User blog:B1mb0w/Alpha Function Code Version 9 (Javascript)

Alpha Function Code Version 9
This version of The Alpha Function has been re-written to use Javascript in Google Sheets. The code is available for anybody to use or copy as you like. This code replaces my last version which used VBA in Microsoft Excel.

The function code is still based on The S Function (Version 2), with a growth rate of \(f_{svo}(n)\).

Version 9 Code (Javascript)
Version 9 has been completely re-written to use Javascript in Google Sheets. Links to the Google Sheet file will added her shortly.

Version 9 has also been 're-calibrated' to let the input parameter range exceed the previous limit of 10,000 and hopefully this will be more interesting. The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. The real number is manipulated by Javascript Code to create a finite sequence of finite integers that represents a unique combination of S and T functions which can be translated into unique finite integers (up to \(f_{svo}(n)\) for any n).

The Alpha Function will still translate unique real numbers into any and every finite integer (up to \(f_{svo}(n)\) for any \(n\)).

Version 9 Examples
Links to the Google Sheet file will added her shortly. Each combination uniquely belongs to an ascending order of all sequences. Therefore each sequence can be assigned a finite ordinal value. The growth rate of these combinations is \(f_{svo}(n)\) for any \(n\)).

The following examples (from Version 8) will be updated as soon as the Google Sheet has been linked to this blog.

\(\alpha(0) = 0\)

\(\alpha(1.5) = 2\)

\(\alpha(1.75) = S(2,1,1) = 4\)

\(\alpha(1.875) = S(2,1,1) + 2 = 6\)

\(\alpha(1.9375) = S(2,1,1) + 3 = 7\)

\(\alpha(2) = S(2,T(0),1) = 8\)

\(\alpha(3) = S(S(2,T(0),1),1,1) = 16\)

\(\alpha(3.12) = S(S(2,T(0),1),1,1) + S(2,1,1) + 3 = 16 + 4 + 3 = 23\)

The growth rate can be seen to accelerate when we start introducing more complex T functions:

\(\alpha(4) = S(2,T(0) + 1,1) = f_{\omega+1}(2) = f_{\omega}(8)\)

\(\alpha(6) = S(S(2,S(T(0),1,1),1),1,1) + S(2,1,1) + 3 = S(S(2,S(T(0),1,1),1),1,1) + 4 + 3\)

\(\alpha(10) = S(2,T(1) + 1,1)\)

\(\alpha(20) = S(2,S(T(1),T(0),1),1) + 3\)

\(\alpha(50) = S(S(2,T(T(0) + 1) + T(0),1),1,1) + 1\)

\(\alpha(100) = S(2,T(T(1)) + 1,1)\)

\(\alpha(500) = S(2,T(T(1) + T(0)),1)\)

Comparing Alpha Function Values
The following examples (from Version 8) will be updated as soon as the Google Sheet has been linked to this blog.

From the above examples, it is interesting to compare:

\(\alpha(3.12) = \alpha(3) + \alpha(1.9375)\)

Here is a more complex example:

\(\alpha(6.76412776412775) = S(S(2,S(T(0),1,1),1),T(0) + S(S(2,T(0) + 1,1),T(0),S(S(8,4 + 3,4 + 3),4 + 2,S(S(8,4 + 3,4 + 2),4 + 2,S(S(8,4 + 3,4 + 1),4 + 2,S(8,4,1))))),1)\)

\(\alpha(6.78) = S(2,S(T(0),1,1) + 1,1)\)

\(\alpha(7.68550368550367) = S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),S(S(2,S(T(0),1,1),1),T(0) + S(S(2,T(0) + 1,1),T(0),S(S(8,4 + 3,4 + 3),4 + 2,S(S(8,4 + 3,4 + 2),4 + 2,S(S(8,4 + 3,4 + 1),4,1)))),1))\)

and

\(\alpha(7.68550368550367) = S(\alpha(6.78),S(T(0),1,1),\alpha(6.76412776412735))\)

Granularity Examples
The following examples (from Version 8) will be updated as soon as the Google Sheet has been linked to this blog.

These examples illustrate the fine detail in real numbers that can be used to access large numbers via the Alpha Function:

\(\alpha(5.85) = S(2,S(T(0),1,1),1)\)

\(\alpha(7.69) = S(2,S(T(0),1,1) + T(0),1)\)

\(\alpha(8.61) = S(2,S(T(0),1,1) + T(0) + 1,1)\)

\(\alpha(9.54) = S(2,T(1),1)\)

\(\alpha(10) = S(2,T(1) + 1,1)\)

\(\alpha(11) = S(\alpha(10),S(T(0),\alpha(9.54) + S(\alpha(7.69),S(T(0),1,1) + 3,S(S(S(S(2,S(T(0),1,1) + 1,1),T(0) + 1,1),T(0),3),6,S(\alpha(5.85),T(0) + 1,1))),1),1)\)

\(\alpha(11.3) = S(S(S(\alpha(10),T(1),7),3,S(S(2,T(0) + 1,1),T(0),1) + 1),1,\alpha(10) + S(\alpha(9.54),S(T(0),7,\alpha(5.85)),1))\)

\(\alpha(11.39) = S(\alpha(10),T(1),S(\alpha(9.54),S(S(S(T(0),5,4),4,2),1,S(S(T(0),3,1),1,S(S(2,T(0) + 1,1),T(0),1)) + 2),1))\)

\(\alpha(11.397) = S(\alpha(10),T(1),S(\alpha(9.54),S(S(T(0),S(\alpha(8.61),T(0) + 5,1) + S(\alpha(7.69),1,2) + 1,1),2,1) + T(0),1))\)

\(\alpha(11.3972) = S(\alpha(10),T(1),S(\alpha(9.54),S(T(0),S(S(S(\alpha(8.61),S(T(0),1,1) + T(0),3),T(0) + 7,1),T(0),1) + 1,S(\alpha(8.61),T(0) + 2,1)),1))\)

or

\(\alpha(11.3972) = … S(S(S(\alpha(8.61),S(T(0),1,1) + T(0),3),T(0) + 7,1),T(0),1) + 1,S(\alpha(8.61),T(0) + 2,1)),1))\)

\(\alpha(11.39723) = ... S(\alpha(8.61),S(T(0),1,1) + T(0),S(\alpha(7.69),1,1) + S(8,5,1) + S(S(8,3,1),2,7)),1),1))\)

\(\alpha(11.397234) = ... S(\alpha(8.61),S(T(0),1,1) + T(0),S(S(\alpha(7.69),S(T(0),1,1) + 4,1),4,S(\alpha(7.69),S(T(0),1,1) + 1,1))),1),1))\)

\(\alpha(11.3972345) = ... S(\alpha(8.61),S(T(0),1,1) + T(0),S(\alpha(7.69),S(T(0),1,1) + S(S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),16),S(S(2,T(0) + 1,1),T(0),16),1),1)),1),1))\)

or

\(\alpha(11.3972345) = ... ... S(S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),16),S(S(2,T(0) + 1,1),T(0),16),1),1)),1),1))\)

\(\alpha(11.397234504) = ... ... S(S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),S(\alpha(5.85),T(0) + 1,1) + \alpha(5.85)),T(0),1),1)),1),1))\)

\(\alpha(11.3972345049) = ... ... S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),S(\alpha(5.85),T(0) + S(S(2,T(0) + 1,1),T(0),1),4)),1)),1),1))\)

or

\(\alpha(11.3972345049) = ... ... ... S(S(2,T(0) + 1,1),T(0),1),4)),1)),1),1))\)

\(\alpha(11.39723450498) = ... ... ... S(S(2,T(0) + 1,1),T(0),S(S(8,6,3),4,1)),1)),1)),1),1))\)

\(\alpha(11.397234504981) = ... ... ... S(S(2,T(0) + 1,1),T(0),S(S(8,7,1),6,2)),1)),1)),1),1))\)

\(\alpha(11.3972345049816) = ... ... ... S(S(2,T(0) + 1,1),T(0),S(8,7,7)),1)),1)),1),1))\)

\(\alpha(11.3972345049817) = S(2,T(1) + T(0),1)\)

Growth Rate of the Alpha Function
The Alpha Function is now 're-calibrated' to accept real number inputs up to TBA at which point the Alpha Function will generate an S Function approaching:

\(\alpha(TBA) = S(2,T^{\omega}(0),1) = \omega\)

In other words, the Alpha Function has been hard-coded to asymptotically reach infinity when TBA.

Further References
Further references to relevant blogs can be found here: User:B1mb0w