User blog:B1mb0w/Beta Function Code Version 4

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

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

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

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:

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Granularity Examples \(\beta(5.1963,3)\) to \(\beta(9,3)\)
Version 3 code correctly transitions from \(\omega^{\omega}\) to \(\omega^{\omega^{\omega}}\) without error:

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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:

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Granularity Examples \(\beta(10.079,4)\) to \(\beta(16,4)\)
These examples transition up to \(\omega\uparrow\uparrow 4\) in base \(v = 4\):

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Undesired Values that are still generated
Further to the above example, here are other ranges of undesired values:

This error will remain for now and I will try to correct it in a future version of the code.

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Granularity Examples near \(\beta(12.2118455,3)\)
The Beta Function successfully generates the correct Veblen ordinals as seen in these two examples:

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Valid Sequence Counts
WORK IN PROGRESS

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

\(\beta(24.7,5) = f_{\omega^3.(f_{2}^3(f_{4}^3(5)).(2^{f_{2}^4(5) + 2})) + f_{(\omega\uparrow\uparrow 2)^{\omega^3.4 + 3}.(\omega^3)}(5)}(f_{(\omega\uparrow\uparrow 4)^{\omega^4.4}.4 + 1}^4(5))\)

\(\beta(24.71,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow f_{3}^4(5).4 + f_{2}^{f_{3}^{f_{\omega}^2(5).(2^{f_{f_{3}^4(5) + 3}(f_{\omega}(5)) + 2}) + 3}(f_{\omega + 1}(5)).8 + 5}(f_{\omega + 1}^2(5)))}}(5)\)

\(\beta(24.72,5) = f_{\omega^2.(f_{(\omega\uparrow\uparrow 4)^{\omega^4.3 + 4}.4 + (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3).(\omega^2 + \omega.3)}}(5))}(f_{(\omega\uparrow\uparrow 4)^{\omega^4.4 + 1} + 3}^2(5))\)

\(\beta(24.73,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow f_{2}(f_{3}^4(5)).(2^{f_{\varphi((\omega\uparrow\uparrow 2)^3.(\omega^2.2 + 4) + 4,\varphi(\varphi(2,\varphi(1,(\varphi(5,\omega^4 + 1)\uparrow\uparrow 2)^{\varphi(3,0)})),0,0),0,0,0)}(5)}))}}(5)\)

\(\beta(24.74,5) = f_{(\omega\uparrow\uparrow 4)^{\omega^4.4 + 1}.((\omega\uparrow\uparrow 2).(\omega^4 + \omega^3.2 + 4) + \omega^3.2 + 4) + (\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^3.(\omega^3.2 + \omega^2 + 1) + (\omega\uparrow\uparrow 2)^2)}(5)\)

\(\beta(24.75,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow f_{4}(5))^3.2 + (\omega\uparrow\uparrow 4)^{\omega^2 + 2}.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3)^2.3 + 2}.(\omega.4 + 4) + \omega.3)}}(5)\)

\(\beta(24.85,5) = f_{(\omega\uparrow\uparrow 4)^{\omega^4.4 + \omega + 2}.(\omega^2.3 + 2) + (\omega\uparrow\uparrow 2)^4.(\omega^3 + 1) + \omega^3.3 + 1}^2(5).(2^{f_{4}(5) + f_{3}^2(5).32})\)

\(\beta(24.86,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow f_{3}^2(f_{4}^2(5)) + 4)^{(\omega\uparrow\uparrow f_{(\varphi((\omega\uparrow\uparrow 3)^2.2 + 3,\omega^4 + 4)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 4).((\omega\uparrow\uparrow 2)^{\omega^4})}}(5))}}}(5)\)

\(\beta(24.87,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow f_{3}^{f_{\varphi(\varphi(\varphi((\omega\uparrow\uparrow 2)^4.3 + 2,(\omega\uparrow\uparrow 4)^{\omega.4 + 2}.((\omega\uparrow\uparrow 3)^{\omega.4 + 3}.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 4)^{\omega^3.4 + \omega.2}}))),0,0),0,0,0)}(5)}(f_{4}^2(5)))}}(5)\)

\(\beta(24.88,5) = f_{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 4)^4.((\omega\uparrow\uparrow 3)^{\omega^4.(f_{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3)^{\omega}}}(5))})}}(f_{(\omega\uparrow\uparrow 4)^{\omega^4.4 + \omega.3 + 4} + (\omega\uparrow\uparrow 2).(\omega^4.2 + 4) + 3}(5))\)

\(\beta(24.89,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow f_{4}^3(5) + f_{4}^2(5).(2^{f_{\varphi(\varphi(2,\varphi((\omega\uparrow\uparrow 4).((\omega\uparrow\uparrow 2)^4.(\omega.3 + 2) + 3) + \omega^3.4 + 2,(\varphi(3,3)\uparrow\uparrow 3)^4),0),0,0,0,0)}(5)}))}}(5)\)

\(\beta(24.9,5) = f_{(\omega\uparrow\uparrow 4)^{\omega^4.4 + \omega^2 + 3}.4 + (\omega\uparrow\uparrow 4)^{\omega^3.3 + 3}.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3).(\omega^4.3 + 1) + (\omega\uparrow\uparrow 2).2 + 2}.2 + \omega)}(5)\)

\(\beta(24.91,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow f_{2}^44(f_{4}^3(5)).4 + 3)^{(\omega\uparrow\uparrow f_{\varphi(\varphi(2,\varphi(f_{6}^{f_{3}(5).(2^{f_{2}^4(5).2 + f_{2}^3(5) + 2}) + 5}(f_{7}^3(5)),0)),0,0)}(5))}}}(5)\)

\(\beta(24.92,5) = f_{\omega^{f_{(\omega\uparrow\uparrow 3)^4.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega^2 + \omega + 2}.(\omega^3.3)}})}(5)}}(f_{(\omega\uparrow\uparrow 4)^{\omega^4.4 + \omega^2.2 + \omega.4}.3 + 3}(5))\)

\(\beta(24.93,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow f_{3}^{f_{4}^4(5) + f_{2}^3(5).(2^{f_{\varphi(\varphi(3,(\omega\uparrow\uparrow 2)^2.3 + 3,(\omega\uparrow\uparrow 4)^2.((\omega\uparrow\uparrow 3)^{\omega.4 + 1}.3)),0,0,0)}(5)})}(f_{4}^3(5)))}}(5)\)

\(\beta(24.94,5) = f_{4}^{f_{4}^2(5).(2^{f_{2}(5).4 + 2}) + 3}(f_{5}(f_{(\omega\uparrow\uparrow 4)^{\omega^4.4 + \omega^2.4 + 4}.2}^4(5))) + f_{4}(5)\)

\(\beta(24.95,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow f_{4}^4(5) + 24)^{\omega^4.2 + \omega^2 + 2}.((\omega\uparrow\uparrow f_{3}^4(5) + 20 + 10 + f_{\varphi((\omega\uparrow\uparrow 3)^3.((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3)^3}),0,0,0)}(5)))}}(5)\)

\(\beta(24.96,5) = f_{f_{2}^3(5)}(f_{(\omega\uparrow\uparrow 4)^{\omega^4.4 + \omega^3 + \omega^2.4}.((\omega\uparrow\uparrow 3)^{\omega^3.3 + 4}.3 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^3.3 + 3}^4(5))\)

\(\beta(24.97,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow f_{2}^2(f_{4}^4(5)) + f_{2}^3(5).(2^{f_{\varphi(1,\varphi(1,0,0),0)}(5)}))}}(5)\)

\(\beta(24.98,5) = f_{(\omega\uparrow\uparrow 4)^{\omega.(f_{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3)^2.4 + (\omega\uparrow\uparrow 3).(\omega^3)}}}(5))}}(f_{(\omega\uparrow\uparrow 4)^{\omega^4.4 + \omega^3.3 + 4}.((\omega\uparrow\uparrow 2)^3.(\omega^3)) + 4}^3(5))\)

\(\beta(24.99,5) = f_{(\omega\uparrow\uparrow 4)^{(\omega\uparrow\uparrow f_{3}^{f_{\varphi((\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3)^{\omega + 4}.(\omega^3.3 + 4) + 1}.(\omega^2.3 + 2) + (\omega\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).3 + \omega^2.2},0)}(5)}(f_{4}^4(5)))}}(5)\)

\(\beta(25,5) = f_{\varphi(1,0)}(5)\)

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