User blog:B1mb0w/Alpha Function Code q1.0

Alpha Function Code (Quantum Function) version 1.0
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 they like.

The function code is still based on The S Function (Version 2), with a growth rate of \(f_{\varphi(1,1,0)}(n)\).

Version 9 Code (Javascript)
Version 9 has been completely re-written to use Javascript in Google Sheets. A link to the first draft Google Sheet file is available here:

First Draft Google Sheet File

Version 9 has also been 're-calibrated' to allow an input parameter range from 0 to 100,000 that should 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.

The Alpha Function translates unique real numbers into any and every finite integer.

Version 9 Examples
Links to the Google Sheet file will added here shortly. Each combination uniquely belongs to an ascending order of all sequences. Therefore each sequence can be assigned a finite ordinal value.

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

\(\alpha(0.0) = 0\)

\(\alpha(0.2) = 2\)

\(\alpha(1.0) = S(2,1,1) = 4\)

\(\alpha(2.0) = S(S(2,1,1),0,2) = 6\)

\(\alpha(3.0) = S(S(2,1,1),0,S(2,0,1)) = 7\)

\(\alpha(3.60) = S(2,T(0),1) = 8\)

\(\alpha(3.95) = S(S(2,T(0),1),1,1) = 16\)

\(\alpha(4.09) = S(S(2,T(0),1),1,2) = 24\)

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

\(\alpha(4.75) = S(2,S(T(0),0,1),1) = f_{\omega+1}(2) = f_{\omega}(8)\)

\(\alpha(6.42) = S(2,S(T(0),1,1),1) = f_{\omega.2}(2) = f_{\epsilon_0}(2)\)

\(\alpha(12.68) = S(2,T(1),1)\)

\(\alpha(28.72) = S(2,S(T(1),T(0),1),1)\)

\(\alpha(78.75) = S(2,T(T(0)),1)\)

\(\alpha(275.0) = S(2,T(T(1)),1)\)

\(\alpha(1305.25) = S(2,T(T(T(0))),1)\)

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

\(\alpha(10.60) = S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1)\)

\(\alpha(11.00) = S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),1,2)\)

\(\alpha(11.30) = S(S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),5,S(S(2,S(T(0),1,1),1),T(0),S(S(S(8,2,1),1,S(8,0,4)),0,3))),1,1)\)

\(\alpha(11.45) = S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),S(S(S(S(2,S(S(T(0),1,1),0,1),1),S(S(2,S(T(0),1,1),1),S(T(0),0,S(16,0,1)),1),1),S(S(S(S(S(2,S(T(0),1,1),1),S(T(0),0,6),S(8,0,1)),T(0),5),1,S(S(8,5,2),0,7)),0,1),1),S(S(2,S(T(0),0,1),1),T(0),2),1),1)\)

\(\alpha(11.48) = S(S(S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),S(S(S(2,S(S(T(0),1,1),0,T(0)),1),S(S(2,S(T(0),0,1),1),0,1),1),0,3),S(S(2,S(S(T(0),1,1),0,T(0)),1),0,3)),1,1),0,S(S(S(2,S(S(T(0),1,1),0,1),1),S(T(0),1,1),S(S(S(2,S(T(0),0,1),1),1,1),0,S(S(8,1,7),0,6))),T(0),1))\)

\(\alpha(11.49) = S(S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),S(S(S(S(2,S(S(T(0),1,1),0,T(0)),1),S(T(0),0,2),1),1,S(S(2,S(S(T(0),1,1),0,T(0)),1),2,1)),0,7),1),0,5)\)

\(\alpha(11.50) = S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),S(S(S(S(S(2,S(S(T(0),1,1),0,T(0)),1),S(S(T(0),1,1),0,1),16),4,S(S(S(S(S(2,S(S(T(0),1,1),0,T(0)),1),S(8,0,2),S(S(2,S(S(T(0),1,1),0,1),1),1,6)),4,7),1,1),0,1)),3,1),1,2),1)\)

\(\alpha(11.51) = S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),T(0),1)\)

\(\alpha(11.52) = S(S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),T(0),1),0,1)\)

\(\alpha(11.55) = S(S(S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),T(0),1),1,2),0,S(S(2,S(S(T(0),1,1),0,T(0)),1),S(T(0),0,3),S(S(S(S(2,S(T(0),0,1),1),T(0),5),S(S(S(S(S(8,7,1),3,7),2,1),1,1),0,1),2),4,1)))\)

\(\alpha(11.70) = S(S(S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),S(T(0),0,1),1),1,1),0,1)\)

\(\alpha(12.00) = S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),S(T(0),0,S(S(S(2,S(T(0),1,1),1),1,1),0,1)),S(S(S(S(2,S(S(T(0),1,1),0,T(0)),1),S(2,S(S(T(0),1,1),0,1),1),7),S(S(2,S(T(0),0,1),1),0,1),4),S(2,S(T(0),0,1),1),S(S(S(2,S(S(T(0),1,1),0,1),1),S(T(0),1,1),2),T(0),8)))\)

\(\alpha(12.7) = S(S(2,T(1),1),0,1)\)

Comparing Alpha Function Values
From the above examples, it is interesting to compare:

\(\alpha(8.9) = S(2,S(S(T(0),1,1),0,T(0)),1)\)

\(\alpha(10.6) = S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1)\)

\(\alpha(11.00) = S(\alpha(10.6),1,2)\)

\(\alpha(11.30) = S(S(\alpha(10.6),5,S(S(2,S(T(0),1,1),1),T(0),S(S(S(8,2,1),1,S(8,0,4)),0,3))),1,1)\)

\(\alpha(11.45) = S(\alpha(10.6),S(S(S(S(2,S(S(T(0),1,1),0,1),1),S(S(2,S(T(0),1,1),1),S(T(0),0,S(16,0,1)),1),1),S(S(S(S(S(2,S(T(0),1,1),1),S(T(0),0,6),S(8,0,1)),T(0),5),1,S(S(8,5,2),0,7)),0,1),1),S(S(2,S(T(0),0,1),1),T(0),2),1),1)\)

\(\alpha(11.48) = S(S(S(\alpha(10.6),S(S(\alpha(8.9),S(S(2,S(T(0),0,1),1),0,1),1),0,3),S(\alpha(8.9),0,3)),1,1),0,S(S(S(2,S(S(T(0),1,1),0,1),1),S(T(0),1,1),S(S(S(2,S(T(0),0,1),1),1,1),0,S(S(8,1,7),0,6))),T(0),1))\)

\(\alpha(11.49) = S(S(\alpha(10.6),S(S(S(\alpha(8.9),S(T(0),0,2),1),1,S(\alpha(8.9),2,1)),0,7),1),0,5)\)

\(\alpha(11.50) = S(\alpha(10.6),S(S(S(S(\alpha(8.9),S(S(T(0),1,1),0,1),16),4,S(S(S(S(\alpha(8.9),S(8,0,2),S(S(2,S(S(T(0),1,1),0,1),1),1,6)),4,7),1,1),0,1)),3,1),1,2),1)\)

\(\alpha(11.51) = S(\alpha(10.6),T(0),1) = f_{\omega}(\alpha(10.6))\)

\(\alpha(11.52) = S(\alpha(11.51),0,1)\)

\(\alpha(11.55) = S(S(\alpha(11.51),1,2),0,S(\alpha(8.9),S(T(0),0,3),S(S(S(S(2,S(T(0),0,1),1),T(0),5),S(S(S(S(S(8,7,1),3,7),2,1),1,1),0,1),2),4,1)))\)

\(\alpha(11.7) = S(S(S(\alpha(10.6),S(T(0),0,1),1),1,1),0,1)\)

\(\alpha(12) = S(\alpha(10.6),S(T(0),0,S(S(S(2,S(T(0),1,1),1),1,1),0,1)),S(S(S(\alpha(8.9),S(2,S(S(T(0),1,1),0,1),1),7),S(S(2,S(T(0),0,1),1),0,1),4),S(2,S(T(0),0,1),1),S(S(S(2,S(S(T(0),1,1),0,1),1),S(T(0),1,1),2),T(0),8)))\)

\(\alpha(12.68) = S(2,T(1),1)\)

Growth Rate of the Alpha Function
The Alpha Function is now 're-calibrated' to accept real number inputs up to 100,000 at which point the Alpha Function will generate an S Function approaching:

\(\alpha(100,000) = S(2,T^{\omega}(0),1) = \omega\)

In other words, the Alpha Function has been hard-coded to asymptotically reach infinity when 100,000.

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