FANDOM


Alpha Function - Sequence Generating Code

The Alpha Function has been defined using program code shown below.

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

This version has been completely replaced by my Version 6 blog. Please refer to that blog instead.


Sequence Generating Code Version 5

Version 5 of the Sequence Generating Code will create long finite integer strings to define large Veblen ordinals and FGH functions (up to the size of SVO). Refer to my other blogs on Unique Ordinal Representation and Program Code Version 4 for more information.


Key Changes to Version 4 Code

Version 5 will extend Program Code Version 4 code in two specific ways:

Extend \(n_0\) sequence to support Veblen ordinals of the form:

  • \(\varphi(\alpha,\varphi(\beta,0) + \gamma)\) where \(\beta > \alpha\) and \(1 < \gamma < \varphi(\beta,0)\)
  • The \(n_0\) sequence, i.e.:
    • \(n(q) = (Q(0:,g_{[Q]}(Q),S,n_1(Q)))\)
  • can only generate Veblen ordinals of the form:
    • \(\varphi(\alpha,\varphi(\beta,0) + 1)\) where \(\beta > \alpha\) and \(1 = \gamma\) by definition

Extend \(f_{\gamma}^U(V)\) sequence to explicitly allow \(U\) and \(V\) to become very big integers. Version 4 implicitly allows \(U\) and \(V\) to be any integer up to \(\omega\), however, in practice, it is rare for Version 4 to generate a number \(> 20\).

  • This is not due to a bug in the code, but instead, the Alpha Number (real number) required to generate the necessary sequences would be so finely tuned to a prohibitive number of decimal places, that, it could not be calculated without computer code to handle high precision numbers, and could not be printed on a blog, even if it was calculated.
  • Therefore Version 5 will use sequences as substitutions to allow \(U\) and \(V\) to grow rapidly (at the growth rate of Small Veblen Ordinal).
  • CORRECTION: This extension is unnecessary. Provided any finite value for \(U\) and \(V\) can be reached (which they can), then it is less important to make them explicitly easier to access. Instead, an extension will be required to make numbers of the form:
    • \(f_1^2(3) < 13 < 14 < 15 < f_1^2(4)\)
  • accessible will be more useful. I.e. the Alpha Function cannot generate a string for the numbers 13 to 15. The closest it can get is either of the two FGH functions which equate to then numbers 12 and 16.
  • An extension of this form will be attempted in Version 6 of this code.


Version 5 Code

We start with the Version 4 generating rules as follows:

\(g(q) = (q,q(0:a,q(1:,m_0(q)),T,g_E(q),g_C(q),g_a(q)))\)

\(m(q) = (p,R,g_F(q),g_{[q]}(q),n_0(q))\)

\(n(q) = (Q,Q(0:,g_{[Q]}(Q),S,n_1(Q)))\)

The \(n(q)\) sequence definition is extended with a further sequence element \(g_X\):

  • \(n_k(q) = (Q(0:,g_{[Q]}(Q),S,g_X,n_{k+1}(Q)))\) and these rules
    • similar generating rules as before
    • \(g_X =\) a generated \(g\) sequence smaller than: \((q-n_{[k-1]})\)


Valid Sequence Counts

There are a large number of Valid Sequences using the above code. A previous estimate of Valid Sequence Counts for Version 4 code estimated the growth rate for the function \(C(n)\) to be:

\(f_{\omega}(n) << C(n) << f_{\omega + 1}(n)\)

Where \(C(n)\) is the number of valid sequences that can be constructed only using integers between 0 to n.

The extension of the code for Version 5 will increase the number of valid sequences. Without proof, I estimate the growth rate to be:

\(f_{\omega + 1}(n) << C(n) << f_{\omega.2}(n)\)


Example Sequences \(\alpha(100)\) to \(\alpha(300)\)

The following example sequences have been generated by the Alpha Function for input values of 100 to 300.

\(\alpha(100) = (<2,<0,0,<0,0>,<0,0>,0>,0,<0,0>,<0,0>,<0,0>>,2,2)\)

\(\alpha(110) = (<2,<0,0,<1,5,<0,1>,<0,2>,<1,1,<0,1>,<1,0,<0,5>,<0,4>,<1,0,<0,4>,<0,4>,<0,3>>>,<0,5>>>,<1,2,<0,4>,<1,0,<0,9>,<0,1>,<0,1>>,<0,2>>,0>,0,<0,2>,<2,<0,0,<0,1>,<0,5>,<1,<0,0>,5,<2,<0,0,<0,1>,<0,4>,<1,<0,0>,2,<0,10>,0>,2,<0,2>,<0,10>,<0,3>>,0>,5,<0,5>,<2,<0,0,<0,1>,<0,0>,0>,4,<0,3>,<2,<0,0,<0,0>,<1,2,<0,2>,<0,0>,<1,2,<0,0>,<1,1,<0,10>,<1,0,<0,2>,<0,1>,<0,0>>,<1,1,<0,8>,<0,0>,<1,1,<0,2>,<0,3>,<0,2>>>>,<1,1,<0,9>,<0,2>,<0,5>>>>,0>,4,<0,5>,<2,<0,0,<0,0>,<0,0>,0>,6,<0,6>,<2,<0,0,<0,0>,<0,0>,0>,3,<0,5>,<0,5>,<0,5>>,<0,5>>,<0,5>>,<0,5>>,<0,5>>,<0,5>>,3,16)\)

\(\alpha(120) = (<2,<0,1,<1,2,<0,1>,<1,1,<0,5>,<1,0,<0,3>,<0,2>,<1,0,<0,0>,<0,1>,<0,2>>>,<0,5>>,<1,0,<0,5>,<0,3>,<1,0,<0,0>,<0,2>,<0,1>>>>,<1,3,<0,5>,<0,2>,<1,1,<0,2>,<1,0,<0,5>,<0,3>,<0,0>>,<1,0,<0,0>,<0,0>,<0,2>>>>,<1,<1,0,<0,4>,<0,0>,<0,5>>,4,<1,5,<0,5>,<0,1>,<0,11>>,0>,5,<0,13>,<0,3>,<0,5>>,7,15)\)

\(\alpha(130) = (<2,<0,2,<0,5>,<0,21>,<1,<0,3>,4,<0,0>,0>,10,<1,8,<0,0>,<1,1,<0,2>,<1,0,<0,1>,<0,3>,<0,0>>,<0,2>>,<0,3>>,<0,3>,<2,<0,2,<0,4>,<1,4,<0,5>,<0,1>,<1,1,<0,1>,<1,0,<0,5>,<0,2>,<0,1>>,<1,1,<0,0>,<0,4>,<1,0,<0,11>,<0,1>,<0,0>>>>>,0>,0,<0,5>,<1,3,<0,0>,<0,4>,<1,2,<0,2>,<1,1,<0,8>,<1,0,<0,0>,<0,3>,<0,1>>,<1,0,<0,11>,<0,2>,<1,0,<0,4>,<0,4>,<0,2>>>>,<1,2,<0,0>,<1,0,<0,6>,<0,4>,<0,5>>,<0,5>>>>,<2,<0,2,<0,1>,<1,3,<0,9>,<1,1,<0,4>,<0,4>,<1,1,<0,0>,<0,4>,<1,0,<0,4>,<0,2>,<1,0,<0,1>,<0,0>,<0,0>>>>>,<0,0>>,0>,0,<0,0>,<0,0>,<0,0>>>>,2,22)\)

\(\alpha(140) = (<2,<0,3,<0,0>,<1,9,<0,1>,<1,3,<0,2>,<1,0,<0,1>,<0,1>,<1,0,<0,0>,<0,5>,<0,2>>>,<1,0,<0,1>,<0,2>,<0,7>>>,<1,5,<0,5>,<0,1>,<0,4>>>,0>,3,<0,4>,<0,3>,<1,4,<0,2>,<1,2,<0,2>,<1,1,<0,6>,<1,0,<0,3>,<0,5>,<1,0,<0,1>,<0,1>,<1,0,<0,0>,<0,1>,<0,0>>>>,<0,0>>,<0,9>>,<1,3,<0,4>,<0,0>,<0,3>>>>,3,15)\)

\(\alpha(150) = (<2,<0,3,<1,2,<0,5>,<0,5>,<1,0,<0,1>,<0,5>,<0,2>>>,<0,3>,0>,1,<0,8>,<1,5,<0,0>,<1,3,<0,5>,<1,0,<0,1>,<0,3>,<0,5>>,<0,1>>,<1,3,<0,0>,<1,0,<0,2>,<0,5>,<0,16>>,<0,3>>>,<1,10,<0,4>,<1,2,<0,3>,<1,1,<0,0>,<1,0,<0,0>,<0,3>,<0,2>>,<0,5>>,<1,1,<0,2>,<1,0,<0,2>,<0,1>,<1,0,<0,1>,<0,3>,<0,4>>>,<0,0>>>,<0,5>>>,7,21)\)

\(\alpha(160) = (<2,<0,4,<0,4>,<0,0>,0>,2,<0,2>,<2,<0,4,<0,0>,<0,5>,0>,1,<0,4>,<2,<0,4,<0,0>,<0,3>,0>,2,<0,4>,<2,<0,4,<0,0>,<0,2>,0>,1,<0,7>,<2,<0,3,<0,5>,<0,5>,0>,2,<0,11>,<2,<0,3,<0,1>,<1,9,<0,2>,<1,5,<0,4>,<0,0>,<0,11>>,<0,1>>,<1,<0,0>,3,<1,4,<0,3>,<0,0>,<0,3>>,0>,5,<0,0>,<2,<0,3,<0,0>,<1,0,<0,0>,<0,1>,<0,4>>,0>,1,<0,10>,<1,4,<0,3>,<0,2>,<1,1,<0,5>,<1,0,<0,5>,<0,5>,<0,8>>,<1,1,<0,1>,<0,2>,<1,0,<0,0>,<0,2>,<0,1>>>>>,<2,<0,0,<1,2,<0,2>,<0,4>,<1,1,<0,4>,<1,0,<0,2>,<0,4>,<1,0,<0,0>,<0,1>,<0,6>>>,<0,1>>>,<1,0,<0,3>,<0,0>,<0,6>>,<1,<1,1,<0,3>,<1,0,<0,0>,<0,0>,<0,0>>,<0,0>>,0,<0,0>,0>,0,<0,0>,<0,0>,<0,0>>>,<0,0>>,<0,0>>,<0,0>>,<0,0>>,<0,0>>,<0,0>>,2,12)\)

\(\alpha(170) = (<2,<0,4,<1,4,<0,5>,<0,3>,<1,1,<0,10>,<0,3>,<0,0>>>,<1,3,<0,4>,<0,2>,<1,0,<0,4>,<0,11>,<1,0,<0,0>,<0,2>,<0,1>>>>,0>,3,<1,0,<0,0>,<0,3>,<0,5>>,<1,5,<0,0>,<1,0,<0,4>,<0,3>,<1,0,<0,2>,<0,11>,<1,0,<0,1>,<0,0>,<1,0,<0,0>,<0,4>,<0,2>>>>>,<1,1,<0,9>,<1,0,<0,4>,<0,2>,<0,1>>,<0,5>>>,<1,11,<0,3>,<1,9,<0,5>,<0,2>,<0,1>>,<1,6,<0,1>,<0,5>,<1,3,<0,2>,<0,2>,<1,0,<0,0>,<0,5>,<0,4>>>>>>,4,15)\)

\(\alpha(180) = (<2,<0,5,<0,5>,<0,0>,<1,<0,2>,0,<0,2>,0>,5,<1,4,<0,7>,<1,0,<0,0>,<0,1>,<0,3>>,<0,2>>,<0,3>,<2,<0,4,<0,2>,<0,2>,0>,4,<0,2>,<2,<0,2,<1,1,<0,10>,<1,0,<0,1>,<0,0>,<1,0,<0,0>,<0,10>,<0,10>>>,<0,14>>,<0,1>,<1,<1,1,<0,5>,<1,0,<0,2>,<0,17>,<0,5>>,<1,1,<0,2>,<1,0,<0,5>,<0,0>,<0,9>>,<0,0>>>,17,<1,1,<0,3>,<0,1>,<1,0,<0,3>,<0,2>,<1,0,<0,0>,<0,5>,<0,5>>>>,1>,<1,<0,0>,4,<1,0,<0,3>,<0,4>,<1,0,<0,0>,<0,5>,<0,0>>>,0>,0,<0,3>,<0,10>,<2,<0,0,<1,11,<0,5>,<0,10>,<1,11,<0,3>,<0,1>,<1,1,<0,0>,<1,0,<0,0>,<0,4>,<0,4>>,<0,2>>>>,<0,2>,<1,<0,3>,3,<1,3,<0,3>,<1,0,<0,0>,<0,0>,<0,0>>,<0,0>>,0>,0,<0,0>,<0,0>,<0,0>>>,<0,0>>>,2,18)\)

\(\alpha(190) = (<2,<0,5,<1,4,<0,10>,<1,1,<0,5>,<1,0,<0,0>,<0,8>,<0,4>>,<0,3>>,<0,0>>,<1,2,<0,4>,<0,3>,<1,0,<0,1>,<0,5>,<0,5>>>,0>,16,<2,<0,0,<0,5>,<1,1,<0,5>,<0,3>,<1,1,<0,3>,<0,3>,<1,1,<0,1>,<0,3>,<1,0,<0,5>,<0,0>,<1,0,<0,2>,<0,6>,<0,2>>>>>>,<1,<0,0>,0,<2,<0,0,<0,2>,<1,10,<0,4>,<1,1,<0,3>,<1,0,<0,4>,<0,5>,<0,4>>,<0,4>>,<1,6,<0,16>,<0,5>,<1,6,<0,4>,<1,3,<0,0>,<1,2,<0,8>,<0,5>,<1,1,<0,0>,<1,0,<0,5>,<0,1>,<1,0,<0,2>,<0,3>,<1,0,<0,0>,<0,0>,<0,0>>>>,<0,0>>>,<0,0>>,<0,0>>>>,0>,0,<0,0>,<0,0>,<0,0>>,0>,0,<0,0>,<0,0>,<0,0>>,<0,0>,<0,0>>,2,17)\)

\(\alpha(200) = (<2,<0,8,<0,3>,<1,0,<0,1>,<0,9>,<1,0,<0,0>,<0,3>,<0,3>>>,0>,5,<0,1>,<2,<0,0,<0,1>,<0,4>,0>,0,<0,9>,<0,3>,<0,1>>,<1,5,<0,0>,<0,7>,<0,3>>>,2,10)\)

\(\alpha(210) = (<2,<0,11,<0,5>,<0,1>,<1,<0,2>,5,<2,<0,3,<0,7>,<1,3,<0,0>,<0,2>,<0,4>>,<1,<0,6>,7,<1,0,<0,0>,<0,1>,<0,3>>,0>,1,<0,3>,<1,0,<0,5>,<0,0>,<0,2>>,<1,8,<0,3>,<0,1>,<1,6,<0,4>,<0,2>,<1,2,<0,4>,<0,1>,<1,1,<0,3>,<1,0,<0,1>,<0,3>,<0,6>>,<0,2>>>>>>,1>,<1,<0,0>,4,<1,5,<0,3>,<0,0>,<1,4,<0,0>,<0,1>,<1,3,<0,0>,<0,2>,<1,0,<0,0>,<0,4>,<0,4>>>>>,0>,3,<0,0>,<0,5>,<0,0>>,11,19)\)

\(\alpha(220) = (<2,<1,0,<0,2>,<1,2,<0,2>,<1,1,<0,3>,<1,0,<0,2>,<0,5>,<1,0,<0,1>,<0,8>,<0,3>>>,<1,1,<0,1>,<1,0,<0,2>,<0,1>,<0,11>>,<0,1>>>,<0,0>>,<1,<0,0>,1,<0,2>,0>,1,<1,0,<0,9>,<0,1>,<0,2>>,<2,<1,0,<0,1>,<0,3>,<1,<0,0>,3,<0,4>,0>,2,<0,5>,<2,<0,3,<1,4,<0,3>,<0,4>,<0,3>>,<0,7>,<1,<1,4,<0,3>,<0,0>,<1,1,<0,4>,<1,0,<0,2>,<0,0>,<1,0,<0,1>,<0,2>,<1,0,<0,0>,<0,3>,<0,5>>>>,<0,1>>>,2,<2,<0,2,<1,9,<0,1>,<1,7,<0,3>,<0,3>,<1,6,<0,4>,<0,4>,<0,11>>>,<0,9>>,<1,6,<0,6>,<1,4,<0,3>,<1,0,<0,0>,<0,0>,<0,0>>,<0,0>>,<0,0>>,0>,0,<0,0>,<0,0>,<0,0>>,0>,0,<0,0>,<0,0>,<0,0>>,<0,0>>,<0,0>>,2,12)\)

\(\alpha(230) = (<2,<1,0,<1,1,<0,2>,<0,6>,<1,0,<0,6>,<0,3>,<1,0,<0,0>,<0,4>,<0,8>>>>,<1,3,<0,9>,<0,2>,<1,3,<0,7>,<1,2,<0,2>,<0,3>,<1,2,<0,1>,<1,1,<0,7>,<1,0,<0,1>,<0,1>,<1,0,<0,0>,<0,5>,<0,5>>>,<1,0,<0,4>,<0,4>,<1,0,<0,2>,<0,0>,<0,1>>>>,<0,2>>>,<1,0,<0,7>,<0,3>,<0,1>>>>,<1,<1,0,<0,3>,<0,3>,<0,2>>,1,<2,<0,2,<0,0>,<1,6,<0,3>,<1,0,<0,4>,<0,1>,<0,5>>,<0,22>>,0>,4,<0,4>,<0,2>,<0,5>>,1>,<1,<1,0,<0,0>,<0,6>,<0,0>>,5,<1,0,<0,3>,<0,5>,<1,0,<0,1>,<0,5>,<0,9>>>,1>,<1,<1,0,<0,0>,<0,4>,<0,3>>,3,<1,3,<0,3>,<1,0,<0,0>,<0,0>,<0,0>>,<0,0>>,0>,0,<0,0>,<0,0>,<0,0>>,2,23)\)

\(\alpha(240) = (<2,<1,0,<1,11,<0,3>,<0,4>,<1,0,<0,4>,<0,0>,<0,4>>>,<0,4>,<1,<1,3,<0,0>,<1,0,<0,0>,<0,1>,<0,1>>,<0,0>>,10,<0,3>,0>,4,<1,5,<0,1>,<0,1>,<0,5>>,<1,0,<0,4>,<0,4>,<1,0,<0,1>,<0,3>,<0,0>>>,<2,<0,7,<0,3>,<0,0>,<1,<0,0>,2,<0,3>,0>,2,<0,5>,<1,9,<0,5>,<1,2,<0,2>,<0,0>,<0,2>>,<1,4,<0,4>,<1,3,<0,14>,<0,5>,<0,1>>,<0,3>>>,<1,4,<0,3>,<1,3,<0,1>,<0,1>,<1,3,<0,0>,<0,5>,<0,4>>>,<1,1,<0,3>,<0,5>,<0,9>>>>>,11,24)\)

\(\alpha(250) = (<2,<1,1,<0,4>,<1,15,<0,0>,<1,12,<0,6>,<1,1,<0,5>,<1,0,<0,3>,<0,0>,<1,0,<0,1>,<0,2>,<1,0,<0,0>,<0,0>,<0,3>>>>,<0,2>>,<0,2>>,<1,0,<0,5>,<0,5>,<0,0>>>,0>,3,<0,1>,<2,<1,0,<0,2>,<0,5>,0>,2,<0,5>,<2,<1,0,<0,1>,<0,4>,0>,2,<0,3>,<1,0,<0,3>,<0,7>,<0,2>>,<0,1>>,<2,<0,3,<1,1,<0,4>,<0,5>,<1,1,<0,3>,<1,0,<0,6>,<0,5>,<0,7>>,<1,1,<0,0>,<1,0,<0,0>,<0,5>,<0,2>>,<1,0,<0,1>,<0,4>,<0,3>>>>>,<1,5,<0,1>,<0,5>,<0,5>>,0>,11,<0,4>,<0,6>,<0,1>>>,<0,6>>,11,17)\)

\(\alpha(260) = (<2,<1,1,<1,3,<0,0>,<0,0>,<0,2>>,<1,10,<0,0>,<0,0>,<0,4>>,<1,<1,2,<0,4>,<0,3>,<0,1>>,8,<2,<0,4,<0,5>,<1,7,<0,2>,<0,5>,<0,0>>,<1,<0,1>,0,<0,0>,1>,<1,<0,0>,1,<1,3,<0,3>,<1,2,<0,2>,<0,9>,<1,2,<0,0>,<0,4>,<0,3>>>,<1,0,<0,15>,<0,9>,<1,0,<0,14>,<0,7>,<0,5>>>>,0>,2,<0,0>,<0,4>,<0,9>>,1>,<1,<1,0,<0,1>,<0,2>,<1,0,<0,0>,<0,3>,<0,0>>>,16,<2,<1,1,<0,2>,<0,0>,0>,5,<0,3>,<0,9>,<2,<0,3,<0,1>,<0,2>,<1,<0,0>,1,<1,2,<0,5>,<0,0>,<1,0,<0,5>,<0,2>,<1,0,<0,3>,<0,13>,<1,0,<0,0>,<0,5>,<0,8>>>>>,0>,2,<0,5>,<2,<0,2,<1,5,<0,1>,<1,2,<0,3>,<1,0,<0,0>,<0,0>,<0,0>>,<0,0>>,<0,0>>,<0,0>,0>,0,<0,0>,<0,0>,<0,0>>,<0,0>>>,0>,0,<0,0>,<0,0>,<0,0>>,2,17)\)

\(\alpha(270) = (<2,<1,2,<0,0>,<1,4,<0,3>,<0,8>,<0,5>>,0>,1,<1,2,<0,0>,<0,5>,<1,1,<0,1>,<1,0,<0,10>,<0,7>,<0,3>>,<0,4>>>,<2,<0,0,<1,0,<0,3>,<0,9>,<0,1>>,<1,2,<0,3>,<0,0>,<1,1,<0,4>,<1,0,<0,5>,<0,3>,<0,4>>,<0,4>>>,0>,0,<0,2>,<0,7>,<1,14,<0,1>,<1,3,<0,5>,<0,14>,<0,0>>,<0,4>>>,<2,<1,1,<0,1>,<0,3>,0>,1,<0,8>,<0,0>,<0,4>>>,13,19)\)

\(\alpha(280) = (<2,<1,2,<0,5>,<0,9>,<1,<0,0>,1,<0,4>,0>,1,<1,10,<0,5>,<1,8,<0,4>,<1,6,<0,3>,<0,7>,<0,5>>,<0,13>>,<0,4>>,<1,1,<0,0>,<1,0,<0,0>,<0,2>,<0,0>>,<1,0,<0,2>,<0,4>,<0,2>>>,<0,1>>,7,19)\)

\(\alpha(290) = (<2,<1,2,<1,2,<0,10>,<0,4>,<0,3>>,<1,5,<0,3>,<0,2>,<0,1>>,<1,<1,2,<0,7>,<0,5>,<1,2,<0,5>,<0,4>,<1,1,<0,3>,<1,0,<0,0>,<0,7>,<0,2>>,<1,0,<0,3>,<0,3>,<0,2>>>>>,5,<1,3,<0,10>,<0,4>,<1,1,<0,0>,<0,0>,<1,0,<0,1>,<0,1>,<0,4>>>>,1>,<1,<1,0,<0,0>,<0,4>,<0,4>>,1,<1,13,<0,10>,<0,5>,<1,13,<0,8>,<1,8,<0,4>,<0,7>,<1,2,<0,1>,<1,0,<0,0>,<0,5>,<0,3>>,<1,2,<0,0>,<1,1,<0,5>,<1,0,<0,1>,<0,0>,<1,0,<0,0>,<0,4>,<0,3>>>,<1,0,<0,8>,<0,1>,<1,0,<0,0>,<0,0>,<0,3>>>>,<0,0>>>>,<1,10,<0,3>,<1,0,<0,0>,<0,0>,<0,0>>,<0,0>>>>,0>,0,<0,0>,<0,0>,<0,0>>,2,14)\)

\(\alpha(300) = (<2,<1,3,<0,0>,<0,3>,0>,5,<2,<0,3,<0,2>,<0,5>,0>,2,<0,6>,<0,5>,<2,<0,1,<1,4,<0,5>,<0,17>,<0,5>>,<0,7>,<1,<1,2,<0,1>,<1,0,<0,5>,<0,1>,<0,5>>,<1,2,<0,0>,<0,2>,<1,0,<0,1>,<0,6>,<0,1>>>>,3,<1,3,<0,0>,<1,1,<0,3>,<1,0,<0,4>,<0,3>,<1,0,<0,1>,<0,4>,<0,5>>>,<1,0,<0,1>,<0,2>,<1,0,<0,0>,<0,4>,<0,1>>>>,<0,1>>,1>,<1,<1,2,<0,0>,<1,1,<0,0>,<1,0,<0,0>,<0,4>,<0,3>>,<1,0,<0,3>,<0,0>,<0,3>>>,<0,2>>,5,<0,0>,0>,7,<0,1>,<1,2,<0,6>,<0,4>,<1,2,<0,1>,<1,0,<0,5>,<0,8>,<0,5>>,<1,1,<0,0>,<1,0,<0,5>,<0,1>,<1,0,<0,2>,<0,3>,<1,0,<0,0>,<0,0>,<0,0>>>>,<0,0>>>>,<0,0>>>,<0,0>,<0,0>>,2,18)\)

Here are the equivalent Veblen and FGH functions generated by the Alpha Function for input values of 100 to 300.

\(\alpha(100) = \varphi(1,0)\)

\(\alpha(110) = \varphi((\omega\uparrow\uparrow 6)^{2}.3 + (\omega\uparrow\uparrow 2)^{2}.(\omega^{6}.5 + \omega^{5}.5 + 3) + 5,(\omega\uparrow\uparrow 3)^{5}.(\omega^{10}.2 + 1) + 2)^{3}.((\varphi^{6}(1,\varphi(2,5) + (\varphi^{3}(1,\varphi(2,4) + 11_*)\uparrow\uparrow 3)^{3}.11 + 3_*)\uparrow\uparrow 6)^{6}.((\varphi(2,0)\uparrow\uparrow 5)^{4}.((\varphi(1,(\omega\uparrow\uparrow 3)^{3} + (\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2)^{11}.(\omega^{3}.2) + (\omega\uparrow\uparrow 2)^{9} + (\omega\uparrow\uparrow 2)^{3}.4 + 2) + (\omega\uparrow\uparrow 2)^{10}.3 + 5)\uparrow\uparrow 5)^{6}.((\varphi(1,0)\uparrow\uparrow 7)^{7}.((\varphi(1,0)\uparrow\uparrow 4)^{6}.6 + 5) + 5) + 5) + 5) + 5) + 5\)

\(\alpha(120) = (\varphi^{5}(\omega^{5} + 5,\varphi^{2}((\omega\uparrow\uparrow 3)^{2}.((\omega\uparrow\uparrow 2)^{6}.(\omega^{4}.3 + \omega.2 + 2) + 5) + \omega^{6}.4 + \omega.3 + 1,(\omega\uparrow\uparrow 4)^{6}.3 + (\omega\uparrow\uparrow 2)^{3}.(\omega^{6}.4) + \omega + 2_*) + (\omega\uparrow\uparrow 6)^{6}.2 + 11_*)\uparrow\uparrow 6)^{14}.4 + 5\)

\(\alpha(130) = (\varphi^{5}(4,\varphi^{3}(6,21_*) + 1_*)\uparrow\uparrow 11)^{(\omega\uparrow\uparrow 9).((\omega\uparrow\uparrow 2)^{3}.(\omega^{2}.4) + 2) + 3}.4 + \varphi^{3}(5,(\omega\uparrow\uparrow 5)^{6}.2 + (\omega\uparrow\uparrow 2)^{2}.(\omega^{6}.3 + 1) + (\omega\uparrow\uparrow 2).5 + \omega^{12}.2_*)^{6}.((\omega\uparrow\uparrow 4).5 + (\omega\uparrow\uparrow 3)^{3}.((\omega\uparrow\uparrow 2)^{9}.(\omega.4 + 1) + \omega^{12}.3 + \omega^{5}.5 + 2) + (\omega\uparrow\uparrow 3).(\omega^{7}.5 + 5) + 5) + \varphi^{3}(2,(\omega\uparrow\uparrow 4)^{10}.((\omega\uparrow\uparrow 2)^{5}.5 + (\omega\uparrow\uparrow 2).5 + \omega^{5}.3 + \omega^{2})_*)\)

\(\alpha(140) = (\varphi^{4}(1,(\omega\uparrow\uparrow 10)^{2}.((\omega\uparrow\uparrow 4)^{3}.(\omega^{2}.2 + \omega.6 + 2) + \omega^{2}.3 + 7) + (\omega\uparrow\uparrow 6)^{6}.2 + 4_*)\uparrow\uparrow 4)^{5}.4 + (\omega\uparrow\uparrow 5)^{3}.((\omega\uparrow\uparrow 3)^{3}.((\omega\uparrow\uparrow 2)^{7}.(\omega^{4}.6 + \omega^{2}.2 + \omega.2)) + 9) + (\omega\uparrow\uparrow 4)^{5} + 3\)

\(\alpha(150) = (\varphi^{4}((\omega\uparrow\uparrow 3)^{6}.6 + \omega^{2}.6 + 2,3_*)\uparrow\uparrow 2)^{9}.((\omega\uparrow\uparrow 6).((\omega\uparrow\uparrow 4)^{6}.(\omega^{2}.4 + 5) + 1) + (\omega\uparrow\uparrow 4).(\omega^{3}.6 + 16) + 3) + (\omega\uparrow\uparrow 11)^{5}.((\omega\uparrow\uparrow 3)^{4}.((\omega\uparrow\uparrow 2).(\omega.4 + 2) + 5) + (\omega\uparrow\uparrow 2)^{3}.(\omega^{3}.2 + \omega^{2}.4 + 4)) + 5\)

\(\alpha(160) = (\varphi^{5}(5,0_*)\uparrow\uparrow 3)^{3}.((\varphi^{5}(1,5_*)\uparrow\uparrow 2)^{5}.((\varphi^{5}(1,3_*)\uparrow\uparrow 3)^{5}.((\varphi^{5}(1,2_*)\uparrow\uparrow 2)^{8}.((\varphi^{4}(6,5_*)\uparrow\uparrow 3)^{12}.((\varphi^{4}(1,\varphi^{4}(2,(\omega\uparrow\uparrow 10)^{3}.((\omega\uparrow\uparrow 6)^{5} + 11) + 1_*) + (\omega\uparrow\uparrow 5)^{4} + 3_*)\uparrow\uparrow 6).((\varphi^{4}(1,\omega.2 + 4_*)\uparrow\uparrow 2)^{11}.((\omega\uparrow\uparrow 5)^{4}.3 + (\omega\uparrow\uparrow 2)^{6}.(\omega^{6}.6 + 8) + (\omega\uparrow\uparrow 2)^{2}.3 + \omega.3 + 1) + \varphi((\omega\uparrow\uparrow 2)^{4}.(\omega),\varphi((\omega\uparrow\uparrow 3)^{3}.5 + (\omega\uparrow\uparrow 2)^{5}.(\omega^{3}.5 + \omega.2 + 6) + 1,\omega^{4} + 6) + 1)))))))\)

\(\alpha(170) = (\varphi^{5}((\omega\uparrow\uparrow 5)^{6}.4 + (\omega\uparrow\uparrow 2)^{11}.4,(\omega\uparrow\uparrow 4)^{5}.3 + \omega^{5}.12 + \omega.3 + 1_*)\uparrow\uparrow 4)^{\omega.4 + 5}.((\omega\uparrow\uparrow 6).(\omega^{5}.4 + \omega^{3}.12 + \omega^{2} + \omega.5 + 2) + (\omega\uparrow\uparrow 2)^{10}.(\omega^{5}.3 + 1) + 5) + (\omega\uparrow\uparrow 12)^{4}.((\omega\uparrow\uparrow 10)^{6}.3 + 1) + (\omega\uparrow\uparrow 7)^{2}.6 + (\omega\uparrow\uparrow 4)^{3}.3 + \omega.6 + 4\)

\(\alpha(180) = (\varphi(3,\varphi^{6}(6,0_*) + 3)\uparrow\uparrow 6)^{(\omega\uparrow\uparrow 5)^{8}.(\omega.2 + 3) + 2}.4 + (\varphi^{5}(3,2_*)\uparrow\uparrow 5)^{3}.(\varphi^{5}(1,\varphi^{18}((\omega\uparrow\uparrow 2)^{6}.(\omega^{3}.18 + 5) + (\omega\uparrow\uparrow 2)^{3}.(\omega^{6} + 9),\varphi^{3}((\omega\uparrow\uparrow 2)^{11}.(\omega^{2} + \omega.11 + 10) + 14,1_*) + (\omega\uparrow\uparrow 2)^{4}.2 + \omega^{4}.3 + \omega.6 + 5_*) + \omega^{4}.5 + \omega.6_*)^{4}.11 + \varphi^{4}(4,\varphi((\omega\uparrow\uparrow 12)^{6}.11 + (\omega\uparrow\uparrow 12)^{4}.2 + (\omega\uparrow\uparrow 2).(\omega.5 + 4) + 2,2) + (\omega\uparrow\uparrow 4)^{4}.(\omega)_*))\)

\(\alpha(190) = (\varphi^{6}((\omega\uparrow\uparrow 5)^{11}.((\omega\uparrow\uparrow 2)^{6}.(\omega.9 + 4) + 3),(\omega\uparrow\uparrow 3)^{5}.4 + \omega^{2}.6 + 5_*)\uparrow\uparrow 17)^{\varphi(1,\varphi(6,(\omega\uparrow\uparrow 2)^{6}.4 + (\omega\uparrow\uparrow 2)^{4}.4 + (\omega\uparrow\uparrow 2)^{2}.4 + \omega^{6} + \omega^{3}.7 + 2) + \varphi(3,(\omega\uparrow\uparrow 11)^{5}.((\omega\uparrow\uparrow 2)^{4}.(\omega^{5}.6 + 4) + 4) + (\omega\uparrow\uparrow 7)^{17}.6 + (\omega\uparrow\uparrow 7)^{5}.((\omega\uparrow\uparrow 4).((\omega\uparrow\uparrow 3)^{9}.6 + (\omega\uparrow\uparrow 2).(\omega^{6}.2 + \omega^{3}.4 + \omega)))))}\)

\(\alpha(200) = (\varphi^{9}(4,\omega^{2}.10 + \omega.4 + 3_*)\uparrow\uparrow 6)^{2}.(\varphi(2,4)^{10}.4 + 1) + (\omega\uparrow\uparrow 6).8 + 3\)

\(\alpha(210) = (\varphi^{5}(1,\varphi^{6}(3,\varphi^{12}(6,1_*) + (\varphi^{8}(7,\varphi^{4}(8,(\omega\uparrow\uparrow 4).3 + 4_*) + \omega.2 + 3_*)\uparrow\uparrow 2)^{4}.(\omega^{6} + 2) + (\omega\uparrow\uparrow 9)^{4}.2 + (\omega\uparrow\uparrow 7)^{5}.3 + (\omega\uparrow\uparrow 3)^{5}.2 + (\omega\uparrow\uparrow 2)^{4}.(\omega^{2}.4 + 6) + 2_*) + (\omega\uparrow\uparrow 6)^{4} + (\omega\uparrow\uparrow 5).2 + (\omega\uparrow\uparrow 4).3 + \omega.5 + 4_*)\uparrow\uparrow 4).6\)

\(\alpha(220) = (\varphi^{2}(1,\varphi^{2}(3_*,(\omega\uparrow\uparrow 3)^{3}.((\omega\uparrow\uparrow 2)^{4}.(\omega^{3}.6 + \omega^{2}.9 + 3) + (\omega\uparrow\uparrow 2)^{2}.(\omega^{3}.2 + 11) + 1)) + 3_*)\uparrow\uparrow 2)^{\omega^{10}.2 + 2}.((\varphi^{4}(1,\varphi^{2}(2_*,3) + 5_*)\uparrow\uparrow 3)^{6}.(\varphi^{3}((\omega\uparrow\uparrow 5)^{4} + (\omega\uparrow\uparrow 2)^{5}.(\omega^{3} + \omega^{2}.3 + \omega.4 + 5) + 1,\varphi^{4}((\omega\uparrow\uparrow 5)^{4}.5 + 3,7_*) + \varphi^{3}((\omega\uparrow\uparrow 10)^{2}.((\omega\uparrow\uparrow 8)^{4}.4 + (\omega\uparrow\uparrow 7)^{5}.5 + 11) + 9,(\omega\uparrow\uparrow 7)^{7}.((\omega\uparrow\uparrow 5)^{4}.(\omega))_*)_*)))\)

\(\alpha(230) = \varphi^{4}(\omega.5 + 3,\varphi^{6}(\omega.7,\varphi^{2}(\omega^{4}.4 + 2,\varphi^{2}((\omega\uparrow\uparrow 2)^{3}.7 + \omega^{7}.4 + \omega.5 + 8_*,(\omega\uparrow\uparrow 4)^{10}.3 + (\omega\uparrow\uparrow 4)^{8}.((\omega\uparrow\uparrow 3)^{3}.4 + (\omega\uparrow\uparrow 3)^{2}.((\omega\uparrow\uparrow 2)^{8}.(\omega^{2}.2 + \omega.6 + 5) + \omega^{5}.5 + \omega^{3} + 1) + 2) + \omega^{8}.4 + 1) + (\varphi^{3}(1,(\omega\uparrow\uparrow 7)^{4}.(\omega^{5}.2 + 5) + 22_*)\uparrow\uparrow 5)^{5}.3 + 5_*) + \omega^{4}.6 + \omega^{2}.6 + 9_*) + (\omega\uparrow\uparrow 4)^{4}.(\omega)_*)\)

\(\alpha(240) = (\varphi^{11}((\omega\uparrow\uparrow 4).(\omega.2 + 1),\varphi^{2}((\omega\uparrow\uparrow 12)^{4}.5 + \omega^{5} + 4_*,4) + 4_*)\uparrow\uparrow 5)^{(\omega\uparrow\uparrow 6)^{2}.2 + 5}.(\omega^{5}.5 + \omega^{2}.4) + (\varphi^{3}(1,\varphi^{8}(4,0_*) + 4_*)\uparrow\uparrow 3)^{6}.((\omega\uparrow\uparrow 10)^{6}.((\omega\uparrow\uparrow 3)^{3} + 2) + (\omega\uparrow\uparrow 5)^{5}.((\omega\uparrow\uparrow 4)^{15}.6 + 1) + 3) + (\omega\uparrow\uparrow 5)^{4}.((\omega\uparrow\uparrow 4)^{2}.2 + (\omega\uparrow\uparrow 4).6 + 4) + (\omega\uparrow\uparrow 2)^{4}.6 + 9\)

\(\alpha(250) = (\varphi^{3}(5_*,(\omega\uparrow\uparrow 16).((\omega\uparrow\uparrow 13)^{7}.((\omega\uparrow\uparrow 2)^{6}.(\omega^{4} + \omega^{2}.3 + \omega + 3) + 2) + 2) + \omega^{6}.6)\uparrow\uparrow 4)^{2}.((\varphi^{2}(3_*,5)\uparrow\uparrow 3)^{6}.((\varphi^{2}(2_*,4)\uparrow\uparrow 3)^{4}.(\omega^{4}.8 + 2) + 1) + (\varphi^{4}((\omega\uparrow\uparrow 2)^{5}.6 + (\omega\uparrow\uparrow 2)^{4}.(\omega^{7}.6 + 7) + (\omega\uparrow\uparrow 2).(\omega.6 + 2) + \omega^{2}.5 + 3,(\omega\uparrow\uparrow 6)^{2}.6 + 5_*)\uparrow\uparrow 12)^{5}.7 + 1) + 6\)

\(\alpha(260) = \varphi^{17}(\omega^{2}.3 + \omega.4,\varphi^{9}((\omega\uparrow\uparrow 3)^{5}.4 + 1,\varphi^{3}((\omega\uparrow\uparrow 4) + 2_*,(\omega\uparrow\uparrow 11) + 4) + (\varphi^{2}(1,\varphi(2,\varphi^{5}(6,(\omega\uparrow\uparrow 8)^{3}.6_*) + 1) + (\omega\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 3)^{3}.10 + (\omega\uparrow\uparrow 3).5 + 3) + \omega^{16}.10 + \omega^{15}.8 + 5_*)\uparrow\uparrow 3).5 + 9_*) + (\varphi^{3}(3_*,0)\uparrow\uparrow 6)^{4}.10 + (\varphi^{2}(1,\varphi^{4}(2,2_*) + (\omega\uparrow\uparrow 3)^{6} + \omega^{6}.3 + \omega^{4}.14 + \omega.6 + 8_*)\uparrow\uparrow 3)^{6}.(\varphi^{3}((\omega\uparrow\uparrow 6)^{2}.((\omega\uparrow\uparrow 3)^{4}.(\omega)),0_*))_*)\)

\(\alpha(270) = (\varphi^{4}(1_*,(\omega\uparrow\uparrow 5)^{4}.9 + 5)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3).6 + (\omega\uparrow\uparrow 2)^{2}.(\omega^{11}.8 + 3) + 4}.(\varphi(\omega^{4}.10 + 1,(\omega\uparrow\uparrow 3)^{4} + (\omega\uparrow\uparrow 2)^{5}.(\omega^{6}.4 + 4) + 4)^{3}.8 + (\omega\uparrow\uparrow 15)^{2}.((\omega\uparrow\uparrow 4)^{6}.15) + 4) + (\varphi^{3}(2_*,3)\uparrow\uparrow 2)^{9} + 4\)

\(\alpha(280) = (\varphi^{2}(1,\varphi^{4}(6_*,9) + 5_*)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 11)^{6}.((\omega\uparrow\uparrow 9)^{5}.((\omega\uparrow\uparrow 7)^{4}.8 + 5) + 13) + 4}.((\omega\uparrow\uparrow 2).(\omega.3) + \omega^{3}.5 + 2) + 1\)

\(\alpha(290) = \varphi^{2}(\omega.5 + 4,\varphi^{6}((\omega\uparrow\uparrow 3)^{8}.6 + (\omega\uparrow\uparrow 3)^{6}.5 + (\omega\uparrow\uparrow 2)^{4}.(\omega.8 + 2) + \omega^{4}.4 + 2,\varphi^{4}((\omega\uparrow\uparrow 3)^{11}.5 + 3_*,(\omega\uparrow\uparrow 6)^{4}.3 + 1) + (\omega\uparrow\uparrow 4)^{11}.5 + (\omega\uparrow\uparrow 2) + \omega^{2}.2 + 4_*) + (\omega\uparrow\uparrow 14)^{11}.6 + (\omega\uparrow\uparrow 14)^{9}.((\omega\uparrow\uparrow 9)^{5}.8 + (\omega\uparrow\uparrow 3)^{2}.(\omega.6 + 3) + (\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2)^{6}.(\omega^{2} + \omega.5 + 3) + \omega^{9}.2 + \omega + 3)) + (\omega\uparrow\uparrow 11)^{4}.(\omega)_*)\)

\(\alpha(300) = (\varphi^{5}(1_*,3)\uparrow\uparrow 6)^{(\varphi^{4}(3,5_*)\uparrow\uparrow 3)^{7}.6 + (\varphi^{6}((\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2).(\omega.5 + 3) + \omega^{4} + 3) + 2,\varphi^{4}((\omega\uparrow\uparrow 3)^{2}.(\omega^{6}.2 + 5) + (\omega\uparrow\uparrow 3).3 + \omega^{2}.7 + 1,\varphi^{2}((\omega\uparrow\uparrow 5)^{6}.18 + 5,7_*) + (\omega\uparrow\uparrow 4).((\omega\uparrow\uparrow 2)^{4}.(\omega^{5}.4 + \omega^{2}.5 + 5) + \omega^{2}.3 + \omega.5 + 1) + 1_*) + 1_*)\uparrow\uparrow 8)^{2}.((\omega\uparrow\uparrow 3)^{7}.5 + (\omega\uparrow\uparrow 3)^{2}.(\omega^{6}.9 + 5) + (\omega\uparrow\uparrow 2).(\omega^{6}.2 + \omega^{3}.4 + \omega))}\)


Granularity Examples of this Version

Version 5 makes it possible to access Veblen functions of the form:

  • \(\varphi(\alpha,\varphi(\beta,0) + \gamma)\) where \(\beta > \alpha\) and \(1 < \gamma < \varphi(\beta,0)\)

Using the following Real Number inputs into the Alpha Function generates these ordinals:

\(\alpha(451.99967) = (\varphi^{6}(3,\varphi^{13}(5_*,3) + 1_*)\uparrow\uparrow 3)^{\varphi(3,(\omega\uparrow\uparrow 5)^{2}.((\omega\uparrow\uparrow 3)^{6}.(\omega^{3} + 5) + 2) + 2).((\omega\uparrow\uparrow 6)^{6}.((\omega\uparrow\uparrow 5).(\omega^{18}.3) + (\omega\uparrow\uparrow 2)^{6}.(\omega.7 + 2) + \omega^{6}.5 + 7))}.(\varphi^{2}(3,\varphi(8,\varphi^{2}(10,5_*) + (\omega\uparrow\uparrow 2).10 + 3) + \varphi^{2}(5,(\omega\uparrow\uparrow 2)^{11}.2 + (\omega\uparrow\uparrow 2)^{3}.3 + \omega^{4}.4 + \omega_*)_*))\)

\(\alpha(451.99970) = (\varphi^{6}(3,\varphi^{13}(5_*,3) + 2_*)\uparrow\uparrow 3)^{\omega^{3}.3 + \omega.10 + 7}.(\varphi^{10}((\omega\uparrow\uparrow 9).((\omega\uparrow\uparrow 5)^{6}.5 + (\omega\uparrow\uparrow 5).((\omega\uparrow\uparrow 4)^{3}.(\omega.5 + 3) + (\omega\uparrow\uparrow 4)^{2}.((\omega\uparrow\uparrow 3)^{6}.((\omega\uparrow\uparrow 2)^{3}.3 + \omega.4 + 2) + (\omega\uparrow\uparrow 2)^{5}.(\omega^{5}.11 + \omega + 4) + \omega^{3}.5 + 5) + 3) + \omega)_*,0))\)

\(\alpha(451.99973) = (\varphi^{6}(3,\varphi^{13}(5_*,3) + 3_*)\uparrow\uparrow 3).((\omega\uparrow\uparrow 4).((\omega\uparrow\uparrow 3)^{6}.2 + 1) + 5) + 5\)

Where the root ordinal remains the same in each case:

\(\varphi^{13}(5_*,3) = \varphi(\varphi(\varphi(\varphi(\varphi(\varphi(\varphi(\varphi(\varphi(\varphi(\varphi(\varphi(\varphi(5,3),3),3),3),3),3),3),3),3),3),3),3),3)\)

But the overall ordinal increases from:

\(\varphi^{6}(3,\varphi^{13}(5_*,3) + 1_*)\)

to

\(\varphi^{6}(3,\varphi^{13}(5_*,3) + 2_*)\)

to

\(\varphi^{6}(3,\varphi^{13}(5_*,3) + 3_*)\)

The progression can be illustrated in more detail with these Alpha input numbers:

\(\alpha(451.99967) = (\varphi^{6}(3,\varphi^{13}(5_*,3) + 1_*)\uparrow\uparrow 3)^{\varphi(3,(\omega\uparrow\uparrow 5)^{2}.((\omega\uparrow\uparrow 3)^{6}.(\omega^{3} + 5) + 2) + 2).((\omega\uparrow\uparrow 6)^{6}.((\omega\uparrow\uparrow 5).(\omega^{18}.3) + (\omega\uparrow\uparrow 2)^{6}.(\omega.7 + 2) + \omega^{6}.5 + 7))}.(\varphi^{2}(3,\varphi(8,\varphi^{2}(10,5_*) + (\omega\uparrow\uparrow 2).10 + 3) + \varphi^{2}(5,(\omega\uparrow\uparrow 2)^{11}.2 + (\omega\uparrow\uparrow 2)^{3}.3 + \omega^{4}.4 + \omega_*)_*))\)

\(\alpha(451.99968) = \varphi(1,\varphi^{6}(3,\varphi^{13}(5_*,3) + 1_*) + (\omega\uparrow\uparrow 3)^{6}.((\omega\uparrow\uparrow 2)^{2}.5 + 4) + (\omega\uparrow\uparrow 3)^{5} + (\omega\uparrow\uparrow 3).5 + 3)^{4}.(\varphi^{2}(1,\varphi^{3}(7_*,1) + (\omega\uparrow\uparrow 3)^{4}.(\omega^{3}.2 + \omega.4 + 1) + (\omega\uparrow\uparrow 2).(\omega^{6}.4 + 5) + \omega^{3}.5 + \omega.6 + 4_*)^{5}.((\varphi^{13}(1,\varphi^{3}(2_*,\omega^{2}.3 + \omega.3 + 16) + 9_*)\uparrow\uparrow 2)^{4}.((\omega\uparrow\uparrow 5).(\omega^{2} + \omega.3 + 1) + 4) + \omega^{7}.7 + \omega^{5}.4 + \omega))\)

\(\alpha(451.99969) = (\varphi^{13}(1,\varphi^{3}(2,\varphi^{6}(3,\varphi^{13}(5_*,3) + 1_*) + (\omega\uparrow\uparrow 6)^{2}.((\omega\uparrow\uparrow 3)^{8}.(\omega^{4}.2 + 3) + (\omega\uparrow\uparrow 2)^{8}.(\omega^{10}.3 + \omega.5 + 2) + 3) + (\omega\uparrow\uparrow 2)^{5}.6 + 1_*) + (\omega\uparrow\uparrow 6).9 + (\omega\uparrow\uparrow 2).10 + \omega^{3}.6 + 2_*)\uparrow\uparrow 4)^{(\varphi((\omega\uparrow\uparrow 3)^{12}.3 + 3,\omega^{5}.2 + \omega^{3}.6 + \omega.4 + 5)\uparrow\uparrow 5)^{5}.8 + (\varphi(\omega^{3}.2 + \omega^{2}.16 + 5,(\omega\uparrow\uparrow 2)^{2}.(\omega^{3}.2 + \omega^{2}.2 + 2) + 1)\uparrow\uparrow 2)^{3}.4 + \varphi(\omega^{2}.7 + \omega.6 + 1,(\omega\uparrow\uparrow 4)^{4}.(\omega))}\)

\(\alpha(451.99970) = (\varphi^{6}(3,\varphi^{13}(5_*,3) + 2_*)\uparrow\uparrow 3)^{\omega^{3}.3 + \omega.10 + 7}.(\varphi^{10}((\omega\uparrow\uparrow 9).((\omega\uparrow\uparrow 5)^{6}.5 + (\omega\uparrow\uparrow 5).((\omega\uparrow\uparrow 4)^{3}.(\omega.5 + 3) + (\omega\uparrow\uparrow 4)^{2}.((\omega\uparrow\uparrow 3)^{6}.((\omega\uparrow\uparrow 2)^{3}.3 + \omega.4 + 2) + (\omega\uparrow\uparrow 2)^{5}.(\omega^{5}.11 + \omega + 4) + \omega^{3}.5 + 5) + 3) + \omega)_*,0))\)

Where the overall ordinal increases from:

\(\varphi^{6}(3,\varphi^{13}(5_*,3) + 1_*)\)

to

\(\varphi(1,\varphi^{6}(3,\varphi^{13}(5_*,3) + 1_*) + \gamma)^{4}\)

to

\(\varphi^{13}(1,\varphi^{3}(2,\varphi^{6}(3,\varphi^{13}(5_*,3) + 1_*) + \delta_*) + \epsilon_*)\)

to

\(\varphi^{6}(3,\varphi^{13}(5_*,3) + 2_*)\)

where the ordinals \(\gamma, \delta, \epsilon\) are large but \(<< \varphi^{13}(5_*,3)\)


Test Bed for Version 5

Below is the test bed and various results using version 5.

\(\alpha(300) = J_8(<2,<1,3,<0,0>,<0,3>,0>,5,<2,<0,3,<0,2>,<0,5>,0>,2,<0,6>,<0,5>,<2,<0,1,<1,4,<0,5>,<0,17>,<0,5>>,<0,7>,<1,<1,2,<0,1>,<1,0,<0,5>,<0,1>,<0,5>>,<1,2,<0,0>,<0,2>,<1,0,<0,1>,<0,6>,<0,1>>>>,3,<1,3,<0,0>,<1,1,<0,3>,<1,0,<0,4>,<0,3>,<1,0,<0,1>,<0,4>,<0,5>>>,<1,0,<0,1>,<0,2>,<1,0,<0,0>,<0,4>,<0,1>>>>,<0,1>>,1>,<1,<1,2,<0,0>,<1,1,<0,0>,<1,0,<0,0>,<0,4>,<0,3>>,<1,0,<0,3>,<0,0>,<0,3>>>,<0,2>>,5,<0,0>,0>,7,<0,1>,<1,2,<0,6>,<0,4>,<1,2,<0,1>,<1,0,<0,5>,<0,8>,<0,5>>,<1,1,<0,0>,<1,0,<0,5>,<0,1>,<1,0,<0,2>,<0,3>,<1,0,<0,0>,<0,0>,<0,0>>>>,<0,0>>>>,<0,0>>>,<0,0>,<0,0>>,2,18)\)

\(\alpha(300) = (\varphi^{5}(1_*,3)\uparrow\uparrow 6)^{(\varphi^{4}(3,5_*)\uparrow\uparrow 3)^{7}.6 + (\varphi^{6}((\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2).(\omega.5 + 3) + \omega^{4} + 3) + 2,\varphi^{4}((\omega\uparrow\uparrow 3)^{2}.(\omega^{6}.2 + 5) + (\omega\uparrow\uparrow 3).3 + \omega^{2}.7 + 1,\varphi^{2}((\omega\uparrow\uparrow 5)^{6}.18 + 5,7_*) + (\omega\uparrow\uparrow 4).((\omega\uparrow\uparrow 2)^{4}.(\omega^{5}.4 + \omega^{2}.5 + 5) + \omega^{2}.3 + \omega.5 + 1) + 1_*) + 1_*)\uparrow\uparrow 8)^{2}.((\omega\uparrow\uparrow 3)^{7}.5 + (\omega\uparrow\uparrow 3)^{2}.(\omega^{6}.9 + 5) + (\omega\uparrow\uparrow 2).(\omega^{6}.2 + \omega^{3}.4 + \omega))}\)

Alpha(22311.1)

\(\alpha(22311.1) = \varphi^{7}(6,(\varphi^{5}(4,9_*)\uparrow\uparrow 5).(\varphi^{2}(1,\varphi^{5}(3,(\omega\uparrow\uparrow 9)^{6}.3 + (\omega\uparrow\uparrow 4)^{4}.6 + 3_*) + \varphi^{2}(1,\varphi^{2}(2,(\omega\uparrow\uparrow 6)^{4}.2 + 5_*) + (\omega\uparrow\uparrow 11)^{3}.4 + 2_*)^{5}.3 + 1_*)^{4}.((\varphi^{3}(1,\varphi^{3}(2,1_*) + \omega^{10}.2 + \omega^{6}.2 + \omega.8 + 4_*)\uparrow\uparrow 6)^{6}.(\varphi((\omega\uparrow\uparrow 2)^{4}.(\omega),\varphi^{2}((\omega\uparrow\uparrow 3)^{11}.6 + 3,(\omega\uparrow\uparrow 3)^{3}.5 + (\omega\uparrow\uparrow 3).(\omega^{4}.4 + \omega^{3}.6 + 4) + (\omega\uparrow\uparrow 2)^{2}.(\omega^{5}.6 + \omega^{4} + 3)_*) + 1)))),0_*,0)\)

\(= \varphi^{7}(6,(\varphi^{5}(4,9_*)\uparrow\uparrow 5).(\varphi^{2}(1,\varphi^{5}(3,(\omega\uparrow\uparrow 9)^{6}.3 + \mu_0 + 3_*) + \varphi^{2}(1,\varphi^{2}(2,\mu_1 + 5_*) + \mu_2 + 2_*)^{5}.3 + 1_*)^{4}.((\varphi^{3}(1,\varphi^{3}(2,1_*) + \omega^{10}.2 + \omega^{6}.2 + \omega.8 + 4_*)\uparrow\uparrow 6)^{6}.(\varphi((\omega\uparrow\uparrow 2)^{4}.(\omega),\varphi^{2}((\omega\uparrow\uparrow 3)^{11}.6 + 3,(\omega\uparrow\uparrow 3)^{3}.5 + (\omega\uparrow\uparrow 3).(\omega^{4}.4 + \omega^{3}.6 + 4) + (\omega\uparrow\uparrow 2)^{2}.(\omega^{5}.6 + \omega^{4} + 3)_*) + 1)))),0_*,0)\)

\(= \varphi^{7}(6,(\varphi^{5}(4,9_*)\uparrow\uparrow 5).(\varphi^{2}(1,\varphi^{5}(3,(\omega\uparrow\uparrow 9)^{6}.3 + \mu_0 + 3_*) + \varphi^{2}(1,\gamma_*)^{5}.3 + 1_*)^{4}.((\varphi^{3}(1,\mu_3 + \omega^{6}.2 + \omega.8 + 4_*)\uparrow\uparrow 6)^{6}.(\varphi(\mu_4,\varphi^{2}(\mu_5,(\omega\uparrow\uparrow 3)^{3}.5 + (\omega\uparrow\uparrow 3).(\omega^{4}.4 + \omega^{3}.6 + 4) + (\omega\uparrow\uparrow 2)^{2}.(\omega^{5}.6 + \omega^{4} + 3)_*) + 1)))),0_*,0)\)

\(= \varphi^{7}(6,(\varphi^{5}(4,9_*)\uparrow\uparrow 5).(\varphi^{2}(1,\varphi^{5}(3,(\omega\uparrow\uparrow 9)^{6}.3 + \mu_0 + 3_*) + \epsilon + 1_*)^{4}.((\varphi^{3}(1,\mu_3 + \omega^{6}.2 + \omega.8 + 4_*)\uparrow\uparrow 6)^{6}.(\varphi(\mu_4,\varphi^{2}(\mu_5,\delta_*) + 1)))),0_*,0)\)

\(= \varphi^{7}(6,(\varphi^{5}(4,9_*)\uparrow\uparrow 5).(\varphi^{2}(1,\varphi^{5}(3,\mu_8 + 3_*) + \epsilon + 1_*)^{4}.((\varphi^{3}(1,\mu_7 + 4_*)\uparrow\uparrow 6)^{6}.\mu_6)),0_*,0)\)

\(= \varphi^{7}(6,(\varphi^{5}(4,9_*)\uparrow\uparrow 5).(\varphi^{2}(1,\mu_9 + 1_*)^{4}.((\varphi^{3}(1,\mu_7 + 4_*)\uparrow\uparrow 6)^{6}.\mu_6)),0_*,0)\)

\(= \varphi^{7}(6,(\varphi^{5}(4,9_*)\uparrow\uparrow 5).(\varphi^{2}(1,\mu_9 + 1_*)^{4}.\mu_{10}),0_*,0)\)

\(= \varphi^{7}(6,(\varphi^{5}(4,9_*)\uparrow\uparrow 5).\mu_{11},0_*,0)\)

\(= \varphi(6,(\varphi^{5}(4,9_*)\uparrow\uparrow 5).\mu_{11},\varphi^6(6,(\varphi^{5}(4,9_*)\uparrow\uparrow 5).\mu_{11},0_*,0),0)\)

\(= \varphi(6,\mu_{12},\varphi(6,\mu_{12},\varphi(6,\mu_{12},\varphi(6,\mu_{12},\varphi(6,\mu_{12},\varphi(6,\mu_{12},\varphi(6,\mu_{12},0,0),0),0),0),0),0),0)\)

Alpha(402653184)

\(\alpha^2(1.94) = \alpha(f_2^2(3)) = \alpha(402653184)\)

\(= J_8(<8,<5,3,<0,7>,<3,<0,0,<2,<2,5,<1,3,<0,2>,<1,2,<0,7>,<1,0,<0,6>,<0,5>,<1,0,<0,2>,<0,3>,<0,2>>>,<1,2,<0,3>,<1,1,<0,1>,<1,0,<0,0>,<0,5>,<0,2>>,<0,0>>,<1,2,<0,0>,<1,1,<0,0>,<0,5>,<1,0,<0,5>,<0,11>,<1,0,<0,2>,<0,8>,<1,0,<0,0>,<0,3>,<0,2>>>>>,<1,1,<0,2>,<1,0,<0,2>,<0,4>,<1,0,<0,0>,<0,6>,<0,2>>>,<1,1,<0,0>,<0,4>,<0,0>>>>>>,<0,3>>,<0,2>,0>,3,<0,0>,<0,1>,<0,3>>,<0,2>,<1,2,<0,4>,<1,1,<0,0>,<0,3>,<0,0>>,<1,2,<0,1>,<0,0>,<0,0>>>,0>,0,<0,0>,<0,0>,<0,0>>,<0,0>,<0,0>,<0,0>,<0,0>,<0,0>,<0,0>,0>,0,<0,0>,<0,0>,<0,0>>,2,12)\)

\(= \varphi^{5}(8,\varphi((\varphi^{7}((\omega\uparrow\uparrow 4)^{3}.((\omega\uparrow\uparrow 3)^{8}.(\omega^{7}.6 + \omega^{3}.4 + 2) + (\omega\uparrow\uparrow 3)^{4}.((\omega\uparrow\uparrow 2)^{2}.(\omega.6 + 2)) + (\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2).6 + \omega^{6}.12 + \omega^{3}.9 + \omega.4 + 2) + (\omega\uparrow\uparrow 2)^{3}.(\omega^{3}.5 + \omega.7 + 2) + (\omega\uparrow\uparrow 2).5) + 3,2)\uparrow\uparrow 4).2 + 3,2,(\omega\uparrow\uparrow 3)^{5}.((\omega\uparrow\uparrow 2).4) + (\omega\uparrow\uparrow 3)^{2}),0_*,0,0,0,0,0)\)

\(= \varphi^{5}(8,\varphi((\varphi^{7}(\mu_0.(\mu_1.(\omega^{7}.6 + \omega^{3}.4 + 2) + (\omega\uparrow\uparrow 3)^{4}.((\omega\uparrow\uparrow 2)^{2}.(\omega.6 + 2)) + (\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2).6 + \omega^{6}.12 + \omega^{3}.9 + \omega.4 + 2) + (\omega\uparrow\uparrow 2)^{3}.(\omega^{3}.5 + \omega.7 + 2) + (\omega\uparrow\uparrow 2).5) + 3,2)\uparrow\uparrow 4).2 + 3,2,(\omega\uparrow\uparrow 3)^{5}.((\omega\uparrow\uparrow 2).4) + (\omega\uparrow\uparrow 3)^{2}),0_*,0,0,0,0,0)\)

\(= \varphi^{5}(8,\varphi((\varphi^{7}(\mu_0.\mu_2 + \mu_3.(\mu_4.(\omega.6 + 2)) + (\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2).6 + \omega^{6}.12 + \omega^{3}.9 + \omega.4 + 2) + (\omega\uparrow\uparrow 2)^{3}.(\omega^{3}.5 + \omega.7 + 2) + (\omega\uparrow\uparrow 2).5) + 3,2)\uparrow\uparrow 4).2 + 3,2,(\omega\uparrow\uparrow 3)^{5}.((\omega\uparrow\uparrow 2).4) + (\omega\uparrow\uparrow 3)^{2}),0_*,0,0,0,0,0)\)

\(= \varphi^{5}(8,\varphi((\varphi^{7}(\mu_5 + \mu_3.\mu_6 + \mu_7.(\mu_8 + \omega^{6}.12 + \omega^{3}.9 + \omega.4 + 2) + (\omega\uparrow\uparrow 2)^{3}.(\omega^{3}.5 + \omega.7 + 2) + (\omega\uparrow\uparrow 2).5) + 3,2)\uparrow\uparrow 4).2 + 3,2,(\omega\uparrow\uparrow 3)^{5}.((\omega\uparrow\uparrow 2).4) + (\omega\uparrow\uparrow 3)^{2}),0_*,0,0,0,0,0)\)

\(= \varphi^{5}(8,\varphi((\varphi^{7}(\mu_9 + \mu_7.(\mu_{10} + \omega.4 + 2) + (\omega\uparrow\uparrow 2)^{3}.(\omega^{3}.5 + \omega.7 + 2) + (\omega\uparrow\uparrow 2).5) + 3,2)\uparrow\uparrow 4).2 + 3,2,(\omega\uparrow\uparrow 3)^{5}.((\omega\uparrow\uparrow 2).4) + (\omega\uparrow\uparrow 3)^{2}),0_*,0,0,0,0,0)\)

\(= \varphi^{5}(8,\varphi((\varphi^{7}(\mu_9 + \mu_7.(\mu_{10} + \omega.4 + 2) + \mu_{11}.(\omega^{3}.5 + \omega.7 + 2) + \mu_{12}) + 3,2)\uparrow\uparrow 4).2 + 3,2,(\omega\uparrow\uparrow 3)^{5}.((\omega\uparrow\uparrow 2).4) + (\omega\uparrow\uparrow 3)^{2}),0_*,0,0,0,0,0)\)

\(= \varphi^{5}(8,\varphi((\varphi^{7}(\mu_9 + \mu_7.(\mu_{10} + \omega.4 + 2) + \mu_{11}.(\omega^{3}.5 + \omega.7 + 2) + \mu_{12}) + 3,2)\uparrow\uparrow 4).2 + 3,2,\mu_{13}.\mu_{14} + (\omega\uparrow\uparrow 3)^{2}),0_*,0,0,0,0,0)\)

\(= \varphi^{5}(8,\varphi((\varphi^{7}(\mu_9 + \mu_7.\mu_{15} + \mu_{16} + \mu_{12}) + 3,2)\uparrow\uparrow 4).2 + 3,2,\mu_{13}.\mu_{14} + (\omega\uparrow\uparrow 3)^{2}),0_*,0,0,0,0,0)\)

\(= \varphi^{5}(8,\varphi((\varphi^{7}(\mu_{17},2)\uparrow\uparrow 4).2 + 3,2,\mu_{18} + \mu_{19}),0_*,0,0,0,0,0)\)

\(= \varphi^{5}(8,\varphi(\mu_{20},2,\mu_{21}),0_*,0,0,0,0,0)\)

\(= \varphi^{5}(8,\mu_{22},0_*,0,0,0,0,0)\)

\(= \varphi(8,\mu_{22},\varphi(8,\mu_{22},\varphi(8,\mu_{22},\varphi(8,\mu_{22},\varphi(8,\mu_{22},0,0,0,0,0,0),0,0,0,0,0),0,0,0,0,0),0,0,0,0,0),0,0,0,0,0)\)

Alpha(44739197927.424)

\(\alpha(44739197927.424) = \varphi^{3}(\omega^{2}.4 + 3,\varphi^{7}(1,\varphi^{6}((\varphi^{3}(6_*,(\omega\uparrow\uparrow 6)^{6}.((\omega\uparrow\uparrow 5).2 + 5) + (\omega\uparrow\uparrow 5)^{4}.6 + 5)\uparrow\uparrow 4)^{3}.((\omega\uparrow\uparrow 4)^{11}) + (\omega\uparrow\uparrow 4).12 + 2,7,0)^{5}.(\varphi((\varphi^{4}(2,\varphi^{10}(6,3_*) + 4_*)\uparrow\uparrow 6)^{4}.((\varphi^{3}(3,(\omega\uparrow\uparrow 4)^{3}.(\omega^{4}.3 + \omega^{3} + 5) + (\omega\uparrow\uparrow 4)^{2}.((\omega\uparrow\uparrow 3)^{11}.2 + (\omega\uparrow\uparrow 2)^{6}.(\omega^{5}.2 + 1) + 4) + 5_*)\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 4)^{4}.(\omega))),0,0)),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\omega^{2}.4 + 3,\varphi^{7}(1,\varphi^{6}((\varphi^{3}(6_*,(\omega\uparrow\uparrow 6)^{6}.((\omega\uparrow\uparrow 5).2 + 5) + (\omega\uparrow\uparrow 5)^{4}.6 + 5)\uparrow\uparrow 4)^{3}.((\omega\uparrow\uparrow 4)^{11}) + (\omega\uparrow\uparrow 4).12 + 2,7,0)^{5}.(\varphi((\varphi^{4}(2,\varphi^{10}(6,3_*) + 4_*)\uparrow\uparrow 6)^{4}.((\varphi^{3}(3,(\omega\uparrow\uparrow 4)^{3}.(\omega^{4}.3 + \omega^{3} + 5) + (\omega\uparrow\uparrow 4)^{2}.((\omega\uparrow\uparrow 3)^{11}.2 + (\omega\uparrow\uparrow 2)^{6}.(\omega^{5}.2 + 1) + 4) + 5_*)\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 4)^{4}.(\omega))),0,0)),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\varphi^{7}(1,\varphi^{6}((\varphi^{3}(6_*,\mu_1.((\omega\uparrow\uparrow 5).2 + 5) + \mu_2 + 5)\uparrow\uparrow 4)^{3}.((\omega\uparrow\uparrow 4)^{11}) + (\omega\uparrow\uparrow 4).12 + 2,7,0)^{5}.(\varphi((\varphi^{4}(2,\varphi^{10}(6,3_*) + 4_*)\uparrow\uparrow 6)^{4}.((\varphi^{3}(3,(\omega\uparrow\uparrow 4)^{3}.(\omega^{4}.3 + \omega^{3} + 5) + (\omega\uparrow\uparrow 4)^{2}.((\omega\uparrow\uparrow 3)^{11}.2 + (\omega\uparrow\uparrow 2)^{6}.(\omega^{5}.2 + 1) + 4) + 5_*)\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 4)^{4}.(\omega))),0,0)),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\varphi^{7}(1,\varphi^{6}((\varphi^{3}(6_*,\mu_1.\mu_3 + \mu_2 + 5)\uparrow\uparrow 4)^{3}.((\omega\uparrow\uparrow 4)^{11}) + \mu_4 + 2,7,0)^{5}.(\varphi((\varphi^{4}(2,\varphi^{10}(6,3_*) + 4_*)\uparrow\uparrow 6)^{4}.((\varphi^{3}(3,\mu_5.(\omega^{4}.3 + \omega^{3} + 5) + (\omega\uparrow\uparrow 4)^{2}.((\omega\uparrow\uparrow 3)^{11}.2 + (\omega\uparrow\uparrow 2)^{6}.(\omega^{5}.2 + 1) + 4) + 5_*)\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 4)^{4}.(\omega))),0,0)),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\varphi^{7}(1,\varphi^{6}((\varphi^{3}(6_*,\mu_6)\uparrow\uparrow 4)^{3}.\mu_7 + \mu_4 + 2,7,0)^{5}.(\varphi((\varphi^{4}(2,\varphi^{10}(6,3_*) + 4_*)\uparrow\uparrow 6)^{4}.((\varphi^{3}(3,\mu_5.(\omega^{4}.3 + \omega^{3} + 5) + \mu_8.(\mu_9 + (\omega\uparrow\uparrow 2)^{6}.(\omega^{5}.2 + 1) + 4) + 5_*)\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 4)^{4}.(\omega))),0,0)),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\varphi^{7}(1,\varphi^{6}(\mu_{10}.\mu_7 + \mu_4 + 2,7,0)^{5}.(\varphi((\varphi^{4}(2,\mu_{11} + 4_*)\uparrow\uparrow 6)^{4}.((\varphi^{3}(3,\mu_5.(\omega^{4}.3 + \omega^{3} + 5) + \mu_8.(\mu_9 + \mu_{12}.(\omega^{5}.2 + 1) + 4) + 5_*)\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 4)^{4}.(\omega))),0,0)),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\varphi^{7}(1,\varphi^{6}(\mu_{13},7,0)^{5}.(\varphi((\mu_{14}\uparrow\uparrow 6)^{4}.((\varphi^{3}(3,\mu_5.\mu_{15} + \mu_8.(\mu_9 + \mu_{12}.(\omega^{5}.2 + 1) + 4) + 5_*)\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 4)^{4}.(\omega))),0,0)),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\varphi^{7}(1,\mu_{16}.(\varphi((\mu_{14}\uparrow\uparrow 6)^{4}.((\varphi^{3}(3,\mu_5.\mu_{15} + \mu_8.\mu_{17} + 5_*)\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 4)^{4}.(\omega))),0,0)),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\varphi^{7}(1,\mu_{16}.(\varphi(\mu_{19}.((\varphi^{3}(3,\mu_5.\mu_{15} + \mu_8.\mu_{17} + 5_*)\uparrow\uparrow 4)^{4}.\mu_{18}),0,0)),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\varphi^{7}(1,\mu_{16}.\varphi(\mu_{19}.((\varphi^{3}(3,\mu_{20} + 5_*)\uparrow\uparrow 4)^{4}.\mu_{18}),0,0),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\varphi^{7}(1,\mu_{16}.\varphi(\mu_{19}.((\mu_{21}\uparrow\uparrow 4)^{4}.\mu_{18},0,0)),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\varphi^{7}(1,\mu_{16}.\varphi(\mu_{19}.(\mu_{22}.\mu_{18}),0,0),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\varphi^{7}(1,\mu_{16}.\varphi(\mu_{23},0,0),0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\varphi^{7}(1,\mu_{16}.\mu_{24},0,0,0_*,0),0_*,0,0,0,0,0,0,0)\)

\(= \varphi^{3}(\mu_0,\mu_{25},0_*,0,0,0,0,0,0,0)\)

\(= \varphi(\mu_0,\mu_{25},\varphi(\mu_0,\mu_{25},\varphi(\mu_0,\mu_{25},0,0,0,0,0,0,0,0),0,0,0,0,0,0,0),0,0,0,0,0,0,0)\)

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