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## $$\alpha_0$$ Ruler Function

The Alpha Function can generate every finite integer up to a very large number. The Alpha Function has a growth rate faster than $$f_{LVO}(n)$$ for any n.

The Alpha Function is 'calibrated' to accept real number inputs up to 10,000 and generate unique S() function outputs representing any and every big number up to the size of $$f_{LVO}(v)$$ for any n.

## Ruler Functions

I have started work on a set of Ruler Functions based on the Alpha function that will be useful to measure the size of very large numbers. Different 'rulers' can be used as required. A rough sketch of how this will look is:

$$\alpha_0$$ Ruler Function

$$\alpha_0(100) = \alpha(31.6) >> f_{svo}(3)$$ approximately

$$\alpha_1$$ Ruler Function

$$\alpha_1(100) = \alpha(5.79955) = f_{\omega + 1}(64) >> g_{64} = G$$ is Graham's number

$$\alpha_2$$ Ruler Function

$$\alpha_2(100) = \alpha($$ TBA $$) = f_3(4) >>$$ Googolplex

$$\alpha_3$$ Ruler Function

$$\alpha_3(100) = \alpha(4.9140375) = f_1^{322}(f_{\omega + 1}(2)) >>$$ Googol

## Examples of $$\alpha_0$$ Ruler Function

The function is 'calibrated' to provide an 'interesting' range of all numbers up to $$f_{svo}(3)$$ (approximately):

$$\alpha_0(0) = 0$$

$$\alpha_0(1) = 1$$

$$\alpha_0(2) = S(2,T(0),1) = 8$$

$$\alpha_0(3) = S(2,T(0),1) + 1 = 8 + 1$$

$$\alpha_0(4) = S(2,T(0),1) + 2 = 8 + 2$$

$$\alpha_0(5) = S(2,T(0),1) + S(2,1,1) = 8 + 4$$

$$\alpha_0(6) = S(S(2,T(0),1),1,1) = 16$$

$$\alpha_0(7) = S(S(2,T(0),1),1,1) + 2 = 16 + 2$$

$$\alpha_0(8) = S(S(2,T(0),1),1,2) = 32$$

$$\alpha_0(9) = S(S(2,T(0),1),1,S(2,1,1)) = S(8,1,4)$$

$$\alpha_0(10) = S(2,T(0) + 1,1) = 2048$$

$$\alpha_0(11) = S(2,T(0) + 1,1) + 3 = 2048 + 3$$

$$\alpha_0(12) = S(S(2,T(0) + 1,1),1,1) + 1 = S(2048,1,1) + 1$$

$$\alpha_0(13) = S(S(2,T(0) + 1,1),1,S(2,T(0),1) + 1) + S(S(2,T(0),1),1,1) + 1 = S(2048,1,8 + 1) + 16 + 1$$

$$\alpha_0(14) = S(S(2,T(0) + 1,1),T(0),1) + S(S(2,T(0),1),1,1) = S(2048,T(0),1) + 16$$

$$\alpha_0(15) = S(S(2,T(0) + 1,1),T(0),2) = S(2048,T(0),2)$$

$$\alpha_0(16) = S(S(S(2,T(0) + 1,1),T(0),S(2,T(0),1) + 1),1,S(2,1,1)) + S(2,1,1) + 2$$

$$= S(S(2048,T(0),8 + 1),1,4) + 4 + 2$$

$$\alpha_0(17) = S(2,S(T(0),1,1),1) + 1$$

$$\alpha_0(18) = S(S(2,S(T(0),1,1),1),1,S(2,1,1)) + 1 = S(S(2,S(T(0),1,1),1),1,4) + 1$$

$$\alpha_0(19) = S(S(2,S(T(0),1,1),1),T(0),S(S(2,T(0),1),1,1) + S(2,T(0),1) + S(2,1,1)) + 1$$

$$= S(S(2,S(T(0),1,1),1),T(0),16 + 8 + 4) + 1$$

$$\alpha_0(20) = S(2,S(T(0),1,1) + 1,1)$$

$$\alpha_0(21) = S(S(2,S(T(0),1,1) + 1,1),1,1) + 2$$

$$\alpha_0(22) = S(S(2,S(T(0),1,1) + 1,1),T(0),S(S(2,S(T(0),1,1),1),1,1) + 1) + S(S(2,T(0),1),1,S(2,1,1) + 2)$$

$$= S(S(2,S(T(0),1,1) + 1,1),T(0),S(S(2,S(T(0),1,1),1),1,1) + 1) + S(8,1,4 + 2)$$

$$\alpha_0(23) = S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),S(2,1,1) + 1) + 2$$

$$= S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),4 + 1) + 2$$

$$\alpha_0(24) = S(2,S(T(0),1,1) + T(0),1) + S(S(S(2,T(0) + 1,1),T(0),S(2,1,1) + 1),1,S(2,T(0) + 1,1) + 1) + 2$$

$$= S(2,S(T(0),1,1) + T(0),1) + S(S(2048,T(0),4 + 1),1,2048 + 1) + 2$$

$$\alpha_0(25) = S(S(2,S(T(0),1,1) + T(0),1),T(0),1) + 2$$

$$\alpha_0(30) = S(2,T(1),1)$$

$$\alpha_0(35) = S(S(2,T(1) + 1,1),T(1),1) + S(2,1,1) = S(S(2,T(1) + 1,1),T(1),1) + 4$$

$$\alpha_0(50) = S(S(S(S(S(2,S(T(1),1,1) + T(0),1),S(T(0),1,1) + 1,1),T(0) + 1,2),T(0),S(2,1,1)),1,1) + 1$$

$$= S(S(S(S(S(2,S(T(1),1,1) + T(0),1),S(T(0),1,1) + 1,1),T(0) + 1,2),T(0),4),1,1) + 1$$

$$\alpha_0(55) = S(S(2,S(T(1),1,T(0)) + 1,1),T(0),1)$$

$$\alpha_0(65) = S(S(2,S(T(1),T(0),1) + T(1) + 1,1),1,3)$$

$$\alpha_0(70) = S(2,S(T(1),T(0),S(T(0),1,1)),1) + 1$$

$$\alpha_0(85) = S(S(2,S(T(1),S(T(0),1,1),S(T(0),1,1) + 1),1),T(0),S(2,1,1) + 1) + S(2,1,1) + 2$$

$$= S(S(2,S(T(1),S(T(0),1,1),S(T(0),1,1) + 1),1),T(0),4 + 1) + 4 + 2$$

$$\alpha_0(90) = S(S(2,S(S(S(T(1),S(T(0),1,1) + 1,S(T(0),1,1) + T(0)),T(0) + 1,S(T(0),1,1)),T(0),1) + 1,1),1,1) + 1$$

$$\alpha_0(95) = S(S(S(2,S(T(1),S(T(0),1,1) + T(0) + 1,1),1),T(1),1),1,S(2,1,1) + 1)$$

$$= S(S(S(2,S(T(1),S(T(0),1,1) + T(0) + 1,1),1),T(1),1),1,4 + 1)$$

$$\alpha_0(99.5) = S(2,T(T(0)),1) + 2$$

$$\alpha_0(99.6) = S(S(2,T(T(0)),1),1,1) + 2$$

$$\alpha_0(99.8) = S(S(S(S(2,T(T(0)),1),S(T(0),1,1) + 1,1),S(T(0),1,1),S(2,1,1) + 1),1,1) + S(2,1,1)$$

$$= S(S(S(S(2,T(T(0)),1),S(T(0),1,1) + 1,1),S(T(0),1,1),4 + 1),1,1) + 4$$

$$\alpha_0(99.92) = S(S(2,T(T(0)),1),S(T(1),T(0),1) + S(T(1),1,S(T(0),1,1)) + S(T(1),1,T(0) + 1) + T(1),2) + 3$$

$$\alpha_0(99.94) = S(S(2,T(T(0)),1),S(S(T(1),T(0) + 1,1),T(0),T(0)),1) + 1$$

$$\alpha_0(99.95) = S(S(2,T(T(0)),1),S(T(1),T(0) + 1,S(T(0),1,1) + T(0) + 1),S(2,T(0) + 1,1) + S(S(2,T(0),1),1,1) + 1)$$

$$= S(S(2,T(T(0)),1),S(T(1),T(0) + 1,S(T(0),1,1) + T(0) + 1),2048 + 16 + 1)$$

$$\alpha_0(99.96) = S(S(S(S(2,T(T(0)),1),S(T(1),S(T(0),1,1) + 1,1),1),T(1) + 1,S(2,1,1) + 1),1,S(2,1,1) + 2)$$

$$= S(S(S(S(2,T(T(0)),1),S(T(1),S(T(0),1,1) + 1,1),1),T(1) + 1,4 + 1),1,4 + 2)$$

$$\alpha_0(99.97) = S(S(S(2,T(T(0)),1),S(S(T(1),S(T(0),1,1) + T(0),1),1,1) + 1,1),1,1) + 2$$

$$\alpha_0(99.98) = S(S(2,T(T(0)),1),S(S(S(T(1),S(T(0),1,1) + T(0) + 1,1),T(0),1),1,T(0)) + S(T(0),1,1),2)$$

$$\alpha_0(100) = S(2,T(T(0)) + 1,1)$$