrow 5)^2.((\varphi(1,1)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^4.(\omega^4.5 + \omega) + \omega^3.2 + 3}.(\varphi(1,1)^{(\varphi(1,0)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^4 + 4) + (\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2)^2.(\omega^3.2 + \omega.3 + 2) + 1) + 2}.((\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3).5 + (\omega\uparrow\uparrow 2)^5.(\omega^4.2 + 2) + \omega.4}.((\omega\uparrow\uparrow 3)))}))}}(6)\)
\(\beta(37.4,6) = f_{(\varphi(1,3)\uparrow\uparrow 5)^3.((\omega\uparrow\uparrow 2)^{\omega^3.3 + \omega + 4}.(\omega^2.2 + \omega.5) + \omega^5 + \omega + 2) + (\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3).5 + 3}.((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^2.2 + \omega^2.4 + \omega.5 + 5}.2 + \omega^5.4 + 2) + (\omega\uparrow\uparrow 5)^5.3 + (\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^3.(\omega^2.3 + \omega.4) + (\omega\uparrow\uparrow 2)}}(6)\)
\(\beta(37.5,6) = f_{\varphi(1,4)^{(\omega\uparrow\uparrow 3)^{\omega^2.3 + 3}.(\omega^4.2) + (\omega\uparrow\uparrow 3)^{\omega^2 + \omega.4 + 1}.2 + \omega^5.2 + 3}.3 + \varphi(1,4)^{(\omega\uparrow\uparrow 3)^{\omega^2.2 + \omega.3 + 1}.(\omega^5.3 + \omega^4.5 + 4) + (\omega\uparrow\uparrow 2)^{\omega^2 + 3}.(\omega^4.5 + 4) + (\omega\uparrow\uparrow 2)^{\omega^2}.5 + 5}.((\omega\uparrow\uparrow 2)^{\omega.3 + 3})}(6)\)
\(\beta(37.6,6) = f_{(\varphi(1,4)\uparrow\uparrow 2)^{(\varphi(1,2)\uparrow\uparrow 4)^3.((\varphi(1,1)\uparrow\uparrow 5)^4.((\omega\uparrow\uparrow 3)^{\omega^2.2 + \omega.5 + 3}.2 + 4) + 4) + 3} + \varphi(1,4)^{(\omega\uparrow\uparrow 5).3 + 4}.((\varphi(1,3)\uparrow\uparrow 5)^{(\varphi(1,1)\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^{\omega^3.5 + \omega.5 + 4} + 2}.4 + 2}.(\varphi(1,3)^{(\omega\uparrow\uparrow 5)^{\omega^5.3 + 2} + (\omega\uparrow\uparrow 3).(\omega^2 + \omega.2 + 3) + (\omega\uparrow\uparrow 2)^4.(\omega^3.3 + \omega)}))}(6)\)
\(\beta(37.7,6) = f_{(\varphi(1,4)\uparrow\uparrow 4)^2.((\omega\uparrow\uparrow (f_{\omega^4.4}^{f_{2}(f_{4}^{3}(6)) + f_{2}^{f_{2}^{4}(6) + f_{1}^{6}(f_{2}(6))}(f_{3}^{4}(6))}(f_{\omega^4.4 + 1}^{2}(6)))))}(f_{(\varphi(1,4)\uparrow\uparrow 4)^3.((\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3).3 + (\omega\uparrow\uparrow 2)^5.3 + \omega.5 + 4}.((\omega\uparrow\uparrow 2)^5.(\omega.4 + 4) + 4) + \omega^4 + \omega.2 + 1) + 5}^{4}(6))\)
\(\beta(37.8,6) = f_{f_{(\omega\uparrow\uparrow 3)^{\omega}}(6)}(f_{(\varphi(1,4)\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3)^{\omega^3 + \omega^2.4 + 4}.2 + 5}.((\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^3.2 + \omega^2.5 + \omega.4 + 2}.2 + (\omega\uparrow\uparrow 2)^{\omega^4.4 + 3}.5 + (\omega\uparrow\uparrow 2)^{\omega^2.2 + 1}.(\omega^4 + \omega.4 + 2) + (\omega\uparrow\uparrow 2)^{\omega^2.2}.(\omega^2.4) + 1) + 2}^{2}(6))\)
\(\beta(37.9,6) = f_{\varphi(1,5)^{(\varphi(1,2)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 4)^{\omega^5.3 + \omega^4.5 + 1}.3 + 5}.3 + (\varphi(1,2)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^4.3 + (\omega\uparrow\uparrow 2)^3.4}.3 + (\varphi(1,1)\uparrow\uparrow 3)^{\omega^4.4 + \omega.4 + 2}.((\omega\uparrow\uparrow 2)^5.2 + 4) + (\omega\uparrow\uparrow 2)^4.(\omega^2.4 + 2) + 2}.((\omega\uparrow\uparrow 5)^4 + 2) + (\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4).5 + 3}.(\omega^3.3 + \omega)}(6)\)
\(\beta(38,6) = f_{(\omega\uparrow\uparrow (f_{(\omega\uparrow\uparrow (f_{2}^{f_{2}^{5}(6) + 4}(f_{3}^{4}(6)) + 1))^{(\omega\uparrow\uparrow 15)^6}}(f_{(\varphi(1,0)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2)^{\omega^4 + 5}.3 + 3) + \omega^4 + 4}.(\varphi(1,0)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^5.(\omega.4 + 4) + 4}.3 + 2}.4 + 5)}^{3}(6))))}(f_{(\varphi(1,5)\uparrow\uparrow 3)^3.(\omega^3.4 + 5) + 1}(6))\)
\(\beta(39,6) = f_{(\omega\uparrow\uparrow 3)^4.(f_{\varphi(1,(\omega\uparrow\uparrow 2)^4.(\omega^2 + 4) + (\omega\uparrow\uparrow 2).(\omega^2.5 + \omega.5 + 3) + \omega^4.3 + \omega^2.5 + 2)^{(\varphi(1,2)\uparrow\uparrow 3)^4.((\varphi(1,1)\uparrow\uparrow 4)^3.3 + 2) + \omega^5.5 + \omega^2.5 + 1}.((\varphi(1,5)\uparrow\uparrow 2)^3.((\omega\uparrow\uparrow 2)^5.(\omega^3.3 + 3) + \omega^4 + \omega.4 + 4) + (\omega\uparrow\uparrow 3)^{\omega})}(6))}(f_{\varphi(1,(\omega\uparrow\uparrow 2)^{\omega^2.4}.5 + 4).(\omega^2.3 + \omega + 1) + 3}^{4}(6))\)
\(\beta(40,6) = f_{(\omega\uparrow\uparrow (f_{\varphi(1,3)}(6)))}(f_{\varphi(1,(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^{\omega^4.2 + \omega^3.3}.(\omega.5 + 3) + 4}.((\omega\uparrow\uparrow 3)^5.2 + (\omega\uparrow\uparrow 2).3 + \omega.2) + (\omega\uparrow\uparrow 2)^{\omega^2.3 + 1}.(\omega^2 + 2) + \omega.4 + 4).2 + (\omega\uparrow\uparrow 2)^4.(\omega^3 + \omega^2.4 + 4) + (\omega\uparrow\uparrow 2)^3.(\omega^2.4) + 1}(6))\)
\(\beta(50,6) = f_{(\varphi(2,\varphi(2,0)^2.(\omega.3 + 5) + 1)\uparrow\uparrow 5)^4.(\varphi(1,5)^{(\varphi(1,3)\uparrow\uparrow 3)^{(\varphi(1,1)\uparrow\uparrow 5)^{\omega^2.4 + 5}.((\varphi(1,0)\uparrow\uparrow 3)^4.2 + (\omega\uparrow\uparrow 2)^2.(\omega.4 + 3) + 4) + 5}.2 + \omega^2 + \omega + 2}.(\varphi(1,4)^{\varphi(1,1)^{(\omega\uparrow\uparrow 2)^{\omega^4.3 + \omega^2.5 + 1}.(\omega^3.3) + \omega^5.2 + 2}.(\varphi(1,0)^{(\omega\uparrow\uparrow 2)^4.3 + (\omega\uparrow\uparrow 2)^3.3 + 1}.((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^3.(\omega)}))}))}(6)\)
\(\beta(60,6) = f_{\varphi(3,(\varphi(2,(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4).5 + (\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.2 + 2}.(\omega^3.2 + 5) + 3}.(\omega^4 + 5) + 5}.(\omega^2.3 + 4) + 2)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 4)^{\omega^3.2 + \omega^2.5 + 3}}.4 + \varphi(1,(\varphi(2,(\omega\uparrow\uparrow 3)^4 + 5)\uparrow\uparrow 2)^4.(\varphi(1,(\omega\uparrow\uparrow 3)^{\omega}))))}(6)\)
\(\beta(70,6) = f_{\varphi(4,(\varphi(1,(\omega\uparrow\uparrow 4).4 + 5)\uparrow\uparrow 3)^{(\varphi(1,0)\uparrow\uparrow 4)^{(\varphi(1,0)\uparrow\uparrow 5)^3.((\varphi(1,0)\uparrow\uparrow 2)^{\omega^2.5 + 3}.5 + 1) + \omega^5.2 + \omega^3.3 + 5}.(\varphi(1,0)^{(\omega\uparrow\uparrow 2).4 + 4} + 4) + (\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^{\omega^4.2 + \omega.5 + 3}.3 + \omega^5.3}.2 + (\omega\uparrow\uparrow 3).4 + 3}.(\varphi(1,\omega^2.2 + 3)))}(6)\)
\(\beta(80,6) = f_{1}^{4}(f_{2}^{2}(f_{(\varphi(5,(\omega\uparrow\uparrow 2)^{\omega^3.4 + \omega^2.4 + \omega.2 + 4}.(\omega.3) + 2)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^5.(\omega.5) + 2}.((\omega\uparrow\uparrow 5)^4 + 1) + (\omega\uparrow\uparrow 4)^3.5 + (\omega\uparrow\uparrow 2)^{\omega^2.4 + \omega.2 + 1}.2}^{3}(6))) + 2\)
\(\beta(90,6) = f_{\varphi(1,\varphi(\omega.3 + 5,1)^{\varphi(2,(\varphi(2,(\omega\uparrow\uparrow 3)^4.(\omega^4.4 + 2) + 2)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 4)^{\omega^5.3 + \omega^3.5 + \omega.5 + 4}.((\omega\uparrow\uparrow 3)^2.((\omega\uparrow\uparrow 2)^{\omega.5 + 2}.(\omega^2.4) + (\omega\uparrow\uparrow 2)^{\omega.3 + 3}.(\omega^5 + \omega^4.3 + 5) + \omega^2.2) + \omega^5.4 + 5) + 3})})}(6)\)
\(\beta(100,6) = f_{\varphi(\omega^4.3 + \omega.2 + 2,\varphi(1,(\varphi(3,(\omega\uparrow\uparrow 2)^{\omega^4.4 + 5}.(\omega^3.4 + 2) + 3)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3) + \omega^4.2 + \omega.2 + 1}.((\omega\uparrow\uparrow 2)^5.(\omega^4 + \omega + 2) + \omega^5 + 5) + \varphi(1,1)^2.((\varphi(1,0)\uparrow\uparrow 5)^{\omega^5.5 + 3}.(\omega^4.2 + 2) + (\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.4 + \omega^3.4 + \omega^2.4 + \omega.5}.4 + (\omega\uparrow\uparrow 2)}})))}(6)\)
\(\beta(200,6) = f_{(\varphi((\omega\uparrow\uparrow 5)^{\omega^3.5 + 1}.2 + 4,0)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3)^5.(\omega^4.2 + \omega^3 + \omega^2.2 + 4) + (\omega\uparrow\uparrow 2)^5.4 + 4}.((\omega\uparrow\uparrow 2)^4.(\omega^3.5 + 4) + 4) + (\varphi(5,1)\uparrow\uparrow 3)^{(\varphi(5,1)\uparrow\uparrow 8)^{\varphi(1,(\varphi(4,(\omega\uparrow\uparrow 2)^2 + 5)\uparrow\uparrow 2)^{\varphi(1,\varphi(3,0)^{(\omega\uparrow\uparrow 5)^4.((\omega\uparrow\uparrow 3))})})}}}(6)\)
\(\beta(300,6) = f_{\varphi(3,\varphi(1,0),(\varphi(1,0,0)\uparrow\uparrow 3)^{(\varphi(1,0,0)\uparrow\uparrow 5)^{(\varphi(1,0,0)\uparrow\uparrow (f_{\varphi((\omega\uparrow\uparrow 5)^3.5 + (\omega\uparrow\uparrow 2)^5.3 + 5,\varphi(\varphi((\omega\uparrow\uparrow 4)^4.(\omega^4.3 + \omega^2.2 + 1) + 3,(\omega\uparrow\uparrow 4)^2.((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega.3 + 4}.(\omega.3 + 3) + (\omega\uparrow\uparrow 2)^3.(\omega^4 + \omega.4 + 4) + \omega})),0,0),0,0)}(6)))}}),0)}(6)\)
\(\beta(400,6) = f_{\varphi(\omega.4 + 5,\varphi(1,0),\varphi(2,0,1)^4 + \varphi(1,(\varphi(2,0,0)\uparrow\uparrow 3)^{(\varphi(2,\omega^3.4 + \omega^2.5 + \omega.2)\uparrow\uparrow 2)^{\varphi(1,\varphi(2,\omega^2 + 4)^{\varphi(1,(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^4.(\omega^4 + \omega.3 + 4) + (\omega\uparrow\uparrow 2)^2.5 + 4} + (\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^{\omega^2}))})}})),0)}(6)\)
\(\beta(500,6) = f_{\varphi((\omega\uparrow\uparrow 3).3 + \omega^4 + \omega^2.5 + \omega.2 + 5,\omega^4.4 + 5,0)^{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2).3 + 5}.((\omega\uparrow\uparrow 2)^5.5 + \omega^2.4 + 5) + (\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{\omega^4 + 4}.5 + 5) + (\omega\uparrow\uparrow 3)^2.3 + 3}.4 + \varphi(1,0,0)}(6)\)
\(\beta(600,6) = f_{\varphi((\omega\uparrow\uparrow 4)^{\omega^2.3 + \omega.5 + 2}.3 + 5,\varphi(3,\varphi(2,1)^{(\omega\uparrow\uparrow 3)^5.3}.((\omega\uparrow\uparrow 5)^2.((\omega\uparrow\uparrow 3)^4.(\omega^5.3 + \omega^4.2 + \omega^2.3 + 2) + 1) + (\omega\uparrow\uparrow 3)^{\omega^2.5 + 5}.4 + \omega^2 + \omega + 5) + (\omega\uparrow\uparrow 5)^5.((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^2.(\omega.4) + (\omega\uparrow\uparrow 2)})),0)}(6)\)
\(\beta(700,6) = f_{\varphi((\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3)^{\omega^5.5 + \omega^2.5 + 2}.(\omega^4.5) + 2}.(\omega^5.2) + 1,\varphi(2,\varphi(1,3,\varphi(1,\varphi((\omega\uparrow\uparrow 4)^4.((\omega\uparrow\uparrow 2)^{\omega^3.4}.5 + 5) + 5,4) + 1)),0),0)}(6)\)
\(\beta(800,6) = f_{\varphi((\varphi(2,(\varphi(2,(\omega\uparrow\uparrow 4).4 + 3)\uparrow\uparrow 3) + (\omega\uparrow\uparrow 4)^2.4 + 5)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 5)^4.4}.4 + 1,\varphi((\omega\uparrow\uparrow 5)^4.((\omega\uparrow\uparrow 2).(\omega^5 + \omega + 1) + \omega^2.4 + 4) + 3,\varphi((\omega\uparrow\uparrow 5).(\omega^2.2 + 4) + (\omega\uparrow\uparrow 3).(\omega^3.3 + \omega),0)),0)}(6)\)
\(\beta(900,6) = f_{\varphi(\varphi(4,\varphi(3,(\varphi(1,(\omega\uparrow\uparrow 5)^4.(\omega^3.2 + \omega.4 + 4) + (\omega\uparrow\uparrow 5)^2.5 + (\omega\uparrow\uparrow 3)^{\omega^4.2 + 2}.4 + (\omega\uparrow\uparrow 2)^{\omega^4 + \omega.3 + 3}.2 + 5)\uparrow\uparrow 2)^{(\varphi(1,(\omega\uparrow\uparrow 5)^2.4 + (\omega\uparrow\uparrow 5).3 + \omega^4.4 + 5)\uparrow\uparrow 5)^3.((\omega\uparrow\uparrow 2)^4 + (\omega\uparrow\uparrow 2)^3.(\omega^3.4 + \omega^2.3 + 4) + 1) + (\omega\uparrow\uparrow 3)^{\omega}})),0,0)}(6)\)
\(\beta(1000,6) = f_{\varphi((\varphi(\omega^4.2 + 5,(\omega\uparrow\uparrow 5)^4.((\omega\uparrow\uparrow 2)^{\omega.3 + 4}.(\omega^3 + 3) + 3) + (\omega\uparrow\uparrow 4)^{\omega + 3}.(\omega^3 + 5) + \omega^4 + 4)\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^5.4 + \omega^5.2 + 2}.(\omega^4.4 + \omega.4 + 4) + (\omega\uparrow\uparrow 3).3 + 1) + \varphi(4,(\omega\uparrow\uparrow 2)^{\omega.3 + 3}),0,0)}(6)\)
\(\beta(1500,6) = f_{\varphi(2,\varphi(1,(\varphi(3,(\omega\uparrow\uparrow 3)^{\omega^2.4}.5 + 5,(\omega\uparrow\uparrow 2)^{\omega^2.4 + \omega.4 + 5}.(\omega^2.4 + \omega.4 + 5) + 3)\uparrow\uparrow 3)^{\varphi(1,(\varphi(2,4,\omega^5.3 + 2)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega.3 + 3}.(\omega^3.3 + \omega + 4) + 4}.(\varphi((\omega\uparrow\uparrow 5)^4.((\omega\uparrow\uparrow 3)),0)))}),0,0)}(6)\)
\(\beta(2000,6) = f_{\varphi(5,\varphi(2,2,4,\varphi(1,\varphi(2,2,4,(\omega\uparrow\uparrow 3)^5.(\omega^2.5 + 3) + 5)^{\varphi((\omega\uparrow\uparrow 4)^2.3 + \omega^2.2 + 2,\omega^5.5 + \omega^4.5 + \omega^2 + 2,(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{\omega^5.2 + \omega^4 + \omega^3.5 + 3}.(\omega^2.5 + 4) + \omega})})),0,0)}(6)\)
\(\beta(2500,6) = f_{\varphi((\omega\uparrow\uparrow 3)^4.4 + 4,(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^5.(\omega^4.4 + 3)} + (\omega\uparrow\uparrow 3)^2.((\omega\uparrow\uparrow 2)^2.(\omega^5 + \omega^4.4 + 4) + 5) + \omega.4 + 5,(\varphi(5,4)\uparrow\uparrow 5)^{(\varphi(5,0)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 5)^4.3 + 2}.5 + 3}.((\omega\uparrow\uparrow 2)^2.(\omega.2 + 1)) + \omega^4 + \omega.4 + 4,\varphi(\omega,0))}(6)\)
\(\beta(3000,6) = f_{\varphi((\omega\uparrow\uparrow 5)^4.((\omega\uparrow\uparrow 2)^{\omega}.(\omega + 2)) + 2,(\omega\uparrow\uparrow 2)^{\omega.4}.3 + \omega^3.2 + \omega^2.5 + \omega.2 + 5,\varphi((\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2).4 + 4}.4 + 5,\varphi((\omega\uparrow\uparrow 2)^{\omega^4.5 + 1}.(\omega^2.5 + 4) + \omega,0,0,0),0,0),0)}(6)\)
\(\beta(3500,6) = f_{\varphi((\varphi(3,3)\uparrow\uparrow 3)^3.((\varphi(1,(\varphi(3,2)\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3)^5.(\omega.5 + 3) + 3}.(\omega^3.5 + 3) + (\omega\uparrow\uparrow 3)^4.3 + \omega^3.5 + 5)\uparrow\uparrow 5)^5 + (\varphi(1,(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^2.2 + 3})\uparrow\uparrow 3)^5.(\varphi(1,(\omega\uparrow\uparrow 2)^{\omega^2.4 + 2}.(\omega + 5) + 5)^3.(\varphi(1,(\omega\uparrow\uparrow 2)^2.(\omega.4) + (\omega\uparrow\uparrow 2))))),0,0,0)}(6)\)
\(\beta(4000,6) = f_{\varphi(\varphi(\omega.4 + 5,\varphi(\omega.4 + 3,(\omega\uparrow\uparrow 5)^3.5 + (\omega\uparrow\uparrow 3).5 + \omega^4.5 + \omega^2.5 + 5)^{(\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2)^{\omega^4.5 + 3}.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.4}.2 + (\omega\uparrow\uparrow 2)^4.(\omega^2.5) + (\omega\uparrow\uparrow 2)^3.(\omega^4 + \omega.4 + 4) + \omega}),0,0,0)}(6)\)
\(\beta(4500,6) = f_{\varphi((\varphi((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^2.(\omega.4 + 3) + (\omega\uparrow\uparrow 2).(\omega^3.3 + \omega.4 + 4) + \omega^4.5 + 4}.(\omega^3.4 + 1) + \omega^5.4 + 3,(\omega\uparrow\uparrow 3)^3.2)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 4)^3.3 + 3}.((\omega\uparrow\uparrow 4)^{\omega^5.3 + 4}.2 + (\omega\uparrow\uparrow 3)^2 + 1) + \varphi(\omega^4.4 + \omega^2.3 + \omega,0),0,0,0)}(6)\)
\(\beta(5000,6) = f_{\varphi((\varphi(1,5,2)\uparrow\uparrow 4)^2.(\varphi(2,\varphi(1,(\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^2.4 + \omega^3.2 + \omega.4))^5.5 + (\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3).3 + \omega^4.3 + 2}.3 + (\omega\uparrow\uparrow 5)^2.2 + (\omega\uparrow\uparrow 5).((\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^4.3 + \omega}}))),0,0,0)}(6)\)
\(\beta(6000,6) = f_{\varphi(\varphi((\omega\uparrow\uparrow 2)^4.2 + \omega^5.5 + \omega^4.5,\varphi(1,\varphi(1,(\varphi(1,3)\uparrow\uparrow 5)^{\varphi(1,3).((\varphi(1,1)\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 3)^{\omega^3.4 + \omega.3}.2 + (\omega\uparrow\uparrow 2)^{\omega^2.5 + \omega.2}.5 + \omega^3.3}.((\varphi(1,1)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 3)^4.(\omega^4.4 + 1) + (\omega\uparrow\uparrow 3)^3.5 + 4}.((\omega\uparrow\uparrow 3)^{\omega})))})),0),0,0,0)}(6)\)
\(\beta(7000,6) = f_{\varphi(\varphi(\varphi(3,\varphi(3,(\omega\uparrow\uparrow 5)^5.(\omega^5.3 + 2) + 5)^{\varphi(1,5)^{\varphi(1,2)^2.((\omega\uparrow\uparrow 5)^3.((\omega\uparrow\uparrow 2)^3.(\omega^3.5 + \omega^2.2 + 4) + (\omega\uparrow\uparrow 2)^2.3 + \omega.2 + 2) + (\omega\uparrow\uparrow 3)^4.2 + (\omega\uparrow\uparrow 3).2 + \omega^2.2 + \omega.3 + 2) + 5}.(\omega.4 + 5) + (\omega\uparrow\uparrow 2)^{\omega.3 + 3}}),0,0),0,0,0)}(6)\)
\(\beta(8000,6) = f_{\varphi(1,\varphi(4,(\varphi(2,5)\uparrow\uparrow 5)^3.(\omega^2.5 + 3) + 2)^4.2 + (\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2).5 + 2}.(\omega^4.2 + 2) + \omega^3.3 + \omega^2 + 1,1,(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{\omega^5.3 + \omega^2.3 + 1}}.3 + \omega^3.3 + \omega.4 + 5,\varphi((\omega\uparrow\uparrow 2)^3.(\omega),0))}(6)\)
\(\beta(9000,6) = f_{\varphi(3,1,(\varphi(\omega^4.2 + 1,(\omega\uparrow\uparrow 2))\uparrow\uparrow 2)^{\varphi(\omega^2 + \omega + 4,(\varphi(\omega.3 + 2,(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2).4 + \omega.2 + 5}.(\omega^5 + \omega.5 + 5) + 5)\uparrow\uparrow 2)^{\varphi(4,\varphi(4,(\omega\uparrow\uparrow 5)^3.5 + (\omega\uparrow\uparrow 4)^{\omega^5.4 + \omega.4 + 2}.2 + \omega^3.3 + \omega))})},0,0)}(6)\)
\(\beta(10000,6) = f_{\varphi(4,\varphi(1,\varphi(1,(\omega\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 3)^{\omega.5}.(\omega.5 + 4) + (\omega\uparrow\uparrow 3)^2.3 + (\omega\uparrow\uparrow 3)}.3 + (\omega\uparrow\uparrow 2)^2.(\omega^2 + 4) + 5}.((\omega\uparrow\uparrow 3)^4.4 + \omega^3.3 + \omega.4 + 4) + (\omega\uparrow\uparrow 3).(\omega + 4) + \omega^4.4 + \omega^2.3 + \omega,0),0),0,0,0)}(6)\)
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